ADVANCES IN MATHEMATICS RESEARCH, VOLUME 8
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ADVANCES IN MATHEMATICS RESEARCH, VOLUME 8

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ADVANCES IN MATHEMATICS RESEARCH EDITOR: GABRIEL OYIBO Advances in Mathematics Research, Volume 8 2009. 978-1-60456-454-9

Advances in Mathematics Research, Volume 7 2007. 1-59454-458-1

Advances in Mathematics Research, Volume 6 2005. 1-59454-032-2-3

Advances in Mathematics Research, Volume 5 2005. 1-59033-799-92

Advances in Mathematics Research, Volume 4 2003. 1-59033-518-X

Advances in Mathematics Research, Volume 3 2003. 1-59033-452-3

Advances in Mathematics Research, Volume 2 2003. 1-59033-430-2

Advances in Mathematics Research, Volume 1 2002. 1-59033-223-7

ADVANCES IN MATHEMATICS RESEARCH, VOLUME 8

ALBERT R. BASWELL EDITOR

Nova Science Publishers, Inc. New York

Copyright © 2009 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Available upon request.

ISBN 978-1-61209-811-1 (eBook)

Published by Nova Science Publishers, Inc. New York

CONTENTS Preface

vii

Chapter 1

The Method of Characteristics for the Numerical Solution of Hyperbolic Differential Equations M. Shoucri

Chapter 2

Negotiating Mathematics and Science School Subject Boundaries: The Role of Aesthetic Understanding Linda Darby

Chapter 3

The Mathematical Basis of Periodicity in Atomic and Molecular Spectroscopy K. Balasubramanian

115

Chapter 4

Mathematical Modelling of Thermo-Mechanical Destruction of Polypropylene G.M. Danilova-Volkovskaya, E.A. Amineva and B.M. Yazyyev

141

Chapter 5

A Design-based Study of a Cognitive Tool for Teaching and Learning the Perimeter of Closed Shapes Siu Cheung Kong

145

Chapter 6

Modeling Asymmetric Consumer Behavior and Demand Equations for Bridging Gaps in Retailing Rajagopal

161

Chapter 7

Higher Education: Federal Science, Technology, Engineering, and Mathematics Programs and Related Trends United States Government Accountability Office

177

Chapter 8

Science, Technology, Engineering, and Mathematics (STEM) Education Issues and Legislative Options Jeffrey J. Kuenzi, Christine M. Matthew and Bonnie F. Mangan

247

1

89

vi

Contents

Chapter 9

On Computational Models for the Hyperspace S. Romaguera

277

Chapter 10

Periodic-Type Solutions of Differential Inclusions Jan Andres

295

Index

355

PREFACE "Advances in Mathematics Research" presents original research results on the leading edge of mathematics research. Each article has been carefully selected in an attempt to present substantial research results across a broad spectrum. The application of the method of characteristics for the numerical solution of hyperbolic type partial differential equations will be presented in Chapter 1. Especial attention will be given to the numerical solution of the Vlasov equation, which is of fundamental importance in the study of the kinetic theory of plasmas, and to other equations pertinent to plasma physics. Examples will be presented with possible combination with fractional step methods in the case of several dimensions. The methods are quite general and can be applied to different equations of hyperbolic type in the field of mathematical physics. Examples for the application of the method of characteristics to fluid equations will be presented, for the numerical solution of the shallow water equations and for the numerical solution of the equations of the incompressible ideal magnetohydrodynamic (MHD) flows in plasmas. A tradition of subject specialisation at the secondary level has resulted in the promotion of pedagogy appropriate for specific areas of content. Chapter 2 explores how the culture of the subject, including traditions of practice, beliefs and basic assumptions, influences teachers as they teach across school mathematics and science. Such negotiation of subject boundaries requires that a teacher understand the language, epistemology and traditions of the subject, and how these things govern what is appropriate for teaching and learning. This research gains insight into relationships between subject culture and pedagogy by examining both teaching practice in the classroom and interrogating teachers’ constructions of what it means to teach and learn mathematics and science. Teachers’ level of confidence with, and commitment to, both the discipline’s subject matter and the pedagogical practices required to present that subject matter is juxtaposed with their views of themselves as teachers operating within different subject cultures. Six teachers from two secondary schools were interviewed and observed over a period of eighteen months. The research involved observing and videoing the teachers’ mathematics and/or science lessons, then interviewing them about their practice and views about school mathematics and science. The focus of this chapter is on the role of the aesthetic, specifically “aesthetic understanding,” in the ways science and mathematics teachers experience, situate themselves within, and negotiate boundaries between the subject cultures of mathematics and science. The chapter outlines teachers’ commitments to the discipline, subject and teaching by exploring three elements of aesthetic understanding: the compelling and dramatic nature of

viii

Albert R. Baswell

understanding (teachers’ motivations and passions); understanding that brings unification or coherence (relationships between disciplinary commitments and knowing how to teach); and perceived transformation of the person (teacher identity and positioning). This research has shown that problems arise for teachers when they lack such aesthetic understanding, and this has implications for teachers who teach subjects for which they have limited background and training Chapter 3 applies combinatorial and group-theoretical relationships to the study of periodicity in atomic and molecular spectroscopy. The relationship between combinatorics and both atomic and molecular energy levels must be intimate since the energy levels arise from the combinatorics of the electronic or nuclear spin configurations or the rotational or vibrational energy levels of molecules. Over the years the authors have done considerable work on the use of combinatorial and group-theoretical methods for molecular spectroscopy [1–15]. The role of group theory [1–40] is evident since the classification of electronic and molecular levels has to be made according to the irreducible representations of the molecular symmetry group of the molecule under consideration. Combinatorics plays a vital role in the enumeration of electronic, nuclear, rotational and vibrational energy levels and wave functions. As can be seen from other chapters in this book, the whole Periodic Table of the elements has a mathematical group-theoretical basis since the electronic shells have their origin in group theory. Indeed, this concept can even be generalized to other particles beyond electrons such as bosons or other fermions that exhibit more spin configurations than just the bi-spin orientations of electrons. Chapter 4 provided mathematical description of the processes of thermonuclear destruction in deformed polypropylene melts; the aim was to use the criterion of destruction estimation in modelling and optimising the processing of polypropylene into products. With the consideration of cognitive inflexibility of learners in computing perimeter of closed shapes, a theory-driven design of a cognitive tool (CT) called the ‘Interactive Perimeter Learning Tool (IPLT)’ for supporting the teaching and learning of the mathematics target topic was developed in Chapter 5. An empirical study in the form of pre-test—post-test reflected that learners of varying mathematical abilities had statistically significant gains in using the IPLT for learning support. The IPLT could effectively address the inflexibility commonly exhibited by learners in learning this topic such as the formation of the abstract association of an irregular closed shape with a regular closed shape. The assertion of teachers on the effectiveness of the IPLT and the enthusiasm of students for using the IPLT for learning reflect that the CT had a pedagogical value in fostering learner-centred learning. Based on the feedback of this study, the IPLT will be refined under the design-based research approach. Chapter 6 attempts to discuss the interdependence of variability in consumer behavior due to intrinsic and extrinsic retail environment which influence the process of determining the choices on products and services. It is argued in the paper that suboptimal choice of consumers affect the demand of the products and services in the long-run and the cause and effect has been explained through the single non-linear equations. A system of demand equations which explains the process of optimization of consumer choice and behavioral adjustment towards gaining a long-term association with the market has also been discussed in the paper. The United States has long been known as a world leader in scientific and technological innovation. To help maintain this advantage, the federal government has spent billions of

Preface

ix

dollars on education programs in the science technology, engineering, and mathematics (STEM) fields for many years. However, concerns have been raised about the nation’s ability to maintain its global technological competitive advantage in the future. Chapter 7 presents information on(1) the number of federal programs funded in fiscal year 2004 that were designed to increase the number of students and graduates pursuing STEM degrees and occupations or improve educational programs in STEM fields, and what agencies report about their effectiveness; (2) how the numbers, percentages, and characteristics of students, graduates, and employees in STEM fields have changed over the years; and (3) factors cited by educators and others as affecting students’ decisions about pursing STEM degrees and occupations, and suggestions that have been made to encourage more participation. There is growing concern that the United States is not preparing a sufficient number of students, teachers, and practitioners in the areas of science, technology, engineering, and mathematics (STEM). A large majority of secondary school students fail to reach proficiency in math and science, and many are taught by teachers lacking adequate subject matter knowledge. When compared to other nations, the math and science achievement of U.S. pupils and the rate of STEM degree attainment appear inconsistent with a nation considered the world leader in scientific innovation. In a recent international assessment of 15-year-old students, the U.S. ranked 28th in math literacy and 24th in science literacy. Moreover, the U.S. ranks 20th among all nations in the proportion of 24-year-olds who earn degrees in natural science or engineering. A recent study by the Government Accountability Office found that 207 distinct federal STEM education programs were appropriated nearly $3 billion in FY2004. Nearly threequarters of those funds and nearly half of the STEM programs were in two agencies — the National Institutes of Health and the National Science Foundation. Still, the study concluded that these programs are highly decentralized and require better coordination. Several pieces of legislation have been introduced in the 109th Congress that address U.S. economic competitiveness in general and support STEM education in particular. These proposals are designed to improve output from the STEM educational pipeline at all levels, and are drawn from several recommendations offered by the scientific and business communities. The objective of Chapter 8 is to provide a useful context for these legislative proposals. To achieve this, the report first presents data on the state of STEM education and then examines the federal role in promoting STEM education. The report concludes with a discussion of selected legislative options currently being considered to improve STEM education. The report will be updated as significant legislative actions occur. Let BX be the continuous poset of formal balls of a metric space (X, d) endowed by the weightable quasi-metric qd induced by d. in Chapter 9 the authors show that the continuous poset B(CX) of formal balls of the space CX of nonempty closed bounded subsets of X endowed by the quasi-metric qHd induced by the Hausdorff metric Hd on CX is isometric to a sup-closed subspace of the space C(BX) of nonempty sup-closed bounded subsets of BX endowed with the Hausdorff quasi-metric qHd. The authors also show that the quasi-metric space (B(CX), qHd) is bicomplete if and only if the metric space (X, d) is complete. Several consequences are derived. In particular, our approach provides an interesting class of weightable quasi-metric spaces for which weightability of the Hausdorff quasi-metric holds on certain paradigmatic subspaces. Moreover, some properties from topological algebra are

x

Albert R. Baswell

discussed; for instance, the authors prove that if (X, d) is a metric monoid (respectively, a metric cone), then (B(CX), qHd) is a quasi-metric monoid (respectively, (B(CcX), qHd ) is a quasi-metric cone, where by Cc(X) the authors denote the family of all convex members of C(X)). The main purpose of Chapter 10 is two-fold: (i) rather than a complete account or a systematic study, the authors would like to indicate a flavour of the theory of periodictype oscillations, and (ii) to present some of our own results for periodic-type solutions of differential equations and inclusions. For (i), the authors preferably selected in Section 4. (Primer of periodic-type oscillations) the related results (including ours) which are easy for formulation while, for (ii), some technicalities had to be involved in Section 5. in order to derive sufficiently general criteria of the effective solvability of given actual problems. Results are, nevertheless, sketched in a form that is convenient for exposition and not necessarily in the greatest generality possible. Our objective is so to give the reader an overall idea of what the standard theory is like as well as to include enough information about its most recent progress. Formally, the focus of the object is simply the determination of the readable text for a wider audience with some parts to yield also a profit for the specialists.

In: Advances in Mathematics Research, Volume 8 Editor: Albert R. Baswell, pp. 1-87

ISBN: 978-1-60456-454-9 © 2009 Nova Science Publishers, Inc.

Chapter 1

THE METHOD OF CHARACTERISTICS FOR THE NUMERICAL SOLUTION OF HYPERBOLIC DIFFERENTIAL EQUATIONS M. Shoucri Institut de Recherche d’Hydro-Québec (IREQ), Varennes, Québec, Canada J3X1S1

Abstract The application of the method of characteristics for the numerical solution of hyperbolic type partial differential equations will be presented. Especial attention will be given to the numerical solution of the Vlasov equation, which is of fundamental importance in the study of the kinetic theory of plasmas, and to other equations pertinent to plasma physics. Examples will be presented with possible combination with fractional step methods in the case of several dimensions. The methods are quite general and can be applied to different equations of hyperbolic type in the field of mathematical physics. Examples for the application of the method of characteristics to fluid equations will be presented, for the numerical solution of the shallow water equations and for the numerical solution of the equations of the incompressible ideal magnetohydrodynamic (MHD) flows in plasmas.

1. Introduction Different types of partial differential equations require different numerical methods of solution. Numerical methods for hyperbolic equations are generally more complicated and difficult to develop compared to the numerical methods applied for parabolic or elliptic type partial differential equations. There has been important advances in the last few decades in the domain of the numerical solution of hyperbolic type partial differential equations using the method of characteristics, when applied to solve the initial value problem for general first order partial differential equations The order of a partial differential equation is the order of the highest-order partial derivative that appears in the equation. Let us consider for example the following simple hyperbolic type advection equation:

2

M. Shoucri

∂f ∂f +c = 0. ∂t ∂x

(1.1)

where c is a constant, sometimes called the velocity of propagation. The characteristic equation to solve Eq.(1.1) is dx / dt = c . The rate at which the solution will propagate along the characteristics is c. If c is a constant, all the points on the solution profile will move at the same speed along the characteristics determined by the solution of dx / dt = c . Let us assume the initial condition x(0) = x 0 . The solution of the characteristic equation gives the characteristic curves x = x0 + ct ( a straight line for the present case where c is a constant), where x0 is the point where each curve intersects the x-axis at t=0 in the x-t plane. If at t=0 we have

f ≡ f ( x 0 ) , x0 = x − ct , then f ( x, t ) = f ( x − ct ) . The function f ( x, t )

remains constant along a characteristic, which can be verified if we differentiate f ( x, t ) along one of these curves to find the rate of change of f along the characteristic:

∂f df ( x(t ), t ) ∂f ( x(t ), t ) dx ∂f ( x(t ), t ) ∂f = + = +c = 0. dt ∂t dt ∂x ∂t ∂x

(1.2)

which verify that f is constant along the characteristic curves. This is the simplest mathematical model of wave propagation. Constant quantities along the characteristic curves are called Riemann invariant [1]. We next consider the variable coefficient advection equation written as follows:

∂f ∂f + g ( x, t ) = 0. ∂t ∂x

(1.3)

The characteristic equation is dx / dt = g ( x, t ) . Again if the value of f at some arbitrary point ( x 0 , t 0 ) is known, the coordinate of the characteristic curve passing through

( x0 , t 0 ) can

be

determined

by

integrating

the

ordinary

differential

equation

dx / dt = g ( x, t ) . The velocity of propagation depends now on the spatial coordinate and time. In the general case an analytic solution is not straightforward and the characteristic curves are not straight lines anymore. Also it will be possible for the characteristic curves to intersect. The solution obtained by following the characteristic curves may contain discontinuities, which can lead to the formation of shocks or rarefaction waves [1]. Numerical techniques can be used to produce good approximations by following the solution computationally with small time-steps . As an example, we can discretize Eq.(1.3) as follows:

f ( x, t + dt ) − f ( x, t ) f ( x, t ) − f ( x − dx, t ) + g ( x, t ) = 0. dt dx

(1.4)

The Method of Characteristics for the Numerical Solution…

f ( x, t + dt ) = f ( x, t ) − g ( x, t ).

3

dt .( f ( x, t ) − f ( x − dx, t ) = f ( x − dx, t ) . (1.5) dx

For a small time-step between t and t+dt, it is possible to write the solution for the characteristic equation between x and x+dx in the form: t + dt

dx =

∫ g ( x(t ′), t ′)dt ′ .

(1.6)

t

Substituting in the right hand side of Eq.(1.5), we get: t + dt

f ( x, t + dt ) = f ( x −

∫ g ( x(t ′), t ′)dt ′, t ) .

(1.7)

t

Eq.(1.5) and Eq.(1.7) indicate that the value of the function f at the time t+dt and at a position x is equal to the value of the function at time t, at the shifted position t + dt

x − dx = x −

∫ g ( x(t ′), t ′)dt ′ . Eq.(1.7) is an implicit equation, and in all but the simplest t

cases different numerical approximations must be used to write an explicit solution. It is the purpose of the present chapter to discuss some of these approximations through examples and numerical methods applied to hyperbolic equations. Some of these approximations have been recently discussed for instance in [2,3]. The value of the function at the shifted position is usually calculated by interpolation from the known values of the function at the neighbouring grid points. In the present chapter cubic splines interpolation will be extensively used to calculate the shifted value in Eq.(1.7), since in several applications and problems they have compared favourably with other methods of interpolation [4]. For the more general case where several dimensions are involved, the fractional step technique allows sometimes the reduction of the multi-dimensional equation to an equivalent set of one dimensional equations [2-5]. The shifts become fractional, i.e. each of the dimension is shifted separately. The specific order, number of shifts and choice of the size of shift-factors depend now on the numerical method. If the fractional step technique cannot be applied, we can use other methods which consist in interpolating in several dimensions using a tensor product of Bsplines [6]. This technique has been extensively applied in the field of meteorology [7,8], where it is called the semi-Lagrangian method (although we prefer to call it the EulerLagrange method, since it essentially uses a fixed Eulerian grid, and uses a corrector or an iterative process to take care of the variation of the velocity along the characteristic curve). We can generalize Eq.(1.3) for a multi-dimensional problem in the following form:

df ∂f ∂f = + G (r, t ). = 0 . dt ∂t ∂r

(1.8)

4

M. Shoucri

which reflects the fact that the function f (r, t ) is constant along the trajectories defined by the characteristic curves :

dr = G (r, t ) . dt

(1.9)

Denoting by r (t ; ri , t n ) the characteristic crossing the grid point ri at tn , we can also write at t = t n :

f (r (t ; ri , t n ), t ) = f (r (t ; ri , t n ), t n ) = f (ri , t n ) .

(1.10)

Replacing t n by t n + Δt and t by t − Δt , results in :

f (ri , t n + Δt ) = f (r (t − Δt ; ri , t n + Δt ), t − Δt ) .

(1.11)

r (t − Δt ; ri , t n + Δt ) is the characteristic which ends up at ri at time t n + Δt . The function value at the time-step t n + Δt and at the grid point ri can be calculated by looking backward to the function value at an interstitial point, prescribed by the characteristic curve at the previous time t − Δt . The starting point at the previous time-step t n − Δt , of the

ri at time t n + Δt , is denoted by curve ending at ~ r = r (t n − Δt; ri , t n + Δt ) (see Fig.(1)). Usually ~r is an intermediate interstitial point

characteristic

which does not coincide with a grid point. The value of the function at ~ r has to be calculated by interpolation. Discretizing Eq.(1.9) of the characteristic curves using a leap-frog scheme, we can write:

r (t n + Δt ) − r (t n − Δt ) ri − ~ r ≡ = G (r (t n ), t n ) . 2Δt 2Δt

(1.12)

Using r (t n ) ≈ (r (t n + Δt ) + r (t n − Δt ) ) / 2 ≡ (ri + ~ r ) / 2 in the right hand side of

r ) / 2 . We solve this Eq.(1.12), results in Δ r = ΔtG (ri − Δ r , t n ) , where Δ r = (ri − ~ equation numerically for Δ r using the Newton iterative scheme :

Δkr+1 = ΔtG (ri − Δkr , t n )

. 0

(1.13)

starting with k=0, Δ r = 0 . Two or three iterations are usually sufficient to converge to precise results. We then calculate the value of f at the position ri at t n + Δt :

The Method of Characteristics for the Numerical Solution…

5

f (ri , t n + Δt ) = f (r (t − Δt ; ri , t n + Δt ), t n − Δt ) = f (ri − 2 * Δ r , t n − Δt ) . (1.14) j +1 Q (x i , y j ; t n +1 )

j

j −1 P (x , y ; t n −1 )

i −1

i

i +1

Figure 1.

The multi-dimensional interpolation in Eqs.(1.13-1.14) will generally involve a tensor product of B-splines. In practice, we will restrict ourselves to problems in two dimensions. In the Fig.(1) we give an example for the case of a two-dimensional space, showing the point of departure P at t n − Δt , where the value of the function f is to be interpolated as in Eq.(1.14) to yield the value of f (ri , t n + Δt ) at the point Q. Similar schemes have been extensively used in problems of meteorology [7,8], and more recently in plasma physics [6,9]. The ideas outlined in this introduction will be applied to selected problems in the present chapter. In section 2 we will present examples where a fractional step method reduces the multi-dimensional problem to an equivalent set of one-dimensional (1D) problems. In section 3 we will present examples where 2D interpolation involves a tensor product of cubic Bsplines. We will emphasize the precision, good performance and numerical stability of the cubic splines interpolation, which have been also previously pointed out in [4,7]. Examples will be taken from the field of plasma physics, especially concerning the numerical solution of the Vlasov equation, of fundamental importance in the kinetic theory of plasmas. Some additional applications in the field of fluid dynamics will be presented in section 4, for the numerical solution of the shallow water equations, and for the numerical solution of the equations of the incompressible ideal magnetohydrodynamic flows in plasmas.

2. The Fractional Step Method Applied to the Vlasov Equation The study of nonlinear processes in kinetic plasmas is heavily based on the numerical solution of the Vlasov equation for the distribution function. The Vlasov equation provides the basic dynamical description of hot plasmas in regimes where the effect of collisions are negligible with respect to those originating from the collective, mean-field electromagnetic interactions. The Vlasov-type equation is an advection equation in phase-space for the distribution function f , of the general form given in Eq.(1.8). Different techniques have been proposed to

6

M. Shoucri

solve this equation. Particle-in-cell (PIC) methods for instance approximate the plasma by a finite number of pseudo-particles and compute their trajectories given by Eq.(1.9). However, the numerical noise in these codes decreases only as 1 / N , where N is the number of pseudo-particles in any particular computational cell. This noise problem becomes important if the physics of interest is in the low density region of phase-space or in the high energy tail of the distribution function. On the other hand the direct numerical solution of the Vlasov equation as a partial differential equation on a fixed grid in phase-space has become an important method for the numerical solution of the Vlasov equation. Interest in Eulerian gridbased Vlasov solvers arises from the very low noise level associated with these methods, and the recent advances of parallel computers have increased the interest in the applications of splitting schemes to higher dimensional problems. The original, ground-breaking publication of Cheng and Knorr [10], which proposed the second-order fractional step scheme or splitting scheme for the solution of the Vlasov-Poisson system, was followed by several publications where this method was successfully applied to one-dimensional (two-dimensional in phasespace) Vlasov- Poisson problems [11-14]. The technique was extended to higher phase-space dimensions [15-19]. An important application using the Eulerian splitting schemes for the Vlasov-Maxwell system of equations has been reported for the study of laser-plasma interaction [20-27 and references therein], and extended to two-dimensional problems [28]. In the work on beat wave current drive [29], a constant magnetic field was introduced in the Vlasov equation. Further applications in the recent work in [30,31] testify to the success of this method in laser-plasma interaction. We also note the application of Eulerian splitting schemes to study two spatial dimension problems of Kelvin-Helmholtz instabilities and higher dimensionality gyrokinetic equations [32-39]. There exists also a variety of other applications using different methods developed for Eulerian grid-based Vlasov solvers [4044]. Of particular interest is the work coupling a Vlasov equation to a Fokker-Planck collision operator presented in [45]. In the present section 2 , we will present selected examples where the fractional step techniques associated with interpolation along the characteristic curves in one dimension are applied for the numerical solution of the Vlasov equation.

2.1. The Fractional Step Method Applied to the Vlasov-Poisson System in One Spatial Dimension The first system we study is the Vlasov-Poisson system in one spatial dimension ( a twodimensional phase space x-v ). The problem is the long time nonlinear evolution of a twostream instability in a collisionless plasma [46,47]. The system in this case evolves to a Bernstein-Greene-Kruskal BKG equilibrium [48] consisting of a stationary structure exhibiting holes or vortices in phase-space. BKG structures with more than one hole are unstable and coalesce until the evolution brings a final stable vortex. This flow of energy of the system during the evolution to the longest wavelength available in the system ( inverse cascade ) is characteristic of two-dimensional systems and has been discussed in several publications ( see for instance [49-50]). We use an Eulerian code associated with a method of fractional step for the integration of the Vlasov equation along the characteristics. The Eulerian method allows accurate resolution of the phase-space on a fixed Eulerian grid. In the present problem the spatial dimension x is assumed to be periodic. The normalized Vlasov

The Method of Characteristics for the Numerical Solution…

7

equation for the electron distribution function f ( x, v, t ) and the Poisson equation for the potential

ϕ (x) are given by: ∂f ∂f ∂f + v − Ex = 0. ∂t ∂x ∂v

(2.1)

∂ 2ϕ = −(1 − ne ) , ∂x 2 ∞

where

ne =

∫ f dv , and E

x

=−

−∞

∂ϕ ∂x

(2.2)

The ions form an immobile background in the present problem. The distance x , the velocity v and the time t are respectively normalized to the Debye length λ De = vth / ω pe , the thermal velocity vth and the inverse plasma frequency

−1 ω pe . Eq.(2.1) is essentially a two-

dimensional advection equation. An important property of this equation is that its characteristics, the particles trajectories dx / dt = v , dv / dt = − E x describe a Hamiltonian flow in phase-space. The particles motion is described by the Hamiltonian:

H=

v2 + ϕ ( x) . 2

(2.3)

The Vlasov Eq.(2.1) can be written in the form:

∂f + [H , f ] = 0 . ∂t The Poisson brackets

(2.4)

[H , f ] = ⎧⎨ ∂H

∂f ∂H ∂f ⎫ − ⎬ ⎩ ∂v ∂x ∂x ∂v ⎭

The distribution function f is constant along the particle trajectories. As a consequence, the integral over the entire phase-space of the distribution function is a constant, as well as the integral of any arbitrary smooth function of f. Thus the evolution of the distribution function f is constrained by a number of constants of motion. Hamiltonian systems like Eq.(2.4) are known to develop increasingly smaller scales during their nonlinear evolution. One way to control these finer structures is to increase resolution. These small structures dissipate when they reach the size of a the grid. We write the initial electron distribution function in the form[46]:

8

M. Shoucri

f ( x, v, t = 0) = A(1 +

ε 1−ξ

) e −ε (1 + α cos(k 0 x)

.

(2.5)

+ β cos(2k 0 x) + γ cos(3k 0 x)) A=

With

2 − 2ξ 2π 3 − 2ξ 1

ε = v 2 / 2 , and ξ is a parameter which characterizes a produced vortex in phase-space. 2π k0 =

L

denotes the fundamental wavenumber, L is the length of the periodic box. We

choose k 0 =

k M2 =

kM , where k M is the maximum wavenumber for instability [46] given by 4

3 − 2ξ 2ξ − 1 , which leads to a box length L = 8π . We choose ξ = 0.90 , which 3 − 2ξ 2ξ − 1

gives k M = 0.816 , L = 30.78λ De and k 0 = 0.204 . We take a cut-off velocity at v max = ±6vth . The distribution function is given at mesh points in the phase-space, with Nx = 128 points in space and Nv = 256 points in velocity space. The time-step is Δt = 0.25ω pe . A method which has second order in time precision [10,11] −1

is obtained by splitting Eq.(2.1) as follows: Step1 -

Solve

∂f ∂f +v = 0 for a step Δt / 2 ∂t ∂x

(2.6) *

- Solve Poisson equation for the electric field which we denote by E x .

Step2 -Solve Step3 -Solve

∂f ∂f − E x* = 0 for a step Δt ∂t ∂v ∂f ∂f +v = 0 for a step Δt / 2 ∂t ∂x

(2.7) (2.8)

In this 2D phase-space problem the shifts become fractional, i.e. each of the dimension of the phase-space is shifted separately. This splitting has the advantage that each of the x or v updates is a linear advection effected by applying successively the shifts :

f a ( x, v, t + Δt / 2) = f ( x − vΔt / 2, v, t ) ,

(2.9)

f b ( x, v, t + Δt ) = f a ( x, v − E x* Δt , t ) ,

(2.10)

The Method of Characteristics for the Numerical Solution…

9

f ( x, v, t + Δt / 2) = f b ( x − vΔt / 2, v, t ) ,

(2.11)

That is, half of the spatial shift is performed first in space. Since v is an independent variable, the shift in Eq.(2.9) is done as in Eq.(1.1) for each value of v (see Appendix A). This *

is followed by solving Poisson equation for the calculation of the electric field E x , which is used for the calculation of the total shift in velocity space where the integral in Eq.(1.7) is approximated as in Eq.(2.10). Poisson equation in Eq.(2.2) is discretized in space as a tridiagonal matrix:

ϕ j −1 − 2ϕ j + ϕ j +1 = −Δx 2 (1 − nej ) .

(2.12)

where Δx = L / N x , the subscript j denotes the grid-point xj . Eq.(2.12) is solved using appropriate boundary conditions ( periodic boundary conditions for the present problem). From

ϕ we calculate E x* (Eq.(2.2)). Finally the second half of the spatial shift is repeated in

Eq.(2.11). It has been shown in [10] that the overall precision of this numerical scheme is

O( Δt 2 ) . We can verify after this sequence that the distribution function f t = ( n + 1)Δt can be written as follows: f

n +1

n +1

at time

( x, v ) = f n ( x * , v * ) 1 E x ( x − vΔt / 2) Δt ) . 2 v * = v + E x ( x − vΔt / 2) x * = x − Δt (v +

(2.13)

On the other hand we can consider the characteristics equations , dx / dt = v ,

dv / dt = − E x , which are the particles trajectory. The integration of these equations between t and t = t + Δt gives the following result :

x(t ) = x(t + Δt ) − Δt (v(t + Δt ) + 1 / 2 E x ( x , t + Δt / 2)Δt ) v(t ) = v(t + Δt ) + ΔtE x ( x , t + Δt / 2)

.

(2.14)

where x = x(t + Δt / 2) . The field E x ( x, t ) in Eq.(2.7) is calculated after the first shift. The density distribution, and therefore E x ( x, t ) , remains unaffected by the second shiht. Thus the field E x ( x , t + Δt / 2) can be approximated by E x ( x − vΔt / 2, t + Δt / 2) . The shifts in Eqs.(2.9-2.11) are calculated using a cubic spline interpolation as defined in the appendices. For the present problem, we used the results in Appendix A.

10

M. Shoucri

Figure 2. The vorticity at t=50

Figure 3. The vorticity at t=100

−1 ω pe .

−1 ω pe .

Figure 4. Time evolution of the first Fourier mode.

The Method of Characteristics for the Numerical Solution…

Figure 5. Time evolution of the second Fourier mode.

Figure 6. The vorticity at t=40

−1 ω pe .

Figure 7. The vorticity at t=50

−1 ω pe .

11

12

M. Shoucri

Figure 8. The vorticity at t=500

−1 ω pe .

We apply the numerical scheme previously discussed to study the evolution of a two-stream instability. We introduce a perturbation on the fundamental wavenumber k 0 by taking

α = 0.001 and β = γ = 0. Only one vortex appears in phase space during the nonlinear plasma evolution ( see Fig.(2)), and the final equilibrium in Fig.(3) consists of a single smooth hole. Fig.(4) and Fig.(5) show the nonlinear evolution of the first and second Fourier modes respectively, showing the initial growth and saturation. In a second experiment, we start with a perturbation in Eq.(2.5) of the three modes, α = 0.001 , β = γ = α / 1.2 . We obtain in the first step the appearance of two vortices in the phase-space shown in Fig.(6) at −1 −1 t = 40ω pe , followed rapidly by the coalescence of the vortices at t = 50ω pe in Fig.(7) (so

the two vortices structure is not stable). Note the tendency of holes to behave as quasiparticles just before coalescence [47]. We finally end up with a single vortex (see Fig.(8)). We note again this tendency of the energy to move to the longest wavelength available in the system [49,50] ( the so called inverse cascade), which is characteristic of two dimensional systems. Small scale vortices can be created in the transient regime, but they rapidly coalesce to give rise to larger vortices, and finally only large scale structures persist. The system selects the longest wavelength allowed by the imposed boundary conditions. Fig.(9) and Fig.(10) show the nonlinear evolution of the first and second Fourier modes respectively, showing the initial growth and saturation. We note that the saturation level decreases the higher the mode. Statistical studies presented in [49,50] for 2D systems predict for two dimensional systems a level of the energy associated with the different Fourier modes of the form E k

2

= 1 /(δ + σk 2 ) ( δ and σ are constants), with energy condensing in the low k

modes (inverse cascade). We note the strong influence of the initial conditions on the plasma evolution, although the final state is generally a single vortex structure. We also note the accurate and stable performance of the noiseless Eulerian numerical code, which provided precise information on the phase-space behaviour of the one-dimensional Vlasov plasma.

The Method of Characteristics for the Numerical Solution…

13

Finally we point to the extension of the fractional step method to a fourth order scheme using a symplectic integrator, recently reported in [42].

Figure 9. Time evolution of the first Fourier mode.

Figure 10. Time evolution of the second Fourier mode.

2.2. The Vlasov-Poisson System in Higher Phase-Space Dimensions: the Problem of the Formation of an Electric Field at a Plasma Edge in a Slab Geometry Further evaluation of the performance of the cubic spline interpolation with respect to other interpolation methods, like the cubic interpolated propagation CIP method and the flux corrected transport method, has been presented in [4] and shows the cubic spline interpolation compares favourably with respect to the other methods. We consider in this section the problem of the charge separation at a plasma edge. This problem, with the calculation of the self-consistent electric field along a steep gradient, is of major importance in many physical problems. In tokamak physics, it is highly relevant to the edge physics associated with the

14

M. Shoucri

high confinement mode (H mode). Two methods will be used to study this problem, and the results obtained will be compared. In the first method presented in this section, Cartesian geometry ( a slab model) will be used at the edge of the plasma, and a fractional step technique associated with 1D interpolation using a cubic spline will be applied. In the second method to be presented in section 3.2, we will discuss the solution of the same problem at the plasma edge using cylindrical coordinates ( r , θ , z ) , with a code which applies a 2D interpolation using a tensor product of cubic B-splines [6,51]. The plasma is assumed to be in front of a floating limiter with the magnetic field being aligned parallel to the limiter surface. Electrons are assumed to be frozen along the magnetic field lines. We compare the electric field with the macroscopic values calculated from the same kinetic codes for the gradient of the ion pressure and the Lorentz force term. We find that along the gradient, these quantities balance exactly the electric field. The inhomogeneous direction in the 1D slab geometry considered is the x direction, normal to the limiter plane (y, z). The constant magnetic field is in the y direction (assumed to represent the toroidal direction), and z represents the poloidal direction. The ions are described by the 1D in space ( three phase-space dimensions) Vlasov equation for the ion distribution function f i ( x, v x , v z , t ) :

∂f i ∂f ∂f ∂f + v x i + (E x − v z ω ci ) i + v x ω ci i = 0 ∂t ∂x ∂v x ∂v z

(2.15)

In Eq.(2.15) time is normalized to the inverse ion plasma frequency ω −pi1 , velocity is normalized to the acoustic velocity c s =

Te / M i (Te is the electron temperature and Mi is

the ion mass), and length is normalized to the Debye length λ De = c s / ω pi , where

ω pi is

the ion plasma frequency. The potential is normalized to Te / e , and the density is normalized to the peak initial central density.

ω ci is the ion cyclotron frequency. We assume deuterons

plasma. The system is solved over a length L = 175

λ De in front of the limiter plate, with an

initial density profile for the ions and electrons (indices i and e denote ions and electrons respectively):

ni = ne = 0.5 (1 + tanh (( x − L / 5) / 7 ))

(2.16)

The initial value of the ion distribution function f i ( x, v x , v z ) is given by: 2

2

e − ( vx + vz ) / 2Ti f i ( x, v x , v z ) = ni ( x ) 2πTi

(2.17)

The magnetized electrons are frozen along the magnetic field lines, with a constant profile given by Eq. (2.16). In this case the electrons cannot move across the magnetic field in

The Method of Characteristics for the Numerical Solution…

15

the gradient region to compensate the charge separation which is built up due to the finite ion orbits. It is important to calculate the ion orbits accurately by using an accurate Eulerian Vlasov code. The larger the ion gyroradius, the bigger the charge separation and the selfconsistent electric field at the edge. (Hence the important role played by even a small fraction of impurity ions). The electric field is calculated from the Poisson equation:

∂ 2ϕ = −(ni − ne ) ∂x 2

;

Ex = −

∂ϕ ∂x

(2.18)

The following parameters are used for deuterium ions:

ω ci = 0 .1 ; ω pi

2Ti ρi 1 = = 10 2 λ De Te ω ci / ω pi

Ti = 1; Te

(2.19)

If we assume an initial Maxwellian distribution for the ions with Tix = Tiz = Ti spatially constant, then the factor 2Ti in the calculation of the gyro-radius in Eq.(2.19) takes into 2 > = < v x2 > + < v z2 > = 2Ti / mi . We assume account that the perpendicular temperature < v ⊥

in the present calculation that the deuterons hitting a wall at x = 0 are collected by a floating limiter. Since the magnetized electrons do not move in the x direction across the magnetic field there is no electron current collected at the floating limiter. Therefore we have at x = 0 the relation :

∂E x = − J xi x = 0 ∂t x = 0

t

E x x = 0 = − ∫ J xi x = 0 dt

or

(2.20)

0

Integrating Eq. (2.18) over the domain (0, L), we get the total charge σ in the system: x

E x x = L − E x x = 0 = ∫ (ni − ne ) dx = σ

(2.21)

0

The difference between the electric fields at the boundaries must be equal to the charge appearing in the system. Equation (2.15) is solved by a method of fractional step, in which the advection term in space is solved first, then the equation in velocity space can be solved either using 2D interpolation with a tensor product of cubic B-spline as discussed in [51] (to be applied in section 3.2) , or by successive 1D cubic spline interpolation as follows: Step1- Solve

∂f ∂f + vx = 0 for a step Δt / 2 ∂t ∂x

(2.22)

16

M. Shoucri n +1 / 2

- Solve Poisson equation for the electric field which we denote by E x

.

Step2- Solve

∂f ∂f + ( E xn+1 / 2 − v zω ci ) = 0 for a step Δt / 2 ∂t ∂v x

(2.23)

Step3- Solve

∂f ∂f + v xω ci = 0 for a step Δt ∂t ∂v z

(2.24)

Step4- Repeat Step2 for a time step Δt / 2 Step5- Repeat Step1 for a time step Δt / 2 This splitting leads to the following successive shifts :

f a ( x, v x , v z , t + Δt / 2) = f ( x − v x Δt / 2, v x , v z , t ) ,

(2.25)

f b ( x, v x , v z , t + Δt / 2) = f a ( x, v x − E xn +1 / 2 Δt / 2 + v zω ci Δt / 2, v z , t ) ,

(2.26)

f c ( x, v x , v z , t + Δt ) = f b ( x, v x , v z − v zω ci Δt , t ) ,

(2.27)

We then repeat Eq.(2.26) and Eq.(2.25) to complete the cycle. We can then verify after this sequence that the distribution function f

n +1

at time t = ( n + 1) Δt can be written in the

following form:

f

n +1

*

*

( x, v x , v z ) = f n ( x * , v x , v z )

(2.28)

where:

x * = x − v x Δt +

*

1 * 2 Δt 2 E x Δt − v z ω ci 2 2

(2.29)

1 v *x = v x − E x* Δt + v zω ci Δt − v xω ci2 Δt 2 2

(2.30)

1 1 v *z = v z − v xω ci Δt + ω ci E x* Δt 2 − v z ω ci2 Δt 2 2 2

(2.31)

where E x = E x ( x − v x Δt / 2, t = nΔt + Δt / 2)

The Method of Characteristics for the Numerical Solution…

17

On the other hand , we can consider the characteristics equations for Eq.(2.15) which describe the particles motion:

dx = vx dt

(2.32)

dv x = E x − v z ω ci dt

(2.33)

dv z = v xω ci dt

(2.34)

By integrating the Eqs.(2.32-2.34) from t n = nΔt to t n +1 = (n + 1)Δt , we get:

x n = x n +1 − v xn +1

Δt Δt − v xn 2 2

v xn = v xn +1 − E xn +1 / 2 Δt + ω ci v zn +1 v zn = v zn +1 − v xn +1ω ci

(2.35)

Δt Δt + ω ci v zn 2 2

Δt Δt − v xnω ci 2 2

(2.36)

(2.37)

Eqs(2.35-2.36) leads to the following solution correct to second order in Δt :

x =x n

v =v n x

n +1

n +1 x

−v

−E

n +1 x

Δt + E

n +1 / 2 x

n +1 / 2 x

Δt + ω ci v

2 Δt 2 n +1 Δt − ω ci v z 2 2

(2.38)

Δt 2 2

(2.39)

n +1 z

Δt − ω v

v zn = v zn +1 − v xn +1ω ci Δt + ω ci E xn +1 / 2

2 ci

n +1 x

Δt 2 Δt 2 − v zn +1ω ci2 2 2

(2.40)

By comparing Eqs.(2.29-2.31) to Eqs.(2.38-2.40), we see that the splitting scheme 2

integrates the distribution function along the characteristics correctly to an order O( Δt ) . ( 2

n +1

Note also that to an order O (Δt ) , v z

v zn ).

in the last term in Eq.(2.40) can be substituted by

18

M. Shoucri

. Figure 11. Plot, for the Cartesian geometry, of the electric field Ex (solid curve), the Lorentz force

+ 0.1 < v z >

(dash- dotted curve), the pressure force ∇Pi

∇Pi / ni + 0.1 < v z >

/ ni (dotted

curve), and the sum

(broken curve). The density ni/2 is is also plotted (dash- three-dots curve,

plotted for reference).

We assume that the gyrating plasma deuterons are allowed to enter or leave at the right boundary. So the electric field at the right boundary x = L must be such that the difference between the electric fields at both boundaries in Eq. (2.21) is equal to the total charge σ appearing in the system. Fig. (11) shows at t = 500 the plot of the electric field E x (solid curve, we concentrate on the region x < 100 to emphasize the gradient region, although the system extends to x = 175). We also plot ni / 2 (dash-three-dots curve) in the same figure for reference. The dash-dotted curve gives the Lorentz force, which in our normalized units is given by < v z > ω ci / ω pi = 0.1 < v z > , and the dotted curve gives the pressure force

∇Pi / ni , Pi = 0.5 ni (Tix + Tiz ) , with: Tix, z ( x ) =

< v x, z > =

1 dv x dv z (v x, z − < v x, z > )2 f i (x, v x , v z ) ni ∫

1 dv x dv z v x , z f i (x, v x , v z ) ; ni ∫

(2.41)

n i ( x ) = ∫ dv x dv z f i (x, v x , v z ) (2.42)

In steady state the transport < v x > vanishes. The broken curve in Fig. (11) gives the sum ∇Pi / ni + 0.1 < v z > , which shows a good agreement along the gradient with the solid curve E x . In the region x < 20 we have small oscillations in space (and time), the accuracy of the curve plotted in this region being degraded by the division by ni , due to the low density ni and large ∇Ti appearing close to the surface.

The Method of Characteristics for the Numerical Solution…

Figure 12. Plot of niEx (solid curve),

+ 0.1ni < v z > , (dash-dotted curve) , ∇Pi

19

(dotted curve), and

∇Pi + 0.1ni < v z > ( broken curve), (ni/10 is also plotted for reference).

Figure 13. Charge (ni –ne).

We plot in Fig. (12) the quantities

ni E x , ∇Pi ,

0.1 ni < v z > and the sum

∇Pi + 0.1 ni < v z > . We note that there is a very nice agreement for the relation ni E x = ∇Pi + 0.1 ni < v z > (here the density ni / 10 is plotted with the dash-three-dots curve to locate the profiles with respect to the gradient). The electric field should interact with the constant magnetic field to give an ExB drift in the poloidal direction ( there is no shear in this drift in the flat part of the electric field, which can explain the absence of turbulence at the plasma edge observed in H-mode tokamaks). The charge σ appearing in the system is calculated by the code and amounts to –0.34197 at t = 500. The charge collected and accumulated at x = 0, which defines E x x = 0 from Eq. (2.20), is 0.34535. The difference between these two numbers is ≈ 0.00338, which is E x x = L from Eq. (2.21). We see also from Figs. (11,12) that inside the plasma at the right boundary, in the flat part of the density where ∇Pi = 0 , the constant electric field is exactly compensated by the Lorentz force due to

20

M. Shoucri

the poloidal drift 0.1 < v z > , while along the gradient the electric field is essentially balanced by ∇Pi / ni (the electric drift is equal and opposite to the diamagnetic drift). Fig. (13) shows

the charge density (ni − ne ) at t = 500, which illustrates how the combined effect of the steep profile at a plasma edge and the large ion orbits (large ratio ρ i / λDe ) leads to a charge separation at a plasma edge along the gradient, when the electrons frozen to the magnetic field cannot move across the field to compensate the charge separation caused by the finite ion gyroradius. Fig. (14) shows the potential. Figs (15) and (16) show the temperatures Tix and Tiz (solid curves). The broken lines represent the pressures ni Tix and ni Tiz which follow closely the curve of the density ni . Thus close to the floating limiter a complex sheath structure is formed which governs the plasma-wall transition.

Figure 14. Potential profile.

Figure 15. Temperature Tx.

The Method of Characteristics for the Numerical Solution…

21

Figure 16. Temperature Tz.

2.3. Vlasov-Maxwell Equations for Laser-Plasma Interaction Two systems of equations for 1D laser-plasma interaction will be discussed in this chapter. In the first one, presented in this section, we consider a linearly polarized electromagnetic wave [21]. This system is solved using a fractional step method and uses a cubic spline interpolation to solve for the advection term in the reduced one dimensional equations. In the second system, to be presented in section 3.3 a fully relativistic code is used [52], and the wave is circularly polarized. In this case, to advance the equations in time, we shall use a tensor product of cubic B-spline for a two dimensional interpolation along the characteristics. Comparison for the results obtained by the two methods will be provided at the end of section 3.3, both from the physical point of view and from the numerical point of view to underline the accuracy of the cubic spline interpolation. In the model we present in this section, a linear polarization of the electromagnetic wave is assumed. Time t is normalized to the inverse electron plasma frequency

−1 ω pe , length is

normalized to l 0 = cω pe , velocity and momentum are normalized respectively to the −1

velocity of light c and to M e c , where M e is the electron rest mass. The one-dimensional Vlasov equations for the electron distribution function f e ( x, p xe , t ) and the ion distribution function f i ( x, p xi , t ) are given by [20]:

∂f e,i ∂t

+ me ,i

∂f e,i

p xe,i 2 1/ 2

(1 + (me,i p xe,i ) )

∂x

∓ ( E x + v ey,i B z )

∂f e,i ∂p xe,i

= 0.

(2.43)

The indices e and i refers to electrons and ions. In our normalized units me = 1 , for the electrons, and mi = M e / M i for the ions. The relativistic correction in this case is given by

22

M. Shoucri

γ e,i = (1 + (me,i p xe,i ) 2 )1 / 2 . For the velocity in the direction normal to the gradient, we have the following relation:

∂v ey,i ∂t

= ∓ m e ,i E y .

(2.44)

The electric field is calculated from the relation E x = −

∂ϕ , and the potential ϕ is ∂x

calculated from Poisson equation:

∂ 2ϕ = f e ( x, p xe ) dp xe − ∫ f i ( x, p xi ) dp xi ∂x 2 ∫ We note also that the canonical momentum

. (2.45)

Py = v ey,i / me ,i ∓ a y = 0 . Hence

v ey,i = ± me,i a y , which when derived with respect to time leads to Eq.(2.44) ( E y = −∂a y / ∂t ). a y = eAy / M e c is the normalized y component of the vector potential. The linearly polarized electromagnetic field propagates in the x direction with an electric field Ey in the y direction, normalized to ω pe M e c / e , and a magnetic field B z = ∂a y / ∂x in the z direction, normalized to

ω pe M e / e . We define the quantity E ± = E y ± B z , which obeys

the equation:

(

∂ ∂ ± ) E ± = − J y = v ey ∫ f e ( x, p xe , t ) dp xe − v iy ∫ f i ( x, p xi , t ) dp xi = v ey ne − v iy ni . (2.46) ∂t ∂x The Hamiltonian associated with this system is given by:

H e ,i =

1 1 + (me,i p xe,i ) 2 me ,i

(

We can write Eq.(2.43) in the form:

[H

e ,i

, f e ,i ] =

∂f e ,i ∂t

)

1/ 2

±ϕ +

1 me,i Ay2 . 2

(2.47)

+ [H e ,i , f e ,i ] = 0 , where the Poisson bracket:

∂H e ,i ∂f e ,i ∂p xe ,i ∂x

−

∂H e ,i ∂f e ,i ∂x ∂p xe,i

.

(2.48)

Eqs.(2.43) are solved by a fractional step [20]. The momentum space is divided into Np cells between − p x max e ,i and + p x max e ,i . The length L of the system is divided into Nx cells.

The Method of Characteristics for the Numerical Solution…

23

The fractional step method involves the following steps to advance Eq.(2.43) in time from tn to tn+1 :

Step1- For a time step Δt / 2 , we calculate

f e*,i ( x, p xe,i , t n +1 / 2 ) = f e,i ( x − (me ,i p xe,i / γ e ,i )Δt / 2, p xe,i , t n ) . Step2- Calculate the fields at time t n +1 / 2 using

(2.49)

fe*,i , then use these values to shift for

Δt

in the direction p xe ,i the distribution functions:

f e*,i* ( x, p xe,i , t n +1 / 2 ) = f e*,i ( x, p xe,i ± ( E xn +1 / 2 + v ey,in +1 / 2 B zn +1 / 2 )Δt , t n +1 / 2 ) . (2.50) Step3- Shift again for a time step Δt / 2 in x space:

f e ,i ( x, p xe,i , t n +1 ) = f e*,*i ( x − (me,i p xe,i / γ e ,i )Δt / 2, p xe,i , t n ) .

(2.51)

The shifts in Eqs.(2.49-2.51) are done using cubic spline interpolation (see Appendix B). The solution of Eq.(2.44) between tn and tn+1 is given by the time centered scheme:

v ey,i ( x, t n +1 ) = v ey,i ( x, t n ) ∓ Δt

E + ( x, t n +1 / 2 ) − E − ( x, t n +1 / 2 ) . 2

(2.52)

Eqs.(2.46) are solved using the centered scheme with Δx = Δt :

E ± ( x ± Δt , t n +1 / 2 ) = E ± ( x, t n −1 / 2 ) − ΔtJ y ( x ± Δt / 2, t n ) . with

J y ( x ± Δt / 2, t n ) =

J y ( x ± Δx, t n ) + J y ( x, t n ) 2

(2.53)

.

We integrate exactly along the vacuum characteristic with Δx = Δt , we can write the following numerical scheme for Eq.(2.53):

1 Δt ( ne ( x, t n +1 / 2 )v ey ( x, t n ) + 2 e ne ( x ∓ Δx, t n −1 / 2 )v y ( x ∓ Δx, t n ) − ni ( x, t n +1 / 2 )v iy ( x, t n ) + . (2.54)

E ± ( x, t n +1 / 2 ) = E ± ( x ∓ Δx, t n −1 / 2 ) +

ni ( x ∓ Δx, t n −1 / 2 )v iy ( x ∓ Δx, t n )) n +1 / 2

The calculation of E x

is done by discretizing Eq.(2.45) using a tridiagonal matrix

similar to Eq.(2.12). The solution of Eq.(2.45) in a finite domain require boundary conditions.

24

M. Shoucri

The system is initially neutral. If a charge Q p appears in the system, it has to disappear through the boundaries. A first approximation used in [21] was to divide this charge equally between the two (left and right) boundaries. A more accurate calculation consists in calculating the charge at the two boundaries by collecting the current hitting these boundaries. Let us assume that Ql , Q p , and Qr are the charges calculated during the simulation respectively at the left boundary, in the plasma, and at the right boundary. Ql can be calculated by collecting the current at the left boundary from the relation :

∂E x ∂t

= −J x x =0

x =0

= − ( J xi − J xe ) x =0 .

(2.55)

t

From which :

and

Ex

J xe,i

x =0

x =0

= −∫ J x

= m e ,i

0

0

p xe,i

∫γ

−∞

x =0

dt = Ql .

(2.56)

f e,i (0, p xe,i , t )dp xe,i

e ,i

t

and a similar expression at the right boundary: E x

∞

and

J xe,i

x= L

= m e ,i ∫ 0

p xe,i

γ e ,i

x=L

= −∫ J x 0

x=L

dt = Qr .

(2.57)

f e,i ( L, p xe,i , t )dp xe,i

Integrating Eq.(2.45) over the domain (0, L) , we get : L

Ex

x=L

− Ex

x =0

= ∫ (ni − ne )dx = Q p

(2.58)

0

where Qp is the charge appearing in the plasma. The charge appearing at the right boundary is

Qr = E x

x=L

, and the charge appearing at the left boundary is Ql = E x

x =0

. Eq.(2.58) is

also written Ql + Q p − Qr = 0 . This relation is usually verified by the code. To take into account any imbalance in this relation, let us consider the electric field E x ( x = x0 ) at a point x0 far to the left of the simulation box and boundaries. The electric field is the sum of the fields from Ql , Q p , and Qr . Since a plate of charge q gives an electric field E x = q / 2

The Method of Characteristics for the Numerical Solution…

25

for a point to the right of the plate ( and E x = − q / 2 for a point to the left ), we can write with the convention of signs in Eq.(2.58) [53]:

E x ( x0 ) = (−Ql − Q p + Qr ) / 2 .

(2.59)

Now if we move from x = x 0 to x = 0 + , just inside the left boundary, Ql is now to our left, so it will contribute to the field by Ql / 2 instead of − Ql / 2 , and the electric field is now given by:

E x ( x = 0 + ) = (Ql − Q p + Qr ) / 2 .

(2.60)

This is the boundary condition used in [53]. Now consider the case when the charges are exactly balanced in the system, then as we mentioned before Ql + Q p − Qr = 0 , or

Q p = Qr − Ql , and substituting in Eq.(2.60), we get E x ( x = 0) = Ql , which is the result in Eq.(2.56). Finally we note that the pump wave is penetrating the plasma at the left boundary at x=0 where we set E ( x = 0) = 2 E0 sin(ω 0 t ) , E ( x = 0) = 0 , for the solution of Eqs.(2.53) . +

−

The normalized wave amplitude E 0 = E am e /(ω epw M e c) ≡

Vosc / c n / nc

amplitude and nc the critical density. This value of

E 0 = (0.00854265λ I 0 ) / n / nc .

We

use

the

, where Eam is the wave

E 0 can also be written

parameters

of

[53],

where

λ = 0.527 μm for the laser, and n/nc =0.032, and the intensity I0=100 (in units of 1014 W/cm2 ). This results in E 0 = 0.25 .

Figure 17. Frequency spectrum of E+.

26

M. Shoucri

Figure 18. Wavenumber spectrum of E+.

We use again the same parameters as in [53]. The laser pump is

ω 0 = 5.59ω pe and the

laser wavenumber is k 0 = 5.5(ω pe / c) . For the scattered mode we have and k SRS = 4.4(ω pe / c) . For the plasma wave we have

ω SRS = 4.478ω pe

ω epw = 1.1124ω pe , and

k epw = 9.86(ω pe / c) . The electron thermal velocity is vTe = 0.026c , Te / Ti = 3.5 . The length of the system is L = 50.265 , and N x = 5000 grid points in space, N v = 256 grid points in velocity space for electrons and 128 for ions. Δx = Δt =0.0105 . We show in +

Fig.(17) the results obtained for the frequency spectrum of the electromagnetic field E at the position x=5 at t=60. We can identify the contribution of the pump and scattered mode at

ω 0 and ω SRS . Fig.(18) shows the wavenumber spectrum for E + at t=60, where again we

Figure 19. Frequency spectrum for the plasma wave.

The Method of Characteristics for the Numerical Solution…

27

Figure 20. Wavenumber spectrum for the plasma wave.

can identify the contribution of the laser pump k 0 = 5.5(ω pe / c) and the scattered mode at

k SRS = 4.4(ω pe / c) . Fig.(19) shows the frequency spectrum of the plasma wave showing the

ω epw = 1.1124ω pe peak and a peak at ω h = 11.18ω pe , i.e. the harmonic of the pump

wave

ω 0 . We can identify in Fig.(20) the wavenumber peak at k epw = 9.86(ω pe / c) for the

plasma wave, followed by a small neighbouring peak at k h = 11(ω pe / c) , i.e. the harmonic of the pump wave k 0 . The harmonic peaks at k h and

ω h in Fig.(19) and (20) result from the

v y B z term in Eq.(2.43). This can also be verified in the results in Fig.(7) of [54] where we see the response of the plasma at the harmonic of the electromagnetic wave frequency, when using a model similar to what has been presented in this section and in Eq.(2.43) applied to the problem of inductive coupling. Indeed, if we assume a linearly polarized wave:

E = (0, E y ,0) , we can write in a linear analysis E y = E 0 cos(ψ ), ψ = (k 0 x − ω 0 t ) . Faraday’s law is:

∂E y ∂B = (0,0,− ). ∂t ∂x

(2.61)

Then B = (0,0, B z ) with B z = B0 cos(ψ ), and B0 = E 0 k 0 / ω 0 . Also from Eq.(2.44)

v = (0, v y ,0) with v y = −v 0 sin(ψ ), and v0 = E 0 / ω 0 . The longitudinal Lorentz force 1 v y B z = − k 0 v 02 sin(2ψ ) . This drive a longitudinal response at the 2nd harmonic of the 2

light wave.

28

M. Shoucri

Figure 21. Contour plot and 3D view for the phase-space of the distribution function from x=5.1 to x=9.8 at

−1 t = 60ω pe .

Figure 22. Longitudinal electric field.

The Method of Characteristics for the Numerical Solution…

29

Figure 23. Charge (ni–ne).

Fig.(21) shows the phase-space structure in contour plot and 3D view of the tail of the electron distribution function between x= 5.1 and x=9.8 at t=60. Note the clear structure of the vortices in Fig.(21), in the low density regions of the phase-space, without numerical noise Fig.(22) shows the electric field E x at t=60 across the simulation box, and Fig.(23) shows the charge (ni –ne ). We note the system is solved for ions and electrons, but the response of the ions at t=60 is still negligible. In the conclusion of this section 2 , we stress the importance of the cubic spline and the good performance of the interpolation technique, having low numerical diffusion and dispersion and high accuracy. We also note the numerical stability of the numerical code.

3. Problems Involving the Interpolation along the Characteristic Curves in Two Dimensions The problems studied in section 2 for the Vlasov equation dealt essentially with the fractional step methods where the interpolation along the characteristic curves was carried out in 1D using a cubic spline. We present in this section examples where the interpolation along the characteristic curves is carried in two dimensions , using a tensor product of cubic B-splines . The integration along the characteristics in higher dimensions applied to Eulerian Vlasov codes has been formulated sometime ago in [15,16], and only recently applied [6,9]. The first example we present in this section is the solution of the guiding-center equations in 2D ( which are the equations of a plasma in a strong magnetic field) to study the Kelvin-Helmholtz instabilities. These equations are isomorphic with the Euler equations that govern 2D inviscid incompressible fluids in hydrodynamics [55-60]. We use this example to introduce in section 3.1 the methods of 2D interpolation discussed in section 1 and in [6,9]. In section 3.2 we will reconsider the problem of the formation of an electric field at a plasma edge, presented in section 2.2 , however we use this time a cylindrical geometry in the Vlasov equation. We will consider next in section 3.3 a case of laser-plasma interaction similar to what has been presented in section 2.3 , treated however with a circularly polarized wave and a fully relativistic Vlasov equation. The results obtained in this section will be compared both from the physical point of view as well as from the numerical point of view with the corresponding

30

M. Shoucri

results in sections 2.2 and 2.3, especially concerning the accuracy of the results and of the numerical techniques using cubic spline interpolation. Finally the problem of solving a reduced set of magnetohydrodynamic equations to study the problem of magnetic reconnection will be presented in section 3.4.

3.1. Solution of the Guiding-Center or Euler Equations In the two-dimensional space r = ( x, y ) , the evolution of a plasma in a strong magnetic field [55-60] B = Be z is governed by the guiding center equation :

∂ρ ∂ρ + VD . = 0. ∂t ∂r with the drift or guiding-center velocity VD = e z ×

(3.1)

∂φ ∂φ , E=− , and φ is calculated ∂r ∂r

from Poisson equation:

Δφ = − ρ The initial condition for the charge density is

(3.2)

ρ (r, t = 0) = ρ 0 (r ) . Eqs.(3.1) and (3.2)

are isomorphic to the Euler equations that govern 2D inviscid incompressible fluids in hydrodynamics, where the charge density corresponds to the flow vorticity, and the potential φ corresponds to the stream function [57]. For a plasma in a strong magnetic field, the particle motion along the magnetic field B and across the magnetic field B are decoupled, the velocity perpendicular to the magnetic field is the guiding-center velocity VD and the Vlasov equation can be reduced to the guiding-center plasma model in Eqs(3.1-3.2). These equations are also the simplest form for the gyro-kinetic or drift-kinetic Vlasov equation, when the kinetic motion along the magnetic field is taken into account and is coupled to the guidingcenter motion across the magnetic field, as for instance in the case when the magnetic field is slightly tilted with respect to e z [32-38,61]. The system in Eqs.(3.1-3.2) is solved on a rectangular domain L x × L y . Periodic boundary conditions are used in the y direction, and Dirichlet boundary conditions are used in the x direction with

φ ( x = 0, L x , y ) = 0 ,

ρ ( x = 0, L x , y ) = 0 . Following the steps of what has been presented in Eqs.(1.8-1.14), Eq.(3.1) can be integrated along the characteristic curves given by the solution of the equations:

dr = VD (r, t ) . dt

(3.3)

The Method of Characteristics for the Numerical Solution… We write that the value of

31

ρ along the characteristic curves is constant (see Eq.(1.11)): ρ (r, t n+1 ) = ρ (r (t n−1 ), t n −1 ) .

(3.4)

We assume the value of r (t = t n +1 ) = ri , where the ri are grid points. A numerical scheme accurate to second order in Δt for the solution of Eq.(3.3) is given by the following leapfrog scheme (similar to Eq.(1.12)):

ri − r (t n−1 ) ⎛ r + r (t n−1 ) ⎞ = VDn ⎜ i ⎟. 2Δt 2 ⎝ ⎠ where VD = e z × n

(3.5)

∂φ (r, t n ) , and E(r, t n ) is computed by solving Poisson equation for ∂r

ρ (r, t n ) at t = t n . To solve Eq.(3.2), we Fourier transform this equation in the periodic y direction. We denote by

ρ k and φ k the Fourier transform of ρ and φ in the periodic y y

y

direction. We have:

φ ( x, y ) = ∑ e

ik y y

φ k ( x ) ; ρ ( x, y ) = ∑ e

ik y y

y

ky

ρ k ( x) y

ky

Which by substituting in Eq.(3.2) gives the following result:

∂ 2φ k y ( x ) ∂x

2

− k y2φ k y ( x) = − ρ k y ( x) .

(3.6)

We can derive the following tridiagonal matrix by discretizing Eq.(3.6) in the x direction [61]:

(1 − C k y )φ k y i +1 − (+2 + 10C k y )φ k y i + (1 − C k y )φ k y i −1 = −

where C k y = conditions for

k y2 Δx 2 12

Δx 2 ( ρ k y i −1 + 10 ρ k yi + ρ k y i +1 ) . (3.7) 12

. The tridiagonal matrix in Eq.(3.7) is solved with Dirichlet boundary

φ k , which is then Fourier transformed back to get φ . Again the implicit y

equation in Eq.(3.5) is solved by iteration as in Eq.(1.13), to calculate Δ r = (ri − r (t n −1 )) / 2 as follows:,

Δkr+1 = ΔtVD (ri − Δkr , t n ) .

32

M. Shoucri From Eqs.(1.14) and (3.4) the new value

.

ρ (ri , t n +1 ) is calculated from the relation:

ρ (ri , t n + Δt ) = ρ (ri − 2 * Δ r , t n − Δt ) .

More explicitly we have to calculate (see Fig.(1))

(3.8)

ρ ( x i − 2 Δ x , y j − 2 Δ y , t n − Δt ) ,

where Δ x and Δ y are obtained by iteration by solving :

where V Dy =

Δkx+1 = ΔtV Dx ( xi − Δkx , y j − Δky , t n )

(3.9)

Δky+1 = ΔtVDy ( xi − Δkx , y j − Δky , t n )

(3.10)

∂φ ∂φ 0 0 ; V Dx = − . We start with Δ x = 0 , Δ y = 0 . Usually two or three ∂x ∂y

iterations are sufficient for convergence. The interpolations in Eqs.(3.8-3.10) are carried out using a 2D cubic B-splines defined as a tensor product of one dimensional cubic B-splines. We first evaluate the coefficients η ij from the values of the function at grid points (details can be found in [6], we have however to introduce appropriate modification to the boundary condition in the direction y, to take into account the periodic boundary condition as in [51]): Nx N y

ρ ( xi , y j ) = ∑∑η ij Bi ( x) B j ( y ) .

(3.11)

i =0 j =0

The cubic B-spline has been defined in Appendix C . Eq.(3.11) generalizes to two dimensions the results presented in Appendix C. Then the value of the function at interstitial points ρ ( xi + Δ x , y j + Δ y ) is given by: 3

3

ρ ( xi + Δ x , y j + Δ y ) = ∑∑η i −κ , j −l bκx bly .

(3.12)

κ =0 l =0

where i ≡ xi , Δ x = x − xi , and j ≡ y j , Δ y = y − y j , bκ are defined in Appendix C, x

and a similar definition holds for bl by substituting y

Δ y for Δ x in the expression of bκx .

We finally note that in [9], in order to accelerate the calculation, a linear interpolation has been used in the intermediate step in Eqs. (3.9-3.10). This turned out however to introduce strong numerical diffusion [43,44]. We apply the previously described numerical method to study the stability of a sinusoidal profile [58,59]. Any function of one space variable is an equilibrium solution to Eqs(3.1-3.2). We consider the sinusoidal profile ρ 0 = sin( x) . From Eq.(3.2) the self-consistent potential

The Method of Characteristics for the Numerical Solution… is given by

φ 0 = sin( x) . A necessary condition for the flow V D 0 =

33

∂φ 0 = cos( x) to have ∂x

an unstable growing solution is that this flow should have a point of inflexion. This is the Rayleigh necessary condition for instability. This instability due to the shear in the velocity flow is called Kelvin-Helmholtz instability. Furthermore if the equilibrium is perturbed with a perturbation of the form e values of

ω such that

ω ky

− i (ωt + k y y )

, k y = 2π / L y , there exists an eigensolution with real

= V D ( x s ) (the so-called neutrally stable eigensolution), where xs

is the point of inflexion, and one can construct unstable solutions for which

ω ky

→ VD ( x s )

ω tends to zero through positive values. In the present case, we consider a domain with L x = 2π , and L y = 10 . We use Nx =Ny =256 and Δt =0.005 . In as the imaginary part of

the domain 0 ≤ x ≤ 2π where the equilibrium flow is defined we have two points of inflexion

ω ky

located

xs = π / 2

at

x s = 3π / 2 .

and

At

these

points

= VD ( x s ) = cos( x s ) = 0 . In this case ω = ω s = 0 , and a neutrally stable solution

satisfying the boundary conditions of zero at x = 0 and x = 2π is given by [58,59] :

x 2

(3.13)

3 2

(3.14)

φ s = sin( ) .;

k ys = We perturb the equilibrium

ρ 0 as follows: x 2

ρ 0 = sin( x ) + ε sin( ) cos(k y y ) .

(3.15)

with k y = 2π / L y =0.628, and ε = 0.015 . Since we are close to a neutrally stable solution, we can use a Taylor expansion to calculate the growth rate of the instability:

ω ky

=

ω ky

+ (k y − k ys ) ω s , k ys

∂ ∂k y

⎛ω ⎜ ⎜k ⎝ y

⎞ ⎟ . ⎟ ⎠ ω s ,k ys

(3.16)

Details of these calculations have been presented in [58,59], and lead to the following result for the real and imaginary parts of ω :

34

M. Shoucri

Re ω 3 = (k ys − k y ) . ky 2

(3.17)

Im ω 3 = (k ys − k y ) . ky 2

(3.18)

Figure 24. Growth and saturation of the potential at the position x=3Lx /4.

Figure 25. Contour plot of the potential at t=80.

In this case k y < k ys , and the system is unstable [58,59]. From Eq.(3.17), the phase velocity

Re ω is equal to 0.2058. We do verify by following the center of the vortex in ky

Figs(25-26) that the phase velocity of the vortex is indeed 0.2 . Fig.(24) shows the growth and oscillation of the potential , monitored at the position x = 3Lx / 4; y = 0 . The theoretical results from Eqs(3.17-3.18) show Im ω = Re ω = 0.129 , in good agreement with the observations taken from the results of Figs.(24), which shows a growth and an oscillation period of 0.124. Figs(25-26) shows the contour plot of the potential at t=80 and 120 during the saturation phase, and the corresponding charge is given in Figs.(27-28) (dotted curves denote negative values). Note the very nice agreement of the theoretical and numerical results.

The Method of Characteristics for the Numerical Solution…

Figure 26. Contour plot of the potential at t=120.

Figure 27. Contour plot of the charge at t=80.

Figure 28. Contour plot of the charge at t=80.

35

36

M. Shoucri

3.2. The Vlasov-Poisson System in Higher Phase-Space Dimensions: Formation of an Electric Field at a Plasma edge in a Cylindrical Geometry We will discuss in this section, using cylindrical geometry, the problem of the formation of an electric field at a plasma edge which has been studied in section 2.2 using a slab geometry. In cylindrical geometry, it is more convenient to use a tensor product of cubic B-spline to interpolate in 2D velocity space, while in section 2.2 we applied a fractional step associated with 1D cubic spline interpolation. Hence we have the opportunity to compare two completely different codes, and to evaluate the performance of the cubic spline for the solution of the same problem. In the present cylindrical geometry, the external magnetic field is in the z direction, and ( r , θ ) is the poloidal plane. The plasma is assumed uniform in the z and θ directions. Electrons are magnetized along the magnetic field, and consequently have a constant profile. The Vlasov and Poisson equations are written for the deuterons distribution function f i (r , v r , vθ , t ) and for the potential ϕ (r ) with the same normalization and parameters as in section 2.2, as follows (see [6] ):

v v ⎞ ∂f v 2 ⎤ ∂f ∂f i ∂f ⎡ ⎛ + v r i + ⎢ E r + vθ ω ci + θ ⎥ i − ⎜ ω ci v r + r θ ⎟ i = 0 r ⎥⎦ ∂v r ⎝ r ⎠ ∂vθ ∂t ∂r ⎢⎣ Er = −

∂φ ; ∂r

1 ∂ ∂φ r = −(ni − ne ) ; ni = r ∂r ∂r

∫f

i

(3.19)

(3.20)

dv r dvθ

We advance Eq. (3.19) for a time step Δt as follows: Step1-We solve for a time step Δt / 2 , using cubic spline interpolation, the equation:

∂f i ∂f + vr i = 0 ∂t ∂r *

(3.21)

the solution is given by f i ( r , v r , vθ , t + Δt / 2) = f i ( r − v r Δt / 2, v r , vθ , t )

(3.22)

n

We calculate the shift in Eq.(3.22) using 1D cubic spline interpolation as discussed *

before. We then solve Poisson equation in Eq.(3.20) to calculate the electric field E r . Step2-We solve next for a time step Δt the equation:

∂f i ⎡ * vθ2 ⎤ ∂f i ⎛ v v ⎞ ∂f + ⎢ E r + vθ ω ci + ⎥ − ⎜ ω ci v r + r θ ⎟ i = 0 ∂t ⎣ r ⎦ ∂v r ⎝ r ⎠ ∂vθ

(3.23)

Splitting Eq.(3.23) is not straightforward as in Cartesian geometry. This equation is solved using 2D interpolation. The characteristics of this equation are given by:

The Method of Characteristics for the Numerical Solution…

v 2 dv v v dv r = E r + vϑ ω ci + ϑ ; ϑ = −v r ω ci − ϑ r dt r dt r

37

(3.24)

The solution of Eqs.(3.23) is given by:

f i ** ( r , vr , vθ , t + Δt ) = f * ( r , v r − 2a, vθ − 2b, t )

(3.25)

The shift in Eq. (3.25) is effected using a tensor product of cubic B-spline [6,51] for the 2D interpolation, as discussed in section 3.1. However, the quantities a and b in Eq.(3.25) are calculated analytically by solving the equations of the characteristics in Eqs.(3.24), since in the present case an analytic solution is possible, similar to what has been presented in Eqs.(2.38-2.40). This solution gives, to an order O ( Δt 2 ) , at a given position r, the expressions: a=

b=−

Δt 2

Δt 2

⎛ ⎜ E r + ω ci ⎜ ⎝

⎛ v v Δt ⎛ ⎜⎜ vθ + ⎜ ω ci v r + r θ 2 ⎝ r ⎝

2 ⎞ v ⎞ ⎞ vθ + Δt θ (ω ci v r + v r vθ / r ) ⎟ ⎟ ⎟⎟ + ⎟ r ⎠⎠ r ⎠

⎛ vv vv ⎜ ω ci vr + r θ + Δt vr ⎛⎜ ω ci vr + r θ ⎞⎟ − Δt ⎜ r ⎠ 2 r r 2 ⎝ ⎝

(3.26)

v 2 ⎞⎞ v ⎞⎛ ⎛ ⎜ ω ci + θ ⎟⎜⎜ E r + vθ ω ci + θ ⎟⎟ ⎟⎟ (3.27) r ⎠⎠ r ⎠⎝ ⎝

( note that an iterative solution of Eqs.(3.24) as in section 3.1 would give the same results to an order O ( Δt 2 ) ). Step3-We then repeat the step in Eq. (3.22) for Δt / 2 to calculate f n +1 from f ** . The initial profiles are the same as in section 2.2, written in the cylindrical geometry as:

ni (r ) = ne (r ) = 0.5 (1 + tanh (( R − r − L / 5) / 7.)) ;

(3.28)

with a similar profile for the frozen electrons. The positive r direction is pointing towards the right, or outside the plasma. R is the plasma radius and L the width of the edge (taken to be 175 as in section 2.2). The initial ion distribution is given by: 2

2

e − ( vr +vθ ) / 2Ti f i (r , v r , vθ ) = ni (r ) 2πTi

(3.29)

We assume that the deuterons hitting the wall surface or limiter at r= R are collected by a floating cylindrical vessel. Since the magnetized electrons do not move in the r direction across the magnetic field, there is no electron current collected at the floating vessel.

38

M. Shoucri

Therefore we have at r= R the relation (we stress that the subscript i denotes here ion contribution):

∂E r = − J ri r = R ∂t r = R

t

E r r = R = − ∫ J ri r = R dt

or

(3.30)

0

Integrating Eq. (3.20) over the domain (R-L, R), we get for the total charge σ appearing in the system: R

R Er

r=R

− ( R − L) E r

r = R− L

=

∫ (n

i

− ne )rdr = σ

(3.31)

R− L

which is the equivalent to Eq.(2.21) obtained for the slab geometry. We assume that the gyrating plasma ions (deuterons) are allowed to enter or leave at the left boundary. The electric fields at the left boundary r = R - L and at the wall r = R must satisfy Eq. (3.31). We use a very large value of R (R = 10000 Debye lengths in the present calculation, so vθ2 / r is negligible), so that the system should behave essentially as a Cartesian system. Indeed we recover the same results as those which have been presented in Cartesian geometry in section 2.2. These results have been presented for cylindrical geometry in Fig.(1-6) in [51], and are identical to Figs(11-16) ( if we take into account the mirroring due to the fact that in the Cartesian geometry and in the cylindrical geometry, the edge gradients are in opposite directions). Fig. (29) shows at t = 500 the plot of the electric field Er (solid curve, we concentrate on the region less than 100 Debye lengths from the boundary to emphasize the edge region). To position the profiles in Fig. (29) with respect to the gradient we also plot − ni / 2 in the same figure. The electric field has the direction of pushing the ions back to the interior of the plasma (and interact with the magnetic field to give a poloidal drift rotation, note also that in the flat part of the electric field, this drift has no shear, which can explain the absence of turbulence). The dash-dotted curve gives the Lorentz force, which in our normalized units is given by − < vθ > ω ci / ω pi , and the dotted curve gives the pressure

force ∇Pi / ni , Pi = 0.5 ni (Tir + Tiθ ) , with the following definition in cylindrical geometry:

Tir ,θ ( r ) =

< v r , θ >=

1 dv r dvθ ( v r ,θ − < v r , θ > ) 2 f i (r , v r , vθ ) ni ∫

1 dv r dvθ v r , θ f i (r, v r , vθ ) ; ni ∫

ni ( r ) = ∫ dv r dvθ f i (r, v r , vθ )

(3.32)

(3.33)

In steady state the transport < v r > vanishes. The broken curve in Fig. (29) gives the sum ∇Pi / ni − 0.1 < vθ > , which shows a very good agreement along the gradient with the solid curve for E r . In the region less than 20 Debye lengths from the wall, we have small

The Method of Characteristics for the Numerical Solution…

39

oscillations in space (and time), the accuracy being degraded by the low density ni and large

∇Ti

appearing

close

to

the

surface.

We

plot

in

Fig. (30)

the

quantities

n i E r , ∇Pi , − 0.1 n i < vθ > and the sum ∇Pi − 0.1n i < vθ > . We see that there is a very nice agreement for the relation ni E r = ∇Pi − 0.1 ni < vθ > (the density − ni / 10 is also plotted to locate the profiles with respect to the gradient). The charge σ / R appearing in the system and calculated by the code from Eq.(3.31) amounts to -0.360 at t = 500. The collected charge calculated from Eq.(3.29) at r = R is 0.364, hence E r r = R = −0.364 . The difference E r r = R −σ / R as calculated from Eq.(3.30) gives for E r r = R − L the value of -0.004, which is very close to the value obtained by the code at R – r = 175 (see Fig. (30)). We see also from Fig. (30) that at the left boundary inside the plasma in the flat part of the density where ∇Pi = 0 , the electric field is compensated by the Lorentz force due to the poloidal drift − 0.1 < vθ > , while along the gradient the electric field is essentially balanced by ∇Pi / ni . Figs. (3-6) in [51] are reproducing essentially Fig.(13-16) of section 2.2 (taking into consideration the mirroring due to the difference in the positive direction). We see that the results in section 2.2 obtained in Cartesian geometry by a fractional step method associated with 1D cubic spline interpolation are the same as those obtained in this section using a cylindrical geometry , and associated with 2D interpolation in velocity space with a tensor product of cubic B-spline. By using two different numerical techniques based on the cubic spline interpolation with two different coordinate systems, we get identic results for the same problem. The curves in Figs(11-16) and in Figs(1-6) of [51] gives essentially identical results. This illustrates the accuracy of the method of characteristics used , and of the cubic spline used for the numerical interpolation.

Figure 29. Plot, for the cylindrical geometry, of the electric field Er (solid curve), the Lorentz force

− 0.1 < vθ > sum ∇Pi

(dash- dotted curve), the pressure force ∇Pi

/ ni − 0.1 < vθ >

plotted for reference).

/ ni (dotted

curve), and the

(broken curve). The density -ni/2 is is also plotted (dash-three-dots curve,

40

M. Shoucri

Figure 30. Plot of niEr (solid curve),

− 0.1ni < vθ > , (dash-dotted curve) , ∇Pi

(dotted curve), and

∇Pi − 0.1ni < vθ > ( broken curve), (-ni/10 is also plotted for reference).

3.3. One-Dimensional Fully Relativistic System for the Problem of LaserPlasma Interaction The problem of laser-plasma interaction treated in section 2.3 with a linear polarization will be repeated in this section with a full relativistic equation with a circular polarization. In the present case the fractional step will not be used , we will rather apply a 2D interpolation using a tensor product of cubic B-splines. When studying similarities or differences in the results, attention will be given to the accurate performance of the cubic spline interpolation. The general form of the Vlasov equation is written for the present problem ( using the same normalization as in section 2.3 ) in a 4D phase-space for the electron distribution function Fe ( x, p xe , p ye , p ze , t ) and the ion distribution function Fi ( x, p xi , p yi , p zi , t ) (one spatial dimension) as follows [52]:

∂Fe ,i ∂t with

+ me ,i

p xe,i ∂Fe ,i

γ e ,i

∂x

∓ (E +

pxB

γ e ,i

)⋅

∂Fe,i ∂p e ,i

= 0.

γ e ,i = (1 + me2,i ( p xe2 ,i + p ye2 ,i + p ze2 ,i ) )

1/ 2

(3.34)

(3.35)

(the upper sign is for electrons and the lower sign for ions, and subscripts e or i denote electrons or ions respectively). Again in our normalized units me = 1 and mi =

Me . Mi

Eq.(3.34) can be reduced to a two-dimensional phase-space Vlasov equation if the canonical momentum Pce,i connected to the particle momentum p e ,i by the relation Pce ,i = p e ,i ∓ a is chosen initially as zero. a = eA / M e c is the normalized vector potential. For a particle in an

The Method of Characteristics for the Numerical Solution…

41

electromagnetic wave propagating in a one-dimensional spatial system, we can write the following Hamiltonian:

H e ,i = where

(

1 1 + me2,i ( Pce,i ± a ) 2 me ,i

)

1/ 2

±ϕ .

(3.36)

ϕ is the electrostatic potential. Choosing the Coulomb gauge ( diva = 0 ) , we have

for the vector potential a = a ⊥ ( x, t ) , and we also have the following relation along the longitudinal direction:

dPcxe,i dt

=−

∂H e,i

(3.37)

∂x

And since there is no transverse dependence :

dPc ⊥e ,i dt

= −∇ ⊥ H e ,i = 0 .

(3.38)

This last equation means Pc ⊥ e ,i = const. We can choose this constant to be zero without loss of generality, which means that initially all particles at a given (x,t) have the same perpendicular momentum p e ,i = ± a ⊥ ( x, t ) . The Hamiltonian now is written:

H e ,i =

1/ 2 1 1 + me2,i p xe2 ,i + me2,i a ⊥2 ( x, t ) ± ϕ ( x, t ) . m e ,i

(

)

(3.39)

The 4D distribution function Fe ,i ( x, p x , p ⊥ , t ) can now be reduced to a 2D distribution function f e ,i ( x, p xe ,i , t ) corresponding to Eq.(3.39):

df e,i dt

=

∂f e,i ∂t

+

∂H e,i ∂f e,i ∂p xe,i ∂x

−

∂H e,i ∂f e,i ∂x ∂p xe,i

= 0.

(3.40)

Which gives the following Vlasov equations for electrons and ions::

∂f e ,i ∂t Where

+ me ,i

p xe,i ∂f e ,i

γ e ,i ∂x

(

+ (∓ E x −

me ,i ∂a ⊥2 ∂f e,i ) = 0. 2γ e ,i ∂x ∂p xe,i

γ e,i = 1 + (me,i p e,i )2 + (me,i a ⊥ )2

)

1/ 2

.

(3.41)

42

M. Shoucri

Ex = −

∂a ∂ϕ and E ⊥ = − ⊥ ∂x ∂t

(3.42)

and Poisson equation is given by Eq.(2.45). The transverse electromagnetic fields E y , B z and E z , B y for the circularly polarized wave obey Maxwell’s equations. With

E ± = E y ± B z and F ± = E z ± B y , we have:

(

∂ ∂ ∂ ∂ ± )E ± = − J y . ; ( ∓ )F ± = − J z ∂t ∂x ∂t ∂x

(3.43)

Which are integrated along their vacuum characteristic x=t. In our normalized units we have the following expressions for the normal current densities:

J ⊥ e ,i = − a ⊥ m e ,i ∫

J ⊥ = J ⊥ e + J ⊥i ;

f e ,i

γ e ,i

dp xe,i .

(3.44)

The numerical scheme to advance Eq.(3.41) from time tn to tn+1 necessitates the ±

±

knowledge of the electromagnetic field E and F at time tn+1/2 . This is done using a scheme similar to Eq.(2.53), where we integrate Eq.(3.43) exactly along the vacuum characteristics with Δx = Δt , to calculate E have a

n +1 ⊥

= a − ΔtE n ⊥

n +1 / 2

calculate E x

n +1 / 2 ⊥

± n +1 / 2

and F

± n +1 / 2

, from which we calculate a

. From Eq.(3.42) we also

n +1 / 2 ⊥

= (a ⊥n +1 + a ⊥n ) / 2 . To

, two methods have been used. A first method calculates E x from f e ,i n

n

using Poisson equation, then we use a Taylor expansion::

Δt ⎛ ∂E ⎞ ⎛ Δt ⎞ = E + ⎜ x ⎟ + 0.5⎜ ⎟ 2 ⎝ ∂t ⎠ ⎝ 2⎠ n

E

n +1 / 2 x

⎛ ∂ Ex ⎜⎜ 2 ⎝ ∂t 2

⎛ ∂E ⎞ with ⎜ x ⎟ = − J xn ; ⎝ ∂t ⎠ n

+∞

J xn = mi

and

p xi

∫γ

−∞

n +1 / 2

n

+∞

n +1 / 2

n

⎛ ∂ 2 Ex ⎞ ⎜⎜ 2 ⎟⎟ ; ⎝ ∂t ⎠

⎞ ⎛ ∂J ⎞ ⎟⎟ = −⎜ x ⎟ ⎝ ∂t ⎠ ⎠

f i n dp xi − me

i

A second method to calculate E x which E x

2

n x

p xe

∫γ

−∞

n

.

(3.45)

f en dp xe

e

is to use Ampère’s equation:

∂E x = − J x , from ∂t

= E xn −1 / 2 − ΔtJ xn . Both methods gave the same results. The boundary

conditions are the same as what has been discussed in section 2.3. Now given f e ,i at mesh n

points (we stress here that the subscript i denotes the ion distribution function), we follow the

The Method of Characteristics for the Numerical Solution… n +1

same steps as in section 3.1 to calculate the new value f e ,i

43

at mesh points from the

relations:

f en,i+1 ( X e ,i ) = f en,i ( X e,i − 2Δ X e ,i ) ; .

(3.46)

where Δ X e,i is the two dimensional vector:

Δ X e ,i =

X e,i

is

the

two

Δt V ( X e ,i - Δ X e ,i , t n +1 / 2 ) . 2

dimensional

(3.47)

X e ,i = (x, p xe,i ) ,

vector

and

⎛ p xe,i me ,i ∂ ( a ⊥( n +1 / 2) ) 2 ⎞ ⎟ . Eq.(3.47) for Δ X e,i is implicit and is Ve ,i = ⎜⎜ me,i , ∓ E xn +1 / 2 − ⎟ 2 ∂ γ γ x , , e i e i ⎝ ⎠ n +1 n solved again iteratively as in Eq.(3.9-3.10). Then f e ,i is calculated by interpolating f e ,i in Eq.(3.46) in the two dimensions ( x, p xe,i ) using a tensor product of cubic B-splines [6] as discussed for Eqs.(3.11-3.12). We use the same parameters as section 2.3 and in [53]. The pump wave is penetrating the plasma at the left boundary at x=0 where we set E ( x = 0) = 2 E0 sin(ω 0 t ) and +

F − ( x = 0) = 2 E0 cos(ω 0 t ) . The laser pump is ω 0 = 5.59ω pe and the laser wavenumber is k 0 = 5.5(ω pe / c ) . For the scattered mode we have

ω SRS = 4.478ω pe

and

k SRS = 4.4(ω pe / c) . For the plasma wave we have ω epw = 1.1124ω pe , and k epw = 9.86(ω pe / c) . The electron thermal velocity is vTe = 0.026c , Te / Ti = 3.5 . The length of the system is L = 50.265 , and N x = 5000 grid points in space, N v = 256 grid points in velocity space for electrons and 128 for ions. Δx = Δt =0.0105 . Figs.(31-37) show the results obtained when using the present fully relativistic model. Fig(31) for the frequency +

spectrum of the electromagnetic wave E at x=5 and t=60 is very close to Fig.(17). We can identify the contribution of the pump and the scattered mode at ω 0 = 5.59ω pe and

ω SRS = 4.478ω pe . Fig.(32) for the wavenumber spectrum of E + is essentially the same as Fig.(18),

we

can

identify

k 0 = 5.5(ω pe / c )

for

the

laser

pump

wave,

and

k SRS = 4.4(ω pe / c) for the scattered mode. Figs.(33) and (34) differ from Fig.(19) and (20) by the absence of the harmonic peaks at

ω h = 11.18ω pe and k h = 11(ω pe / c) . These

harmonic peaks in Fig.(19) and (20) result from the v y B z term in Eq.(2.54) as we explained at the end of section 2.3.

44

M. Shoucri

Figure 31. Frequency spectrum of E+

Figure 32. Wavenumber spectrum of E+.

In circular polarization we have for the pump wave in a linear analysis, following the same notation as at the end of section 2.3, E = E 0 (0, cosψ , sin ψ ) ,

ψ = (k 0 x − ω 0 t ) .

Faragay’s law is:

∂E y ∂E ∂B ). = (0, z ,− ∂x ∂t ∂x ∂a ⊥ and p ⊥ = a ⊥ , we get ∂t p ⊥ = p 0 (0,− sin ψ , cosψ ) . We thus see that pxB is identically zero, p and B being

which gives B = B0 (0,− sin ψ , cosψ ) . From E ⊥ = −

parallel. So in this case there is no 2nd harmonic longitudinal response to the leading order.

The Method of Characteristics for the Numerical Solution…

45

Figure 33. Frequency. spectrum of the plasma.

Figure 34. Wavenumber spectrum of the plasma.

We

(

can

assume

γ e,i ≈ 1 + (me,i p e,i )2

)

1/ 2

following

Eq.(3.41)

that

for

a⊥ → 0 ,

we

have

and the approximation in Eq.(2.43) in section 2.3 becomes valid.

Indeed, for the parameters in [53] used in this chapter, we have estimated at the end of section 2.3 that E 0 = 0.25 . From Eq.(3.42) we can estimate that the amplitude a 0 of a ⊥ is

a 0 = E 0 / ω 0 = 0.044 , which is small. However, Fig.(35) shows the phase-space structure between x=5.1 and x=9.8 at t=60. It shows the vortices structure more important than in Fig.(21), due to the fact that the amplitude of the electric field in the present case where a circularly polarized wave is used, is now

E x2 + E y2 = E0 2 (if the amplitude is reduced to

E0 , then the vortices are similar to Fig.(21)) . Note again the clear picture of the vortices, in the low density region of the phase-space, with very little numerical noise appearing. Figs(36) and (37) show respectively the electric field and the charge (ni –ne ) across the box. Both figures show an enhanced value compared to what is presented in Figs.(22) and (23).

46

M. Shoucri

Figure 35. Contour plot and 3D view for the phase-space of the distribution function from x=5.1 to x=9.8 at

−1 t = 60ω pe .

Figure 36. Longitudinal electric field.

The Method of Characteristics for the Numerical Solution…

47

Figure 37. Charge (nI – ne ).

So we have been able using the technique of cubic spline interpolation to get results from two different models for laser-plasma interaction, using two different numerical codes, which show similarities and differences in the physics associated with the scattering results. Differences observed have been explained as essentially due to different physics associated with the two models, and not to numerical problems. For the model used in section 2.3 with a linearly polarized wave, we used a method of fractional step associated with 1D interpolation using cubic spline, and in the method used in the present section with circular polarization a 2D interpolation in velocity space using a tensor product of cubic B-splines has been used.

3.4. Numerical Solution of a Reduced Model for the Collisionless Magnetic Reconnection In the ideal magnetohydrodynamic (MHD) plasma description, the magnetic field is frozen in the plasma, and its flux through a surface moving with the plasma remains constant. This conservation of the magnetic topology requires that if two plasma elements are initially connected by a magnetic field line, they remain connected by a magnetic field line at any subsequent time, and it constrains the plasma dynamics by making configurations with lower magnetic energy but different topological connection inaccessible. Magnetic field reconnection removes these constraints. It is an important process in high temperature magnetically confined plasma. In this process, the magnetic configuration undergoes a topological rearrangement in a relatively short time, during which the magnetic energy is converted into heat and into kinetic flow energy. Typical situations are in tokamak plasma configurations and in solar flares and coronal loops mass ejections, when strong magnetic fields are present. In the magnetopause it allows particles from the solar wind to enter the magnetosphere. In the present work, we consider a dissipationless two-dimensional configuration with a strong superimposed homogeneous magnetic field perpendicular to the reconnection plane. In the limit of a small ion gyroradius, this two-dimensional system gives a two-fluid equations model where small scale effects related to the electron temperature and electron inertia are retained, but magnetic curvature effects are neglected. We consider a 2D configuration with a strong magnetic field in the ignorable z direction,

48

M. Shoucri

B = B0 e z + ∇ψ x e z , where B0 is constant and ψ ( x, y, t ) is the magnetic flux function. The dimensionless governing equations, normalized to the Alfvén time

τ A and to the

equilibrium scale length Leq , are Hamiltonian and can be cast in a Lagrangian invariant form [62,63,64], similar to what has been presented in section 3.1:

∂G± + [φ ± , G± ] = 0 ; G± = ψ − d e2 ∇ 2ψ ± d e ρ s ∇ 2ϕ ∂t The Poisson brackets

[A, B] = e z .∇Ax∇B ,

(3.48)

and the Lagrangian invariants G ± are

conserved fields advected along the characteristic curves, x ± (t ) :

dx ± (t ) / dt = υ ± ( x ± , t ) , υ ± ( x ± , t ) = e z x ∇φ ± where

φ ± = ϕ ± ( ρ s / d e )ψ . d e

(3.49)

is the electron collisionless skin depth and

ρ s = (M e / M i )1 / 2 v the / ω ci is the ion sound Larmor radius, where vthe is the electron

ω ci is the ion cyclotron frequency. The magnetic flux ψ and the plasma stream function ϕ are given by:

thermal velocity and

ψ − d e2 ∇ 2ψ = (G+ + G− ) / 2 ; d e ρ s ∇ 2ϕ = (G+ − G− ) / 2 If we compare with Eq.(2.4), we see that

φ±

(3.50)

play the role of the single particle

Hamiltonian, and that the two Eqs.(3.48) have the form of 1D Vlasov equations, with x and y playing the role of the coordinate and the conjugate momentum for the equivalent ‘ distribution functions’ G ± of two ‘particle ‘ species with opposite charges in the Poissontype equation for

ϕ , and equal charges in the Yukawa-type equation for ψ

[62,63,64].

We apply a method of integration along the characteristics for the numerical solution of Eq(3.48), similar to what has been discussed in section 1 and for the numerical solution of Eqs.(3.1-3.2). To advance Eq.(3.48) in time, Eq.(3.49) are solved iteratively to determine the departure point of the characteristics ( similar to Eq.(3.8)) , and the values of G ± at these departure points are calculated by a two-dimensional interpolation using a tensor product of cubic B-splines, as discussed for Eqs.(3.9) (see Fig.(1)).The departure point of the characteristics is calculated from the expressions:

Δkx+±1 = Δtυ x ± ( xi − Δkx ± , y j − Δky ± ; t n )

(3.51)

Δky+±1 = Δtυ y ± ( xi − Δkx ± , y j − Δky ± ; t n )

(3.52)

The Method of Characteristics for the Numerical Solution…

49

∂φ ∂φ ± 0 0 ; υ x ± = − ± . We start with Δ x ± = 0 , Δ y ± = 0 , and two or three ∂x ∂y iterations in Eqs.(3.51-3.52) are sufficient for convergence. The solutions G ± at t = (n + 1)Δt are calculated from the expression: where

υ y± =

G ± ( xi , y j , t n +1 ) = G ± ( xi − 2Δ x ± , y j − 2Δ y ± , t n − Δt )

(3.53)

Figure 38. Magnetic flux.

Figure 39. G+ at time t=30.

The code in [62-64] is a finite difference code using filtering and dissipation to remove small scale features which develop. No small scales filtering or dissipation is added to the present code, as is done in [62,63]. Instead, we use a fine grid of N x xN y = 2048 x512 to resolve small details, and stop the calculation when the small details are of the order of the

50

M. Shoucri

Figure 40. G+ at time t=35.

Figure 41. G- at time t=30.

grid size.. We consider an initial equilibrium perturbation

ψ (t = 0) = 1 / cosh 2 ( x) + δψ ( x, y ) , and the

δψ = − 10 − 4 exp( − x 2 /( 2 d e2 )) cos y

d e = ρ s = 0 .2 .

The

equations

are

integrated

is

the

initial

numerically

in

perturbation. the

spatial

domain − 2π < x < 2π , 0 < y < 2π . The domain is periodic in the y direction, and we apply Dirichlet boundary conditions in the x direction. The solution of Eq.(3.48) is followed by a solution of Eqs.(3.50) to determine ψ and ϕ , and these quantities are used to calculate

φ ± , to repeat again the integration of Eq.(3.48). The solution of Eqs.(3.50) is done by Fourier transforming in the periodic y direction, then discretizing the equations in the x direction and solving the resulting tridiagonal system with appropriate boundary conditions, and then Fourier transforming back (details have been presented in section 3.1 , Eqs(3.6-3.7)). During the evolution of the reconnection process, we see in Fig.(38) in the contours of the magnetic flux a magnetic island generated and growing in the linear phase and early non-linear phase, in which the process exhibit a quasi-explosive behaviour. In the full nonlinear regime, equilibrium is reached, the island growth saturates and remains more or less unchanged. The

The Method of Characteristics for the Numerical Solution…

Figure 42. Stream function. ϕ .

Figure 43. Stream function ϕ .

Figure 44.

φ+

at t=30.

51

52

M. Shoucri

Figure 45. Current J at t =30.

contours of G+ in Fig.(39) at t=30 and in Fig.(40) at t=35 show the formation of a vortex structure. A similar vortex structure is developed also for the invariant G − , which is advected in the opposite direction with respect to G + . Asymptotic states for 2D systems showing the formation of vortex structures has been discussed in [49,50], who showed that energy should move to the largest scale available in the system, showing the formation of a large vortex, similar to the 2D results we obtained in the previous examples. The model preserves parity. If we choose the initial values such that ψ (− x) = ψ ( x) , and ϕ ( − x) = ϕ ( x) , these relations imply G + (− x, y ) = G− ( x, y ),

φ + (− x, y ) = −φ − ( x, y ),

which are maintained and

accurately verified by the code. Fig.(41) for G− ( x, y ) at t=30 shows how this symmetry is well reproduced by the code ( to be compared with Fig.(39)). Fig.(42) shows the stream function ϕ at t=30 and Fig.(43) at t=35. Fig.(44) shows the function φ + at t=30. In Fig.(45) we have a 3D view and a contour plot of the current J = −∇ ⊥ψ at t=30. Note the important 2

The Method of Characteristics for the Numerical Solution…

53

Figure 46. Current J at t =35.

Figure 47. Plot of

ln(δψ ( x = L x / 2, y = 0))

against time.

peak structure of the current around the X-point. We note also in Fig.(46) for the contour plot of the current at t=35 the fine scale structures which develop, and which make further calculation difficult, even with the 2048x512 grid points we have. Magnetic reconnection leads to the development of increasingly narrow current and vorticity layers. To avoid this difficulty, a numerical diffusion term was added in [62,63] to smooth the solution and push further in time. But this is done at the expense of eliminating some details , as for instance the thin filament current peaking at the X-point in Fig.(45), which has not been observed in [62,63,64]. Finally we stress again the symmetry in the solution reproduced with great −3

precision by the code. The time-step used for this calculation was Δt = 10 .

54

M. Shoucri

Figure 48. Plot , against time, of the difference between each term of energy, as defined in Eq.(3.54), and the corresponding value at t=0, divided by the Total energy E(0).

Finally

in Fig.(47) a curve showing the evolution of ln(δψ ( x = L x / 2, y = 0)) against time, which shows the growth and saturation of the perturbation

we

present

δψ . And in Fig.(48), the curves present the evolution of the different energies. 2

∫ dxdy ∇ψ (dotted curve), which is decaying, is transformed mainly into plasma kinetic energy ∫ dxdy ∇ϕ (broken curve), into electron parallel kinetic energy ∫ dxdyd J (two-dashes-dot curve), and into electron internal energy ∫ dxdyρ U (dash-

The magnetic energy

2

2 e

2

2 s

2

two-dots curve). The total energy E (full curve) is given by:

(

E = ∫ dxdy ∇ψ

2

+ d e2 J 2 + ρ s2U 2 + ∇ϕ

2

)/ 2

(3.54)

(The quantities plotted in Fig.(48) are the difference between each term of energy as defined in Eq.(3.54), and the corresponding value at t=0, divided by the total energy E(0) at t=0). The extension of this method to the 3D reduced model [63] for collisionless magnetic reconnection is outlined as follows. The 3D equation:

∂G± + [φ ± , G± ] = ∂t is solved using a fractional step method :

∂ (φ ± ∓

ρs

de ∂z

G± ) (3.55)

The Method of Characteristics for the Numerical Solution…

55

Step1-Solve for a step Δt / 2 the equation:

∂G± + [φ ± , G± ] = 0 ∂t

(3.56)

Step2-Solve for a step Δt the equation :

∂G± = ∂t

∂ (φ ± ∓

ρs

de ∂z

G± ) ;

(3.57)

Step3-Repeat Step1-. Eq.(3.56) is solved with the same method discussed in section 1 and 3.1. Eq.(3.57) can be solved by Fourier transform in the periodic z-direction.

4. Application of the Method of Characteristics to Fluid Equations We have already mentioned in section 3.1 that the 2D guiding-center equations in a magnetized plas are isomorphic to the Euler equations that govern the 2D inviscid incompressible fluids in hydrodynamics. In section 3.4, a reduced set of fluid-like equations has been applied to study magnetic reconnection. We present in this section some additional applications in the field of fluid dynamics, for the numerical solution of the shallow water equations, and for the numerical solution of the equations of the incompressible ideal magnetohydrodynamic flows in plasmas.

4.1. Numerical Solution of the Shallow Water Equations The shallow water equations are of great importance since they are widely applied for the study of atmospheric weather prediction and oceanic dynamics. They are the simplest equations which describe both slow flows and fast gravity wave oscillations, the two main categories of fluid motion present in the more complicated primitive equations, and which are commonly used for atmospheric, oceanic and climate modeling. A method of fractional step for the numerical solution of the shallow water equations has been recently presented in [3]. It consists of splitting the equations and successively integrating in every direction along the characteristics using the Riemann invariants of the equations, which are constant quantities along the characteristics. The integration is stepped up in time using cubic spline interpolation to advance the advection terms along the characteristics. It has also the great advantage of solving the shallow water equations without the iterative steps involved in the multidimensional interpolation problem, and the iteration associated with the intermediate step of solving a Helmholtz equation, which is usually the case in other methods like the semiLagrangian or Euler-Lagrange method [7,8,65 and references therein]. The absence of iterative steps in the present method reduces considerably the numerical diffusion, and makes it suitable for problems in which small time steps and grid sizes are required, as for instance

56

M. Shoucri

the problem of the calculation of the potential vorticity field we study in the present section, where steep gradients and fine scale structures develop, and more generally for regional climate modeling problems. The linear analysis (unpublished) of the shallow water equations for the fractional step method shows the method is unconditionally stable, reproducing exactly the frequency of the slow mode, while the frequencies of the fast modes are exact to second order. We present in this section a new application of the fractional step method to the shallow water equations to study the evolution of a complex flow typical of atmospheric or oceanic situations, namely the nonlinear instability of a zonal jet, similar to what has been presented in [66]. As pointed out in [66], there is well established observations that even in the presence of relatively smooth, large scale flows, tracer fields in the atmosphere and the ocean develop fine scale structures. A tracer of particular significance is the potential vorticity, which develops steep gradients and evolves into thin filaments whose numerical study demands a resolution with small grid sizes and whose evolution requires small timesteps. Several methods have been discussed in [66] for the numerical solution of the potential vorticity of a zonal jet. The semi-Lagangian or Euler-Lagrange method requires iterations at each time-step to interpolate along the characteristics, and includes also an intermediate step for solving by iteration a Helmholtz equation [7,8,65,66]. This double-iterative numerical method can be computationally prohibitive if done on small grid sizes and with small timesteps, as recently pointed out in [66], and results in an important numerical diffusion difficult to evaluate or control. Other methods to solve the shallow water equations include the pseudospectral method [66], which requires the addition of an explicit hyperdiffusion for the numerical stability. This ad hoc addition of hyperdiffusion seriously degrades the solution accuracy. In the results presented in [66], the complex filamentary structures and steep gradients surrounding most vortices are substantially smoothed out in the pseudospectral and semi-Lagrange methods, this later one does worse than the pseudospectral method because the numerical diffusion occurs through repeated iterations and interpolations and is thus not directly controllable. In the contour-advective semi-Lagrangian method presented in [66], the potential vorticity is discretized by level sets separated by contours that are advected in a fully Lagrangian way. This allows one to maintain potential vorticity gradients that are steep, however the small scales in the potential vorticity are removed with contour surgery, by topologically reconnecting contours and eliminating very fine scale filamentary structures. This contour surgery is, of course, an ad hoc procedure as much as the hyperdiffusion used in the pseudospectral method. All the three previously discussed methods (semi_Lagrangian, pseudospectral and contour-advected semi-Lagrangian) require the knowledge of the calculated variables at three time levels, and require different time filtering to damp highfrequency modes and small-scale high frequency gravity waves, otherwise the numerical scheme is unstable. In the fractional step method we present in this section, no iterations are required since only two time levels are used to advance the equations in time, and no time filtering is required. So the numerical diffusion is minimal. The fractional step method applied to the shallow water equations has been recently presented and applied to a climate modeling problem [3], and compared favourably when applied with small grid sizes and timesteps with respect to the semi-Lagrangian method. Further evaluation of the performance of the fractional step method applied to the shallow water equations has been recently presented in [67]. It is the purpose of the present work to apply this fractional step method to follow on an Eulerian grid the evolution of quantities like the height and the velocity field, which are relatively broader in scale, while reconstructing and capturing at each time-step the complex

The Method of Characteristics for the Numerical Solution…

57

filamentary and fine scale structures and the steep gradients associated with the corresponding potential vorticity. In other words the potential vorticity is post-processed at each time step from the height and velocity field obtained from the direct solution of the shallow water equations, as it has been recently reported [68]. In the present section, the results obtained will be compared with results obtained by directly integrating the potential vorticity equation on the same Eulerian grid. These direct integrations which follow a quantity developing steep gradients and fine scale structures on an Eulerian grid develop naturally numerical noise, as it will be shown at the end of this section. We write the two-dimensional shallow water equations in their simplest form in terms of the height h and the velocities (u , v ) respectively along the x and y directions.

∂h ∂uh ∂vh + + =0 ; ∂t ∂x ∂y

(4.1)

∂u ∂u ∂u ∂h ∂v ∂v ∂v ∂h +u +v + g = f v ; +u +v + g =−f u ∂t ∂x ∂y ∂x ∂t ∂x ∂y ∂y

(4.2)

f is the Coriolis parameter and g is the gravitational field. We write the geopotential

φ = gh ,

and we use the following time-centered scheme [68] to integrate by fractional step Eqs(4.1,4.2): Step 1 - solve for Δt 2 the equations in the x direction:

∂u ∂u ∂φ +u + =0 ; ∂t ∂x ∂x

(4.3)

∂φ ∂φ ∂u +u +φ =0 ∂t ∂x ∂x

(4.4)

∂v ∂v +u =0 ; ∂t ∂x

(4.5)

)

(4.6)

Eqs(4.3,4.4) are rewritten:

(

∂R x ± ∂R x ± + u± φ =0 ; ∂t ∂x where R x ± = u ± 2

φ are the Riemann invariants [1]. The solution of Eq. (4.6) at t + Δt 2

is written as follows:

R x ± ( x, y, t + Δt / 2 ) = R x ± ( x ± , y, t )

(4.7)

58 where x ± = x −

M. Shoucri t + Δt / 2

∫ (u ±

φ ) dt . The solution of Eq.(4.5) for v at t + Δt 2 is written:

t

v( x, y, t + Δt / 2) = v( x −

t + Δt / 2

∫ udt , y, t )

(4.8)

t

t

Q (m Δx ; (n + 1/2) Δt)

(n + 1/2) Δt

n Δt O (m - 1) Δx η Δx

ξ Δx

P

m Δx

x

Figure 49. Details for the calculation if Eq.(4.12).

To find the value of the function at t + Δt / 2 at the arrival grid points, the right hand sides of Eqs.(4.7,4.8) imply finding the value of the function at time t = nΔt at the departure point of the characteristic at the shifted position. This value is obtained using cubic spline interpolation from the values of the function at the neighboring grid points at t = nΔt . To avoid iterations, we show as an example how the integral in Eq.(4.8) is approximated (the same technique is applied to approximate the other integrals in Eq.(4.7)). We write:

ξΔx =

t + Δt / 2

∫ u( x(t ′), t ′)dt ′

(4.9)

t

Δx is the grid size in the x direction, and from Eq.(4.9) ξΔx gives the distance of the departure point P of the characteristic from the grid point at x = mΔx (the point P in Fig.(49) is located between the grid points x = (m − 1)Δx and x = mΔx , and the characteristic through the point P reaches the grid point Q (mΔx, (n + 1 / 2)Δt ) at t = (n + 1 / 2)Δt ). The vertical axis in Fig.(49) is time. The velocity u at the point P (denoted by u P ), can be written as a linear interpolation at time t = nΔt of the value of u at the grid point at x = mΔx (denoted by u m ), and the value of u at the grid point

x = (m − 1)Δx (denoted by u m −1 ) :

The Method of Characteristics for the Numerical Solution…

59

u p = u m−1 (1 − η ) + u mη = u m−1ξ + u m (1 − ξ ) where

(4.10)

η + ξ = 1 . The distance OP in Fig.(49) is ηΔx . Usually u at the point Q is unknown.

We use Taylor expansion for Δt / 2 :

u Q = u m + (u m − u mn −1 / 2 ) = 2u m − u mn −1 / 2 (the values of u without superscript denotes the time t = nΔt ). We can approximate the integral in Eq.(4.9) as follows:

1 2

ξΔx = (u P + u Q )

Δt 2

(4.11)

We substitute in Eq.(4.11) for u P and u Q . We get for the shifted value in Eq.(4.9):

1 Δt (3u m − u mn −1 / 2 ) 2 ξΔx = 2 1 u − u m −1 Δt 1+ ( m ) 2 Δx 2

(4.12)

n −1 / 2

This result reduces to the one in [68] if we approximate u m

by u m in eq.(4.12).

Eq.(4.12) gives an explicit approximation for the value to be shifted in Eq.(4.8). The same technique can be applied to the integrals in Eq.(4.7), so the calculation of the integrals in Eqs.(4.7,4.8) using the approximation of Eq.(4.12) remains explicit, and the interpolated values in Eqs.(4.7,4.8) are calculated using a cubic spline interpolation. No iteration is implied in this calculation. The results we present] show that this approximation is sufficient and good. Step 2 - use the results of Step 1 to solve for Δt 2 the equations in the y direction:

∂v ∂v ∂φ +v + =0 ∂t ∂y ∂y

(4.13)

∂φ ∂φ ∂v +v +φ =0 ∂t ∂y ∂y

(4.14)

∂u ∂u +v =0 ; ∂t ∂y

(4.15)

Eqs.(4.13) and (4.14) are rewritten:

60

M. Shoucri

∂R y ± ∂t

(

+ v± φ

) ∂R∂y

y±

=0 ;

φ are the Riemann invariants. The solution of Eq. (4.16) is written:

where R y ± = v ± 2

Δt ⎞ ⎛ R y ± ⎜ x, y , t + ⎟ = R y ± (x, y ± , t ) 2⎠ ⎝ where y ± = y −

(4.16)

t + Δt / 2

∫ (v ±

(4.17)

φ )dt . The solution for u in Eq.(4.15) is calculated in a similar

t

way to Eq. (4.8).

u ( x, y, t + Δt / 2) = u ( x, y −

t + Δt / 2

∫ vdt, t )

(4.18)

t

The calculation of the integrals in Eqs.(4.17) and (4.18) is effected in a similar way as explained for Eq.(4.9),by substituting y for x. Step 3 - use the results at the end of Step 2 to solve the source terms for Δt:

∂u − fv = 0 ∂t

;

∂v + fu = 0 ∂t

(4.19)

If we denote by Uo and Vo the values of u and v at the end of Step 2, the values of u and v after Δt in Step 3 are given by:

u ( x, y, t + Δt ) = U o ( x, y ) cos( fΔt ) + Vo ( x, y ) sin( fΔt ) ;

(4.20)

v( x, y, t + Δt ) = Vo ( x, y ) cos( fΔt ) − U o ( x, y ) sin( fΔt )

(4.21)

Step 4 use the results at the end of Step 3 to solve for Δt 2 the equations in the y direction (as in Step 2) Step 5 use the results at the end of Step 4 to solve for Δt 2 the equations in the x direction (as in Step 1) This entire cycle will advance the solution by one time-step Δt . We have mentioned that in Eqs.(4.7),(4.8) and Eqs.(4.17-4.18) the value of the function at the points of departure of the characteristics ( the shifted value) are calculated from the values of the function at the grid points using a cubic spline interpolation. We use a simple cubic spline defined over three grid points, calculated by writing that the function, its first and second derivatives are continuous at the grid points. Details are given in Appendix B. Testing this cubic spline polynomial

The Method of Characteristics for the Numerical Solution…

61

against other methods [4] has shown that this cubic polynomial has very low numerical diffusion compared to other polynomials. The code developed for the present problem is less than 500 fortran lines. At every time step, we reconstruct the potential vorticity q from the calculated values of h, u and v using the relations:

q=

∂v ∂u f +ζ ;ζ = − ∂x ∂y h

(4.22)

The derivatives ∂v / ∂x and ∂u / ∂y are calculated from the values of u and v using cubic splines. If we operate on the first equation of Eq.(4.2) by ∂ / ∂y , and on the second equation of Eq.(4.2) by ∂ / ∂x , we can derive with the help of Eq.(4.1) the following equation for the potential vorticity q :

∂q ∂q ∂q +u +v =0 ∂t ∂x ∂y

(4.23)

which simply states that q is constant along the characteristics :

dx dy =u ; =v dt dt

(4.24)

As we mentioned in the introduction, it is generally difficult to find a method for the direct integration of Eq.(4.23), because the potential vorticity generally develops steep gradients and finescale structures. In any Eulerian code, this will require a large number of grid points and small time-steps. In the method we present in this paper, we solve directly for the relatively large scale variables , height and velocity in Eqs.(4.1,4.2), and the potential vorticity is accurately calculated (post-processed) at each time-step using Eq.(4.22), capturing the small scale key features and steep gradients associated with the solution. We apply the numerical scheme presented for the numerical solution of Eqs.(4.1,4.2), and for the problem of post processing the potential vorticity from this solution. The initial flow consists of a perturbed unstable zonal jet which rapidly becomes very complex, and is specified by prescribing the initial potential vorticity as follows ( we follow the notation of [66]):

q ( x, y,0) = q + Q sgn( yˆ )(a − || yˆ | − a |)

(4.25)

for | yˆ |< 2a (the vertical lines indicate the absolute value). Q is the amplitude of the potential vorticity, q is the mean potential vorticity, 2a is the distance from the minimum to maximum potential vorticity, and:

yˆ = y + c m sin mx + c n sin nx

(4.26)

62

M. Shoucri

where yˆ is the perturbed y coordinate, used to perturb the jet. We use for the present test the same parameters as in [66]. Equilibrium height h = 1 , the Coriolis factor f = 4π ,

a = 0.5 , h Q / f = 1 , Q = 4π = q . The deformation radius is LR = gh / f = 0.5 , and

g = (2π ) 2 . A doubly periodic domain which spans the range (−π , π ) covers about 12.5 deformation radii in each direction. We take m=2 , n=3, c 2 = −0.1 , c3 = 0.1 to perturb the q profile . We use a slightly different method to calculate the initial velocities and height h. We assume initially v=0 . We balance the second of Eq.(4.2) initially:

fu = − g

from which

f

∂h ∂y

∂2h ∂u = −g 2 ∂y ∂y

Figure 50. Initial profiles for the shallow water problem

h / 2π

(4.27)

(4.28)

q / 4π

(full curve) , u (broken curve),

(dash-dot curve).

We substitute for ∂u / ∂y from Eq.(4.28) into Eq.(4.22) and solve numerically for the initial value of h .Fig.(50) shows the initial equilibrium profiles ( uniform in x ) for q / 4π ( full curve) , u ( broken curve) and h / 2π ( dash-dot curve). These initial values of u,v and h are used in Eqs.(1,2) to start the evolution of the system. In Figs.(51) and (52) we show respectively the velocities u and v at t=8, and in Fig.(53) we show the contour and 3D view of the geopotential φ . Figs.(51-53) show structures which are generally broader in scale than the potential vorticity. Fig.(54) shows the the potential vorticity q , calculated at t=8 from Eq.(4.22). The instability in the potential vorticity has developed and the initial zonal jet has evolved into vortices and fine structures with steep gradients. These fine scale structures are

The Method of Characteristics for the Numerical Solution…

Fig 51. Velocity u .

Figure 52. Velocity v.

63

64

M. Shoucri

Figure 53. Geopotential

φ.

Figure 54. Potential vorticity q.

The Method of Characteristics for the Numerical Solution…

65

inevitably generated by the forward enstrophy cascade. We note how the steep potential vorticity gradients and the small scale features are nicely captured and reconstructed using Eq.(4.22), from the solution of the relatively broader scale height and velocities, by post processing these relatively smooth large scale flows with the help of Eq.(4.22). These calculations are done with 200x200 grid points, and a time-step Δt = Δx / 20 . This time-step has been chosen after few tests to determine the time-step at which the solution appears to converge and become independent of Δt ( the solution obtained with Δt = Δx / 10 is essentially identical to what we are presenting here). The computation CPU time on a sunblade 1000 workstation of 750 Mhz was 64 minutes to reach t=8, and the required memory for the code was 4.5 Mbytes.

Figure 55. Potential vorticity q(fractional step method).

We present for comparison in Figs(55) the solution for the potential vorticity q obtained by the direct integration of Eq.(4.23) with the initial value in Eq.(4.25) by a fractional step method, and by a semi-Lagrangian method in Fig.(56). For the fractional step method, we follow the steps of the techniques presented in section 2 . To advance Eq.(4.23) in for a timestep Δt , we use the following sequence : Step1 Solve for Δt / 2 the equation :

∂q ∂q +u =0 ∂t ∂x

(4.29)

66

M. Shoucri Step2 Solve for Δt the equation:

∂q ∂q +v =0 ∂t ∂y

(4.30)

Step3 Repeat Step1 for Δt / 2 .

Figure 56. Potential vorticity q (semi-Lagrangian method).

We use the same values of u and v calculated from Eqs.(4.2). Eqs(4.29-4.30) are solved as described for Eq.(4.5), (4.6) or (4.15) . The other method used for the direct solution of Eq.(4.23) is the semi-Lagrangian or Euler-Lagrange method. As described in sections 1 and 3.1, we calculate the displacements Δ x and Δ y along the characteristic curves by solving iteratively the equations:

Δkx+1 = Δtu ( x − Δkx , y − Δky , t n )

(4.31)

The Method of Characteristics for the Numerical Solution…

Δky+1 = Δtv ( x − Δkx , y − Δky , t n )

67

(4.32)

where u and v are calculated from Eqs.(4.2) using the same method we previously discussed. 0

0

We start with Δ x = 0 and Δ y = 0 . Usually two or three iterations are necessary to get convergence. Then the function q is advanced from t n − Δt to t n + Δt using the relation:

q ( x, y, t n + Δt ) = q ( x − 2Δ x , y − 2Δ y , t n − Δt )

(4.33)

As explained in section 3, the interpolations in Eqs.(4.31-4.33) are done using a tensor product of cubic B-spline [6] ] (with appropriate modification to the boundary conditions to take into account the periodicity as for instance in [51]). We used the same Eulerian grid as for the solution of Eqs.(4.1-4.2). Figs(55) and (56) show noisy figures compared to what has been presented in Fig.(53), obtained from the direct solution of Eqs.(4.1-4.2) and Eq.(4.22). This noisy behaviour is to be expected since for the fractional step and semi-Lagrangian methods we have the difficult challenge to follow on an Eulerian grid the potential vorticity of a zonal jet, a quantity which develops steep gradients and fine scale structures. Finally the extension of the method to three dimensional problems is straightforward, requiring only the addition of an extra fractional step in the third dimension to what is presented in this section. The Appendix in [68] outlines an example for a 3D problem.

4.2. Two-Dimensional Magnetohydrodynamic Flows We present another example for the application of the method of characteristics for the numerical solution of fluid equations, namely the equations of two-dimensional incompressible magnetohydrodynamic flows in plasmas. These equations play an important role in the understanding of strong turbulence properties in high Reynolds number conducting fluids, which have important effects on the reconnection of the magnetic field and changes of flow topology [69-72]. As discussed in section 3.4, magnetic reconnection is a fundamental process which allows magnetized plasmas to convert the energy stored in the field lines into kinetic energy of the plasma. In ideal MHD, the frozen-in flux condition prohibits the magnetic field topology to change. Thus reconnection depends on a non-ideal mechanism responsible, in the region where the topology change takes place, for the dynamics of a diffusion process which creates a mechanism that breaks the magnetic field frozen in the plasma. Hence the importance of a solution to the pertinent equations where numerical diffusion is controlled to the minimum. We have already presented in section 3.4 an application of the methods of characteristics for the numerical solution of a reduced set of MHD equations. We extend this method to the set of two-dimensional fluid ideal MHD equations usually applied to study incompressible MHD turbulence. There is an abundant literature for the numerical solution of these equations [69-72], based essentially on finite difference schemes. Our intention is to apply the method of characteristics to the solution of these equations. The pertinent incompressible magnetohydrodynamic equations can be written in the form:

68

M. Shoucri

∂z ± + z ∓ .∇z ± = −∇p + vΔz ± . div z ± =0 ∂t

(4.34)

z ± denotes the Elsässer variables z ± = u ± B , where u is the velocity, B the magnetic field, and p the total pressure. Here we assume a magnetic Prandtl number equal to one. We use the same parameters as in [69,70]. The two-dimensional MHD equations in Eq.(4.34) are solved in a rectangular box of size L x = L y = 2π with periodic boundary conditions. We use as initial conditions for the magnetic flux function ψ and the velocity streamfunction

ϕ

the following expressions :

1 [cos(2 x + 2.3) + cos( y + 4.1)], . 3 ϕ ( x, y ) = cos( x + 1.4) + cos( y + 0.5)

ψ ( x, y ) =

(4.35)

B = e z x∇ψ and u = e z x∇ϕ . These initial conditions introduced in [69] show a stronger tendency to generate turbulent small scale structures than the Orszag-Tang vortex. They are made less symmetric by means of arbitrary phases. They have also been used in [70]. The numerical scheme applied to Eq.(4.34) is the following: Step1 solve for a time step Δt / 2 the equation :

∂z ± + z ∓ .∇z ± = 0 . ∂t

(4.36)

This equation is solved in 2D using a tensor product of cubic B-spline for interpolation, as described in section 1 and applied for the problems presented in section 3. We next calculate the pressure p by taking the divergence of Eq.(4.34). This gives the following equation:

div (z ∓ .∇z ± ) = −Δp .

(4.37)

This equation is solved for p using a fast Fourier transform algorithm, since we have periodic boundary conditions. Step2 solve for a time step Δt the equation:

∂z ± = −∇p + vΔz ± . ∂t

(4.38)

∇p is treated explicitly. The diffusion term is treated by an alternate direction implicit scheme.

The Method of Characteristics for the Numerical Solution…

69

Step3 repeat Step1 for a time step Δt / 2 . The quantities u

B=

B are calculated from the relations : u =

and

z+ + z− , 2

z+ − z− . 2 +

We write for reference the explicit form for the solution of Eq.(4.36) for z x = u x + B x :

∂z x+ ∂z + ∂z + + z x− x + z −y x = 0 . ∂t ∂x ∂y

(4.39)

The characteristic equations for Eq.(4.39) are given by:

dx = z x− , dt

dy = z −y . dt

(4.40)

We calculate the displacement Δ x and Δ y as explained in the previous sections, using the following iterations:

Δkx+1 =

Δt − z x ( xi − Δkx , y j − Δky , t n + Δt / 4) 4

(4.41)

Δky+1 =

Δt − z y ( xi − Δkx , y j − Δky , t n + Δt / 4) 4

(4.42)

−

−

The values of z x and z y at t n + Δt / 4 can be calculated by a predictor-corrector technique. Two or three iterations are necessary for the convergence in Eqs.(4.41-4.42), and a tensor product of cubic B-spline is used for the interpolation [6] (with appropriate modification to the boundary conditions to take into account the periodicity as for instance in +

[51]). Then the value of z x is advanced in time for Δt / 2 as indicated in Step1 using the relation:

z +x ( xi , y j , t n + Δt / 2) = z x+ ( xi − 2Δ x , y j − 2Δy, t n ) .

(4.43)

Again a tensor product of cubic B-spline is used for the interpolation [6] in Eq.(4.43). The same method is applied to the other variables in Eq.(4.36).

70

M. Shoucri

Figure 57. Current J at t=0.

Figure 58. Current J at t=1.

Figure 59. Current J at t=2.

The solution we present has been obtained using 512x512 grid points with the kinematic −3

viscosity v = 10 in Eq.(4.34). One has to be careful when v is different from zero since any kind of smoothing may artificially inhibit the energy transfer to small scales and slow down magnetic reconnection and the associated instabilities. Comparison with the case v = 0 has shown very close results up to t=3. However, at this stage the growth of the current becomes very big and the system goes to a numerical instability for longer runs with v = 0 . The nonlinear dynamics indeed leads to the formation, near neutral X points, of magnetic current sheets corresponding to strongly sheared magnetic field configurations. The finite value of v keeps the growth of the different variables under control. A discussion of the effect of v on the solution can be found in [71]. (We note that the claim in [70] that the first

The Method of Characteristics for the Numerical Solution…

71

simulation presented is done with v = 0 is probably due to the presence of an important numerical diffusion in the code). Figs(57-61) show the current density J respectively at t=0,1,2,3 and 6. From Eq.(4.35) we have:

Bx = −

∂ψ ; ∂y

By =

∂ψ ; ∂x

J = − Δψ = − ( −

Figure 60. Current J at t=3.

Figure 61. Current J at t=6.

Figure 62. Vorticity at t=0.

∂B x ∂B y + ) . ∂x ∂y

(4.44)

72

M. Shoucri

Figure 63. Vorticity at t=1.

Figure 64. Vorticity at t=2.

Figure 65. Vorticity at t=6.

The results in Figs.(57-61) are very close to what is presented in Fig.(2) of [70], showing the formation of current sheets. Figs.(62-65) show the vorticity U respectively at t=0,1,2 and 6. From Eqs.(4.35) we have:

ux = −

∂ϕ ; ∂y

uy =

∂ϕ ; ∂x

U = Δϕ = (−

∂u x ∂u y + ). ∂x ∂y

(4.45)

The Method of Characteristics for the Numerical Solution…

73

Figure 66. ux at t=2.

Figure 67. ux at t=6.

Figure 68. uy at t=2.

At t=0, B x , B y , u x , and u y are calculated from Eq.(4.35). Figs.(66-69) show u x , and

u y at t=2 and t=6, and Figs.(70-73) show B x and B y at t=2 and t=6. In Figs.(74-77) the magnetic flux function

ψ at t=0,1,2 and 6 is presented. Note in Figs.(75-76) at t=1 and t=2

how, in the regions where the magnetic vortices are pushed towards each others or squeezed between each others, intense currents are created in the corresponding current density plots in Figs.(58-59). Fig.(78) presents the time evolution of the total enstrophy ( which is the sum of the kinetic and magnetic enstrophies):

(

2

2

)

W = ∫ ∇xu + ∇xB dxdy .

(4.46)

74

M. Shoucri

Figure 69. uy at t=6.

Figure 70. Bx at t=2.

Figure 71. Bx at t=6.

In Fig.(79) we present the time evolution of the total energy (which is the sum of the kinetic and magnetic energies):

E=

(

)

1 2 2 u + B dxdy . ∫ 2

(4.47)

The Method of Characteristics for the Numerical Solution… The time evolution of the velocity-magnetic field correlation [70]

75

ρ = H / E , where the

∫

cross-correlation H = u.Bdxdy , is presented in Fig.(80). The distribution of kinetic and magnetic energies among the different scales is described by the kinetic and magnetic energy spectra:

E u (k ) =

∑

uˆ (k ´ )

∑

Bˆ (k ´ )

2

.

(4.48)

.

(4.49)

k ≤ k ´ < k +1

E M (k ) =

2

k ≤ k ´ < k +1

Fig.(81) shows on a logarithmic scale the kinetic ( log10 E ( log 10 E

M

u

solid line) and magnetic

dashed line) energy spectra at t=6 (plotted against log10 k , with k varying

between 1 and 100, showing a slight dominance of the magnetic energy over the kinetic energy over most of the scale lengths). It is beyond the scope of the present work to repeat what has been presented in [69-72]. We intended in this section to outline the pertinent steps for one more application of the method of characteristics (which generally shows low numerical diffusion) to the equations of ideal MHD flows.

Figure 72. By at t=2.

Figure 73. By at t=6.

76

M. Shoucri

Figure 74. Flux function ψ at t=0.

Figure 75. Flux function ψ at t=1.

Figure 76. Flux function ψ at t=2.

Figure 77. Flux function ψ at t=6.

The Method of Characteristics for the Numerical Solution…

Figure 78. Time evolution of the enstrophy.

Figure 79. Time evolution of the energy.

Figure 80. Time evolution of the correlation coefficient

Figure 81. Kinetic (solid line) and magnetic (dashed line) energy spectra at t=6.

77

78

M. Shoucri

5. Conclusion We have presented in the appendices some simple cubic spline relations which we have applied for interpolation in several problems, showing how hyperbolic type differential equations are solved using the method of characteristics. The values of the functions which remain constant along the characteristic curves are stepped-up in time using cubic spline interpolation. Results illustrating the performance of the cubic spline when applied to interpolation in Eulerian grid-based solvers have been presented. Comparison of the cubic spline interpolation with other methods [4] have shown that the cubic spline interpolation compares favourably with the other methods. Since the ground breaking work of Cheng and Knorr [10] which applied the fractional step method to the one-dimensional Vlasov equation, there has been important applications of the method of characteristics for the numerical solution of the kinetic equations of plasmas, and especially for extending these methods to higher dimensions [15-18]. A historical overview on several applications of these methods in the field of the kinetic equations of plasmas has been recently given in a Vlasovia workshop [73]. This workshop included also recent applications on massively parallel computers, especially in the field of the numerical solution of gyro-kinetic equations [74-77], which testify for the impressive advances and applications of these methods. We mention also the work in [78,79]. It is beyond the scope of the present chapter to review all these works. The intention in this chapter was to present the essential elements of the interesting technique of the method of characteristics associated with cubic splines interpolation, with appropriate selected examples to illustrate the performance, the accuracy, the powerful and efficient tools which Eulerian grid-based solvers can provide for the numerical solution of plasmas kinetic equations and fluid equations: long time evolution of 1D BGK modes, charge separation at a plasma edge involving higher phase-space dimensionality, laser-plasma interaction, magnetic reconnection and the shallow water equations. In section 2.2 and section 3.2 for instance, we have presented the problem of the formation of a charge separation and an electric field at a plasma edge. In Cartesian geometry in section 2.2 a method of fractional step associated with 1D cubic spline interpolation along the characteristic curves has been applied, while in cylindrical geometry in section 3.2 a two dimensional interpolation using a tensor product of cubic B-spline [51] has been applied in velocity space. The results from these two different codes are identical (can be superposed if we take into consideration a mirroring due to the opposite positive direction in the two codes). Another comparison has been presented in section 2.3 and 3.3, where two different models for laser-plasma interaction have been discussed. The model in section 2.3 uses a linear polarization for the electromagnetic wave, and applies a method of fractional step for the numerical solution of the equations, associated with one dimensional cubic spline interpolation. The model in section 3.3 is fully relativistic with circular polarization, and applies a two dimensional interpolation with a tensor product of cubic B-spline. Again the two different codes are providing similar results, the only difference reflects and underlines the difference in the physical models. Several recent applications of similar codes for problems of laser-plasma interaction have been recently reported [30,31,52], which testify for the success of these methods, together with different other applications of the method of characteristics for the kinetic equations of plasmas which we have rapidly reviewed. In section 3.4 for the numerical solution of the problem of magnetic reconnection, a thin filament current has been observed at the X-point in Fig.(45)

The Method of Characteristics for the Numerical Solution…

79

which was not previously observed in [62-64] where an artificial numerical diffusion was added. The method of characteristics has also been applied for the numerical solution of fluid equations. At the same time where [10] was published, the application of the method of characteristics in 1D fluid equations associated with cubic spline interpolation was also published in [80]. In plasma physics, one of the early applications of this method to fluid equations has been for the numerical solution of the coupled mode equations [81-82]. There is an abundant theoretical literature on the method of characteristics applied to fluid equations (see for instance [1,83]). 2D interpolation using tensor product of cubic B-splines is commonly applied for the numerical solution of the weather forecast equations in what is known as the semi-Lagrangian method [7,8,65,66]. In section 4.1 we have presented an interesting application to the problem of the calculation of the potential vorticity of a zonal jet from the solution of the shallow water equations, reproducing on an Eulerian grid the fine scale structures and the steep gradients of the potential vorticity without numerical noise. And the problem of magnetic reconnection using the equations of the incompressible ideal MHD flows in plasmas has been studied in section 4.2.

Acknowledgments The tutoring and fruitful discussions with Professor Georg Knorr are gratefully acknowledged. I would like also to acknowledge the fruitful discussions with Drs. Hartmut Gerhauser and Karl-Heinz Finken on the problem of the formation of an electric field at a plasma edge, and the collaboration with Dr. Erich Pohn on problems of two-dimensional interpolation used in the present chapter, as well as with Dr. David Strozzi for problems of laser-plasma interaction presented in section 2.3.

Appendix A The Shift Operator Using the Cubic Spline Let us assume 0 < Δ < 1 , with a uniform grid size normalized to 1 , and Δ is a constant. We

y j , with the function use a Taylor expansion to calculate the shifted value y j ( x j + Δ) = ~ y = f (x) and the notation f ( x j ) = f j : 1 y j ( x j + Δ) = ~ y j = f j + p j Δ + s j Δ2 + g j Δ3 2

(A.1)

p j , s j and g j are respectively the derivative, second derivative and third derivative of the function f (x) at the grid point j ≡ x j . We write that the function, its derivative and second derivative are continous at every grid point, we get the following cubic spline relations on a uniform grid [84,85]:

p j −1 + 4 p j + p j +1 = 3( f j +1 − f j −1 )

(A.2)

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M. Shoucri

s j −1 + 4s j + s j +1 = 6( f j −1 − 2 f j + f j −1 )

(A.3)

g j −1 + 4 g j + g j +1 = − f j −1 + 3 f j − 3 f j +1 + f j + 2

(A.4)

We can verify after substitution from Eqs.(A.2-A.4) in Eq.(A.1) the following relation:

~y + ~y + ~y = Af + Bf + Cf + Df j −1 j j +1 j −1 j j +1 j+2

(A.5)

A = (1 − Δ) 3

(A.6)

B = 4 − 3Δ2 (1 + (1 − Δ))

(A.7)

C = 4 − 3(1 − Δ) 2 (1 + Δ)

(A.8)

D = Δ3

(A.9)

We verify that for Δ = 0 we have from Eq.(A.5):

i.e.

~ y j −1 + ~ yj + ~ y j +1 = f j −1 + 4 f j + f j +1 ~ yj = fj

and for Δ = 1 we verify that:

i.e.

~y + ~y + ~ y j +1 = f j + 4 f j +1 + f j + 2 j −1 j ~ y j = f j +1

as it should be. The inversion of the tridiagonal matrix in Eq.(A.5) with appropriate boundary conditions y j . The calculation of y j ( x j − Δ) = ~ y j with 0 < Δ < 1 is done in a similar determines ~ way and leads to the following relation :

~ y j −1 + ~ yj + ~ y j +1 = Af j +1 + Bf j + Cf j −1 + Df j − 2

Appendix B Interpolation Using the Cubic Spline In the general case of a variable grid size, and for an equation of the form:

(A.10)

The Method of Characteristics for the Numerical Solution…

81

∂f ∂f + υ ( x) =0 ∂t ∂x

(B.1)

The interpolation with a cubic spline polynomial is treated as follows. We assume that y j ( x) is a cubic polynomial in x j , x j +1 such that y j ( x j ) = f ( x j ) = f j , and

{

}

y j ( x j +1 ) = f ( x j +1 ) = f j +1 . We denote by s j the second derivative at the point x j , and we

{

}

set Δx j = x j +1 − x j . We can write for the second derivative in x j , x j +1 the following linear interpolation:

y ′j′ ( x) = s j

x j +1 − x

Δx j

+ s j +1

x − xj

(B.2)

Δx j

so that y ′j′ ( x j ) = s j and y ′j′ ( x j +1 ) = s j +1 . Integrating twice Eq.(B.2), we get:

y j ( x) =

sj

6 Δx j

( x j +1 − x) 3 +

s j +1

6 Δx j

( x − x j ) 3 + a j ( x j +1 − x) + b j ( x − x j )

(B.3)

With y j ( x j ) = f ( x j ) = f j , and y j ( x j +1 ) = f ( x j +1 ) = f j +1 , we get:

aj =

fj

Δx j

− sj

Δx j 6

; bj =

f j +1

Δx j

− s j +1

Δx j

(B.4)

6

and

y j ( x) =

sj

6 Δx j

⎛ fj Δx j (x − x j )3 + ⎜ − sj ⎜ Δx 6Δx j 6 ⎝ j Δx j ⎞ ⎟( x − x j ) 6 ⎟⎠

( x j +1 − x) 3 +

⎛ f j +1 +⎜ − s j +1 ⎜ Δx j ⎝

s j +1

⎞ ⎟( x j +1 − x ) ⎟ ⎠

(B.5)

We write that the derivative y ′j ( x) is continuous at x = x j : y ′j ( x j ) = y ′j −1 ( x j ) , we get the following relation:

⎛ f j +1 − f i f j − f j −1 ⎞ ⎟ Δx j s j +1 + 2(Δx j + Δx j −1 ) s j + Δx j −1 s j −1 = 6⎜ − ⎜ Δx ⎟ x Δ j j −1 ⎝ ⎠

(B.6)

82

M. Shoucri

Inverting the tridiagonal matrix in Eq.(B.6) with proper boundary conditions determine the s j to be used in the cubic polynomial given in Eq.(B.5), which can also be rewritten in the form:

y j ( x) = A j f j + B j f j +1 + C j s j + D j s j +1 Where A j =

x j +1 − x

Δx j

; Bj =

x − xj

Δx j

; Cj =

Δx 2j 6

(B.5)

A j ( A 2j − 1) ; D j =

Δx 2j 6

B j ( B 2j − 1) .

For a value of y j ( x) at x = x j + Δ j where x j < x < x j +1 , (as for instance in Eq.(B.1) when Δ j = υ ( x j )Δt ,

υ ( x j ) < 0) we have : A j = 1 −

Δj Δx j

and B j =

Δj Δx j

. It is

straightforward to derive from the previous results the expression for y j ( x) at x = x j − Δ j , where x j −1 < x < x j .(as for instance in Eq.(B.1) when Δ j = υ ( x j ) Δt ,

υ ( x j ) > 0) . In this

case we have, with Δx j −1 = x j − x j −1 , we have:

y j ( x) = A j f j + B j f j −1 + C j s j + D j s j −1 Where A j = 1 − B j ; B j =

Δj Δx j −1

; Cj =

Δx 2j −1 6

2 j

A j ( A − 1) ; D j =

(B.6)

Δx 2j −1 6

B j ( B 2j − 1) .

(note that in the previous results, we assumed , υ ( x j ) Δt < Δx j , but the results can be generalized without difficulty to arbitrary values of

υ ( x j ) Δt ).

Appendix C Interpolation Using the Cubic B-spline To interpolate using a cubic B-spline, we write the function f(x) as follows:

f ( x) =

N x −1

∑γ j = −2

where the B j ( x) are defined as follows [84]:

j

B j ( x)

(C.1)

The Method of Characteristics for the Numerical Solution…

⎧( x − x j ) 3 ⎪ 2 3 1 ⎪1 + 3( x − x j +1 ) + 3( x − x j +1 ) − 3( x − x j ) B j ( x) = ⎨ 6 ⎪1 + 3( x j +3 − x) + 3( x j +3 − x) 2 − 3( x j +3 − x) 3 ⎪( x j + 4 − x) 3 ⎩

83

x j ≤ x < x j +1 x j +1 ≤ x < x j + 2 , (C.2) x j + 2 ≤ x < x j +3 x j +3 ≤ x < x j + 4

and B j (x) is equal to zero otherwise. Note that in this case the cubic polynomial is defined using four grid points. Because of the local definition of each B-spline, only 4 summands of Eq.(C.1) are non-zero. The calculation of the coefficients for the B-spline interpolation is performed as follows. We write for the given function value at the grid points xi :

f (x j ) =

N x −1

∑γ

j = −2

j

B j (x j )

(C.3)

which results in the equation:

γ j −3 + 4γ j − 2 + γ j −1 = 6 f j ; for j=1,…….Nx

(C.4)

where f j ( x) = f j , and we assume as boundary condition that the derivative is equal to zero at the boundaries : γ 0 = γ − 2 ,

γN

x −1

= γ N x −3 . We use the recursive ansatz:

γ j = γ j +1 X j + H j

(C.5)

which inserted into Eq.(C4) yields by comparison the coefficients:

Xj =−

1 ; X − 2 = −2 4 + X j −1

H j = X j ( H j −1 − 6 f j + 2 ) ; H − 2 = 3 f1

(C.6)

(C.7)

The values of X − 2 , H − 2 are obtained by considering the recursive ansatz with j = 2 and the left boundary condition

γ 0 = γ − 2 . The starting value of the recursion is obtained

using the right boundary condition:

γN

x −1

=

H N x −3 + H N x − 2 X N x −3

1 − X N s − 2 X N x −3

(C.8)

84

M. Shoucri Once

the

coefficients

γj

are

known,

~ f = f ( x j + Δ x ) are now calculated as follows : ~ 3 f = ∑γ κ =0

j −κ

arbitrary

bκx

interstitial

function

values

(C.9)

where j ≡ x j , Δ x = x − x j , and :

1 b0x = Δ 3x ; 6 1 b1x = (1 + 3(Δ x + Δ 2x − Δ 3x ) ) 6 1 b2x = (1 + 3((1 − Δ x ) + (1 − Δ x ) 2 − (1 − Δ x )3 ) ) ; 6 1 3 b3x = (1 − Δ x ) 6

(C.10)

References [1] Abbott, B.A. An Introduction to the Method of Characteristics; Thames and Hudson: London, 1966 [2] Pohn, E.; Shoucri, M.; Kamelander, G. Comp. Phys. Comm. 2005, 166, 81-93 [3] Shoucri, M. Comp. Phys. Comm. 2004, 164, 396-401 [4] Pohn, E.; Shoucri, M.; Kamelander, G. Comp. Phys. Comm. 2001, 137, 380-395;ibid 2001, 137, 396-404 [5] Yanenko, N.N. The Method of Fractional Steps, Springer-Verlag, New York, 1971 [6] Shoucri, M., Gerhauser, H., Finken, K-H. Comp. Phys. Comm. 2003, 154, 65-75 [7] Makar, P.A., Karpik, S.R. Mon. Wea. Rev. 1996, 124, 182-199 [8] Staniforth, A., Côté, J. Mon. Wea. Rev. 1991, 119, 2206-2223 [9] Sonnendrücker, E., Roche, J., Bertrand, P., Ghizzo, A. J. Comp. Phys. 1998, 149, 201 [10] Cheng, C.Z., Knorr, G. J. Comp. Phys. 1976, 22, 330-351 [11] Shoucri, M., Gagné, R. J. Comp. Phys. 1997, 24, 445-449; Phys. Fluids 1978, 21, 11681175, IEEE Plasma Science 1978, PS-6, 245-248; J. Comp. Phys. 1978, 27, 315-322 [12] Shoucri, M. Phys. Fluids 1978, 21, 1359-1365; ibid 1979, 22, 2038; ibid 1980, 23, 2030; J. de Physique 1979, 40, 38-39 [13] Shoucri, M., Storey, O. 1986, 29, 262-265; Simon, A., Short, R.W. Phys Fluids 1988, 31, 217 [14] Bertrand, P., Ghizzo, A., Feix, M., Fijalkow, E., Mineau, P., Suh, N.D., Shoucri, M. In Nonlinear Phenomena in Vlasov Plasmas; Doveil, F.;Ed.; Proc. Cargèse Workshop; Les Editions de Physique: Les Ulis, France, 1988; 109-125 [15] Cheng, C.Z. J. Comp. Phys. 1977, 24, 348-360

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[16] Shoucri, M., Gagné, R. J. Comp. Phys. 1978, 27, 315-322; Shoucri, M. In Modeling and Simulation; Proc. 10th Annual Pittsburgh Conference; Publisher: Instrument Society of America, Pittsburgh, 1979; Vol. 10; 1187-1192 [17] Shoucri, M. IEEE Plasma Science, 1979, PS-7, 69-72 [18] Johnson, L.E. J. Plasma Phys. 1980, 23, 433-452 [19] Rickman, J.D. IEEE Plasma Science 1982, PS-10, 45-56 [20] Ghizzo, A., Bertrand, P., Shoucri, M., Johnston, T., Feix, M., Fijalkow, E. J. Comp. Phys. 1990, 90, 431-457 [21] Bertrand, P., Ghizzo, A., Johnston, T., Shoucri, M., Fijalkow, E., Feix, M. Phys. Fluids 1990, B2, 1028-1037 [22] Johnston, T., Bertrand, P., Ghizzo, A., Shoucri, M., Fijalkow, E., Feix, M. Phys. Fluids 1992, B4, 2523-2537 [23] Ghizzo, A., Shoucri, M., Bertrand, P., Johnston, T., Lebas, J. J. Comp. Phys. 1993, 108, 373-376 [24] Shoucri, M., Bertrand, P., Ghizzo, A., Lebas, J., Johnston, T., Feix, M., Fijalkow, E. Phys. Letters A, 1991, 156, 76-80 [25] Ghizzo, A., Bertrand, P., Lebas, J., Shoucri, M., Johnston, T. J., Fijalkow, E., Feix, M.R. J. Comp. Phys. 1992, 102, 417-422 [26] Bertrand, P., Ghizzo, A., Karttunen, S., Pättikangas, T., Salomaa, R., Shoucri, M. Phys. Plasmas 1995, 2, 3115-3129; Physical Rev. E 1994, 49, 5656-5659 [27] Ghizzo, A., Bertrand, P., Bégué, M.L., Johnston, T., Shoucri, M. IEEE Plasma Science 1996, 24, 370-378 [28] Bégué, M.L., Ghizzo, A., Bertrand, P. J. Comp. Phys. 1999, 151, 458-478 [29] Ghizzo, A., Bertrand, P., Shoucri, M., Johnston, T., Fijalkow, E., Feix, M.R., Demchenko, V.V., Nucl. Fusion 1992, 32, 45-65 [30] Brunner, S., Valeo, E. Phys. Rev. Lett. 2004, 93, 145003-1 -145003-4 [31] Strozzi, D., Shoucri, M., Bers, A., Williams, E., Langdon, A.B. J. Plasma Physics. 2006, 72, 1299-1302 [32] Manfredi, G., Shoucri, M., Feix, M., Bertrand, P., Fijalkow, E., Ghizzo, A. J. Comp. Phys. 1995, 121, 298-313 [33] Ghizzo, A., Bertrand, P., Shoucri, M., Fijalkow, E., Feix, M. Phys. Fluids 1993, B5, 4312-4326 [34] Manfredi, G., Shoucri, Shkarofsky, I., Ghizzo, A., Bertrand, P., Fijalkow, E., Feix, M., Karttunen, S., Pattikangas, T., Salomaa, R. Fusion Tech. 1996, 29, 244-260 [35] Manfredi, G., Shoucri, M., Dendy, R.O., Ghizzo, A., Bertrand, P. Phys. Plasmas 1996, 3, 202-217 [36] Manfredi, G., Shoucri, M., Bertrand, P., Ghizzo, A., Lebas, J.,Knorr, G., Sonnendrücker, E., Bürbaumer, H., Entler, W., Kamelander, G. Phys. Scripta 1998, 58, 159-175 [37] Shoucri, M., Lebas, J., Knorr, G., Bertrand, P., Ghizzo, A., Manfredi, G., Christopher, I., Phys. Scripta 1997, 55, 617-627; ibid 1998, 57, 283-285 [38] Shoucri, M., Manfredi, G., Bertrand, P., Ghizzo, A., Knorr, G. J. Plasma Phys 1999, 61, 191 [39] Watanabe, T.-H., Sugama, H., Sato, T. J. Phys. Soc. Japan, 2001, 70, 3565-3576 [40] Watanabe, T.-H., Sugama, H. Trans. Theory Stat. Phys. 2005, 34, 287-309 [41] Arber, T.D., Vann, R.G.L., J. Comp. Phys. 2002, 180, 339

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[42] Pohn, E., Shoucri, M., Kamelander, G. Comp. Phys. Comm. 2005, 166, 81-93; J. Plasma Physics. 2006, 72, 1139-1143 [43] Pohn, E., Shoucri, M. Proc. Vlasovia workshop (to be published in Commun. Nonlinear Sci. Numer. Simul.) 2007 [44] Shoucri, M. Czech. J. Phys. 2001, 51, 1139-1151 [45] Batishchev, O., Shoucri, M., Batishcheva, A., Shkarofsky, I. J. Plasma Physics 1999, 61, 347- 364 [46] Ghizzo, A., Izrar, B., Bertrand, P., Fijalkow, E., Feix, M., Shoucri, M. Phys. Fluids 1988, 31, 72-82 [47] Ghizzo, A., Izrar, B., Bertrand, P., Feix, M.R., Fijalkow, E., Shoucri, M. Phys. Lett. A 1987, 120, 191-195 [48] Bernstein, I.B., Greene, S.M., Kruskal, M.D. Phys.Rev. 1957, 108, 546-554 [49] Knorr, G. Plasma Phys. 1977, 19, 529-538 [50] Knorr, G., Pecseli, H.L. J. Plasma Phys. 1989, 41, 157-170 [51] Reproduced from Comp. Phys Comm., Vol. 164; Shoucri, M., Gerhauser, H., Finken, K.H., Study of the Generation of a Charge Separation and Electric field at a Plasma Edge using Eulerian Vlasov Codes in Cylindrical Geometry, p. 139-141, Copyright 2004, with permission from Elsevier. [52] Huot, F., Ghizzo A., Bertrand, P., Sonnendrücker, E., et al J. Comp. Phys. 2003, 185, 512-531 [53] Strozzi, D., Shoucri, M., Bers, A. Comp. Phys. Comm. 2004, 164, 156-159 [54] Shoucri, M., Matte, J.-P., Côté, A. J. Phys. D : Appl. Phys. 2003, 36,2083-2088 [55] Joyce, G., Montgomery, D. J. Plasma Phys. 1973, 10, 107-120 [56] Knorr, G. Plasma Phys.1974, 5, 423-434 [57] Marchand, R., Shoucri, M. J. Plasma Phys. 2001, 65, 151-160 [58] Shoucri, M. Int. J. Num. Methods Eng. 1981, 17, 1525-1538 [59] Shoucri, M., Knorr G. Plasma Phys. 1976, 18, 187-204 [60] Krane, B., Christopher, I., Shoucri, M., Knorr, G. Phys. Rev. Lett. 1998, 80, 4422-4425 [61] Ghizzo, A., Bertrand, B., Shoucri, M., Fijalkow, E., Feix, M.R. J. Comp. Phys. 1993, 108, 105- 121 [62] Grasso, D., Califano, F., Pegoraro, F., Porcelli, F. Phys. Rev. Lett. 2001, 86, 5051-5054 [63] Grasso, D., Borgogno D., Califano, F., Farina, D., Pegoraro, F., Porcelli, F. Comp. Phys. Comm. 2004, 164, 23-28 [64] Pegoraro, F., Liseikina, T., Echkina, E.Yu. Trans. Theory Stat. Phys. 2005, 34, 243-259 [65] Durran, D.R. Numerical Methods for Wave Equations in Geophysical Fluid Dynamics; Text in Applied Mathematics 32; Springer: New-York, N.Y., 1998 [66] Dritschel, D.G., Polvani, L., Mohebalhojeh, A.R. Mon. Wea. Rev. 1999, 127, 1551-1565 [67] Imai, Y., Aoki, T., Shoucri, M. J. Appl. Meteo. Climat. 2007, 46, 388-395 [68] Reproduced from Comp. Phys. Comm., Vol. 176; Shoucri, M., Numerical Solution of the Shallow Water Equations with a Fractional Step Method, p. 23-32, Copyright 2007, with permission from Elsevier. [69] Biskamp, D., Welter, H. Phys. Fluids 1989, B1, 1964-1979 [70] Grauer, R., Marliani, C. Phys. Plasmas, 1995, 2, 41-47 [71] Politano, H., Pouquet, A., Sulem, P.L. Phys. Fluids 1989, B1, 2330-2339 [72] Pouquet, A., Sulem, P.L., Meneguzzi, M. Phys. Fluids, 1988, 2635-2642

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[73] Shoucri, M. Proc. Vlasovia Workshop, ( to be published in Comm. Nonlinear Sci. Numer. Simul.) 2007 [74] Grandgirard, V., Brunetti, M., Bertrand, P., Besse, N., Garbet, X., Ghendrih, P., Manfredi, G., Sarazin, Y., Sauter, O., Sonnendrücker, E., Vaclavick, J., Villard, L. J. Comp. Phys. 2006, 217, 395-423 [75] Jenko, J. Comp. Phys. Comm. 2000, 125, 196-209 [76] Candy, J., Waltz, R.E. J. Comp. Phys. 2003, 186, 545-581 [77] Idomura, Y., Watanabe, T.-H., Sugama, H. C. R. Physique 2006, 7, 650-669 [78] Nakamura, T., Yabe, T. Comp. Phys. Comm. 1999, 120, 122-135 [79] Mangeney, A., Califano, F., Cavazzoni, C., Travnicek, P. J. Comp. Phys. 2002, 179, 475-490 [80] Purnell, D.K. Mon. Wea. Rev. 1976, 104, 42-48 [81] Johnston, T.W., Picard, G., Matte, J.P., Fuchs, V., Shoucri, M. Plasma Phys. Cont. Fusion 1985, 27, 473-485 [82] Reveillé, T., Bertrand, P., Ghizzo, A., Lebas, J., Johnston, T.W., Shoucri, M. Phys. Fluids 1992, B4, 2665-2668 [83] Toro, E.F. Riemann Solvers and Numerical Methods for Fluid Dynamics; Springer: Berlin, 1999 [84] de Boor, C. A Practical Guide to Splines; Applied Mathematics 27; Springer-Verlag: New-York, N.Y., 1978 [85] Ahlberg, J.H., Nilson, E.N., Walsh, J.L. The Theory of Splines and their Applications; Academic Press: New York, N.Y., 1967

In: Advances in Mathematics Research, Volume 8 Editor: Albert R. Baswell, pp. 89-114

ISBN: 978-1-60456-454-9 © 2009 Nova Science Publishers, Inc.

Chapter 2

NEGOTIATING MATHEMATICS AND SCIENCE SCHOOL SUBJECT BOUNDARIES: THE ROLE OF AESTHETIC UNDERSTANDING Linda Darby Faculty of Education, Deakin University, Australia

Abstract A tradition of subject specialisation at the secondary level has resulted in the promotion of pedagogy appropriate for specific areas of content. This chapter explores how the culture of the subject, including traditions of practice, beliefs and basic assumptions, influences teachers as they teach across school mathematics and science. Such negotiation of subject boundaries requires that a teacher understand the language, epistemology and traditions of the subject, and how these things govern what is appropriate for teaching and learning. This research gains insight into relationships between subject culture and pedagogy by examining both teaching practice in the classroom and interrogating teachers’ constructions of what it means to teach and learn mathematics and science. Teachers’ level of confidence with, and commitment to, both the discipline’s subject matter and the pedagogical practices required to present that subject matter is juxtaposed with their views of themselves as teachers operating within different subject cultures. Six teachers from two secondary schools were interviewed and observed over a period of eighteen months. The research involved observing and videoing the teachers’ mathematics and/or science lessons, then interviewing them about their practice and views about school mathematics and science. The focus of this chapter is on the role of the aesthetic, specifically “aesthetic understanding,” in the ways science and mathematics teachers experience, situate themselves within, and negotiate boundaries between the subject cultures of mathematics and science. The chapter outlines teachers’ commitments to the discipline, subject and teaching by exploring three elements of aesthetic understanding: the compelling and dramatic nature of understanding (teachers’ motivations and passions); understanding that brings unification or coherence (relationships between disciplinary commitments and knowing how to teach); and perceived transformation of the person (teacher identity and positioning). This research has shown that problems arise for teachers when they lack such aesthetic understanding, and this has implications for teachers who teach subjects for which they have limited background and training.

90

Linda Darby

Introduction Science and mathematics are often closely associated during discussions about teaching and learning. This is reflected in the common expectation that teachers trained in either junior secondary mathematics or science will teach both at some time in their career. This suggests an assumption that mathematics and science have elements in common, such as common ways of thinking. This, in turn, implies assumptions about what might be common in terms of pedagogies appropriate for the two subjects (see, for example, Beane, 1995; Berlin & White, 1995). Little research, however, investigates how teachers internalize and deal with these assumptions. As disciplines, mathematics and science are distinguishable epistemologically and methodologically, and these differences are represented in the subject matter, pedagogies and purposes associated with their respective school versions. These differences place demands on teachers as they make decisions about what needs to be taught, the methods used, and the value that the subjects might have for students. The subjects are recognizably different; as are the ways students and teachers have been traditionally perceived in relation to those subjects. The distinctive nature of school subjects is described by Goodson (1993, p.31), who explaining how the organisational structure of the subject influences the ways teachers relate to the subjects and their students: [the] “subject” is the major reference point in the work of the contemporary secondary school: the information and knowledge transmitted in schools is formally selected and organised through subjects. The teacher is identified by the pupils and relates to them mainly through her or his subject specialisation.

Research is needed to understand how teachers experience the different demands that school mathematics and science place on teaching and learning. Of particular interest is how teachers construct for themselves these two subjects, and factors that influence the way teachers negotiate the boundaries that exists within the secondary school context. The research reported in this chapter explores how teachers’ experiences with the subjects influence them as they teach across mathematics and science. Negotiating subject boundaries requires that a teacher understand the language, epistemology and traditions of the subject, and how these things govern what is appropriate for teaching and learning. Teachers are, in a sense, inducted into the culture of the subjects by way of their own experiences of doing, using, learning and teaching mathematics and science. This research gains insight into the subject cultures of secondary mathematics and science from the perspective of the teacher and his/her classroom practice, focusing on the personal aspects of teaching, including how teachers see themselves as teachers, learners and participants with respect to mathematics and science. Teachers’ level of confidence with, and commitment to, both the discipline’s subject matter and the pedagogical moves required to present that subject matter is juxtaposed with their views of themselves as teachers operating within different subject cultures. This chapter begins by comparing mathematics and science as forms of education. The differences and similarities between the two subjects are explored so as to describe the cultural traditions that the teachers participating in this research are likely to have been exposed to. A section follows that explores the relationship between the individual and culture in the context of education. This leads into a discussion of the role that aesthetics (in

Negotiating Mathematics and Science School Subject Boundaries

91

the tradition of Dewey) has played in education. Theory surrounding the notions of aesthetic experience and aesthetic understanding is discussed in terms of student learning, but I open these discussions to include questions about the role of aesthetics in the relationship between subject culture and pedagogy. The research and my findings follow, using the framework of aesthetic understanding.

Comparing Mathematics and Science as Secondary School Subjects The academic disciplines of mathematics and science are represented as school subjects; however, the nature of these school versions do not, and perhaps cannot, necessarily mirror the academic versions. The foundational knowledge of mathematics and science are translated and organised for the purpose of meeting educational outcomes. In schools, mathematics is often presented as a requirement for adult life where students recognise the utility of mathematics. Crockcroft states that mathematics should also be presented as a subject to enjoy through the use of mathematics puzzles and problems that students can engage with. In addition, mathematics education takes on a role of developing “the powers of ‘abstraction’ and ‘generalization’ and their expression in algebraic form on which higher level mathematics depends… [All] students should have opportunity to gain insight, however slight, into the generalised nature of mathematics and the logical process on which it depends” (Crockcroft, 1982, p.67). Where work in science is about relating scientific evidence to scientific theory (Board of Studies, 2000a), science education allows students to be exposed to scientific ideas through participating in processes employed by scientists (cf Gunstone & White, 2000). Through learning and applying science, students are empowered as members of society: “Science education contributes to developing scientifically and technologically literate citizens who will be able to make more informed decisions about their lifestyle and the kind of society in which they wish to live” (Board of Studies, 2000b, p.5). Like mathematics education, science education serves a mechanistic purpose in the lifelong learning of students. Despite this apparent similarly in the purpose of the two subjects, Siskin’s (1994) research revealed differences in discursive patterns and dominant themes in subjects as teachers talk about their work. Siskin states that these dominant themes are worth exploring because they “translate into systematically different conceptions of the tasks of teaching and learning” (p.162). How, then do mathematics and science compare? An initial point of comparison draws on the assumption of Siskin (1994) that mathematics is a single discipline, whereas science is a cluster of disciplines, that is chemistry, biology, physics, geology. This difference has a number of implications for teaching. Where mathematics is characterised by an “ordered progression from place to place through a sequence of steps” and different levels (Siskin, 1994, p.170), science is characterised by a progression through disciplinary routes. Subsequently, in mathematics, Siskin (1994) claims that teachers develop general agreement about “what counts as knowledge, and how it is organised and produced” (p.170). Counter to these claims of general agreement, Schoenfeld (2004) states that, as with other

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subject areas, controversies exist about the epistemological foundations of the mathematics discipline, particularly “what constitutes ‘thinking mathematically’, which is presumably the goal of mathematics instruction” (p.243). Despite these controversies, mathematics has often been and continues to be characterised by incremental learning, “a slow systematic and progressive movement from the simple to the complex” (Hargreaves, 1994, p.139). Mathematics activities are, therefore, often seen as “a sequential progression through a series of topics, each of which is a prerequisite to what follows” (Sherin, Mendez, & Louis, 2004, p.208), p.208). The hierarchical nature of the way the mathematical curriculum is organised makes mathematics difficult to teach and learn (Crockcroft, 1982). With this as a teaching model, Siskin claims that “mathematics teachers value testing, placement, and tracking as the means of assigning students to the right rungs during their progress up the ladder” (p.170). Tracking is presented as a point of difference between mathematics and teachers of other subjects: where tracking is viewed by mathematics teachers as a means of meeting student learning needs, tracking is viewed by teachers from other subjects as simply “convoluted” and extraneous. One of the consequences of having widespread agreement on the content and sequence – what Siskin (1994) calls “the tight paradigm of mathematics” – is that teachers are able to learn the routines, and thereby follow the same curriculum. Homogeneity across the subject often results, Siskin asserts, such that mathematics instruction in a department can be somewhat similar. This view of homogeneity is observed by Reys (2001) who notes that in America at least there exists a generally agreed upon core body of basic knowledge such that mathematics texts from different publishers are almost indistinguishable. The best sellers are emulated by other publishers – deviation from the “norm” (best seller) results in low book sales, thereby limiting motivation to change textbooks dramatically to address the reformed American Standards-based curriculum. In 1986, Dorfler and McLone were of an opinion congruent with Reys and Siskin stating that “the material content of school mathematics is to a high degree internationally standardised. Deviations from this standard are only minor and depend on the educational system, local traditions and influences and perhaps special local demands” (p.58). This view dominates accounts of how subject matter is organized in school mathematics. On the other hand, according to Siskin (1994), the multi-disciplinary nature of school science “brings together not different ways of knowing the same content, but the same scientific method used to know different topics” (p.174). It is beyond the scope of this chapter to explore the debate surrounding either the nature of science as represented in schools, or how representative the “scientific method” is of the way scientists operate. Suffice to say that, according to Schoenfeld (2004), claims to a scientific method that permeates all the scientific disciplines is overstated, suggesting that different disciplines of science, such as physics and biology are more disparate in both theory and method than are anthropology and sociology. Many writers in science education prefer to focus on the nature of science as the underlying thread of school science curriculum and pedagogy, recognising the disciplines of science as adopting many methods but being subject to some basic tenets of science, although the question of what these tenets should be are still unresolved, (see for example Longbottom & Butler, 1999; MacDonald, 1996). Studies have investigated how the nature of science is represented, with more recent research promoting the importance of explicit instruction of and participation in the nature of science in science classrooms (Hart, Mulhall, Berry, Loughran, & Gunstone, 2000). Such learning experiences can be achieved by providing

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authentic science experiences (Rahm, Miller, Hartley, & Moore, 2003; Roth & McGinn, 1998) where students are considered as nonscientists participating in the scientific community of practice by engaging in “habits of thought” cognisant with scientific thinking (Trumbull, Bonney, Bascom, & Cabral, 2000, p.1). A second area of comparison relates to the nature of the knowledge underpinning the subjects of mathematics and science. Science is characterised by Siskin (1994) as being less abstract and more activist than mathematics. The Victorian Mathematics Curriculum and Standards Frameworks II (Board of Studies, 2000a) states that “[b]ecause mathematical knowledge is about relationships between things, it is inherently an abstract discipline. This abstractness makes it applicable in a wide variety of situations, but present particular challenges to teachers and learners” (p.5, emphasis in the original). However, application in mathematics has an important place in applying concepts and skills in the process of problem solving, where problems are contextualised for students in both familiar and unfamiliar everyday situations (Crockcroft, 1982). By comparison the Science Curriculum and Standards Frameworks II (Board of Studies, 2000) talks about application of scientific knowledge and making connections between the science community and society. For example, where mathematics patterns are taken out of context, such as tile patterns on a bathroom wall, patterns in science are dealt with in real life contexts. Scientists then “do something with it” that places the theory into practice, what Siskin (1994) calls “activist” and “making a difference”. This can be linked to the application of scientific principles to real life contexts: “science knowledge is characterised by a complexity of application of conceptions to the real world, and to classroom activities” (Tyler et al., 1999, p.211). With the variety of disciplines and the phenomena associated with those disciplines comes a “rich conceptual base” that adds complexity to planning in science (p.211). This can be contrasted with the contestable notion of an agreed conceptual sequence associated with mathematics curriculum. The National Council of Teachers of Mathematics (2000) identified that one of the call marks of an effective mathematics teacher is having an understanding of the “big ideas of mathematics and [being] able to represent mathematics as a coherent and connected enterprise” (p.17). Rico and Shulman (2004) mention the long-standing dispute over the way science should be represented, arguing for a divergence from the entrenched and much criticised “science-asfacts” model towards “science as ‘doing’, investigating, conducting research, actively seeking solutions to yet-solved problems” (p.162). Rico and Shulman state that these poor models are unfortunately perpetuated by commercially produced materials that “emphasise facts, formulae, demonstration, and vocabulary” (p.162). Processes of science are seen to be addons rather than necessary for learning science content. Studies of science textbooks show that often only the basic facts are covered and that they introduce more new vocabulary than foreign language textbooks. Van den Berg (2000) claims that American textbooks tend to have little educational value partly because the experiments are dictated by tradition, and utilize largely recipe style procedures. Traditional modes of instruction in mathematics have also been criticised and researched. Siskin’s (1994) analysis of how mathematics teachers expressed their knowledge of their discipline demonstrated above reinforces a content driven focus in school mathematics. In 1988, the Victorian Ministry of Education launched a curriculum framework that dealt with the pervading problem of classroom approaches that failed to encourage students in their mathematical learnings: “the type of mathematics that has tended to be offered to students in

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the past has become abstract at too early a stage” (p.11). Underlying their recommendations was the need to broaden students experience of mathematics so as to develop skills, concepts, applications and processes which allow meaningful participation in society” (p.12). Schoenfeld (2004) and Sherin et al. (2004) reiterate this paradigmatic shift in current mathematics curriculum where both content and process are essential for mathematical understanding. A teacher participating in Sherin et al.’s research on Fostering a Community of Learners (FCL) reported that when he began to rethink mathematics as content and process, the classroom discourse was transformed with greater emphasis given to students sharing and responding to each other’s ideas. This emphasis on teacher instruction is indicative of the general movement in mathematics education reform where effective mathematics instruction is reconceptualized as “a human construction based on historical efforts to solve particular problems, accepted modes of discourse and validation that are essentially social in nature” (Tytler, et al., 1999). In 1988 the Crockcroft Report described six elements of successful mathematics teaching: exposition, discussion, practical work, practice, problem solving and investigational work. Clearly, this seemingly new emphasis on content and process has been evident in the literature but maybe not so apparent in the classroom. In many ways science and mathematics can be differentiated. In addition to the arguments presented thus far, mathematics and science can be further characterised by their degree of reliance on equipment (“materials of the trade”), and by the type of clientele they attract, for example, science has a somewhat contestable masculine image as evident by a seemingly lack of female science teachers and students electing to continue with science beyond the post-compulsory years (Siskin, 1994). However, the fact remains that teachers, educators and researcher often closely align science and mathematics because they apparently share “linear ways of approaching things, step-by-step procedures, quantitative methods, and a mature paradigm” (Siskin, 1994, p.174). While research exists that explores how teachers represent their disciplines through classroom practices, there has been very little research that takes a trans-disciplinary approach to such exploration. The question remains, what is it about the subjects that affords and constrains particular teaching and learning practices? And from the persepective of the teacher, how do teachers’ experience of these cultural traditions shape their sense of themselves, their students and their practice? This type of information is valuable, especially at a time when the traditional boundaries between subjects are being challenged.

Relationship between Subject Culture and the Individual I am approaching this relationship between subject culture and pedagogy from the individual teacher’s perspective, recognising that, although there may be a (or a number of) subject culture(s) that these teachers are operating within and contributing to, the teachers respond to this in their own way dependent on the sum of their personal beliefs, experiences, knowledge etc. Borrowing from cultural theory relating to cultural organization and leadership, I am framing subject culture as those patterns of “shared basic assumptions that the group learned as it solved its problems of external adaptation and internal integration” (Schein, 1992). These assumptions work well enough to be considered valid and are taught to new members during enculturation. In the teaching context, enculturation involves a lifetime of experiences of learning, practicing and teaching the subject. If the “group” here refers to all science teachers

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across all schools, then subject culture refers to those shared basic assumptions that govern the dominance of certain “subject paradigms” (what should be taught) and “subject pedagogies” (how this should be taught) (Ball & Lacey, 1980). These basic assumptions act as signposts and guidelines for teaching and learning the subject. Paechter (1991) prefers to use the term “subject subculture” to recognise that every school is likely to have their own consensual view about the nature of the subject, the way it should be taught, the role of the teacher, and what might be expected of the students. Schwab (1969) refers to this consensus as unity, which he sees as important in providing opportunities for group action (see also Ball and Lacey, 1980). Schwab also impresses the importance of diversity of practice and beliefs amongst teachers. This view acknowledges that teachers will bring with them their own interpretation of teaching the subject. Similarly, Goodson (1985) argues that teachers have a personalised concept of a subject and what constitutes the practice of teaching. This perspective on subject culture supports the assumption that a teacher’s construction of the subject (including what and how it is taught) and the role of the teacher and learner, is mediated by a teacher’s lens of personal beliefs, knowledge and experiences. It makes sense then, that the effect of the subject culture on shaping pedagogy is mediated by a lens of personal beliefs about what constitutes the subject, teaching and learning. Consequently, decisions about teaching and learning are likely to be based on experiences of the subject cultures and from life. Such experiences are likely to evoke in the teacher a response that is not only cognitive, but also affective.

The Aesthetic in Education The aesthetic became important to my explorations of subject culture and pedagogy when I became attentive to how teachers constructed themselves in relation to the subject. Teachers recognised that their interest in the topic under instruction had a strong bearing on how they taught. Subsequently, my interests turned to exploring the idea that teaching and knowing how to teach involves both cognitive and affective dimensions. Zembylas (2005b) recognises that emotion and cognition are inextricably linked in the process of student learning. I assert that the same can be said for teachers in their development as mathematics or science teachers. Increasing attention is being given to the affective domain as researchers explore its centrality in the learning of mathematics (Bishop, 1991; Sinclair, 2004), learning of science (Alsop, Ibrahim, & Kurucz, 2006; Chandrasekhar, 1990; Zembylas, 2005b) and learning in general (Beijaard, Meijer, & Verloop, 2004; Ivie, 1999; Pajares, 1992; Schwab, 1978; Zembylas, 2005a). The affective domain is often separated from cognition (Sinclair, 2004). Aesthetics is part of the affective domain, as are beliefs, values, attitudes, emotions and feelings, self-concept and identity (Schuck & Grootenboer, 2004). Educational research into the nature and importance of the aesthetic has centred predominantly on its role in learning (Gadanidis & Hoogland, 2002; Girod, Rau, & Schepige, 2003; Wickman, 2006). Other research focuses on the role of the aesthetic in the activity, psychology and affective response of scientists and mathematicians to their discipline, often with the intent of informing mathematics and science teaching of that which provokes an aesthetic response (Burton, 2002, 2004; Sinclair, 2004). For example, Sinclair (2004)

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explains that the aesthetic has long been claimed to play a central role in developing and appreciating mathematics. Recognition of the beauty of mathematics stems from the Ancient Greeks who believed in the affinity between mathematics and beauty based on its order, symmetry, harmony and elegance. This is often called the aesthetic of mathematics, but such an aesthetic is often removed from the mathematics curriculum (Doxiadis, 2003) and the mathematics story is shortened to a sequence of steps that can result in students failing to experience the pleasure of the process (Gadanidis & Hoogland, 2002). In science also, the words beauty, inspiring, artful and passion are often used by scientists to describe their work (Girod, Rau & Schepige, 2003). “The scientist does not study nature because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful…intellectual beauty is what makes intelligence sure and strong” (Poincare, 1946, quoted in Girod et al., 2003, p. 575). Science educators draw from the discipline of science the important ideas, behaviours and dispositions that should be presented to students. If science is characterised as being analytic, logical, objective and methodical, this is then translated in classrooms as requiring students to be removed critical observers of objects, events and the world. By comparison, Girod et al. (2003) make the point that some scientists “portray science with an opposing personality—one that draws us in, begs our curiosity, passion, and emotion” (p.575), which, if translated to the classroom, they claim can improve the quality of the learning experience. These portrayals of science and mathematics as eliciting an affective response such as curiosity and the pleasure of the process are in contrast to the objects of science and mathematics that “are amenable to a rational and cognitive inquiry” (Wickman, 2006, xii). Understanding these contrasting positions comes from Dewey’s theory of aesthetic experience. Dewey breaks down false binaries such as objective and subjective, logic and intuition, thought and feeling, mind and heart, and think and feel. Wickman explains that in an aesthetic experience the inner emotional world is continuous with the outer world, meaning that one cannot think of one without the other. The cognitive (factual, what is the case) cannot be conceived of without the normative (values, what ought to be) in an aesthetic experience (which is evaluative). In keeping with this epistemology, Girod et al. (2003) claim that “from the perspective of aesthetic understanding, science learning is something to be swept-up in, yielded to, and experienced. Learning in this way joins cognition, affect, and action in productive and powerful ways” (p.575-576). Limited research seeks to clarify the role of the aesthetic in teachers’ work (see Ivie, 1999), however, teaching is often referred to as an artistry (see, for example, Rubin, 1985). This chapter focuses on the role of the aesthetic, specifically aesthetic understanding, in the relationship between subject culture and pedagogy. I frame this in terms of not so much what and how the teachers learn, but how their aesthetic understanding relating to teaching mathematics and science can give insight into how teachers negotiate boundaries between the subjects of mathematics and science and their enacted subject cultures. In particular, the aims of the chapter are to: • •

Focus on how teachers construct themselves as teachers of a subject for which they have a level of commitment and about which they hold beliefs and values; and Explore the degree to which and in what manner the teacher has an aesthetic response, as part of their personal response to the subject cultures within which they teach.

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To do this, I use the framework of aesthetic understanding from Girod et al. (2003) to explore how the teachers’ construction of the subject and teaching is not simply cognitive but has an aesthetic dimension. “Aesthetic understanding is a rich network of conceptual knowledge combined with a deep appreciation for the beauty and power of ideas that literally transform one’s experiences and perceptions of the world” (p.578).

Methodology Past experience in qualitative research (see, for example, Darby, 2005b) led me to a methodology consistent with the belief that we construct meaning through interaction with our social setting. Consequently, the meaning gained from this research is considered to be a co-construction between the participants and myself as researcher. Such an emphasis on research refers to “qualitative research,” an umbrella term used by Merriam (1998) relating to orientations to inquiry focussing on understanding and explaining the meaning of social phenomena. Qualitative inquiry evolved out of recognition that human beings are chasms of complexity that could not be understood through the positivistic process of scientific experimentation that simply tests scientific theory. The positivist approach demands an adherence to procedures that are reproducible, based on refutable knowledge claims, and controlled for researcher errors or bias (Gall et al.,1999). Cohen and others (2000) describe positivism as being “characterised by its claim that science provides us with the clearest possible ideal of knowledge” (p.9). It is objective and quantifiable. Reality is considered to be stable, observable and measurable (Merriam, 1998). Used within the social sciences, the methodological procedures used to investigate social phenomena mirror those used in the natural sciences, and the end-product is expressed as laws or law-like generalisations akin to those established for the description of natural phenomena (Cohen, et al., 2000). Bogdan and Bilken broaden the scope of what qualifies as research to reflect a paradigmatic shift from positivism towards qualitative research, also referred to as postpositivism, or “anti-positivism”, and that Cohen and others (2000) consider to be naturalistic. Bogdan and Bilken’s (1992) qualitative research mode emphasises “description, induction, grounded theory, and the study of people’s understanding” (p.ix). In this view, there is a rejection of the positivist view of an objective observer of phenomena on the basis that the behaviour of individuals “can only be understood by the researcher sharing their frame of reference: understanding the world around them has to come from the inside, not the outside” (p.19). A key philosophical assumption underlying this form of inquiry is that “reality is constructed by individuals interacting with their social worlds” (Merriam, 1998, p.6). It is these constructions of reality, or meaning perspectives of individuals, that my research into teacher pedagogy was interested in accessing and understanding. This research explores the complex ways that teachers construct for themselves their ideas about teaching and learning, and the factors involved in the way these constructions may appear to be manifested within the classroom setting. The qualitative paradigm qualifies this subjective knowledge of teachers, the emic1 or insiders’ perspective, as worthy of investigation, useful and informing of educative practice, and provides a vehicle for understanding the complexity 1

as distinct from and preferred over the etic or outsiders’ perspective.

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of the social setting within which the teachers are situated. What’s more, qualitative research has the potential to provide rich and meaningful information to the body of educational research as it characteristically builds abstractions, concepts, hypotheses or theories through an inductive process rather than simply testing existing theory (Merriam, 1998). Such a view is consistent with a constructivist approach to research as described by Guba and Lincoln (such as 1998, 1981). The research reported in this chapter uses the discourse of the classroom and interviews to understand how teachers have constructed for themselves knowledge about teaching and learning, and the various factors that are brought to bear on both the development of their knowledge and beliefs and the manifestation of such beliefs in the classroom. My research is most suitably called constructivist as I have attempted to understand and reconstruct the constructions held by both the participants and the researcher. Constructivists operate according to the premise that knowledge and truth is constructed, not discovered by the mind; and that reality is both expressed in a variety of symbols and language systems, and “stretched and shaped to fit purposeful acts of intentional human agents” (Schwandt, 2000, p.236). Constructivist inquiry, Guba and Lincoln claim, “denotes an alternative paradigm whose breakaway assumption is to move from ontological realism to ontological relativism” (p.203). According to Guba and Lincoln (1998) a relativist ontology claims that “[r]ealities are apprehensible in the form of multiple, intangible mental constructions, socially and experientially based, local and specific in nature (although elements are often shared among many individuals and even across cultures), and dependent for their form and content on the individual person or groups holding the construction” (p.206). Constructions are considered to be “more or less informed and/or sophisticated” (p.206), rather than absolutely “true.” Furthermore, constructions and the realities associated with them are subject to change as the constructors are more informed and sophisticated. This results in the potential for multiple and sometimes conflicting social realities of the human intellect. My research focuses on how the mathematics and science teachers are constructing for themselves pedagogy while operating within and in response to the social setting of mathematics and/or science education, making the constructivist paradigm suitable. Although the research is focused more closely on the individual teacher’s constructions, I used my interactions with the teachers, the setting, and the literature to assist me in constructing a broader picture of the teachers’ cultural setting. These act as the social setting for the research, a setting cushioned in a socially mediated subjective language that, through this act of research, for me has meaning through my experience of it.

Research Methods The research reported in this chapter forms part of a doctoral study associated with a Deakin University ARC Linkage Project with the Victorian Department of Education and Training called Improving Middle Years Mathematics and Science (IMYMS)2. Six teachers of mathematics and/or science from two schools (School A and School B), teaching across 2

The IMYMS Project is being undertaken by Russell Tytler and Susie Groves of Deakin University, and Annette Gough of RMIT and funded by an Australian Research Council Linkage Grant, and linkage partner, Victorian Department of Education and Training. Funding was granted in 2003. My Ph. D. is one of two Australian Postgraduate Awards (Industry) associated with this grant.

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Years 7 to 10, participated in a dialogue with me and each other over a period of about one year in order to understand differences between the subject cultures of mathematics and science. A variety of qualitative methods were selected that would support and feed into this dialogue. These methods are outlined below. Two sequences of lessons in mathematics and/or science were observed for each teacher in order to gain some insight into the general practice of the teachers. Two of these lessons on two separate occasions were videoed, one mathematics and one science lesson for three teacher (Simon*, Pauline*, Ian^), two science lessons for two teachers (Donna*, James^), and two mathematics lessons for one teacher (Rose*). (* indicates teachers from School A; ^ indicates teachers from School B.) The video footage of both lessons on both occasions were returned to each teacher for personal viewing with a set of questions to guide their attention and reflection (a modified video stimulated recall process). A “reflective interview” with each teacher followed the private viewing on both occasions. The first interview explored teacher’s response to the video and the questions, and explored teacher background with, commitments to, and beliefs about the subjects, as well as exploring any lines of inquiry that were emerging from preliminary analyses of classroom observations or prior interviews (involved all teachers) (see Darby, 2004, for an explanation of the reflective interview and modified stimulated recall process). The second interview was preceded by an informal discussion with the teacher about the aims and big ideas represented in the unit of which the videoed lessons were a part, then the reflective interview asked teachers to explain how this lesson fitted within the unit sequence (involved only Simon, Donna and Rose; Pauline participated in the informal discussion but not the second reflective interview). A focus group discussion involving the four teachers from School A followed the first round of videoing and reflective interviewing, with discussion based around three statements arising from data analysis. Each statement was accompanied by feedback to each teacher that included excerpts from their reflective interviews that contributed to the development of the statement, and supportive experpts from literature that expand on or correlate with the teachers’ ideas. The statements were:

Statement 1: Mathematics and science place different demands on teachers and students. For example, a student absent from mathematics for an extended period of time is at a greater disadvantage than a student absent from science for an equal amount of time. Is this necessarily the case? Are there parts of learning and teaching in mathematics and in science for which this is not really true? Statement 2: a. There are some practices that are translated readily from mathematics to science and vice versa. b. There are some practices in science that really should be used more often in mathematics, and vice versa. c. There are some practices that cannot be translated because the subjects are very different. What are your views on this? Statement 3: The influences on teachers' treatment of content in their teaching, and their attitude to the subject, are in the following order: 1. school, personal and work experiences in relation to subject interests; 2. their undergraduate degree experience; 3. conversations and

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interaction with other teachers; 4. experiences of teaching the subject; 5. curriculum documents and direction by the subject department; and 6. professional development. To what extent is this true for you?

Teacher Profiles This chapter draws on data from three of the teachers from School A: Donna, Pauline and Rose. School A is a co-education school offering Years 7-12 in a provincial city in Victoria. These teachers were selected by the Head of Science to participate in this “video study.”

Donna Donna was in her fourth and fifth year of teaching during the project. Donna originally went through high school with the intention of becoming a veterinarian but then decided to explore her interests in zoology and ecology through a Bachelor of Science. Prior to doing a Graduate Diploma of Education in 1999, Donna had been working at a tourism park as an education officer, taking tour groups on possum prowls and conducting other environmental activities. She also worked at a horse-riding place and managed school and other groups, and has been involved in dolphin research. School A is Donna’s second school. Throughout her teaching career, she has taught junior science at all year levels, Year 11 and 12 Biology, and some junior mathematics.

Pauline Pauline was in her second and third year of teaching during the project. She completed a three-year Bachelor of Science majoring in physics, then enrolled in a two year teaching degree that prepared her to teach Prep to Year 12. Her methods were general science and senior physics, but she was also qualified to teach mathematics to Year 12. Pauline chose the combination of science and mathematics due to the demand for science and mathematics teachers. School A was the second school she has taught at. At both schools she has been teaching junior mathematics and science, and Year 11 and 12 Further Mathematics and Physics.

Rose Rose has been a mathematics teacher for about 15 years. Rose went to university to complete a Science Education degree, where she studied mathematics, statistics, chemistry and physics. She had no interest in the science, however, only doing it because she thought she had to. Although she has taught science, she chose fairly early in her career to teach only mathematics. Since completing her training, Rose has taught at various schools. During the project, Rose assumed the role of Head of Junior Mathematics. She taught mathematics at all year levels.

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Looking for the Aesthetic in the Relationship between Subject Culture and Pedagogy Girod et al. (2003) describe aesthetic understanding as being “transformative,” “unifying” and “compelling and dramatic” (p.578). These three aspects of aesthetic understanding are described below using excerpts from the interviews of Pauline, Donna and Rose. I use the teachers’ reflections to explore, firstly, how the subject culture frames the development of these three components of aesthetic understanding, and how the teachers’ aesthetic understanding of the subject guides how they teach. I then explore how the application of this framework helps to understand the relationship between pedagogy, which is underpinned by theoretical and perspectival frameworks in relation to teaching and learning (van Manen, 1990), and cultural practices of the subject, which the teachers participate in and contribute to.

Compelling and Dramatic Nature of Understanding This aspect of aesthetic understanding recognises that aesthetic experiences are steeped in emotion. Aesthetic experience “…quickens us from the slackness of routine and enables us to forget ourselves in the delight of experiencing the world about us in its varied qualities and forms” (Dewey, 1934/1980, quoted in Girod et al., 2003). In such experiences, emotion, cognition and action are fused. So when Rose says to her students at the beginning of the year “I love mathematics and by the end of the year I want you to really like mathematics too” she is demonstrating her passion for mathematics, that there is something about mathematics that compels her into further engagement with it. It is this that she wants to share with the students so that they can appreciate mathematics in the same way. Rose explains that she is interested in mathematics because it is logical and “it appeals to my logical brain.” A passion for the subject is evident here, a passion for the content matter, but also for teaching the content. In the focus group discussion I asked the teachers what passion is and what it looks like in mathematics compared to science. Rose shared with me during the focus group discussion an experience she had during a lesson where she and a small group of students were working together on a different task to the rest of the class. “And I was so engrossed,” Rose exclaimed, “I didn’t realise the class had finished! And I turned around and they were all sitting back in their chairs, but my kids were so engrossed in what they were doing and really happy.” Donna replied, “That’s what passionate looks like in mathematics!” Rose’s passion for promoting student engagement with the subject is recognisably an experience of “flow” where, simply put, a person is so engrossed in a task that they lose all sense of time (Csikszentmihalyi, 1997). During the focus group discussion, various teachers explained how passion for the subject (or discipline) as distinct ways of knowing and bodies of knowledge are evidential in the classroom: “You’re interested in it. Enjoy it. If you enjoy something then you’re going to impart that enjoyment onto your students” (Rose); and “You can see that [teachers] know their stuff and are passionate about mathematics” (Donna).

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Teachers’ lack of passion about the subject was also considered to be evident to students: “I think kids pick up on it when you don’t enjoy it. If you’re teaching something you don’t particularly enjoy, it seems like they muck up more. I dunno, maybe we’re all suffering together!” (Pauline). Many authors assert the importance of students seeing that teachers are passionate about their subject (see, for example, Darby, 2005b; Education Training Committee, 2006; Lane, 2006; Palmer, 1998). During the focus group discussion I asked teachers: if passion for the subject is so important what happens when teachers teach outside of their subject area? Donna explained that in these instances, a general passion for teaching students is important. As Donna explains below, this passion is rooted in that which first lured them into teaching: DONNA: What got you here in the first place, your passion for teaching.You may not be happy about it, but you’ve still got the basic passion for teaching to try and do the right thing by the kids and you go out of the way to make sure, no matter what subject it is, that you’re teaching them the best way you can…. It comes down to that you’re teaching people, not the subject.

This suggests that a passion for teaching is related to the activity of teaching students, separate from the content matter under instruction. In this case, the passion emerges out of a desire to engage with students.

Aesthetic, Passion and the Subject Three forms of passion are evident above: a passion for the subject matter, a passion for promoting student engagement with the subject, and a passion for teaching in general. This multi-dimensional framing of what drives teachers is represented by Day (2004): To be passionate about teaching is not only to express enthusiasm but also to enact it in a principled, value-led, intelligent way. All effective teachers have a passion for their subject, passion for their pupils and a passionate belief that who they are and how they teach can make a difference in their pupils’ lives, both in the moment of teaching and the days, weeks, months and even years afterwards. Passion is associated with enthusiasm, caring, commitment, and hope, which are themselves key characteristics of effectiveness in teaching. (p.12)

As indicated above by Donna, this sense of care can be perceived of as a passion for teaching in general and as separate from the subject matter. This is likely to be important for those teachers with a teaching allotment that includes a subject for which they have limited experience, training and commitment, and more than likely, passion. The question here is where the passion lies for the teacher: in the act of relating with students (as stated by Donna), or in the act of engaging students with subject matter that the teacher believes is valuable, whether it be process or conceptual (as demonstrated by Rose’s commitments in teaching mathematics). A question remains as to whether a teacher can be effective at engaging students in the subject matter if they have little passion, or even appreciation, for the subject. Rose believes that teacher interest is vital: “If you’re not interested in something, you shouldn’t teach it!” Day also describes the importance of teachers sharing with students a commitment to the subject they are teaching:

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When students can appreciate their teacher as someone who is passionately committed to a field of study and to upholding high standards within it, it is much easier for them to take their work seriously. Getting them to learn then becomes a matter of inspiration by example rather than by enforcement and obedience. (p.15)

The Education and Training Committee’s (2006) inquiry into the promotion of mathematics and science education in Victoria supported this view saying that when promoting student engagement there is a “need for teachers to be passionate and deeply knowledgeable about their subject area” (p.172). Following this view, a passion for teaching is more likely to be coloured by a teacher’s conceptual and aesthetic commitments to the subjects they teach; therefore, passion for teaching, at least at the secondary level, is less likely to be seen as generic, but more likely subject specific. Research by Siskin (1994) into the culture of subject departments in secondary schools found that what mattered for the teachers in her study was “not simply that they teach, but what they teach” (p.155, emphasis in original). Neumann (2006) asserts that, in the context of scholarship in higher education, “passion illuminates the complexity of both teaching and research, showing that what resides at the heart of both is the learning of a particular subject” (p.413, italics in original). Subject here refers not necessarily to a school subject or discipline but a subject of thought on which a conversation can be focused. In the classroom, the teacher makes the focus of conversation the ideas of mathematics or science, however, how they represent these ideas depends on the teachers beliefs about what the subject can offer the students. For Rose, mathematics offers training in logic and a potentially enjoyable endeavour. A passion for teaching remains, then, to be coloured by the teacher’s conceptualisation of the subject. According to this view, pedagogy is influenced by an inextricable link between the way teachers see their students and the subject: teachers have an understanding of what students need in order to make the subject matter have meaning. “Teachers understand and value their subjects for what they offer students, and understand their students through the metaphors and assumptions of the subjects” (Siskin, 1994, p.158). Consequently, pedagogical knowledge is tied to how the teacher understands the knowledge of the subject. Conversely, the content knowledge of teachers as representations of the epistemology of the subject is transformed in a way that meets the perceived learning needs of the students. Overarching both of these relationships, however, is the teachers’ aesthetic commitments to and appreciations for the subject.

Learning that Brings Unification or Coherence to Aspects of the World or the Subject This aspect of aesthetic experience acknowledges that “it is not possible to divide in a vital experience the practical, emotional, and intellectual from one another” (Dewey, 1934/1980, quoted in Girod et al., 2003, p.578). Experience is complete and results in deep meaning because the experience retains its value and wholeness, and this coherence can be used to guide future experiences. According to Girod et al., an “aesthetic understanding depends on developing a similar coherence of parts, pieces, ideas, and concepts” (p.578). This is evident in the classroom when the learning of individual parts of a concept brings greater understanding of the entire concept.

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The teachers in my study referred to this element of aesthetic understanding when they talked about planning for different subjects. Donna explains here that, she has a stronger grounding in biological science due to personal experiences with the subject matter, the discipline, and the type of thinking required, the manifestation of which is a more intuitive approach to teaching science than mathematics: DONNA: I don’t have a big mathematics background, so I have to spend a bit of time thinking about what could be available and what I could do; whereas with a science background, I think of things just because I’m experienced in that area. So I suppose it might depend on how much mathematics you’ve done or what resources you’ve been exposed to, what you might know of... I do a lot more prep for a topic like physics than I would for chemistry or biology. I’m teaching a nine ten combined class in biology, and I’m finding that, like I do my normal prep but I can just go off in class and say, I did this and I’ve got this example, and we’ve been having great class discussions and fun activities. I wouldn’t have the confidence doing that with a physics topic. So I might spend a lot more time researching it, I might check a few things with another teacher. But I wouldn’t have that flamboyance in a topic that, because I haven’t done physics at all, apart from bits and pieces of it.

Donna compares her teaching of biology to that of physics, both sciences and underpinned by a common philosophy of what constitutes knowledge, but distinct in terms of the nature of the phenomena being represented. Donna’s coherent and unified picture of biological science stems from her experiences of learning biology and working with these science concepts in the natural world. Physics, however, is perhaps as foreign for her as any other subject that has not been encountered in any meaningful way. It is for this reason that her teaching of biology requires less planning and research compared to her teaching of physics or mathematics.

Aesthetic, Coherence and the Subject In Donna’s reflection, there is a degree of understanding of the connections between ideas and content, but also how the content is used in a way that is appropriate for student learning. Knowledge of the content matter and the knowledge required to teach this knowledge is evident. The knowledge that Donna refers to can be aligned with Shulman’s knowledge domains that he introduced in 1986 and 1987 to emphasise the domain-specificity of knowledge. Shulman distinguishes between subject matter knowledge, pedagogical content knowledge and pedagogical knowledge. “Subject matter knowledge”, also called content knowledge, is the knowledge that teachers have about the content considered appropriate for teaching. Donna explained that such knowledge is related to the extent of her background with respect to the subject. Having a limited background in physics has meant that she has less content knowledge, and which results in her having to do more preparation for her lesson planning. “Pedagogical content knowledge” (PCK) adds to this dimension of subject matter the knowledge required for teaching it to students, and includes the “ways of representing and formulating the subject that makes it comprehensible to others” (Shulman, 1986, p.10). PCK refers to that conglomeration of teacher knowledge that transforms the subject matter in a way that is sensitive to the needs and requirements of the learners.

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In Donna’s case, she recognises that her pedagogical content knowledge, that is, knowing how to make the content understandable for students, is limited by her deficient subject matter knowledge of physics and mathematics. In comparison, she attributes her ability to teach biology to her “background” of experiences, and that this allows her to more meaningfully transform subject matter knowledge using classroom strategies where her richer understanding of the subject narrative can be shared with students. At first blush it appears that knowledge of content, resources and strategies for teaching accounts for her greater confidence with the teaching of biological science. The “flamboyance” she refers to hints to something other than knowledge, such as an intuitive sense of how to use the science ideas and her experiences to draw students into thinking, talking and engaging with the ideas: “You can think of different ways to get it across to the kids.” She has feelings of “comfort” and confidence in her ability to bring the subject of biology to life for her students. “To know something,” states van Manen (1982, p.295), “is to know what that something is in the way that it is and speaks to us.” That which first appears cognitive takes on an intuitive nature, and this becomes part of what teachers do but may not know that they do or why they do it.

Perceived Transformation of the Person and the World Donna’s description of her teaching above exudes a sense of pride in what she knows and how she can share this with students in an engaging way. There is passion, no doubt, but she has also “developed a sense of self in which the pride of the craft [is] the key” (Palmer, 1998, p.14). A person is transformed by what they have experienced and what they have come to know out of that experience. “Knowing changes the individual as well as the individual’s world” (Girod et al., 2003, p. 578). The transformative nature of aesthetic understanding can lead to identity formation and personal positioning. A person can say “I am the type of person that looks at the world in this way.” In the context of my study, this relates to how teachers position themselves as teachers of a subject, and how this positioning stems from their experiences of teaching, learning and participating in mathematics and science. I describe two teachers here, Rose and Pauline, to demonstrate how they position themselves in relation to the subject based on their level of competence and confidence with teaching the subject.

Rose’s Transformation Rose’s experiences and interests shape the way she sees herself. Rose stated a number of times that she describes herself as a teacher of students, not a teacher of the subject: “I see myself as a teacher first, not a mathematics teacher… I’d been looking after little kids from when I was this high. I just loved looking after kids.” During a different interview she mentioned that being a mother has made her more attentive to the support needs of her students such that she is more attentive to the overall well-being of her students. ROSE: I believe I do a lot more instructing than some people and I also do a lot more student helping… I think it is because I’m a Mum. I taught for 4 ½ years before I had children and then I came back to teaching and I reckon I was a lot better teacher, because I relate to it… I don’t think it’s an us and them, I think its an us together.

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She situates herself not necessarily outside of being a mathematics teacher, but prefers to identify herself as someone who has strong beliefs about the centrality of the student in the teaching-learning interface. This was demonstrated also when, on viewing her videoed lessons, she said that: “I looked for how the kids were working ‘cause that’s interesting. What I said and how I responded to the kids. To their needs. That’s what I look for.” Over the years, Rose has developed an understanding of her role as teacher that has transformed her into a person that is attentive to the needs of her students. The nature of the subject shapes her pedagogical response, such that a sense of care compels her to teach mathematics in a way that makes it less threatening for students: ROSE: I want them to enjoy mathematics. Because mathematics is a threatening subject, it is so threatening because it is so sequential…[At the start of the year] there was hardly anyone that liked mathematics, some of them thought they were good at it, but hardly any of them liked it. You ask them now they have come right round because they enjoy it.

Rose refers here to a mathematics curriculum characterized by incremental learning (Hargreaves, 1994) and sequential progression (Sherin, Mendez & Louis, 2004). Because she understands the threat that such a curriculum structure might pose for students, her sense of care for the students compels her to employ actions that remove the threat and make her view of “mathematics-as-enjoyable” more accessible and in the realm of possibility.

Pauline’s Identity Crisis as She Negotiates Subject Boundaries Pauline spoke of a rich science background with interests and studies in physics, and many engaging and interesting experiences in relation to science. In order to get a sense of how Pauline situates herself in relation to mathematics and science, I need to first reconstruct Pauline in relation to the previous two aspects. Evident in the following quote is a confidence in how she expresses an appreciation for the purpose of science in her own and her students’ lives, as well as what it means to be passionate about science: PAULINE: I find my knowledge of Science extends to everything. It extends to when I go to the Doctor and I talk about my health … everything I do is informed by my science knowledge, and I just think that scientific literacy is so important for kids to get the most out of themselves, out of their world… I like collecting [stories]. I don’t think I have enough. I like telling stories and getting the kids’ stories out as well. And I have found that when I studied science they were the things that got me excited when a teacher told me a really interesting story and I don’t know if mine are interesting or not, but I know that they were the sort of things that got my interest going in science and why I wanted to do more.

In comparison, limited expertise in mathematics teaching makes it difficult for Pauline to be confident in her abilities, and she defers to a label of science teacher rather than mathematics teacher, as evident in the following quotes: “I am not really experienced enough or done enough PD [Professional Development] to know better ways of doing it. A major part of my PD plan, especially for middle years, is doing more PD and finding better ways to teach stuff ‘cause I don’t like the way I teach Mathematics at the moment” “I think I am a crap mathematics teacher.”

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“It is funny. I feel more confident teaching science than I do mathematics, even though I have been teaching both for the same amount of time” “I have always felt Mathematics is kind of my fall-back method. Whereas if I was asked to describe myself I would describe myself as a Science teacher, first and foremost.”

Quite clearly, Pauline has a stronger sense of herself in relation to science teaching than mathematics teaching. She attributes this partly to her limited background experience with mathematics: “Well my mathematics method is just a thing on paper that says that I did mathematics to second year at Uni. There was nothing that I did in my teaching degree that prepared me for teaching mathematics. The only preparation that I had was my rounds.” She laments at not knowing how to make mathematics learning more interesting for her students because of her limited intuitive sense of what will work in the classroom, she is less capable of finding resources and knowing what to look for, and she has a limited sense of how to be passionate about teaching the subject in a way that will profit student learning at the junior level. She enjoys teaching mathematics at the senior level because she enjoys toiling over problems with the students, but she is unable to do this as much at the junior level. These limitations to her knowledge led her to the conclusion that she is less comfortable with the label of mathematics teacher than she is with that of science teacher.

Aesthetic, Identity and the Subject In Beijaard’s (1995) research into the interplay between the private and public in developing identity, he makes a distinction between role and identity – hope and courage, care and compassion, he asserts, are associated with identity, not role. In the above example, Pauline appears to accept the role of mathematics and science teacher and the associated activities that are assumed as part of this role, but her identity arises out of her history of caring for and committing to science as an area of study. Further, in Pauline’s description above, she attributes her lack of confidence in mathematics teaching with lacking the knowledge of how to teach. Earlier, Donna recognised that her teaching of biology is benefited by knowing what activities will work and when. Day (2004), however, points out that knowing what and how to teach is not limited to cognitive engagement. He states that “good teaching can never be reduced to technique or competence” (p.15). Good teachers, he asserts, tend not to describe themselves only in terms of technical competence, but also acknowledge that “teaching and learning is work that involves the emotions and intellect of self and student” (p.64). This difference between a competence view and the aesthetic was demonstrated by Pauline’s appraisal of herself as a mathematics teacher and a science teacher. Her deficit view in relation to mathematics that she attributes to limited technical competence is based on limited knowledge of what and how to teach, and her hope lays in future professional development to provide useful strategies for teaching. By comparison, her appraisal of her competence and confidence in science was laden with meaningful experiences and stories from a history of engaging with the subject. Pauline exhibited a richer sense of herself in relation to her science teaching, one that is positive and based not solely on competence, but she also aligns herself with science teaching at an emotional level. Her knowledge of how and what to teach is ‘continuous with’ her aesthetic response, meaning that one cannot think of one without the other.

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Insights and Implications The previous analysis and discussion have explored the idea that a teacher’s aesthetic understanding of and response to the subject determines: where their passions lie with respect to teaching the subject, to what extent they have a coherent and intuitive sense of what is required to teach the subject, and how the teacher is transformed by what they know as they develop an identity in relation to the subject. These discussions are valuable in understanding the relationship between subject culture and pedagogy for two reasons. These reasons are dicussed below.

Appreciation for the Aesthetic in the Teaching Act The first is that a framework of aesthetic understanding helps to clarify and assign some level of importance to the role of the aesthetic in the teaching of subject matter to students. A teacher who can be regarded as having an appreciative aesthetic understanding of the subject: • • •

is compelled by and passionate about the subject and students engaging with the subject; has a coherent, unified and intuitive sense of what the subject is about and how to bring it to life for students; and has been transformed by what they know and believe in a way that aligns them to personally and professionally identify with the subject.

Being attentive to the aesthetic when evaluating teaching redirects the question from simply asking, what does the teacher know and believe about the subject and what is required to teach it? Instead, the question becomes, how does what the teacher know and believe affect her sense of who she is in relation to the subject, and how does this personal positioning spill out into the classroom? The analysis has shown that a teacher with an appreciative aesthetic understanding of a subject see themselves, the subject matter, their teaching and their students in relation to the subject. Even Rose, who labelled herself as a teacher of students rather than a teacher of the subject, expressed her sense of care in the context of, and in response to, the nature of the subject and what was required for students to learn. The student is central to her conceptualisation of the subject. She was unable to describe what the subject is like without including stories about her interactions with students on a personal level, and in relation to how the students learn in the subjects. By talking about how she interacts with students and the students’ learning needs, Rose gives clues as to her values and aesthetic commitments to the subject, which is viewed through a lens of what the subject offers her students as well as what it offers herself as learner, practitioner and teacher of the subject. A common emphasis in current science education reform is to draw on and respond to student interests in selecting contexts for teaching science-related content. Pivotal in achieving this end is giving teachers space within the curriculum to inject their own interests, hobbies and expertise in constructing such contexts. Tytler (2007) provides examples of innovation occurring in schools where “teachers with serious interests [felt] that they were being given permission to import these into the classroom” (p.57-58):

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In one school a teacher with no previous history of innovation was encouraged by the SIS coordinator, who knew of his interest in winemaking, to initiate a Chemistry of wine making unit. The school is now producing award-winning wines. (p.52, italics in original)

These types of stories, Tytler (2007) asserts, exemplify a re-imagined science education for Australia. Hence, teachers’ interests are highlighted as important in the development of local content and approaches. In these situations, teachers are more likely to possess an aesthetic understanding that is deeply rooted in teachers’ experiences, and where the subject matter has personal meaning for the teacher. Pedagogical practices can be enriched by a deep understanding of the associated content, which, provided the learning needs and interests of students are taken into account, provide a strong foundation for knowing what value it might have for students and how such contexts could be generative of new interests for students.

The Aesthetic in the Negotiation of Subject Boundaries Secondly, examining teachers from the perspective of aesthetic understanding provides insight into what is involved for teachers, aesthetically, as they move between subjects and their enacted subject cultures. Such insights are particularly pertinent at present when a shortage of suitably qualified mathematics and science teachers is resulting in a relatively high percentage of teachers teaching out-of-field, that is, teaching a subject for which they lack tertiary training, and arguably, limited experience, commitment and, aesthetic understanding. A survey involving 8.2% of teachers of junior science in Australia (Harris, Jensz, & Baldwin, 2005) showed that 16% of respondents lacked a minor in any university science discipline, while 8% had not studied any tertiary science. Similarly, a survey of mathematics teachers from 30% of Australian schools (Harris & Jensz, 2006) showed that 20% of teachers of junior mathematics had not studied mathematics beyond first year university, while 8% had no tertiary training in mathematics. Other reports in the media reflect similar or higher proportions of teachers teaching outside their fields of expertise (Rodd, 2007; Topsfield, 2007). The figures are even more startling for teachers beginning their careers. Unfortunately, these teachers are more likely to be asked to teach out-of-field than their experienced colleagues (Ingersoll, 1998). A recent study of beginning teachers in Australia showed that 40.1 % of teachers nationally and 57% in Victoria had taught subjects outside their qualifications (Rodd, 2007). While it is acknowledged that tertiary training will not automatically result in effective teaching, the major concern both nationally and internationally is that without solid tertiary experience in the discipline, teachers lack content knowledge, and without studies in the teaching of a subject, teachers are not equipped with the variety of methods and teaching skills required to teach the subject effectively (Darling-Hammond, 2000; Education Training Committee, 2006; Ingersoll, 1998; Thomas, 2000). The data reported in this chapter suggests that a teacher teaching out-of-field, whether it be a science teacher teaching mathematics (in the case of Pauline) or a biologist teaching physics (in the case of Donna), potentially has limited or unappreciative aesthetic understanding of what the subject can offer his/her students. This has implications especially when the history of engagement with the subject has been negative, restricted to poor traditional learning experiences, or limited. Reliance on traditional teaching approaches may result, as may a lack of “flamboyance” in the way the subject is presented, with a potential

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outcome of not demonstrating for students what it looks like to appreciate the subject. Also teachers teaching outside of their disciplines, such as a mathematics teacher teaching science, may bring with them a sense of what constitutes good teaching appropriate for one subject that may seem inappropriate in another. A theoretical framework of aesthetic understanding, therefore, helps to identify the barriers, disconnections, and lacking appreciations that may prevent teachers who are not trained in the discipline from personally engaging with the subject, which, inevitably impacts negatively on the quality of teaching. The problem for the “untrained” mathematics or science teacher is not simply a lack of content knowledge, but this framework of aesthetic understanding gives significance to the importance of teachers being committed to the subject, being able to identify with it, and knowing how to bring the subject matter alive for students. Tertiary training is considered to be the most effective determinant of whether a teacher is suitable for teaching a subject. Having a background in a discipline, it is assumed, equips teachers with the disciplinary knowledge to draw on in their teaching, but it also equips teachers with an appreciation and enthusiasm for the subject that can be transmitted to students (Darby, 2005a, 2005b), something that is a quality of effective teachers and potentially lacking for teachers teaching out-of-field (Ingvarson, Beavis, Bishop, Peck, & Elsworth, 2004). However, other research shows that, while a teacher’s practice is dependent on the experiences that the teacher has had with the subject or discipline, these experiences are not necessarily related to exposure at university level. For example, other factors, such as career trajectory (Siskin, 1994) and professional development focusing on changes to teacher beliefs (Russell Tytler, Smith, & Grover, 1999), have been found to be cogent in determining how teachers approach teaching and learning. There is an assumption here that teachers can be inducted into the culture of a subject through their experiences, and that, with further training, teachers can improve their competence and confidence in teaching a subject for which they have previously had limited background. Competence refers to teachers’ development of knowledge and skills that are: subject-specific, such as content knowledge (CK) and pedagogical content knowledge (PCK); and generic, including pedagogical knowledge (PK) (Shulman, 1986a, 1987). Confidence relates to teachers’ attitudes (Ernest, 1989; Koballa, 1988), agency and self-efficacy (Boaler & Greeno, 2000), professional identity (Connelly & Clandinin, 1999) and aesthetic understanding as is described in this chapter. Allowing inexperienced teachers of the subject to have an aesthetic experience of the subject matter through targeted professional development may allow them to see themselves and their identity in relation to subject matter ideas.

Conclusion The analysis teases out what it can mean for a teacher to be compelled by and passionate about the subject and students engaging with the subject, to have a coherent and unified sense of what the subject is about and how to bring it to life for students, and to be transformed by what he/she knows and believes in a way that aligns them to personally and professionally identify with the subject. The teachers’ construction of the subject, their students and teaching is not simply cognitive but has an aesthetic dimension. An implication of this is that teachers who teach outside of their subject area—their subject area typically being dependent on whether they are “mathematics- or science-trained”—may be lacking an appreciative

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aesthetic understanding. Their aesthetic response to the content matter and how to teach it may be unlike that of someone who has an appreciative aesthetic understanding of the subject. Such teachers may: attempt to bring in a style appropriate for a subject that has a different set of demands; have a limited set of experiences with relevant phenomena, processes, ways of thinking and attitudes that can feed into their teaching; and fail to exhibit a passion for the subject and what the subject can do for their students. Consequently, any efforts to improve mathematics and science education should be aware that allowing teachers to experience the subject in a way that results in aesthetic appreciation for the beauty and elegance of mathematics and science is just as valuable as them developing conceptual and pedagogical knowledge associated with the subject. A teacher may then experience content in ways that allow them to more clearly see themselves in relation to subject matter ideas.

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Schwab, J. J. (1969). College curricula and student protest. Chicago: University of Chicago Press. Schwab, J. J. (1978). Eros and education: A discussion of one aspect of discussion. In I. Westbury & N. Wilkof (Eds.), Science Curriculum and Liberal Education (pp. 105-132). Chicago: University of Chicago Press. Sherin, M. G., Mendez, E. P., & Louis, D. A. (2004). A discipline apart: The challenge of 'Fostering a Community of Learners' in mathematics classrooms. Journal of Curriculum Studies, 36(2), 207-232. Shulman, L. S. (1986a). Paradigms and research programs in the study of teaching: A contemporary perspective. In M. C. Wittrock (Ed.), Handbook of research on teaching (3 ed., pp. 3-36). New York: Macmillan. Shulman, L. S. (1986b). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4-14. Shulman, L. S. (1987). Knowledge and teaching: Foundations of the new reform. Harvard Educational Review, 57(1), 1-22. Sinclair, N. (2004). The roles of the aesthetic in mathematical inquiry. Mathematical Thinking and Learning, 6(3), 261-284. Siskin, L. S. (1994). Realms of knowledge: Academic departments in secondary schools. London: The Falmer Press. Thomas, J. (2000, October). Mathematical science in Australia: Looking for a future. Retrieved January, 2007, from http://www.FASTS.org Topsfield, J. (2007, February 1). Labour pledges HECS cut for maths, science. The Age, p. 5. Trumbull, D. J., Bonney, R., Bascom, D., & Cabral, A. (2000). Thinking scientifically during participation in a citizen-science project. Science Education, 84, 265-275. Tytler, R. (2007). Re-imagining science education: Engaging students in science for Australia's future. Camberwell: Australian Council for Educational Research. Tytler, R., Smith, R., & Grover, P. (1999). A comparison of professional development models for teachers of primary mathematics and science. Asia-Pacific Journal of Teacher Education, 27(3), 193-214. Van den Berg, E. (2000). More on the quality of texts. Science Education International, 11(2), 19-21. Van Manen, M. (1982). Phenomenological pedagogy. Curriculum Inquiry, 12(3). Van Manen, M. (1990). Researching lived experience; Human science for an action sensitive pedagogy. London: The Althouse Press. Wickman, P. (2006). Aesthetic experience in science education: Learning and meaningmaking as situated talk and action. Mahwah, N.J.: Lawrence Erlbaum Associates, Inc. Zembylas, M. (2005a). Discursive practices, genealogies, and emotional rules: A poststructuralist view on emotion and identity in teaching. Teaching and Teacher Education, 21, 935-948. Zembylas, M. (2005b). Three perspectives on linking the cognitive and the emotional in science learning: Conceptual change, socio-constructivism and poststructuralism. Studies in Science Education, 41, 91-116.

In: Advances in Mathematics Research, Volume 8 Editor: Albert R. Baswell, pp. 115-140

ISBN: 978-1-60456-454-9 © 2009 Nova Science Publishers, Inc.

Chapter 3

THE MATHEMATICAL BASIS OF PERIODICITY IN ATOMIC AND MOLECULAR SPECTROSCOPY K. Balasubramanian Center for Image Processing and Integrated Computing, University of California Davis, Livermore, California 94550; Chemistry and Applied Material Science Directorate, Lawrence Livermore National Laboratory, University of California, Livermore, California 94550; and Glenn T. Seaborg Center, Lawrence Berkeley Laboratory, University of California, Berkeley, California 94720, USA

Introduction This chapter applies combinatorial and group-theoretical relationships to the study of periodicity in atomic and molecular spectroscopy. The relationship between combinatorics and both atomic and molecular energy levels must be intimate since the energy levels arise from the combinatorics of the electronic or nuclear spin configurations or the rotational or vibrational energy levels of molecules. Over the years we have done considerable work on the use of combinatorial and group-theoretical methods for molecular spectroscopy [1–15]. The role of group theory [1–40] is evident since the classification of electronic and molecular levels has to be made according to the irreducible representations of the molecular symmetry group of the molecule under consideration. Combinatorics plays a vital role in the enumeration of electronic, nuclear, rotational and vibrational energy levels and wave functions. As can be seen from other chapters in this book, the whole Periodic Table of the elements has a mathematical group-theoretical basis since the electronic shells have their origin in group theory. Indeed, this concept can even be generalized to other particles beyond electrons such as bosons or other fermions that exhibit more spin configurations than just the bi-spin orientations of electrons. It has been shown that Einstein’s special theory of relativity is quite important for classifying the energy levels of very heavy atoms and molecules that contain very heavy atoms [41–48]. This is because to keep balance with the increased electrostatic attraction in heavier nuclei having a large number of protons, the core electrons of such heavy atoms must move with considerably faster average speeds. We have shown, for example, that the

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averaged speed of the 1s electron of heavier atoms such as gold is about 60% of the speed of light. Consequently, ordinary quantum mechanics does not hold, and one needs to invoke relativistic quantum mechanics to deal with such heavy atoms and with molecules containing very heavy atoms. We have defined relativistic effects as the difference in the observable properties of electrons as a consequence of using the correct speed of light compared to the classical infinite speed. Mathematically, the introduction of relativity results in a double group symmetry owing to the spin-orbit coupling term [41], which is a relativistic term in the Hamiltonian. This is a natural symmetry consequence of the LS spin-orbit operator, which changes sign upon rotation by 360º. Thus, the periodicity of the identity operation, which is normally envisaged as a rotation through 360º, is no longer the identity operation of the group. This is illustrated in Figure 1 with a Möbius strip, which exemplifies the double group symmetry. As one completes a 360º rotation along the Möbius surface there is a sign change since one goes from the inside of the surface to the outside. This requires the introduction of a new operation R in the normal point group of a molecule that corresponds to the rotation by 360º which is not equal to E, the identity operation. Hence we have to make use of the double group and double-valued representations in both atomic and molecular spectroscopy.

Figure 1. A Möbius strip exemplifying the double group relativistic periodicity. The introduction of spin-orbit coupling into the relativistic Hamiltonian changes the periodicity of the normal point group symmetry into a double group symmetry, as rotation through 360° is not the identity operation. Note that the Möbius strip changes sign in this operation. Generalization of this to other complex phases results in Berry’s phase, where rotation through 360° may yield exp(2πi/n), thus resulting in other kinds of periodicity.

The double group consists of twice the number of operations as the normal point group but is not a simple direct product of the normal point group and another group. This is a consequence of the fact that only some of the conjugacy classes of the normal point group generate new conjugacy classes upon multiplication by the operation R. The other conjugacy classes, which are called two-sided operations, such as the C2 rotations, double in order instead of generating new classes. This is because such operations when multiplied by R

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become equivalent to the operation, and thus the new operations belong to the same conjugacy class as the corresponding old operations. This feature complicates double group theory and the resulting periodicity of the double group. A characteristic feature is the generation of even-dimensional double-valued representations that characterize half-integral quantum numbers. We shall discuss this in one of the ensuing sections. In this chapter we shall consider the mathematical basis of atomic periodicity and spectroscopy with the use of group theory and combinatorics. We shall also consider the combinatorics of unitary groups and Young diagrams and their connections to the electronic spin functions. We shall also discuss molecular periodicity by considering the combinatorial basis of molecular electronic states. We describe the double groups and the periodicity arising from the classification of states in the double group. We expound on the combinatorics and periodicity pertinent to the rotational levels, nuclear spin functions, and rovibronic levels of molecules and give some examples.

Combinatorial Periodicity in Molecular Electronic and Atomic Spectroscopy As might be expected, the classification of atomic states and thus the Periodic Table of the elements are based on combinatorial and group-theoretical considerations. An interesting related combinatorial problem has to do with the graphical unitary group approach to manyelectron configuration and correlation problems [4, 49]. The associated fermionic algebras involving Young diagrams and the symmetric permutation group approach have been discussed previously [4, 49]. It would also be interesting to consider cases of similar enumerations for other bosons or even fermions that are more than spin 1⁄2 particles. Such cases, while possibly not applicable to electronic systems, are applicable to nuclear spin species, and would have considerable group-theoretical value. There are several applications of group theory to atomic states. An early application by Curl and Kilpatrick [16] showed that the Schur functions of the symmetric groups Sn can be used as generating functions for atomic term symbols. The method involves replacement of cycle index polynomial terms by the various ml and ms symbol powers for generating functions of the atomic states that transform according to the irreducible representations of the Sn group. From this the authors were able to establish a periodic connection to the combinatorial enumeration of atomic term symbols even for complicated cases such as those for f7 shells. Balasubramanian [4] has established the connection between the graphical unitary group approach for electronic configurations and the Schur function algebra of the Sn groups, which play an important role in the Periodic Table in terms of the classification of various spin multiplets and the term symbols of electronic states. We shall consider this in some detail before enumerating the atomic states. The electronic states arising from the many-electron configurations have certain periodicities and patterns as enumerated by the Schur functions of the symmetric groups Sn. The representation theory of the symmetric group is well known [17, 20, 50], and we will not repeat it here. The irreducible representations of Sn may be characterized by Young diagrams for the various partitions of the integer n, denoted by [n]. The states of many particles (including

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bosons and fermions) that possess multiple spin orientations can be represented by generalized Young Tableaus (GYTs). For example, Figure 2 shows all of the possible GYTs for the partitions of six occupied by six particles that have three spin orientations (for example, a spin-1 particle such as the bosonic deuterium nucleus) with the possibility that two have the first kind of spin orientation, two have second kind and the last two particles have the third kind. We have denoted this [122232] shape as shown in Figure 2.

1

1

1

2

2 3

2 3

1

3

3

1

1

2 2

1

1

2 2

3

3

3

3

2

1

1

2

2

1

1

3

3

2

1 1

3

2

2 3 3

3

1 1

2

1

2 3

3

2 2 3

2

1

1

2

3

1 1 2

2

3

3

2

3

1 3

Figure 2. Generalized Young Tableaus (GYTs) for the partition of six for a spin 1 boson (e.g., deuterium) corresponding to the spin distribution of two particles with the first spin orientation, two with the second orientation, and two with the third or [122232] shape.

As can be seen from Figure 2, the GYTs have numbers in any column in strictly ascending order while the numbers in any row must be in non-decreasing order. These tableaus represent the nuclear spin functions that transform according to the particular irreducible representation that the diagram represents. It is interesting to note that for a spin-1 particle such GYTs can have at the most three rows and, likewise for electrons, which are spin 1⁄2 particles, the GYTs can have at the most two rows. In general for a spin-j particle there can be at most only 2j + 1 rows in the GYTs. The enumeration of the GYTs for the various shapes of the spin distributions is a fundamental problem that is common to electronic and nuclear structures. In the context of many-electron spin functions, the graphical unitary group approach requires the enumeration of Gel’fand states which are the GYTs containing two rows. The results also have some interesting periodicity trends in the mathematical sense. These GYTs and the associated spin

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multiplets and spin states can be enumerated by polynomials called the Schur functions of the symmetric group Sn. The Schur function corresponding to a partition λ of n is denoted by {λ} and is defined in the following way: 1 {λ} = n!

∑χλ(g)s1b s2 b …snbn 1

(1)

2

g∈G

where χλ(g) is value of the character for g in the group G = Sn corresponding to the irreducible representation [λ] of the group Sn. To illustrate this, the Schur function corresponding to the partition 4 + 1 + 1 is given by the Schur function, {6;4,1,1}, shown below: 1

{6;4,1,1} = 120 [10s16 + 30 s14s2 + 40 s13s3 – 90 s12s22 – 120 s1s1s3 – 30 s23 + 40 s32 + 120 s6] (2) The Schur function is the generator for the GYTs, and is obtained by replacing every sk in the Schur function or S-function by ∑λik. The coefficient of a typical term λ1a1 λ2a1… λmam in i

the generating function thus obtained yields the number of GYTs with the shape [1a12a1…mam ]. The GYT generators are so powerful that they also enumerate the atomic states when applied to electronic spin functions which are GYTs with only two rows. As an illustration of GYT generation, let us consider the partition 2 + 1 for three particles. Let the particle under consideration be a spin-1 boson, which has three spin orientations that we depict symbolically as λ1, λ2 and λ3. The S-function in this case is given as: 1

{3;2,1} = 6 [2 s13 – 2 s3]

(3)

The GF of the GYTs for a spin 1 particle is thus given as: {λ1,λ2,λ3;2,1} = λ12λ2 + λ1λ22 + λ2λ32 + λ22λ3 + λ1λ32+ λ12λ3 + 2 λ1λ2λ3

(4)

The above generating function thus obtained from the S-function generates all of the GYTs shown below in Figure 3 for all possible spin distributions or shapes for the partition 2 + 1. 1

1

1

2

2

2

2

1

3

2 3

3

2 2 3

1 3

1

1

3

3

1 2 3

Figure 3. All possible GYTs corresponding to the partition 2 + 1 as enumerated by the S-function {λ1,λ2,λ3;2,1}.

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The enumeration technique can be applied to GYTs of any shape belonging to any particle with any spin shape and spin distribution. The method is not restricted to just spin 1 or spin 1⁄2 particles. We can use the above method to generate all of the possible spin states for a manyelectron system or all of the possible atomic spectral energy levels for a given open-shell electronic configuration. First, we illustrate the method for obtaining all of the possible electronic spin states. The GYTs for electrons may contain at most two rows since there are only two possible distinct spin orientations for an electron (α and β) and thus there cannot be more than two rows. Accordingly, only certain partitions are allowed for an electron. This means that the GYTs can be formed only by the integers 1 and 2. Each spin distribution or spin shape then contains representations that are sums of the GYTs with the appropriate shape. For example for a system of six electrons with five spins up and one spin down there are exactly two GYTs as shown in Figure 4.

1

1

1

1

1

2

1

1

1

1

1

2 Figure 4. The GYTs for six electrons with five spin ups and one spin down.

Figure 5. The many-electron spin multiplets for an even number of electrons; there are exactly N cells and at most two rows for the spin functions.

The [1a12a1] GYTs enumerate states with a total spin quantum number Mz = (a1 – a2)/2. Consequently, once the GYTs are sorted out according to their total Mz values we obtain the spin multiplets for the many-electron systems. A neat set of periodic spin multiplets are obtained for such many-electronic systems. These are shown in Figures 5 and 6, respectively, for even and odd numbers of electrons. This important result was made possible by use of the periodic S-functions.. The method of S-functions is powerful and general in that it can be applied to more than electrons. Thus, the same method can be applied to other particles that have integral spins

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such as the deuterium nuclear spin functions or to the cases with multinomial spin distributions. In such cases the diagrams become more complex with many more rows depending on the particles. For a spin 1 particle the diagrams have three rows at most. For a spin j particle the diagrams will have up to (2j + 1) rows yielding an array of complex spin multiplets. Next we demonstrate the periodic power of the S-function for enumerating the possible electronic states of an atom [16] which are well known as atomic term symbols in atomic spectroscopy. The method is completely analogous to generating the GYTs and manyelectron spin multiplets demonstrated above. The only difference is that we obtain a generating function for the different ML projections and spin projections and the total function must comply with the Pauli Exclusion Principle, as electrons are fermions. The method can be applied from simple cases, e. g., main group open-shells such as p2, p3, to more complex lanthanides and actinides that have fm open-shell f-electrons (Table 1). For example, consider the most complex half-filled 5f7 shells. The possible electronic states sorted according to the total spin and total angular momentum in compliance with Pauli’s Principle are given by: 2

S(2), 2P(5), 2D(7), 2F(10), 2G(10), 2H(9), 2I(9), 2J(7), 2K(4),2M(2), 2N, 2O S(2), 4P(2), 4D(6), 4F(5), 4G(7), 4H(5), 4I(5), 4J(3), 4K(3),4M, 4N 6 6 S, P, 6D, 6F, 6G, 6H, 6I 8 S 4

The mathematical aspect of periodicity in atomic states is dependent on the orbital angular momentum of the electrons and spins as exemplified by the S-function generator used above for the generation of atomic term symbols. Table 1. All possible atomic term symbols for all actinides and lanthanides. Shell f1/f13 f2/f12 f3/f11 f4/f10

f5/f9

f6/f8

f7

States 2

F S 1D 1G 1I 3P 3F 3H 2 2 P D(2) 2F(2) 2G(2) 2H(2) 2I 2J 2K 4S 4D 4F 4G 4I 1 S(2), 1D(4), 1F(1), 1G(4), 1H(2), 1I(3), 1J, 1K(2), 1M, 3 P(3) 3D(2) 3F(4) 3G(3) 3H(4) 3I(2) 3J(2) 3K 3L 5 5 S D 5F 5G 5I 2 P(4), 2D(5), 2F(7), 2G(6), 2H(7), 2I(5), 2J(5), 2K(3),2L(2), 2M 2N 4 4 S P(2) 4D(3) 4F(4) 4G(4) 4H(3) 4I(3) 4J(2) 4K 4L 6 6 6 P F H 1 S(4), 1P. 1D(6), 1F(4), 1G(8), 1H(4), 1I(7), 1J(3), 1K(4), 1L (2), 1M(2), 1O 3 P(6) 3D(5) 3F(9) 3G(7) 3H(9) 3I(6) 3J(6) 3K(3) 3L(3) 3M 3N 5 5 5 S P D(3) 5F(2) 5G(3) 5H(2) 5I(2) 5J 5K 7 F 2 S(2), 2P(5), 2D(7), 2F(10), 2G(10), 2H(9), 2I(9), 2J(7), 2K(5) 2L(4),2M(2), 2 N, 2O 4 S(2), 4P(2), 4D(6), 4F(5), 4G(7), 4H(5), 4I(5), 4J(3), 4K(3),4M, 4N 6 6 S, P, 6D, 6F, 6G, 6H, 6I 8 S 1

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Figure 6. The many-electron spin multiplets for an odd number of electrons; there are exactly N cells and at most two rows for the spin functions.

Yet another aspects of periodicity involves the molecular electronic states. The electronic configurations themselves consist of two parts, namely the spin part that was generated using the S-functions and space types that can also be generated using multinomial generators. In certain cases, as shown by the author, the orbital degeneracy can bring out additional symmetry. A space type can be imagined as a distribution of electrons in boxes such that a permutation of electrons within a box does not generate a new space type and the boxes themselves can be permuted if the orbitals are degenerate. Such groups are called wreath product groups. Balasubramanian [7] used this group theory combined with combinatorial multinomial generating functions to generate electronic space types. This can be illustrated for the benzene delocalized orbital π electrons. The periodic generating function for the number of space types of an n-orbital electronic configuration is given by: F = (1 + w + w2)n

(3)

where the coefficient of wm gives the number of space types with m electrons distributed among these n orbitals. For the case of benzene with six π electrons distributed among six orbitals we seek the coefficient of w6 with n = 6 in the above generating function. This is given by: 6 6 6 1 + ⎛⎝3⎞⎠ + ⎛⎝ 2 2 2 ⎞⎠ + ⎛⎝ 4 1 1 ⎞⎠ = 141

(4)

These 141 space types of benzene enumerated here are divided into equivalence classes of space types according to the symmetry equivalence from the wreath product groups induced by orbital degeneracy. As is well known, the six π orbitals of benzene are divided into 1 + 2 + 2 + 1 equivalence classes of orbitals. Thus, switching of the orbitals in the second and third set leads to equivalences, and the electrons can themselves be switched in each orbital. The result is a direct product of wreath product groups as shown below:

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S2 × S2[S2] × S2[S2] × S2 The cycle index polynomial of the totally symmetric representation of the above group generates the equivalence classes of the space types from the well-known Pólya Theorem [1, 52–60]. Consequently, the cycle index and the generating functions for the case of benzene are as follows. P=

{ (s

1 2 2 1

2

} { (s

+ s2)

1 4 8 1

}

+ 2 s12s2 + 3 s22 + 2 s4)

2

(5)

4 2 4 3 2 2 2 4 1 ⎧2 (1 + w + w ) + 2·2 (1 + w + w ) (1 + w + w )⎫2 ⎬ F = 28 {22(1 + w + w2)2}⎨ 2 2 4 2 4 8 +3·2 (1 + w + w ) + 2·2(1 + w + w )

⎩

⎭

(6)

The coefficient of w6 in the above generating function can be seen to be 58, which suggests that for benzene 141 space types are divided into 58 equivalence classes. Table 2 gives the number of equivalence classes of the space types for the various atoms that exhibit equivalence among the p orbitals. Table 2. Equivalence classes of the space types for the electronic configurations of atoms that have degenerate p orbitals. System He Li Be B C N O F

Total No Space Types 45 156 414 882 1554 23-4 2907 3139

Equivalence Classes 17 42 86 148 223 295 349 368

In summary, we have shown that the electronic configurations of molecules and atoms can be simplified using the mathematical periodicity of the spin functions and space types. The former case was accomplished using the S-functions of the symmetric permutation groups Sn while the latter case was simplified using the wreath product configuration symmetry groups.

Combinatorial Periodicity in Molecular and NMR Spectroscopies The concept of mathematical periodicity as described by the orbit structure of a permutation finds important applications in molecular and nuclear spin spectrsocopies. The orbit structure of a permutation comprises several cycles such that in each cycle a set of nuclei is visited followed by a return to the starting point. The cyclic structure of the permutation (12345)(678)(9,10) is illustrated in Figure 7. The orbit structure in Figure 7 determines the nuclear spin statistical weights of the rotational levels of molecules.

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Thus, from the periodic orbit structure in Figure 7, one determines a polynomial s5s3s2, because we have one orbit of length 5, one orbit of length 3 and one orbit of length 2. The periodicity and the length of the period associated with each such orbit then determine a generating function for the nuclear spin statistical weights of the energy levels. This concept can also be used in NMR and ESR spectrsocopies where the periodicity and the length of the orbits determine the NMR spin energy levels and thus the NMR spectra associated with the molecules. Although in ordinary NMR only Zeeman-allowed transitions are observed and thus only those transitions with changes of a single spin, multiple quantum NMR offers a powerful tool to probe into transitions involving multiple spin quantum numbers. Thus, all NMR interaction energy levels can be probed.

Figure 7. Periodic orbit structure for the permutation (12345)(678)(9,10).

We shall start with an application of permutational periodic structure in molecular spectroscopy. Indeed, the rotational energy levels of a molecule themselves have periodicities based on their point groups. We illustrate this with an icosahedral cluster, namely N20 [37] and C60 [29–33] systems. Consider the highly energetic regular dodecahedral N20 cluster [37], which exhibits icosahedral symmetry analogous to that in the fullerene C20. Since 14N is a spin 1 particle it exhibits an interesting generating function and nuclear spin species distribution. The generalized character cycle indices for all of the irreducible representations for the N20 cluster with Ih symmetry are shown in Table 3. These were constructed using the orbit structures of permutations as demonstrated in Figure 7. The cycle indices for the various irreducible representations were obtained by multiplying the periodic orbit structures of each permutation by the corresponding character values. Note that the resulting polynomials are the same for the T1g and T2g representations and likewise the T1u and T2u representations since the orbit structures multiplied by their character values become identical owing to accidental degeneracy. We have used our generalization [1, 5–6, 56] of Pólya’s Theorem for all characters to seek generating functions for the nuclear spin species of 14N. Note that since the 14N nucleus is a spin 1 particle, we replace every xk in the cycle index in Table 3 by λk + μk + νk where the symbols λ, μ and ν stand for –1, 0 and 1 spin projections of the spin 1 14N nucleus. The resulting generating functions for the nuclear spin functions are shown in Table 4. The generating functions shown in Table 4 have two parts, one consisting of coefficients and the other of the trinomial λiμjνk. We do not show k since k = 20 – i + j) and it can thus be deduced from the values of i and j. To illustrate how the generating functions in Table 4 are

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obtained, let us consider the T1g or the T2g representation. From Table 3 we obtain the GCCI for this representation and we make the substitution given by x

GF = PGx (xk → λk + µk + νk)

(7)

The above substitution yields the following expression: 1

GFT1g = 120 [3(λ + μ + ν)20 + 12(λ5 + μ5 + ν5)4 – 12(λ2 + μ2 + ν2)10 + 12(λ10 + μ10 + ν10)2 – 15(λ + μ + ν)4(λ2 + μ2 + ν2)8]

(8)

Table 3. The GCCIs for the dodecahedral N20 cluster. N20 Order Ag Au T1g=T2g T1u=T2u

120 1 1 1 3 3

54 24 1 1 1 ⁄2 1 ⁄2

1236 20 1 1 0 0

210 15 1 1 –1 –1

210 1 1 –1 3 –3

102 24 1 –1 1 ⁄2 –1⁄2

263 20 1 –1 0 0

1428 15 1 –1 –1 1

Gg Gu Hg Hu

4 4 5 5

–1 –1 0 0

1 1 –1 –1

0 0 1 1

4 –4 5 –5

–1 1 0 0

1 –1 –1 1

0 0 1 –1

Figure 8. Nuclear frequency spin spectrum for the Ag representation of N20.

When Equation (8) is simplified into a trinomial it has several terms with coefficients for each term. Table 4 shows the coefficients and powers of λ and μ for the term. The power of ν is simply 20 – (i +j) and is thus not shown. The actual computations of the generating

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functions for N20 (and for C60 discussed subsequently) were carried out using computer code in quadruple precision developed by Balasubramanian [8, 9]. It is important to employ a quadruple precision arithmetic especially for C60, as the coefficients grow astronomically and thus any lower precision results in errors. An interesting consequence of the periodicity is that the g and u representations differ in some of their coefficients so significantly that one can say that there is inversion contrast in combinatorics. For example, the coefficient of the term λ9μ6ν5 in Table 4 for the Ag representation is 647706 while the corresponding coefficient for the Au representation is 645606. Moreover, the first non-zero coefficient for the Au representation is for the (18,2,0) partition, which means that at least 18 colors of one kind and two colors of another kind are needed to induce chirality in the binomial distribution. A purely trinomial term has two chiral colorings for the lowest order term, i. e., the (18,1,1) term in Table 4 has a coefficient of two for Au. Table 4. Generating functions for the dodecahedral N20 cluster.

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Table 4. Generating functions for the dodecahedral N20 cluster. Continued

The coefficients thus enumerated in Table 4 can be sorted according to their total MF values where the term l has the projection –1, m has the projection 0, and v has the projection +1. Thus the term λiμjνk in Table 4 represents a total nuclear spin quantum number MF of (–i + k). When these coefficients are sorted according to their total MF values, they separate into nuclear spin multiplets with MF values ranging typically from –I, –I+1, –I+2,….0,….I–2, I–1, I. Such a multiplet would represent a nuclear spin multiplet with a multiplicity of 2I + 1. In this way for each irreducible representation the nuclear spin multiplets are separated according to their multiplicities and the results are shown in Table 5 for N20. As can be seen from Table 5, the frequencies of the spin multiplets corresponding to the g and u representations differ even for the singlet spin states. For example, the 1Ag state has a frequency of 113035 while the 1Au state has a frequency of 112444. There is a similar difference in the triplet state and most of the spin multiplets. This means that the parity can be contrasted even in low spin nuclear states. The corresponding rovibronic levels will also be populated with appreciable differences in the populations. From the nuclear spin multiplets

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we can also obtain the total nuclear spin statistical weights by the use of the Pauli Principle. Since 14N nuclei are bosons, the overall wavefunction, which is a product of the rovibronic wavefunction and nuclear spin function, must be symmetric or must transform as the Ag irreducible representation. The frequency of each representation is obtained by adding the product of 2S + 1 and the frequency. The results are shown as a footnote in Table 5. On the basis of this, the frequencies shown in this footnote are themselves the nuclear spin statistical weights for N20 (see Figure 8). Table 5. Nuclear spin species for the N20 cluster

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Other irreducible representations have similar spectra comparable to that in Figure 8, except that the intensities of the peaks vary. The Ag representation is particularly important as it gives the number of lines in multiple quantum NMR spectra. The frequencies of other irreducible representations determine the intensities of the lines in the spectra. The multiple quantum NMR spectra usually contain structural information for the (n – 2) quantum as this value exhibits dipolar couplings that contrast the structure. For the present case n – 2 corresponds to the spin multiplet 2S + 1 = 37. These multiplets have the frequencies 5, 2, 2, 7, 11, 1, 5, 5, 6, for the Ag, T1g, T2g, Gg, Hg, Au, T1u, T2u, Gu, and Hu representations, respectively. Thus, the dodecahedral N20 cluster exhibits interesting mathematical periodicity in spectroscopic terms. Next we consider the C60 cluster [29–33] as another example that demonstrates mathematical periodicity and its applications. The GCCIs of C60 are constructed analogously to those of N20 discussed above. The fact that C60 has 60 vertices would of course divide the permutation of 60 vertices into various periodic orbits. The nuclear spin species thus obtained using the GCCIs are shown in Table 6. As seen from Table 6, the frequencies grow astronomically as expected. This is because of the combinatorial explosion of the coefficients in the generating functions even though these functions are binomials. The binomial expansion is due to the fact that 13C60 is comprised of 13C nuclei, which exhibit only two spin orientations, as they are spin 1⁄2 particles. The same is true of C60H60, as protons are spin 1⁄2 particles and 12C has no nuclear spin. Again a major contrast is that the g and u representations have different frequencies due to the difference in the periodicity of the permutation multiplied by the character value for these representations. This feature manifests itself as contrasting frequencies for the g and u irreducible representations. We note that earlier work had an error in the spin statistical weights of C60 [31] primarily owing to the arithmetical precision but this was subsequently corrected [30, 32]. The relative differences between the g and u parities are especially significant for high-spin nuclear multiplets. For example, for the 2S + 1 = 57 spin multiplet of 13 C6, the frequencies of the Ag, T1g, T2g, Gg, Hg, Au, T1u, T2u, Gu, and Hu representations are 22, 36, 36, 58, 80, 14, 42, 42, 56, and 70, respectively. Similarly for 2S + 1 = 55 the frequency of the Ag representation is 280 while it is 260 for Au. Consequently, the contrast in the g and u spin populations can be seen experimentally if high-spin nuclear states can be excited. Table 6. Nuclear spin multiplets for 13C60 or C60H60. Frequency of the irreducible representation Ag: 9607679885269312 Spin multiplets and their frequencies for Ag: 2S+1 1 7 13 19 25 31 37 43 49 55 61

Frequency 31791575566072 150988619146706 105558807981090 31605175642230 4481735502630 298734989924 8805633300 101874363 372752 280 1

2S+1 3 9 15 21 27 33 39 45 51 57

Frequency 89413728633564 149756091280506 76925432220000 17892025439775 1980110898945 101492436960 2227563126 18110340 41528 22

2S+1 5 11 17 23 29 35 41 47 53 59

Frequency 13095954950748 13219208028055 5141513084676 933143835273 80345370985 3139590568 50512570 280174 388

130

K. Balasubramanian Table 6. Continued

Frequency of the irreducible representation T1g: 28823036970926496 Spin multiplets and their frequencies for T1g : 2S+1 1 7 13 19 25 31 37 43 49 55 61

Frequency 95374646372040 452965902231668 316676363633175 94815530686980 13445194549380 896204629630 26416344630 305608974 1114158 804 0

2S+1 3 9 15 21 27 33 39 45 51 57

Frequency 268241251090167 449268197030424 230776308338940 53676052490265 5940332333550 304475471640 6682635360 54304371 124257 36

2S+1 5 11 17 23 29 35 41 47 53 59

Frequency 39287856402727 39657626655407 15424535176554 2799431557098 241035603798 9418755979 151524170 840285 1123

Frequency of the irreducible representation Gg: 38430716856193728 Spin multiplets and their frequencies for Gg: 2S+1 1 7 13 19 25 31 37 43 49 55 61

Frequency 127166221937640 603954521378374 422235171614265 126420706329465 17926930052010 1194939619444 35221977930 407483337 1486916 1084 0

2S+1 3 9 15 21 27 33 39 45 51 57

Frequency 357654979723731 599024288311326 307701740558940 71568077929785 7920443232495 405967908600 8910198522 72414711 165779 58

2S+1 5 11 17 23 29 35 41 47 53 59

Frequency 52383811353475 52876834683423 20566048261230 3732575392371 321380974795 12558346548 202036737 1120460 1512

Frequency of the irreducible representation Hg 48038396740938240 Spin multiplets and their frequencies for Hg: 2S+1 1 7 13 19 25 31 37 43 49 55 61

Frequency 158957797411208 754943140441100 527793979532265 158025881932935 22408665535200 1493674601616 44027608785 509357130 1859568 1354 0

2S+1 3 9 15 21 27 33 39 45 51 57

Frequency 447068708357295 748780379591832 384627172778940 89460103369560 9900554131440 507460345560 11137761648 90525051 207307 80

2S+1 5 11 17 23 29 35 41 47 53 59

Frequency 65479766312622 66096042717787 25707561349782 4665719229588 401726346555 15697937361 252549365 1400644 1902

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Table 6. Continued Frequency of the irreducible representation Au: 9607678793631424 Spin multiplets and their frequencies for Au: 2S+1 1 7 13 19 25 31 37 43 49 55 61

Frequency 31791571643468 150988613640506 105558798039270 31605170531130 4481732871390 298734348764 8805495420 101861196 371694 260 0

2S+1 3 9 15 21 27 33 39 45 51 57

Frequency 89413727296344 149756080818726 76925425313100 17892020535870 1980109351620 101491992360 2227502850 18103410 41266 14

2S+1 5 11 17 23 29 35 41 47 53 59

Frequency 13095954114986 13219207292373 5141512318638 933143526111 80345252581 3139568730 50509098 279955 377

Frequency of the irreducible representation T1u: 28823037990981216 Spin multiplets and their frequencies for T1u: 2S+1 1 7 13 19 25 31 37 43 49 55 61

Frequency 95374639953380 452965915721858 316676367808710 94815537801090 13445196226770 896205406510 26416442910 305623968 1114942 826

2S+1 3 9 15 21 27 33 39 45 51 57 0

Frequency 268241262122232 449268199508214 230776318887660 53676055391160 5940334271070 304475780520 6682705140 54309474 124548 42

2S+1 5 11 17 23 29 35 41 47 53 59

Frequency 39287856269005 39657627967718 15424535578410 2799431961645 241035683183 9418781778 151526692 840531 1132

Frequency of the irreducible representation Gu: 38430716784610624 Spin multiplets and their frequencies for Gu: 2S+1 1 7 13 19 25 31 37 43 49 55 61

Frequency 127166211596396 603954529362364 422235165847980 126420708332460 17926929098160 1194939755164 35221938330 407485164 1486642 1086 0

2S+1 3 9 15 21 27 33 39 45 51 57

Frequency 2S+1 357654989418576 5 599024280327336 11 307701744200760 17 71568075926790 23 7920443622690 29 405967772880 35 8910208020 41 72412884 47 165808 53 56 59

Frequency 52383810383991 52876835260051 20566047897048 3732575487756 321380935775 12558350508 202035787 1120487 1509

132

K. Balasubramanian Table 6. Continued

Frequency of the irreducible representation Hu: 48038395577718272 Spin multiplets and their frequencies for Hu 2S+1 1 7 13 19 25 31 37 43 49 55 61

Frequency 158957783147612 754943142918890 527793963824370 158025878824830 22408661950230 1493674096176 44027431350 509345790 1858246 1336 0

2S+1 3 9 15 21 27 33 39 45 51 57

Frequency 2S+1 447068716714920 5 748780361146062 11 384627169513860 17 89460096462660 23 9900552974310 29 507459765240 35 11137710870 41 90516294 47 207074 53 70 59

Frequency 65479764507375 66096042558712 25707560219562 4665719015799 401726189132 15697919478 252544942 1400452 1887

Table 7. Correlations of the rotational levels of C60: the nuclear spin statistical weights J = 0 to 30

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133

Table 7 shows the correlation of the rotational levels for C60 from J = 0 to 30 with the corresponding weights only in the rotational subgroup I. Note that for purposes of comparing with experimental results one must use the nuclear spin frequencies given in Table 6, but the statistical weights in Table 7 in factored form yield the orders of magnitude. All levels in Table 7 are of g symmetry since the J states can correlate into g levels. (1) The irreducible representations for J > 31 are given by q[A + 3T1 + 3T2 +4G + 5H] + Γ(r), where q is the quotient obtained by dividing J by 30 and r is the remainder. Γ(r) is the set of irreducible representations spanned by J = r listed in this Table (see text for further discussion). Note that since nuclear spin statistical weights are the same for g and u symmetries, we do not show g or u. (2) f = 19 215 358 678 900 736 for C60H60; f = 706 519 304 586 988 199 183 738 259 for C60D60. Each correlation in Table 7 was obtained using the mathematical method of subduction. As can be seen from Table 7, we have a very interesting periodicity among rotational levels. The correlations for the rotational levels with J > 31 have a periodic relation to the levels with J < 30. This is another mathematical manifestation of periodicity. The relations for all J > 30 are as follows: D(J)↓Ih = q(D(30)↓Ih – A) + (D(r) ↓Ih), q = [J/30], r = J – 30[J/30]

(9)

where the function within square brackets is the greatest integer contained in the brackets and thus q and r are quotients and remainders obtained by dividing J by 30. The term D(30) stands for the subduced representations for J = 30 that are displayed in Table 7. To illustrate this, the J = 195 rotational level contains the following representations: D(195)↓Ih = 6(Ag + 3T1g + 3T2g + 4Gg + 5Hg) + (Ag + 2T1g + 2T2g + 2Gg + 2Hg)≡ 7Ag + 20T1g + 20T2g + 26Gg + 32Hg

(10)

The above concept of the periodicity of the rotational levels of C60 is illustrated in Figure 9. It is worthy of note that the nuclear spin statistical weights of the rotational levels vary approximately as (2J + 1) due to large nuclear spin statistical weights.

J=0 J=1

J=30 * * * * . *

J=2

.J=3 Figure 9. Periodicity of the rotational levels of buckminsterfullerene, C60.

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Periodicity of Double Groups and Electronic States The concept of the double group [17, 24–27, 42, 51] is required when the normal periodicity resulting from rotation through 360° breaks down, as demonstrated for the Möbius strip. This happens when half-integral states are considered. For example, the rovibronic states of openshell systems with an odd number of open-shell electrons exhibit half-integral spin states due to an odd number of open-shell electrons and thus we need a new concept of periodicity. This is also the case when spin-orbit coupling is introduced into the Hamiltonian.

Table 8. Character table for the Ih2 double group.

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Table 8. Character table for the Ih2 double group. (Continued)

This difficulty was circumvented by Bethe through the concept of the double group. He introduced a new operation called R that changes the sign for the rotation through 360° for half-integral states and yet retains the same symmetry for the integral states, as shown above for the C60 integral rotational levels. Since the periodicity and the group structure are quite different for the double group, we provide a few examples of double group character tables and correlation tables. Most of the character tables appear in books such as those of Hamermesh [17] or Altmann and Herzig [24] for the double groups. Balasubramanian [51] developed the

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K. Balasubramanian

character table for the icosahedral double group denoted by Ih2 that is shown in Table 8. Note that the operation R introduces a few new conjugacy classes for the Ih2 double group while other conjugacy classes just double in their orders. This is a consequence of the fact that certain operations are called two-valued operations and these operations when multiplied by R become equivalent, and thus belong to the same conjugacy class. However, other operations, such as C5 and RC5, become inequivalent, and thus belong to different conjugacy classes. The new irreducible representations in the double group are called two-valued representations and they are always even dimensional and correspond to half-integral representations. The number of such representations equals the number of new conjugacy classes, as demonstrated in Table 8. These are called E1g(1/2), Gg(3/2), Ig(5/2), E2g(7/2), with the corresponding u representations. The correlation table for the half-integral states of the Ih2 double group is shown in Table 9. Note that the corresponding table for the integral representations has already been discussed for C60 (Table 7). As can be seen from Table 9, the half-integral spin or rovibronic states all correlate only into double-valued representations, which are all even dimensional. As a result, the representation corresponding to 1⁄2 is a degenerate two-dimensional irreducible representation. The quartet state with s = 3⁄2 is also four-fold degenerate and s = 5⁄2 is likewise the six-fold degenerate I representation in the double group. The first case which splits into two irreducible representations is the s = 7⁄2 case. The periodicity is reduced in the double-valued representation to half as s = 31⁄2 is related to s = 1⁄2 by periodicity. All higher s values are obtained using a periodic relation as shown in Table 9. Table 9. Periodic correlation table for the half-integral states of the Ih2 double group. Irreducible Representationsa

s ⁄2 3 ⁄2 5 ⁄2 7 ⁄2 9 ⁄2 11 ⁄2 13 ⁄2 15 ⁄2 17 ⁄2 19 ⁄2 21 ⁄2 23 ⁄2 25 ⁄2 27 ⁄2 29 ⁄2 31 ⁄2 1

1

E1g′( ⁄2) Gg′(3⁄2) Ig′(5⁄2) E2g′(7⁄2) + Ig′(5⁄2) Gg′(3⁄2) + Ig′(5⁄2) E1g′(1⁄2) + Gg′(3⁄2) + Ig′(5⁄2) E1g′(1⁄2) + Gg′(3⁄2) + Ig′ (5⁄2) + E2g′(7⁄2) Gg′ (3⁄2) + 2 Ig′(5⁄2) G′g(3⁄2) + 2 Ig′(5⁄2) + E2g′(7⁄2) E1g′(1⁄2) + Gg′(3⁄2) + 2 Ig′(5⁄2)+ E2g′(7⁄2) E1g′(1⁄2) + 2 Gg′(3⁄2) + 2 Ig′(5⁄2) E1g′(1⁄2) + 2 Gg′(3⁄2) + 2 Ig′(5⁄2) + E2g′(7⁄2) E1g′(1⁄2) + Gg′(3⁄2) + 3 Ig′(5⁄2) + E2g′(7⁄2) 2 Gg′(3⁄2) + 3 Ig′(5⁄2) + E2g′(7⁄2) E1g′(1⁄2) + 2 Gg′(3⁄2) + 3 Ig′(5⁄2) + E2g′(7⁄2) 2 E1g′(1⁄2) + 2 Gg′(3⁄2) + 3 Ig′(5⁄2) + E2g′(7⁄2) a

Ds = q{E1g′(1⁄2) + 2 Gg′(3⁄2) + 3 Ig′(5⁄2) + E2g′(7⁄2)} + Ds′, 2s + 1 q = 30 , s′ = s – 15q, if s > 31⁄2

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We have also collected the correlation tables [42] for the octahedral double group Oh2 in Table 10, the correlation table for the Td2 in Table 11, and the correlation table for the D6h2 in Table 12. These correlation tables all demonstrate interesting mathematical periodicity for the rotational or rovibronic levels. The octahedral integral rotational levels exhibit a period of 12 analogous to that for the tetrahedral group. However, the half-integral spin states or rovibronic states exhibit a period of six both in the octahedral and tetrahedral double groups. The D6h2 double group exhibits a different periodic trend as seen from Table 12. The periodicity of six is same for both the half-valued and integral representations. Thus, the periodicity trends exhibited by the double groups are quite interesting. These correlation tables are quite valuable in obtaining the rovibronic levels of molecules with both an odd and even number of electrons. It is important to obtain the overall rovibronic correlation as opposed to individual rotational correlations owing to the fact that the total wavefunction may become a half-integral representation, especially for systems with an odd number of electrons. Furthermore, for molecules containing very heavy atoms spin-orbit effects become quite significant, and thus the coupling of the spin with orbital angular momentum splits the electronic states into spin-orbit states. The exact manner in which these states are split by spin-orbit coupling is given by the double group correlation tables shown here. Table 10. Periodic correlation table for the half-integral states of the Oh2 double group. Irreducible Representations in the Oh2 Groupa

s 0 1 2 3 4 5

A1g T1g Eg + T2g A2g + T1g + T2g A1g + Eg + T1g + T2g Eg + 2T1g + T2 + T2g

6 ⁄2

A1g + A2g + Eg + T1g + 2T2g E1g′(1⁄2)

1 3

⁄2 ⁄2 7 ⁄2 9 ⁄2 11 ⁄2 6 + s′ 5

12n + s′

Gg′(3⁄2) E2g′(5⁄2) + Gg′(3⁄2) E1g′(1⁄2) + E2g′(5⁄2) + Gg′(3⁄2) E1g′(1⁄2) + 2 Gg′(3⁄2) E1g′(1⁄2) + E2g′(5⁄2) + 2 Gg′(3⁄2) E1g′(1⁄2) + E2g′(5⁄2) + 2 Gg′(3⁄2) + terms of s′ but interchange E1g′(1⁄2) with E2g′(5⁄2) 2n {(E1g′(1⁄2) + E2g′(5⁄2) + 2 Gg′(3⁄2)} + terms for s′

a

Terms for other integral s values are found using the formula: D(12n+s′) = Ds′ + n(A1g + A2g + 2 Eg + 3 T1g + 3 T2g), s′ < 12.

The concept of periodicity can be extended to cases beyond the double groups. Such cases would involve Berry’s phase where a rotation through 360° would yield a complex number, exp(2πi/n), for an integer n > 2. The symmetry exhibited by such systems could be quite intriguing. It is hoped that this chapter will stimulate future investigations into Berry’s phase.

138

K. Balasubramanian Table 11. Periodic correlation table for the half-integral states of the Td2 group.

a

s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 ⁄2 3 ⁄2 5 ⁄2 7 ⁄2 9 ⁄2 11 ⁄2 6 + s′ 2n + s′

Irreducible Representationsa A1 T1 E + T2 A2 + T1 + T2 A1 + E + T1 + T2 E + 2 T1 + T2 A1 + A2 + E + T1 + 2 T2 A2 + E + 2 T1 + 2 T2 A1 + 2E + 2 T1 + 2 T2 A1 + A2 + E + 3 T1 + 2 T2 A1 + A2 + 2E + 2 T1 + 3 T2 A2 + 2E + 3 T1 + 3 T2 2 A1 + A2 + 2 E + 3 T1 + 3 T2 A1 + A2 + 2 E + 4 T1 + 3 T2 A1 + A2 + 3 E + 3 T1 + 4 T1 A1 + 2 A2 + 2 E + 4 T1 + 4 T1 E1/2 G3/2 G3/2 + E5/2 E1/2 + G3/2 + E5/2 E1/2 + 2 G3/2 E1/2 + E5/2+ 2 G3/2 E1/2 + E5/2 + 2 G3/2 + terms for s′, but interchange E1/2 and E5/2 2n(E1/2 + E5/2)

Other integral spin states are correlated using the formula

D(12n+s'′ = Ds′ + n(A1 + A2 + 2 E + 3T1 + 3T2), s′ < 12.

Table 12. Periodic correlation table for the half-integral states of the D6h2 double group s 0 1 2 3 4 5 6

1

⁄2 ⁄2 5 ⁄2 7 ⁄2 9 ⁄2 11 ⁄2 3

a

6n + s′

Irreducible Representationsa A1g A2g + E1g A1g + E1g + E2g A2g + B1g + B2g + E1g + E2g A1g + B1g + B2g + E1g + 2 E2g A2g + B1g + B2g + 2 E1g + 2 E2g 2 A1g + A2g + B1g + B2g + 2 E1g + 2 E2g E1g′(1⁄2) E1g′(1⁄2) + E3g′(3⁄2) E1g′(1⁄2) + E2g′(5⁄2) + E3g′(3⁄2) E1g′(1⁄2) + 2 E2g′(5⁄2) + E3g′(3⁄2) E1g′(1⁄2) + 2 E2g′(5⁄2) + 2 E3g′(3⁄2) 2 E1g′(1⁄2) + 2 E2g′(5⁄2) + 2 E3g′(3⁄2) 6n{E1g′(1⁄2) + E2g′(5⁄2) + E3g′(3⁄2)} + terms of s′, s′ < 6

Terms for other integral s values may be found by using

D(12n+s′) = Ds′ + n(A1g + A2g + B1g + B2g + 2 E1g + 2 E2g), s′ < 6

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Acknowledgement This research was performed under the auspices of the US Department of Energy by the University of California, LLNL under contract number W-7405-Eng-48 while the work at UC Davis was supported by the National Science Foundation.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]

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K. Balasubramanian, Chem. Rev., 85, 599 (1985). K. Balasubramanian, J. Chem. Phys., 72, 665 (1980). K. Balasubramanian, J. Chem. Phys., 73, 3321 (1980). K. Balasubramanian, Theor. Chim. Acta, 59, 237 (1981). K. Balasubramanian, J. Chem. Phys., 74, 6824 (1981). K. Balasubramanian, J. Chem. Phys., 75, 4572 (1981). K. Balasubramanian, Int. J. Quant. Chem., 20, 1255 (1981). K. Balasubramanian, J. Comput. Chem., 3, 69 (1982). K. Balasubramanian, J. Comput. Chem., 3, 75 (1982). K. Balasubramanian, H. Strauss, and K. S. Pitzer, J. Mol. Spectrosc., 93, 447 (1982). K. Balasubramanian, J. Phys. Chem., 86, 4668 (1982). K. Balasubramanian, J. Chem. Phys, 78, 6358 (1983). K. Balasubramanian, J. Chem. Phys, 78, 6369 (1983). K. Balasubramanian, Group Theory of Non-rigid Molecules and its Applications,” Elsevier Publishing Co., 23, 149–168 (1983). K. Balasubramanian, Theor. Chim. Acta, 78, 31 (1990). R. F. Curl, Jr., and J. E. Kilpatrick, Amer. J. Phys. 28, 357 (1960) M. Hamermesh, Group Theory and its Physical Applications, Addison Wesley, Reading MA, 1964 F. A. Cotton, Chemical Applications of Group Theory, Wiley Interscience, New York, NY, 1971 P. R. Bunker, “Molecular Symmetry and Spectroscopy,Academic Press, New York, NY, 1979 D. E. Littlewood, Theory of Group Characters and Matrix Representations of Groups, , Oxford, New York, NY, 1958 H. Weyl, Theory of Groups and Quantum Mechanics, Dover Publications, New York, NY, 1950 M. Tinkham, Group Theory and Quantum Mechanics, McGraw-Hill, New York, NY 1964 B. R. Judd, Operator Techniques in Atomic Spectroscopy, Princeton University Press, Princeton,, NJ 1998 S L Altmann and P Herzig, Point-Group Theory Tables, Clarendon Press, Oxford, 1994. P. Pyykkö and H. Toivonen, Tables of Representation and Rotation Matrices for The Relativistic Irreducible Representations of 38 Point Groups, Acta Academiae Aboensis, Ser B, 43, 1 (1983) H.T. Toivonen and P. Pyykkö, Int. J. Quant. Chem., 11, 697 (1977)

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[27] H.T. Toivonen and P. Pyykkö, Relativistic Molecular Orbitals and Representation Matrices for the Double Groups T and Th, Department of Physical Chemistry, Åbo Akademi, Finland, Report No. B79 (1977), 11 pp. [28] K. Balasubramanian, J. Mag. Res., 91, 45 (1991). [29] K. Balasubramanian, Chem. Phys. Lett., 183, 292 (1991). [30] K. Balasubramanian, Chem. Phys. Lett., 200, 649 (1992). [31] W. G. Harter and T. C. Reimer, Chem. Phys. Lett., 194, 230 (1992). [32] W. G. Harter and T. C. Reimer, Chem. Phys. Lett., 198, 429E (1992). [33] W. G. Harter and D. E. Weeks, J. Chem. Phys., 90, 4727 (1989). [34] K. Balasubramanian, J. Chem. Phys., 95, 8273 (1991). [35] K. Balasubramanian and T. R. Dyke, J. Phys. Chem., 88, 4688 (1984). [36] K. Balasubramanian, J. Mol. Spectroscopy 157, 254 (1993). [37] K. Balasubramanian, Chem. Phys. Lett., 202, 271(1993). [38] K. Balasubramanian, J. Phys. Chem., 97, 8736 (1993) [39] K. Balasubramanian, Mol. Phys., 80, 655 (1993). [40] K. Balasubramanian, J. Chem. Phys., in press. [41] K. Balasubramanian, Relativistic Effects in Chemistry, Part B: Applications, Wileyinterscience, New York, NY, p. 527, 1997. [42] K. Balasubramanian, Relativistic Effects in Chemistry, Part A: Theory and Techniques, Wiley-Interscience, New York, NY, p. 301, 1997. [43] K. S. Pitzer, Accts. Chem. Res., 12, 271 (1979). [44] P. Pyykkö and J. P. Desclaux, Accts. Chem. Res., 12, 276 (1979). [45] P. Pyykkö, Adv. Quant. Chem., ll, 353 (1978). [46] K. Balasubramanian, J. Phys. Chem., 93, 6585 (1989). [47] P. Pyykkö Ed. Proceedings of the Symposium on Relativistic Effects in Quantum Chemistry; Int. J. Quantum Chem., 25 (1984). [48] P. Pyykkö, Relativistic Theory of Atoms and Molecules, Springer Verlag: Berlin and New York, Part I 1986 Part II 1993, Part 3 2000. For comprehensive list of references up to 2002 see http://www.csc.fi/rtam/. [49] J. Paldus, Theoretical Chemistry: Advances and Perspectives, H. Eyring and D. J. Henderson, Eds, Academic Press, New York, NY, 1976 [50] X. Y. Liu and Balasubramanian, J. Comput. Chem., 10, 417 (1989). [51] K. Balasubramanian, Chem. Phys. Lett., 260, 476 (1996). [52] G. Pólya, Acta Math, 68, 145 (1937) [53] K. Balasubramanian, Theor. Chim. Acta, 53,129 (1979) [54] G. Pólya and R.C. Read, Combinatorial Enumeration of Groups, Graphs, and Chemical Compounds, Springer, New York, NY, 1987. [55] J. H. Redfield, Amer. J. Math. 49, 433 (1927). [56] K. Balasubramanian, J. Math. Chem. 14, 113 (1993). [57] D. H. Rouvray, Chem. Soc. Rev. 3, 355 (1974). [58] D. H. Rouvray, Endeavour, 34, 28 (1975) [59] A.T. Balaban, Chemical Applications of Graph Theory, Academic Press, New York, NY, 1976. [60] R. Read, in “Graph Theory and Applications”, Y. Alavi et al. eds., Lecture Notes in Mathematics, 303, 243 (1972) Springer, 1972.

In: Advances in Mathematics Research, Volume 8 Editor: Albert R. Baswell, pp. 141-144

ISBN: 978-1-60456-454-9 © 2009 Nova Science Publishers, Inc.

Chapter 4

MATHEMATICAL MODELLING OF THERMOMECHANICAL DESTRUCTION OF POLYPROPYLENE G.M. Danilova-Volkovskaya, E.A. Amineva1 and B.M. Yazyyev2 1

Rostov-on-Don Agricultural Machinery State Academy; 344023, Strana Sovetov Street, 1, Rostov-on-Don. 2 Ushakov Naval State Academy 353900, Lenin Avenue, 93, Novorossiysk

Abstract There has been provided mathematical description of the processes of thermonuclear destruction in deformed polypropylene melts; the aim was to use the criterion of destruction estimation in modelling and optimising the processing of polypropylene into products.

Keywords: Thermo-mechanical destruction, polypropylene, molecular mass, effective viscosity.

During processing polypropylene melts under the action of transverse strain there occur strain-chemical conversions which can result in both decrease and increase in their molecular masses; the mechanical effect on the rapidity and level of the occurring processes is considerably more prominent than the mere contribution of thermal and thermal-oxidative breakdown. These data necessitate studying the process of polymer destruction. For this purpose it would be most effective to apply the criterion of assessment of the intensity with which destructive processes happen in polymer melts. If the destruction is observant from the initial value of molecular mass М0 to a certain finite value М∞, then at point of time t the chain group with molecular mass М0 - Мt (where Мt is the average value of molecular mass at a given point of time) is involved in the process. It is natural to assume that the rate of destruction in a unit time is proportional to the whole number of breakdowns in macromolecules up to the destruction limit. These assumptions enable us to propose an expression for calculating the rate of destruction process:

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d (( M t − M ∞ ) / M t = − Kdt , (M t − M ∞ ) / M ∞ The integration of this expression results in:

ln

Mt − M∞ = − Kt + e , M∞

(1)

Since at t=0 Мt = M0, then:

С = ln

Mt − M∞ M∞

,

(2)

If we substitute (1) with (2) after some transformations we get:

ln

From here:

Mt − M∞ = − Kt , M∞

М t = ( M 0 − M ∞ )e kt + M ∞ ,

As value М0 - М∞ is constant for the polymer of the given molecular mass, we can designate it as A; after substitution we get:

ln

from here:

Mt − M∞ = − Kt M0 − M∞

М t = A ⋅ e − kt + M ∞ , where K is the rate constant depending on the chemical

nature of a polymer and, in particular, on how close macromolecular chains are packed. Each criterion obtained from the given expressions represents a concept of one of the interrelated consequences of thermo-mechanical destruction process: decrease in molecular weight, the number of macromolecular breakdowns, and the approach to the possible level of macromolecular destructions. The merit of the criteria is that their values do not depend on the initial molecular weight [1-3]. Paper 20 dwells on the ideas allowing us to advance in the quantitative assessment of thermo-mechanic destruction degree. Taking these data as a basis we can propose an expression for calculating the degree of thermo-mechanic destruction in the form of:

Mathematical Modelling of Thermo-Mechanical Destruction of Polypropylene

ϕ а1 =

1 η 0 − kt ⋅ ⋅e , а ηt

143

(3)

where a is the constant of proportionality which is equal to 3.105. On the other hand:

ϕ а1 = (η а ,τ 1, 2 ,η 0 , it ) ,

(4)

where ηа is the effective viscosity of a material melt, τ1,2 are transverse strains during processing. Combining the defining parameters of equation (3) and modifying this equation into a dimensionless form, it is possible to demonstrate that criterion φ1а, is the function of only two parameters ηа and τ1,2. Comparing (3) and (4) enables the following expression for the criterion of thermomechanic destruction degree to be proposed:

⎛ τ ⋅t ⎞ ϕ а1 = f ⎜⎜η 0 , 1, 2 ⎟⎟ , ηa ⎠ ⎝

(5)

The direct application of this expression in order to estimate the degree of thermomechanic destruction in connection with polymer processing is hindered because the process rate constant depends on the temperature and intensity of thermo-mechanical impact on a material. Consequently, of significant interest is the issue of selecting an attribute for characterizing the degree of destruction. Most researchers consider it worthwhile to simply use viscosity variable (ηа) or characteristic viscosity variable. Here is proposed the criterion for the rate of thermo-mechanical destruction in the polymeric system Ψ11: −τ 12 ⋅t 1 ⎡η 0 ηa ⎤ Ψ = ⋅⎢ е ⎥, a ⎢⎣η a ⎥⎦ 1 1

(6)

where τ12 are strain rate tangents. This relation is helpful because it provides an opportunity for the quantitative assessment of polymer thermo-mechanical destruction rate in dependence with the thermo-mechanical impact regime during processing. Analyzing the data obtained when testing the samples of extrusion products made of polypropylene, the conducted research on their molecular-weight properties, and the calculated values of the criterion for the destruction processes rate, we concluded that the processes of attachment and bifurcation correspond to the values of Ψ11 = 1, while the processes of destruction correspond to Ψ11= - 1.

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Assuming that the effective viscosity in a polypropylene melt is sensitive to changes in molecular mass and in chain-length distribution and taking into consideration the specific character of the thermo-mechanical impact developing during extrusion, it is proposed to calculate the intensity of destruction processes from the latter expression. The advantage of the criterion is that it does not require defining the molecular mass of a polymer. Comparing the values of Ψ11, obtained at testing PP samples processed under various technological regimes and calculated with the aid of a mathematical model allows us to propose applying the criterion to the estimation of physical and chemical transformations occurring in a polymer at modifying the parameters of thermo-mechanical impact. Taking into consideration Ψ11 values, we have found the optimal regime when PP is under extrusion processed into products with improved deformation-strength properties [4].

Conclusions There has been provided mathematical description of the processes of thermonuclear destruction in deformed polypropylene melts; the aim was to use the criterion of destruction estimation in modelling and optimising the processing of polypropylene into products.

References [1] Olroyd J.G. On the formulation of rheological equation of stat. - Trans. Roy. Soc., 1970, A 200, N 1063, p. 523 -527. [2] De Witt T., Mezner .W. A rheological equation of state which predicts non-Newtonian viscosity, normal stresses and dynamics module. J. Appl.Phys., 1985, v. 26, p. 889-892. [3] Baramboymb I.K. Mechanochemistry of high-molecular substances. – 3rd edition. Moscow. The Chemistry publishing house, 1978, p. 34. [4] Danilova-Volkovskaya G.M. The effect of processing parameters and modifiers on the properties of polypropylene and PP-based composite materials. — Doctoral Thesis, (technical sciences). 2005, p. 273.

In: Advances in Mathematics Research, Volume 8 Editor: Albert R. Baswell, pp. 145-160

ISBN: 978-1-60456-454-9 © 2009 Nova Science Publishers, Inc.

Chapter 5

A DESIGN-BASED STUDY OF A COGNITIVE TOOL FOR TEACHING AND LEARNING THE PERIMETER OF CLOSED SHAPES Siu Cheung Kong* Department of Mathematics, Science, Social Sciences and Technology The Hong Kong Institute of Education, Hong Kong

Abstract With the consideration of cognitive inflexibility of learners in computing perimeter of closed shapes, a theory-driven design of a cognitive tool (CT) called the ‘Interactive Perimeter Learning Tool (IPLT)’ for supporting the teaching and learning of the mathematics target topic was developed in this study. An empirical study in the form of pre-test—post-test reflected that learners of varying mathematical abilities had statistically significant gains in using the IPLT for learning support. The IPLT could effectively address the inflexibility commonly exhibited by learners in learning this topic such as the formation of the abstract association of an irregular closed shape with a regular closed shape. The assertion of teachers on the effectiveness of the IPLT and the enthusiasm of students for using the IPLT for learning reflect that the CT had a pedagogical value in fostering learner-centred learning. Based on the feedback of this study, the IPLT will be refined under the design-based research approach.

Keywords: cognitive inflexibility, cognitive tool, design-based research, mathematics, perimeter.

Introduction Perimeter is a linear measure. It refers to the distance around a shape. The general computation of perimeter involves the application of elementary arithmetic, in which the operation of addition is applied to calculate the perimeter of a shape by adding all the length *

E-mail address: [email protected]

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of the sides together. The centrepiece of this procedural knowledge relates to the concept of line segments. The study of perimeter is one of the major components of the learning dimension ‘measure’ in the primary mathematics curriculum in Hong Kong. This study domain covers four topics: the ‘introduction to perimeter,’ the ‘perimeter of squares,’ the ‘perimeter of rectangles’ and the ‘perimeter of closed shapes.’ Among these four topics, the ‘perimeter of closed shapes’ is considered to be the relatively complicated topic in the subject domain. This topic focuses on the concepts of lines and shapes as well as the knowledge about how to find the perimeter of irregular closed shapes, and develops upon the fundamental understanding of the topics ‘perimeter,’ the ‘perimeter of squares,’ and the ‘perimeter of rectangles.’ The centrepiece of finding perimeter of closed shapes is the correct combination of line segments on the shape border. The effective teaching and learning of the ‘perimeter of closed shapes’ should focus on the development of this conceptual strategy in an incremental approach with four stages. The first stage is the acquisition of fundamental concepts. Direct process of providing definitions of ‘perimeter,’ such as asking students to finger-outline the border of the irregular shapes, should be adopted. The second stage is the reinforcement for students of the general computation method of perimeter with sufficient information. Cases in which the lengths of all line segments of the irregular shapes are given should be designed for students to consolidate their operation of finding perimeter by adding the lengths of all the sides of a shape. The third stage is the generalisation of the computation of perimeter of closed shapes with the just necessary information. Cases in which only the lengths of certain line segments of a variety of symmetric irregular closed shapes are given, such as the shapes ‘T’ and ‘+,’ should be designed for students to associate an irregular closed shape with a regular closed shape and thus formulate the general conceptual abstractions of line movement and shape conversion for finding perimeter of closed shapes. The fourth stage is the application and transfer of knowledge about ‘perimeter of closed shapes.’ Cases in which only the lengths of certain line segments of a number of complex irregular closed shapes are given, such as the shapes ‘U’ and ‘H,’ should be designed for students to link and apply the basic knowledge about ‘perimeter’ by including the line segments inside the converted shapes and hence develop flexible and transferable procedural knowledge about finding perimeter of closed shapes. However, the main goal of current traditional pedagogical practices for this subject topic commonly focuses on the automation of procedural skills rather than the understanding of important measurement ideas of the ‘perimeter of closed shapes.’ This leads to the cognitive inflexibility that students commonly exhibit at the third and fourth stages in learning this topic.

Cognitive Inflexibility Cognitive inflexibility is a kind of performance problems associated with automation. It refers to the failure to detect a new situation and the consequent demand for a change at the level of control of knowledge to restate the routine problem-solving strategy (Cañas, Antolí, Fajardo & Salmerón, 2005; Cañas, Fajardo & Salmerón, 2006). It proposes the existence of a failure in the evaluation of the situation that leads to a failure in its execution. The occurrence of this problem is due to the automation processes that begin after extensive practice within a

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particular type of tasks. When people repeatedly put a strategy which has shown to be effective in previous situations into practice, they incline to insistently apply the same strategy in new situations even after the conditions of the situations have changed and the strategy is no longer appropriate. There are four types of behaviour in connection with cognitive inflexibility (Cañas et al., 2006). The first one is cognitive blockade, which refers to the tendency to continue with an initial problem-solving strategy in situations where it is rational to select an alternative problem-solving strategy. It denotes the failure to make a global diagnosis of the situation, which is a problem due to the insistence of problem-solvers on focusing solely upon the concrete aspects of the situation and their inability to re-evaluate the new situation. The second one is cognitive hysteresis, which is the tendency to adhere to a problem-solving strategy after evidence has proven it to be a mistake. It represents the failure to discern an erroneous diagnosis, which is a problem due to the problem-solvers cannot judge the situation with new pieces of evidence. The third one is functional fixation, which refers to the tendency to consider only the available objects in problem-solving as known in terms of its more common function. It represents the failure to use the elements that have taken a new form for problem-solving, which is a problem due to the problem-solvers fix assignment of an object to a category and hence make the properties that are assigned to that object become conditioned. The fourth one is functional reduction, which is the tendency to solve a problem by adopting strategies that address a single cause regardless of all other possible influencing variables. It denotes the failure to adapt behaviour to the changed environmental conditions, which is a problem due to the problem-solvers consider only in part the causes of a phenomenon. The major type of cognitive inflexibility involved in the learning of the ‘perimeter of closed shapes’ is cognitive blockade. Students in this case generally show a limitation in developing a holistic comprehension of the strategies for finding perimeter of closed shapes. There are two inadequacies of students in developing the conceptual knowledge about and computational skills in the target subject. First, students lack the ability to develop the relational knowledge about perimeters of polygons in regular and irregular shapes. In the calculation of perimeter of an irregular closed shape, an abstract association of an irregular closed shape with a regular closed shape is involved. It is a strategy to coordinate side lengths and collections of side lengths for the computation of perimeter (Barrett, Clements, Klanderman, Pennisi & Polaki, 2006). For example, to find the perimeter of the irregular closed shape ‘T,’ imaginary steps to move lines (see Figure 1) must be made to convert the irregular closed shape into a regular closed shape in the form of a rectangle. The perimeter can then be computed easily by applying the knowledge and concept of the ‘perimeter of rectangles.’ It is commonly found that students find it difficult to understand the abovementioned abstract association within several lessons. Students generally encounter the cognitive constraint on the configuration of shape and the movement of line segments. Such students exhibit the cognitive inflexibility that they insistently focus on the irregularity in spatial pattern of the irregular closed shape. The students perceive that the shape configuration is fixed and the line segments are immovable. They have no realisation of the flexibility in changing the shape configuration by conceptually moving the position of line segments. The students are thus unable to formulate an immediate perceptual conversion from an irregular closed shape into a regular closed shape.

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Figure 1. Steps showing the movement of lines to find the formula for calculating the perimeter of the irregular closed shape ‘T.’

Figure 2. Example showing the movement of lines to find the formula for calculating the perimeter of the irregular closed shape ‘U.’

Second, students lack the ability to restructure the procedural knowledge about perimeter of irregular closed shapes. In the computation of perimeter of an irregular closed shape, a complete inclusion of subdivisions of continuous linear units should be entailed. It is a strategy to coordinate length attributes around a perimeter and further integrate the fundament concepts of line segments into the calculation of perimeter (Barrett et al., 2006). For instance, to find the perimeter of the irregular closed shape ‘U,’ a regular closed shape in the form of a rectangle with two vertical straight lines inside will be transformed from the original shape by imaginary steps of line movements (see Figure 2). The calculation of perimeter should include the line segments inside the resulted shape apart from the border of the resulted shape.

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It is often found that students are puzzled as to the need for the aforementioned complete inclusion of line segments. Students generally lack the intention to recall the fundamental knowledge and concept of perimeter. Such students exhibit the cognitive inflexibility that they insistently continue with the routine procedures for finding perimeters. They have no flexibility in linking the re-interpretation of the new situation and the application of relevant prior knowledge of the target subject. The students are thus unable to modify their relevant knowledge and adapt their problem-solving strategies according to the new task requirements. To address the learning problems caused by cognitive inflexibility, previous studies have recommended the adoption of multiple organisational schemes for presenting subject matter in instruction and the multiple classification schemes for knowledge representation. It is suggested that subject topics cover knowledge that will have to be used in many different ways that cannot all be anticipated in advance and involve cases or examples of knowledge application that typically involves the simultaneous interactive involvement of multiple complex conceptual structures should be introduced under the instructional approach of multiple knowledge representations (Cañas et al., 2006; Spiro, Collins, Thota & Feltovich, 2003). The rationale behind this instructional approach is that knowledge that will have to be used in many ways must be taught and mentally represented in many ways (Spiro, Feltovich, Jacobson & Coulson, 1991). The instructional approach of multiple knowledge representations focuses on the reorganisation of knowledge to explain how students adapt to new situations. It addresses the irregularity and variation in training over a fixed repetition of steps. By designing different learning scenarios for training different problem-solving strategies in different sequence orders, this instructional approach is considered to be appropriate for improving cognitive flexibility in learning situations (Cañas et al., 2006). To realise this instructional approach, ample resources for demonstrating content diversity, interlinking practices for applying and transferring knowledge, and flexible tools for exploring multiple problem-solving strategies in different contexts should be provided. In this regard, a cognitive tool which addressed the two types of cognitive inflexibility in the learning of the ‘perimeter of closed shapes’ was designed in this study to assist students in the concept and strategy development for the topic ‘perimeter of closed shapes.’

The Study This study adopted the design-based research approach to designing a cognitive tool in supporting the learning and teaching of the mathematics topic, the ‘perimeter of closed shapes.’ Design-based research is a fundamental mode of scholarly inquiry that is useful across many academic disciplines, and has become an increasingly accepted approach to theoretical and empirical study in the field of education in the past decade (Bell, 2004). Design-based research is an attempt to combine theory-driven design with empirical studies of learning environments (Bell, 2004; Design-Based Research Collective, 2003; Hoadley, 2004). It aims to design and explore a whole range of innovations. The most common type of design-based research combines software design and studies in education (Hawkins & Collins, 1992; Hoadley, 2002). The study reported herein combined a theory-driven design of a cognitive tool and an empirical study on the learning context involving the use of the cognitive tool in a real classroom setting.

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The Theory-Driven Design of the Cognitive Tool Cognitive tools (CTs) are both mental and computational devices that can be used to support the cognitive processes of learners (Derry & LaJoie, 1993; Kommers, Jonassen & Mayes, 1992; Kong & Kwok, 2005). A web-based CT entitled the ‘Interactive Perimeter Learning Tool (IPLT)’ was designed in this study to assist students with the abstraction of conceptual and procedural knowledge about the ‘perimeter of closed shapes.’ It was designed as a mechanism for supporting the instructional approach of multiple knowledge representations. By using different representations of mathematical concepts, the CT could provide students with many opportunities for developing their own intuitive, computational, and conceptual knowledge (Moyer, 2001). As mentioned, the key strategy to find perimeter of closed shapes is the combination of line segments on the border of irregular shapes. The two common problems of students in calculating perimeter of closed shapes, viz. the inflexibility in forming an abstract association of an irregular closed shape with a regular closed shape and the inflexibility in making a complete inclusion of line segments of shape border, are closely related to the failure of students to realise this important step. In this regard, the IPLT aimed at providing learning support in the concept development concerning the concepts of lines as well as shapes and the strategy development regarding the combination of line segments of shape border in finding perimeter of closed shapes. The IPLT is a graphical tool for the display of graphical representation of irregular closed shapes. It consisted of a regular plane on which a set of irregular closed shapes was shown on the interface one at a time. Three features were designed to cover the important concepts of the topic and address the common inadequacies of students in learning the target subject. The first feature was the design of movable line segments of shape border. This feature was designed for students to develop the concepts of lines and shapes and the strategies of combining line segments in the calculation of perimeter of closed shapes. This feature enabled students to freely move the line segments of shape border of a closed shape by clicking line segments and dropping them on the designated positions (see Figure 3). Students were allowed to move the vertical lines leftwards and rightwards, and move the horizontal lines upwards and downwards. The line segments were not allowed to be rotated. The second feature was the provision of the just necessary information for the calculation of perimeter of closed shapes. This feature was designed to address the inadequacy of students in forming an abstract association of an irregular closed shape with a regular closed shape in the computation of perimeter of closed shapes. On each display of closed shapes, the horizontal distance between the leftmost and the rightmost vertices and the vertical distance between the highest and the lowest vertices of the closed shapes were displayed in fixed positions (see Figure 3). For the complicated closed shapes, measurements of the width between certain sides of the shape were shown in fixed positions. This feature stimulated students to convert the irregular closed shape that was displayed on the IPLT into a regular closed shape by just providing the lengths of the vertical and horizontal sides, which is the information normally given for the calculation of perimeter of regular closed shapes. This feature could thus help students to form an abstract association of an irregular closed shape with a regular closed shape. The third feature was the design of returnable graphical representations of the closed shapes. This feature was designed in response to the inadequacy of students in making a

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complete inclusion of line segments of shape border in the calculation of perimeter of closed shapes. A ‘Reset’ button (see Figure 3) was provided in the bottom right-hand corner of the interface. Students could press this button to get back the original graphical representation of a closed shape at their convenience when they were lost in the perimeter task after converting the original shape into the resulted shape. The IPLT was of benefit to students in the learning of the ‘perimeter of closed shapes.’ The IPLT could help students to rectify their common learning problems at the third and fourth stages of the target subject by providing students with many more chances to actively participate in the free exploration of perimeter in different contexts which involved a variety of representations and to fully generalise the important concepts of the subject topic. Building on the basic conceptual and procedural knowledge about perimeter that has been learnt in the classroom learning, with the use of the IPLT, students could freely explore the abstract association of an irregular closed shape with a regular closed shape at the skill-based and rule-based levels and realise the complete inclusion of line segments of shape border at the knowledge-based level. In this regard, the IPLT could facilitate students to make many potential combinations of relevant prior knowledge cognitively available in the calculation of perimeter of closed shapes.

The Empirical Study To investigate the potential of the IPLT for supporting the learning and teaching of the mathematics topic, the ‘perimeter of closed shapes,’ an empirical study was conducted in a real classroom setting. Researchers have suggested that the major criteria for the evaluation of a CT include the learning achievement of students after using the CT, the justification of teachers for the use of the CT in teaching, and the preference of students for the use of the CT in learning (Hawkins & Collins, 1992). In this respect, three specific research questions were investigated in this study. The display of the horizontal distance between the leftmost and the rightmost vertices and the vertical distance between the highest and the lowest vertices of the closed shapes for stimulating the abstract association of an irregular closed shape with a regular closed shape.

The movable line segments of shape border for converting an irregular shape into a regular shape.

A ‘Reset’ button for returning the resulted shape to the original shape. Figure 3. An interface of the IPLT.

1. What is the degree of the IPLT in catering for the diverse needs of students with varying learning abilities? 2. What are the opinions of teachers on the use of the IPLT in teaching? 3. What are the views of students on the use of the IPLT in learning?

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Two classes of Primary Four students in a primary school in Hong Kong were invited to participate in this study. Table 1 shows the profile of the two classes of Primary Four students who participated in this study. The two classes of students had varying levels of learning ability in mathematics, which were reflected in a test held just before the study. The corresponding test results are shown in the ‘mathematics mean score’ row in Table 1. The use of the IPLT was incorporated into the normal teaching lessons of ‘perimeter of closed shapes’ in the invited school, in which each lesson lasted for 35 minutes. The teaching materials for students of the two participating classes included PowerPoint materials, the IPLT, and a number of activity worksheets. One class (the At-riskIPLTGp class) worked with the desktop version of the IPLT with desktop computers individually in the computer room during a double lesson; while another class (the EliteIPLTGp class) worked with the mobile version of the IPLT with personal digital assistants (PDAs), in groups of five to six, during a triple lesson. Each student was assigned a computing tool to access the IPLT which was located in a web server. The teachers of these two classes mainly asked the students to use the IPLT to complete the learning tasks specified on the activity worksheets. Table 1. Profile of the two classes of Primary Four students EliteIPLTGp

At-riskIPLTGp

Number of students

Profile

36

25

Ratio of boys to girls

14 : 22

16 : 9

8.97 (S.D. = 0. 17)

9.16 (S.D. = 0.37)

70.00 (S.D. = 11.45)

52.56 (S.D. = 15.37)

Mean age in years Mathematics mean score (max = 100)

Table 2. The major learning and teaching activities for students with the use of the IPLT 1.

2. 3.

4. 5.

Learning and teaching activities Instruction in the target subject: The teacher made use of PowerPoint presentation to demonstrate the movement of line segments of shape border to calculate perimeter of closed shapes. Brief of the functions of the IPLT: The teacher introduced the use of the IPLT. Exploration of the target subject and completion of activity worksheets: Students were asked to use the IPLT to explore the way of finding perimeter of closed shapes by completing activity worksheets. Relevant guidance and probing questions were offered occasionally by the teacher during these sessions. Answer check: Students were requested by the teacher to give answers and corresponding explanations. Class discussions: Class discussions were conducted for students to consolidate knowledge.

Table 2 summarises the major learning and teaching activities for students with the use of the IPLT. The learning and teaching activities for the students focused on the student-centred exploration of the movement of line segments of shape border in finding perimeter of closed shapes. The majority of class time was spent on the use of the IPLT for the completion of the activity worksheets. Initially, teachers used PowerPoint files to demonstrate the movement of line segments of shape border for the calculation of perimeter of closed shapes. The use of the

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IPLT was subsequently introduced to students by teachers. Students were asked to explore the way of finding perimeter of closed shapes and complete five questions on the activity worksheets with the use of the IPLT. Guidance and probing questions were given by teachers. In the answer-checking sessions, teachers asked some students to explain their answers rather than telling students the answers directly. Class discussions followed for knowledge consolidation of students.

Evaluation Methods Quantitative and qualitative methods were adopted in the empirical study to collect data on the potentials of the IPLT to assist in the teaching and learning of the ‘perimeter of closed shapes’ with regard to the learning achievement of students after using the IPLT, the justification of teachers for the use of the IPLT in teaching, and the preference of students for the use of the IPLT in learning. To study the impact of the IPLT on the learning achievement of the students, a set of pretest—post-test instruments was designed to garner quantitative data concerning the learning benefits to students in terms of academic results after working with the IPLT. Under the pretest—post-test design, a pair of identical tests was incorporated in the teaching process for all of the students who participated in this study in order to measure the knowledge of students about finding perimeter of closed shapes before and after learning the target topic. A test paper consisting of eight questions about the calculation of perimeter of closed shapes was designed for the pre-test and the post-test. Figure 4 shows a sample question in the test paper. To investigate the justification of the teachers for the use of the IPLT in teaching, semistructured, individual interviews with the teachers who used the IPLT were conducted after the teaching period to gather qualitative data regarding their opinions of the use of this CT in terms of application situation and teaching effect. Five questions about perception of teaching benefits and suggestions for further improvement of the IPLT were designed for the interviews. The responses were processed by content analysis of the interview records. To study the preference of students for the use of the IPLT in learning, a questionnaire was distributed to the students in this study to collect qualitative data reflective of their views on the use of this CT in terms of application situation and learning effect. Students were asked to indicate their level of agreement with four statements about user-friendliness, learning benefits and usage preference in relation to the IPLT. The mean rating of each statement on a 5-point Likert scale, from 1 = ‘strongly disagree’ to 5 = ‘strongly agree,’ and the corresponding standard deviation were calculated.

Results and Discussions Learning Outcome of Students from Pre-Test—Post-Test Instruments This section reports the learning achievement of students with varying mathematical abilities after using the IPLT for leaning. Table 3 and Figure 5 show the effects of the IPLT on the elite students who participated in the study. The paired t-test result in Table 3 indicates that the mean difference between the pre-test and post-test measures of the elite students is

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significantly different. In this case, the elite class had a statistically significant gain in the knowledge of the target subject after the teaching period.

The graphical representation of an irregular closed shape with the length marks.

The answer spaces for the calculation expression and answer of the perimeter of the closed shape. Figure 4. A sample question in the test paper.

Table 3. Mean, standard deviation and paired t-test of the pre-test and post-test measures for the elite students who participated in the study Group EliteIPLTGp

Number of students 36

Pre-test Mean (S.D.) 3.47

(2.06)

Post-test Mean (S.D.) 6.53

(1.28)

Paired t-test -10.16***

t(35)

*** p < .001.

7 6 5 4 3 2 1 0

EliteIPLTGp

Pre-test

Post-test

Figure 5. Effects of the IPLT on the elite students on their knowledge and concepts of finding perimeter of closed shapes.

Table 4 and Figure 6 show the effects of the IPLT on the at-risk students who participated in the study. The paired t-test result in Table 4 indicates that the mean difference between the pre-test and post-test measures of the at-risk students is significantly different. In other words,

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the at-risk class had a statistically significant gain in the knowledge of the target subject after the teaching period. In summary, the findings show that students had a statistically significant gain in the learning of ‘perimeter of closed shapes’ with the use of the IPLT. This reflects that the integration of the CT with the traditional teaching materials in classroom instruction was effective for learning the knowledge of finding the perimeter of closed shapes. The IPLT could cater for the diverse needs of students of differing learning abilities. Table 4. Mean, standard deviation and paired t-test of the pre-test and post-test measures for the at-risk students who participated in the study Group

Number of students

At-riskIPLTGp

25

Pre-test Mean (S.D.) 3.56

(1.96)

Post-test Mean (S.D.) 5.76

Paired t-test

(1.76) -5.75***

t(24)

*** p < .001.

7 6 5 4 3 2 1 0

At-riskIPLTGp

Pre-test

Post-test

Figure 6. Effects of the IPLT on the at-risk students on their knowledge and concepts of finding perimeter of closed shapes.

Feedback of Teachers from Interviews Table 5 shows the key points that the teachers made during the interviews about the use of the IPLT for teaching the ‘perimeter of closed shapes.’ The teacher of the EliteIPLTGp class pointed out that it was easy for students to use the IPLT. He asserted that the use of the IPLT fostered teacher-student and student-student interactions in the classroom setting. The teacher observed that his students were highly involved in the learning activities with the use of the IPLT. He pointed out that the mobile version of the IPLT in PDAs allowed him to walk around the classroom for checking the learning progress of students and providing instant feedback to individual students. The teacher showed his appreciation of the capabilities of the IPLT to move line segments of shape border and reset graphical representations of closed shapes because these features of the IPLT made the teaching of the target subject more efficient and facilitated the organisation of relevant in-class leaning activities. The teacher also agreed that the IPLT could effectively address the inadequacies of students in learning the ways of finding perimeters of closed shapes, in particular for their inflexibility concerning

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the formation of the abstract association of an irregular closed shape with a regular closed shape. The teacher suggested that the lines that displayed on the IPLT should be thinner so as to facilitate students to move line segments and form a regular closed shape for each attempt. The teacher of the At-riskIPLTGp class also agreed that the IPLT was easy for students to use and strengthened the teacher-student and student-student interactions in the classroom environment. She noticed that her students were eager to use the IPLT in solving problems on the activity worksheets. The teacher thought that the IPLT was favourable for learning the target subject because the IPLT allowed students to move the lines freely and explore the shapes and perimeters in many different ways. The teacher pointed out that the IPLT was helpful for students to solve the learning problems of students in learning the target subject, particularly for their inflexibility concerning the realisation of a complete inclusion of line segments of shape border in the calculation of perimeters of closed shapes. The teacher proposed that the lines and the numbers displayed on the IPLT should be thinner and larger respectively for clear reference. In addition, the teacher suggested that there should be a simultaneous movement of line segments and their corresponding length marks for easy reference. The teacher further pointed out that the mobile version of the IPLT should be adopted so as to enhance the teacher-student interactions and increase efficiency of classroom logistics in terms of equipment allocation. Table 5. Key points that the teachers made during the interviews about the use of the IPLT for teaching the ‘perimeter of closed shapes’ Question theme EliteIPLTGp The IPLT was easy for students to Yes. use. The IPLT facilitated interaction Yes. between students and teacher. The mobile version of the IPLT allowed the teacher to walk around the classroom for checking the learning progress of students and providing instant feedback to individual students. The IPLT facilitated students to Yes. discuss the ways of finding perimeter of closed shapes. The IPLT assisted students to Yes. understand the ways of finding Enabled students to move line perimeter of closed shapes. segments of an irregular closed shape to form a regular closed shape. Aspects to be improved regarding Lines that displayed on the IPLT the IPLT. should be thinner.

At-riskIPLTGp Yes. Yes.

Yes. Yes. Enabled students to move the lines freely and explore the shapes and perimeters in different ways. Lines that displayed on the IPLT should be thinner. Numbers that displayed on the IPLT should be larger. Length marks should move along with the lines.

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Table 6. Evaluation results of the student questionnaire survey on the use of the IPLT for learning the ‘perimeter of closed shapes’ Evaluation item This application is easy to use. I can use this application to discuss the ways of finding perimeter of closed shapes with my classmates. The application helps me to understand the ways of finding perimeter of closed shapes. I like to use this application for doing exercise.

EliteIPLTGp At-riskIPLTGp M (S.D.) M (S.D.) 4.50 (0.51) 4.64 (0.48)

t-test 1.07

4.41 (0.61)

4.60 (0.57)

1.21

4.29 (0.68)

4.52 (0.75)

1.17

4.56 (0.56)

4.48 (0.70)

-0.46

Remarks: 1 = strongly disagree; 2 = disagree; 3 = neutral; 4 = agree; 5 = strongly agree.

In summary, the teachers expressed their positive views on the potential integration of the IPLT with traditional pedagogical practices to achieve optimal learning outcome. They exhibited a strong preference for and a high degree of involvement in the use of the IPLT for teaching. The teachers were satisfied with the user-friendliness of the IPLT. They thought that the use of the IPLT in classroom could enhance the teacher-student and student-student interactions. The teachers also asserted the helpfulness of the IPLT for students to understand the ways of and address the cognitive inflexibility in finding perimeter of closed shapes. They indicated that students participated actively in the classroom activities with the use of the IPLT.

Feedback of Students from Questionnaire Survey The students who used the IPLT showed their strong preference for using this CT for learning the target subject. They found that the IPLT was easy to use. The students indicated that they could use the IPLT in the discussions about the subject matter and the completion of relevant exercises. The IPLT was considered as a very helpful learning tool for understanding the ways of finding perimeter of closed shapes. Table 6 shows the evaluation results of the student questionnaire survey on the use of the IPLT for learning the ‘perimeter of closed shapes.’ The t-test shows that none of the tests of equality of means for the two classes could be rejected. This reflects that students in both classes had the same perception of the IPLT. In summary, the students had a positive perception of the IPLT regarding its effectiveness on, importance in, and appealing effect on learning. They were motivated to use the IPLT for acquiring and discussing the subject knowledge in a learner-centred approach. The students also showed a high degree of involvement in using the IPLT for the completion of learning activities.

Implications of the Empirical Study The evaluation results from the empirical study reveal that there is a potential integration of the IPLT into traditional pedagogical practices to achieve optimal learning outcome. Three implications in connection with the development and implementation of the designed CT were drawn based on the abovementioned evaluation results.

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The first implication is related to the pedagogical value of the designed CT. In the student survey, the students showed their great enthusiasm for using the IPLT in learning the target subject. This reveals that the IPLT had a potential to act as a good mediator to facilitate learner-centred learning and induce a constructively qualitative change in the nature and depth of a range of educational processes. This result concurs with the findings of previous studies, that is, that the integration of computer-supported CTs within a learner-centred learning environment offers substantial potential to support the learning process and enhance the learning effect of students because such CTs enable students to regulate the amount and sequence of available resources and explore the gist of subject knowledge according to their individual needs (Hawkins & Collins, 1992; Iiyoshi, Hannafin & Wang, 2005; Jonassen & Reeves, 1996; Kong & Kwok, 2005). The second implication relates to the pedagogical use of the designed CT. In the interviews, the teachers asserted that the attempt at incorporating the use of the IPLT with traditional pedagogical practices was successful. One of the teachers, whose students used the desktop version of the IPLT during class time, further indicated her preference for the mobile version rather than the desktop version of the IPLT because she thought that the former could create an encouraging ambience for student-student discussions about the subject knowledge without location constraint. To cater to the different pedagogical styles of teachers, the provision of both mobile and desktop versions of the IPLT as practised in this study is recommended. The final implication concerns the improvement work for the designed CT. The designbased research approach is a strategy for designing cognitive artefacts by eliciting information about aspects to be improved regarding the CT from users. According to the feedback from the teacher interviews, the lines and the numbers that are displayed on the IPLT should be thinner and larger, respectively, in order to reduce the visual hindrance to students to find the total length of the line segments. In addition, the corresponding length marks of line segments should be made movable so that students are empowered with greater flexibility and autonomy over the use of the IPLT for explorative learning.

Conclusion Primary school students commonly exhibit cognitive inflexibility in applying knowledge about line segments and shapes for the formulation of abstract association of an irregular closed shape with a regular closed shape and for the realisation of a complete inclusion of line segments of shape border in the calculation of perimeter of closed shapes. This study adopted the design-based research approach in developing a theory-driven design of a cognitive tool (CT) entitled the ‘Interactive Perimeter Learning Tool (IPLT)’ and used the CT in an empirical study for supporting the teaching and learning of the ‘perimeter of closed shapes’ in primary mathematics classroom. The evaluation results show that the incorporation of the CT in traditional pedagogical practices facilitated the teaching and learning of the ‘perimeter of closed shapes’. The students attained statistically significant achievement in the learning outcome with the use of the IPLT. In the teaching period, the students and teachers who participated in the study showed their enthusiasm for using the IPLT in learning and teaching the target subject. The teachers recognised the effectiveness of the IPLT on addressing the cognitive inflexibility

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exhibited by students in learning the target subject. The students asserted the helpfulness of the IPLT in supporting the learning of the target subject in a learner-centred approach. These findings shed light on the future study of the designed CT in two aspects. The first aspect relates to the refinement of the designed CT. Since the application of the IPLT in classroom instruction helped to promote the active learning of student, it is worthwhile to continue the improvement work of this CT, especially for the better graphic display in order to facilitate the exploration of shapes and perimeters in different ways. The second aspect concerns the evaluation of the designed CT. There are two issues about the evaluation. The first is about the concern of shifting the evaluation focus from measuring the academic results of the students to understanding the actual changes in the learning process. Discourse analyses of the interactions between the students and the cognitive artefacts and among students are suggested for further evaluation of the pedagogical use of the IPLT in the real classroom setting. This helps to comprehensively delve into the potential of the IPLT to act as a mediator to foster learner-centred learning. The second issue relates to the integration of the IPLT with traditional pedagogical practices. Further investigation about the most effective pedagogical integration of this CT with traditional classroom teaching in the aspects of, for example, the proportion of class time that should be allocated for learning with the IPLT to that for learning without the IPLT, and the sequence of the adoption of the IPLT, viz. at the initial, middle or final stage of the teaching period, in classroom instruction should be conducted. This helps to look into the ways to capitalise on the use of the CT in supporting the classroom teaching of the target subject.

References Barrett, J. E., Clements, D. H., Klanderman, D., Pennisi, S. J., Polaki, M. V. (2006). Students’ coordination of geometric reasoning and measuring strategies on a fixed perimeter task: developing mathematical understanding of linear measurement. Journal for Research in Mathematics Education, 37(3), 187-221. Bell, P. (2004). On the theoretical breadth of design-based research in education. Educational Psychologist, 39(4), 243-253. Cañas, J. J., Antolí, A., Fajardo, I., & Salmerón, L. (2005). Cognitive inflexibility and the development and use of strategies for solving complex dynamic problems: effects of different types of training. Theoretical Issues in Ergonomics Science, 6(1), 95-108. Cañas, J. J., Fajardo, I., & Salmerón, L. (2006). Cognitive flexibility. In W. Karwowski (Ed.), International encyclopedia of ergonomics and human factors (pp. 297-301). Boca Raton, FL: Taylor & Francis. Derry, S. J., & LaJoie, S. P. (1993). A middle camp for (un)intelligent instructional computing: an introduction. In S. P. LaJoie & S. J. Derry (Eds.), Computers as cognitive tools (pp. 1-11). NJ: Lawrence Erlbaum Associates. Design-Based Research Collective. (2003). Design-based research: an emerging paradigm for educational inquiry. Educational Researcher, 32(1), 5-8. Hawkins, J., & Collins, A. (1992). Design-experiments for infusing technology into learning. Educational Technology, 32(9), 63-67. Hoadley, C. M. (2004). Methodological alignment in design-based research. Educational Psychologist, 39(4), 203-212.

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Hoadley, C. P. (2002). Creating context: design-based research in creating and understanding CSCL. In G. Stahl (Ed.), Computer support for collaborative learning 2002 (pp. 453462). Mahwah, NJ: Erlbaum. Iiyoshi, T., Hannafin, M. J., and Wang, F. (2005). Cognitive tools and student-centred learning: rethinking tools, functions and applications. Educational Media International, 42(4), 281-296. Jonassen, D. H., & Reeves, T. C. (1996). Learning with technology: using computer as cognitive tools. In D. H. Jonassen (Ed.), Handbook of research on educational communication and technology (pp. 693-719). NY: Scholastic Press. Kommers, P., Jonassen, D. H., & Mayes, T. (1992). Cognitive tools for learning. Heidelberg FRG: Springer-Verlag. Kong, S. C., & Kwok, L. F. (2005). A cognitive tool for teaching the addition/subtraction of common fractions: a model of affordances. Computers and Education, 45(2), 245-265. Moyer, P. S. (2001). Using representations to explore perimeter and area. Teaching Children Mathematics, 8(1), 52-59. Spiro, R. J., Collins, B. P., Thota, J. J., & Feltovich, P. J. (2003). Cognitive flexibility theory: hypermedia for complex learning, adaptive knowledge application, and experience acceleration. Educational Technology, 43(5), 5-10. Spiro, R. J., Feltovich, P. J., Jacobson, M. J., & Coulson, R. L. (1991). Cognitive flexibility, constructivism, and hypertext: random access instruction for advanced knowledge acquisition in ill-structured domains. Educational Technology, 31(5), 24-33.

In: Advances in Mathematics Research, Volume 8 Editor: Albert R. Baswell, pp. 161-176

ISBN: 978-1-60456-454-9 © 2009 Nova Science Publishers, Inc.

Chapter 6

MODELING ASYMMETRIC CONSUMER BEHAVIOR AND DEMAND EQUATIONS 1 FOR BRIDGING GAPS IN RETAILING Rajagopal* Department of Marketing, Business Division, Monterrey Institute of Technology and Higher Education, ITESM Mexico City Campus, Tlalpan, Mexico

Introduction In growing competitive markets the large and reputed firms are developing strategies to move into the provision of innovative combinations of products and services as 'high-value integrated solutions' tailored to each customer's needs than simply 'moving downstream' into services. Such firms are developing innovative combinations of service capabilities such as operations, business consultancy and finance required to provide complete solutions to each customer's needs in order to augment the customer value towards the innovative or new products. It has been argued that provision of integrated solutions is attracting firms traditionally based in manufacturing and services to occupy a new base in the value stream centered on 'systems integration' using internal or external sources of product designing, supply and customer focused promotion (Davies,2004). Besides organizational perspectives of enhancing customer value, the functional variables like pricing play a significant role in developing the customer perceptions towards new products.

1

*

Author expresses his gratitude to the anonymous referees for their valuable suggestion to improve the paper. Author also acknowledges the support provided by Amritanshu Rajagopal, student of Industrial and Systems Engineering of ITESM, Mexico City Campus in data collection, translation of questionnaires in Spanish language, computing the data, developing Tables and figures in this study. E-mail address: [email protected] Home page: http://www.geocities.com/prof_rajagopal/homepage.html. PhD (India) FRSA (London), Professor, Department of Marketing, Business Division, Monterrey Institute of Technology and Higher Education, ITESM, Mexico City Campus, 222, Calle del Puente, Tlalpan, Mexico DF 14380.

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Consumers plan to spread consumption of their total resources evenly over the remainder of their life span at a given time. Such consumption smoothing occurs though current demand for the products and services and buying power may fluctuate over time. Thus, borrowing allows consumers with lower initial assets but higher expected future income to make utilitymaximizing consumption choices within the overall, lifetime budget constraint. On the contrary, consumers may not be very capable of taking lifetime budget constraints into account when making repeated consumption choices that are distributed over time and are influenced by the self reference criterion (see Herrnstein and Prelec, 1992). It has been observed in previous studies that value to expenditure ratios increase consumer sensitivity in volume of buying and driving repeat buying decisions for the regular and high-tech products (Carroll and Dunn 1997). Consumers often have enough insight towards limiting their choices by employing self-rationing strategies. Consumers may show suboptimal behavior while adapting to the self-rationing strategies to limit their choice which may induce asymmetric consumer behavior in the long run (Loewenstein, 1996). This paper attempts to discuss the interdependence of variability in consumer behavior due to intrinsic and extrinsic retail environment which influence the process of determining the choices on products and services. It is argued in the paper that suboptimal choice of consumers affect the demand of the products and services in the long-run and the cause and effect has been explained through the single non-linear equations. A system of demand equations which explains the process of optimization of consumer choice and behavioral adjustment towards gaining a long-term association with the market has also been discussed in the paper.

Related Contributions Customer Value and Choice Probabilities The prospect theory developed by Tversky and Kahnman (1981) towards framing decisions and understanding the dynamics of choices of consumers reveals that the consumers exercise options in consuming products to optimize their satisfaction and ultimate value. Value measurements have been used as one of the principal tools to assess the trend of consumer behavior for non-conventional products. Value syndrome influences the individual and group decisions in retail and bulk deals, and conditionalizes the decision process of consumers. Conditional consumption behavior suggests that consumption depends heavily on the utility function and on the source of uncertainty (Carroll and Kimball, 1996 and Deaton 1992). Repeat buying behavior of customers is largely determined by the values acquired on the product. The attributes, awareness, trial, availability and repeat (AATAR) factors influence the customers towards making re-buying decisions in reference to the marketing strategies of the firm. Decision of customers on repeat buying is also affected by the level of satisfaction derived on the products and number of customers attracted towards buying the same product, as a behavioral determinant (Rajagopal, 2005). A study using market-level data for the yogurt category developed an econometric model derived from a game-theoretic perspective explicitly considers firms' use of product-line length as a competitive tool (Dragnska and Jain, 2005). On demand side, the study analytically establishes link between customer choice and the length of the product line and includes a measure of line length in the utility function to

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investigate customer preference for variety using a brand-level discrete-choice model. The study reveals that supply side is characterized by price and line length competition between oligopolistic firms. Each successive purchase decision is relatively unimportant to an individual consumer, which may be derived from the economic and relational variables associated with the product or services. The formal model of price competition analyzed here is derived from that of Chintagunta and Rao (1996), who similarly consider a dynamic duopoly with adaptive consumers. Besides price, brand and quality are other variables which drive consumers’ perceptions on choice mapping and repeat consumption. Bergemann and Välikmäki (1996) examine the effect of strategic pricing on the rate of information acquisition by a buyer which reveals that optimal learning allows consumers to analyze their choice pattern. Erev and Haruvy (2001) also consider the implications of adaptive learning by consumers but again firms have fixed pricing policies. Recent theoretical explanations for sustained dominance include network effects, increasing returns to scale and learning by doing. There is now considerable evidence to explain consumer choice behavior such as in retail buying the choices are determined by an exogenous random process (Erev and Roth, 1998; Camerer and Ho, 1999; Erev and Barron, 2001). Consumer as a decision maker is endowed with propensities and values for each choice that is made. There are some critical issues associated to the price sensitive consumer behavior, whether customers are equally price-sensitive while purchasing products for functional (e.g. purchasing frozen vegetables, toiletries or paper towels) versus hedonic (e.g. purchasing a high technology computer or a video camera) consumption situations and whether perceived value derived during consuming the product influences price sensitivity. It may also be stated that higher price volatility makes consumers more sensitive to gains and less sensitive to losses, while intense price promotion by competing brands makes consumers more sensitive to losses but does not influence consumers’ sensitivity to gains (Han et.al, 2001).

Behavioral Asymmetry and Customer Choice Value of a customer may be defined in reference to a firm as the expected performance measures are based on key assumptions concerning retention rate and profit margin and the customer value also tracks market value of these firms over time. Value of all customers is determined by the acquisition rate and cost of acquiring new customers (Guptaet al, 2003). A long standing approach to examining how consumers react to price and income changes estimates a set of demand equations for the main commodities and bases deductions on coefficient values. At its simplest, economic demand theory assumes that consumers choose to allocate their limited spending power to purchases of goods to maximize their own satisfaction. This assumption of rational economic behavior imposes substantial constraints (aggregation, homogeneity, symmetry and negativity) on the specification of a system of equations. Consumer behavior is largely driven by tangible and intangible factors which include product attributes, pricing, willingness to pay (disposable income), product attractiveness and related variables. Value and pricing models have been developed for many different products, services and assets. Some of them are extensions and refinements of conventional models on value driven pricing theories (Gamrowski & Rachev, 1999; Pedersen, 2000). There have been some other models developed and calibrated addressing specific issues such as model for household assets demand (Perraudin & Sorensen, 2000).

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Key marketing variables such as price, brand name, and product attributes affect customers' judgment processes and derive inference on its quality dimensions leading to customer satisfaction. The experimental study conducted indicates that customers use price and brand name differently to judge the quality dimensions and measure the degree of satisfaction (Brucks et.al., 2000). Most importantly, these are expected to raise their spending and association with the products and services of the company with increasing levels of satisfactions that attribute, to values of customers (Reichheld and Sasser, 1990). However, it has been observed that low perceived use value; comparative advantages over physical attributes and economic gains of the product make significant impact on determining customer value for the relatively new products. Motivational forces are commonly accepted to have a key influencing role in the explanation of shopping behavior. Personal shopping motives, values and perceived shopping alternatives are often considered independent inputs into a choice model, it is argued that shopping motives influence the perception of retail store attributes as well as the attitude towards retail stores (Morschett et.al, 2005). In retail self-service store where consumer exercises in-store brand options, both service and merchandise quality exert significant influence on store performance, measured by sales growth and customer growth, and their impact is mediated by customer satisfaction. Liberal environment of self-service stores for merchandise decisions, service quality and learning about competitive brands are the major attributes of retail self-service stores (Babakus et.al, 2004). Retail self-service stores offer an environment of three distinct dimensions of emotions e.g. pleasantness, arousal and dominance. Customer value gap may be defined as the negative driver, which lowers the returns on the aggregate customer value. This is an important variable, which needs to be carefully examined by a firm and measure its impact on the profitability of the firm in reference to spatial (coverage of the market) and temporal (over time) market dimension (Marjolein and Verspagen, 1999).

Organizational Influences on Customer Values Another study explores qualitatively the understanding of the importance of intangibles as performance drivers in reference to Swedish organizations using a combination of evolutionary theory, knowledge-based theory and organizational learning. The study reveals that customer values are created towards new products through individual perceptions, and organizational and relational competence (Johanson et.al., 2001). The firms need to ascertain a continuous organizational learning process with respect to value creation chain and measure performance of the new products introduced in the market. In growing competitive markets the large and reputed firms are developing strategies to move into the provision of innovative combinations of products and services as 'high-value integrated solutions' tailored to each customer's needs than simply 'moving downstream' into services. Such firms are developing innovative combinations of service capabilities such as operations, business consultancy and finance required to provide complete solutions to each customer's needs in order to augment customer value towards the innovative or new products. It has been argued that provision of integrated solutions is attracting firms traditionally based in manufacturing and services to occupy a new base in the value stream cantered on 'systems integration' using internal or external sources of product designing, supply and customer focused promotion (Davies,2004). Besides organizational perspectives of enhancing the customer value, the

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functional variables like pricing play a significant role in developing customer perceptions towards the new products. Analysis of the perceived values of customers towards new products is a complex issue. Despite considerable research in the field of measuring customer values in the recent past, it is still not clear how value interacts with marketing related constructs. However there exists the need for evolving a comprehensive application model determining the interrelationship between customer satisfaction and customer value, which may help in reducing the ambiguities surrounding both concepts. One of the studies in this regard discusses two alternative models yielding empirically tested results in a cross-sectional survey with purchasing managers in Germany. The first model suggests a direct impact of perceived value on the purchasing managers' intentions. In the second model, perceived value is mediated by satisfaction. This research suggests that value and satisfaction can be conceptualized and measured as two distinct, yet complementary constructs (Eggert and Ulaga, 2002). Improving customer value through faster response times for new products is a significant way to gain competitive advantage. In the globalization process many approaches to new product development emerge, which exhibit an internal focus and view the new product development process as terminating with product launch. However, it is process output that really counts, such as customer availability. A study proposes that with shortening product life cycles it should pay to get the product into the market as quickly as possible, and indicates that these markets should be defined on an international basis. The results of the study reveals that greater new product commercial success is significantly associated with a more ambitious and speedier launch into overseas markets as the process of innovation is only complete when potential customers on a world scale are introduced effectively to the new product (Oakley, 1996). Retail sales performance and customer value approach are conceptually and methodically analogous. Both concepts calculate the value of a particular decision unit by analytical attributes forecast and the risk-adjusted value parameters. However, virtually no scholarly attention has been devoted to the question if any of these components of the shareholder value could be determined in a more market oriented way using individual customer lifetime values (Rajagopal, 2005). Value of a customer may be defined in reference to a firm as the expected performance measures are based on key assumptions concerning retention rate and profit margin and the customer value also tracks market value of these firms over time. Value of all customers is determined by the acquisition rate and cost of acquiring new customers (Gupta, Lehmann and Stuart, 2003).

Objectives and Design of Model This paper emerges from a new indirect utility function and derives the corresponding system of equations, relating commodity demands to prices, income and customer values that satisfy the customer behavior by utility maximization (aggregation, homogeneity, symmetry and negativity). One of the assumptions made in developing the asymmetric model for measuring the consumer behavior and derived values is that consumers are determined by the sub-sets of tangible and intangible variables which include product attributes, awareness, product attractiveness, price, disposable income and competitive advantage. The analytical construct

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in this paper has been derived using the Linear Expenditure System (LES) and Linear Equations in Demand Estimation (LEDE) to measure the variability in consumer behavior. This paper analyzes the belief-based or reinforcement learning attributes in forming typical consumer behavior through inter-personal communication, point of purchase information, advertisement or corporate image may have a significant impact on market organization. The model of dynamic oligopoly, where consumers learn about the relative perceived values of the different brands has also been discussed in the paper. The basic premise of the model is towards reinforcement type behavioral model, where more familiar products have a greater probability of being selected. Consequently, consumers can get locked into inferior choices without considering the novelty factor in reference to product attributes or brand. Such lock-in behavior may be cyclical and asymmetric in context to personality traits and demand for the product. Such situation may become significant when firms influence consumer opinion in the short run as consumers’ initial estimate of a firm’s quality is high (or low), it has an incentive to charge above (or below) the myopic price in order to slow (or speed up) learning and behavioral adaptation of consumer towards the competing products.

Construct of Model Choice Variability and Demand Equation Ofek Elie (2002) discussed that the values of product and service are not always the same and are subject to value life cycle that governs customer preferences in the long-run. If customers prefer the product and service for N periods with Q as value perceived by the customer, the value may be determined as Q>N, where Q and N both are exogenous variables. If every customer receives higher perceived values for each of his buying, the value added product q ≥ Q, where ‘q’ refers to the change in the quality brought by innovation or up-graded technology. Customer may refrain from buying the products if q ≤ Q, that does not influence his buying decisions. However, a strong referral ‘R’ may lead to influence the customer values, with an advantage factor β that may be explained by price or quality factor. In view of the above discussion it may be assumed that customer preferences have high variability that grows the behavioral factors in retail buyers’ decisions (Rajagopal, 2006a): N

(

)

Dbn = ∑ β t μ C t , Zˆ + β N +1Qt t =1

(1)

Dbn is expressed as initial buying decision of the customers, C represents t ˆ consumption, Z is a vector of customer attributes (viz. preferential variables) and Qt is the Where,

value perceived by the customer. Customer behavior is largely derived from the customer value and it has a dynamic attribute that plays a key role in buying and is an intangible factor to be considered in all marketing and selling functions. The value equation for customer satisfaction may be expressed as a function of all value drivers wherein each driver contains the parameters that

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directly or indirectly offer competitive advantages to the customers and enhance the customer value.

V ′ = Ks , Km , Kd , Kc

[∏ {V (x, t , q, p )}]

(2)

In the above equation V ′ is a specific customer value driver, K are constants for supplies(Ks), margins (Km), distribution (Kd), and cost to customers (Kc); x is volume, t is time, q is quality and p denotes price. Perceived customer value (V) is a function of price (p)

∏

and non-price factors including quality (q) and volume (x) in a given time t. Hence has been used as a multiplication operator in the above equation. Quality of the product and volume are closely associated with the customer values. Total utility for the conventional products goes up due to economy of scale as the quality is also increased simultaneously (∂v/∂x>0). The ∂v customer value is enhanced by offering larger volume of product at a competitive price in a given time (∂v/∂p>0) and (∂v/∂t>0). Conventional products create lower values to the customers (∂v/∂x<0) while innovative products irrespective of price advantages, enhance the customer value (∂v/∂x>0). Value addition in the conventional products provides lesser enhancement in customer satisfaction as compared to the innovative products. Such transition in the customer value, due to shift in technology may be expressed as:

⎡ ⎤ Tp Vhj′ = a ⎢∑ (1+ j ′+ i ) ⎥ + b(X j ) ⎢⎣ (1 + V p ) ⎥⎦

(3)

V′

In this equation hj represents enhancements in customer value over the transition from conventional to innovative products, a and b are constants, Tp denotes high-tech and highvalue products, Vp represents value of product performance that leads to enhance the customer

value, volume is denoted by X and ( j ) is the period during which customer value is measured (Rajagopal, 2006b). In reference to optimization theory it may be stated that maximizing a valid or direct utility function subject to a budget constraint or alternatively, appealing to duality theory and commencing from a cost or indirect utility function, influences on consumer behavior (Shida, 2001). In the latter case, let U (p, y) be the indirect utility function, where p is a vector of prices and y is income. For validity, U (p, y) should be homogeneous of degree zero in income and prices (p), non-decreasing in y, non-increasing in p, and convex or quasi-convex in p. The demand equations can be obtained through:

′

qi =

∂U ∂U / ∂p ∂y

(4)

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Rajagopal In the above equation

(qi ) is denoted as demand for the product. Simple utility function

y⎤ ⎡ ⎢⎣U = P ⎥⎦ derived by generic consumer behavior may be understood as wherein P may be expressed as geometric mean of prices which derives dynamic consumer behavior with the

(α )

j over the products in a given retail variability factor of competitive advantage environment and j is the vector adding to unity over n commodities. Such condition of consumer behavior may be explained as:

log P = ∑ (α j log p j )

(5)

The above equation helps in deriving the demand equation as below:

⎛pq ⎞ wi = ⎜⎜ i i ⎟⎟ = α j ⎝ y ⎠

(6)

(w )

i represents the individual expenditure limits or disposable income for buying wherein, the products. However, these equations limit consumer responses to changes in prices, competitive advantages or disposable income to maintaining consistency in buying behavior due to change in the elasticity of aforesaid variables. While such a consumption pattern might sometimes be plausible, it adds to the asymmetric behavior of consumers in retail buying. Propensity of consumption during leisure season may be largely determined as a driver of retail attractions in terms of appealing sales promotions and availability of innovative products. Choice of consumers is thus established by the propensity of consumption in the array of innovative products in the retail stores. The propensity of consumption of buyers may

be denoted by

(θ

lim 0 − ∞

0−∞ ] θ = [θ lim jt ... n

) from a j

, which is measured in reference to frequency of buying

th

store in a given time t. The choice pattern of buyers in shopping during holiday season may be derived as:

x i (t ) =

exp{βθ i (t )}

∑ exp{βθ n

j =1

In the above equation exponential expression

j

(t )} (7)

{xi (t )} is the probability of buying strategy i at time t. In the

β represents the degree of value optimization on buying. At higher

levels of β , consumer will have higher probabilities of buying with increased propensity.

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Consumer Choice for New Products According to the customer choice model of Giannakas and Fulton (2002), individuals are assumed to consume optimum one unit of product of their interest within a given time. The construct of the model of customer behavior, assumes that prior to choice of new product,

U

vi customer i derives perceived use value, for t conventional or new products having willingness to pay for the conventional product on the perceived used value of customer i at price p. In order to explain the customer preference to the product and estimating the brand value in reference to this study it may be derived that customer obtains the perceived use

value

(U

vi

− pt

) from consuming conventional product. The customer also exercises his

option of buying a high value substitute (new products) at an alternate price

(

)

pa

where p ≥ p . Hence without availability of new products, the customer value Cs may be derived as: a

t

[(

)] (

)(

Csit = lim 0−∞ U vi − p t , U vi − p a = U vi − p t

)

(8)

Following the scenario when customers get access to new products in the market,

E

customers enhance the perceived use value of the new product by factor vi . This parameter is subjective to the customer decision in view of their preferences towards consuming organic products. However, due to lack of awareness, advertising and sales promotions, many customers may not be able to establish their preferences explicitly towards synthetic and conventional products. If α represents the market segment for new products, the customers

(αE )

vi . Accordingly, the customer value would access the products and perceive its value by after the new products are made available in the market segment may be described as:

[(

)

(

)

CsiAP = lim 0−∞ U vi − p t − αE vi , U vi − p a , Bivg Wherein, the expression

]

(9)

Cs iAP denotes the enhancement of customer value for new Bivg

products in reference to the advertising and promotion strategies of the firms and represents the brand value of new products perceived by the customers. Further,

(Cs )comprising word of ) and point of sales promotions (Cs ). Hence, the

be understood as a function of interpersonal communication

(Cs

ma

IP i

i mouth and referrals, commercials factors influencing the enhancement of customer value may be expressed as:

(

CsiAP may

Cs iAP = f Cs iIP , Cs ima , Cs ips

)

ps i

lim 0 − ∞

(10)

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Rajagopal

Motivational forces are commonly accepted to have a key influencing role in the explanation of shopping behavior. Personal shopping motives, values and perceived shopping alternatives are often considered independent inputs into a choice model. It is argued that shopping motives influence the perception of retail store attributes as well as the attitude towards retail stores (Morschett et.al, 2005). Lliberal environment of the self-service stores for merchandise decisions, service quality and learning about competitive brands are the major attributes of retail self-service stores (Babakus et.al, 2004). Retail self-service stores offer an environment of three distinct dimensions of emotions e.g. pleasantness, arousal and dominance. The change in the customer value observed among the synthetic and new products in reference to advertising and promotional strategies used by the firms, may be described as:

CsiAP −t = CstAP − Csit

(11)

N

The model assumes that if rs number of customers in a given retail store has preferred to use new products; the change in the customer value may be derived as:

ΔCs iAP −t =

Nrs1 ... Nrs n

∑ ΔCs i =1

AP −t i

(12)

Customers choose the product which offers maximum utility in reference to the price, awareness and promotional advantage over other conventional products. Hence, the customer value for new products may be expressed as:

(

Evi −t Evi ΔC Nrs − Cs it =1 = α C i

Evi

)

(13)

Where, C represents the customer value in total derived by all factors. Value and pricing models have been developed for many different products, services and assets. Some of them are extensions and refinements of convention models on value driven pricing theories (Gamrowski and Rachev, 1999; Pedersen, 2000). There have also been some models that are developed and calibrated addressing specific issues such as model for household assets demand (Perraudin and Sorensen, 2000). Key marketing variables such as price, brand name, and product attributes affect customers' judgment processes and derive inference on its quality dimensions leading to customer satisfaction. The experimental study conducted indicates that customers use price and brand name differently to judge the quality dimensions and measure the degree of satisfaction (Brucks et.al. 2000). Value of corporate brand endorsement across different products and product lines, and at lower levels of the brand hierarchy also needs to be assessed as a customer value driver. Use of corporate brand endorsement either as a name identifier or logo identifies the product with the company, and provides reassurance for the customer (Rajagopal and Sanchez, 2004).

Modeling Asymmetric Consumer Behavior and Demand Equations…

171

Cost of acquiring new products would be the difference in traditional good price,

(C ) sc

variation in the perceived use value and search cost as indicated by i for each customer. Hence, appreciation of customer value to obtain new products may be expressed as:

[(

)

ΔCiKrs −t = α Evi + p a + Cisc − Csit

]

(14)

Krs i

C

represents the cost of acquiring the new products from a given retail store. Where, Competitive advantage of a firm is also measurable from the perspective of product attractiveness to generate new customers. Given the scope of retail networks, a feasible value structure for customers may be reflected in repeat buying behavior ( Rˆ ) that explains the relationship of the customer value with the product and associated marketing strategies. The impact of such customer value attributes in a given situation may be described as: n

∑ Cs

Nsr =1

AP i

= Rˆ = C Evi (15)

Repeat buying behavior of customers is largely determined by the values acquired on the product. Decision of customers on repeat buying is also affected by the level of satisfaction derived on the products and number of customers attracted towards buying the same product, as a behavioral determinant (Rajagopal, 2005). New product attractiveness may comprise the product features including improved attributes, use of advance technology, innovativeness, extended product applications, brand augmentation, perceived use value, competitive advantages, corporate image, product advertisements, sales and services policies associated therewith, which contribute in building sustainable customer values towards making buying decisions on the new products (Rajagopal, 2006a). Attributes of the new products lead to satisfaction to the customers and is

F

reflected through the product attractiveness ( x ). It has been observed that the new products have been considered as new and experimental products in Mexico by a significant number of consumers. Hence product attractiveness variables in the following equation as:

[

(Fx )

may be explained along the associated

(

Fx = ∏ αE vi Cs tAP , q, Z xi , p a

)]

t

(16)

Z

Wherein, q denotes quality of the product and xi represents services offered by the retailers towards prospecting and retaining customers who intend to buy the new products. Customer value may also be negative or low if attributes are not built into the new product to maximize the customer value as per the estimation of the firm. Perceived use value of customers by market segments appreciation p

a

αEvi is a function of advertising and promotion CsiAP , price

and retailer services

Z xi in a given time t and

∏

has been used as a

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Rajagopal

multiplication operator in the above equation. The quality of the product and volume are closely associated with the customer values. Introduction of new technological products makes it important for marketers to understand how innovators or first adopters respond to persuasion cues. It has been observed in a study that innovativeness and perceived product newness which are the major constituents of new product attractiveness were independent constructs that had independent effects on customer's attitude toward the brand and purchase intent for the new product (Lafferty and Goldsmith, 2004).

Customer Value Enhancement

M ( i1 +i2 +i3 +...in ) j

t A firm may introduce new product with high investment in terms of product attributes (i1), distribution (i2), promotion (i3) and other related factors (…in) related with gaining competitive advantage in a given time (t) in the jth market. Let us assume that s is the

V

estimated market coverage for the new product, the customer value ( np ) may be initially positive and high, resulting into deeper market penetration (with s increasing). This may be described as:

M t(i1 +i2 +i3 +...) j =

However, Vnp ≤

∂s =k ∂t

(17)

∂v may become negative following product competition within the ∂t

product line due to the product overlap strategy of the firm. In the above equation, volume of buying is represented by ∂v in a given time t. To augment the customer value and enhance market coverage for the new products in potential markets the firm may optimize the product line [s ] pt by pruning the slow moving products in the jth chain in h market in order to j ,h

reposition them in new market. The opportunity cost in moving the slow performance products may be derived by inputting the values of V´ from equation (ii) as:

[s ]

j ,h Pt

⎡ ∂v ⎤ =⎢ ⎥ ⎣ ∂t ⎦

j ,h

+ ∏{V ( x, t , q, p )} (18)

Hence to enhance market coverage for the new product with enhancing customer value for the new product of the firm, the strategy may be described as:

[

]

s = ∫ k + {s}Pt ∂t + β t R j ,h

(19)

Where in s is market coverage of the new product, k is investment on market functions derived in a given time [equation (vi)] and R is the referral factor influencing the customer values with an advantage factor coefficient

β in time t. The products constituting the optimal

Modeling Asymmetric Consumer Behavior and Demand Equations…

173

P

product line of the firm in a given time are represented by t in the above equation. The firm may measure the customer value shocks accordingly and shield the uncertainties occurring to the estimated market coverage due to declining customer values for the new products. As is common the new products are susceptible to such value shocks in view of the companies’ own product line strategy. Let us assume that new product attractiveness is

Fx and initial product market Vnp

M t( i1 +i2 +i3 +...in ) j , perceived customer value of the new product is and C competitive advantage driver for the customer is at at a given time, we get the following investment is

equation.

[

]

Fx = ∑t M t(i1 + i2 +...in ) j (Vnp )(C at ) jh

(20)

Hence, the above equation may be further simplified as,

Fx = M tin , j

Where in

∂v ′ ∂b ′ ∂s ∂v (Vnp )(C at ) = M tin , j = M tin, j ∂s ∂t ∂s ∂t

(21)

M ti n , j denotes the initial investment made by the firm for introducing new

products, V ′ represents the volume of penetration of new product in a given market in time t

with estimated market coverage s and b′ expresses the volume of repeat buying during the period the new product was penetrated in the market by the firm. The total quality for new products goes up due to economy of scale as the quality is also increased simultaneously

∂ s > 0) and (∂ b′ ∂ s > 0) . In reference to the new products x, competitive products (∂ ∂ < 0) while innovative products irrespective of create lower values to the customers v x

(∂ v

(∂

∂ > 0)

price advantages, enhance the customer value v x . Value addition in the competitive products provides lesser enhancement in customer satisfaction as compared to the innovative products if the new products have faster penetration, re-buying attributes and market coverage.

∫ s∂s = ∫ V

Therefore

np

+ C at

(22)

V

C

at In the above equation np denotes customer value for the new product and represents the competitive advantage at a given time. The prospect theory laid by Tversky and Khanman (1981) proposes that the intensity of

G =g

(∂

∂

)

pt x p gains plays strategic role in value enhancement as xt . In this situation t represents the period wherein promotional strategies were implemented to enhance customer

174

Rajagopal

g

values in reference to product specific gains ( pt ). However, in order to measure relationship/variability between repeat buying behavior and customer value, it would be appropriate to determine the cumulative decision weights ( w ) and substitute it in the equation (1) to get the following equation:

[

G xt = w∑ g pt (r j m j ) + β n +1Qt jh

k =1

] (23)

The customer value however may be the driver function of gains on buying decision on new products and the influencing variables such as perceived use value and referrals.

Conclusion and Managerial Implications Existing theoretical and methodological issues are reviewed in this study and a new framework has been proposed for future research in measuring customer value in specific reference to the new products as launching innovative and high technology products is a continuous process for the firms in the present competitive markets. The assumptions and methodology employed in this paper are quite different from those of conventional behavioral models. It would be interesting to analyze the robustness of convergence of the model in reference to asymmetric behavioral patterns of consumer and demand orientation. Some of the existing empirical evidences, both from laboratory and field consumer data, seem to give enhanced scope for analyzing the asymmetric behavior model discussed in the paper that predicts suboptimal consumer behavior even in the long run in view of shifts in retailing strategies. The framework for measuring customer values discussed in this paper provides analytical dimensions for establishing the long run customer relationship by the firm and to optimize its profit levels. To test some hypotheses concerning homogeneity in consumer behavior and its impact on derived demand for the product or services may be required to be taken-up with additional parameters to estimate model. While following the LES data to estimate demand functions, the results may not aggregate over consumers’ asymmetric behavior, particularly in reference to expenditures. Hence, individual expenditures on goods over a group of consumers of varying incomes produce different equations with income when replaced by utility based qualitative parameters. However, it is believed that utility maximization, and the consequent constraints on demand equations, pertain strictly to individuals and consequently consider their application at aggregate level provided the demand equations aggregate over consumers qualitative parameters. It is necessary to standardize the qualitative parameters before applying in this model. One of the challenges for a firm is to incorporate and validate the preferences of customers into the design of new products and services in order to maximize customer value. An augmented and sustainable customer value builds loyalty towards the product and brand and helps to stabilize customer behavior. Systematically explored concepts in the field of customer value and market driven approach towards new products would be beneficial for a company to derive long term profit optimization strategy over the period. Hence, a comprehensive framework for estimating both the value of a customer and profit optimization

Modeling Asymmetric Consumer Behavior and Demand Equations…

175

need to be developed. On a tactical level, managers need to consider as what is the optimum spread of customers on a matrix of product attractiveness and market coverage. The model discussed in this paper provides a holistic view of the customer behavior driven by the value matrices associated with product attractiveness, market coverage, brand and point-ofpurchase services offered to the customers. Analysis of these variables would help strategists/managers to develop appropriate strategies to enhance customer value for the new products and optimize profit of the firm.

References Babakus E, Bienstock C C and Van Scotter J R (2004), Linking perceived quality and customer satisfaction to store traffic and revenue growth, Decision Sciences, 35 (4), 713737 Bergemann D, Välimäki J (1996) Learning and strategic pricing, Econometrica, 64, 11251149 Brucks M, Zeithaml V A and Naylor G (2000), Price and brand name as indicators of quality dimensions of customer durables, Journal of Academy of Marketing Science, 28 (3), 359-374 Carroll D and Dunn W E (1997), Unemployment expectations, NBER, Working Paper # 6081 Carroll C D and Kimball M S (1996), On the concavity of the consumption function, Econometrics, 64(4), 981-992 Chintagunta, P.K., Rao, P.V. (1996), Pricing strategies in a dynamic duopoly: a differential game model, Management Science, 42, 1501-14 Davies A (2004), Moving base into high-value integrated solutions: A value stream approach, Industrial and Corporate Change, 13 (5), October, 727-756 Deaton A (1992), Understanding competition, Oxford University Press, Oxford Erev I and Barron G (2001), On adaptation, maximization and reinforcement learning among cognitive strategies, Working Paper, Columbia University Erev I and Roth A E(1998), Predicting how people play games: reinforcement learning in experimental games with unique, mixed strategy equilibria, American Economic Review, 88, 848-881 Gramrowski B and Rachev S (1999), A testable version of the pareto-stable CAPM, Mathematical and Computer Modeling, Vol. 29, 61-81. Herrnstein, R and Dražen P (1992), “Melioration,” in George L and Jon E (Eds), Choice over time, New York: Russell Sage, 235-263. Loewenstein, G (1996), Out of control: Visceral influences on behavior, Organizational behavior and Human Decision Processes, 65 (March), 272-292. Marjolein C and Verspagen B (1999), Spatial distance in a technology gap model, Maastricht Economic Research Institute on Innovation and Technology (MERIT), Working Paper No. 021 Morschett, D; Swoboda, B and Foscht, T (2005), Perception of store attributes and overall attitude towards grocery retailers: The role of shopping motives, The International Review of Retail, Distribution and Consumer Research, 15 (4), 423-447 Ofek E (2002), Customer profitability and lifetime value, Harvard Business School, Note, August, 1-9 (Publication reference 9-503-019)

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Pederson C S(2000), Sparsing risk and return in CAPM: A general utility based model, European Journal of Operational Research, Vol. 123 (3), 628-639. Perraudin W R M and Sorensen B E (2000), The demand of risky assets: Sample selection and household portfolios, Journal of Econometrics, Vol. 97, 117-144 Rajagopal (2006a), Measuring customer value gaps: An empirical study in mexican retail markets, Economic Issues, 11(1), March, 19-40 Rajagopal (2006b), Measuring customer value and market dynamics for newproducts of a firm: An analytical construct for gaining competitive advantage, Global Business and Economics Review, 8 (3-4), 187-205 Reichheld F F and Sasser W E (1990), Zero defections: Quality comes to services, Harvard Business Review, Sep-Oct, pp 105-111 Shida M (2000), Fundamental theorems of morse theory for optimization on manifolds with corners, Journal of Optimization Theory and Applications, 106 (3), September, 683-688 Tversky A and Kahnman D (1981), The framing decisions and psychology of hoice, Science, No.211, 453-458

In: Advances in Mathematics Research, Volume 8 Editor: Albert R. Baswell, pp. 177-246

ISBN: 978-1-60456-454-9 © 2009 Nova Science Publishers, Inc.

Chapter 7

HIGHER EDUCATION: FEDERAL SCIENCE, TECHNOLOGY, ENGINEERING, AND MATHEMATICS PROGRAMS AND RELATED TRENDS* United States Government Accountability Office

Why This Study? The United States has long been known as a world leader in scientific and technological innovation. To help maintain this advantage, the federal government has spent billions of dollars on education programs in the science technology, engineering, and mathematics (STEM) fields for many years. However, concerns have been raised about the nation’s ability to maintain its global technological competitive advantage in the future. This report presents information on(1) the number of federal programs funded in fiscal year 2004 that were designed to increase the number of students and graduates pursuing STEM degrees and occupations or improve educational programs in STEM fields, and what agencies report about their effectiveness; (2) how the numbers, percentages, and characteristics of students, graduates, and employees in STEM fields have changed over the years; and (3) factors cited by educators and others as affecting students’ decisions about pursing STEM degrees and occupations, and suggestions that have been made to encourage more participation. GAO received written and/or technical comments from several agencies. While one agency, the National Science Foundation, raised several questions about the findings, the others generally agreed with the findings and conclusion and several agencies commended GAO for this work.

*

Excerpted from http://www.gao.gov/new.items/d06114.pdf

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United States Government Accountability Office

Abbreviations BEST BLS CGS CLF COS CPS DHS EPA HHS HRSA IPEDS NASA NCES NCLBA NIH NPSAS NSF NSTC SAO SEVIS STEM

Building Engineering and Science Talent Bureau of Labor Statistics Council of Graduate Schools civilian labor force Committee on Science Current Population Survey Department of Homeland Security Environmental Protection Agency Health and Human Services Health Resources and Services Administration Integrated Postsecondary Education Data System National Aeronautics and Space Administration National Center for Education Statistics No Child Left Behind Act National Institutes of Health National Postsecondary Student Aid Study National Science Foundation National Science and Technology Council Security Advisory Opinion Student and Exchange Visitor Information System science, technology, engineering, and mathematics

The United States has long been known as a world leader in scientific and technological innovation. To help maintain this advantage, the federal government has spent billions of dollars on education programs in the science, technology, engineering, and mathematics (STEM) fields for many years. Some of these programs were designed to increase the numbers of women and minorities pursuing degrees in STEM fields. In addition, for many years, thousands of international students came to the United States to study and work in STEM fields. However, concerns have been raised about the nation’s ability to maintain its global technological competitive advantage in the future. In spite of the billions of dollars spent to encourage students and graduates to pursue studies in STEM fields or improve STEM educational programs, the percentage of United States students earning bachelor’s degrees in STEM fields has been relatively constant—about a third of bachelor’s degrees— since 1977. Furthermore, after the events of September 11, 2001, the United States established several new systems and processes to help enhance border security. In some cases, implementation of these new systems and processes, which established requirements for several federal agencies, higher education institutions, and potential students, made it more difficult for international students to enter this country to study and work. In the last few years, many reports and news articles have been published, and several bills have been introduced in Congress that address issues related to STEM education and occupations. This report presents information on (1) the number of federal civilian education programs funded in fiscal year 2004 that were designed to increase the numbers of students and graduates pursuing STEM degrees and occupations or improve educational programs in

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179

STEM fields and what agencies report about their effectiveness; (2) how the numbers, percentages, and characteristics of students, graduates, and employees in STEM fields have changed over the years; and (3) factors cited by educators and others as influencing people’s decisions about pursuing STEM degrees and occupations, and suggestions that have been made to encourage greater participation in STEM fields. To determine the number of programs designed to increase the numbers of students and graduates pursuing STEM degrees and occupations, we identified 15 federal departments and agencies as having STEM programs, and we developed and conducted a survey asking each department or agency to provide information on its education programs, including information about their effectiveness [1]. We received responses from 14 of them, the Department of Defense did not participate, and we determined that at least 13 agencies had STEM education programs during fiscal year 2004 that met our criteria. To describe how the numbers of students, graduates, and employees in STEM fields have changed, we analyzed and reported data from the Department of Education’s (Education) National Center for Education Statistics (NCES) and the Department of Labor’s (Labor) Bureau of Labor Statistics (BLS). Specifically, as shown in table 1, we used the National Postsecondary Student Aid Study (NPSAS) and the Integrated Postsecondary Education Data System (IPEDS) from NCES and the Current Population Survey (CPS) data from BLS. We assessed the data for reliability and reasonableness and found them to be sufficiently reliable for the purposes of this report. To obtain perspectives on the factors that influence people’s decisions about pursuing STEM degrees and occupations, and to obtain suggestions for encouraging greater participation in STEM fields, we interviewed educators and administrators in eight colleges and universities (the University of California Los Angeles and the University of Southern California in California; Clark Atlanta University, Georgia Institute of Technology, and Spelman College in Georgia; the University of Illinois; Purdue University in Indiana; and Pennsylvania State University). We selected these colleges and universities to include a mix of public and private institutions, provide geographic diversity, and include a few minorityserving institutions, including one (Spelman College) that serves only women students. Table 1. Sources of Data, Data Obtained, Time Span of Data, and Years Analyzed Department

Agency

Database

Data obtained

Education

NCES

NPSAS

College student enrollment Graduation/degrees

Time span of data 9 years

Years analyzed

Academic years 1995-1996 and 2003-2004 Education NCES IPEDS 9 years Academic years 1994-1995 and 2002-2003 Labor BLS CPS Employment 10 years Calendar years 1994 through 2003 Sources: NCES’s National Postsecondary Student Aid Study (NPSAS) and Integrated Postsecondary Education Data System (IPEDS) and BLS’s Current Population Survey (CPS) data. Note: Enrollment and employment information are based on sample data and are subject to sampling error. The 95-percent confidence intervals for student enrollment estimates are contained in appendix V of this report. Percentage estimates for STEM employment have 95-percent confidence intervals of within +/- 6 percentage points and other employment estimates (such as wages and salaries) have confidence intervals of within +/- 10 percent of the estimate, unless otherwise noted. See appendixes I, V, and VI for additional information.

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United States Government Accountability Office

In addition, most of the institutions had large total numbers of students, including international students, enrolled in STEM fields. We also asked officials from the eight universities to identify current students to whom we could send an e-mail survey. We received responses from 31 students from five of these institutions. In addition, we interviewed federal agency officials and representatives from associations and education organizations, and analyzed reports on various topics related to STEM education and occupations. Appendix I contains a more detailed discussion of our scope and methodology. We conducted our work between October 2004 and October 2005 in accordance with generally accepted government auditing standards.

Results in Brief Officials from 13 federal civilian agencies reported having 207 education programs funded in fiscal year 2004 that were designed to increase the numbers of students and graduates pursuing STEM degrees and occupations or improve educational programs in STEM fields, but they reported little about the effectiveness of these programs. The 13 agencies reported spending about $2.8 billion in fiscal year 2004 for these programs. According to the survey responses, the National Institutes of Health (NIH) and the National Science Foundation (NSF) sponsored 99 of the 207 programs and spent about $2 billion of the approximate $2.8 billion. The program costs ranged from $4,000 for a national scholars program sponsored by the Department of Agriculture (USDA) to about $547 million for an NIH program that is designed to develop and enhance research training opportunities for individuals in biomedical, behavioral, and clinical research by supporting training programs at institutions of higher learning. Officials reported that most of the 207 programs had multiple goals, and many were targeted to multiple groups. For example, 2 programs were identified as having one goal of attracting and preparing students at any education level to pursue coursework in STEM areas, while 112 programs had this as one of multiple goals. Agency officials also reported that evaluations were completed or under way for about half of the programs, and most of the completed evaluations reported that the programs had been effective and achieved established goals. However, some programs that have not been evaluated have operated for many years. While the total numbers of students, graduates, and employees have increased in STEM fields, changes in the numbers and percentages of women, minorities and international students varied during the periods reviewed. From the 1995-1996 academic year to the 20032004 academic year, the number of students increased in STEM fields by 21 percent—more than the 11 percent increase in non-STEM fields. Also, students enrolled in STEM fields increased from 21 percent to 23 percent of all students. Changes in the numbers and percentages of domestic minority students varied by group. For example, the number of African American students increased 69 percent and the number of Hispanic students increased 33 percent. The total number of graduates in STEM fields increased by 8 percent from the 1994-1995 academic year to the 2002-2003 academic year, while graduates in nonSTEM fields increased 30 percent. Further, the numbers of graduates decreased in at least four of eight STEM fields at each education level. The total number of domestic minority graduates in STEM fields increased, and international graduates continued to earn about onethird or more of the master’s and doctoral degrees in three fields. Moreover, from 1994 to

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181

2003, employment increased by 23 percent in STEM fields as compared with 17 percent in non-STEM fields. African American employees continued to be less than 10 percent of all STEM employees, and there was no statistically significant change in the percentage of women employees. Educators and others cited several factors as influencing students’ decisions about pursuing STEM degrees and occupations, and they suggested many ways to encourage more participation in STEM fields. Studies, education experts, university officials, and others cited teacher quality at the kindergarten through 12th grade levels and students’ high school preparation in mathematics and science courses as major factors that influence domestic students’ decisions about pursuing STEM degrees and occupations. In addition, university officials, students, and studies identified mentoring as a key factor for women and minorities. Also, according to university officials, education experts, and reports, international students’ decisions about pursuing STEM degrees and occupations in the United States are influenced by yet other factors, including more stringent visa requirements and increased educational opportunities outside the United States. We have reported that several aspects of the visa process have been improved, but further steps could be taken. In order to promote participation in the STEM fields, officials at most of the eight universities visited and current students offered suggestions that focused on four areas: teacher quality, mathematics and science preparation and courses, outreach to underrepresented groups, and the federal role in STEM education. The students who responded to our e-mail survey generally agreed with most of the suggestions and expressed their desires for better mathematics and science preparation for college. However, before adopting such suggestions, it is important to know the extent to which existing STEM education programs are appropriately targeted and making the best use of available federal resources. We received written comments on a draft of this report from the Department of Commerce, the Department of Health and Human Services, and the National Science and Technology Council. These agencies generally agreed with our findings and conclusions. We also received written comments from the National Science Foundation which questioned our findings related to program evaluations, interagency collaboration, and the methodology we used to support our findings on the factors that influenced decisions about pursing STEM fields. Also, the National Science Foundation provided information to clarify examples cited in the report, stated that the data categories were not clear, and commented on the graduate level enrollment data we used. We revised the report to acknowledge that the National Science Foundation uses a variety of mechanisms to evaluate its programs and we added a bibliography that identifies the reports and research used during the course of this review to address the comment about our methodology related to the factors that influenced decisions about pursuing STEM fields. We also revised the report to clarify the examples and the data categories and to explain the reasons for selecting the enrollment data we used. However, we did not make changes to address the comment related to interagency collaboration for the reason explained in the agency comments section of this report. The written comments are reprinted in appendixes VII, VIII, IX, and X. In addition, we received technical comments from the Departments of Commerce, Health and Human Services, Homeland Security, Labor, and Transportation, and the Environmental Protection Agency and National Aeronautics and Space Administration, which we incorporated when appropriate.

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Background STEM includes many fields of study and occupations. Based on the National Science Foundations’ categorization of STEM fields, we developed STEM fields of study from NCES’s National Postsecondary Student Aid Study (NPSAS) and Integrated Postsecondary Education Data System (IPEDS), and identified occupations from BLS’s Current Population Survey (CPS). Using these data sources, we developed nine STEM fields for students, eight STEM fields for graduates, and four broad STEM fields for occupations. Table 2 lists these STEM fields and occupations and examples of subfields. Additional information on STEM occupations is provided in appendix I. Many of the STEM fields require completion of advanced courses in mathematics or science, subjects that are introduced and developed at the kindergarten through 12th grade level, and the federal government has taken steps to help improve achievement in these and other subjects. Enacted in 2002, the No Child Left Behind Act (NCLBA) seeks to improve the academic achievement of all of the nation’s school-aged children. NCLBA requires that states develop and implement academic content and achievement standards in mathematics, science and the reading or language arts. All students are required to participate in statewide assessments during their elementary and secondary school years. Improving teacher quality is another goal of NCLBA as a strategy to raise student academic achievement. Specifically, all teachers teaching core academic subjects must be highly qualified by the end of the 20052006 school year [2]. NCLBA generally defines highly qualified teachers as those that have (1) a bachelor’s degree, (2) state certification, and (3) subject area knowledge for each academic subject they teach. The federal government also plays a role in coordinating federal science and technology issues. The National Science and Technology Council (NSTC) was established in 1993 and is the principal means for the Administration to coordinate science and technology among the diverse parts of the federal research and development areas. One objective of NSTC is to establish clear national goals for federal science and technology investments in areas ranging from information technologies and health research to improving transportation systems and strengthening fundamental research. NSTC is responsible for preparing research and development strategies that are coordinated across federal agencies in order to accomplish these multiple national goals. In addition, the federal government, universities and colleges, and others have developed programs to provide opportunities for all students to pursue STEM education and occupations [3]. Additional steps have been taken to increase the numbers of women, minorities, and students with disadvantaged backgrounds in the STEM fields, such as providing additional academic and research opportunities. According to the 2000 Census, 52 percent of the total U.S. population 18 and over were women; in 2003, members of racial or ethnic groups constituted from 0.5 percent to 12.6 percent of the civilian labor force (CLF), as shown in table 3. In addition to domestic students, international students have pursued STEM degrees and worked in STEM occupations in the United States. To do so, international students and scholars must obtain visas [4]. International students who wish to study in the United States must first apply to a Student and Exchange Visitor Information System (SEVIS) certified school. In order to enroll students from other nations, U.S. colleges and universities must be

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certified by the Student and Exchange Visitor Program within the Department of Homeland Security’s Immigration and Customs Enforcement organization. Table 2. List of STEM Fields Based on NCES’s NPSAS and IPEDS Data and BLS’s CPS Data Enrollment–NCES’ NPSAS data Agricultural sciences

Biological sciences

Degrees–NCES’IP EDS data

Occupations–BLS’ CPS data

Biological/agricultural sciences Botany Zoology Dairy Forestry Poultry Wildlife management

Science Agricultural and food scientists Astronomers and physicists Atmospheric and space scientists Biological scientists Chemists and materials scientists Environmental scientists and geoscientists Nurses Psychologists Sociologists Urban and regional planners

Earth, atmospheric, and ocean sciences Geology Geophysics and seismology Physical sciences

Psychology

P sychol

Social sciences

Technology

T echnol

E ngineerin

Computer sciences Mathematics

ogy Clinical Social Social sciences Political science Sociology

Enrollment–NCES’ NPSAS data Engineering

Physical sciences Chemistry Physics

ogy Solar Automotive engineering

Degrees–NCES’IP EDS data

g Aerospace, aeronautical, and astronautical Architectural Chemical Civil Electrical, electronics, and communication Nuclear Mathematics/computer sciences Actuarial science Applied mathematics Mathematical statistics Operations research Data processing Programming

Technology Clinical laboratory technologists and technicians Diagnostic-related technologists and technicians Medical, dental, and ophthalmic laboratory technicians

Occupations–BLS’ CPS data Engineering Architects, except naval Aerospace engineers Chemical engineers Civil engineers Electrical and electronic engineers Nuclear engineers

Mathematics and computer sciences Computer scientists and systems analysts Computer programmers Computer software engineers Actuaries Mathematicians Statisticians

Sources: NCES for NPSAS and IPEDS data; CPS for occupations. Note: This table is not designed to show a direct relationship from enrollment to occupation, but to provide examples of majors, degrees, and occupations in STEM fields from the three sources of data.

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Table 3. Percentage of the U.S. Population for Selected Racial or Ethnic Groups in the Civilian Labor Force, Calendar Years 1994 and 2003 Percentage of U.S. population Percentage of U.S. population in the CLF, 1994 in the CLF, 2003 Race or ethnicity Hispanic or Latino origin Black or African American Asian American Indian or Alaska Native

8.9 10.8 2.8 0.5

12.6 10.7 4.4 0.5

Source: GAO calculations based upon March 1994 and March2003 CPS data.

As of February 2004, nearly 9,000 technical schools and colleges and universities had been certified. SEVIS, is an Internet-based system that maintains data on international students and exchange visitors before and during their stay in the United States. Upon admitting a student, the school enters the student’s name and other information into the SEVIS database. At this time the student may apply for a student visa. In some cases, a Security Advisory Opinion (SAO) from the Department of State (State) may be needed to determine whether or not to issue a visa to the student. SAOs are required for a number of reasons, including concerns that a visa applicant may engage in the illegal transfer of sensitive technology. An SAO based on technology transfer concerns is known as Visas Mantis and, according to State officials, is the most common type of SAO applied to science applicants [5]. In April 2004, the Congressional Research Service reported that State maintains a technology alert list that includes 16 sensitive areas of study. The list was produced in an effort to help the United States prevent the illegal transfer of controlled technology and includes chemical and biotechnology engineering, missile technology, nuclear technology, robotics, and advanced computer technology [6]. Many foreign workers enter the United States annually through the H-1B visa program, which assists U.S. employers in temporarily filling specialty occupations [7]. Employed workers may stay in the United States on an H-1B visa for up to 6 years. The current cap on the number of H-1B visas that can be granted is 65,000. The law exempts certain workers, however, from this cap, including those who are employed or have accepted employment in specified positions. Moreover, up to 20,000 exemptions are allowed for those holding a master’s degree or higher.

More than 200 Federal Education Programs are Designed to Increase the Numbers of Students and Graduates or Improve Educational Programs in STEM Fields, but Most Have Not Been Evaluated Officials from 13 federal civilian agencies reported having 207 education programs funded in fiscal year 2004 that were specifically established to increase the numbers of students and graduates pursuing STEM degrees and occupations, or improve educational programs in STEM fields, but they reported little about the effectiveness of these programs [8]. These 13 federal agencies reported spending about $2.8 billion for their STEM education

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programs. Taken together, NIH and NSF sponsored nearly half of the programs and spent about 71 percent of the funds. In addition, agencies reported that most of the programs had multiple goals, and many were targeted to multiple groups. Although evaluations have been done or were under way for about half of the programs, little is known about the extent to which most STEM programs are achieving their desired results. Coordination among the federal STEM education programs has been limited. However, in 2003, the National Science and Technology Council formed a subcommittee to address STEM education and workforce policy issues across federal agencies.

Federal Civilian Agencies Reported Sponsoring over 200 STEM Education Programs and Spending Billions in Fiscal Year 2004 Officials from 13 federal civilian agencies provided information on 207 STEM education programs funded in fiscal year 2004. The numbers of programs ranged from 51 to 1 per agency with two agencies, NIH and NSF, sponsoring nearly half of the programs—99 of 207. Table 4 provides a summary of the numbers of programs by agency, and appendix II contains a list of the 207 STEM education programs and funding levels for fiscal year 2004 by agency. Table 4. Number of STEM Education Programs Reported by Federal Civilian Agencies Federal agency Department of Health and Human Services/ National Institutes of Health National Science Foundation Department of Energy Environmental Protection Agency Department of Agriculture Department of Commerce Department of the Interior National Aeronautics and Space Administration Department of Education Department of Transportation Department of Health and Human Services/Health Resources and Services Administration Department of Health and Human Services/Indian Health Service Department of Homeland Security Total

Source: GAO survey responses from 13 federal agencies.

Number of STEM education programs 51 48 26 21 16 13 13 5 4 4 3 2 1 207

Federal civilian agencies reported that approximately $2.8 billion was spent on STEM education programs in fiscal year 2004 [9]. The funding levels for STEM education programs among the agencies ranged from about $998 million to about $4.7 million. NIH and NSF accounted for about 71 percent of the total—about $2 billion of the approximate $2.8 billion. NIH spent about $998 million in fiscal year 2004, about 3.6 percent of its $28 billion

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appropriation, and NSF spent about $997 million, which represented 18 percent of its appropriation. Four other agencies, some with a few programs, spent about 23 percent of the total: $636 million. For example, the National Aeronautics and Space Administration (NASA) spent about $231 million on 5 programs and the Department of Education (Education) spent about $221 million on 4 programs during fiscal year 2004. Figure 1 shows the 6 federal civilian agencies that used the most funds for STEM education programs and the funds used by the remaining 7 agencies. Dollars in millions 1,200 998

1,000

997

800 600 400 231

221

200

154

121 63

0

s s h e n n n y d n d on er t e al t io io c nc io an atio an ati at th ct gen ie dat itu He s r r s c e t t c t t e u ic s llO ns of rc nis ro A lS un ut ini A Ed ou mi na F o lP alI na dm s a o n o d i t e io er A at R sA en A e at N m N lth ice al pac a ro n i e v v io S H er at En S N

Figure 1. Amounts Funded by Agencies for STEM-Related Federal Education Programs in Fiscal Year 2004.

Table 5. Funding Levels for Federal STEM Education Programs in Fiscal Year 2004

Program funding levels Less than $1 million $1 million to $5 million $5.1 million to $10 million $10.1 million to $50 million More than $50 million Total

Numbers of STEM education programs 93 51 19 31 13 207

Source: GAO survey responses from 13 federal agencies.

Percentage of total STEM education programs 45 25 9 15 6 100

The funding reported for individual STEM education programs varied significantly, and many of the programs have been funded for more than 10 years. The funding ranged from $4,000 for an USDA-sponsored program that offered scholarships to U.S. citizens seeking bachelor’s degrees at Hispanic-serving institutions, to about $547 million for a NIH grant program that is designed to develop and enhance research training opportunities for individuals in biomedical, behavioral, and clinical research by supporting training programs at institutions of higher education. As shown in table 5, most programs were funded at $5

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million or less and 13 programs were funded at more than $50 million in fiscal year 2004. About half of the STEM education programs were first funded after 1998. The oldest program began in 1936, and 72 programs are over 10 years old [10]. Appendix III describes the STEM education programs that received funding of $10 million or more during fiscal year 2004 or 2005 [11].

Federal Agencies Reported Most STEM Programs Had Multiple Goals and Were Targeted to Multiple Groups Agencies reported that most of the STEM education programs had multiple goals. Survey respondents reported that 80 percent (165 of 207) of the education programs had multiple goals, with about half of these identifying four or more goals for individual programs [12]. Moreover, according to the survey responders, few programs had a single goal. For example, 2 programs were identified as having one goal of attracting and preparing students at any education level to pursue coursework in the STEM areas, while 112 programs identified this as one of multiple goals. Table 6 shows the program goals and numbers of STEM programs aligned with them. Table 6. Program Goals and Numbers of STEM Programs with One or Multiple Goals Program goal

Programs with only this goal

Attract and prepare students at any education level to pursue coursework in STEM areas Attract students to pursue degrees (2-year through Ph.D.) and postdoctoral appointments Provide growth and research opportunities for college and graduate students in STEM fields Attract graduates to pursue careers in STEM fields Improve teacher education in STEM areas Improve or expand the capacity of institutions to promote or foster STEM fields Source: GAO survey responses from 13 federal agencies

2

Programs with Total programs multiple goals with this goal including this goal and other goal(s) 112 114

6

131

137

3

100

103

17 8 3

114 65 87

131 73 90

The STEM education programs provided financial assistance to students, educators, and institutions. According to the survey responses, 131 programs provided financial support for students or scholars, and 84 programs provided assistance for teacher and faculty development [13]. Many of the programs provided financial assistance to multiple beneficiaries, as shown in table 7. Most of the programs were not targeted to a specific group but aimed to serve a wide range of students, educators, and institutions. Of the 207 programs, 54 were targeted to 1 group and 151 had multiple target groups [14] In addition, many programs were targeted to the same group. For example, while 12 programs were aimed solely at graduate students, 88 other programs had graduate students as one of multiple target groups. Fewer programs were targeted to elementary and secondary teachers and kindergarten through 12th grade students

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than to other target groups. Table 8 summarizes the numbers of STEM programs targeted to one group and multiple groups. Table 7. Numbers of STEM Programs with One or Multiple Types of Assistance and Beneficiaries Type of assistance

Programs that Programs that provide Total programs provide only this this type and other that provide this type of assistance types of assistance type of assistance 54 77 131

Financial support for students or scholars Institutional support to improve educational quality 6 Support for teacher and faculty development 12 Institutional physical infrastructure support 1 Source: GAO survey responses from 13 federal agencies

70 72 26

76 84 27

Table 8. Numbers of STEM Programs Targeted to One Group and Multiple Groups Targeted to Targeted to this Total programs Targeted group only this group and other groups targeted to this group Kindergarten through grade 12 students Elementary school students 0 28 28 Middle or junior high school students 1 33 34 High school students 3 50 53 Undergraduate students 2-year college students 1 57 58 4-year college students 4 92 96 Graduate students and postdoctoral scholars Graduate students 12 88 100 Postdoctoral scholars 12 58 70 Teachers, college faculty and instructional staff Elementary school teachers 0 39 39 Secondary school teachers 3 47 50 College faculty or instructional staff 4 75 79 Institutions 5 77 82 Source: GAO survey responses from13 federal agencies.

Some programs limited participation to certain groups. According to survey respondents, U.S. citizenship was required to be eligible for 53 programs, and an additional 75 programs were open only to U.S. citizens or permanent residents [15]. About one-fourth of the programs had no citizenship requirement, and 24 programs allowed noncitizens or permanent residents to participate in some cases. According to a NSF official, students receiving scholarships or fellowships through NSF programs must be U.S. citizens or permanent residents. In commenting on a draft of this report, NSF reported that these restrictions are considered to be an effective strategy to support its goal of creating a diverse, competitive, and globally-engaged U.S. workforce of scientists, engineers, technologists, and wellprepared citizens. Officials at two universities said that some research programs are not open to non-citizens. Such restrictions may reflect concerns about access to sensitive areas. In addition to these restrictions, some programs are designed to increase minority representation in STEM fields. For example, NSF sponsors a program called Opportunities for Enhancing

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Diversity in the Geosciences to increase participation by African Americans, Hispanic Americans, Native Americans (American Indians and Alaskan Natives), Native Pacific Islanders (Polynesians or Micronesians), and persons with disabilities.

Agency Officials Reported That Evaluations Were Completed or under Way for about Half of the Federal Programs Evaluations had been completed or were under way for about half of the STEM education programs. Agency officials responded that evaluations were completed for 55 of the 207 programs and that for 49 programs, evaluations were under way at the time we conducted our survey. Agency officials provided us documentation for evaluations of 43 programs, and most of the completed evaluations reviewed reported that the programs met their objectives or goals. For example, a March 2004 report on the outcomes and impacts of NSF’s Minority Postdoctoral Research Fellowships program concluded that there was strong qualitative and quantitative evidence that this program is meeting its broad goal of preparing scientists from those ethnic groups that are significantly underrepresented in tenured U.S. science and engineering professorships and for positions of leadership in industry and government. However, evaluations had not been done for 103 programs, some of which have been operating for many years. Of these, it may have been too soon to expect evaluations for about 32 programs that were initially funded in fiscal year 2002 or later. However, of the remaining 71 programs, 17 have been operating for over 15 years and have not been evaluated. In commenting on a draft of this report NSF noted that all of its programs undergo evaluation and that it uses a variety of mechanisms for program evaluation. We reported in 2003 that several agencies used various strategies to develop and improve evaluations [16]. Evaluations play an important role in improving program operations and ensuring an efficient use of federal resources. Although some of the STEM education programs are small in terms of their funding levels, evaluations can be designed to consider the size of the program and the costs associated with measuring outcomes and collecting data.

A Subcommittee Was Established in 2003 to Help Coordinate STEM Education Programs among Federal Agencies Coordination of federal STEM education programs has been limited. In January 2003 the National Science and Technology Council (NSTC), Committee on Science (COS), established a subcommittee on education and workforce development. The purpose of the subcommittee is to advise and assist COS and NSTC on policies, procedures, and programs relating to STEM education and workforce development. According to its charter, the subcommittee will address education and workforce policy issues and research and development efforts that focus on STEM education issues at all levels, as well as current and projected STEM workforce needs, trends, and issues. The members include representatives from 20 agencies and offices—the 13 agencies that responded to our survey as well as the Departments of Defense, State, and Justice, and the Office of Science and Technology Policy, the Office of Management and Budget, the Domestic Policy Council, and the National Economic Council. The subcommittee has working groups on (1) human capacity in STEM

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areas, (2) minority programs, (3) effective practices for assessing federal efforts, and (4) issues affecting graduate and postdoctoral researchers. The Human Capacity in STEM working group is focused on three strategic initiatives: defining and assessing national STEM needs, including programs and research projects; identifying and analyzing the available data regarding the STEM workforce; and creating and implementing a comprehensive national response that enhances STEM workforce development. NSTC reported that as of June 2005 the subcommittee had a number of accomplishments and projects under way that related to attracting students to STEM fields. For example, it has (1) surveyed federal agency education programs designed to increase the participation of women and underrepresented minorities in STEM studies; (2) inventoried federal fellowship programs for graduate students and postdoctoral fellows; and (3) coordinated the Excellence in Science, Technology, Engineering, and Mathematics Education Week activities, which provide an opportunity for the nation’s schools to focus on improving mathematics and science education. In addition, the subcommittee is developing a Web site for federal educational resources in STEM fields and a set of principles that agencies would use in setting levels of support for graduate and postdoctoral fellowships and traineeships.

Numbers of Students, Graduates, and Employees in STEM Fields Generally Increased, but Percentage Changes Varied While the total numbers of students, graduates, and employees have increased in STEM fields, percentage changes for women, minorities, and international students varied during the periods reviewed.

Students

Graduates

Employees

UNIVERSITY

1995-1996 to 2003-2004

1994-1995 to 2002-2003 Total increase in STEM was less than non-STEM

1994 - 2003

Percentage increase was greater in STEM than non-STEM

Decrease at doctoral level in most fields

Increase was greater in STEM than non-STEM

Increase mostly at bachelor’s and master’s level

Increase in percentages of women in most fields

No significant change in percentage of women

Increase in percentage of women

No change in percentages of minorities at master’s or doctoral levels

African Americans continued to be less than 10 percent of the total

International graduates continued to earn about one-third or more of master’s and Ph.D.s in three fields

Median annual wages and salaries increased in all fields

Increase in minority students but percentage changes varied by race/ethnicity Increase in international students at bachelor’s level

Source: GAO analysis of CPS, IPEDS, and NPSAS data; graphics in part by Art Explosion.

Figure 2. Key Changes in Students, Graduates, and Employees in STEM Fields.

The increase in the percentage of students in STEM fields was greater than the increase in non-STEM fields, but the change in percentage of graduates in STEM fields was less than the percentage change in non-STEM fields. Moreover, employment increased more in STEM fields than in non-STEM fields. Further, changes in the percentages of minority students

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varied by race or ethnic group, international graduates continued to earn about a third or more of the advanced degrees in three STEM fields, and there was no statistically significant change in the percentage of women employees. Figure 2 summarizes key changes in the students, graduates, and employees in STEM fields.

Numbers of Students in STEM Fields Grew, but This Increase Varied by Education Level and Student Characteristics Total enrollments of students in STEM fields have increased, and the percentage change was greater for STEM fields than non-STEM fields, but the percentage of students in STEM fields remained about the same. From the 1995-1996 academic year to the 2003-2004 academic year, total enrollments in STEM fields increased 21 percent—more than the 11 percent enrollment increase in non-STEM fields. The number of students enrolled in STEM fields represented 23 percent of all students enrolled during the 2003-2004 academic year, a modest increase from the 21 percent these students constituted in the 1995-1996 academic year. Table 9 summarizes the changes in overall enrollment across all education levels from the 19951996 academic year to the 2003-2004 academic year. The increase in the numbers of students in STEM fields is mostly a result of increases at the bachelor’s and master’s levels. Of the total increase of about 865,000 students in STEM fields, about 740,000 was due to the increase in the numbers of students at the bachelor’s and master’s levels. See table 23 in appendix IV for additional information on the estimated numbers of students in STEM fields in academic years 1995-1996 and 2003-2004. Table 9. Estimated Changes in the Numbers and Percentages of Students in the STEM and Non-STEM Fields across All Education Levels, Academic Years 1995-1996 and 2003-2004 Academic year1995-1996 STEM Non- STEM

Academic year 2003-2004 STEM Non- STEM

Enrollment measures Students enrolled (in thousands) 4,132 15,243 4,997 16,883 Percentage of total enrollment 21 79 23 77 Source: GAO calculations based upon NPSAS data. Note: The totals for STEM and non-STEM enrollment include students in bachelor’s, master’s, and doctoral programs as well as students enrolled in certificate, associate’s, other undergraduate, first- professional degree, and post-bachelor’s or post-master’s certificate programs. The percentage changes between the 1995-1996 and 2003-2004 academic years for STEM and non-STEM students are statistically significant. See appendix V for confidence intervals associated with these estimates.

The percentage of students in STEM fields who are women increased from the 19951996 academic year to the 2003-2004 academic year, and in the 2003-2004 academic year women students constituted at least 50 percent of the students in 3 STEM fields—biological sciences, psychology, and social sciences. However, in the 2003-2004 academic year, men students continued to outnumber women students in STEM fields, and men constituted an estimated 54 percent of the STEM students overall. In addition, men constituted at least 76 percent of the students enrolled in computer sciences, engineering, and technology [17]. See tables 24 and 25 in appendix IV for additional information on changes in the numbers and

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percentages of women students in the STEM fields for academic years 1995-1996 and 20032004. While the numbers of domestic minority students in STEM fields also increased, changes in the percentages of minority students varied by racial or ethnic group. For example, Hispanic students increased 33 percent, from the 1995-1996 academic year to the 2003-2004 academic year. In comparison, the number of African American students increased about 69 percent. African American students increased from 9 to 12 percent of all students in STEM fields while Asian/Pacific Islander students continued to constitute about 7 percent. Table 10 shows the numbers and percentages of minority students in STEM fields for the 1995-1996 academic year and the 2003-2004 academic year. Table 10. Estimated Percentage Changes in the Numbers and Percentages of Domestic Minority Students in STEM fields for All Education Levels for Academic Years 19951996 and 2003-2004

Source: GAO calculations based upon NPSAS data. Note: All percentage changes are statistically significant. See appendix V for confidence intervals associated with these estimates.

From the 1995-1996 academic year to the 2003-2004 academic year, the number of international students in STEM fields increased by about 57 percent solely because of an increase at the bachelor’s level. The numbers of international students in STEM fields at the master’s and doctoral levels declined, with the largest decline occurring at the doctoral level. Table 11 shows the numbers and percentage changes in international students from the 19951996 academic year to the 2003-2004 academic year. Table 11. Estimated Changes in Numbers of International Students in STEM fields by Education Levels from the 1995-1996 Academic Year to the 2003-2004 Academic Year Percentage Number of international Number of international students, 1995-1996 students, 2003-2004 change Bachelor’s 31,858 139,875 +339 Master’s 40,025 22,384 -44 Doctoral 36,461 7,582 -79 Total 108,344 169,841 +57 Source: GAO calculations based upon NPSAS data. Note: Changes in enrollment between the 1995-1996 and 2003-2004 academic years are significant at the 95 percent confidence level for international students and for all education levels. See appendix V for confidence intervals associated with these estimates. Education level

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According to the Institute of International Education, from the 2002-2003 academic year to the 2003-2004 academic year, the number of international students declined for the first time in over 30 years, and that was the second such decline since the 1954-1955 academic year, when the institute began collecting and reporting data on international students [18]. Moreover, in November 2004, the Council of Graduate Schools (CGS) reported a 6 percent decline in first-time international graduate student enrollment from 2003 to 2004. Following a decade of steady growth, CGS also reported that the number of first-time international students studying in the United States decreased between 6 percent and 10 percent for 3 consecutive years.

Total Numbers of Graduates with STEM Degrees Increased, but Numbers Decreased in Some Fields, and Percentages of Minority Graduates at the Master’s and Doctoral Levels Did Not Change The number of graduates with degrees in STEM fields increased by 8 percent from the 19941995 academic year to the 2002-2003 academic year. However, during this same period the number of graduates with degrees in non-STEM fields increased by 30 percent. From academic year 1994-1995 to academic year 2002-2003, the percentage of graduates with STEM degrees decreased from 32 percent to 28 percent of total graduates. Table 12 provides data on the changes in the numbers and percentages of graduates in STEM and non-STEM fields. Table 12. Numbers of Graduates and Percentage Changes in STEM and Non-STEM Fields across All Degree Levels from the1994-1995 Academic Year to the 2002-2003 Academic Year STEM fields

Graduation measures

Non-STEM fields Percentage Percentage 1994-1995 2002-2003 1994-1995 2002-2003 change change 519 560 +8 1,112 1,444 +30 32 28 -4 68 72 +4

Graduates (in thousands) Percentage of total graduates Source: GAO calculations based upon IPEDS data.

Decreases in the numbers of graduates occurred in some STEM fields at each education level, but particularly at the doctoral level. The numbers of graduates with bachelor’s degrees decreased in four of eight STEM fields, the numbers with master’s degrees decreased in five of eight fields, and the numbers with doctoral degrees decreased in six of eight STEM fields. At the doctoral level, these declines ranged from 14 percent in mathematics/computer sciences to 74 percent in technology. Figure 3 shows the percentage change in graduates with degrees in STEM fields from the 1994-1995 academic year to the 2002-2003 academic year.

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Percent change 90 72

70 48

50

25

30 14

10

12

11

6

0

-10 -14

-18

-6

-8

-30

-11

-14 -15

-14

-18

1

-29 -35

-41

-50

-44 -53

-70

Te ch n

olo

gy

s sc ial So c

Ps y

ch o

ien

ce

log y

s cie n ls sic a Ph y

cs /co ma ti Ma t he

ce

mp sc uter ien ce s

g rin ine e En g

Ea r a n t h, a d o tm ce o s an ph sc eric ien , ce s

Bi olo

gic

al/ ag ri

cu l sc tura ien l ce s

-74

Bachelor’s Master’s Ph.D.s

Figure 3. Percentage Changes in Bachelor’s, Master’s, and Doctoral Graduates in STEM Fields from Academic Year 1994-1995 to Academic Year 2002-2003.

From the 1994-1995 academic year to the 2002-2003 academic year, the total number of women graduates increased in four of the eight fields, and the percentages of women earning degrees in STEM fields increased in six of the eight fields at all three educational levels. Conversely, the total number of men graduates decreased, and the percentages of men graduates declined in six of the eight fields at all three levels from the 1994-1995 academic year to the 2002-2003 academic year. However, men continued to constitute over 50 percent of the graduates in five of eight fields at all three education levels. Table 13 summarizes the numbers of graduates by gender, level, and field. Table 26 in appendix IV provides additional data on the percentages of men and women graduates by STEM field and education level. The total numbers of domestic minority graduates in STEM fields increased, although the percentage of minority graduates with STEM degrees at the master’s or doctoral level did not change from the 1994-1995 academic year to the 2002-2003 academic year. For example, while the number of Native American graduates increased 37 percent, Native American graduates remained less than 1 percent of all STEM graduates at the master’s and doctoral levels. Table 14 shows the percentages and numbers of domestic minority graduates for the 1994-1995 academic year and the 2002-2003 academic year. International students earned about one-third or more of the degrees at both the master’s and doctoral levels in several fields in the 1994-1995 and the 2002-2003 academic years. For example, in academic year 2002-2003, international students earned between 45 percent and 57 percent of all degrees in engineering and mathematics/computer sciences at the master’s and doctoral levels. However, at each level there were changes in the numbers and percentages of international graduates. At the master’s level, the total number of international graduates increased by about 31 percent from the 1994-1995 academic year to the 2002-2003 academic year; while the number of graduates decreased in four of the fields and the percentages of international graduates declined in three fields. At the doctoral level, the total

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number of international graduates decreased by 12 percent, while the percentage of international graduates increased or remained the same in all fields. Table 15 shows the numbers and percentages of international graduates in STEM fields. Table 13. Numbers and Percentage Changes in Men and Women Graduates with STEM Degrees by Education Level and Field for Academic Years 1994-1995 and 2002-2003 Education level

STEM field

Number of men graduates

1994-1995 2002-2003 Bachelor’s Biological/agricu 36,108 23,266 level ltural sciences Earth, 2,954 2,243 atmospheric, and ocean sciences Engineering 52,562 48,214 Mathematics and 25,258 46,381 computer sciences Physical sciences 9,607 8,739 Psychology 19,664 18,616 Social sciences 56,643 63,465 Technology 14,349 9,174 Master’s Biological/agricu 4,768 2,413 level ltural sciences Earth, 1,032 805 atmospheric, and ocean sciences Engineering 24,031 20,258 Mathematics and 10,398 14,531 computer sciences Physical sciences 2,958 2,350 Psychology 4,013 3,645 Social sciences 11,952 11,057 Technology 927 467 Doctoral Biological/agricu 3,616 1,526 level ltural sciences Earth, 488 315 atmospheric, and ocean sciences Engineering 5,401 4,159 Mathematics and 1,690 1,378 computer sciences Physical sciences 2,939 2,396 Psychology 1,529 1,380 Social sciences 2,347 2,111 Technology 24 7 Source: GAO calculations based upon IPEDS data.

Percentage change in men graduates

Number of women graduates

-36

1994-1995 2002-2003 35,648 35,546

Percentage change in women graduates

-24

1,524

1,626

+7

-8 +84

10,960 13,651

11,709 20,436

+7 +50

-9 -5 +12 -36 -49

5,292 53,010 56,624 1,602 4,340

6,222 64,470 77,701 1,257 2,934

+18 +22 +37 -22 -32

-22

451

552

+22

-16 +40

4,643 4,474

5,271 7,517

+14 +68

-21 -9 -7 -50 -58

1,283 10,319 11,398 222 2,160

1,299 12,433 13,674 173 1,161

+1 +20 +20 -22 -46

-35

134

125

-7

-23 -18

728 434

839 439

+15 +1

-18 -10 -10 -71

922 2,511 1,463 3

892 3,086 1,729 0

-3 +23 +18 -100

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Table 14. Numbers and Percentage Changes in Domestic Minority Graduates in STEM Fields by Education Levels and Race or Ethnicity for Academic Years 1994-1995 and 2002-2003

Source: GAO calculations based upon IPEDS data.

Table 15. Changes in Numbers and Percentages of International Graduates in STEM fields at the Master’s and Doctoral Degree Levels, 1994-1995 and 2002-2003 Academic Years

Source: GAO calculations based upon IPEDS data.

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STEM Employment Rose, but the Percentage of Women Remained About the Same and Minorities Continued to be Underrepresented While the total number of STEM employees increased, this increase varied across STEM fields. Employment increased by 23 percent in STEM fields as compared to 17 percent in non-STEM fields from calendar year 1994 to calendar year 2003. Employment increased by 78 percent in the mathematics/computer sciences field and by 20 percent in the science field over this period. The changes in number of employees in the engineering and technology fields were not statistically significant. Employment estimates from 1994 to 2003 in the STEM fields are shown in figure 4. Number of employees (in millions) 3.5

3.0

2.5

2.0

1.5

1.0

0.5

0 1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

Calendar year Science Technology Engineering Mathematics/computer sciences

Figure 4. Estimated Numbers of Employees in STEM Fields from Calendar Years 1994 through 2003.

From calendar years 1994 to 2003, the estimated number of women employees in STEM fields increased from about 2.7 million to about 3.5 million. Overall, there was not a statistically significant change in the percentage of women employees in the STEM fields. Table 16 shows the numbers and percentages of men and women employed in the STEM fields for calendar years 1994 and 2003. In addition, the estimated number of minorities employed in the STEM fields as well as the percentage of total STEM employees they constituted increased, but African American and Hispanic employees remain underrepresented relative to their percentages in the civilian labor force [19]. Between 1994 and 2003, the estimated number of African American employees increased by about 44 percent, the estimated numbers of Hispanic employees increased by 90 percent, as did the estimated numbers of other minorities employed in STEM fields [20]. In calendar year 2003, African Americans comprised about 8.7 percent of STEM employees compared to about 10.7 percent of the CLF. Similarly, Hispanic employees

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comprised about 10 percent of STEM employees in calendar year 2003, compared to about 12.6 percent of the CLF. Table 17 shows the estimated percentages of STEM employees by selected racial or ethnic groups in 1994 and 2003. Table 16. Estimated Numbers and Percentages of Employees in STEM Fields by Gender in Calendar Years 1994 and2003(numbers in thousands) STEM field

1994 2003 Men Women Men Women Number Percent Number Percent Number Percent Number Percent 792 32 1,711 68 829 28 2,179 72 955 68 445 32 1,050 71 425 29 1,658 92 *141 8 1,572 90 *169 10

Science Technology Engineering Mathematics/ computer sciences 1,056 71 432 29 1,952 74 695 26 Total 4,461 62 2,729 38 5,404 61 3,467 39 Source: GAO calculations based upon CPS data. Note: Estimated employee numbers noted by an asterisk have a 95 percent confidence interval of within +/25 percent of the estimate itself. All other estimated employee numbers have a 95 percent confidence interval of within +/- 16 percent of the estimate. See appendix VI for confidence intervals associated with these estimates. Calculations of percentages and numbers may differ due to rounding.

International employees have filled hundreds of thousands of positions, many in STEM fields, through the H-1B visa program. However, the numbers and types of occupations have changed over the years. We reported that while the limit for the H-1B program was 115,000 in 1999, the number of visas approved exceeded the limit by more than 20,000 because of problems with the system used to track the data [21]. Available data show that in 1999, the majority of the approved occupations were in STEM fields. Specifically, an estimated 60 percent of the positions approved in fiscal year 1999 were related to information technology and 5 percent were for electrical/electronics engineering. By 2002, the limit for the H-1B program had increased to 195,000, but the number approved, 79,000, did not reach this limit. In 2003, we reported that the number of approved H-1B petitions in certain occupations had declined. For example, the number of approvals for systems analysis/programming positions declined by 106,671 from 2001 to 2002 [22]. Table 17. Estimated Percentages of STEM Employees by Selected Racial or Ethnic Group for Calendar Years 1994 and 2003

Race or ethnicity Black or African American Hispanic or Latino origin Other minoritiesa

Percentage of total STEM employees, 1994 7.5 5.7 4.5

Percentage of total STEM employees, 2003 8.7 10.0 6.9

Source: GAO calculations based upon CPS data. Note: Estimated percentages have 95 percent confidence intervals of +/- 1 percentage point. Changes for African Americans between calendar years 1994 and 2003 were not statistically significant at the 95percent confidence level. Differences for Hispanic or Latino origin and other minorities were statistically significant. See appendix VI for confidence intervals associated with these estimates. aOther minorities include Asian/Pacific Islanders and American Indian or Alaska Native.

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Although the estimated total number of employees in STEM fields increased from 1994 to 2003, according to an NSF report, many with STEM degrees were not employed in these occupations. In 2004, NSF reported that about 67 percent of employees with degrees in science or engineering were employed in fields somewhat or not at all related to their degree [23]. Specifically, 70 percent of employees with bachelor’s degrees, 51 percent with master’s degrees, and 54 percent with doctoral degrees reported that their employment was somewhat or not at all related to their degree in science or engineering. In addition to increases in the numbers of employees in STEM fields, inflation-adjusted median annual wages and salaries increased in all four STEM fields over the 10-year period (1994 to 2003). These increases ranged from 6 percent in science to 15 percent in engineering. Figure 5 shows trends in median annual wages and salaries for STEM fields.

University Officials and Others Cited Several Factors That Influence Decisions about Participation in STEM Fields and Suggested Ways to Encourage Greater Participation University officials, researchers, and students identified several factors that influenced students’ decisions about pursuing STEM degrees and occupations, and they suggested some ways to encourage more participation in STEM fields. Specifically, university officials said and researchers reported that the quality of teachers in kindergarten through 12th grades and the levels of mathematics and science courses completed during high school affected students’ success in and decisions about STEM fields. In addition, several sources noted that mentoring played a key role in the participation of women and minorities in STEM fields. Annual wages and salaries (in thousands of dollars) 70

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Figure 5. Estimated Median Annual Wages and Salaries in STEM Fields for Calendar Years 1994 through 2003.

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Current students from five universities we visited generally agreed with these observations, and several said that having good mathematics and science instruction was important to their overall educational success. International students’ decisions about participating in STEM education and occupations were affected by opportunities outside the United States and the visa process. To encourage more student participation in the STEM fields, university officials, researchers, and others have made several suggestions, and four were made repeatedly. These suggestions focused on teacher quality, high school students’ math and science preparation, outreach activities, and the federal role in STEM education.

Teacher Quality and Mathematics and Science Preparation Were Cited as Key Factors Affecting Domestic Students’ STEM Participation Decisions University officials frequently cited teacher quality as a key factor that affected domestic students’ interest in and decisions about pursuing STEM degrees and occupations. Officials at all eight universities we visited expressed the view that a student’s experience from kindergarten through the 12th grades played a large role in influencing whether the student pursued a STEM degree. Officials at one university we visited said that students pursuing STEM degrees have associated their interests with teachers who taught them good skills in mathematics or excited them about science. On the other hand, officials at many of the universities we visited told us that some teachers were unqualified and unable to impart the subject matter, causing students to lose interest in mathematics and science. For example, officials at one university we visited said that some elementary and secondary teachers do not have sufficient training to effectively teach students in the STEM fields and that this has an adverse effect on what students learn in these fields and reduces the interest and enthusiasm students express in pursuing coursework in high school, degree programs in college, or careers in these areas. Teacher quality issues, in general, have been cited in past reports by Education. In 2002, Education reported that in the 1999-2000 school year, 14 to 22 percent of middle-grade students taking English, mathematics, and science were in classes led by teachers without a major, minor, or certification in these subjects—commonly referred to as “out-of-field” teachers [24]. Also, approximately 30 to 40 percent of the middle-grade students in biology/life science, physical science, or English as a second language/bilingual education classes had teachers lacking these credentials. At the high school level, 17 percent of students enrolled in physics and 36 percent of those enrolled in geology/earth/space science were in classes instructed by out-of-field teachers. The percentages of students taught by out-of-field teachers were significantly higher when the criteria used were teacher certification and a major in the subject taught. For example, 45 percent of the high school students enrolled in biology/life science and approximately 30 percent of those enrolled in mathematics, English, and social science classes had out-of-field teachers. During the 2002-2003 school year, Education reported that the number and distribution of teachers on waivers—which allowed prospective teachers in classrooms while they completed their formal training—was problematic. Also, states reported that the problem of underprepared teachers was worse on average in districts that serve large proportions of high-poverty children—the percentage of teachers on waivers was larger in high-poverty school districts than all other school districts

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in 39 states. Moreover, in 2004, Education reported that 48 of the 50 states granted waivers [25]. In addition to teacher quality, students’ high school preparation in mathematics and science was cited by university officials and others as affecting students’ success in collegelevel courses and their decisions about pursuing STEM degrees and occupations. University officials at six of the eight universities we visited cited students’ ability to opt out of mathematics and science courses during high school as a factor that influenced whether they would participate and succeed in the STEM fields during undergraduate and graduate school. University officials said, for example, that because many students had not taken higher-level mathematics and science courses such as calculus and physics in high school, they were immediately behind other students who were better prepared. In July 2005, on the basis of findings from the 2004 National Assessment of Educational Progress, the National Center for Education Statistics reported that 17 percent of the 17-year-olds reported that they had taken calculus, and this represents the highest percentage in any previous assessment year [26]. In a study that solicited the views of several hundred students who had left the STEM fields, researchers found that the effects of inadequate high school preparation contributed to college students’ decisions to leave the science fields [27]. These researchers found that approximately 40 percent of those college students who left the science fields reported some problems related to high school science preparation. The underpreparation was often linked to problems such as not understanding calculus; lack of laboratory experience or exposure to computers, and no introduction to theoretical material or to analytic modes of thought. Further, 12 current students we interviewed said they were not adequately prepared for college mathematics or science. For example, one student stated that her high school courses had been limited because she attended an all-girls school where the curriculum catered to students who were not interested in STEM, and so it had been difficult to obtain the courses that were of interest to her. Several other factors were mentioned during our interviews with university officials, students, and others as influencing decisions about participation in STEM fields. These factors included relatively low pay in STEM fields, additional tuition costs to obtain STEM degrees, lack of commitment on the part of some students to meet the rigorous academic demands, and the inability of some professors in STEM fields to effectively impart their knowledge to students in the classrooms. For example, officials from five universities said that low pay in STEM fields relative to other fields such as law and business dissuaded students from pursuing STEM degrees in some areas. Also, in a study that solicited the views of college students who left the STEM fields as well as those who continued to pursue STEM degrees, researchers found that students experienced greater financial difficulties in obtaining their degrees because of the extra time needed to obtain degrees in certain STEM fields. Researchers also noted that poor teaching at the university level was the most common complaint among students who left as well as those who remained in STEM fields. Students reported that faculty do not like to teach, do not value teaching as a professional activity, and therefore lack any incentive to learn to teach effectively [28]. Finally, 11 of the students we interviewed commented about the need for professors in STEM fields to alter their methods and to show more interest in teaching to retain students’ attention.

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Mentoring Cited as a Key Factor Affecting Women’s and Minorities’ STEM Participation Decisions University officials and students said that mentoring is important for all students but plays a vital role in the academic experiences of women and minorities in the STEM fields. Officials at seven of the eight universities discussed the important role that mentors play, especially for women and minorities in STEM fields. For example, one professor said that mentors helped students by advising them on the best track to follow for obtaining their degrees and achieving professional goals. Also, four students we interviewed—three women and one man—expressed the importance of mentors. Specifically, while all four students identified mentoring as critical to academic success in the STEM fields, two students expressed their satisfaction since they had mentors, while the other two students said that it would have been helpful to have had someone who could have been a mentor or role model. Studies have also reported that mentors play a significant role in the success of women and minorities in the STEM fields. In 2004, some of the women students and faculty with whom we talked reported a strong mentor was a crucial part in the academic training of some of the women participating in sciences, and some women had pursued advanced degrees because of the encouragement and support of mentors [29]. In September 2000, a congressional commission reported that women were adversely affected throughout the STEM education pipeline and career path by a lack of role models and mentors [30]. For example, the report found that girls rejection of mathematics and science may be partially driven by teachers, parents, and peers when they subtly, and not so subtly, steer girls away from the informal technical pastimes (such as working on cars, fixing bicycles, and changing hardware on computers) and science activities (such as science fairs and clubs) that too often were still thought of as the province of boys. In addition, the commission reported that a greater proportion of women switched out of STEM majors than men, relative to their representation in the STEM major population. Reasons cited for the higher attrition rate among women students included lack of role models, distaste for the competitive nature of science and engineering education, and inability to obtain adequate academic guidance or advice. Further, according to the report, women’s retention and graduation in STEM graduate programs were affected by their interaction with faculty, integration into the department (versus isolation), and other factors, including whether there were role models, mentors, and women faculty.

International Students’ STEM Participation Decisions Were Affected by Opportunities Outside the United States and the Visa Process Officials at seven of the eight universities visited, along with education policy experts, told us that competition from other countries for top international students, and educational or work opportunities, affected international students’ decisions about studying in the United States. They told us that other countries, including Canada, Australia, New Zealand, and the United Kingdom, had seized the opportunity since September 11 to compete against the United States for international students who were among the best students in the world, especially in the STEM fields. Also, university officials told us that students from several countries, including China and India, were being recruited to attend universities and get jobs in their

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own countries. In addition, education organizations and associations have reported that global competition for the best science and engineering students and scholars is under way. One organization, NAFSA: Association of International Educators reported that the international student market has become highly competitive, and the United States is not competing as well as other countries [31]. According to university officials, international students’ decisions about pursuing STEM degrees and occupations in the United States were also influenced by the perceived unwelcoming attitude of Americans and the visa process. Officials from three of the universities said that the perceived unwelcoming attitude of Americans had affected the recruitment of international students to the United States. Also, officials at six of the eight universities visited expressed their concern about the impact of the tightened visa procedures and/or increased security measures since September 11 on international graduate school enrollments. For example, officials at one university stated that because of the time needed to process visas, a few students had missed their class start dates. Officials from one university told us that they were being more proactive in helping new international students navigate the visa system, to the extent possible. While some university officials acknowledged that visa processing had significantly improved, since 2003 several education associations have requested further changes in U.S. visa policies because of the lengthy procedures and time needed to obtain approval to enter the country. We have reported on various aspects of the visa process, made several recommendations, and noted that some improvements have been made. In October 2002 we cited the need for a clear policy on how to balance national security concerns with the desire to facilitate legitimate travel when issuing visas and we made several recommendations to help improve the visa process [32]. In 2003, we reported that the Departments of State, Homeland Security, and Justice could more effectively manage the visa function if they had clear and comprehensive policies and procedures and increased agency coordination and information sharing [33]. In February 2004 and February 2005, we reported on the State Department’s efforts to improve the program for issuing visas to international science students and scholars. In 2004 we found that the time to adjudicate a visa depended largely on whether an applicant had to undergo a security check known as Visas Mantis, which is designed to protect against sensitive technology transfers. Based on a random sample of Visas Mantis cases for science students and scholars, it took State an average of 67 days to complete the process [34]. In 2005, we reported a significant decline in Visas Mantis processing times and in the number of cases pending more than 60 days [35]. We also reported that, in some cases, science students and scholars can obtain a visa within 24 hours. We have also issued several reports on SEVIS operations. In June 2004 we noted that when SEVIS began operating, significant problems were reported [36]. For example, colleges and universities and exchange programs had trouble gaining access to the system, and when access was obtained, these users’ sessions would “time out” before they could complete their tasks. In that report we also noted that SEVIS performance had improved, but that several key system performance requirements were not being measured. In March 2005, we reported that the Department of Homeland Security (DHS) had taken steps to address our recommendations and that educational organizations generally agreed that SEVIS performance had continued to improve [37]. However, educational organizations continued to cite problems, which they believe created hardships for students and exchange visitors.

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Several Suggestions Were Made to Encourage More Participation in the STEM Fields To increase the number of students entering STEM fields, officials from seven universities and others stated that teacher quality needs to improve. Officials of one university said that kindergarten through 12th grade classrooms need teachers who are knowledgeable in the mathematics and science content areas. As previously noted, Education has reported on the extent to which classes have been taught by teachers with little or no content knowledge in the STEM fields. The Congressional Commission on the Advancement of Women and Minorities reported that teacher effectiveness is the most important element in a good education [38]. The commission also suggested that boosting teacher effectiveness can do more to improve education than any other single factor. States are taking action to meet NCLBA’s requirement of having all teachers of core academic subjects be highly qualified by the end of the 2005-2006 school year. University officials and some students suggested that better preparation and mandatory courses in mathematics and science were needed for students during their kindergarten through 12th grade school years. Officials from five universities suggested that mandatory mathematics and science courses, especially in high school, may lead to increased student interest and preparation in the STEM fields. With a greater interest and depth of knowledge, students would be better prepared and more inclined to pursue STEM degrees in college. Further, nearly half of the students who replied to this question suggested that students needed additional mathematics and science training prior to college. However, adding mathematics and science classes has resource implications, since more teachers in these subjects would be needed. Also this change could require curriculum policy changes that would take time to implement. More outreach, especially to women and minorities from kindergarten through the 12th grade, was suggested by university officials, students, and other organizations. Officials from six of the universities we visited suggested that increased outreach activities are needed to help create more interest in mathematics and science for younger students. For example, at one university we visited, officials told us that through inviting students to their campuses or visiting local schools, they have provided some students with opportunities to engage in science laboratories and hands-on activities that foster interest and excitement for students and can make these fields more relevant in their lives. Officials from another university told us that these experiences were especially important for women and minorities who might not have otherwise had these opportunities. The current students we interviewed also suggested more outreach activities. Specifically, two students said that outreach was needed to further stimulate students’ interest in the STEM fields. One organization, Building Engineering and Science Talent (BEST), suggested that research universities increase their presence in prekindergarten through 12th grade mathematics and science education in order to strengthen domestic students’ interests and abilities. BEST reported that one model producing results entailed universities adopting students from low-income school districts from 7th through 12th grades and providing them advanced instruction in algebra, chemistry, physics, and trigonometry. However, officials at one university told us that because of limited resources, their efforts were constrained and only a few students would benefit from this type of outreach.

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Furthermore, university officials from the eight schools and other education organizations made suggestions regarding the role of the federal government. University officials suggested that the federal government could enhance its role in STEM education by providing more effective leadership through developing and implementing a national agenda for STEM education and increasing federal funding for academic research. Officials at six universities suggested that the federal government undertake a new initiative modeled after the National Defense Education Act of 1958, enacted in response to the former Soviet Union’s achievement in its space program, which provided new funding for mathematics and science education and training at all education levels. In June 2005, CGS called for a renewed commitment to graduate education by the federal government through actions such as providing funds to support students trained at the doctoral level in the sciences, technology, engineering, and mathematics; expanding U.S. citizen participation in doctoral study in selected fields through graduate support awarded competitively to universities across the country; requiring recruitment, outreach, and mentoring activities that promote greater participation and success, especially for underrepresented groups; and fostering interdisciplinary research preparation. In August 2003, the National Science Board recommended that the federal government direct substantial new support to students and institutions in order to improve success in science and engineering studies by domestic undergraduate students from all demographic groups. According to this report, such support could include scholarships and other forms of financial assistance to students, incentives to institutions to expand and improve the quality of their science and engineering programs in areas in which degree attainment is insufficient, financial support to community colleges to increase the success of students in transferring to 4-year science and engineering programs, and expanded funding for programs that best succeed in graduating underrepresented minorities and women in science and engineering. BEST also suggested that the federal government allocate additional resources to expand the mathematics and science education opportunities for underrepresented groups. However, little is known about how well federal resources have been used in the past. Changes that would require additional federal funds would likely have an impact on other federal programs, given the nation’s limited resources and growing fiscal imbalance, and changing the federal role could take several years.

Concluding Observations While the total numbers of STEM graduates have increased, some fields have experienced declines, especially at the master’s and doctoral levels. Given the trends in the numbers and percentages of students pursuing STEM degrees, particularly advanced degrees, and recent developments that have influenced international students’ decisions about pursuing degrees in the United States, it is uncertain whether the number of STEM graduates will be sufficient to meet future academic and employment needs and help the country maintain its technological competitive advantage. Moreover, it is too early to tell if the declines in international graduate student enrollments will continue in the future. In terms of employment, despite some gains, the percentage of women in the STEM workforce has not changed significantly, minority employees remain underrepresented, and many with degrees in STEM fields are not employed in STEM occupations.

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To help improve the trends in the numbers of students, graduates, and employees in STEM fields, university officials and others made several suggestions, such as increasing the federal commitment to STEM education programs. However, before making changes, it is important to know the extent to which existing STEM education programs are appropriately targeted and making the best use of available federal resources. Additionally, in an era of limited financial resources and growing federal deficits, information about the effectiveness of these programs can help guide policy makers and program managers.

Agency Comments and Our Evaluation We received written comments on a draft of this report from Commerce, the Department of Health and Human Services (HHS), NSF, and NSTC. These comments are reprinted in appendixes VII, VIII, IX, and X, respectively. We also received technical comments from the Departments of Commerce, Health and Human Services, Homeland Security, Labor, and Transportation; and the Environmental Protection Agency and National Aeronautics and Space Administration, which we incorporated when appropriate. In commenting on a draft of this report, Commerce, HHS, and NSTC commended GAO for this work. Commerce explicitly concurred with several findings and agreed with our overall conclusion. However, Commerce suggested that we revise the conclusion to point out that despite overall increases in STEM students, the numbers of graduates in certain fields have declined. We modified the concluding observations to make this point. HHS agreed with our conclusion that it is important to evaluate ongoing programs to determine the extent to which they are achieving their desired results. The comments from NSTC cited improvements made to help ensure that international students, exchange visitors, and scientists are able to apply for and receive visas in a timely manner. We did not make any changes to the report since we had cited another GAO product that discussed such improvements in the visa process. NSF commented about several of our findings. NSF stated that our program evaluations finding may be misleading largely because the type of information GAO requested and accepted from agencies was limited to program level evaluations and did not include evaluations of individual underlying projects. NSF suggested that we include information on the range of approaches used to assure program effectiveness. Our finding is based on agency officials’ responses to a survey question that did not limit or stipulate the types of evaluations that could have been included. To help improve the trends in the numbers of students, graduates, and employees in STEM fields, university officials and others made several suggestions, such as increasing the federal commitment to STEM education programs. However, before making changes, it is important to know the extent to which existing STEM education programs are appropriately targeted and making the best use of available federal resources. Additionally, in an era of limited financial resources and growing federal deficits, information about the effectiveness of these programs can help guide policy makers We received written comments on a draft of this report from Commerce, the Department of Health and Human Services (HHS), NSF, and NSTC. These comments are reprinted in appendixes VII, VIII, IX, and X, respectively. We also received technical comments from the Departments of Commerce, Health and Human Services, Homeland Security, Labor, and Transportation; and the Environmental Protection

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Agency and National Aeronautics and Space Administration, which we incorporated when appropriate. In commenting on a draft of this report, Commerce, HHS, and NSTC commended GAO for this work. Commerce explicitly concurred with several findings and agreed with our overall conclusion. However, Commerce suggested that we revise the conclusion to point out that despite overall increases in STEM students, the numbers of graduates in certain fields have declined. We modified the concluding observations to make this point. HHS agreed with our conclusion that it is important to evaluate ongoing programs to determine the extent to which they are achieving their desired results. The comments from NSTC cited improvements made to help ensure that international students, exchange visitors, and scientists are able to apply for and receive visas in a timely manner. We did not make any changes to the report since we had cited another GAO product that discussed such improvements in the visa process. NSF commented about several of our findings. NSF stated that our program evaluations finding may be misleading largely because the type of information GAO requested and accepted from agencies was limited to program level evaluations and did not include evaluations of individual underlying projects. NSF suggested that we include information on the range of approaches used to assure program effectiveness. Our finding is based on agency officials’ responses to a survey question that did not limit or stipulate the types of evaluations that could have been included. Nonetheless, we modified the report to acknowledge that NSF uses various approaches to evaluate its programs. NSF criticized the methodology we used to support our finding on the factors that influence decisions about pursuing STEM fields and suggested that we make it clearer in the body of the report that the findings are based on interviews with educators and administrators from 8 colleges and universities, and responses from 31 students. Also, NSF suggested that we improve the report by including corroborating information from reports and studies. Our finding was not limited to interviews at the 8 colleges and universities and responses from 31 current students but was also based on interviews with numerous representatives and policy experts from various organizations as well as findings from research and reports—which are cited in the body of the report. Using this approach, we were able to corroborate the testimonial evidence with data from reports and research as well as to determine whether information in the reports and research remained accurate by seeking the views of those currently teaching or studying in STEM fields. As NSF noted, this approach yielded reasonable observations. Additional information about our methodology is listed in appendix I, and we added a bibliography that identifies the reports and research used during the course of this review. NSF also commented that the report mentions the NSTC efforts for interagency collaboration, but does not mention other collaboration efforts such as the Federal Interagency Committee on Education and the Federal Interagency Coordinating Council. NSF also pointed out that interagency collaboration occurs at the program level. We did not modify the report in response to this comment. In conducting our work, we determined that the NSTC effort was the primary mechanism for interagency collaboration focused on STEM programs. The coordinating groups cited by NSF are focused on different issues. The Federal Interagency Committee on Education was established to coordinate the federal programs, policies, and practices affecting education broadly, and the Federal Interagency Coordinating

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Council was established to minimize duplication of programs and activities relating to children with disabilities. In addition, NSF provided information to clarify examples related to their programs that we cited in the report, stated that some data categories were not clear, and commented on the graduate level enrollment data we used in the report. NSF pointed out that while its program called Opportunities for Enhancing Diversity in the Geosciences is designed to increase participation by minorities, it does not limit eligibility to minorities. Also, NSF noted that while the draft report correctly indicated that students receiving scholarships or fellowships from NSF must be U.S. citizens or permanent residents, the reason given for limiting participation in these programs in the draft report was not accurate. According to NSF, these restrictions are considered to be an effective strategy to support its goal of creating a diverse, competitive and globally engaged U.S. workforce of scientists, engineers, technologists and well prepared citizens. We revised the report to reflect these changes. Further, NSF commented that the data categories were not clear, particularly the technology degrees and occupations, and that the data did not include associate degrees. We added information that lists all of the occupations included in the analysis, and we added footnotes to clarify which data included associate degrees and which ones did not. In addition, NSF commented that the graduate level enrollment data for international students based on NPSAS data are questionable in comparison with other available data and that this may be because the NPSAS data include a relatively small sample for graduate education. We considered using NPSAS and other data but decided to use the NPSAS data for two reasons: NPSAS data were more comprehensive and more current. Specifically, the NPSAS data were available through the 2003-2004 academic year and included numbers and characteristics of students enrolled for all degree fields—STEM and non-STEM—for all education levels, and citizenship information.

Appendix I: Objectives, Scope, and Methodology Objectives The objectives of our study were to determine (1) the number of federal civilian education programs funded in fiscal year 2004 that were specifically designed to increase the number of students and graduates pursuing science, technology, engineering, and mathematics (STEM) degrees and occupations, or improve educational programs in STEM fields, and what agencies report about their effectiveness; (2) how the numbers, percentages, and characteristics of students, graduates, and employees in STEM fields have changed over the years; and (3) factors cited by educators and others as influencing people’s decisions about pursuing STEM degrees and occupations, and suggestions to encourage greater participation in STEM fields.

Scope and Methodology In conducting our review, we used multiple methodologies. We (1) conducted a survey of federal departments and agencies that sponsored education programs specifically designed to

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increase the number of students and graduates pursuing STEM degrees and occupations or improve educational programs in STEM fields; (2) obtained and analyzed data, including the most recent data available, on students, graduates, and employees in STEM fields and occupations; (3) visited eight colleges and universities; (4) reviewed reports and studies; and (5) interviewed agency officials, representatives and policy experts from various organizations, and current students. We conducted our work between October 2004 and October 2005 in accordance with generally accepted government auditing standards.

Survey To provide Congress with a better understanding of what programs federal agencies were supporting to increase the nation’s pool of scientists, technologists, engineers, and mathematicians, we designed a survey to determine (1) the number of federal education programs (prekindergarten through postdoctorate) designed to increase the quantity of students and graduates pursuing STEM degrees and occupations or improve the educational programs in STEM fields and (2) what agencies reported about the effectiveness of these programs. The survey asked the officials to describe the goals, target population, and funding levels for fiscal years 2003, 2004, and 2005 of such programs. In addition, the officials were asked when the programs began and if the programs had been or were being evaluated. We identified the agencies likely to support STEM education programs by reviewing the Catalog of Federal Domestic Assistance and the Department of Education’s Eisenhower National Clearinghouse, Guidebook of Federal Resources for K-12 Mathematics and Science, 2004-05. Using these resources, we identified 15 agencies with STEM education programs. The survey was conducted via e-mail using an ActiveX enabled MSWord attachment. A contact point was designated for each agency, and questionnaires were sent to that individual. One questionnaire was completed for each program the agency sponsored. Agency officials were asked to provide confirming documentation for their responses whenever possible. The questionnaire was forwarded to agencies on February 15, 2005, and responses were received through early May 2005. We received 244 completed surveys and determined that 207 of them met the criteria for STEM programs. The following agencies participated in our survey: the Departments of Agriculture, Commerce, Education, Energy, Homeland Security, Interior, Labor, and Transportation. In addition, the Health Resources and Services Administration, Indian Health Service, and National Institutes of Health, all part of Health and Human Services, took part in the survey. Also participating were the U.S. Environmental Protection Agency; the National Aeronautics and Space Administration; and the National Science Foundation. Labor’s programs did not meet our criteria for 2004 and the Department of Defense (DOD) did not submit a survey. According to DOD officials, DOD needed 3 months to complete the survey and therefore could not provide responses within the time frames of our work. We obtained varied amounts of documentation from 13 civilian agencies for the 207 STEM education programs funded in 2004 and information about the effectiveness of some programs. Because we administered the survey to all of the known federal agencies sponsoring STEM education programs, our results are not subject to sampling error. However, the practical difficulties of conducting any survey may introduce other types of errors, commonly referred to as nonsampling errors. For example, differences in how a particular question is

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interpreted, the sources of information available to respondents in answering a question, or the types of people who do not respond can introduce unwanted variability into the survey results. We included steps in the development of the survey, the collection of data, and the editing and analysis of data for the purpose of minimizing such nonsampling errors. To reduce nonsampling error, the questionnaire was reviewed by survey specialists and pretested in person with three officials from agencies familiar with STEM education programs to develop a questionnaire that was relevant, easy to comprehend, unambiguous, and unbiased. We made changes to the content and format of the questionnaire based on the specialists’ reviews and the results of the pretests. To further reduce nonsampling error, data for this study returned electronically were entered directly into the instrument by the respondents and converted into a database for analysis. Completed questionnaires returned as hard copy were keypunched, and a sample of these records was verified by comparing them with their corresponding questionnaires, and any errors were corrected. When the data were analyzed, a second, independent analyst checked all computer programs. Finally, to assess the reliability of key data obtained from our survey about some of the programs, we compared the responses with the documentation provided, or we independently researched the information from other publicly available sources.

Analyses of Student, Graduate, and Employee Data To determine how the numbers and characteristics of students, graduates, and employees in STEM fields have changed, we obtained and analyzed data from the Department of Education (Education) and the Department of Labor. Specifically, we analyzed the National Postsecondary Student Aid Study (NPSAS) data and the Integrated Postsecondary Education Data System (IPEDS) data from the Department of Education’s National Center for Education Statistics (NCES), and we analyzed data from the Department of Labor’s Bureau of Labor Statistics’ (BLS) Current Population Survey (CPS). Based on National Science Foundation’s categorization of STEM fields, we developed STEM fields of study from NPSAS and IPEDS, and identified occupations from the CPS. Using these data sources, we developed nine STEM fields for students, eight STEM fields for graduates, and four broad STEM fields for occupations. For our data reliability assessment, we reviewed agency documentation on the data sets and conducted electronic tests of the files. On the basis of these reviews, we determined that the required data elements from NPSAS, IPEDS and CPS were sufficiently reliable for our purposes. These data sources, type, time span, and years analyzed are shown in table 18. NPSAS is a comprehensive nationwide study designed to determine how students and their families pay for postsecondary education, and to describe some demographic and other characteristics of those enrolled. The study is based on a nationally representative sample of students in postsecondary education institutions, including undergraduate, graduate, and firstprofessional students. The NPSAS has been conducted every several years since the 19861987 academic year. For this report, we analyzed the results of the NPSAS survey for the 1995-1996 academic year and the 2003-2004 academic year to compare student enrollment and demographic characteristics between these two periods for the nine STEM fields and non-STEM fields.

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Table 18. Sources of Data, Data Obtained, Time Span of Data, and Years Analyzed Department

Agency

Database

Data obtained

Time span Years analyzed of data 9 years Academic years 1995-1996 and 2003-2004

Education

NCES

NPSAS

College student enrollment

Education

NCES

IPEDS

Graduation/degrees 9 years

Academic years 1994-1995 and 2002-2003

Labor

BLS

CPS

Employment

Calendar years 1994 through 2003

Sources: NPSAS, IPEDS, and CPS data.

10 years

Because the NPSAS sample is a probability sample of students, the sample is only one of a large number of samples that might have been drawn. Since each sample could have provided different estimates, confidence in the precision of the particular sample’s results is expressed as a 95-percent confidence interval (for example, plus or minus 4 percentage points). This is the interval that would contain the actual population value for 95 percent of the samples that could have been drawn. As a result, we are 95 percent confident that each of the confidence intervals in this report will include the true values in the study population. NPSAS estimates used in this report and the upper and lower bounds of the 95 percent confidence intervals for each estimate relied on in this report are presented in appendix V. IPEDS is a single, comprehensive system designed to encompass all institutions and educational organizations whose primary purpose is to provide postsecondary education. IPEDS is built around a series of interrelated surveys to collect institution-level data in such areas as enrollments, program completions, faculty, staff, and finances. For this report, we analyzed the results of IPEDS data for the 1994-1995 academic year and the 2002-2003 academic year to compare the numbers and characteristics of graduates with degrees in eight STEM fields and non-STEM fields. To analyze changes in employees in STEM and non-STEM fields, we obtained employment estimates from BLS’s Current Population Survey March supplement for 1995 through 2004 (calendar years 1994 through 2003). The CPS is a monthly survey of households conducted by the U.S. Census Bureau (Census) for BLS. The CPS provides a comprehensive body of information on the employment and unemployment experience of the nation’s population, classified by age, sex, race, and a variety of other characteristics. A more complete description of the survey, including sample design, estimation, and other methodology can be found in the CPS documentation prepared by Census and BLS [1]. This March supplement (the Annual Demographic Supplement) is specifically designed to estimate family characteristics, including income from all sources and occupation and industry classification of the job held longest during the previous year. It is conducted during the month of March each year because it is believed that since March is the month before the deadline for filing federal income tax returns, respondents would be more likely to report income more accurately than at any other point during the year [2].

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United States Government Accountability Office Table 19. Classification codes and Occupations, 2002-2003 Science

T echnol

ogy

Engineering

Mathematics/Computer Science

1600 – Agricultural and food scientists

1540 – Drafters

1300 – Architects, except naval

1610 – Biological scientists

1550 – Engineering technicians, except drafters

1310 – Surveyors, cartographers, 1010 – Computer and photogrammetrists programmers

1640 – Conservation scientists and foresters

1560 – Surveying and mapping technicians

1320 – Aerospace engineers

1020 – Computer software engineers

1650 – Medical scientists

1900 – Agricultural and food science technicians

1330 – Agricultural engineers

1040 – Computer support specialists

1700 – Astronomers and physicists

1910 – Biological technicians

1340 – Biomedical engineers

1060 – Database administrators

1710 – Atmospheric and space scientists

1920 – Chemical technicians

1350 – Chemical engineers

1100 – Network and computer systems administrators

1720 – Chemists and materials scientists

1930 – Geological and petroleum technicians

1360 – Civil engineers

1110 – Network systems and data communications analysts

1740 – Environmental scientists and geoscientists

1940 – Nuclear technicians

1400 – Computer hardware engineers

1200 – Actuaries

1760 – Physical scientists, all other

1960 – Other life, physical, and social science technicians

1410 – Electrical and electronic engineers

1210 – Mathematicians

1800 – Economists

3300 – Clinical laboratory technologists and technicians

1420 – Environmental engineers

1220 – Operations research analysts

1810 – Market and survey researchers

7010 – Computer, automated teller 1430 – Industrial engineers, and office machine repairers including health and safety

1230 – Statisticians

1820 – Psychologists

8760 – Medical, dental, and ophthalmic laboratory technicians

1240 – Miscellaneous mathematical science occupations

1440 – Marine engineers and naval architects

1830 – Sociologists

1450 – Materials engineers

1840 – Urban and regional planners

1460 – Mechanical engineers

1860 – Miscellaneous social scientists and related workers

1500 – Mining and geological engineers, including mining safety engineers

2010 – Social workers

1510 – Nuclear engineers

3130 – Registered nurses

1520 – Petroleum engineers

6010 – Agricultural inspectors

1530 – Engineers, all other

1000 – Computer scientists and systems analysts

We used the CPS data to produce estimates on (1) four STEM fields, (2) men and women, (3) two separate minority groups (Black or African American, and Hispanic or Latino origin), and (4) median annual wages and salaries. The measures of median annual wages and salaries could include bonuses, but do not include noncash benefits such as health insurance or pensions. CPS salary reported in March of each year was for the longest held position actually worked the year before and reported by the worker himself (or a knowledgeable member of the household). Tables 19 and 20 list the classification codes and occupations included in our analysis of CPS data over a 10-year period (1994-2003). In developing the STEM groups, we considered the occupational requirements and educational attainment of individuals in certain occupations. We also excluded doctors and other health care providers except registered nurses. During the period of review, some codes and occupation titles were changed; we worked with BLS officials to identify variations in codes and occupations and accounted for these changes where appropriate and possible. Because the CPS is a probability sample based on random selections, the sample is only one of a large number of samples that might have been drawn. Since each sample could have provided different estimates, confidence in the precision of the particular sample’s results is expressed as a 95 percent confidence interval (e.g., plus or minus 4 percentage points). This is the interval that would contain the actual population value for 95 percent of the samples that could have been drawn. As a result, we are 95 percent confident that each of the confidence intervals in this report will include the true values in the study population. We use the CPS

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general variance methodology to estimate this sampling error and report it as confidence intervals. Percentage estimates we produce from the CPS data have 95 percent confidence intervals of plus or minus 6 percentage points or less. Estimates other than percentages have 95 percent confidence intervals of no more than plus or minus 10 percent of the estimate itself, unless otherwise noted. Consistent with the CPS documentation guidelines, we do not produce estimates based on the March supplement data for populations of less than 75,000. Table 20. Classification codes and occupations, 1994-2001 Science

T echnol

ogy

Engineering

Mathematics/Computer Science

069 – Physicists and astronomers

203 – Clinical laboratory technologists and technicians

043 – Architects

064 – Computer systems analysts and scientists

073 – Chemists, except biochemists

213 – Electrical and electronic technicians

044 – Aerospace engineers

065 – Operations and systems researchers and analysts

074 – Atmospheric and space 214 – Industrial engineering scientists technicians

045 – Metallurgical and materials engineers

066 – Actuaries

075 – Geologists and geodesists

215 – Mechanical engineering technicians

046 – Mining engineers

067 – Statisticians

076 – Physical scientists, n.e.c.

216 – Engineering technicians, n.e.c.

047 – Petroleum engineers

068 – Mathematical scientists, n.e.c.

077 – Agricultural and food scientists

217 – Drafting occupations

048 – Chemical engineers

229 – Computer programmers

078 – Biological and life scientists

218 – Surveying and mapping technicians

049 – Nuclear engineers

079 – Forestry and conservation scientists

223 – Biological technicians

053 – Civil engineers

083 – Medical scientists

224 – Chemical technicians

054 – Agricultural engineers

095 – Registered Nurses

225 – Science technicians, n.e.c.

055 – Electrical and electronic engineers

166 – Economists

235 – Technicians, n.e.c.

056 – Industrial engineers

167 – Psychologists

525 – Data processing equipment repairers

057 – Mechanical engineers

168 – Sociologists

058 – Marine and naval architects

169 – Social scientists, n.e.c.

059 – Engineers, n.e.c.

173 – Urban planners

063 – Surveyors and mapping scientists

174 – Social workers 489 – Inspectors, agricultural products

Note: For occupations not elsewhere classified (n.e.c.).

GAO’s internal control procedures provide reasonable assurance that our data analyses are appropriate for the purposes we are using them. These procedures include, but are not limited to, having skilled staff perform the analyses, supervisory review by senior analysts, and indexing/referencing (confirming that the analyses are supported by the underlying audit documentation) activities.

College and University Visits We interviewed administrators and professors during site visits to eight colleges and universities—the University of California at Los Angeles and the University of Southern California in California; Clark Atlanta University, Georgia Institute of Technology, and Spelman College in Georgia; the University of Illinois; Purdue University in Indiana; and Pennsylvania State University. These colleges and universities were selected based on the following factors: large numbers of domestic and international students in STEM fields, a

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mix of public and private institutions, number of doctoral degrees conferred, and some geographic diversity. We also selected three minority-serving colleges and universities, one of which serves only women students. Clark Atlanta University and Spelman College were selected, in part, because of their partnerships with the College of Engineering at the Georgia Institute of Technology. During these visits we asked the university officials about factors that influenced whether people pursue a STEM education or occupations and suggestions for addressing those factors that may influence participation. For example, we asked university officials to identify (1) issues related to the education pipeline; (2) steps taken by their university to alleviate some of the conditions that may discourage student participation in STEM areas; and (3) the federal role, if any, in attracting and retaining domestic students in STEM fields. We also obtained documents on programs they sponsored to help support STEM students and graduates.

Reviews of Reports and Studies We reviewed several articles, reports, and books related to trends in STEM enrollment and factors that have an effect on people’s decisions to pursue STEM fields. For two studies, we evaluated the methodological soundness using common social science and statistical practices. We examined each study’s methodology, including its limitations, data sources, analyses, and conclusions. •

Talking about Leaving: Why Undergraduates Leave the Sciences, by Elaine Seymour and Nancy Hewitt.3 This study used interviews and focus groups/group interviews at selected universities to identify self-reported reasons for changing majors from science, mathematics, or engineering. The study had four primary objectives: (1) to identify sources of qualitative differences in educational experiences of science, mathematics, and engineering students at higher educational institutions of different types; (2) to identify differences in structure, culture, and pedagogy of science, mathematics, and engineering departments and the impact on student retention; (3) to compare and contrast causes of science, mathematics, and engineering students’ attrition by race/ethnicity and gender; and (4) to estimate the relative importance of factors found to contribute to science, mathematics, and engineering students’ attrition. The researchers selected seven universities to represent the types of colleges and universities that supply most of the nations’ scientists, mathematicians, and engineers. The types of institutions were selected to test whether there are differences in educational experiences, culture and pedagogy, race/ethnicity and gender attrition, and reasons for attrition by type of institution. Because the selection of students was not strictly random and because there is no documentation that the data were weighted to reflect the proportions of types of students selected, it is not possible to determine confidence intervals. Thus it is not possible to say which differences are statistically significant. The findings are now more than a decade old and thus might not reflect current pedagogy and other factors about the educational experience, students, or the socioeconomic environment. It is important to note that the quantitative results of this study are based on the views of one constituency or

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•

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stakeholder—students. Views of faculty, school administrators, graduates, professional associations, and employers are not included. NCES’s Qualifications of the Public School Teacher Workforce: Prevalence of Out-ofField Teaching, 1987-1988 to 1999-2000 report. This study is an analysis based upon the Schools and Staffing Survey for 1999-2000. The report was issued in 2004 by the Institute of Education Sciences, U.S. Department of Education. NCES’s Schools and Staffing Survey (SASS) is a representative sample of U.S. schools, districts, principals, and teachers. The report focusing on teacher’s qualifications uses data from the district and teacher portion of SASS. The 1999-2000 SASS included a nationally representative sample of public schools and universe of all public charter schools with students in any of grades 1 through 12 and in operation in school year 1999-2000. The 1999-2000 SASS administration also included nationally representative samples of teachers in the selected public and public charter schools who taught students in grades kindergarten through 12 in school year 1999-2000. There were 51,811 public school teachers in the sample and 42,086 completed public school teacher interviews. In addition, there are 3,617 public charter school teachers in the sample with 2,847 completed interviews. The overall weighted teacher response rate was 76.7 percent for public school teachers and 71.8 percent for public charter school teachers. NCES has strong standards for carrying out educational surveys. The Office of Management and Budget vetted the questionnaire and sample design. The Census Bureau carried out survey quality control and data editing. One potential limitation is the amount of time it takes the Census Bureau to get the data from field collection to public release, but this is partly due to the thoroughness of the data quality steps followed. The SASS survey meets GAO standards for use as evidence in a report.

Interviews We interviewed officials from 13 federal agencies with STEM education programs to obtain information about the STEM programs and their views on related topics, including factors that influence students’ decisions about pursuing STEM degrees and occupations, and the extent of coordination among the federal agencies. We also interviewed officials from the National Science and Technology Council to discuss coordination efforts. In addition, we interviewed representatives and policy experts from various organizations. These organizations were the American Association for the Advancement of Science, the Commission on Professionals in Science and Technology, the Council of Graduate Schools, NAFSA: Association of International Educators, the National Academies, and the Council on Competitiveness. We also conducted interviews via e-mail with 31 students. We asked officials from the eight universities visited to identify students to complete our e-mail interviews, and students who completed the interviews attended five of the colleges we visited. Of the 31 students: 16 attended Purdue University, 6 attended the University of Southern California, 6 attended Spelman College, 2 attended the University of California Los Angeles, and 1 attended the Georgia Institute of Technology. In addition, 19 students were undergraduates and 12 were graduate students; 19 students identified themselves as women and 12 students identified themselves as men. Of the 19 undergraduate students, 9 said that they plan to pursue graduate work in a STEM field.

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Appendix II: List of 207 Federal STEM Education Programs Based on surveys submitted by officials representing the 13 civilian federal agencies, table 21 contains a list of the 207 science, technology, engineering, and mathematics (STEM) education programs funded in fiscal year 2004. Table 21. Federal STEM Education Programs Funded in FY 2004 Program Program name number Department of Agriculture 1. 1890 Institution Teaching and Research Capacity Building Grants Program 2. Higher Education Challenge Grants Program 3. Hispanic-Serving Institutions Education Grants Program 4. Alaska Native-Serving and Native Hawaiian-Serving Institutions Education Grants Program 5. Food and Agricultural Sciences National Needs Graduate and Postdoctoral Fellowships Grants Program 6. Tribal Colleges Endowment Program 7. Tribal Colleges Education Equity Grants Program 8. Tribal Colleges Research Grant Program 9. Higher Education Multicultural Scholars Program 10. International Science and Education Competitive Grants Program 11. Secondary and Two-Year Postsecondary Agricultural Education Challenge Grants Program 12. Agriculture in the Classroom 13. Career Intern Program 14. Veterinary Medical Doctoral Program 15. 1890 National Scholars Program 16. Hispanic Scholars Program Department of Commerce 17. Educational Partnership Program with Minority Serving Institutions 18. National Marine Sanctuaries Education Program 19. National Sea Grant College Program 20. Chesapeake Bay Watershed Education and Training Program 21. Coral Reef Conservation Program 22. Exploration, Education and Outreach 23. National Estuarine Research Reserve Graduate Research Fellowship Program 24. Bay Watershed Education and Training Hawaii Program 25. Monterey Bay Watershed Education and Training Program 26. Dr. Nancy Foster Scholarship Program 27. EstuaryLive 28. Teacher at Sea Program 29. High School-High Tech Department of Education 30. Mathematics and Science Partnerships Program 31. Upward Bound Math and Science Program 32. Graduate Assistance in Areas of National Need 33. Minority Science and Engineering Improvement Program

Fiscal year 04 funding $11.4 million $4.6 million $4.6 million $3 million $2.9 million $1.9 million $1.7 million $1.1 million $986,000 $859,000 $839,000 $623,000 $272,000 $140,000 $16,000 $4,000 $7.4 million $4.4 million $4 million $2.5 million $1.8 million $1.3 million $1 million $500,000 $500,000 $494,000 $115,000 $95,000 $11,000 $149 million $32.8 million $30.6 million $8.9 million

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Table 21. Continued Program Program name number Department of Energy 34. Science Undergraduate Laboratory Internship 35. Computational Science Graduate Fellowship 36. Global Change Education Program 37. Laboratory Science Teacher Professional Development 38. National Science Bowl 39. Community College Institute of Science and Technology 40. Albert Einstein Distinguished Educator Fellowship 41. QuarkNet 42. Fusion Energy Sciences Fellowship Program 43. Pre-Service Teacher Fellowships 44. National Undergraduate Fellowship Program in Plasma Physics and Fusion Energy Sciences 45. Fusion Energy Postdoctoral Research Program 46. Faculty and Student Teams 47. Advancing Precollege Science and Mathematics Education 48. Pan American Advanced Studies Institute 49. Trenton Community Partnership 50. Fusion/Plasma Education 51. National Middle School Science Bowl 52. Research Project on the Recruitment, Retention, and Promotion of Women in the Chemical Sciences 53. Used Energy Related Laboratory Equipment 54. Plasma Physics Summer Institute for High School Physics Teachers 55. Pre-Service Teacher Program 56. Wonders of Physics Traveling Show 57. Hampton University Graduate Studies 58. Contemporary Physics Education Project 59. Cooperative Education Program Environmental Protection Agency 60. Science to Achieve Results Research Grants Program 61. Science to Achieve Results Graduate Fellowship Program 62. Post-Doctoral Fellows Environmental Research Growth Opportunities 63. Intern Program 64. Environmental Science and Engineering Fellows Program 65. Greater Research Opportunities Graduate Fellowship Program 66. Environmental Risk and Impact in Communities of Color and Economically Disadvantaged Communities 67. Research Internship for Students in Ecology 68. National Network for Environmental Management Studies Fellowship Program 69. Cooperative Agreements for Training Cooperative Partnerships 70. University of Cincinnati/EPA Research Training Grant 71. P3 Award: National Student Design Competition for Sustainability 72. Environmental Protection Agency and the Hispanic Association of Colleges and Universities Cooperative Agreement 73. Environmental Science Program 74. Environmental Career Organization’s Internship Program 75. EPA—Cincinnati Research Apprenticeship Program 76. Environmental Protection Internship Program Summer Training Initiative 77. Tribal Lands Environmental Science Scholarship Program

Fiscal year 04 funding $2.5 million $2 million $1.4 million $1 million $702,000 $605,000 $600,000 $575,000 $555,000 $510,000 $300,000 $243,000 $215,000 $209,000 $200,000 $200,000 $125,000 $100,000 $100,000 $80,000 $78,000 $45,000 $45,000 $40,000 $23,000 $17,000 $93.3 million $10 million $7.4 million $3 million $2.5 million $1.5 million $824,000 $698,000 $589,000 $352,000 $300,000 $150,000 $121,000 $100,000 $89,000 $75,000 $72,000 $60,000

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United States Government Accountability Office Table 21. Continued

Program Program name number 78. Internship Program for University of Arizona Engineering Students 79. Teacher Professional Development Workshop for Teachers Grade 6-12 80. Saturday Academy, Apprenticeships in Science and Engineering Program Department of Health and Human Services/Health Resources and Services Administration 81. Scholarships for Disadvantaged Students Program 82. Nursing Workforce Diversity 83. Faculty Loan Repayment Program Department of Health and Human Services/Indian Health Service 84. Indian Health Professions Scholarship 85. Health Professions Scholarship Program for Indians Department of Health and Human Services/National Institutes of Health 86. Ruth L. Kirschstein National Research Service Award Institutional Research Training Grants 87. Ruth L. Kirschstein National Research Service Awards for Individual Postdoctoral Fellows 88. Research Supplements to Promote Diversity in Health-Related Research 89. Postdoctoral Visiting Fellow Program 90. Clinical Research Loan Repayment Program 91. Ruth L. Kirschstein National Research Service Awards for Individual Predoctoral Fellows, Predoctoral Minority Students, and Predoctoral Students with Disabilities 92. Minority Access to Research Careers Program 93. Postdoctoral Intramural Research Training Award Program 94. Science Education Partnership Award 95. Pediatric Research Loan Repayment Program 96. Post-baccalaureate Intramural Research Training Award Program 97. Ruth L. Kirschstein National Research Service Award Short-Term Institutional Research Training Grants 98. Health Disparities Research Loan Repayment Program 99. Graduate Program Partnerships 100. Student Intramural Research Training Award Program 101. Career Opportunities in Research Education and Training Honors Undergraduate Research Training Grant 102. General Research Loan Repayment Program 103. Ruth L. Kirschstein National Research Service Awards for Individual M.D./Ph.D. Predoctoral Fellows 104. Science Education Drug Abuse Partnership Award 105. Pharmacology Research Associate Training Program 106. Technical Intramural Research Training Award 107. Fellowships in Cancer Epidemiology and Genetics 108. Clinical Research Loan Repayment Program for Individuals from Disadvantaged Backgrounds 109. Contraception and Infertility Research Loan Repayment Program 110. Medical Infomatics Training Program 111. Undergraduate Scholarship Program for Individuals from Disadvantaged Backgrounds 112. Curriculum Supplement Series 113. National Science Foundation and the National Institute of Biomedical Imaging and Bioengineering 114. Summer Institute for Training in Biostatistics

Fiscal year 04 funding $50,000 $18,000 $6,000 $45.5 million $16 million $1.1 million $8.1 million $3.7 million $546.9 million $72.6 million $70 million $64.8 million $40.6 million $33.8 million $30.7 million $30.2 million $16 million $15.9 million $9.1 million $9 million $8.7 million $7.4 million $6.3 million $5 million $4.9 million $4.7 million $3.1 million $2.7 million $1.9 million $1.8 million $1.7 million $1 million $853,000 $838,000 $788,000 $782,000 $694,000

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Table 21. Continued Program Fiscal year Program name number 04 funding 115. Summer Institute on Design and Conduct of Randomized Clinical Trials $622,000 Involving Behavioral Interventions 116. Clinical Research Loan Repayment Program for Individuals from Disadvantaged $551,000 Background 117. Clinical Research Training Program $407,000 118. NIH Academy $385,000 119. Health Communications Internship Program $340,000 120. NIH/National Institute of Standards and Technology Joint Postdoctoral Program $338,000 121. Summer Genetics Institute $323,000 122. AIDS Research Loan Repayment Program $271,000 123. Intramural NIAID Research Opportunities $271,000 124. Cancer Research Interns in Residence $250,000 125. Comparative Molecular Pathology Research Training Program $199,000 126. Office of Research on Women’s Health-funded Programs with the Office of $179,000 Intramural Research 127. Summer Institute for Social Work Research $144,000 128. Office of Research on Women’s Health-funded Programs with the Office of $119,000 Intramural Training and Education 129. CCR/JHU Master of Science in Biotechnology Concentration in Molecular $111,000 Targets and Drug Discovery Technologies 130. Introduction to Cancer Research Careers $96,000 131. Fellows Award for Research Excellence Program $61,000 132. Office of Research on Women’s Health-funded Programs Supplements to $60,000 Promote Reentry into Biomedical and Behavioral Research Careers 133. Translational Research in Clinical Oncology $28,000 134. National Institute of Environmental Health Sciences Office of Fellows’ Career $20,000 Development 135. Mobilizing for Action to Address the Unequal Burden of Cancer: NIH Research $10,000 and Training Opportunities 136. Sallie Rosen Kaplan Fellowship for Women in Cancer Research $5,000 Department of Homeland Security 137. Scholars and Fellows Program $4.7 million Department of the Interior 138. Cooperative Research Units Program $15.3 million 139. Water Resources Research Act Program $6.4 million 140. U.S. Geological Survey Mendenhall Postdoctoral Research Fellowship Program $3.5 million 141. Student Educational Employment Program $1.8 million 142. EDMAP Component of the National Cooperative Geologic Mapping Program $490,000 143. Student Career Experience Program $177,000 144. Cooperative Development Energy Program $60,000 145. Diversity Employment Program $30,000 146. Cooperative Agreement with Langston University $15,000 147. Mathematics, Science, and Engineering Academy $15,000 148. Shorebird Sister Schools Program $15,000 149. Build a Bridge Contest $14,000 150. VIVA Technology $8,000 National Aeronautics and Space Administration 151. Minority University Research Education Program $106.6 million 152. Higher Education $77.4 million 153. Elementary and Secondary Education $31.3 million

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United States Government Accountability Office Table 21. Continued

Program Program name number 154. E-Education 155. Informal Education National Science Foundation 156. Math and Science Partnership Program 157. Graduate Research Fellowship Program 158. Integrative Graduate Education and Research Traineeship Program 159. Teacher Professional Continuum 160. Research Experiences for Undergraduates 161. Graduate Teaching Fellows in K-12 Education 162. Advanced Technological Education 163. Course, Curriculum, and Laboratory Improvement 164. Research on Learning and Education 165. Computer Science, Engineering, and Mathematics Scholarships 166. Louis Stokes Alliances for Minority Participation 167. Centers for Learning and Teaching 168. Instructional Materials Development 169. Science, Technology, Engineering, and Mathematics Talent Expansion Program 170. Historically Black Colleges and Universities Undergraduate Program 171. Interagency Education Research Initiative 172. Information Technology Experiences for Students and Teachers 173. Enhancing the Mathematical Sciences Workforce in the 21st Century 174. Centers of Research Excellence in Science and Technology 175. ADVANCE: Increasing the Participation and Advancement of Women in Academic Science and Engineering Careers 176. Federal Cyber Service: Scholarship for Service 177. Alliances for Graduate Education and the Professoriate 178. Research on Gender in Science and Engineering 179. Tribal Colleges and Universities Program 180. Model Institutions for Excellence 181. Grants for the Department-Level Reform of Undergraduate Engineering Education 182. Robert Noyce Scholarship Program 183. Research Experiences for Teachers 184. Nanoscale Science and Engineering Education 185. Research in Disabilities Education 186. Opportunities for Enhancing Diversity in the Geosciences 187. Mathematical Sciences Postdoctoral Research Fellowships 188. Minority Postdoctoral Research Fellowships and Supporting Activities 189. Partnerships for Research and Education in Materials 190. Undergraduate Research Centers 191. Centers for Ocean Science Education Excellence 192. Undergraduate Mentoring in Environmental Biology 193. Director’s Award for Distinguished Teaching Scholars 194. Astronomy and Astrophysics Postdoctoral Fellowship Program 195. Geoscience Education 196. Internships in Public Science Education 197. Discovery Corps Fellowship Program 198. East Asia and Pacific Summer Institutes for U.S. Graduate Students 199. Pan-American Advanced Studies Institutes 200. Distinguished International Postdoctoral Research Fellowships

Fiscal year 04 funding $9.7 million $5.5 million $138.7 million $96 million $67.7 million $61.5 million $51.7 million $49.8 million $45.9 million $40.7 million $39.4 million $33.9 million $33.3 million $30.8 million $29.3 million $25 million $23.8 million $23.6 million $20.9 million $20.6 million $19.8 million $19.4 million $15.8 million $15.3 million $10 million $10 million $9.7 million $8.2 million $8 million $5.8 million $4.8 million $4.6 million $4 million $3.7 million $3.2 million $3 million $3 million $2.8 million $2.2 million $1.8 million $1.6 million $1.5 million $1.2 million $1.1 million $1 million $800,000 $788,000

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Table 21. Continued Program Program name number 201. Postdoctoral Fellowships in Polar Regions Research 202. Arctic Research and Education 203. Developing Global Scientists and Engineers Department of Transportation 204. University Transportation Centers Program 205. Dwight David Eisenhower Transportation Fellowship Program 206. Summer Transportation Institute 207. Summer Transportation Internship Program for Diverse Groups

Fiscal year 04 funding $667,000 $300,000 $172,000 $32.5 million $2 million $2 million $925,000

Source: GAO survey responses from 13 federal agencies.

Appendix III: Federal STEM Education Programs Funded at $10 Million or More The federal civilian agencies reported that the following science, technology, engineering, and mathematics (STEM) education programs were funded with at least $10 million in either fiscal year 2004 or 2005. However, programs that received $10 million or more in fiscal year 2004 but were unfunded for fiscal year 2005 were excluded from table 22. Agency officials also provided the program descriptions in table 22. Table 22. Federal STEM Education Programs Funded at $10 Million or More during Fiscal Year 2004 or Fiscal Year2005 Funding (in millions of dollars)

Program

Description

a

First year

2004

2005

1990

$11.4

$12.5

Department of Agriculture 1890 Institution Teaching and Is intended to strengthen teaching and research programs in the Research Capacity Building food and agricultural sciences by building the institutional capacities Grants Program of the 1890 Land-Grant Institutions and Tuskegee University and West Virginia State University through cooperative linkages with federal and nonfederal entities. The program supports projects that strengthen teaching programs in the food and agricultural sciences in the targeted educational need areas of curriculum design and materials development, faculty preparation and enhancement of teaching, student experiential learning, and student recruitment and retention.

Department of Education Mathematics and Science Partnerships Program

Is intended to increase the academic achievement of students in mathematics and science by enhancing the content knowledge and teaching skills of classroom teachers. Partnerships are between high-need school districts and the science, technology, engineering, and mathematics faculties of institutions of higher education.

2002

$149

$180

Upward Bound Math and Science Program

Designed to prepare low-income, first-generation college students for postsecondary education programs that lead to careers in the fields of math and science.

1990

$32.8

$32.8

Graduate Assistance in Areas Provides fellowships in academic areas of national need to assist of National Need graduate students with excellent academic records who demonstrate financial need and plan to pursue the highest degree available in their courses of study.

1988

$30.6

$30.4

1995

$93.3

$80.1

1995

$10

EnvironmentalP rotection Agency Science to Achieve Results Research Grants Program

Funds research grants in numerous environmental science and engineering disciplines. The program engages the nation’s best scientists and engineers in targeted research. The grant program is currently focused on the health effects of particulate matter, drinking water, water quality, global change, ecosystem assessment and restoration, human health risk assessment, endocrine disrupting chemicals, pollution prevention and new technologies, children’s health, and socio-economic research.

Science to Achieve Results The purpose of this fellowship program is to encourage promising Graduate Fellowship Program students to obtain advanced degrees and pursue careers in environmentally related fields.

$10

222

United States Government Accountability Office Table 22. Continued Funding (in millions of dollars)

Program

Description

a

First year

2004

2005 Not avail.

Department of Health and HumanS ervices/Health Resources andS ervices Administration Scholarships for Disadvantaged Students Program

Funds are awarded to accredited schools of allopathic medicine, osteopathic medicine, dentistry, optometry, pharmacy, podiatric medicine, veterinary medicine, nursing, public health, chiropractic, or allied health, and schools offering graduate programs in behavioral and mental health practice. Priority is given to schools based on the proportion of graduating students going into primary care, the proportion of underrepresented minority students enrolled, and graduates working in medically underserved communities. Schools select qualified students and provide scholarships that cannot exceed tuition and reasonable educational and living expenses.

1991

$45.5

Nursing Workforce Diversity

To increase nursing education opportunities for individuals who are from disadvantaged backgrounds (including racial and ethnic minorities underrepresented among registered nurses) by providing student stipends, pre-entry preparation, and retention activities.

1989

$16

$16

Department of Health and HumanS ervices/NationalI nstitutes of Health Ruth L. Kirschstein National Research Service Award Institutional Research Training Grants

Is designed to develop and enhance research training opportunities for individuals in biomedical, behavioral, and clinical research by supporting training programs at institutions of higher education. These institutional training grants allow the director of the program to select the trainees and to develop a curriculum of study and research experiences necessary to provide high-quality research training. The grant helps offset the cost of stipends and tuition for the appointed trainees. Graduate students, postdoctoral trainees, and short-term research training for health professional students can be supported by this grant.

1975

$546.9

Not avail.

Ruth L. Kirschstein National Research Service Awards for Individual Postdoctoral Fellows

To support the advanced training of individual students who have recently received doctoral degrees. This phase of research education and training is performed under the direct supervision of a sponsor who is an active investigator in the area of the proposed research. The training is designed to enhance the fellow’s understanding of the health-related sciences and extend his/her potential to become a productive scientist who can perform research in biomedical, behavioral, or clinical fields.

1975

$72.6

Not avail.

Research Supplements to Promote Diversity in HealthRelated Research

To improve the diversity of the research workforce by recruiting and supporting students, postdoctoral fellows, and eligible investigators from groups that have been shown to be underrepresented, such as individuals from underrepresented racial and ethnic groups, individuals with disabilities, and individuals from disadvantaged backgrounds.

1989

$70

Postdoctoral Visiting Fellow Program

To provide advanced practical biomedical research experience to individuals who are foreign nationals and are 1 to 5 years beyond obtaining their Ph.D. or professional doctorate (e.g., M.D., DDS, etc.).

1950

$64.8

Program

Description

Clinical Research Loan Repayment Program

$70

$70.7

Funding (in millions of dollars)

a

First year

2004

2005

To attract health professionals to careers in clinical research. Clinical research is defined as “patient-oriented clinical research conducted with human subjects, or research on the causes and consequences of disease in human populations involving material of human origin (such as tissue specimens and cognitive phenomena) for which an investigator or colleague directly interacts with human subjects in an outpatient or inpatient setting to clarify a problem in human physiology, pathophysiology or disease, or epidemiologic or behavioral studies, outcomes research or health services research, or developing new technologies, therapeutic interventions, or clinical trials.”

2002

$40.6

$42.6

Ruth L. Kirschstein National Research Service Awards for Individual Predoctoral Fellows, Predoctoral Minority Students, and Predoctoral Students with Disabilities

Provides predoctoral fellowships to students who are candidates for doctoral degrees and are performing dissertation research and training under the supervision of a mentor who is an active and established investigator in the area of the proposed research. The applicant and mentor must provide evidence of potential for a productive research career based upon the quality of previous research training, academic record, and training program. The applicant and mentor must propose a research project that will enhance the student’s ability to understand and perform scientific research. The training program should be carried out in a research environment that includes appropriate resources and is demonstrably committed to the student’s training.

1975

$33.8

Not avail.

Minority Access to Research Careers Program

Offers special research training support to 4-year colleges, universities, and health professional schools with substantial enrollments of minorities such as African Americans, Hispanic Americans, Native Americans (including Alaska Natives), and natives of U.S. Pacific Islands. Individual fellowships are also provided for graduate students and faculty.

1972

$30.7

$30.7

Postdoctoral Intramural Research Training Award Program

To provide advanced practical biomedical research experience to individuals who are 1 to 5 years beyond obtaining their Ph.D. or professional doctorate (e.g., M.D., DDS, etc.).

1986

$30.2

$33.3

Science Education Partnership Award

Provides funds for the development, implementation, and evaluation of innovative kindergarten through 12th grade (K-12) science education programs, teaching materials, and science center/museum programs. This program supports partnerships linking biomedical, clinical researchers, and behavioral scientists with K-12 teachers and schools, museum and science educators, media experts, and other interested organizations.

1992

$16

$16

Pediatric Research Loan Repayment Program

A program to attract health professionals to careers in pediatric research. Qualified pediatric research is defined as “research directly related to diseases, disorders, and other conditions in children.”

2002

$15.9

$16

Higher Education: Federal Science, Technology, Engineering…

223

Table 22. Continued Funding (in millions of dollars)a

Program

Description

First year

Post-baccalaureate Intramural Research Training Award Program

To provide (1) recent college graduates (graduated no more than 2 years prior to activation of traineeship), an introduction early in their careers to biomedical research fields; encourage their pursuit of professional careers in biomedical research; and allow additional time to pursue successful application to either graduate or medical school programs or (2) students who have been accepted into graduate, other doctoral, or medical degree programs, and who have written permission from their school to delay entrance for up to 1 year.

2004

2005

1996

$9.1

$12.3

Provides scholarships for undergraduate and fellowships for graduate students pursuing degrees in mission-relevant fields and postdoctoral fellowships for their contributions to Department of Homeland Security research projects. Students receive professional mentoring and complete a summer internship to connect academic interests with homeland security initiatives. Postdoctoral scholars are also mentored by DHS scientists.

2003

$4.7

$10.7

The program links graduate science training with the research needs of state and federal agencies, and provides students with one-onone mentoring by federal research scientists working on both applied and basic research needs of interest to the program. Program cooperators and partners provide graduate training opportunities and support.

1936

$15.3

$15

To build the capacity of community colleges to train in high-growth, high-demand industries and to actually train workers in those industries through partnerships that also include workforce investment boards and employers.

2005

$0

$250

2002

$106.6

$73.6

Department of HomelandS ecurity University Programs

Department of theI nterior Cooperative Research Units Program

Department of Labor Community College/Community Based Job Training Grant Initiative

National Aeronautics andS pace Administration Minority University Research Education Program

To expand and advance NASA’s scientific and technological base through collaborative efforts with Historically Black Colleges and Universities (HBCU) and other minority universities (OMU), including Hispanic-serving institutions and Tribal colleges and universities. This program also provides K-12 awards to build and support successful pathways for students to progress to the next level of mathematics and science, through a college preparatory curriculum, and enrollment in college. Higher-education awards are also given that seek to improve the rate at which underrepresented minorities are awarded degrees in STEM disciplines through increased research training and exposure to cutting-edge technologies that better prepare them to enter STEM graduate programs, the NASA workforce pipeline, and employment in NASA-related industries.

Program

Description

Post-baccalaureate Intramural Research Training Award Program

To provide (1) recent college graduates (graduated no more than 2 years prior to activation of traineeship), an introduction early in their careers to biomedical research fields; encourage their pursuit of professional careers in biomedical research; and allow additional time to pursue successful application to either graduate or medical school programs or (2) students who have been accepted into graduate, other doctoral, or medical degree programs, and who have written permission from their school to delay entrance for up to 1 year.

Funding (in millions of dollars)a

First year

2004

2005

1996

$9.1

$12.3

Provides scholarships for undergraduate and fellowships for graduate students pursuing degrees in mission-relevant fields and postdoctoral fellowships for their contributions to Department of Homeland Security research projects. Students receive professional mentoring and complete a summer internship to connect academic interests with homeland security initiatives. Postdoctoral scholars are also mentored by DHS scientists.

2003

$4.7

$10.7

The program links graduate science training with the research needs of state and federal agencies, and provides students with one-onone mentoring by federal research scientists working on both applied and basic research needs of interest to the program. Program cooperators and partners provide graduate training opportunities and support.

1936

$15.3

$15

To build the capacity of community colleges to train in high-growth, high-demand industries and to actually train workers in those industries through partnerships that also include workforce investment boards and employers.

2005

$0

$250

2002

$106.6

$73.6

Department of HomelandS ecurity University Programs

Department of theI nterior Cooperative Research Units Program

Department of Labor Community College/Community Based Job Training Grant Initiative

National Aeronautics andS pace Administration Minority University Research Education Program

To expand and advance NASA’s scientific and technological base through collaborative efforts with Historically Black Colleges and Universities (HBCU) and other minority universities (OMU), including Hispanic-serving institutions and Tribal colleges and universities. This program also provides K-12 awards to build and support successful pathways for students to progress to the next level of mathematics and science, through a college preparatory curriculum, and enrollment in college. Higher-education awards are also given that seek to improve the rate at which underrepresented minorities are awarded degrees in STEM disciplines through increased research training and exposure to cutting-edge technologies that better prepare them to enter STEM graduate programs, the NASA workforce pipeline, and employment in NASA-related industries.

224

United States Government Accountability Office Table 22. Continued Funding (in millions of dollars)

Program

Description

Higher Education

a

First year

2004

2005

The Higher Education Program focuses on supporting institutions of higher education in strengthening their research capabilities and providing opportunities that attract and prepare increasing numbers of students for NASA-related careers. The research conducted by the institutions will contribute to the research needs of NASA’s Mission Directorates. The student projects serve as a major link in the student pipeline for addressing NASA’s human capital strategies and the President’s management agenda by helping to build, sustain, and effectively deploy the skilled, knowledgeable, diverse, and high-performing workforce needed to meet the current and emerging needs of government and its citizens.

2002

$77.4

$62.4

Elementary and Secondary Education

To increase the rigor of STEM experiences provided to K-12 students through workshops, summer internships, and classroom activities; provide high-quality professional development to teachers in STEM through NASA programs; develop technological avenues through the NASA Web site that will allow families to have common experiences with learning about space exploration; encourage inquiry teaching in K-12 classrooms; improve the content and focus of grade level/science team meetings in NASA Explorer Schools; and share the knowledge gained through the Educator Astronaut Program with teachers, students, and families.

2002

$31.3

$23.2

Informal Education

The principal purpose of the informal education program is to support projects designed to increase public interest in, understanding of, and engagement in STEM activities. The goal of all informal education programs is an informed citizenry that has access to the ideas of science and engineering and understands its role in enhancing the quality of life and the health, prosperity, welfare, and security of the nation. Informal learning is self-directed, voluntary, and motivated mainly by intrinsic interests, curiosity, exploration, and social interaction.

2002

$5.5

$10.2

Math and Science Partnership (MSP)Program

The MSP is a major research and development effort that supports innovative partnerships to improve kindergarten through grade 12 student achievement in mathematics and science. MSP projects are expected to both raise the achievement levels of all students and significantly reduce achievement gaps in the mathematics and science performance of diverse student populations. Successful projects serve as models that can be widely replicated in educational practice to improve the mathematics and science achievement of all the nation’s students.

2002

$138.7

$79.4

Graduate Research Fellowship Program (GRFP)

The purpose of the GRFP is to ensure the vitality of the scientific and technological workforce in the United States and to reinforce its diversity. The program recognizes and supports outstanding graduate students in the relevant science and engineering disciplines who are pursuing research-based master’s and doctoral degrees. NSF fellows are expected to become knowledge experts who can contribute significantly to research, teaching, and innovations in science and engineering.

1952

$96

$96.6

Program

Description

Integrative Graduate Education and Research Traineeship Program

NationalS cienceF oundation

a

Funding (in millions of dollars)

First year

2004

2005

This program provides support to universities for student positions in interdisciplinary areas of science and engineering. Traineeships focus on multidisciplinary and intersectoral research opportunities and prepare future faculty in effective teaching methods, applications of advanced educational technologies, and student mentoring techniques.

1998

$67.7

$69

Teacher Professional Continuum

The program addresses critical issues and needs regarding the recruitment, preparation, induction, retention, and lifelong development of kindergarten through grade 12 STEM teachers. Its goals are to improve the quality and coherence of teacher learning experiences across the continuum through research that informs teaching practice and the development of innovative resources for the professional development of kindergarten through grade 12 STEM teachers.

2004

$61.5

$60.2

Research Experiences for Undergraduates

This program supports active participation by undergraduate students in research projects in any of the areas of research funded by the National Science Foundation. The program seeks to involve students in meaningful ways in all kinds of research—whether disciplinary, interdisciplinary, or educational in focus—linked to the efforts of individual investigators, research groups, centers, and national facilities. Particular emphasis is given to the recruitment of women, minorities, and persons with disabilities.

1987

$51.7

$51.1

Graduate Teaching Fellows in This program supports fellowships and associated training that K-12 Education enable graduate students in NSF-supported STEM disciplines to acquire additional skills that will broadly prepare them for professional and scientific careers. Through interactions with teachers, graduate students can improve communication and teaching skills while enriching STEM instruction in kindergarten through grade 12 schools. This program also provides institutions of higher education with an opportunity to make a permanent change in their graduate programs by including partnerships with schools in a manner that will mutually benefit faculties and students.

1999

$49.8

$49.9

Advanced Technological Education (ATE)

With an emphasis on 2-year colleges, the ATE program focuses on the education of technicians for the high-technology fields that drive our nation’s economy. The program involves partnerships between academic institutions and employers to promote improvement in the education of science and engineering technicians at the undergraduate and secondary school levels. The ATE program supports curriculum development, professional development of college faculty and secondary school teachers, career pathways to 2-year colleges from secondary schools and from 2-year colleges to 4-year institutions, and other activities. The program also invites proposals focusing on applied research relating to technician education.

1994

$45.9

$45.1

Course, Curriculum, and Laboratory Improvement

This program emphasizes projects that build on prior work and contribute to the knowledge base of undergraduate STEM education research and practice. In addition, projects should contribute to building a community of scholars who work in related areas of undergraduate education.

1999

$40.7

$40.6

Higher Education: Federal Science, Technology, Engineering…

225

Table 22. Continued Funding (in millions of dollars)

Program

Description

Research on Learning and Education

a

First year

2004

2005

The program seeks to capitalize on important developments across a wide range of fields related to human learning and to STEM education. It supports research across a continuum that includes (1) the biological basis of human learning; (2) behavioral, cognitive, affective, and social aspects of STEM learning; (3) STEM learning in formal and informal educational settings; (4) STEM policy research; and (5) the diffusion of STEM innovations.

2000

$39.4

$38.2

Computer Science, Engineering, and Mathematics Scholarships

This program supports scholarships for academically talented, financially needy students, enabling them to enter the hightechnology workforce following completion of an associate, baccalaureate, or graduate-level degree in computer science, computer technology, engineering, engineering technology, or mathematics. Academic institutions apply for awards to support scholarship activities and are responsible for selecting scholarship recipients, reporting demographic information about student scholars, and managing the project at the institution.

1999

$33.9

$75

Louis Stokes Alliances for Minority Participation

The program is aimed at increasing the quality and quantity of students successfully completing STEM baccalaureate degree programs and increasing the number of students interested in, academically qualified for, and matriculated into programs of graduate study. It also supports sustained and comprehensive approaches that facilitate achievement of the long-term goal of increasing the number of students who earn doctorates in STEM, particularly those from populations underrepresented in STEM fields.

1991

$33.3

$35

Centers for Learning and Teaching

The program focuses on the advanced preparation of STEM educators, as well as the establishment of meaningful partnerships among education stakeholders, especially Ph.D.-granting institutions, school systems, and informal education performers. Its goals are to renew and diversify the cadre of leaders in STEM education; to increase the number of kindergarten through undergraduate educators capable of delivering high-quality STEM instruction and assessment; and to conduct research into STEM education issues of national import, such as the nature of learning, teaching strategies, and reform policies and outcomes.

2000

$30.8

$28.4

Instructional Materials Development

This program contains three components. It supports (1) the creation and substantial revision of comprehensive curricula and supplemental materials that are research-based, enhance classroom instruction, and reflect standards for science, mathematics, and technology education developed by professional organizations; (2) the creation of tools for assessing student learning that are tied to nationally developed standards and reflect the most current thinking on how students learn mathematics and science; and (3) research for development of this program and projects.

1983

$29.3

$28.5

Program

Description

Science, Technology, Engineering, and Mathematics Talent Expansion Program

The program seeks to increase the number of students (U.S. citizens or permanent residents) receiving associate or baccalaureate degrees in established or emerging fields within STEM. Type 1 proposals that provide for full implementation efforts at academic institutions are solicited. Type 2 proposals that support educational research projects on associate or baccalaureate degree attainment in STEM are also solicited.

Historically Black Colleges and Universities (HBCU) Undergraduate Program

Interagency Education Research Initiative

Funding (in millions of dollars)

First year

a

2004

2005

2002

$25

$25.3

This program provides awards to enhance the quality of STEM instructional and outreach programs at HBCUs as a means to broaden participation in the nation’s STEM workforce. Project strategies include curriculum enhancement, faculty professional development, undergraduate research, academic enrichment, infusion of technology to enhance STEM instruction, collaborations with research institutions and industry, and other activities that meet institutional needs.

1998

$23.8

$25.2

This is a collaborative effort with the U.S. Department of Education. The goal is to support scientific research that investigates the effectiveness of educational interventions in reading, mathematics, and the sciences as they are implemented in varied school settings with diverse student populations.

1999

$23.6

$13.8

Information Technology The program is designed to increase the opportunities for students Experiences for Students and and teachers to learn about, experience, and use information Teachers technologies within the context of STEM, including information technology courses. It is in direct response to the concern about shortages of technology workers in the United States. It has two components: (1) youth-based projects with strong emphasis on career and educational paths and (2) comprehensive projects for students and teachers.

2003

$20.9

$25

Enhancing the Mathematical Sciences Workforce in the 21st Century

The long-range goal of this program is to increase the number of U.S. citizens, nationals, and permanent residents who are well prepared in the mathematical sciences and who pursue careers in the mathematical sciences and in other NSF-supported disciplines.

2004

$20.6

$20.7

Centers of Research Excellence in Science and Technology

This program makes resources available to significantly enhance the research capabilities of minority-serving institutions through the establishment of centers that effectively integrate education and research. It promotes the development of new knowledge, enhancements of the research productivity of individual faculty, and an expanded diverse student presence in STEM disciplines.

1987

$19.8

$15.9

ADVANCE: Increasing the Participation and Advancement of Women in Academic Science and Engineering Careers

The program goal is to increase the representation and advancement of women in academic science and engineering careers, thereby contributing to the development of a more diverse science and engineering workforce. Members of underrepresented minority groups and individuals with disabilities are especially encouraged to apply.

2001

$19.4

$19.8

226

United States Government Accountability Office Table 22. Continued Funding (in millions of dollars)a

Program

Description

First year

Federal Cyber Service: Scholarship for Service

2004

2005

This program seeks to increase the number of qualified students entering the fields of information assurance and computer security and to increase the capacity of the United States’ higher education enterprise to continue to produce professionals in these fields to meet the needs of our increasingly technological society. The program has two tracks: provides funds to colleges and universities to (1) award scholarships to students to pursue academic programs in the information assurance and computer security fields for the final 2 years of undergraduate study, or for 2 years of master’s-level study, or for the final 2 years of Ph.D.-level study, and (2) improve the quality and increase the production of information assurance and computer security professionals.

2001

$15.8

$14.1

Alliances for Graduate Education and the Professoriate

This program is intended to increase significantly the number of domestic students receiving doctoral degrees in STEM, with special emphasis on those population groups underrepresented in these fields. The program is interested in increasing the number of minorities who will enter the professoriate in these disciplines. Specific objectives are to develop (1) and implement innovative models for recruiting, mentoring, and retaining minority students in STEM doctoral programs, and (2) effective strategies for identifying and supporting underrepresented minorities who want to pursue academic careers.

1998

$15.3

$14.8

Research on Gender in Science and Engineering

The program seeks to broaden the participation of girls and women in all fields of STEM education by supporting research, dissemination of research, and extension services in education that will lead to a larger and more diverse domestic science and engineering workforce. Typical projects will contribute to the knowledge base addressing gender-related differences in learning and in the educational experiences that affect student interest, performance, and choice of careers, and how pedagogical approaches and teaching styles, curriculum, student services, and institutional culture contribute to causing or closing gender gaps that persist in certain fields.

1993

$10

$9.8

Tribal Colleges and Universities Program

This program provides awards to enhance the quality of STEM instructional and outreach programs, with special attention to the use of information technologies at Tribal colleges and universities, Alaskan Native-serving institutions, and Native Hawaiian-serving institutions. Support is available for the implementation of comprehensive institutional approaches to strengthen STEM teaching and learning in ways that improve access to, retention within, and graduation from STEM programs, particularly those that have a strong technological foundation. Through this program, assistance is provided to eligible institutions in their efforts to bridge the digital divide and prepare students for careers in information technology, science, mathematics, and engineering fields.

2001

$10

$9.8

Program

Description

Funding (in millions of dollars)

a

First year

2004

2005

1998

$32.5

$32.5

Department of Transportation University Transportation Centers Program (UTC)

The UTC program’s mission is to advance U.S. technology and expertise in the many disciplines comprising transportation through the mechanisms of education, research, and technology transfer at university-based centers of excellence. The UTC program’s goals include (1) developing a multidisciplinary program of coursework and experiential learning that reinforces the transportation theme of the center; (2) increasing the numbers of students, faculty, and staff who are attracted to and substantially involved in the undergraduate, graduate, and professional programs of the center; and (3) having students, faculty, and staff who reflect the growing diversity of the U.S. workforce and are substantially involved in the undergraduate, graduate, and professional programs of the center. Source: GAO survey responses from 13 federal agencies.

a The dollar amounts for fiscal years 2004 and 2005 contain actual and estimated program funding levels.

Appendix IV: Data on Students and Graduates in STEM Fields Table 23 provides estimates for the numbers of students in science, technology, engineering, and mathematics (STEM) fields by education level for the 1995-1996 and 2003-2004 academic years. Tables 24 and 25 provide additional information regarding students in STEM fields by gender for the 1995-1996 and 2003-2004 academic years. Table 26 provides

Higher Education: Federal Science, Technology, Engineering…

227

additional information regarding graduates in STEM fields by gender for the 1994-1995 and 2002-2003 academic years. Appendix V contains confidence intervals for these estimates. Table 23. Estimated Numbers of Students in STEM Fields by Education Level for Academic Years 1995-1996 and 2003-2004 Education level/STEM field

Academic year 1995-1996

Academic year 2003-2004

Percentage change

Bachelor’s level Total 2,218,510 2,876,721 30 Agricultural sciences 101,885 87,025 b Biological sciences 407,336 351,595 -14 Computer sciences 261,139 456,303 75 Engineering 363,504 422,230 16 Mathematics 57,133 64,307 b Physical sciences 107,832 129,207 b Psychology 309,810 409,827 32 Social sciences 536,487 825,495 54 Technology 73,384 130,733 78 Master’s level Total 321,293 403,200 25 Agricultural sciences a 12,977 a Biological sciences 34,701 19,467 -44 Computer sciences 49,071 58,939 b Engineering 66,296 90,234 b Mathematics a 12,531 a Physical sciences a 22,008 a Psychology 30,008 31,918 b Social sciences 82,177 144,895 76 Technology a 10,231 a Doctoral level Total 217,395 198,504 b Agricultural sciences a 5,983 a Biological sciences a 33,884 a Computer sciences a 9,196 a Engineering 32,181 35,687 b Mathematics a 9,412 a Physical sciences 38,058 24,973 b Psychology 30,291 33,994 b Social sciences 54,092 42,464 b Technology a 2,912 a Source: GAO calculations based upon NPSAS data. Note: Enrollment totals differ from those cited in table 9 because table 9 includes students enrolled in certificate, associate’s, other undergraduate, first-professional degree, and post-bachelor’s or postmaster’s certificate programs. a Sample sizes are insufficient to accurately produce estimates. b Changes between academic years 1995-1996 and 2003-2004 are not statistically significant at the 95-percent confidence level. See table 30 for significance of percentage changes.

228

United States Government Accountability Office

Table 24. Estimated Percentages of Students by Gender and STEM Field for Academic Years 1995-1996 and 2003-2004 Male Female Agricultural Percent: 1995-1996 Percent: 2003-2004 Percent: 1995-1996 Percent: 2003-2004 sciences Total 58 55 42 45 Bachelor’s 56 54 44 46 Master’s a a a a Doctorate a 61 a 39 Biological sciences Total 46 42 54 58 Bachelor’s 45 42 55 58 Master’s a 26 a 74 Doctorate a 50 a 50 Computer sciences Total 67 76 33 24 Bachelor’s 69 77 31 23 Master’s a 69 a 31 Doctorate a 72 a 28 Engineering Total 83 83 17 17 Bachelor’s 83 83 17 17 Master’s a 81 a 19 Doctorate a 78 a 22 Mathematics Total 62 55 38 45 Bachelor’s 57 54 43 46 Master’s a a a a Doctorate a 68 a 32 Physical sciences Total 62 56 38 44 Bachelor’s 56 53 44 47 Master’s a a a a Doctorate a 68 a 32 Psychology Total 26 26 74 74 Bachelor’s 26 26 74 74 Master’s a 21 a 79 Doctorate a 30 a 70 Social sciences Total 54 41 46 59 Bachelor’s 52 42 48 58 Master’s 51 35 49 65 Doctorate 83 46 17 54 Technology Total 89 81 11 19 Bachelor’s 88 81 12 19 Master’s a a a a Doctorate a a a a Source: GAO calculations based upon NPSAS data. a Sample sizes are insufficient to accurately produce estimates.

Higher Education: Federal Science, Technology, Engineering…

229

Table 25. Estimated Number of Women Students and Percentage Change by Education Level and STEM Field for Academic Years 1995-1996 and 2003-2004 Number of women students Education level/STEM field

1995-1996

2003-2004

Percentage change in women students b b b b b b +33 +84 +184 a a a a a a b +133 a a a a a a a a +143 a

Bachelor’s level Agricultural sciences 44,444 39,702 Biological sciences 222,323 203,038 Computer sciences 82,013 104,824 Engineering 59,985 70,353 Mathematics 24,597 29,791 Physical sciences 47,421 60,203 Psychology 229,772 304,712 Social sciences 258,023 475,544 Technology 8,871 25,227 Master’s level Agricultural sciences a a Biological sciences a 14,415 Computer sciences a 18,000 Engineering a 17,042 Mathematics a 5,562 Physical sciences a 8,497 Psychology 23,857 25,342 Social sciences 40,395 94,169 Technology a 1,280 Doctoral level Agricultural sciences a 2,353 Biological sciences a 17,074 Computer sciences a 2,556 Engineering a 7,868 Mathematics a 3,042 Physical sciences a 8,105 Psychology a 23,843 Social sciences 9,440 22,931 Technology a 692 Source: GAO calculations based upon NPSAS data. a Sample sizes are insufficient to accurately produce estimates. b Changes between academic years 1995-1996 and 2003-2004 are not statistically significant at the95-percent confidence level. See table 29 for confidence intervals.

Table 26. Comparisons in the Percentage of STEM Graduates by Field and Gender for Academic Years 1994-1995 and 2002-2003 Percentage Percentage Percentage Percentage graduates, men, graduates, men, graduates, women graduates, women 1994-1995 2002-2003 1994-1995 2002-2003

STEM Degree/field Bachelor’s degree Biological/agricultural sciences Earth, atmospheric, and ocean sciences Engineering Mathematics and computer sciences Physical sciences Psychology Social sciences

50 66

40 58

50 34

60 42

83 65 64 27 50

80 69 58 22 45

17 35 36 73 50

20 31 42 78 55

230

United States Government Accountability Office Table 26. Continued Percentage Percentage Percentage Percentage graduates, men, graduates, men, graduates, women graduates, women 1994-1995 2002-2003 1994-1995 2002-2003

Technology 90 Master’s degree Biological/agricultural sciences 52 Earth, atmospheric, and ocean 70 sciences Engineering 84 Mathematics and computer sciences 70 Physical sciences 70 Psychology 28 Social sciences 51 Technology 81 Doctoral degree Biological/agricultural sciences 63 Earth, atmospheric, and ocean 78 sciences Engineering 88 80 Mathematics and computer sciences Physical sciences 76 Psychology 38 Social sciences 62 Technology 89 Source: GAO calculations based upon IPEDS data.

88

10

12

45 59

48 30

55 41

79 66 64 23 45 73

16 30 30 72 49 19

21 34 36 77 55 27

57 72

37 22

43 28

83 76 73 31 55 100

12 20 24 62 38 11

17 24 27 69 45 0

Appendix V: Confidence Intervals for Estimates of Students at the Bachelor’s, Master’s, and Doctoral Levels Because the National Postsecondary Student Aid Study (NPSAS) sample is a probability sample of students, the sample is only one of a large number of samples that might have been Table 27. Estimated Changes in the Numbers and Percentages of Students in the STEM and Non-STEM Fields across All Education Levels, Academic Years 1995-1996 and 2003-2004 (95 percent confidence intervals) Lower and upper bounds of 95 percent confidence interval

STEM field

Non-STEM field

Lower bound: number of students: 1995-1996

3,941,589

14,885,171

Upper bound: number of students: 1995-1996

4,323,159

15,601,065

Lower bound: percentage of students: 1995-1996

20

78

Upper bound: percentage of students: 1995-1996

22

80

Lower bound: number of students: 2003-2004

4,911,850

16,740,049

Upper bound: number of students: 2003-2004

5,082,515

17,025,326

Lower bound: percentage of students: 2003-2004

22

77

Upper bound: percentage of students: 2003-2004

23

78

Lower bound: percentage change: 1995/96-2003/04

15

8

Upper bound: percentage change: 1995/96-2003/04

26.9

13.5

Source: GAO calculations based upon 1995-1996 and 2003-2004 NPSAS data. Note: The totals for STEM and non-STEM enrollments include students in addition to the bachelor’s, master’s, and doctorate education levels. These totals also include students enrolled in certificate, associate’s, other undergraduate, first-professional degree, and post-bachelor’s or post-master’s certificate programs. The percentage changes between the 1995-1996 and 2003-2004 academic years for STEM and non-STEM students are statistically significant.

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231

drawn. Since each sample could have provided different estimates, confidence in the precision of the particular sample’s results is expressed as a 95-percent confidence interval (for example, plus or minus 4 percentage points). This is the interval that would contain the actual population value for 95 percent of the samples that could have been drawn. As a result, we are 95 percent confident that each of the confidence intervals in this report will include the true values in the study population. The upper and lower bounds of the 95 percent confidence intervals for each estimate relied on in this report are presented in the following tables. Table 28. Numbers of Students by Education Level in all STEM Fields for Academic Years 1995-1996 and 2003-2004 (95 percent confidence intervals) Total

Agricultural Sciences

Biological Sciences

Computer Sciences

Engineering

Mathematics

Physical Sciences

Psychology

Social Sciences

Technology

Total

Bachelors

Masters

Doctorate

Lower bound: Number of Students: 1995-1996

2,633,867

2,114,316

271,208

171,824

Upper bound: Number of Students: 1995-1996

2,880,529

2,322,704

377,821

271,230

Lower bound: Number of Students: 2003-2004

3,411,004

2,819,206

366,141

185,230

Upper bound: Number of Students: 2003-2004

3,545,844

2,934,236

442,938

212,471

Lower bound: Number of Students: 1995-1996

93,346

78,241

a

a a

Upper bound: Number of Students: 1995-1996

151,132

130,144

a

Lower bound: Number of Students: 2003-2004

93,543

76,472

7,296

4,661

Upper bound: Number of Students: 2003-2004

119,613

98,590

21,202

7,553

Lower bound: Number of Students: 1995-1996

416,315

360,553

18,883

a

Upper bound: Number of Students: 1995-1996

524,615

454,119

57,066

a

Lower bound: Number of Students: 2003-2004

383,277

330,834

13,728

30,401

Upper bound: Number of Students: 2003-2004

427,502

372,355

26,694

37,367

Lower bound: Number of Students: 1995-1996

275,804

224,616

31,634

a

Upper bound: Number of Students: 1995-1996

363,084

297,662

71,242

a

Lower bound: Number of Students: 2003-2004

495,359

428,927

47,669

7,427

Upper bound: Number of Students: 2003-2004

554,747

483,679

70,210

11,243

Lower bound: Number of Students: 1995-1996

411,868

321,464

45,912

16,620

Upper bound: Number of Students: 1995-1996

516,391

405,544

90,768

54,155

Lower bound: Number of Students: 2003-2004

514,794

400,252

63,632

32,113

Upper bound: Number of Students: 2003-2004

583,058

444,208

116,835

39,261

Lower bound: Number of Students: 1995-1996

68,083

42,910

a

a a

Upper bound: Number of Students: 1995-1996

119,165

74,456

a

Lower bound: Number of Students: 2003-2004

75,705

55,314

7,869

7,687

Upper bound: Number of Students: 2003-2004

97,848

74,318

18,867

11,392

Lower bound: Number of Students: 1995-1996

139,416

87,966

a

21,279

Upper bound: Number of Students: 1995-1996

214,274

130,658

a

60,546

Lower bound: Number of Students: 2003-2004

160,895

116,479

14,944

22,043

Upper bound: Number of Students: 2003-2004

192,534

142,894

31,092

27,903

Lower bound: Number of Students: 1995-1996

327,359

271,188

17,600

16,929

Upper bound: Number of Students: 1995-1996

416,804

348,432

47,037

48,601

Lower bound: Number of Students: 2003-2004

449,858

385,660

24,218

27,846

Upper bound: Number of Students: 2003-2004

502,696

433,995

41,116

40,142

Lower bound: Number of Students: 1995-1996

608,199

478,659

60,792

33,489

Upper bound: Number of Students: 1995-1996

742,107

594,315

103,562

79,414

Lower bound: Number of Students: 2003-2004

974,279

791,462

125,457

38,291

Total

Bachelors

Masters

Doctorate

Upper bound: Number of Students: 2003-2004

1,052,506

859,527

164,333

46,636

Lower bound: Number of Students: 1995-1996

63,910

57,446

a

a

Upper bound: Number of Students: 1995-1996

104,308

92,251

a

a

Lower bound: Number of Students: 2003-2004

130,347

118,492

5,556

1,814

Upper bound: Number of Students: 2003-2004

158,418

143,848

17,158

4,421

Source: GAO calculations based upon 1995-1996 and 2003-2004 NPSAS data. a Sample sizes are insufficient to accurately produce estimates.

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Table 29. Estimated Numbers and Percentage Changes in Women Students in STEM Fields, Academic Years 1995-1996 and2003-2004 (95 percent confidence intervals)

Source: GAO calculations based upon NPSAS data. a Sample sizes are insufficient to accurately produce estimates.

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Table 30. Estimated Percentage Changes in Bachelor’s, Master’s, and Doctoral Students in STEM Fields, Academic Years 1995-1996 and 2003-2004 (95 percent confidence intervals)

Lower and upper bounds of 95 percent confidence interval STEM fields

Percentage change in academic years 1995-1996 and 2003-2004

Total

Bachelor’s

Master’s

Doctoral

Agricultural sciences

Lower bound: percentage change

-34.8

-38.7

a

a

11.9

9.5

a

a

no

no

a

a

Lower bound: percentage change

-24.4

-24.8

-79.6

a

Upper bound: percentage change

-2.6

-2.5

-8.3

a

Upper bound: percentage change Statistically significant Biological sciences

yes

yes

yes

a

Lower bound: percentage change

41.1

48.1

-34.8

a

Upper bound: percentage change

89.5

101.3

75

a

Statistically significant

yes

yes

no

a

Lower bound: percentage change

3.5

1.4

-27.5

-55.4

Upper bound: percentage change

33.8

30.9

99.7

77.2

yes

yes

no

no

Lower bound: percentage change

-33.5

-21.8

a

a

Upper bound: percentage change

23

46.9

a

a a

Statistically significant Computer sciences

Engineering

Statistically significant Mathematics

no

no

a

Lower bound: percentage change

-21.7

-6.6

a

-70.2

Upper bound percentage change

24.4

46.3

a

1.4

no

no

a

no

Lower bound: percentage change

11.7

14

-51.2

-48.8

Upper bound: percentage change

45.4

50.5

63.9

73.3

yes

yes

no

no

Lower bound: percentage change

34.6

36.1

24.7

-59.3

Upper bound: percentage change

66.5

71.6

127.9

16.3

yes

yes

yes

no

Lower bound: percentage change

30

33.4

a

a

Upper bound: percentage change

119.6

122.9

a

a

yes

yes

a

a

Lower bound: percentage change

20

23.1

1.8

-29.5

Upper bound: percentage change

32.3

36.3

49.2

12.1

yes

yes

yes

no

Statistically significant Physical sciences

Statistically significant Psychology

Statistically significant Social sciences

Statistically significant Technology

Statistically significant

Total

Statistically significant

Source: GAO calculations based upon 1995-1996 and 2003-2004 NPSAS data. a Sample sizes are insufficient to accurately produce estimates.

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Table 31. Estimates of STEM Students by Gender and Field for Academic Years 19951996 and 2003-2004 (95 percent confidence intervals)

Source: GAO calculations based upon 1995-1996 and 2003-2004 NPSAS data. a Sample sizes are insufficient to accurately produce estimates.

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Table 32. Estimates of Students for Selected Racial or Ethnic Groups in STEM Fields for All Education Levels and Fields for the Academic Years 1995-1996 and 2002-2003 (95 percent confidence intervals) Lower bound: number of students, academic year, 1995-1996

Race or ethnicity African American Hispanic

285,3

Asian/Pacific Islander

Upper bound: number of students, academic year, 1995-1996

Lower bound: number of students, academic year, 2003-2004

Upper bound: number of students, academic year, 2003-2004

303,832

416,502

577,854

639,114

81

446,621

461,738

515,423 367,377

247,347

330,541

322,738

Native American

11,464

28,103

30,064

47,694

Other/multiple minorities

17,708

44,434

150,264

183,174

Source: GAO Calculations based upon 1995-1996 and 2003-2004 NPSAS data.

Table 33. Estimates of International Students in STEM Fields by Education Levels for Academic Years 1995-1996 and 2003-2004 (95 percent confidence intervals)

Education level Total Bachelor’s

Lower bound: number of students, 1995-1996

Upper bound: number of students, 1995-1996

Lower bound: number of students, 2003-2004

2

142,192

154,466

186,322

12

102

4

47,684

125,950

154,911

155

523

80,81 20,25

Upper bound: number of students, 2003-2004

Lower bound: Upper bound: percentage percentage change change

Master’s

23,06

3

64,587

16,359

29,899

-76

-13

Doctoral

20,52

5

59,861

5,168

10,735

-90

-68

Source: GAO calculations based upon 1995-1996 and 2003-2004 NPSAS data.

Appendix VI: Confidence Intervals for Estimates of STEM Employment by Gender, Race or Ethnicity, and Wages and Salaries The current population survey (CPS) was used to obtain estimates about employees and wages and salaries in science, technology, engineering, and mathematics (STEM) fields. Because the current population survey (CPS) is a probability sample based on random selections, the sample is only one of a large number of samples that might have been drawn. Since each sample could have provided different estimates, confidence in the precision of the particular sample’s results is expressed as a 95 percent confidence interval (e.g., plus or minus 4 percentage points). This is the interval that would contain the actual population value for 95 percent of the samples that could have been drawn. As a result, we are 95 percent confident that each of the confidence intervals in this report will include the true values in the study population. We use the CPS general variance methodology to estimate this sampling error and report it as confidence intervals. Percentage estimates we produce from the CPS

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United States Government Accountability Office

data have 95 percent confidence intervals of plus or minus 6 percentage points or less. Estimates other than percentages have 95 percent confidence intervals of no more than plus or minus 10 percent of the estimate itself, unless otherwise noted. Consistent with the CPS documentation guidelines, we do not produce estimates based on the March supplement data for populations of less than 75,000. Table 34. Estimated Total Number of Employees by STEM Field between Calendar Years 1994 and 2003

STEM fields Science

Lower bound: calendar year 1994

Upper bound: calendar year 1994

Lower bound: calendar year 2003

Upper bound: calendar year 2003

Statistically significant

2,349,605

2,656,451

2,874,347

3,143,071

yes

Technology

1,285,

321

1,515,671

1,379,375

1,568,189

no

Engineering

1,668,

514

1,929,240

1,638,355

1,843,427

no

2,520,858

2,773,146

yes

Mathematics/ computer sciences

1,369,047

1,606,

395

Table 35. Estimated Numbers of Employees in STEM Fields by Gender for Calendar Years 1994 and 2003

STEM fields Science

Lower Lower Upper Upper Lower Upper Lower Upper bound: bound: bound: bound: bound: bound: bound: bound: calendar calendar calendar calendar calendar calendar calendar calendar year year year year year year year year 1994, 1994, 2003, 2003, Statistically 1994, 1994, 2003, 2003, Statistically women women women women significant men men men men significant 925,548

no

Technology

1,594,527 1,827,685 2,031,124 2,327,390 385,433

505,329

357,805

489,899

yes no

941,960 1,157,900

no

Engineering

107,109

174,669

126,947

210,407

no 1,538,198 1,777,778 1,440,510 1,703,920

no

Mathematics/ computer sciences

372,953

491,053

610,649

779,525

yes

708,673

875,171

863,785 1,046,445

733,358

959,765 1,151,681 1,805,505 2,098,325

yes

Table 36. Estimated Changes in STEM Employment by Gender for Calendar Years 1994 and 2003

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Table 37. Estimated Percentages of STEM Employees for Selected Racial or Ethnic Groups for Calendar Years 1994 and 2003

Table 38. Estimated Changes in Median Annual Wages and Salaries in the STEM Fields for Calendar Years 1994 and 2003

Appendix VII: Comments from the Department of Commerce

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Appendix VIII: Comments from the Department of Health and Human Services

239

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Appendix IX: Comments from the National Science Foundation

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Appendix X: Comments from the National Science and Technology Council

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Acknowledgments Cornelia M. Ashby, Carolyn M. Taylor, Assistant Director; Tim Hall, Analyst in Charge; Mark Ward; Dorian Herring; Patricia Bundy; Paula Bonin; Scott Heacock; Wilfred Holloway; Lise Levie; John Mingus; Mark Ramage; James Rebbe; and Monica Wolford made key contributions to this report.

Bibliography Congressional Research Service, Foreign Students in the United States: Policies and Legislation, RL31146, January 24, 2003, Washington, D.C. Congressional Research Service, Immigration: Legislative Issues on Nonimmigrant Professional Specialty (H-1B) Workers, RL30498, May 5, 2005, Washington, D.C. Congressional Research Service, Monitoring Foreign Students in the United States: The Student and Exchange Visitor Information System (SEVIS), RL32188, October 20, 2004, Washington, D.C. Congressional Research Service, Science, Engineering, and Mathematics Education: Status and Issues, 98-871 STM, April 27, 2004, Washington, D.C. Council on Competitiveness, Innovate America, December 2004, Washington, D.C. Council of Graduate Schools, NDEA 21: A Renewed Commitment to Graduate Education, June 2005, Washington, D.C. Institute of International Education, Open Doors: Report on International Educational Exchange, 2004, New York. Jackson, Shirley Ann, The Quiet Crisis: Falling Short in Producing American Scientific and Technical Talent, Building Engineering and Science Talent, September 2002, San Diego, California. NAFSA: Association of International Educators, In America’s Interest: Welcoming International Students, Report of the Strategic Task Force on International Student Access, January 14, 2003, Washington, D.C. NAFSA: Association of International Educators, Toward an International Education Policy for the United States: International Education in an Age of Globalism and Terrorism, May 2003, Washington, D.C. National Center for Education Statistics, Qualifications of the Public School Teacher Workforce: Prevalence of Out-of-Field Teaching 1987-88 to 1999-2000, May 2002, revised August 2004, Washington, D.C. National Science Foundation, The Science and Engineering Workforce Realizing America’s Potential, National Science Board, August 14, 2003, Arlington, Virginia. National Science Foundation, Science and Engineering Indicators, 2004, Volume 1, National Science Board, January 15, 2004, Arlington, Virginia. Report of the Congressional Commission on the Advancement of Women and Minorities in Science, Engineering and Technology Development. Land of Plenty: Diversity as America’s Competitive Edge in Science, Engineering, and Technology, September 2000. A Report to the Nation from the National Commission on Mathematics and Science Teaching for the 21st Century, Before It’s Too Late, September 27, 2000.

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Seymour, Elaine, and Nancy M. Hewitt, Talking about Leaving: Why Undergraduates Leave the Sciences, Westview Press, 1997, Boulder, Colorado. The National Academies, Policy Implications of International Graduate Students and Postdoctoral Scholars in the United States, 2005, Washington, D.C. U.S. Department of Education, National Center for Education Statistics, Institute of Education Sciences, The Nation’s Report Card, NAEP 2004: Trends in Academic Progress, July 2005, Washington, D.C. U.S. Department of Education, The Secretary’s Third Annual Report on Teacher Quality, Office of Postsecondary Education, 2004, Washington, D.C. U.S. Department of Homeland Security, 2003 Yearbook of Immigration Statistics, Office of Immigration Statistics, September 2004, Washington, D.C.

References [1] [2] [3] [4]

[5] [6] [7]

[8]

For the purposes of this report, we will use the term “agency” when referring to any of the 13 federal departments and agencies that responded to our survey. Core subjects include English, reading or language arts, mathematics, science, foreign languages, civics and government, economics, arts, history, and geography. Other federal programs that are not specifically designed to attract students to STEM education and occupations, such as Pell Grants, may provide financial assistance to students who obtain degrees in STEM fields. There are several types of visas that authorize people to study and work in the United States. F, or student, visas, are for study at 2- and 4-year colleges and universities and other academic institutions; the exchange visitor, or J, visas are for people who will be participating in a cultural exchange program; and M visas are for nonacademic study at institutions, such as vocational and technical schools. In addition, H-1B visas allow noncitizens to work in the United States. GAO, Border Security: Streamlined Visas Mantis Program Has Lowered Burden on Foreign Science Students and Scholars, but Further Refinements Needed, GAO-05-198 (Washington, D.C.: Feb. 18, 2005). Congressional Research Service, Science, Engineering, and Mathematics Education: Status and Issues, 98-871 STM, April 27, 2004, Washington, D.C. A specialty occupation is defined as one that requires the application of a body of highly specialized knowledge, and the attainment of at least a bachelor’s degree (or its equivalent), and the possession of a license or other credential to practice the occupation if required. GAO asked agencies to include STEM and related education programs with one or more of the following as a primary objective: (1) attract and prepare students at any education level to pursue coursework in STEM areas, (2) attract students to pursue degrees (2-year degrees through post doctoral) in STEM fields, (3) provide growth and research opportunities for college and graduate students in STEM fields, such as working with researchers and/or conducting research to further their education, (4) attract graduates to pursue careers in STEM fields, (5) improve teacher (pre-service, inservice, and postsecondary) education in STEM areas, and (6) improve or expand the capacity of institutions to promote or foster STEM fields.

Higher Education: Federal Science, Technology, Engineering… [9] [10] [11] [12] [13] [14] [15]

[16] [17]

[18] [19]

[20] [21] [22] [23] [24] [25] [26] [27] [28] [29]

245

The program funding levels, as provided by agency officials, contain both actual and estimated amounts for fiscal year 2004. Six survey respondents did not include the date the program was initially funded. Fiscal year 2005 funding levels were not available for all of the 207 STEM education programs. Three survey respondents did not identify the program goals. One survey respondent did not identify the type of assistance supported by the program. Two survey respondents did not identify the group targeted by the program. Lawful permanent residents, also commonly referred to as immigrants, are legally accorded the privilege of residing permanently in the United States. They may be issued immigrant visas by the Department of State overseas or adjusted to permanent resident status by the Department of Homeland Security in the United States. GAO, Program Evaluation: An Evaluation Culture and Collaborative Partnerships Help Build Agency Capacity,G AO-03-454 (Washington, D.C.: May 2, 2003). In 2004, we reported on women’s participation in federally funded science programs. Among other issues, this report discussed priorities pertaining to compliance with provisions of Title IX of the Education Amendments of 1972. For additional information, see GAO, Gender Issues: Women’s Participation in the Sciences Has Increased, but Agencies Need to Do More to Ensure Compliance with Title IX, GAO04-639, (Washington, D.C.: July 22, 2004). Institute of International Education, Open Doors: Report on International Educational Exchange, 2004, New York. On the basis of March 2004 CPS estimates, the Pew Hispanic Research Center reported that over 10 million unauthorized immigrants resided in the United States and that people of Hispanic and Latino origin constituted a significant portion of these unauthorized immigrants. Other minorities include Asian/Pacific Islanders and American Indian or Alaska Native. GAO, H-1B Foreign Workers: Better Controls Needed to Help Employers and Protect Workers, GAO/HEHS-00-157 (Washington, D.C.: Sept. 7, 2000). GAO, H-1B Foreign Workers: Better Tracking Needed to Help Determine H-1B Program’s Effects on U.S. Workforce, GAO-03-883 (Washington, D.C.: Sept. 10, 2003). National Science Foundation, Science and Engineering Indicators, 2004, Volume 1, National Science Board, January 15, 2004. National Center for Education Statistics, Qualifications of the Public School Teacher Workforce: Prevalence of Out-of-Field Teaching 1987-88 to 1999-2000, May 2002, revised August 2004,Washington, D.C. U.S. Department of Education, The Secretary’s Third Annual Report on Teacher Quality, Office of Postsecondary Education, 2004, Washington, D.C. U.S. Department of Education, National Center for Education Statistics, Institute of Education Sciences, The Nation’s Report Card, NAEP 2004: Trends in Academic Progress, July 2005, Washington, D.C. 27 Seymour, Elaine, and Nancy M. Hewitt, Talking about Leaving: Why Undergraduates Leave the Sciences, Westview Press, 1997, Boulder, Colorado. Seymour and Hewitt. GAO-04-639.

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[30] Report of the Congressional Commission on the Advancement of Women and Minorities in Science, Engineering and Technology Development, Land of Plenty: Diversity as America’s Competitive Edge in Science, Engineering, and Technology, September 2000. [31] NAFSA: Association of International Educators, In America’s Interest: Welcoming International Students, Report of the Strategic Task Force on International Student Access, January 14, 2003, Washington, D.C. [32] GAO, Border Security: Visa Process Should Be Strengthened as an Antiterrorism Tool, GAO-03-132NI (Washington, D.C.: Oct. 21, 2002). [33] GAO, Border Security: New Policies and Increased Interagency Coordination Needed to Improve Visa Process, GAO-03-1013T (Washington, D. C.: July 15, 2003). [34] GAO, Border Security: Improvements Needed to Reduce Time Taken to Adjudicate Visas for Science Students and Scholars, GAO-04-371 (Washington, D.C.: Feb. 25, 2004). [35] GAO-05-198. [36] GAO, Homeland Security: Performance of Information System to Monitor Foreign Students and Exchange Visitors Has Improved, but Issues Remain, GAO-04-690 (Washington, D.C.: June 18, 2004). [37] GAO, Homeland Security: Performance of Foreign Student and Exchange Visitor Information System Continues to Improve, but Issues Remain, GAO-05-440T (Washington, D.C.: March 17, 2005). [38] Report of the Congressional Commission on the Advancement of Women and Minorities in Science, Engineering and Technology Development, Land of Plenty: Diversity as America’s Competitive Edge in Science, Engineering, and Technology, September 2000.

Appendix I [1] [2] [3]

See Technical Paper 63RV:Current Population Survey—Design and Methodology, issued Mar. 2002. Electronic version available at http://www.censusgov/prod/ 2002pubs/tp63rv.pdf. See Technical Paper 63RV, page 11-4. Seymour, Elaine, and Nancy M. Hewitt, Talking about Leaving: Why Undergraduates Leave the Sciences, Westview Press, 1997, Boulder, Colorado.

In: Advances in Mathematics Research, Volume 8 Editor: Albert R. Baswell, pp. 247-275

ISBN: 978-1-60456-454-9 © 2009 Nova Science Publishers, Inc.

Chapter 8

SCIENCE, TECHNOLOGY, ENGINEERING, AND MATHEMATICS (STEM) EDUCATION ISSUES * AND LEGISLATIVE OPTIONS Jeffrey J. Kuenzi, Christine M. Matthew and Bonnie F. Mangan

Abstract There is growing concern that the United States is not preparing a sufficient number of students, teachers, and practitioners in the areas of science, technology, engineering, and mathematics (STEM). A large majority of secondary school students fail to reach proficiency in math and science, and many are taught by teachers lacking adequate subject matter knowledge. When compared to other nations, the math and science achievement of U.S. pupils and the rate of STEM degree attainment appear inconsistent with a nation considered the world leader in scientific innovation. In a recent international assessment of 15-year-old students, the U.S. ranked 28th in math literacy and 24th in science literacy. Moreover, the U.S. ranks 20th among all nations in the proportion of 24-year-olds who earn degrees in natural science or engineering. A recent study by the Government Accountability Office found that 207 distinct federal STEM education programs were appropriated nearly $3 billion in FY2004. Nearly threequarters of those funds and nearly half of the STEM programs were in two agencies — the National Institutes of Health and the National Science Foundation. Still, the study concluded that these programs are highly decentralized and require better coordination. Several pieces of legislation have been introduced in the 109th Congress that address U.S. economic competitiveness in general and support STEM education in particular. These proposals are designed to improve output from the STEM educational pipeline at all levels, and are drawn from several recommendations offered by the scientific and business communities.

*

Excerpted from CRS Report RL33434 dated July 26, 2006.

248

Jeffrey J. Kuenzi, Christine M. Matthew and Bonnie F. Mangan The objective of this report is to provide a useful context for these legislative proposals. To achieve this, the report first presents data on the state of STEM education and then examines the federal role in promoting STEM education. The report concludes with a discussion of selected legislative options currently being considered to improve STEM education. The report will be updated as significant legislative actions occur.

Introduction There is growing concern that the United States is not preparing a sufficient number of students, teachers, and professionals in the areas of science, technology, engineering, and mathematics (STEM).1 Although the most recent National Assessment of Educational Progress (NAEP) results show improvement in U.S. pupils’ knowledge of math and science, the large majority of students still fail to reach adequate levels of proficiency. When compared to other nations, the achievement of U.S. pupils appears inconsistent with the nation’s role as a world leader in scientific innovation. For example, among the 40 countries participating in the 2003 Program for International Student Assessment (PISA), the U.S. ranked 28th in math literacy and 24th in science literacy. Some attribute poor student performance to an inadequate supply of qualified teachers. This appears to be the case with respect to subject-matter knowledge: many U.S. math and science teachers lack an undergraduate major or minor in those fields — as many as half of those teaching in middle school math. Indeed, post-secondary degrees in math and physical science have steadily decreased in recent decades as a proportion of all STEM degrees awarded. While degrees in some STEM fields (particularly biology and computer science) have increased in recent decades, the overall proportion of STEM degrees awarded in the United States has historically remained at about 17% of all postsecondary degrees awarded. Meanwhile, many other nations have seen rapid growth in postsecondary educational attainment — with particularly high growth in the number of STEM degrees awarded. According to the National Science Foundation, the United States currently ranks 20th among all nations in the proportion of 24-year-olds who earn degrees in natural science or engineering. Once a leader in STEM education, the United States is now far behind many countries on several measures. What has been the federal role in promoting STEM education? A recent study by the Government Accountability Office (GAO) found that 207 distinct federal STEM education programs were appropriated nearly $3 billion in FY2004.2 Nearly three-quarters of those funds supported 99 programs in two agencies — the National Institutes of Health (NIH) and the National Science Foundation (NSF). Most of the 207 programs had multiple goals, 1

In 2005 and early 2006, at least six major reports were released by highly respected U.S. academic, scientific, and business organizations on the need to improve science and mathematics education: The Education Commission of the States, Keeping America Competitive: Five Strategies To Improve Mathematics and Science Education, July 2005; The Association of American Universities, National Defense Education and Innovation Initiative, Meeting America’s Economic and Security Challenges in the 21st Century, January 2006; The National Academy of Sciences, Committee on Science, Engineering, and Public Policy, Rising Above the Gathering Storm: Energizing and Employing America for a Brighter Economic Future, February 2006; The National Summit on Competitiveness, Statement of the National Summit on Competitiveness: Investing in U.S. Innovation, December 2005; The Business Roundtable, Tapping America’s Potential: The Education for Innovation Initiative, July 2005; the Center for Strategic and International Studies, Waiting for Sputnik, 2005. 2 U.S. Government Accountability Office, Federal Science, Technology, Engineering, and Mathematics Programs and Related Trends, GAO-06-114, Oct. 2005.

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provided multiple types of assistance, and were targeted at multiple groups. The study concluded that these programs are highly decentralized and could benefit from stronger coordination, while noting that the creation of the National Science and Technology Council in 1993 was a step in the right direction.3 Several pieces of legislation have been introduced in the 109th Congress that would support STEM education in the United States. Many of the proposals in these bills have been influenced by the recommendations of several reports recently issued by the scientific, business, and policy-making communities. Of particular influence has been a report issued by the National Academy of Sciences (NAS), Rising Above the Gathering Storm: Energizing and Employing America for a Brighter Economic Future — also known as the “Augustine” report. Many of the recommendations appearing in the NAS report are also contained in the Administration’s American Competitiveness Initiative.4 Among the report’s many recommendations, five are targeted at improving STEM education. These five recommendations seek to increase the supply of new STEM teachers, improve the skills of current STEM teachers, enlarge the pre-collegiate pipeline, increase postsecondary degree attainment, and enhance support for graduate and early-career research. The purpose of this report is to put these legislative proposals into a useful context. The first section analyzes data from various sources to build a more thorough understanding of the status of STEM education in the United States. The second section looks at the federal role in promoting STEM education, providing a broad overview of nearly all of the programs in federal agencies and a detailed look at a few selected programs. Finally, the third section discusses legislative options currently being considered to improve STEM education. This discussion focuses primarily on the proposals that have seen congressional action to date.

STEM Education in the United States Elementary and Secondary Education Assessments of Math and Science Knowledge National-level assessment of U.S. students’ knowledge of math and science is a relatively recent phenomenon, and assessments in other countries that provide for international comparisons are even more recent. Yet the limited information available thus far is beginning to reveal results that concern many individuals interested in the U.S. educational system and the economy’s future competitiveness. The most recent assessments show improvement in U.S. pupils’ knowledge of math and science; however, the large majority still fail to reach adequate levels of proficiency. Moreover, when compared to other nations, the achievement of U.S. students is seen by many as inconsistent with the nation’s role as a world leader in scientific innovation. 3

4

These points were reiterated by Cornelia M. Ashby, Director of GAO’s Education, Workforce, and Income Security Team. Her testimony can be found at [http://edworkforce. house.gov/hearings/109th/ fc/competitiveness050306/wl5306.htm], as well as on the GAO website at [http://www.gao.gov/ new.items/d06702t.pdf]. Office of Science and Technology Policy, Domestic Policy Council, American Competitiveness Initiative — Leading the World In Innovation, Feb. 2006.

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The National Assessment of Educational Progress (NAEP) is the only nationally representative, continuing assessment of elementary and secondary students’ math and science knowledge. Since 1969, NAEP has assessed students from both public and nonpublic schools at grades 4, 8, and 12. Students’ performance on the assessment is measured on a 0500 scale, and beginning in 1990 has been reported in terms of the percentages of students attaining three achievement levels: basic, proficient, and advanced.5 Proficient is the level identified by the National Assessment Governing Board as the degree of academic achievement that all students should reach, and “represents solid academic performance. Students reaching this level have demonstrated competency over challenging subject matter.” In contrast, the board states that “Basic denotes partial mastery of the knowledge and skills that are fundamental for proficient work at a given grade.”6 The most recent NAEP administration occurred in 2005. Figure 1 displays the available results from the NAEP math tests administered between 1990 and 2005. Although the proportion of 4th and 8th grade students achieving the proficient level or above has been increasing each year, overall math performance has been quite low. The percentage performing at the basic level has not improved in 15 years. About two in five students continue to achieve only partial mastery of math. In 2005, only about one-third of 4th and 8th grade students performed at the proficient level in math — 36% and 30%, respectively.7 The remainder of students — approximately 20% of 4th graders and just over 30% of 8th graders — scored below the basic level. For 12th grade students, the most recently published NAEP results are from the 2000 assessments.8 Only 17% of 12th grade students performed at the proficient or higher level on the math assessment that year.9 This figure was only slightly higher than the previous two assessments in 1996 (16%) and 1992 (15%), but was significantly higher, in statistical terms, than the 12% reported proficient in 1990. Progress aside, it appears that very few students graduate from U.S. high schools with math skills considered adequate. More than half of all 12th grade students performed below even the basic level in each assessment year except 1996. Similarly low levels of achievement have been found with regard to knowledge of science. Less than one-third of 4th and 8th grade students and less than one-fifth of 12th grade students score at or above proficient in science. In 2005, the percentage of 4th, 8th, and 12th grade students scoring proficient or above was 29%, 29%, and 18%, respectively; compared to 27%, 30%, and 18% in 2000 and 28%, 29%, and 21% in 1996.10

5

For more information on NAEP and other assessments, see CRS Report RL31407, Educational Testing: Implementation of ESEA Title I-A Requirements Under the No Child Left Behind Act, by Wayne C. Riddle. 6 The National Assessment Governing Board is an independent, bipartisan group created by Congress in 1988 to set policy for the NAEP. More information on the board and NAEP achievement levels can be found at [http://www.nagb.org/]. 7 U.S. Department of Education, National Center for Education Statistics, The Nation’s Report Card: Mathematics 2005, (NCES 2006-453), Oct. 2005, p. 3. 8 The reporting delay for the 2005 grade 12 math assessments is due, in part, to substantial changes made in the assessment framework, and will not include comparisons to results from previous years. 9 U.S. Department of Education, National Center for Education Statistics, The Nation’s Report Card: Mathematics 2000 (NCES 2001-517) Aug. 2001, Figure B. 10 U.S. Department of Education, National Center for Education Statistics, The Nation’s Report Card: Science 2005 (NCES 2006-466) May 2006, Figures 4, 14, and 24.

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Source: U.S. Department of Education, National Center for Education Statistics, The Nation’s Report Card, various years.

Figure 1. Percentages of Students Scoring Basic and Proficient in Math, Selected Years: 1990-2005.

U.S. Students Compared to Students in Other Nations Another relatively recent development in the area of academic assessment has been the effort by a number of nations to produce reliable cross-national comparison data.11 The Trends in International Mathematics and Science Study (TIMSS) assesses achievement in these subjects at grades 4 and 8 among students in several countries around the world. TIMSS has been administered to 4th grade students on two occasions (1995 and 2003) and to 8th grade students on three occasions (1995, 1999, and 2003). In the latest administration, 25 countries participated in assessments of their 4th grade students, and 45 countries participated in assessments of their 8th grade students. Unlike NAEP, TIMSS results are reported only in terms of numerical scores, not achievement levels.

11

More information on the development of this assessment can be found in archived CRS Report 86-683, Comparison of the Achievement of American Elementary and Secondary Pupils with Those Abroad — The Examinations Sponsored by the International Association for the Evaluation of Educational Achievement (IEA), by Wayne C. Riddle (available on request).

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U.S. 4th grade pupils outscored the international average on the most recent TIMSS assessment.12 The international average score for all countries participating in the 2003 4th grade TIMSS was 495 in math and 489 in science.13 The average score for U.S. students was 518 in math and 536 in science. U.S. 4th grade students outscored students in 13 of the 24 countries participating in the math assessment in 2003. In science, U.S. students outperformed students in 16 of the 24 countries. Among the 10 Organization for Economic Co-operation and Development (OECD) member states participating in the 2003 TIMSS, U.S. 4th grade students ranked fourth in math and tied for second in science. U.S. 8th grade pupils also outscored the international average. Among 8th grade students, the international average on the 2003 TIMSS was 466 in math and 473 in science. The average score for U.S. students was 504 in math and 527 in science. Among the 44 countries participating in the 8th grade assessments in 2003, U.S. students outscored students in 25 countries in math and 32 countries in science. Twelve OECD countries participated in the 8th grade TIMSS in 2003 — five outscored the United States in math and three outscored the United States in science. TIMSS previously assessed students at grade 4 in 1995 and grade 8 in 1995 and 1999. Although there was no measurable difference between U.S. 4th graders’ average scores in 1995 and 2003, the standing of the United States declined relative to that of the 14 other countries participating in both math and science assessments. In math, U.S. 4th graders outperformed students in nine of these countries in 1995, on average, compared to six countries in 2003. In science, U.S. 4th graders outperformed students in 13 of these countries in 1995, on average, compared to eight countries in 2003. Among 8th graders, U.S. scores increased on both the math and science assessments between 1995 and 2003. The increase in scores translated into a higher ranking of the United States relative to other countries. In math, 12 of the 21 participating countries outscored U.S. 8th graders in 1995, while seven did so in 2003. In science, 15 of the 21 participating countries outscored U.S. 8th graders in 1995, while 10 did so in 2003. Table 1 displays the 2003 TIMSS math and science scores of 4th and 8th grade students by country (scores in bold are higher than the U.S. score). The Program for International Student Assessment (PISA) is an OECD-developed effort to measure, among other things, mathematical and scientific literacy among students 15 years of age — i.e., roughly at the end of their compulsory education.14 In 2003, U.S. students scored an average of 483 on math literacy —behind 23 of the 29 OECD member states that participated and behind four of the 11 non-OECD countries. The average U.S. student scored 491 on science literacy —behind 19 of the 29 OECD countries and behind three of the 11 non-OECD countries. Table 2 displays the 2003 PISA scores on math and science literacy by country (scores in bold are higher than the U.S. score). 12

Performance on the 1995 TIMSS assessment was normalized on a scale in which the average was set at 500 and the standard deviation at 100. Each country was weighted so that its students contributed equally to the mean and standard deviation of the scale. To provide trend estimates, subsequent TIMSS assessments are pegged to the 1995 average. 13 All the TIMSS results in this report were taken from, Patrick Gonzales, Juan Carlos Guzmán, Lisette Partelow, Erin Pahlke, Leslie Jocelyn, David Kastberg, and Trevor Williams, Highlights From the Trends in International Mathematics and Science Study (TIMSS) 2003 (NCES 2005 — 005), Dec. 2004. 14 Like the TIMSS, PISA results are normalized on a scale with 500 as the average score, and results are not reported in terms of achievement levels. In 2003, PISA assessments were administered in just over 40 countries.

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Table 1. TIMSS Scores by Grade and Country/Jurisdiction, 2003

International average United States United Kingdom Tunisia Sweden South Africa Slovenia Slovak Republic Singapore Serbia Scotland Saudi Arabia Russian Federation Romania Philippines Palestinian National Authority Norway New Zealand Netherlands Morocco Moldova, Republic of Malaysia Macedonia, Republic of Lithuania Lebanon Latvia Korea, Republic of Jordan Japan Italy Israel Iran, Islamic Republic of Indonesia Hungary Hong Kong SAR Ghana Estonia Egypt Cyprus Chinese Taipei Chile Bulgaria Botswana Belgium-Flemish Bahrain Australia Armenia

4th Math 495 518 531 339 — — 479 — 594 — 490 — 532 — 358 — 451 493 540 347 504 — — 534 — 536 — — 565 503 — 389 — 529 575 — — — 510 564 — — — 551 — 499 456

Grade Science 489 536 540 314 — — 490 — 565 — 502 — 526 — 332 — 466 520 525 304 496 — — 512 — 532 — — 543 516 — 414 — 530 542 — — — 480 551 — — — 518 — 521 437

8th Math 466 504 — 410 499 264 493 508 605 477 498 332 508 475 378 390 461 494 536 387 460 508 435 502 433 508 589 424 570 484 496 411 411 529 586 276 531 406 459 585 387 476 366 537 401 505 478

Grade Science 473 527 — 404 524 244 520 517 578 468 512 398 514 470 377 435 494 520 536 396 472 510 449 519 393 512 558 475 552 491 488 453 420 543 556 255 552 421 441 571 413 479 365 516 438 527 461

Source: U.S. Department of Education, National Center for Education Statistics, Highlights From the Trends in International Mathematics and Science Study (TIMSS) 2003, NCES 2005-005, Dec. 2004.

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OECD average United States Turkey Switzerland Sweden Spain Slovak Republic Portugal Poland Norway New Zealand Netherlands Mexico Luxembourg Korea, Republic of Japan Italy Ireland Iceland Hungary Greece Germany France Finland Denmark Czech Republic Canada Belgium Austria Australia Non-OECD Countries Uruguay United Kingdom Tunisia Thailand Serbia and Montenegro Russian Federation Macao SAR Liechtenstein Latvia Indonesia Hong Kong SAR

Math 500 483 423 527 509 485 498 466 490 495 524 538 385 493 542 534 466 503 515 490 445 503 511 544 514 517 533 529 506 524

Science 500 491 434 513 506 487 495 468 498 484 521 524 405 483 538 548 487 505 495 503 481 502 511 548 475 523 519 509 491 525

422 508 359 417 437 468 527 536 483 360 550

438 518 385 429 436 489 525 525 489 395 540

Source: U.S. Department of Education, National Center for Education Statistics, International Outcomes of Learning in Mathematics Literacy and Problem Solving, NCES 2005-003, Dec. 2004.

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Math and Science Teacher Quality Many observers look to the nation’s teaching force as a source of national shortcomings in student math and science achievement. A recent review of the research on teacher quality conducted over the last 20 years revealed that, among those who teach math and science, having a major in the subject taught has a significant positive impact on student achievement.15 Unfortunately, many U.S. math and science teachers lack this credential. The Schools and Staffing Survey (SASS) is the only nationally representative survey that collects detailed data on teachers’ preparation and subject assignments.16 The most recent administration of the survey for which public data are available took place during the 19992000 school year. That year, there were just under 3 million teachers in U.S. schools, about evenly split between the elementary and secondary levels. Among the nation’s 1.4 million public secondary school teachers, 13.7% reported math as their main teaching assignment and 11.4% reported science as their main teaching assignment.17 Nearly all public secondary school math and science teachers held at least a baccalaureate degree (99.7%), and most had some form of state teaching certification (86.2%) at the time of the survey.18 However, many of those who taught middle school (classified as grades 5-8) math and science lacked an undergraduate or graduate major or minor in the subject they taught. Among middle-school teachers, 51.5% of those who taught math and 40.0% of those who taught science did not have a major or minor in these subjects. By contrast, few of those who taught high school (classified as grades 9-12) math or science lacked an undergraduate or graduate major or minor in that subject. Among high school teachers, 14.5% of those who taught math and 11.2% of those who taught science did not have a major or minor in these subjects.19. Table 3 displays these statistics for teachers in eight subject areas. Table 3. Percentage of Middle and High School Teachers Lacking a Major or Minor in Subject Taught, 1999-2000. English Foreign language Mathematics Science Social science ESL/bilingual education Arts and music Physical/health education

Middle School 44.8% 27.2% 51.5% 40.0% 29.6% 57.6% 6.8% 12.6%

High School 13.3% 28.3% 14.5% 11.2% 10.5% 59.4% 6.1% 9.5%

Source: U.S. Department of Education, National Center for Education Statistics, Qualifications of the Public School Teacher Workforce: Prevalence of Out-of-Field Teaching 1987-88 to 1999-2000, NCES 2002603, May 2002.

15

Michael B. Allen, Eight Questions on Teacher Preparation: What Does the Research Say?, Education Commission of the States, July 2003. 16 The sample is drawn from the Department of Education Common Core of Data, which contains virtually every school in the country. 17 U.S. Department of Education, Digest of Education Statistics, 2004, NCES 2005-025, Oct. 2005, Table 67. 18 CRS analysis of Schools and Staffing Survey data, Mar. 29, 2006. 19 U.S. Department of Education, Qualifications of the Public School Teacher Workforce, May 2002, Tables B-11 and B-12.

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Given the link between teachers’ undergraduate majors and student achievement in math and science, these data appear to comport with some of the NAEP findings discussed earlier. Recall that those assessments revealed that only about one-third of 4th and 8th grade students performed at the proficient or higher level in math and science. On the other hand, at the high school level, the data seem to diverge. While four-fifths of math and science teachers at this level have a major in the subject, only two-fifths of high school students scored proficient or above on the NAEP in those subjects.

Postsecondary Education STEM Degrees Awarded in the United States The number of students attaining STEM postsecondary degrees in the U.S. more than doubled between 1960 and 2000; however, as a proportion of degrees in all fields, STEM degree awards have stagnated during this period.20 In the 2002-2003 academic year, more than 2.5 million degrees were awarded by postsecondary institutions in the United States.21 That year, just under 16% (399,465) of all degrees were conferred in STEM fields; all STEM degrees comprised 14.6% of associate degrees, 16.7% of baccalaureate degrees, 12.9% of master’s degrees, and 34.8% of doctoral degrees.22 Table 4 displays the distribution of degrees granted by academic level and field of study. At the associate and baccalaureate levels, the number of STEM degrees awarded was roughly equivalent to the number awarded in business. In 2002-2003, 92,640 associate degrees and 224,911 baccalaureate degrees were awarded in STEM fields, compared to 102,157 and 293,545, respectively, in business. However, nearly twice as many master’s degrees were granted in business (127,545) as in STEM (65,897), and an even larger number of master’s degrees were awarded in education (147,448). At the doctoral level, STEM plays a larger role. Doctoral degrees awarded in STEM fields account for more than one-third of all degrees awarded at this level. Education is the only field in which more doctoral degrees (6,835) were awarded than in the largest three STEM fields — biology, engineering, and the physical sciences (5,003, 5,333, and 3,858, respectively). Specialization within STEM fields also varies by academic level. Engineering was among the most common STEM specialties at all levels of study in 2002-2003. Biology was a common specialization at the baccalaureate and doctoral levels, but not at the master’s level. Computer science was common at all but the doctoral level. Physical sciences was a common specialization only at the doctoral level.

20

Through various “completions” surveys of postsecondary institutions administered annually since 1960, ED enumerates the number of degrees earned in each field during the previous academic year. 21 U.S. Department of Education, National Center for Education Statistics, Digest of Education Statistics, 2004, NCES 2005-025, Oct. 2005, Table 169. 22 Includes Ph.D., Ed.D., and comparable degrees at the doctoral level, but excludes first-professional degrees, such as M.D., D.D.S., and law degrees.

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Table 4. Degrees Conferred by Degree-Granting Institutions by Academic Level and Field of Study, 2002-2003. All fields STEM fields, total STEM, percentage of all fields Biological and biomedical sciences Computer and information sciences Engineering and engineering technologies Mathematics and statistics Physical sciences and science technologies

Associate 632,912 92,640 14.6% 1,496 46,089 42,133 732 2,190

Baccalaureate Master’s 1,348,503 512,645 224,911 65,897 16.7% 12.9% 60,072 6,990 57,439 19,503 76,967 30,669 12,493 17,940

3,626 5,109

Doctoral 46,024 16,017 34.8% 5,003 816 5,333

Total 2,540,084 399,465 15.7% 73,561 123,847 155,102

1,007 3,858

17,858 29,097

Non-STEM fields, total 540,272 1,123,592 446,748 30,007 2,140,619 Business 102,157 293,545 127,545 1,251 524,498 Education 11,199 105,790 147,448 6,835 271,272 English language and literature/letters 896 53,670 7,413 1,246 63,225 Foreign languages and area studies 1,176 23,530 4,558 1,228 30,492 Liberal arts and sciences, general 216,814 40,221 3,312 78 260,425 studies, and humanities Philosophy, theology, and religious 804 18,270 6,677 1,983 27,734 studies/vocations Psychology 1,784 78,613 17,123 4,831 102,351 Social sciences 5,422 115,488 12,109 2,989 136,008 History 316 27,730 2,525 861 31,432 Other 199,704 366,735 118,038 8,705 693,182 Source: U.S. Department of Education, National Center for Education Statistics, Digest of Education Statistics, 2004, NCES2005-025, Oct. 2005, Table 249-252.

Figure 2 displays the trends in STEM degrees awarded over the last three decades (excluding associate degrees). The solid line represents the number of STEM degrees awarded as a proportion of the total number of degrees awarded in all fields of study. The flat line indicates that the ratio of STEM degrees to all degrees awarded has historically hovered at around 17%. The bars represent the number of degrees awarded in each STEM sub-field as a proportion of all STEM degrees awarded. The top two segments of each bar reveal a consistent decline, since 1970, in the number of degrees awarded in math and the physical sciences. The bottom segment of each bar shows a history of fluctuation in the number of degrees awarded in biology over the last 30 years. The middle two segments in the figure represent the proportion of degrees awarded in engineering and computer science. The figure reveals a steady decline in the proportion of STEM degrees awarded in engineering since 1980, and a steady increase in computer science degrees (except for a contraction that occurred in the late 1980s following a rapid expansion in the early 1980s).

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Figure 2. STEM Degrees Awarded, 1970-2003.

U.S. Degrees Awarded to Foreign Students The increased presence of foreign students in graduate science and engineering programs and in the scientific workforce has been and continues to be of concern to some in the scientific community. Enrollment of U.S. citizens in graduate science and engineering programs has not kept pace with that of foreign students in these programs. According to the National Science Foundation (NSF) Survey of Earned Doctorates, foreign students earned one-third of all doctoral degrees awarded in 2003. Doctoral degrees awarded to foreign students were concentrated in STEM fields. The NSF reports that foreign students earned “more than half of those [awarded] in engineering, 44% of those in mathematics and computer science, and 35% of those in the physical sciences.”23 Many of these degree recipients remain in the United States to work. The same NSF report indicates that 53% of those who earned a doctorate in 1993 remained in the U.S. as of 1997, and 61% of the 1998 cohort were still working in the United States in 2003. In addition to the number of foreign students in graduate science and engineering programs, a significant number of university faculty in the scientific disciplines are foreign, and foreign doctorates are employed in large numbers by industry.24 23

National Science Board, Science and Engineering Indicators, 2006, (NSB 06-1). Arlington, VA: National Science Foundation, Jan. 2006, p. O-15. 24 For more information on issues related to foreign students and foreign technical workers, see the following: CRS Report 97-746, Foreign Science and Engineering Presence in U.S. Institutions and the Labor Force, by Christine M. Matthews; CRS Report RL31973, Programs Funded by the H-1B Visa Education and Training Fee and Labor Market Conditions for Information Technology (IT) Workers, by Linda Levine; and CRS Report RL30498, Immigration: Legislative Issues on Nonimmigrant Professional Specialty (H-1B) Workers, by Ruth Ellen Wasem.

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International Postsecondary Educational Attainment The United States has one of the highest rates of postsecondary educational attainment in the world. In 2003, the most recent academic year for which international data are available, 38% of the U.S. population aged 25-64 held a postsecondary degree — 9% at the tertiary-type B (vocational level) and 29% at the tertiary-type A (university level) or above. The OECD compiled comparison data from 30 OECD member states and 13 other nations. Three countries (Canada, Israel, and the Russian Federation) had larger shares at the two tertiary levels combined; however, all three had lower rates at the tertiary-type A level. At the tertiary-type A level, only one country (Norway) had a rate as high as the United States. The average for OECD member states was 16% at tertiary-type A and 8% at tertiary-type B.25 China and India were not included in the OECD data. Reliable information on postsecondary educational attainment is very difficult to obtain for these countries. The World Bank estimates that, in 1998, tertiary enrollment of the population between 18 and 24 years old was 6% in China and 8% in India, up from 1.7% and 5.2%, respectively, in 1980.26 Based on measures constructed by faculty at the Center for International Development (CID), the National Science Foundation (NSF) has generated an estimate of the distribution of the world’s population that possesses a tertiary education.27 The NSF estimates that the number of people in the world who had a tertiary education more than doubled from 73 million in 1980 to 194 million in 2000. Moreover, the two fastest-growing countries were China and India. China housed 5.4% of the world’s tertiary degree holders in 1980, and India had 4.1%; by 2000, the share in these countries was 10.5% and 7.7%, respectively. Indeed, as Figure 3 indicates, China and India were the only countries to substantially increase their share of the world’s tertiary degree-holders during that period.

International Comparisons in STEM Education The NSF has compiled data for many countries on the share of first university degrees awarded in STEM fields.28 According to these data, the United States has one of the lowest rates of STEM to non-STEM degree production in the world. In 2002, STEM degrees accounted for 16.8% of all first university degrees awarded in the United States (the same NCES figure reported at the outset of this section). The international average for the ratio of STEM to non-STEM degrees was 26.4% in 2002. Table 5 displays the field of first university degrees for regions and countries that award more than 200,000 university degrees annually. 25

Organization for Economic Co-operation and Development, Education at a Glance, OECD Indicators 2005, Paris, France, Sept. 2005. The OECD compiles annual data from national labor force surveys on educational attainment for the 30 OECD member countries, as well as 13 non-OECD countries that participate in the World Education Indicators (WEI) program. More information on sources and methods can be found at [http://www.oecd.org/ dataoecd/36/39/35324864.pdf]. 26 The World Bank, Constructing Knowledge Societies: new challenges for tertiary education, Washington, D.C., October 2002. Available at [http://siteresources.worldbank .org/EDUCATION/Resources/2782001099079877269/547664-1099079956815/ ConstructingKnowledgeSocieties.pdf]. 27 Unlike the OECD data, which are based on labor-force surveys of households and individuals, the CID data are based on the United Nations Educational, Scientific and Cultural Organization (UNESCO) census and survey data of the entire population. Documentation describing methodology as well as data files for the CID data is available at [http://www.cid.harvard.edu/ciddata/ciddata.html].

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Source: National Science Foundation, Science and Engineering Indicators, 2006, Volume 1, Arlington, VA, NSB 06-01, Jan. 2006.

Figure 3. Population 15 Years Old or Older With Tertiary Education by Country, 1980 and 2000.

28

First university degrees are those designated Level 5A by the International Standard Classification of Education (ISCED 97), and usually require less than five years to complete. More information on this classification and the ISCED is available at [http://www.unesco. org/education/information/nfsunesco/doc/isced_1997.htm].

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Among these nations, only Brazil awards a smaller share (15.5%) of STEM degrees than the United States. By contrast, the world leaders in the proportion of STEM degrees awarded are Japan (64.0%) and China (52.1%). Although the U.S. ranks near the bottom in the proportion of STEM degrees, it ranks third (behind Japan and China) in the absolute number of STEM degrees awarded. Table 5. Field of First University Degree, by Selected Region and Country, 2002 or the Most Recent Year Available Region/Country All regions Asia China India Japan South Korea Middle East Europe France Spain United Kingdom Central/Eastern Europe Russia North/Central America Mexico United States South America Brazil

All fields 9,057,193 3,224,593 929,598 750,000 548,897 239,793 445,488 2,682,448 309,009 211,979 282,380 1,176,898 554,814 1,827,226 321,799 1,305,730 543,805 395,988

STEM Fields 2,395,238 1,073,369 484,704 176,036 351,299 97,307 104,974 713,274 83,984 55,418 72,810 319,188 183,729 341,526 80,315 219,175 96,724 61,281

Percent STEM 26.4% 33.3% 52.1% 23.5% 64.0% 40.6% 23.6% 26.6% 27.2% 26.1% 25.8% 27.1% 33.1% 18.7% 25.0% 16.8% 17.8% 15.5%

Source: National Science Foundation, Science and Engineering Indicators, 2006, Volume 1, Arlington, VA, NSB 06-01, January 2006, Table 2-37.

Federal Programs that Promote STEM Education Government Accountability Office Study According to a 2005 Government Accountability Office (GAO) survey of 13 federal civilian agencies, in FY2004 there were 207 federal education programs designed to increase the number of students studying in STEM fields and/or improve the quality of STEM education.29 About $2.8 billion was appropriated for these programs that year, and about 71% ($2 billion) of those funds supported 99 programs in two agencies. In 2004, the National Institutes of Health (NIH) received $998 million that funded 51 programs, and the National Science Foundation (NSF) received $997 million that funded 48 programs. Seven of the 13 29

U.S. Government Accountability Office, Federal Science, Technology, Engineering, and Mathematics Programs and Related Trends, GAO-06-114, Oct. 2005. The GAO study does not include programs in the Department of Defense because the department decided not to participate. Other programs were omitted from the report for various reasons; typically because they did not meet the GAO criteria for a STEM-related educational program (according to an Apr. 26, 2006 conversation with the report’s lead author, Tim Hall).

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agencies had more than five STEM-related education programs. In addition to the NIH and NSF, only three other agencies received more than $100 million for STEM-related education programs. In FY2004, the National Aeronautics and Space Administration (NASA) received $231 million that funded five programs, the U.S. Department of Education (ED) received $221 million that funded four programs, and the Environmental Protection Agency (EPA) received $121 million that funded 21 programs. The GAO study found that most of the 207 programs had multiple goals, provided multiple types of assistance, and were targeted at multiple groups. The analysis identified six major program goals, four main types of assistance, and 11 target groups. The findings revealed that federal STEM education programs are heavily geared toward attracting college graduates into pursuing careers in STEM fields by providing financial assistance at the graduate and postdoctoral levels. Moreover, improving K-12 teacher education in STEM areas was the least frequent of the major goals, improving infrastructure was the least frequent of the main types of assistance, and elementary and secondary students were the least frequent group targeted by federal STEM education programs.30 The major goals of these programs were found by GAO to be the following (the number of programs with this goal is shown in parentheses): • • • • • •

attract and prepare students at all educational levels to pursue coursework in STEM areas (114), attract students to pursue STEM postsecondary degrees (two-year through Ph.D.) and postdoctoral appointments (137), provide growth and research opportunities for college and graduate students in STEM fields (103), attract graduates to pursue careers in STEM fields (131), improve teacher education in STEM areas (73), and improve or expand the capacity of institutions to promote STEM fields (90).

The four main types of assistance provided by these programs were as follows (the number of programs providing this service is shown in parentheses): • • • •

financial support for students or scholars (131), institutional support to improve educational quality (76), support for teacher and faculty development (84), and institutional physical infrastructure support (27).

The 11 target groups served by these programs were the following (the number of programs targeting them is shown in parentheses): • • • 30

elementary school students (28), middle school students (34), high school students (53),

Attrition rates among college students majoring in STEM fields combined with the growth of foreign students in U.S. graduate STEM programs suggest that pre-college STEM education may be a major source of the nation’s difficulty in this area.

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two-year college students (58), four-year college students (96), graduate students (100), postdoctoral scholars (70), elementary school teachers (39), secondary school teachers (50), college faculty or instructional staff (79), and institutions (82).

Description of Selected Federal STEM Programs The 2005 GAO report did not discuss federal STEM programs in detail (a very brief description of programs funded at $10 million or more is contained in Appendix III of the report). This section describes the kinds of activities the largest of these programs support, and how they operate at the federal, state, and/or local levels.31.

NIH National Research Service Awards The NIH was appropriated $998 million in FY2004 in support of its 51 STEM educational programs. Nearly two-thirds ($653 million) of those funds went to three programs under the National Research Service Awards (NRSA), first funded in 1975.32 Most of these funds ($547 million) went to one program, the NRSA Institutional Research Training Grants, which provides pre- and postdoctoral fellowships in health-related fields. An additional $73 million went to NRSA Individual Postdoctoral Fellowship Grants and $34 million went to NRSA Predoctoral Fellowship Grants. The Training Grants are awarded to institutions to develop or enhance research training opportunities for individuals, selected by the institution, who are training for careers in specified areas of interest to the institution or principal investigator. The Fellowship Grants are awarded directly to individuals from various organizations within the NIH (e.g., the National Institute on Aging) to support the particular research interests of the individual receiving the award. NRSA grant applicants must be U.S. citizens or nationals, or permanent resident aliens of the United States — individuals on temporary or student visas are not eligible. Predoctoral trainees must have received a baccalaureate degree by the starting date of their appointment, and must be training at the postbaccalaureate level and be enrolled in a program leading to a Ph.D. in science or in an equivalent research doctoral degree program. Health-profession students who wish to interrupt their studies for a year or more to engage in full-time research training before completing their professional degrees are also eligible. Postdoctoral trainees must have received, as of the beginning date of their appointment, a Ph.D., M.D., or comparable doctoral degree from an accredited domestic or foreign institution. Institutional grants are made for a five-year period. Trainee appointments are normally made in 12-month increments, although short-term (two- to three-month) awards are available. No individual 31

Additional program descriptions are available in the CRS congressional distribution memorandum, Federally Sponsored Programs for K-12 Science, Mathematics, and Technology Education, by Bonnie F. Mangan, available upon request. 32 More information on the NRSA program is available at [http://grants.nih.gov/ training/nrsa.htm].

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trainee may receive more than five years of aggregate NRSA support at the predoctoral level or three years of support at the postdoctoral level, including any combination of support from institutional training grants and individual fellowship awards. The annual stipend for predoctoral trainees in 2005 was about $12,000, and the postdoctoral stipend was between $20,000 and $32,000 (depending on years of experience). In FY2004, Training Grants were awarded to 293 institutions in all but six states. A total of 2,356 grants were awarded, which funded nearly 9,000 predoctoral fellowships and nearly 5,500 postdoctoral fellowships. The Fellowship Grant programs supported around 2,500 preand postdoctoral students in 2004. The large majority of the Training Grants were awarded through the National Institute of General Medical Sciences.

NSF Graduate Research Fellowships The largest of the NSF STEM education programs — the Graduate Research Fellowships ($97 million in FY2005) — is also one of the longest-running federal STEM programs (enacted in 1952). The purpose of this program is to increase the size and diversity of the U.S. workforce in science and engineering. The program provides three years of support to approximately 1,000 graduate students annually in STEM disciplines who are pursuing research-based master’s and doctoral degrees, with additional focus on women in engineering and computer and information sciences. In 2006, 907 awards were given to graduate students studying in nine major fields at 150 instituions. Applicants must be U.S. citizens or nationals, or permanent resident aliens of the United States; must have completed no more than twelve months of full-time graduate study at the time of their application; and must be pursuing an advanced degree in a STEM field supported by the National Science Foundation.33 The fellows’ affiliated institution receives a $40,500 award — $30,000 for a 12-month stipend and $10,500 for an annual cost-ofeducation allowance. These awards are for a maximum of three years and usable over a fiveyear period, and provide a one-time $1,000 International Research Travel Allowance. All discipline-based review panels, made up of professors, researchers, and others respected in their fields, convene for three days each year to read and evaluate applications in their areas of expertise. In 2005, there were 29 such panels made up of more than 500 experts.

NSF Mathematics and Science Partnerships The Mathematics and Science Partnerships program was the NSF’s second-largest program in FY2005 ($79 million in FY2005) and was the agency’s largest program in FY2004 ($139 million). Since its inception in 2002, this program has awarded grants that support four types of projects (the number of awards is shown in parentheses): • •

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Comprehensive Partnership projects (12) to implement change in mathematics and science education across the K-12 continuum; Targeted Partnership projects (28) to improve K-12 student achievement in a narrower grade range or disciplinary focus in mathematics and/or science;

A list of NSF-supported fields of study can be found at [http://www.nsf.gov/pubs/2005/nsf05601/ nsf05601.htm#study].

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Institute Partnership projects (8) to focus on improving middle and high school mathematics and science through the development of school-based intellectual leaders and master teachers; and Research, Evaluation and Technical Assistance projects (22) to build research, evaluation, and infrastructure capacity for the MSP.

One of the Comprehensive Partnership projects is between the Baltimore County Public Schools (BCPS) and the University of Maryland, Baltimore County (UMBC). The two main goals of the UMBC-BCPS STEM Partnership are to (1) facilitate the implementation, testing, refinement, and dissemination of promising practices for improving STEM student achievement, and (2) improve teacher quality and retention in selected high-need elementary, middle, and high schools in Baltimore County Public Schools. Centered on creating and evaluating performance-based pre-service (internship) teacher education programs and sustainable professional development programs for teachers and administrators, the project is designed to increase K-12 student achievement in STEM areas by increasing teacher and administrator knowledge. Ongoing assessments of student work and the differentiation of instruction based upon these assessments serve to evaluate and refine instruction, curricula and assessments, professional development programs, administrative leadership strategies, and directions for overall school improvement in STEM areas. UMBC and BCPS collaboration is facilitated by the creation of the Center for Excellence in STEM Education, where UMBC faculty and BCPS teachers and administrators develop projects to serve the needs of the BCPS district and the university. At the center, faculty and teachers work together to simultaneously improve the university’s STEM and teacher education departments and the teaching and learning culture in the BCPS. One of the Targeted Partnership grants supports the Promoting Reflective Inquiry in Mathematics Education Partnership, which includes Black Hills State University, Technology and Innovations in Education (TIE) of the Black Hills Special Services Cooperative, and the Rapid City School District in South Dakota. The overall goal of the partnership is aimed at improving achievement in mathematics for all students in Rapid City schools, with a particular goal of reducing the achievement gap between Native American and non-Native American students. The project seeks to improve the professional capacity and sustain the quality of K-12 in-service teachers of mathematics in the Rapid City School District, and student teachers of mathematics from Black Hills State University in order to provide effective, inquiry-based mathematics instruction. Objectives include reducing the number of high school students taking non-college preparatory mathematics, increasing the number of students taking upper level mathematics, and increasing student performance on college entrance exams. To accomplish these goals, the project provides 100 hours of professional development in combination with content-based workshops at the district level, and buildingbased activities involving modeling of effective lessons, peer mentoring and coaching, and lesson study. Mathematics education and discipline faculty from Black Hills State University are involved in district-wide professional development activities. A cadre of building-based Mathematics Lead Teachers convenes learning teams composed of mathematics teachers, mathematics student teachers, school counselors, and building administrators to identify key issues in mathematics curriculum and instruction.

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NSF Research Experiences for Undergraduates The Research Experiences for Undergraduates (REU) program is the largest of the NSF STEM education programs that supports active research participation by undergraduate students ($51 million in FY2005). REU projects involve students in research through two avenues. REU Sites are based on independent proposals to initiate and conduct projects that engage a number of students in research. REU Supplements are requested for ongoing NSFfunded research projects or are included as a component of proposals for new or renewal NSF grants or cooperative agreements. REU projects may be based in a single discipline or academic department, or on interdisciplinary or multi-department research opportunities with a coherent intellectual theme. Undergraduate student participants in either Sites or Supplements must be citizens or permanent residents of the United States or its possessions. Students apply directly to REU Sites (rather that to the NSF) to participate in the program. One of the grantees under this program is the REU Site in Microbiology at the University of Iowa. The goals of this project are to (1) recruit and select bright students, including women, individuals with diverse backgrounds with respect to geographic origin and ethnicity, and students from non-Ph.D.-granting institutions where research possibilities are limited; (2) involve students in basic, experimental research in microbiology; (3) expose students to a broad range of bioscience research; (4) develop each student’s critical-thinking skills; and (5) develop each student’s ability to record, analyze, and present scientific information. The student participants are integrated into faculty research programs and expected to perform like beginning graduate students. Informal faculty-student discussions and weekly seminars supplement laboratory research. Weekly informal lunches, two picnics, and a banquet facilitate social and scientific interactions. At the end of each summer’s program, the students prepare oral presentations to be given at a Summer Program Symposium. Each student also prepares a written research report under the guidance of a mentor.

NASA Minority University Research Education Program Nearly half of the funds ($106 million of $231 million) appropriated for NASA’s STEM education programs in FY2004 went to the Minority University Research Education Program (MUREP). MUREP supports grants to expand and advance NASA’s scientific and technological base through collaborative efforts with Historically Black Colleges and Universities (HBCUs) and other minority universities, including Hispanic-serving institutions and tribal colleges and universities. The program provides (1) K-12 awards to build and support successful pathways for students to progress to the next level of mathematics and science through a college preparatory curriculum and enrollment in college; (2) highereducation awards to improve the rate at which underrepresented minorities are awarded degrees in STEM disciplines; and (3) partnership awards to higher-education institutions and school districts that improve K-12 STEM teaching. One of the partnership programs, the Minority University Mathematics, Science and Technology Awards for Teacher and Curriculum Enhancement Program, supports collaborative efforts between universities and school districts to increase the number and percentage of state-certified STEM teachers in schools with high percentages of disadvantaged students. Grant awards range from $50,000 to $200,000 annually for each of three years of support, for a total of up to $600,000. A longstanding grant funded under this

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program involves a partnership between Florida International University and Miami-Dade County Public Schools. Students from three middle schools and one high school attend mathematics, science, and technology classes for half of each day at the university and spend the other half of the day in their home school. University faculty, graduate students, and preservice secondary mathematics teachers work with district teachers in providing the at-risk students with standards-based curriculum and instruction.

ED Mathematics and Science Partnerships Three-quarters of the STEM program funds in the Department of Education ($149 million of $221 million) in FY2004 went to the Mathematics and Science Partnership (MSP) program. The MSP is intended to increase the academic achievement of students in mathematics and science by enhancing the content knowledge and teaching skills of classroom teachers. These partnerships — between state education agencies, high-need school districts, and STEM faculty in institutions of higher education — are supported by state-administered formula grants and carried out in collaboration with the NSF-MSP program. Partnerships must use their grants for one or more of several specific activities. Among them are the following: • • • •

• • •

professional development to improve math and science teachers’ subject knowledge; activities to promote strong teaching skills among these teachers and teacher educators; math and science summer workshops or institutes with academic-year followup; recruitment of math, science, and engineering majors to teaching jobs through signing and performance incentives, stipends for alternative certification, and scholarships for advanced course work; development and redesign of more rigorous, standards-aligned math and science curricula; distance-learning programs for math and science teachers; and opportunities for math and science teachers to have contact with working mathematicians, scientists, and engineers.

A review of projects funded in 2004 revealed that most grantees focus on math (as opposed to science) instruction in middle schools, and provide professional development to roughly 46 teachers over a period of about 21 months.34 The survey found that most projects link content to state standards, and that algebra, geometry, and problem-solving are the top three math topics addressed by professional development activities. Most projects administer content knowledge tests to teachers, conduct observations, and make pre-and post-test comparisons. About half of the projects develop their own tests for teachers, and most rely on state tests of academic achievement to measure student knowledge.

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Analysts at the Brookings Institution conducted a survey of 266 winning MSP projects from 41 states. Results of the survey are available at [http://www.ed.gov/programs/mathsci/ proposalreview.doc].

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Proposals to Improve STEM Education Several pieces of legislation have been introduced in the 109th Congress with the purpose of improving STEM education in the United States. Many of the proposals in these bills have been influenced by the recommendations of several reports recently issued by leading academic, scientific, and business organizations (mentioned in the introduction of this report).35 These recommendations, particularly those from the business community, are not limited to the educational system. This report does not discuss these non-educational policy recommendations (e.g., immigration policies that affect the supply of foreign workers to fill U.S. demand in STEM occupations). The concluding section of this report discusses STEM education policy recommendations in detail, as well as selected pieces of legislation that have been introduced in this area.

Recommendations by the Scientific Community The recommendations to improve federal STEM policy concern every aspect of the educational pipeline. All of the recent reports issuing STEM education policy recommendations focus on five areas: improving elementary and secondary preparation in math and science, recruiting new elementary and secondary math and science teachers, retooling current math and science teachers, increasing the number of undergraduate STEM degrees awarded, and supporting graduate and early-career research. As mentioned at the outset of this report, one report that has been of particular influence in the STEM debate is from the National Academy of Sciences (NAS) — Rising Above the Gathering Storm. This influence is perhaps due to the clear targets and concrete programs laid out in the report. The NAS report’s five recommendations to improve STEM education follow. • • • • •

quadruple middle- and high-school math and science course-taking by 2010, recruit 10,000 new math and science teachers per year, strengthen the skills of 250,000 current math and science teachers, increase the number of STEM baccalaureate degrees awarded, and support graduate and early-career research in STEM fields.

To enlarge the pipeline of future STEM degree recipients, NAS sets a goal of quadrupling the number of middle and high school students taking Advanced Placement (AP) or International Baccalaureate (IB) math or science courses, from the current 1.1 million to 4.5 million by 2010. NAS further sets a goal of increasing the number of students who pass either the AP or IB tests to 700,000 by 2010. To enlarge the pipeline, NAS also supports the 35

The Education Commission of the States, Keeping America Competitive: Five Strategies To Improve Mathematics and Science Education, July 2005; The Association of American Universities, National Defense Education and Innovation Initiative, Meeting America’s Economic and Security Challenges in the 21st Century, Jan. 2006; The National Academy of Sciences, Committee on Science, Engineering, and Public Policy, Rising Above the Gathering Storm: Energizing and Employing America for a Brighter Economic Future, Feb. 2006; The National Summit on Competitiveness, Statement of the National Summit on Competitiveness: Investing in U.S. Innovation, Dec. 2005; The Business Roundtable, Tapping America’s Potential: The Education for Innovation Initiative, July 2005; The Center for Strategic and International Studies, Waiting for Sputnik, 2005.

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expansion of programs such as statewide specialty high schools for STEM immersion and inquiry-based learning through laboratory experience, summer internships, and other research opportunities. To recruit 10,000 new STEM teachers, NAS advocates the creation of a competitive grant program to award merit-based scholarships to obtain a four-year STEM degree in conjunction with certification as a K-12 mathematics or science teacher. These $10,000 to $20,000 awards could be used only for educational expenses and would require a five-year service commitment. An additional $10,000 annual bonus would be awarded to participating teachers in underserved schools in inner cities and rural areas. In further support of this scholarship program, NAS recommends that five-year, $1 million matching grants be awarded to postsecondary institutions to encourage the creation of programs that integrate the obtainment of a STEM bachelor’s degree with teacher certification. NAS proposes four approaches to achieving the goal of strengthening the skills of 250,000 current STEM teachers. First, NAS proposes that matching grants be awarded to support the establishment of state and regional summer institutes for STEM teachers modeled after the Merck Institute for Science Education. Second, NAS proposes that additional grants go to postsecondary institutions that support STEM master’s degree programs for current STEM teachers (with or without STEM bachelor’s degrees) modeled after the University of Pennsylvania Science Teachers Institute. Third, NAS proposes that programs be created to train current teachers to provide AP, IB, and pre-AP or pre-IB instruction modeled after the Advanced Placement Initiative and the Laying the Foundation programs. Fourth, NAS proposes the creation of a national panel to collect, evaluate, and develop rigorous K-12 STEM curricula modeled after Project Lead the Way. To increase STEM bachelor’s degree attainment, NAS proposes providing 25,000 new scholarships each year. These Undergraduate Scholar Awards in Science, Technology, Engineering, and Mathematics (USA-STEM) would be distributed to each state in proportion with its population, and awarded to students based on competitive national exams. The $20,000 scholarships could only go to U.S. citizens, and could only be used for the payment of tuition and fees in pursuit of a STEM degree at a U.S. postsecondary institution. To increase graduate study in areas of national need, including STEM, NAS proposes the creation of 5,000 new fellowships each year to U.S. citizens pursuing doctoral degrees. The fellowships would be administered by the National Science Foundation, which would also draw on the advice of several federal agencies in determining the areas of need. An annual stipend of $30,000 would be accompanied by an additional $20,000 annually to cover the cost of tuition and fees. These fellowships would also be portable, so that students could choose to study at a particular institution without the influence of faculty research grants.

Legislation in the 109th Congress Several bills containing STEM education-related proposals have been introduced in the 109th Congress, and have also seen additional legislative action. Some of these bills have already been passed by Congress and signed into law by the President. The National Aeronautics and Space Administration Authorization Act of 2005 (P.L. 109-155) directed the Administrator to develop, expand, and evaluate educational outreach programs in science and space that serve elementary and secondary schools. The National Defense Authorization Act of 2006 (P.L.

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109-163) made permanent the Science, Mathematics and Research for Transformation pilot program initiated by the Defense Act of 2005 (P.L. 108-375) to address deficiencies of scientists and engineers in the national security workforce. The Deficit Reduction Act of 2005 (P.L. 109-171) established the Academic Competitiveness Grants and the National Science and Mathematics Access to Retain Talent Grants programs, which supplement Pell Grants for students studying mathematics, technology, engineering, critical foreign languages, and physical, life, and computer sciences. The act also established the Academic Competitiveness Council, chaired by the Secretary of Education and charged with identifying and evaluating all federal STEM programs, and recommending reforms to improve program integration and coordination. Additional bills that have been introduced in the 109th Congress that would make substantial changes or additions to current federal STEM education policy include two intended to reauthorize the Higher Education Act (HEA), as well as several pieces of socalled “competitiveness” legislation. On February 28, 2006, the Senate Committee on Health, Education, Labor, and Pensions reported S. 1614, the Higher Education Amendments of 2005 (S.Rept. 109-218). On March 30, 2006, the House passed H.R. 609, the College Access and Opportunity Act of 2005. On April 24, 2006, the Senate Committee on Energy and Natural Resources reported S. 2197, Protecting America’s Competitive Edge Through Energy Act of 2006 (S.Rept. 109-249). A companion bill, S. 2198, Protecting America’s Competitive Edge Through Education and Research Act of 2006, has been the subject of two hearings (February 28, 2006 and March 1, 2006). Another bill that would make substantial additions to federal STEM education policy is S. 2109, the National Innovation Act. On June 22, 2006, the House Committee on Science reported two bills — H.R. 5358, the Science and Mathematics Education for Competitiveness Act, and H.R. 5356, the Early Career Research Act.

Secondary School Math and Science Preparation S. 2197 would provide experiential-based learning opportunities for students by establishing a summer internship program for middle school and secondary school students at the National Laboratories funded by the Department of Energy (DOE). Language in the bill requires that 40% of the participants be from low-income families. The bill also requires that the participants be from schools where teachers are teaching “out-of-field,” hold temporary certification, or have a high turnover rate. For this purpose, S. 2197 would authorize appropriations of $50 million for each of five fiscal years — FY2007 through FY2011. S. 2109 would increase support for science education through the NSF. The bill would authorize the following amounts for expansion of science, mathematics, engineering, and technology talent under the NSF Authorization Act of 2002 (P.L. 107-368): FY2007, $35 million; FY2008, $50 million; FY2009, $100 million; and FY2010, $150 million. S. 2109 would also promote innovation-based experiential learning. This bill would allow NSF to award grants to local education agencies (LEAs) for implementation of innovation-based experiential learning. A total of 500 elementary or middle schools and 500 secondary schools would participate. Funding would total $10 million in FY2007 and $20 million each for FY2008 and FY2009.

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Recruiting and Retaining New STEM Teachers Provisions in S. 2198 would direct the NSF to provide two types of support for future science and mathematics teachers. One such award would be four-year fellowships in the amount of $10,000 annually to individuals who complete a baccalaureate degree in science, engineering, or mathematics, with concurrent teacher certification. A requirement would be that these individuals teach as full-time mathematics, science, or elementary school teachers in highneed elementary and secondary schools. Additional support for teacher recruitment would be through scholarships for science and mathematics teachers. The NSF Director would award merit-based scholarships of up to $20,000 per year for not more than four years to students majoring in science, mathematics, and engineering education who pursue concurrent teacher certification to assist students in paying their college education expenses. S. 2198 would also authorize institutional grants to provide an integrated course of study in mathematics, science, engineering, or teacher education that leads to a baccalaureate degree in the STEM disciplines with concurrent teacher certification. The awards would total $1 million per year for a period of five years. Matching funds in predetermined amounts would be required from non-federal sources — not less than 25% of the amount for the first year, not less than 35% for the second year, and not less than 50% of the amount of the grant award for each succeeding year of the grant. S. 2198 would provide further institutional grants to develop part-time, three-year master’s degree programs in science and mathematics education for teacher enhancement. Eligible participants must collaborate with a teacher preparation program of an institution of higher education. The competitively awarded grants are not to exceed $1 million. Priority would be given to applicants who consult with LEAs, use online technology, and develop innovative efforts directed at reducing shortages of science and mathematics teachers in lowincome urban or rural areas. H.R. 609 and S. 1614 would expand and extend the current loan forgiveness program for STEM teachers. Currently, HEA Title IV, Section 428J and 465, as amended by the Taxpayer-Teacher Protection Act of 2004 (P.L. 108-409), provides a higher maximum debt relief for qualified math and science secondary school teachers; up to $17,500 in loan forgiveness compared to $5,000 for other eligible teachers. However, only teachers who were new borrowers between October 1, 1998 and October 1, 2005 are eligible. Both S. 1614 and H.R. 609 would extend eligibility for loan forgiveness of up to $17,500 to qualified math and science secondary school teachers who were new borrowers after October 1, 1998; i.e., they would extend eligibility beyond the October 1, 2005 limit set by P.L. 108-409.36 H.R. 609 would establish the Mathematics and Science Incentive Program to provide eligible math and science teachers relief from interest payments on student loans in return for working in high-need schools. Participating teachers would serve at least five years and could not receive more than $5,000 in total relief. S. 1614 would allow state and partnership grantees under the HEA Title II, Teacher Enhancement Grant program to provide scholarships to students who later teach in the areas of math or science. H.R. 609 would create an Adjunct Teacher Corps that would award grants to LEAs or other educational organizations (private or public) to recruit professionals with math and 36

More information on teacher loan forgiveness can be found in CRS Report RL32516, Student Loan Forgiveness Programs, by Gail McCallion.

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science skills to serve as adjunct teachers. Grants could be used to develop outreach programs, fund signing bonuses, and compensate outside entities for the costs associated with allowing employees to serve. Grantees would have to match the federal funds received dollarfor-dollar.

Upgrading the STEM Skills of Current Teachers Another type of award provided in S. 2198 would be in the form of a fellowship for five years, in the amount of $10,000 annually, to teachers who have successfully completed a master’s degree in science or mathematics education, and who undertake increased responsibilities such as teacher mentoring and other leadership activities. S. 2198 would also direct the Secretary of Education to award grants to nonprofit entities that partner with local school districts for the training of teachers who will lead Advanced Placement or International Baccalaureate (AP-IB) and pre-AP-IB programs in science and mathematics. The grantees must demonstrate an ability to serve not fewer than 10,000 children from lowincome families. S. 2198 would also support a National Clearinghouse on Mathematics and Science Teaching Materials. The Secretary of Education would establish a national panel, after consultation with the National Academies, to collect proven effective K-12 mathematics and science teaching materials, and create a clearinghouse of such materials for dissemination to states and school districts. The bill would authorize the appropriation of $20 million for each fiscal year — FY2007 through FY2011. S. 2197 would provide for assistance to speciality schools for science and mathematics. The bill would require that funds and staff of the National Laboratories be made available to assist in teaching courses at statewide speciality schools with comprehensive programs in science, mathematics, and engineering. Both S. 1614 and H.R. 609 would target the HEA, Title II, Teacher Enhancement Grant program to math and science teachers. Under the current program, partnership grantees may provide professional development to improve teachers’ content knowledge; however, no particular subject areas are specified. Both bills would support efforts by partnership grantees to increase the number of math and science teachers and to provide opportunities for clinical experience in the areas of math and science. Both bills would also allow state grantees to support bonus pay for math and science teachers. H.R. 5358 would rename the current NSF Math and Science Partnership program that supports teacher training partnerships between LEAs and either IHEs or eligible nonprofit organizations. Grantees would operate teacher institutes that provide intensive content instruction in science and mathematics, as well as induction programs for new teachers, professional development, and training in the use of technology and laboratory equipment. Grants would be awarded for a period of five years at between $75,000 and $2 million per year. The bill would authorize appropriations of $50 million for each of the fiscal years 2007 through 2011.

Increase STEM Baccalaureate Degree Attainment S. 2198 would provide support for a Future American Scientist Scholarships program. This program would provide 25,000 new competitive merit-based undergraduate scholarships to

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students who are U.S. citizens. The scholarships would be awarded in the amount of $20,000 per year. The bill would authorize an appropriation of $375 million for FY2007; $750 million for FY2008; $1.125 billion for FY2009; and $1.5 billion annually for FY2010 through FY2013. S. 1614 would create a new Mathematics and Science Scholars Program that would award $1,000 for each of up to two years to eligible undergraduate students majoring in mathematics, science, technology, or engineering. To be eligible, students would have to complete a “rigorous secondary school curriculum in mathematics and science” as determined by the state. In determining priority for the scholarship winners, the governor of the state may take into consideration (1) the student’s regional or geographic location, (2) whether the student attended school in a high-need area, (3) attended a low-performing school, or (4) is a member of a group under-represented in STEM fields. H.R. 609 would amend the current Robert C. Byrd Honors Scholarship Program to focus these awards on students pursuing a major in studies leading to a baccalaureate, master’s, or doctoral degree in mathematics, engineering, or the physical, life, or computer sciences. These scholarships would be for a period of up to five years, and would be equal to the students’ unmet financial need (i.e., the cost of attendance minus any non-loan aid). At least 50% of the cost of these scholarships would have to come from non-federal funds. H.R. 5358 would amend the current NSF Robert Noyce Scholarship Program, which provides scholarships to undergraduates majoring in STEM fields. Students would be eligible for a $10,000 scholarship annually (up from the current $7,500), and would be required to teach math or science in a high-need school. Recipients must serve two years as a teacher for every year they received a scholarship, with a maximum of four years of service. The bill would also make special considerations for students and working professionals attending parttime. The bill authorizes $50 million for this program in FY2007, $70 million in FY2008, and $90 million for FY2009 through FY2011. H.R. 5358 would establish a Talent Expansion Program, as well as Centers for Undergraduate Education in Science, Mathematics, and Engineering, which would award grants to IHEs intended to improve and expand course-taking in STEM fields. The former program would be authorized at $40 million for FY2007, $45 million for FY2008, and $50 million for FY2009 through FY2011. The latter program would be authorized at $4 million for FY2007, $10 million for FY2008 through FY2011.

Graduate Research and Early-Career Scholarship S. 2198 would provide funding for graduate research fellowships in the critical fields of science, mathematics, and engineering. The bill would establish a fellowship program to provide tuition and financial support for eligible students pursuing master’s and doctoral degrees in science, mathematics, and engineering, and other areas of national need. S. 2198 would authorize appropriations of $225 million for FY2007, $450 million for FY2008, and $675 million for FY2009 through FY2013. Both S. 1614 and H.R. 609 would amend the Graduate Assistance in Areas of National Need program (HEA, Title VII, Section 711) to encourage study in STEM fields. The current program provides grants to institutions of higher education that award graduate degrees in areas of national need. This program also funds fellowships to graduate students pursuing doctoral degrees in areas of national need at eligible institutions. The Secretary of Education,

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in consultation with federal agencies and nonprofit organizations, is given the authority to determine the areas of national need. S. 1614 would require the Secretary to consult with specific federal and nonprofit agencies and organizations (including NAS) in determining the areas of national need. H.R. 609 would require that in determining the areas of national need, the Secretary shall prioritize “math and science teachers, special education teachers, and teachers who provide instruction for limited English proficient individuals.” S. 2109 proposes expanded graduate fellowship and graduate traineeship programs in the NSF. It would authorize the appropriation of $34 million each year for FY2007 through FY2011 for expansion of the Graduate Research Fellowship Program in NSF. Such funding would allow for an additional 250 fellowships to be awarded to U.S. citizens each year during the five-year period. The total number of fellowships to be awarded is 1,250. The bill also provides $57 million each year for a period of five years for the expansion of the Integrative Graduate Education and Research Traineeship program in NSF. The support would provide an additional 250 grants per year to U.S. citizens, for a total of 1,250 awards. S. 2109 would direct the Secretary of the Department of Defense (DOD) to utilize appropriations for expansion of the Science, Mathematics, and Research for Transformation Defense Scholarship Program (SMART). The bill would provide support for an additional 160 doctoral degrees and 60 master’s degrees for FY2007 through FY2011. The Secretary of Defense would be charged with expanding by 200 the number of participants in the National Defense Science and Engineering Graduate Fellowship program each fiscal year from FY2007 through FY2011. S. 2109 calls for the establishment of a program to award traineeships to undergraduate and graduate students pursuing studies in areas important to DOD in science, mathematics, and engineering. The selected programs should expose students to multidisciplinary studies, innovation-oriented studies, and academic, private sector, or government laboratories and research. Awardees would be required to work for DOD for 10 years following the completion of the degree program. The bill would authorize appropriations in the amount of $11.1 million each year for FY2007 through FY2011. S. 2109 would establish a clearinghouse of successful professional science master’s degree programs, and make the program elements available to colleges and universities. Grants would be awarded for pilot programs at four-year institutions that foster improvement of professional science master’s degree programs. The bill directs that preference be given to institutions that obtain two-thirds of their funding from non-federal support. A maximum of 200 grants would be awarded to four-year institutions for one three-year term, with renewal possible for a maximum of two additional years. Language included in the bill states that performance benchmarks be developed prior to the beginning of the program. The amount of $20 million would be made available for evaluation and reporting of the pilot program. S. 2197 would authorize research grants for early-career scientists and engineers. The participants must have completed their degrees no more than 10 years before the awarding of the grants. Not less than 65 grants per year would be awarded to outstanding early-career researchers to support other researchers in the DOE National Laboratories, and federally funded research and development centers. The grants would be awarded for a period of five years in duration, and at a level of $100,000 for each year of the grant period. The bill would authorize the appropriation of $6.5 million in FY2007, $13 million in FY2008, $19.5 million in FY2009, $26 million in FY2010, and $32.5 million in FY2011. H.R. 5356 would authorize two new programs (one carried out by the NSF Director and the other by the DOE Under Secretary of Science) that would award grants to scientists and

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engineers at the early stages of their careers at IHEs and other research institutions. Five-year, merit-based grants are to be awarded at a minimum of $80,000 per year to support innovative projects that integrate research and education.

Federal Program Coordination Under S. 2198, the Director of the Office of Science and Technology Policy would establish a standing committee on education in mathematics, science, and engineering. The responsibility of the committee would include the development of national goals for support of the disciplines by the federal government. Language in the bill stipulates that there should be public comment on the national goals. In addition, plans should be periodically reviewed and updated by the Director. S. 2197 calls for the creation of a position of Director of Mathematics, Science and Engineering Education, and charges that the Director administer and oversee programs at the DOE. In conjunction with the NAS, the Director would assess, after five years, the performance of science and mathematics programs at DOE. The bill would establish a Mathematics, Science, and Engineering Education fund, using 0.3% of funding made available to DOE for research, education, demonstration, and commercial application. H.R. 609 would establish Mathematics and Science Education Coordinating Council Grants to assist states in coordinating math- and science-related activities supported by the Elementary and Secondary Education Act Title II, Part B, Mathematics and Science Partnerships Program, and the HEA Title II, Teacher Quality Enhancement Program.

In: Advances in Mathematics Research, Volume 8 ISBN 978-1-60456-454-9 c 2009 Nova Science Publishers, Inc. Editor: Albert R. Baswell, pp. 277-294

Chapter 9

O N C OMPUTATIONAL M ODELS FOR THE H YPERSPACE Salvador Romaguera∗ Instituto Universitario de Matem´atica Pura y Aplicada, Universidad Polit´ecnica de Valencia, 46022 Valencia, Spain.

Abstract Let BX be the continuous poset of formal balls of a metric space (X, d) endowed by the weightable quasi-metric qd induced by d. We show that the continuous poset B(CX) of formal balls of the space CX of nonempty closed bounded subsets of X endowed by the quasi-metric qHd induced by the Hausdorff metric Hd on CX is isometric to a sup-closed subspace of the space C(BX) of nonempty sup-closed bounded subsets of BX endowed with the Hausdorff quasi-metric Hqd . We also show that the quasi-metric space (B(CX), qHd ) is bicomplete if and only if the metric space (X, d) is complete. Several consequences are derived. In particular, our approach provides an interesting class of weightable quasi-metric spaces for which weightability of the Hausdorff quasi-metric holds on certain paradigmatic subspaces. Moreover, some properties from topological algebra are discussed; for instance, we prove that if (X, d) is a metric monoid (respectively, a metric cone), then (B(CX), qHd ) is a quasi-metric monoid (respectively, (B(Cc X), qHd ) is a quasi-metric cone, where by Cc (X) we denote the family of all convex members of C(X)).

MSC (2000): 06B35, 54B20, 54E35, 54E50, 54F05, 54H12 Keywords: computational model, metric space, hyperspace, Hausdorff metric, formal ball, domain, complete, weightable quasi-metric, partial metric, quasi-metric monoid.

1.

Introduction

What is a computational model? Following Edalat and S¨underhauf [8, Introduction], and informally speaking, by a computational model we could mean a mathematical structure ∗

E-mail address: [email protected] The author thanks the support of the Spanish Ministry of Education and Science, and FEDER, under grant MTM2006-14925-C02-01.

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which is constructed in an easy way and serves useful in performing certain computations on the structure. Motivated by the fact that both (continuous) domains and metric spaces are the basic mathematical structures in computer science, some authors have formalized the idea of “computational model” by connecting the domain theory with the theory of metric spaces and related topological spaces (see [11, 19, 24, 31, etc]). On the other hand, it is well known that the Hausdorff metric constitutes an efficient tool in several branches of Mathematics and Computer Science, such as convex analysis ( [4, 27]), fractals ( [9]), image processing ( [16,33,35]) , denotational semantics of programming languages ( [1–3]), asymptotic complexity of algorithms ( [28]), etc. It then suggests the natural problem of establishing connections between computational models for hyperspaces of a metric space (X, d), endowed with the Hausdorff metric, and the corresponding hyperspaces of computational models for (X, d). Here we investigate this problem. In particular, we study the case that the computational model is constructed from the continuous poset BX of formal balls of (X, d) as discussed by Edalat and Heckmann in [7] and by Heckmann in [14]; more precisely, we prove that the continuous poset B(CX) of formal balls of the hyperspace CX of nonempty closed bounded subsets of X endowed by the weightable quasi-metric qHd induced by the Hausdorff metric Hd on CX, is isometric to a sup-closed subspace of the hyperspace C(BX) of nonempty sup-closed bounded subsets of BX endowed with the Hausdorff quasi-metric Hqd , where qd denotes the weightable quasi-metric on BX induced by the metric d. We also prove that the quasi-metric space (B(CX), qHd ) is bicomplete if and only if the metric space (X, d) is complete. Several consequences are derived. In particular, it follows that the Hausdorff quasi-metric Hqd is weightable on the subspace of C(BX) which is isometric to (B(CX), qHd ). Moreover, some properties from topological algebra are explored; for instance, we show that if (X, d) is a metric monoid, then (B(CX), qHd ) is a quasi-metric monoid. The paper is organized as follows. In Section 2 we recall the basic concepts and results from domain theory and from the theory of metric spaces and their generalizations, respectively, that will be used later on. In Sections 3 and 4 we prove our main results and, finally, we present our conclusion in the light of the obtained results.

2.

Background

In the following the letters N and R+ will denote the set of natural numbers and the set of nonnegative real numbers, respectively. Our basic reference for Domain Theory is [13]. Let us recall that a partial order (or simply an order) on a nonempty set D is a binary relation ≤ on D such that for each x, y, z ∈ D : (i) x ≤ x (reflexivity); (ii) if x ≤ y and y ≤ z, then x ≤ z (transitivity); (iii) if x ≤ y and y ≤ x, then x = y (antisymmetry). A partially ordered set, or poset for short, is a nonempty set D equipped with a partial order ≤; it will be denoted by (D, ≤) or simply by D if no confusion arises. A subset A of a poset D is directed provided that it is nonempty and every finite subset of A has an upper bound in A. The least upper bound of a subset A of D is denoted by sup A if it exists.

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A poset D is said to be directed complete, and is called a dcpo, if every directed subset of D has a least upper bound. A point x in D is called maximal if condition x ≤ y implies x = y. The set of all maximal points of D will be denoted by max D. Let D be a poset and x, y ∈ D; we say that x is way below y, in symbols x ≪ y, if for each directed subset A of D for which sup A exists, the relation y ≤ sup A implies the existence of some z ∈ A with x ≤ z. A poset D is called continuous if for each x ∈ D, the set x ⇓= {u ∈ D : u ≪ x} is directed and x = sup(x ⇓). A continuous poset which is also a dcpo is called a continuous domain or, simply, a domain. A subset B of a poset D is a basis for D if for each x ∈ D, the set xB ⇓= {u ∈ B : u ≪ x} is directed and x = sup(xB ⇓). Recall that a poset has a basis if and only if it is continuous. Therefore, a dcpo has a basis if and only if it is a domain. A dcpo having a countable basis is said to be an ω-continuous domain [7]. The Scott topology σ(D) of a dcpo (D, ≤) is constructed as follows [13, Chapter II]: A subset U of D is open with respect to σ(D) provided that: (i) U =↑ U, where ↑ U = {y ∈ D : x ≤ y for some x ∈ U }; (ii) for each directed subset A of D with sup D ∈ U, it follows that D ∩ U 6= ∅. If (D, ≤) is a domain, then the sets x ⇑, x ∈ D, form an open base for σ(D), where x ⇑= {y ∈ D : x ≪ y} (see [13, Proposition II-1.10]). Moreover, σ(D) has a countable base if and only if (D, ≤) is an ω-continuous domain. In case that (D, ≤) is a continuous poset it is possible to show yet that the sets x ⇑, x ∈ D, form an open base for a topology on D, which is also called the Scott topology of (D, ≤) and it is also denoted by σ(D) (see, for instance, [7, p. 58]). If A is a subset of D, we denote by σ(D)|A the restriction of σ(D) to A. Our basic reference for general topology is [10] and for quasi-metric spaces they are [12] and [21]. By a quasi-metric on a set X we mean a function q : X × X → R+ such that for all x, y, z ∈ X : (i) x = y ⇔ q(x, y) = q(y, x) = 0; (ii) q(x, z) ≤ q(x, y) + q(y, z). A quasi-metric space is a pair (X, q) such that X is a set and q is a quasi-metric on X. Each quasi-metric q on X induces a T0 topology τq on X which has as a base the family of open balls {Bq (x, r) : x ∈ X, ε > 0}, where Bq (x, ε) = {y ∈ X : q(x, y) < ε} for all x ∈ X and ε > 0. Given a quasi-metric q on X, then the function q −1 defined on X × X by q −1 (x, y) = q(y, x), is also a quasi-metric on X, called the conjugate of q, and the function q s defined on X × X by q s (x, y) = max{q(x, y), q −1 (x, y)} is a metric on X. A subset A of a quasi-metric space (X, q) is called sup-closed if A is closed in the metric space (X, q s ). A quasi-metric space (X, q) is said to be bicomplete if (X, q s ) is a complete metric space. In this case, we say that q is a bicomplete quasi-metric on X. The notion of a partial metric space, and its equivalent weightable quasi-metric space, was introduced by Matthews in [25] as a part of the study of denotational semantics of dataflow networks.

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Let us recall that a partial metric [25] on a set X is a function p : X × X → R+ such that for all x, y, z ∈ X : (i) x = y ⇔ p(x, x) = p(x, y) = p(y, y); (ii) p(x, x) ≤ p(x, y); (iii) p(x, y) = p(y, x); (iv) p(x, z) ≤ p(x, y) + p(y, z) − p(y, y). A partial metric space is a pair (X, p) such that X is a set and p is a partial metric on X. Each partial metric p on X induces a T0 -topology τp on X which has as a base the family of open p-balls {Bp (x, ε) : x ∈ X, ε > 0}, where Bp (x, ε) = {y ∈ X : p(x, y) < ε + p(x, x)} for all x ∈ X and ε > 0. A quasi-metric space (X, q) is called weightable if there exists a function w : X → R+ such that for all x, y ∈ X, q(x, y) + w(x) = q(y, x) + w(y). The function w is said to be a weighting function for (X, q) and the quasi-metric q is weightable by the function w. The relationship between partial metric spaces and weightable quasi-metric spaces is given in the following result. Theorem A [25]. a) Let (X, p) be a partial metric space. Then, the function q(p) : X × X → R+ defined by q(p)(x, y) = p(x, y) − p(x, x) for all x, y ∈ X, is a weightable quasi-metric on X with weighting function w given by w(x) = p(x, x) for all x ∈ X. Furthermore τp = τq(p) . b) Conversely, if (X, q) is a weightable quasi-metric space with weighting function w, then the function p(q) : X × X → R+ defined by p(q)(x, y) = q(x, y) + w(x) is a partial metric on X. Furthermore τq = τp(q) . Note that if p is a partial metric on a set X, we have p(q(p)) = p. According to [25, Definition 5.2], a sequence (xn )n in a partial metric space (X, p) is called a Cauchy sequence if there exists limn,m p(xn , xm ). A partial metric space (X, p) is said to be complete [25, Definition 5.3] if every Cauchy sequence (xn )n in X converges, with respect to τp , to a point x ∈ X such that p(x, x) = limn,m p(xn , xm ). The relationship between complete partial metric spaces and bicomplete weightable quasi-metric spaces is given in the following result. Theorem B [26]. A partial metric space (X, p) is complete if and only if (X, q(p)) is a bicomplete quasi-metric space. The following are easy but useful examples of posets. Example 1. a) It is well known that if (X, q) is a quasi-metric space, then the binary relation ≤q defined on X by x ≤q y ⇔ q(x, y) = 0, is a partial order on X. Hence (X, ≤q ) is a poset. b) Similarly, if (X, p) is a partial metric space, then the binary relation ≤p defined on X by x ≤p y ⇔ p(x, y) = p(x, x), is a partial order on X. Hence (X, ≤p ) is a poset [14, 25, 30, etc]. Therefore, if (X, p) is a partial metric space, one has ≤p =≤q(p) . We conclude this section by recalling the construction of the Hausdorff (quasi-)metric of a (quasi-)metric space.

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Recall that given a metric space (X, d), we denote by CX the family of all nonempty closed bounded subsets of X. For each pair A, B ∈ CX put Hd (A, B) = max{Hd+ (A, B), Hd− (A, B)}, where Hd+ (A, B) = supb∈B d(A, b) and Hd− (A, B) = supa∈A d(a, B). Then Hd is a metric on CX which is called the Hausdorff metric of (X, d) (see, for instance, [4]), and we refer to (CX, Hd ) as an hyperspace ( [15, p. 125]) . This notion admits a natural extension to the setting of quasi-metric spaces (compare [5, 22]). Let (X, q) be a quasi-metric space. Denote by CX the family of all nonempty closed bounded subsets of the metric space (X, q s ). For each pair A, B ∈ CX put Hq (A, B) = max{Hq+ (A, B), Hq− (A, B)}, where Hq+ (A, B) = supb∈B q(A, b) and Hq− (A, B) = supa∈A q(a, B). Then Hq is a quasi-metric on CX which is called the Hausdorff quasi-metric of (X, q), and we refer to (CX, Hq ) as an hyperspace.

3.

The Weightable Quasi-metric Space (B(CX), qHd )

In this section we shall prove some results of the structure of the quasi-metric space (B(CX), qHd ) which were announced in Section 1. First, we give formal notions of a computational model and of a quantifiable computational model, respectively. Definition 1 (compare [11, 19, 24]). A computational model for a topological space (X, τ ) is a pair (D, φ) such that D is a domain and φ is an homeomorphism from (X, τ ) onto (max D, σ(D)|max D ). Definition 2. A quantifiable computational model for a metric space (X, d) is a pair (D, φ) such that: (i) D is a domain equipped with a bicomplete quasi-metric q satisfying: (i1 ) the topology induced by q coincides with the Scott topology inherited from D; (i2 ) the partial order ≤q induced by q coincides with the partial order of D. (ii) φ is an isometry from (X, d) to (D, q) with max D = φ(X). In the last decade several authors ( [6–8, 11, 14, 19, 20, 23, 24, 31, 32, 34, etc]) have constructed computational models for various topological structures. In particular, Lawson proved in [23] that a metrizable space X is a Polish space if and only if there is an ωcontinuous domain D such that: (i) (X, τ ) is homeomorphic to (max D, σ(D)|max D ), and (ii) σ(D)|max D coincides with the Lawson topology on max D. (Let us recall that a Polish space is a separable metrizable topological space that admits a compatible complete metric, and that a topological space is said to be separable if it has a countable dense subset). Therefore, each Polish space has a computational model. In [7], Edalat and Heckmann gave a more direct and explicit construction of an ωcontinuous domain for any Polish space satisfying conditions (i) and (ii) above, with the help of the notion of a formal ball. In fact, they proved, among other results that each complete metric space, has a computational model. Later on, Heckmann ( [14]) improved this result by essentially showing that each complete metric space has a quantifiable computational model.

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Next we recall the main results from [7] and [14] because our approach is based on them. Given a metric space (X, d), define BX := {(x, r) : x ∈ X, r ∈ R+ }. Then, each pair (x, r) ∈ BX is called a formal ball in (X, d). Denote by ⊑d the binary relation on BX given by (x, r) ⊑d (y, s) ⇐⇒ d(x, y) ≤ r − s. Then (BX, ⊑d ) is a poset. Furthermore (BX, ⊑d ) is a domain if and only if the metric space (X, d) is complete. It is also proved in [7] that for each metric space (X, d), (BX, ⊑d ) is a continuous poset such that max BX = {(x, 0) : x ∈ X}. Moreover, the mapping φ : X → BX given by φ(x) = (x, 0) for all x ∈ X, is an homeomorphism from (X, τd ) onto (max BX, σ(BX)|max BX ). In [14] Heckmann essentially proved, among others, the following crucial result: Given a (complete) metric space (X, d), the function pd defined on BX × BX by pd ((x, r), (y, s)) = max{d(x, y), |r − s|} + r + s, is a (complete) partial metric on BX. Furthermore, the topology induced by pd coincides with the Scott topology on BX, and the partial order ≤pd induced by pd coincides with ⊑d on BX. The weightable quasi-metric q(pd ) induced by pd will be simply denoted by qd , and, by Theorem A, we have qd ((x, r), (y, s)) = max{d(x, y), |r − s|} + s − r, for all (x, r), (y, s) ∈ BX. We will refer to qd as the weightable quasi-metric induced by d. Remark 1. Taking into account the preceding observations and Theorem B, we deduce that if (X, d) is a complete metric space, then the pair (BX, φ) is a quantifiable computational model for (X, d), where BX is equipped with the bicomplete quasi-metric qd induced by d, and φ : X → BX is given by φ(x) = (x, 0) for all x ∈ X. From the above constructions we also deduce that for a metric space (X, d), the pair (B(CX), qHd ) is a weightable quasi-metric space where B(CX) is the continuous poset of formal balls of the metric hyperspace (CX, Hd ) and qHd is the weightable quasi-metric on B(CX) induced by Hd , i.e., qHd ((A, r), (B, s)) = max{Hd (A, B), |r − s|} + s − r, for all (A, r), (B, s) ∈ B(CX). On the other hand, we can construct the quasi-metric hyperspace (C(BX), Hqd ), where C(BX) is the family of all nonempty subsets of BX that are closed and bounded in the metric space (BX, (qd )s ).

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In the light of these facts, it suggests the natural question of obtaining the precise relationship between the weightable quasi-metric space (B(CX), qHd ) and the quasi-metric hyperspace (C(BX), Hqd ). In Theorem 1 below we answer this question. Let us recall that an isometry from a quasi-metric space (X, q) to a quasi-metric space (Y, q ′ ) is a mapping f : X → Y such that q ′ (f (x), f (y)) = q(x, y) for all x, y ∈ X. (Note that every isometry is one-to-one.) The quasi-metric spaces (X, q) and (Y, q ′ ) are said to be isometric if there is an isometry from (X, q) onto (Y, q ′ ). Theorem 1. Let (X, d) be a metric space. Then (B(CX), qHd ) is isometric to a supclosed subspace of (C(BX), Hqd ), via the isometry Φ : (B(CX), qHd ) → (C(BX), Hqd ) given by Φ((A, r)) = A × {r}, for all A ∈ CX and r ∈ R+ . Proof. We first show that Φ is well-defined. Indeed, let A ∈ CX and r ∈ R+ , and let (y, s) ∈ BX such that (y, s) belongs to the closure of A × {r} in (BX, (qd )s ). Then, there is a sequence (an )n in A such that lim(qd )s ((y, s), (an , r)) = 0. n

Hence limn {max{d(y, an ), |r − s|}} = 0, so y ∈ A and r = s. Thus A × {r} ∈ C(BX) and consequently Φ is well-defined. Now let (A, r), (B, s) ∈ B(CX). We easily deduce the following relations inf a∈A {max{d(a, b), |r − s|} + s − r} = max{inf a∈A d(a, b), |r − s|} + s − r, and supb∈B {max{d(A, b), |r − s|} + s − r} = max{supb∈B d(A, b), |r − s|} + s − r. Therefore Hq+d (Φ((A, r)), Φ((B, s))) = sup qd ((A × {r}), (b, s)) b∈B

= sup{ inf qd ((a, r), (b, s))} b∈B a∈A = sup inf {max{d(a, b), |r − s|} + s − r} b∈B a∈A = sup max{ inf d(a, b), |r − s|} + s − r b∈B

a∈A

= sup {max{d(A, b), |r − s|} + s − r} b∈B

= max{sup d(A, b), |r − s|} + s − r =

b∈B max{Hd+ (A, B), |r

− s|} + s − r.

Similarly, we show that Hq−d (Φ((A, r)), Φ((B, s))) = max{Hd− (A, B), |r − s|} + s − r.

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Hence Hqd (Φ((A, r)), Φ((B, s))) = max max{Hd+ (A, B), |r − s|} + s − r, max{Hd− (A, B), |r − s|} + s − r = max max{Hd+ (A, B), Hd− (A, B)}, |r − s| + s − r

= max{Hd (A, B), |r − s|} + s − r = qHd ((A, r), (B, s)).

Consequently Φ is an isometry from (B(CX), qHd ) to (C(BX), Hqd ). Finally, we show that the set Φ(B(CX)) is closed in the metric space (C(BX), (Hqd )s ). Let U ∈ C(BX) and let ((An , rn ))n be a sequence in B(CX) such that lim(Hqd )s (U, An × {rn }) = 0. n

Then for each (x, sx ) ∈ U there are sequences (an )n , (bn )n , with an , bn ∈ An for all n ∈ N, such that lim {{max{d(x, an ), |sx − rn |} + rn − sx } = 0,

and

n

lim {{max{d(x, bn ), |sx − rn |} + sx − rn } = 0. n

Therefore limn rn = sx (and also limn d(x, an ) = limn d(x, bn ) = 0). Thus, if we put s = limn rn and B = {x ∈ X : (x, s) ∈ U }, we have that U = B × {s}. Since U is closed in the metric space (BX, (qd )s ), it easily follows that B is closed in (X, d). Thus U ∈ Φ(B(CX)), and hence Φ(B(CX)) is closed in (C(BX), (Hqd )s ). This concludes the proof. The preceding result is interesting because from the weightable quasi-metric space (BX, qd ) one obtains a nice subspace of the quasi-metric hyperspace (C(BX), Hqd ), namely (Φ(B(CX)), Hqd ), which is also weightable. The following example shows that, nevertheless, (C(BX), Hqd ) is not weightable, in general. Example 2. Let X = {a} and let d be the trivial metric on X, i.e., d(a, a) = 0. We show that (C(BX), Hqd ) is not weightable. Assume the contrary. Then, there exists a weighting function W for (C(BX), Hqd ). We have the following relations qd ((a, 0), (a, 1)) = max{d(a, a), 1} + 1 = 2,

and

qd ((a, 1), (a, 0)) = max{d(a, a), 1} − 1 = 0. Then Hqd ({(a, 0)}, {(a, 1)}) = qd ((a, 0), (a, 1)) = 2, and Hqd ({(a, 1)}, {(a, 0)}) = qd ((a, 1), (a, 0)) = 0. Hence (1)

2 + W ({(a, 0)}) = 0 + W ({(a, 1)}).

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On the other hand, if B = {(a, 0), (a, 1)}, we obtain Hqd ({(a, 0)}, B) = max{Hq+d ({(a, 0)}, B), Hq−d ({(a, 0)}, B)} = max sup qd ({(a, 0)}, b), qd ({(a, 0)}, B) b∈B

= 2, and Hqd (B, {(a, 0)}) = max{Hq+d (B, {(a, 0)}), Hq−d (B, {(a, 0)})} = max qd (B, {(a, 0)}), sup qd (b, {(a, 0)}) b∈B

= 0. Hence (2)

2 + W ({(a, 0}) = 0 + W (B).

Moreover, we have Hqd ({(a, 1)}, B) = max{Hq+d ({(a, 1)}, B), Hq−d ({(a, 1)}, B)} = max sup qd ({(a, 1)}, b), qd ({(a, 1)}, B) b∈B

= 0, and Hqd (B, {(a, 1)}) = max{Hq+d (B, {(a, 1)}), Hq−d (B, {(a, 1)})} = max qd (B, {(a, 1)}), sup qd (b, {(a, 1)}) b∈B

= 2. Hence (3)

0 + W ({(a, 1}) = 2 + W (B).

From (1) and (2) it follows that W ({(a, 1)}) = W (B), which contradicts relation (3). We conclude that (C(BX), Hqd ) is not weightable. Theorem 2. For a metric space (X, d) the following are equivalent. (1) (X, d) is complete. (2) (B(CX), qHd ) is weightable and bicomplete. (3) (Φ(B(CX)), Hqd ) is weightable and bicomplete. Proof. (1) =⇒ (2). If (X, d) is complete, then the hyperspace (CX, Hd ) is complete ( [4, Theorem 3.2.4]). So, by Remark 1, the weightable quasi-metric space (B(CX), qHd ) is bicomplete.

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(2) =⇒ (3). It is an obvious consequence of Theorem 1. (3) =⇒ (1). Let (xn )n be a Cauchy sequence in (X, d). Then ({xn } × {0})n is a Cauchy sequence in (Φ(B(CX)), (Hqd )s ). Let A ∈ CX and r ∈ R+ such that limn (Hqd )s (A × {r}, {xn } × {0}) = 0. (Of course, r = 0). Choose a ∈ A. Clearly limn d(a, xn ) = 0. We conclude that (X, d) is complete. (Note that, in particular, A = {a}). Observe that if (X, d) is a metric space, then for each A, B ∈ CX and r, s ∈ R+ we have Hqd (A × {r}, B × {s}) = 0 ⇔ qHd ((A, r), (B, s)) = 0 ⇔ Hd (A, B) ≤ r − s, and hence A × {r} ≤Hqd B × {s} ⇔ (A, r) ⊑Hd (B, s). Consequently, from the fact that (B(CX), ⊑Hd ) is a continuous poset, it follows that (Φ(B(CX)), ≤Hqd ) is a continuous poset, where ≤Hqd is the partial order induced by Hqd (see Example 1 a)). Moreover, if (X, d) is complete, then (B(CX), ⊑Hd ) is a domain, and, thus, (Φ(B(CX)), ≤Hqd ) is also a domain. So, by Theorem 2, (Φ(B(CX)), φ) is a quantifiable computational model for (CX, Hd ), where φ(A) = A × {0} for all A ∈ CX.

4.

Properties from Topological Algebra

In this section we obtain some properties of (B(CX), qHd ) from topological algebra. Actually, we shall do this approach in a more general setting where the space BX of formal balls is replaced by a product space X × Y with X and Y be metric monoids or metric cones, respectively. In this direction, our first result generalizes Heckmann’s construction of the partial metric pd given in Section 3. Proposition 1. Let (X, d) and (Y, e) be metric spaces. Fix u0 ∈ Y. Then the function pd,e : (X × Y ) × (X × Y ) → R+ given by pd,e ((x, u), (y, v)) = max{d(x, y), e(u, v)} + e(u0 , u) + e(u0 , v), for all x, y ∈ X and u, v ∈ Y, is a partial metric on X × Y. Proof. Let x, y, z ∈ X and u, v, w ∈ Y. Then pd,e ((x, u), (y, v)) = pd,e ((x, u), (x, u)) = pd,e ((y, v), (y, v)) ⇐⇒ max{d(x, y), e(u, v)} + e(u0 , u) + e(u0 , v) = 2e(u0 , u) = 2e(u0 , v) ⇐⇒ (x, u) = (y, v).

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Moreover pd,e ((x, u), (x, u)) = 2e(u0 , u) ≤ e(u0 , u) + e(u0 , v) + e(v, u) ≤ pd,e ((x, u), (y, v)), and, obviously, pd,e ((x, u), (y, v)) = pd,e ((y, v), (x, u)). Finally pd,e ((x, u), (y, v)) = max{d(x, y), e(u, v)} + e(u0 , u) + e(u0 , v) ≤ max{d(x, z), e(u, w)} + max{d(z, y), e(w, v)} + e(u0 , u) + e(u0 , v) = pd,e ((x, u), (z, w)) + pd,e ((z, w), (y, v)) − pd,e ((z, w), (z, w)). We conclude that (X × Y, pd,e ) is a partial metric space. Note that if (Y, e) is the set R+ endowed with the Euclidean metric and put u0 = 0, then pd,e is exactly the partial metric pd on BX constructed in Section 3. Remark 2. The partial metric pd,e of Theorem 3 satisfies pd,e ((x, u), (x, u)) = 2e(u0 , u), for all x ∈ X and u ∈ Y. Hence, the weightable quasi-metric q(pd,e ) induced by pd,e , has the function w given by w((x, u)) = 2e(u0 , u), for all x ∈ X and u ∈ Y , as a weighting function. Furthermore, we have q(pd,e )((x, u), (y, v)) = max{d(x, y), e(u, v)} + e(u0 , v) − e(u0 , u), and hence

q(pd,e ) + (q(pd,e ))−1 2

((x, u), (y, v)) = max{d(x, y), e(u, v)},

for all x, y ∈ X and u, v ∈ Y. Therefore, the metric (q(pd,e )) + (q(pd,e ))−1 )/2 coincides with the product metric d × e on X × Y. The following result generalizes Theorem 2 to this context. Proposition 2. Let (X,d) and (Y,e) be complete metric spaces. Then (X × Y, q(pd,e )) is a bicomplete quasi-metric space. Proof. By Remark 2 it follows that (X × Y, (q(pd,e )) + (q(pd,e ))−1 )/2) is a complete metric space, and thus (X, (q(pd,e ))s ) is a complete metric space. We conclude that (X, q(pd,e )) is bicomplete.

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Let us recall that a monoid is a semigroup (X, ∗) having neutral element. If (X, ∗) and (Y, ⋆) are (commutative) monoids, then (X × Y, ∗⋆)) is a (commutative) monoid, where (x, u) ∗ ⋆(y, v) = (x ∗ y, u ⋆ v) for all x, y ∈ X, u, v ∈ Y. By a (quasi-)metric monoid we mean a triple (X, ∗, d) such that (X, ∗) is a monoid and d is a (quasi-)metric on X such that d(x ∗ a, y ∗ b) ≤ d(x, y) + d(a, b) for all x, y, a, b ∈ X, or equivalently, d(x ∗ z, y ∗ z) ≤ d(x, y) and d(z ∗ x, z ∗ y) ≤ d(x, y) for all x, y, z ∈ X (see, for instance, [30]). According to [30, Definition 1], by a partial metric monoid we mean a triple (X, ∗, p) such that (X, ∗) is a monoid and p is a partial metric on X such that p(x ∗ a, y ∗ b) ≤ p(x, y) + p(a, b) for all x, y, a, b ∈ X. Lemma 1. Let (X, ∗, d) and (Y, ⋆, e) be metric monoids. Then (X × Y, ∗⋆, pd,e ) is a partial metric monoid whenever the point u0 of Proposition 1 is the neutral element of (Y, ⋆). Proof. First note that if u0 is the neutral element of (Y, ⋆), we have e(u0 , u ⋆ v) ≤ e(u0 , u) + e(u0 , v) for all u, v ∈ Y. Now let x, y, a, b ∈ X and u, v, s, t ∈ Y. Then, by Proposition 1, pd,e ((x, u) ∗ ⋆(y, v), (a, s) ∗ ⋆(b, t)) = pd,e ((x ∗ y, u ⋆ v), (a ∗ b, s ⋆ t)) = max {d(x ∗ y, a ∗ b), e(u ⋆ v, s ⋆ t)} + e(u0 , u ⋆ v) + e(u0 , s ⋆ t) ≤ max {d(x, a) + d(y, b), e(u, s) + e(v, t)} + e(u0 , u) + e(u0 , v) + e(u0 , s) + e(u0 , t) ≤ max {d(x, a), e(u, s)} + max {d(y, b), e(v, t)} + e(u0 , u) + e(u0 , v) +e(u0 , s) + e(u0 , t) = pd,e ((x, u), (a, s)) + pd,e ((y, v), (b, t)). We have shown that (X × Y, ∗⋆, pd,e ) is a partial metric monoid. Lemma 2. Let (X, ∗, d) and (Y, ⋆, e) be metric monoids. Then (X × Y, ∗⋆, q(pd,e )) is a quasi-metric monoid whenever the point u0 of Proposition 1 is the neutral element of (Y, ⋆) and e(u0 , u ⋆ v) = e(u0 , u) + e(u0 , v) for all u, v ∈ Y . Proof. Let x, y, a, b ∈ X and u, v, s, t ∈ Y. From Theorem A, Lemma 1 and Remark 2, we deduce the following relations q(pd,e )((x, u) ∗ ⋆(y, v), (a, s) ∗ ⋆(b, t)) = q(pd,e )((x ∗ y, u ⋆ v), (a ∗ b, s ⋆ t)) = pd,e ((x ∗ y, u ⋆ v), (a ∗ b, s ⋆ t)) − pd,e ((x ∗ y, u ⋆ v), (x ∗ y, u ⋆ v)) ≤ pd,e ((x, u), (a, s)) + pd,e ((y, v), (b, t)) − 2e(u0 , u ⋆ v) = pd,e ((x, u), (a, s)) + pd,e ((y, v), (b, t)) − 2e(u0 , u) − 2e(u0 , v) = q(pd,e )((x, u), (a, s)) + q(pd,e )((y, v), (b, t)). We have shown that (X × Y, ∗⋆, q(pd,e )) is a quasi-metric monoid.

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If we denote by + the usual addition on R+ , then the monoid (R+ , +), endowed with the Euclidean metric, has the structure of a metric monoid. From this fact and Lemmas 1 and 2, we immediately deduce the following result. Proposition 3. Let (X, ∗, d) be a metric monoid. Then (BX, ∗+, pd ) is a partial metric monoid and (BX, ∗+, qd ) is a quasi-metric monoid. Following [17], a cone (a semilinear space in [29]) is a triple (X, ∗, ◦) such that (X, ∗) is a commutative monoid with neutral element 0 and ◦ is a function from R+ × X to X which satisfies for all r, s ∈ R+ and x, y ∈ X : (i) r◦(s◦x) = (rs)◦x; (ii) (r+s)◦x = r◦x∗s◦x; (iii) r ◦ (x ∗ y) = r ◦ x ∗ r ◦ y; (iv) 1 ◦ x = x; (v) 0 ◦ x = 0 (see [18] for related structures). If (X, ∗, ◦) and (Y, ⋆, ·) are cones, then (X × Y, ∗⋆, ◦·) is a cone, where we define r ◦ ·(x, u) = (r ◦ x, r · u) for all x ∈ X, u ∈ Y and r ∈ R+ . By a (quasi-)metric cone we mean a quadruple (X, ∗, ◦, d) such that (X, ∗, ◦) is a cone and d is a (quasi-)metric on X for which d(x∗a, y∗b) ≤ d(x, y)+d(a, b) and d(r◦x, r◦y) ≤ rd(x, y) for all x, y, a, b ∈ X and r ∈ R+ . By a partial metric cone we mean a quadruple (X, ∗, ◦, p) such that (X, ∗, ◦) is a cone and p is a partial metric on X for which p(x ∗ a, y ∗ b) ≤ p(x, y) + p(a, b) and p(r ◦ x, r ◦ y) ≤ rp(x, y) for all x, y, a, b ∈ X and r ∈ R+ . Lemma 3. Let (X, ∗, ◦, d) and (Y, ⋆, ·, e) be metric cones. Then (X × Y, ∗⋆, ◦·, pd,e ) is a partial metric cone whenever the point u0 of Proposition 1 is the neutral element of (Y, ⋆, ·). Proof. By virtue of Lemma 1 we only need to prove that for each x, y ∈ X, u, v ∈ Y and r ∈ R+ , it follows that pd,e (r ◦ ·(x, u), r ◦ ·(y, v)) ≤ rpd,e ((x, u), (y, v)). In fact, pd,e (r ◦ ·(x, u), r ◦ ·(y, v)) = pd,e ((r ◦ x, r · u), (r ◦ y, r · v)) = max {d(r ◦ x, r ◦ y), e(r · u, r · v)} + e(u0 , r · u) + e(u0 , r · v) ≤ max {rd(x, y), re(u, v)} + re(u0 , u) + re(u0 , v) = r [max {d(x, y), e(u, v)} + e(u0 , u) + e(u0 , v)] = rpd,e ((x, u), (y, v)). We have shown that (X × Y, ∗⋆, ◦·, pd,e ) is a partial metric cone. Lemma 4. Let (X, ∗, ◦, d) and (Y, ⋆, ·, e) be metric cones. If u0 is the neutral element of (Y, ⋆, ·). Then (X × Y, ∗⋆, ◦·, q(pd,e )) is a quasi-metric cone whenever the point u0 of Proposition 1 is the neutral element of (Y, ⋆, ·) and e(u0 , u ⋆ v) = e(u0 , u) + e(u0 , v), e(u0 , r · u) = re(u0 , u), for all u, v ∈ Y and r ∈ R+ . Proof. By virtue of Lemma 2 we only need to prove that for each x, y ∈ X, u, v ∈ Y and r ∈ R+ , it follows that q(pd,e )(r ◦ ·(x, u), r ◦ ·(y, v)) ≤ rq(pd,e )((x, u), (y, v)). In fact,

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from Theorem A, Lemma 3 and Remark 2, we obtain q(pd,e )(r ◦ ·(x, u), r ◦ ·(y, v)) = q(pd,e )((r ◦ x, r · u), (r ◦ y, r · v)) = pd,e ((r ◦ x, r · u), (r ◦ y, r · v)) − 2e(u0 , r · u) ≤ rpd,e ((x, u), (y, v)) + re(u0 , v) − 2re(u0 , u) = rq(pd,e )((x, u), (y, v)). We have shown that (X × Y, ∗⋆, ·◦, q(pd,e )) is a quasi-metric cone. If we denote by · the usual multiplication on R+ , then the cone (R+ , +, ·), endowed with the Euclidean metric, has the structure of a metric cone. From this fact and Lemmas 3 and 4, we immediately deduce the following result. Proposition 4. Let (X, ∗, ◦, d) be a metric cone. Then (BX, ∗+, ◦·, pd ) is a partial metric cone and (BX, ∗+, ◦·, qd ) is a quasi-metric cone. In order to apply Propositions 3 and 4 to B(CX) and B(Cc X), respectively, where by Cc X we denote the family of all nonempty closed bounded convex subsets of a metric cone (X, ∗, ◦, d), we need to establish some properties of the structure of CX and Cc X (related results may be found, for instance, in [4, p. 89 and 91] and in [27]). Let (X, ∗, d) be a metric monoid with neutral element 0. For each A, B ∈ CX, define A ∗ B = {a ∗ b : a ∈ A, b ∈ B}, and A ⊛ B = A ∗ B. Then, we have the following. Lemma 5. (CX, ⊛, Hd ) is a metric monoid. Proof. Let A, B, C ∈ CX. Obviously A ⊛ B is a nonempty closed subset of (X, d). Moreover, it is bounded because A and B are bounded and (X, ∗, d) is a metric monoid. Thus A ⊛ B ∈ CX. Furthermore (A ⊛ B) ⊛ C = (A ⊛ B) ∗ C = A ∗ B ∗ C = A ∗ B ∗ C = A ∗ B ∗ C = A ∗ (B ⊛ C) = A ⊛ (B ⊛ C). Since for each A ∈ CX, A ⊛ {0} = A, it follows that (CX, ⊛) is a monoid. If, in addition, (X, ∗) is commutative then it is obvious that (CX, ⊛) is also commutative. Next we show that Hd+ (A ⊛ C, B ⊛ C) ≤ Hd+ (A, B). Indeed, choose an arbitrary ε > 0. Let x ∈ B ⊛ C. Then, there exist b ∈ B and c ∈ C such that d(b ∗ c, x) < ε. Now let a ∈ A such that d(a, b) < ε + d(A, b). Thus d(A ⊛ C, x) ≤ d(A ∗ C, x) ≤ d(a ∗ c, x) ≤ d(a ∗ c, b ∗ c) + d(b ∗ c, x) < 2ε + d(A, b) ≤ 2ε + Hd+ (A, B). Hence Hd+ (A ⊛ C, B ⊛ C) ≤ Hd+ (A, B). Similarly we show the following inequalities: Hd+ (C ⊛ A, C ⊛ B) ≤ Hd+ (A, B), Hq− (A ⊛ C, B ⊛ C) ≤ Hq− (A, B) and Hq− (C ⊛ A, C ⊛ B) ≤ Hq− (A, B). We conclude that (CX, ⊛, Hd ) is a metric monoid.

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Theorem 3. Let (X, ∗, d) be a metric monoid. Then (B(CX), ⊛+, pHd ) is a partial metric monoid and (B(CX), ⊛+, qHd ) is a quasi-metric monoid. Proof. Since, by Lemma 5, (CX, ⊛, Hd ) is a metric monoid, it follows from Proposition 3 that (B(CX), ⊛+, pHd ) is a partial metric monoid and (B(CX), ⊛+, qHd ) is a quasi-metric monoid. If (X, ∗, ◦, d) is a metric cone and for each A, B ∈ Cc X and r ∈ R+ we define A ⊛ B = A ∗ B, and r ◦ A = {r ◦ a : a ∈ A}, we have the following. Lemma 6. (Cc X, ⊛, ◦, Hd ) is a metric cone. Proof. We first show that (Cc X, ⊛, ◦) is a cone. It is easily seen that for each A, B ∈ Cc X and r ∈ R+ , A ⊛ B ∈ Cc X and r ◦ A ∈ Cc X, so, by virtue of Lemma 5, we only need to prove that for each A, B ∈ Cc X and r, s ∈ R+ . the conditions (i)-(v) of the notion of a cone given above, hold. In fact, it is clear that r ◦ (s ◦ A) = (rs) ◦ A. Moreover r ◦ (A ⊛ B) = r ◦ A ∗ B = r ◦ (A ∗ B) = r ◦ A ∗ r ◦ B = r ◦ A ⊛ r ◦ B. On the other hand, by convexity of A, we obtain (r + s) ◦ A = r ◦ A ∗ s ◦ A, so (r + s) ◦ A ⊆ r ◦ A ⊛ s ◦ A. Now let x ∈ r ◦ A ⊛ s ◦ A. Then there exist two sequences (an )n , (bn )n , in A such that limn d(x, r ◦ an ∗ s ◦ bn ) = 0. Therefore 1 1 lim d ◦ x, ◦ (r ◦ an ∗ s ◦ bn ) = 0. n r+s r+s 1 ◦ x ∈ A, so x ∈ (r + s) ◦ A. We conclude Since A is closed convex, we deduce that r+s that (r + s) ◦ A = r ◦ A ⊛ s ◦ A. Finally, it is obvious that 1 ◦ A = A and 0 ◦ A = 0. Hence (Cc X, ⊛, ◦) is a cone. Now from the fact that for each A, B ∈ Cc X and r ∈ R+ , we have

d(r ◦ A, r ◦ b) ≤ rd(A, b),

and d(r ◦ a, r ◦ B) ≤ r(d(a, B),

for all a ∈ A, b ∈ B, it follows that Hd (r ◦ A, r ◦ B) ≤ rHd (A, B). By Lemma 5 we conclude that (Cc X, ⊛, ◦, Hd ) is a metric cone. Theorem 4. Let (X, ∗, ◦, d) be a metric cone. Then (B(CX), ⊛+, ◦·, pHd ) is a partial metric cone and (B(CX), ⊛+, ◦·, qHd ) is a quasi-metric cone. Proof. Since, by Lemma 6, (CX, ⊛, ◦, Hd ) is a metric cone, it follows from Proposition 4 that (B(CX), ⊛+, ◦·, pHd ) is a partial metric cone and (B(CX), ⊛+, ◦·, qHd ) is a quasimetric cone.

5.

Conclusion

It is known that for a metric space (X, d), the continuous poset BX of formal balls can be endowed with a weightable quasi-metric qd whose induced topology coincides with

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the Scott topology. We have analyzed the continuous poset B(CX) of the hyperspace (CX, Hd ). We have proved that the weightable quasi-metric space (B(CX), qHd ) is isometric to a sup-dense subspace of the hyperspace (C(BX), Hqd ). If (X, d) is complete, then this subspace provides a quantifiable computational model for (CX, Hd ). Some properties from topological algebra have been also discussed. In particular, if (X, d) is a metric monoid, then (B(CX), qHd ) admits a structure of quasi-metric monoid, and if (X, d) is a metric cone, then (B(Cc X), qHd ) admits a structure of quasi-metric cone.

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In: Advances in Mathematics Research, Volume 8 ISBN 978-1-60456-454-9 c 2009 Nova Science Publishers, Inc. Editor: Albert R. Baswell, pp. 295-353

Chapter 10

P ERIODIC -T YPE S OLUTIONS OF D IFFERENTIAL I NCLUSIONS Jan Andres∗ Dept. of Math. Analysis, Fac. of Science, Palack´y University, Tomkova 40, 779 00 Olomouc–Hejˇc´ın, Czech Republic

1.

Introduction

Periodicity is one of the most influental phenomena for everybody’s life. However, its pure form occurs in nature rather rarely. For instance, periodic structure in three independent directions (lattice symmetry) which is typical for crystals is never purely periodic in a mathematical (ideal) sense. Moreover, in the early 80’s, certain materials were found with diffraction patterns as those for crystals, but with other symmetries that are not commensurate with lattice symmetry. These new substances, called quasicrystals, have much to do with (aperiodic) Penrose tiling and quasi-periodicity (or even with almost-periodicity; cf. [Me1], [Me2]). Regular time repetitions, like clock measurements, are similarly always periodic only with some accuracy. That is why mathematical models deal also with various sorts of (generalized) periodicity which can be rather far from its original meaning (cf. e.g. [A4], [AKZ], [Fe], [JMe], [LF], [LLY], [LY], [Ru]). Periodic dynamics (in a pure or generalized sense) are usually described by solutions of differential or difference equations or inclusions. In order to extend the notion of a dynamical system in an appropriate way, the “terminus technicus” periodic process was introduced (cf. e.g. [AG1] and the references therein). Although many monographs have been already written about free and forced linear as well as nonlinear oscillations (see e.g. [H1], [Md], [Mi]), especially the theory of almost-periodic oscillations is far from to be built in a satisfactory level (cf. [AG1], [BMC], [HM], [J3], [Mw2], [SY]). Moreover, there are many sorts of almost-periodicity and some of them are a bit curious (cf. [ABG], [Be], [Bs], [Le]). It is interesting that the creator of the classical theory of almost-periodic functions, Harald Bohr [Bh], came to (uniformly) almost-periodic functions not in order to extend the notion of a ∗

E-mail address: 6198959214).

[email protected]

Supported by the Council of Czech Government (MSM

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periodic function, but because of his investigation of Dirichlet series. There is still a lot to be also said about quasi-periodic (cf. [BC1], [BC2], [CZ], [Lv], [Or], [SM]), anti-periodic (cf. [AAP1], [AAP2], [AF], [AP], [FMT]) and derivo-periodic (cf. [A2], [A3], [AGMP], [Mw1]) oscillations. Hence, the main purpose of our contribution is two-fold: (i) rather than a complete account or a systematic study, we would like to indicate a flavour of the theory of periodictype oscillations, and (ii) to present some of our own results for periodic-type solutions of differential equations and inclusions. For (i), we preferably selected in Section 4. (Primer of periodic-type oscillations) the related results (including ours) which are easy for formulation while, for (ii), some technicalities had to be involved in Section 5. in order to derive sufficiently general criteria of the effective solvability of given actual problems. Results are, nevertheless, sketched in a form that is convenient for exposition and not necessarily in the greatest generality possible. Our objective is so to give the reader an overall idea of what the standard theory is like as well as to include enough information about its most recent progress. Formally, the focus of the object is simply the determination of the readable text for a wider audience with some parts to yield also a profit for the specialists. Although Section 3. (Periodic-type maps, multivalued maps and their selections) contains a lot of a new material, eventually of an independent interest, its choice was tendentiously determined by the needs of the main Sections 4. and 5.. Not to break the context, we also recall in Preliminaries several useful facts about multivalued maps and their selections. In Concluding remarks, possible generalizations, extensions and improvements are only indicated.

2.

Preliminaries

In the entire text, by a multivalued map ϕ : X ⊸ Y , we mean the one with at least nonempty values, i.e. ϕ : X → 2Y \ {0}. It is convenient to recall the definitions of multivalued maps which will be under consideration and their basic properties. In particular, the existence of appropriate single-valued selections will be of our interest here. For more details, see [AG1], [HP], [Ry]. D EFINITION 2.1 A map ϕ : X ⊸ Y , where X, Y are metric spaces, is said to be upper semicontinuous (u.s.c.) if, for every open U ⊂ Y , the set {x ∈ X | ϕ(x) ⊂ U } is open in X. It is said to be lower semicontinuous (l.s.c.) if, for every open U ⊂ Y , the set {x ∈ X | ϕ(x) ∩ U 6= ∅} is open in X. If it is both u.s.c. and l.s.c., then it is called continuous. Obviously, in the single-valued case, if f : X → Y is u.s.c. or l.s.c., then it is continuous. Moreover, the compact-valued map ϕ : X ⊸ Y is continuous if and only if it is Hausdorff-continuous, i.e. continuous w.r.t. the metric d in X and the Hausdorff-metric dH in {B ⊂ Y | B is nonempty and bounded}, where dH (A, B) := inf{ε > 0 | A ⊂ Oε (B) and B ⊂ Oε (A)} and Oε (B) := {x ∈ X | ∃y ∈ B : d(x, y) < ε}. Every u.s.c. map ϕ : X ⊸ Y with closed values has a closed graph Γϕ , but not vice versa. Nevertheless, if the graph Γϕ of a compact map ϕ : X ⊸ Y is closed, then ϕ is u.s.c.

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The following proposition will be important for the existence of single-valued continuous selections (see, e.g. [HP, Corollary 1.4.8]). P ROPOSITION 2.1 Let X be a paracompact (e.g. metric) space and Y be a Banach space. If ϕ : X ⊸ Y is l.s.c. with convex closed values, then given [xi , yi ] ∈ Γϕ , i = 1, . . . , n, there exist a single-valued continuous selection f ⊂ ϕ of ϕ such that f (xi ) = yi , i = 1, . . . , n. We say that ϕ : X ⊸ Y is Lipschitzian or Lipschitz-continuous with constant L (w.r.t. the metric d in X and the Hausdorff-metric dH in {B ⊂ Y | B is nonempty and bounded}) if dH (ϕ(x1 ), ϕ(x2 )) ≤ Ld(x1 , x2 ), for all x1 , x2 ∈ X. For metric X and Banach Y , Lipschitz-continuous ϕ : X ⊸ Y with bounded, closed, convex values admits a Lipschitz-selection f ⊂ ϕ if and only if Y is finite-dimensional.√The Lipschitz constant of f is not necessarily the same as for ϕ. For Y = Rn , it is n( 12 3+ 5 1)L, where L is a constant of ϕ. For more details, see [HP, pp. 101–105]. Besides semicontinuous maps, measurable and semi-Carath´eodory maps will be also of importance. Hence, assume that Y = (Y, d) is a separable metric space and (Ω, U, ν) is a measurable space, i.e. a set Ω equipped with σ-algebra U of subsets and a countably additive measure ν on U. A typical example is when Ω is a bounded domain in Rn , equipped with the Lebesque measure. D EFINITION 2.2 A map ϕ : Ω ⊸ Y is called strongly measurable if there exists a sequence of step multivalued maps ϕn : Ω ⊸ Y such that dH (ϕn (ω), ϕ(ω)) → 0, for almost all (a.a.) ω ∈ Ω, as n → ∞. In the single-valued case, one can simply replace multivalued step maps by single-valued step maps and dH (ϕn (ω), ϕ(ω)) by d (ϕn (ω), ϕ(ω)). A map ϕ : Ω ⊸ Y is called measurable if {ω ∈ Ω | ϕ(ω) ⊂ V } ∈ U, for each open V ⊂Y. A map ϕ : Ω ⊸ Y is called weakly measurable if {ω ∈ Ω | ϕ(ω) ⊂ V } ∈ U, for each closed V ⊂ Y . Obviously, if ϕ is strongly measurable, then it is measurable and if ϕ is measurable, then it is also weakly measurable. If ϕ has compact values, then the notions of measurability and weak measurability coincide. In separable Banach spaces Y , the notions of strong measurability and measurability coincide for multivalued maps with compact values as well as for single-valued maps (see [KOZ, Theorem 1.3.1 on pp. 45–49]). If Y is a not necessarily separable Banach space, then a strongly measurable map ϕ : Ω ⊸ Y with compact values has a single-valued strongly measurable selection (see e.g. [De, Proposition 3.4(b) on pp. 25– 26]). Furthermore, if Y is a separable complete space, then every measurable ϕ : Ω ⊸ Y with closed values has, according to the following Kuratowski–Ryll-Nardzewski theorem (see e.g. [AG1, Theorem 3.49 in Chapter I.3]), a single-valued measurable selection. P ROPOSITION 2.2 (Kuratowski–Ryll-Nardzewski) Let Ω be as above and Y be a separable complete space. Then every measurable map ϕ : Ω ⊸ Y with closed values has a singlevalued measurable selection. Now, let Ω = [0, a] be equipped with the Lebesque measure and X, Y be Banach. D EFINITION 2.3 A map ϕ : [0, a] × X ⊸ Y with nonempty, compact and convex values is called u-Carath´eodory (resp. l-Carath´eodory, resp. Carath´eodory) if it satisfies (i) t ⊸ ϕ(t, x) is strongly measurable, for every x ∈ X,

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(ii) x ⊸ ϕ(t, x) is u.s.c. (resp. l.s.c., resp. continuous), for almost all t ∈ [0, a], (iii) kykY ≤ r(t)(1 + kxkX ), for every (t, x) ∈ [0, a] × X, y ∈ ϕ(t, x), where r : [0, a] → [0, ∞) is an integrable function. For our needs (semi-) Carath´eodory maps will be employed only in Euclidean spaces. Moreover, for X = Rm and Y = Rn , one can state the following proposition. P ROPOSITION 2.3 (i) Carath´eodory maps are product-measurable (measurable as the whole (t, x) ⊸ ϕ(t, x)), i.e. w.r.t. minimal σ-algebra U[0,a] ⊗ B(Rm ), generated by U[0,a] × B(Rm ), where U[0,a] is the σ-algebra of subsets of [0, a], and B(Rm ) stands for the Borel sets of Rm , (ii) they possess a single-valued Carath´eodory selection f ⊂ ϕ. It need not be so for u-Carath´eodory or l-Carath´eodory maps. Nevertheless, for uCarath´eodory maps, we have at least (again X = Rm and Y = Rn ). P ROPOSITION 2.4 u-Carath´eodory maps (in the sense of Definition 2.3) are weakly superpositionally measurable, i.e. the composition ϕ(t, q(t)) admits, for every q ∈ C([0, a], Rm ), a single-valued measurable selection. If they are still product-measurable, then they are also superpositionally measurable, i.e. the composition ϕ(t, q(t)) is measurable, for every q ∈ C([0, a], Rm ). R EMARK 2.1 If X, Y are separable Banach spaces and ϕ : X ⊸ Y is a Carath´eodory mapping, then ϕ is also superpositionally measurable, i.e. ϕ(t, q(t)) is measurable, for every q ∈ C([0, a], X) (see [KOZ, Theorem 1.3.4 on p. 56]). Under the same assumptions, Proposition 2.3 can be appropriately generalized (see [KOZ, Proposition 7.9 on p. 229 and Proposition 7.23 on pp. 234–235]). If ϕ : X ⊸ Y is only u-Carath´eodory and X, Y are (not necessarily separable) Banach spaces, then ϕ is weakly superpositionally measurable, i.e. ϕ(t, q(t)) admits a single-valued measurable selection, for every q ∈ C([0, a], X) (see e.g. [De, Proposition 3.5 on pp. 26– 27] or [KOZ, Theorem 1.3.5 on pp. 57–58]). For l-Carath´eodory maps, single-valued Carath´eodory selections can be guaranteed, under suitable restrictions (cf. [Ry] and the references therein). Nevertheless, since lCarath´eodory maps will not be employed in the sequel, the related statements are omitted here.

3. 3.1.

Periodic-Type Maps, Multivalued Maps and Their Selections Periodic Maps

Function f , defined on Rn and having values in an arbitrary set S, is called periodic if there exists a nonzero vector (0 6=) ω ∈ Rn such that f (x + ω) = f (x), for all x ∈ Rn . Any ω ∈ Rn satisfying this equality is called a period of f . The set P of all periods of f generates a subgroup in Rn which (by the hypothesis) does not degenerate to 0 (cf. [Bo]). Sometimes (e.g. when f is not defined on the whole Rn ), we can restrict ourselves to a subset X ⊂ Rn , e.g. for X = Rn+ . The equality f (x + ω) ≡ f (x) then means f (x + ω) = f (x), for all x ∈ X, where f (x) and f (x + ω) are defined. If f : Rn ⊸ S or f : X ⊸ S

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is multivalued with nonempty values, i.e. f : Rn → 2S \ {0} or f : X → 2S \ {0}, then by a periodic (multivalued) function f , we often mean only that f (x) ⊂ f (x + ω), provided each component of ω is positive, for all x, where f (x) and f (x + ω) are defined. If f : Rn → H is a continuous map from Rn into a Hausdorff topological space H, then the group P of its periods is closed. If V ⊂ P is the biggest vector subspace of P , then f is constant on any class (mod V ). Thus, f can be defined in this way by its restriction on a subspace which is complementary to V . That is why it is enough to consider continuous periodic functions with a discrete group of periods P . If rank P = p, then f is called pperiodic1 and any system of p vectors generating P is called the main system of periods of f . For two different main systems of periods, one can be obtained from the other by a linear transformation with integer coefficients whose determinant has an absolute value equal to 1. If P is a closed subgroup of the group Rn and π : Rn → Rn /P is a canonical projection, then g → g ◦π is a bijective mapping of all functions from Rn /P into S whose periods are contained in P . If S is a topological space, then for continuity of g it is necessary and sufficient g ◦ π to be continuous. For more details, concerning the algebraic aspects of periodic maps, see e.g. [Bo]. If S is a vector space and f : Rn → S is single-valued, then the basic equality f (x + ω) = f (x) can be equivalently rewritten into f (x + ω) − f (x) = 0. If S = (S, d) is a metric space, then we can still write d(f (x), f (x + ω)) = 0. On the other hand, if S = (S, d0 ) is only pseudometric, i.e. if d0 (f (x), f (x + ω)) = 0 does not necessarily imply that f (x) = f (x + ω), then the set {f : Rn → S | d0 (f (x), f (x + ω)) = 0} can be rather far from the set of periodic functions. For multivalued maps f : Rn ⊸ S, the situation is even more delicate. If A, B ⊂ S are subsets of a vector space, then defining A − B := {a − b | a ∈ A, b ∈ B}, the equality f (x + ω) − f (x) = 0 can never be satisfied, provided f is not single-valued. If S = (S, d) is a metric space, then dist(f (x), f (x + ω)) = inf[d(y1 , y2 ) | y1 ∈ f (x), y2 ∈ f (x + ω)] = 0, for the set distance, can imply rather curious possibilities, but including the case f (x) ⊂ f (x + ω). On the other hand, the equality f (x) = f (x + ω) can be expressed equivalently by means of the Hausdorff metric dH (for its definition, see Preliminaries) as dH (f (x), f (x+ω)) = 0 while, for the inclusion f (x) ⊂ f (x+ω), we have not equivalently that dH (f (x), f (x + ω)) ≥ 0. Moreover, if f : Rn ⊸ B, where B is a Banach space, is a periodic lower semicontinuous (l.s.c.) map with convex closed values, then it admits a single-valued continuous periodic selection f0 ⊂ f with the same period (see Proposition 2.1). If f : Rn ⊸ S, 1

By a p-periodic map f , we usually mean in a completely different way that f (x + p) ≡ f (x), where p ∈ Rn is the minimal period. The different meaning can be, however, easily recognized from the context. If the minimal period is 0, the function f is constant.

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where S is a separable complete space, is a measurable periodic map with closed values, then according to the well-known Kuratowski–Ryll–Nardzewski theorem (cf. Proposition 2.2) f admits a single-valued measurable periodic selection f0 ⊂ f with the same period. If f : R → Rn is (almost everywhere) differentiable and ω-periodic, then f˙ is also ω-periodic. Denoting therefore, as usual, CP (R, Rn ) := {f ∈ C (R, Rn ) | f (x) ≡ f (x + ω)}, C (k)P (R, Rn ) := {f ∈ C (k) (R, Rn ) | f (x) ≡ f (x + ω)}, we have C (k)P (R, Rn ) = {f ∈ C (k) (R, Rn ) | f (j) (x) ≡ f (j) (x + ω), j = 0, . . . , k}. If we endow CP (R, Rn ) by the norm kf k1 := maxx∈[0,ω] |f (x)| and C (k)P (R, Rn ) by Pk (j) (x)|, then the spaces (CP (R, Rn ), k · k ) and the norm kf k2 := 1 j=0 maxx∈[0,ω] |f (C (k)P (R, Rn ), k · k2 ) are Banach. Similarly, denoting AC (k−1)P (R, Rn ) := {f ∈ AC (k−1) (R, Rn ) | f (x) ≡ f (x + ω), } where AC (k−1) (R, Rn ) is a space of functions f : R → Rn whose (k − 1)th derivatives f (k−1) (x) are absolutely continuous, and endowing it by the norm kf k3 :=

k−1 X j=0

max |f

x∈[0,ω]

(j)

(x)| +

Z

ω

|f (k) (x)| dx,

0

the space (AC (k−1)P (R, Rn ), k · k3 ) is Banach as well, etc. The basic problem of the Fourier harmonic analysis consists in representation of (continuous) periodic functions by means of series of trigonometric functions of the form ̺1 cos T x resp. ̺2 sin T x. More precisely, let f : R → R be a function such that f (x + ω) ≡ f (x) and assume that the following Dirichlet conditions are satisfied: (i) f is single-valued and finite, for every x ∈ [0, ω] (observe that this condition already follows from the notation f : R → R), (ii) f admits in [0, ω] at most a finite number of (finite) discontinuities, (iii) f admits in [0, ω] a finite number of local extremal points. Then f can be uniquely expressed by means of the Fourier series as follows: ∞

X 1 f (x) = A0 + (An cos nT x + Bn sin nT x), 2 n=1

where T =

2π ω ,

2 An = ω

Z

0

ω

f (x) cos nT x dx,

2 Bn = ω

Z

ω

f (x) sin nT x dx.

0

This series is convergent in each point x = x0 ∈ [0, ω] to the value 1 [f (x0 + 0) + f (x0 − 0)]. 2

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In fact, an ω-periodic f can be equivalently expressed by means of the series consisting only of trigonometric functions of one type, namely ∞ p X 1 Bn 2 2 f (x) = A0 + An + Bn cos nT x − arctan 2 An n=1

resp. f (x) =

∞ p X 1 Bn 1 A2n + Bn2 sin nT x + π − arctan , A0 + 2 2 An n=1

where the Fourier coefficients An , Bn are as above. If an ω-periodic f is additionally even or odd, then simply ∞

X 1 f (x) = A0 + An cos nT x 2 n=1

or f (x) =

∞ X

Bn sin nT x,

n=1

respectively. For more details concerning trigonometric series and the Fourier analysis, see e.g. [Z1, Z2]. For abstract harmonic analysis, see e.g. the monograph in two volumes [HR1, HR2]. For the classification and approximation of periodic functions, see [St].

3.2.

Anti-periodic Maps

Function f , defined on Rn and having values in a vector space V (unlike for periodic maps, we must also consider function −f which requires values in a vector space) is called antiperiodic if there exists a nonzero vector (0 6=) ω ∈ Rn such that f (x + ω) = −f (x), for all x ∈ Rn . Any ω ∈ Rn satisfying this equality is called an anti-period of f . Obviously, every anti-periodic function with anti-period ω ∈ Rn is 2ω-periodic, but not vice versa. The class of anti-periodic functions can be therefore regarded as a special subclass of periodic functions. As for periodic functions, if not otherwise stated, by an ω-anti-periodic function, we shall mean the one with the minimal anti-period ω ∈ Rn . A generalization appears in the context of Bloch waves and the Floquet theory (studied in Section 4.2. below), where solutions of given differential equations are typically of the form x(t + T ) ≡ ̺x(t), where ̺ = e(p/q)πi ; p, q ∈ Z, because, for ̺ = 1 (p = 0), we obtain a T -periodic solution, while for ̺ = −1 (p = q 6= 0), we obtain a T -anti-periodic solution. Functions of this form are therefore sometimes called periodic in the sense of Bloch. Another generalization might appear in this context, namely functions satisfying the functional equality f (x + ω) = α(x)f (x), where α is a suitable function. These maps are sometimes also rather incorrectly called as quasi-periodic. We saw in Section 3.1. that odd ω-periodic functions f can be expressed in the form f (x) =

∞ X

n=1

Bn sin nT x,

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where T =

2π ω

and Bn =

2 ω

Z

ω

f (x) sin nT x dx.

0

Hence, if the graph of an ω-anti-periodic function f (x + c) is, for some c = c0 ∈ Rn , still symmetric w.r.t. the origin 0, then the function f (x + c0 ) is an odd ω-anti-periodic as well as odd 2ω-periodic by which f can be expressed in the form ∞ X

f (x) =

Bn sin

n=1

where 1 Bn = ω

Z

2ω+c0

c0

π (x − c0 ), ω

π f (x) sin n x dx. ω

Otherwise, the related Fourier series includes also the terms An cos n ωπ x, as for general 2ω-periodic functions, in Section 3.1.. If V is a vector space and f : Rn → V is single-valued, then the basic equality f (x + ω) = −f (x) can be rewritten into f (x + ω) + f (x) = 0. If V = (V, d) is a metric vector space, then we can still write d(−f (x), f (x + ω)) = 0. On the other hand, if V = (V, d0 ) is only a pseudometric vector space, i.e. if d0 (−f (x), f (x + ω)) = 0 does not necessarily imply that f (x + ω) = −f (x), then the set {f : Rn → V | d0 (−f (x), f (x + ω)) = 0} can be rather far from the set of anti-periodic functions. For multivalued maps f : Rn ⊸ V , the situation is again more delicate. The equality f (x) + f (x + ω) = 0 can never be satisfied, provided f is not single-valued. On the other hand, the equality f (x+ω) = −f (x) can be expressed equivalently by means of the Hausdorff metric dH (for its definition, see Preliminaries) as dH (−f (x), f (x + ω)) = 0. By anti-periodic (multivalued) functions, we can only mean, similarly as in Section 3.1., that −f (x) ⊂ f (x + ω), provided each component of ω ∈ Rn is positive, for all x, where f (x) and f (x + ω) are defined. This implies that dist(−f (x), f (x + ω)) = inf[d(y1 , y2 ) | y1 ∈ −f (x), y2 ∈ f (x + ω)] = 0, but the reverse implication does not obviously hold. We also have dH (−f (x), f (x + ω)) ≥ 0, but the reverse implication means something completely different. If f : Rn ⊸ B, where B is a Banach space, is an anti-periodic lower semicontinuous (l.s.c.) map with convex closed values, then it admits a single-valued continuous antiperiodic selection f0 ⊂ f with the same anti-period (see Proposition 2.1). If f : Rn ⊸ B, where B is a separable Banach space, is a measurable anti-periodic map with closed values then, according to the well-known Kuratowski–Ryll–Nardzewski theorem (cf. Proposition 2.2), f admits a single-valued measurable anti-periodic selection f0 ⊂ f with the same anti-period. If f : R → Rn is (almost everywhere) differentiable and ω-anti-periodic, then f˙ is also ω-anti-periodic. Denoting therefore CAP (R, Rn ) := {f ∈ C (R, Rn ) | −f (x) ≡ f (x + ω)}, C (k)AP (R, Rn ) := {f ∈ C (k) (R, Rn ) | −f (x) ≡ f (x + ω)}, AC (k−1)AP (R, Rn ) := {f ∈ AC (k−1) (R, Rn ) | −f (x) ≡ f (x + ω)},

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the spaces (CAP (R, Rn ), k · k1 ), (C (k)AP (R, Rn ), k · k2 ), (AC (k−1)AP (R, Rn ), k · k3 ) are Banach, provided the norms k · k1 , k · k2 , k · k3 are defined as in Section 3.1..

3.3.

Quasi-periodic Maps

D EFINITION 3.1 A continuous function f : R → Rn is said to be k-period quasiperiodic if it takes the form f (x) = f (ω1 x, ω2 x, . . . , ωk x), where f is 1-periodic separately in its k arguments, i.e. f (x1 , . . . , xj , . . . , xk ) ≡ f (x1 , . . . , xj+1 , . . . , xk ), for every j ∈ {1, . . . , k}, and ω1 , ω2 , . . . , ωk are incommensurate frequencies, namely if n1 ω1 + n2 ω2 + · · · + nk ωk = 0 holds only when all the integers nj ∈ Z, j = 1, . . . , k, are zero, i.e. n1 = n2 = . . . = nk = 0. R EMARK 3.1 As for periodic functions, Definition 3.1 can be easily extended in a natural way to k-period quasi-periodic maps f : Rn → S, where S is an arbitrary set. For our needs, it is however enough to be restricted to real vector functions as in Definition 3.1. A more general class of (not necessarily continuous) quasi-periodic functions was studied in [BP]. √ E XAMPLE 3.1 Function f (x) = sin(2πx) + sin(2 2πx) is obviously 2-period quasi√ √ ω1 2 1 √ periodic with frequencies ω1 = 1 and ω2 = 2, because ω2 = 2 = 2 6∈ Q. R EMARK 3.2 Of course, 1-period quasi-periodic functions are periodic in the usual sense. Furthermore, it is well-known that quasi-periodic functions are related to the problem of invariant tori and to the celebrated KAM theory (cf. e.g. [BHS]). Every (continuous) quasi-periodic function is well-known (cf. e.g. [NB, p. 232]) to be uniformly almost-periodic (for the definition of a uniform almost-periodicity, see the following section). The reverse statement does not hold in general. Nevertheless, we know necessary and sufficient additional conditions under which a uniformly almost-periodic function becomes quasi-periodic. For Nakajima’s proof of the following theorem, see e.g. [Yo, pp. 30–34]. T HEOREM 3.1 Function f : R → Rn is k-period quasi-periodic if and only if it is uniformly almost-periodic and its module (spectrum) has a finite integer basis, namely f (x) =

X m

m1 mk am exp 2πix + ··· + , ω1 ωk

where ω1 , . . . , ωk are some reals and m = (m1 , . . . , mk ), for m1 , m2 , . . . , mk ∈ Z. In [MP], the authors found necessary and sufficient conditions under which a function f : R → R can be represented as a sum of n periodic functions. Defining, for given constants h0 , . . . , hn > 0, the differences ∆(h0 )f (x) := f (x + h0 ) − f (x) and ∆(h0 , h1 , . . . , hn )f (x) := ∆(h0 , . . . , hn−1 )(∆(hn ))f (x), their theorem reads as follows.

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T HEOREM 3.2 ([MP]) A function f : R → R is the sum of finitely many periodic functions if and only if there exist positive numbers h0 , . . . , hn−1 such that hi /hj ∈ Q, for i 6= j, and ∆(h0 , . . . , hn−1 )f = 0. Recalling that a function f : R → R is Darboux if, for an arbitrary interval I ⊂ R, the image f (I) is connected, the authors of [NW] specified Theorem 3.2 for Darboux summands as follows. T HEOREM 3.3 ([NW]) Assume, for given constants h0 , . . . , hn−1 > 0, that hi /hj ∈ Q, for i 6= j. A function f : R → R is the sum of n ∈ N Darboux functions f0 , . . . , fn−1 of periods h0 , . . . , hn−1 , respectively, if and only if ∆(h0 , . . . , hn−1 )f = 0. Recalling that a function f : R → R is Marczewski measurable if, for each perfect (i.e. closed with no isolated points) set P ⊂ R, there exists a perfect subset Q ⊂ P such that the restriction f |Q is continuous, the authors of [NW] still specified Theorem 3.2 for Marczewski measurable summands as follows. T HEOREM 3.4 ([NW]) Assume, for given constants h0 , . . . , hn−1 > 0, that hi /hj ∈ Q, for i 6= j. A function f : R → R is the sum of n ∈ N Marczewski measurable periodic functions f0 , . . . , fn−1 of periods h0 , . . . , hn−1 , respectively, if and only if f is Marczewski measurable and ∆(h0 , . . . , hn−1 )f = 0. R EMARK 3.3 As was proved in [NW], although the identity map f (x) = x cannot be represented as the sum of a finite number of Lebesgue measurable functions or periodic functions with the Baire property (i.e. their domain is R \ M , where M ⊂ R is a meager (first Bair category) set and f −1 (W ) ∩ R differs from an open set by a meager set in R, for every open subset W of R, by which they are, according to the Kuratowski theorem, continuous on R \ M ), it can be written as the sum of two periodic Marczewski measurable functions. Finite sums of periodic functions form the most typical subclass of quasi-periodic functions. If the summands are multivalued lower semicontinuous periodic maps with convex closed values or measurable periodic maps with closed values then, as pointed out in Section 3.1., there exist single-valued periodic continuous or measurable selections, respectively, forming the single-valued summands. Moreover, it has again meaning to consider such multivalued summands only on a subset X ⊂ R, e.g. for X = [0, ∞), and with the inclusion periodicity property, namely fj (x) ⊂ fj (x + hj ), j = 0, . . . , n − 1, for all x ∈ X. Unfortunately, quasi-periodic functions are well-known to form a vector space which is not closed w.r.t. the uniform convergence on R (cf. e.g. [F3]). Since the Banach space of uniformly almost-periodic functions has not this handicap, that is perhaps why it is preferebly studied. Since (apart from the special case of periodic summands) we shall not deal in this paper with quasi-periodic solutions, let us finally mention its relationship to partial differential equations, as explained e.g. in [OT]. Consider the equation x˙ = f (ω1 t, . . . , ωk t, x), (3.1) where f : R2 → R is continuous and f (·, x) is a (continuous) k-periodic quasi-periodic function (i.e. ω1 , . . . , ωk are linearly independent reals over the rationals Q), for every x ∈ R. Then x(t) = x(ω1 t, . . . , ωk t), where x(t1 . . . , tk ) is 1-periodic in each tj , j = 1, . . . , k,

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is a quasi-periodic solution of (3.1) if and only if x(t) satisfies the partial differential equation, as its equation of characteristics, ω1

∂x ∂x + · · · + ωk = f (t1 , . . . , tk , x), ∂t1 ∂tk

(3.2)

in a distributional sense. In [OT], the authors did not consider (3.2) in a distributional sense, but in a classical setting, for which they interpreted the operator on the left-hand side of (3.2) as the directional derivative. This approach lead them to quasi-periodic solutions of (3.1) in a weaker sense (close to almost automorphic solutions; cf. [NG], [SY]), but still with significant properties, allowing them to preserve a version of the Massera transformation theorem, studied in Section 4.3. below.

3.4.

Almost-periodic Maps

The theory of almost-periodic (a.p.) functions was created by H. Bohr in the Twenties, but it was restricted to the class of uniformly continuous functions. Let us, therefore, consider it firstly as a subspace of the space C (R, R) of all continuous functions, defined on R and with the values in R. Let us recall that a set X ⊂ R is said to be relatively dense (r.d.) if there exists a number l > 0 s.t. every interval [a, a + l] contains at least one point of X. D EFINITION 3.2 (Bohr-type definition) A function f ∈ C (R, R) is said to be uniformly almost-periodic (u.a.p.) if, for every ǫ > 0, there corresponds a r.d. set {τ }ǫ s.t. sup |f (x + τ ) − f (x)| < ǫ x∈R

∀τ ∈ {τ }ǫ .

Each number τ ∈ {τ }ǫ is called an ǫ-uniformly almost-period (or a uniformly ǫtranslation number) of f . The class Cap of u.a.p. functions has the following important properties (see e.g. [Bh, Bs]): • Every u.a.p. function is uniformly continuous. • Every u.a.p. function is uniformly bounded. • If a sequence of u.a.p. functions fn converges uniformly in R to a function f , then f is u.a.p., too. In other words, the set of u.a.p. functions is closed w.r.t. the uniform convergence. Since it is a closed subset of the Banach space Cb := C ∩L∞ (i.e. the space of bounded continuous functions, endowed with the sup-norm), it is Banach, too. It is easy to show that the space is even a commutative Banach algebra, w.r.t. the usual product of functions. D EFINITION 3.3 (normality or Bochner-type definition) A function f ∈ C (R, R) is called uniformly normal if, for every sequence {hi } of real numbers, there corresponds a subsequence {hni } s.t. the sequence of functions {f (x + hni )} is uniformly convergent. The numbers hi are called translation numbers and the functions f hi (x) := f (x + hi ) are called translates.

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In other words, f is uniformly normal if the set of translates is precompact in Cb , i.e. if it contains a fundamental (Cauchy) subsequence. Obviously, in complete spaces, it is equivalent to say that the set of translates is precompact or relatively compact (i.e. the closure is compact). Since every trigonometric polynomial P (x) =

n X

ak eiλk x

(ak ∈ R; λk ∈ R)

k=1

is well-known to be u.a.p., every function f , obtained as the limit of a uniformly convergent sequence of trigonometric polynomials, is u.a.p. Thus, it is natural to introduce the third definition of a.p. (continuous) functions. D EFINITION 3.4 (approximation) We call Cap (R, R) the (Banach) space obtained as the closure of the space P(R, R) of all trigonometric polynomials in the space Cb , endowed with the sup-norm. R EMARK 3.4 Equivalently (cf. [Bh], [Bs]), function f belongs to Cap (R, R) if, for any ǫ > 0, there exists a trigonometric polynomial Tǫ s.t. sup |f (x) − Tǫ (x)| < ǫ . x∈R

It is easy to show that Cap , like C , is invariant under translations, that is Cap contains, together with f , the functions f t (x) := f (x + t) ∀t ∈ R. The Definitions 3.2, 3.3 and 3.4, are equivalent (see e.g. [Bh], [Bs]): T HEOREM 3.5 A continuous function f is u.a.p. if and only if it is uniformly normal and if and only if it belongs to Cap (R, R). For every function f , we will call by the mean value of f the number 1 T →∞ 2T

M [f ] = lim

Z

T

f (x) dx.

−T

The mean value of every u.a.p. function f exists and (cf. [Bh], [Bs]) 1 (a) M [f ] = lim T →∞ T

Z

1 (b) M [f ] = lim T →∞ 2T

T

0

Z

1 f (x) dx = lim T →∞ T

Z

0

f (x) dx,

−T

a+T

f (x) dx;

(3.3)

uniformly w.r.t. a ∈ R.

a−T

R EMARK 3.5 Every even function satisfies (3.3), while necessary condition for an odd function to be u.a.p. is that M [f ] = 0. Furthermore, since, for every u.a.p. function f and for every real number λ, the function f (x)e−iλx is a u.a.p. function, the number a(λ, f ) := M [f (x)e−iλx ] always exists.

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For every u.a.p. function f , there always exists at most a countable infinite set of values λ (called the Bohr–Fourier exponents or frequencies) for which a(λ, f ) 6= 0 (see e.g. [Bh], [Bs]). The numbers a(λ, f ) are called the Bohr–Fourier coefficients and the set σ(f ) := {λn | a(λn , f ) 6= 0} is called the spectrumP of f . The formal series n a(λn , f )e−iλx is called the Bohr–Fourier series of f and we write X f (x) ∼ a(λn , f )e−iλx . n

In [F2], the author found necessary and sufficient conditions in order a uniformly a.p. function f ∈ C (R, R) to be uniformly approximated on R by continuous periodic functions with an arbitrary accuracy. It is so if and only if f has a one-point basis, i.e. that there exists a number r ∈ R such that σ(f ) ⊂ rQ = {rq | q ∈ Q}. If σ(f ) has a finite integer basis, then f is, according to Theorem 3.1, quasi-periodic, and vice versa. Every nonconstant continuous ω-periodic function f has a one-point basis {2π/ω} which, however, need not be integer. On the other hand, there also exist a.p. functions with a one-point basis which are not periodic. In fact, according to [F2], function f can have values in a Banach space. In [F1], uniformly a.p. functions which are (uniform) limits of sequences of continuous periodic functions were considered with a special respect to the structure of their Bohr– Fourier exponents. Let us note that A. S. Besicovitch [Bs] has shown that there are infinitely many trigonometric series convergent, in every finite interval, to any continuous bounded function of a bounded variation and, therefore, that the sum-function of an everywhere convergent trigonometric series is not necessarily uniformly almost-periodic. For more details like the connection between the Bohr–Fourier exponents and almostperiods, see e.g. [ABG] and the references therein. In order to deal with locally Lebesgue integrable almost-periodic functions, let us introduce the following Stepanov, Weyl and Besicovitch (pseudo)norms and distances Z x+L 1 p 1 p p (Stepanov) kf kS := sup |f (t)| dt , DS p (f, g) = kf − gkS p ; L L L x∈R L x (Weyl) kf kW p := lim kf kS p , DW p (f, g) = limL→∞ DS p (f, g); L

L→∞

(Besicovitch) kf kB p

1 := lim sup 2T T →∞

L

Z

T

p

|f (t)| dt

−T

1

p

, DB p (f, g) = kf − gkB p ,

where f, g ∈ Lploc (R, R), p ∈ N and L > 0. It is without any loss of generality to fix L > 0 as L = 1 (cf. [ABG], [Bh], [Bs]). Hence, taking Gp successively as S p (= S1p ) or W p or B p , we can generalize Definitions 3.2, 3.3 and 3.4 as follows. D EFINITION 3.5 (Bohr-type definition) A function f ∈ Lploc (R, R) is said to have the Gpap property if, for every ǫ > 0, there corresponds a r.d. set {τ }ǫ s.t. DGp (f (x + τ ), f (x)) < ǫ;

∀τ ∈ {τ }ǫ .

308

Jan Andres Each number τ ∈ {τ }ǫ is called an ǫ-Gp -almost-period of f .

D EFINITION 3.6 (normality or Bochner-type definition) A function f ∈ Lploc (R, R) is called Gp -normal if the family of functions {f h }, defined as f h (t) = f (t + h), where h ∈ R is an arbitrary number, is Gp -precompact, i.e. if for each sequence {f hi }, we can choose a fundamental subsequence. D EFINITION 3.7 (approximation) We denote by G p (R, R) the space obtained as the closure in BG p := {f ∈ Lploc (R, R) | kf kGp < ∞} of the space P(R, R) of all trigonometric polynomials w.r.t. the Gp -norm. p We can still define, rather curiously, the class e-Wap , e-W p -normal and e-W p of equiWeyl a.p. functions by means of the Stepanov (pseudo)metric as follows.

D EFINITION 3.8 (Bohr-type definition) A function f ∈ Lploc (R, R) is said to be equip almost-periodic in the sense of Weyl (e-Wap ) if, for every ǫ > 0, there corresponds a r.d. set {τ }ǫ and a number L0 = L0 (ǫ) s.t.

Z x+L 1 p 1 p sup |f (t + τ ) − f (t)| dt < ǫ; x∈R L x

∀L ≥ L0 (ǫ).

Each number τ ∈ {τ }ǫ is called an ǫ-equi-Weyl-almost-period (or equi-Weyl ǫtranslation number of f ). D EFINITION 3.9 (equi-W p -normality) A function f ∈ Lploc (R, R) is said to be equi-W p normal if the family of functions {f h } (h is an arbitrary real number) is SLp -precompact, for sufficiently large L, i.e. if for each sequence {f hi }, we can choose an SLp -fundamental subsequence, for a sufficiently large L. D EFINITION 3.10 (approximation) We will denote by equi-W p (R, R) the space obtained as the closure in BS p := {f ∈ Lploc (R, R) | f b ∈ L∞ (R, Lp ([0, 1]))} of the space of all trigonometric polynomials w.r.t. to the SLp -norm, for a sufficiently large L, i.e. for every f ∈ e − W p and for every ǫ > 0, there exist L0 = L0 (ǫ) and a trigonometric polynomial Tǫ s.t. DS p (f, Tǫ ) < ǫ L

∀L ≥ L0 (ǫ).

p Although the classes Sap , S p -normal and S p are, as in Theorem 3.5, equivalent, it is not so for other classes. The hierarchy of almost-periodic function spaces was established in [ABG] in the form of the following Table 1.

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Table 1. a. periods

normal

Cap

Bohr Stepanov equi-Weyl

⇔

u. normal

⇑ ⇓r

⇑ ⇓r

p Sap

⇔

S p -normal ⇑ ⇓r

p e-Wap

e-W p -normal

⇑ ⇓r

............

⇐ ;

p Wap

Weyl

⇑ ⇓r

Besicovitch

⇑ ⇓r

u.a.p.

⇔

Sp m

⇐ ; ⇐ ;

Wp ⇑ ⇓r

............

B p -normal

e-W p ⇑ ⇓r

............

W p -normal ............

⇐ ;

p Bap

⇑ ⇓r

⇔

⇑ ⇓r

⇑ ⇓r ⇔

approx.

⇐ ;

Bp

For uniformly continuous bounded (ucb) functions (w.r.t. the sup-norm), like solutions of differential equations and inclusions studied in Section 5.4. below, the following relations take place for classes of a.p. functions in Table 1 (cf. [ABG]): p {Cap }ucb = {u-normal}ucb = {u.a.p.}ucb = {Sap }ucb = {S p -normal}ucb = {S p }ucb p = {e- W p }ucb ⊂ {e-Wap }ucb = {e-W p -normal}ucb p = {W p }ucb ⊂ {Wap }ucb = {W p -normal}ucb p ⊂ {Bap }ucb = {B p -normal}ucb ⊃ {B p }ucb .

Modifying the examples in [ABG], one can check that the inclusions in the second and p p p third lines are strict, i.e. that {Wap }ucb 6⊂ {e-Wap }ucb 6⊂ {Sap }ucb . Replacing the Euclidean distances |.−.| in Definitions 3.2, 3.5 and 3.8 by the Hausdorff metric dH (·, ·) defined in Preliminaries, we can extend in a natural correct way the Bohrp p p p type definitions of classes Cap , Sap , e-Wap , Wap , Bap of a.p. functions to those of a.p. multivalued functions with nonempty compact values as follows (cf. [ABG], [AG1]). D EFINITION 3.11 (Bohr-type definition) A (Hausdorff) continuous multivalued function ϕ : R ⊸ R with nonempty compact values is said to be uniformly-almost-periodic (u.a.p.) if, for every ǫ > 0, there corresponds a r.d. set {τ }ǫ s.t. sup dH ϕ(t + τ ), ϕ(t) < ǫ ∀t ∈ {τ }ǫ . t∈R

Furthermore, a measurable multivalued function ϕ : R ⊸ R with nonempty compact p p p p p values is said to have the Gap -property, where Gap is Sap or Wap or Bap , if, for every ǫ > 0, there corresponds a r.d. set {τ }ǫ s.t. ∀t ∈ {τ }ǫ : Z sup x∈R

or

x

x+1

dH

p 1 p ϕ(t + τ ), ϕ(t) dt < ǫ,

Z x+L p 1 p 1 lim sup dH ϕ(t + τ ), ϕ(t) dt < ǫ, L→∞ x∈R L x

310 or

Jan Andres p 1 Z T p 1 lim sup dH ϕ(t + τ ), ϕ(t) dt < ǫ, 2T −T T →∞

respectively. At last, a measurable multivalued function ϕ : R ⊸ R with nonempty compact values p is said to be equi-almost-periodic in the sense of Weyl (e-Wap ) if, for every ǫ > 0, there corresponds a r.d. set {τ }ǫ and a number L0 = L0 (ǫ) s.t. Z x+L p 1 p 1 sup dH ϕ(t + τ ), ϕ(t) dt < ǫ, ∀t ∈ {τ }ǫ , ∀L ≥ L0 (ǫ). x∈R L x Each number τ ∈ {τ }ǫ is called a respective ǫ-almost-period (or a respective ǫtranslation number of ϕ). p p Although multivalued Sap and e-Wap maps ϕ were shown to admit a single-valued p and e-Wap selection f ⊂ ϕ in [DS], [D1] and [D2], a multivalued u.a.p. map need not possess, rather curiously, a single-valued u.a.p. selection, as pointed out in [BVLL]. Moreover, the values of ϕ can be even in a complete metric space.

p Sap

R EMARK 3.6 The replacement of the Euclidean distances |.−.| in Bochner-type definitions 3.3, 3.6 and 3.9 by the Hausdorff metric dH (·, ·) leads also to natural correct definitions of u. normal, S p -normal, e-W p -normal, W p -normal and B p -normal measurable (in case of u. normal: (Hausdorff-) continuous) multivalued functions with nonempty compact values. On the other hand, it is not so for the classes of a.p. functions defined by means of approximation by trigonometric polynomials, because the arbitrary accuracy requirement for these approximations would reduce multivalued maps to single-valued functions, only. It is, therefore, more natural to define, as in [D3], the class of B p -multivalued measurable functions with nonempty compact values, when replacing the Euclidean distances |. − .| by p the Hausdorff metric dH (·, ·), jointly with trigonometric polynomials by Sap -multivalued maps. For B p -multivalued maps ϕ, defined in this way, the existence of a single-valued B p selection f ⊂ ϕ, was proved in [D3]. The values of ϕ can be also in a complete metric space.

3.5.

Derivo-periodic Maps

A function f is said to be derivo-periodic if f˙(t) ≡ f˙(t + ω), for some ω > 0. If f˙ is measurable, then we can obviously assume, without any loss of generality, that f˙(t) = f˙(t + ω), almost everywhere (a.e.). By a function, we will understand here a single-valued one, i.e. f : R → R, or a multivalued one, i.e. f : R ⊸ R (or, equivalently, f : R → 2R \ {0}, but (for the sake of simplicity) always with R as its domain as well as its range. Besides the standard derivative (for a single-valued function), we will also consider the one in the sense of F. S. De Blasi [DB] (for multivalued functions). It is well-known (cf. [Fa, p. 235]) that a continuously differentiable (single-valued) function is derivo-periodic if and only if it takes the form f (t) = f0 (t) + αt, where f0 is periodic and α ∈ R. More precisely, for f ∈ C 1 (R, R), we have: f˙(t) = f˙(t + ω), for some ω > 0 ⇐⇒ f (t) = f0 (t) + αt,

(3.4)

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311

where f0 (t) = f0 (t+ω). It follows immediately that a primitive function (an antiderivative) to such f˙ is ω-periodic if and only if it is bounded, i.e. α = 0. Physically, derivo-periodic functions can correspond to a motion with a periodic velocity, a subsynchronous level of performance of the motor or a motion of particles in a sinusoidal potential related to a free-electron laser. For many references concerning derivoperiodic motions and their applications in astronomy, engineering, laser physics, etc., see [AG1]. Particularly, in quantum physics, a “slalom orbit” of an electron beam should be described, in view of the Heisenberg Uncertainty Principle, by means of a multivalued function with a periodic single-valued derivative. This requirement can be satisfied for De Blasi-like differentiable multivalued functions, as pointed out in [ABP], because according to the result of our former Ph.D. student L. J¨utner (cf. also [AG1, Theorem 2.18 in Appendix 2]), a De Blasi-like differentiable function is always a sum of a single-valued continuous function having right-hand side and left-hand side derivatives plus a multivalued constant. Furthermore, it is well-known that if right-hand side or left-hand side derivatives of a single-valued part are continuous, then this single-valued part becomes continuously differentiable, and so satisfies (3.4). On the other hand, a natural question arises, namely how much regular must be f in order (3.4) to be satisfied? It is intuitively clear that this has a lot to do with the Fundamental Theorem of Calculus (the Newton-Leibniz formula), because the simplest proof of (3.4) relies on it (see [Fa, p. 235] or [AG1, p. 657]). In this light, the usage of the Newton integral or the Lebesgue integral allows us to replace the C 1 -class in (3.4) by differentiable or absolutely continuous functions, respectively; of course, satisfying (3.4) almost everywhere, in the latter case. As we will see, relation (3.4) holding a.e. can be verified for the related class of ACG∗ functions defined below. Since a continuous (single-valued) function f with right-hand side or left-hand side derivatives f˙+ or f˙− is known to belong to the ACG∗ -class, the singlevalued part of De Blasi-like differentiable multivalued function satisfies (3.4), a.e. Consequently, a multivalued analogy of (3.4) holds for De Blasi-like differentiable multivalued functions. Hence, let us start with the class of generalized absolutely continuous in the restricted sense (ACG∗ -) functions. Usually, the definitions of AC∗ and ACG∗ -functions are given on a subset of a closed (bounded) interval in R. For our needs, we extend the notion of ACG∗ -functions onto an arbitrary subset of R in the following definition. D EFINITION 3.12 A function f : [a, b] → R is said to be AC∗ on S ⊂ [a, b] if, for every ǫ > 0, there exists δ > 0 such that, for any subpartition P = {[aj , bj ]}sj=1 of [a, b] with aj , bj ∈ S, for every j = 1, . . . , s, s X j=1

|bj − aj | < δ ⇒

s X

ω(f ; [aj , bj ]) < ǫ,

j=1

where ω denotes the oscillation of f on the interval [aj , bj ] : ω(f ; [aj , bj ]) := sup{|f (y) − f (x)| | x, y ∈ [aj , bj ]}. Furthermore, a function f : J → R, J ⊂ R, is said to be ACG∗ on S ⊂ J if f is continuous on S and S is a countable union of sets on which f is AC∗ .

312

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In [ABP], we established the following characterization of a.e. derivo-periodic functions for ACG∗ -functions. T HEOREM 3.6 If f : R → R is ACG∗ on R, then the following conditions are equivalent: (i) f˙(t) = f˙(t + ω), ω > 0, for a.a. t ∈ R, (ii) f˙+ (t) = f˙+ (t + ω) or f˙− (t) = f˙− (t + ω), ω > 0, for a.a. t ∈ R, (iii) f (t) = f0 (t) + αt, for all t ∈ R, where α ∈ R, f0 is ω-periodic, ω > 0, and ACG∗ on R. R EMARK 3.7 The class of functions having derivative everywhere, but on a set with zero (Lebesgue) measure on which have negligible variation, coincides on any closed interval with the class of ACG∗ -functions. It follows from definitions that a function f : R → R is ACG∗ on R if and only if it is ACG∗ on [k, k + 1], for every k ∈ Z. Also, a function f : R → R has derivative everywhere, but on a set with zero (Lebesgue) measure on which have negligible variation if and only if this holds on [k, k + 1], for every k ∈ Z. Therefore, Theorem 3.6 can be equivalently expressed in terms of functions having derivative everywhere, but on a set with zero measure on which have negligible variation. This was done by means of the Kurzweil–Henstock integral in [ABP], where the precise definitions and more details can be found. Now, we shall proceed to De Blasi-like differentiable multivalued functions. The original notion introduced in [DB] reduces in R into the following definition. D EFINITION 3.13 A multivalued function ϕ : R ⊸ R (i.e. ϕ : R → 2R \ {∅}) is said to be De Blasi-like differentiable at t ∈ R if there exists (a single-valued(!); cf. [ABP], [AG1]) mapping Dt ϕ : R → R such that Dt ϕ is positively homogeneous (i.e. Dt ϕ(λs) = λDt ϕ(s), λ ≥ 0, for all s ∈ R) and a number δ > 0 such that dH (ϕ(t + h), ϕ(t) + Dt ϕ(h)) = o(h),

whenever |h| ≤ δ,

where o(h) denotes a nonnegative function such that limh→0 o(h)/|h| = 0, dH (·, ·) stands for the Hausdorff metric (cf. Preliminaries). Dt ϕ is called the differential of ϕ at t. Of course, ϕ is said to be De Blasi-like differentiable on J ⊂ R (or simply, for J = R, De Blasi-like differentiable) if it is so at every point t ∈ R. R EMARK 3.8 According to an important statement in [AG1, Theorem (A2.18)]), we can define equivalently a De Blasi-like differentiable function ϕ : R ⊸ R as a sum of a singlevalued continuous function f : R → R having (standard) right-hand side and left-hand side derivatives plus a bounded interval (i.e., a multivalued constant) {C}, namely ϕ = f +{C}. R EMARK 3.9 It is well-known that a function f : [a, b] → R having right-hand side and left-hand side derivatives, for all t ∈ E ⊂ [a, b], admits derivatives with at most countably many exceptions on E ⊂ [a, b]. Moreover, continuous functions which are differentiable for all, but countably many t ∈ E ⊂ [a, b] are known to be there ACG∗ . Thus, continuous functions f : [a, b] → R having right-hand side and left-hand side derivatives, for all t ∈ E ⊂ [a, b], are there ACG∗ . For more details, see [ABP] and the references therein. Since the single-valued part f of De Blasi-like differentiable multivalued function ϕ : R ⊸ R (see Remark 3.8) is, according to Remark 3.9, ACG∗ on any closed interval, and

Periodic-Type Solutions of Differential Inclusions

313

since (cf. [ABP] or [AG1, Theorem (A2.20)]) Dt ϕ(t) ≡ Dt ϕ(t + ω) ⇐⇒ f˙+ (t) ≡ f˙+ (t + ω) and f˙− (t) ≡ f˙− (t + ω),

(3.5)

we arrive by means of Theorem 3.6 and Remark 3.8 at the following result. T HEOREM 3.7 Let ϕ : R ⊸ R be a De Blasi-like differentiable multivalued function in the sense of Definition 3.13. Then ϕ is derivo-periodic with period ω > 0, i.e. (3.5) holds, if and only if there exist α ∈ R and an ω-periodic continuous function f0 : R → R such that ϕ(t) = [f0 (t) + αt] + {C},

for all t ∈ R,

where {C} is a bounded interval (a multivalued constant). R EMARK 3.10 For a single-valued De Blasi-like differentiable function ϕ = f (i.e. {C} = {0}), Theorem 3.7 represents only a particular case of Theorem 3.6. On the other hand, Theorem 3.7 generalizes the statement that a differentiable (in a usual sense) function f : R → R is derivo-periodic if and only if f (t) = f0 (t) + αt with a (differentiable) periodic function f0 : R → R. In view of the above results, the primitives of periodic derivatives become, under natural assumptions, periodic if and only if they are bounded. According to the well-known Bohl–Bohr theorem (see e.g. [Bs]), the same is true in the class of uniformly continuously differentiable functions w.r.t. (uniformly or Bohr-type) almost-periodic (a.p.) functions. For the definitions of a.p. functions and their properties; see the foregoing section. Thus, a natural question arises, whether or not (?) the almost-periodicity of a (uniformly continuous) derivative implies that the function itself takes the form of a sum of an a.p. function plus some linear part. The following arguments demonstrate that this must be answered negatively in general. Since there are examples (see e.g. [J1], [J2], [JM] or, more recently, [OT]) showing the existence of a.p. functions with a zero mean value whose primitives are unbounded, they cannot be (according to the mentioned Bohl–Bohr theorem) a.p. Moreover, if x˙ 0 (t) is a.p. with a zero mean value, namely ¯˙0 := lim 1 x T →∞ 2T

Z

T

−T

1 x˙ 0 (t) dt = lim T →∞ T

Z

T

x0 (t) = 0, t→∞ t

x˙ 0 (t) dt = lim 0

and such that x0 (t) is (as in the mentioned examples) unbounded, we can conclude that x0 (t) is not a.p., and because of x0 (t) = o(t), it does not contain any linear part. A bit more ¯˙ generally, let x(t) ˙ be a.p. having not necessarily a zero mean value. Then x(t) ˙ = x˙ 0 (t) + x, where again Z 1 T ¯˙ = 0. ¯ x(t) ˙ dt ∈ R and x x˙ := lim T →∞ T 0 Thus, integrating x(t) ˙ from 0 to t, we obtain that ¯˙ + C, x(t) = x0 (t) + xt where C is a suitable real constant. Since x0 (t) need not be a.p. and, as a consequence of x0 (t) = o(t), no linear part can be extracted from it, there is no chance, in general, for such

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¯˙ + C and an a.p. function, as claimed an x(t) to be a sum of the present linear function xt above. Since the mentioned example in [JM] concerns in fact quasi-periodic functions with only two basic frequencies, the same is also true for this subclass.

4. 4.1.

Primer of Periodic-Type Oscillations Linear Systems with Constant Coefficients

Consider the linear system x ∈ Rn ,

x˙ + Ax = p(t),

(4.6)

where A is an (n × n)-matrix with (real) constant entries and p : R → Rn is a (measurable) locally Lebesgue integrable n-vector function. It is well-known (see e.g. [Fa, Chapter 2.1]) that system (4.6) possesses a unique ωperiodic solution if and only if σ(A) ∩ 2πiZ/ω = ∅, where σ(A) is a spectrum of A, i.e. the set of eigenvalues of A, provided p(t) ≡ p(t + ω). If p is anti-periodic, p(t) ≡ −p(t + ω), it is also half-periodic, p(t) ≡ p(t + 2ω), but not vice versa. That is why the spectral condition cannot be used for the characterization of existence and uniqueness result about anti-periodic solutions. On the other hand, if σ(A)∩πiZ/ω = ∅, then system (4.6) admits a unique ω-anti-periodic solution, provided p(t) ≡ −p(t + ω). It follows from the Fredholm alternative, which is usually called in the context of ODEs as Conti’s lemma, that system (4.6) has a unique 2qω-periodic solution if and only if the homogeneous system x˙ + Ax = 0,

x ∈ Rn ,

(4.7)

has only the trivial 2qω-periodic solution. Unlike for periodic solutions, the system x˙ = p(t) possesses the only ω-anti-periodic solution, provided p(t) ≡ −p(t + ω). Because of the superposition principle, system x˙ + Ax =

k X

pj (t),

x ∈ Rn ,

(4.8)

j=1

where pj : R → Rn are (measurable) locally Lebesgue integrable n-vector functions such that pj (t) ≡ pj (t+ωj ), where ωj are linearly independent reals over the rationals Q, admits a k-period-quasi-periodic solution, provided σ(A) ∩ 2πiZ/ωj = ∅, for each j = 1, . . . , k. If p : R → Rn is an essentially bounded Stepanov-almost-periodic n-vector function then, according to the well-known Bohr–Neugebauer-type result (cf. e.g. [Ra], and the references therein), every entirely bounded solution of system (4.6), with an arbitrary real matrix A, is uniformly-almost-periodic. Analogous results also hold for almost-periodic solutions in a more general sense (Weyl, Besicovitch); cf. [Ra]. Now, let A be a regular real (n × n)-matrix and f ∈ ACG ∗ (R, Rn ) be ω-derivo-periodic, i.e. p(t) ˙ ≡ p(t ˙ + ω). Then (4.6) admits a unique (smooth) ω-derivoperiodic solution if and only if σ(A) ∩ 2πiZ/ω = ∅ (for more details and the definition of the ACG ∗ -class of generalized absolutely continuous functions (see [ABP] and cf. Section 3.5.). The same is true for a unique, but this time an absolutely continuous (i.e.

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Carath´eodory) ω-derivo-periodic solution of (4.6) if p takes the form p(t) = p0 (t) + αt, where p0 : R → Rn is, more generally, a (measurable) locally Lebesgue integrable n-vector function such that p0 (t) ≡ p0 (t + ω) and α ∈ Rn is a real n-vector. All the above existence (not uniqueness) results can be extended, on the basis of selection theorems, to linear differential inclusions x˙ + Ax ∈ P (t),

x ∈ Rn ,

(4.9)

resp. x˙ + Ax ∈

k X

Pj (t),

x ∈ Rn .

(4.10)

j=1

Since every measurable multivalued map with closed values possesses, according to the Kuratowski–Ryll-Nardzewski theorem (cf. Proposition 2.2) (single-valued) measurable selections, the same is true for ω-periodic and ω-anti-periodic multivalued maps w.r.t. measurable ω-periodic and ω-anti-periodic selections. That is why the above existence conclusions for ω-periodic and ω-anti-periodic solutions can be extended to (4.9) as well as those for k-period-quasi-periodic solutions to (4.10). Furthermore, since Stepanov-almost-periodic multivalued maps with compact convex values possess (single-valued) Stepanov-almost-periodic selections (cf. Section 3.4.), the analogous extension holds for (4.9) w.r.t. uniformly-almost-periodic multivalued solutions. Since, rather curiously, uniformly-almost-periodic multivalued maps with compact convex values need not possess, according to the observation in [BVLL] (cf. Section 3.4.), (single-valued) uniformly-almost-periodic selections, we cannot obtain in this way smooth uniformly-almost-periodic solutions of (4.9). At last, the similar extension can be done for (smooth) ω-derivo-periodic solutions of (4.9), provided P : R ⊸ Rn is a DeBlasi-like differentiable multivalued map whose derivative is ω-periodic (for more details, see [ABP]). The same is true for (4.9) w.r.t. Carath´eodory ω-derivo-periodic solutions, provided P takes the from P (t) = P0 (t) + αt, where P0 is a measurable multivalued map with closed values such that P0 (t) ≡ P0 (t + ω) and α ∈ Rn is a real n-vector.

4.2.

Linear Systems with Time-Variable Coefficients

Now, consider the linear system x˙ + A(t)x = p(t),

x ∈ Rn ,

(4.11)

where A is this time an (n × n)-matrix with (real) time-variable entries which are (measurable) locally Lebesgue integrable and p : R → Rn is again a (measurable) locally Lebesgue integrable n-vector function. We shall discuss the possibility of further extension of results from the foregoing section. It follows from the Floquet theory (see e.g. [Fa, Chapters 2.2 and 2.3], [YS1], [YS2]) that system (4.6) has a unique (harmonic) ω-periodic solution if and only if 1 is not a Floquet (characteristic) multiplier of A, provided A(t) ≡ A(t + ω) and p(t) ≡ p(t + ω). Let us recall that by Floquet multipliers we mean, as usual, eigenvalues of the monodromy

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Jan Andres

matrix X(ω), where X(t) = Φ(t) exp(Λt) is the fundamental matrix of solutions to the homogeneous system x˙ + A(t)x = 0, x ∈ Rn , (4.12) with Φ(0) = I, Φ(t) ≡ Φ(t + ω), and Λ is a constant matrix. More generally, system (4.11) has a unique (subharmonic) 2qω-periodic solution if and only if exp( pq πi), where p and q > 0 are integers, is not a characteristic multiplier of A, provided A(t) ≡ A(t + ω) and p(t) ≡ p(t + 2qω). Thus, it has a unique ω-anti-periodic solution if −1 is not a characteristic multiplier of A, provided A(t) ≡ A(t + ω) and p(t) ≡ −p(t + ω). However, since it also has a unique ω-anti-periodic solution, when A(t) ≡ 0 and p(t) ≡ −p(t + ω), the multiplier condition cannot be used for the “only if” part. According to the Massera transformation theorem [Ma] (cf. [Yo, Theorem 15.3]), if system (4.11) admits a bounded solution in the future, then it also has a (harmonic) ωperiodic solution, provided A(t) ≡ A(t + ω) and p(t) ≡ p(t + ω). Unfortunately, the analogous deduction from the above conclusions for k-period-quasiperiodic solutions of the system x˙ + A(t)x =

k X

pj (t),

x ∈ Rn ,

j=1

is impossible, because the period of A should be contradictionally an integer multiple of all linearly independent over Q periods of pj , j = 1, . . . , k. There is also no direct analogy of Bohr–Neugebauer-type or Massera type results for almost-periodic A and p. The appropriate theory for (4.11) with almost-periodic A and p is due to J. Favard (cf. e.g. [Yo, ChapterIII.18]). Nevertheless, if the trivial solution of the homogeneous system (4.12) is uniformly asymptotically stable, then there is a unique (smooth) uniformly-almost-periodic solution of (4.11) which is globally uniformly asymptotically stable, provided A and p are uniformly-almost-periodic (cf. [Yo, Theorem 19.4]). In order system (4.11), where A(t) ≡ A(t + ω), to have an ω-derivo-periodic solution, p should take the special form p(t) = p0 (t) + A(t)αt, where p0 (t) ≡ p0 (t + ω) and α ∈ Rn ˙ is an n-vector. But since then p(t + ω) = p(t) + A(t)αω and p(t ˙ + ω) = p(t) ˙ + A(t)αω, provided A and p are differentiable, p is ω-periodic resp. ω-derivo-periodic if and only if ˙ A(t)α = 0 resp. A(t)α = 0. Therefore, system (4.11), where A is differentiable with A(t) ≡ A(t + ω) and p is ω-derivo-periodic, can have a pure (α 6= 0) ω-derivo-periodic solution only if A(t) ≡ A is a constant matrix. As concerns the extensions of the existence results (sufficiency criteria) to differential inclusions of the form x˙ + A(t)x ∈ P (t), x ∈ Rn , where A is the same as above and P : R ⊸ Rn is at least a measurable multivalued map with closed values, the situation is quite analogous to the foregoing section, because these multivalued extensions rely on the same selection theorems for P . Since a uniformlyalmost-periodic P with compact convex values need not admit, as already mentioned, a uniformly-almost-periodic selection, but only Stepanov-almost-periodic selection, it would be nice to show the existence of a uniformly-almost-periodic solution of (4.11), provided A

Periodic-Type Solutions of Differential Inclusions

317

is uniformly-almost-periodic such that the trivial solution of (4.12) is uniformly asymptotically stable and p is an essentially bounded Stepanov-almost-periodic n-vector function. For a constant matrix A(t) ≡ A, it is true.

4.3.

Nonlinear Scalar Equations

Consider the nonlinear scalar equation x˙ = f (t, x),

x ∈ Rn ,

(4.13)

where f : R × R → R is a Carath´eodory function (cf. Preliminaries). J. L. Massera [Ma] (cf. [Yo, Theorem 15.3]) proved, for a continuous f such that f (t, x) ≡ f (t + ω, x), that the existence of a solution of (4.13) which is bounded in the future implies, under the uniqueness assumption, the existence of a (harmonic) ω-periodic solution of (4.13). This Massera’s theorem was improved in [OO1] (cf. [DOO]), for (4.13) with L1 -Carath´eodory function f on [0, ω] × R, by showing that, for the same conclusion, the uniqueness assumption can be omitted. If equation (4.13), where f (t, x) ≡ f (t+ω, x) satisfies the L1 -Carath´eodory conditions on [0, ω]×R, admits an nω-periodic solution with n > 1 then, for every k ∈ N, there exists a kω-periodic solution of (4.13). Moreover, the set of all (subharmonic) k-periodic solutions of (4.13) has a topological dimension at least k as a subset of L∞ (R). This remarkable multiplicity results was obtained in [OO2] (for an alternative proof, see [AFP1], [AFP2]). Under the same assumptions, there still exists an infinite-dimensional subset of L∞ (R) of uniformly-almost-periodic solutions of (4.13) which are not nω-periodic, for any n ∈ N (see [OO1, Theorem 3.6]). If f (·, x) : R → R is quasi-periodic, uniformly w.r.t. x ∈ R, then the boundedness of solutions need not imply the existence of quasi-periodic (and all the worse, uniformlyalmost-periodic) solutions of (4.13). Z. Opial (cf. [Yo, p. 181], [OT]) has constructed f such that f (·, x) is 2-period-quasi-periodic and all solutions of (4.13) are bounded in the future, but none is uniformly-almost-periodic. A. M. Fink and P. O. Frederickson (cf. [Yo, p. 181], [OT]) have modified Opial’s example by showing that even uniform ultimate boundedness (i.e. dissipativity: lim supt→∞ |x(t)| < D, for all solutions x(t) of (4.13), where D is a suitable common constant) is insufficient to imply the existence of a uniformly-almostperiodic (and so, quasi-periodic) solution of (4.13), provided again f (·, x) is 2-period-quasiperiodic, uniformly w.r.t. x ∈ R. Nevertheless, it was proved in [OT] that the existence of a bounded solution of a quasi-periodic in time equation (4.13) implies the existence of a quasi-periodic solution in a certain weaker (close to almost-automorphic) sense.

4.4.

Nonlinear Planar Systems

Consider the nonlinear planar system x˙ = f (t, x),

x ∈ R2 ,

(4.14)

where f : R × R2 → R2 is a continuous function such that f (t, x) ≡ f (t + ω, x). J. L. Massera [Ma] (cf. [Yo, Theorem 15.5]) established a theorem saying that, under the uniqueness assumption, if all solutions of (4.14) exist in the future and at least one of

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Jan Andres

them is bounded (on the half-line), then there exists a (harmonic) ω-periodic solution of (4.14). He also gave in [Ma] and example (cf. [Yo, Example 15.1]) showing that the sole existence of a bounded solution (i.e. without assuming the existence of all solutions on the half-line) need not imply ω-periodic solutions. Nevertheless, in particular, Lagrange-stable systems (4.14) imply the existence of harmonics. In this case, the further information about the structure of solutions of (4.14) can be deduced from the result in [Mu]. It follows from slightly improved results of T. Matsuoka (cf. [A1, Theorem 1.2], [AG1, Chapter III.9]) that, under the uniqueness assumption, three (harmonic) ω-periodic solutions of (4.14) imply “generically” the existence of infinitely many (subharmonic) kωperiodic solutions of (4.14), k ∈ N. The “genericity” is however understood in terms of the Artin braid group theory, i.e. with the exception of certain simplest braids, representing the three given harmonics. The natural way how to prove at least three harmonics of (4.14) is by means of the Nielsen fixed point theory. In [A1], [AG1], we constructed in this way an example of system (4.14) possessing at least three harmonics. To combine these results for obtaining infinitely many subharmonics by excluding the related exceptional braids is, however, a difficult task.

4.5.

Nonlinear Systems in Rn

J. L. Massera constructed in [Ma] an example that, for n > 2, there is no analogy of his transformation theorem in [Ma] for planar systems. According to the example of S.-N. Chow in (cf. [Yo, pp. 177–180]), even the existence of a bounded uniformly asymptotically stable solution does not necessarily imply the existence of a harmonic of a Lagrange-stable (i.e. all solutions are bounded) systems in R3 . Consider the nonlinear system x˙ = f (t, x),

x ∈ Rn ,

(4.15)

where f : R × Rn → Rn is a Carath´eodory function (cf. Preliminaries). Paraphrasing the planar result of M. L. Cartwright, T. Yoshizawa asserted (see [Yo, Theorem 15.8] and cf. [H2]) that, for a continuous f such that f (t, x) ≡ f (t + ω, x), dissipative system (4.15) implies, under the uniqueness condition, the existence of a (harmonic) ω-periodic solution. In [AG2], we have slightly improved this result, namely that the uniformly dissipative, not necessarily uniquely solvable, systems (4.15) imply the existence of harmonics, provided f is Carath´eodory and f (t, x) ≡ f (t + ω, x). More generally, (4.15) can be replaced, for the same conclusion, by the inclusion x˙ ∈ F (t, x), where F : R × Rn ⊸ Rn is an upperCarath´eodory multivalued map (cf. Preliminaries) with compact convex values such that F (t, x) ≡ F (t + ω, x). Let us note that, under the assumptions of Yoshizawa’s theorem, uniquely solvable dissipative systems are, according to the result of N. Pavel (cf. [AG1], [Yo]), uniformly dissipative, i.e. ∀D1 > 0 ∃△t > 0 : [t0 ∈ R, |x(t0 )| < D1 , t ≥ t0 + △t] ⇒ |x(t)| ≤ D2 , where D2 > 0 is a common constant, for all solutions x(t) of (4.15).

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319

On the basis of the statements from the foregoing Sections 4.1. and 4.2., we can already obtain, by means of the standard fixed point theorems, a lot of classical results about periodic-type solutions of nonlinear systems. Let us demonstrate it on two simple examples for anti-periodic and almost-periodic oscillations. Since the homogeneous system x = 0, x ∈ Rn , has only the trivial anti-periodic solutions, there is exactly one ω-anti-periodic solution of the nonhomogeneous system x˙ = p(t), where p : R → Rn is a (measurable) locally Lebesgue integrable n-vector function such that p(t) ≡ −p(t + ω). Therefore, applying the Schauder linearization technique and the standard Schauder fixed point theorem (cf. [A1], [AG1]), one can immediately obtain an ω-anti-periodic solution of the form 1 x(t) = 2

Z

0

f (s, x(s)) ds + −ω

Z

t

f (s, x(s)) ds 0

to (4.15), where f (t, x) ≡ −f (t + ω, x), provided |f (t, x)| ≤ α|x| + β holds, for all (t, x) ∈ [0, ω] × Rn with a sufficiently small constant α. Similarly, since the homogeneous system (4.7), where the real parts of all eigenvalues of A are nonzero, has only the trivial entirely bounded solution, there is exactly one uniformlyalmost-periodic solution of the nonhomogeneous system (4.6), where p : R → Rn is a uniformly-almost-periodic n-vector function. Therefore, applying the Schauder linearization technique and the standard Banach fixed point theorem (cf. [AG1]), one can immediately obtain a unique uniformly-almost-periodic solution of the form Z ∞ G(t − s)f (s, x(s)) ds x(t) = −∞

to the system x˙ + Ax = f (t, x),

(4.16)

where f (·, x) : R → Rn is uniformly-almost-periodic, uniformly w.r.t. x ∈ Rn , provided f (t, ·) : Rn → Rn is Lipschitz-continuous with a sufficiently small Lipschitz constant. This well-known theorem was probably proved for the first time by G. I. Birjuk in 1954 (cf. also [H1]). Because of the mentioned Kuratowski–Ryll-Nardzewski selection theorem (cf. Proposition 2.2), the first example can be directly extended to system x˙ ∈ f (t, x) + P (t), where P : R ⊸ Rn is a (multivalued) bounded, measurable map with closed values such that P (t) ≡ −P (t + ω). On the other hand, since the uniformly-almost-periodic multivalued map P : R → Rn with compact convex values need not possess single-valued uniformlyalmost-periodic selections, but only Stepanov-almost-periodic selections, a multivalued extension of the second example is not so straightforward (see [AG1, Chapter III.103] and cf. Section 5.4. below). Finally, consider again the semilinear system (4.16) and assume that, for given ω > 0 and α ∈ Rn , A is a real (n × n)-matrix such that σ(A) ∩ 2πiZ/ω = ∅ and f takes the special form f (t, x) := f0 (t, x) + (tA + E)α, where f0 : R × Rn → Rn is an n-vector bounded Carath´eodory function such that f0 (t, x + αt) ≡ f0 (t + ω, x + α(t + ω)),

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e.g. f0 (t, x) ≡ f0 (t + ω, x) ≡ f0 (t, x + αω). We will show that, under these assumptions, system (4.16) possesses an ω-derivo-periodic solution. One can readily check that x(t) = x0 (t) + αt is an ω-derivo-periodic solution of (4.16) if and only if x0 (t) is an ω-periodic solution of the functional system x˙ + Ax = f0 (t, x + αt),

x ∈ Rn .

Since, furthermore, f1 (t + ω, x) ≡ f1 (t, x) := f0 (t, x + αt) ≡ f0 (t + ω, x + α(t + ω)), the functional system really admits, by the analogous arguments as above, an ω-periodic solution, and subsequently system (4.16) has a desired ω-derivo-periodic solution, as claimed. In particular, system x˙ + Ax = f (x) + p(t),

x ∈ Rn ,

where A is the same as above, admits an ω-derivo-periodic solution, provided f : Rn → Rn is a continuous n-vector function such that f (x) ≡ f (x + αω) and p : R → Rn takes the special form p(t) = p0 (t) + (tA + E)α, where p0 : R → Rn is an n-vector measurable bounded function such that p0 (t) ≡ p0 (t + ω), because f0 (t + ω, x) ≡ f0 (t, x) := f (x + αt) ≡ f (x + α(t + ω)). Since, according Kuratowski–Ryll-Nardzewski theorem (cf. Proposition 2.2) a measurable, bounded, ω-periodic map P : R ⊸ Rn with closed values possesses a single valued measurable, bounded, ω-periodic selection p0 ⊂ P0 , the same is true for the inclusion x˙ + Ax ∈ f (x) + P (t),

x ∈ Rn ,

where A, f are the same as above and P takes the special form P (t) = P0 (t) + (tA + E)α, provided P0 : R ⊸ Rn is measurable bounded multivalued map with closed values such that P0 (t) ≡ P0 (t + ω). Further extensions are available, e.g. for the inclusion of the form x˙ + Ax ∈ F (x) + P (t),

x ∈ Rn ,

provided F : Rn ⊸ Rn is an l.s.c. map with convex compact values such that F (x) ≡ F (x + αω). On the other hand, since u.s.c. maps F with convex compact values need not possesses continuous selections, the related extension is not so straightforward.

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5.

321

General Theorems for Periodic-Type Solutions

5.1.

Bounded Solutions

Firstly, we are interested in the existence of an entirely bounded solution to the semilinear differential inclusion x˙ + A(t)x ∈ F (t, x),

for a.a. t ∈ R, x ∈ Rn ,

(5.17)

with A ∈ L(Rn ), where L(Rn ) denotes the space of linear continuous transformations in Rn , and a set-valued transformation F . Our assumptions concerning the inclusion (5.17) will be the following: (A1) A : R → L(Rn ) is a (measurable) locally Lebesgue integrable matrix. (A2) Assume that x˙ + A(t) x = 0 (5.18) admits a regular exponential dichotomy (cf. Remark 5.1 below). Denote by G Green’s function for (5.18). (F) Let F : R × Rn ⊸ Rn be a u-Carath´eodory set-valued map (cf. Definition 2.3) such that |F (t, x)| ≤ m(t) + K|x|, for a.a. t ∈ R, x ∈ Rn , (5.19) where K ≥ 0 is a sufficiently small constant and m ∈ L1loc (R) is such that, for a constant M , Z t+1 m(s) ds t ∈ R < M. sup t

The following theorem is a special (finite-dimensional) case of Theorem 5.1 in [A1] (cf. also [AG1, Theorem III.5.33]). T HEOREM 5.1 Under the assumptions (A1), (A2), (F), the semilinear differential inclusion (5.17) admits an entirely bounded solution of the form x(t) =

Z

∞

G(t, s)f (s, x(s)) ds,

f ⊂ F.

−∞

R EMARK 5.1 Condition (A2) is satisfied, provided there exists a projection matrix P (P = P 2 ) and constants k > 0, λ > 0 such that ( |X(t)P X −1 (s)| ≤ k exp(−λ(t − s)), for s ≤ t, (5.20) −1 |X(t)(I − P )X (s)| ≤ k exp(−λ(s − t)), for t ≤ s, where X(t) is the fundamental matrix of (5.18), satisfying X(0) = I, i.e., the unit matrix. If A in (A1) is a piece-wise continuous and periodic, then it is well-known that (5.20) takes place, whenever all the associated Floquet multipliers lie off the unit cycle. If A in (A1) is (continuous and) almost-periodic, then it is enough (see [Pa]) that (5.20) holds only on a half-line [t0 , ∞) or even on a sufficiently long finite interval.

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R EMARK 5.2 If A : R → L(Rn ) is a linear, bounded operator whose spectrum does not intersect the imaginary axis, then the constant K in (F) can be easily taken as K < 1/C(A), where ( Z ∞ e−A(t−s) P− , for t > s, (5.21) |G(t, s)| ds ≤ C(A), G(t, s) = sup e−A(t−s) P+ , for t < s t∈R −∞

and P− , P+ stand for the corresponding spectral projections to the invariant subspaces of A. The existence of bounded solutions can be also obtained in the sequential way by means of the following proposition (for its proof, see e.g. [A1, Proposition 4.5] or [AG1, Proposition III.1.37]).

P ROPOSITION 5.1 Let F : R × Rn ⊸ Rn be a u-Carath´eodory mapping with nonempty, compact and convex values such that |F (t, x)| ≤ α(t) + β(t)|x|, for all (t, x) ∈ R × Rn , where α, β are locally Lebesgue integrable functions in R. Then, for every x0 ∈ Rn , there exists a solution x ∈ ACloc (R, Rn ) of the Cauchy problem ( x˙ ∈ F (t, x), for a.a. t ∈ (−∞, ∞), x ∈ Rn , x(0) = x0 . Let {xm (t)} be a sequence of absolutely continuous functions such that (i) for every m ∈ N, xm ∈ AC ([−m, m], Rn ) is a solution of x˙ ∈ F (t, x),

for a.a. t ∈ [−m, m], x ∈ Rn ,

(ii) sup{|xm (t)| | m ∈ N, t ∈ [−m, m]} := M < ∞ and xm (t) ∈ D ⊂ Rn , for every t ∈ [−m, m], where D is a given closed subdomain of Rn . Then there exists an entirely bounded solution x ∈ ACloc (R, Rn ) of the inclusion x˙ ∈ F (t, x),

for a.a. t ∈ (−∞, ∞),

such that sup |x(t)| ≤ M (< ∞) and x(t) ∈ D, for all t ∈ R. t∈R

Hence, consider still the boundary value problems of the type ( x˙ + A(t)x ∈ F (t, x), for a.a. t ∈ [0, τ ], x ∈ Rn , Lx = Θ,

(5.22)

where (i) A : [0, τ ] → L(Rn ) is a measurable linear operator such that |A(t)| ≤ γ(t), for all t ∈ [0, τ ] and some integrable function γ : [0, τ ] → [0, ∞), (ii) the associated homogeneous problem ( x˙ + A(t)x = 0, for a.a. t ∈ [0, τ ], x ∈ Rn , Lx = 0

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has only the trivial solution, (iii) F : [0, τ ] × Rn ⊸ Rn is a u-Carath´eodory mapping with nonempty, compact and convex values (cf. Definition 2.3), (iv) there are two nonnegative Lebesgue-integrable functions δ1 , δ2 : [0, τ ] → [0, ∞) such that |F (t, x)| ≤ δ1 (t) + δ2 (t)|x|,

for a.a. t ∈ [0, τ ] and all x ∈ Rn ,

where |F (t, x)| = sup{|y| | y ∈ F (t, x)}. P ROPOSITION 5.2 Consider problem (5.22) with (i)–(iv) above and let G : [0, τ ] × Rn × Rn × [0, 1] → Rn be a product-measurable u-Carath´eodory map (cf. Definition 2.3 and Proposition 2.3) such that G(t, c, c, 1) ⊂ F (t, c),

for all (t, c) ∈ [0, τ ] × Rn .

Assume, furthermore, that (v) there exists a (bounded) retract Q of C([0, τ ], Rn ) such that Q \ ∂Q is nonempty (open) and such that G(t, x, q(t), λ) is Lipschitzian in x with a sufficiently small Lipschitz constant (cf. Preliminaries), for a.a. t ∈ [0, τ ] and each (q, λ) ∈ Q × [0, 1], (vi) there exists a Lebesgue integrable function α : [0, τ ] → [0, ∞) such that |G(t, x(t), q(t), λ)| ≤ α(t),

a.e. in [0, τ ],

for any (x, q, λ) ∈ ΓT (i.e. from the graph of T ), where T denotes the set-valued map which assigns, to any (q, λ) ∈ Q × [0, 1], the set of solutions of ( x˙ + A(t)x ∈ G(t, x, q(t), λ), for a.a. t ∈ [0, 1], x ∈ Rn , Lx = Θ, (vii) T (Q × {0}) ⊂ Q holds and the boundary ∂Q of Q is fixed point free w.r.t. T , for every (q, λ) ∈ Q × [0, 1]. Then problem (5.22) has a solution. R EMARK 5.3 Rescaling t in (5.22), the interval [0, τ ] can be obviously replaced in Proposition 5.2 by any compact interval J, e.g. J = [−m, m], m ∈ N. Therefore, the second part of Proposition 5.1 can be still applied for obtaining an entirely bounded solution, as claimed. E XAMPLE 5.1 Consider problem (5.22), on the interval [−m, m], m ∈ N. Assume that the appropriate conditions (i)–(iv) are satisfied. Taking (for a product-measurable F : [−m, m] × Rn ⊸ Rn ) G(t, q(t)) = F (t, q(t)),

for q ∈ Q,

where Q = {µ ∈ C([−m, m], Rn ) | maxt∈[−m,m] |µ(t)| ≤ D} and D > 0 is a sufficiently big constant which will be specified below, we can see that (v) holds trivially. Furthermore, according to (iv), we get |G(t, q(t))| ≤ δ1 (t) + δ2 (t)D,

for a.a. t ∈ [−m, m],

(5.23)

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i.e. (vi) holds as well with α(t) = δ1 (t) + δ2 (t)D. At last, the associated linear problem ( x˙ + A(t)x ∈ F (t, q(t)), for a.a. t ∈ [−m, m], x ∈ Rn , Lx = Θ has, according to the Fredholm alternative (Conti’s lemma), for every q ∈ Q, a nonempty set of solutions of the form Z m T (q) = H(t, s)f (s, q(s)) ds, −m

where H is the related Green function and f ⊂ F is a measurable selection. For more details, see [A1] or [AG1, Chapter III.5]. Therefore, in order to apply Proposition 5.2 for the solvability of (5.22), we only need to show (cf. (vii)) that T (Q) ⊂ Q (and that ∂Q is fixed point free w.r.t. T , for every q ∈ Q, which is, however, not necessary here). Hence, in view of (5.23), we have that Z m H(t, s)f (s, q(s)) ds max |T (q)| = max t∈[−m,m] t∈[−m,m] −m Z m ≤ max |H(t, s)|(δ1 (s) + δ2 (s)D) ds t∈[−m,m] −m Z m Z m δ2 (t) dt , δ1 (t) dt + D = max |H(t, s)| t,s∈[−m,m]

−m

−m

and subsequently the above requirement holds for D≥ provided Z

maxt,s∈[−m,m] |H(t, s)| 1−

Rm

−m δ1 (t) dt Rm , maxt,s∈[−m,m] |H(t, s)| −m δ2 (t) dt

m

δ2 (t) dt < −m

1 maxt,s∈[−m,m] |H(t, s)|

,

m ∈ N.

(5.24)

(5.25)

(Observe that for D strictly bigger than the above quantity, ∂Q becomes fixed point free). The problem (5.22) is solvable on the interval J = [−m, m], for every m ∈ N, and subsequently the inclusion (5.17) admits, according to the second part of Proposition 5.2, an entirely bounded solution x(t) such that supt∈(−∞,∞) |x(t)| ≤ D. If, instead of (5.23), we have |F (t, x)| ≤ δ1 + δ2 D′ ,

for a.a. t ∈ (−∞, ∞) and |x| ≤ D′ ,

where D′ satisfies, for every m ∈ N, the inequality Rm δ max 1 t∈[−m,m] −m |H(t, s)| ds Rm , D′ ≥ 1 − δ2 maxt∈[−m,m] −m |H(t, s)| ds provided

δ2 <

maxt∈[−m,m]

1 Rm

−m |H(t, s)| ds

,

(5.23’)

(5.24’)

(5.25’)

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then problem (5.22) is again solvable on the interval J = [−m, m], for every m ∈ N, and subsequently the inclusion (5.17) admits, according to the second part of Proposition 5.2, an entirely bounded solution x(t) such that supt∈(−∞,∞) |x(t)| ≤ D′ . R EMARK 5.4 Observe that the growth restriction (5.23) with conditions (5.24), (5.25), resp. the growth restriction (5.23’) with conditions (5.24’), (5.25’), are of the same quality as the growth restriction (5.19) with condition (5.21) in Theorem 5.1 . If problem (5.22) takes the special form (m ∈ N) ( x˙ + A(t)x ∈ F (t, x), for a.a. t ∈ [−m, m], x ∈ Rn , x(−m) = M x(m), where M is a regular (n × n)-matrix,

(5.26)

usually called the Floquet problem, then the bound sets approach allows us to say something about the localization of solution of (5.26). Therefore, if the solution values a located, for every m ∈ N, in a given set K ⊂ Rn , the second part of Proposition 5.2 guarantees the existence of an entirely bounded solution of (5.17). The conditions for the solvability of (5.26) were established in [AMT3]. T HEOREM 5.2 Assume that A : R → L(Rn ) is an essentially bounded, locally Lebesgue integrable matrix and F : R × Rn ⊸ Rn is a u-Carath´eodory mapping with nonempty, convex, compact values. Assume, furthermore, that 1) the homogeneous periodic problem ( x˙ + A(t)x = 0, for a.a. t ∈ [−m, m], x ∈ Rn , x(−m) = x(m), has, for every m ∈ N, only the trivial solution, 2) there exists a u-Carath´eodory mapping G : R × Rn × Rn × [0, 1] ⊸ Rn such that, for all (t, x, y) ∈ R × R2n and λ ∈ [0, 1], it follows: G(t, x, y, 0) = G0 (t, x), G(t, y, y, 1) ⊂ F (t, y); |w| ≤ s(t)(1 + |x| + |y| + λ), with w ∈ G(t, x, y, λ) and s ∈ L1loc (R), 3) there exists a bounded, non-empty and open K ⊂ Rn , whose closure is a retract of Rn , and a function V : Rn → R of class C 2 such that V (x) ≤ 0 on K, V (x) = 0 and ∇V (x) 6= 0, for every x ∈ ∂K, and <∇V (x), w> ≤ 0, for all t ∈ R, x ∈ ∂K, λ ∈ (0, 1], w ∈ G(t, x, x, λ) − A(t)x, 4) G(t, ·, y, λ) is Lipschitzian with a sufficiently small Lipschitz constant L, for every (t, y, λ) ∈ R × K × [0, 1], 5) for each m ∈ N, the set of solutions of problem ( x˙ + A(t)x ∈ G0 (t, x), for a.a. t ∈ [−m, m], x ∈ Rn , (5.27) x(m) = x(−m),

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is a subset of C([−m, m], K). Then inclusion (5.22) admits a bounded solution x(t) of the form Z ∞ G(t, s)f (s, x(s)) ds, f ⊂ F, x(t) = −∞

such that x(t) ∈ K, for all t ∈ R. Proof. Consider problem (5.26) with M = I, i.e. the periodic boundary value problem ( x˙ + A(t)x ∈ F (t, x), for a.a. t ∈ [−m, m], x ∈ Rn , (5.28) x(m) = x(−m), whose associated homogeneous problem has, according to 1), only the trivial solution. According to [AMT3, Theorem 1] (cf. Theorem 5.5 in Section 5. below), problem (5.28) admits, for every m ∈ N, a solution xm (t) such that xm (t) ∈ K, for all t ∈ [−m, m], m ∈ N. According to the second part of Proposition 5.1, inclusion (5.17) admits an entirely bounded solution x(t) of the form Z ∞ G(t, s)f (s, x(s)) ds, f ⊂ F x(t) = −∞

(cf. Remark 5.3 and Theorem 5.1) with x(t) ∈ K, for all t ∈ R. R EMARK 5.5 If G in Theorem 5.2 is globally u.s.c., then the bounding function V can be e.g. only locally Lipschitzian and its derivative can be replaced by the Dini derivatives. For more details, see [AMT1] (cf. also [AG1, Chapter III.8 (a)]). R EMARK 5.6 If G in Theorem 5.2 is u-Carath´eodory and the bounding function does not belong to the class C 2 , then the related conclusions must be assumed in some neighbourhood of the boundary ∂K of K. For more details, see [AMT2] (cf. also [AG1, Chapter III.8 (b)]). As an application of a modified, in view of Remark 5.5, Theorem 5.2, let us give the following example. E XAMPLE 5.2 At first, consider the family of periodic problems ( x˙ + A(t)x ∈ F1 (t, x) + F2 (t, x), x(−m) = x(m), m ∈ N,

(5.29)

where x = (x1 , . . . , xn ), A = (aij )i,j=1,...,n : R → L(Rn ) is a bounded continuous matrix, F = F1 + F2 = (f11 , . . . , f1n ) + (f21 , . . . , f2n ), F1 , F2 : R × Rn ⊸ Rn are globally u.s.c. multivalued functions which are bounded in t ∈ (−∞, ∞), for every x ∈ Rn , and linearly bounded in x ∈ Rn , for every t ∈ (−∞, ∞). Let A be such that the homogeneous problem ( x˙ + A(t)x = 0, x(−m) = x(m),

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has, for every m ∈ N, only the trivial solution. Assume, furthermore, that there exist positive constants Ri , i = 1, . . . , n, such that ± [aii (t)(±Ri ) − λf1i (t, x(±Ri )) − λf2i (t, x(±Ri ))] >

n X j=1 j6=i

Rj

sup

|aij (t)|,

t∈(−∞,∞)

i = 1, . . . n, t ∈ (−∞, ∞), where x(±Ri ) = (x1 , . . . , xi−1 , ±Ri , xi+1 , . . . , xn ), |xj | ≤ Rj , j 6= i, λ ∈ (0, 1], and that F1 (t, ·) is Lipschitzian with a sufficiently small constant L, for every t ∈ (−∞, ∞) (cf. Preliminaries). In order to apply Theorem 5.1 for the solvability of (5.29), let us still consider the enlarged family of problems ( x˙ + A(t)x ∈ λF1 (t, x) + λF2 (t, q(t)), λ ∈ [0, 1], (5.30) x(m) = x(−m), where q ∈ Q = {e q ∈ C([−m, m], Rn ) | qe(t) ∈ K, for all t ∈ [−m, m]}. Observe that if ξ ∈ ∂K, then ξ = ξ(±Ri ) = (ξ1 , . . . , ξi−1 , ±Ri , ξi+1 , . . . , ξn ), for some i and |ξj | ≤ Rj for all j 6= i. Therefore, let us define for (5.30) the locally Lipschitzian bounding function composed of Vξ (x) = ±xi − Ri , i = 1, . . . , n, where ξ = ξ(±Ri ) ∈ ∂K. One can easily check that, under the above assumptions, we have: ad 3) Vξ (ξ) = 0 and Vξ (x) ≤ 0, for x ∈ K, and (∇Vξ (ξ)(λF1 (t, ξ) + λF2 (t, ξ) − A(t)ξ)) = n X = ±λ[f1i (t, ξ) + f2i (t, ξ)] ∓ aij (t)ξj < 0, j=1

for all t ∈ [−m, m], m ∈ N, where λ ∈ (0, 1]. Since the conditions 1), 2), 4), 5) are satisfied either by hypotheses or trivially, inclusion x˙ + A(t)x ∈ F1 (t, x) + F2 (t, q(t)),

for a.a. t ∈ (−∞, ∞), x ∈ Rn ,

(5.31)

admits, according to Theorem 5.2, a bounded solution x(t) such that x(t) ∈ K, for all t ∈ R. Via anti-periodic problems, we can still obtain the following theorem (cf. [AMT3, Theorem 2]). T HEOREM 5.3 Assume that A(t) ≡ 0 and let conditions 2)–4) in Theorem 5.2 be satisfied with K ⊂ Rn symmetric w.r.t. the origin (observe that condition 1) holds automatically). Let, furthermore, the sets of solutions of problem ( x˙ ∈ G0 (t, x), for a.a. t ∈ [−m, m], (5.32) x(m) = −x(−m), be, for each m ∈ N, a subset of C([−m, m], K). Then inclusion (5.17) admits a bounded solution x(t) such that x(t) ∈ K, for all t ∈ R.

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Proof. Given m ∈ N, let us consider problem (5.26) with M = −I, and A ≡ 0, i.e. the anti-periodic boundary value problem ( x˙ ∈ F (t, x), for a.a. t ∈ [−m, m], x ∈ Rn , (5.33) x(m) = −x(−m), whose associated homogeneous problem has only the trivial solution. In this case, the invariance of ∂K w.r.t. M = −I is equivalent to the symmetry of the origin. According to [AMT3, Theorem 1] (cf. Theorem 5.8 in Section 5.3. below), problem (5.33) admits, for each m ∈ N, a solution xm (·) such that xm (t) ∈ K, for all t ∈ [−m, m]. The conclusion follows by means of the second part of Proposition 5.2. R EMARK 5.7 A typical case occurs when G(t, x, y, λ) = λF (t, y) + (1 − λ)G0 (t, x), where G0 is a u-Carath´eodory multivalued mapping. If, in particular, G0 (t, x) ≡ 0 then, according to condition 1), the set of solutions of (5.32) is a subset of C([−m, m], K) if and only if K is a neighbourhood retract of the origin. E XAMPLE 5.3 Consider the differential inclusion x˙ ∈ F1 (t, x) + F2 (t, x),

for a.a. t ∈ R, x ∈ Rn ,

(5.34)

where F1 , F2 : R × Rn ⊸ Rn are u-Carath´eodory maps and F1 (t, ·) is Lipschitzian, with a sufficiently small Lipschitz constant, for all t ∈ R. Assume, furthermore, the existence of positive constants R such that <x, w> ≤ 0, for all t ∈ R, x ∈ Rn , with |x| = R, and w ∈ F1 (t, x) + F2 (t, x). Take the u-Carath´eodory map G(t, x, y, λ) = λ(F1 (t, x) + F2 (t, y)), and put K as the ball B0R , i.e. K = B0R . Since K is a neighbourhood of the origin and G0 ≡ 0, the set of solutions of (5.32) is, for all m, a subset of C([−m, m], K) (see Remark 5.7). Furthermore, since all assumptions of Theorem 5.2 can be satisfied by means of C 2 -function V (x) = x2 − R2 , the inclusion (5.34) admits, according to Theorem 5.2, a bounded solution x(t) such that x(t) ∈ K, for all t ∈ R.

5.2.

Periodic Solutions

Consider inclusion (5.17), where A(t) ≡ A(t + τ ) and F (t, x) ≡ F (t + τ, x), for some τ > 0. Modifying slightly the proof of Theorem 5.1, we can give immediately the following theorem. T HEOREM 5.4 Let A : [0, τ ] → L(Rn ) be a Lebesgue integrable (n × n)-matrix function whose Floquet multipliers are different from 1 (cf. Section 4.2.). Let F : [0, τ ] × Rn ⊸ Rn be a u-Carath´eodory set-valued map such that |F (t, x)| ≤ m(t) + K|x|,

for a.a. t ∈ [0, τ ], x ∈ Rn ,

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where K ≥ 0 is a sufficiently small constant and m ∈ L1 ([0, τ ]). Then inclusion (5.17) admits a τ -periodic solution x(t) of the form Z τ G(t, s)f (s, x(s)) ds, f ⊂ F. x(t) = 0

R EMARK 5.8 Constant K in Theorem 5.4 can be taken, similarly as in Remark 5.2, as K < 1/C(A), where Z τ max |G(t, s)| ds ≤ C(A). t∈[0,τ ] 0

More generally, as a particular case of Proposition 5.2, we can give immediately the following corollary. C OROLLARY 5.1 Consider problem ( x(t) ˙ + A(t)x(t) ∈ F (t, x(t)), x(0) = x(τ ),

for a.a. t ∈ [0, τ ],

(5.35)

where F (t, x) ≡ F (t + τ, x) satisfies conditions (iii) and (iv) in Proposition 5.2. Let G : [0, τ ] × Rn × Rn × [0, 1] ⊸ Rn be a product-measurable u-Carath´eodory map (cf. Definition 2.3 and Proposition 2.3) such that G(t, c, c, 0) ≡ −A(t)c and G(t, c, c, 1) ⊂ F (t, c),

for all (t, c) ∈ [0, τ ] × Rn .

Assume that A : [0, τ ] → L(Rn ) is an essentially bounded τ -periodic (n × n)-matrix, satisfying condition (i) in Proposition 5.2, whose Floquet multipliers are different from 1. Let, furthermore, conditions (v), (vi) in Proposition 5.2 be satisfied with 0 ∈ Q (instead of T (Q) × {0} ⊂ Q, in condition (vii)), where Lx = x(0) − x(τ ) and Θ = 0. Then inclusion (5.17) admits a τ -periodic solution. Proof. Since 1 is not a Floquet multiplier of A, the homogeneous problem ( x˙ + A(t) = 0, for a.a. t ∈ [0, τ ], x ∈ Rn , x(0) = x(τ )

(5.36)

admits only the trivial solution, by which condition (ii) in Proposition 5.2 holds. Thus, for G(t, c, c, 0) ≡ −A(t)c, all assumptions of Proposition 5.2, where Lx = x(0) − x(τ ) and Θ = 0, are satisfied, because 0 ∈ Q implies that T (Q) × {0} ⊂ Q in condition (vii). The assertion, therefore, follows directly from Proposition 5.2. R EMARK 5.9 Example 5.1 can be easily specified for the periodic problem, when taking Lx = x(0) − x(τ ) and Θ = 0, with the same growth restrictions (5.23), (5.24), (5.25) resp. (5.23’), (5.24’), (5.25’). If Floquet problem (5.26) takes (for M = I) the special form (5.35), then the bound sets approach allows us to say something about the localization of solutions. The following theorem is a special case (for M = I) of [AMT3, Theorem 1].

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T HEOREM 5.5 Let A : [0, τ ] → L(Rn ) and F : [0, τ ] × Rn ⊸ Rn be the same as in Corollary 5.1. Assume that the following conditions are satisfied: 1) there exists a u-Carath´eodory mapping G : [0, τ ] × R2n × [0, 1] ⊸ Rn such that, for all t ∈ [0, τ ], x, y ∈ Rn and λ ∈ [0, 1], it holds G(t, x, y, 0) = G0 (t, x), G(t, y, y, 1) ⊂ F (t, y); |w| ≤ s(t)(1 + |x| + |y| + λ), with w ∈ G(t, x, y, λ) and s ∈ L1 ([0, τ ]), 2) there exists a nonempty, open and bounded set K ⊂ Rn , whose closure is a retract of Rn , and a C 2 -function V : Rn → R such that V (x) ≤ 0 on K, V (x) = 0 and ∇V (x) 6= 0 on ∂K, and <∇V (x), w> ≤ 0, (5.37) for all t ∈ [0, τ ], x ∈ ∂K, λ ∈ (0, 1], and w ∈ G(t, x, x, λ) − A(t)x, where <·, ·> denotes the inner product, 3) G(t, ·, y, λ) is Lipschitzian with a sufficiently small Lipschitz constant L (cf. Preliminaries), for each (t, y, λ) ∈ [0, τ ] × K × [0, 1], 4) for each solution x(t) of problem ( x˙ + A(t)x ∈ G0 (t, x), for a.a. t ∈ [0, τ ], x ∈ Rn , (5.38) x(b) = x(a) it holds x(t) ∈ K, for all t ∈ [0, τ ]. Then inclusion (5.17) admits a τ -periodic solution Z τ G(t, s)f (s, x(s)) ds, x(t) =

f ⊂ F,

0

such that x(t) ∈ K, for all t ∈ (−∞, ∞). R EMARK 5.10 A typical case occurs when G(t, x, y, λ) = λF (t, y) + (1 − λ)G0 (t, x), where G0 is a u-Carath´eodory multivalued mapping. If, in particular, G0 (t, x) ≡ 0 then, because of the assumption that 1 is not a Floquet multiplier of A, homogeneous problem (5.36) has only the trivial solution, by which condition 4) is satisfied if and only if 0 ∈ K. R EMARK 5.11 Remarks 5.5 and 5.6 from the foregoing section hold here fully as well. As an application of a modified, in view of Remark 5.11, Theorem 5.5, let us give the following example. E XAMPLE 5.4 Consider the periodic problem ( x˙ + A(t)x ∈ F1 (t, x) + F2 (t, x), x(0) = x(τ ),

(5.39)

where x = (x1 , . . . , xn ), matrix A = (aij )i,j=1,...,n : [0, τ ] → L(Rn ) is continuous and such that 1 is not its Floquet multiplier, F = F1 + F2 = (f11 , . . . , f1n ) + (f21 , . . . , f2n ), F1 , F2 : [0, τ ] × Rn ⊸ Rn are globally upper semicontinuous multivalued functions which

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are bounded in t ∈ [0, τ ], for every x ∈ Rn , and linearly bounded in x ∈ Rn , for every t ∈ [0, τ ]. Assume, furthermore, that there exist positive constants Ri , i = 1, . . . , n such that ± [aii (t)(±Ri ) − λf1i (t, x(±Ri )) − λf2i (t, x(±Ri ))] >

n X

Rj max |aij (t)|,

j=1 j6=i

t∈[0,τ ]

i = 1, . . . n, t ∈ [0, τ ], where x(±Ri ) = (x1 , . . . , xi−1 , ±Ri , xi+1 , . . . , xn ), |xj | ≤ Rj , j 6= i, λ ∈ (0, 1], and that F1 (t, ·) is Lipschitzian with a sufficiently small constant L, for every t ∈ [0, τ ] (cf. Preliminaries). In order to apply Theorem 5.5 for the solvability of (5.39), let us still consider the enlarged family of problems ( x˙ + A(t)x ∈ λF1 (t, x) + λF2 (t, q(t)), λ ∈ [0, 1], (5.40) x(0) = x(τ ), where q ∈ Q = {e q ∈ C([0, τ ], Rn ) | qe(t) ∈ K, for all t ∈ [0, τ ]}. Observe that if ξ ∈ ∂K, then ξ = ξ(±Ri ) = (ξ1 , . . . , ξi−1 , ±Ri , ξi+1 , . . . , ξn ), for some i and |ξj | ≤ Rj for all j 6= i. Therefore, let us define for (5.40) the locally Lipschitzian bounding function composed of Vξ (x) = ±xi − Ri , i = 1, . . . , n, where ξ = ξ(±Ri ) ∈ ∂K. One can easily check that, under the above assumptions, we have: ad 2) Vξ (ξ) = 0 and Vξ (x) ≤ 0, for x ∈ K, and (∇Vξ (ξ)(λF1 (t, ξ) + λF2 (t, ξ) − A(t)ξ)) = n X aij (t)ξj < 0, = ±λ[f1i (t, ξ) + f2i (t, ξ)] ∓ j=1

for all t ∈ [0, τ ], where λ ∈ (0, 1]. Since conditions 1), 3) are satisfied by hypotheses and condition 4) follows trivially from the fact that 1 is not a Floquet multiplier of A, inclusion x˙ + A(t)x ∈ F1 (t, x) + F2 (t, q(t)) admits, according to Theorem 5.5, a τ -periodic solution x(t) such that x(t) ∈ K, for all t ∈ R. For the completeness, applying the Nielsen fixed point theory developed in [AG1], we can establish the following slight modification of the multiplicity results in [A1, Theorem 6.1] and [AG1, Theorem III.6.4]. T HEOREM 5.6 Consider boundary value problem (5.17) on the interval J = [0, τ ]. Assume 2 that A : J → Rn is a single-valued essentially bounded, Lebesgue measurable (n × n)matrix such that 1 is not its Floquet multiplier and F : J × R ⊸ Rn is a u-Carath´eodory

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product-measurable mapping with nonempty, compact and convex values (cf. Definition 2.3 and Proposition 2.3) satisfying |F (t, x)| ≤ µ(t)(|x| + 1),

(5.41)

where µ : J → [0, ∞) is a suitable Lebesgue integrable bounded function. Then inclusion (5.17) has at least N = N (r|T (Q) ◦ T (·)) solutions (for the definition of the Nielsen number N , see [AG1, Chapter I.10]), provided there exists a closed connected subset Q of C(J, Rn ) with a finitely generated abelian fundamental group such that (i) T (Q) is bounded, (ii) T : Q ⊸ U is retractible onto Q with a retraction r, where U is an open subset of C(J, Rn ) containing Q, i.e. there is a continuous retraction r : U → Q and p ∈ U \ Q with r(p) = q implies that p 6∈ T (q), (iii) T (Q) ⊂ {x ∈ AC (J, Rn ) | x(0) = x(τ )}, where T (q) denotes the set of (existing) solutions to (5.17). R EMARK 5.12 In the single-valued case, we can obviously assume the unique solvability of the associated linearized problem. Moreover, Q need not then have a finitely generated abelian fundamental group. In the multivalued case, the latter is true, provided Q is compact and T (Q) ⊂ Q. For more details, see [A1] and [AG1, Chapter III.6]. R EMARK 5.13 The Nielsen number N ∈ N ∪ {0} is a homotopy invariant guaranteeing the existence of at least N τ -periodic solutions of inclusion (5.17). Its computation is always a difficult task (cf. [AG1, Chapter III.6]). R EMARK 5.14 We constructed a nontrivial example of a planar (i.e. in R2 ) differential inclusion with at least three τ -periodic solutions in [A1, Theorem 6.2] and [AG1, Theorem III.6.17]. This result can be, however, alternatively obtained by different techniques. Finding the multiplicity criteria which can be obtained by Theorem 5.6, but not alternatively by different techniques, is an open problem. For functional differential equations and inclusions, we have found such criteria in [AF1] and [AF2].

5.3.

Anti-periodic Solutions

Consider inclusion (5.17), where this time A(t) ≡ A(t + τ ) and F (t, x) ≡ −F (t + τ, x), for some τ > 0. Modifying slightly the proof of Theorem 5.1, we can give immediately the following theorem. T HEOREM 5.7 Let either A : [0, τ ] → L(Rn ) be a Lebesgue integrable (n × n)-matrix function whose Floquet multipliers are different from −1 (cf. Section 4.2.) or A(t) ≡ 0. Let F : [0, τ ] × Rn ⊸ Rn be a u-Carath´eodory set-valued map such that |F (t, x)| ≤ m(t) + K|x|,

for a.a. t ∈ [0, τ ], x ∈ Rn ,

where K ≥ 0 is a sufficiently small constant and m ∈ L1 ([0, τ ]). Then inclusion (5.17) admits a τ -anti-periodic solution x(t) of the form Z τ G(t, s)f (s, x(s)) ds, f ⊂ F, x(t) = 0

Periodic-Type Solutions of Differential Inclusions or

1 x(t) = 2

Z

0

f (s, x(s)) ds + −τ

Z

333

t

f (s, x(s)) ds,

f ⊂ F,

0

respectively. R EMARK 5.15 Constant K in Theorem 5.7 can be taken either as K < 1/C(A), where Z τ max |G(t, s)| ds ≤ C(A), t∈[0,τ ] 0

or as K < 1/τ , respectively. More generally, as a particular case of Proposition 5.2, we can give immediately the following corollary. C OROLLARY 5.2 Consider problem ( x˙ + A(t)x ∈ F (t, x), for a.a. t ∈ [0, τ ], x ∈ Rn , x(0) = −x(τ ),

(5.42)

where A : [0, τ ] → L(Rn ) is either an essentially bounded τ -periodic (n × n)-matrix, satisfying condition (i) Proposition 5.2, whose Floquet multipliers are different from −1, or A(t) ≡ 0, and F (t, x) ≡ −F (t+τ, −x) satisfies conditions (iii) and (iv) in Proposition 5.2. Let G : [0, τ ] × Rn × Rn × [0, 1] ⊸ Rn be a product-measurable u-Carath´eodory map (cf. Definition 2.3 and Proposition 2.3) such that G(t, c, c, 0) ≡ −A(t)c and G(t, c, c, 1) ⊂ F (t, c),

for all (t, c) ∈ [0, τ ] × Rn .

Assume that conditions (v), (vi) in Proposition 5.2 hold with 0 ∈ Q (instead of T (Q) × {0} ⊂ Q, in condition (vii)), where Lx = x(0) + x(τ ) and Θ = 0. Then inclusion (5.17) admits a τ -anti-periodic (2τ -periodic) solution. Proof. Since −1 is not a Floquet multiplier of A or A(t) ≡ 0, the homogeneous problem ( x˙ + A(t) = 0, for a.a. t ∈ [0, τ ], x ∈ Rn , (5.43) x(0) = −x(τ ) admits only the trivial solution, by which condition (ii) in Proposition 5.2 holds. Thus, G(t, c, c, 0) ≡ −A(t)c, and all assumptions of Proposition 5.2, where Lx = x(0) − x(τ ) and Θ = 0, are satisfied, because 0 ∈ Q implies that T (Q × {0}) ⊂ Q, in condition (vii). The assertion, therefore, follows directly from Proposition 5.2. R EMARK 5.16 Example 5.1 can be easily specified for the anti-periodic problem, when taking Lx = x(0) + x(τ ) and Θ = 0, either with the same growth restrictions (5.23), (5.24), (5.25) resp. (5.23’), (5.24’), (5.25’), or (when A(t) ≡ 0) with Rτ Z τ 2 0 Rδ1 (t) dt D≥ 2 δ2 (t) dt < , where τ 3 0 3 − 0 δ2 (t) dt resp.

D≥

2 3τ

δ1 , − δ2

where δ2 <

2 . 3τ

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In the case of ODEs, Corollary 5.2 can be still improved, when A(t) ≡ 0, as follows (cf. [A1, Corollary 5.4] and [AG1, Chapter III.5.38]). C OROLLARY 5.3 Consider problem ( x˙ = f (t, x), for a.a. t ∈ [0, τ ], x ∈ Rn , x(0) = −x(τ ), where f (t, x) ≡ −f (t + τ, −x) is a Carath´eodory function. Let g : [0, τ ] × Rn × Rn × [0, 1] → Rn be a Carath´eodory function such that g(t, c, c, 1) = f (t, c),

for all (t, c) ∈ [0, τ ] × Rn .

Assume that (i) there exists a bounded retract Q of C([0, τ ], Rn ) such that Q\∂Q is nonempty (open) and such that g(t, x, q(t), λ) satisfies |g(t, x, q(t), λ) − g(t, y, q(t), λ)| ≤ p(t)|x − y|,

x, y ∈ Rn

for a.a. t ∈ [0, τ ] and each (q, λ) ∈ Q × [0, 1], where p : [0, τ ] → [0, ∞) is a Lebesgue integrable function with Z τ p(t) dt ≤ π, 0

(ii) there exists a Lebesgue integrable function α : [0, τ ] → [0, ∞) such that |g(t, x(t), q(t), λ)| ≤ α(t),

a.e. in [0, τ ],

for any (x, q, λ) ∈ ΓT , where T denotes the set-valued map which assigns, to any (q, λ) ∈ Q × [0, 1], the set of solutions of ( x(t) ˙ = g(t, x(t), q(t), λ), for a.a. t ∈ [0, τ ], x(0) = −x(τ ), (iii) T (Q × {0}) ⊂ Q holds and ∂Q is fixed point free w.r.t. T , for every (q, λ) ∈ Q × [0, 1]. Then the equation x˙ = f (t, x) admits a τ -anti-periodic (2τ -periodic) solution. R EMARK 5.17 As in Corollary 5.2, the requirement T (Q × {0}) ⊂ Q in condition (vii) reduces to {0} ⊂ Q, provided g(t, x, q, λ) = λg(t, x, λ), λ ∈ [0, 1]. If Floquet problem (5.26) takes (for M = −I) the special form (5.42), then the bound sets approach allows us to say something about the localization of solutions. The following theorem is a special case (for M = −I) of [AMT3, Theorem 1]. T HEOREM 5.8 Let A : [0, τ ] → L(Rn ) and F : [0, τ ] × Rn ⊸ Rn be the same as in Corollary 5.2. Assume that the following conditions are satisfied: 1) there exists a u-Carath´eodory mapping G : [0, τ ] × R2n × [0, 1] ⊸ Rn such that, for all t ∈ [0, τ ], x, y ∈ Rn and λ ∈ [0, 1], it holds G(t, x, y, 0) = G0 (t, x), G(t, y, y, 1) ⊂ F (t, y); |w| ≤ s(t)(1 + |x| + |y| + λ),

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335

with w ∈ G(t, x, y, λ) and s ∈ L1 ([0, τ ]), 2) there exists a nonempty, open and bounded set K ⊂ Rn , whose closure is a retract of Rn , and whose boundary ∂K is symmetric w.r.t. the origin 0 ∈ Rn , and a C 2 -function V : Rn → R such that V (x) ≤ 0 on K, V (x) = 0, ∇V (x) 6= 0 on ∂K, and <∇V (x), w> ≤ 0,

(5.44)

for all t ∈ [0, τ ], x ∈ ∂K, λ ∈ (0, 1], and w ∈ G(t, x, x, λ) − A(t)x, where <·, ·> denotes the inner product, 3) G(t, ·, y, λ) is Lipschitzian with a sufficiently small Lipschitz constant L, for each (t, y, λ) ∈ [0, τ ] × K × [0, 1], 4) for each solution x(t) of problem ( x˙ + A(t)x ∈ G0 (t, x), for a.a. t ∈ [0, τ ], x ∈ Rn , (5.45) x(b) = −x(a), it holds x(t) ∈ K, for all t ∈ [0, τ ]. Then inclusion (5.17) admits a τ -anti-periodic solution Z τ G(t, s)f (s, x(s)) ds, f ⊂ F, x(t) = 0

resp. 1 x(t) = 2

Z

0

f (s, x(s)) ds + −τ

Z

t

f (s, x(s)) ds,

f ⊂ F,

0

such that x(t) ∈ K, for all t ∈ (−∞, ∞). R EMARK 5.18 If A(t) ≡ 0 and F = f resp. G = g is single-valued, then the Lipschitz constant L in condition 3) can be taken, according to condition (i) in Corollary 5.3, as π L R τ < τ or can be replaced by the Lipschitzian function p : [0, τ ] → [0, ∞) such that 0 p(t)dt = π.

R EMARK 5.19 A typical case occurs when G(t, x, y, λ) = λF (t, y) + (1 − λ)G0 (t, x), where G0 is u-Carath´eodory multivalued mapping. If, in particular, G0 (t, x) ≡ 0 then, because of the assumption that either −1 is not a Floquet multiplier of A or A(t) ≡ 0, homogeneous problem (5.43) has only the trivial solution, by which condition 4) is automatically satisfied, namely 0 ∈ K. R EMARK 5.20 Remarks 5.5 and 5.6 from Section 5.1. hold here fully as well. As the first application of Theorem 5.8 (with A(t) ≡ 0), let us give the following example. E XAMPLE 5.5 Consider the differential inclusion ( x˙ ∈ F1 (t, x) + F2 (t, x), F1 (t, x) ≡ −F1 (t + τ, −x), F2 (t, x) ≡ −F2 (t + τ, −x),

(5.46)

where F1 , F2 : R × Rn ⊸ Rn are upper-Carath´eodory maps and F1 (t, ·) is Lipschitzian, with a sufficiently small Lipschitz constant L (in the single-valued case, it is enough, in

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Jan Andres

view of Remark 5.18, to take L ≤ πτ ), for all t ∈ R. Assume, furthermore, the existence of a positive constants R such that <x, w> ≤ 0, for all t ∈ R, x ∈ Rn , with |x| = R, and w ∈ F1 (t, x) + F2 (t, x). Take the u-Carath´eodory map G(t, x, y, λ) := λ(F1 (t, x) + F2 (t, y)), and put K as the ball B0R , i.e. K = B0R . Since K is a neighbourhood of the origin and G0 ≡ 0, the set of solutions of (5.43) is, with A(t) ≡ 0, a subset of C([0, τ ], K) (see Remark 5.19). Furthermore, since all assumptions of Theorem 5.8 can be satisfied by means of C 2 -function V (x) = |x|2 − R2 , the inclusion (5.46) admits, according to Theorem 5.8, a τ -anti-periodic solution x(t) such that x(t) ∈ K, for all t ∈ R. As the second application of a modified, in view of Remark 5.20, Theorem 5.8 (with A(t) 6≡ 0), let us give the following example. E XAMPLE 5.6 Consider the anti-periodic problem x˙ + A(t)x ∈ F1 (t, x) + F2 (t, x), F1 (t, x) ≡ −F1 (t + τ, −x), F2 (t, x) ≡ −F2 (t + τ, −x), x(0) = −x(τ ),

(5.47)

where x = (x1 , . . . , xn ), matrix A = (aij )i,j=1,...,n : [0, τ ] → L(Rn ) is continuous with A(t) ≡ A(t + τ ), and such that −1 is not its Floquet multiplier, F = F1 + F2 = (f11 , . . . , f1n ) + (f21 , . . . , f2n ), F1 , F2 : [0, τ ] × Rn ⊸ Rn are globally upper semicontinuous multivalued functions with nonempty, convex, compact values which are bounded in t ∈ [0, τ ], for every x ∈ Rn , and linearly bounded in x ∈ Rn , for all t ∈ [0, τ ]. Assume, furthermore, that there exist positive constants Ri , i = 1, . . . , n, such that ± [aii (t)(±Ri ) − λf1i (t, x(±Ri )) − λf2i (t, x(±Ri ))] >

n X j=1 j6=i

Rj max |aij (t)|, t∈[0,τ ]

i = 1, . . . n, t ∈ (0, τ ), where x(±Ri ) = (x1 , . . . , xi−1 , ±Ri , xi+1 , . . . , xn ), |xj | ≤ Rj , j 6= i, λ ∈ (0, 1], n h i X ±λf1i (0, x(±Ri )) ± λf2i (0, x(±Ri )) ∓ aii (0)(±Ri ) < − |aij (0)|Rj j=1 j6=i

and n h i X ±λf1i (τ, −x(±Ri )) ± λf2i (τ, −x(±Ri )) ± aii (τ )(±Ri ) < − |aij (τ )|Rj , j=1 j6=i

i=1,. . . ,n, where λ ∈ (0, 1], and that F1 (t, ·) is Lipschitzian with a sufficiently small constant L, for every t ∈ [0, τ ] (cf. Preliminaries).

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337

In order to apply Theorem 5.8 for the solvability of (5.47), let us still consider the enlarged family of problems ( x˙ + A(t)x ∈ λF1 (t, x) + λF2 (t, q(t)), x(0) = −x(τ ),

λ ∈ [0, 1],

(5.48)

where q ∈ Q = {e q ∈ C([0, τ ], Rn ) | qe(t) ∈ K, for all t ∈ [0, τ ]}. Observe that if ξ ∈ ∂K, then ξ = ξ(±Ri ) = (ξ1 , . . . , ξi−1 , ±Ri , ξi+1 , . . . , ξn ), for some i and |ξj | ≤ Rj for all j 6= i. Therefore, let us define for (5.48) the locally Lipschitzian bounding function composed of Vξ (x) = ±xi − Ri , i = 1, . . . , n, where ξ = ξ(±Ri ) ∈ ∂K. One can easily check that, under the above assumptions, we have: ad 2) Vξ (ξ) = 0 and Vξ (x) ≤ 0, for x ∈ K, (∇Vξ (ξ)(λF1 (t, ξ) + λF2 (t, ξ) − A(t)ξ)) = = ±λf1i (t, x(±Ri )) ± λf2i (t, x(±Ri )) ∓ aii (t)(±Ri ) ∓ for all t ∈ (0, τ ), where λ ∈ (0, 1],

n X

aij (t)ξj < 0,

j=1 j6=i

(∇Vξ (ξ)(λF1 (0, ξ) + λF2 (0, ξ) − A(0)ξ)) = = ±λf1i (0, x(±Ri )) ± λf2i (0, x(±Ri )) ∓ aii (0)(±Ri ) ∓

n X

aij (0)ξj < 0,

j=1 j6=i

where λ ∈ (0, 1], and

(∇V−ξ (−ξ)(λF1 (τ, −ξ) + λF2 (τ, −ξ) + A(τ )ξ)) = = ∓λf1i (τ, −x(±Ri )) ∓ λf2i (τ, −x(±Ri )) ∓ aii (τ )(±Ri ) ∓

n X

aij (τ )ξj < 0,

j=1 j6=i

where λ ∈ (0, 1].

Since conditions 1), 3) are satisfied by hypotheses and condition 4) follows trivially from fact that −1 is not a Floquet multiplier of A, inclusion x˙ + A(t)x ∈ F1 (t, x) + F2 (t, x), admits, according to Theorem 5.8, a τ -anti-periodic solution x(t) such that x(t) ∈ K, for all t ∈ R. As the third application of a modified, in view of Remark 5.20, Theorem 5.8, let us give the following example.

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E XAMPLE 5.7 Consider the differential inclusion ( x˙ + A(t)x ∈ F1 (t, x) + F2 (t, x), F1 (t, x) ≡ −F1 (t + τ, −x), F2 (t, x) ≡ −F2 (t + τ, −x),

(5.49)

where matrix A = (aij )i,j=1,...,n : [0, τ ] → L(Rn ) is either Lebesgue-integrable essentially bounded function with A(t) ≡ A(t + τ ) such that −1 is not its Floquet multiplier or A(t) ≡ 0, F1 , F2 : [0, τ ] × Rn ⊸ Rn are upper-Carath´eodory multivalued functions with nonempty convex and compact values such that there exist c1 and c2 ∈ L1 ([0, τ ]), satisfying |F1 (t, 0)| ≤ c1 (t), |F2 (t, x)| ≤ c2 (t),

for all t ∈ [0, τ ], for all (t, x) ∈ [0, τ ] × K,

(K is defined below) and F1 (t, · ) is Lipschitzian, with a sufficiently small Lipschitz constant L, for all t ∈ R (⇒ |F1 (t, x)| ≤ L|x| + |F1 (t, 0)| ≤ L|x| + c1 (t) ≤ (L + c1 (t))(1 + |x|), for all (t, x) ∈ [0, τ ] × R). Assume, furthermore, the existence of positive constants ε and Rj , j = 1, . . . , n, such that K = Πnj=1 (−Rj , Rj ), ∂Kj = {ξ ∈ ∂K | ξj = ±Rj } and Q = {q ∈ C([0, τ ]) | q(t) ∈ K, for all t ∈ [0, τ ]}, and take, for all j = 1, . . . , n, ξ ∈ ∂Kj , x ∈ K ∩ Bξε , t ∈ [0, τ ], q ∈ Q, λ ∈ (0, 1], and w ∈ −A(t)x + λ[F1 (t, x) + F2 (t, x)], satisfying (sign ξj · wj ) < 0. Let us consider the family of multivalued functions defined as G(t, x, q, λ) = λ(F1 (t, x) + F2 (t, q)) which, recalling the growth conditions imposed on F1 and F2 and the boundedness of K, satisfy the conditions 1), 3) of Theorem 5.8. Moreover, condition 4) is trivially satisfied, because the only solution of (5.45) with G0 ≡ 0 is x = 0 ∈ int Q = Q \ ∂Q. Taking the locally Lipschitzian bounding function V : Rn → R as composed of Vξj (xj ) = sign ξj · xj − Rj , i = 1, . . . , n, we have Vξ (ξ) = 0 and Vξ (x) ≤ 0, for x ∈ K. Moreover (∇Vξ (x), w) = sign ξj · wj < 0, for ξ ∈ ∂Kj , x ∈ K ∩ Bξε , w ∈ −A(t)x + λ[F1 (t, x) + F2 (t, x)], λ ∈ (0, 1], j = 1, . . . , n. Thus condition 2) holds, too. A modified version, in view of Remark 5.20, Theorem 5.8 therefore implies the existence of τ -anti-periodic solution x(t) of inclusion (5.49) such that x(t) ∈ K, for all t ∈ R.

5.4.

Almost-periodic Solutions

Consider the inclusion x˙ + Ax ∈ F (x) + P (t),

for a.a. t ∈ R, x ∈ Rn ,

(5.50)

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339

where A is an (n×n)-matrix with real constant entries whose eigenvalues have nonzero real parts, F : Rn ⊸ Rn is Lipschitzian mapping with convex, compact values and P : R ⊸ Rn is an essentially bounded measurable map with closed values which will be successively assumed to be almost-periodic in the sense of Stepanov, Weyl and Besicovitch. Since such F possesses a (single-valued) Lipschitzian selection√ f ⊂ F , whose constant is, however, not necessarily the same, but can be taken as Ln(12 3/5 + 1), where L is a constant of F (see Preliminaries), and since such P possesses (single-valued) almostperiodic selections p ⊂ P in the sense of Stepanov, Weyl and Besicovitch, respectively (cf. Section 3.4.), we shall firstly consider the system x˙ + Ax = f (x) + p(t),

for a.a. t ∈ R, x ∈ Rn ,

(5.51)

√ where f : Rn → Rn is a Lipschitzian map with the constant L0 = Ln(12 3/5 + 1) and P : R → Rn is an essentially bounded function which is almost-periodic in a given sense (Stepanov, Weyl, Besicovitch). We already know from Section 5.1. that if L0 is sufficiently small (cf. Example 5.1), then system (5.51) admits an entirely bounded solution x(t) of the form (cf. Remark 5.1)

x(t) =

Z

∞

G(t − s)[f (x(s)) + p(s)] ds, where sup t∈R

−∞

Z

∞

|G(t − s)| ds ≤ −∞

2k . λ

(5.52)

In fact, there is a unique bounded solution of (5.51). If p is Stepanov almost-periodic (p ∈ Sap ) (cf. Section 3.4.), then in order to prove that x ∈ Sap , we need the Bochner transform xb of x, defined as xb (t) := x(t + η),

η ∈ [0, 1], t ∈ R,

and satisfying xb ∈ C(R, L([0, 1])n ), whenever x ∈ Lloc (R, Rn ) which is trivially satisfied by the hypothesis that x(t) is a solution of (5.51). Moreover, n o Sap = x ∈ Lloc (R, Rn ) | xb ∈ Cap (R, L([0, 1])n ) and kxkSap = kxb kBC(R,L([0,1])n ) . For more details, see e.g. [AG1, Appendix 1]. Since x(t + τ ) =

Z

∞

−∞

G(t − s)[f (x(s − t)) + p(s − t)] ds,

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Jan Andres

we obtain (cf. (5.52)) kxb (t + τ ) − xb (t)kBC =

Z ∞

b G (t − s)[f (x(s + τ )) − f (x(s)) + p(s + τ ) − p(s)] ds =

−∞ BC Z ∞ b b b b |G(t − s)| kf (x(s + τ )) − f (x(s)) + p (s + τ ) − p (s)kBC ds ≤ sup t∈R Z−∞ ∞ |G(t − s)| ds kf b (x(t + τ )) − f b (x(t)) + pb (t + τ ) − pb (t)kBC ≤ sup t∈R

−∞

2k b ≤ kf (x(t + τ )) − f b (x(t)) + pb (t + τ ) − pb (t)kBC λ i 2k h L0 kxb (t + τ ) − xb (t)kBC + kpb (t + τ ) − pb (t)kBC , ≤ λ

i.e. kxb (t + τ ) − xb (t)kBC ≤ provided L0 <

kpb (t + τ ) − pb (t)kBC , 1/ 2k λ − L0

1 C(A) .

Thus, if kpb (t + τ ) − pb (t)kBC < ε, i.e. if τ is an ε-almost-period of p, then it is also an ε(1/ 2k λ − L0 )-almost-period of x, by which the solution x(t) of (5.51) is almost-periodic in the sense of Stepanov. In fact, since x(t) is also bounded and uniformly continuous, it is uniformly almost-periodic (cf. Section 3.4.). We can give the first theorem for almost-periodic solutions of (5.50). T HEOREM 5.9 Let A, F, P be as above. If the Lipschitz constant L of F still satisfies the λ inequality L ≤ 2kn(12√λ 3/5+1) (⇒ L0 < 2k ), where P is almost-periodic in the sense of Stepanov, then inclusion (5.50) admits a uniformly almost-periodic solution. R EMARK 5.21 For a uniformly almost-periodic (multivalued) map P , we cannot improve Theorem 5.9 in the sense that the solution is classical (smooth), because as pointed out in Section 3.4., P need not possess a uniformly almost-periodic selection. Theorem 5.9 can be slightly improved, on the basis of Theorem 5.2 in Section 5.1., as follows. C OROLLARY 5.4 Let the assumptions of Theorem 5.2 in Section 5.1. be satisfied with A(t) ≡ A, whose eigenvalues have nonzero real parts, and G(t, x, y, λ) = λ[F (y) + P (t)]. λ If F is Lipschitzian with constant L < 2kn(12√ , on the set K, and P is almost3/5+1) periodic in the sense of Stepanov, then inclusion (5.50) admits a uniformly almost-periodic solution x(t) such that x(t) ∈ K, for all t ∈ R. Consider again (5.50), where A, F are as above, but P be this time almostperiodic in the sense of Weyl. More precisely, let P be the sum P = P0 + p1 , where P0 is a multivalued e-Wap -perturbation of a single-valued Wap -function p1 . Since P0 possesses a single-valued e-Wap -selection p0 (cf. Section 3.4.) which is (e-)W -uniformly continuous (cf. [AG1, Appendix 1]), it can be proved by the similar arguments as in [AG1, p. 564] that the sum p = p0 + p1 : R → Rn is also a Wap -function (i.e. p ∈ Wap ). For definitions and more details, see Section 3.4..

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Hence, consider also √ (5.51), where f ⊂ F is a Lipschitzian selection of F with the constant L0 = Ln(12 3/5 + 1) and p ∈ Wap . System (5.51) admits, by the same reasons as above, a (unique) entirely bounded solution x(t) of the form (5.52). Since again Z ∞ G(t − s)[f (x(s − t)) + p(s − t)] ds, x(t + τ ) = −∞

and so Z

a+l

|x(t + τ ) − x(t)| dt = Z a+l Z ∞ dt |G(t − s)|[f (x(s + τ )) − f (x(s)) + p(s + τ ) − p(s)] ds = a

−∞ ∞

a

≤

Z

a+l

dt

a

Z

|G(t − s)| |f (x(s + τ )) − f (x(s)) + p(s + τ ) − p(s)| ds,

−∞

we can proceed similarly as above (this time, without using the Bochner transform, but applying the Fubini theorem; for more details, see [AG1, pp. 548–549]) to obtain Z a+l 2εk 1 |x(t + τ ) − x(t)| dt < , lim sup l→∞ a∈R l λ − 2L0 k a λ provided L0 < 2k , where τ is an ε-almost-periodic of p in the Wap -pseudometric and λ, k are suitable constants related to A (cf. (5.52)). Thus, x(t) is Wap -almost-periodic (i.e. x ∈ Wap ). In fact, since it is bounded and uniformly continuous, it is also Wap -uniformly-continuous, and subsequently W -normal (for more details, see [AG1, Appendix 1] and cf. Section 3.4.). We can, therefore, give the second theorem for almost-periodic solutions of (5.50).

T HEOREM 5.10 Let A, F, P be as above. If the Lipschitz constant L of F still satisfies λ λ ), where λ, k are suitable constants related the inequality L < 2kn(12√ (⇒ L0 < 2k 3/5+1) to A (cf. (5.52)), and P = P0 + p1 , where P0 is a multivalued e-Wap -perturbation of a single-valued Wap -function p1 , then inclusion (5.50) admits a W -normal solution. R EMARK 5.22 W -normality is a bit more than the Wap -property, but a bit less than the e-Wap -property (see Table 1 in Section 3.4.). It is, therefore, a question whether the e-Wap property of P implies the existence of an e-W -normal solution of (5.50). Theorem 5.10 can be slightly improved, on the basis of Theorem 5.2 in Section 5.1., as follows. C OROLLARY 5.5 Let the assumptions of Theorem 5.2 in Section 5.1. be satisfied with A(t) ≡ A, whose eigenvalues have nonzero real parts, and G(t, x, y, λ) = λ[F (y) + P (t)]. λ If F is Lipschitzian with constant L < 2kn(12√ , on the set K, and P = P0 + p1 , 3/5+1) where P0 is a multivalued e-Wap -perturbation of a single-valued Wap -function p1 , then inclusion (5.50) admits a W -normal solution x(t) such that x(t) ∈ K, for all t ∈ R. Finally, consider (5.50), where A, F are as above, but P be this time almost-periodic in the sense of Besicovitch. More precisely, let P be the sum P = P0 + p1 , where P0 is a multivalued B-perturbation of a single-valued Bap -function p1 . Since P0 possesses

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a single-valued B-selection p0 (cf. Section 3.4.) which is Bap -uniformly continuous (cf. [AG1, Appendix 1]), it can be proved by the similar arguments as in [AG1, p. 564] that the sum p = p0 + p1 : R → Rn is also a Bap -function (i.e. p ∈ Bap ). For definitions and more details, see Section 3.4.. Hence, consider also √ (5.51), where f ⊂ F is a Lipschitzian selection of F with the constant L0 = Ln(12 3/5 + 1) and p ∈ Bap . System (5.51) admits, by the same reasons as above, a (unique) entirely bounded solution x(t) of the form (5.52). Since again Z ∞ G(t − s)[f (x(s − t)) + p(s − t)] ds, x(t + τ ) = −∞

and so Z

T

|x(t + τ ) − x(t)| dt = Z T Z ∞ |G(t − s)|[f (x(s + τ )) − f (x(s)) + p(s + τ ) − p(s)] ds dt = −T

−∞ ∞

−T T

≤

Z

−T

dt

Z

|G(t − s)| |f (x(s + τ )) − f (x(s)) + p(s + τ ) − p(s)| ds,

−∞

we can proceed similarly as above (for more details, see [AG1, pp. 550–551]) to obtain Z T 2εk 1 |x(t + τ ) − x(t)| dt < , lim T →∞ 2T λ − 2L0 k −T λ provided L0 < 2k , where τ is an ε-almost-periodic of p in the Bap -pseudometric and λ, k are suitable constants related to A (cf. (5.52)). Thus, x(t) is Bap -almost-periodic (i.e. x ∈ Bap ). In fact, since it is bounded and uniformly continuous, it is also Bap -uniformly-continuous, and subsequently B-normal (for more details, see [AG1, Appendix 1] and cf. Section 3.4.). We can, therefore, give the third theorem for almost-periodic solutions of (5.50).

T HEOREM 5.11 Let A, F, P be as above. If the Lipschitz constant L of F still satisfies the λ inequality L < 2kn(12√λ 3/5+1) (⇒ L0 < 2k ), where λ, k are suitable constants related to A (cf. (5.52)), and P = P0 + p1 , where P0 is a multivalued B-perturbation of a single-valued Bap -function p1 , then inclusion (5.50) admits a B-normal solution. R EMARK 5.23 B-normality is a bit more than the Bap -property, but a bit less than the Bproperty (see Table 1 in Section 3.4.). It is, therefore, a question whether the B-property of P implies the existence of a B-solution of (5.50). Theorem 5.11 can be slightly improved, on the basis of Theorem 5.2 in Section 5.1., as follows. C OROLLARY 5.6 Let the assumptions of Theorem 5.2 in Section 5.1. be satisfied with A(t) ≡ A, whose eigenvalues have nonzero real parts, and G(t, x, y, λ) = λ[F (y) + P (t)]. λ , on the set K, and P = P0 + p1 , If F is Lipschitzian with constant L < 2kn(12√ 3/5+1) where P0 is a multivalued B-perturbation of a single-valued Bap -function p1 , then inclusion (5.50) admits a B-normal solution x(t) such that x(t) ∈ K, for all t ∈ R.

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R EMARK 5.24 In the single-valued case, Lipschitz constant L can be taken as L = L0 < λ 2k . The proved bounded almost-periodic solutions are then unique (in the case of corollaries, on K). R EMARK 5.25 Theorems 5.10 and 5.11 are slight generalizations of the analogous results in [AG1, Chapter III.10], because multivalued perturbations P0 in the sum P = P0 + p1 were there assumed almost-periodic in the sense of Stepanov, everywhere. Corollaries 5.4, 5.5, 5.6 are formally new.

5.5.

Derivo-periodic Solutions

Consider the inclusion x˙ ∈ F (x) + P (t),

for a.a. t ∈ R, x ∈ Rn ,

(5.53)

where x = (x1 , . . . , xn ) and F (x) = (f1 (x), . . . , fn (x)). D EFINITION 5.1 We say that x ∈ ACloc (R, Rn ) is an ω-derivo-periodic solution of (5.53) if x(t) ˙ = x(t ˙ + ω), for a.a. t ∈ R. Assume, we have a suitable notion of a multivalued derivative (see e.g. [AG1, Appendix 2], say D∗ , or multivalued partial derivatives Dx∗j (w.r.t. xj ), j = 1, . . . , n, such that F : Rn ⊸ Rn is αω-D∗ -periodic, α ∈ Rn , ω > 0, i.e. D∗ F (x) ≡ D∗ F (x + αω),

(5.54)

where D∗ F means the Jacobi matrix, namely D∗ F = (D∗ (fi )xj )ni,j=1 . H YPOTHESIS 5.1 The αω-D∗ -periodicity of F , i.e. (5.54), implies that F can be written as F (x) = F0 (x) − Ax,

(5.55)

F0 (x) ≡ F0 (x + αω),

(5.56)

where F0 is αω-periodic, i.e. and A is a suitable (n × n)-matrix. If so, then (5.53) with an αω-D∗ -periodic F would take the following quasi-linear form: x˙ + Ax ∈ F0 (x) + P (t),

(5.57)

where F0 satisfies (5.56). Moreover, x0 (t) is obviously an ω-periodic solution of x˙ + Ax ∈ F0 (x + αt) + P (t),

(5.58)

where F0 satisfies (5.56) and P is ω-periodic, i.e. P (t) ≡ P (t + ω),

(5.59)

if and only if x(t) = x0 (t) + αt is an ω-derivo-periodic solution of x˙ + Ax ∈ F0 (x) + [P (t) + (tA + I)α], where I denotes the unit matrix.

(5.60)

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H YPOTHESIS 5.2 [P (t) + (tA + I)α] is ω-D∗ -periodic, i.e. D∗ {P (t + ω) + [(t + ω)A + I]α} ≡ D∗ [P (t) + (tA + I)α] = D∗ P (t) + Aα ≡ D∗ P (t + ω) + Aα

(5.61)

holds, provided (5.59) takes place. More important is, however, the “reverse” formulation: x(t) = x0 (t) + αt is an ωderivo-periodic solution of (5.57) if and only if x0 (t) is an ω-periodic solution of x˙ + Ax ∈ F0 (x + αt) + [P (t) − (tA + I)α],

(5.62)

where F0 satisfies (5.56). H YPOTHESIS 5.3 An ω-D∗ -periodic P implies that P (t) = P0 (t) + (tA + I)α

(5.63)

holds with an ω-periodic P0 , i.e. P0 (t) ≡ P0 (t + ω).

(5.64)

If so, then (5.62) would take the form x˙ + Ax ∈ F0 (x + αt) + P0 (t),

(5.65)

where, in view of (5.56), F0 (x + α(t + ω)) ≡ F0 (x + αt) =: F1 (t, x) ≡ F1 (t + ω, x). For (5.65), we can easily find sufficient conditions for the existence of at least one ωperiodic solution, e.g. if σ(A) ∩ 2πiZ/ω = ∅ (see Sections 4.5. and 4.1.), provided a u.s.c. F0 with nonempty, convex and compact values is αω-periodic, i.e. (5.56), and a measurable, essentially bounded P0 with nonempty, convex and compact values is ω-periodic, i.e. (5.64). Therefore, we can give immediately P ROPOSITION 5.3 If Hypothesis 5.3 is satisfied for a D∗ -differentiable, ω-D∗ -periodic P , i.e. if (5.63) is implied, then (5.65) with the same α can be written in the form of (5.62), and subsequently (5.57) admits an ω-derivo-periodic solution, provided σ(A) ∩ 2πiZ/ω = ∅ and (5.56) takes place for a u.s.c. multivalued function F0 with nonempty, convex and compact values. P ROPOSITION 5.4 Let the assumptions of Proposition 5.3 be satisfied. If Hypothesis 5.1 is still satisfied, for an αω-D∗ -periodic, D∗ -differentiable multivalued function F with nonempty, convex and compact values and with the same A as in (5.57), then even inclusion (5.53) admits an ω-derivo-periodic solution. Although we have to our disposal several different notions of multivalued derivatives (see [AG1, Appendix 2]), to satisfy Hypothesis 5.1 or 5.3, which are assumed in Propositions 5.3 and 5.4, only the usage of a multivalued derivative due to F. S. De Blasi, introduced in Section 3.5., is available.

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Moreover, F in Hypothesis 5.1 must have a special form to satisfy (5.55), e.g. such that D∗ (Fi (x))xj = D∗ (Fi (xj ))xj ,

for all i, j = 1, . . . , n.

(5.66)

As pointed out in Remark 3.8 in Section 3.5., in Rn , we can define equivalently a De Blasi-like differentiable function as a sum of single-valued continuous function having right-hand side and left-hand side derivatives plus a multivalued constant. Thus, according to Theorem 3.7 in Section 3.5., a De Blasi-like differentiable multivalued function F with (5.66) takes the form (5.55) with F0 satisfying (5.56) if and only if (5.54) takes place jointly with Fi (αj ω) = Fi (0) − aij αj ,

for all i, j = 1, . . . , n,

(5.67)

where A = (aij )ni,j=1 and α = (α1 , . . . , αn ). Similarly, a De Blasi-like differentiable multivalued function P takes the form (5.63) with P0 satisfying (5.64) if and only if P is ω-D∗ -periodic, i.e. D∗ P (t) ≡ D∗ P (t + ω),

(5.68)

P (ω) = P (0) + (ωA + I)α.

(5.69)

jointly with Hence, we are ready to give the following three theorems, when applying Propositions 5.3 and 5.4. T HEOREM 5.12 Let σ(A) ∩ 2πiZ/ω = ∅ hold for a real (n × n)-matrix A. Assume, furthermore, that F0 : Rn ⊸ Rn is a u.s.c. multivalued function with nonempty, convex and compact values which is αω-periodic, i.e. (5.56), where α ∈ Rn , ω > 0. Let, at last, P : R ⊸ Rn be a De Blasi-like differentiable multivalued function with nonempty, convex and compact values which is ω-derivo-periodic, i.e. (5.68), satisfying (5.69), where I is a unit matrix. Then the inclusion (5.57) admits an ω-derivo-periodic solution in the sense of Definition 5.1. T HEOREM 5.13 Let F : Rn ⊸ Rn be a De Blasi-like differentiable multivalued function with nonempty, convex and compact values which is αω-derivo-periodic, i.e. (5.54), satisfying (5.66) and (5.67), where A is a real (n × n)-matrix such that σ(A) ∩ 2πiZ/ω = ∅, I is a unit matrix and α ∈ Rn , ω > 0. Then the inclusion x˙ ∈ F (x) + [P (t) + (tA + I)α]

(5.70)

admits an ω-derivo-periodic solution in the sense of Definition 5.1, provided an essentially bounded measurable multivalued function P : R ⊸ Rn with nonempty, convex and compact values is ω-periodic, i.e. (5.59). T HEOREM 5.14 Let F : Rn ⊸ Rn be a De Blasi-like continuously differentiable multivalued function with nonempty, convex and compact values which is αω-derivo-periodic, i.e. (5.54), satisfying (5.66) and (5.67), where A is a real (n × n)-matrix such that σ(A) ∩ 2πiZ/ω = ∅, I is a unit matrix and α ∈ Rn , ω > 0. Assume, furthermore, that P : R ⊸ Rn is a De Blasi-like differentiable multivalued function with nonempty, convex and compact values which is ω-derivo-periodic, i.e. (5.68), satisfying (5.69). Then the inclusion (5.53) admits an ω-derivo-periodic solution in the sense of Definition 5.1.

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Without applying Propositions 5.3 and 5.4, we can give immediately the following statement. C OROLLARY 5.7 Let A be a real (n × n)-matrix such that σ(A) ∩ 2πiZ/ω = ∅. Assume that F0 : Rn ⊸ Rn is a u.s.c. multivalued map with nonempty, convex and compact values satisfying (5.56) and P : R ⊸ Rn is an essentialy bounded measurable multivalued function with nonempty, convex and compact values which is ω-periodic, i.e. (5.59). Then the inclusion (5.60) admits an ω-derivo-periodic solution in the sense of Definition 5.1. R EMARK 5.26 All the above statements were obtained, via appropriate transformations, by means of the results for periodic solutions in Section 5.2.. It would be much more interesting, but also more delicate, to find sufficient conditions for the existence of ω-derivoperiodic solutions for inclusions of the form (5.57), where F0 satisfies (5.56) and P satisfies (5.59). Such criteria are known, for for pendulum-type equations with suitable instance, 0 −1 constants a, b, where n = 2, A = 0 a , F = (0, −b sin x1 )T , P = (0, p(t))T . For the related references, see e.g. Remarks and comments to [AG1, Chapter III.11].

6.

Concluding Remarks

• Sufficient conditions for the existence of periodic-type solutions of differential systems can be significantly improved for those of higher-order (scalar) differential equations and inclusions (for bounded, periodic and anti-periodic solutions, cf. [AG1, Chapter III.5]; for almost-periodic solutions, cf. [AG1, Chapter III.10], [ABR1], [ABR2]; for derivo-periodic solutions, cf. [A2]). • Periodic-type investigations have been extended to second-order vector differential equations and inclusions (for bounded, periodic and anti-periodic solutions, cf. [AKM]; for anti-periodic solutions, cf. [AAP2]; for quasi-periodic solutions, cf. [BC2]; for almostperiodic solutions, cf. [BMC]). • Appropriate generalizations have been formulated in abstract spaces (for boun-ded, periodic and anti-periodic solutions, cf. [A1], [AMT4], [AG1, Chapter III.5]; for periodic solutions, cf. [HM], [KOZ]; for anti-periodic solutions, cf. [AAP1], [AF], [AP]; for almostperiodic solutions, cf. [AG1, Chapter III.10], [NG], [Za]). • For implicit differential equations, the study of periodic solutions can be transformed to periodic problems for (explicit) differential inclusions (cf. [AG1, Chapter III.11], [FGK], [FK]). • Periodic solutions of random differential equations and inclusions can be investigated in a deterministic way (cf. [AG1, Chapter III.4], [A5]). • Recently, an enormous activity has been spent to the research of difference equations and equations on time scales. It is natural to consider in this frame periodic-type oscillations to (possibly multivalued) discrete models. Some results have been already obtained especially for periodic and almost-periodic solutions of difference equations (cf. e.g. the related papers published in Journal of Difference Equations and Applications). • Obviously, many problems of periodic-type remain open. Although not formulated explicitly, some of them could be easily recognized in our text.

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INDEX 1 1G, 121

4 4G, 121, 133

A Aβ, 8, 315, 316, 317, 321, 322, 323, 324, 325, 326, 327, 328, 329, 330, 331, 332, 333, 334, 335, 336, 337, 338, 340, 341, 342 academic, 91, 149, 153, 159, 180, 182, 191, 192, 193, 194, 201, 202, 204, 205, 208, 210, 211, 221, 222, 223, 224, 225, 226, 227, 229, 230, 233, 235, 244, 248, 250, 251, 256, 259, 266, 267, 268, 274 academic performance, 250 academic success, 202 access, 152, 160, 169, 188, 203, 224, 226 accidental, 124 accuracy, 18, 21, 29, 30, 39, 56, 78, 295, 307, 310 achievement, ix, 151, 153, 158, 182, 205, 221, 224, 225, 247, 248, 249, 250, 251, 252, 255, 265, 267 acoustic, 14 activation, 223 ad hoc, 56 adaptation, 94, 166, 175 adjustment, viii, 162 administration, 215, 250, 251, 255 administrative, 265 administrators, 179, 207, 212, 213, 215, 265 adult, 91 advertisement, 166 advertisements, 171 advertising, 169, 170, 171 aeronautical, 183 aesthetics, 90, 91 affective dimension, 95 Africa, 253

African American, 180, 181, 184, 189, 190, 192, 197, 198, 212, 222, 235 African Americans, 189, 190, 197, 198, 222 Ag, 125, 126, 128, 129, 133 age, 112, 152, 211, 252 agents, 98 aggregation, 163, 165 agricultural, 183, 213, 221, 229, 230 aid, 144, 273 AIDS, 219 Alaska, 184, 198, 216, 222, 245 Alaska Natives, 222 Alaskan Native, 189, 226 Albert Einstein, 217 algorithm, 68 aliens, 263, 264 alternative, 98, 147, 165, 267, 314, 317, 324 alternatives, 164, 170 American Association for the Advancement of Science, 215 American Competitiveness Initiative, 249 American Indian, 184, 189, 198, 245 American Indians, 189 amplitude, 25, 45, 61 AMS, 348 Amsterdam, 347, 349 analysts, 183, 212, 213 angular momentum, 121, 137 anomalous, 349 anthropology, 92 ants, 216 appendix, 179, 182, 185, 191, 192, 194, 198, 207, 211 application, vii, 1, 6, 56, 67, 75, 79, 93, 101, 117, 124, 143, 145, 146, 149, 153, 157, 159, 160, 165, 174, 223, 244, 264, 275, 326, 330, 335, 336, 337, 347 applied research, 224 appropriations, 270, 272, 273, 274 Arabia, 253 Arctic, 221 arithmetic, 126, 145 Arizona, 218 Armenia, 253

356

Index

arousal, 164, 170 Asia, 220, 261 Asian, 184, 192, 198, 235, 245 assessment, ix, 141, 142, 143, 201, 210, 221, 225, 247, 249, 250, 251, 252 assets, 162, 163, 170, 176 assignment, 147, 255 assumptions, vii, 89, 90, 94, 95, 103, 141, 163, 165, 174, 298, 313, 317, 318, 320, 321, 327, 328, 329, 331, 333, 336, 337, 340, 341, 342, 344 astronomy, 311 asymptotic, 278, 349 asymptotically, 316, 317, 318 Athens, 112 atmosphere, 56 atoms, 115, 116, 123, 137 attachment, 143, 209 attitudes, 95, 110, 111, 112 attractiveness, 163, 165, 171, 172, 173, 175 auditing, 180, 209 Australasia, 113 Australia, 109, 112, 114, 202, 253, 254 Australian Research Council, 98 Austria, 254 authenticity, 113 authority, 274 automation, 146 autonomy, 158 availability, 162, 165, 168, 169 awareness, 162, 165, 169, 170

B Bahrain, 253 Banach spaces, 292, 297, 298, 350 barriers, 110 basic research, 223 beginning teachers, 109 behavior, viii, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 174, 175 behaviours, 96 Belgium, 254 beliefs, vii, 89, 94, 95, 96, 98, 99, 103, 106, 110, 112, 113 benchmarks, 274 benefits, 153, 212 benzene, 122, 123 bias, 97 bifurcation, 143 bilingual, 200, 255 binomial distribution, 126 biotechnology, 184 bipartisan, 250 Bohr, 295, 305, 307, 309, 348 bonus, 269, 272 border security, 178 borrowers, 271 borrowing, 162

boson, 118, 119 bosons, viii, 115, 117, 118, 128 Botswana, 253 boundary conditions, 9, 12, 23, 30, 33, 50, 67, 68, 69, 80, 82 boundary value problem, 322, 326, 328, 331, 347, 348 bounded solution, 314, 316, 317, 318, 319, 321, 322, 323, 324, 325, 326, 327, 328, 339, 341, 342 bounds, 211, 230, 231, 233 boys, 152, 202 braids, 318 brain, 101 Brazil, 261 breakdown, 141 broad spectrum, vii Bulgaria, 253 Business Roundtable, 248, 268 buyer, 163

C calculus, 201 Canada, 1, 202, 254, 259 candidates, 222 CAP, 302, 303 capacity, 187, 189, 190, 216, 221, 223, 226, 244, 245, 262, 265 cast, 48 categorization, 182, 210 category a, 147 category d, 162 cation, 222 Cauchy problem, 322 CCR, 219 cell, 6 Census, 182, 211, 215 Census Bureau, 211, 215 Central America, 261 certificate, 191, 227, 230 certification, 182, 200, 255, 267, 269, 270, 271 chaos, 352 characteristic viscosity, 143 charge density, 20, 30 chemicals, 221 children, 105, 182, 200, 208, 221, 222, 272 Chile, 253 China, 202, 259, 261 chiral, 126 chirality, 126 Cincinnati, 217 citizens, 91, 186, 188, 208, 224, 225, 258, 263, 264, 266, 269, 273, 274 citizenship, 188, 208 civilian, 178, 180, 182, 184, 185, 186, 197, 208, 209, 216, 221, 261 classes, 116, 122, 123, 136, 152, 157, 200, 204, 267, 308, 309, 310

Index classical, 116, 295, 305, 319, 340 classification, viii, 115, 117, 149, 211, 212, 260, 301 classroom, vii, 89, 90, 93, 94, 96, 97, 98, 99, 101, 103, 105, 107, 108, 112, 149, 151, 155, 156, 157, 158, 159, 221, 224, 225, 267 classroom practice, 90 classroom teacher, 221, 267 classroom teachers, 221, 267 classrooms, 92, 96, 111, 113, 114, 200, 201, 204, 224 closure, 283, 306, 308, 325, 330, 335 Co, 139, 216, 246, 252, 259 codes, 6, 14, 29, 36, 47, 78, 212, 213 cognition, 95, 96, 101 cognitive, viii, 95, 96, 97, 105, 107, 110, 114, 145, 146, 147, 149, 150, 157, 158, 159, 160, 175, 222, 225 cognitive process, 150 cognitive tool, viii, 145, 149, 158, 160 coherence, viii, 89, 103, 224 cohort, 258 collaboration, 79, 181, 207, 265, 267 college students, 188, 201, 221, 262, 263 colleges, 179, 182, 184, 203, 205, 207, 209, 213, 214, 215, 222, 223, 224, 226, 244, 274 collisions, 5 Colorado, 244, 245, 246 colors, 126 Columbia, 175 Columbia University, 175 combinatorics, viii, 115, 117, 126 combined effect, 20 commercials, 169 commodity, 165 communication, 160, 166, 169, 183, 224 communities, ix, 113, 222, 247, 249 community, 93, 205, 223, 224, 258, 268 comparative advantage, 164 compassion, 107 competence, 105, 107, 110, 164 competency, 250 competition, 163, 172, 175, 202, 203 competitive advantage, ix, 165, 167, 168, 171, 172, 173, 176, 177, 178, 205 competitive markets, 161, 164, 174 competitiveness, 249, 270 complexity, 93, 97, 103, 278, 293 compliance, 121, 245 components, 101, 146, 165, 225 composition, 298 comprehension, 147 compulsory education, 252 computation, 65, 145, 146, 147, 148, 150, 332 computer science, 183, 191, 194, 197, 198, 199, 225, 229, 230, 236, 248, 257, 258, 270, 273, 278 computer systems, 212 computer technology, 184, 225 computing, viii, 145, 152, 159, 161 concrete, 147, 268

357

confidence, vii, 89, 90, 104, 105, 106, 107, 110, 179, 191, 192, 198, 211, 212, 213, 214, 227, 229, 230, 231, 232, 233, 234, 235, 236 confidence interval, 179, 191, 192, 198, 211, 212, 213, 214, 227, 229, 230, 231, 232, 233, 234, 235, 236 confidence intervals, 179, 191, 192, 198, 211, 213, 214, 227, 229, 230, 231, 232, 233, 234, 235, 236 configuration, 47, 117, 120, 122, 123, 147 confinement, 14 confusion, 278 Congress, ix, 178, 209, 247, 249, 250, 268, 269, 270 consensus, 95 conservation, 47, 213 consolidation, 153 constraints, 47, 162, 163, 174 construction, 94, 95, 97, 98, 110, 280, 281, 286 constructivist, 98 consumer choice, viii, 162, 163 consumers, viii, 162, 163, 165, 166, 168, 171, 174 consumption, 162, 163, 166, 168, 175 consumption function, 175 content analysis, 153 continuity, 299 control, 7, 56, 70, 146, 175, 213, 292 convergence, 32, 49, 67, 69, 174, 304, 305 conversion, 146, 147 convex, x, 167, 277, 278, 290, 291, 293, 297, 299, 302, 304, 315, 316, 318, 319, 320, 322, 323, 325, 332, 336, 338, 339, 344, 345, 346 coordination, ix, 159, 203, 215, 247, 249, 270 correlation, 75, 77, 117, 133, 135, 136, 137, 138 correlation coefficient, 77 correlations, 133, 137 costs, 180, 189, 201, 272 Coulomb, 41 Coulomb gauge, 41 coupling, 6, 27, 116, 134, 137 course work, 267 coverage, 164, 172, 173, 175 CPU, 65 CRC, 351, 353 credentials, 200 critical density, 25 CRM, 352 cross-sectional, 165 CRS, 247, 250, 251, 255, 258, 263, 271 crystals, 295 cues, 172 cultural practices, 101 culture, vii, 89, 90, 91, 94, 95, 96, 101, 103, 108, 110, 112, 113, 214, 226 curiosity, 96, 224 Current Population Survey (CPS), 178, 179, 182, 183, 184, 190, 198, 210, 211, 212, 213, 235, 236, 245, 246 curriculum, 92, 93, 94, 96, 100, 106, 108, 111, 146, 201, 204, 221, 222, 223, 224, 225, 226, 265, 266, 267, 273

358

Index

curriculum development, 224 customer preferences, 166 customers, 162, 163, 164, 165, 166, 167, 169, 170, 171, 173, 174, 175 cycles, 123 cyclotron, 14, 48 Cyprus, 253 Czech Republic, 254, 295

D data analysis, 99 data collection, 161 data communication, 212 data set, 210 database, 184, 210 debt, 271 decisions, ix, 90, 91, 95, 162, 164, 166, 170, 171, 176, 177, 179, 181, 199, 200, 201, 202, 203, 205, 207, 208, 214, 215 deduction, 316 deficit, 107 Deficit Reduction Act, 270 deficits, 206 definition, 32, 38, 83, 299, 302, 303, 305, 306, 307, 308, 309, 311, 312, 314, 332 deformation, 62 degenerate, 122, 123, 136, 298 demand, viii, 100, 112, 146, 162, 163, 166, 167, 168, 170, 174, 176, 268 demographic characteristics, 210 Denmark, 254 density, 6, 9, 14, 18, 19, 20, 29, 39, 45, 71, 73 dentistry, 222 Department of Agriculture, 180, 185, 216, 221 Department of Commerce, 185, 216, 237 Department of Defense (DOD), 179, 274 Department of Education, 98, 179, 185, 186, 209, 210, 215, 216, 221, 225, 244, 245, 250, 251, 253, 254, 255, 256, 257, 262, 267 Department of Energy (DOE), 139, 185, 217, 270 Department of Health and Human Services, 181, 185, 206, 218, 239 Department of Homeland Security, 178, 185, 203, 219, 244, 245 Department of State, 184, 245 Department of the Interior, 185, 219 Department of Transportation, 185, 221, 226 derivatives, 60, 61, 300, 311, 312, 313, 326, 343, 344, 345 derived demand, 174 designers, 113 desire, 102, 203 destruction, viii, 141, 142, 143, 144 destruction processes, 143, 144 destructive process, 141 deviation, 92 dichotomy, 321

differential equations, x, 296, 301, 304, 309, 332, 346, 347, 348, 349, 350, 351, 352 differentiation, 265 diffraction, 295 diffusion, 29, 32, 53, 55, 56, 61, 67, 68, 71, 75, 79, 225 diffusion process, 67 digital divide, 226 dimensionality, 6, 78 Dirichlet boundary conditions, 30, 50 Dirichlet condition, 300 disadvantaged students, 266 discipline, vii, 89, 90, 91, 92, 93, 95, 96, 101, 103, 104, 109, 110, 114, 265, 266 discourse, 94, 98 Discovery, 219, 220 dispersion, 29 displacement, 69 disposable income, 163, 165, 168 dissipative system, 318, 348 distribution, 5, 6, 7, 8, 9, 14, 15, 16, 17, 21, 23, 28, 29, 36, 37, 40, 41, 42, 46, 48, 75, 118, 120, 122, 124, 126, 144, 167, 172, 200, 256, 259, 263 distribution function, 5, 6, 7, 8, 9, 14, 16, 17, 21, 23, 28, 29, 40, 41, 42, 46, 48 divergence, 68, 93 diversity, 95, 149, 179, 214, 222, 224, 226, 264 division, 18 doctors, 212 domain-specificity, 104 dominance, 75, 95, 163, 164, 170 draft, 181, 188, 189, 206, 207, 208 drinking, 221 duality, 167 duopoly, 163, 175 duplication, 208 duration, 274 dynamical system, 295

E earth, 200 East Asia, 220 Eastern Europe, 261 ecology, 100 economic competitiveness, ix, 247 economics, 244 ecosystem, 221 Education, 89, 95, 98, 100, 102, 103, 109, 111, 112, 113, 114, 145, 159, 160, 161, 178, 179, 181, 182, 183, 184, 185, 186, 187, 189, 190, 191, 192, 193, 195, 196, 197, 199, 200, 201, 203, 204, 205, 207, 209, 210, 211, 213, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224, 225, 226, 227, 229, 230, 231, 233, 235, 237, 239, 241, 243, 244, 245, 248, 249, 250, 251, 253, 254, 255, 256, 257, 258, 259, 260, 261, 262, 263, 265, 266, 267, 268, 269, 270, 271, 272, 273, 274, 275

359

Index education reform, 108 educational attainment, 259 educational institutions, 214 educational process, 158 educational programs, ix, 177, 178, 180, 208, 209 educational quality, 188, 262 educational research, 113, 225 educational settings, 225 educational system, 92, 249, 268 educators, ix, 94, 96, 177, 179, 187, 207, 208, 225, 267 Egypt, 253 elasticity, 168 electric field, 8, 9, 13, 14, 15, 16, 18, 19, 20, 22, 24, 25, 28, 29, 36, 38, 39, 45, 46, 78, 79 electromagnetic, 5, 21, 22, 26, 27, 41, 42, 43, 78 electromagnetic fields, 42 electromagnetic wave, 21, 27, 41, 43, 78 electron, 7, 14, 15, 21, 26, 29, 37, 40, 43, 47, 48, 54, 116, 117, 120, 121, 311 electron beam, 311 electronic systems, 117 electrons, viii, 14, 15, 20, 21, 26, 29, 37, 40, 41, 43, 115, 116, 118, 120, 121, 122, 134, 137 Elementary and Secondary Education Act, 275 elementary school, 262, 263, 271 emotion, 95, 96, 101, 114 emotional, 96, 103, 107, 114 emotionality, 111 emotions, 95, 107, 164, 170 employees, ix, 177, 179, 180, 181, 190, 191, 197, 198, 199, 205, 206, 208, 209, 210, 211, 235, 272 employers, 184, 215, 223, 224 employment, 179, 181, 184, 190, 199, 205, 211, 223 empowered, 91, 158 encouragement, 202 enculturation, 94 endocrine, 221 energy, viii, 6, 12, 47, 52, 54, 67, 70, 75, 77, 115, 120, 124 energy transfer, 70 engagement, 101, 102, 103, 107, 109, 112, 224 English as a second language, 200 enrollment, 179, 181, 183, 191, 192, 208, 211, 214, 223, 259, 266 enterprise, 93, 226 enthusiasm, viii, 102, 110, 145, 158, 200 environment, viii, 156, 162, 164, 168, 170, 214, 222 environmental conditions, 147 Environmental Protection Agency, 178, 181, 185, 206, 217, 262 epistemological, 92 epistemology, vii, 89, 90, 96, 103 equality, 157, 298, 299, 301, 302 equilibrium, 6, 12, 32, 33, 48, 50, 62 equipment, 94, 156, 213, 272 ergonomics, 159 ESL, 255 ESR, 124

estimating, 169, 174 Estonia, 253 ethnic groups, 182, 189, 198, 222 ethnicity, 184, 190, 198, 214, 235, 266 Euclidean space, 298 Euler equations, 29, 30, 55 Eulerian, 3, 6, 12, 15, 29, 56, 57, 61, 67, 78, 79, 86 Europe, 261 evolution, 6, 7, 10, 11, 12, 13, 30, 50, 54, 56, 62, 73, 74, 75, 77, 78, 347 execution, 146 exercise, 157, 162 expenditures, 174 expertise, 106, 108, 109, 226, 264 exposure, 110, 201, 223 extrusion, 143, 144

F failure, 146, 147, 150 family, x, 211, 277, 279, 280, 281, 282, 290, 308, 326, 327, 331, 337, 338 FCL, 94 fear, 113 February, 114, 184, 203, 209, 248, 270 federal funds, 205, 272 federal government, viii, 177, 178, 182, 205, 275 feedback, viii, 99, 145, 155, 156, 158 feelings, 95, 105 fees, 269 fermions, viii, 115, 117, 118, 121 filament, 53, 78 finance, 161, 164 financial resources, 206 financial support, 187, 205, 262, 273 Finland, 140, 254 firms, 161, 162, 163, 164, 165, 166, 169, 170, 174 first-time, 193 fixation, 147 flexibility, 147, 149, 158, 159, 160 floating, 14, 15, 20, 37 flow, 6, 7, 30, 33, 47, 56, 61, 67, 101, 112, 292 fluid, vii, 1, 5, 55, 67, 78, 79 focus group, 99, 101, 102, 214 focus groups, 214 focusing, 90, 110, 147, 215, 224 food, 183, 212, 213, 221 foreign language, 93, 270 foreign nation, 222 foreign nationals, 222 Forestry, 183, 213 forgiveness, 271 Fourier, 10, 11, 12, 13, 31, 50, 55, 68, 300, 301, 302, 307, 349 Fourier analysis, 301 fractals, 278, 292 framing, 94, 102, 162, 176 France, 84, 254, 259, 261, 348

360

Index

fullerene, 124 function values, 84 funding, 185, 186, 187, 189, 205, 209, 216, 217, 218, 219, 220, 221, 226, 245, 273, 274, 275 funds, ix, 185, 186, 205, 222, 226, 247, 248, 261, 263, 266, 267, 271, 272, 273

G games, 175 gauge, 41 gender, 194, 214, 226, 227 gender gap, 226 gene, 278, 296, 343, 346 generalization, 91, 124, 301 generalizations, 278, 296, 343, 346 generation, 117, 119, 121 generators, 119, 122 geography, 244 geology, 91, 200 Georgia, 179, 213, 214, 215 Germany, 165, 254 girls, 152, 202, 226 globalization, 165 goals, 113, 180, 182, 185, 187, 189, 202, 209, 224, 225, 226, 245, 248, 262, 265, 266, 275 gold, 116 government, viii, 177, 178, 180, 182, 189, 205, 209, 224, 244, 274, 275 Government Accountability Office (GAO), ix, 177, 178, 180, 182, 184, 185, 186, 187, 188, 190, 191, 192, 193, 194, 195, 196, 198, 200, 202, 204, 206, 207, 208, 210, 212, 213, 214, 215, 216, 218, 220, 221, 222, 224, 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 238, 240, 242, 244, 245, 246, 247, 248, 249, 261, 262, 263 grades, 199, 200, 204, 215, 250, 251, 255 graduate education, 205, 208 graduate students, 187, 190, 215, 221, 222, 223, 224, 244, 262, 263, 264, 266, 267, 273, 274 grants, 221, 222, 263, 264, 265, 266, 267, 269, 270, 271, 272, 273, 274, 275 graph, 296, 302, 323 gravitational field, 57 gravity, 55, 56 Greece, 254 grounding, 104 groups, 98, 100, 117, 122, 123, 124, 135, 137, 152, 180, 181, 182, 185, 187, 188, 189, 198, 205, 207, 212, 222, 224, 225, 226, 249, 262 growth, 12, 33, 34, 50, 54, 70, 111, 114, 164, 175, 187, 193, 244, 248, 262, 325, 329, 333, 338 growth rate, 33 guidance, 152, 202, 266 guidelines, 95, 213, 236

H H1, 295, 319, 350 H2, 318, 350 Hamiltonian, 7, 22, 41, 48, 116, 134 hanging, 112 hardships, 203 harm, 96 harmonics, 318 harmony, 96 Harvard, 114, 175, 176 Hawaii, 216 health, 106, 182, 212, 221, 222, 224, 255 Health and Human Services (HHS), 178, 181, 185, 206, 207, 218, 239 health education, 255 health effects, 221 health services, 222 heart, 96, 103 heat, 47 hedonic, 163 height, 56, 57, 61, 62, 65 Heisenberg, 311 Heisenberg Uncertainty Principle, 311 Helmholtz equation, 55, 56 high school, 100, 181, 199, 200, 201, 204, 250, 255, 256, 262, 265, 267, 268, 269 high tech, 162, 163, 167, 174 high temperature, 47 higher education, 103, 178, 186, 214, 221, 222, 224, 226, 267, 271, 273 Higher Education Act (HEA), 270 Hilbert, 347, 348 Hilbert space, 347, 348 hips, 272 Hispanic, 180, 184, 189, 192, 197, 198, 212, 216, 217, 222, 235, 245 holistic, 147, 175 Homeland Security, 178, 181, 185, 203, 206, 209, 219, 223, 244, 245, 246 homeomorphic, 281 homogeneity, 92, 163, 165, 174 Hong Kong, 145, 146, 152, 253, 254 House, 111, 113, 270 household, 163, 170, 176, 212 households, 211, 259 human, 94, 97, 98, 159, 189, 221, 222, 224, 225 human capital, 224 human subjects, 222 Hungary, 253, 254 hydrodynamics, 29, 30, 55 hyperbolic, vii, 1, 3, 78 hypermedia, 160 hypertext, 160 hypothesis, 298, 339 hysteresis, 147

Index

I ice, 178, 180, 182, 184, 186, 188, 190, 192, 194, 196, 198, 200, 202, 204, 206, 208, 210, 212, 214, 216, 218, 220, 222, 224, 226, 228, 230, 232, 234, 236, 238, 240, 242, 244 icosahedral, 124, 136 id, 80, 179, 202, 204 identity, viii, 89, 95, 105, 107, 108, 110, 111, 112, 114, 116, 304 IEA, 251 Illinois, 179, 213 images, 293 immersion, 269 immigrants, 245 immigration, 268 Immigration and Customs Enforcement, 183 implementation, 157, 178, 222, 225, 226, 265, 270 in situ, 147 incentive, 166, 201 incentives, 205, 267 inclusion, 148, 149, 150, 151, 156, 158, 299, 304, 318, 320, 321, 322, 324, 325, 326, 327, 328, 329, 330, 331, 332, 333, 335, 336, 337, 338, 340, 341, 342, 343, 344, 345, 346 income, 162, 163, 165, 167, 168, 174, 211, 271, 272 income tax, 211 incomes, 174 incompressible, vii, 1, 5, 29, 30, 55, 67, 79 increasing returns, 163 indexing, 213 India, 161, 202, 259, 261 Indian, 185, 209, 218 Indian Health Service, 209, 218 Indiana, 179, 213 Indians, 218 indicators, 175 indices, 14, 21, 124 individual perception, 164 individual students, 155, 156, 222 Indonesia, 253, 254 induction, 97, 224, 272 industry, 189, 211, 225, 258 inequality, 324, 340, 341, 342 inertia, 47 infinite, 116, 307 Information System, 246 Information Technology (IT), 198, 220, 225, 258 infrastructure, 188, 262, 265 inherited, 281 innovation, viii, ix, 108, 109, 165, 166, 175, 177, 178, 247, 248, 249, 268, 270 insight, vii, 89, 90, 91, 96, 99, 109, 162 inspectors, 212 inspiration, 103 instabilities, 6, 29, 70 instability, 6, 8, 12, 33, 56, 62, 70

361

institutions, 178, 179, 180, 186, 187, 205, 210, 211, 214, 221, 222, 223, 224, 225, 226, 244, 256, 262, 263, 264, 266, 267, 269, 273, 274, 275 institutions of higher education, 186, 221, 222, 267, 272, 273, 275 instruction, 92, 93, 94, 95, 102, 113, 149, 155, 159, 160, 200, 204, 224, 225, 265, 267, 269, 272, 274 instruments, 153 insurance, 212 intangible, 98, 163, 165, 166 Integrated Postsecondary Education Data, 178, 179 integration, 6, 9, 29, 48, 50, 55, 61, 65, 94, 111, 142, 155, 157, 158, 159, 161, 164, 202, 270 intellect, 98, 107 intelligence, 96 intensity, 25, 141, 143, 144, 173 intentions, 165 interaction, 6, 21, 29, 40, 47, 78, 79, 97, 100, 124, 156, 202, 224 interactions, 5, 98, 108, 155, 156, 157, 159, 224, 266 Interactive Perimeter Learning Tool, viii, 145, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159 interdependence, viii, 162 interdisciplinary, 205, 224, 266 interface, 106, 150, 151 International Baccalaureate, 268, 272 international students, 178, 180, 181, 182, 190, 192, 193, 194, 202, 203, 205, 206, 207, 208, 213 internship, 223, 265, 270 interpersonal communication, 169 interpretation, 95 interstitial, 4, 32, 84 interval, 198, 211, 212, 231, 235, 304, 305, 307, 311, 312, 313, 321, 323, 324, 325, 331 interview, 99, 105, 153 interviews, 98, 99, 101, 153, 155, 156, 158, 201, 207, 214, 215 intrinsic, viii, 162, 224 intuition, 96 invariants, 48, 55, 57, 60 inversion, 80, 126 investment, 172, 173, 223 ions, vii, viii, 7, 14, 15, 21, 26, 29, 38, 40, 41, 43, 67, 78, 79, 89, 101, 114, 181, 206, 207, 211, 225, 338 IPEDS, 178, 179, 182, 183, 190, 193, 195, 196, 210, 211, 230 Iran, 253 Ireland, 254 Islamic, 253 island, 50 isolation, 202 Israel, 253, 259, 350 Italy, 253, 254 iteration, 31, 32, 55, 56, 59 iterative solution, 37

362

Index

J January, 112, 113, 114, 189, 243, 245, 246, 248, 261 Japan, 85, 253, 254, 261 jobs, 202, 267 Jordan, 253 Josephson junction, 348 judge, 147, 164, 170 judgment, 164, 170 junior high, 188 junior high school, 188 justification, 151, 153

K K-12, 209, 220, 222, 223, 224, 262, 263, 264, 265, 266, 269, 272 Kelvin-Helmholtz instability, 33 kindergarten, 181, 182, 187, 199, 200, 204, 215, 222, 224, 225 kinetic energy, 54, 67 kinetic equations, 78 Korea, 253, 254

L L1, 321, 325, 329, 330, 332, 335, 338 labor, 178, 182, 197, 259 labor force, 178, 182, 197, 259 lack of confidence, 107 Lagrangian, 48, 55, 56 language, vii, 89, 90, 93, 98, 161, 182, 244, 255, 257 laser, 25, 26, 27, 43, 311 Latino, 184, 198, 212, 245 lattice, 295 Latvia, 253, 254 law, 27, 44, 184, 201, 256, 269 laws, 97 lead, 2, 33, 105, 166, 171, 204, 221, 226, 261, 272, 305 leadership, 94, 113, 189, 205, 265, 272 learners, viii, 90, 93, 104, 145, 150 learning, vii, viii, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 101, 103, 104, 105, 106, 107, 108, 109, 110, 111, 113, 114, 145, 146, 147, 149, 150, 151, 152, 153, 155, 156, 157, 158, 159, 160, 163, 164, 166, 170, 180, 221, 224, 225, 226, 265, 269, 270 learning culture, 265 learning environment, 149, 158 learning process, 158, 159, 164 learning task, 152 Lebanon, 253 legislation, ix, 247, 249, 268, 270 legislative, ix, 248, 249, 269 legislative proposals, ix, 248, 249 leisure, 168

lens, 95, 108 lesson plan, 104 life cycle, 165, 166 life span, 162 lifelong learning, 91 lifestyle, 91 lifetime, 94, 162, 165, 175 Likert scale, 153 limitation, 147, 215 limitations, 107, 214 Lincoln, 98 linear, 8, 21, 27, 32, 40, 44, 50, 56, 58, 78, 81, 94, 145, 148, 159, 295, 299, 313, 314, 315, 321, 322, 324 linear function, 314 linkage, 98 links, 223 literacy, ix, 106, 247, 248, 252 Lithuania, 253 loans, 271 localization, 325, 329, 334 location, 158, 273 logistics, 156 London, 84, 111, 112, 114, 161 long-term, viii, 162, 225 Los Angeles, 179, 213, 215 losses, 163 love, 101 low-income, 204, 221, 270 loyalty, 174 Luxembourg, 254

M Macao, 254 Macedonia, 253 macromolecular chains, 142 macromolecules, 141 magnetic, 6, 14, 15, 19, 20, 22, 29, 30, 36, 37, 38, 47, 48, 50, 54, 55, 67, 68, 70, 73, 74, 75, 77, 78, 79 magnetic field, 6, 14, 19, 22, 29, 30, 36, 37, 38, 47, 67, 70 magnetosphere, 47 Malaysia, 253 management, 183, 224 manifold, 176 manifolds, 176 manufacturing, 161, 164 mapping, 163, 212, 213, 282, 283, 298, 299, 312, 322, 323, 325, 328, 330, 332, 334, 335, 339 market, viii, 162, 163, 164, 165, 166, 169, 171, 172, 173, 174, 175, 176, 203 market penetration, 172 market segment, 169, 171 market value, 163, 165 marketing, 162, 164, 165, 166, 170, 171 markets, 165, 172, 176

363

Index Markov, 351 Maryland, 265 masculine image, 94 Massachusetts, 292 Massera type, 316 mastery, 250 mathematicians, 95, 209, 214, 267 mathematics, vii, viii, ix, 89, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 109, 110, 111, 112, 113, 114, 145, 146, 149, 151, 152, 158, 177, 178, 181, 182, 183, 190, 193, 194, 197, 199, 200, 201, 202, 204, 205, 208, 214, 216, 221, 223, 224, 225, 226, 235, 244, 247, 248, 258, 264, 265, 266, 267, 269, 270, 271, 272, 273, 274, 275 mathematics education, 91, 94, 112, 113, 248, 271, 272 Matrices, 139, 140 matrix, 9, 23, 31, 80, 82, 175, 314, 316, 317, 321, 325, 326, 330, 331, 336, 338, 343, 345 measurement, 146, 159 measures, 153, 154, 155, 163, 165, 191, 193, 203, 212, 248, 259, 292 media, 109, 222 median, 199, 212 medicine, 222 Mediterranean, 112 melt, 143, 144 melts, viii, 141, 144 memory, 65 men, 191, 194, 195, 197, 202, 212, 215, 229, 230, 236 mental health, 222 mentor, 202, 222, 266 mentoring, 181, 199, 202, 205, 223, 224, 226, 265, 272 merchandise, 164, 170 Merck, 269 merit-based, 269, 271, 272, 275 metaphors, 103 methodological procedures, 97 metric, ix, x, 174, 277, 278, 279, 280, 281, 282, 283, 284, 285, 286, 287, 288, 289, 290, 291, 292, 293, 294, 296, 297, 299, 302, 308, 309, 310, 312 metric spaces, 278, 280, 286, 287, 292, 293, 294, 296 Mexico, 161, 171, 254, 261 Mexico City, 161 MHD, vii, 1, 47, 67, 68, 75, 79 Middle East, 261 middle schools, 267, 270 mining, 212 Ministry of Education, 93, 277 minorities, 178, 180, 181, 182, 190, 197, 198, 199, 202, 204, 205, 208, 222, 223, 224, 226, 235, 245, 266 minority, 179, 180, 188, 190, 192, 194, 205, 212, 222, 223, 225, 226, 266 minority groups, 212, 225

minority students, 180, 190, 192, 222, 226 mirror, 91, 97 misleading, 206, 207 MIT, 292 ML, 121 modeling, 55, 56, 265 models, 47, 78, 93, 114, 163, 165, 170, 174, 202, 224, 226, 278, 281, 292, 293, 295, 346 Moldova, 253 molecular mass, 141, 142, 144 molecular weight, 142 molecules, viii, 115, 116, 117, 123, 124, 137 momentum, 21, 22, 40, 41, 48, 121, 137 monodromy, 315 monograph, 301 monoids, 286, 288, 294 monotone, 348 Montenegro, 254 Morocco, 253 Moscow, 144, 350, 351, 352 motion, 7, 17, 30, 55, 311 motivation, 92 motives, 164, 170, 175 motors, 351 mouth, 169 movement, 92, 94, 146, 147, 148, 152, 156 MSC, 277 MSP, 224, 265, 267 multidimensional, 55 multidisciplinary, 224, 226, 274 multiplication, 116, 167, 172, 290 multiplicity, 127, 317, 331, 332 multiplier, 315, 316, 329, 330, 331, 333, 335, 336, 337, 338 music, 255 myopic, 166

N nation, ix, 177, 178, 182, 190, 205, 209, 211, 221, 224, 225, 247, 248, 249, 255, 262 national, 180, 182, 190, 203, 205, 221, 224, 225, 255, 259, 269, 270, 272, 273, 274, 275 National Academy of Sciences (NAS), 248, 249, 268, 269, 274, 275 National Aeronautics and Space Administration, 178, 185, 186, 207, 209, 219, 223, 224, 262, 266 National Assessment of Educational Progress, 201, 250 National Center for Education Statistics (NCES), 178, 179, 182, 183, 210, 211, 215, 243, 244, 245, 250, 251, 252, 253, 254, 255, 256, 257, 259 National Defense Authorization Act, 269 National Institute of Standards and Technology, 219 National Institutes of Health, ix, 178, 180, 185, 186, 209, 218, 219, 247, 248, 261, 262, 263

364

Index

National Postsecondary Student Aid Study (NPSAS), 178, 179, 182, 183, 190, 191, 192, 208, 210, 211, 227, 228, 229, 230, 231, 232, 233, 234, 235 National Science and Technology Council, 178, 182, 189, 249 National Science Foundation, ix, 139, 177, 178, 180, 181, 185, 218, 220, 224, 240, 243, 245, 247, 248, 259, 260, 261, 264, 269 national security, 203, 270 Native American, 189, 194, 222, 235, 265 Native Americans, 189, 222 Native Hawaiian, 216, 226 natural, ix, 97, 104, 116, 141, 247, 248, 278, 281, 283, 303, 306, 309, 310, 311, 313, 318, 346 natural science, ix, 97, 247, 248 natural sciences, 97 negativity, 163, 165 negotiation, vii, 89 Netherlands, 253, 254 network, 97, 163 New York, 84, 87, 112, 113, 114, 139, 140, 175, 243, 245, 292, 293, 348, 349, 350, 351 New Zealand, 202, 253, 254 Newton, 4, 311 Nielsen, 318, 331, 332, 347, 350 NMR, 123, 124, 129 No Child Left Behind, 178, 182, 250 noise, 6, 29, 45, 57, 79 non-citizens, 188 nonlinear, viii, 5, 6, 7, 12, 50, 56, 70, 162, 295, 317, 318, 319, 347, 348, 351 nonlinear dynamics, 70 non-linear equations, viii, 162 nonlinear systems, 319, 348 non-Newtonian, 144 normal, 14, 22, 42, 104, 116, 134, 144, 152, 305, 306, 309, 310 normalization, 36, 40 normed linear space, 293 norms, 303, 307 Norway, 253, 254, 259 novelty, 166 nuclear, viii, 115, 117, 118, 121, 123, 124, 127, 128, 129, 132, 133, 184 nuclei, 115, 123, 128, 129 nucleus, 118, 124 nurses, 212, 222 nursing, 222

O obedience, 103 observations, 34, 56, 99, 200, 206, 207, 267, 282 occupational, 212 OECD, 252, 254, 259 Office of Management and Budget, 189 Office of Science and Technology Policy, 189, 249, 275

oligopoly, 166 Oncology, 219 one dimension, 3, 6, 21, 32, 78 online, 271 operator, 6, 116, 167, 172, 305, 322 ophthalmic, 183, 212 optical, 347 optical systems, 347 optimization, viii, 162, 167, 168, 174, 176 oral, 266 oral presentations, 266 orbit, 123, 124, 311 ordinary differential equations, 347 organ, 113 organic, 169 organization, 94, 166, 183, 203, 204 organizations, 164, 180, 203, 204, 205, 207, 209, 211, 215, 222, 225, 248, 263, 268, 271, 272, 274 orientation, 118, 174 oscillation, 34, 311 oscillations, x, 18, 39, 55, 295, 296, 319, 346, 347, 348, 349, 350 outpatient, 222 outreach programs, 225, 226, 269

P Pacific, 189, 192, 198, 220, 222, 235, 245, 349, 350 Pacific Islander, 192, 198, 235, 245 Pacific Islanders, 198, 245 paper, viii, 61, 107, 111, 112, 113, 142, 153, 154, 161, 162, 163, 165, 166, 174, 175, 246, 278, 304 parabolic, 1 parallel computers, 6, 78 parameter, 8, 57, 169 parents, 202 Paris, 259, 349 Parliament, 112 partial differential equations, vii, 1 particles, viii, 7, 9, 17, 41, 47, 115, 117, 118, 119, 120, 121, 129, 311 particulate matter, 221 partition, 118, 119, 126 partnership, 113, 265, 266, 267, 271, 272 partnerships, 214, 222, 223, 224, 225, 267, 272 pathophysiology, 222 pathways, 223, 224, 266 PDAs, 152, 155 pedagogical, vii, viii, 89, 90, 103, 104, 105, 106, 110, 111, 145, 146, 157, 158, 159, 226 pedagogies, 90, 95 pedagogy, vii, 89, 91, 92, 94, 95, 96, 97, 98, 101, 103, 108, 112, 114, 214 pediatric, 222 peer, 265 peers, 202 Pell Grants, 244, 270 pendulum, 351

Index Pennsylvania, 179, 213, 269 pensions, 212 perception, 153, 157, 164, 170 perceptions, 97, 111, 112, 161, 163, 165 performance, 5, 12, 13, 29, 36, 40, 56, 78, 146, 163, 164, 165, 167, 172, 203, 224, 226, 248, 250, 265, 267, 274, 275, 311 performers, 225 periodic, 6, 8, 9, 31, 32, 50, 55, 62, 68, 117, 120, 121, 122, 124, 129, 133, 136, 137, 295, 296, 298, 299, 300, 301, 303, 304, 307, 310, 311, 313, 314, 316, 318, 321, 325, 326, 329, 330, 339, 340, 344, 346, 348, 349, 350, 351, 352 Periodic Table, viii, 115, 117 periodicity, viii, 67, 69, 115, 116, 117, 118, 121, 122, 123, 124, 126, 129, 133, 134, 135, 136, 137, 295, 304, 349, 351 permanent resident, 188, 208, 225, 245, 263, 264, 266 personal, 90, 94, 95, 96, 99, 104, 105, 108, 109, 152 personality, 96, 166 personality traits, 166 persons with disabilities, 189, 224 persuasion, 172 perturbation, 12, 33, 50, 54 perturbations, 343 Petroleum, 212, 213 phase space, 6, 12 Philippines, 253 philosophical, 97 philosophy, 104 physical sciences, 256 physicists, 183, 212 physics, vii, 1, 5, 6, 13, 47, 79, 91, 92, 100, 104, 105, 106, 109, 200, 201, 204, 311 physiology, 222 pilot programs, 274 PISA, 248, 252, 254 planar, 317, 318, 332 planning, 93, 104 plasma, vii, 1, 5, 6, 7, 12, 13, 14, 18, 19, 20, 21, 24, 25, 26, 27, 29, 30, 36, 37, 38, 39, 43, 45, 47, 48, 54, 67, 78, 79 plasma physics, 1, 5, 79 play, 48, 67, 96, 117, 161, 165, 175, 189, 202 pleasure, 96 Poisson, 6, 7, 8, 9, 15, 16, 22, 30, 31, 36, 42, 48 Poisson equation, 7, 8, 9, 15, 16, 22, 30, 31, 36, 42 Poland, 254 polarization, 21, 40, 44, 47, 78 policy makers, 206 pollution, 221 polygons, 147 polymer, 141, 142, 143, 144 polymer destruction, 141 polymer melts, 141 polynomial, 60, 61, 81, 82, 83, 117, 123, 124, 306, 308 polynomials, 61, 119, 124, 306, 308, 310

365

polypropylene, viii, 141, 143, 144 poor, 93, 109, 201, 248 population, 182, 184, 202, 209, 211, 212, 226, 231, 235, 259, 269 population group, 226 portfolios, 176 Portugal, 254 positivism, 97 positivist, 97 postsecondary education, 210, 211, 221, 248, 259 poststructuralism, 114 power, 97, 121, 125, 162, 163 powers, 91, 117, 125 Prandtl, 68 prediction, 55 preference, 151, 153, 157, 158, 163, 169, 274 pressure, 14, 18, 38, 39, 68 prevention, 221 price competition, 163 prices, 165, 167, 168 pricing policies, 163 primary care, 222 primary school, 152 primitives, 313 prior knowledge, 149, 151 priorities, 245 private, 99, 107, 179, 214, 271, 274 proactive, 203 probability, 166, 168, 211, 212, 230, 235 probe, 124 problem solving, 94 problem-solving, 146, 147, 149, 267 problem-solving strategies, 149 procedural knowledge, 146, 148, 150, 151 product attributes, 163, 164, 165, 170 product design, 161, 164 product market, 173 product performance, 167 production, 226, 259 productivity, 225 professional careers, 223 professional development, 100, 107, 110, 114, 224, 225, 265, 267, 272 profit, x, 107, 163, 165, 174, 175, 296 profit margin, 163 profitability, 164, 175 program, 180, 181, 184, 186, 187, 188, 189, 198, 203, 205, 206, 207, 208, 209, 211, 221, 222, 223, 224, 225, 226, 244, 245, 259, 261, 262, 263, 264, 266, 267, 269, 270, 271, 272, 273, 274 programming, 198, 278, 292 programming languages, 292 promote, 159, 181, 187, 205, 224, 244, 262, 267, 270 promote innovation, 270 propagation, 2, 13 property, 7, 304, 307, 341, 342 proportionality, 143 proposition, 297, 298, 322 prosperity, 224

366

Index

protons, 115, 129 pruning, 172 pseudo, 307, 308 psychology, 95, 112, 176, 191 public, 107, 179, 214, 215, 222, 224, 250, 253, 255, 271, 275 public health, 222 public interest, 224 public schools, 215 publishers, 92 pupils, ix, 90, 102, 247, 248, 249, 252

Q qualifications, 109, 215 qualitative differences, 214 qualitative research, 97, 98 quality control, 215 quality of life, 224 quantum, 116, 117, 120, 124, 127, 129, 311, 347 quantum mechanics, 116 quasi-linear, 343 quasiparticles, 12 quasi-periodic, 296, 301, 303, 304, 305, 307, 314, 317, 346, 349, 352 questionnaire, 153, 157, 209, 210, 215 questionnaires, 161, 209, 210

R race, 190, 191, 211, 214 radius, 37, 48, 62 random, 160, 163, 203, 212, 214, 235, 346 random access, 160 range, 62, 149, 158, 187, 206, 207, 225, 264, 266, 310 Rayleigh, 33 reading, 182, 225, 244 real numbers, 278, 305 realism, 98 reality, 97, 98 reasoning, 159 recall, 99, 149, 278, 280, 281, 282, 283, 288, 296, 305, 315 recalling, 280, 338 recognition, 97 recruiting, 222, 226, 268 recursion, 83 reduction, 3, 147 referees, 161 reflection, 99, 104 reflexivity, 278 reforms, 270 regional, 56, 183, 212, 269, 273 registered nurses, 212, 222 regular, viii, 124, 145, 146, 147, 148, 150, 151, 156, 158, 162, 311, 314, 321, 325

reinforcement, 146, 166, 175 reinforcement learning, 166, 175 rejection, 97, 202 relationship, viii, 90, 91, 94, 96, 101, 108, 115, 171, 174, 183, 280, 283, 304 relationships, vii, viii, 89, 93, 103, 115 relativity, 116 reliability, 179, 210 repetitions, 295 representative samples, 215 research, vii, viii, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 100, 103, 104, 107, 110, 111, 112, 113, 114, 139, 143, 145, 149, 151, 158, 159, 160, 165, 174, 180, 181, 182, 183, 186, 187, 188, 189, 190, 204, 205, 207, 212, 221, 222, 223, 224, 225, 226, 244, 249, 255, 262, 263, 265, 266, 268, 269, 273, 274, 275, 346 research and development, 175, 182, 224, 274 researchers, 95, 113, 143, 190, 199, 200, 201, 212, 213, 214, 222, 244, 264, 274 resolution, 6, 7, 56 resources, 104, 105, 107, 149, 158, 162, 181, 189, 190, 204, 205, 206, 209, 222, 224, 225 response time, 165 responsibilities, 272 retail, viii, 162, 163, 164, 166, 168, 170, 171, 176 retention, 163, 165, 202, 214, 221, 222, 224, 226, 265 returns, 164, 211 revenue, 175 Reynolds, 67 Reynolds number, 67 Rhode Island, 350 risk, 176, 221 risk assessment, 221 robotics, 184 robustness, 174 Romania, 253 rotations, 116, 347 routines, 92 rural, 269, 271 rural areas, 269, 271 Russia, 261 Russian, 253, 254, 259, 294, 349, 350, 351, 352

S safety, 212 salaries, 179, 190, 199, 212, 235 salary, 212 sales, 92, 164, 165, 168, 169, 171 sample, 153, 154, 179, 203, 208, 210, 211, 212, 215, 230, 231, 235, 255 sample design, 211, 215 sampling, 179, 209, 213, 235 sampling error, 179, 209, 213 SAR, 253, 254

Index satisfaction, 162, 163, 164, 165, 166, 167, 170, 171, 173, 175, 202 saturation, 12, 34, 54 Saturday, 218 Saudi Arabia, 253 scalar, 317, 346 scattering, 47 scholarship, 103, 113, 225, 269, 273 Scholarship Program, 216, 217, 218, 220, 273, 274 scholarships, 186, 188, 205, 208, 222, 223, 225, 226, 267, 269, 271, 272, 273 school, vii, ix, 89, 90, 91, 92, 93, 95, 98, 99, 100, 103, 108, 109, 111, 112, 113, 114, 152, 158, 181, 182, 184, 188, 190, 199, 200, 201, 203, 204, 205, 215, 221, 222, 223, 224, 225, 244, 247, 248, 250, 255, 256, 262, 263, 265, 266, 267, 268, 269, 270, 271, 272, 273 Schools and Staffing Survey (SASS), 215, 255 science education, 91, 92, 98, 103, 108, 109, 111, 112, 113, 114, 190, 204, 205, 264, 270 science educators, 222 science literacy, ix, 247, 248, 252 science teaching, 95, 107, 113, 272 scientific community, 93 scientific knowledge, 93 scientific method, 92 scientific theory, 91, 97 scientists, 91, 92, 95, 96, 183, 188, 189, 206, 207, 208, 209, 212, 213, 214, 221, 222, 223, 267, 270, 274 scores, 251, 252 search, 171 second language, 200 secondary school students, ix, 247, 270 secondary schools, vii, 89, 103, 112, 113, 114, 224, 269, 270, 271 secondary students, 250, 262 secondary teachers, 187, 200 security, 178, 203, 223, 224, 226, 270 selecting, 108, 143, 181, 225 self-concept, 95 self-efficacy, 110 semantics, 278, 279, 292 semigroup, 288 Senate, 270 sensitivity, 162, 163 separation, 13, 15, 20, 78 September 11, 178, 202, 203 Serbia, 253, 254 series, 92, 211, 296, 300, 301, 302, 307 service quality, 164, 170 services, viii, 161, 162, 163, 164, 170, 171, 174, 175, 176, 222, 226 sex, 211 shape, viii, 94, 105, 118, 119, 120, 145, 146, 147, 148, 150, 151, 152, 154, 155, 156, 158 shaping, 95 shareholder value, 165 shares, 259

367

sharing, 94, 97, 102, 203 shear, 19, 33, 38 Shell, 121 shocks, 2, 173 short run, 166 shortage, 109 short-term, 222, 263 sign, 40, 116, 135, 338 signs, 25 simulation, 24, 29, 71 Singapore, 253 SIS, 109 skills, 93, 94, 109, 110, 146, 147, 200, 221, 224, 249, 250, 266, 267, 268, 269, 272 skin, 48 Slovenia, 253 smoothing, 70, 162 social construct, 111 social phenomena, 97 social sciences, 191 socioeconomic, 214 sociology, 92, 111 software, 149, 183, 212 solar, 47 solar wind, 47 solutions, x, 33, 49, 93, 161, 164, 175, 295, 296, 301, 304, 305, 309, 314, 315, 316, 317, 318, 319, 323, 324, 325, 327, 328, 329, 332, 334, 336, 340, 341, 342, 343, 346, 347, 348, 349, 350, 351, 352 South Africa, 253 South America, 261 South Dakota, 265 South Korea, 261 Soviet Union, 205 space exploration, 224 Spain, 254, 261, 277 spatial, 2, 6, 9, 40, 41, 50, 147, 164 special education, 274 special theory of relativity, 115 specialisation, vii, 89, 90 specialization, 256 species, 48, 117, 124, 128, 129 spectroscopy, viii, 115, 116, 117, 121, 124 spectrum, vii, 25, 26, 27, 43, 44, 45, 125, 303, 307, 314, 322 speed, 2, 116, 166 speed of light, 116 spin, viii, 115, 117, 118, 119, 120, 121, 122, 123, 124, 125, 127, 128, 129, 132, 133, 134, 136, 137, 138 spin-1, 118, 119 sponsor, 222 Sputnik, 248, 268 SRS, 26, 43 stability, 5, 29, 32, 56 stabilize, 174 stages, 146, 151, 275 stakeholder, 215 stakeholders, 225

368

Index

standard deviation, 153, 154, 155, 252 standards, 93, 103, 111, 113, 180, 182, 209, 215, 225, 267 State Department, 203 statistics, 100, 183, 255, 257 steady state, 18, 38 STEM, ix, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 221, 223, 224, 225, 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 244, 245, 247, 248, 249, 251, 253, 255, 256, 257, 258, 259, 261, 262, 263, 264, 265, 266, 267, 268, 269, 270, 271, 272, 273, 275 STEM fields, ix, 177, 178, 179, 180, 181, 182, 183, 184, 187, 188, 190, 191, 192, 193, 194, 195, 197, 198, 199, 200, 201, 202, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 227, 233, 236, 244, 248, 256, 257, 258, 259, 261, 262, 268, 273 STM, 243, 244 strain, 141, 143 strains, 143 strategic, 163, 173, 175, 190 strategies, 105, 107, 111, 147, 149, 150, 159, 161, 162, 164, 169, 170, 171, 173, 174, 175, 182, 189, 224, 225, 226, 265 stress, 29, 38, 42, 53 student achievement, 112, 224, 256, 264, 265 Student and Exchange Visitor Information System, 178, 182, 184, 203, 243 student enrollment, 179, 193, 205, 210 student populations, 224, 225 student retention, 214 student teacher, 265 students, viii, ix, 90, 91, 92, 93, 94, 95, 96, 99, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 177, 178, 179, 180, 181, 182, 184, 187, 188, 190, 191, 192, 193, 194, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 213, 214, 215, 221, 222, 223, 224, 225, 226, 227, 229, 230, 235, 244, 247, 248, 249, 250, 251, 252, 256, 258, 261, 262, 263, 264, 265, 266, 267, 268, 269, 270, 271, 273, 274 subjective, 96, 97, 98, 169 substances, 144, 295 substitution, 80, 125, 142 subtraction, 160 suffering, 102 summer, 223, 224, 266, 267, 269, 270 superposition, 314 supervision, 222 supplemental, 225 supply, 161, 163, 164, 214, 248, 249, 268 supporting institutions, 224 surgery, 56 Sweden, 253, 254 switching, 122

Switzerland, 254 symbols, 98, 117, 121, 124, 279 symmetry, viii, 52, 53, 96, 115, 116, 122, 123, 124, 133, 135, 137, 163, 165, 295, 328 symplectic, 13 synchronous, 351 syndrome, 162 systems, 6, 7, 12, 21, 39, 52, 98, 113, 120, 124, 134, 137, 161, 164, 178, 182, 183, 198, 212, 213, 225, 292, 299, 318, 346, 347, 348, 349, 350, 351

T talent, 270 tangible, 163, 165 target population, 209 targets, 268 taste, 202 Taylor expansion, 33, 42, 59, 79 teacher effectiveness, 204 teacher instruction, 94 teacher preparation, 271 teacher training, 272 teachers, vii, viii, ix, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 107, 108, 109, 110, 111, 112, 113, 114, 145, 151, 152, 153, 155, 156, 157, 158, 182, 188, 199, 200, 202, 204, 215, 222, 224, 225, 247, 248, 249, 255, 256, 263, 265, 266, 267, 268, 269, 270, 271, 272, 274 teaching, vii, viii, 89, 90, 91, 92, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 145, 146, 149, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 182, 201, 207, 221, 222, 224, 225, 226, 248, 255, 265, 266, 267, 270, 272 teaching process, 111, 153 teaching strategies, 225 technician, 224 technicians, 183, 212, 213, 224 technology, ix, 113, 159, 160, 163, 166, 167, 171, 174, 175, 177, 178, 182, 184, 191, 193, 197, 203, 205, 208, 216, 221, 225, 226, 235, 247, 248, 267, 270, 271, 272, 273 technology gap, 175 technology transfer, 184, 203, 226 temperature, 14, 15, 47, 143 temporal, 164 tertiary education, 259 testimony, 249 textbooks, 92, 93 Thailand, 254 theology, 257 theory, vii, viii, x, 1, 5, 92, 93, 94, 96, 97, 98, 115, 117, 122, 160, 162, 163, 164, 167, 173, 176, 278, 292, 293, 295, 296, 301, 303, 305, 315, 316, 318, 331, 347 therapeutic interventions, 222 thermo-mechanical, 142, 143, 144

369

Index thermonuclear, viii, 141, 144 thinking, 90, 92, 93, 104, 105, 111, 225 threat, 106 threatening, 106 TIE, 265 time, 2, 3, 4, 6, 7, 8, 9, 14, 16, 18, 21, 22, 23, 29, 36, 39, 42, 47, 48, 49, 50, 53, 54, 55, 56, 57, 58, 59, 61, 65, 68, 69, 73, 74, 75, 78, 79, 90, 94, 99, 101, 104, 107, 141, 150, 152, 158, 159, 162, 163, 164, 165, 167, 168, 169, 171, 172, 173, 175, 184, 189, 193, 201, 203, 204, 209, 210, 215, 223, 255, 264, 273, 295, 314, 315, 317, 319, 332, 340, 341, 346 tin, 173 tissue, 222 Title I-A, 250 Title IX of the Education Amendments of 1972, 245 tokamak, 13, 47 topological, ix, 47, 277, 278, 281, 286, 292, 293, 299, 317, 349 topological structures, 281 topology, 47, 67, 279, 281, 282, 291, 292, 293 total energy, 54, 74 tourism, 100 tracking, 92 trade, 94 tradition, vii, 89, 91, 93 traffic, 175 trainees, 222, 263, 264 training, viii, 89, 100, 102, 103, 109, 110, 149, 159, 180, 186, 200, 202, 204, 205, 222, 223, 224, 263, 264, 272 training programs, 180, 186, 222 trajectory, 9, 110 trans, 305, 306 transfer, 70, 146, 184, 226 transformation, viii, 89, 299, 305, 316, 318, 321 transformations, 142, 144, 321, 346 transition, 20, 167 transitions, 124 translation, 161, 305 transport, 13, 18, 38 transportation, 182, 226 travel, 203 trend, 137, 162, 252 trial, 162 tribal, 266 tribal colleges, 266 tuition, 201, 222, 269, 273 Tunisia, 253, 254 turbulence, 19, 38, 67 turbulent, 68 Turkey, 254 turnover, 270 tutoring, 79 two-dimensional, 5, 6, 30, 40, 47, 48, 57, 67, 68, 79, 136, 351 two-dimensional space, 5, 30

U U.S. Geological Survey, 219 uncertainty, 162 Uncertainty Principle, 311 undergraduate, 99, 191, 201, 205, 210, 215, 223, 224, 225, 226, 227, 230, 248, 255, 256, 266, 268, 272, 273, 274 undergraduate education, 224 undergraduates, 215, 273 unemployment, 211 UNESCO, 259 unification, viii, 89 uniform, 36, 62, 79, 303, 304, 305, 307, 317 United Kingdom, 253, 254, 261 United Nations, 259 United States, viii, ix, 177, 178, 180, 181, 182, 184, 186, 188, 190, 192, 193, 194, 196, 198, 200, 202, 203, 204, 205, 206, 208, 210, 212, 214, 216, 218, 220, 222, 224, 225, 226, 228, 230, 232, 234, 236, 238, 240, 242, 243, 244, 245, 246, 247, 248, 249, 252, 253, 254, 256, 258, 259, 261, 263, 266, 268 universe, 215 universities, 113, 179, 180, 181, 182, 184, 188, 200, 201, 202, 203, 204, 205, 207, 209, 213, 214, 215, 222, 223, 224, 226, 244, 266, 274 Uruguay, 254 USDA, 180

V vacuum, 23, 42 Valencia, 277 validation, 94 validity, 167 values, 3, 14, 23, 32, 33, 34, 48, 52, 58, 59, 60, 61, 62, 66, 69, 78, 82, 83, 84, 95, 96, 108, 111, 120, 124, 127, 136, 137, 138, 142, 143, 144, 162, 163, 164, 165, 166, 167, 170, 171, 172, 173, 174, 211, 212, 231, 235, 296, 297, 298, 299, 300, 301, 302, 304, 305, 307, 309, 310, 315, 316, 318, 319, 320, 322, 323, 325, 332, 336, 338, 339, 344, 345, 346 variability, viii, 162, 166, 168, 174, 210 variable, 2, 9, 32, 80, 143, 164 variables, 56, 61, 68, 69, 70, 147, 161, 163, 164, 165, 166, 168, 170, 171, 174, 175 variance, 213, 235 variation, 3, 149, 171, 307, 312 vector, 22, 40, 41, 43, 166, 167, 168, 298, 299, 301, 302, 303, 304, 346, 348 vegetables, 163 velocity, 2, 3, 7, 8, 9, 14, 15, 21, 22, 26, 30, 33, 34, 36, 39, 43, 47, 48, 56, 57, 58, 61, 68, 78, 311 veterinary medicine, 222 Victoria, 100, 103, 109, 112 visa, 181, 184, 198, 200, 202, 203, 206, 207, 246, 258

370

Index

visa system, 203 visas, 182, 184, 198, 203, 206, 207, 244, 245, 263 viscosity, 70, 141, 143, 144 vocabulary, 93 vocational, 244, 259 volatility, 163 vortex, 6, 8, 12, 34, 52, 68 vortices, 6, 12, 29, 45, 56, 62, 73

W wages, 179, 190, 199, 212, 235 water, vii, 1, 5, 55, 56, 57, 62, 78, 79, 221 water quality, 221 wave propagation, 2 weather prediction, 55 web, 152 web-based, 150 welfare, 224 well-being, 105 wind, 47 wine, 109 winemaking, 109

winning, 267 women, 178, 179, 180, 181, 182, 190, 191, 192, 194, 195, 197, 199, 202, 204, 205, 212, 214, 215, 224, 225, 226, 229, 230, 236, 245, 264, 266 workers, 184, 212, 213, 223, 225, 258, 268 workforce, 185, 188, 189, 190, 205, 208, 222, 223, 224, 225, 226, 258, 264, 270 working groups, 189 workstation, 65 World Bank, 259 writing, 60

Y yield, x, 5, 116, 133, 137, 296 yogurt, 162

Z zoology, 100

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ADVANCES IN MATHEMATICS RESEARCH EDITOR: GABRIEL OYIBO Advances in Mathematics Research, Volume 8 2009. 978-1-60456-454-9

Advances in Mathematics Research, Volume 7 2007. 1-59454-458-1

Advances in Mathematics Research, Volume 6 2005. 1-59454-032-2-3

Advances in Mathematics Research, Volume 5 2005. 1-59033-799-92

Advances in Mathematics Research, Volume 4 2003. 1-59033-518-X

Advances in Mathematics Research, Volume 3 2003. 1-59033-452-3

Advances in Mathematics Research, Volume 2 2003. 1-59033-430-2

Advances in Mathematics Research, Volume 1 2002. 1-59033-223-7

ADVANCES IN MATHEMATICS RESEARCH, VOLUME 8

ALBERT R. BASWELL EDITOR

Nova Science Publishers, Inc. New York

Copyright © 2009 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Available upon request.

ISBN 978-1-61209-811-1 (eBook)

Published by Nova Science Publishers, Inc. New York

CONTENTS Preface

vii

Chapter 1

The Method of Characteristics for the Numerical Solution of Hyperbolic Differential Equations M. Shoucri

Chapter 2

Negotiating Mathematics and Science School Subject Boundaries: The Role of Aesthetic Understanding Linda Darby

Chapter 3

The Mathematical Basis of Periodicity in Atomic and Molecular Spectroscopy K. Balasubramanian

115

Chapter 4

Mathematical Modelling of Thermo-Mechanical Destruction of Polypropylene G.M. Danilova-Volkovskaya, E.A. Amineva and B.M. Yazyyev

141

Chapter 5

A Design-based Study of a Cognitive Tool for Teaching and Learning the Perimeter of Closed Shapes Siu Cheung Kong

145

Chapter 6

Modeling Asymmetric Consumer Behavior and Demand Equations for Bridging Gaps in Retailing Rajagopal

161

Chapter 7

Higher Education: Federal Science, Technology, Engineering, and Mathematics Programs and Related Trends United States Government Accountability Office

177

Chapter 8

Science, Technology, Engineering, and Mathematics (STEM) Education Issues and Legislative Options Jeffrey J. Kuenzi, Christine M. Matthew and Bonnie F. Mangan

247

1

89

vi

Contents

Chapter 9

On Computational Models for the Hyperspace S. Romaguera

277

Chapter 10

Periodic-Type Solutions of Differential Inclusions Jan Andres

295

Index

355

PREFACE "Advances in Mathematics Research" presents original research results on the leading edge of mathematics research. Each article has been carefully selected in an attempt to present substantial research results across a broad spectrum. The application of the method of characteristics for the numerical solution of hyperbolic type partial differential equations will be presented in Chapter 1. Especial attention will be given to the numerical solution of the Vlasov equation, which is of fundamental importance in the study of the kinetic theory of plasmas, and to other equations pertinent to plasma physics. Examples will be presented with possible combination with fractional step methods in the case of several dimensions. The methods are quite general and can be applied to different equations of hyperbolic type in the field of mathematical physics. Examples for the application of the method of characteristics to fluid equations will be presented, for the numerical solution of the shallow water equations and for the numerical solution of the equations of the incompressible ideal magnetohydrodynamic (MHD) flows in plasmas. A tradition of subject specialisation at the secondary level has resulted in the promotion of pedagogy appropriate for specific areas of content. Chapter 2 explores how the culture of the subject, including traditions of practice, beliefs and basic assumptions, influences teachers as they teach across school mathematics and science. Such negotiation of subject boundaries requires that a teacher understand the language, epistemology and traditions of the subject, and how these things govern what is appropriate for teaching and learning. This research gains insight into relationships between subject culture and pedagogy by examining both teaching practice in the classroom and interrogating teachers’ constructions of what it means to teach and learn mathematics and science. Teachers’ level of confidence with, and commitment to, both the discipline’s subject matter and the pedagogical practices required to present that subject matter is juxtaposed with their views of themselves as teachers operating within different subject cultures. Six teachers from two secondary schools were interviewed and observed over a period of eighteen months. The research involved observing and videoing the teachers’ mathematics and/or science lessons, then interviewing them about their practice and views about school mathematics and science. The focus of this chapter is on the role of the aesthetic, specifically “aesthetic understanding,” in the ways science and mathematics teachers experience, situate themselves within, and negotiate boundaries between the subject cultures of mathematics and science. The chapter outlines teachers’ commitments to the discipline, subject and teaching by exploring three elements of aesthetic understanding: the compelling and dramatic nature of

viii

Albert R. Baswell

understanding (teachers’ motivations and passions); understanding that brings unification or coherence (relationships between disciplinary commitments and knowing how to teach); and perceived transformation of the person (teacher identity and positioning). This research has shown that problems arise for teachers when they lack such aesthetic understanding, and this has implications for teachers who teach subjects for which they have limited background and training Chapter 3 applies combinatorial and group-theoretical relationships to the study of periodicity in atomic and molecular spectroscopy. The relationship between combinatorics and both atomic and molecular energy levels must be intimate since the energy levels arise from the combinatorics of the electronic or nuclear spin configurations or the rotational or vibrational energy levels of molecules. Over the years the authors have done considerable work on the use of combinatorial and group-theoretical methods for molecular spectroscopy [1–15]. The role of group theory [1–40] is evident since the classification of electronic and molecular levels has to be made according to the irreducible representations of the molecular symmetry group of the molecule under consideration. Combinatorics plays a vital role in the enumeration of electronic, nuclear, rotational and vibrational energy levels and wave functions. As can be seen from other chapters in this book, the whole Periodic Table of the elements has a mathematical group-theoretical basis since the electronic shells have their origin in group theory. Indeed, this concept can even be generalized to other particles beyond electrons such as bosons or other fermions that exhibit more spin configurations than just the bi-spin orientations of electrons. Chapter 4 provided mathematical description of the processes of thermonuclear destruction in deformed polypropylene melts; the aim was to use the criterion of destruction estimation in modelling and optimising the processing of polypropylene into products. With the consideration of cognitive inflexibility of learners in computing perimeter of closed shapes, a theory-driven design of a cognitive tool (CT) called the ‘Interactive Perimeter Learning Tool (IPLT)’ for supporting the teaching and learning of the mathematics target topic was developed in Chapter 5. An empirical study in the form of pre-test—post-test reflected that learners of varying mathematical abilities had statistically significant gains in using the IPLT for learning support. The IPLT could effectively address the inflexibility commonly exhibited by learners in learning this topic such as the formation of the abstract association of an irregular closed shape with a regular closed shape. The assertion of teachers on the effectiveness of the IPLT and the enthusiasm of students for using the IPLT for learning reflect that the CT had a pedagogical value in fostering learner-centred learning. Based on the feedback of this study, the IPLT will be refined under the design-based research approach. Chapter 6 attempts to discuss the interdependence of variability in consumer behavior due to intrinsic and extrinsic retail environment which influence the process of determining the choices on products and services. It is argued in the paper that suboptimal choice of consumers affect the demand of the products and services in the long-run and the cause and effect has been explained through the single non-linear equations. A system of demand equations which explains the process of optimization of consumer choice and behavioral adjustment towards gaining a long-term association with the market has also been discussed in the paper. The United States has long been known as a world leader in scientific and technological innovation. To help maintain this advantage, the federal government has spent billions of

Preface

ix

dollars on education programs in the science technology, engineering, and mathematics (STEM) fields for many years. However, concerns have been raised about the nation’s ability to maintain its global technological competitive advantage in the future. Chapter 7 presents information on(1) the number of federal programs funded in fiscal year 2004 that were designed to increase the number of students and graduates pursuing STEM degrees and occupations or improve educational programs in STEM fields, and what agencies report about their effectiveness; (2) how the numbers, percentages, and characteristics of students, graduates, and employees in STEM fields have changed over the years; and (3) factors cited by educators and others as affecting students’ decisions about pursing STEM degrees and occupations, and suggestions that have been made to encourage more participation. There is growing concern that the United States is not preparing a sufficient number of students, teachers, and practitioners in the areas of science, technology, engineering, and mathematics (STEM). A large majority of secondary school students fail to reach proficiency in math and science, and many are taught by teachers lacking adequate subject matter knowledge. When compared to other nations, the math and science achievement of U.S. pupils and the rate of STEM degree attainment appear inconsistent with a nation considered the world leader in scientific innovation. In a recent international assessment of 15-year-old students, the U.S. ranked 28th in math literacy and 24th in science literacy. Moreover, the U.S. ranks 20th among all nations in the proportion of 24-year-olds who earn degrees in natural science or engineering. A recent study by the Government Accountability Office found that 207 distinct federal STEM education programs were appropriated nearly $3 billion in FY2004. Nearly threequarters of those funds and nearly half of the STEM programs were in two agencies — the National Institutes of Health and the National Science Foundation. Still, the study concluded that these programs are highly decentralized and require better coordination. Several pieces of legislation have been introduced in the 109th Congress that address U.S. economic competitiveness in general and support STEM education in particular. These proposals are designed to improve output from the STEM educational pipeline at all levels, and are drawn from several recommendations offered by the scientific and business communities. The objective of Chapter 8 is to provide a useful context for these legislative proposals. To achieve this, the report first presents data on the state of STEM education and then examines the federal role in promoting STEM education. The report concludes with a discussion of selected legislative options currently being considered to improve STEM education. The report will be updated as significant legislative actions occur. Let BX be the continuous poset of formal balls of a metric space (X, d) endowed by the weightable quasi-metric qd induced by d. in Chapter 9 the authors show that the continuous poset B(CX) of formal balls of the space CX of nonempty closed bounded subsets of X endowed by the quasi-metric qHd induced by the Hausdorff metric Hd on CX is isometric to a sup-closed subspace of the space C(BX) of nonempty sup-closed bounded subsets of BX endowed with the Hausdorff quasi-metric qHd. The authors also show that the quasi-metric space (B(CX), qHd) is bicomplete if and only if the metric space (X, d) is complete. Several consequences are derived. In particular, our approach provides an interesting class of weightable quasi-metric spaces for which weightability of the Hausdorff quasi-metric holds on certain paradigmatic subspaces. Moreover, some properties from topological algebra are

x

Albert R. Baswell

discussed; for instance, the authors prove that if (X, d) is a metric monoid (respectively, a metric cone), then (B(CX), qHd) is a quasi-metric monoid (respectively, (B(CcX), qHd ) is a quasi-metric cone, where by Cc(X) the authors denote the family of all convex members of C(X)). The main purpose of Chapter 10 is two-fold: (i) rather than a complete account or a systematic study, the authors would like to indicate a flavour of the theory of periodictype oscillations, and (ii) to present some of our own results for periodic-type solutions of differential equations and inclusions. For (i), the authors preferably selected in Section 4. (Primer of periodic-type oscillations) the related results (including ours) which are easy for formulation while, for (ii), some technicalities had to be involved in Section 5. in order to derive sufficiently general criteria of the effective solvability of given actual problems. Results are, nevertheless, sketched in a form that is convenient for exposition and not necessarily in the greatest generality possible. Our objective is so to give the reader an overall idea of what the standard theory is like as well as to include enough information about its most recent progress. Formally, the focus of the object is simply the determination of the readable text for a wider audience with some parts to yield also a profit for the specialists.

In: Advances in Mathematics Research, Volume 8 Editor: Albert R. Baswell, pp. 1-87

ISBN: 978-1-60456-454-9 © 2009 Nova Science Publishers, Inc.

Chapter 1

THE METHOD OF CHARACTERISTICS FOR THE NUMERICAL SOLUTION OF HYPERBOLIC DIFFERENTIAL EQUATIONS M. Shoucri Institut de Recherche d’Hydro-Québec (IREQ), Varennes, Québec, Canada J3X1S1

Abstract The application of the method of characteristics for the numerical solution of hyperbolic type partial differential equations will be presented. Especial attention will be given to the numerical solution of the Vlasov equation, which is of fundamental importance in the study of the kinetic theory of plasmas, and to other equations pertinent to plasma physics. Examples will be presented with possible combination with fractional step methods in the case of several dimensions. The methods are quite general and can be applied to different equations of hyperbolic type in the field of mathematical physics. Examples for the application of the method of characteristics to fluid equations will be presented, for the numerical solution of the shallow water equations and for the numerical solution of the equations of the incompressible ideal magnetohydrodynamic (MHD) flows in plasmas.

1. Introduction Different types of partial differential equations require different numerical methods of solution. Numerical methods for hyperbolic equations are generally more complicated and difficult to develop compared to the numerical methods applied for parabolic or elliptic type partial differential equations. There has been important advances in the last few decades in the domain of the numerical solution of hyperbolic type partial differential equations using the method of characteristics, when applied to solve the initial value problem for general first order partial differential equations The order of a partial differential equation is the order of the highest-order partial derivative that appears in the equation. Let us consider for example the following simple hyperbolic type advection equation:

2

M. Shoucri

∂f ∂f +c = 0. ∂t ∂x

(1.1)

where c is a constant, sometimes called the velocity of propagation. The characteristic equation to solve Eq.(1.1) is dx / dt = c . The rate at which the solution will propagate along the characteristics is c. If c is a constant, all the points on the solution profile will move at the same speed along the characteristics determined by the solution of dx / dt = c . Let us assume the initial condition x(0) = x 0 . The solution of the characteristic equation gives the characteristic curves x = x0 + ct ( a straight line for the present case where c is a constant), where x0 is the point where each curve intersects the x-axis at t=0 in the x-t plane. If at t=0 we have

f ≡ f ( x 0 ) , x0 = x − ct , then f ( x, t ) = f ( x − ct ) . The function f ( x, t )

remains constant along a characteristic, which can be verified if we differentiate f ( x, t ) along one of these curves to find the rate of change of f along the characteristic:

∂f df ( x(t ), t ) ∂f ( x(t ), t ) dx ∂f ( x(t ), t ) ∂f = + = +c = 0. dt ∂t dt ∂x ∂t ∂x

(1.2)

which verify that f is constant along the characteristic curves. This is the simplest mathematical model of wave propagation. Constant quantities along the characteristic curves are called Riemann invariant [1]. We next consider the variable coefficient advection equation written as follows:

∂f ∂f + g ( x, t ) = 0. ∂t ∂x

(1.3)

The characteristic equation is dx / dt = g ( x, t ) . Again if the value of f at some arbitrary point ( x 0 , t 0 ) is known, the coordinate of the characteristic curve passing through

( x0 , t 0 ) can

be

determined

by

integrating

the

ordinary

differential

equation

dx / dt = g ( x, t ) . The velocity of propagation depends now on the spatial coordinate and time. In the general case an analytic solution is not straightforward and the characteristic curves are not straight lines anymore. Also it will be possible for the characteristic curves to intersect. The solution obtained by following the characteristic curves may contain discontinuities, which can lead to the formation of shocks or rarefaction waves [1]. Numerical techniques can be used to produce good approximations by following the solution computationally with small time-steps . As an example, we can discretize Eq.(1.3) as follows:

f ( x, t + dt ) − f ( x, t ) f ( x, t ) − f ( x − dx, t ) + g ( x, t ) = 0. dt dx

(1.4)

The Method of Characteristics for the Numerical Solution…

f ( x, t + dt ) = f ( x, t ) − g ( x, t ).

3

dt .( f ( x, t ) − f ( x − dx, t ) = f ( x − dx, t ) . (1.5) dx

For a small time-step between t and t+dt, it is possible to write the solution for the characteristic equation between x and x+dx in the form: t + dt

dx =

∫ g ( x(t ′), t ′)dt ′ .

(1.6)

t

Substituting in the right hand side of Eq.(1.5), we get: t + dt

f ( x, t + dt ) = f ( x −

∫ g ( x(t ′), t ′)dt ′, t ) .

(1.7)

t

Eq.(1.5) and Eq.(1.7) indicate that the value of the function f at the time t+dt and at a position x is equal to the value of the function at time t, at the shifted position t + dt

x − dx = x −

∫ g ( x(t ′), t ′)dt ′ . Eq.(1.7) is an implicit equation, and in all but the simplest t

cases different numerical approximations must be used to write an explicit solution. It is the purpose of the present chapter to discuss some of these approximations through examples and numerical methods applied to hyperbolic equations. Some of these approximations have been recently discussed for instance in [2,3]. The value of the function at the shifted position is usually calculated by interpolation from the known values of the function at the neighbouring grid points. In the present chapter cubic splines interpolation will be extensively used to calculate the shifted value in Eq.(1.7), since in several applications and problems they have compared favourably with other methods of interpolation [4]. For the more general case where several dimensions are involved, the fractional step technique allows sometimes the reduction of the multi-dimensional equation to an equivalent set of one dimensional equations [2-5]. The shifts become fractional, i.e. each of the dimension is shifted separately. The specific order, number of shifts and choice of the size of shift-factors depend now on the numerical method. If the fractional step technique cannot be applied, we can use other methods which consist in interpolating in several dimensions using a tensor product of Bsplines [6]. This technique has been extensively applied in the field of meteorology [7,8], where it is called the semi-Lagrangian method (although we prefer to call it the EulerLagrange method, since it essentially uses a fixed Eulerian grid, and uses a corrector or an iterative process to take care of the variation of the velocity along the characteristic curve). We can generalize Eq.(1.3) for a multi-dimensional problem in the following form:

df ∂f ∂f = + G (r, t ). = 0 . dt ∂t ∂r

(1.8)

4

M. Shoucri

which reflects the fact that the function f (r, t ) is constant along the trajectories defined by the characteristic curves :

dr = G (r, t ) . dt

(1.9)

Denoting by r (t ; ri , t n ) the characteristic crossing the grid point ri at tn , we can also write at t = t n :

f (r (t ; ri , t n ), t ) = f (r (t ; ri , t n ), t n ) = f (ri , t n ) .

(1.10)

Replacing t n by t n + Δt and t by t − Δt , results in :

f (ri , t n + Δt ) = f (r (t − Δt ; ri , t n + Δt ), t − Δt ) .

(1.11)

r (t − Δt ; ri , t n + Δt ) is the characteristic which ends up at ri at time t n + Δt . The function value at the time-step t n + Δt and at the grid point ri can be calculated by looking backward to the function value at an interstitial point, prescribed by the characteristic curve at the previous time t − Δt . The starting point at the previous time-step t n − Δt , of the

ri at time t n + Δt , is denoted by curve ending at ~ r = r (t n − Δt; ri , t n + Δt ) (see Fig.(1)). Usually ~r is an intermediate interstitial point

characteristic

which does not coincide with a grid point. The value of the function at ~ r has to be calculated by interpolation. Discretizing Eq.(1.9) of the characteristic curves using a leap-frog scheme, we can write:

r (t n + Δt ) − r (t n − Δt ) ri − ~ r ≡ = G (r (t n ), t n ) . 2Δt 2Δt

(1.12)

Using r (t n ) ≈ (r (t n + Δt ) + r (t n − Δt ) ) / 2 ≡ (ri + ~ r ) / 2 in the right hand side of

r ) / 2 . We solve this Eq.(1.12), results in Δ r = ΔtG (ri − Δ r , t n ) , where Δ r = (ri − ~ equation numerically for Δ r using the Newton iterative scheme :

Δkr+1 = ΔtG (ri − Δkr , t n )

. 0

(1.13)

starting with k=0, Δ r = 0 . Two or three iterations are usually sufficient to converge to precise results. We then calculate the value of f at the position ri at t n + Δt :

The Method of Characteristics for the Numerical Solution…

5

f (ri , t n + Δt ) = f (r (t − Δt ; ri , t n + Δt ), t n − Δt ) = f (ri − 2 * Δ r , t n − Δt ) . (1.14) j +1 Q (x i , y j ; t n +1 )

j

j −1 P (x , y ; t n −1 )

i −1

i

i +1

Figure 1.

The multi-dimensional interpolation in Eqs.(1.13-1.14) will generally involve a tensor product of B-splines. In practice, we will restrict ourselves to problems in two dimensions. In the Fig.(1) we give an example for the case of a two-dimensional space, showing the point of departure P at t n − Δt , where the value of the function f is to be interpolated as in Eq.(1.14) to yield the value of f (ri , t n + Δt ) at the point Q. Similar schemes have been extensively used in problems of meteorology [7,8], and more recently in plasma physics [6,9]. The ideas outlined in this introduction will be applied to selected problems in the present chapter. In section 2 we will present examples where a fractional step method reduces the multi-dimensional problem to an equivalent set of one-dimensional (1D) problems. In section 3 we will present examples where 2D interpolation involves a tensor product of cubic Bsplines. We will emphasize the precision, good performance and numerical stability of the cubic splines interpolation, which have been also previously pointed out in [4,7]. Examples will be taken from the field of plasma physics, especially concerning the numerical solution of the Vlasov equation, of fundamental importance in the kinetic theory of plasmas. Some additional applications in the field of fluid dynamics will be presented in section 4, for the numerical solution of the shallow water equations, and for the numerical solution of the equations of the incompressible ideal magnetohydrodynamic flows in plasmas.

2. The Fractional Step Method Applied to the Vlasov Equation The study of nonlinear processes in kinetic plasmas is heavily based on the numerical solution of the Vlasov equation for the distribution function. The Vlasov equation provides the basic dynamical description of hot plasmas in regimes where the effect of collisions are negligible with respect to those originating from the collective, mean-field electromagnetic interactions. The Vlasov-type equation is an advection equation in phase-space for the distribution function f , of the general form given in Eq.(1.8). Different techniques have been proposed to

6

M. Shoucri

solve this equation. Particle-in-cell (PIC) methods for instance approximate the plasma by a finite number of pseudo-particles and compute their trajectories given by Eq.(1.9). However, the numerical noise in these codes decreases only as 1 / N , where N is the number of pseudo-particles in any particular computational cell. This noise problem becomes important if the physics of interest is in the low density region of phase-space or in the high energy tail of the distribution function. On the other hand the direct numerical solution of the Vlasov equation as a partial differential equation on a fixed grid in phase-space has become an important method for the numerical solution of the Vlasov equation. Interest in Eulerian gridbased Vlasov solvers arises from the very low noise level associated with these methods, and the recent advances of parallel computers have increased the interest in the applications of splitting schemes to higher dimensional problems. The original, ground-breaking publication of Cheng and Knorr [10], which proposed the second-order fractional step scheme or splitting scheme for the solution of the Vlasov-Poisson system, was followed by several publications where this method was successfully applied to one-dimensional (two-dimensional in phasespace) Vlasov- Poisson problems [11-14]. The technique was extended to higher phase-space dimensions [15-19]. An important application using the Eulerian splitting schemes for the Vlasov-Maxwell system of equations has been reported for the study of laser-plasma interaction [20-27 and references therein], and extended to two-dimensional problems [28]. In the work on beat wave current drive [29], a constant magnetic field was introduced in the Vlasov equation. Further applications in the recent work in [30,31] testify to the success of this method in laser-plasma interaction. We also note the application of Eulerian splitting schemes to study two spatial dimension problems of Kelvin-Helmholtz instabilities and higher dimensionality gyrokinetic equations [32-39]. There exists also a variety of other applications using different methods developed for Eulerian grid-based Vlasov solvers [4044]. Of particular interest is the work coupling a Vlasov equation to a Fokker-Planck collision operator presented in [45]. In the present section 2 , we will present selected examples where the fractional step techniques associated with interpolation along the characteristic curves in one dimension are applied for the numerical solution of the Vlasov equation.

2.1. The Fractional Step Method Applied to the Vlasov-Poisson System in One Spatial Dimension The first system we study is the Vlasov-Poisson system in one spatial dimension ( a twodimensional phase space x-v ). The problem is the long time nonlinear evolution of a twostream instability in a collisionless plasma [46,47]. The system in this case evolves to a Bernstein-Greene-Kruskal BKG equilibrium [48] consisting of a stationary structure exhibiting holes or vortices in phase-space. BKG structures with more than one hole are unstable and coalesce until the evolution brings a final stable vortex. This flow of energy of the system during the evolution to the longest wavelength available in the system ( inverse cascade ) is characteristic of two-dimensional systems and has been discussed in several publications ( see for instance [49-50]). We use an Eulerian code associated with a method of fractional step for the integration of the Vlasov equation along the characteristics. The Eulerian method allows accurate resolution of the phase-space on a fixed Eulerian grid. In the present problem the spatial dimension x is assumed to be periodic. The normalized Vlasov

The Method of Characteristics for the Numerical Solution…

7

equation for the electron distribution function f ( x, v, t ) and the Poisson equation for the potential

ϕ (x) are given by: ∂f ∂f ∂f + v − Ex = 0. ∂t ∂x ∂v

(2.1)

∂ 2ϕ = −(1 − ne ) , ∂x 2 ∞

where

ne =

∫ f dv , and E

x

=−

−∞

∂ϕ ∂x

(2.2)

The ions form an immobile background in the present problem. The distance x , the velocity v and the time t are respectively normalized to the Debye length λ De = vth / ω pe , the thermal velocity vth and the inverse plasma frequency

−1 ω pe . Eq.(2.1) is essentially a two-

dimensional advection equation. An important property of this equation is that its characteristics, the particles trajectories dx / dt = v , dv / dt = − E x describe a Hamiltonian flow in phase-space. The particles motion is described by the Hamiltonian:

H=

v2 + ϕ ( x) . 2

(2.3)

The Vlasov Eq.(2.1) can be written in the form:

∂f + [H , f ] = 0 . ∂t The Poisson brackets

(2.4)

[H , f ] = ⎧⎨ ∂H

∂f ∂H ∂f ⎫ − ⎬ ⎩ ∂v ∂x ∂x ∂v ⎭

The distribution function f is constant along the particle trajectories. As a consequence, the integral over the entire phase-space of the distribution function is a constant, as well as the integral of any arbitrary smooth function of f. Thus the evolution of the distribution function f is constrained by a number of constants of motion. Hamiltonian systems like Eq.(2.4) are known to develop increasingly smaller scales during their nonlinear evolution. One way to control these finer structures is to increase resolution. These small structures dissipate when they reach the size of a the grid. We write the initial electron distribution function in the form[46]:

8

M. Shoucri

f ( x, v, t = 0) = A(1 +

ε 1−ξ

) e −ε (1 + α cos(k 0 x)

.

(2.5)

+ β cos(2k 0 x) + γ cos(3k 0 x)) A=

With

2 − 2ξ 2π 3 − 2ξ 1

ε = v 2 / 2 , and ξ is a parameter which characterizes a produced vortex in phase-space. 2π k0 =

L

denotes the fundamental wavenumber, L is the length of the periodic box. We

choose k 0 =

k M2 =

kM , where k M is the maximum wavenumber for instability [46] given by 4

3 − 2ξ 2ξ − 1 , which leads to a box length L = 8π . We choose ξ = 0.90 , which 3 − 2ξ 2ξ − 1

gives k M = 0.816 , L = 30.78λ De and k 0 = 0.204 . We take a cut-off velocity at v max = ±6vth . The distribution function is given at mesh points in the phase-space, with Nx = 128 points in space and Nv = 256 points in velocity space. The time-step is Δt = 0.25ω pe . A method which has second order in time precision [10,11] −1

is obtained by splitting Eq.(2.1) as follows: Step1 -

Solve

∂f ∂f +v = 0 for a step Δt / 2 ∂t ∂x

(2.6) *

- Solve Poisson equation for the electric field which we denote by E x .

Step2 -Solve Step3 -Solve

∂f ∂f − E x* = 0 for a step Δt ∂t ∂v ∂f ∂f +v = 0 for a step Δt / 2 ∂t ∂x

(2.7) (2.8)

In this 2D phase-space problem the shifts become fractional, i.e. each of the dimension of the phase-space is shifted separately. This splitting has the advantage that each of the x or v updates is a linear advection effected by applying successively the shifts :

f a ( x, v, t + Δt / 2) = f ( x − vΔt / 2, v, t ) ,

(2.9)

f b ( x, v, t + Δt ) = f a ( x, v − E x* Δt , t ) ,

(2.10)

The Method of Characteristics for the Numerical Solution…

9

f ( x, v, t + Δt / 2) = f b ( x − vΔt / 2, v, t ) ,

(2.11)

That is, half of the spatial shift is performed first in space. Since v is an independent variable, the shift in Eq.(2.9) is done as in Eq.(1.1) for each value of v (see Appendix A). This *

is followed by solving Poisson equation for the calculation of the electric field E x , which is used for the calculation of the total shift in velocity space where the integral in Eq.(1.7) is approximated as in Eq.(2.10). Poisson equation in Eq.(2.2) is discretized in space as a tridiagonal matrix:

ϕ j −1 − 2ϕ j + ϕ j +1 = −Δx 2 (1 − nej ) .

(2.12)

where Δx = L / N x , the subscript j denotes the grid-point xj . Eq.(2.12) is solved using appropriate boundary conditions ( periodic boundary conditions for the present problem). From

ϕ we calculate E x* (Eq.(2.2)). Finally the second half of the spatial shift is repeated in

Eq.(2.11). It has been shown in [10] that the overall precision of this numerical scheme is

O( Δt 2 ) . We can verify after this sequence that the distribution function f t = ( n + 1)Δt can be written as follows: f

n +1

n +1

at time

( x, v ) = f n ( x * , v * ) 1 E x ( x − vΔt / 2) Δt ) . 2 v * = v + E x ( x − vΔt / 2) x * = x − Δt (v +

(2.13)

On the other hand we can consider the characteristics equations , dx / dt = v ,

dv / dt = − E x , which are the particles trajectory. The integration of these equations between t and t = t + Δt gives the following result :

x(t ) = x(t + Δt ) − Δt (v(t + Δt ) + 1 / 2 E x ( x , t + Δt / 2)Δt ) v(t ) = v(t + Δt ) + ΔtE x ( x , t + Δt / 2)

.

(2.14)

where x = x(t + Δt / 2) . The field E x ( x, t ) in Eq.(2.7) is calculated after the first shift. The density distribution, and therefore E x ( x, t ) , remains unaffected by the second shiht. Thus the field E x ( x , t + Δt / 2) can be approximated by E x ( x − vΔt / 2, t + Δt / 2) . The shifts in Eqs.(2.9-2.11) are calculated using a cubic spline interpolation as defined in the appendices. For the present problem, we used the results in Appendix A.

10

M. Shoucri

Figure 2. The vorticity at t=50

Figure 3. The vorticity at t=100

−1 ω pe .

−1 ω pe .

Figure 4. Time evolution of the first Fourier mode.

The Method of Characteristics for the Numerical Solution…

Figure 5. Time evolution of the second Fourier mode.

Figure 6. The vorticity at t=40

−1 ω pe .

Figure 7. The vorticity at t=50

−1 ω pe .

11

12

M. Shoucri

Figure 8. The vorticity at t=500

−1 ω pe .

We apply the numerical scheme previously discussed to study the evolution of a two-stream instability. We introduce a perturbation on the fundamental wavenumber k 0 by taking

α = 0.001 and β = γ = 0. Only one vortex appears in phase space during the nonlinear plasma evolution ( see Fig.(2)), and the final equilibrium in Fig.(3) consists of a single smooth hole. Fig.(4) and Fig.(5) show the nonlinear evolution of the first and second Fourier modes respectively, showing the initial growth and saturation. In a second experiment, we start with a perturbation in Eq.(2.5) of the three modes, α = 0.001 , β = γ = α / 1.2 . We obtain in the first step the appearance of two vortices in the phase-space shown in Fig.(6) at −1 −1 t = 40ω pe , followed rapidly by the coalescence of the vortices at t = 50ω pe in Fig.(7) (so

the two vortices structure is not stable). Note the tendency of holes to behave as quasiparticles just before coalescence [47]. We finally end up with a single vortex (see Fig.(8)). We note again this tendency of the energy to move to the longest wavelength available in the system [49,50] ( the so called inverse cascade), which is characteristic of two dimensional systems. Small scale vortices can be created in the transient regime, but they rapidly coalesce to give rise to larger vortices, and finally only large scale structures persist. The system selects the longest wavelength allowed by the imposed boundary conditions. Fig.(9) and Fig.(10) show the nonlinear evolution of the first and second Fourier modes respectively, showing the initial growth and saturation. We note that the saturation level decreases the higher the mode. Statistical studies presented in [49,50] for 2D systems predict for two dimensional systems a level of the energy associated with the different Fourier modes of the form E k

2

= 1 /(δ + σk 2 ) ( δ and σ are constants), with energy condensing in the low k

modes (inverse cascade). We note the strong influence of the initial conditions on the plasma evolution, although the final state is generally a single vortex structure. We also note the accurate and stable performance of the noiseless Eulerian numerical code, which provided precise information on the phase-space behaviour of the one-dimensional Vlasov plasma.

The Method of Characteristics for the Numerical Solution…

13

Finally we point to the extension of the fractional step method to a fourth order scheme using a symplectic integrator, recently reported in [42].

Figure 9. Time evolution of the first Fourier mode.

Figure 10. Time evolution of the second Fourier mode.

2.2. The Vlasov-Poisson System in Higher Phase-Space Dimensions: the Problem of the Formation of an Electric Field at a Plasma Edge in a Slab Geometry Further evaluation of the performance of the cubic spline interpolation with respect to other interpolation methods, like the cubic interpolated propagation CIP method and the flux corrected transport method, has been presented in [4] and shows the cubic spline interpolation compares favourably with respect to the other methods. We consider in this section the problem of the charge separation at a plasma edge. This problem, with the calculation of the self-consistent electric field along a steep gradient, is of major importance in many physical problems. In tokamak physics, it is highly relevant to the edge physics associated with the

14

M. Shoucri

high confinement mode (H mode). Two methods will be used to study this problem, and the results obtained will be compared. In the first method presented in this section, Cartesian geometry ( a slab model) will be used at the edge of the plasma, and a fractional step technique associated with 1D interpolation using a cubic spline will be applied. In the second method to be presented in section 3.2, we will discuss the solution of the same problem at the plasma edge using cylindrical coordinates ( r , θ , z ) , with a code which applies a 2D interpolation using a tensor product of cubic B-splines [6,51]. The plasma is assumed to be in front of a floating limiter with the magnetic field being aligned parallel to the limiter surface. Electrons are assumed to be frozen along the magnetic field lines. We compare the electric field with the macroscopic values calculated from the same kinetic codes for the gradient of the ion pressure and the Lorentz force term. We find that along the gradient, these quantities balance exactly the electric field. The inhomogeneous direction in the 1D slab geometry considered is the x direction, normal to the limiter plane (y, z). The constant magnetic field is in the y direction (assumed to represent the toroidal direction), and z represents the poloidal direction. The ions are described by the 1D in space ( three phase-space dimensions) Vlasov equation for the ion distribution function f i ( x, v x , v z , t ) :

∂f i ∂f ∂f ∂f + v x i + (E x − v z ω ci ) i + v x ω ci i = 0 ∂t ∂x ∂v x ∂v z

(2.15)

In Eq.(2.15) time is normalized to the inverse ion plasma frequency ω −pi1 , velocity is normalized to the acoustic velocity c s =

Te / M i (Te is the electron temperature and Mi is

the ion mass), and length is normalized to the Debye length λ De = c s / ω pi , where

ω pi is

the ion plasma frequency. The potential is normalized to Te / e , and the density is normalized to the peak initial central density.

ω ci is the ion cyclotron frequency. We assume deuterons

plasma. The system is solved over a length L = 175

λ De in front of the limiter plate, with an

initial density profile for the ions and electrons (indices i and e denote ions and electrons respectively):

ni = ne = 0.5 (1 + tanh (( x − L / 5) / 7 ))

(2.16)

The initial value of the ion distribution function f i ( x, v x , v z ) is given by: 2

2

e − ( vx + vz ) / 2Ti f i ( x, v x , v z ) = ni ( x ) 2πTi

(2.17)

The magnetized electrons are frozen along the magnetic field lines, with a constant profile given by Eq. (2.16). In this case the electrons cannot move across the magnetic field in

The Method of Characteristics for the Numerical Solution…

15

the gradient region to compensate the charge separation which is built up due to the finite ion orbits. It is important to calculate the ion orbits accurately by using an accurate Eulerian Vlasov code. The larger the ion gyroradius, the bigger the charge separation and the selfconsistent electric field at the edge. (Hence the important role played by even a small fraction of impurity ions). The electric field is calculated from the Poisson equation:

∂ 2ϕ = −(ni − ne ) ∂x 2

;

Ex = −

∂ϕ ∂x

(2.18)

The following parameters are used for deuterium ions:

ω ci = 0 .1 ; ω pi

2Ti ρi 1 = = 10 2 λ De Te ω ci / ω pi

Ti = 1; Te

(2.19)

If we assume an initial Maxwellian distribution for the ions with Tix = Tiz = Ti spatially constant, then the factor 2Ti in the calculation of the gyro-radius in Eq.(2.19) takes into 2 > = < v x2 > + < v z2 > = 2Ti / mi . We assume account that the perpendicular temperature < v ⊥

in the present calculation that the deuterons hitting a wall at x = 0 are collected by a floating limiter. Since the magnetized electrons do not move in the x direction across the magnetic field there is no electron current collected at the floating limiter. Therefore we have at x = 0 the relation :

∂E x = − J xi x = 0 ∂t x = 0

t

E x x = 0 = − ∫ J xi x = 0 dt

or

(2.20)

0

Integrating Eq. (2.18) over the domain (0, L), we get the total charge σ in the system: x

E x x = L − E x x = 0 = ∫ (ni − ne ) dx = σ

(2.21)

0

The difference between the electric fields at the boundaries must be equal to the charge appearing in the system. Equation (2.15) is solved by a method of fractional step, in which the advection term in space is solved first, then the equation in velocity space can be solved either using 2D interpolation with a tensor product of cubic B-spline as discussed in [51] (to be applied in section 3.2) , or by successive 1D cubic spline interpolation as follows: Step1- Solve

∂f ∂f + vx = 0 for a step Δt / 2 ∂t ∂x

(2.22)

16

M. Shoucri n +1 / 2

- Solve Poisson equation for the electric field which we denote by E x

.

Step2- Solve

∂f ∂f + ( E xn+1 / 2 − v zω ci ) = 0 for a step Δt / 2 ∂t ∂v x

(2.23)

Step3- Solve

∂f ∂f + v xω ci = 0 for a step Δt ∂t ∂v z

(2.24)

Step4- Repeat Step2 for a time step Δt / 2 Step5- Repeat Step1 for a time step Δt / 2 This splitting leads to the following successive shifts :

f a ( x, v x , v z , t + Δt / 2) = f ( x − v x Δt / 2, v x , v z , t ) ,

(2.25)

f b ( x, v x , v z , t + Δt / 2) = f a ( x, v x − E xn +1 / 2 Δt / 2 + v zω ci Δt / 2, v z , t ) ,

(2.26)

f c ( x, v x , v z , t + Δt ) = f b ( x, v x , v z − v zω ci Δt , t ) ,

(2.27)

We then repeat Eq.(2.26) and Eq.(2.25) to complete the cycle. We can then verify after this sequence that the distribution function f

n +1

at time t = ( n + 1) Δt can be written in the

following form:

f

n +1

*

*

( x, v x , v z ) = f n ( x * , v x , v z )

(2.28)

where:

x * = x − v x Δt +

*

1 * 2 Δt 2 E x Δt − v z ω ci 2 2

(2.29)

1 v *x = v x − E x* Δt + v zω ci Δt − v xω ci2 Δt 2 2

(2.30)

1 1 v *z = v z − v xω ci Δt + ω ci E x* Δt 2 − v z ω ci2 Δt 2 2 2

(2.31)

where E x = E x ( x − v x Δt / 2, t = nΔt + Δt / 2)

The Method of Characteristics for the Numerical Solution…

17

On the other hand , we can consider the characteristics equations for Eq.(2.15) which describe the particles motion:

dx = vx dt

(2.32)

dv x = E x − v z ω ci dt

(2.33)

dv z = v xω ci dt

(2.34)

By integrating the Eqs.(2.32-2.34) from t n = nΔt to t n +1 = (n + 1)Δt , we get:

x n = x n +1 − v xn +1

Δt Δt − v xn 2 2

v xn = v xn +1 − E xn +1 / 2 Δt + ω ci v zn +1 v zn = v zn +1 − v xn +1ω ci

(2.35)

Δt Δt + ω ci v zn 2 2

Δt Δt − v xnω ci 2 2

(2.36)

(2.37)

Eqs(2.35-2.36) leads to the following solution correct to second order in Δt :

x =x n

v =v n x

n +1

n +1 x

−v

−E

n +1 x

Δt + E

n +1 / 2 x

n +1 / 2 x

Δt + ω ci v

2 Δt 2 n +1 Δt − ω ci v z 2 2

(2.38)

Δt 2 2

(2.39)

n +1 z

Δt − ω v

v zn = v zn +1 − v xn +1ω ci Δt + ω ci E xn +1 / 2

2 ci

n +1 x

Δt 2 Δt 2 − v zn +1ω ci2 2 2

(2.40)

By comparing Eqs.(2.29-2.31) to Eqs.(2.38-2.40), we see that the splitting scheme 2

integrates the distribution function along the characteristics correctly to an order O( Δt ) . ( 2

n +1

Note also that to an order O (Δt ) , v z

v zn ).

in the last term in Eq.(2.40) can be substituted by

18

M. Shoucri

. Figure 11. Plot, for the Cartesian geometry, of the electric field Ex (solid curve), the Lorentz force

+ 0.1 < v z >

(dash- dotted curve), the pressure force ∇Pi

∇Pi / ni + 0.1 < v z >

/ ni (dotted

curve), and the sum

(broken curve). The density ni/2 is is also plotted (dash- three-dots curve,

plotted for reference).

We assume that the gyrating plasma deuterons are allowed to enter or leave at the right boundary. So the electric field at the right boundary x = L must be such that the difference between the electric fields at both boundaries in Eq. (2.21) is equal to the total charge σ appearing in the system. Fig. (11) shows at t = 500 the plot of the electric field E x (solid curve, we concentrate on the region x < 100 to emphasize the gradient region, although the system extends to x = 175). We also plot ni / 2 (dash-three-dots curve) in the same figure for reference. The dash-dotted curve gives the Lorentz force, which in our normalized units is given by < v z > ω ci / ω pi = 0.1 < v z > , and the dotted curve gives the pressure force

∇Pi / ni , Pi = 0.5 ni (Tix + Tiz ) , with: Tix, z ( x ) =

< v x, z > =

1 dv x dv z (v x, z − < v x, z > )2 f i (x, v x , v z ) ni ∫

1 dv x dv z v x , z f i (x, v x , v z ) ; ni ∫

(2.41)

n i ( x ) = ∫ dv x dv z f i (x, v x , v z ) (2.42)

In steady state the transport < v x > vanishes. The broken curve in Fig. (11) gives the sum ∇Pi / ni + 0.1 < v z > , which shows a good agreement along the gradient with the solid curve E x . In the region x < 20 we have small oscillations in space (and time), the accuracy of the curve plotted in this region being degraded by the division by ni , due to the low density ni and large ∇Ti appearing close to the surface.

The Method of Characteristics for the Numerical Solution…

Figure 12. Plot of niEx (solid curve),

+ 0.1ni < v z > , (dash-dotted curve) , ∇Pi

19

(dotted curve), and

∇Pi + 0.1ni < v z > ( broken curve), (ni/10 is also plotted for reference).

Figure 13. Charge (ni –ne).

We plot in Fig. (12) the quantities

ni E x , ∇Pi ,

0.1 ni < v z > and the sum

∇Pi + 0.1 ni < v z > . We note that there is a very nice agreement for the relation ni E x = ∇Pi + 0.1 ni < v z > (here the density ni / 10 is plotted with the dash-three-dots curve to locate the profiles with respect to the gradient). The electric field should interact with the constant magnetic field to give an ExB drift in the poloidal direction ( there is no shear in this drift in the flat part of the electric field, which can explain the absence of turbulence at the plasma edge observed in H-mode tokamaks). The charge σ appearing in the system is calculated by the code and amounts to –0.34197 at t = 500. The charge collected and accumulated at x = 0, which defines E x x = 0 from Eq. (2.20), is 0.34535. The difference between these two numbers is ≈ 0.00338, which is E x x = L from Eq. (2.21). We see also from Figs. (11,12) that inside the plasma at the right boundary, in the flat part of the density where ∇Pi = 0 , the constant electric field is exactly compensated by the Lorentz force due to

20

M. Shoucri

the poloidal drift 0.1 < v z > , while along the gradient the electric field is essentially balanced by ∇Pi / ni (the electric drift is equal and opposite to the diamagnetic drift). Fig. (13) shows

the charge density (ni − ne ) at t = 500, which illustrates how the combined effect of the steep profile at a plasma edge and the large ion orbits (large ratio ρ i / λDe ) leads to a charge separation at a plasma edge along the gradient, when the electrons frozen to the magnetic field cannot move across the field to compensate the charge separation caused by the finite ion gyroradius. Fig. (14) shows the potential. Figs (15) and (16) show the temperatures Tix and Tiz (solid curves). The broken lines represent the pressures ni Tix and ni Tiz which follow closely the curve of the density ni . Thus close to the floating limiter a complex sheath structure is formed which governs the plasma-wall transition.

Figure 14. Potential profile.

Figure 15. Temperature Tx.

The Method of Characteristics for the Numerical Solution…

21

Figure 16. Temperature Tz.

2.3. Vlasov-Maxwell Equations for Laser-Plasma Interaction Two systems of equations for 1D laser-plasma interaction will be discussed in this chapter. In the first one, presented in this section, we consider a linearly polarized electromagnetic wave [21]. This system is solved using a fractional step method and uses a cubic spline interpolation to solve for the advection term in the reduced one dimensional equations. In the second system, to be presented in section 3.3 a fully relativistic code is used [52], and the wave is circularly polarized. In this case, to advance the equations in time, we shall use a tensor product of cubic B-spline for a two dimensional interpolation along the characteristics. Comparison for the results obtained by the two methods will be provided at the end of section 3.3, both from the physical point of view and from the numerical point of view to underline the accuracy of the cubic spline interpolation. In the model we present in this section, a linear polarization of the electromagnetic wave is assumed. Time t is normalized to the inverse electron plasma frequency

−1 ω pe , length is

normalized to l 0 = cω pe , velocity and momentum are normalized respectively to the −1

velocity of light c and to M e c , where M e is the electron rest mass. The one-dimensional Vlasov equations for the electron distribution function f e ( x, p xe , t ) and the ion distribution function f i ( x, p xi , t ) are given by [20]:

∂f e,i ∂t

+ me ,i

∂f e,i

p xe,i 2 1/ 2

(1 + (me,i p xe,i ) )

∂x

∓ ( E x + v ey,i B z )

∂f e,i ∂p xe,i

= 0.

(2.43)

The indices e and i refers to electrons and ions. In our normalized units me = 1 , for the electrons, and mi = M e / M i for the ions. The relativistic correction in this case is given by

22

M. Shoucri

γ e,i = (1 + (me,i p xe,i ) 2 )1 / 2 . For the velocity in the direction normal to the gradient, we have the following relation:

∂v ey,i ∂t

= ∓ m e ,i E y .

(2.44)

The electric field is calculated from the relation E x = −

∂ϕ , and the potential ϕ is ∂x

calculated from Poisson equation:

∂ 2ϕ = f e ( x, p xe ) dp xe − ∫ f i ( x, p xi ) dp xi ∂x 2 ∫ We note also that the canonical momentum

. (2.45)

Py = v ey,i / me ,i ∓ a y = 0 . Hence

v ey,i = ± me,i a y , which when derived with respect to time leads to Eq.(2.44) ( E y = −∂a y / ∂t ). a y = eAy / M e c is the normalized y component of the vector potential. The linearly polarized electromagnetic field propagates in the x direction with an electric field Ey in the y direction, normalized to ω pe M e c / e , and a magnetic field B z = ∂a y / ∂x in the z direction, normalized to

ω pe M e / e . We define the quantity E ± = E y ± B z , which obeys

the equation:

(

∂ ∂ ± ) E ± = − J y = v ey ∫ f e ( x, p xe , t ) dp xe − v iy ∫ f i ( x, p xi , t ) dp xi = v ey ne − v iy ni . (2.46) ∂t ∂x The Hamiltonian associated with this system is given by:

H e ,i =

1 1 + (me,i p xe,i ) 2 me ,i

(

We can write Eq.(2.43) in the form:

[H

e ,i

, f e ,i ] =

∂f e ,i ∂t

)

1/ 2

±ϕ +

1 me,i Ay2 . 2

(2.47)

+ [H e ,i , f e ,i ] = 0 , where the Poisson bracket:

∂H e ,i ∂f e ,i ∂p xe ,i ∂x

−

∂H e ,i ∂f e ,i ∂x ∂p xe,i

.

(2.48)

Eqs.(2.43) are solved by a fractional step [20]. The momentum space is divided into Np cells between − p x max e ,i and + p x max e ,i . The length L of the system is divided into Nx cells.

The Method of Characteristics for the Numerical Solution…

23

The fractional step method involves the following steps to advance Eq.(2.43) in time from tn to tn+1 :

Step1- For a time step Δt / 2 , we calculate

f e*,i ( x, p xe,i , t n +1 / 2 ) = f e,i ( x − (me ,i p xe,i / γ e ,i )Δt / 2, p xe,i , t n ) . Step2- Calculate the fields at time t n +1 / 2 using

(2.49)

fe*,i , then use these values to shift for

Δt

in the direction p xe ,i the distribution functions:

f e*,i* ( x, p xe,i , t n +1 / 2 ) = f e*,i ( x, p xe,i ± ( E xn +1 / 2 + v ey,in +1 / 2 B zn +1 / 2 )Δt , t n +1 / 2 ) . (2.50) Step3- Shift again for a time step Δt / 2 in x space:

f e ,i ( x, p xe,i , t n +1 ) = f e*,*i ( x − (me,i p xe,i / γ e ,i )Δt / 2, p xe,i , t n ) .

(2.51)

The shifts in Eqs.(2.49-2.51) are done using cubic spline interpolation (see Appendix B). The solution of Eq.(2.44) between tn and tn+1 is given by the time centered scheme:

v ey,i ( x, t n +1 ) = v ey,i ( x, t n ) ∓ Δt

E + ( x, t n +1 / 2 ) − E − ( x, t n +1 / 2 ) . 2

(2.52)

Eqs.(2.46) are solved using the centered scheme with Δx = Δt :

E ± ( x ± Δt , t n +1 / 2 ) = E ± ( x, t n −1 / 2 ) − ΔtJ y ( x ± Δt / 2, t n ) . with

J y ( x ± Δt / 2, t n ) =

J y ( x ± Δx, t n ) + J y ( x, t n ) 2

(2.53)

.

We integrate exactly along the vacuum characteristic with Δx = Δt , we can write the following numerical scheme for Eq.(2.53):

1 Δt ( ne ( x, t n +1 / 2 )v ey ( x, t n ) + 2 e ne ( x ∓ Δx, t n −1 / 2 )v y ( x ∓ Δx, t n ) − ni ( x, t n +1 / 2 )v iy ( x, t n ) + . (2.54)

E ± ( x, t n +1 / 2 ) = E ± ( x ∓ Δx, t n −1 / 2 ) +

ni ( x ∓ Δx, t n −1 / 2 )v iy ( x ∓ Δx, t n )) n +1 / 2

The calculation of E x

is done by discretizing Eq.(2.45) using a tridiagonal matrix

similar to Eq.(2.12). The solution of Eq.(2.45) in a finite domain require boundary conditions.

24

M. Shoucri

The system is initially neutral. If a charge Q p appears in the system, it has to disappear through the boundaries. A first approximation used in [21] was to divide this charge equally between the two (left and right) boundaries. A more accurate calculation consists in calculating the charge at the two boundaries by collecting the current hitting these boundaries. Let us assume that Ql , Q p , and Qr are the charges calculated during the simulation respectively at the left boundary, in the plasma, and at the right boundary. Ql can be calculated by collecting the current at the left boundary from the relation :

∂E x ∂t

= −J x x =0

x =0

= − ( J xi − J xe ) x =0 .

(2.55)

t

From which :

and

Ex

J xe,i

x =0

x =0

= −∫ J x

= m e ,i

0

0

p xe,i

∫γ

−∞

x =0

dt = Ql .

(2.56)

f e,i (0, p xe,i , t )dp xe,i

e ,i

t

and a similar expression at the right boundary: E x

∞

and

J xe,i

x= L

= m e ,i ∫ 0

p xe,i

γ e ,i

x=L

= −∫ J x 0

x=L

dt = Qr .

(2.57)

f e,i ( L, p xe,i , t )dp xe,i

Integrating Eq.(2.45) over the domain (0, L) , we get : L

Ex

x=L

− Ex

x =0

= ∫ (ni − ne )dx = Q p

(2.58)

0

where Qp is the charge appearing in the plasma. The charge appearing at the right boundary is

Qr = E x

x=L

, and the charge appearing at the left boundary is Ql = E x

x =0

. Eq.(2.58) is

also written Ql + Q p − Qr = 0 . This relation is usually verified by the code. To take into account any imbalance in this relation, let us consider the electric field E x ( x = x0 ) at a point x0 far to the left of the simulation box and boundaries. The electric field is the sum of the fields from Ql , Q p , and Qr . Since a plate of charge q gives an electric field E x = q / 2

The Method of Characteristics for the Numerical Solution…

25

for a point to the right of the plate ( and E x = − q / 2 for a point to the left ), we can write with the convention of signs in Eq.(2.58) [53]:

E x ( x0 ) = (−Ql − Q p + Qr ) / 2 .

(2.59)

Now if we move from x = x 0 to x = 0 + , just inside the left boundary, Ql is now to our left, so it will contribute to the field by Ql / 2 instead of − Ql / 2 , and the electric field is now given by:

E x ( x = 0 + ) = (Ql − Q p + Qr ) / 2 .

(2.60)

This is the boundary condition used in [53]. Now consider the case when the charges are exactly balanced in the system, then as we mentioned before Ql + Q p − Qr = 0 , or

Q p = Qr − Ql , and substituting in Eq.(2.60), we get E x ( x = 0) = Ql , which is the result in Eq.(2.56). Finally we note that the pump wave is penetrating the plasma at the left boundary at x=0 where we set E ( x = 0) = 2 E0 sin(ω 0 t ) , E ( x = 0) = 0 , for the solution of Eqs.(2.53) . +

−

The normalized wave amplitude E 0 = E am e /(ω epw M e c) ≡

Vosc / c n / nc

amplitude and nc the critical density. This value of

E 0 = (0.00854265λ I 0 ) / n / nc .

We

use

the

, where Eam is the wave

E 0 can also be written

parameters

of

[53],

where

λ = 0.527 μm for the laser, and n/nc =0.032, and the intensity I0=100 (in units of 1014 W/cm2 ). This results in E 0 = 0.25 .

Figure 17. Frequency spectrum of E+.

26

M. Shoucri

Figure 18. Wavenumber spectrum of E+.

We use again the same parameters as in [53]. The laser pump is

ω 0 = 5.59ω pe and the

laser wavenumber is k 0 = 5.5(ω pe / c) . For the scattered mode we have and k SRS = 4.4(ω pe / c) . For the plasma wave we have

ω SRS = 4.478ω pe

ω epw = 1.1124ω pe , and

k epw = 9.86(ω pe / c) . The electron thermal velocity is vTe = 0.026c , Te / Ti = 3.5 . The length of the system is L = 50.265 , and N x = 5000 grid points in space, N v = 256 grid points in velocity space for electrons and 128 for ions. Δx = Δt =0.0105 . We show in +

Fig.(17) the results obtained for the frequency spectrum of the electromagnetic field E at the position x=5 at t=60. We can identify the contribution of the pump and scattered mode at

ω 0 and ω SRS . Fig.(18) shows the wavenumber spectrum for E + at t=60, where again we

Figure 19. Frequency spectrum for the plasma wave.

The Method of Characteristics for the Numerical Solution…

27

Figure 20. Wavenumber spectrum for the plasma wave.

can identify the contribution of the laser pump k 0 = 5.5(ω pe / c) and the scattered mode at

k SRS = 4.4(ω pe / c) . Fig.(19) shows the frequency spectrum of the plasma wave showing the

ω epw = 1.1124ω pe peak and a peak at ω h = 11.18ω pe , i.e. the harmonic of the pump

wave

ω 0 . We can identify in Fig.(20) the wavenumber peak at k epw = 9.86(ω pe / c) for the

plasma wave, followed by a small neighbouring peak at k h = 11(ω pe / c) , i.e. the harmonic of the pump wave k 0 . The harmonic peaks at k h and

ω h in Fig.(19) and (20) result from the

v y B z term in Eq.(2.43). This can also be verified in the results in Fig.(7) of [54] where we see the response of the plasma at the harmonic of the electromagnetic wave frequency, when using a model similar to what has been presented in this section and in Eq.(2.43) applied to the problem of inductive coupling. Indeed, if we assume a linearly polarized wave:

E = (0, E y ,0) , we can write in a linear analysis E y = E 0 cos(ψ ), ψ = (k 0 x − ω 0 t ) . Faraday’s law is:

∂E y ∂B = (0,0,− ). ∂t ∂x

(2.61)

Then B = (0,0, B z ) with B z = B0 cos(ψ ), and B0 = E 0 k 0 / ω 0 . Also from Eq.(2.44)

v = (0, v y ,0) with v y = −v 0 sin(ψ ), and v0 = E 0 / ω 0 . The longitudinal Lorentz force 1 v y B z = − k 0 v 02 sin(2ψ ) . This drive a longitudinal response at the 2nd harmonic of the 2

light wave.

28

M. Shoucri

Figure 21. Contour plot and 3D view for the phase-space of the distribution function from x=5.1 to x=9.8 at

−1 t = 60ω pe .

Figure 22. Longitudinal electric field.

The Method of Characteristics for the Numerical Solution…

29

Figure 23. Charge (ni–ne).

Fig.(21) shows the phase-space structure in contour plot and 3D view of the tail of the electron distribution function between x= 5.1 and x=9.8 at t=60. Note the clear structure of the vortices in Fig.(21), in the low density regions of the phase-space, without numerical noise Fig.(22) shows the electric field E x at t=60 across the simulation box, and Fig.(23) shows the charge (ni –ne ). We note the system is solved for ions and electrons, but the response of the ions at t=60 is still negligible. In the conclusion of this section 2 , we stress the importance of the cubic spline and the good performance of the interpolation technique, having low numerical diffusion and dispersion and high accuracy. We also note the numerical stability of the numerical code.

3. Problems Involving the Interpolation along the Characteristic Curves in Two Dimensions The problems studied in section 2 for the Vlasov equation dealt essentially with the fractional step methods where the interpolation along the characteristic curves was carried out in 1D using a cubic spline. We present in this section examples where the interpolation along the characteristic curves is carried in two dimensions , using a tensor product of cubic B-splines . The integration along the characteristics in higher dimensions applied to Eulerian Vlasov codes has been formulated sometime ago in [15,16], and only recently applied [6,9]. The first example we present in this section is the solution of the guiding-center equations in 2D ( which are the equations of a plasma in a strong magnetic field) to study the Kelvin-Helmholtz instabilities. These equations are isomorphic with the Euler equations that govern 2D inviscid incompressible fluids in hydrodynamics [55-60]. We use this example to introduce in section 3.1 the methods of 2D interpolation discussed in section 1 and in [6,9]. In section 3.2 we will reconsider the problem of the formation of an electric field at a plasma edge, presented in section 2.2 , however we use this time a cylindrical geometry in the Vlasov equation. We will consider next in section 3.3 a case of laser-plasma interaction similar to what has been presented in section 2.3 , treated however with a circularly polarized wave and a fully relativistic Vlasov equation. The results obtained in this section will be compared both from the physical point of view as well as from the numerical point of view with the corresponding

30

M. Shoucri

results in sections 2.2 and 2.3, especially concerning the accuracy of the results and of the numerical techniques using cubic spline interpolation. Finally the problem of solving a reduced set of magnetohydrodynamic equations to study the problem of magnetic reconnection will be presented in section 3.4.

3.1. Solution of the Guiding-Center or Euler Equations In the two-dimensional space r = ( x, y ) , the evolution of a plasma in a strong magnetic field [55-60] B = Be z is governed by the guiding center equation :

∂ρ ∂ρ + VD . = 0. ∂t ∂r with the drift or guiding-center velocity VD = e z ×

(3.1)

∂φ ∂φ , E=− , and φ is calculated ∂r ∂r

from Poisson equation:

Δφ = − ρ The initial condition for the charge density is

(3.2)

ρ (r, t = 0) = ρ 0 (r ) . Eqs.(3.1) and (3.2)

are isomorphic to the Euler equations that govern 2D inviscid incompressible fluids in hydrodynamics, where the charge density corresponds to the flow vorticity, and the potential φ corresponds to the stream function [57]. For a plasma in a strong magnetic field, the particle motion along the magnetic field B and across the magnetic field B are decoupled, the velocity perpendicular to the magnetic field is the guiding-center velocity VD and the Vlasov equation can be reduced to the guiding-center plasma model in Eqs(3.1-3.2). These equations are also the simplest form for the gyro-kinetic or drift-kinetic Vlasov equation, when the kinetic motion along the magnetic field is taken into account and is coupled to the guidingcenter motion across the magnetic field, as for instance in the case when the magnetic field is slightly tilted with respect to e z [32-38,61]. The system in Eqs.(3.1-3.2) is solved on a rectangular domain L x × L y . Periodic boundary conditions are used in the y direction, and Dirichlet boundary conditions are used in the x direction with

φ ( x = 0, L x , y ) = 0 ,

ρ ( x = 0, L x , y ) = 0 . Following the steps of what has been presented in Eqs.(1.8-1.14), Eq.(3.1) can be integrated along the characteristic curves given by the solution of the equations:

dr = VD (r, t ) . dt

(3.3)

The Method of Characteristics for the Numerical Solution… We write that the value of

31

ρ along the characteristic curves is constant (see Eq.(1.11)): ρ (r, t n+1 ) = ρ (r (t n−1 ), t n −1 ) .

(3.4)

We assume the value of r (t = t n +1 ) = ri , where the ri are grid points. A numerical scheme accurate to second order in Δt for the solution of Eq.(3.3) is given by the following leapfrog scheme (similar to Eq.(1.12)):

ri − r (t n−1 ) ⎛ r + r (t n−1 ) ⎞ = VDn ⎜ i ⎟. 2Δt 2 ⎝ ⎠ where VD = e z × n

(3.5)

∂φ (r, t n ) , and E(r, t n ) is computed by solving Poisson equation for ∂r

ρ (r, t n ) at t = t n . To solve Eq.(3.2), we Fourier transform this equation in the periodic y direction. We denote by

ρ k and φ k the Fourier transform of ρ and φ in the periodic y y

y

direction. We have:

φ ( x, y ) = ∑ e

ik y y

φ k ( x ) ; ρ ( x, y ) = ∑ e

ik y y

y

ky

ρ k ( x) y

ky

Which by substituting in Eq.(3.2) gives the following result:

∂ 2φ k y ( x ) ∂x

2

− k y2φ k y ( x) = − ρ k y ( x) .

(3.6)

We can derive the following tridiagonal matrix by discretizing Eq.(3.6) in the x direction [61]:

(1 − C k y )φ k y i +1 − (+2 + 10C k y )φ k y i + (1 − C k y )φ k y i −1 = −

where C k y = conditions for

k y2 Δx 2 12

Δx 2 ( ρ k y i −1 + 10 ρ k yi + ρ k y i +1 ) . (3.7) 12

. The tridiagonal matrix in Eq.(3.7) is solved with Dirichlet boundary

φ k , which is then Fourier transformed back to get φ . Again the implicit y

equation in Eq.(3.5) is solved by iteration as in Eq.(1.13), to calculate Δ r = (ri − r (t n −1 )) / 2 as follows:,

Δkr+1 = ΔtVD (ri − Δkr , t n ) .

32

M. Shoucri From Eqs.(1.14) and (3.4) the new value

.

ρ (ri , t n +1 ) is calculated from the relation:

ρ (ri , t n + Δt ) = ρ (ri − 2 * Δ r , t n − Δt ) .

More explicitly we have to calculate (see Fig.(1))

(3.8)

ρ ( x i − 2 Δ x , y j − 2 Δ y , t n − Δt ) ,

where Δ x and Δ y are obtained by iteration by solving :

where V Dy =

Δkx+1 = ΔtV Dx ( xi − Δkx , y j − Δky , t n )

(3.9)

Δky+1 = ΔtVDy ( xi − Δkx , y j − Δky , t n )

(3.10)

∂φ ∂φ 0 0 ; V Dx = − . We start with Δ x = 0 , Δ y = 0 . Usually two or three ∂x ∂y

iterations are sufficient for convergence. The interpolations in Eqs.(3.8-3.10) are carried out using a 2D cubic B-splines defined as a tensor product of one dimensional cubic B-splines. We first evaluate the coefficients η ij from the values of the function at grid points (details can be found in [6], we have however to introduce appropriate modification to the boundary condition in the direction y, to take into account the periodic boundary condition as in [51]): Nx N y

ρ ( xi , y j ) = ∑∑η ij Bi ( x) B j ( y ) .

(3.11)

i =0 j =0

The cubic B-spline has been defined in Appendix C . Eq.(3.11) generalizes to two dimensions the results presented in Appendix C. Then the value of the function at interstitial points ρ ( xi + Δ x , y j + Δ y ) is given by: 3

3

ρ ( xi + Δ x , y j + Δ y ) = ∑∑η i −κ , j −l bκx bly .

(3.12)

κ =0 l =0

where i ≡ xi , Δ x = x − xi , and j ≡ y j , Δ y = y − y j , bκ are defined in Appendix C, x

and a similar definition holds for bl by substituting y

Δ y for Δ x in the expression of bκx .

We finally note that in [9], in order to accelerate the calculation, a linear interpolation has been used in the intermediate step in Eqs. (3.9-3.10). This turned out however to introduce strong numerical diffusion [43,44]. We apply the previously described numerical method to study the stability of a sinusoidal profile [58,59]. Any function of one space variable is an equilibrium solution to Eqs(3.1-3.2). We consider the sinusoidal profile ρ 0 = sin( x) . From Eq.(3.2) the self-consistent potential

The Method of Characteristics for the Numerical Solution… is given by

φ 0 = sin( x) . A necessary condition for the flow V D 0 =

33

∂φ 0 = cos( x) to have ∂x

an unstable growing solution is that this flow should have a point of inflexion. This is the Rayleigh necessary condition for instability. This instability due to the shear in the velocity flow is called Kelvin-Helmholtz instability. Furthermore if the equilibrium is perturbed with a perturbation of the form e values of

ω such that

ω ky

− i (ωt + k y y )

, k y = 2π / L y , there exists an eigensolution with real

= V D ( x s ) (the so-called neutrally stable eigensolution), where xs

is the point of inflexion, and one can construct unstable solutions for which

ω ky

→ VD ( x s )

ω tends to zero through positive values. In the present case, we consider a domain with L x = 2π , and L y = 10 . We use Nx =Ny =256 and Δt =0.005 . In as the imaginary part of

the domain 0 ≤ x ≤ 2π where the equilibrium flow is defined we have two points of inflexion

ω ky

located

xs = π / 2

at

x s = 3π / 2 .

and

At

these

points

= VD ( x s ) = cos( x s ) = 0 . In this case ω = ω s = 0 , and a neutrally stable solution

satisfying the boundary conditions of zero at x = 0 and x = 2π is given by [58,59] :

x 2

(3.13)

3 2

(3.14)

φ s = sin( ) .;

k ys = We perturb the equilibrium

ρ 0 as follows: x 2

ρ 0 = sin( x ) + ε sin( ) cos(k y y ) .

(3.15)

with k y = 2π / L y =0.628, and ε = 0.015 . Since we are close to a neutrally stable solution, we can use a Taylor expansion to calculate the growth rate of the instability:

ω ky

=

ω ky

+ (k y − k ys ) ω s , k ys

∂ ∂k y

⎛ω ⎜ ⎜k ⎝ y

⎞ ⎟ . ⎟ ⎠ ω s ,k ys

(3.16)

Details of these calculations have been presented in [58,59], and lead to the following result for the real and imaginary parts of ω :

34

M. Shoucri

Re ω 3 = (k ys − k y ) . ky 2

(3.17)

Im ω 3 = (k ys − k y ) . ky 2

(3.18)

Figure 24. Growth and saturation of the potential at the position x=3Lx /4.

Figure 25. Contour plot of the potential at t=80.

In this case k y < k ys , and the system is unstable [58,59]. From Eq.(3.17), the phase velocity

Re ω is equal to 0.2058. We do verify by following the center of the vortex in ky

Figs(25-26) that the phase velocity of the vortex is indeed 0.2 . Fig.(24) shows the growth and oscillation of the potential , monitored at the position x = 3Lx / 4; y = 0 . The theoretical results from Eqs(3.17-3.18) show Im ω = Re ω = 0.129 , in good agreement with the observations taken from the results of Figs.(24), which shows a growth and an oscillation period of 0.124. Figs(25-26) shows the contour plot of the potential at t=80 and 120 during the saturation phase, and the corresponding charge is given in Figs.(27-28) (dotted curves denote negative values). Note the very nice agreement of the theoretical and numerical results.

The Method of Characteristics for the Numerical Solution…

Figure 26. Contour plot of the potential at t=120.

Figure 27. Contour plot of the charge at t=80.

Figure 28. Contour plot of the charge at t=80.

35

36

M. Shoucri

3.2. The Vlasov-Poisson System in Higher Phase-Space Dimensions: Formation of an Electric Field at a Plasma edge in a Cylindrical Geometry We will discuss in this section, using cylindrical geometry, the problem of the formation of an electric field at a plasma edge which has been studied in section 2.2 using a slab geometry. In cylindrical geometry, it is more convenient to use a tensor product of cubic B-spline to interpolate in 2D velocity space, while in section 2.2 we applied a fractional step associated with 1D cubic spline interpolation. Hence we have the opportunity to compare two completely different codes, and to evaluate the performance of the cubic spline for the solution of the same problem. In the present cylindrical geometry, the external magnetic field is in the z direction, and ( r , θ ) is the poloidal plane. The plasma is assumed uniform in the z and θ directions. Electrons are magnetized along the magnetic field, and consequently have a constant profile. The Vlasov and Poisson equations are written for the deuterons distribution function f i (r , v r , vθ , t ) and for the potential ϕ (r ) with the same normalization and parameters as in section 2.2, as follows (see [6] ):

v v ⎞ ∂f v 2 ⎤ ∂f ∂f i ∂f ⎡ ⎛ + v r i + ⎢ E r + vθ ω ci + θ ⎥ i − ⎜ ω ci v r + r θ ⎟ i = 0 r ⎥⎦ ∂v r ⎝ r ⎠ ∂vθ ∂t ∂r ⎢⎣ Er = −

∂φ ; ∂r

1 ∂ ∂φ r = −(ni − ne ) ; ni = r ∂r ∂r

∫f

i

(3.19)

(3.20)

dv r dvθ

We advance Eq. (3.19) for a time step Δt as follows: Step1-We solve for a time step Δt / 2 , using cubic spline interpolation, the equation:

∂f i ∂f + vr i = 0 ∂t ∂r *

(3.21)

the solution is given by f i ( r , v r , vθ , t + Δt / 2) = f i ( r − v r Δt / 2, v r , vθ , t )

(3.22)

n

We calculate the shift in Eq.(3.22) using 1D cubic spline interpolation as discussed *

before. We then solve Poisson equation in Eq.(3.20) to calculate the electric field E r . Step2-We solve next for a time step Δt the equation:

∂f i ⎡ * vθ2 ⎤ ∂f i ⎛ v v ⎞ ∂f + ⎢ E r + vθ ω ci + ⎥ − ⎜ ω ci v r + r θ ⎟ i = 0 ∂t ⎣ r ⎦ ∂v r ⎝ r ⎠ ∂vθ

(3.23)

Splitting Eq.(3.23) is not straightforward as in Cartesian geometry. This equation is solved using 2D interpolation. The characteristics of this equation are given by:

The Method of Characteristics for the Numerical Solution…

v 2 dv v v dv r = E r + vϑ ω ci + ϑ ; ϑ = −v r ω ci − ϑ r dt r dt r

37

(3.24)

The solution of Eqs.(3.23) is given by:

f i ** ( r , vr , vθ , t + Δt ) = f * ( r , v r − 2a, vθ − 2b, t )

(3.25)

The shift in Eq. (3.25) is effected using a tensor product of cubic B-spline [6,51] for the 2D interpolation, as discussed in section 3.1. However, the quantities a and b in Eq.(3.25) are calculated analytically by solving the equations of the characteristics in Eqs.(3.24), since in the present case an analytic solution is possible, similar to what has been presented in Eqs.(2.38-2.40). This solution gives, to an order O ( Δt 2 ) , at a given position r, the expressions: a=

b=−

Δt 2

Δt 2

⎛ ⎜ E r + ω ci ⎜ ⎝

⎛ v v Δt ⎛ ⎜⎜ vθ + ⎜ ω ci v r + r θ 2 ⎝ r ⎝

2 ⎞ v ⎞ ⎞ vθ + Δt θ (ω ci v r + v r vθ / r ) ⎟ ⎟ ⎟⎟ + ⎟ r ⎠⎠ r ⎠

⎛ vv vv ⎜ ω ci vr + r θ + Δt vr ⎛⎜ ω ci vr + r θ ⎞⎟ − Δt ⎜ r ⎠ 2 r r 2 ⎝ ⎝

(3.26)

v 2 ⎞⎞ v ⎞⎛ ⎛ ⎜ ω ci + θ ⎟⎜⎜ E r + vθ ω ci + θ ⎟⎟ ⎟⎟ (3.27) r ⎠⎠ r ⎠⎝ ⎝

( note that an iterative solution of Eqs.(3.24) as in section 3.1 would give the same results to an order O ( Δt 2 ) ). Step3-We then repeat the step in Eq. (3.22) for Δt / 2 to calculate f n +1 from f ** . The initial profiles are the same as in section 2.2, written in the cylindrical geometry as:

ni (r ) = ne (r ) = 0.5 (1 + tanh (( R − r − L / 5) / 7.)) ;

(3.28)

with a similar profile for the frozen electrons. The positive r direction is pointing towards the right, or outside the plasma. R is the plasma radius and L the width of the edge (taken to be 175 as in section 2.2). The initial ion distribution is given by: 2

2

e − ( vr +vθ ) / 2Ti f i (r , v r , vθ ) = ni (r ) 2πTi

(3.29)

We assume that the deuterons hitting the wall surface or limiter at r= R are collected by a floating cylindrical vessel. Since the magnetized electrons do not move in the r direction across the magnetic field, there is no electron current collected at the floating vessel.

38

M. Shoucri

Therefore we have at r= R the relation (we stress that the subscript i denotes here ion contribution):

∂E r = − J ri r = R ∂t r = R

t

E r r = R = − ∫ J ri r = R dt

or

(3.30)

0

Integrating Eq. (3.20) over the domain (R-L, R), we get for the total charge σ appearing in the system: R

R Er

r=R

− ( R − L) E r

r = R− L

=

∫ (n

i

− ne )rdr = σ

(3.31)

R− L

which is the equivalent to Eq.(2.21) obtained for the slab geometry. We assume that the gyrating plasma ions (deuterons) are allowed to enter or leave at the left boundary. The electric fields at the left boundary r = R - L and at the wall r = R must satisfy Eq. (3.31). We use a very large value of R (R = 10000 Debye lengths in the present calculation, so vθ2 / r is negligible), so that the system should behave essentially as a Cartesian system. Indeed we recover the same results as those which have been presented in Cartesian geometry in section 2.2. These results have been presented for cylindrical geometry in Fig.(1-6) in [51], and are identical to Figs(11-16) ( if we take into account the mirroring due to the fact that in the Cartesian geometry and in the cylindrical geometry, the edge gradients are in opposite directions). Fig. (29) shows at t = 500 the plot of the electric field Er (solid curve, we concentrate on the region less than 100 Debye lengths from the boundary to emphasize the edge region). To position the profiles in Fig. (29) with respect to the gradient we also plot − ni / 2 in the same figure. The electric field has the direction of pushing the ions back to the interior of the plasma (and interact with the magnetic field to give a poloidal drift rotation, note also that in the flat part of the electric field, this drift has no shear, which can explain the absence of turbulence). The dash-dotted curve gives the Lorentz force, which in our normalized units is given by − < vθ > ω ci / ω pi , and the dotted curve gives the pressure

force ∇Pi / ni , Pi = 0.5 ni (Tir + Tiθ ) , with the following definition in cylindrical geometry:

Tir ,θ ( r ) =

< v r , θ >=

1 dv r dvθ ( v r ,θ − < v r , θ > ) 2 f i (r , v r , vθ ) ni ∫

1 dv r dvθ v r , θ f i (r, v r , vθ ) ; ni ∫

ni ( r ) = ∫ dv r dvθ f i (r, v r , vθ )

(3.32)

(3.33)

In steady state the transport < v r > vanishes. The broken curve in Fig. (29) gives the sum ∇Pi / ni − 0.1 < vθ > , which shows a very good agreement along the gradient with the solid curve for E r . In the region less than 20 Debye lengths from the wall, we have small

The Method of Characteristics for the Numerical Solution…

39

oscillations in space (and time), the accuracy being degraded by the low density ni and large

∇Ti

appearing

close

to

the

surface.

We

plot

in

Fig. (30)

the

quantities

n i E r , ∇Pi , − 0.1 n i < vθ > and the sum ∇Pi − 0.1n i < vθ > . We see that there is a very nice agreement for the relation ni E r = ∇Pi − 0.1 ni < vθ > (the density − ni / 10 is also plotted to locate the profiles with respect to the gradient). The charge σ / R appearing in the system and calculated by the code from Eq.(3.31) amounts to -0.360 at t = 500. The collected charge calculated from Eq.(3.29) at r = R is 0.364, hence E r r = R = −0.364 . The difference E r r = R −σ / R as calculated from Eq.(3.30) gives for E r r = R − L the value of -0.004, which is very close to the value obtained by the code at R – r = 175 (see Fig. (30)). We see also from Fig. (30) that at the left boundary inside the plasma in the flat part of the density where ∇Pi = 0 , the electric field is compensated by the Lorentz force due to the poloidal drift − 0.1 < vθ > , while along the gradient the electric field is essentially balanced by ∇Pi / ni . Figs. (3-6) in [51] are reproducing essentially Fig.(13-16) of section 2.2 (taking into consideration the mirroring due to the difference in the positive direction). We see that the results in section 2.2 obtained in Cartesian geometry by a fractional step method associated with 1D cubic spline interpolation are the same as those obtained in this section using a cylindrical geometry , and associated with 2D interpolation in velocity space with a tensor product of cubic B-spline. By using two different numerical techniques based on the cubic spline interpolation with two different coordinate systems, we get identic results for the same problem. The curves in Figs(11-16) and in Figs(1-6) of [51] gives essentially identical results. This illustrates the accuracy of the method of characteristics used , and of the cubic spline used for the numerical interpolation.

Figure 29. Plot, for the cylindrical geometry, of the electric field Er (solid curve), the Lorentz force

− 0.1 < vθ > sum ∇Pi

(dash- dotted curve), the pressure force ∇Pi

/ ni − 0.1 < vθ >

plotted for reference).

/ ni (dotted

curve), and the

(broken curve). The density -ni/2 is is also plotted (dash-three-dots curve,

40

M. Shoucri

Figure 30. Plot of niEr (solid curve),

− 0.1ni < vθ > , (dash-dotted curve) , ∇Pi

(dotted curve), and

∇Pi − 0.1ni < vθ > ( broken curve), (-ni/10 is also plotted for reference).

3.3. One-Dimensional Fully Relativistic System for the Problem of LaserPlasma Interaction The problem of laser-plasma interaction treated in section 2.3 with a linear polarization will be repeated in this section with a full relativistic equation with a circular polarization. In the present case the fractional step will not be used , we will rather apply a 2D interpolation using a tensor product of cubic B-splines. When studying similarities or differences in the results, attention will be given to the accurate performance of the cubic spline interpolation. The general form of the Vlasov equation is written for the present problem ( using the same normalization as in section 2.3 ) in a 4D phase-space for the electron distribution function Fe ( x, p xe , p ye , p ze , t ) and the ion distribution function Fi ( x, p xi , p yi , p zi , t ) (one spatial dimension) as follows [52]:

∂Fe ,i ∂t with

+ me ,i

p xe,i ∂Fe ,i

γ e ,i

∂x

∓ (E +

pxB

γ e ,i

)⋅

∂Fe,i ∂p e ,i

= 0.

γ e ,i = (1 + me2,i ( p xe2 ,i + p ye2 ,i + p ze2 ,i ) )

1/ 2

(3.34)

(3.35)

(the upper sign is for electrons and the lower sign for ions, and subscripts e or i denote electrons or ions respectively). Again in our normalized units me = 1 and mi =

Me . Mi

Eq.(3.34) can be reduced to a two-dimensional phase-space Vlasov equation if the canonical momentum Pce,i connected to the particle momentum p e ,i by the relation Pce ,i = p e ,i ∓ a is chosen initially as zero. a = eA / M e c is the normalized vector potential. For a particle in an

The Method of Characteristics for the Numerical Solution…

41

electromagnetic wave propagating in a one-dimensional spatial system, we can write the following Hamiltonian:

H e ,i = where

(

1 1 + me2,i ( Pce,i ± a ) 2 me ,i

)

1/ 2

±ϕ .

(3.36)

ϕ is the electrostatic potential. Choosing the Coulomb gauge ( diva = 0 ) , we have

for the vector potential a = a ⊥ ( x, t ) , and we also have the following relation along the longitudinal direction:

dPcxe,i dt

=−

∂H e,i

(3.37)

∂x

And since there is no transverse dependence :

dPc ⊥e ,i dt

= −∇ ⊥ H e ,i = 0 .

(3.38)

This last equation means Pc ⊥ e ,i = const. We can choose this constant to be zero without loss of generality, which means that initially all particles at a given (x,t) have the same perpendicular momentum p e ,i = ± a ⊥ ( x, t ) . The Hamiltonian now is written:

H e ,i =

1/ 2 1 1 + me2,i p xe2 ,i + me2,i a ⊥2 ( x, t ) ± ϕ ( x, t ) . m e ,i

(

)

(3.39)

The 4D distribution function Fe ,i ( x, p x , p ⊥ , t ) can now be reduced to a 2D distribution function f e ,i ( x, p xe ,i , t ) corresponding to Eq.(3.39):

df e,i dt

=

∂f e,i ∂t

+

∂H e,i ∂f e,i ∂p xe,i ∂x

−

∂H e,i ∂f e,i ∂x ∂p xe,i

= 0.

(3.40)

Which gives the following Vlasov equations for electrons and ions::

∂f e ,i ∂t Where

+ me ,i

p xe,i ∂f e ,i

γ e ,i ∂x

(

+ (∓ E x −

me ,i ∂a ⊥2 ∂f e,i ) = 0. 2γ e ,i ∂x ∂p xe,i

γ e,i = 1 + (me,i p e,i )2 + (me,i a ⊥ )2

)

1/ 2

.

(3.41)

42

M. Shoucri

Ex = −

∂a ∂ϕ and E ⊥ = − ⊥ ∂x ∂t

(3.42)

and Poisson equation is given by Eq.(2.45). The transverse electromagnetic fields E y , B z and E z , B y for the circularly polarized wave obey Maxwell’s equations. With

E ± = E y ± B z and F ± = E z ± B y , we have:

(

∂ ∂ ∂ ∂ ± )E ± = − J y . ; ( ∓ )F ± = − J z ∂t ∂x ∂t ∂x

(3.43)

Which are integrated along their vacuum characteristic x=t. In our normalized units we have the following expressions for the normal current densities:

J ⊥ e ,i = − a ⊥ m e ,i ∫

J ⊥ = J ⊥ e + J ⊥i ;

f e ,i

γ e ,i

dp xe,i .

(3.44)

The numerical scheme to advance Eq.(3.41) from time tn to tn+1 necessitates the ±

±

knowledge of the electromagnetic field E and F at time tn+1/2 . This is done using a scheme similar to Eq.(2.53), where we integrate Eq.(3.43) exactly along the vacuum characteristics with Δx = Δt , to calculate E have a

n +1 ⊥

= a − ΔtE n ⊥

n +1 / 2

calculate E x

n +1 / 2 ⊥

± n +1 / 2

and F

± n +1 / 2

, from which we calculate a

. From Eq.(3.42) we also

n +1 / 2 ⊥

= (a ⊥n +1 + a ⊥n ) / 2 . To

, two methods have been used. A first method calculates E x from f e ,i n

n

using Poisson equation, then we use a Taylor expansion::

Δt ⎛ ∂E ⎞ ⎛ Δt ⎞ = E + ⎜ x ⎟ + 0.5⎜ ⎟ 2 ⎝ ∂t ⎠ ⎝ 2⎠ n

E

n +1 / 2 x

⎛ ∂ Ex ⎜⎜ 2 ⎝ ∂t 2

⎛ ∂E ⎞ with ⎜ x ⎟ = − J xn ; ⎝ ∂t ⎠ n

+∞

J xn = mi

and

p xi

∫γ

−∞

n +1 / 2

n

+∞

n +1 / 2

n

⎛ ∂ 2 Ex ⎞ ⎜⎜ 2 ⎟⎟ ; ⎝ ∂t ⎠

⎞ ⎛ ∂J ⎞ ⎟⎟ = −⎜ x ⎟ ⎝ ∂t ⎠ ⎠

f i n dp xi − me

i

A second method to calculate E x which E x

2

n x

p xe

∫γ

−∞

n

.

(3.45)

f en dp xe

e

is to use Ampère’s equation:

∂E x = − J x , from ∂t

= E xn −1 / 2 − ΔtJ xn . Both methods gave the same results. The boundary

conditions are the same as what has been discussed in section 2.3. Now given f e ,i at mesh n

points (we stress here that the subscript i denotes the ion distribution function), we follow the

The Method of Characteristics for the Numerical Solution… n +1

same steps as in section 3.1 to calculate the new value f e ,i

43

at mesh points from the

relations:

f en,i+1 ( X e ,i ) = f en,i ( X e,i − 2Δ X e ,i ) ; .

(3.46)

where Δ X e,i is the two dimensional vector:

Δ X e ,i =

X e,i

is

the

two

Δt V ( X e ,i - Δ X e ,i , t n +1 / 2 ) . 2

dimensional

(3.47)

X e ,i = (x, p xe,i ) ,

vector

and

⎛ p xe,i me ,i ∂ ( a ⊥( n +1 / 2) ) 2 ⎞ ⎟ . Eq.(3.47) for Δ X e,i is implicit and is Ve ,i = ⎜⎜ me,i , ∓ E xn +1 / 2 − ⎟ 2 ∂ γ γ x , , e i e i ⎝ ⎠ n +1 n solved again iteratively as in Eq.(3.9-3.10). Then f e ,i is calculated by interpolating f e ,i in Eq.(3.46) in the two dimensions ( x, p xe,i ) using a tensor product of cubic B-splines [6] as discussed for Eqs.(3.11-3.12). We use the same parameters as section 2.3 and in [53]. The pump wave is penetrating the plasma at the left boundary at x=0 where we set E ( x = 0) = 2 E0 sin(ω 0 t ) and +

F − ( x = 0) = 2 E0 cos(ω 0 t ) . The laser pump is ω 0 = 5.59ω pe and the laser wavenumber is k 0 = 5.5(ω pe / c ) . For the scattered mode we have

ω SRS = 4.478ω pe

and

k SRS = 4.4(ω pe / c) . For the plasma wave we have ω epw = 1.1124ω pe , and k epw = 9.86(ω pe / c) . The electron thermal velocity is vTe = 0.026c , Te / Ti = 3.5 . The length of the system is L = 50.265 , and N x = 5000 grid points in space, N v = 256 grid points in velocity space for electrons and 128 for ions. Δx = Δt =0.0105 . Figs.(31-37) show the results obtained when using the present fully relativistic model. Fig(31) for the frequency +

spectrum of the electromagnetic wave E at x=5 and t=60 is very close to Fig.(17). We can identify the contribution of the pump and the scattered mode at ω 0 = 5.59ω pe and

ω SRS = 4.478ω pe . Fig.(32) for the wavenumber spectrum of E + is essentially the same as Fig.(18),

we

can

identify

k 0 = 5.5(ω pe / c )

for

the

laser

pump

wave,

and

k SRS = 4.4(ω pe / c) for the scattered mode. Figs.(33) and (34) differ from Fig.(19) and (20) by the absence of the harmonic peaks at

ω h = 11.18ω pe and k h = 11(ω pe / c) . These

harmonic peaks in Fig.(19) and (20) result from the v y B z term in Eq.(2.54) as we explained at the end of section 2.3.

44

M. Shoucri

Figure 31. Frequency spectrum of E+

Figure 32. Wavenumber spectrum of E+.

In circular polarization we have for the pump wave in a linear analysis, following the same notation as at the end of section 2.3, E = E 0 (0, cosψ , sin ψ ) ,

ψ = (k 0 x − ω 0 t ) .

Faragay’s law is:

∂E y ∂E ∂B ). = (0, z ,− ∂x ∂t ∂x ∂a ⊥ and p ⊥ = a ⊥ , we get ∂t p ⊥ = p 0 (0,− sin ψ , cosψ ) . We thus see that pxB is identically zero, p and B being

which gives B = B0 (0,− sin ψ , cosψ ) . From E ⊥ = −

parallel. So in this case there is no 2nd harmonic longitudinal response to the leading order.

The Method of Characteristics for the Numerical Solution…

45

Figure 33. Frequency. spectrum of the plasma.

Figure 34. Wavenumber spectrum of the plasma.

We

(

can

assume

γ e,i ≈ 1 + (me,i p e,i )2

)

1/ 2

following

Eq.(3.41)

that

for

a⊥ → 0 ,

we

have

and the approximation in Eq.(2.43) in section 2.3 becomes valid.

Indeed, for the parameters in [53] used in this chapter, we have estimated at the end of section 2.3 that E 0 = 0.25 . From Eq.(3.42) we can estimate that the amplitude a 0 of a ⊥ is

a 0 = E 0 / ω 0 = 0.044 , which is small. However, Fig.(35) shows the phase-space structure between x=5.1 and x=9.8 at t=60. It shows the vortices structure more important than in Fig.(21), due to the fact that the amplitude of the electric field in the present case where a circularly polarized wave is used, is now

E x2 + E y2 = E0 2 (if the amplitude is reduced to

E0 , then the vortices are similar to Fig.(21)) . Note again the clear picture of the vortices, in the low density region of the phase-space, with very little numerical noise appearing. Figs(36) and (37) show respectively the electric field and the charge (ni –ne ) across the box. Both figures show an enhanced value compared to what is presented in Figs.(22) and (23).

46

M. Shoucri

Figure 35. Contour plot and 3D view for the phase-space of the distribution function from x=5.1 to x=9.8 at

−1 t = 60ω pe .

Figure 36. Longitudinal electric field.

The Method of Characteristics for the Numerical Solution…

47

Figure 37. Charge (nI – ne ).

So we have been able using the technique of cubic spline interpolation to get results from two different models for laser-plasma interaction, using two different numerical codes, which show similarities and differences in the physics associated with the scattering results. Differences observed have been explained as essentially due to different physics associated with the two models, and not to numerical problems. For the model used in section 2.3 with a linearly polarized wave, we used a method of fractional step associated with 1D interpolation using cubic spline, and in the method used in the present section with circular polarization a 2D interpolation in velocity space using a tensor product of cubic B-splines has been used.

3.4. Numerical Solution of a Reduced Model for the Collisionless Magnetic Reconnection In the ideal magnetohydrodynamic (MHD) plasma description, the magnetic field is frozen in the plasma, and its flux through a surface moving with the plasma remains constant. This conservation of the magnetic topology requires that if two plasma elements are initially connected by a magnetic field line, they remain connected by a magnetic field line at any subsequent time, and it constrains the plasma dynamics by making configurations with lower magnetic energy but different topological connection inaccessible. Magnetic field reconnection removes these constraints. It is an important process in high temperature magnetically confined plasma. In this process, the magnetic configuration undergoes a topological rearrangement in a relatively short time, during which the magnetic energy is converted into heat and into kinetic flow energy. Typical situations are in tokamak plasma configurations and in solar flares and coronal loops mass ejections, when strong magnetic fields are present. In the magnetopause it allows particles from the solar wind to enter the magnetosphere. In the present work, we consider a dissipationless two-dimensional configuration with a strong superimposed homogeneous magnetic field perpendicular to the reconnection plane. In the limit of a small ion gyroradius, this two-dimensional system gives a two-fluid equations model where small scale effects related to the electron temperature and electron inertia are retained, but magnetic curvature effects are neglected. We consider a 2D configuration with a strong magnetic field in the ignorable z direction,

48

M. Shoucri

B = B0 e z + ∇ψ x e z , where B0 is constant and ψ ( x, y, t ) is the magnetic flux function. The dimensionless governing equations, normalized to the Alfvén time

τ A and to the

equilibrium scale length Leq , are Hamiltonian and can be cast in a Lagrangian invariant form [62,63,64], similar to what has been presented in section 3.1:

∂G± + [φ ± , G± ] = 0 ; G± = ψ − d e2 ∇ 2ψ ± d e ρ s ∇ 2ϕ ∂t The Poisson brackets

[A, B] = e z .∇Ax∇B ,

(3.48)

and the Lagrangian invariants G ± are

conserved fields advected along the characteristic curves, x ± (t ) :

dx ± (t ) / dt = υ ± ( x ± , t ) , υ ± ( x ± , t ) = e z x ∇φ ± where

φ ± = ϕ ± ( ρ s / d e )ψ . d e

(3.49)

is the electron collisionless skin depth and

ρ s = (M e / M i )1 / 2 v the / ω ci is the ion sound Larmor radius, where vthe is the electron

ω ci is the ion cyclotron frequency. The magnetic flux ψ and the plasma stream function ϕ are given by:

thermal velocity and

ψ − d e2 ∇ 2ψ = (G+ + G− ) / 2 ; d e ρ s ∇ 2ϕ = (G+ − G− ) / 2 If we compare with Eq.(2.4), we see that

φ±

(3.50)

play the role of the single particle

Hamiltonian, and that the two Eqs.(3.48) have the form of 1D Vlasov equations, with x and y playing the role of the coordinate and the conjugate momentum for the equivalent ‘ distribution functions’ G ± of two ‘particle ‘ species with opposite charges in the Poissontype equation for

ϕ , and equal charges in the Yukawa-type equation for ψ

[62,63,64].

We apply a method of integration along the characteristics for the numerical solution of Eq(3.48), similar to what has been discussed in section 1 and for the numerical solution of Eqs.(3.1-3.2). To advance Eq.(3.48) in time, Eq.(3.49) are solved iteratively to determine the departure point of the characteristics ( similar to Eq.(3.8)) , and the values of G ± at these departure points are calculated by a two-dimensional interpolation using a tensor product of cubic B-splines, as discussed for Eqs.(3.9) (see Fig.(1)).The departure point of the characteristics is calculated from the expressions:

Δkx+±1 = Δtυ x ± ( xi − Δkx ± , y j − Δky ± ; t n )

(3.51)

Δky+±1 = Δtυ y ± ( xi − Δkx ± , y j − Δky ± ; t n )

(3.52)

The Method of Characteristics for the Numerical Solution…

49

∂φ ∂φ ± 0 0 ; υ x ± = − ± . We start with Δ x ± = 0 , Δ y ± = 0 , and two or three ∂x ∂y iterations in Eqs.(3.51-3.52) are sufficient for convergence. The solutions G ± at t = (n + 1)Δt are calculated from the expression: where

υ y± =

G ± ( xi , y j , t n +1 ) = G ± ( xi − 2Δ x ± , y j − 2Δ y ± , t n − Δt )

(3.53)

Figure 38. Magnetic flux.

Figure 39. G+ at time t=30.

The code in [62-64] is a finite difference code using filtering and dissipation to remove small scale features which develop. No small scales filtering or dissipation is added to the present code, as is done in [62,63]. Instead, we use a fine grid of N x xN y = 2048 x512 to resolve small details, and stop the calculation when the small details are of the order of the

50

M. Shoucri

Figure 40. G+ at time t=35.

Figure 41. G- at time t=30.

grid size.. We consider an initial equilibrium perturbation

ψ (t = 0) = 1 / cosh 2 ( x) + δψ ( x, y ) , and the

δψ = − 10 − 4 exp( − x 2 /( 2 d e2 )) cos y

d e = ρ s = 0 .2 .

The

equations

are

integrated

is

the

initial

numerically

in

perturbation. the

spatial

domain − 2π < x < 2π , 0 < y < 2π . The domain is periodic in the y direction, and we apply Dirichlet boundary conditions in the x direction. The solution of Eq.(3.48) is followed by a solution of Eqs.(3.50) to determine ψ and ϕ , and these quantities are used to calculate

φ ± , to repeat again the integration of Eq.(3.48). The solution of Eqs.(3.50) is done by Fourier transforming in the periodic y direction, then discretizing the equations in the x direction and solving the resulting tridiagonal system with appropriate boundary conditions, and then Fourier transforming back (details have been presented in section 3.1 , Eqs(3.6-3.7)). During the evolution of the reconnection process, we see in Fig.(38) in the contours of the magnetic flux a magnetic island generated and growing in the linear phase and early non-linear phase, in which the process exhibit a quasi-explosive behaviour. In the full nonlinear regime, equilibrium is reached, the island growth saturates and remains more or less unchanged. The

The Method of Characteristics for the Numerical Solution…

Figure 42. Stream function. ϕ .

Figure 43. Stream function ϕ .

Figure 44.

φ+

at t=30.

51

52

M. Shoucri

Figure 45. Current J at t =30.

contours of G+ in Fig.(39) at t=30 and in Fig.(40) at t=35 show the formation of a vortex structure. A similar vortex structure is developed also for the invariant G − , which is advected in the opposite direction with respect to G + . Asymptotic states for 2D systems showing the formation of vortex structures has been discussed in [49,50], who showed that energy should move to the largest scale available in the system, showing the formation of a large vortex, similar to the 2D results we obtained in the previous examples. The model preserves parity. If we choose the initial values such that ψ (− x) = ψ ( x) , and ϕ ( − x) = ϕ ( x) , these relations imply G + (− x, y ) = G− ( x, y ),

φ + (− x, y ) = −φ − ( x, y ),

which are maintained and

accurately verified by the code. Fig.(41) for G− ( x, y ) at t=30 shows how this symmetry is well reproduced by the code ( to be compared with Fig.(39)). Fig.(42) shows the stream function ϕ at t=30 and Fig.(43) at t=35. Fig.(44) shows the function φ + at t=30. In Fig.(45) we have a 3D view and a contour plot of the current J = −∇ ⊥ψ at t=30. Note the important 2

The Method of Characteristics for the Numerical Solution…

53

Figure 46. Current J at t =35.

Figure 47. Plot of

ln(δψ ( x = L x / 2, y = 0))

against time.

peak structure of the current around the X-point. We note also in Fig.(46) for the contour plot of the current at t=35 the fine scale structures which develop, and which make further calculation difficult, even with the 2048x512 grid points we have. Magnetic reconnection leads to the development of increasingly narrow current and vorticity layers. To avoid this difficulty, a numerical diffusion term was added in [62,63] to smooth the solution and push further in time. But this is done at the expense of eliminating some details , as for instance the thin filament current peaking at the X-point in Fig.(45), which has not been observed in [62,63,64]. Finally we stress again the symmetry in the solution reproduced with great −3

precision by the code. The time-step used for this calculation was Δt = 10 .

54

M. Shoucri

Figure 48. Plot , against time, of the difference between each term of energy, as defined in Eq.(3.54), and the corresponding value at t=0, divided by the Total energy E(0).

Finally

in Fig.(47) a curve showing the evolution of ln(δψ ( x = L x / 2, y = 0)) against time, which shows the growth and saturation of the perturbation

we

present

δψ . And in Fig.(48), the curves present the evolution of the different energies. 2

∫ dxdy ∇ψ (dotted curve), which is decaying, is transformed mainly into plasma kinetic energy ∫ dxdy ∇ϕ (broken curve), into electron parallel kinetic energy ∫ dxdyd J (two-dashes-dot curve), and into electron internal energy ∫ dxdyρ U (dash-

The magnetic energy

2

2 e

2

2 s

2

two-dots curve). The total energy E (full curve) is given by:

(

E = ∫ dxdy ∇ψ

2

+ d e2 J 2 + ρ s2U 2 + ∇ϕ

2

)/ 2

(3.54)

(The quantities plotted in Fig.(48) are the difference between each term of energy as defined in Eq.(3.54), and the corresponding value at t=0, divided by the total energy E(0) at t=0). The extension of this method to the 3D reduced model [63] for collisionless magnetic reconnection is outlined as follows. The 3D equation:

∂G± + [φ ± , G± ] = ∂t is solved using a fractional step method :

∂ (φ ± ∓

ρs

de ∂z

G± ) (3.55)

The Method of Characteristics for the Numerical Solution…

55

Step1-Solve for a step Δt / 2 the equation:

∂G± + [φ ± , G± ] = 0 ∂t

(3.56)

Step2-Solve for a step Δt the equation :

∂G± = ∂t

∂ (φ ± ∓

ρs

de ∂z

G± ) ;

(3.57)

Step3-Repeat Step1-. Eq.(3.56) is solved with the same method discussed in section 1 and 3.1. Eq.(3.57) can be solved by Fourier transform in the periodic z-direction.

4. Application of the Method of Characteristics to Fluid Equations We have already mentioned in section 3.1 that the 2D guiding-center equations in a magnetized plas are isomorphic to the Euler equations that govern the 2D inviscid incompressible fluids in hydrodynamics. In section 3.4, a reduced set of fluid-like equations has been applied to study magnetic reconnection. We present in this section some additional applications in the field of fluid dynamics, for the numerical solution of the shallow water equations, and for the numerical solution of the equations of the incompressible ideal magnetohydrodynamic flows in plasmas.

4.1. Numerical Solution of the Shallow Water Equations The shallow water equations are of great importance since they are widely applied for the study of atmospheric weather prediction and oceanic dynamics. They are the simplest equations which describe both slow flows and fast gravity wave oscillations, the two main categories of fluid motion present in the more complicated primitive equations, and which are commonly used for atmospheric, oceanic and climate modeling. A method of fractional step for the numerical solution of the shallow water equations has been recently presented in [3]. It consists of splitting the equations and successively integrating in every direction along the characteristics using the Riemann invariants of the equations, which are constant quantities along the characteristics. The integration is stepped up in time using cubic spline interpolation to advance the advection terms along the characteristics. It has also the great advantage of solving the shallow water equations without the iterative steps involved in the multidimensional interpolation problem, and the iteration associated with the intermediate step of solving a Helmholtz equation, which is usually the case in other methods like the semiLagrangian or Euler-Lagrange method [7,8,65 and references therein]. The absence of iterative steps in the present method reduces considerably the numerical diffusion, and makes it suitable for problems in which small time steps and grid sizes are required, as for instance

56

M. Shoucri

the problem of the calculation of the potential vorticity field we study in the present section, where steep gradients and fine scale structures develop, and more generally for regional climate modeling problems. The linear analysis (unpublished) of the shallow water equations for the fractional step method shows the method is unconditionally stable, reproducing exactly the frequency of the slow mode, while the frequencies of the fast modes are exact to second order. We present in this section a new application of the fractional step method to the shallow water equations to study the evolution of a complex flow typical of atmospheric or oceanic situations, namely the nonlinear instability of a zonal jet, similar to what has been presented in [66]. As pointed out in [66], there is well established observations that even in the presence of relatively smooth, large scale flows, tracer fields in the atmosphere and the ocean develop fine scale structures. A tracer of particular significance is the potential vorticity, which develops steep gradients and evolves into thin filaments whose numerical study demands a resolution with small grid sizes and whose evolution requires small timesteps. Several methods have been discussed in [66] for the numerical solution of the potential vorticity of a zonal jet. The semi-Lagangian or Euler-Lagrange method requires iterations at each time-step to interpolate along the characteristics, and includes also an intermediate step for solving by iteration a Helmholtz equation [7,8,65,66]. This double-iterative numerical method can be computationally prohibitive if done on small grid sizes and with small timesteps, as recently pointed out in [66], and results in an important numerical diffusion difficult to evaluate or control. Other methods to solve the shallow water equations include the pseudospectral method [66], which requires the addition of an explicit hyperdiffusion for the numerical stability. This ad hoc addition of hyperdiffusion seriously degrades the solution accuracy. In the results presented in [66], the complex filamentary structures and steep gradients surrounding most vortices are substantially smoothed out in the pseudospectral and semi-Lagrange methods, this later one does worse than the pseudospectral method because the numerical diffusion occurs through repeated iterations and interpolations and is thus not directly controllable. In the contour-advective semi-Lagrangian method presented in [66], the potential vorticity is discretized by level sets separated by contours that are advected in a fully Lagrangian way. This allows one to maintain potential vorticity gradients that are steep, however the small scales in the potential vorticity are removed with contour surgery, by topologically reconnecting contours and eliminating very fine scale filamentary structures. This contour surgery is, of course, an ad hoc procedure as much as the hyperdiffusion used in the pseudospectral method. All the three previously discussed methods (semi_Lagrangian, pseudospectral and contour-advected semi-Lagrangian) require the knowledge of the calculated variables at three time levels, and require different time filtering to damp highfrequency modes and small-scale high frequency gravity waves, otherwise the numerical scheme is unstable. In the fractional step method we present in this section, no iterations are required since only two time levels are used to advance the equations in time, and no time filtering is required. So the numerical diffusion is minimal. The fractional step method applied to the shallow water equations has been recently presented and applied to a climate modeling problem [3], and compared favourably when applied with small grid sizes and timesteps with respect to the semi-Lagrangian method. Further evaluation of the performance of the fractional step method applied to the shallow water equations has been recently presented in [67]. It is the purpose of the present work to apply this fractional step method to follow on an Eulerian grid the evolution of quantities like the height and the velocity field, which are relatively broader in scale, while reconstructing and capturing at each time-step the complex

The Method of Characteristics for the Numerical Solution…

57

filamentary and fine scale structures and the steep gradients associated with the corresponding potential vorticity. In other words the potential vorticity is post-processed at each time step from the height and velocity field obtained from the direct solution of the shallow water equations, as it has been recently reported [68]. In the present section, the results obtained will be compared with results obtained by directly integrating the potential vorticity equation on the same Eulerian grid. These direct integrations which follow a quantity developing steep gradients and fine scale structures on an Eulerian grid develop naturally numerical noise, as it will be shown at the end of this section. We write the two-dimensional shallow water equations in their simplest form in terms of the height h and the velocities (u , v ) respectively along the x and y directions.

∂h ∂uh ∂vh + + =0 ; ∂t ∂x ∂y

(4.1)

∂u ∂u ∂u ∂h ∂v ∂v ∂v ∂h +u +v + g = f v ; +u +v + g =−f u ∂t ∂x ∂y ∂x ∂t ∂x ∂y ∂y

(4.2)

f is the Coriolis parameter and g is the gravitational field. We write the geopotential

φ = gh ,

and we use the following time-centered scheme [68] to integrate by fractional step Eqs(4.1,4.2): Step 1 - solve for Δt 2 the equations in the x direction:

∂u ∂u ∂φ +u + =0 ; ∂t ∂x ∂x

(4.3)

∂φ ∂φ ∂u +u +φ =0 ∂t ∂x ∂x

(4.4)

∂v ∂v +u =0 ; ∂t ∂x

(4.5)

)

(4.6)

Eqs(4.3,4.4) are rewritten:

(

∂R x ± ∂R x ± + u± φ =0 ; ∂t ∂x where R x ± = u ± 2

φ are the Riemann invariants [1]. The solution of Eq. (4.6) at t + Δt 2

is written as follows:

R x ± ( x, y, t + Δt / 2 ) = R x ± ( x ± , y, t )

(4.7)

58 where x ± = x −

M. Shoucri t + Δt / 2

∫ (u ±

φ ) dt . The solution of Eq.(4.5) for v at t + Δt 2 is written:

t

v( x, y, t + Δt / 2) = v( x −

t + Δt / 2

∫ udt , y, t )

(4.8)

t

t

Q (m Δx ; (n + 1/2) Δt)

(n + 1/2) Δt

n Δt O (m - 1) Δx η Δx

ξ Δx

P

m Δx

x

Figure 49. Details for the calculation if Eq.(4.12).

To find the value of the function at t + Δt / 2 at the arrival grid points, the right hand sides of Eqs.(4.7,4.8) imply finding the value of the function at time t = nΔt at the departure point of the characteristic at the shifted position. This value is obtained using cubic spline interpolation from the values of the function at the neighboring grid points at t = nΔt . To avoid iterations, we show as an example how the integral in Eq.(4.8) is approximated (the same technique is applied to approximate the other integrals in Eq.(4.7)). We write:

ξΔx =

t + Δt / 2

∫ u( x(t ′), t ′)dt ′

(4.9)

t

Δx is the grid size in the x direction, and from Eq.(4.9) ξΔx gives the distance of the departure point P of the characteristic from the grid point at x = mΔx (the point P in Fig.(49) is located between the grid points x = (m − 1)Δx and x = mΔx , and the characteristic through the point P reaches the grid point Q (mΔx, (n + 1 / 2)Δt ) at t = (n + 1 / 2)Δt ). The vertical axis in Fig.(49) is time. The velocity u at the point P (denoted by u P ), can be written as a linear interpolation at time t = nΔt of the value of u at the grid point at x = mΔx (denoted by u m ), and the value of u at the grid point

x = (m − 1)Δx (denoted by u m −1 ) :

The Method of Characteristics for the Numerical Solution…

59

u p = u m−1 (1 − η ) + u mη = u m−1ξ + u m (1 − ξ ) where

(4.10)

η + ξ = 1 . The distance OP in Fig.(49) is ηΔx . Usually u at the point Q is unknown.

We use Taylor expansion for Δt / 2 :

u Q = u m + (u m − u mn −1 / 2 ) = 2u m − u mn −1 / 2 (the values of u without superscript denotes the time t = nΔt ). We can approximate the integral in Eq.(4.9) as follows:

1 2

ξΔx = (u P + u Q )

Δt 2

(4.11)

We substitute in Eq.(4.11) for u P and u Q . We get for the shifted value in Eq.(4.9):

1 Δt (3u m − u mn −1 / 2 ) 2 ξΔx = 2 1 u − u m −1 Δt 1+ ( m ) 2 Δx 2

(4.12)

n −1 / 2

This result reduces to the one in [68] if we approximate u m

by u m in eq.(4.12).

Eq.(4.12) gives an explicit approximation for the value to be shifted in Eq.(4.8). The same technique can be applied to the integrals in Eq.(4.7), so the calculation of the integrals in Eqs.(4.7,4.8) using the approximation of Eq.(4.12) remains explicit, and the interpolated values in Eqs.(4.7,4.8) are calculated using a cubic spline interpolation. No iteration is implied in this calculation. The results we present] show that this approximation is sufficient and good. Step 2 - use the results of Step 1 to solve for Δt 2 the equations in the y direction:

∂v ∂v ∂φ +v + =0 ∂t ∂y ∂y

(4.13)

∂φ ∂φ ∂v +v +φ =0 ∂t ∂y ∂y

(4.14)

∂u ∂u +v =0 ; ∂t ∂y

(4.15)

Eqs.(4.13) and (4.14) are rewritten:

60

M. Shoucri

∂R y ± ∂t

(

+ v± φ

) ∂R∂y

y±

=0 ;

φ are the Riemann invariants. The solution of Eq. (4.16) is written:

where R y ± = v ± 2

Δt ⎞ ⎛ R y ± ⎜ x, y , t + ⎟ = R y ± (x, y ± , t ) 2⎠ ⎝ where y ± = y −

(4.16)

t + Δt / 2

∫ (v ±

(4.17)

φ )dt . The solution for u in Eq.(4.15) is calculated in a similar

t

way to Eq. (4.8).

u ( x, y, t + Δt / 2) = u ( x, y −

t + Δt / 2

∫ vdt, t )

(4.18)

t

The calculation of the integrals in Eqs.(4.17) and (4.18) is effected in a similar way as explained for Eq.(4.9),by substituting y for x. Step 3 - use the results at the end of Step 2 to solve the source terms for Δt:

∂u − fv = 0 ∂t

;

∂v + fu = 0 ∂t

(4.19)

If we denote by Uo and Vo the values of u and v at the end of Step 2, the values of u and v after Δt in Step 3 are given by:

u ( x, y, t + Δt ) = U o ( x, y ) cos( fΔt ) + Vo ( x, y ) sin( fΔt ) ;

(4.20)

v( x, y, t + Δt ) = Vo ( x, y ) cos( fΔt ) − U o ( x, y ) sin( fΔt )

(4.21)

Step 4 use the results at the end of Step 3 to solve for Δt 2 the equations in the y direction (as in Step 2) Step 5 use the results at the end of Step 4 to solve for Δt 2 the equations in the x direction (as in Step 1) This entire cycle will advance the solution by one time-step Δt . We have mentioned that in Eqs.(4.7),(4.8) and Eqs.(4.17-4.18) the value of the function at the points of departure of the characteristics ( the shifted value) are calculated from the values of the function at the grid points using a cubic spline interpolation. We use a simple cubic spline defined over three grid points, calculated by writing that the function, its first and second derivatives are continuous at the grid points. Details are given in Appendix B. Testing this cubic spline polynomial

The Method of Characteristics for the Numerical Solution…

61

against other methods [4] has shown that this cubic polynomial has very low numerical diffusion compared to other polynomials. The code developed for the present problem is less than 500 fortran lines. At every time step, we reconstruct the potential vorticity q from the calculated values of h, u and v using the relations:

q=

∂v ∂u f +ζ ;ζ = − ∂x ∂y h

(4.22)

The derivatives ∂v / ∂x and ∂u / ∂y are calculated from the values of u and v using cubic splines. If we operate on the first equation of Eq.(4.2) by ∂ / ∂y , and on the second equation of Eq.(4.2) by ∂ / ∂x , we can derive with the help of Eq.(4.1) the following equation for the potential vorticity q :

∂q ∂q ∂q +u +v =0 ∂t ∂x ∂y

(4.23)

which simply states that q is constant along the characteristics :

dx dy =u ; =v dt dt

(4.24)

As we mentioned in the introduction, it is generally difficult to find a method for the direct integration of Eq.(4.23), because the potential vorticity generally develops steep gradients and finescale structures. In any Eulerian code, this will require a large number of grid points and small time-steps. In the method we present in this paper, we solve directly for the relatively large scale variables , height and velocity in Eqs.(4.1,4.2), and the potential vorticity is accurately calculated (post-processed) at each time-step using Eq.(4.22), capturing the small scale key features and steep gradients associated with the solution. We apply the numerical scheme presented for the numerical solution of Eqs.(4.1,4.2), and for the problem of post processing the potential vorticity from this solution. The initial flow consists of a perturbed unstable zonal jet which rapidly becomes very complex, and is specified by prescribing the initial potential vorticity as follows ( we follow the notation of [66]):

q ( x, y,0) = q + Q sgn( yˆ )(a − || yˆ | − a |)

(4.25)

for | yˆ |< 2a (the vertical lines indicate the absolute value). Q is the amplitude of the potential vorticity, q is the mean potential vorticity, 2a is the distance from the minimum to maximum potential vorticity, and:

yˆ = y + c m sin mx + c n sin nx

(4.26)

62

M. Shoucri

where yˆ is the perturbed y coordinate, used to perturb the jet. We use for the present test the same parameters as in [66]. Equilibrium height h = 1 , the Coriolis factor f = 4π ,

a = 0.5 , h Q / f = 1 , Q = 4π = q . The deformation radius is LR = gh / f = 0.5 , and

g = (2π ) 2 . A doubly periodic domain which spans the range (−π , π ) covers about 12.5 deformation radii in each direction. We take m=2 , n=3, c 2 = −0.1 , c3 = 0.1 to perturb the q profile . We use a slightly different method to calculate the initial velocities and height h. We assume initially v=0 . We balance the second of Eq.(4.2) initially:

fu = − g

from which

f

∂h ∂y

∂2h ∂u = −g 2 ∂y ∂y

Figure 50. Initial profiles for the shallow water problem

h / 2π

(4.27)

(4.28)

q / 4π

(full curve) , u (broken curve),

(dash-dot curve).

We substitute for ∂u / ∂y from Eq.(4.28) into Eq.(4.22) and solve numerically for the initial value of h .Fig.(50) shows the initial equilibrium profiles ( uniform in x ) for q / 4π ( full curve) , u ( broken curve) and h / 2π ( dash-dot curve). These initial values of u,v and h are used in Eqs.(1,2) to start the evolution of the system. In Figs.(51) and (52) we show respectively the velocities u and v at t=8, and in Fig.(53) we show the contour and 3D view of the geopotential φ . Figs.(51-53) show structures which are generally broader in scale than the potential vorticity. Fig.(54) shows the the potential vorticity q , calculated at t=8 from Eq.(4.22). The instability in the potential vorticity has developed and the initial zonal jet has evolved into vortices and fine structures with steep gradients. These fine scale structures are

The Method of Characteristics for the Numerical Solution…

Fig 51. Velocity u .

Figure 52. Velocity v.

63

64

M. Shoucri

Figure 53. Geopotential

φ.

Figure 54. Potential vorticity q.

The Method of Characteristics for the Numerical Solution…

65

inevitably generated by the forward enstrophy cascade. We note how the steep potential vorticity gradients and the small scale features are nicely captured and reconstructed using Eq.(4.22), from the solution of the relatively broader scale height and velocities, by post processing these relatively smooth large scale flows with the help of Eq.(4.22). These calculations are done with 200x200 grid points, and a time-step Δt = Δx / 20 . This time-step has been chosen after few tests to determine the time-step at which the solution appears to converge and become independent of Δt ( the solution obtained with Δt = Δx / 10 is essentially identical to what we are presenting here). The computation CPU time on a sunblade 1000 workstation of 750 Mhz was 64 minutes to reach t=8, and the required memory for the code was 4.5 Mbytes.

Figure 55. Potential vorticity q(fractional step method).

We present for comparison in Figs(55) the solution for the potential vorticity q obtained by the direct integration of Eq.(4.23) with the initial value in Eq.(4.25) by a fractional step method, and by a semi-Lagrangian method in Fig.(56). For the fractional step method, we follow the steps of the techniques presented in section 2 . To advance Eq.(4.23) in for a timestep Δt , we use the following sequence : Step1 Solve for Δt / 2 the equation :

∂q ∂q +u =0 ∂t ∂x

(4.29)

66

M. Shoucri Step2 Solve for Δt the equation:

∂q ∂q +v =0 ∂t ∂y

(4.30)

Step3 Repeat Step1 for Δt / 2 .

Figure 56. Potential vorticity q (semi-Lagrangian method).

We use the same values of u and v calculated from Eqs.(4.2). Eqs(4.29-4.30) are solved as described for Eq.(4.5), (4.6) or (4.15) . The other method used for the direct solution of Eq.(4.23) is the semi-Lagrangian or Euler-Lagrange method. As described in sections 1 and 3.1, we calculate the displacements Δ x and Δ y along the characteristic curves by solving iteratively the equations:

Δkx+1 = Δtu ( x − Δkx , y − Δky , t n )

(4.31)

The Method of Characteristics for the Numerical Solution…

Δky+1 = Δtv ( x − Δkx , y − Δky , t n )

67

(4.32)

where u and v are calculated from Eqs.(4.2) using the same method we previously discussed. 0

0

We start with Δ x = 0 and Δ y = 0 . Usually two or three iterations are necessary to get convergence. Then the function q is advanced from t n − Δt to t n + Δt using the relation:

q ( x, y, t n + Δt ) = q ( x − 2Δ x , y − 2Δ y , t n − Δt )

(4.33)

As explained in section 3, the interpolations in Eqs.(4.31-4.33) are done using a tensor product of cubic B-spline [6] ] (with appropriate modification to the boundary conditions to take into account the periodicity as for instance in [51]). We used the same Eulerian grid as for the solution of Eqs.(4.1-4.2). Figs(55) and (56) show noisy figures compared to what has been presented in Fig.(53), obtained from the direct solution of Eqs.(4.1-4.2) and Eq.(4.22). This noisy behaviour is to be expected since for the fractional step and semi-Lagrangian methods we have the difficult challenge to follow on an Eulerian grid the potential vorticity of a zonal jet, a quantity which develops steep gradients and fine scale structures. Finally the extension of the method to three dimensional problems is straightforward, requiring only the addition of an extra fractional step in the third dimension to what is presented in this section. The Appendix in [68] outlines an example for a 3D problem.

4.2. Two-Dimensional Magnetohydrodynamic Flows We present another example for the application of the method of characteristics for the numerical solution of fluid equations, namely the equations of two-dimensional incompressible magnetohydrodynamic flows in plasmas. These equations play an important role in the understanding of strong turbulence properties in high Reynolds number conducting fluids, which have important effects on the reconnection of the magnetic field and changes of flow topology [69-72]. As discussed in section 3.4, magnetic reconnection is a fundamental process which allows magnetized plasmas to convert the energy stored in the field lines into kinetic energy of the plasma. In ideal MHD, the frozen-in flux condition prohibits the magnetic field topology to change. Thus reconnection depends on a non-ideal mechanism responsible, in the region where the topology change takes place, for the dynamics of a diffusion process which creates a mechanism that breaks the magnetic field frozen in the plasma. Hence the importance of a solution to the pertinent equations where numerical diffusion is controlled to the minimum. We have already presented in section 3.4 an application of the methods of characteristics for the numerical solution of a reduced set of MHD equations. We extend this method to the set of two-dimensional fluid ideal MHD equations usually applied to study incompressible MHD turbulence. There is an abundant literature for the numerical solution of these equations [69-72], based essentially on finite difference schemes. Our intention is to apply the method of characteristics to the solution of these equations. The pertinent incompressible magnetohydrodynamic equations can be written in the form:

68

M. Shoucri

∂z ± + z ∓ .∇z ± = −∇p + vΔz ± . div z ± =0 ∂t

(4.34)

z ± denotes the Elsässer variables z ± = u ± B , where u is the velocity, B the magnetic field, and p the total pressure. Here we assume a magnetic Prandtl number equal to one. We use the same parameters as in [69,70]. The two-dimensional MHD equations in Eq.(4.34) are solved in a rectangular box of size L x = L y = 2π with periodic boundary conditions. We use as initial conditions for the magnetic flux function ψ and the velocity streamfunction

ϕ

the following expressions :

1 [cos(2 x + 2.3) + cos( y + 4.1)], . 3 ϕ ( x, y ) = cos( x + 1.4) + cos( y + 0.5)

ψ ( x, y ) =

(4.35)

B = e z x∇ψ and u = e z x∇ϕ . These initial conditions introduced in [69] show a stronger tendency to generate turbulent small scale structures than the Orszag-Tang vortex. They are made less symmetric by means of arbitrary phases. They have also been used in [70]. The numerical scheme applied to Eq.(4.34) is the following: Step1 solve for a time step Δt / 2 the equation :

∂z ± + z ∓ .∇z ± = 0 . ∂t

(4.36)

This equation is solved in 2D using a tensor product of cubic B-spline for interpolation, as described in section 1 and applied for the problems presented in section 3. We next calculate the pressure p by taking the divergence of Eq.(4.34). This gives the following equation:

div (z ∓ .∇z ± ) = −Δp .

(4.37)

This equation is solved for p using a fast Fourier transform algorithm, since we have periodic boundary conditions. Step2 solve for a time step Δt the equation:

∂z ± = −∇p + vΔz ± . ∂t

(4.38)

∇p is treated explicitly. The diffusion term is treated by an alternate direction implicit scheme.

The Method of Characteristics for the Numerical Solution…

69

Step3 repeat Step1 for a time step Δt / 2 . The quantities u

B=

B are calculated from the relations : u =

and

z+ + z− , 2

z+ − z− . 2 +

We write for reference the explicit form for the solution of Eq.(4.36) for z x = u x + B x :

∂z x+ ∂z + ∂z + + z x− x + z −y x = 0 . ∂t ∂x ∂y

(4.39)

The characteristic equations for Eq.(4.39) are given by:

dx = z x− , dt

dy = z −y . dt

(4.40)

We calculate the displacement Δ x and Δ y as explained in the previous sections, using the following iterations:

Δkx+1 =

Δt − z x ( xi − Δkx , y j − Δky , t n + Δt / 4) 4

(4.41)

Δky+1 =

Δt − z y ( xi − Δkx , y j − Δky , t n + Δt / 4) 4

(4.42)

−

−

The values of z x and z y at t n + Δt / 4 can be calculated by a predictor-corrector technique. Two or three iterations are necessary for the convergence in Eqs.(4.41-4.42), and a tensor product of cubic B-spline is used for the interpolation [6] (with appropriate modification to the boundary conditions to take into account the periodicity as for instance in +

[51]). Then the value of z x is advanced in time for Δt / 2 as indicated in Step1 using the relation:

z +x ( xi , y j , t n + Δt / 2) = z x+ ( xi − 2Δ x , y j − 2Δy, t n ) .

(4.43)

Again a tensor product of cubic B-spline is used for the interpolation [6] in Eq.(4.43). The same method is applied to the other variables in Eq.(4.36).

70

M. Shoucri

Figure 57. Current J at t=0.

Figure 58. Current J at t=1.

Figure 59. Current J at t=2.

The solution we present has been obtained using 512x512 grid points with the kinematic −3

viscosity v = 10 in Eq.(4.34). One has to be careful when v is different from zero since any kind of smoothing may artificially inhibit the energy transfer to small scales and slow down magnetic reconnection and the associated instabilities. Comparison with the case v = 0 has shown very close results up to t=3. However, at this stage the growth of the current becomes very big and the system goes to a numerical instability for longer runs with v = 0 . The nonlinear dynamics indeed leads to the formation, near neutral X points, of magnetic current sheets corresponding to strongly sheared magnetic field configurations. The finite value of v keeps the growth of the different variables under control. A discussion of the effect of v on the solution can be found in [71]. (We note that the claim in [70] that the first

The Method of Characteristics for the Numerical Solution…

71

simulation presented is done with v = 0 is probably due to the presence of an important numerical diffusion in the code). Figs(57-61) show the current density J respectively at t=0,1,2,3 and 6. From Eq.(4.35) we have:

Bx = −

∂ψ ; ∂y

By =

∂ψ ; ∂x

J = − Δψ = − ( −

Figure 60. Current J at t=3.

Figure 61. Current J at t=6.

Figure 62. Vorticity at t=0.

∂B x ∂B y + ) . ∂x ∂y

(4.44)

72

M. Shoucri

Figure 63. Vorticity at t=1.

Figure 64. Vorticity at t=2.

Figure 65. Vorticity at t=6.

The results in Figs.(57-61) are very close to what is presented in Fig.(2) of [70], showing the formation of current sheets. Figs.(62-65) show the vorticity U respectively at t=0,1,2 and 6. From Eqs.(4.35) we have:

ux = −

∂ϕ ; ∂y

uy =

∂ϕ ; ∂x

U = Δϕ = (−

∂u x ∂u y + ). ∂x ∂y

(4.45)

The Method of Characteristics for the Numerical Solution…

73

Figure 66. ux at t=2.

Figure 67. ux at t=6.

Figure 68. uy at t=2.

At t=0, B x , B y , u x , and u y are calculated from Eq.(4.35). Figs.(66-69) show u x , and

u y at t=2 and t=6, and Figs.(70-73) show B x and B y at t=2 and t=6. In Figs.(74-77) the magnetic flux function

ψ at t=0,1,2 and 6 is presented. Note in Figs.(75-76) at t=1 and t=2

how, in the regions where the magnetic vortices are pushed towards each others or squeezed between each others, intense currents are created in the corresponding current density plots in Figs.(58-59). Fig.(78) presents the time evolution of the total enstrophy ( which is the sum of the kinetic and magnetic enstrophies):

(

2

2

)

W = ∫ ∇xu + ∇xB dxdy .

(4.46)

74

M. Shoucri

Figure 69. uy at t=6.

Figure 70. Bx at t=2.

Figure 71. Bx at t=6.

In Fig.(79) we present the time evolution of the total energy (which is the sum of the kinetic and magnetic energies):

E=

(

)

1 2 2 u + B dxdy . ∫ 2

(4.47)

The Method of Characteristics for the Numerical Solution… The time evolution of the velocity-magnetic field correlation [70]

75

ρ = H / E , where the

∫

cross-correlation H = u.Bdxdy , is presented in Fig.(80). The distribution of kinetic and magnetic energies among the different scales is described by the kinetic and magnetic energy spectra:

E u (k ) =

∑

uˆ (k ´ )

∑

Bˆ (k ´ )

2

.

(4.48)

.

(4.49)

k ≤ k ´ < k +1

E M (k ) =

2

k ≤ k ´ < k +1

Fig.(81) shows on a logarithmic scale the kinetic ( log10 E ( log 10 E

M

u

solid line) and magnetic

dashed line) energy spectra at t=6 (plotted against log10 k , with k varying

between 1 and 100, showing a slight dominance of the magnetic energy over the kinetic energy over most of the scale lengths). It is beyond the scope of the present work to repeat what has been presented in [69-72]. We intended in this section to outline the pertinent steps for one more application of the method of characteristics (which generally shows low numerical diffusion) to the equations of ideal MHD flows.

Figure 72. By at t=2.

Figure 73. By at t=6.

76

M. Shoucri

Figure 74. Flux function ψ at t=0.

Figure 75. Flux function ψ at t=1.

Figure 76. Flux function ψ at t=2.

Figure 77. Flux function ψ at t=6.

The Method of Characteristics for the Numerical Solution…

Figure 78. Time evolution of the enstrophy.

Figure 79. Time evolution of the energy.

Figure 80. Time evolution of the correlation coefficient

Figure 81. Kinetic (solid line) and magnetic (dashed line) energy spectra at t=6.

77

78

M. Shoucri

5. Conclusion We have presented in the appendices some simple cubic spline relations which we have applied for interpolation in several problems, showing how hyperbolic type differential equations are solved using the method of characteristics. The values of the functions which remain constant along the characteristic curves are stepped-up in time using cubic spline interpolation. Results illustrating the performance of the cubic spline when applied to interpolation in Eulerian grid-based solvers have been presented. Comparison of the cubic spline interpolation with other methods [4] have shown that the cubic spline interpolation compares favourably with the other methods. Since the ground breaking work of Cheng and Knorr [10] which applied the fractional step method to the one-dimensional Vlasov equation, there has been important applications of the method of characteristics for the numerical solution of the kinetic equations of plasmas, and especially for extending these methods to higher dimensions [15-18]. A historical overview on several applications of these methods in the field of the kinetic equations of plasmas has been recently given in a Vlasovia workshop [73]. This workshop included also recent applications on massively parallel computers, especially in the field of the numerical solution of gyro-kinetic equations [74-77], which testify for the impressive advances and applications of these methods. We mention also the work in [78,79]. It is beyond the scope of the present chapter to review all these works. The intention in this chapter was to present the essential elements of the interesting technique of the method of characteristics associated with cubic splines interpolation, with appropriate selected examples to illustrate the performance, the accuracy, the powerful and efficient tools which Eulerian grid-based solvers can provide for the numerical solution of plasmas kinetic equations and fluid equations: long time evolution of 1D BGK modes, charge separation at a plasma edge involving higher phase-space dimensionality, laser-plasma interaction, magnetic reconnection and the shallow water equations. In section 2.2 and section 3.2 for instance, we have presented the problem of the formation of a charge separation and an electric field at a plasma edge. In Cartesian geometry in section 2.2 a method of fractional step associated with 1D cubic spline interpolation along the characteristic curves has been applied, while in cylindrical geometry in section 3.2 a two dimensional interpolation using a tensor product of cubic B-spline [51] has been applied in velocity space. The results from these two different codes are identical (can be superposed if we take into consideration a mirroring due to the opposite positive direction in the two codes). Another comparison has been presented in section 2.3 and 3.3, where two different models for laser-plasma interaction have been discussed. The model in section 2.3 uses a linear polarization for the electromagnetic wave, and applies a method of fractional step for the numerical solution of the equations, associated with one dimensional cubic spline interpolation. The model in section 3.3 is fully relativistic with circular polarization, and applies a two dimensional interpolation with a tensor product of cubic B-spline. Again the two different codes are providing similar results, the only difference reflects and underlines the difference in the physical models. Several recent applications of similar codes for problems of laser-plasma interaction have been recently reported [30,31,52], which testify for the success of these methods, together with different other applications of the method of characteristics for the kinetic equations of plasmas which we have rapidly reviewed. In section 3.4 for the numerical solution of the problem of magnetic reconnection, a thin filament current has been observed at the X-point in Fig.(45)

The Method of Characteristics for the Numerical Solution…

79

which was not previously observed in [62-64] where an artificial numerical diffusion was added. The method of characteristics has also been applied for the numerical solution of fluid equations. At the same time where [10] was published, the application of the method of characteristics in 1D fluid equations associated with cubic spline interpolation was also published in [80]. In plasma physics, one of the early applications of this method to fluid equations has been for the numerical solution of the coupled mode equations [81-82]. There is an abundant theoretical literature on the method of characteristics applied to fluid equations (see for instance [1,83]). 2D interpolation using tensor product of cubic B-splines is commonly applied for the numerical solution of the weather forecast equations in what is known as the semi-Lagrangian method [7,8,65,66]. In section 4.1 we have presented an interesting application to the problem of the calculation of the potential vorticity of a zonal jet from the solution of the shallow water equations, reproducing on an Eulerian grid the fine scale structures and the steep gradients of the potential vorticity without numerical noise. And the problem of magnetic reconnection using the equations of the incompressible ideal MHD flows in plasmas has been studied in section 4.2.

Acknowledgments The tutoring and fruitful discussions with Professor Georg Knorr are gratefully acknowledged. I would like also to acknowledge the fruitful discussions with Drs. Hartmut Gerhauser and Karl-Heinz Finken on the problem of the formation of an electric field at a plasma edge, and the collaboration with Dr. Erich Pohn on problems of two-dimensional interpolation used in the present chapter, as well as with Dr. David Strozzi for problems of laser-plasma interaction presented in section 2.3.

Appendix A The Shift Operator Using the Cubic Spline Let us assume 0 < Δ < 1 , with a uniform grid size normalized to 1 , and Δ is a constant. We

y j , with the function use a Taylor expansion to calculate the shifted value y j ( x j + Δ) = ~ y = f (x) and the notation f ( x j ) = f j : 1 y j ( x j + Δ) = ~ y j = f j + p j Δ + s j Δ2 + g j Δ3 2

(A.1)

p j , s j and g j are respectively the derivative, second derivative and third derivative of the function f (x) at the grid point j ≡ x j . We write that the function, its derivative and second derivative are continous at every grid point, we get the following cubic spline relations on a uniform grid [84,85]:

p j −1 + 4 p j + p j +1 = 3( f j +1 − f j −1 )

(A.2)

80

M. Shoucri

s j −1 + 4s j + s j +1 = 6( f j −1 − 2 f j + f j −1 )

(A.3)

g j −1 + 4 g j + g j +1 = − f j −1 + 3 f j − 3 f j +1 + f j + 2

(A.4)

We can verify after substitution from Eqs.(A.2-A.4) in Eq.(A.1) the following relation:

~y + ~y + ~y = Af + Bf + Cf + Df j −1 j j +1 j −1 j j +1 j+2

(A.5)

A = (1 − Δ) 3

(A.6)

B = 4 − 3Δ2 (1 + (1 − Δ))

(A.7)

C = 4 − 3(1 − Δ) 2 (1 + Δ)

(A.8)

D = Δ3

(A.9)

We verify that for Δ = 0 we have from Eq.(A.5):

i.e.

~ y j −1 + ~ yj + ~ y j +1 = f j −1 + 4 f j + f j +1 ~ yj = fj

and for Δ = 1 we verify that:

i.e.

~y + ~y + ~ y j +1 = f j + 4 f j +1 + f j + 2 j −1 j ~ y j = f j +1

as it should be. The inversion of the tridiagonal matrix in Eq.(A.5) with appropriate boundary conditions y j . The calculation of y j ( x j − Δ) = ~ y j with 0 < Δ < 1 is done in a similar determines ~ way and leads to the following relation :

~ y j −1 + ~ yj + ~ y j +1 = Af j +1 + Bf j + Cf j −1 + Df j − 2

Appendix B Interpolation Using the Cubic Spline In the general case of a variable grid size, and for an equation of the form:

(A.10)

The Method of Characteristics for the Numerical Solution…

81

∂f ∂f + υ ( x) =0 ∂t ∂x

(B.1)

The interpolation with a cubic spline polynomial is treated as follows. We assume that y j ( x) is a cubic polynomial in x j , x j +1 such that y j ( x j ) = f ( x j ) = f j , and

{

}

y j ( x j +1 ) = f ( x j +1 ) = f j +1 . We denote by s j the second derivative at the point x j , and we

{

}

set Δx j = x j +1 − x j . We can write for the second derivative in x j , x j +1 the following linear interpolation:

y ′j′ ( x) = s j

x j +1 − x

Δx j

+ s j +1

x − xj

(B.2)

Δx j

so that y ′j′ ( x j ) = s j and y ′j′ ( x j +1 ) = s j +1 . Integrating twice Eq.(B.2), we get:

y j ( x) =

sj

6 Δx j

( x j +1 − x) 3 +

s j +1

6 Δx j

( x − x j ) 3 + a j ( x j +1 − x) + b j ( x − x j )

(B.3)

With y j ( x j ) = f ( x j ) = f j , and y j ( x j +1 ) = f ( x j +1 ) = f j +1 , we get:

aj =

fj

Δx j

− sj

Δx j 6

; bj =

f j +1

Δx j

− s j +1

Δx j

(B.4)

6

and

y j ( x) =

sj

6 Δx j

⎛ fj Δx j (x − x j )3 + ⎜ − sj ⎜ Δx 6Δx j 6 ⎝ j Δx j ⎞ ⎟( x − x j ) 6 ⎟⎠

( x j +1 − x) 3 +

⎛ f j +1 +⎜ − s j +1 ⎜ Δx j ⎝

s j +1

⎞ ⎟( x j +1 − x ) ⎟ ⎠

(B.5)

We write that the derivative y ′j ( x) is continuous at x = x j : y ′j ( x j ) = y ′j −1 ( x j ) , we get the following relation:

⎛ f j +1 − f i f j − f j −1 ⎞ ⎟ Δx j s j +1 + 2(Δx j + Δx j −1 ) s j + Δx j −1 s j −1 = 6⎜ − ⎜ Δx ⎟ x Δ j j −1 ⎝ ⎠

(B.6)

82

M. Shoucri

Inverting the tridiagonal matrix in Eq.(B.6) with proper boundary conditions determine the s j to be used in the cubic polynomial given in Eq.(B.5), which can also be rewritten in the form:

y j ( x) = A j f j + B j f j +1 + C j s j + D j s j +1 Where A j =

x j +1 − x

Δx j

; Bj =

x − xj

Δx j

; Cj =

Δx 2j 6

(B.5)

A j ( A 2j − 1) ; D j =

Δx 2j 6

B j ( B 2j − 1) .

For a value of y j ( x) at x = x j + Δ j where x j < x < x j +1 , (as for instance in Eq.(B.1) when Δ j = υ ( x j )Δt ,

υ ( x j ) < 0) we have : A j = 1 −

Δj Δx j

and B j =

Δj Δx j

. It is

straightforward to derive from the previous results the expression for y j ( x) at x = x j − Δ j , where x j −1 < x < x j .(as for instance in Eq.(B.1) when Δ j = υ ( x j ) Δt ,

υ ( x j ) > 0) . In this

case we have, with Δx j −1 = x j − x j −1 , we have:

y j ( x) = A j f j + B j f j −1 + C j s j + D j s j −1 Where A j = 1 − B j ; B j =

Δj Δx j −1

; Cj =

Δx 2j −1 6

2 j

A j ( A − 1) ; D j =

(B.6)

Δx 2j −1 6

B j ( B 2j − 1) .

(note that in the previous results, we assumed , υ ( x j ) Δt < Δx j , but the results can be generalized without difficulty to arbitrary values of

υ ( x j ) Δt ).

Appendix C Interpolation Using the Cubic B-spline To interpolate using a cubic B-spline, we write the function f(x) as follows:

f ( x) =

N x −1

∑γ j = −2

where the B j ( x) are defined as follows [84]:

j

B j ( x)

(C.1)

The Method of Characteristics for the Numerical Solution…

⎧( x − x j ) 3 ⎪ 2 3 1 ⎪1 + 3( x − x j +1 ) + 3( x − x j +1 ) − 3( x − x j ) B j ( x) = ⎨ 6 ⎪1 + 3( x j +3 − x) + 3( x j +3 − x) 2 − 3( x j +3 − x) 3 ⎪( x j + 4 − x) 3 ⎩

83

x j ≤ x < x j +1 x j +1 ≤ x < x j + 2 , (C.2) x j + 2 ≤ x < x j +3 x j +3 ≤ x < x j + 4

and B j (x) is equal to zero otherwise. Note that in this case the cubic polynomial is defined using four grid points. Because of the local definition of each B-spline, only 4 summands of Eq.(C.1) are non-zero. The calculation of the coefficients for the B-spline interpolation is performed as follows. We write for the given function value at the grid points xi :

f (x j ) =

N x −1

∑γ

j = −2

j

B j (x j )

(C.3)

which results in the equation:

γ j −3 + 4γ j − 2 + γ j −1 = 6 f j ; for j=1,…….Nx

(C.4)

where f j ( x) = f j , and we assume as boundary condition that the derivative is equal to zero at the boundaries : γ 0 = γ − 2 ,

γN

x −1

= γ N x −3 . We use the recursive ansatz:

γ j = γ j +1 X j + H j

(C.5)

which inserted into Eq.(C4) yields by comparison the coefficients:

Xj =−

1 ; X − 2 = −2 4 + X j −1

H j = X j ( H j −1 − 6 f j + 2 ) ; H − 2 = 3 f1

(C.6)

(C.7)

The values of X − 2 , H − 2 are obtained by considering the recursive ansatz with j = 2 and the left boundary condition

γ 0 = γ − 2 . The starting value of the recursion is obtained

using the right boundary condition:

γN

x −1

=

H N x −3 + H N x − 2 X N x −3

1 − X N s − 2 X N x −3

(C.8)

84

M. Shoucri Once

the

coefficients

γj

are

known,

~ f = f ( x j + Δ x ) are now calculated as follows : ~ 3 f = ∑γ κ =0

j −κ

arbitrary

bκx

interstitial

function

values

(C.9)

where j ≡ x j , Δ x = x − x j , and :

1 b0x = Δ 3x ; 6 1 b1x = (1 + 3(Δ x + Δ 2x − Δ 3x ) ) 6 1 b2x = (1 + 3((1 − Δ x ) + (1 − Δ x ) 2 − (1 − Δ x )3 ) ) ; 6 1 3 b3x = (1 − Δ x ) 6

(C.10)

References [1] Abbott, B.A. An Introduction to the Method of Characteristics; Thames and Hudson: London, 1966 [2] Pohn, E.; Shoucri, M.; Kamelander, G. Comp. Phys. Comm. 2005, 166, 81-93 [3] Shoucri, M. Comp. Phys. Comm. 2004, 164, 396-401 [4] Pohn, E.; Shoucri, M.; Kamelander, G. Comp. Phys. Comm. 2001, 137, 380-395;ibid 2001, 137, 396-404 [5] Yanenko, N.N. The Method of Fractional Steps, Springer-Verlag, New York, 1971 [6] Shoucri, M., Gerhauser, H., Finken, K-H. Comp. Phys. Comm. 2003, 154, 65-75 [7] Makar, P.A., Karpik, S.R. Mon. Wea. Rev. 1996, 124, 182-199 [8] Staniforth, A., Côté, J. Mon. Wea. Rev. 1991, 119, 2206-2223 [9] Sonnendrücker, E., Roche, J., Bertrand, P., Ghizzo, A. J. Comp. Phys. 1998, 149, 201 [10] Cheng, C.Z., Knorr, G. J. Comp. Phys. 1976, 22, 330-351 [11] Shoucri, M., Gagné, R. J. Comp. Phys. 1997, 24, 445-449; Phys. Fluids 1978, 21, 11681175, IEEE Plasma Science 1978, PS-6, 245-248; J. Comp. Phys. 1978, 27, 315-322 [12] Shoucri, M. Phys. Fluids 1978, 21, 1359-1365; ibid 1979, 22, 2038; ibid 1980, 23, 2030; J. de Physique 1979, 40, 38-39 [13] Shoucri, M., Storey, O. 1986, 29, 262-265; Simon, A., Short, R.W. Phys Fluids 1988, 31, 217 [14] Bertrand, P., Ghizzo, A., Feix, M., Fijalkow, E., Mineau, P., Suh, N.D., Shoucri, M. In Nonlinear Phenomena in Vlasov Plasmas; Doveil, F.;Ed.; Proc. Cargèse Workshop; Les Editions de Physique: Les Ulis, France, 1988; 109-125 [15] Cheng, C.Z. J. Comp. Phys. 1977, 24, 348-360

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[16] Shoucri, M., Gagné, R. J. Comp. Phys. 1978, 27, 315-322; Shoucri, M. In Modeling and Simulation; Proc. 10th Annual Pittsburgh Conference; Publisher: Instrument Society of America, Pittsburgh, 1979; Vol. 10; 1187-1192 [17] Shoucri, M. IEEE Plasma Science, 1979, PS-7, 69-72 [18] Johnson, L.E. J. Plasma Phys. 1980, 23, 433-452 [19] Rickman, J.D. IEEE Plasma Science 1982, PS-10, 45-56 [20] Ghizzo, A., Bertrand, P., Shoucri, M., Johnston, T., Feix, M., Fijalkow, E. J. Comp. Phys. 1990, 90, 431-457 [21] Bertrand, P., Ghizzo, A., Johnston, T., Shoucri, M., Fijalkow, E., Feix, M. Phys. Fluids 1990, B2, 1028-1037 [22] Johnston, T., Bertrand, P., Ghizzo, A., Shoucri, M., Fijalkow, E., Feix, M. Phys. Fluids 1992, B4, 2523-2537 [23] Ghizzo, A., Shoucri, M., Bertrand, P., Johnston, T., Lebas, J. J. Comp. Phys. 1993, 108, 373-376 [24] Shoucri, M., Bertrand, P., Ghizzo, A., Lebas, J., Johnston, T., Feix, M., Fijalkow, E. Phys. Letters A, 1991, 156, 76-80 [25] Ghizzo, A., Bertrand, P., Lebas, J., Shoucri, M., Johnston, T. J., Fijalkow, E., Feix, M.R. J. Comp. Phys. 1992, 102, 417-422 [26] Bertrand, P., Ghizzo, A., Karttunen, S., Pättikangas, T., Salomaa, R., Shoucri, M. Phys. Plasmas 1995, 2, 3115-3129; Physical Rev. E 1994, 49, 5656-5659 [27] Ghizzo, A., Bertrand, P., Bégué, M.L., Johnston, T., Shoucri, M. IEEE Plasma Science 1996, 24, 370-378 [28] Bégué, M.L., Ghizzo, A., Bertrand, P. J. Comp. Phys. 1999, 151, 458-478 [29] Ghizzo, A., Bertrand, P., Shoucri, M., Johnston, T., Fijalkow, E., Feix, M.R., Demchenko, V.V., Nucl. Fusion 1992, 32, 45-65 [30] Brunner, S., Valeo, E. Phys. Rev. Lett. 2004, 93, 145003-1 -145003-4 [31] Strozzi, D., Shoucri, M., Bers, A., Williams, E., Langdon, A.B. J. Plasma Physics. 2006, 72, 1299-1302 [32] Manfredi, G., Shoucri, M., Feix, M., Bertrand, P., Fijalkow, E., Ghizzo, A. J. Comp. Phys. 1995, 121, 298-313 [33] Ghizzo, A., Bertrand, P., Shoucri, M., Fijalkow, E., Feix, M. Phys. Fluids 1993, B5, 4312-4326 [34] Manfredi, G., Shoucri, Shkarofsky, I., Ghizzo, A., Bertrand, P., Fijalkow, E., Feix, M., Karttunen, S., Pattikangas, T., Salomaa, R. Fusion Tech. 1996, 29, 244-260 [35] Manfredi, G., Shoucri, M., Dendy, R.O., Ghizzo, A., Bertrand, P. Phys. Plasmas 1996, 3, 202-217 [36] Manfredi, G., Shoucri, M., Bertrand, P., Ghizzo, A., Lebas, J.,Knorr, G., Sonnendrücker, E., Bürbaumer, H., Entler, W., Kamelander, G. Phys. Scripta 1998, 58, 159-175 [37] Shoucri, M., Lebas, J., Knorr, G., Bertrand, P., Ghizzo, A., Manfredi, G., Christopher, I., Phys. Scripta 1997, 55, 617-627; ibid 1998, 57, 283-285 [38] Shoucri, M., Manfredi, G., Bertrand, P., Ghizzo, A., Knorr, G. J. Plasma Phys 1999, 61, 191 [39] Watanabe, T.-H., Sugama, H., Sato, T. J. Phys. Soc. Japan, 2001, 70, 3565-3576 [40] Watanabe, T.-H., Sugama, H. Trans. Theory Stat. Phys. 2005, 34, 287-309 [41] Arber, T.D., Vann, R.G.L., J. Comp. Phys. 2002, 180, 339

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[42] Pohn, E., Shoucri, M., Kamelander, G. Comp. Phys. Comm. 2005, 166, 81-93; J. Plasma Physics. 2006, 72, 1139-1143 [43] Pohn, E., Shoucri, M. Proc. Vlasovia workshop (to be published in Commun. Nonlinear Sci. Numer. Simul.) 2007 [44] Shoucri, M. Czech. J. Phys. 2001, 51, 1139-1151 [45] Batishchev, O., Shoucri, M., Batishcheva, A., Shkarofsky, I. J. Plasma Physics 1999, 61, 347- 364 [46] Ghizzo, A., Izrar, B., Bertrand, P., Fijalkow, E., Feix, M., Shoucri, M. Phys. Fluids 1988, 31, 72-82 [47] Ghizzo, A., Izrar, B., Bertrand, P., Feix, M.R., Fijalkow, E., Shoucri, M. Phys. Lett. A 1987, 120, 191-195 [48] Bernstein, I.B., Greene, S.M., Kruskal, M.D. Phys.Rev. 1957, 108, 546-554 [49] Knorr, G. Plasma Phys. 1977, 19, 529-538 [50] Knorr, G., Pecseli, H.L. J. Plasma Phys. 1989, 41, 157-170 [51] Reproduced from Comp. Phys Comm., Vol. 164; Shoucri, M., Gerhauser, H., Finken, K.H., Study of the Generation of a Charge Separation and Electric field at a Plasma Edge using Eulerian Vlasov Codes in Cylindrical Geometry, p. 139-141, Copyright 2004, with permission from Elsevier. [52] Huot, F., Ghizzo A., Bertrand, P., Sonnendrücker, E., et al J. Comp. Phys. 2003, 185, 512-531 [53] Strozzi, D., Shoucri, M., Bers, A. Comp. Phys. Comm. 2004, 164, 156-159 [54] Shoucri, M., Matte, J.-P., Côté, A. J. Phys. D : Appl. Phys. 2003, 36,2083-2088 [55] Joyce, G., Montgomery, D. J. Plasma Phys. 1973, 10, 107-120 [56] Knorr, G. Plasma Phys.1974, 5, 423-434 [57] Marchand, R., Shoucri, M. J. Plasma Phys. 2001, 65, 151-160 [58] Shoucri, M. Int. J. Num. Methods Eng. 1981, 17, 1525-1538 [59] Shoucri, M., Knorr G. Plasma Phys. 1976, 18, 187-204 [60] Krane, B., Christopher, I., Shoucri, M., Knorr, G. Phys. Rev. Lett. 1998, 80, 4422-4425 [61] Ghizzo, A., Bertrand, B., Shoucri, M., Fijalkow, E., Feix, M.R. J. Comp. Phys. 1993, 108, 105- 121 [62] Grasso, D., Califano, F., Pegoraro, F., Porcelli, F. Phys. Rev. Lett. 2001, 86, 5051-5054 [63] Grasso, D., Borgogno D., Califano, F., Farina, D., Pegoraro, F., Porcelli, F. Comp. Phys. Comm. 2004, 164, 23-28 [64] Pegoraro, F., Liseikina, T., Echkina, E.Yu. Trans. Theory Stat. Phys. 2005, 34, 243-259 [65] Durran, D.R. Numerical Methods for Wave Equations in Geophysical Fluid Dynamics; Text in Applied Mathematics 32; Springer: New-York, N.Y., 1998 [66] Dritschel, D.G., Polvani, L., Mohebalhojeh, A.R. Mon. Wea. Rev. 1999, 127, 1551-1565 [67] Imai, Y., Aoki, T., Shoucri, M. J. Appl. Meteo. Climat. 2007, 46, 388-395 [68] Reproduced from Comp. Phys. Comm., Vol. 176; Shoucri, M., Numerical Solution of the Shallow Water Equations with a Fractional Step Method, p. 23-32, Copyright 2007, with permission from Elsevier. [69] Biskamp, D., Welter, H. Phys. Fluids 1989, B1, 1964-1979 [70] Grauer, R., Marliani, C. Phys. Plasmas, 1995, 2, 41-47 [71] Politano, H., Pouquet, A., Sulem, P.L. Phys. Fluids 1989, B1, 2330-2339 [72] Pouquet, A., Sulem, P.L., Meneguzzi, M. Phys. Fluids, 1988, 2635-2642

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[73] Shoucri, M. Proc. Vlasovia Workshop, ( to be published in Comm. Nonlinear Sci. Numer. Simul.) 2007 [74] Grandgirard, V., Brunetti, M., Bertrand, P., Besse, N., Garbet, X., Ghendrih, P., Manfredi, G., Sarazin, Y., Sauter, O., Sonnendrücker, E., Vaclavick, J., Villard, L. J. Comp. Phys. 2006, 217, 395-423 [75] Jenko, J. Comp. Phys. Comm. 2000, 125, 196-209 [76] Candy, J., Waltz, R.E. J. Comp. Phys. 2003, 186, 545-581 [77] Idomura, Y., Watanabe, T.-H., Sugama, H. C. R. Physique 2006, 7, 650-669 [78] Nakamura, T., Yabe, T. Comp. Phys. Comm. 1999, 120, 122-135 [79] Mangeney, A., Califano, F., Cavazzoni, C., Travnicek, P. J. Comp. Phys. 2002, 179, 475-490 [80] Purnell, D.K. Mon. Wea. Rev. 1976, 104, 42-48 [81] Johnston, T.W., Picard, G., Matte, J.P., Fuchs, V., Shoucri, M. Plasma Phys. Cont. Fusion 1985, 27, 473-485 [82] Reveillé, T., Bertrand, P., Ghizzo, A., Lebas, J., Johnston, T.W., Shoucri, M. Phys. Fluids 1992, B4, 2665-2668 [83] Toro, E.F. Riemann Solvers and Numerical Methods for Fluid Dynamics; Springer: Berlin, 1999 [84] de Boor, C. A Practical Guide to Splines; Applied Mathematics 27; Springer-Verlag: New-York, N.Y., 1978 [85] Ahlberg, J.H., Nilson, E.N., Walsh, J.L. The Theory of Splines and their Applications; Academic Press: New York, N.Y., 1967

In: Advances in Mathematics Research, Volume 8 Editor: Albert R. Baswell, pp. 89-114

ISBN: 978-1-60456-454-9 © 2009 Nova Science Publishers, Inc.

Chapter 2

NEGOTIATING MATHEMATICS AND SCIENCE SCHOOL SUBJECT BOUNDARIES: THE ROLE OF AESTHETIC UNDERSTANDING Linda Darby Faculty of Education, Deakin University, Australia

Abstract A tradition of subject specialisation at the secondary level has resulted in the promotion of pedagogy appropriate for specific areas of content. This chapter explores how the culture of the subject, including traditions of practice, beliefs and basic assumptions, influences teachers as they teach across school mathematics and science. Such negotiation of subject boundaries requires that a teacher understand the language, epistemology and traditions of the subject, and how these things govern what is appropriate for teaching and learning. This research gains insight into relationships between subject culture and pedagogy by examining both teaching practice in the classroom and interrogating teachers’ constructions of what it means to teach and learn mathematics and science. Teachers’ level of confidence with, and commitment to, both the discipline’s subject matter and the pedagogical practices required to present that subject matter is juxtaposed with their views of themselves as teachers operating within different subject cultures. Six teachers from two secondary schools were interviewed and observed over a period of eighteen months. The research involved observing and videoing the teachers’ mathematics and/or science lessons, then interviewing them about their practice and views about school mathematics and science. The focus of this chapter is on the role of the aesthetic, specifically “aesthetic understanding,” in the ways science and mathematics teachers experience, situate themselves within, and negotiate boundaries between the subject cultures of mathematics and science. The chapter outlines teachers’ commitments to the discipline, subject and teaching by exploring three elements of aesthetic understanding: the compelling and dramatic nature of understanding (teachers’ motivations and passions); understanding that brings unification or coherence (relationships between disciplinary commitments and knowing how to teach); and perceived transformation of the person (teacher identity and positioning). This research has shown that problems arise for teachers when they lack such aesthetic understanding, and this has implications for teachers who teach subjects for which they have limited background and training.

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Introduction Science and mathematics are often closely associated during discussions about teaching and learning. This is reflected in the common expectation that teachers trained in either junior secondary mathematics or science will teach both at some time in their career. This suggests an assumption that mathematics and science have elements in common, such as common ways of thinking. This, in turn, implies assumptions about what might be common in terms of pedagogies appropriate for the two subjects (see, for example, Beane, 1995; Berlin & White, 1995). Little research, however, investigates how teachers internalize and deal with these assumptions. As disciplines, mathematics and science are distinguishable epistemologically and methodologically, and these differences are represented in the subject matter, pedagogies and purposes associated with their respective school versions. These differences place demands on teachers as they make decisions about what needs to be taught, the methods used, and the value that the subjects might have for students. The subjects are recognizably different; as are the ways students and teachers have been traditionally perceived in relation to those subjects. The distinctive nature of school subjects is described by Goodson (1993, p.31), who explaining how the organisational structure of the subject influences the ways teachers relate to the subjects and their students: [the] “subject” is the major reference point in the work of the contemporary secondary school: the information and knowledge transmitted in schools is formally selected and organised through subjects. The teacher is identified by the pupils and relates to them mainly through her or his subject specialisation.

Research is needed to understand how teachers experience the different demands that school mathematics and science place on teaching and learning. Of particular interest is how teachers construct for themselves these two subjects, and factors that influence the way teachers negotiate the boundaries that exists within the secondary school context. The research reported in this chapter explores how teachers’ experiences with the subjects influence them as they teach across mathematics and science. Negotiating subject boundaries requires that a teacher understand the language, epistemology and traditions of the subject, and how these things govern what is appropriate for teaching and learning. Teachers are, in a sense, inducted into the culture of the subjects by way of their own experiences of doing, using, learning and teaching mathematics and science. This research gains insight into the subject cultures of secondary mathematics and science from the perspective of the teacher and his/her classroom practice, focusing on the personal aspects of teaching, including how teachers see themselves as teachers, learners and participants with respect to mathematics and science. Teachers’ level of confidence with, and commitment to, both the discipline’s subject matter and the pedagogical moves required to present that subject matter is juxtaposed with their views of themselves as teachers operating within different subject cultures. This chapter begins by comparing mathematics and science as forms of education. The differences and similarities between the two subjects are explored so as to describe the cultural traditions that the teachers participating in this research are likely to have been exposed to. A section follows that explores the relationship between the individual and culture in the context of education. This leads into a discussion of the role that aesthetics (in

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the tradition of Dewey) has played in education. Theory surrounding the notions of aesthetic experience and aesthetic understanding is discussed in terms of student learning, but I open these discussions to include questions about the role of aesthetics in the relationship between subject culture and pedagogy. The research and my findings follow, using the framework of aesthetic understanding.

Comparing Mathematics and Science as Secondary School Subjects The academic disciplines of mathematics and science are represented as school subjects; however, the nature of these school versions do not, and perhaps cannot, necessarily mirror the academic versions. The foundational knowledge of mathematics and science are translated and organised for the purpose of meeting educational outcomes. In schools, mathematics is often presented as a requirement for adult life where students recognise the utility of mathematics. Crockcroft states that mathematics should also be presented as a subject to enjoy through the use of mathematics puzzles and problems that students can engage with. In addition, mathematics education takes on a role of developing “the powers of ‘abstraction’ and ‘generalization’ and their expression in algebraic form on which higher level mathematics depends… [All] students should have opportunity to gain insight, however slight, into the generalised nature of mathematics and the logical process on which it depends” (Crockcroft, 1982, p.67). Where work in science is about relating scientific evidence to scientific theory (Board of Studies, 2000a), science education allows students to be exposed to scientific ideas through participating in processes employed by scientists (cf Gunstone & White, 2000). Through learning and applying science, students are empowered as members of society: “Science education contributes to developing scientifically and technologically literate citizens who will be able to make more informed decisions about their lifestyle and the kind of society in which they wish to live” (Board of Studies, 2000b, p.5). Like mathematics education, science education serves a mechanistic purpose in the lifelong learning of students. Despite this apparent similarly in the purpose of the two subjects, Siskin’s (1994) research revealed differences in discursive patterns and dominant themes in subjects as teachers talk about their work. Siskin states that these dominant themes are worth exploring because they “translate into systematically different conceptions of the tasks of teaching and learning” (p.162). How, then do mathematics and science compare? An initial point of comparison draws on the assumption of Siskin (1994) that mathematics is a single discipline, whereas science is a cluster of disciplines, that is chemistry, biology, physics, geology. This difference has a number of implications for teaching. Where mathematics is characterised by an “ordered progression from place to place through a sequence of steps” and different levels (Siskin, 1994, p.170), science is characterised by a progression through disciplinary routes. Subsequently, in mathematics, Siskin (1994) claims that teachers develop general agreement about “what counts as knowledge, and how it is organised and produced” (p.170). Counter to these claims of general agreement, Schoenfeld (2004) states that, as with other

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subject areas, controversies exist about the epistemological foundations of the mathematics discipline, particularly “what constitutes ‘thinking mathematically’, which is presumably the goal of mathematics instruction” (p.243). Despite these controversies, mathematics has often been and continues to be characterised by incremental learning, “a slow systematic and progressive movement from the simple to the complex” (Hargreaves, 1994, p.139). Mathematics activities are, therefore, often seen as “a sequential progression through a series of topics, each of which is a prerequisite to what follows” (Sherin, Mendez, & Louis, 2004, p.208), p.208). The hierarchical nature of the way the mathematical curriculum is organised makes mathematics difficult to teach and learn (Crockcroft, 1982). With this as a teaching model, Siskin claims that “mathematics teachers value testing, placement, and tracking as the means of assigning students to the right rungs during their progress up the ladder” (p.170). Tracking is presented as a point of difference between mathematics and teachers of other subjects: where tracking is viewed by mathematics teachers as a means of meeting student learning needs, tracking is viewed by teachers from other subjects as simply “convoluted” and extraneous. One of the consequences of having widespread agreement on the content and sequence – what Siskin (1994) calls “the tight paradigm of mathematics” – is that teachers are able to learn the routines, and thereby follow the same curriculum. Homogeneity across the subject often results, Siskin asserts, such that mathematics instruction in a department can be somewhat similar. This view of homogeneity is observed by Reys (2001) who notes that in America at least there exists a generally agreed upon core body of basic knowledge such that mathematics texts from different publishers are almost indistinguishable. The best sellers are emulated by other publishers – deviation from the “norm” (best seller) results in low book sales, thereby limiting motivation to change textbooks dramatically to address the reformed American Standards-based curriculum. In 1986, Dorfler and McLone were of an opinion congruent with Reys and Siskin stating that “the material content of school mathematics is to a high degree internationally standardised. Deviations from this standard are only minor and depend on the educational system, local traditions and influences and perhaps special local demands” (p.58). This view dominates accounts of how subject matter is organized in school mathematics. On the other hand, according to Siskin (1994), the multi-disciplinary nature of school science “brings together not different ways of knowing the same content, but the same scientific method used to know different topics” (p.174). It is beyond the scope of this chapter to explore the debate surrounding either the nature of science as represented in schools, or how representative the “scientific method” is of the way scientists operate. Suffice to say that, according to Schoenfeld (2004), claims to a scientific method that permeates all the scientific disciplines is overstated, suggesting that different disciplines of science, such as physics and biology are more disparate in both theory and method than are anthropology and sociology. Many writers in science education prefer to focus on the nature of science as the underlying thread of school science curriculum and pedagogy, recognising the disciplines of science as adopting many methods but being subject to some basic tenets of science, although the question of what these tenets should be are still unresolved, (see for example Longbottom & Butler, 1999; MacDonald, 1996). Studies have investigated how the nature of science is represented, with more recent research promoting the importance of explicit instruction of and participation in the nature of science in science classrooms (Hart, Mulhall, Berry, Loughran, & Gunstone, 2000). Such learning experiences can be achieved by providing

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authentic science experiences (Rahm, Miller, Hartley, & Moore, 2003; Roth & McGinn, 1998) where students are considered as nonscientists participating in the scientific community of practice by engaging in “habits of thought” cognisant with scientific thinking (Trumbull, Bonney, Bascom, & Cabral, 2000, p.1). A second area of comparison relates to the nature of the knowledge underpinning the subjects of mathematics and science. Science is characterised by Siskin (1994) as being less abstract and more activist than mathematics. The Victorian Mathematics Curriculum and Standards Frameworks II (Board of Studies, 2000a) states that “[b]ecause mathematical knowledge is about relationships between things, it is inherently an abstract discipline. This abstractness makes it applicable in a wide variety of situations, but present particular challenges to teachers and learners” (p.5, emphasis in the original). However, application in mathematics has an important place in applying concepts and skills in the process of problem solving, where problems are contextualised for students in both familiar and unfamiliar everyday situations (Crockcroft, 1982). By comparison the Science Curriculum and Standards Frameworks II (Board of Studies, 2000) talks about application of scientific knowledge and making connections between the science community and society. For example, where mathematics patterns are taken out of context, such as tile patterns on a bathroom wall, patterns in science are dealt with in real life contexts. Scientists then “do something with it” that places the theory into practice, what Siskin (1994) calls “activist” and “making a difference”. This can be linked to the application of scientific principles to real life contexts: “science knowledge is characterised by a complexity of application of conceptions to the real world, and to classroom activities” (Tyler et al., 1999, p.211). With the variety of disciplines and the phenomena associated with those disciplines comes a “rich conceptual base” that adds complexity to planning in science (p.211). This can be contrasted with the contestable notion of an agreed conceptual sequence associated with mathematics curriculum. The National Council of Teachers of Mathematics (2000) identified that one of the call marks of an effective mathematics teacher is having an understanding of the “big ideas of mathematics and [being] able to represent mathematics as a coherent and connected enterprise” (p.17). Rico and Shulman (2004) mention the long-standing dispute over the way science should be represented, arguing for a divergence from the entrenched and much criticised “science-asfacts” model towards “science as ‘doing’, investigating, conducting research, actively seeking solutions to yet-solved problems” (p.162). Rico and Shulman state that these poor models are unfortunately perpetuated by commercially produced materials that “emphasise facts, formulae, demonstration, and vocabulary” (p.162). Processes of science are seen to be addons rather than necessary for learning science content. Studies of science textbooks show that often only the basic facts are covered and that they introduce more new vocabulary than foreign language textbooks. Van den Berg (2000) claims that American textbooks tend to have little educational value partly because the experiments are dictated by tradition, and utilize largely recipe style procedures. Traditional modes of instruction in mathematics have also been criticised and researched. Siskin’s (1994) analysis of how mathematics teachers expressed their knowledge of their discipline demonstrated above reinforces a content driven focus in school mathematics. In 1988, the Victorian Ministry of Education launched a curriculum framework that dealt with the pervading problem of classroom approaches that failed to encourage students in their mathematical learnings: “the type of mathematics that has tended to be offered to students in

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the past has become abstract at too early a stage” (p.11). Underlying their recommendations was the need to broaden students experience of mathematics so as to develop skills, concepts, applications and processes which allow meaningful participation in society” (p.12). Schoenfeld (2004) and Sherin et al. (2004) reiterate this paradigmatic shift in current mathematics curriculum where both content and process are essential for mathematical understanding. A teacher participating in Sherin et al.’s research on Fostering a Community of Learners (FCL) reported that when he began to rethink mathematics as content and process, the classroom discourse was transformed with greater emphasis given to students sharing and responding to each other’s ideas. This emphasis on teacher instruction is indicative of the general movement in mathematics education reform where effective mathematics instruction is reconceptualized as “a human construction based on historical efforts to solve particular problems, accepted modes of discourse and validation that are essentially social in nature” (Tytler, et al., 1999). In 1988 the Crockcroft Report described six elements of successful mathematics teaching: exposition, discussion, practical work, practice, problem solving and investigational work. Clearly, this seemingly new emphasis on content and process has been evident in the literature but maybe not so apparent in the classroom. In many ways science and mathematics can be differentiated. In addition to the arguments presented thus far, mathematics and science can be further characterised by their degree of reliance on equipment (“materials of the trade”), and by the type of clientele they attract, for example, science has a somewhat contestable masculine image as evident by a seemingly lack of female science teachers and students electing to continue with science beyond the post-compulsory years (Siskin, 1994). However, the fact remains that teachers, educators and researcher often closely align science and mathematics because they apparently share “linear ways of approaching things, step-by-step procedures, quantitative methods, and a mature paradigm” (Siskin, 1994, p.174). While research exists that explores how teachers represent their disciplines through classroom practices, there has been very little research that takes a trans-disciplinary approach to such exploration. The question remains, what is it about the subjects that affords and constrains particular teaching and learning practices? And from the persepective of the teacher, how do teachers’ experience of these cultural traditions shape their sense of themselves, their students and their practice? This type of information is valuable, especially at a time when the traditional boundaries between subjects are being challenged.

Relationship between Subject Culture and the Individual I am approaching this relationship between subject culture and pedagogy from the individual teacher’s perspective, recognising that, although there may be a (or a number of) subject culture(s) that these teachers are operating within and contributing to, the teachers respond to this in their own way dependent on the sum of their personal beliefs, experiences, knowledge etc. Borrowing from cultural theory relating to cultural organization and leadership, I am framing subject culture as those patterns of “shared basic assumptions that the group learned as it solved its problems of external adaptation and internal integration” (Schein, 1992). These assumptions work well enough to be considered valid and are taught to new members during enculturation. In the teaching context, enculturation involves a lifetime of experiences of learning, practicing and teaching the subject. If the “group” here refers to all science teachers

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across all schools, then subject culture refers to those shared basic assumptions that govern the dominance of certain “subject paradigms” (what should be taught) and “subject pedagogies” (how this should be taught) (Ball & Lacey, 1980). These basic assumptions act as signposts and guidelines for teaching and learning the subject. Paechter (1991) prefers to use the term “subject subculture” to recognise that every school is likely to have their own consensual view about the nature of the subject, the way it should be taught, the role of the teacher, and what might be expected of the students. Schwab (1969) refers to this consensus as unity, which he sees as important in providing opportunities for group action (see also Ball and Lacey, 1980). Schwab also impresses the importance of diversity of practice and beliefs amongst teachers. This view acknowledges that teachers will bring with them their own interpretation of teaching the subject. Similarly, Goodson (1985) argues that teachers have a personalised concept of a subject and what constitutes the practice of teaching. This perspective on subject culture supports the assumption that a teacher’s construction of the subject (including what and how it is taught) and the role of the teacher and learner, is mediated by a teacher’s lens of personal beliefs, knowledge and experiences. It makes sense then, that the effect of the subject culture on shaping pedagogy is mediated by a lens of personal beliefs about what constitutes the subject, teaching and learning. Consequently, decisions about teaching and learning are likely to be based on experiences of the subject cultures and from life. Such experiences are likely to evoke in the teacher a response that is not only cognitive, but also affective.

The Aesthetic in Education The aesthetic became important to my explorations of subject culture and pedagogy when I became attentive to how teachers constructed themselves in relation to the subject. Teachers recognised that their interest in the topic under instruction had a strong bearing on how they taught. Subsequently, my interests turned to exploring the idea that teaching and knowing how to teach involves both cognitive and affective dimensions. Zembylas (2005b) recognises that emotion and cognition are inextricably linked in the process of student learning. I assert that the same can be said for teachers in their development as mathematics or science teachers. Increasing attention is being given to the affective domain as researchers explore its centrality in the learning of mathematics (Bishop, 1991; Sinclair, 2004), learning of science (Alsop, Ibrahim, & Kurucz, 2006; Chandrasekhar, 1990; Zembylas, 2005b) and learning in general (Beijaard, Meijer, & Verloop, 2004; Ivie, 1999; Pajares, 1992; Schwab, 1978; Zembylas, 2005a). The affective domain is often separated from cognition (Sinclair, 2004). Aesthetics is part of the affective domain, as are beliefs, values, attitudes, emotions and feelings, self-concept and identity (Schuck & Grootenboer, 2004). Educational research into the nature and importance of the aesthetic has centred predominantly on its role in learning (Gadanidis & Hoogland, 2002; Girod, Rau, & Schepige, 2003; Wickman, 2006). Other research focuses on the role of the aesthetic in the activity, psychology and affective response of scientists and mathematicians to their discipline, often with the intent of informing mathematics and science teaching of that which provokes an aesthetic response (Burton, 2002, 2004; Sinclair, 2004). For example, Sinclair (2004)

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explains that the aesthetic has long been claimed to play a central role in developing and appreciating mathematics. Recognition of the beauty of mathematics stems from the Ancient Greeks who believed in the affinity between mathematics and beauty based on its order, symmetry, harmony and elegance. This is often called the aesthetic of mathematics, but such an aesthetic is often removed from the mathematics curriculum (Doxiadis, 2003) and the mathematics story is shortened to a sequence of steps that can result in students failing to experience the pleasure of the process (Gadanidis & Hoogland, 2002). In science also, the words beauty, inspiring, artful and passion are often used by scientists to describe their work (Girod, Rau & Schepige, 2003). “The scientist does not study nature because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful…intellectual beauty is what makes intelligence sure and strong” (Poincare, 1946, quoted in Girod et al., 2003, p. 575). Science educators draw from the discipline of science the important ideas, behaviours and dispositions that should be presented to students. If science is characterised as being analytic, logical, objective and methodical, this is then translated in classrooms as requiring students to be removed critical observers of objects, events and the world. By comparison, Girod et al. (2003) make the point that some scientists “portray science with an opposing personality—one that draws us in, begs our curiosity, passion, and emotion” (p.575), which, if translated to the classroom, they claim can improve the quality of the learning experience. These portrayals of science and mathematics as eliciting an affective response such as curiosity and the pleasure of the process are in contrast to the objects of science and mathematics that “are amenable to a rational and cognitive inquiry” (Wickman, 2006, xii). Understanding these contrasting positions comes from Dewey’s theory of aesthetic experience. Dewey breaks down false binaries such as objective and subjective, logic and intuition, thought and feeling, mind and heart, and think and feel. Wickman explains that in an aesthetic experience the inner emotional world is continuous with the outer world, meaning that one cannot think of one without the other. The cognitive (factual, what is the case) cannot be conceived of without the normative (values, what ought to be) in an aesthetic experience (which is evaluative). In keeping with this epistemology, Girod et al. (2003) claim that “from the perspective of aesthetic understanding, science learning is something to be swept-up in, yielded to, and experienced. Learning in this way joins cognition, affect, and action in productive and powerful ways” (p.575-576). Limited research seeks to clarify the role of the aesthetic in teachers’ work (see Ivie, 1999), however, teaching is often referred to as an artistry (see, for example, Rubin, 1985). This chapter focuses on the role of the aesthetic, specifically aesthetic understanding, in the relationship between subject culture and pedagogy. I frame this in terms of not so much what and how the teachers learn, but how their aesthetic understanding relating to teaching mathematics and science can give insight into how teachers negotiate boundaries between the subjects of mathematics and science and their enacted subject cultures. In particular, the aims of the chapter are to: • •

Focus on how teachers construct themselves as teachers of a subject for which they have a level of commitment and about which they hold beliefs and values; and Explore the degree to which and in what manner the teacher has an aesthetic response, as part of their personal response to the subject cultures within which they teach.

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To do this, I use the framework of aesthetic understanding from Girod et al. (2003) to explore how the teachers’ construction of the subject and teaching is not simply cognitive but has an aesthetic dimension. “Aesthetic understanding is a rich network of conceptual knowledge combined with a deep appreciation for the beauty and power of ideas that literally transform one’s experiences and perceptions of the world” (p.578).

Methodology Past experience in qualitative research (see, for example, Darby, 2005b) led me to a methodology consistent with the belief that we construct meaning through interaction with our social setting. Consequently, the meaning gained from this research is considered to be a co-construction between the participants and myself as researcher. Such an emphasis on research refers to “qualitative research,” an umbrella term used by Merriam (1998) relating to orientations to inquiry focussing on understanding and explaining the meaning of social phenomena. Qualitative inquiry evolved out of recognition that human beings are chasms of complexity that could not be understood through the positivistic process of scientific experimentation that simply tests scientific theory. The positivist approach demands an adherence to procedures that are reproducible, based on refutable knowledge claims, and controlled for researcher errors or bias (Gall et al.,1999). Cohen and others (2000) describe positivism as being “characterised by its claim that science provides us with the clearest possible ideal of knowledge” (p.9). It is objective and quantifiable. Reality is considered to be stable, observable and measurable (Merriam, 1998). Used within the social sciences, the methodological procedures used to investigate social phenomena mirror those used in the natural sciences, and the end-product is expressed as laws or law-like generalisations akin to those established for the description of natural phenomena (Cohen, et al., 2000). Bogdan and Bilken broaden the scope of what qualifies as research to reflect a paradigmatic shift from positivism towards qualitative research, also referred to as postpositivism, or “anti-positivism”, and that Cohen and others (2000) consider to be naturalistic. Bogdan and Bilken’s (1992) qualitative research mode emphasises “description, induction, grounded theory, and the study of people’s understanding” (p.ix). In this view, there is a rejection of the positivist view of an objective observer of phenomena on the basis that the behaviour of individuals “can only be understood by the researcher sharing their frame of reference: understanding the world around them has to come from the inside, not the outside” (p.19). A key philosophical assumption underlying this form of inquiry is that “reality is constructed by individuals interacting with their social worlds” (Merriam, 1998, p.6). It is these constructions of reality, or meaning perspectives of individuals, that my research into teacher pedagogy was interested in accessing and understanding. This research explores the complex ways that teachers construct for themselves their ideas about teaching and learning, and the factors involved in the way these constructions may appear to be manifested within the classroom setting. The qualitative paradigm qualifies this subjective knowledge of teachers, the emic1 or insiders’ perspective, as worthy of investigation, useful and informing of educative practice, and provides a vehicle for understanding the complexity 1

as distinct from and preferred over the etic or outsiders’ perspective.

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of the social setting within which the teachers are situated. What’s more, qualitative research has the potential to provide rich and meaningful information to the body of educational research as it characteristically builds abstractions, concepts, hypotheses or theories through an inductive process rather than simply testing existing theory (Merriam, 1998). Such a view is consistent with a constructivist approach to research as described by Guba and Lincoln (such as 1998, 1981). The research reported in this chapter uses the discourse of the classroom and interviews to understand how teachers have constructed for themselves knowledge about teaching and learning, and the various factors that are brought to bear on both the development of their knowledge and beliefs and the manifestation of such beliefs in the classroom. My research is most suitably called constructivist as I have attempted to understand and reconstruct the constructions held by both the participants and the researcher. Constructivists operate according to the premise that knowledge and truth is constructed, not discovered by the mind; and that reality is both expressed in a variety of symbols and language systems, and “stretched and shaped to fit purposeful acts of intentional human agents” (Schwandt, 2000, p.236). Constructivist inquiry, Guba and Lincoln claim, “denotes an alternative paradigm whose breakaway assumption is to move from ontological realism to ontological relativism” (p.203). According to Guba and Lincoln (1998) a relativist ontology claims that “[r]ealities are apprehensible in the form of multiple, intangible mental constructions, socially and experientially based, local and specific in nature (although elements are often shared among many individuals and even across cultures), and dependent for their form and content on the individual person or groups holding the construction” (p.206). Constructions are considered to be “more or less informed and/or sophisticated” (p.206), rather than absolutely “true.” Furthermore, constructions and the realities associated with them are subject to change as the constructors are more informed and sophisticated. This results in the potential for multiple and sometimes conflicting social realities of the human intellect. My research focuses on how the mathematics and science teachers are constructing for themselves pedagogy while operating within and in response to the social setting of mathematics and/or science education, making the constructivist paradigm suitable. Although the research is focused more closely on the individual teacher’s constructions, I used my interactions with the teachers, the setting, and the literature to assist me in constructing a broader picture of the teachers’ cultural setting. These act as the social setting for the research, a setting cushioned in a socially mediated subjective language that, through this act of research, for me has meaning through my experience of it.

Research Methods The research reported in this chapter forms part of a doctoral study associated with a Deakin University ARC Linkage Project with the Victorian Department of Education and Training called Improving Middle Years Mathematics and Science (IMYMS)2. Six teachers of mathematics and/or science from two schools (School A and School B), teaching across 2

The IMYMS Project is being undertaken by Russell Tytler and Susie Groves of Deakin University, and Annette Gough of RMIT and funded by an Australian Research Council Linkage Grant, and linkage partner, Victorian Department of Education and Training. Funding was granted in 2003. My Ph. D. is one of two Australian Postgraduate Awards (Industry) associated with this grant.

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Years 7 to 10, participated in a dialogue with me and each other over a period of about one year in order to understand differences between the subject cultures of mathematics and science. A variety of qualitative methods were selected that would support and feed into this dialogue. These methods are outlined below. Two sequences of lessons in mathematics and/or science were observed for each teacher in order to gain some insight into the general practice of the teachers. Two of these lessons on two separate occasions were videoed, one mathematics and one science lesson for three teacher (Simon*, Pauline*, Ian^), two science lessons for two teachers (Donna*, James^), and two mathematics lessons for one teacher (Rose*). (* indicates teachers from School A; ^ indicates teachers from School B.) The video footage of both lessons on both occasions were returned to each teacher for personal viewing with a set of questions to guide their attention and reflection (a modified video stimulated recall process). A “reflective interview” with each teacher followed the private viewing on both occasions. The first interview explored teacher’s response to the video and the questions, and explored teacher background with, commitments to, and beliefs about the subjects, as well as exploring any lines of inquiry that were emerging from preliminary analyses of classroom observations or prior interviews (involved all teachers) (see Darby, 2004, for an explanation of the reflective interview and modified stimulated recall process). The second interview was preceded by an informal discussion with the teacher about the aims and big ideas represented in the unit of which the videoed lessons were a part, then the reflective interview asked teachers to explain how this lesson fitted within the unit sequence (involved only Simon, Donna and Rose; Pauline participated in the informal discussion but not the second reflective interview). A focus group discussion involving the four teachers from School A followed the first round of videoing and reflective interviewing, with discussion based around three statements arising from data analysis. Each statement was accompanied by feedback to each teacher that included excerpts from their reflective interviews that contributed to the development of the statement, and supportive experpts from literature that expand on or correlate with the teachers’ ideas. The statements were:

Statement 1: Mathematics and science place different demands on teachers and students. For example, a student absent from mathematics for an extended period of time is at a greater disadvantage than a student absent from science for an equal amount of time. Is this necessarily the case? Are there parts of learning and teaching in mathematics and in science for which this is not really true? Statement 2: a. There are some practices that are translated readily from mathematics to science and vice versa. b. There are some practices in science that really should be used more often in mathematics, and vice versa. c. There are some practices that cannot be translated because the subjects are very different. What are your views on this? Statement 3: The influences on teachers' treatment of content in their teaching, and their attitude to the subject, are in the following order: 1. school, personal and work experiences in relation to subject interests; 2. their undergraduate degree experience; 3. conversations and

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interaction with other teachers; 4. experiences of teaching the subject; 5. curriculum documents and direction by the subject department; and 6. professional development. To what extent is this true for you?

Teacher Profiles This chapter draws on data from three of the teachers from School A: Donna, Pauline and Rose. School A is a co-education school offering Years 7-12 in a provincial city in Victoria. These teachers were selected by the Head of Science to participate in this “video study.”

Donna Donna was in her fourth and fifth year of teaching during the project. Donna originally went through high school with the intention of becoming a veterinarian but then decided to explore her interests in zoology and ecology through a Bachelor of Science. Prior to doing a Graduate Diploma of Education in 1999, Donna had been working at a tourism park as an education officer, taking tour groups on possum prowls and conducting other environmental activities. She also worked at a horse-riding place and managed school and other groups, and has been involved in dolphin research. School A is Donna’s second school. Throughout her teaching career, she has taught junior science at all year levels, Year 11 and 12 Biology, and some junior mathematics.

Pauline Pauline was in her second and third year of teaching during the project. She completed a three-year Bachelor of Science majoring in physics, then enrolled in a two year teaching degree that prepared her to teach Prep to Year 12. Her methods were general science and senior physics, but she was also qualified to teach mathematics to Year 12. Pauline chose the combination of science and mathematics due to the demand for science and mathematics teachers. School A was the second school she has taught at. At both schools she has been teaching junior mathematics and science, and Year 11 and 12 Further Mathematics and Physics.

Rose Rose has been a mathematics teacher for about 15 years. Rose went to university to complete a Science Education degree, where she studied mathematics, statistics, chemistry and physics. She had no interest in the science, however, only doing it because she thought she had to. Although she has taught science, she chose fairly early in her career to teach only mathematics. Since completing her training, Rose has taught at various schools. During the project, Rose assumed the role of Head of Junior Mathematics. She taught mathematics at all year levels.

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Looking for the Aesthetic in the Relationship between Subject Culture and Pedagogy Girod et al. (2003) describe aesthetic understanding as being “transformative,” “unifying” and “compelling and dramatic” (p.578). These three aspects of aesthetic understanding are described below using excerpts from the interviews of Pauline, Donna and Rose. I use the teachers’ reflections to explore, firstly, how the subject culture frames the development of these three components of aesthetic understanding, and how the teachers’ aesthetic understanding of the subject guides how they teach. I then explore how the application of this framework helps to understand the relationship between pedagogy, which is underpinned by theoretical and perspectival frameworks in relation to teaching and learning (van Manen, 1990), and cultural practices of the subject, which the teachers participate in and contribute to.

Compelling and Dramatic Nature of Understanding This aspect of aesthetic understanding recognises that aesthetic experiences are steeped in emotion. Aesthetic experience “…quickens us from the slackness of routine and enables us to forget ourselves in the delight of experiencing the world about us in its varied qualities and forms” (Dewey, 1934/1980, quoted in Girod et al., 2003). In such experiences, emotion, cognition and action are fused. So when Rose says to her students at the beginning of the year “I love mathematics and by the end of the year I want you to really like mathematics too” she is demonstrating her passion for mathematics, that there is something about mathematics that compels her into further engagement with it. It is this that she wants to share with the students so that they can appreciate mathematics in the same way. Rose explains that she is interested in mathematics because it is logical and “it appeals to my logical brain.” A passion for the subject is evident here, a passion for the content matter, but also for teaching the content. In the focus group discussion I asked the teachers what passion is and what it looks like in mathematics compared to science. Rose shared with me during the focus group discussion an experience she had during a lesson where she and a small group of students were working together on a different task to the rest of the class. “And I was so engrossed,” Rose exclaimed, “I didn’t realise the class had finished! And I turned around and they were all sitting back in their chairs, but my kids were so engrossed in what they were doing and really happy.” Donna replied, “That’s what passionate looks like in mathematics!” Rose’s passion for promoting student engagement with the subject is recognisably an experience of “flow” where, simply put, a person is so engrossed in a task that they lose all sense of time (Csikszentmihalyi, 1997). During the focus group discussion, various teachers explained how passion for the subject (or discipline) as distinct ways of knowing and bodies of knowledge are evidential in the classroom: “You’re interested in it. Enjoy it. If you enjoy something then you’re going to impart that enjoyment onto your students” (Rose); and “You can see that [teachers] know their stuff and are passionate about mathematics” (Donna).

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Teachers’ lack of passion about the subject was also considered to be evident to students: “I think kids pick up on it when you don’t enjoy it. If you’re teaching something you don’t particularly enjoy, it seems like they muck up more. I dunno, maybe we’re all suffering together!” (Pauline). Many authors assert the importance of students seeing that teachers are passionate about their subject (see, for example, Darby, 2005b; Education Training Committee, 2006; Lane, 2006; Palmer, 1998). During the focus group discussion I asked teachers: if passion for the subject is so important what happens when teachers teach outside of their subject area? Donna explained that in these instances, a general passion for teaching students is important. As Donna explains below, this passion is rooted in that which first lured them into teaching: DONNA: What got you here in the first place, your passion for teaching.You may not be happy about it, but you’ve still got the basic passion for teaching to try and do the right thing by the kids and you go out of the way to make sure, no matter what subject it is, that you’re teaching them the best way you can…. It comes down to that you’re teaching people, not the subject.

This suggests that a passion for teaching is related to the activity of teaching students, separate from the content matter under instruction. In this case, the passion emerges out of a desire to engage with students.

Aesthetic, Passion and the Subject Three forms of passion are evident above: a passion for the subject matter, a passion for promoting student engagement with the subject, and a passion for teaching in general. This multi-dimensional framing of what drives teachers is represented by Day (2004): To be passionate about teaching is not only to express enthusiasm but also to enact it in a principled, value-led, intelligent way. All effective teachers have a passion for their subject, passion for their pupils and a passionate belief that who they are and how they teach can make a difference in their pupils’ lives, both in the moment of teaching and the days, weeks, months and even years afterwards. Passion is associated with enthusiasm, caring, commitment, and hope, which are themselves key characteristics of effectiveness in teaching. (p.12)

As indicated above by Donna, this sense of care can be perceived of as a passion for teaching in general and as separate from the subject matter. This is likely to be important for those teachers with a teaching allotment that includes a subject for which they have limited experience, training and commitment, and more than likely, passion. The question here is where the passion lies for the teacher: in the act of relating with students (as stated by Donna), or in the act of engaging students with subject matter that the teacher believes is valuable, whether it be process or conceptual (as demonstrated by Rose’s commitments in teaching mathematics). A question remains as to whether a teacher can be effective at engaging students in the subject matter if they have little passion, or even appreciation, for the subject. Rose believes that teacher interest is vital: “If you’re not interested in something, you shouldn’t teach it!” Day also describes the importance of teachers sharing with students a commitment to the subject they are teaching:

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When students can appreciate their teacher as someone who is passionately committed to a field of study and to upholding high standards within it, it is much easier for them to take their work seriously. Getting them to learn then becomes a matter of inspiration by example rather than by enforcement and obedience. (p.15)

The Education and Training Committee’s (2006) inquiry into the promotion of mathematics and science education in Victoria supported this view saying that when promoting student engagement there is a “need for teachers to be passionate and deeply knowledgeable about their subject area” (p.172). Following this view, a passion for teaching is more likely to be coloured by a teacher’s conceptual and aesthetic commitments to the subjects they teach; therefore, passion for teaching, at least at the secondary level, is less likely to be seen as generic, but more likely subject specific. Research by Siskin (1994) into the culture of subject departments in secondary schools found that what mattered for the teachers in her study was “not simply that they teach, but what they teach” (p.155, emphasis in original). Neumann (2006) asserts that, in the context of scholarship in higher education, “passion illuminates the complexity of both teaching and research, showing that what resides at the heart of both is the learning of a particular subject” (p.413, italics in original). Subject here refers not necessarily to a school subject or discipline but a subject of thought on which a conversation can be focused. In the classroom, the teacher makes the focus of conversation the ideas of mathematics or science, however, how they represent these ideas depends on the teachers beliefs about what the subject can offer the students. For Rose, mathematics offers training in logic and a potentially enjoyable endeavour. A passion for teaching remains, then, to be coloured by the teacher’s conceptualisation of the subject. According to this view, pedagogy is influenced by an inextricable link between the way teachers see their students and the subject: teachers have an understanding of what students need in order to make the subject matter have meaning. “Teachers understand and value their subjects for what they offer students, and understand their students through the metaphors and assumptions of the subjects” (Siskin, 1994, p.158). Consequently, pedagogical knowledge is tied to how the teacher understands the knowledge of the subject. Conversely, the content knowledge of teachers as representations of the epistemology of the subject is transformed in a way that meets the perceived learning needs of the students. Overarching both of these relationships, however, is the teachers’ aesthetic commitments to and appreciations for the subject.

Learning that Brings Unification or Coherence to Aspects of the World or the Subject This aspect of aesthetic experience acknowledges that “it is not possible to divide in a vital experience the practical, emotional, and intellectual from one another” (Dewey, 1934/1980, quoted in Girod et al., 2003, p.578). Experience is complete and results in deep meaning because the experience retains its value and wholeness, and this coherence can be used to guide future experiences. According to Girod et al., an “aesthetic understanding depends on developing a similar coherence of parts, pieces, ideas, and concepts” (p.578). This is evident in the classroom when the learning of individual parts of a concept brings greater understanding of the entire concept.

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The teachers in my study referred to this element of aesthetic understanding when they talked about planning for different subjects. Donna explains here that, she has a stronger grounding in biological science due to personal experiences with the subject matter, the discipline, and the type of thinking required, the manifestation of which is a more intuitive approach to teaching science than mathematics: DONNA: I don’t have a big mathematics background, so I have to spend a bit of time thinking about what could be available and what I could do; whereas with a science background, I think of things just because I’m experienced in that area. So I suppose it might depend on how much mathematics you’ve done or what resources you’ve been exposed to, what you might know of... I do a lot more prep for a topic like physics than I would for chemistry or biology. I’m teaching a nine ten combined class in biology, and I’m finding that, like I do my normal prep but I can just go off in class and say, I did this and I’ve got this example, and we’ve been having great class discussions and fun activities. I wouldn’t have the confidence doing that with a physics topic. So I might spend a lot more time researching it, I might check a few things with another teacher. But I wouldn’t have that flamboyance in a topic that, because I haven’t done physics at all, apart from bits and pieces of it.

Donna compares her teaching of biology to that of physics, both sciences and underpinned by a common philosophy of what constitutes knowledge, but distinct in terms of the nature of the phenomena being represented. Donna’s coherent and unified picture of biological science stems from her experiences of learning biology and working with these science concepts in the natural world. Physics, however, is perhaps as foreign for her as any other subject that has not been encountered in any meaningful way. It is for this reason that her teaching of biology requires less planning and research compared to her teaching of physics or mathematics.

Aesthetic, Coherence and the Subject In Donna’s reflection, there is a degree of understanding of the connections between ideas and content, but also how the content is used in a way that is appropriate for student learning. Knowledge of the content matter and the knowledge required to teach this knowledge is evident. The knowledge that Donna refers to can be aligned with Shulman’s knowledge domains that he introduced in 1986 and 1987 to emphasise the domain-specificity of knowledge. Shulman distinguishes between subject matter knowledge, pedagogical content knowledge and pedagogical knowledge. “Subject matter knowledge”, also called content knowledge, is the knowledge that teachers have about the content considered appropriate for teaching. Donna explained that such knowledge is related to the extent of her background with respect to the subject. Having a limited background in physics has meant that she has less content knowledge, and which results in her having to do more preparation for her lesson planning. “Pedagogical content knowledge” (PCK) adds to this dimension of subject matter the knowledge required for teaching it to students, and includes the “ways of representing and formulating the subject that makes it comprehensible to others” (Shulman, 1986, p.10). PCK refers to that conglomeration of teacher knowledge that transforms the subject matter in a way that is sensitive to the needs and requirements of the learners.

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In Donna’s case, she recognises that her pedagogical content knowledge, that is, knowing how to make the content understandable for students, is limited by her deficient subject matter knowledge of physics and mathematics. In comparison, she attributes her ability to teach biology to her “background” of experiences, and that this allows her to more meaningfully transform subject matter knowledge using classroom strategies where her richer understanding of the subject narrative can be shared with students. At first blush it appears that knowledge of content, resources and strategies for teaching accounts for her greater confidence with the teaching of biological science. The “flamboyance” she refers to hints to something other than knowledge, such as an intuitive sense of how to use the science ideas and her experiences to draw students into thinking, talking and engaging with the ideas: “You can think of different ways to get it across to the kids.” She has feelings of “comfort” and confidence in her ability to bring the subject of biology to life for her students. “To know something,” states van Manen (1982, p.295), “is to know what that something is in the way that it is and speaks to us.” That which first appears cognitive takes on an intuitive nature, and this becomes part of what teachers do but may not know that they do or why they do it.

Perceived Transformation of the Person and the World Donna’s description of her teaching above exudes a sense of pride in what she knows and how she can share this with students in an engaging way. There is passion, no doubt, but she has also “developed a sense of self in which the pride of the craft [is] the key” (Palmer, 1998, p.14). A person is transformed by what they have experienced and what they have come to know out of that experience. “Knowing changes the individual as well as the individual’s world” (Girod et al., 2003, p. 578). The transformative nature of aesthetic understanding can lead to identity formation and personal positioning. A person can say “I am the type of person that looks at the world in this way.” In the context of my study, this relates to how teachers position themselves as teachers of a subject, and how this positioning stems from their experiences of teaching, learning and participating in mathematics and science. I describe two teachers here, Rose and Pauline, to demonstrate how they position themselves in relation to the subject based on their level of competence and confidence with teaching the subject.

Rose’s Transformation Rose’s experiences and interests shape the way she sees herself. Rose stated a number of times that she describes herself as a teacher of students, not a teacher of the subject: “I see myself as a teacher first, not a mathematics teacher… I’d been looking after little kids from when I was this high. I just loved looking after kids.” During a different interview she mentioned that being a mother has made her more attentive to the support needs of her students such that she is more attentive to the overall well-being of her students. ROSE: I believe I do a lot more instructing than some people and I also do a lot more student helping… I think it is because I’m a Mum. I taught for 4 ½ years before I had children and then I came back to teaching and I reckon I was a lot better teacher, because I relate to it… I don’t think it’s an us and them, I think its an us together.

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She situates herself not necessarily outside of being a mathematics teacher, but prefers to identify herself as someone who has strong beliefs about the centrality of the student in the teaching-learning interface. This was demonstrated also when, on viewing her videoed lessons, she said that: “I looked for how the kids were working ‘cause that’s interesting. What I said and how I responded to the kids. To their needs. That’s what I look for.” Over the years, Rose has developed an understanding of her role as teacher that has transformed her into a person that is attentive to the needs of her students. The nature of the subject shapes her pedagogical response, such that a sense of care compels her to teach mathematics in a way that makes it less threatening for students: ROSE: I want them to enjoy mathematics. Because mathematics is a threatening subject, it is so threatening because it is so sequential…[At the start of the year] there was hardly anyone that liked mathematics, some of them thought they were good at it, but hardly any of them liked it. You ask them now they have come right round because they enjoy it.

Rose refers here to a mathematics curriculum characterized by incremental learning (Hargreaves, 1994) and sequential progression (Sherin, Mendez & Louis, 2004). Because she understands the threat that such a curriculum structure might pose for students, her sense of care for the students compels her to employ actions that remove the threat and make her view of “mathematics-as-enjoyable” more accessible and in the realm of possibility.

Pauline’s Identity Crisis as She Negotiates Subject Boundaries Pauline spoke of a rich science background with interests and studies in physics, and many engaging and interesting experiences in relation to science. In order to get a sense of how Pauline situates herself in relation to mathematics and science, I need to first reconstruct Pauline in relation to the previous two aspects. Evident in the following quote is a confidence in how she expresses an appreciation for the purpose of science in her own and her students’ lives, as well as what it means to be passionate about science: PAULINE: I find my knowledge of Science extends to everything. It extends to when I go to the Doctor and I talk about my health … everything I do is informed by my science knowledge, and I just think that scientific literacy is so important for kids to get the most out of themselves, out of their world… I like collecting [stories]. I don’t think I have enough. I like telling stories and getting the kids’ stories out as well. And I have found that when I studied science they were the things that got me excited when a teacher told me a really interesting story and I don’t know if mine are interesting or not, but I know that they were the sort of things that got my interest going in science and why I wanted to do more.

In comparison, limited expertise in mathematics teaching makes it difficult for Pauline to be confident in her abilities, and she defers to a label of science teacher rather than mathematics teacher, as evident in the following quotes: “I am not really experienced enough or done enough PD [Professional Development] to know better ways of doing it. A major part of my PD plan, especially for middle years, is doing more PD and finding better ways to teach stuff ‘cause I don’t like the way I teach Mathematics at the moment” “I think I am a crap mathematics teacher.”

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“It is funny. I feel more confident teaching science than I do mathematics, even though I have been teaching both for the same amount of time” “I have always felt Mathematics is kind of my fall-back method. Whereas if I was asked to describe myself I would describe myself as a Science teacher, first and foremost.”

Quite clearly, Pauline has a stronger sense of herself in relation to science teaching than mathematics teaching. She attributes this partly to her limited background experience with mathematics: “Well my mathematics method is just a thing on paper that says that I did mathematics to second year at Uni. There was nothing that I did in my teaching degree that prepared me for teaching mathematics. The only preparation that I had was my rounds.” She laments at not knowing how to make mathematics learning more interesting for her students because of her limited intuitive sense of what will work in the classroom, she is less capable of finding resources and knowing what to look for, and she has a limited sense of how to be passionate about teaching the subject in a way that will profit student learning at the junior level. She enjoys teaching mathematics at the senior level because she enjoys toiling over problems with the students, but she is unable to do this as much at the junior level. These limitations to her knowledge led her to the conclusion that she is less comfortable with the label of mathematics teacher than she is with that of science teacher.

Aesthetic, Identity and the Subject In Beijaard’s (1995) research into the interplay between the private and public in developing identity, he makes a distinction between role and identity – hope and courage, care and compassion, he asserts, are associated with identity, not role. In the above example, Pauline appears to accept the role of mathematics and science teacher and the associated activities that are assumed as part of this role, but her identity arises out of her history of caring for and committing to science as an area of study. Further, in Pauline’s description above, she attributes her lack of confidence in mathematics teaching with lacking the knowledge of how to teach. Earlier, Donna recognised that her teaching of biology is benefited by knowing what activities will work and when. Day (2004), however, points out that knowing what and how to teach is not limited to cognitive engagement. He states that “good teaching can never be reduced to technique or competence” (p.15). Good teachers, he asserts, tend not to describe themselves only in terms of technical competence, but also acknowledge that “teaching and learning is work that involves the emotions and intellect of self and student” (p.64). This difference between a competence view and the aesthetic was demonstrated by Pauline’s appraisal of herself as a mathematics teacher and a science teacher. Her deficit view in relation to mathematics that she attributes to limited technical competence is based on limited knowledge of what and how to teach, and her hope lays in future professional development to provide useful strategies for teaching. By comparison, her appraisal of her competence and confidence in science was laden with meaningful experiences and stories from a history of engaging with the subject. Pauline exhibited a richer sense of herself in relation to her science teaching, one that is positive and based not solely on competence, but she also aligns herself with science teaching at an emotional level. Her knowledge of how and what to teach is ‘continuous with’ her aesthetic response, meaning that one cannot think of one without the other.

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Insights and Implications The previous analysis and discussion have explored the idea that a teacher’s aesthetic understanding of and response to the subject determines: where their passions lie with respect to teaching the subject, to what extent they have a coherent and intuitive sense of what is required to teach the subject, and how the teacher is transformed by what they know as they develop an identity in relation to the subject. These discussions are valuable in understanding the relationship between subject culture and pedagogy for two reasons. These reasons are dicussed below.

Appreciation for the Aesthetic in the Teaching Act The first is that a framework of aesthetic understanding helps to clarify and assign some level of importance to the role of the aesthetic in the teaching of subject matter to students. A teacher who can be regarded as having an appreciative aesthetic understanding of the subject: • • •

is compelled by and passionate about the subject and students engaging with the subject; has a coherent, unified and intuitive sense of what the subject is about and how to bring it to life for students; and has been transformed by what they know and believe in a way that aligns them to personally and professionally identify with the subject.

Being attentive to the aesthetic when evaluating teaching redirects the question from simply asking, what does the teacher know and believe about the subject and what is required to teach it? Instead, the question becomes, how does what the teacher know and believe affect her sense of who she is in relation to the subject, and how does this personal positioning spill out into the classroom? The analysis has shown that a teacher with an appreciative aesthetic understanding of a subject see themselves, the subject matter, their teaching and their students in relation to the subject. Even Rose, who labelled herself as a teacher of students rather than a teacher of the subject, expressed her sense of care in the context of, and in response to, the nature of the subject and what was required for students to learn. The student is central to her conceptualisation of the subject. She was unable to describe what the subject is like without including stories about her interactions with students on a personal level, and in relation to how the students learn in the subjects. By talking about how she interacts with students and the students’ learning needs, Rose gives clues as to her values and aesthetic commitments to the subject, which is viewed through a lens of what the subject offers her students as well as what it offers herself as learner, practitioner and teacher of the subject. A common emphasis in current science education reform is to draw on and respond to student interests in selecting contexts for teaching science-related content. Pivotal in achieving this end is giving teachers space within the curriculum to inject their own interests, hobbies and expertise in constructing such contexts. Tytler (2007) provides examples of innovation occurring in schools where “teachers with serious interests [felt] that they were being given permission to import these into the classroom” (p.57-58):

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In one school a teacher with no previous history of innovation was encouraged by the SIS coordinator, who knew of his interest in winemaking, to initiate a Chemistry of wine making unit. The school is now producing award-winning wines. (p.52, italics in original)

These types of stories, Tytler (2007) asserts, exemplify a re-imagined science education for Australia. Hence, teachers’ interests are highlighted as important in the development of local content and approaches. In these situations, teachers are more likely to possess an aesthetic understanding that is deeply rooted in teachers’ experiences, and where the subject matter has personal meaning for the teacher. Pedagogical practices can be enriched by a deep understanding of the associated content, which, provided the learning needs and interests of students are taken into account, provide a strong foundation for knowing what value it might have for students and how such contexts could be generative of new interests for students.

The Aesthetic in the Negotiation of Subject Boundaries Secondly, examining teachers from the perspective of aesthetic understanding provides insight into what is involved for teachers, aesthetically, as they move between subjects and their enacted subject cultures. Such insights are particularly pertinent at present when a shortage of suitably qualified mathematics and science teachers is resulting in a relatively high percentage of teachers teaching out-of-field, that is, teaching a subject for which they lack tertiary training, and arguably, limited experience, commitment and, aesthetic understanding. A survey involving 8.2% of teachers of junior science in Australia (Harris, Jensz, & Baldwin, 2005) showed that 16% of respondents lacked a minor in any university science discipline, while 8% had not studied any tertiary science. Similarly, a survey of mathematics teachers from 30% of Australian schools (Harris & Jensz, 2006) showed that 20% of teachers of junior mathematics had not studied mathematics beyond first year university, while 8% had no tertiary training in mathematics. Other reports in the media reflect similar or higher proportions of teachers teaching outside their fields of expertise (Rodd, 2007; Topsfield, 2007). The figures are even more startling for teachers beginning their careers. Unfortunately, these teachers are more likely to be asked to teach out-of-field than their experienced colleagues (Ingersoll, 1998). A recent study of beginning teachers in Australia showed that 40.1 % of teachers nationally and 57% in Victoria had taught subjects outside their qualifications (Rodd, 2007). While it is acknowledged that tertiary training will not automatically result in effective teaching, the major concern both nationally and internationally is that without solid tertiary experience in the discipline, teachers lack content knowledge, and without studies in the teaching of a subject, teachers are not equipped with the variety of methods and teaching skills required to teach the subject effectively (Darling-Hammond, 2000; Education Training Committee, 2006; Ingersoll, 1998; Thomas, 2000). The data reported in this chapter suggests that a teacher teaching out-of-field, whether it be a science teacher teaching mathematics (in the case of Pauline) or a biologist teaching physics (in the case of Donna), potentially has limited or unappreciative aesthetic understanding of what the subject can offer his/her students. This has implications especially when the history of engagement with the subject has been negative, restricted to poor traditional learning experiences, or limited. Reliance on traditional teaching approaches may result, as may a lack of “flamboyance” in the way the subject is presented, with a potential

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outcome of not demonstrating for students what it looks like to appreciate the subject. Also teachers teaching outside of their disciplines, such as a mathematics teacher teaching science, may bring with them a sense of what constitutes good teaching appropriate for one subject that may seem inappropriate in another. A theoretical framework of aesthetic understanding, therefore, helps to identify the barriers, disconnections, and lacking appreciations that may prevent teachers who are not trained in the discipline from personally engaging with the subject, which, inevitably impacts negatively on the quality of teaching. The problem for the “untrained” mathematics or science teacher is not simply a lack of content knowledge, but this framework of aesthetic understanding gives significance to the importance of teachers being committed to the subject, being able to identify with it, and knowing how to bring the subject matter alive for students. Tertiary training is considered to be the most effective determinant of whether a teacher is suitable for teaching a subject. Having a background in a discipline, it is assumed, equips teachers with the disciplinary knowledge to draw on in their teaching, but it also equips teachers with an appreciation and enthusiasm for the subject that can be transmitted to students (Darby, 2005a, 2005b), something that is a quality of effective teachers and potentially lacking for teachers teaching out-of-field (Ingvarson, Beavis, Bishop, Peck, & Elsworth, 2004). However, other research shows that, while a teacher’s practice is dependent on the experiences that the teacher has had with the subject or discipline, these experiences are not necessarily related to exposure at university level. For example, other factors, such as career trajectory (Siskin, 1994) and professional development focusing on changes to teacher beliefs (Russell Tytler, Smith, & Grover, 1999), have been found to be cogent in determining how teachers approach teaching and learning. There is an assumption here that teachers can be inducted into the culture of a subject through their experiences, and that, with further training, teachers can improve their competence and confidence in teaching a subject for which they have previously had limited background. Competence refers to teachers’ development of knowledge and skills that are: subject-specific, such as content knowledge (CK) and pedagogical content knowledge (PCK); and generic, including pedagogical knowledge (PK) (Shulman, 1986a, 1987). Confidence relates to teachers’ attitudes (Ernest, 1989; Koballa, 1988), agency and self-efficacy (Boaler & Greeno, 2000), professional identity (Connelly & Clandinin, 1999) and aesthetic understanding as is described in this chapter. Allowing inexperienced teachers of the subject to have an aesthetic experience of the subject matter through targeted professional development may allow them to see themselves and their identity in relation to subject matter ideas.

Conclusion The analysis teases out what it can mean for a teacher to be compelled by and passionate about the subject and students engaging with the subject, to have a coherent and unified sense of what the subject is about and how to bring it to life for students, and to be transformed by what he/she knows and believes in a way that aligns them to personally and professionally identify with the subject. The teachers’ construction of the subject, their students and teaching is not simply cognitive but has an aesthetic dimension. An implication of this is that teachers who teach outside of their subject area—their subject area typically being dependent on whether they are “mathematics- or science-trained”—may be lacking an appreciative

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aesthetic understanding. Their aesthetic response to the content matter and how to teach it may be unlike that of someone who has an appreciative aesthetic understanding of the subject. Such teachers may: attempt to bring in a style appropriate for a subject that has a different set of demands; have a limited set of experiences with relevant phenomena, processes, ways of thinking and attitudes that can feed into their teaching; and fail to exhibit a passion for the subject and what the subject can do for their students. Consequently, any efforts to improve mathematics and science education should be aware that allowing teachers to experience the subject in a way that results in aesthetic appreciation for the beauty and elegance of mathematics and science is just as valuable as them developing conceptual and pedagogical knowledge associated with the subject. A teacher may then experience content in ways that allow them to more clearly see themselves in relation to subject matter ideas.

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Chandrasekhar, S. (1990). Truth and beauty: Aesthetics and motivations in science. Chicago: University of Chicago Press. Connelly, F. M., & Clandinin, D. J. (1999). Shaping a professional identity: Stories of experience. New York, N.Y.: Teachers College Press. Crockcroft, W. H. (1982). Mathematics counts. Report of the Committee of Inquiry inot the Teaching of Mathematics in Schools under the chairmanship of Dr W.H. Crockroft. London: Her Majesty's Stationary Office. Csikszentmihalyi, M. (1997). Finding flow: The psychology of engagement with everyday life. New York: Basic Books. Darby, L. (2004). The inner and outer teacher: Constructing stories of subject, classroom and identity. Paper presented at the Symposium on Contemporary Approaches to Research in Mathematics, Science and Environmental Education, Deakin University, Burwood, December. Darby, L. (2005). Having stories to tell: Negotiating subject boundaries in mathematics and science. Paper presented at the Australasian Association for Research in Science Education, Hamilton, NZ, July. Darby, L. (2005b). Science students' perceptions of engaging pedagogy. Research in Science Education, 35, 425-445. Darling-Hammond, L. (2000). Teacher quality and student achievement: A review of state policy evidence. Educational Policy Analysis Archives, 8(1), [Online]. Day, C. (2004). A passion for teaching. London: Routledge Falmer. Doxiadis, A. (2003). Embedding mathematics in the soul: Narrative as a force in mathematics education. Opening address to the Third Mediterranean Conference of Mathematics Education, Athens, January 3, 2003. Retrieved August 2006, 2006, from www.apostolosdioxiadis.com Education Training Committee. (2006). Inquiry into the promotion of mathematics and science education. Melbourne: Parliament of Victoria. Ernest, P. (1989). The knowledge, beliefs and attitudes of the mathematics teacher: A model. Journal of Education for Teaching, 15(1), 113-133. Gadanidis, G., & Hoogland, C. (2002). Mathematics as story. Retrieved August, 2006, from http://publish.edu.uwo.ca/george.gadanidis/pdf/math-as-story.pdf Girod, M., Rau, C., & Schepige, A. (2003). Appreciating the beauty of science ideas: Teaching for aesthetic understanding. Science Education, 87, 574-587. Gunstone, R. F., & White, R. T. (2000). Goals, methods and achievements of research in science education. In R. Millar, J. Leach & J. Osborne (Eds.), Improving science edcuation: The contribution of research (pp. 293-307). Buckingham: Open University Press. Hargreaves, A. (1994). Changing teachers, changing times: Teachers' work and culture in the postmodern age. London: Cassell. Harris, K.-L., & Jensz, F. (2006). The preparation of mathematics teachers in Australia. Meeting the demand for suitably qualified mathematics teachers in secondary schools. Melbourne: Centre of the Study of Higher Education, The University of Melbourne. Harris, K.-L., Jensz, F., & Baldwin, G. (2005). Who's teaching science? Meeting the demand for qualified science teachers in Australian secondary schools. Melbourne: Centre for the Study of Higher Education, The University of Melbourne.

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Hart, C., Mulhall, P., Berry, A., Loughran, J., & Gunstone, R. F. (2000). What is the purpose of this experiment? Or can students learn something from doing experiments? Journal of Research in Science Teaching, 37(7), 655-675. Ingersoll, R. M. (1998). The problem of out-of-field teaching. Phi Delta Kappan, 79(10), 773-776. Ingvarson, L., Beavis, A., Bishop, A. J., Peck, R., & Elsworth, G. (2004). Investigation of effective mathematics teaching and learning in Australian secondary schools. Melbourne: Australian Council for Educational Research. Ivie, S. D. (1999). Aesthetics: The authentic knowledge base for teacher education. Journal of Philosophy and History of Education, 49, 97-104. Koballa, T. R. (1988). Attitude and related concepts in science education. Science Education, 72(2), 115-125. Lane, B. (2006, Wednesday July 19). A tough subject, minus fear. The Australian, p. 25. Longbottom, J. E., & Butler, P. H. (1999). Why teach science? Setting rational goals for science education. Science Education, 83(4), 473-492. MacDonald, D. (1996). Making both the nature of science and science subject matter explicit intents of science teaching. Journal of Science Teacher Education, 7(3), 183-196. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: NCTM. Neumann, A. (2006). Professing passion: Emotion in the scholarship of professors at research universities. American Educational Research Journal, 43(3), 381-424. Paechter, C. (1991). Subcultural retreat: Negotiating the Design and Technology Curriculum. Paper presented at the British Educational Research Association Annual Conference. Pajares, M. F. (1992). Teachers' beliefs and educational research: Cleaning up a messy construct. Review of Educational Research, 62(3), 307-332. Palmer, P. J. (1998). The courage to teach. Danvers, M.A.: Jossey-Bass. Rahm, J., Miller, H. C., Hartley, L., & Moore, J. C. (2003). The value of an emergent notion of authenticity: Example from two student/teacher-scientist partnership programs. Journal of Research in Science Teaching, 40(8), 737-756. Reys, R. E. (2001). Curricular controversy in the math wars: A battle without winners. Phi Delta Kappan, 255-258. Rico, S. A., & Shulman, J. H. (2004). Invertebrates and organ systems: science instruction and 'Fostering a Community of Learners'. Journal of Curriculum Studies, 36(2), 159-181. Rodd, D. (2007, January 30). Teachers' doubt casts cloud over classrooms. The Age, p. 3. Roth, W., & McGinn, M. K. (1998). Knowing, researching, and reporting science education: lessons from science and technology studies. Journal of Research in Science Teaching, 35(2), 213-235. Rubin, L. J. (1985). Artistry in teaching. New York: Random House. Schein, E. (1992). Organizational culture and leadership (2nd ed.). San Fransisco: JosseyBass. Schoenfeld, A. H. (2004). Multiple learning communities: Students, teachers, instructional designers, and researchers. Journal of Curriculum Studies, 36(2), 237-255. Schuck, S., & Grootenboer, P. (2004). Affective issues in mathematics education. In B. Perry, G. Anthony & C. Diezman (Eds.), Research in mathematics education in Australasia 2000-2003 (pp. 53-74). Flaxton: Post Pressed.

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Schwab, J. J. (1969). College curricula and student protest. Chicago: University of Chicago Press. Schwab, J. J. (1978). Eros and education: A discussion of one aspect of discussion. In I. Westbury & N. Wilkof (Eds.), Science Curriculum and Liberal Education (pp. 105-132). Chicago: University of Chicago Press. Sherin, M. G., Mendez, E. P., & Louis, D. A. (2004). A discipline apart: The challenge of 'Fostering a Community of Learners' in mathematics classrooms. Journal of Curriculum Studies, 36(2), 207-232. Shulman, L. S. (1986a). Paradigms and research programs in the study of teaching: A contemporary perspective. In M. C. Wittrock (Ed.), Handbook of research on teaching (3 ed., pp. 3-36). New York: Macmillan. Shulman, L. S. (1986b). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4-14. Shulman, L. S. (1987). Knowledge and teaching: Foundations of the new reform. Harvard Educational Review, 57(1), 1-22. Sinclair, N. (2004). The roles of the aesthetic in mathematical inquiry. Mathematical Thinking and Learning, 6(3), 261-284. Siskin, L. S. (1994). Realms of knowledge: Academic departments in secondary schools. London: The Falmer Press. Thomas, J. (2000, October). Mathematical science in Australia: Looking for a future. Retrieved January, 2007, from http://www.FASTS.org Topsfield, J. (2007, February 1). Labour pledges HECS cut for maths, science. The Age, p. 5. Trumbull, D. J., Bonney, R., Bascom, D., & Cabral, A. (2000). Thinking scientifically during participation in a citizen-science project. Science Education, 84, 265-275. Tytler, R. (2007). Re-imagining science education: Engaging students in science for Australia's future. Camberwell: Australian Council for Educational Research. Tytler, R., Smith, R., & Grover, P. (1999). A comparison of professional development models for teachers of primary mathematics and science. Asia-Pacific Journal of Teacher Education, 27(3), 193-214. Van den Berg, E. (2000). More on the quality of texts. Science Education International, 11(2), 19-21. Van Manen, M. (1982). Phenomenological pedagogy. Curriculum Inquiry, 12(3). Van Manen, M. (1990). Researching lived experience; Human science for an action sensitive pedagogy. London: The Althouse Press. Wickman, P. (2006). Aesthetic experience in science education: Learning and meaningmaking as situated talk and action. Mahwah, N.J.: Lawrence Erlbaum Associates, Inc. Zembylas, M. (2005a). Discursive practices, genealogies, and emotional rules: A poststructuralist view on emotion and identity in teaching. Teaching and Teacher Education, 21, 935-948. Zembylas, M. (2005b). Three perspectives on linking the cognitive and the emotional in science learning: Conceptual change, socio-constructivism and poststructuralism. Studies in Science Education, 41, 91-116.

In: Advances in Mathematics Research, Volume 8 Editor: Albert R. Baswell, pp. 115-140

ISBN: 978-1-60456-454-9 © 2009 Nova Science Publishers, Inc.

Chapter 3

THE MATHEMATICAL BASIS OF PERIODICITY IN ATOMIC AND MOLECULAR SPECTROSCOPY K. Balasubramanian Center for Image Processing and Integrated Computing, University of California Davis, Livermore, California 94550; Chemistry and Applied Material Science Directorate, Lawrence Livermore National Laboratory, University of California, Livermore, California 94550; and Glenn T. Seaborg Center, Lawrence Berkeley Laboratory, University of California, Berkeley, California 94720, USA

Introduction This chapter applies combinatorial and group-theoretical relationships to the study of periodicity in atomic and molecular spectroscopy. The relationship between combinatorics and both atomic and molecular energy levels must be intimate since the energy levels arise from the combinatorics of the electronic or nuclear spin configurations or the rotational or vibrational energy levels of molecules. Over the years we have done considerable work on the use of combinatorial and group-theoretical methods for molecular spectroscopy [1–15]. The role of group theory [1–40] is evident since the classification of electronic and molecular levels has to be made according to the irreducible representations of the molecular symmetry group of the molecule under consideration. Combinatorics plays a vital role in the enumeration of electronic, nuclear, rotational and vibrational energy levels and wave functions. As can be seen from other chapters in this book, the whole Periodic Table of the elements has a mathematical group-theoretical basis since the electronic shells have their origin in group theory. Indeed, this concept can even be generalized to other particles beyond electrons such as bosons or other fermions that exhibit more spin configurations than just the bi-spin orientations of electrons. It has been shown that Einstein’s special theory of relativity is quite important for classifying the energy levels of very heavy atoms and molecules that contain very heavy atoms [41–48]. This is because to keep balance with the increased electrostatic attraction in heavier nuclei having a large number of protons, the core electrons of such heavy atoms must move with considerably faster average speeds. We have shown, for example, that the

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averaged speed of the 1s electron of heavier atoms such as gold is about 60% of the speed of light. Consequently, ordinary quantum mechanics does not hold, and one needs to invoke relativistic quantum mechanics to deal with such heavy atoms and with molecules containing very heavy atoms. We have defined relativistic effects as the difference in the observable properties of electrons as a consequence of using the correct speed of light compared to the classical infinite speed. Mathematically, the introduction of relativity results in a double group symmetry owing to the spin-orbit coupling term [41], which is a relativistic term in the Hamiltonian. This is a natural symmetry consequence of the LS spin-orbit operator, which changes sign upon rotation by 360º. Thus, the periodicity of the identity operation, which is normally envisaged as a rotation through 360º, is no longer the identity operation of the group. This is illustrated in Figure 1 with a Möbius strip, which exemplifies the double group symmetry. As one completes a 360º rotation along the Möbius surface there is a sign change since one goes from the inside of the surface to the outside. This requires the introduction of a new operation R in the normal point group of a molecule that corresponds to the rotation by 360º which is not equal to E, the identity operation. Hence we have to make use of the double group and double-valued representations in both atomic and molecular spectroscopy.

Figure 1. A Möbius strip exemplifying the double group relativistic periodicity. The introduction of spin-orbit coupling into the relativistic Hamiltonian changes the periodicity of the normal point group symmetry into a double group symmetry, as rotation through 360° is not the identity operation. Note that the Möbius strip changes sign in this operation. Generalization of this to other complex phases results in Berry’s phase, where rotation through 360° may yield exp(2πi/n), thus resulting in other kinds of periodicity.

The double group consists of twice the number of operations as the normal point group but is not a simple direct product of the normal point group and another group. This is a consequence of the fact that only some of the conjugacy classes of the normal point group generate new conjugacy classes upon multiplication by the operation R. The other conjugacy classes, which are called two-sided operations, such as the C2 rotations, double in order instead of generating new classes. This is because such operations when multiplied by R

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become equivalent to the operation, and thus the new operations belong to the same conjugacy class as the corresponding old operations. This feature complicates double group theory and the resulting periodicity of the double group. A characteristic feature is the generation of even-dimensional double-valued representations that characterize half-integral quantum numbers. We shall discuss this in one of the ensuing sections. In this chapter we shall consider the mathematical basis of atomic periodicity and spectroscopy with the use of group theory and combinatorics. We shall also consider the combinatorics of unitary groups and Young diagrams and their connections to the electronic spin functions. We shall also discuss molecular periodicity by considering the combinatorial basis of molecular electronic states. We describe the double groups and the periodicity arising from the classification of states in the double group. We expound on the combinatorics and periodicity pertinent to the rotational levels, nuclear spin functions, and rovibronic levels of molecules and give some examples.

Combinatorial Periodicity in Molecular Electronic and Atomic Spectroscopy As might be expected, the classification of atomic states and thus the Periodic Table of the elements are based on combinatorial and group-theoretical considerations. An interesting related combinatorial problem has to do with the graphical unitary group approach to manyelectron configuration and correlation problems [4, 49]. The associated fermionic algebras involving Young diagrams and the symmetric permutation group approach have been discussed previously [4, 49]. It would also be interesting to consider cases of similar enumerations for other bosons or even fermions that are more than spin 1⁄2 particles. Such cases, while possibly not applicable to electronic systems, are applicable to nuclear spin species, and would have considerable group-theoretical value. There are several applications of group theory to atomic states. An early application by Curl and Kilpatrick [16] showed that the Schur functions of the symmetric groups Sn can be used as generating functions for atomic term symbols. The method involves replacement of cycle index polynomial terms by the various ml and ms symbol powers for generating functions of the atomic states that transform according to the irreducible representations of the Sn group. From this the authors were able to establish a periodic connection to the combinatorial enumeration of atomic term symbols even for complicated cases such as those for f7 shells. Balasubramanian [4] has established the connection between the graphical unitary group approach for electronic configurations and the Schur function algebra of the Sn groups, which play an important role in the Periodic Table in terms of the classification of various spin multiplets and the term symbols of electronic states. We shall consider this in some detail before enumerating the atomic states. The electronic states arising from the many-electron configurations have certain periodicities and patterns as enumerated by the Schur functions of the symmetric groups Sn. The representation theory of the symmetric group is well known [17, 20, 50], and we will not repeat it here. The irreducible representations of Sn may be characterized by Young diagrams for the various partitions of the integer n, denoted by [n]. The states of many particles (including

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bosons and fermions) that possess multiple spin orientations can be represented by generalized Young Tableaus (GYTs). For example, Figure 2 shows all of the possible GYTs for the partitions of six occupied by six particles that have three spin orientations (for example, a spin-1 particle such as the bosonic deuterium nucleus) with the possibility that two have the first kind of spin orientation, two have second kind and the last two particles have the third kind. We have denoted this [122232] shape as shown in Figure 2.

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Figure 2. Generalized Young Tableaus (GYTs) for the partition of six for a spin 1 boson (e.g., deuterium) corresponding to the spin distribution of two particles with the first spin orientation, two with the second orientation, and two with the third or [122232] shape.

As can be seen from Figure 2, the GYTs have numbers in any column in strictly ascending order while the numbers in any row must be in non-decreasing order. These tableaus represent the nuclear spin functions that transform according to the particular irreducible representation that the diagram represents. It is interesting to note that for a spin-1 particle such GYTs can have at the most three rows and, likewise for electrons, which are spin 1⁄2 particles, the GYTs can have at the most two rows. In general for a spin-j particle there can be at most only 2j + 1 rows in the GYTs. The enumeration of the GYTs for the various shapes of the spin distributions is a fundamental problem that is common to electronic and nuclear structures. In the context of many-electron spin functions, the graphical unitary group approach requires the enumeration of Gel’fand states which are the GYTs containing two rows. The results also have some interesting periodicity trends in the mathematical sense. These GYTs and the associated spin

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multiplets and spin states can be enumerated by polynomials called the Schur functions of the symmetric group Sn. The Schur function corresponding to a partition λ of n is denoted by {λ} and is defined in the following way: 1 {λ} = n!

∑χλ(g)s1b s2 b …snbn 1

(1)

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where χλ(g) is value of the character for g in the group G = Sn corresponding to the irreducible representation [λ] of the group Sn. To illustrate this, the Schur function corresponding to the partition 4 + 1 + 1 is given by the Schur function, {6;4,1,1}, shown below: 1

{6;4,1,1} = 120 [10s16 + 30 s14s2 + 40 s13s3 – 90 s12s22 – 120 s1s1s3 – 30 s23 + 40 s32 + 120 s6] (2) The Schur function is the generator for the GYTs, and is obtained by replacing every sk in the Schur function or S-function by ∑λik. The coefficient of a typical term λ1a1 λ2a1… λmam in i

the generating function thus obtained yields the number of GYTs with the shape [1a12a1…mam ]. The GYT generators are so powerful that they also enumerate the atomic states when applied to electronic spin functions which are GYTs with only two rows. As an illustration of GYT generation, let us consider the partition 2 + 1 for three particles. Let the particle under consideration be a spin-1 boson, which has three spin orientations that we depict symbolically as λ1, λ2 and λ3. The S-function in this case is given as: 1

{3;2,1} = 6 [2 s13 – 2 s3]

(3)

The GF of the GYTs for a spin 1 particle is thus given as: {λ1,λ2,λ3;2,1} = λ12λ2 + λ1λ22 + λ2λ32 + λ22λ3 + λ1λ32+ λ12λ3 + 2 λ1λ2λ3

(4)

The above generating function thus obtained from the S-function generates all of the GYTs shown below in Figure 3 for all possible spin distributions or shapes for the partition 2 + 1. 1

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Figure 3. All possible GYTs corresponding to the partition 2 + 1 as enumerated by the S-function {λ1,λ2,λ3;2,1}.

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The enumeration technique can be applied to GYTs of any shape belonging to any particle with any spin shape and spin distribution. The method is not restricted to just spin 1 or spin 1⁄2 particles. We can use the above method to generate all of the possible spin states for a manyelectron system or all of the possible atomic spectral energy levels for a given open-shell electronic configuration. First, we illustrate the method for obtaining all of the possible electronic spin states. The GYTs for electrons may contain at most two rows since there are only two possible distinct spin orientations for an electron (α and β) and thus there cannot be more than two rows. Accordingly, only certain partitions are allowed for an electron. This means that the GYTs can be formed only by the integers 1 and 2. Each spin distribution or spin shape then contains representations that are sums of the GYTs with the appropriate shape. For example for a system of six electrons with five spins up and one spin down there are exactly two GYTs as shown in Figure 4.

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2 Figure 4. The GYTs for six electrons with five spin ups and one spin down.

Figure 5. The many-electron spin multiplets for an even number of electrons; there are exactly N cells and at most two rows for the spin functions.

The [1a12a1] GYTs enumerate states with a total spin quantum number Mz = (a1 – a2)/2. Consequently, once the GYTs are sorted out according to their total Mz values we obtain the spin multiplets for the many-electron systems. A neat set of periodic spin multiplets are obtained for such many-electronic systems. These are shown in Figures 5 and 6, respectively, for even and odd numbers of electrons. This important result was made possible by use of the periodic S-functions.. The method of S-functions is powerful and general in that it can be applied to more than electrons. Thus, the same method can be applied to other particles that have integral spins

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such as the deuterium nuclear spin functions or to the cases with multinomial spin distributions. In such cases the diagrams become more complex with many more rows depending on the particles. For a spin 1 particle the diagrams have three rows at most. For a spin j particle the diagrams will have up to (2j + 1) rows yielding an array of complex spin multiplets. Next we demonstrate the periodic power of the S-function for enumerating the possible electronic states of an atom [16] which are well known as atomic term symbols in atomic spectroscopy. The method is completely analogous to generating the GYTs and manyelectron spin multiplets demonstrated above. The only difference is that we obtain a generating function for the different ML projections and spin projections and the total function must comply with the Pauli Exclusion Principle, as electrons are fermions. The method can be applied from simple cases, e. g., main group open-shells such as p2, p3, to more complex lanthanides and actinides that have fm open-shell f-electrons (Table 1). For example, consider the most complex half-filled 5f7 shells. The possible electronic states sorted according to the total spin and total angular momentum in compliance with Pauli’s Principle are given by: 2

S(2), 2P(5), 2D(7), 2F(10), 2G(10), 2H(9), 2I(9), 2J(7), 2K(4),2M(2), 2N, 2O S(2), 4P(2), 4D(6), 4F(5), 4G(7), 4H(5), 4I(5), 4J(3), 4K(3),4M, 4N 6 6 S, P, 6D, 6F, 6G, 6H, 6I 8 S 4

The mathematical aspect of periodicity in atomic states is dependent on the orbital angular momentum of the electrons and spins as exemplified by the S-function generator used above for the generation of atomic term symbols. Table 1. All possible atomic term symbols for all actinides and lanthanides. Shell f1/f13 f2/f12 f3/f11 f4/f10

f5/f9

f6/f8

f7

States 2

F S 1D 1G 1I 3P 3F 3H 2 2 P D(2) 2F(2) 2G(2) 2H(2) 2I 2J 2K 4S 4D 4F 4G 4I 1 S(2), 1D(4), 1F(1), 1G(4), 1H(2), 1I(3), 1J, 1K(2), 1M, 3 P(3) 3D(2) 3F(4) 3G(3) 3H(4) 3I(2) 3J(2) 3K 3L 5 5 S D 5F 5G 5I 2 P(4), 2D(5), 2F(7), 2G(6), 2H(7), 2I(5), 2J(5), 2K(3),2L(2), 2M 2N 4 4 S P(2) 4D(3) 4F(4) 4G(4) 4H(3) 4I(3) 4J(2) 4K 4L 6 6 6 P F H 1 S(4), 1P. 1D(6), 1F(4), 1G(8), 1H(4), 1I(7), 1J(3), 1K(4), 1L (2), 1M(2), 1O 3 P(6) 3D(5) 3F(9) 3G(7) 3H(9) 3I(6) 3J(6) 3K(3) 3L(3) 3M 3N 5 5 5 S P D(3) 5F(2) 5G(3) 5H(2) 5I(2) 5J 5K 7 F 2 S(2), 2P(5), 2D(7), 2F(10), 2G(10), 2H(9), 2I(9), 2J(7), 2K(5) 2L(4),2M(2), 2 N, 2O 4 S(2), 4P(2), 4D(6), 4F(5), 4G(7), 4H(5), 4I(5), 4J(3), 4K(3),4M, 4N 6 6 S, P, 6D, 6F, 6G, 6H, 6I 8 S 1

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Figure 6. The many-electron spin multiplets for an odd number of electrons; there are exactly N cells and at most two rows for the spin functions.

Yet another aspects of periodicity involves the molecular electronic states. The electronic configurations themselves consist of two parts, namely the spin part that was generated using the S-functions and space types that can also be generated using multinomial generators. In certain cases, as shown by the author, the orbital degeneracy can bring out additional symmetry. A space type can be imagined as a distribution of electrons in boxes such that a permutation of electrons within a box does not generate a new space type and the boxes themselves can be permuted if the orbitals are degenerate. Such groups are called wreath product groups. Balasubramanian [7] used this group theory combined with combinatorial multinomial generating functions to generate electronic space types. This can be illustrated for the benzene delocalized orbital π electrons. The periodic generating function for the number of space types of an n-orbital electronic configuration is given by: F = (1 + w + w2)n

(3)

where the coefficient of wm gives the number of space types with m electrons distributed among these n orbitals. For the case of benzene with six π electrons distributed among six orbitals we seek the coefficient of w6 with n = 6 in the above generating function. This is given by: 6 6 6 1 + ⎛⎝3⎞⎠ + ⎛⎝ 2 2 2 ⎞⎠ + ⎛⎝ 4 1 1 ⎞⎠ = 141

(4)

These 141 space types of benzene enumerated here are divided into equivalence classes of space types according to the symmetry equivalence from the wreath product groups induced by orbital degeneracy. As is well known, the six π orbitals of benzene are divided into 1 + 2 + 2 + 1 equivalence classes of orbitals. Thus, switching of the orbitals in the second and third set leads to equivalences, and the electrons can themselves be switched in each orbital. The result is a direct product of wreath product groups as shown below:

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S2 × S2[S2] × S2[S2] × S2 The cycle index polynomial of the totally symmetric representation of the above group generates the equivalence classes of the space types from the well-known Pólya Theorem [1, 52–60]. Consequently, the cycle index and the generating functions for the case of benzene are as follows. P=

{ (s

1 2 2 1

2

} { (s

+ s2)

1 4 8 1

}

+ 2 s12s2 + 3 s22 + 2 s4)

2

(5)

4 2 4 3 2 2 2 4 1 ⎧2 (1 + w + w ) + 2·2 (1 + w + w ) (1 + w + w )⎫2 ⎬ F = 28 {22(1 + w + w2)2}⎨ 2 2 4 2 4 8 +3·2 (1 + w + w ) + 2·2(1 + w + w )

⎩

⎭

(6)

The coefficient of w6 in the above generating function can be seen to be 58, which suggests that for benzene 141 space types are divided into 58 equivalence classes. Table 2 gives the number of equivalence classes of the space types for the various atoms that exhibit equivalence among the p orbitals. Table 2. Equivalence classes of the space types for the electronic configurations of atoms that have degenerate p orbitals. System He Li Be B C N O F

Total No Space Types 45 156 414 882 1554 23-4 2907 3139

Equivalence Classes 17 42 86 148 223 295 349 368

In summary, we have shown that the electronic configurations of molecules and atoms can be simplified using the mathematical periodicity of the spin functions and space types. The former case was accomplished using the S-functions of the symmetric permutation groups Sn while the latter case was simplified using the wreath product configuration symmetry groups.

Combinatorial Periodicity in Molecular and NMR Spectroscopies The concept of mathematical periodicity as described by the orbit structure of a permutation finds important applications in molecular and nuclear spin spectrsocopies. The orbit structure of a permutation comprises several cycles such that in each cycle a set of nuclei is visited followed by a return to the starting point. The cyclic structure of the permutation (12345)(678)(9,10) is illustrated in Figure 7. The orbit structure in Figure 7 determines the nuclear spin statistical weights of the rotational levels of molecules.

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Thus, from the periodic orbit structure in Figure 7, one determines a polynomial s5s3s2, because we have one orbit of length 5, one orbit of length 3 and one orbit of length 2. The periodicity and the length of the period associated with each such orbit then determine a generating function for the nuclear spin statistical weights of the energy levels. This concept can also be used in NMR and ESR spectrsocopies where the periodicity and the length of the orbits determine the NMR spin energy levels and thus the NMR spectra associated with the molecules. Although in ordinary NMR only Zeeman-allowed transitions are observed and thus only those transitions with changes of a single spin, multiple quantum NMR offers a powerful tool to probe into transitions involving multiple spin quantum numbers. Thus, all NMR interaction energy levels can be probed.

Figure 7. Periodic orbit structure for the permutation (12345)(678)(9,10).

We shall start with an application of permutational periodic structure in molecular spectroscopy. Indeed, the rotational energy levels of a molecule themselves have periodicities based on their point groups. We illustrate this with an icosahedral cluster, namely N20 [37] and C60 [29–33] systems. Consider the highly energetic regular dodecahedral N20 cluster [37], which exhibits icosahedral symmetry analogous to that in the fullerene C20. Since 14N is a spin 1 particle it exhibits an interesting generating function and nuclear spin species distribution. The generalized character cycle indices for all of the irreducible representations for the N20 cluster with Ih symmetry are shown in Table 3. These were constructed using the orbit structures of permutations as demonstrated in Figure 7. The cycle indices for the various irreducible representations were obtained by multiplying the periodic orbit structures of each permutation by the corresponding character values. Note that the resulting polynomials are the same for the T1g and T2g representations and likewise the T1u and T2u representations since the orbit structures multiplied by their character values become identical owing to accidental degeneracy. We have used our generalization [1, 5–6, 56] of Pólya’s Theorem for all characters to seek generating functions for the nuclear spin species of 14N. Note that since the 14N nucleus is a spin 1 particle, we replace every xk in the cycle index in Table 3 by λk + μk + νk where the symbols λ, μ and ν stand for –1, 0 and 1 spin projections of the spin 1 14N nucleus. The resulting generating functions for the nuclear spin functions are shown in Table 4. The generating functions shown in Table 4 have two parts, one consisting of coefficients and the other of the trinomial λiμjνk. We do not show k since k = 20 – i + j) and it can thus be deduced from the values of i and j. To illustrate how the generating functions in Table 4 are

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125

obtained, let us consider the T1g or the T2g representation. From Table 3 we obtain the GCCI for this representation and we make the substitution given by x

GF = PGx (xk → λk + µk + νk)

(7)

The above substitution yields the following expression: 1

GFT1g = 120 [3(λ + μ + ν)20 + 12(λ5 + μ5 + ν5)4 – 12(λ2 + μ2 + ν2)10 + 12(λ10 + μ10 + ν10)2 – 15(λ + μ + ν)4(λ2 + μ2 + ν2)8]

(8)

Table 3. The GCCIs for the dodecahedral N20 cluster. N20 Order Ag Au T1g=T2g T1u=T2u

120 1 1 1 3 3

54 24 1 1 1 ⁄2 1 ⁄2

1236 20 1 1 0 0

210 15 1 1 –1 –1

210 1 1 –1 3 –3

102 24 1 –1 1 ⁄2 –1⁄2

263 20 1 –1 0 0

1428 15 1 –1 –1 1

Gg Gu Hg Hu

4 4 5 5

–1 –1 0 0

1 1 –1 –1

0 0 1 1

4 –4 5 –5

–1 1 0 0

1 –1 –1 1

0 0 1 –1

Figure 8. Nuclear frequency spin spectrum for the Ag representation of N20.

When Equation (8) is simplified into a trinomial it has several terms with coefficients for each term. Table 4 shows the coefficients and powers of λ and μ for the term. The power of ν is simply 20 – (i +j) and is thus not shown. The actual computations of the generating

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functions for N20 (and for C60 discussed subsequently) were carried out using computer code in quadruple precision developed by Balasubramanian [8, 9]. It is important to employ a quadruple precision arithmetic especially for C60, as the coefficients grow astronomically and thus any lower precision results in errors. An interesting consequence of the periodicity is that the g and u representations differ in some of their coefficients so significantly that one can say that there is inversion contrast in combinatorics. For example, the coefficient of the term λ9μ6ν5 in Table 4 for the Ag representation is 647706 while the corresponding coefficient for the Au representation is 645606. Moreover, the first non-zero coefficient for the Au representation is for the (18,2,0) partition, which means that at least 18 colors of one kind and two colors of another kind are needed to induce chirality in the binomial distribution. A purely trinomial term has two chiral colorings for the lowest order term, i. e., the (18,1,1) term in Table 4 has a coefficient of two for Au. Table 4. Generating functions for the dodecahedral N20 cluster.

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Table 4. Generating functions for the dodecahedral N20 cluster. Continued

The coefficients thus enumerated in Table 4 can be sorted according to their total MF values where the term l has the projection –1, m has the projection 0, and v has the projection +1. Thus the term λiμjνk in Table 4 represents a total nuclear spin quantum number MF of (–i + k). When these coefficients are sorted according to their total MF values, they separate into nuclear spin multiplets with MF values ranging typically from –I, –I+1, –I+2,….0,….I–2, I–1, I. Such a multiplet would represent a nuclear spin multiplet with a multiplicity of 2I + 1. In this way for each irreducible representation the nuclear spin multiplets are separated according to their multiplicities and the results are shown in Table 5 for N20. As can be seen from Table 5, the frequencies of the spin multiplets corresponding to the g and u representations differ even for the singlet spin states. For example, the 1Ag state has a frequency of 113035 while the 1Au state has a frequency of 112444. There is a similar difference in the triplet state and most of the spin multiplets. This means that the parity can be contrasted even in low spin nuclear states. The corresponding rovibronic levels will also be populated with appreciable differences in the populations. From the nuclear spin multiplets

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we can also obtain the total nuclear spin statistical weights by the use of the Pauli Principle. Since 14N nuclei are bosons, the overall wavefunction, which is a product of the rovibronic wavefunction and nuclear spin function, must be symmetric or must transform as the Ag irreducible representation. The frequency of each representation is obtained by adding the product of 2S + 1 and the frequency. The results are shown as a footnote in Table 5. On the basis of this, the frequencies shown in this footnote are themselves the nuclear spin statistical weights for N20 (see Figure 8). Table 5. Nuclear spin species for the N20 cluster

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Other irreducible representations have similar spectra comparable to that in Figure 8, except that the intensities of the peaks vary. The Ag representation is particularly important as it gives the number of lines in multiple quantum NMR spectra. The frequencies of other irreducible representations determine the intensities of the lines in the spectra. The multiple quantum NMR spectra usually contain structural information for the (n – 2) quantum as this value exhibits dipolar couplings that contrast the structure. For the present case n – 2 corresponds to the spin multiplet 2S + 1 = 37. These multiplets have the frequencies 5, 2, 2, 7, 11, 1, 5, 5, 6, for the Ag, T1g, T2g, Gg, Hg, Au, T1u, T2u, Gu, and Hu representations, respectively. Thus, the dodecahedral N20 cluster exhibits interesting mathematical periodicity in spectroscopic terms. Next we consider the C60 cluster [29–33] as another example that demonstrates mathematical periodicity and its applications. The GCCIs of C60 are constructed analogously to those of N20 discussed above. The fact that C60 has 60 vertices would of course divide the permutation of 60 vertices into various periodic orbits. The nuclear spin species thus obtained using the GCCIs are shown in Table 6. As seen from Table 6, the frequencies grow astronomically as expected. This is because of the combinatorial explosion of the coefficients in the generating functions even though these functions are binomials. The binomial expansion is due to the fact that 13C60 is comprised of 13C nuclei, which exhibit only two spin orientations, as they are spin 1⁄2 particles. The same is true of C60H60, as protons are spin 1⁄2 particles and 12C has no nuclear spin. Again a major contrast is that the g and u representations have different frequencies due to the difference in the periodicity of the permutation multiplied by the character value for these representations. This feature manifests itself as contrasting frequencies for the g and u irreducible representations. We note that earlier work had an error in the spin statistical weights of C60 [31] primarily owing to the arithmetical precision but this was subsequently corrected [30, 32]. The relative differences between the g and u parities are especially significant for high-spin nuclear multiplets. For example, for the 2S + 1 = 57 spin multiplet of 13 C6, the frequencies of the Ag, T1g, T2g, Gg, Hg, Au, T1u, T2u, Gu, and Hu representations are 22, 36, 36, 58, 80, 14, 42, 42, 56, and 70, respectively. Similarly for 2S + 1 = 55 the frequency of the Ag representation is 280 while it is 260 for Au. Consequently, the contrast in the g and u spin populations can be seen experimentally if high-spin nuclear states can be excited. Table 6. Nuclear spin multiplets for 13C60 or C60H60. Frequency of the irreducible representation Ag: 9607679885269312 Spin multiplets and their frequencies for Ag: 2S+1 1 7 13 19 25 31 37 43 49 55 61

Frequency 31791575566072 150988619146706 105558807981090 31605175642230 4481735502630 298734989924 8805633300 101874363 372752 280 1

2S+1 3 9 15 21 27 33 39 45 51 57

Frequency 89413728633564 149756091280506 76925432220000 17892025439775 1980110898945 101492436960 2227563126 18110340 41528 22

2S+1 5 11 17 23 29 35 41 47 53 59

Frequency 13095954950748 13219208028055 5141513084676 933143835273 80345370985 3139590568 50512570 280174 388

130

K. Balasubramanian Table 6. Continued

Frequency of the irreducible representation T1g: 28823036970926496 Spin multiplets and their frequencies for T1g : 2S+1 1 7 13 19 25 31 37 43 49 55 61

Frequency 95374646372040 452965902231668 316676363633175 94815530686980 13445194549380 896204629630 26416344630 305608974 1114158 804 0

2S+1 3 9 15 21 27 33 39 45 51 57

Frequency 268241251090167 449268197030424 230776308338940 53676052490265 5940332333550 304475471640 6682635360 54304371 124257 36

2S+1 5 11 17 23 29 35 41 47 53 59

Frequency 39287856402727 39657626655407 15424535176554 2799431557098 241035603798 9418755979 151524170 840285 1123

Frequency of the irreducible representation Gg: 38430716856193728 Spin multiplets and their frequencies for Gg: 2S+1 1 7 13 19 25 31 37 43 49 55 61

Frequency 127166221937640 603954521378374 422235171614265 126420706329465 17926930052010 1194939619444 35221977930 407483337 1486916 1084 0

2S+1 3 9 15 21 27 33 39 45 51 57

Frequency 357654979723731 599024288311326 307701740558940 71568077929785 7920443232495 405967908600 8910198522 72414711 165779 58

2S+1 5 11 17 23 29 35 41 47 53 59

Frequency 52383811353475 52876834683423 20566048261230 3732575392371 321380974795 12558346548 202036737 1120460 1512

Frequency of the irreducible representation Hg 48038396740938240 Spin multiplets and their frequencies for Hg: 2S+1 1 7 13 19 25 31 37 43 49 55 61

Frequency 158957797411208 754943140441100 527793979532265 158025881932935 22408665535200 1493674601616 44027608785 509357130 1859568 1354 0

2S+1 3 9 15 21 27 33 39 45 51 57

Frequency 447068708357295 748780379591832 384627172778940 89460103369560 9900554131440 507460345560 11137761648 90525051 207307 80

2S+1 5 11 17 23 29 35 41 47 53 59

Frequency 65479766312622 66096042717787 25707561349782 4665719229588 401726346555 15697937361 252549365 1400644 1902

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Table 6. Continued Frequency of the irreducible representation Au: 9607678793631424 Spin multiplets and their frequencies for Au: 2S+1 1 7 13 19 25 31 37 43 49 55 61

Frequency 31791571643468 150988613640506 105558798039270 31605170531130 4481732871390 298734348764 8805495420 101861196 371694 260 0

2S+1 3 9 15 21 27 33 39 45 51 57

Frequency 89413727296344 149756080818726 76925425313100 17892020535870 1980109351620 101491992360 2227502850 18103410 41266 14

2S+1 5 11 17 23 29 35 41 47 53 59

Frequency 13095954114986 13219207292373 5141512318638 933143526111 80345252581 3139568730 50509098 279955 377

Frequency of the irreducible representation T1u: 28823037990981216 Spin multiplets and their frequencies for T1u: 2S+1 1 7 13 19 25 31 37 43 49 55 61

Frequency 95374639953380 452965915721858 316676367808710 94815537801090 13445196226770 896205406510 26416442910 305623968 1114942 826

2S+1 3 9 15 21 27 33 39 45 51 57 0

Frequency 268241262122232 449268199508214 230776318887660 53676055391160 5940334271070 304475780520 6682705140 54309474 124548 42

2S+1 5 11 17 23 29 35 41 47 53 59

Frequency 39287856269005 39657627967718 15424535578410 2799431961645 241035683183 9418781778 151526692 840531 1132

Frequency of the irreducible representation Gu: 38430716784610624 Spin multiplets and their frequencies for Gu: 2S+1 1 7 13 19 25 31 37 43 49 55 61

Frequency 127166211596396 603954529362364 422235165847980 126420708332460 17926929098160 1194939755164 35221938330 407485164 1486642 1086 0

2S+1 3 9 15 21 27 33 39 45 51 57

Frequency 2S+1 357654989418576 5 599024280327336 11 307701744200760 17 71568075926790 23 7920443622690 29 405967772880 35 8910208020 41 72412884 47 165808 53 56 59

Frequency 52383810383991 52876835260051 20566047897048 3732575487756 321380935775 12558350508 202035787 1120487 1509

132

K. Balasubramanian Table 6. Continued

Frequency of the irreducible representation Hu: 48038395577718272 Spin multiplets and their frequencies for Hu 2S+1 1 7 13 19 25 31 37 43 49 55 61

Frequency 158957783147612 754943142918890 527793963824370 158025878824830 22408661950230 1493674096176 44027431350 509345790 1858246 1336 0

2S+1 3 9 15 21 27 33 39 45 51 57

Frequency 2S+1 447068716714920 5 748780361146062 11 384627169513860 17 89460096462660 23 9900552974310 29 507459765240 35 11137710870 41 90516294 47 207074 53 70 59

Frequency 65479764507375 66096042558712 25707560219562 4665719015799 401726189132 15697919478 252544942 1400452 1887

Table 7. Correlations of the rotational levels of C60: the nuclear spin statistical weights J = 0 to 30

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133

Table 7 shows the correlation of the rotational levels for C60 from J = 0 to 30 with the corresponding weights only in the rotational subgroup I. Note that for purposes of comparing with experimental results one must use the nuclear spin frequencies given in Table 6, but the statistical weights in Table 7 in factored form yield the orders of magnitude. All levels in Table 7 are of g symmetry since the J states can correlate into g levels. (1) The irreducible representations for J > 31 are given by q[A + 3T1 + 3T2 +4G + 5H] + Γ(r), where q is the quotient obtained by dividing J by 30 and r is the remainder. Γ(r) is the set of irreducible representations spanned by J = r listed in this Table (see text for further discussion). Note that since nuclear spin statistical weights are the same for g and u symmetries, we do not show g or u. (2) f = 19 215 358 678 900 736 for C60H60; f = 706 519 304 586 988 199 183 738 259 for C60D60. Each correlation in Table 7 was obtained using the mathematical method of subduction. As can be seen from Table 7, we have a very interesting periodicity among rotational levels. The correlations for the rotational levels with J > 31 have a periodic relation to the levels with J < 30. This is another mathematical manifestation of periodicity. The relations for all J > 30 are as follows: D(J)↓Ih = q(D(30)↓Ih – A) + (D(r) ↓Ih), q = [J/30], r = J – 30[J/30]

(9)

where the function within square brackets is the greatest integer contained in the brackets and thus q and r are quotients and remainders obtained by dividing J by 30. The term D(30) stands for the subduced representations for J = 30 that are displayed in Table 7. To illustrate this, the J = 195 rotational level contains the following representations: D(195)↓Ih = 6(Ag + 3T1g + 3T2g + 4Gg + 5Hg) + (Ag + 2T1g + 2T2g + 2Gg + 2Hg)≡ 7Ag + 20T1g + 20T2g + 26Gg + 32Hg

(10)

The above concept of the periodicity of the rotational levels of C60 is illustrated in Figure 9. It is worthy of note that the nuclear spin statistical weights of the rotational levels vary approximately as (2J + 1) due to large nuclear spin statistical weights.

J=0 J=1

J=30 * * * * . *

J=2

.J=3 Figure 9. Periodicity of the rotational levels of buckminsterfullerene, C60.

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K. Balasubramanian

Periodicity of Double Groups and Electronic States The concept of the double group [17, 24–27, 42, 51] is required when the normal periodicity resulting from rotation through 360° breaks down, as demonstrated for the Möbius strip. This happens when half-integral states are considered. For example, the rovibronic states of openshell systems with an odd number of open-shell electrons exhibit half-integral spin states due to an odd number of open-shell electrons and thus we need a new concept of periodicity. This is also the case when spin-orbit coupling is introduced into the Hamiltonian.

Table 8. Character table for the Ih2 double group.

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Table 8. Character table for the Ih2 double group. (Continued)

This difficulty was circumvented by Bethe through the concept of the double group. He introduced a new operation called R that changes the sign for the rotation through 360° for half-integral states and yet retains the same symmetry for the integral states, as shown above for the C60 integral rotational levels. Since the periodicity and the group structure are quite different for the double group, we provide a few examples of double group character tables and correlation tables. Most of the character tables appear in books such as those of Hamermesh [17] or Altmann and Herzig [24] for the double groups. Balasubramanian [51] developed the

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K. Balasubramanian

character table for the icosahedral double group denoted by Ih2 that is shown in Table 8. Note that the operation R introduces a few new conjugacy classes for the Ih2 double group while other conjugacy classes just double in their orders. This is a consequence of the fact that certain operations are called two-valued operations and these operations when multiplied by R become equivalent, and thus belong to the same conjugacy class. However, other operations, such as C5 and RC5, become inequivalent, and thus belong to different conjugacy classes. The new irreducible representations in the double group are called two-valued representations and they are always even dimensional and correspond to half-integral representations. The number of such representations equals the number of new conjugacy classes, as demonstrated in Table 8. These are called E1g(1/2), Gg(3/2), Ig(5/2), E2g(7/2), with the corresponding u representations. The correlation table for the half-integral states of the Ih2 double group is shown in Table 9. Note that the corresponding table for the integral representations has already been discussed for C60 (Table 7). As can be seen from Table 9, the half-integral spin or rovibronic states all correlate only into double-valued representations, which are all even dimensional. As a result, the representation corresponding to 1⁄2 is a degenerate two-dimensional irreducible representation. The quartet state with s = 3⁄2 is also four-fold degenerate and s = 5⁄2 is likewise the six-fold degenerate I representation in the double group. The first case which splits into two irreducible representations is the s = 7⁄2 case. The periodicity is reduced in the double-valued representation to half as s = 31⁄2 is related to s = 1⁄2 by periodicity. All higher s values are obtained using a periodic relation as shown in Table 9. Table 9. Periodic correlation table for the half-integral states of the Ih2 double group. Irreducible Representationsa

s ⁄2 3 ⁄2 5 ⁄2 7 ⁄2 9 ⁄2 11 ⁄2 13 ⁄2 15 ⁄2 17 ⁄2 19 ⁄2 21 ⁄2 23 ⁄2 25 ⁄2 27 ⁄2 29 ⁄2 31 ⁄2 1

1

E1g′( ⁄2) Gg′(3⁄2) Ig′(5⁄2) E2g′(7⁄2) + Ig′(5⁄2) Gg′(3⁄2) + Ig′(5⁄2) E1g′(1⁄2) + Gg′(3⁄2) + Ig′(5⁄2) E1g′(1⁄2) + Gg′(3⁄2) + Ig′ (5⁄2) + E2g′(7⁄2) Gg′ (3⁄2) + 2 Ig′(5⁄2) G′g(3⁄2) + 2 Ig′(5⁄2) + E2g′(7⁄2) E1g′(1⁄2) + Gg′(3⁄2) + 2 Ig′(5⁄2)+ E2g′(7⁄2) E1g′(1⁄2) + 2 Gg′(3⁄2) + 2 Ig′(5⁄2) E1g′(1⁄2) + 2 Gg′(3⁄2) + 2 Ig′(5⁄2) + E2g′(7⁄2) E1g′(1⁄2) + Gg′(3⁄2) + 3 Ig′(5⁄2) + E2g′(7⁄2) 2 Gg′(3⁄2) + 3 Ig′(5⁄2) + E2g′(7⁄2) E1g′(1⁄2) + 2 Gg′(3⁄2) + 3 Ig′(5⁄2) + E2g′(7⁄2) 2 E1g′(1⁄2) + 2 Gg′(3⁄2) + 3 Ig′(5⁄2) + E2g′(7⁄2) a

Ds = q{E1g′(1⁄2) + 2 Gg′(3⁄2) + 3 Ig′(5⁄2) + E2g′(7⁄2)} + Ds′, 2s + 1 q = 30 , s′ = s – 15q, if s > 31⁄2

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We have also collected the correlation tables [42] for the octahedral double group Oh2 in Table 10, the correlation table for the Td2 in Table 11, and the correlation table for the D6h2 in Table 12. These correlation tables all demonstrate interesting mathematical periodicity for the rotational or rovibronic levels. The octahedral integral rotational levels exhibit a period of 12 analogous to that for the tetrahedral group. However, the half-integral spin states or rovibronic states exhibit a period of six both in the octahedral and tetrahedral double groups. The D6h2 double group exhibits a different periodic trend as seen from Table 12. The periodicity of six is same for both the half-valued and integral representations. Thus, the periodicity trends exhibited by the double groups are quite interesting. These correlation tables are quite valuable in obtaining the rovibronic levels of molecules with both an odd and even number of electrons. It is important to obtain the overall rovibronic correlation as opposed to individual rotational correlations owing to the fact that the total wavefunction may become a half-integral representation, especially for systems with an odd number of electrons. Furthermore, for molecules containing very heavy atoms spin-orbit effects become quite significant, and thus the coupling of the spin with orbital angular momentum splits the electronic states into spin-orbit states. The exact manner in which these states are split by spin-orbit coupling is given by the double group correlation tables shown here. Table 10. Periodic correlation table for the half-integral states of the Oh2 double group. Irreducible Representations in the Oh2 Groupa

s 0 1 2 3 4 5

A1g T1g Eg + T2g A2g + T1g + T2g A1g + Eg + T1g + T2g Eg + 2T1g + T2 + T2g

6 ⁄2

A1g + A2g + Eg + T1g + 2T2g E1g′(1⁄2)

1 3

⁄2 ⁄2 7 ⁄2 9 ⁄2 11 ⁄2 6 + s′ 5

12n + s′

Gg′(3⁄2) E2g′(5⁄2) + Gg′(3⁄2) E1g′(1⁄2) + E2g′(5⁄2) + Gg′(3⁄2) E1g′(1⁄2) + 2 Gg′(3⁄2) E1g′(1⁄2) + E2g′(5⁄2) + 2 Gg′(3⁄2) E1g′(1⁄2) + E2g′(5⁄2) + 2 Gg′(3⁄2) + terms of s′ but interchange E1g′(1⁄2) with E2g′(5⁄2) 2n {(E1g′(1⁄2) + E2g′(5⁄2) + 2 Gg′(3⁄2)} + terms for s′

a

Terms for other integral s values are found using the formula: D(12n+s′) = Ds′ + n(A1g + A2g + 2 Eg + 3 T1g + 3 T2g), s′ < 12.

The concept of periodicity can be extended to cases beyond the double groups. Such cases would involve Berry’s phase where a rotation through 360° would yield a complex number, exp(2πi/n), for an integer n > 2. The symmetry exhibited by such systems could be quite intriguing. It is hoped that this chapter will stimulate future investigations into Berry’s phase.

138

K. Balasubramanian Table 11. Periodic correlation table for the half-integral states of the Td2 group.

a

s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 ⁄2 3 ⁄2 5 ⁄2 7 ⁄2 9 ⁄2 11 ⁄2 6 + s′ 2n + s′

Irreducible Representationsa A1 T1 E + T2 A2 + T1 + T2 A1 + E + T1 + T2 E + 2 T1 + T2 A1 + A2 + E + T1 + 2 T2 A2 + E + 2 T1 + 2 T2 A1 + 2E + 2 T1 + 2 T2 A1 + A2 + E + 3 T1 + 2 T2 A1 + A2 + 2E + 2 T1 + 3 T2 A2 + 2E + 3 T1 + 3 T2 2 A1 + A2 + 2 E + 3 T1 + 3 T2 A1 + A2 + 2 E + 4 T1 + 3 T2 A1 + A2 + 3 E + 3 T1 + 4 T1 A1 + 2 A2 + 2 E + 4 T1 + 4 T1 E1/2 G3/2 G3/2 + E5/2 E1/2 + G3/2 + E5/2 E1/2 + 2 G3/2 E1/2 + E5/2+ 2 G3/2 E1/2 + E5/2 + 2 G3/2 + terms for s′, but interchange E1/2 and E5/2 2n(E1/2 + E5/2)

Other integral spin states are correlated using the formula

D(12n+s'′ = Ds′ + n(A1 + A2 + 2 E + 3T1 + 3T2), s′ < 12.

Table 12. Periodic correlation table for the half-integral states of the D6h2 double group s 0 1 2 3 4 5 6

1

⁄2 ⁄2 5 ⁄2 7 ⁄2 9 ⁄2 11 ⁄2 3

a

6n + s′

Irreducible Representationsa A1g A2g + E1g A1g + E1g + E2g A2g + B1g + B2g + E1g + E2g A1g + B1g + B2g + E1g + 2 E2g A2g + B1g + B2g + 2 E1g + 2 E2g 2 A1g + A2g + B1g + B2g + 2 E1g + 2 E2g E1g′(1⁄2) E1g′(1⁄2) + E3g′(3⁄2) E1g′(1⁄2) + E2g′(5⁄2) + E3g′(3⁄2) E1g′(1⁄2) + 2 E2g′(5⁄2) + E3g′(3⁄2) E1g′(1⁄2) + 2 E2g′(5⁄2) + 2 E3g′(3⁄2) 2 E1g′(1⁄2) + 2 E2g′(5⁄2) + 2 E3g′(3⁄2) 6n{E1g′(1⁄2) + E2g′(5⁄2) + E3g′(3⁄2)} + terms of s′, s′ < 6

Terms for other integral s values may be found by using

D(12n+s′) = Ds′ + n(A1g + A2g + B1g + B2g + 2 E1g + 2 E2g), s′ < 6

The Mathematical Basis of Periodicity in Atomic and Molecular Spectroscopy

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Acknowledgement This research was performed under the auspices of the US Department of Energy by the University of California, LLNL under contract number W-7405-Eng-48 while the work at UC Davis was supported by the National Science Foundation.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]

[26]

K. Balasubramanian, Chem. Rev., 85, 599 (1985). K. Balasubramanian, J. Chem. Phys., 72, 665 (1980). K. Balasubramanian, J. Chem. Phys., 73, 3321 (1980). K. Balasubramanian, Theor. Chim. Acta, 59, 237 (1981). K. Balasubramanian, J. Chem. Phys., 74, 6824 (1981). K. Balasubramanian, J. Chem. Phys., 75, 4572 (1981). K. Balasubramanian, Int. J. Quant. Chem., 20, 1255 (1981). K. Balasubramanian, J. Comput. Chem., 3, 69 (1982). K. Balasubramanian, J. Comput. Chem., 3, 75 (1982). K. Balasubramanian, H. Strauss, and K. S. Pitzer, J. Mol. Spectrosc., 93, 447 (1982). K. Balasubramanian, J. Phys. Chem., 86, 4668 (1982). K. Balasubramanian, J. Chem. Phys, 78, 6358 (1983). K. Balasubramanian, J. Chem. Phys, 78, 6369 (1983). K. Balasubramanian, Group Theory of Non-rigid Molecules and its Applications,” Elsevier Publishing Co., 23, 149–168 (1983). K. Balasubramanian, Theor. Chim. Acta, 78, 31 (1990). R. F. Curl, Jr., and J. E. Kilpatrick, Amer. J. Phys. 28, 357 (1960) M. Hamermesh, Group Theory and its Physical Applications, Addison Wesley, Reading MA, 1964 F. A. Cotton, Chemical Applications of Group Theory, Wiley Interscience, New York, NY, 1971 P. R. Bunker, “Molecular Symmetry and Spectroscopy,Academic Press, New York, NY, 1979 D. E. Littlewood, Theory of Group Characters and Matrix Representations of Groups, , Oxford, New York, NY, 1958 H. Weyl, Theory of Groups and Quantum Mechanics, Dover Publications, New York, NY, 1950 M. Tinkham, Group Theory and Quantum Mechanics, McGraw-Hill, New York, NY 1964 B. R. Judd, Operator Techniques in Atomic Spectroscopy, Princeton University Press, Princeton,, NJ 1998 S L Altmann and P Herzig, Point-Group Theory Tables, Clarendon Press, Oxford, 1994. P. Pyykkö and H. Toivonen, Tables of Representation and Rotation Matrices for The Relativistic Irreducible Representations of 38 Point Groups, Acta Academiae Aboensis, Ser B, 43, 1 (1983) H.T. Toivonen and P. Pyykkö, Int. J. Quant. Chem., 11, 697 (1977)

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[27] H.T. Toivonen and P. Pyykkö, Relativistic Molecular Orbitals and Representation Matrices for the Double Groups T and Th, Department of Physical Chemistry, Åbo Akademi, Finland, Report No. B79 (1977), 11 pp. [28] K. Balasubramanian, J. Mag. Res., 91, 45 (1991). [29] K. Balasubramanian, Chem. Phys. Lett., 183, 292 (1991). [30] K. Balasubramanian, Chem. Phys. Lett., 200, 649 (1992). [31] W. G. Harter and T. C. Reimer, Chem. Phys. Lett., 194, 230 (1992). [32] W. G. Harter and T. C. Reimer, Chem. Phys. Lett., 198, 429E (1992). [33] W. G. Harter and D. E. Weeks, J. Chem. Phys., 90, 4727 (1989). [34] K. Balasubramanian, J. Chem. Phys., 95, 8273 (1991). [35] K. Balasubramanian and T. R. Dyke, J. Phys. Chem., 88, 4688 (1984). [36] K. Balasubramanian, J. Mol. Spectroscopy 157, 254 (1993). [37] K. Balasubramanian, Chem. Phys. Lett., 202, 271(1993). [38] K. Balasubramanian, J. Phys. Chem., 97, 8736 (1993) [39] K. Balasubramanian, Mol. Phys., 80, 655 (1993). [40] K. Balasubramanian, J. Chem. Phys., in press. [41] K. Balasubramanian, Relativistic Effects in Chemistry, Part B: Applications, Wileyinterscience, New York, NY, p. 527, 1997. [42] K. Balasubramanian, Relativistic Effects in Chemistry, Part A: Theory and Techniques, Wiley-Interscience, New York, NY, p. 301, 1997. [43] K. S. Pitzer, Accts. Chem. Res., 12, 271 (1979). [44] P. Pyykkö and J. P. Desclaux, Accts. Chem. Res., 12, 276 (1979). [45] P. Pyykkö, Adv. Quant. Chem., ll, 353 (1978). [46] K. Balasubramanian, J. Phys. Chem., 93, 6585 (1989). [47] P. Pyykkö Ed. Proceedings of the Symposium on Relativistic Effects in Quantum Chemistry; Int. J. Quantum Chem., 25 (1984). [48] P. Pyykkö, Relativistic Theory of Atoms and Molecules, Springer Verlag: Berlin and New York, Part I 1986 Part II 1993, Part 3 2000. For comprehensive list of references up to 2002 see http://www.csc.fi/rtam/. [49] J. Paldus, Theoretical Chemistry: Advances and Perspectives, H. Eyring and D. J. Henderson, Eds, Academic Press, New York, NY, 1976 [50] X. Y. Liu and Balasubramanian, J. Comput. Chem., 10, 417 (1989). [51] K. Balasubramanian, Chem. Phys. Lett., 260, 476 (1996). [52] G. Pólya, Acta Math, 68, 145 (1937) [53] K. Balasubramanian, Theor. Chim. Acta, 53,129 (1979) [54] G. Pólya and R.C. Read, Combinatorial Enumeration of Groups, Graphs, and Chemical Compounds, Springer, New York, NY, 1987. [55] J. H. Redfield, Amer. J. Math. 49, 433 (1927). [56] K. Balasubramanian, J. Math. Chem. 14, 113 (1993). [57] D. H. Rouvray, Chem. Soc. Rev. 3, 355 (1974). [58] D. H. Rouvray, Endeavour, 34, 28 (1975) [59] A.T. Balaban, Chemical Applications of Graph Theory, Academic Press, New York, NY, 1976. [60] R. Read, in “Graph Theory and Applications”, Y. Alavi et al. eds., Lecture Notes in Mathematics, 303, 243 (1972) Springer, 1972.

In: Advances in Mathematics Research, Volume 8 Editor: Albert R. Baswell, pp. 141-144

ISBN: 978-1-60456-454-9 © 2009 Nova Science Publishers, Inc.

Chapter 4

MATHEMATICAL MODELLING OF THERMOMECHANICAL DESTRUCTION OF POLYPROPYLENE G.M. Danilova-Volkovskaya, E.A. Amineva1 and B.M. Yazyyev2 1

Rostov-on-Don Agricultural Machinery State Academy; 344023, Strana Sovetov Street, 1, Rostov-on-Don. 2 Ushakov Naval State Academy 353900, Lenin Avenue, 93, Novorossiysk

Abstract There has been provided mathematical description of the processes of thermonuclear destruction in deformed polypropylene melts; the aim was to use the criterion of destruction estimation in modelling and optimising the processing of polypropylene into products.

Keywords: Thermo-mechanical destruction, polypropylene, molecular mass, effective viscosity.

During processing polypropylene melts under the action of transverse strain there occur strain-chemical conversions which can result in both decrease and increase in their molecular masses; the mechanical effect on the rapidity and level of the occurring processes is considerably more prominent than the mere contribution of thermal and thermal-oxidative breakdown. These data necessitate studying the process of polymer destruction. For this purpose it would be most effective to apply the criterion of assessment of the intensity with which destructive processes happen in polymer melts. If the destruction is observant from the initial value of molecular mass М0 to a certain finite value М∞, then at point of time t the chain group with molecular mass М0 - Мt (where Мt is the average value of molecular mass at a given point of time) is involved in the process. It is natural to assume that the rate of destruction in a unit time is proportional to the whole number of breakdowns in macromolecules up to the destruction limit. These assumptions enable us to propose an expression for calculating the rate of destruction process:

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d (( M t − M ∞ ) / M t = − Kdt , (M t − M ∞ ) / M ∞ The integration of this expression results in:

ln

Mt − M∞ = − Kt + e , M∞

(1)

Since at t=0 Мt = M0, then:

С = ln

Mt − M∞ M∞

,

(2)

If we substitute (1) with (2) after some transformations we get:

ln

From here:

Mt − M∞ = − Kt , M∞

М t = ( M 0 − M ∞ )e kt + M ∞ ,

As value М0 - М∞ is constant for the polymer of the given molecular mass, we can designate it as A; after substitution we get:

ln

from here:

Mt − M∞ = − Kt M0 − M∞

М t = A ⋅ e − kt + M ∞ , where K is the rate constant depending on the chemical

nature of a polymer and, in particular, on how close macromolecular chains are packed. Each criterion obtained from the given expressions represents a concept of one of the interrelated consequences of thermo-mechanical destruction process: decrease in molecular weight, the number of macromolecular breakdowns, and the approach to the possible level of macromolecular destructions. The merit of the criteria is that their values do not depend on the initial molecular weight [1-3]. Paper 20 dwells on the ideas allowing us to advance in the quantitative assessment of thermo-mechanic destruction degree. Taking these data as a basis we can propose an expression for calculating the degree of thermo-mechanic destruction in the form of:

Mathematical Modelling of Thermo-Mechanical Destruction of Polypropylene

ϕ а1 =

1 η 0 − kt ⋅ ⋅e , а ηt

143

(3)

where a is the constant of proportionality which is equal to 3.105. On the other hand:

ϕ а1 = (η а ,τ 1, 2 ,η 0 , it ) ,

(4)

where ηа is the effective viscosity of a material melt, τ1,2 are transverse strains during processing. Combining the defining parameters of equation (3) and modifying this equation into a dimensionless form, it is possible to demonstrate that criterion φ1а, is the function of only two parameters ηа and τ1,2. Comparing (3) and (4) enables the following expression for the criterion of thermomechanic destruction degree to be proposed:

⎛ τ ⋅t ⎞ ϕ а1 = f ⎜⎜η 0 , 1, 2 ⎟⎟ , ηa ⎠ ⎝

(5)

The direct application of this expression in order to estimate the degree of thermomechanic destruction in connection with polymer processing is hindered because the process rate constant depends on the temperature and intensity of thermo-mechanical impact on a material. Consequently, of significant interest is the issue of selecting an attribute for characterizing the degree of destruction. Most researchers consider it worthwhile to simply use viscosity variable (ηа) or characteristic viscosity variable. Here is proposed the criterion for the rate of thermo-mechanical destruction in the polymeric system Ψ11: −τ 12 ⋅t 1 ⎡η 0 ηa ⎤ Ψ = ⋅⎢ е ⎥, a ⎢⎣η a ⎥⎦ 1 1

(6)

where τ12 are strain rate tangents. This relation is helpful because it provides an opportunity for the quantitative assessment of polymer thermo-mechanical destruction rate in dependence with the thermo-mechanical impact regime during processing. Analyzing the data obtained when testing the samples of extrusion products made of polypropylene, the conducted research on their molecular-weight properties, and the calculated values of the criterion for the destruction processes rate, we concluded that the processes of attachment and bifurcation correspond to the values of Ψ11 = 1, while the processes of destruction correspond to Ψ11= - 1.

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Assuming that the effective viscosity in a polypropylene melt is sensitive to changes in molecular mass and in chain-length distribution and taking into consideration the specific character of the thermo-mechanical impact developing during extrusion, it is proposed to calculate the intensity of destruction processes from the latter expression. The advantage of the criterion is that it does not require defining the molecular mass of a polymer. Comparing the values of Ψ11, obtained at testing PP samples processed under various technological regimes and calculated with the aid of a mathematical model allows us to propose applying the criterion to the estimation of physical and chemical transformations occurring in a polymer at modifying the parameters of thermo-mechanical impact. Taking into consideration Ψ11 values, we have found the optimal regime when PP is under extrusion processed into products with improved deformation-strength properties [4].

Conclusions There has been provided mathematical description of the processes of thermonuclear destruction in deformed polypropylene melts; the aim was to use the criterion of destruction estimation in modelling and optimising the processing of polypropylene into products.

References [1] Olroyd J.G. On the formulation of rheological equation of stat. - Trans. Roy. Soc., 1970, A 200, N 1063, p. 523 -527. [2] De Witt T., Mezner .W. A rheological equation of state which predicts non-Newtonian viscosity, normal stresses and dynamics module. J. Appl.Phys., 1985, v. 26, p. 889-892. [3] Baramboymb I.K. Mechanochemistry of high-molecular substances. – 3rd edition. Moscow. The Chemistry publishing house, 1978, p. 34. [4] Danilova-Volkovskaya G.M. The effect of processing parameters and modifiers on the properties of polypropylene and PP-based composite materials. — Doctoral Thesis, (technical sciences). 2005, p. 273.

In: Advances in Mathematics Research, Volume 8 Editor: Albert R. Baswell, pp. 145-160

ISBN: 978-1-60456-454-9 © 2009 Nova Science Publishers, Inc.

Chapter 5

A DESIGN-BASED STUDY OF A COGNITIVE TOOL FOR TEACHING AND LEARNING THE PERIMETER OF CLOSED SHAPES Siu Cheung Kong* Department of Mathematics, Science, Social Sciences and Technology The Hong Kong Institute of Education, Hong Kong

Abstract With the consideration of cognitive inflexibility of learners in computing perimeter of closed shapes, a theory-driven design of a cognitive tool (CT) called the ‘Interactive Perimeter Learning Tool (IPLT)’ for supporting the teaching and learning of the mathematics target topic was developed in this study. An empirical study in the form of pre-test—post-test reflected that learners of varying mathematical abilities had statistically significant gains in using the IPLT for learning support. The IPLT could effectively address the inflexibility commonly exhibited by learners in learning this topic such as the formation of the abstract association of an irregular closed shape with a regular closed shape. The assertion of teachers on the effectiveness of the IPLT and the enthusiasm of students for using the IPLT for learning reflect that the CT had a pedagogical value in fostering learner-centred learning. Based on the feedback of this study, the IPLT will be refined under the design-based research approach.

Keywords: cognitive inflexibility, cognitive tool, design-based research, mathematics, perimeter.

Introduction Perimeter is a linear measure. It refers to the distance around a shape. The general computation of perimeter involves the application of elementary arithmetic, in which the operation of addition is applied to calculate the perimeter of a shape by adding all the length *

E-mail address: [email protected]

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Siu Cheung Kong

of the sides together. The centrepiece of this procedural knowledge relates to the concept of line segments. The study of perimeter is one of the major components of the learning dimension ‘measure’ in the primary mathematics curriculum in Hong Kong. This study domain covers four topics: the ‘introduction to perimeter,’ the ‘perimeter of squares,’ the ‘perimeter of rectangles’ and the ‘perimeter of closed shapes.’ Among these four topics, the ‘perimeter of closed shapes’ is considered to be the relatively complicated topic in the subject domain. This topic focuses on the concepts of lines and shapes as well as the knowledge about how to find the perimeter of irregular closed shapes, and develops upon the fundamental understanding of the topics ‘perimeter,’ the ‘perimeter of squares,’ and the ‘perimeter of rectangles.’ The centrepiece of finding perimeter of closed shapes is the correct combination of line segments on the shape border. The effective teaching and learning of the ‘perimeter of closed shapes’ should focus on the development of this conceptual strategy in an incremental approach with four stages. The first stage is the acquisition of fundamental concepts. Direct process of providing definitions of ‘perimeter,’ such as asking students to finger-outline the border of the irregular shapes, should be adopted. The second stage is the reinforcement for students of the general computation method of perimeter with sufficient information. Cases in which the lengths of all line segments of the irregular shapes are given should be designed for students to consolidate their operation of finding perimeter by adding the lengths of all the sides of a shape. The third stage is the generalisation of the computation of perimeter of closed shapes with the just necessary information. Cases in which only the lengths of certain line segments of a variety of symmetric irregular closed shapes are given, such as the shapes ‘T’ and ‘+,’ should be designed for students to associate an irregular closed shape with a regular closed shape and thus formulate the general conceptual abstractions of line movement and shape conversion for finding perimeter of closed shapes. The fourth stage is the application and transfer of knowledge about ‘perimeter of closed shapes.’ Cases in which only the lengths of certain line segments of a number of complex irregular closed shapes are given, such as the shapes ‘U’ and ‘H,’ should be designed for students to link and apply the basic knowledge about ‘perimeter’ by including the line segments inside the converted shapes and hence develop flexible and transferable procedural knowledge about finding perimeter of closed shapes. However, the main goal of current traditional pedagogical practices for this subject topic commonly focuses on the automation of procedural skills rather than the understanding of important measurement ideas of the ‘perimeter of closed shapes.’ This leads to the cognitive inflexibility that students commonly exhibit at the third and fourth stages in learning this topic.

Cognitive Inflexibility Cognitive inflexibility is a kind of performance problems associated with automation. It refers to the failure to detect a new situation and the consequent demand for a change at the level of control of knowledge to restate the routine problem-solving strategy (Cañas, Antolí, Fajardo & Salmerón, 2005; Cañas, Fajardo & Salmerón, 2006). It proposes the existence of a failure in the evaluation of the situation that leads to a failure in its execution. The occurrence of this problem is due to the automation processes that begin after extensive practice within a

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particular type of tasks. When people repeatedly put a strategy which has shown to be effective in previous situations into practice, they incline to insistently apply the same strategy in new situations even after the conditions of the situations have changed and the strategy is no longer appropriate. There are four types of behaviour in connection with cognitive inflexibility (Cañas et al., 2006). The first one is cognitive blockade, which refers to the tendency to continue with an initial problem-solving strategy in situations where it is rational to select an alternative problem-solving strategy. It denotes the failure to make a global diagnosis of the situation, which is a problem due to the insistence of problem-solvers on focusing solely upon the concrete aspects of the situation and their inability to re-evaluate the new situation. The second one is cognitive hysteresis, which is the tendency to adhere to a problem-solving strategy after evidence has proven it to be a mistake. It represents the failure to discern an erroneous diagnosis, which is a problem due to the problem-solvers cannot judge the situation with new pieces of evidence. The third one is functional fixation, which refers to the tendency to consider only the available objects in problem-solving as known in terms of its more common function. It represents the failure to use the elements that have taken a new form for problem-solving, which is a problem due to the problem-solvers fix assignment of an object to a category and hence make the properties that are assigned to that object become conditioned. The fourth one is functional reduction, which is the tendency to solve a problem by adopting strategies that address a single cause regardless of all other possible influencing variables. It denotes the failure to adapt behaviour to the changed environmental conditions, which is a problem due to the problem-solvers consider only in part the causes of a phenomenon. The major type of cognitive inflexibility involved in the learning of the ‘perimeter of closed shapes’ is cognitive blockade. Students in this case generally show a limitation in developing a holistic comprehension of the strategies for finding perimeter of closed shapes. There are two inadequacies of students in developing the conceptual knowledge about and computational skills in the target subject. First, students lack the ability to develop the relational knowledge about perimeters of polygons in regular and irregular shapes. In the calculation of perimeter of an irregular closed shape, an abstract association of an irregular closed shape with a regular closed shape is involved. It is a strategy to coordinate side lengths and collections of side lengths for the computation of perimeter (Barrett, Clements, Klanderman, Pennisi & Polaki, 2006). For example, to find the perimeter of the irregular closed shape ‘T,’ imaginary steps to move lines (see Figure 1) must be made to convert the irregular closed shape into a regular closed shape in the form of a rectangle. The perimeter can then be computed easily by applying the knowledge and concept of the ‘perimeter of rectangles.’ It is commonly found that students find it difficult to understand the abovementioned abstract association within several lessons. Students generally encounter the cognitive constraint on the configuration of shape and the movement of line segments. Such students exhibit the cognitive inflexibility that they insistently focus on the irregularity in spatial pattern of the irregular closed shape. The students perceive that the shape configuration is fixed and the line segments are immovable. They have no realisation of the flexibility in changing the shape configuration by conceptually moving the position of line segments. The students are thus unable to formulate an immediate perceptual conversion from an irregular closed shape into a regular closed shape.

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Figure 1. Steps showing the movement of lines to find the formula for calculating the perimeter of the irregular closed shape ‘T.’

Figure 2. Example showing the movement of lines to find the formula for calculating the perimeter of the irregular closed shape ‘U.’

Second, students lack the ability to restructure the procedural knowledge about perimeter of irregular closed shapes. In the computation of perimeter of an irregular closed shape, a complete inclusion of subdivisions of continuous linear units should be entailed. It is a strategy to coordinate length attributes around a perimeter and further integrate the fundament concepts of line segments into the calculation of perimeter (Barrett et al., 2006). For instance, to find the perimeter of the irregular closed shape ‘U,’ a regular closed shape in the form of a rectangle with two vertical straight lines inside will be transformed from the original shape by imaginary steps of line movements (see Figure 2). The calculation of perimeter should include the line segments inside the resulted shape apart from the border of the resulted shape.

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It is often found that students are puzzled as to the need for the aforementioned complete inclusion of line segments. Students generally lack the intention to recall the fundamental knowledge and concept of perimeter. Such students exhibit the cognitive inflexibility that they insistently continue with the routine procedures for finding perimeters. They have no flexibility in linking the re-interpretation of the new situation and the application of relevant prior knowledge of the target subject. The students are thus unable to modify their relevant knowledge and adapt their problem-solving strategies according to the new task requirements. To address the learning problems caused by cognitive inflexibility, previous studies have recommended the adoption of multiple organisational schemes for presenting subject matter in instruction and the multiple classification schemes for knowledge representation. It is suggested that subject topics cover knowledge that will have to be used in many different ways that cannot all be anticipated in advance and involve cases or examples of knowledge application that typically involves the simultaneous interactive involvement of multiple complex conceptual structures should be introduced under the instructional approach of multiple knowledge representations (Cañas et al., 2006; Spiro, Collins, Thota & Feltovich, 2003). The rationale behind this instructional approach is that knowledge that will have to be used in many ways must be taught and mentally represented in many ways (Spiro, Feltovich, Jacobson & Coulson, 1991). The instructional approach of multiple knowledge representations focuses on the reorganisation of knowledge to explain how students adapt to new situations. It addresses the irregularity and variation in training over a fixed repetition of steps. By designing different learning scenarios for training different problem-solving strategies in different sequence orders, this instructional approach is considered to be appropriate for improving cognitive flexibility in learning situations (Cañas et al., 2006). To realise this instructional approach, ample resources for demonstrating content diversity, interlinking practices for applying and transferring knowledge, and flexible tools for exploring multiple problem-solving strategies in different contexts should be provided. In this regard, a cognitive tool which addressed the two types of cognitive inflexibility in the learning of the ‘perimeter of closed shapes’ was designed in this study to assist students in the concept and strategy development for the topic ‘perimeter of closed shapes.’

The Study This study adopted the design-based research approach to designing a cognitive tool in supporting the learning and teaching of the mathematics topic, the ‘perimeter of closed shapes.’ Design-based research is a fundamental mode of scholarly inquiry that is useful across many academic disciplines, and has become an increasingly accepted approach to theoretical and empirical study in the field of education in the past decade (Bell, 2004). Design-based research is an attempt to combine theory-driven design with empirical studies of learning environments (Bell, 2004; Design-Based Research Collective, 2003; Hoadley, 2004). It aims to design and explore a whole range of innovations. The most common type of design-based research combines software design and studies in education (Hawkins & Collins, 1992; Hoadley, 2002). The study reported herein combined a theory-driven design of a cognitive tool and an empirical study on the learning context involving the use of the cognitive tool in a real classroom setting.

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The Theory-Driven Design of the Cognitive Tool Cognitive tools (CTs) are both mental and computational devices that can be used to support the cognitive processes of learners (Derry & LaJoie, 1993; Kommers, Jonassen & Mayes, 1992; Kong & Kwok, 2005). A web-based CT entitled the ‘Interactive Perimeter Learning Tool (IPLT)’ was designed in this study to assist students with the abstraction of conceptual and procedural knowledge about the ‘perimeter of closed shapes.’ It was designed as a mechanism for supporting the instructional approach of multiple knowledge representations. By using different representations of mathematical concepts, the CT could provide students with many opportunities for developing their own intuitive, computational, and conceptual knowledge (Moyer, 2001). As mentioned, the key strategy to find perimeter of closed shapes is the combination of line segments on the border of irregular shapes. The two common problems of students in calculating perimeter of closed shapes, viz. the inflexibility in forming an abstract association of an irregular closed shape with a regular closed shape and the inflexibility in making a complete inclusion of line segments of shape border, are closely related to the failure of students to realise this important step. In this regard, the IPLT aimed at providing learning support in the concept development concerning the concepts of lines as well as shapes and the strategy development regarding the combination of line segments of shape border in finding perimeter of closed shapes. The IPLT is a graphical tool for the display of graphical representation of irregular closed shapes. It consisted of a regular plane on which a set of irregular closed shapes was shown on the interface one at a time. Three features were designed to cover the important concepts of the topic and address the common inadequacies of students in learning the target subject. The first feature was the design of movable line segments of shape border. This feature was designed for students to develop the concepts of lines and shapes and the strategies of combining line segments in the calculation of perimeter of closed shapes. This feature enabled students to freely move the line segments of shape border of a closed shape by clicking line segments and dropping them on the designated positions (see Figure 3). Students were allowed to move the vertical lines leftwards and rightwards, and move the horizontal lines upwards and downwards. The line segments were not allowed to be rotated. The second feature was the provision of the just necessary information for the calculation of perimeter of closed shapes. This feature was designed to address the inadequacy of students in forming an abstract association of an irregular closed shape with a regular closed shape in the computation of perimeter of closed shapes. On each display of closed shapes, the horizontal distance between the leftmost and the rightmost vertices and the vertical distance between the highest and the lowest vertices of the closed shapes were displayed in fixed positions (see Figure 3). For the complicated closed shapes, measurements of the width between certain sides of the shape were shown in fixed positions. This feature stimulated students to convert the irregular closed shape that was displayed on the IPLT into a regular closed shape by just providing the lengths of the vertical and horizontal sides, which is the information normally given for the calculation of perimeter of regular closed shapes. This feature could thus help students to form an abstract association of an irregular closed shape with a regular closed shape. The third feature was the design of returnable graphical representations of the closed shapes. This feature was designed in response to the inadequacy of students in making a

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complete inclusion of line segments of shape border in the calculation of perimeter of closed shapes. A ‘Reset’ button (see Figure 3) was provided in the bottom right-hand corner of the interface. Students could press this button to get back the original graphical representation of a closed shape at their convenience when they were lost in the perimeter task after converting the original shape into the resulted shape. The IPLT was of benefit to students in the learning of the ‘perimeter of closed shapes.’ The IPLT could help students to rectify their common learning problems at the third and fourth stages of the target subject by providing students with many more chances to actively participate in the free exploration of perimeter in different contexts which involved a variety of representations and to fully generalise the important concepts of the subject topic. Building on the basic conceptual and procedural knowledge about perimeter that has been learnt in the classroom learning, with the use of the IPLT, students could freely explore the abstract association of an irregular closed shape with a regular closed shape at the skill-based and rule-based levels and realise the complete inclusion of line segments of shape border at the knowledge-based level. In this regard, the IPLT could facilitate students to make many potential combinations of relevant prior knowledge cognitively available in the calculation of perimeter of closed shapes.

The Empirical Study To investigate the potential of the IPLT for supporting the learning and teaching of the mathematics topic, the ‘perimeter of closed shapes,’ an empirical study was conducted in a real classroom setting. Researchers have suggested that the major criteria for the evaluation of a CT include the learning achievement of students after using the CT, the justification of teachers for the use of the CT in teaching, and the preference of students for the use of the CT in learning (Hawkins & Collins, 1992). In this respect, three specific research questions were investigated in this study. The display of the horizontal distance between the leftmost and the rightmost vertices and the vertical distance between the highest and the lowest vertices of the closed shapes for stimulating the abstract association of an irregular closed shape with a regular closed shape.

The movable line segments of shape border for converting an irregular shape into a regular shape.

A ‘Reset’ button for returning the resulted shape to the original shape. Figure 3. An interface of the IPLT.

1. What is the degree of the IPLT in catering for the diverse needs of students with varying learning abilities? 2. What are the opinions of teachers on the use of the IPLT in teaching? 3. What are the views of students on the use of the IPLT in learning?

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Two classes of Primary Four students in a primary school in Hong Kong were invited to participate in this study. Table 1 shows the profile of the two classes of Primary Four students who participated in this study. The two classes of students had varying levels of learning ability in mathematics, which were reflected in a test held just before the study. The corresponding test results are shown in the ‘mathematics mean score’ row in Table 1. The use of the IPLT was incorporated into the normal teaching lessons of ‘perimeter of closed shapes’ in the invited school, in which each lesson lasted for 35 minutes. The teaching materials for students of the two participating classes included PowerPoint materials, the IPLT, and a number of activity worksheets. One class (the At-riskIPLTGp class) worked with the desktop version of the IPLT with desktop computers individually in the computer room during a double lesson; while another class (the EliteIPLTGp class) worked with the mobile version of the IPLT with personal digital assistants (PDAs), in groups of five to six, during a triple lesson. Each student was assigned a computing tool to access the IPLT which was located in a web server. The teachers of these two classes mainly asked the students to use the IPLT to complete the learning tasks specified on the activity worksheets. Table 1. Profile of the two classes of Primary Four students EliteIPLTGp

At-riskIPLTGp

Number of students

Profile

36

25

Ratio of boys to girls

14 : 22

16 : 9

8.97 (S.D. = 0. 17)

9.16 (S.D. = 0.37)

70.00 (S.D. = 11.45)

52.56 (S.D. = 15.37)

Mean age in years Mathematics mean score (max = 100)

Table 2. The major learning and teaching activities for students with the use of the IPLT 1.

2. 3.

4. 5.

Learning and teaching activities Instruction in the target subject: The teacher made use of PowerPoint presentation to demonstrate the movement of line segments of shape border to calculate perimeter of closed shapes. Brief of the functions of the IPLT: The teacher introduced the use of the IPLT. Exploration of the target subject and completion of activity worksheets: Students were asked to use the IPLT to explore the way of finding perimeter of closed shapes by completing activity worksheets. Relevant guidance and probing questions were offered occasionally by the teacher during these sessions. Answer check: Students were requested by the teacher to give answers and corresponding explanations. Class discussions: Class discussions were conducted for students to consolidate knowledge.

Table 2 summarises the major learning and teaching activities for students with the use of the IPLT. The learning and teaching activities for the students focused on the student-centred exploration of the movement of line segments of shape border in finding perimeter of closed shapes. The majority of class time was spent on the use of the IPLT for the completion of the activity worksheets. Initially, teachers used PowerPoint files to demonstrate the movement of line segments of shape border for the calculation of perimeter of closed shapes. The use of the

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IPLT was subsequently introduced to students by teachers. Students were asked to explore the way of finding perimeter of closed shapes and complete five questions on the activity worksheets with the use of the IPLT. Guidance and probing questions were given by teachers. In the answer-checking sessions, teachers asked some students to explain their answers rather than telling students the answers directly. Class discussions followed for knowledge consolidation of students.

Evaluation Methods Quantitative and qualitative methods were adopted in the empirical study to collect data on the potentials of the IPLT to assist in the teaching and learning of the ‘perimeter of closed shapes’ with regard to the learning achievement of students after using the IPLT, the justification of teachers for the use of the IPLT in teaching, and the preference of students for the use of the IPLT in learning. To study the impact of the IPLT on the learning achievement of the students, a set of pretest—post-test instruments was designed to garner quantitative data concerning the learning benefits to students in terms of academic results after working with the IPLT. Under the pretest—post-test design, a pair of identical tests was incorporated in the teaching process for all of the students who participated in this study in order to measure the knowledge of students about finding perimeter of closed shapes before and after learning the target topic. A test paper consisting of eight questions about the calculation of perimeter of closed shapes was designed for the pre-test and the post-test. Figure 4 shows a sample question in the test paper. To investigate the justification of the teachers for the use of the IPLT in teaching, semistructured, individual interviews with the teachers who used the IPLT were conducted after the teaching period to gather qualitative data regarding their opinions of the use of this CT in terms of application situation and teaching effect. Five questions about perception of teaching benefits and suggestions for further improvement of the IPLT were designed for the interviews. The responses were processed by content analysis of the interview records. To study the preference of students for the use of the IPLT in learning, a questionnaire was distributed to the students in this study to collect qualitative data reflective of their views on the use of this CT in terms of application situation and learning effect. Students were asked to indicate their level of agreement with four statements about user-friendliness, learning benefits and usage preference in relation to the IPLT. The mean rating of each statement on a 5-point Likert scale, from 1 = ‘strongly disagree’ to 5 = ‘strongly agree,’ and the corresponding standard deviation were calculated.

Results and Discussions Learning Outcome of Students from Pre-Test—Post-Test Instruments This section reports the learning achievement of students with varying mathematical abilities after using the IPLT for leaning. Table 3 and Figure 5 show the effects of the IPLT on the elite students who participated in the study. The paired t-test result in Table 3 indicates that the mean difference between the pre-test and post-test measures of the elite students is

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significantly different. In this case, the elite class had a statistically significant gain in the knowledge of the target subject after the teaching period.

The graphical representation of an irregular closed shape with the length marks.

The answer spaces for the calculation expression and answer of the perimeter of the closed shape. Figure 4. A sample question in the test paper.

Table 3. Mean, standard deviation and paired t-test of the pre-test and post-test measures for the elite students who participated in the study Group EliteIPLTGp

Number of students 36

Pre-test Mean (S.D.) 3.47

(2.06)

Post-test Mean (S.D.) 6.53

(1.28)

Paired t-test -10.16***

t(35)

*** p < .001.

7 6 5 4 3 2 1 0

EliteIPLTGp

Pre-test

Post-test

Figure 5. Effects of the IPLT on the elite students on their knowledge and concepts of finding perimeter of closed shapes.

Table 4 and Figure 6 show the effects of the IPLT on the at-risk students who participated in the study. The paired t-test result in Table 4 indicates that the mean difference between the pre-test and post-test measures of the at-risk students is significantly different. In other words,

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the at-risk class had a statistically significant gain in the knowledge of the target subject after the teaching period. In summary, the findings show that students had a statistically significant gain in the learning of ‘perimeter of closed shapes’ with the use of the IPLT. This reflects that the integration of the CT with the traditional teaching materials in classroom instruction was effective for learning the knowledge of finding the perimeter of closed shapes. The IPLT could cater for the diverse needs of students of differing learning abilities. Table 4. Mean, standard deviation and paired t-test of the pre-test and post-test measures for the at-risk students who participated in the study Group

Number of students

At-riskIPLTGp

25

Pre-test Mean (S.D.) 3.56

(1.96)

Post-test Mean (S.D.) 5.76

Paired t-test

(1.76) -5.75***

t(24)

*** p < .001.

7 6 5 4 3 2 1 0

At-riskIPLTGp

Pre-test

Post-test

Figure 6. Effects of the IPLT on the at-risk students on their knowledge and concepts of finding perimeter of closed shapes.

Feedback of Teachers from Interviews Table 5 shows the key points that the teachers made during the interviews about the use of the IPLT for teaching the ‘perimeter of closed shapes.’ The teacher of the EliteIPLTGp class pointed out that it was easy for students to use the IPLT. He asserted that the use of the IPLT fostered teacher-student and student-student interactions in the classroom setting. The teacher observed that his students were highly involved in the learning activities with the use of the IPLT. He pointed out that the mobile version of the IPLT in PDAs allowed him to walk around the classroom for checking the learning progress of students and providing instant feedback to individual students. The teacher showed his appreciation of the capabilities of the IPLT to move line segments of shape border and reset graphical representations of closed shapes because these features of the IPLT made the teaching of the target subject more efficient and facilitated the organisation of relevant in-class leaning activities. The teacher also agreed that the IPLT could effectively address the inadequacies of students in learning the ways of finding perimeters of closed shapes, in particular for their inflexibility concerning

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the formation of the abstract association of an irregular closed shape with a regular closed shape. The teacher suggested that the lines that displayed on the IPLT should be thinner so as to facilitate students to move line segments and form a regular closed shape for each attempt. The teacher of the At-riskIPLTGp class also agreed that the IPLT was easy for students to use and strengthened the teacher-student and student-student interactions in the classroom environment. She noticed that her students were eager to use the IPLT in solving problems on the activity worksheets. The teacher thought that the IPLT was favourable for learning the target subject because the IPLT allowed students to move the lines freely and explore the shapes and perimeters in many different ways. The teacher pointed out that the IPLT was helpful for students to solve the learning problems of students in learning the target subject, particularly for their inflexibility concerning the realisation of a complete inclusion of line segments of shape border in the calculation of perimeters of closed shapes. The teacher proposed that the lines and the numbers displayed on the IPLT should be thinner and larger respectively for clear reference. In addition, the teacher suggested that there should be a simultaneous movement of line segments and their corresponding length marks for easy reference. The teacher further pointed out that the mobile version of the IPLT should be adopted so as to enhance the teacher-student interactions and increase efficiency of classroom logistics in terms of equipment allocation. Table 5. Key points that the teachers made during the interviews about the use of the IPLT for teaching the ‘perimeter of closed shapes’ Question theme EliteIPLTGp The IPLT was easy for students to Yes. use. The IPLT facilitated interaction Yes. between students and teacher. The mobile version of the IPLT allowed the teacher to walk around the classroom for checking the learning progress of students and providing instant feedback to individual students. The IPLT facilitated students to Yes. discuss the ways of finding perimeter of closed shapes. The IPLT assisted students to Yes. understand the ways of finding Enabled students to move line perimeter of closed shapes. segments of an irregular closed shape to form a regular closed shape. Aspects to be improved regarding Lines that displayed on the IPLT the IPLT. should be thinner.

At-riskIPLTGp Yes. Yes.

Yes. Yes. Enabled students to move the lines freely and explore the shapes and perimeters in different ways. Lines that displayed on the IPLT should be thinner. Numbers that displayed on the IPLT should be larger. Length marks should move along with the lines.

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Table 6. Evaluation results of the student questionnaire survey on the use of the IPLT for learning the ‘perimeter of closed shapes’ Evaluation item This application is easy to use. I can use this application to discuss the ways of finding perimeter of closed shapes with my classmates. The application helps me to understand the ways of finding perimeter of closed shapes. I like to use this application for doing exercise.

EliteIPLTGp At-riskIPLTGp M (S.D.) M (S.D.) 4.50 (0.51) 4.64 (0.48)

t-test 1.07

4.41 (0.61)

4.60 (0.57)

1.21

4.29 (0.68)

4.52 (0.75)

1.17

4.56 (0.56)

4.48 (0.70)

-0.46

Remarks: 1 = strongly disagree; 2 = disagree; 3 = neutral; 4 = agree; 5 = strongly agree.

In summary, the teachers expressed their positive views on the potential integration of the IPLT with traditional pedagogical practices to achieve optimal learning outcome. They exhibited a strong preference for and a high degree of involvement in the use of the IPLT for teaching. The teachers were satisfied with the user-friendliness of the IPLT. They thought that the use of the IPLT in classroom could enhance the teacher-student and student-student interactions. The teachers also asserted the helpfulness of the IPLT for students to understand the ways of and address the cognitive inflexibility in finding perimeter of closed shapes. They indicated that students participated actively in the classroom activities with the use of the IPLT.

Feedback of Students from Questionnaire Survey The students who used the IPLT showed their strong preference for using this CT for learning the target subject. They found that the IPLT was easy to use. The students indicated that they could use the IPLT in the discussions about the subject matter and the completion of relevant exercises. The IPLT was considered as a very helpful learning tool for understanding the ways of finding perimeter of closed shapes. Table 6 shows the evaluation results of the student questionnaire survey on the use of the IPLT for learning the ‘perimeter of closed shapes.’ The t-test shows that none of the tests of equality of means for the two classes could be rejected. This reflects that students in both classes had the same perception of the IPLT. In summary, the students had a positive perception of the IPLT regarding its effectiveness on, importance in, and appealing effect on learning. They were motivated to use the IPLT for acquiring and discussing the subject knowledge in a learner-centred approach. The students also showed a high degree of involvement in using the IPLT for the completion of learning activities.

Implications of the Empirical Study The evaluation results from the empirical study reveal that there is a potential integration of the IPLT into traditional pedagogical practices to achieve optimal learning outcome. Three implications in connection with the development and implementation of the designed CT were drawn based on the abovementioned evaluation results.

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The first implication is related to the pedagogical value of the designed CT. In the student survey, the students showed their great enthusiasm for using the IPLT in learning the target subject. This reveals that the IPLT had a potential to act as a good mediator to facilitate learner-centred learning and induce a constructively qualitative change in the nature and depth of a range of educational processes. This result concurs with the findings of previous studies, that is, that the integration of computer-supported CTs within a learner-centred learning environment offers substantial potential to support the learning process and enhance the learning effect of students because such CTs enable students to regulate the amount and sequence of available resources and explore the gist of subject knowledge according to their individual needs (Hawkins & Collins, 1992; Iiyoshi, Hannafin & Wang, 2005; Jonassen & Reeves, 1996; Kong & Kwok, 2005). The second implication relates to the pedagogical use of the designed CT. In the interviews, the teachers asserted that the attempt at incorporating the use of the IPLT with traditional pedagogical practices was successful. One of the teachers, whose students used the desktop version of the IPLT during class time, further indicated her preference for the mobile version rather than the desktop version of the IPLT because she thought that the former could create an encouraging ambience for student-student discussions about the subject knowledge without location constraint. To cater to the different pedagogical styles of teachers, the provision of both mobile and desktop versions of the IPLT as practised in this study is recommended. The final implication concerns the improvement work for the designed CT. The designbased research approach is a strategy for designing cognitive artefacts by eliciting information about aspects to be improved regarding the CT from users. According to the feedback from the teacher interviews, the lines and the numbers that are displayed on the IPLT should be thinner and larger, respectively, in order to reduce the visual hindrance to students to find the total length of the line segments. In addition, the corresponding length marks of line segments should be made movable so that students are empowered with greater flexibility and autonomy over the use of the IPLT for explorative learning.

Conclusion Primary school students commonly exhibit cognitive inflexibility in applying knowledge about line segments and shapes for the formulation of abstract association of an irregular closed shape with a regular closed shape and for the realisation of a complete inclusion of line segments of shape border in the calculation of perimeter of closed shapes. This study adopted the design-based research approach in developing a theory-driven design of a cognitive tool (CT) entitled the ‘Interactive Perimeter Learning Tool (IPLT)’ and used the CT in an empirical study for supporting the teaching and learning of the ‘perimeter of closed shapes’ in primary mathematics classroom. The evaluation results show that the incorporation of the CT in traditional pedagogical practices facilitated the teaching and learning of the ‘perimeter of closed shapes’. The students attained statistically significant achievement in the learning outcome with the use of the IPLT. In the teaching period, the students and teachers who participated in the study showed their enthusiasm for using the IPLT in learning and teaching the target subject. The teachers recognised the effectiveness of the IPLT on addressing the cognitive inflexibility

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exhibited by students in learning the target subject. The students asserted the helpfulness of the IPLT in supporting the learning of the target subject in a learner-centred approach. These findings shed light on the future study of the designed CT in two aspects. The first aspect relates to the refinement of the designed CT. Since the application of the IPLT in classroom instruction helped to promote the active learning of student, it is worthwhile to continue the improvement work of this CT, especially for the better graphic display in order to facilitate the exploration of shapes and perimeters in different ways. The second aspect concerns the evaluation of the designed CT. There are two issues about the evaluation. The first is about the concern of shifting the evaluation focus from measuring the academic results of the students to understanding the actual changes in the learning process. Discourse analyses of the interactions between the students and the cognitive artefacts and among students are suggested for further evaluation of the pedagogical use of the IPLT in the real classroom setting. This helps to comprehensively delve into the potential of the IPLT to act as a mediator to foster learner-centred learning. The second issue relates to the integration of the IPLT with traditional pedagogical practices. Further investigation about the most effective pedagogical integration of this CT with traditional classroom teaching in the aspects of, for example, the proportion of class time that should be allocated for learning with the IPLT to that for learning without the IPLT, and the sequence of the adoption of the IPLT, viz. at the initial, middle or final stage of the teaching period, in classroom instruction should be conducted. This helps to look into the ways to capitalise on the use of the CT in supporting the classroom teaching of the target subject.

References Barrett, J. E., Clements, D. H., Klanderman, D., Pennisi, S. J., Polaki, M. V. (2006). Students’ coordination of geometric reasoning and measuring strategies on a fixed perimeter task: developing mathematical understanding of linear measurement. Journal for Research in Mathematics Education, 37(3), 187-221. Bell, P. (2004). On the theoretical breadth of design-based research in education. Educational Psychologist, 39(4), 243-253. Cañas, J. J., Antolí, A., Fajardo, I., & Salmerón, L. (2005). Cognitive inflexibility and the development and use of strategies for solving complex dynamic problems: effects of different types of training. Theoretical Issues in Ergonomics Science, 6(1), 95-108. Cañas, J. J., Fajardo, I., & Salmerón, L. (2006). Cognitive flexibility. In W. Karwowski (Ed.), International encyclopedia of ergonomics and human factors (pp. 297-301). Boca Raton, FL: Taylor & Francis. Derry, S. J., & LaJoie, S. P. (1993). A middle camp for (un)intelligent instructional computing: an introduction. In S. P. LaJoie & S. J. Derry (Eds.), Computers as cognitive tools (pp. 1-11). NJ: Lawrence Erlbaum Associates. Design-Based Research Collective. (2003). Design-based research: an emerging paradigm for educational inquiry. Educational Researcher, 32(1), 5-8. Hawkins, J., & Collins, A. (1992). Design-experiments for infusing technology into learning. Educational Technology, 32(9), 63-67. Hoadley, C. M. (2004). Methodological alignment in design-based research. Educational Psychologist, 39(4), 203-212.

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Hoadley, C. P. (2002). Creating context: design-based research in creating and understanding CSCL. In G. Stahl (Ed.), Computer support for collaborative learning 2002 (pp. 453462). Mahwah, NJ: Erlbaum. Iiyoshi, T., Hannafin, M. J., and Wang, F. (2005). Cognitive tools and student-centred learning: rethinking tools, functions and applications. Educational Media International, 42(4), 281-296. Jonassen, D. H., & Reeves, T. C. (1996). Learning with technology: using computer as cognitive tools. In D. H. Jonassen (Ed.), Handbook of research on educational communication and technology (pp. 693-719). NY: Scholastic Press. Kommers, P., Jonassen, D. H., & Mayes, T. (1992). Cognitive tools for learning. Heidelberg FRG: Springer-Verlag. Kong, S. C., & Kwok, L. F. (2005). A cognitive tool for teaching the addition/subtraction of common fractions: a model of affordances. Computers and Education, 45(2), 245-265. Moyer, P. S. (2001). Using representations to explore perimeter and area. Teaching Children Mathematics, 8(1), 52-59. Spiro, R. J., Collins, B. P., Thota, J. J., & Feltovich, P. J. (2003). Cognitive flexibility theory: hypermedia for complex learning, adaptive knowledge application, and experience acceleration. Educational Technology, 43(5), 5-10. Spiro, R. J., Feltovich, P. J., Jacobson, M. J., & Coulson, R. L. (1991). Cognitive flexibility, constructivism, and hypertext: random access instruction for advanced knowledge acquisition in ill-structured domains. Educational Technology, 31(5), 24-33.

In: Advances in Mathematics Research, Volume 8 Editor: Albert R. Baswell, pp. 161-176

ISBN: 978-1-60456-454-9 © 2009 Nova Science Publishers, Inc.

Chapter 6

MODELING ASYMMETRIC CONSUMER BEHAVIOR AND DEMAND EQUATIONS 1 FOR BRIDGING GAPS IN RETAILING Rajagopal* Department of Marketing, Business Division, Monterrey Institute of Technology and Higher Education, ITESM Mexico City Campus, Tlalpan, Mexico

Introduction In growing competitive markets the large and reputed firms are developing strategies to move into the provision of innovative combinations of products and services as 'high-value integrated solutions' tailored to each customer's needs than simply 'moving downstream' into services. Such firms are developing innovative combinations of service capabilities such as operations, business consultancy and finance required to provide complete solutions to each customer's needs in order to augment the customer value towards the innovative or new products. It has been argued that provision of integrated solutions is attracting firms traditionally based in manufacturing and services to occupy a new base in the value stream centered on 'systems integration' using internal or external sources of product designing, supply and customer focused promotion (Davies,2004). Besides organizational perspectives of enhancing customer value, the functional variables like pricing play a significant role in developing the customer perceptions towards new products.

1

*

Author expresses his gratitude to the anonymous referees for their valuable suggestion to improve the paper. Author also acknowledges the support provided by Amritanshu Rajagopal, student of Industrial and Systems Engineering of ITESM, Mexico City Campus in data collection, translation of questionnaires in Spanish language, computing the data, developing Tables and figures in this study. E-mail address: [email protected] Home page: http://www.geocities.com/prof_rajagopal/homepage.html. PhD (India) FRSA (London), Professor, Department of Marketing, Business Division, Monterrey Institute of Technology and Higher Education, ITESM, Mexico City Campus, 222, Calle del Puente, Tlalpan, Mexico DF 14380.

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Consumers plan to spread consumption of their total resources evenly over the remainder of their life span at a given time. Such consumption smoothing occurs though current demand for the products and services and buying power may fluctuate over time. Thus, borrowing allows consumers with lower initial assets but higher expected future income to make utilitymaximizing consumption choices within the overall, lifetime budget constraint. On the contrary, consumers may not be very capable of taking lifetime budget constraints into account when making repeated consumption choices that are distributed over time and are influenced by the self reference criterion (see Herrnstein and Prelec, 1992). It has been observed in previous studies that value to expenditure ratios increase consumer sensitivity in volume of buying and driving repeat buying decisions for the regular and high-tech products (Carroll and Dunn 1997). Consumers often have enough insight towards limiting their choices by employing self-rationing strategies. Consumers may show suboptimal behavior while adapting to the self-rationing strategies to limit their choice which may induce asymmetric consumer behavior in the long run (Loewenstein, 1996). This paper attempts to discuss the interdependence of variability in consumer behavior due to intrinsic and extrinsic retail environment which influence the process of determining the choices on products and services. It is argued in the paper that suboptimal choice of consumers affect the demand of the products and services in the long-run and the cause and effect has been explained through the single non-linear equations. A system of demand equations which explains the process of optimization of consumer choice and behavioral adjustment towards gaining a long-term association with the market has also been discussed in the paper.

Related Contributions Customer Value and Choice Probabilities The prospect theory developed by Tversky and Kahnman (1981) towards framing decisions and understanding the dynamics of choices of consumers reveals that the consumers exercise options in consuming products to optimize their satisfaction and ultimate value. Value measurements have been used as one of the principal tools to assess the trend of consumer behavior for non-conventional products. Value syndrome influences the individual and group decisions in retail and bulk deals, and conditionalizes the decision process of consumers. Conditional consumption behavior suggests that consumption depends heavily on the utility function and on the source of uncertainty (Carroll and Kimball, 1996 and Deaton 1992). Repeat buying behavior of customers is largely determined by the values acquired on the product. The attributes, awareness, trial, availability and repeat (AATAR) factors influence the customers towards making re-buying decisions in reference to the marketing strategies of the firm. Decision of customers on repeat buying is also affected by the level of satisfaction derived on the products and number of customers attracted towards buying the same product, as a behavioral determinant (Rajagopal, 2005). A study using market-level data for the yogurt category developed an econometric model derived from a game-theoretic perspective explicitly considers firms' use of product-line length as a competitive tool (Dragnska and Jain, 2005). On demand side, the study analytically establishes link between customer choice and the length of the product line and includes a measure of line length in the utility function to

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investigate customer preference for variety using a brand-level discrete-choice model. The study reveals that supply side is characterized by price and line length competition between oligopolistic firms. Each successive purchase decision is relatively unimportant to an individual consumer, which may be derived from the economic and relational variables associated with the product or services. The formal model of price competition analyzed here is derived from that of Chintagunta and Rao (1996), who similarly consider a dynamic duopoly with adaptive consumers. Besides price, brand and quality are other variables which drive consumers’ perceptions on choice mapping and repeat consumption. Bergemann and Välikmäki (1996) examine the effect of strategic pricing on the rate of information acquisition by a buyer which reveals that optimal learning allows consumers to analyze their choice pattern. Erev and Haruvy (2001) also consider the implications of adaptive learning by consumers but again firms have fixed pricing policies. Recent theoretical explanations for sustained dominance include network effects, increasing returns to scale and learning by doing. There is now considerable evidence to explain consumer choice behavior such as in retail buying the choices are determined by an exogenous random process (Erev and Roth, 1998; Camerer and Ho, 1999; Erev and Barron, 2001). Consumer as a decision maker is endowed with propensities and values for each choice that is made. There are some critical issues associated to the price sensitive consumer behavior, whether customers are equally price-sensitive while purchasing products for functional (e.g. purchasing frozen vegetables, toiletries or paper towels) versus hedonic (e.g. purchasing a high technology computer or a video camera) consumption situations and whether perceived value derived during consuming the product influences price sensitivity. It may also be stated that higher price volatility makes consumers more sensitive to gains and less sensitive to losses, while intense price promotion by competing brands makes consumers more sensitive to losses but does not influence consumers’ sensitivity to gains (Han et.al, 2001).

Behavioral Asymmetry and Customer Choice Value of a customer may be defined in reference to a firm as the expected performance measures are based on key assumptions concerning retention rate and profit margin and the customer value also tracks market value of these firms over time. Value of all customers is determined by the acquisition rate and cost of acquiring new customers (Guptaet al, 2003). A long standing approach to examining how consumers react to price and income changes estimates a set of demand equations for the main commodities and bases deductions on coefficient values. At its simplest, economic demand theory assumes that consumers choose to allocate their limited spending power to purchases of goods to maximize their own satisfaction. This assumption of rational economic behavior imposes substantial constraints (aggregation, homogeneity, symmetry and negativity) on the specification of a system of equations. Consumer behavior is largely driven by tangible and intangible factors which include product attributes, pricing, willingness to pay (disposable income), product attractiveness and related variables. Value and pricing models have been developed for many different products, services and assets. Some of them are extensions and refinements of conventional models on value driven pricing theories (Gamrowski & Rachev, 1999; Pedersen, 2000). There have been some other models developed and calibrated addressing specific issues such as model for household assets demand (Perraudin & Sorensen, 2000).

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Key marketing variables such as price, brand name, and product attributes affect customers' judgment processes and derive inference on its quality dimensions leading to customer satisfaction. The experimental study conducted indicates that customers use price and brand name differently to judge the quality dimensions and measure the degree of satisfaction (Brucks et.al., 2000). Most importantly, these are expected to raise their spending and association with the products and services of the company with increasing levels of satisfactions that attribute, to values of customers (Reichheld and Sasser, 1990). However, it has been observed that low perceived use value; comparative advantages over physical attributes and economic gains of the product make significant impact on determining customer value for the relatively new products. Motivational forces are commonly accepted to have a key influencing role in the explanation of shopping behavior. Personal shopping motives, values and perceived shopping alternatives are often considered independent inputs into a choice model, it is argued that shopping motives influence the perception of retail store attributes as well as the attitude towards retail stores (Morschett et.al, 2005). In retail self-service store where consumer exercises in-store brand options, both service and merchandise quality exert significant influence on store performance, measured by sales growth and customer growth, and their impact is mediated by customer satisfaction. Liberal environment of self-service stores for merchandise decisions, service quality and learning about competitive brands are the major attributes of retail self-service stores (Babakus et.al, 2004). Retail self-service stores offer an environment of three distinct dimensions of emotions e.g. pleasantness, arousal and dominance. Customer value gap may be defined as the negative driver, which lowers the returns on the aggregate customer value. This is an important variable, which needs to be carefully examined by a firm and measure its impact on the profitability of the firm in reference to spatial (coverage of the market) and temporal (over time) market dimension (Marjolein and Verspagen, 1999).

Organizational Influences on Customer Values Another study explores qualitatively the understanding of the importance of intangibles as performance drivers in reference to Swedish organizations using a combination of evolutionary theory, knowledge-based theory and organizational learning. The study reveals that customer values are created towards new products through individual perceptions, and organizational and relational competence (Johanson et.al., 2001). The firms need to ascertain a continuous organizational learning process with respect to value creation chain and measure performance of the new products introduced in the market. In growing competitive markets the large and reputed firms are developing strategies to move into the provision of innovative combinations of products and services as 'high-value integrated solutions' tailored to each customer's needs than simply 'moving downstream' into services. Such firms are developing innovative combinations of service capabilities such as operations, business consultancy and finance required to provide complete solutions to each customer's needs in order to augment customer value towards the innovative or new products. It has been argued that provision of integrated solutions is attracting firms traditionally based in manufacturing and services to occupy a new base in the value stream cantered on 'systems integration' using internal or external sources of product designing, supply and customer focused promotion (Davies,2004). Besides organizational perspectives of enhancing the customer value, the

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functional variables like pricing play a significant role in developing customer perceptions towards the new products. Analysis of the perceived values of customers towards new products is a complex issue. Despite considerable research in the field of measuring customer values in the recent past, it is still not clear how value interacts with marketing related constructs. However there exists the need for evolving a comprehensive application model determining the interrelationship between customer satisfaction and customer value, which may help in reducing the ambiguities surrounding both concepts. One of the studies in this regard discusses two alternative models yielding empirically tested results in a cross-sectional survey with purchasing managers in Germany. The first model suggests a direct impact of perceived value on the purchasing managers' intentions. In the second model, perceived value is mediated by satisfaction. This research suggests that value and satisfaction can be conceptualized and measured as two distinct, yet complementary constructs (Eggert and Ulaga, 2002). Improving customer value through faster response times for new products is a significant way to gain competitive advantage. In the globalization process many approaches to new product development emerge, which exhibit an internal focus and view the new product development process as terminating with product launch. However, it is process output that really counts, such as customer availability. A study proposes that with shortening product life cycles it should pay to get the product into the market as quickly as possible, and indicates that these markets should be defined on an international basis. The results of the study reveals that greater new product commercial success is significantly associated with a more ambitious and speedier launch into overseas markets as the process of innovation is only complete when potential customers on a world scale are introduced effectively to the new product (Oakley, 1996). Retail sales performance and customer value approach are conceptually and methodically analogous. Both concepts calculate the value of a particular decision unit by analytical attributes forecast and the risk-adjusted value parameters. However, virtually no scholarly attention has been devoted to the question if any of these components of the shareholder value could be determined in a more market oriented way using individual customer lifetime values (Rajagopal, 2005). Value of a customer may be defined in reference to a firm as the expected performance measures are based on key assumptions concerning retention rate and profit margin and the customer value also tracks market value of these firms over time. Value of all customers is determined by the acquisition rate and cost of acquiring new customers (Gupta, Lehmann and Stuart, 2003).

Objectives and Design of Model This paper emerges from a new indirect utility function and derives the corresponding system of equations, relating commodity demands to prices, income and customer values that satisfy the customer behavior by utility maximization (aggregation, homogeneity, symmetry and negativity). One of the assumptions made in developing the asymmetric model for measuring the consumer behavior and derived values is that consumers are determined by the sub-sets of tangible and intangible variables which include product attributes, awareness, product attractiveness, price, disposable income and competitive advantage. The analytical construct

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in this paper has been derived using the Linear Expenditure System (LES) and Linear Equations in Demand Estimation (LEDE) to measure the variability in consumer behavior. This paper analyzes the belief-based or reinforcement learning attributes in forming typical consumer behavior through inter-personal communication, point of purchase information, advertisement or corporate image may have a significant impact on market organization. The model of dynamic oligopoly, where consumers learn about the relative perceived values of the different brands has also been discussed in the paper. The basic premise of the model is towards reinforcement type behavioral model, where more familiar products have a greater probability of being selected. Consequently, consumers can get locked into inferior choices without considering the novelty factor in reference to product attributes or brand. Such lock-in behavior may be cyclical and asymmetric in context to personality traits and demand for the product. Such situation may become significant when firms influence consumer opinion in the short run as consumers’ initial estimate of a firm’s quality is high (or low), it has an incentive to charge above (or below) the myopic price in order to slow (or speed up) learning and behavioral adaptation of consumer towards the competing products.

Construct of Model Choice Variability and Demand Equation Ofek Elie (2002) discussed that the values of product and service are not always the same and are subject to value life cycle that governs customer preferences in the long-run. If customers prefer the product and service for N periods with Q as value perceived by the customer, the value may be determined as Q>N, where Q and N both are exogenous variables. If every customer receives higher perceived values for each of his buying, the value added product q ≥ Q, where ‘q’ refers to the change in the quality brought by innovation or up-graded technology. Customer may refrain from buying the products if q ≤ Q, that does not influence his buying decisions. However, a strong referral ‘R’ may lead to influence the customer values, with an advantage factor β that may be explained by price or quality factor. In view of the above discussion it may be assumed that customer preferences have high variability that grows the behavioral factors in retail buyers’ decisions (Rajagopal, 2006a): N

(

)

Dbn = ∑ β t μ C t , Zˆ + β N +1Qt t =1

(1)

Dbn is expressed as initial buying decision of the customers, C represents t ˆ consumption, Z is a vector of customer attributes (viz. preferential variables) and Qt is the Where,

value perceived by the customer. Customer behavior is largely derived from the customer value and it has a dynamic attribute that plays a key role in buying and is an intangible factor to be considered in all marketing and selling functions. The value equation for customer satisfaction may be expressed as a function of all value drivers wherein each driver contains the parameters that

Modeling Asymmetric Consumer Behavior and Demand Equations…

167

directly or indirectly offer competitive advantages to the customers and enhance the customer value.

V ′ = Ks , Km , Kd , Kc

[∏ {V (x, t , q, p )}]

(2)

In the above equation V ′ is a specific customer value driver, K are constants for supplies(Ks), margins (Km), distribution (Kd), and cost to customers (Kc); x is volume, t is time, q is quality and p denotes price. Perceived customer value (V) is a function of price (p)

∏

and non-price factors including quality (q) and volume (x) in a given time t. Hence has been used as a multiplication operator in the above equation. Quality of the product and volume are closely associated with the customer values. Total utility for the conventional products goes up due to economy of scale as the quality is also increased simultaneously (∂v/∂x>0). The ∂v customer value is enhanced by offering larger volume of product at a competitive price in a given time (∂v/∂p>0) and (∂v/∂t>0). Conventional products create lower values to the customers (∂v/∂x<0) while innovative products irrespective of price advantages, enhance the customer value (∂v/∂x>0). Value addition in the conventional products provides lesser enhancement in customer satisfaction as compared to the innovative products. Such transition in the customer value, due to shift in technology may be expressed as:

⎡ ⎤ Tp Vhj′ = a ⎢∑ (1+ j ′+ i ) ⎥ + b(X j ) ⎢⎣ (1 + V p ) ⎥⎦

(3)

V′

In this equation hj represents enhancements in customer value over the transition from conventional to innovative products, a and b are constants, Tp denotes high-tech and highvalue products, Vp represents value of product performance that leads to enhance the customer

value, volume is denoted by X and ( j ) is the period during which customer value is measured (Rajagopal, 2006b). In reference to optimization theory it may be stated that maximizing a valid or direct utility function subject to a budget constraint or alternatively, appealing to duality theory and commencing from a cost or indirect utility function, influences on consumer behavior (Shida, 2001). In the latter case, let U (p, y) be the indirect utility function, where p is a vector of prices and y is income. For validity, U (p, y) should be homogeneous of degree zero in income and prices (p), non-decreasing in y, non-increasing in p, and convex or quasi-convex in p. The demand equations can be obtained through:

′

qi =

∂U ∂U / ∂p ∂y

(4)

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Rajagopal In the above equation

(qi ) is denoted as demand for the product. Simple utility function

y⎤ ⎡ ⎢⎣U = P ⎥⎦ derived by generic consumer behavior may be understood as wherein P may be expressed as geometric mean of prices which derives dynamic consumer behavior with the

(α )

j over the products in a given retail variability factor of competitive advantage environment and j is the vector adding to unity over n commodities. Such condition of consumer behavior may be explained as:

log P = ∑ (α j log p j )

(5)

The above equation helps in deriving the demand equation as below:

⎛pq ⎞ wi = ⎜⎜ i i ⎟⎟ = α j ⎝ y ⎠

(6)

(w )

i represents the individual expenditure limits or disposable income for buying wherein, the products. However, these equations limit consumer responses to changes in prices, competitive advantages or disposable income to maintaining consistency in buying behavior due to change in the elasticity of aforesaid variables. While such a consumption pattern might sometimes be plausible, it adds to the asymmetric behavior of consumers in retail buying. Propensity of consumption during leisure season may be largely determined as a driver of retail attractions in terms of appealing sales promotions and availability of innovative products. Choice of consumers is thus established by the propensity of consumption in the array of innovative products in the retail stores. The propensity of consumption of buyers may

be denoted by

(θ

lim 0 − ∞

0−∞ ] θ = [θ lim jt ... n

) from a j

, which is measured in reference to frequency of buying

th

store in a given time t. The choice pattern of buyers in shopping during holiday season may be derived as:

x i (t ) =

exp{βθ i (t )}

∑ exp{βθ n

j =1

In the above equation exponential expression

j

(t )} (7)

{xi (t )} is the probability of buying strategy i at time t. In the

β represents the degree of value optimization on buying. At higher

levels of β , consumer will have higher probabilities of buying with increased propensity.

Modeling Asymmetric Consumer Behavior and Demand Equations…

169

Consumer Choice for New Products According to the customer choice model of Giannakas and Fulton (2002), individuals are assumed to consume optimum one unit of product of their interest within a given time. The construct of the model of customer behavior, assumes that prior to choice of new product,

U

vi customer i derives perceived use value, for t conventional or new products having willingness to pay for the conventional product on the perceived used value of customer i at price p. In order to explain the customer preference to the product and estimating the brand value in reference to this study it may be derived that customer obtains the perceived use

value

(U

vi

− pt

) from consuming conventional product. The customer also exercises his

option of buying a high value substitute (new products) at an alternate price

(

)

pa

where p ≥ p . Hence without availability of new products, the customer value Cs may be derived as: a

t

[(

)] (

)(

Csit = lim 0−∞ U vi − p t , U vi − p a = U vi − p t

)

(8)

Following the scenario when customers get access to new products in the market,

E

customers enhance the perceived use value of the new product by factor vi . This parameter is subjective to the customer decision in view of their preferences towards consuming organic products. However, due to lack of awareness, advertising and sales promotions, many customers may not be able to establish their preferences explicitly towards synthetic and conventional products. If α represents the market segment for new products, the customers

(αE )

vi . Accordingly, the customer value would access the products and perceive its value by after the new products are made available in the market segment may be described as:

[(

)

(

)

CsiAP = lim 0−∞ U vi − p t − αE vi , U vi − p a , Bivg Wherein, the expression

]

(9)

Cs iAP denotes the enhancement of customer value for new Bivg

products in reference to the advertising and promotion strategies of the firms and represents the brand value of new products perceived by the customers. Further,

(Cs )comprising word of ) and point of sales promotions (Cs ). Hence, the

be understood as a function of interpersonal communication

(Cs

ma

IP i

i mouth and referrals, commercials factors influencing the enhancement of customer value may be expressed as:

(

CsiAP may

Cs iAP = f Cs iIP , Cs ima , Cs ips

)

ps i

lim 0 − ∞

(10)

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Rajagopal

Motivational forces are commonly accepted to have a key influencing role in the explanation of shopping behavior. Personal shopping motives, values and perceived shopping alternatives are often considered independent inputs into a choice model. It is argued that shopping motives influence the perception of retail store attributes as well as the attitude towards retail stores (Morschett et.al, 2005). Lliberal environment of the self-service stores for merchandise decisions, service quality and learning about competitive brands are the major attributes of retail self-service stores (Babakus et.al, 2004). Retail self-service stores offer an environment of three distinct dimensions of emotions e.g. pleasantness, arousal and dominance. The change in the customer value observed among the synthetic and new products in reference to advertising and promotional strategies used by the firms, may be described as:

CsiAP −t = CstAP − Csit

(11)

N

The model assumes that if rs number of customers in a given retail store has preferred to use new products; the change in the customer value may be derived as:

ΔCs iAP −t =

Nrs1 ... Nrs n

∑ ΔCs i =1

AP −t i

(12)

Customers choose the product which offers maximum utility in reference to the price, awareness and promotional advantage over other conventional products. Hence, the customer value for new products may be expressed as:

(

Evi −t Evi ΔC Nrs − Cs it =1 = α C i

Evi

)

(13)

Where, C represents the customer value in total derived by all factors. Value and pricing models have been developed for many different products, services and assets. Some of them are extensions and refinements of convention models on value driven pricing theories (Gamrowski and Rachev, 1999; Pedersen, 2000). There have also been some models that are developed and calibrated addressing specific issues such as model for household assets demand (Perraudin and Sorensen, 2000). Key marketing variables such as price, brand name, and product attributes affect customers' judgment processes and derive inference on its quality dimensions leading to customer satisfaction. The experimental study conducted indicates that customers use price and brand name differently to judge the quality dimensions and measure the degree of satisfaction (Brucks et.al. 2000). Value of corporate brand endorsement across different products and product lines, and at lower levels of the brand hierarchy also needs to be assessed as a customer value driver. Use of corporate brand endorsement either as a name identifier or logo identifies the product with the company, and provides reassurance for the customer (Rajagopal and Sanchez, 2004).

Modeling Asymmetric Consumer Behavior and Demand Equations…

171

Cost of acquiring new products would be the difference in traditional good price,

(C ) sc

variation in the perceived use value and search cost as indicated by i for each customer. Hence, appreciation of customer value to obtain new products may be expressed as:

[(

)

ΔCiKrs −t = α Evi + p a + Cisc − Csit

]

(14)

Krs i

C

represents the cost of acquiring the new products from a given retail store. Where, Competitive advantage of a firm is also measurable from the perspective of product attractiveness to generate new customers. Given the scope of retail networks, a feasible value structure for customers may be reflected in repeat buying behavior ( Rˆ ) that explains the relationship of the customer value with the product and associated marketing strategies. The impact of such customer value attributes in a given situation may be described as: n

∑ Cs

Nsr =1

AP i

= Rˆ = C Evi (15)

Repeat buying behavior of customers is largely determined by the values acquired on the product. Decision of customers on repeat buying is also affected by the level of satisfaction derived on the products and number of customers attracted towards buying the same product, as a behavioral determinant (Rajagopal, 2005). New product attractiveness may comprise the product features including improved attributes, use of advance technology, innovativeness, extended product applications, brand augmentation, perceived use value, competitive advantages, corporate image, product advertisements, sales and services policies associated therewith, which contribute in building sustainable customer values towards making buying decisions on the new products (Rajagopal, 2006a). Attributes of the new products lead to satisfaction to the customers and is

F

reflected through the product attractiveness ( x ). It has been observed that the new products have been considered as new and experimental products in Mexico by a significant number of consumers. Hence product attractiveness variables in the following equation as:

[

(Fx )

may be explained along the associated

(

Fx = ∏ αE vi Cs tAP , q, Z xi , p a

)]

t

(16)

Z

Wherein, q denotes quality of the product and xi represents services offered by the retailers towards prospecting and retaining customers who intend to buy the new products. Customer value may also be negative or low if attributes are not built into the new product to maximize the customer value as per the estimation of the firm. Perceived use value of customers by market segments appreciation p

a

αEvi is a function of advertising and promotion CsiAP , price

and retailer services

Z xi in a given time t and

∏

has been used as a

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Rajagopal

multiplication operator in the above equation. The quality of the product and volume are closely associated with the customer values. Introduction of new technological products makes it important for marketers to understand how innovators or first adopters respond to persuasion cues. It has been observed in a study that innovativeness and perceived product newness which are the major constituents of new product attractiveness were independent constructs that had independent effects on customer's attitude toward the brand and purchase intent for the new product (Lafferty and Goldsmith, 2004).

Customer Value Enhancement

M ( i1 +i2 +i3 +...in ) j

t A firm may introduce new product with high investment in terms of product attributes (i1), distribution (i2), promotion (i3) and other related factors (…in) related with gaining competitive advantage in a given time (t) in the jth market. Let us assume that s is the

V

estimated market coverage for the new product, the customer value ( np ) may be initially positive and high, resulting into deeper market penetration (with s increasing). This may be described as:

M t(i1 +i2 +i3 +...) j =

However, Vnp ≤

∂s =k ∂t

(17)

∂v may become negative following product competition within the ∂t

product line due to the product overlap strategy of the firm. In the above equation, volume of buying is represented by ∂v in a given time t. To augment the customer value and enhance market coverage for the new products in potential markets the firm may optimize the product line [s ] pt by pruning the slow moving products in the jth chain in h market in order to j ,h

reposition them in new market. The opportunity cost in moving the slow performance products may be derived by inputting the values of V´ from equation (ii) as:

[s ]

j ,h Pt

⎡ ∂v ⎤ =⎢ ⎥ ⎣ ∂t ⎦

j ,h

+ ∏{V ( x, t , q, p )} (18)

Hence to enhance market coverage for the new product with enhancing customer value for the new product of the firm, the strategy may be described as:

[

]

s = ∫ k + {s}Pt ∂t + β t R j ,h

(19)

Where in s is market coverage of the new product, k is investment on market functions derived in a given time [equation (vi)] and R is the referral factor influencing the customer values with an advantage factor coefficient

β in time t. The products constituting the optimal

Modeling Asymmetric Consumer Behavior and Demand Equations…

173

P

product line of the firm in a given time are represented by t in the above equation. The firm may measure the customer value shocks accordingly and shield the uncertainties occurring to the estimated market coverage due to declining customer values for the new products. As is common the new products are susceptible to such value shocks in view of the companies’ own product line strategy. Let us assume that new product attractiveness is

Fx and initial product market Vnp

M t( i1 +i2 +i3 +...in ) j , perceived customer value of the new product is and C competitive advantage driver for the customer is at at a given time, we get the following investment is

equation.

[

]

Fx = ∑t M t(i1 + i2 +...in ) j (Vnp )(C at ) jh

(20)

Hence, the above equation may be further simplified as,

Fx = M tin , j

Where in

∂v ′ ∂b ′ ∂s ∂v (Vnp )(C at ) = M tin , j = M tin, j ∂s ∂t ∂s ∂t

(21)

M ti n , j denotes the initial investment made by the firm for introducing new

products, V ′ represents the volume of penetration of new product in a given market in time t

with estimated market coverage s and b′ expresses the volume of repeat buying during the period the new product was penetrated in the market by the firm. The total quality for new products goes up due to economy of scale as the quality is also increased simultaneously

∂ s > 0) and (∂ b′ ∂ s > 0) . In reference to the new products x, competitive products (∂ ∂ < 0) while innovative products irrespective of create lower values to the customers v x

(∂ v

(∂

∂ > 0)

price advantages, enhance the customer value v x . Value addition in the competitive products provides lesser enhancement in customer satisfaction as compared to the innovative products if the new products have faster penetration, re-buying attributes and market coverage.

∫ s∂s = ∫ V

Therefore

np

+ C at

(22)

V

C

at In the above equation np denotes customer value for the new product and represents the competitive advantage at a given time. The prospect theory laid by Tversky and Khanman (1981) proposes that the intensity of

G =g

(∂

∂

)

pt x p gains plays strategic role in value enhancement as xt . In this situation t represents the period wherein promotional strategies were implemented to enhance customer

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Rajagopal

g

values in reference to product specific gains ( pt ). However, in order to measure relationship/variability between repeat buying behavior and customer value, it would be appropriate to determine the cumulative decision weights ( w ) and substitute it in the equation (1) to get the following equation:

[

G xt = w∑ g pt (r j m j ) + β n +1Qt jh

k =1

] (23)

The customer value however may be the driver function of gains on buying decision on new products and the influencing variables such as perceived use value and referrals.

Conclusion and Managerial Implications Existing theoretical and methodological issues are reviewed in this study and a new framework has been proposed for future research in measuring customer value in specific reference to the new products as launching innovative and high technology products is a continuous process for the firms in the present competitive markets. The assumptions and methodology employed in this paper are quite different from those of conventional behavioral models. It would be interesting to analyze the robustness of convergence of the model in reference to asymmetric behavioral patterns of consumer and demand orientation. Some of the existing empirical evidences, both from laboratory and field consumer data, seem to give enhanced scope for analyzing the asymmetric behavior model discussed in the paper that predicts suboptimal consumer behavior even in the long run in view of shifts in retailing strategies. The framework for measuring customer values discussed in this paper provides analytical dimensions for establishing the long run customer relationship by the firm and to optimize its profit levels. To test some hypotheses concerning homogeneity in consumer behavior and its impact on derived demand for the product or services may be required to be taken-up with additional parameters to estimate model. While following the LES data to estimate demand functions, the results may not aggregate over consumers’ asymmetric behavior, particularly in reference to expenditures. Hence, individual expenditures on goods over a group of consumers of varying incomes produce different equations with income when replaced by utility based qualitative parameters. However, it is believed that utility maximization, and the consequent constraints on demand equations, pertain strictly to individuals and consequently consider their application at aggregate level provided the demand equations aggregate over consumers qualitative parameters. It is necessary to standardize the qualitative parameters before applying in this model. One of the challenges for a firm is to incorporate and validate the preferences of customers into the design of new products and services in order to maximize customer value. An augmented and sustainable customer value builds loyalty towards the product and brand and helps to stabilize customer behavior. Systematically explored concepts in the field of customer value and market driven approach towards new products would be beneficial for a company to derive long term profit optimization strategy over the period. Hence, a comprehensive framework for estimating both the value of a customer and profit optimization

Modeling Asymmetric Consumer Behavior and Demand Equations…

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need to be developed. On a tactical level, managers need to consider as what is the optimum spread of customers on a matrix of product attractiveness and market coverage. The model discussed in this paper provides a holistic view of the customer behavior driven by the value matrices associated with product attractiveness, market coverage, brand and point-ofpurchase services offered to the customers. Analysis of these variables would help strategists/managers to develop appropriate strategies to enhance customer value for the new products and optimize profit of the firm.

References Babakus E, Bienstock C C and Van Scotter J R (2004), Linking perceived quality and customer satisfaction to store traffic and revenue growth, Decision Sciences, 35 (4), 713737 Bergemann D, Välimäki J (1996) Learning and strategic pricing, Econometrica, 64, 11251149 Brucks M, Zeithaml V A and Naylor G (2000), Price and brand name as indicators of quality dimensions of customer durables, Journal of Academy of Marketing Science, 28 (3), 359-374 Carroll D and Dunn W E (1997), Unemployment expectations, NBER, Working Paper # 6081 Carroll C D and Kimball M S (1996), On the concavity of the consumption function, Econometrics, 64(4), 981-992 Chintagunta, P.K., Rao, P.V. (1996), Pricing strategies in a dynamic duopoly: a differential game model, Management Science, 42, 1501-14 Davies A (2004), Moving base into high-value integrated solutions: A value stream approach, Industrial and Corporate Change, 13 (5), October, 727-756 Deaton A (1992), Understanding competition, Oxford University Press, Oxford Erev I and Barron G (2001), On adaptation, maximization and reinforcement learning among cognitive strategies, Working Paper, Columbia University Erev I and Roth A E(1998), Predicting how people play games: reinforcement learning in experimental games with unique, mixed strategy equilibria, American Economic Review, 88, 848-881 Gramrowski B and Rachev S (1999), A testable version of the pareto-stable CAPM, Mathematical and Computer Modeling, Vol. 29, 61-81. Herrnstein, R and Dražen P (1992), “Melioration,” in George L and Jon E (Eds), Choice over time, New York: Russell Sage, 235-263. Loewenstein, G (1996), Out of control: Visceral influences on behavior, Organizational behavior and Human Decision Processes, 65 (March), 272-292. Marjolein C and Verspagen B (1999), Spatial distance in a technology gap model, Maastricht Economic Research Institute on Innovation and Technology (MERIT), Working Paper No. 021 Morschett, D; Swoboda, B and Foscht, T (2005), Perception of store attributes and overall attitude towards grocery retailers: The role of shopping motives, The International Review of Retail, Distribution and Consumer Research, 15 (4), 423-447 Ofek E (2002), Customer profitability and lifetime value, Harvard Business School, Note, August, 1-9 (Publication reference 9-503-019)

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Pederson C S(2000), Sparsing risk and return in CAPM: A general utility based model, European Journal of Operational Research, Vol. 123 (3), 628-639. Perraudin W R M and Sorensen B E (2000), The demand of risky assets: Sample selection and household portfolios, Journal of Econometrics, Vol. 97, 117-144 Rajagopal (2006a), Measuring customer value gaps: An empirical study in mexican retail markets, Economic Issues, 11(1), March, 19-40 Rajagopal (2006b), Measuring customer value and market dynamics for newproducts of a firm: An analytical construct for gaining competitive advantage, Global Business and Economics Review, 8 (3-4), 187-205 Reichheld F F and Sasser W E (1990), Zero defections: Quality comes to services, Harvard Business Review, Sep-Oct, pp 105-111 Shida M (2000), Fundamental theorems of morse theory for optimization on manifolds with corners, Journal of Optimization Theory and Applications, 106 (3), September, 683-688 Tversky A and Kahnman D (1981), The framing decisions and psychology of hoice, Science, No.211, 453-458

In: Advances in Mathematics Research, Volume 8 Editor: Albert R. Baswell, pp. 177-246

ISBN: 978-1-60456-454-9 © 2009 Nova Science Publishers, Inc.

Chapter 7

HIGHER EDUCATION: FEDERAL SCIENCE, TECHNOLOGY, ENGINEERING, AND MATHEMATICS PROGRAMS AND RELATED TRENDS* United States Government Accountability Office

Why This Study? The United States has long been known as a world leader in scientific and technological innovation. To help maintain this advantage, the federal government has spent billions of dollars on education programs in the science technology, engineering, and mathematics (STEM) fields for many years. However, concerns have been raised about the nation’s ability to maintain its global technological competitive advantage in the future. This report presents information on(1) the number of federal programs funded in fiscal year 2004 that were designed to increase the number of students and graduates pursuing STEM degrees and occupations or improve educational programs in STEM fields, and what agencies report about their effectiveness; (2) how the numbers, percentages, and characteristics of students, graduates, and employees in STEM fields have changed over the years; and (3) factors cited by educators and others as affecting students’ decisions about pursing STEM degrees and occupations, and suggestions that have been made to encourage more participation. GAO received written and/or technical comments from several agencies. While one agency, the National Science Foundation, raised several questions about the findings, the others generally agreed with the findings and conclusion and several agencies commended GAO for this work.

*

Excerpted from http://www.gao.gov/new.items/d06114.pdf

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Abbreviations BEST BLS CGS CLF COS CPS DHS EPA HHS HRSA IPEDS NASA NCES NCLBA NIH NPSAS NSF NSTC SAO SEVIS STEM

Building Engineering and Science Talent Bureau of Labor Statistics Council of Graduate Schools civilian labor force Committee on Science Current Population Survey Department of Homeland Security Environmental Protection Agency Health and Human Services Health Resources and Services Administration Integrated Postsecondary Education Data System National Aeronautics and Space Administration National Center for Education Statistics No Child Left Behind Act National Institutes of Health National Postsecondary Student Aid Study National Science Foundation National Science and Technology Council Security Advisory Opinion Student and Exchange Visitor Information System science, technology, engineering, and mathematics

The United States has long been known as a world leader in scientific and technological innovation. To help maintain this advantage, the federal government has spent billions of dollars on education programs in the science, technology, engineering, and mathematics (STEM) fields for many years. Some of these programs were designed to increase the numbers of women and minorities pursuing degrees in STEM fields. In addition, for many years, thousands of international students came to the United States to study and work in STEM fields. However, concerns have been raised about the nation’s ability to maintain its global technological competitive advantage in the future. In spite of the billions of dollars spent to encourage students and graduates to pursue studies in STEM fields or improve STEM educational programs, the percentage of United States students earning bachelor’s degrees in STEM fields has been relatively constant—about a third of bachelor’s degrees— since 1977. Furthermore, after the events of September 11, 2001, the United States established several new systems and processes to help enhance border security. In some cases, implementation of these new systems and processes, which established requirements for several federal agencies, higher education institutions, and potential students, made it more difficult for international students to enter this country to study and work. In the last few years, many reports and news articles have been published, and several bills have been introduced in Congress that address issues related to STEM education and occupations. This report presents information on (1) the number of federal civilian education programs funded in fiscal year 2004 that were designed to increase the numbers of students and graduates pursuing STEM degrees and occupations or improve educational programs in

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STEM fields and what agencies report about their effectiveness; (2) how the numbers, percentages, and characteristics of students, graduates, and employees in STEM fields have changed over the years; and (3) factors cited by educators and others as influencing people’s decisions about pursuing STEM degrees and occupations, and suggestions that have been made to encourage greater participation in STEM fields. To determine the number of programs designed to increase the numbers of students and graduates pursuing STEM degrees and occupations, we identified 15 federal departments and agencies as having STEM programs, and we developed and conducted a survey asking each department or agency to provide information on its education programs, including information about their effectiveness [1]. We received responses from 14 of them, the Department of Defense did not participate, and we determined that at least 13 agencies had STEM education programs during fiscal year 2004 that met our criteria. To describe how the numbers of students, graduates, and employees in STEM fields have changed, we analyzed and reported data from the Department of Education’s (Education) National Center for Education Statistics (NCES) and the Department of Labor’s (Labor) Bureau of Labor Statistics (BLS). Specifically, as shown in table 1, we used the National Postsecondary Student Aid Study (NPSAS) and the Integrated Postsecondary Education Data System (IPEDS) from NCES and the Current Population Survey (CPS) data from BLS. We assessed the data for reliability and reasonableness and found them to be sufficiently reliable for the purposes of this report. To obtain perspectives on the factors that influence people’s decisions about pursuing STEM degrees and occupations, and to obtain suggestions for encouraging greater participation in STEM fields, we interviewed educators and administrators in eight colleges and universities (the University of California Los Angeles and the University of Southern California in California; Clark Atlanta University, Georgia Institute of Technology, and Spelman College in Georgia; the University of Illinois; Purdue University in Indiana; and Pennsylvania State University). We selected these colleges and universities to include a mix of public and private institutions, provide geographic diversity, and include a few minorityserving institutions, including one (Spelman College) that serves only women students. Table 1. Sources of Data, Data Obtained, Time Span of Data, and Years Analyzed Department

Agency

Database

Data obtained

Education

NCES

NPSAS

College student enrollment Graduation/degrees

Time span of data 9 years

Years analyzed

Academic years 1995-1996 and 2003-2004 Education NCES IPEDS 9 years Academic years 1994-1995 and 2002-2003 Labor BLS CPS Employment 10 years Calendar years 1994 through 2003 Sources: NCES’s National Postsecondary Student Aid Study (NPSAS) and Integrated Postsecondary Education Data System (IPEDS) and BLS’s Current Population Survey (CPS) data. Note: Enrollment and employment information are based on sample data and are subject to sampling error. The 95-percent confidence intervals for student enrollment estimates are contained in appendix V of this report. Percentage estimates for STEM employment have 95-percent confidence intervals of within +/- 6 percentage points and other employment estimates (such as wages and salaries) have confidence intervals of within +/- 10 percent of the estimate, unless otherwise noted. See appendixes I, V, and VI for additional information.

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In addition, most of the institutions had large total numbers of students, including international students, enrolled in STEM fields. We also asked officials from the eight universities to identify current students to whom we could send an e-mail survey. We received responses from 31 students from five of these institutions. In addition, we interviewed federal agency officials and representatives from associations and education organizations, and analyzed reports on various topics related to STEM education and occupations. Appendix I contains a more detailed discussion of our scope and methodology. We conducted our work between October 2004 and October 2005 in accordance with generally accepted government auditing standards.

Results in Brief Officials from 13 federal civilian agencies reported having 207 education programs funded in fiscal year 2004 that were designed to increase the numbers of students and graduates pursuing STEM degrees and occupations or improve educational programs in STEM fields, but they reported little about the effectiveness of these programs. The 13 agencies reported spending about $2.8 billion in fiscal year 2004 for these programs. According to the survey responses, the National Institutes of Health (NIH) and the National Science Foundation (NSF) sponsored 99 of the 207 programs and spent about $2 billion of the approximate $2.8 billion. The program costs ranged from $4,000 for a national scholars program sponsored by the Department of Agriculture (USDA) to about $547 million for an NIH program that is designed to develop and enhance research training opportunities for individuals in biomedical, behavioral, and clinical research by supporting training programs at institutions of higher learning. Officials reported that most of the 207 programs had multiple goals, and many were targeted to multiple groups. For example, 2 programs were identified as having one goal of attracting and preparing students at any education level to pursue coursework in STEM areas, while 112 programs had this as one of multiple goals. Agency officials also reported that evaluations were completed or under way for about half of the programs, and most of the completed evaluations reported that the programs had been effective and achieved established goals. However, some programs that have not been evaluated have operated for many years. While the total numbers of students, graduates, and employees have increased in STEM fields, changes in the numbers and percentages of women, minorities and international students varied during the periods reviewed. From the 1995-1996 academic year to the 20032004 academic year, the number of students increased in STEM fields by 21 percent—more than the 11 percent increase in non-STEM fields. Also, students enrolled in STEM fields increased from 21 percent to 23 percent of all students. Changes in the numbers and percentages of domestic minority students varied by group. For example, the number of African American students increased 69 percent and the number of Hispanic students increased 33 percent. The total number of graduates in STEM fields increased by 8 percent from the 1994-1995 academic year to the 2002-2003 academic year, while graduates in nonSTEM fields increased 30 percent. Further, the numbers of graduates decreased in at least four of eight STEM fields at each education level. The total number of domestic minority graduates in STEM fields increased, and international graduates continued to earn about onethird or more of the master’s and doctoral degrees in three fields. Moreover, from 1994 to

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2003, employment increased by 23 percent in STEM fields as compared with 17 percent in non-STEM fields. African American employees continued to be less than 10 percent of all STEM employees, and there was no statistically significant change in the percentage of women employees. Educators and others cited several factors as influencing students’ decisions about pursuing STEM degrees and occupations, and they suggested many ways to encourage more participation in STEM fields. Studies, education experts, university officials, and others cited teacher quality at the kindergarten through 12th grade levels and students’ high school preparation in mathematics and science courses as major factors that influence domestic students’ decisions about pursuing STEM degrees and occupations. In addition, university officials, students, and studies identified mentoring as a key factor for women and minorities. Also, according to university officials, education experts, and reports, international students’ decisions about pursuing STEM degrees and occupations in the United States are influenced by yet other factors, including more stringent visa requirements and increased educational opportunities outside the United States. We have reported that several aspects of the visa process have been improved, but further steps could be taken. In order to promote participation in the STEM fields, officials at most of the eight universities visited and current students offered suggestions that focused on four areas: teacher quality, mathematics and science preparation and courses, outreach to underrepresented groups, and the federal role in STEM education. The students who responded to our e-mail survey generally agreed with most of the suggestions and expressed their desires for better mathematics and science preparation for college. However, before adopting such suggestions, it is important to know the extent to which existing STEM education programs are appropriately targeted and making the best use of available federal resources. We received written comments on a draft of this report from the Department of Commerce, the Department of Health and Human Services, and the National Science and Technology Council. These agencies generally agreed with our findings and conclusions. We also received written comments from the National Science Foundation which questioned our findings related to program evaluations, interagency collaboration, and the methodology we used to support our findings on the factors that influenced decisions about pursing STEM fields. Also, the National Science Foundation provided information to clarify examples cited in the report, stated that the data categories were not clear, and commented on the graduate level enrollment data we used. We revised the report to acknowledge that the National Science Foundation uses a variety of mechanisms to evaluate its programs and we added a bibliography that identifies the reports and research used during the course of this review to address the comment about our methodology related to the factors that influenced decisions about pursuing STEM fields. We also revised the report to clarify the examples and the data categories and to explain the reasons for selecting the enrollment data we used. However, we did not make changes to address the comment related to interagency collaboration for the reason explained in the agency comments section of this report. The written comments are reprinted in appendixes VII, VIII, IX, and X. In addition, we received technical comments from the Departments of Commerce, Health and Human Services, Homeland Security, Labor, and Transportation, and the Environmental Protection Agency and National Aeronautics and Space Administration, which we incorporated when appropriate.

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Background STEM includes many fields of study and occupations. Based on the National Science Foundations’ categorization of STEM fields, we developed STEM fields of study from NCES’s National Postsecondary Student Aid Study (NPSAS) and Integrated Postsecondary Education Data System (IPEDS), and identified occupations from BLS’s Current Population Survey (CPS). Using these data sources, we developed nine STEM fields for students, eight STEM fields for graduates, and four broad STEM fields for occupations. Table 2 lists these STEM fields and occupations and examples of subfields. Additional information on STEM occupations is provided in appendix I. Many of the STEM fields require completion of advanced courses in mathematics or science, subjects that are introduced and developed at the kindergarten through 12th grade level, and the federal government has taken steps to help improve achievement in these and other subjects. Enacted in 2002, the No Child Left Behind Act (NCLBA) seeks to improve the academic achievement of all of the nation’s school-aged children. NCLBA requires that states develop and implement academic content and achievement standards in mathematics, science and the reading or language arts. All students are required to participate in statewide assessments during their elementary and secondary school years. Improving teacher quality is another goal of NCLBA as a strategy to raise student academic achievement. Specifically, all teachers teaching core academic subjects must be highly qualified by the end of the 20052006 school year [2]. NCLBA generally defines highly qualified teachers as those that have (1) a bachelor’s degree, (2) state certification, and (3) subject area knowledge for each academic subject they teach. The federal government also plays a role in coordinating federal science and technology issues. The National Science and Technology Council (NSTC) was established in 1993 and is the principal means for the Administration to coordinate science and technology among the diverse parts of the federal research and development areas. One objective of NSTC is to establish clear national goals for federal science and technology investments in areas ranging from information technologies and health research to improving transportation systems and strengthening fundamental research. NSTC is responsible for preparing research and development strategies that are coordinated across federal agencies in order to accomplish these multiple national goals. In addition, the federal government, universities and colleges, and others have developed programs to provide opportunities for all students to pursue STEM education and occupations [3]. Additional steps have been taken to increase the numbers of women, minorities, and students with disadvantaged backgrounds in the STEM fields, such as providing additional academic and research opportunities. According to the 2000 Census, 52 percent of the total U.S. population 18 and over were women; in 2003, members of racial or ethnic groups constituted from 0.5 percent to 12.6 percent of the civilian labor force (CLF), as shown in table 3. In addition to domestic students, international students have pursued STEM degrees and worked in STEM occupations in the United States. To do so, international students and scholars must obtain visas [4]. International students who wish to study in the United States must first apply to a Student and Exchange Visitor Information System (SEVIS) certified school. In order to enroll students from other nations, U.S. colleges and universities must be

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certified by the Student and Exchange Visitor Program within the Department of Homeland Security’s Immigration and Customs Enforcement organization. Table 2. List of STEM Fields Based on NCES’s NPSAS and IPEDS Data and BLS’s CPS Data Enrollment–NCES’ NPSAS data Agricultural sciences

Biological sciences

Degrees–NCES’IP EDS data

Occupations–BLS’ CPS data

Biological/agricultural sciences Botany Zoology Dairy Forestry Poultry Wildlife management

Science Agricultural and food scientists Astronomers and physicists Atmospheric and space scientists Biological scientists Chemists and materials scientists Environmental scientists and geoscientists Nurses Psychologists Sociologists Urban and regional planners

Earth, atmospheric, and ocean sciences Geology Geophysics and seismology Physical sciences

Psychology

P sychol

Social sciences

Technology

T echnol

E ngineerin

Computer sciences Mathematics

ogy Clinical Social Social sciences Political science Sociology

Enrollment–NCES’ NPSAS data Engineering

Physical sciences Chemistry Physics

ogy Solar Automotive engineering

Degrees–NCES’IP EDS data

g Aerospace, aeronautical, and astronautical Architectural Chemical Civil Electrical, electronics, and communication Nuclear Mathematics/computer sciences Actuarial science Applied mathematics Mathematical statistics Operations research Data processing Programming

Technology Clinical laboratory technologists and technicians Diagnostic-related technologists and technicians Medical, dental, and ophthalmic laboratory technicians

Occupations–BLS’ CPS data Engineering Architects, except naval Aerospace engineers Chemical engineers Civil engineers Electrical and electronic engineers Nuclear engineers

Mathematics and computer sciences Computer scientists and systems analysts Computer programmers Computer software engineers Actuaries Mathematicians Statisticians

Sources: NCES for NPSAS and IPEDS data; CPS for occupations. Note: This table is not designed to show a direct relationship from enrollment to occupation, but to provide examples of majors, degrees, and occupations in STEM fields from the three sources of data.

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Table 3. Percentage of the U.S. Population for Selected Racial or Ethnic Groups in the Civilian Labor Force, Calendar Years 1994 and 2003 Percentage of U.S. population Percentage of U.S. population in the CLF, 1994 in the CLF, 2003 Race or ethnicity Hispanic or Latino origin Black or African American Asian American Indian or Alaska Native

8.9 10.8 2.8 0.5

12.6 10.7 4.4 0.5

Source: GAO calculations based upon March 1994 and March2003 CPS data.

As of February 2004, nearly 9,000 technical schools and colleges and universities had been certified. SEVIS, is an Internet-based system that maintains data on international students and exchange visitors before and during their stay in the United States. Upon admitting a student, the school enters the student’s name and other information into the SEVIS database. At this time the student may apply for a student visa. In some cases, a Security Advisory Opinion (SAO) from the Department of State (State) may be needed to determine whether or not to issue a visa to the student. SAOs are required for a number of reasons, including concerns that a visa applicant may engage in the illegal transfer of sensitive technology. An SAO based on technology transfer concerns is known as Visas Mantis and, according to State officials, is the most common type of SAO applied to science applicants [5]. In April 2004, the Congressional Research Service reported that State maintains a technology alert list that includes 16 sensitive areas of study. The list was produced in an effort to help the United States prevent the illegal transfer of controlled technology and includes chemical and biotechnology engineering, missile technology, nuclear technology, robotics, and advanced computer technology [6]. Many foreign workers enter the United States annually through the H-1B visa program, which assists U.S. employers in temporarily filling specialty occupations [7]. Employed workers may stay in the United States on an H-1B visa for up to 6 years. The current cap on the number of H-1B visas that can be granted is 65,000. The law exempts certain workers, however, from this cap, including those who are employed or have accepted employment in specified positions. Moreover, up to 20,000 exemptions are allowed for those holding a master’s degree or higher.

More than 200 Federal Education Programs are Designed to Increase the Numbers of Students and Graduates or Improve Educational Programs in STEM Fields, but Most Have Not Been Evaluated Officials from 13 federal civilian agencies reported having 207 education programs funded in fiscal year 2004 that were specifically established to increase the numbers of students and graduates pursuing STEM degrees and occupations, or improve educational programs in STEM fields, but they reported little about the effectiveness of these programs [8]. These 13 federal agencies reported spending about $2.8 billion for their STEM education

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programs. Taken together, NIH and NSF sponsored nearly half of the programs and spent about 71 percent of the funds. In addition, agencies reported that most of the programs had multiple goals, and many were targeted to multiple groups. Although evaluations have been done or were under way for about half of the programs, little is known about the extent to which most STEM programs are achieving their desired results. Coordination among the federal STEM education programs has been limited. However, in 2003, the National Science and Technology Council formed a subcommittee to address STEM education and workforce policy issues across federal agencies.

Federal Civilian Agencies Reported Sponsoring over 200 STEM Education Programs and Spending Billions in Fiscal Year 2004 Officials from 13 federal civilian agencies provided information on 207 STEM education programs funded in fiscal year 2004. The numbers of programs ranged from 51 to 1 per agency with two agencies, NIH and NSF, sponsoring nearly half of the programs—99 of 207. Table 4 provides a summary of the numbers of programs by agency, and appendix II contains a list of the 207 STEM education programs and funding levels for fiscal year 2004 by agency. Table 4. Number of STEM Education Programs Reported by Federal Civilian Agencies Federal agency Department of Health and Human Services/ National Institutes of Health National Science Foundation Department of Energy Environmental Protection Agency Department of Agriculture Department of Commerce Department of the Interior National Aeronautics and Space Administration Department of Education Department of Transportation Department of Health and Human Services/Health Resources and Services Administration Department of Health and Human Services/Indian Health Service Department of Homeland Security Total

Source: GAO survey responses from 13 federal agencies.

Number of STEM education programs 51 48 26 21 16 13 13 5 4 4 3 2 1 207

Federal civilian agencies reported that approximately $2.8 billion was spent on STEM education programs in fiscal year 2004 [9]. The funding levels for STEM education programs among the agencies ranged from about $998 million to about $4.7 million. NIH and NSF accounted for about 71 percent of the total—about $2 billion of the approximate $2.8 billion. NIH spent about $998 million in fiscal year 2004, about 3.6 percent of its $28 billion

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appropriation, and NSF spent about $997 million, which represented 18 percent of its appropriation. Four other agencies, some with a few programs, spent about 23 percent of the total: $636 million. For example, the National Aeronautics and Space Administration (NASA) spent about $231 million on 5 programs and the Department of Education (Education) spent about $221 million on 4 programs during fiscal year 2004. Figure 1 shows the 6 federal civilian agencies that used the most funds for STEM education programs and the funds used by the remaining 7 agencies. Dollars in millions 1,200 998

1,000

997

800 600 400 231

221

200

154

121 63

0

s s h e n n n y d n d on er t e al t io io c nc io an atio an ati at th ct gen ie dat itu He s r r s c e t t c t t e u ic s llO ns of rc nis ro A lS un ut ini A Ed ou mi na F o lP alI na dm s a o n o d i t e io er A at R sA en A e at N m N lth ice al pac a ro n i e v v io S H er at En S N

Figure 1. Amounts Funded by Agencies for STEM-Related Federal Education Programs in Fiscal Year 2004.

Table 5. Funding Levels for Federal STEM Education Programs in Fiscal Year 2004

Program funding levels Less than $1 million $1 million to $5 million $5.1 million to $10 million $10.1 million to $50 million More than $50 million Total

Numbers of STEM education programs 93 51 19 31 13 207

Source: GAO survey responses from 13 federal agencies.

Percentage of total STEM education programs 45 25 9 15 6 100

The funding reported for individual STEM education programs varied significantly, and many of the programs have been funded for more than 10 years. The funding ranged from $4,000 for an USDA-sponsored program that offered scholarships to U.S. citizens seeking bachelor’s degrees at Hispanic-serving institutions, to about $547 million for a NIH grant program that is designed to develop and enhance research training opportunities for individuals in biomedical, behavioral, and clinical research by supporting training programs at institutions of higher education. As shown in table 5, most programs were funded at $5

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million or less and 13 programs were funded at more than $50 million in fiscal year 2004. About half of the STEM education programs were first funded after 1998. The oldest program began in 1936, and 72 programs are over 10 years old [10]. Appendix III describes the STEM education programs that received funding of $10 million or more during fiscal year 2004 or 2005 [11].

Federal Agencies Reported Most STEM Programs Had Multiple Goals and Were Targeted to Multiple Groups Agencies reported that most of the STEM education programs had multiple goals. Survey respondents reported that 80 percent (165 of 207) of the education programs had multiple goals, with about half of these identifying four or more goals for individual programs [12]. Moreover, according to the survey responders, few programs had a single goal. For example, 2 programs were identified as having one goal of attracting and preparing students at any education level to pursue coursework in the STEM areas, while 112 programs identified this as one of multiple goals. Table 6 shows the program goals and numbers of STEM programs aligned with them. Table 6. Program Goals and Numbers of STEM Programs with One or Multiple Goals Program goal

Programs with only this goal

Attract and prepare students at any education level to pursue coursework in STEM areas Attract students to pursue degrees (2-year through Ph.D.) and postdoctoral appointments Provide growth and research opportunities for college and graduate students in STEM fields Attract graduates to pursue careers in STEM fields Improve teacher education in STEM areas Improve or expand the capacity of institutions to promote or foster STEM fields Source: GAO survey responses from 13 federal agencies

2

Programs with Total programs multiple goals with this goal including this goal and other goal(s) 112 114

6

131

137

3

100

103

17 8 3

114 65 87

131 73 90

The STEM education programs provided financial assistance to students, educators, and institutions. According to the survey responses, 131 programs provided financial support for students or scholars, and 84 programs provided assistance for teacher and faculty development [13]. Many of the programs provided financial assistance to multiple beneficiaries, as shown in table 7. Most of the programs were not targeted to a specific group but aimed to serve a wide range of students, educators, and institutions. Of the 207 programs, 54 were targeted to 1 group and 151 had multiple target groups [14] In addition, many programs were targeted to the same group. For example, while 12 programs were aimed solely at graduate students, 88 other programs had graduate students as one of multiple target groups. Fewer programs were targeted to elementary and secondary teachers and kindergarten through 12th grade students

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than to other target groups. Table 8 summarizes the numbers of STEM programs targeted to one group and multiple groups. Table 7. Numbers of STEM Programs with One or Multiple Types of Assistance and Beneficiaries Type of assistance

Programs that Programs that provide Total programs provide only this this type and other that provide this type of assistance types of assistance type of assistance 54 77 131

Financial support for students or scholars Institutional support to improve educational quality 6 Support for teacher and faculty development 12 Institutional physical infrastructure support 1 Source: GAO survey responses from 13 federal agencies

70 72 26

76 84 27

Table 8. Numbers of STEM Programs Targeted to One Group and Multiple Groups Targeted to Targeted to this Total programs Targeted group only this group and other groups targeted to this group Kindergarten through grade 12 students Elementary school students 0 28 28 Middle or junior high school students 1 33 34 High school students 3 50 53 Undergraduate students 2-year college students 1 57 58 4-year college students 4 92 96 Graduate students and postdoctoral scholars Graduate students 12 88 100 Postdoctoral scholars 12 58 70 Teachers, college faculty and instructional staff Elementary school teachers 0 39 39 Secondary school teachers 3 47 50 College faculty or instructional staff 4 75 79 Institutions 5 77 82 Source: GAO survey responses from13 federal agencies.

Some programs limited participation to certain groups. According to survey respondents, U.S. citizenship was required to be eligible for 53 programs, and an additional 75 programs were open only to U.S. citizens or permanent residents [15]. About one-fourth of the programs had no citizenship requirement, and 24 programs allowed noncitizens or permanent residents to participate in some cases. According to a NSF official, students receiving scholarships or fellowships through NSF programs must be U.S. citizens or permanent residents. In commenting on a draft of this report, NSF reported that these restrictions are considered to be an effective strategy to support its goal of creating a diverse, competitive, and globally-engaged U.S. workforce of scientists, engineers, technologists, and wellprepared citizens. Officials at two universities said that some research programs are not open to non-citizens. Such restrictions may reflect concerns about access to sensitive areas. In addition to these restrictions, some programs are designed to increase minority representation in STEM fields. For example, NSF sponsors a program called Opportunities for Enhancing

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Diversity in the Geosciences to increase participation by African Americans, Hispanic Americans, Native Americans (American Indians and Alaskan Natives), Native Pacific Islanders (Polynesians or Micronesians), and persons with disabilities.

Agency Officials Reported That Evaluations Were Completed or under Way for about Half of the Federal Programs Evaluations had been completed or were under way for about half of the STEM education programs. Agency officials responded that evaluations were completed for 55 of the 207 programs and that for 49 programs, evaluations were under way at the time we conducted our survey. Agency officials provided us documentation for evaluations of 43 programs, and most of the completed evaluations reviewed reported that the programs met their objectives or goals. For example, a March 2004 report on the outcomes and impacts of NSF’s Minority Postdoctoral Research Fellowships program concluded that there was strong qualitative and quantitative evidence that this program is meeting its broad goal of preparing scientists from those ethnic groups that are significantly underrepresented in tenured U.S. science and engineering professorships and for positions of leadership in industry and government. However, evaluations had not been done for 103 programs, some of which have been operating for many years. Of these, it may have been too soon to expect evaluations for about 32 programs that were initially funded in fiscal year 2002 or later. However, of the remaining 71 programs, 17 have been operating for over 15 years and have not been evaluated. In commenting on a draft of this report NSF noted that all of its programs undergo evaluation and that it uses a variety of mechanisms for program evaluation. We reported in 2003 that several agencies used various strategies to develop and improve evaluations [16]. Evaluations play an important role in improving program operations and ensuring an efficient use of federal resources. Although some of the STEM education programs are small in terms of their funding levels, evaluations can be designed to consider the size of the program and the costs associated with measuring outcomes and collecting data.

A Subcommittee Was Established in 2003 to Help Coordinate STEM Education Programs among Federal Agencies Coordination of federal STEM education programs has been limited. In January 2003 the National Science and Technology Council (NSTC), Committee on Science (COS), established a subcommittee on education and workforce development. The purpose of the subcommittee is to advise and assist COS and NSTC on policies, procedures, and programs relating to STEM education and workforce development. According to its charter, the subcommittee will address education and workforce policy issues and research and development efforts that focus on STEM education issues at all levels, as well as current and projected STEM workforce needs, trends, and issues. The members include representatives from 20 agencies and offices—the 13 agencies that responded to our survey as well as the Departments of Defense, State, and Justice, and the Office of Science and Technology Policy, the Office of Management and Budget, the Domestic Policy Council, and the National Economic Council. The subcommittee has working groups on (1) human capacity in STEM

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areas, (2) minority programs, (3) effective practices for assessing federal efforts, and (4) issues affecting graduate and postdoctoral researchers. The Human Capacity in STEM working group is focused on three strategic initiatives: defining and assessing national STEM needs, including programs and research projects; identifying and analyzing the available data regarding the STEM workforce; and creating and implementing a comprehensive national response that enhances STEM workforce development. NSTC reported that as of June 2005 the subcommittee had a number of accomplishments and projects under way that related to attracting students to STEM fields. For example, it has (1) surveyed federal agency education programs designed to increase the participation of women and underrepresented minorities in STEM studies; (2) inventoried federal fellowship programs for graduate students and postdoctoral fellows; and (3) coordinated the Excellence in Science, Technology, Engineering, and Mathematics Education Week activities, which provide an opportunity for the nation’s schools to focus on improving mathematics and science education. In addition, the subcommittee is developing a Web site for federal educational resources in STEM fields and a set of principles that agencies would use in setting levels of support for graduate and postdoctoral fellowships and traineeships.

Numbers of Students, Graduates, and Employees in STEM Fields Generally Increased, but Percentage Changes Varied While the total numbers of students, graduates, and employees have increased in STEM fields, percentage changes for women, minorities, and international students varied during the periods reviewed.

Students

Graduates

Employees

UNIVERSITY

1995-1996 to 2003-2004

1994-1995 to 2002-2003 Total increase in STEM was less than non-STEM

1994 - 2003

Percentage increase was greater in STEM than non-STEM

Decrease at doctoral level in most fields

Increase was greater in STEM than non-STEM

Increase mostly at bachelor’s and master’s level

Increase in percentages of women in most fields

No significant change in percentage of women

Increase in percentage of women

No change in percentages of minorities at master’s or doctoral levels

African Americans continued to be less than 10 percent of the total

International graduates continued to earn about one-third or more of master’s and Ph.D.s in three fields

Median annual wages and salaries increased in all fields

Increase in minority students but percentage changes varied by race/ethnicity Increase in international students at bachelor’s level

Source: GAO analysis of CPS, IPEDS, and NPSAS data; graphics in part by Art Explosion.

Figure 2. Key Changes in Students, Graduates, and Employees in STEM Fields.

The increase in the percentage of students in STEM fields was greater than the increase in non-STEM fields, but the change in percentage of graduates in STEM fields was less than the percentage change in non-STEM fields. Moreover, employment increased more in STEM fields than in non-STEM fields. Further, changes in the percentages of minority students

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varied by race or ethnic group, international graduates continued to earn about a third or more of the advanced degrees in three STEM fields, and there was no statistically significant change in the percentage of women employees. Figure 2 summarizes key changes in the students, graduates, and employees in STEM fields.

Numbers of Students in STEM Fields Grew, but This Increase Varied by Education Level and Student Characteristics Total enrollments of students in STEM fields have increased, and the percentage change was greater for STEM fields than non-STEM fields, but the percentage of students in STEM fields remained about the same. From the 1995-1996 academic year to the 2003-2004 academic year, total enrollments in STEM fields increased 21 percent—more than the 11 percent enrollment increase in non-STEM fields. The number of students enrolled in STEM fields represented 23 percent of all students enrolled during the 2003-2004 academic year, a modest increase from the 21 percent these students constituted in the 1995-1996 academic year. Table 9 summarizes the changes in overall enrollment across all education levels from the 19951996 academic year to the 2003-2004 academic year. The increase in the numbers of students in STEM fields is mostly a result of increases at the bachelor’s and master’s levels. Of the total increase of about 865,000 students in STEM fields, about 740,000 was due to the increase in the numbers of students at the bachelor’s and master’s levels. See table 23 in appendix IV for additional information on the estimated numbers of students in STEM fields in academic years 1995-1996 and 2003-2004. Table 9. Estimated Changes in the Numbers and Percentages of Students in the STEM and Non-STEM Fields across All Education Levels, Academic Years 1995-1996 and 2003-2004 Academic year1995-1996 STEM Non- STEM

Academic year 2003-2004 STEM Non- STEM

Enrollment measures Students enrolled (in thousands) 4,132 15,243 4,997 16,883 Percentage of total enrollment 21 79 23 77 Source: GAO calculations based upon NPSAS data. Note: The totals for STEM and non-STEM enrollment include students in bachelor’s, master’s, and doctoral programs as well as students enrolled in certificate, associate’s, other undergraduate, first- professional degree, and post-bachelor’s or post-master’s certificate programs. The percentage changes between the 1995-1996 and 2003-2004 academic years for STEM and non-STEM students are statistically significant. See appendix V for confidence intervals associated with these estimates.

The percentage of students in STEM fields who are women increased from the 19951996 academic year to the 2003-2004 academic year, and in the 2003-2004 academic year women students constituted at least 50 percent of the students in 3 STEM fields—biological sciences, psychology, and social sciences. However, in the 2003-2004 academic year, men students continued to outnumber women students in STEM fields, and men constituted an estimated 54 percent of the STEM students overall. In addition, men constituted at least 76 percent of the students enrolled in computer sciences, engineering, and technology [17]. See tables 24 and 25 in appendix IV for additional information on changes in the numbers and

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percentages of women students in the STEM fields for academic years 1995-1996 and 20032004. While the numbers of domestic minority students in STEM fields also increased, changes in the percentages of minority students varied by racial or ethnic group. For example, Hispanic students increased 33 percent, from the 1995-1996 academic year to the 2003-2004 academic year. In comparison, the number of African American students increased about 69 percent. African American students increased from 9 to 12 percent of all students in STEM fields while Asian/Pacific Islander students continued to constitute about 7 percent. Table 10 shows the numbers and percentages of minority students in STEM fields for the 1995-1996 academic year and the 2003-2004 academic year. Table 10. Estimated Percentage Changes in the Numbers and Percentages of Domestic Minority Students in STEM fields for All Education Levels for Academic Years 19951996 and 2003-2004

Source: GAO calculations based upon NPSAS data. Note: All percentage changes are statistically significant. See appendix V for confidence intervals associated with these estimates.

From the 1995-1996 academic year to the 2003-2004 academic year, the number of international students in STEM fields increased by about 57 percent solely because of an increase at the bachelor’s level. The numbers of international students in STEM fields at the master’s and doctoral levels declined, with the largest decline occurring at the doctoral level. Table 11 shows the numbers and percentage changes in international students from the 19951996 academic year to the 2003-2004 academic year. Table 11. Estimated Changes in Numbers of International Students in STEM fields by Education Levels from the 1995-1996 Academic Year to the 2003-2004 Academic Year Percentage Number of international Number of international students, 1995-1996 students, 2003-2004 change Bachelor’s 31,858 139,875 +339 Master’s 40,025 22,384 -44 Doctoral 36,461 7,582 -79 Total 108,344 169,841 +57 Source: GAO calculations based upon NPSAS data. Note: Changes in enrollment between the 1995-1996 and 2003-2004 academic years are significant at the 95 percent confidence level for international students and for all education levels. See appendix V for confidence intervals associated with these estimates. Education level

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According to the Institute of International Education, from the 2002-2003 academic year to the 2003-2004 academic year, the number of international students declined for the first time in over 30 years, and that was the second such decline since the 1954-1955 academic year, when the institute began collecting and reporting data on international students [18]. Moreover, in November 2004, the Council of Graduate Schools (CGS) reported a 6 percent decline in first-time international graduate student enrollment from 2003 to 2004. Following a decade of steady growth, CGS also reported that the number of first-time international students studying in the United States decreased between 6 percent and 10 percent for 3 consecutive years.

Total Numbers of Graduates with STEM Degrees Increased, but Numbers Decreased in Some Fields, and Percentages of Minority Graduates at the Master’s and Doctoral Levels Did Not Change The number of graduates with degrees in STEM fields increased by 8 percent from the 19941995 academic year to the 2002-2003 academic year. However, during this same period the number of graduates with degrees in non-STEM fields increased by 30 percent. From academic year 1994-1995 to academic year 2002-2003, the percentage of graduates with STEM degrees decreased from 32 percent to 28 percent of total graduates. Table 12 provides data on the changes in the numbers and percentages of graduates in STEM and non-STEM fields. Table 12. Numbers of Graduates and Percentage Changes in STEM and Non-STEM Fields across All Degree Levels from the1994-1995 Academic Year to the 2002-2003 Academic Year STEM fields

Graduation measures

Non-STEM fields Percentage Percentage 1994-1995 2002-2003 1994-1995 2002-2003 change change 519 560 +8 1,112 1,444 +30 32 28 -4 68 72 +4

Graduates (in thousands) Percentage of total graduates Source: GAO calculations based upon IPEDS data.

Decreases in the numbers of graduates occurred in some STEM fields at each education level, but particularly at the doctoral level. The numbers of graduates with bachelor’s degrees decreased in four of eight STEM fields, the numbers with master’s degrees decreased in five of eight fields, and the numbers with doctoral degrees decreased in six of eight STEM fields. At the doctoral level, these declines ranged from 14 percent in mathematics/computer sciences to 74 percent in technology. Figure 3 shows the percentage change in graduates with degrees in STEM fields from the 1994-1995 academic year to the 2002-2003 academic year.

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Percent change 90 72

70 48

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-18

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-74

Bachelor’s Master’s Ph.D.s

Figure 3. Percentage Changes in Bachelor’s, Master’s, and Doctoral Graduates in STEM Fields from Academic Year 1994-1995 to Academic Year 2002-2003.

From the 1994-1995 academic year to the 2002-2003 academic year, the total number of women graduates increased in four of the eight fields, and the percentages of women earning degrees in STEM fields increased in six of the eight fields at all three educational levels. Conversely, the total number of men graduates decreased, and the percentages of men graduates declined in six of the eight fields at all three levels from the 1994-1995 academic year to the 2002-2003 academic year. However, men continued to constitute over 50 percent of the graduates in five of eight fields at all three education levels. Table 13 summarizes the numbers of graduates by gender, level, and field. Table 26 in appendix IV provides additional data on the percentages of men and women graduates by STEM field and education level. The total numbers of domestic minority graduates in STEM fields increased, although the percentage of minority graduates with STEM degrees at the master’s or doctoral level did not change from the 1994-1995 academic year to the 2002-2003 academic year. For example, while the number of Native American graduates increased 37 percent, Native American graduates remained less than 1 percent of all STEM graduates at the master’s and doctoral levels. Table 14 shows the percentages and numbers of domestic minority graduates for the 1994-1995 academic year and the 2002-2003 academic year. International students earned about one-third or more of the degrees at both the master’s and doctoral levels in several fields in the 1994-1995 and the 2002-2003 academic years. For example, in academic year 2002-2003, international students earned between 45 percent and 57 percent of all degrees in engineering and mathematics/computer sciences at the master’s and doctoral levels. However, at each level there were changes in the numbers and percentages of international graduates. At the master’s level, the total number of international graduates increased by about 31 percent from the 1994-1995 academic year to the 2002-2003 academic year; while the number of graduates decreased in four of the fields and the percentages of international graduates declined in three fields. At the doctoral level, the total

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number of international graduates decreased by 12 percent, while the percentage of international graduates increased or remained the same in all fields. Table 15 shows the numbers and percentages of international graduates in STEM fields. Table 13. Numbers and Percentage Changes in Men and Women Graduates with STEM Degrees by Education Level and Field for Academic Years 1994-1995 and 2002-2003 Education level

STEM field

Number of men graduates

1994-1995 2002-2003 Bachelor’s Biological/agricu 36,108 23,266 level ltural sciences Earth, 2,954 2,243 atmospheric, and ocean sciences Engineering 52,562 48,214 Mathematics and 25,258 46,381 computer sciences Physical sciences 9,607 8,739 Psychology 19,664 18,616 Social sciences 56,643 63,465 Technology 14,349 9,174 Master’s Biological/agricu 4,768 2,413 level ltural sciences Earth, 1,032 805 atmospheric, and ocean sciences Engineering 24,031 20,258 Mathematics and 10,398 14,531 computer sciences Physical sciences 2,958 2,350 Psychology 4,013 3,645 Social sciences 11,952 11,057 Technology 927 467 Doctoral Biological/agricu 3,616 1,526 level ltural sciences Earth, 488 315 atmospheric, and ocean sciences Engineering 5,401 4,159 Mathematics and 1,690 1,378 computer sciences Physical sciences 2,939 2,396 Psychology 1,529 1,380 Social sciences 2,347 2,111 Technology 24 7 Source: GAO calculations based upon IPEDS data.

Percentage change in men graduates

Number of women graduates

-36

1994-1995 2002-2003 35,648 35,546

Percentage change in women graduates

-24

1,524

1,626

+7

-8 +84

10,960 13,651

11,709 20,436

+7 +50

-9 -5 +12 -36 -49

5,292 53,010 56,624 1,602 4,340

6,222 64,470 77,701 1,257 2,934

+18 +22 +37 -22 -32

-22

451

552

+22

-16 +40

4,643 4,474

5,271 7,517

+14 +68

-21 -9 -7 -50 -58

1,283 10,319 11,398 222 2,160

1,299 12,433 13,674 173 1,161

+1 +20 +20 -22 -46

-35

134

125

-7

-23 -18

728 434

839 439

+15 +1

-18 -10 -10 -71

922 2,511 1,463 3

892 3,086 1,729 0

-3 +23 +18 -100

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Table 14. Numbers and Percentage Changes in Domestic Minority Graduates in STEM Fields by Education Levels and Race or Ethnicity for Academic Years 1994-1995 and 2002-2003

Source: GAO calculations based upon IPEDS data.

Table 15. Changes in Numbers and Percentages of International Graduates in STEM fields at the Master’s and Doctoral Degree Levels, 1994-1995 and 2002-2003 Academic Years

Source: GAO calculations based upon IPEDS data.

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STEM Employment Rose, but the Percentage of Women Remained About the Same and Minorities Continued to be Underrepresented While the total number of STEM employees increased, this increase varied across STEM fields. Employment increased by 23 percent in STEM fields as compared to 17 percent in non-STEM fields from calendar year 1994 to calendar year 2003. Employment increased by 78 percent in the mathematics/computer sciences field and by 20 percent in the science field over this period. The changes in number of employees in the engineering and technology fields were not statistically significant. Employment estimates from 1994 to 2003 in the STEM fields are shown in figure 4. Number of employees (in millions) 3.5

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2.0

1.5

1.0

0.5

0 1994

1995

1996

1997

1998

1999

2000

2001

2002

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Calendar year Science Technology Engineering Mathematics/computer sciences

Figure 4. Estimated Numbers of Employees in STEM Fields from Calendar Years 1994 through 2003.

From calendar years 1994 to 2003, the estimated number of women employees in STEM fields increased from about 2.7 million to about 3.5 million. Overall, there was not a statistically significant change in the percentage of women employees in the STEM fields. Table 16 shows the numbers and percentages of men and women employed in the STEM fields for calendar years 1994 and 2003. In addition, the estimated number of minorities employed in the STEM fields as well as the percentage of total STEM employees they constituted increased, but African American and Hispanic employees remain underrepresented relative to their percentages in the civilian labor force [19]. Between 1994 and 2003, the estimated number of African American employees increased by about 44 percent, the estimated numbers of Hispanic employees increased by 90 percent, as did the estimated numbers of other minorities employed in STEM fields [20]. In calendar year 2003, African Americans comprised about 8.7 percent of STEM employees compared to about 10.7 percent of the CLF. Similarly, Hispanic employees

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comprised about 10 percent of STEM employees in calendar year 2003, compared to about 12.6 percent of the CLF. Table 17 shows the estimated percentages of STEM employees by selected racial or ethnic groups in 1994 and 2003. Table 16. Estimated Numbers and Percentages of Employees in STEM Fields by Gender in Calendar Years 1994 and2003(numbers in thousands) STEM field

1994 2003 Men Women Men Women Number Percent Number Percent Number Percent Number Percent 792 32 1,711 68 829 28 2,179 72 955 68 445 32 1,050 71 425 29 1,658 92 *141 8 1,572 90 *169 10

Science Technology Engineering Mathematics/ computer sciences 1,056 71 432 29 1,952 74 695 26 Total 4,461 62 2,729 38 5,404 61 3,467 39 Source: GAO calculations based upon CPS data. Note: Estimated employee numbers noted by an asterisk have a 95 percent confidence interval of within +/25 percent of the estimate itself. All other estimated employee numbers have a 95 percent confidence interval of within +/- 16 percent of the estimate. See appendix VI for confidence intervals associated with these estimates. Calculations of percentages and numbers may differ due to rounding.

International employees have filled hundreds of thousands of positions, many in STEM fields, through the H-1B visa program. However, the numbers and types of occupations have changed over the years. We reported that while the limit for the H-1B program was 115,000 in 1999, the number of visas approved exceeded the limit by more than 20,000 because of problems with the system used to track the data [21]. Available data show that in 1999, the majority of the approved occupations were in STEM fields. Specifically, an estimated 60 percent of the positions approved in fiscal year 1999 were related to information technology and 5 percent were for electrical/electronics engineering. By 2002, the limit for the H-1B program had increased to 195,000, but the number approved, 79,000, did not reach this limit. In 2003, we reported that the number of approved H-1B petitions in certain occupations had declined. For example, the number of approvals for systems analysis/programming positions declined by 106,671 from 2001 to 2002 [22]. Table 17. Estimated Percentages of STEM Employees by Selected Racial or Ethnic Group for Calendar Years 1994 and 2003

Race or ethnicity Black or African American Hispanic or Latino origin Other minoritiesa

Percentage of total STEM employees, 1994 7.5 5.7 4.5

Percentage of total STEM employees, 2003 8.7 10.0 6.9

Source: GAO calculations based upon CPS data. Note: Estimated percentages have 95 percent confidence intervals of +/- 1 percentage point. Changes for African Americans between calendar years 1994 and 2003 were not statistically significant at the 95percent confidence level. Differences for Hispanic or Latino origin and other minorities were statistically significant. See appendix VI for confidence intervals associated with these estimates. aOther minorities include Asian/Pacific Islanders and American Indian or Alaska Native.

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Although the estimated total number of employees in STEM fields increased from 1994 to 2003, according to an NSF report, many with STEM degrees were not employed in these occupations. In 2004, NSF reported that about 67 percent of employees with degrees in science or engineering were employed in fields somewhat or not at all related to their degree [23]. Specifically, 70 percent of employees with bachelor’s degrees, 51 percent with master’s degrees, and 54 percent with doctoral degrees reported that their employment was somewhat or not at all related to their degree in science or engineering. In addition to increases in the numbers of employees in STEM fields, inflation-adjusted median annual wages and salaries increased in all four STEM fields over the 10-year period (1994 to 2003). These increases ranged from 6 percent in science to 15 percent in engineering. Figure 5 shows trends in median annual wages and salaries for STEM fields.

University Officials and Others Cited Several Factors That Influence Decisions about Participation in STEM Fields and Suggested Ways to Encourage Greater Participation University officials, researchers, and students identified several factors that influenced students’ decisions about pursuing STEM degrees and occupations, and they suggested some ways to encourage more participation in STEM fields. Specifically, university officials said and researchers reported that the quality of teachers in kindergarten through 12th grades and the levels of mathematics and science courses completed during high school affected students’ success in and decisions about STEM fields. In addition, several sources noted that mentoring played a key role in the participation of women and minorities in STEM fields. Annual wages and salaries (in thousands of dollars) 70

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Figure 5. Estimated Median Annual Wages and Salaries in STEM Fields for Calendar Years 1994 through 2003.

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Current students from five universities we visited generally agreed with these observations, and several said that having good mathematics and science instruction was important to their overall educational success. International students’ decisions about participating in STEM education and occupations were affected by opportunities outside the United States and the visa process. To encourage more student participation in the STEM fields, university officials, researchers, and others have made several suggestions, and four were made repeatedly. These suggestions focused on teacher quality, high school students’ math and science preparation, outreach activities, and the federal role in STEM education.

Teacher Quality and Mathematics and Science Preparation Were Cited as Key Factors Affecting Domestic Students’ STEM Participation Decisions University officials frequently cited teacher quality as a key factor that affected domestic students’ interest in and decisions about pursuing STEM degrees and occupations. Officials at all eight universities we visited expressed the view that a student’s experience from kindergarten through the 12th grades played a large role in influencing whether the student pursued a STEM degree. Officials at one university we visited said that students pursuing STEM degrees have associated their interests with teachers who taught them good skills in mathematics or excited them about science. On the other hand, officials at many of the universities we visited told us that some teachers were unqualified and unable to impart the subject matter, causing students to lose interest in mathematics and science. For example, officials at one university we visited said that some elementary and secondary teachers do not have sufficient training to effectively teach students in the STEM fields and that this has an adverse effect on what students learn in these fields and reduces the interest and enthusiasm students express in pursuing coursework in high school, degree programs in college, or careers in these areas. Teacher quality issues, in general, have been cited in past reports by Education. In 2002, Education reported that in the 1999-2000 school year, 14 to 22 percent of middle-grade students taking English, mathematics, and science were in classes led by teachers without a major, minor, or certification in these subjects—commonly referred to as “out-of-field” teachers [24]. Also, approximately 30 to 40 percent of the middle-grade students in biology/life science, physical science, or English as a second language/bilingual education classes had teachers lacking these credentials. At the high school level, 17 percent of students enrolled in physics and 36 percent of those enrolled in geology/earth/space science were in classes instructed by out-of-field teachers. The percentages of students taught by out-of-field teachers were significantly higher when the criteria used were teacher certification and a major in the subject taught. For example, 45 percent of the high school students enrolled in biology/life science and approximately 30 percent of those enrolled in mathematics, English, and social science classes had out-of-field teachers. During the 2002-2003 school year, Education reported that the number and distribution of teachers on waivers—which allowed prospective teachers in classrooms while they completed their formal training—was problematic. Also, states reported that the problem of underprepared teachers was worse on average in districts that serve large proportions of high-poverty children—the percentage of teachers on waivers was larger in high-poverty school districts than all other school districts

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in 39 states. Moreover, in 2004, Education reported that 48 of the 50 states granted waivers [25]. In addition to teacher quality, students’ high school preparation in mathematics and science was cited by university officials and others as affecting students’ success in collegelevel courses and their decisions about pursuing STEM degrees and occupations. University officials at six of the eight universities we visited cited students’ ability to opt out of mathematics and science courses during high school as a factor that influenced whether they would participate and succeed in the STEM fields during undergraduate and graduate school. University officials said, for example, that because many students had not taken higher-level mathematics and science courses such as calculus and physics in high school, they were immediately behind other students who were better prepared. In July 2005, on the basis of findings from the 2004 National Assessment of Educational Progress, the National Center for Education Statistics reported that 17 percent of the 17-year-olds reported that they had taken calculus, and this represents the highest percentage in any previous assessment year [26]. In a study that solicited the views of several hundred students who had left the STEM fields, researchers found that the effects of inadequate high school preparation contributed to college students’ decisions to leave the science fields [27]. These researchers found that approximately 40 percent of those college students who left the science fields reported some problems related to high school science preparation. The underpreparation was often linked to problems such as not understanding calculus; lack of laboratory experience or exposure to computers, and no introduction to theoretical material or to analytic modes of thought. Further, 12 current students we interviewed said they were not adequately prepared for college mathematics or science. For example, one student stated that her high school courses had been limited because she attended an all-girls school where the curriculum catered to students who were not interested in STEM, and so it had been difficult to obtain the courses that were of interest to her. Several other factors were mentioned during our interviews with university officials, students, and others as influencing decisions about participation in STEM fields. These factors included relatively low pay in STEM fields, additional tuition costs to obtain STEM degrees, lack of commitment on the part of some students to meet the rigorous academic demands, and the inability of some professors in STEM fields to effectively impart their knowledge to students in the classrooms. For example, officials from five universities said that low pay in STEM fields relative to other fields such as law and business dissuaded students from pursuing STEM degrees in some areas. Also, in a study that solicited the views of college students who left the STEM fields as well as those who continued to pursue STEM degrees, researchers found that students experienced greater financial difficulties in obtaining their degrees because of the extra time needed to obtain degrees in certain STEM fields. Researchers also noted that poor teaching at the university level was the most common complaint among students who left as well as those who remained in STEM fields. Students reported that faculty do not like to teach, do not value teaching as a professional activity, and therefore lack any incentive to learn to teach effectively [28]. Finally, 11 of the students we interviewed commented about the need for professors in STEM fields to alter their methods and to show more interest in teaching to retain students’ attention.

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Mentoring Cited as a Key Factor Affecting Women’s and Minorities’ STEM Participation Decisions University officials and students said that mentoring is important for all students but plays a vital role in the academic experiences of women and minorities in the STEM fields. Officials at seven of the eight universities discussed the important role that mentors play, especially for women and minorities in STEM fields. For example, one professor said that mentors helped students by advising them on the best track to follow for obtaining their degrees and achieving professional goals. Also, four students we interviewed—three women and one man—expressed the importance of mentors. Specifically, while all four students identified mentoring as critical to academic success in the STEM fields, two students expressed their satisfaction since they had mentors, while the other two students said that it would have been helpful to have had someone who could have been a mentor or role model. Studies have also reported that mentors play a significant role in the success of women and minorities in the STEM fields. In 2004, some of the women students and faculty with whom we talked reported a strong mentor was a crucial part in the academic training of some of the women participating in sciences, and some women had pursued advanced degrees because of the encouragement and support of mentors [29]. In September 2000, a congressional commission reported that women were adversely affected throughout the STEM education pipeline and career path by a lack of role models and mentors [30]. For example, the report found that girls rejection of mathematics and science may be partially driven by teachers, parents, and peers when they subtly, and not so subtly, steer girls away from the informal technical pastimes (such as working on cars, fixing bicycles, and changing hardware on computers) and science activities (such as science fairs and clubs) that too often were still thought of as the province of boys. In addition, the commission reported that a greater proportion of women switched out of STEM majors than men, relative to their representation in the STEM major population. Reasons cited for the higher attrition rate among women students included lack of role models, distaste for the competitive nature of science and engineering education, and inability to obtain adequate academic guidance or advice. Further, according to the report, women’s retention and graduation in STEM graduate programs were affected by their interaction with faculty, integration into the department (versus isolation), and other factors, including whether there were role models, mentors, and women faculty.

International Students’ STEM Participation Decisions Were Affected by Opportunities Outside the United States and the Visa Process Officials at seven of the eight universities visited, along with education policy experts, told us that competition from other countries for top international students, and educational or work opportunities, affected international students’ decisions about studying in the United States. They told us that other countries, including Canada, Australia, New Zealand, and the United Kingdom, had seized the opportunity since September 11 to compete against the United States for international students who were among the best students in the world, especially in the STEM fields. Also, university officials told us that students from several countries, including China and India, were being recruited to attend universities and get jobs in their

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own countries. In addition, education organizations and associations have reported that global competition for the best science and engineering students and scholars is under way. One organization, NAFSA: Association of International Educators reported that the international student market has become highly competitive, and the United States is not competing as well as other countries [31]. According to university officials, international students’ decisions about pursuing STEM degrees and occupations in the United States were also influenced by the perceived unwelcoming attitude of Americans and the visa process. Officials from three of the universities said that the perceived unwelcoming attitude of Americans had affected the recruitment of international students to the United States. Also, officials at six of the eight universities visited expressed their concern about the impact of the tightened visa procedures and/or increased security measures since September 11 on international graduate school enrollments. For example, officials at one university stated that because of the time needed to process visas, a few students had missed their class start dates. Officials from one university told us that they were being more proactive in helping new international students navigate the visa system, to the extent possible. While some university officials acknowledged that visa processing had significantly improved, since 2003 several education associations have requested further changes in U.S. visa policies because of the lengthy procedures and time needed to obtain approval to enter the country. We have reported on various aspects of the visa process, made several recommendations, and noted that some improvements have been made. In October 2002 we cited the need for a clear policy on how to balance national security concerns with the desire to facilitate legitimate travel when issuing visas and we made several recommendations to help improve the visa process [32]. In 2003, we reported that the Departments of State, Homeland Security, and Justice could more effectively manage the visa function if they had clear and comprehensive policies and procedures and increased agency coordination and information sharing [33]. In February 2004 and February 2005, we reported on the State Department’s efforts to improve the program for issuing visas to international science students and scholars. In 2004 we found that the time to adjudicate a visa depended largely on whether an applicant had to undergo a security check known as Visas Mantis, which is designed to protect against sensitive technology transfers. Based on a random sample of Visas Mantis cases for science students and scholars, it took State an average of 67 days to complete the process [34]. In 2005, we reported a significant decline in Visas Mantis processing times and in the number of cases pending more than 60 days [35]. We also reported that, in some cases, science students and scholars can obtain a visa within 24 hours. We have also issued several reports on SEVIS operations. In June 2004 we noted that when SEVIS began operating, significant problems were reported [36]. For example, colleges and universities and exchange programs had trouble gaining access to the system, and when access was obtained, these users’ sessions would “time out” before they could complete their tasks. In that report we also noted that SEVIS performance had improved, but that several key system performance requirements were not being measured. In March 2005, we reported that the Department of Homeland Security (DHS) had taken steps to address our recommendations and that educational organizations generally agreed that SEVIS performance had continued to improve [37]. However, educational organizations continued to cite problems, which they believe created hardships for students and exchange visitors.

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Several Suggestions Were Made to Encourage More Participation in the STEM Fields To increase the number of students entering STEM fields, officials from seven universities and others stated that teacher quality needs to improve. Officials of one university said that kindergarten through 12th grade classrooms need teachers who are knowledgeable in the mathematics and science content areas. As previously noted, Education has reported on the extent to which classes have been taught by teachers with little or no content knowledge in the STEM fields. The Congressional Commission on the Advancement of Women and Minorities reported that teacher effectiveness is the most important element in a good education [38]. The commission also suggested that boosting teacher effectiveness can do more to improve education than any other single factor. States are taking action to meet NCLBA’s requirement of having all teachers of core academic subjects be highly qualified by the end of the 2005-2006 school year. University officials and some students suggested that better preparation and mandatory courses in mathematics and science were needed for students during their kindergarten through 12th grade school years. Officials from five universities suggested that mandatory mathematics and science courses, especially in high school, may lead to increased student interest and preparation in the STEM fields. With a greater interest and depth of knowledge, students would be better prepared and more inclined to pursue STEM degrees in college. Further, nearly half of the students who replied to this question suggested that students needed additional mathematics and science training prior to college. However, adding mathematics and science classes has resource implications, since more teachers in these subjects would be needed. Also this change could require curriculum policy changes that would take time to implement. More outreach, especially to women and minorities from kindergarten through the 12th grade, was suggested by university officials, students, and other organizations. Officials from six of the universities we visited suggested that increased outreach activities are needed to help create more interest in mathematics and science for younger students. For example, at one university we visited, officials told us that through inviting students to their campuses or visiting local schools, they have provided some students with opportunities to engage in science laboratories and hands-on activities that foster interest and excitement for students and can make these fields more relevant in their lives. Officials from another university told us that these experiences were especially important for women and minorities who might not have otherwise had these opportunities. The current students we interviewed also suggested more outreach activities. Specifically, two students said that outreach was needed to further stimulate students’ interest in the STEM fields. One organization, Building Engineering and Science Talent (BEST), suggested that research universities increase their presence in prekindergarten through 12th grade mathematics and science education in order to strengthen domestic students’ interests and abilities. BEST reported that one model producing results entailed universities adopting students from low-income school districts from 7th through 12th grades and providing them advanced instruction in algebra, chemistry, physics, and trigonometry. However, officials at one university told us that because of limited resources, their efforts were constrained and only a few students would benefit from this type of outreach.

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Furthermore, university officials from the eight schools and other education organizations made suggestions regarding the role of the federal government. University officials suggested that the federal government could enhance its role in STEM education by providing more effective leadership through developing and implementing a national agenda for STEM education and increasing federal funding for academic research. Officials at six universities suggested that the federal government undertake a new initiative modeled after the National Defense Education Act of 1958, enacted in response to the former Soviet Union’s achievement in its space program, which provided new funding for mathematics and science education and training at all education levels. In June 2005, CGS called for a renewed commitment to graduate education by the federal government through actions such as providing funds to support students trained at the doctoral level in the sciences, technology, engineering, and mathematics; expanding U.S. citizen participation in doctoral study in selected fields through graduate support awarded competitively to universities across the country; requiring recruitment, outreach, and mentoring activities that promote greater participation and success, especially for underrepresented groups; and fostering interdisciplinary research preparation. In August 2003, the National Science Board recommended that the federal government direct substantial new support to students and institutions in order to improve success in science and engineering studies by domestic undergraduate students from all demographic groups. According to this report, such support could include scholarships and other forms of financial assistance to students, incentives to institutions to expand and improve the quality of their science and engineering programs in areas in which degree attainment is insufficient, financial support to community colleges to increase the success of students in transferring to 4-year science and engineering programs, and expanded funding for programs that best succeed in graduating underrepresented minorities and women in science and engineering. BEST also suggested that the federal government allocate additional resources to expand the mathematics and science education opportunities for underrepresented groups. However, little is known about how well federal resources have been used in the past. Changes that would require additional federal funds would likely have an impact on other federal programs, given the nation’s limited resources and growing fiscal imbalance, and changing the federal role could take several years.

Concluding Observations While the total numbers of STEM graduates have increased, some fields have experienced declines, especially at the master’s and doctoral levels. Given the trends in the numbers and percentages of students pursuing STEM degrees, particularly advanced degrees, and recent developments that have influenced international students’ decisions about pursuing degrees in the United States, it is uncertain whether the number of STEM graduates will be sufficient to meet future academic and employment needs and help the country maintain its technological competitive advantage. Moreover, it is too early to tell if the declines in international graduate student enrollments will continue in the future. In terms of employment, despite some gains, the percentage of women in the STEM workforce has not changed significantly, minority employees remain underrepresented, and many with degrees in STEM fields are not employed in STEM occupations.

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To help improve the trends in the numbers of students, graduates, and employees in STEM fields, university officials and others made several suggestions, such as increasing the federal commitment to STEM education programs. However, before making changes, it is important to know the extent to which existing STEM education programs are appropriately targeted and making the best use of available federal resources. Additionally, in an era of limited financial resources and growing federal deficits, information about the effectiveness of these programs can help guide policy makers and program managers.

Agency Comments and Our Evaluation We received written comments on a draft of this report from Commerce, the Department of Health and Human Services (HHS), NSF, and NSTC. These comments are reprinted in appendixes VII, VIII, IX, and X, respectively. We also received technical comments from the Departments of Commerce, Health and Human Services, Homeland Security, Labor, and Transportation; and the Environmental Protection Agency and National Aeronautics and Space Administration, which we incorporated when appropriate. In commenting on a draft of this report, Commerce, HHS, and NSTC commended GAO for this work. Commerce explicitly concurred with several findings and agreed with our overall conclusion. However, Commerce suggested that we revise the conclusion to point out that despite overall increases in STEM students, the numbers of graduates in certain fields have declined. We modified the concluding observations to make this point. HHS agreed with our conclusion that it is important to evaluate ongoing programs to determine the extent to which they are achieving their desired results. The comments from NSTC cited improvements made to help ensure that international students, exchange visitors, and scientists are able to apply for and receive visas in a timely manner. We did not make any changes to the report since we had cited another GAO product that discussed such improvements in the visa process. NSF commented about several of our findings. NSF stated that our program evaluations finding may be misleading largely because the type of information GAO requested and accepted from agencies was limited to program level evaluations and did not include evaluations of individual underlying projects. NSF suggested that we include information on the range of approaches used to assure program effectiveness. Our finding is based on agency officials’ responses to a survey question that did not limit or stipulate the types of evaluations that could have been included. To help improve the trends in the numbers of students, graduates, and employees in STEM fields, university officials and others made several suggestions, such as increasing the federal commitment to STEM education programs. However, before making changes, it is important to know the extent to which existing STEM education programs are appropriately targeted and making the best use of available federal resources. Additionally, in an era of limited financial resources and growing federal deficits, information about the effectiveness of these programs can help guide policy makers We received written comments on a draft of this report from Commerce, the Department of Health and Human Services (HHS), NSF, and NSTC. These comments are reprinted in appendixes VII, VIII, IX, and X, respectively. We also received technical comments from the Departments of Commerce, Health and Human Services, Homeland Security, Labor, and Transportation; and the Environmental Protection

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Agency and National Aeronautics and Space Administration, which we incorporated when appropriate. In commenting on a draft of this report, Commerce, HHS, and NSTC commended GAO for this work. Commerce explicitly concurred with several findings and agreed with our overall conclusion. However, Commerce suggested that we revise the conclusion to point out that despite overall increases in STEM students, the numbers of graduates in certain fields have declined. We modified the concluding observations to make this point. HHS agreed with our conclusion that it is important to evaluate ongoing programs to determine the extent to which they are achieving their desired results. The comments from NSTC cited improvements made to help ensure that international students, exchange visitors, and scientists are able to apply for and receive visas in a timely manner. We did not make any changes to the report since we had cited another GAO product that discussed such improvements in the visa process. NSF commented about several of our findings. NSF stated that our program evaluations finding may be misleading largely because the type of information GAO requested and accepted from agencies was limited to program level evaluations and did not include evaluations of individual underlying projects. NSF suggested that we include information on the range of approaches used to assure program effectiveness. Our finding is based on agency officials’ responses to a survey question that did not limit or stipulate the types of evaluations that could have been included. Nonetheless, we modified the report to acknowledge that NSF uses various approaches to evaluate its programs. NSF criticized the methodology we used to support our finding on the factors that influence decisions about pursuing STEM fields and suggested that we make it clearer in the body of the report that the findings are based on interviews with educators and administrators from 8 colleges and universities, and responses from 31 students. Also, NSF suggested that we improve the report by including corroborating information from reports and studies. Our finding was not limited to interviews at the 8 colleges and universities and responses from 31 current students but was also based on interviews with numerous representatives and policy experts from various organizations as well as findings from research and reports—which are cited in the body of the report. Using this approach, we were able to corroborate the testimonial evidence with data from reports and research as well as to determine whether information in the reports and research remained accurate by seeking the views of those currently teaching or studying in STEM fields. As NSF noted, this approach yielded reasonable observations. Additional information about our methodology is listed in appendix I, and we added a bibliography that identifies the reports and research used during the course of this review. NSF also commented that the report mentions the NSTC efforts for interagency collaboration, but does not mention other collaboration efforts such as the Federal Interagency Committee on Education and the Federal Interagency Coordinating Council. NSF also pointed out that interagency collaboration occurs at the program level. We did not modify the report in response to this comment. In conducting our work, we determined that the NSTC effort was the primary mechanism for interagency collaboration focused on STEM programs. The coordinating groups cited by NSF are focused on different issues. The Federal Interagency Committee on Education was established to coordinate the federal programs, policies, and practices affecting education broadly, and the Federal Interagency Coordinating

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Council was established to minimize duplication of programs and activities relating to children with disabilities. In addition, NSF provided information to clarify examples related to their programs that we cited in the report, stated that some data categories were not clear, and commented on the graduate level enrollment data we used in the report. NSF pointed out that while its program called Opportunities for Enhancing Diversity in the Geosciences is designed to increase participation by minorities, it does not limit eligibility to minorities. Also, NSF noted that while the draft report correctly indicated that students receiving scholarships or fellowships from NSF must be U.S. citizens or permanent residents, the reason given for limiting participation in these programs in the draft report was not accurate. According to NSF, these restrictions are considered to be an effective strategy to support its goal of creating a diverse, competitive and globally engaged U.S. workforce of scientists, engineers, technologists and well prepared citizens. We revised the report to reflect these changes. Further, NSF commented that the data categories were not clear, particularly the technology degrees and occupations, and that the data did not include associate degrees. We added information that lists all of the occupations included in the analysis, and we added footnotes to clarify which data included associate degrees and which ones did not. In addition, NSF commented that the graduate level enrollment data for international students based on NPSAS data are questionable in comparison with other available data and that this may be because the NPSAS data include a relatively small sample for graduate education. We considered using NPSAS and other data but decided to use the NPSAS data for two reasons: NPSAS data were more comprehensive and more current. Specifically, the NPSAS data were available through the 2003-2004 academic year and included numbers and characteristics of students enrolled for all degree fields—STEM and non-STEM—for all education levels, and citizenship information.

Appendix I: Objectives, Scope, and Methodology Objectives The objectives of our study were to determine (1) the number of federal civilian education programs funded in fiscal year 2004 that were specifically designed to increase the number of students and graduates pursuing science, technology, engineering, and mathematics (STEM) degrees and occupations, or improve educational programs in STEM fields, and what agencies report about their effectiveness; (2) how the numbers, percentages, and characteristics of students, graduates, and employees in STEM fields have changed over the years; and (3) factors cited by educators and others as influencing people’s decisions about pursuing STEM degrees and occupations, and suggestions to encourage greater participation in STEM fields.

Scope and Methodology In conducting our review, we used multiple methodologies. We (1) conducted a survey of federal departments and agencies that sponsored education programs specifically designed to

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increase the number of students and graduates pursuing STEM degrees and occupations or improve educational programs in STEM fields; (2) obtained and analyzed data, including the most recent data available, on students, graduates, and employees in STEM fields and occupations; (3) visited eight colleges and universities; (4) reviewed reports and studies; and (5) interviewed agency officials, representatives and policy experts from various organizations, and current students. We conducted our work between October 2004 and October 2005 in accordance with generally accepted government auditing standards.

Survey To provide Congress with a better understanding of what programs federal agencies were supporting to increase the nation’s pool of scientists, technologists, engineers, and mathematicians, we designed a survey to determine (1) the number of federal education programs (prekindergarten through postdoctorate) designed to increase the quantity of students and graduates pursuing STEM degrees and occupations or improve the educational programs in STEM fields and (2) what agencies reported about the effectiveness of these programs. The survey asked the officials to describe the goals, target population, and funding levels for fiscal years 2003, 2004, and 2005 of such programs. In addition, the officials were asked when the programs began and if the programs had been or were being evaluated. We identified the agencies likely to support STEM education programs by reviewing the Catalog of Federal Domestic Assistance and the Department of Education’s Eisenhower National Clearinghouse, Guidebook of Federal Resources for K-12 Mathematics and Science, 2004-05. Using these resources, we identified 15 agencies with STEM education programs. The survey was conducted via e-mail using an ActiveX enabled MSWord attachment. A contact point was designated for each agency, and questionnaires were sent to that individual. One questionnaire was completed for each program the agency sponsored. Agency officials were asked to provide confirming documentation for their responses whenever possible. The questionnaire was forwarded to agencies on February 15, 2005, and responses were received through early May 2005. We received 244 completed surveys and determined that 207 of them met the criteria for STEM programs. The following agencies participated in our survey: the Departments of Agriculture, Commerce, Education, Energy, Homeland Security, Interior, Labor, and Transportation. In addition, the Health Resources and Services Administration, Indian Health Service, and National Institutes of Health, all part of Health and Human Services, took part in the survey. Also participating were the U.S. Environmental Protection Agency; the National Aeronautics and Space Administration; and the National Science Foundation. Labor’s programs did not meet our criteria for 2004 and the Department of Defense (DOD) did not submit a survey. According to DOD officials, DOD needed 3 months to complete the survey and therefore could not provide responses within the time frames of our work. We obtained varied amounts of documentation from 13 civilian agencies for the 207 STEM education programs funded in 2004 and information about the effectiveness of some programs. Because we administered the survey to all of the known federal agencies sponsoring STEM education programs, our results are not subject to sampling error. However, the practical difficulties of conducting any survey may introduce other types of errors, commonly referred to as nonsampling errors. For example, differences in how a particular question is

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interpreted, the sources of information available to respondents in answering a question, or the types of people who do not respond can introduce unwanted variability into the survey results. We included steps in the development of the survey, the collection of data, and the editing and analysis of data for the purpose of minimizing such nonsampling errors. To reduce nonsampling error, the questionnaire was reviewed by survey specialists and pretested in person with three officials from agencies familiar with STEM education programs to develop a questionnaire that was relevant, easy to comprehend, unambiguous, and unbiased. We made changes to the content and format of the questionnaire based on the specialists’ reviews and the results of the pretests. To further reduce nonsampling error, data for this study returned electronically were entered directly into the instrument by the respondents and converted into a database for analysis. Completed questionnaires returned as hard copy were keypunched, and a sample of these records was verified by comparing them with their corresponding questionnaires, and any errors were corrected. When the data were analyzed, a second, independent analyst checked all computer programs. Finally, to assess the reliability of key data obtained from our survey about some of the programs, we compared the responses with the documentation provided, or we independently researched the information from other publicly available sources.

Analyses of Student, Graduate, and Employee Data To determine how the numbers and characteristics of students, graduates, and employees in STEM fields have changed, we obtained and analyzed data from the Department of Education (Education) and the Department of Labor. Specifically, we analyzed the National Postsecondary Student Aid Study (NPSAS) data and the Integrated Postsecondary Education Data System (IPEDS) data from the Department of Education’s National Center for Education Statistics (NCES), and we analyzed data from the Department of Labor’s Bureau of Labor Statistics’ (BLS) Current Population Survey (CPS). Based on National Science Foundation’s categorization of STEM fields, we developed STEM fields of study from NPSAS and IPEDS, and identified occupations from the CPS. Using these data sources, we developed nine STEM fields for students, eight STEM fields for graduates, and four broad STEM fields for occupations. For our data reliability assessment, we reviewed agency documentation on the data sets and conducted electronic tests of the files. On the basis of these reviews, we determined that the required data elements from NPSAS, IPEDS and CPS were sufficiently reliable for our purposes. These data sources, type, time span, and years analyzed are shown in table 18. NPSAS is a comprehensive nationwide study designed to determine how students and their families pay for postsecondary education, and to describe some demographic and other characteristics of those enrolled. The study is based on a nationally representative sample of students in postsecondary education institutions, including undergraduate, graduate, and firstprofessional students. The NPSAS has been conducted every several years since the 19861987 academic year. For this report, we analyzed the results of the NPSAS survey for the 1995-1996 academic year and the 2003-2004 academic year to compare student enrollment and demographic characteristics between these two periods for the nine STEM fields and non-STEM fields.

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Table 18. Sources of Data, Data Obtained, Time Span of Data, and Years Analyzed Department

Agency

Database

Data obtained

Time span Years analyzed of data 9 years Academic years 1995-1996 and 2003-2004

Education

NCES

NPSAS

College student enrollment

Education

NCES

IPEDS

Graduation/degrees 9 years

Academic years 1994-1995 and 2002-2003

Labor

BLS

CPS

Employment

Calendar years 1994 through 2003

Sources: NPSAS, IPEDS, and CPS data.

10 years

Because the NPSAS sample is a probability sample of students, the sample is only one of a large number of samples that might have been drawn. Since each sample could have provided different estimates, confidence in the precision of the particular sample’s results is expressed as a 95-percent confidence interval (for example, plus or minus 4 percentage points). This is the interval that would contain the actual population value for 95 percent of the samples that could have been drawn. As a result, we are 95 percent confident that each of the confidence intervals in this report will include the true values in the study population. NPSAS estimates used in this report and the upper and lower bounds of the 95 percent confidence intervals for each estimate relied on in this report are presented in appendix V. IPEDS is a single, comprehensive system designed to encompass all institutions and educational organizations whose primary purpose is to provide postsecondary education. IPEDS is built around a series of interrelated surveys to collect institution-level data in such areas as enrollments, program completions, faculty, staff, and finances. For this report, we analyzed the results of IPEDS data for the 1994-1995 academic year and the 2002-2003 academic year to compare the numbers and characteristics of graduates with degrees in eight STEM fields and non-STEM fields. To analyze changes in employees in STEM and non-STEM fields, we obtained employment estimates from BLS’s Current Population Survey March supplement for 1995 through 2004 (calendar years 1994 through 2003). The CPS is a monthly survey of households conducted by the U.S. Census Bureau (Census) for BLS. The CPS provides a comprehensive body of information on the employment and unemployment experience of the nation’s population, classified by age, sex, race, and a variety of other characteristics. A more complete description of the survey, including sample design, estimation, and other methodology can be found in the CPS documentation prepared by Census and BLS [1]. This March supplement (the Annual Demographic Supplement) is specifically designed to estimate family characteristics, including income from all sources and occupation and industry classification of the job held longest during the previous year. It is conducted during the month of March each year because it is believed that since March is the month before the deadline for filing federal income tax returns, respondents would be more likely to report income more accurately than at any other point during the year [2].

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United States Government Accountability Office Table 19. Classification codes and Occupations, 2002-2003 Science

T echnol

ogy

Engineering

Mathematics/Computer Science

1600 – Agricultural and food scientists

1540 – Drafters

1300 – Architects, except naval

1610 – Biological scientists

1550 – Engineering technicians, except drafters

1310 – Surveyors, cartographers, 1010 – Computer and photogrammetrists programmers

1640 – Conservation scientists and foresters

1560 – Surveying and mapping technicians

1320 – Aerospace engineers

1020 – Computer software engineers

1650 – Medical scientists

1900 – Agricultural and food science technicians

1330 – Agricultural engineers

1040 – Computer support specialists

1700 – Astronomers and physicists

1910 – Biological technicians

1340 – Biomedical engineers

1060 – Database administrators

1710 – Atmospheric and space scientists

1920 – Chemical technicians

1350 – Chemical engineers

1100 – Network and computer systems administrators

1720 – Chemists and materials scientists

1930 – Geological and petroleum technicians

1360 – Civil engineers

1110 – Network systems and data communications analysts

1740 – Environmental scientists and geoscientists

1940 – Nuclear technicians

1400 – Computer hardware engineers

1200 – Actuaries

1760 – Physical scientists, all other

1960 – Other life, physical, and social science technicians

1410 – Electrical and electronic engineers

1210 – Mathematicians

1800 – Economists

3300 – Clinical laboratory technologists and technicians

1420 – Environmental engineers

1220 – Operations research analysts

1810 – Market and survey researchers

7010 – Computer, automated teller 1430 – Industrial engineers, and office machine repairers including health and safety

1230 – Statisticians

1820 – Psychologists

8760 – Medical, dental, and ophthalmic laboratory technicians

1240 – Miscellaneous mathematical science occupations

1440 – Marine engineers and naval architects

1830 – Sociologists

1450 – Materials engineers

1840 – Urban and regional planners

1460 – Mechanical engineers

1860 – Miscellaneous social scientists and related workers

1500 – Mining and geological engineers, including mining safety engineers

2010 – Social workers

1510 – Nuclear engineers

3130 – Registered nurses

1520 – Petroleum engineers

6010 – Agricultural inspectors

1530 – Engineers, all other

1000 – Computer scientists and systems analysts

We used the CPS data to produce estimates on (1) four STEM fields, (2) men and women, (3) two separate minority groups (Black or African American, and Hispanic or Latino origin), and (4) median annual wages and salaries. The measures of median annual wages and salaries could include bonuses, but do not include noncash benefits such as health insurance or pensions. CPS salary reported in March of each year was for the longest held position actually worked the year before and reported by the worker himself (or a knowledgeable member of the household). Tables 19 and 20 list the classification codes and occupations included in our analysis of CPS data over a 10-year period (1994-2003). In developing the STEM groups, we considered the occupational requirements and educational attainment of individuals in certain occupations. We also excluded doctors and other health care providers except registered nurses. During the period of review, some codes and occupation titles were changed; we worked with BLS officials to identify variations in codes and occupations and accounted for these changes where appropriate and possible. Because the CPS is a probability sample based on random selections, the sample is only one of a large number of samples that might have been drawn. Since each sample could have provided different estimates, confidence in the precision of the particular sample’s results is expressed as a 95 percent confidence interval (e.g., plus or minus 4 percentage points). This is the interval that would contain the actual population value for 95 percent of the samples that could have been drawn. As a result, we are 95 percent confident that each of the confidence intervals in this report will include the true values in the study population. We use the CPS

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general variance methodology to estimate this sampling error and report it as confidence intervals. Percentage estimates we produce from the CPS data have 95 percent confidence intervals of plus or minus 6 percentage points or less. Estimates other than percentages have 95 percent confidence intervals of no more than plus or minus 10 percent of the estimate itself, unless otherwise noted. Consistent with the CPS documentation guidelines, we do not produce estimates based on the March supplement data for populations of less than 75,000. Table 20. Classification codes and occupations, 1994-2001 Science

T echnol

ogy

Engineering

Mathematics/Computer Science

069 – Physicists and astronomers

203 – Clinical laboratory technologists and technicians

043 – Architects

064 – Computer systems analysts and scientists

073 – Chemists, except biochemists

213 – Electrical and electronic technicians

044 – Aerospace engineers

065 – Operations and systems researchers and analysts

074 – Atmospheric and space 214 – Industrial engineering scientists technicians

045 – Metallurgical and materials engineers

066 – Actuaries

075 – Geologists and geodesists

215 – Mechanical engineering technicians

046 – Mining engineers

067 – Statisticians

076 – Physical scientists, n.e.c.

216 – Engineering technicians, n.e.c.

047 – Petroleum engineers

068 – Mathematical scientists, n.e.c.

077 – Agricultural and food scientists

217 – Drafting occupations

048 – Chemical engineers

229 – Computer programmers

078 – Biological and life scientists

218 – Surveying and mapping technicians

049 – Nuclear engineers

079 – Forestry and conservation scientists

223 – Biological technicians

053 – Civil engineers

083 – Medical scientists

224 – Chemical technicians

054 – Agricultural engineers

095 – Registered Nurses

225 – Science technicians, n.e.c.

055 – Electrical and electronic engineers

166 – Economists

235 – Technicians, n.e.c.

056 – Industrial engineers

167 – Psychologists

525 – Data processing equipment repairers

057 – Mechanical engineers

168 – Sociologists

058 – Marine and naval architects

169 – Social scientists, n.e.c.

059 – Engineers, n.e.c.

173 – Urban planners

063 – Surveyors and mapping scientists

174 – Social workers 489 – Inspectors, agricultural products

Note: For occupations not elsewhere classified (n.e.c.).

GAO’s internal control procedures provide reasonable assurance that our data analyses are appropriate for the purposes we are using them. These procedures include, but are not limited to, having skilled staff perform the analyses, supervisory review by senior analysts, and indexing/referencing (confirming that the analyses are supported by the underlying audit documentation) activities.

College and University Visits We interviewed administrators and professors during site visits to eight colleges and universities—the University of California at Los Angeles and the University of Southern California in California; Clark Atlanta University, Georgia Institute of Technology, and Spelman College in Georgia; the University of Illinois; Purdue University in Indiana; and Pennsylvania State University. These colleges and universities were selected based on the following factors: large numbers of domestic and international students in STEM fields, a

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mix of public and private institutions, number of doctoral degrees conferred, and some geographic diversity. We also selected three minority-serving colleges and universities, one of which serves only women students. Clark Atlanta University and Spelman College were selected, in part, because of their partnerships with the College of Engineering at the Georgia Institute of Technology. During these visits we asked the university officials about factors that influenced whether people pursue a STEM education or occupations and suggestions for addressing those factors that may influence participation. For example, we asked university officials to identify (1) issues related to the education pipeline; (2) steps taken by their university to alleviate some of the conditions that may discourage student participation in STEM areas; and (3) the federal role, if any, in attracting and retaining domestic students in STEM fields. We also obtained documents on programs they sponsored to help support STEM students and graduates.

Reviews of Reports and Studies We reviewed several articles, reports, and books related to trends in STEM enrollment and factors that have an effect on people’s decisions to pursue STEM fields. For two studies, we evaluated the methodological soundness using common social science and statistical practices. We examined each study’s methodology, including its limitations, data sources, analyses, and conclusions. •

Talking about Leaving: Why Undergraduates Leave the Sciences, by Elaine Seymour and Nancy Hewitt.3 This study used interviews and focus groups/group interviews at selected universities to identify self-reported reasons for changing majors from science, mathematics, or engineering. The study had four primary objectives: (1) to identify sources of qualitative differences in educational experiences of science, mathematics, and engineering students at higher educational institutions of different types; (2) to identify differences in structure, culture, and pedagogy of science, mathematics, and engineering departments and the impact on student retention; (3) to compare and contrast causes of science, mathematics, and engineering students’ attrition by race/ethnicity and gender; and (4) to estimate the relative importance of factors found to contribute to science, mathematics, and engineering students’ attrition. The researchers selected seven universities to represent the types of colleges and universities that supply most of the nations’ scientists, mathematicians, and engineers. The types of institutions were selected to test whether there are differences in educational experiences, culture and pedagogy, race/ethnicity and gender attrition, and reasons for attrition by type of institution. Because the selection of students was not strictly random and because there is no documentation that the data were weighted to reflect the proportions of types of students selected, it is not possible to determine confidence intervals. Thus it is not possible to say which differences are statistically significant. The findings are now more than a decade old and thus might not reflect current pedagogy and other factors about the educational experience, students, or the socioeconomic environment. It is important to note that the quantitative results of this study are based on the views of one constituency or

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•

215

stakeholder—students. Views of faculty, school administrators, graduates, professional associations, and employers are not included. NCES’s Qualifications of the Public School Teacher Workforce: Prevalence of Out-ofField Teaching, 1987-1988 to 1999-2000 report. This study is an analysis based upon the Schools and Staffing Survey for 1999-2000. The report was issued in 2004 by the Institute of Education Sciences, U.S. Department of Education. NCES’s Schools and Staffing Survey (SASS) is a representative sample of U.S. schools, districts, principals, and teachers. The report focusing on teacher’s qualifications uses data from the district and teacher portion of SASS. The 1999-2000 SASS included a nationally representative sample of public schools and universe of all public charter schools with students in any of grades 1 through 12 and in operation in school year 1999-2000. The 1999-2000 SASS administration also included nationally representative samples of teachers in the selected public and public charter schools who taught students in grades kindergarten through 12 in school year 1999-2000. There were 51,811 public school teachers in the sample and 42,086 completed public school teacher interviews. In addition, there are 3,617 public charter school teachers in the sample with 2,847 completed interviews. The overall weighted teacher response rate was 76.7 percent for public school teachers and 71.8 percent for public charter school teachers. NCES has strong standards for carrying out educational surveys. The Office of Management and Budget vetted the questionnaire and sample design. The Census Bureau carried out survey quality control and data editing. One potential limitation is the amount of time it takes the Census Bureau to get the data from field collection to public release, but this is partly due to the thoroughness of the data quality steps followed. The SASS survey meets GAO standards for use as evidence in a report.

Interviews We interviewed officials from 13 federal agencies with STEM education programs to obtain information about the STEM programs and their views on related topics, including factors that influence students’ decisions about pursuing STEM degrees and occupations, and the extent of coordination among the federal agencies. We also interviewed officials from the National Science and Technology Council to discuss coordination efforts. In addition, we interviewed representatives and policy experts from various organizations. These organizations were the American Association for the Advancement of Science, the Commission on Professionals in Science and Technology, the Council of Graduate Schools, NAFSA: Association of International Educators, the National Academies, and the Council on Competitiveness. We also conducted interviews via e-mail with 31 students. We asked officials from the eight universities visited to identify students to complete our e-mail interviews, and students who completed the interviews attended five of the colleges we visited. Of the 31 students: 16 attended Purdue University, 6 attended the University of Southern California, 6 attended Spelman College, 2 attended the University of California Los Angeles, and 1 attended the Georgia Institute of Technology. In addition, 19 students were undergraduates and 12 were graduate students; 19 students identified themselves as women and 12 students identified themselves as men. Of the 19 undergraduate students, 9 said that they plan to pursue graduate work in a STEM field.

216

United States Government Accountability Office

Appendix II: List of 207 Federal STEM Education Programs Based on surveys submitted by officials representing the 13 civilian federal agencies, table 21 contains a list of the 207 science, technology, engineering, and mathematics (STEM) education programs funded in fiscal year 2004. Table 21. Federal STEM Education Programs Funded in FY 2004 Program Program name number Department of Agriculture 1. 1890 Institution Teaching and Research Capacity Building Grants Program 2. Higher Education Challenge Grants Program 3. Hispanic-Serving Institutions Education Grants Program 4. Alaska Native-Serving and Native Hawaiian-Serving Institutions Education Grants Program 5. Food and Agricultural Sciences National Needs Graduate and Postdoctoral Fellowships Grants Program 6. Tribal Colleges Endowment Program 7. Tribal Colleges Education Equity Grants Program 8. Tribal Colleges Research Grant Program 9. Higher Education Multicultural Scholars Program 10. International Science and Education Competitive Grants Program 11. Secondary and Two-Year Postsecondary Agricultural Education Challenge Grants Program 12. Agriculture in the Classroom 13. Career Intern Program 14. Veterinary Medical Doctoral Program 15. 1890 National Scholars Program 16. Hispanic Scholars Program Department of Commerce 17. Educational Partnership Program with Minority Serving Institutions 18. National Marine Sanctuaries Education Program 19. National Sea Grant College Program 20. Chesapeake Bay Watershed Education and Training Program 21. Coral Reef Conservation Program 22. Exploration, Education and Outreach 23. National Estuarine Research Reserve Graduate Research Fellowship Program 24. Bay Watershed Education and Training Hawaii Program 25. Monterey Bay Watershed Education and Training Program 26. Dr. Nancy Foster Scholarship Program 27. EstuaryLive 28. Teacher at Sea Program 29. High School-High Tech Department of Education 30. Mathematics and Science Partnerships Program 31. Upward Bound Math and Science Program 32. Graduate Assistance in Areas of National Need 33. Minority Science and Engineering Improvement Program

Fiscal year 04 funding $11.4 million $4.6 million $4.6 million $3 million $2.9 million $1.9 million $1.7 million $1.1 million $986,000 $859,000 $839,000 $623,000 $272,000 $140,000 $16,000 $4,000 $7.4 million $4.4 million $4 million $2.5 million $1.8 million $1.3 million $1 million $500,000 $500,000 $494,000 $115,000 $95,000 $11,000 $149 million $32.8 million $30.6 million $8.9 million

Higher Education: Federal Science, Technology, Engineering…

217

Table 21. Continued Program Program name number Department of Energy 34. Science Undergraduate Laboratory Internship 35. Computational Science Graduate Fellowship 36. Global Change Education Program 37. Laboratory Science Teacher Professional Development 38. National Science Bowl 39. Community College Institute of Science and Technology 40. Albert Einstein Distinguished Educator Fellowship 41. QuarkNet 42. Fusion Energy Sciences Fellowship Program 43. Pre-Service Teacher Fellowships 44. National Undergraduate Fellowship Program in Plasma Physics and Fusion Energy Sciences 45. Fusion Energy Postdoctoral Research Program 46. Faculty and Student Teams 47. Advancing Precollege Science and Mathematics Education 48. Pan American Advanced Studies Institute 49. Trenton Community Partnership 50. Fusion/Plasma Education 51. National Middle School Science Bowl 52. Research Project on the Recruitment, Retention, and Promotion of Women in the Chemical Sciences 53. Used Energy Related Laboratory Equipment 54. Plasma Physics Summer Institute for High School Physics Teachers 55. Pre-Service Teacher Program 56. Wonders of Physics Traveling Show 57. Hampton University Graduate Studies 58. Contemporary Physics Education Project 59. Cooperative Education Program Environmental Protection Agency 60. Science to Achieve Results Research Grants Program 61. Science to Achieve Results Graduate Fellowship Program 62. Post-Doctoral Fellows Environmental Research Growth Opportunities 63. Intern Program 64. Environmental Science and Engineering Fellows Program 65. Greater Research Opportunities Graduate Fellowship Program 66. Environmental Risk and Impact in Communities of Color and Economically Disadvantaged Communities 67. Research Internship for Students in Ecology 68. National Network for Environmental Management Studies Fellowship Program 69. Cooperative Agreements for Training Cooperative Partnerships 70. University of Cincinnati/EPA Research Training Grant 71. P3 Award: National Student Design Competition for Sustainability 72. Environmental Protection Agency and the Hispanic Association of Colleges and Universities Cooperative Agreement 73. Environmental Science Program 74. Environmental Career Organization’s Internship Program 75. EPA—Cincinnati Research Apprenticeship Program 76. Environmental Protection Internship Program Summer Training Initiative 77. Tribal Lands Environmental Science Scholarship Program

Fiscal year 04 funding $2.5 million $2 million $1.4 million $1 million $702,000 $605,000 $600,000 $575,000 $555,000 $510,000 $300,000 $243,000 $215,000 $209,000 $200,000 $200,000 $125,000 $100,000 $100,000 $80,000 $78,000 $45,000 $45,000 $40,000 $23,000 $17,000 $93.3 million $10 million $7.4 million $3 million $2.5 million $1.5 million $824,000 $698,000 $589,000 $352,000 $300,000 $150,000 $121,000 $100,000 $89,000 $75,000 $72,000 $60,000

218

United States Government Accountability Office Table 21. Continued

Program Program name number 78. Internship Program for University of Arizona Engineering Students 79. Teacher Professional Development Workshop for Teachers Grade 6-12 80. Saturday Academy, Apprenticeships in Science and Engineering Program Department of Health and Human Services/Health Resources and Services Administration 81. Scholarships for Disadvantaged Students Program 82. Nursing Workforce Diversity 83. Faculty Loan Repayment Program Department of Health and Human Services/Indian Health Service 84. Indian Health Professions Scholarship 85. Health Professions Scholarship Program for Indians Department of Health and Human Services/National Institutes of Health 86. Ruth L. Kirschstein National Research Service Award Institutional Research Training Grants 87. Ruth L. Kirschstein National Research Service Awards for Individual Postdoctoral Fellows 88. Research Supplements to Promote Diversity in Health-Related Research 89. Postdoctoral Visiting Fellow Program 90. Clinical Research Loan Repayment Program 91. Ruth L. Kirschstein National Research Service Awards for Individual Predoctoral Fellows, Predoctoral Minority Students, and Predoctoral Students with Disabilities 92. Minority Access to Research Careers Program 93. Postdoctoral Intramural Research Training Award Program 94. Science Education Partnership Award 95. Pediatric Research Loan Repayment Program 96. Post-baccalaureate Intramural Research Training Award Program 97. Ruth L. Kirschstein National Research Service Award Short-Term Institutional Research Training Grants 98. Health Disparities Research Loan Repayment Program 99. Graduate Program Partnerships 100. Student Intramural Research Training Award Program 101. Career Opportunities in Research Education and Training Honors Undergraduate Research Training Grant 102. General Research Loan Repayment Program 103. Ruth L. Kirschstein National Research Service Awards for Individual M.D./Ph.D. Predoctoral Fellows 104. Science Education Drug Abuse Partnership Award 105. Pharmacology Research Associate Training Program 106. Technical Intramural Research Training Award 107. Fellowships in Cancer Epidemiology and Genetics 108. Clinical Research Loan Repayment Program for Individuals from Disadvantaged Backgrounds 109. Contraception and Infertility Research Loan Repayment Program 110. Medical Infomatics Training Program 111. Undergraduate Scholarship Program for Individuals from Disadvantaged Backgrounds 112. Curriculum Supplement Series 113. National Science Foundation and the National Institute of Biomedical Imaging and Bioengineering 114. Summer Institute for Training in Biostatistics

Fiscal year 04 funding $50,000 $18,000 $6,000 $45.5 million $16 million $1.1 million $8.1 million $3.7 million $546.9 million $72.6 million $70 million $64.8 million $40.6 million $33.8 million $30.7 million $30.2 million $16 million $15.9 million $9.1 million $9 million $8.7 million $7.4 million $6.3 million $5 million $4.9 million $4.7 million $3.1 million $2.7 million $1.9 million $1.8 million $1.7 million $1 million $853,000 $838,000 $788,000 $782,000 $694,000

Higher Education: Federal Science, Technology, Engineering…

219

Table 21. Continued Program Fiscal year Program name number 04 funding 115. Summer Institute on Design and Conduct of Randomized Clinical Trials $622,000 Involving Behavioral Interventions 116. Clinical Research Loan Repayment Program for Individuals from Disadvantaged $551,000 Background 117. Clinical Research Training Program $407,000 118. NIH Academy $385,000 119. Health Communications Internship Program $340,000 120. NIH/National Institute of Standards and Technology Joint Postdoctoral Program $338,000 121. Summer Genetics Institute $323,000 122. AIDS Research Loan Repayment Program $271,000 123. Intramural NIAID Research Opportunities $271,000 124. Cancer Research Interns in Residence $250,000 125. Comparative Molecular Pathology Research Training Program $199,000 126. Office of Research on Women’s Health-funded Programs with the Office of $179,000 Intramural Research 127. Summer Institute for Social Work Research $144,000 128. Office of Research on Women’s Health-funded Programs with the Office of $119,000 Intramural Training and Education 129. CCR/JHU Master of Science in Biotechnology Concentration in Molecular $111,000 Targets and Drug Discovery Technologies 130. Introduction to Cancer Research Careers $96,000 131. Fellows Award for Research Excellence Program $61,000 132. Office of Research on Women’s Health-funded Programs Supplements to $60,000 Promote Reentry into Biomedical and Behavioral Research Careers 133. Translational Research in Clinical Oncology $28,000 134. National Institute of Environmental Health Sciences Office of Fellows’ Career $20,000 Development 135. Mobilizing for Action to Address the Unequal Burden of Cancer: NIH Research $10,000 and Training Opportunities 136. Sallie Rosen Kaplan Fellowship for Women in Cancer Research $5,000 Department of Homeland Security 137. Scholars and Fellows Program $4.7 million Department of the Interior 138. Cooperative Research Units Program $15.3 million 139. Water Resources Research Act Program $6.4 million 140. U.S. Geological Survey Mendenhall Postdoctoral Research Fellowship Program $3.5 million 141. Student Educational Employment Program $1.8 million 142. EDMAP Component of the National Cooperative Geologic Mapping Program $490,000 143. Student Career Experience Program $177,000 144. Cooperative Development Energy Program $60,000 145. Diversity Employment Program $30,000 146. Cooperative Agreement with Langston University $15,000 147. Mathematics, Science, and Engineering Academy $15,000 148. Shorebird Sister Schools Program $15,000 149. Build a Bridge Contest $14,000 150. VIVA Technology $8,000 National Aeronautics and Space Administration 151. Minority University Research Education Program $106.6 million 152. Higher Education $77.4 million 153. Elementary and Secondary Education $31.3 million

220

United States Government Accountability Office Table 21. Continued

Program Program name number 154. E-Education 155. Informal Education National Science Foundation 156. Math and Science Partnership Program 157. Graduate Research Fellowship Program 158. Integrative Graduate Education and Research Traineeship Program 159. Teacher Professional Continuum 160. Research Experiences for Undergraduates 161. Graduate Teaching Fellows in K-12 Education 162. Advanced Technological Education 163. Course, Curriculum, and Laboratory Improvement 164. Research on Learning and Education 165. Computer Science, Engineering, and Mathematics Scholarships 166. Louis Stokes Alliances for Minority Participation 167. Centers for Learning and Teaching 168. Instructional Materials Development 169. Science, Technology, Engineering, and Mathematics Talent Expansion Program 170. Historically Black Colleges and Universities Undergraduate Program 171. Interagency Education Research Initiative 172. Information Technology Experiences for Students and Teachers 173. Enhancing the Mathematical Sciences Workforce in the 21st Century 174. Centers of Research Excellence in Science and Technology 175. ADVANCE: Increasing the Participation and Advancement of Women in Academic Science and Engineering Careers 176. Federal Cyber Service: Scholarship for Service 177. Alliances for Graduate Education and the Professoriate 178. Research on Gender in Science and Engineering 179. Tribal Colleges and Universities Program 180. Model Institutions for Excellence 181. Grants for the Department-Level Reform of Undergraduate Engineering Education 182. Robert Noyce Scholarship Program 183. Research Experiences for Teachers 184. Nanoscale Science and Engineering Education 185. Research in Disabilities Education 186. Opportunities for Enhancing Diversity in the Geosciences 187. Mathematical Sciences Postdoctoral Research Fellowships 188. Minority Postdoctoral Research Fellowships and Supporting Activities 189. Partnerships for Research and Education in Materials 190. Undergraduate Research Centers 191. Centers for Ocean Science Education Excellence 192. Undergraduate Mentoring in Environmental Biology 193. Director’s Award for Distinguished Teaching Scholars 194. Astronomy and Astrophysics Postdoctoral Fellowship Program 195. Geoscience Education 196. Internships in Public Science Education 197. Discovery Corps Fellowship Program 198. East Asia and Pacific Summer Institutes for U.S. Graduate Students 199. Pan-American Advanced Studies Institutes 200. Distinguished International Postdoctoral Research Fellowships

Fiscal year 04 funding $9.7 million $5.5 million $138.7 million $96 million $67.7 million $61.5 million $51.7 million $49.8 million $45.9 million $40.7 million $39.4 million $33.9 million $33.3 million $30.8 million $29.3 million $25 million $23.8 million $23.6 million $20.9 million $20.6 million $19.8 million $19.4 million $15.8 million $15.3 million $10 million $10 million $9.7 million $8.2 million $8 million $5.8 million $4.8 million $4.6 million $4 million $3.7 million $3.2 million $3 million $3 million $2.8 million $2.2 million $1.8 million $1.6 million $1.5 million $1.2 million $1.1 million $1 million $800,000 $788,000

Higher Education: Federal Science, Technology, Engineering…

221

Table 21. Continued Program Program name number 201. Postdoctoral Fellowships in Polar Regions Research 202. Arctic Research and Education 203. Developing Global Scientists and Engineers Department of Transportation 204. University Transportation Centers Program 205. Dwight David Eisenhower Transportation Fellowship Program 206. Summer Transportation Institute 207. Summer Transportation Internship Program for Diverse Groups

Fiscal year 04 funding $667,000 $300,000 $172,000 $32.5 million $2 million $2 million $925,000

Source: GAO survey responses from 13 federal agencies.

Appendix III: Federal STEM Education Programs Funded at $10 Million or More The federal civilian agencies reported that the following science, technology, engineering, and mathematics (STEM) education programs were funded with at least $10 million in either fiscal year 2004 or 2005. However, programs that received $10 million or more in fiscal year 2004 but were unfunded for fiscal year 2005 were excluded from table 22. Agency officials also provided the program descriptions in table 22. Table 22. Federal STEM Education Programs Funded at $10 Million or More during Fiscal Year 2004 or Fiscal Year2005 Funding (in millions of dollars)

Program

Description

a

First year

2004

2005

1990

$11.4

$12.5

Department of Agriculture 1890 Institution Teaching and Is intended to strengthen teaching and research programs in the Research Capacity Building food and agricultural sciences by building the institutional capacities Grants Program of the 1890 Land-Grant Institutions and Tuskegee University and West Virginia State University through cooperative linkages with federal and nonfederal entities. The program supports projects that strengthen teaching programs in the food and agricultural sciences in the targeted educational need areas of curriculum design and materials development, faculty preparation and enhancement of teaching, student experiential learning, and student recruitment and retention.

Department of Education Mathematics and Science Partnerships Program

Is intended to increase the academic achievement of students in mathematics and science by enhancing the content knowledge and teaching skills of classroom teachers. Partnerships are between high-need school districts and the science, technology, engineering, and mathematics faculties of institutions of higher education.

2002

$149

$180

Upward Bound Math and Science Program

Designed to prepare low-income, first-generation college students for postsecondary education programs that lead to careers in the fields of math and science.

1990

$32.8

$32.8

Graduate Assistance in Areas Provides fellowships in academic areas of national need to assist of National Need graduate students with excellent academic records who demonstrate financial need and plan to pursue the highest degree available in their courses of study.

1988

$30.6

$30.4

1995

$93.3

$80.1

1995

$10

EnvironmentalP rotection Agency Science to Achieve Results Research Grants Program

Funds research grants in numerous environmental science and engineering disciplines. The program engages the nation’s best scientists and engineers in targeted research. The grant program is currently focused on the health effects of particulate matter, drinking water, water quality, global change, ecosystem assessment and restoration, human health risk assessment, endocrine disrupting chemicals, pollution prevention and new technologies, children’s health, and socio-economic research.

Science to Achieve Results The purpose of this fellowship program is to encourage promising Graduate Fellowship Program students to obtain advanced degrees and pursue careers in environmentally related fields.

$10

222

United States Government Accountability Office Table 22. Continued Funding (in millions of dollars)

Program

Description

a

First year

2004

2005 Not avail.

Department of Health and HumanS ervices/Health Resources andS ervices Administration Scholarships for Disadvantaged Students Program

Funds are awarded to accredited schools of allopathic medicine, osteopathic medicine, dentistry, optometry, pharmacy, podiatric medicine, veterinary medicine, nursing, public health, chiropractic, or allied health, and schools offering graduate programs in behavioral and mental health practice. Priority is given to schools based on the proportion of graduating students going into primary care, the proportion of underrepresented minority students enrolled, and graduates working in medically underserved communities. Schools select qualified students and provide scholarships that cannot exceed tuition and reasonable educational and living expenses.

1991

$45.5

Nursing Workforce Diversity

To increase nursing education opportunities for individuals who are from disadvantaged backgrounds (including racial and ethnic minorities underrepresented among registered nurses) by providing student stipends, pre-entry preparation, and retention activities.

1989

$16

$16

Department of Health and HumanS ervices/NationalI nstitutes of Health Ruth L. Kirschstein National Research Service Award Institutional Research Training Grants

Is designed to develop and enhance research training opportunities for individuals in biomedical, behavioral, and clinical research by supporting training programs at institutions of higher education. These institutional training grants allow the director of the program to select the trainees and to develop a curriculum of study and research experiences necessary to provide high-quality research training. The grant helps offset the cost of stipends and tuition for the appointed trainees. Graduate students, postdoctoral trainees, and short-term research training for health professional students can be supported by this grant.

1975

$546.9

Not avail.

Ruth L. Kirschstein National Research Service Awards for Individual Postdoctoral Fellows

To support the advanced training of individual students who have recently received doctoral degrees. This phase of research education and training is performed under the direct supervision of a sponsor who is an active investigator in the area of the proposed research. The training is designed to enhance the fellow’s understanding of the health-related sciences and extend his/her potential to become a productive scientist who can perform research in biomedical, behavioral, or clinical fields.

1975

$72.6

Not avail.

Research Supplements to Promote Diversity in HealthRelated Research

To improve the diversity of the research workforce by recruiting and supporting students, postdoctoral fellows, and eligible investigators from groups that have been shown to be underrepresented, such as individuals from underrepresented racial and ethnic groups, individuals with disabilities, and individuals from disadvantaged backgrounds.

1989

$70

Postdoctoral Visiting Fellow Program

To provide advanced practical biomedical research experience to individuals who are foreign nationals and are 1 to 5 years beyond obtaining their Ph.D. or professional doctorate (e.g., M.D., DDS, etc.).

1950

$64.8

Program

Description

Clinical Research Loan Repayment Program

$70

$70.7

Funding (in millions of dollars)

a

First year

2004

2005

To attract health professionals to careers in clinical research. Clinical research is defined as “patient-oriented clinical research conducted with human subjects, or research on the causes and consequences of disease in human populations involving material of human origin (such as tissue specimens and cognitive phenomena) for which an investigator or colleague directly interacts with human subjects in an outpatient or inpatient setting to clarify a problem in human physiology, pathophysiology or disease, or epidemiologic or behavioral studies, outcomes research or health services research, or developing new technologies, therapeutic interventions, or clinical trials.”

2002

$40.6

$42.6

Ruth L. Kirschstein National Research Service Awards for Individual Predoctoral Fellows, Predoctoral Minority Students, and Predoctoral Students with Disabilities

Provides predoctoral fellowships to students who are candidates for doctoral degrees and are performing dissertation research and training under the supervision of a mentor who is an active and established investigator in the area of the proposed research. The applicant and mentor must provide evidence of potential for a productive research career based upon the quality of previous research training, academic record, and training program. The applicant and mentor must propose a research project that will enhance the student’s ability to understand and perform scientific research. The training program should be carried out in a research environment that includes appropriate resources and is demonstrably committed to the student’s training.

1975

$33.8

Not avail.

Minority Access to Research Careers Program

Offers special research training support to 4-year colleges, universities, and health professional schools with substantial enrollments of minorities such as African Americans, Hispanic Americans, Native Americans (including Alaska Natives), and natives of U.S. Pacific Islands. Individual fellowships are also provided for graduate students and faculty.

1972

$30.7

$30.7

Postdoctoral Intramural Research Training Award Program

To provide advanced practical biomedical research experience to individuals who are 1 to 5 years beyond obtaining their Ph.D. or professional doctorate (e.g., M.D., DDS, etc.).

1986

$30.2

$33.3

Science Education Partnership Award

Provides funds for the development, implementation, and evaluation of innovative kindergarten through 12th grade (K-12) science education programs, teaching materials, and science center/museum programs. This program supports partnerships linking biomedical, clinical researchers, and behavioral scientists with K-12 teachers and schools, museum and science educators, media experts, and other interested organizations.

1992

$16

$16

Pediatric Research Loan Repayment Program

A program to attract health professionals to careers in pediatric research. Qualified pediatric research is defined as “research directly related to diseases, disorders, and other conditions in children.”

2002

$15.9

$16

Higher Education: Federal Science, Technology, Engineering…

223

Table 22. Continued Funding (in millions of dollars)a

Program

Description

First year

Post-baccalaureate Intramural Research Training Award Program

To provide (1) recent college graduates (graduated no more than 2 years prior to activation of traineeship), an introduction early in their careers to biomedical research fields; encourage their pursuit of professional careers in biomedical research; and allow additional time to pursue successful application to either graduate or medical school programs or (2) students who have been accepted into graduate, other doctoral, or medical degree programs, and who have written permission from their school to delay entrance for up to 1 year.

2004

2005

1996

$9.1

$12.3

Provides scholarships for undergraduate and fellowships for graduate students pursuing degrees in mission-relevant fields and postdoctoral fellowships for their contributions to Department of Homeland Security research projects. Students receive professional mentoring and complete a summer internship to connect academic interests with homeland security initiatives. Postdoctoral scholars are also mentored by DHS scientists.

2003

$4.7

$10.7

The program links graduate science training with the research needs of state and federal agencies, and provides students with one-onone mentoring by federal research scientists working on both applied and basic research needs of interest to the program. Program cooperators and partners provide graduate training opportunities and support.

1936

$15.3

$15

To build the capacity of community colleges to train in high-growth, high-demand industries and to actually train workers in those industries through partnerships that also include workforce investment boards and employers.

2005

$0

$250

2002

$106.6

$73.6

Department of HomelandS ecurity University Programs

Department of theI nterior Cooperative Research Units Program

Department of Labor Community College/Community Based Job Training Grant Initiative

National Aeronautics andS pace Administration Minority University Research Education Program

To expand and advance NASA’s scientific and technological base through collaborative efforts with Historically Black Colleges and Universities (HBCU) and other minority universities (OMU), including Hispanic-serving institutions and Tribal colleges and universities. This program also provides K-12 awards to build and support successful pathways for students to progress to the next level of mathematics and science, through a college preparatory curriculum, and enrollment in college. Higher-education awards are also given that seek to improve the rate at which underrepresented minorities are awarded degrees in STEM disciplines through increased research training and exposure to cutting-edge technologies that better prepare them to enter STEM graduate programs, the NASA workforce pipeline, and employment in NASA-related industries.

Program

Description

Post-baccalaureate Intramural Research Training Award Program

To provide (1) recent college graduates (graduated no more than 2 years prior to activation of traineeship), an introduction early in their careers to biomedical research fields; encourage their pursuit of professional careers in biomedical research; and allow additional time to pursue successful application to either graduate or medical school programs or (2) students who have been accepted into graduate, other doctoral, or medical degree programs, and who have written permission from their school to delay entrance for up to 1 year.

Funding (in millions of dollars)a

First year

2004

2005

1996

$9.1

$12.3

Provides scholarships for undergraduate and fellowships for graduate students pursuing degrees in mission-relevant fields and postdoctoral fellowships for their contributions to Department of Homeland Security research projects. Students receive professional mentoring and complete a summer internship to connect academic interests with homeland security initiatives. Postdoctoral scholars are also mentored by DHS scientists.

2003

$4.7

$10.7

The program links graduate science training with the research needs of state and federal agencies, and provides students with one-onone mentoring by federal research scientists working on both applied and basic research needs of interest to the program. Program cooperators and partners provide graduate training opportunities and support.

1936

$15.3

$15

To build the capacity of community colleges to train in high-growth, high-demand industries and to actually train workers in those industries through partnerships that also include workforce investment boards and employers.

2005

$0

$250

2002

$106.6

$73.6

Department of HomelandS ecurity University Programs

Department of theI nterior Cooperative Research Units Program

Department of Labor Community College/Community Based Job Training Grant Initiative

National Aeronautics andS pace Administration Minority University Research Education Program

To expand and advance NASA’s scientific and technological base through collaborative efforts with Historically Black Colleges and Universities (HBCU) and other minority universities (OMU), including Hispanic-serving institutions and Tribal colleges and universities. This program also provides K-12 awards to build and support successful pathways for students to progress to the next level of mathematics and science, through a college preparatory curriculum, and enrollment in college. Higher-education awards are also given that seek to improve the rate at which underrepresented minorities are awarded degrees in STEM disciplines through increased research training and exposure to cutting-edge technologies that better prepare them to enter STEM graduate programs, the NASA workforce pipeline, and employment in NASA-related industries.

224

United States Government Accountability Office Table 22. Continued Funding (in millions of dollars)

Program

Description

Higher Education

a

First year

2004

2005

The Higher Education Program focuses on supporting institutions of higher education in strengthening their research capabilities and providing opportunities that attract and prepare increasing numbers of students for NASA-related careers. The research conducted by the institutions will contribute to the research needs of NASA’s Mission Directorates. The student projects serve as a major link in the student pipeline for addressing NASA’s human capital strategies and the President’s management agenda by helping to build, sustain, and effectively deploy the skilled, knowledgeable, diverse, and high-performing workforce needed to meet the current and emerging needs of government and its citizens.

2002

$77.4

$62.4

Elementary and Secondary Education

To increase the rigor of STEM experiences provided to K-12 students through workshops, summer internships, and classroom activities; provide high-quality professional development to teachers in STEM through NASA programs; develop technological avenues through the NASA Web site that will allow families to have common experiences with learning about space exploration; encourage inquiry teaching in K-12 classrooms; improve the content and focus of grade level/science team meetings in NASA Explorer Schools; and share the knowledge gained through the Educator Astronaut Program with teachers, students, and families.

2002

$31.3

$23.2

Informal Education

The principal purpose of the informal education program is to support projects designed to increase public interest in, understanding of, and engagement in STEM activities. The goal of all informal education programs is an informed citizenry that has access to the ideas of science and engineering and understands its role in enhancing the quality of life and the health, prosperity, welfare, and security of the nation. Informal learning is self-directed, voluntary, and motivated mainly by intrinsic interests, curiosity, exploration, and social interaction.

2002

$5.5

$10.2

Math and Science Partnership (MSP)Program

The MSP is a major research and development effort that supports innovative partnerships to improve kindergarten through grade 12 student achievement in mathematics and science. MSP projects are expected to both raise the achievement levels of all students and significantly reduce achievement gaps in the mathematics and science performance of diverse student populations. Successful projects serve as models that can be widely replicated in educational practice to improve the mathematics and science achievement of all the nation’s students.

2002

$138.7

$79.4

Graduate Research Fellowship Program (GRFP)

The purpose of the GRFP is to ensure the vitality of the scientific and technological workforce in the United States and to reinforce its diversity. The program recognizes and supports outstanding graduate students in the relevant science and engineering disciplines who are pursuing research-based master’s and doctoral degrees. NSF fellows are expected to become knowledge experts who can contribute significantly to research, teaching, and innovations in science and engineering.

1952

$96

$96.6

Program

Description

Integrative Graduate Education and Research Traineeship Program

NationalS cienceF oundation

a

Funding (in millions of dollars)

First year

2004

2005

This program provides support to universities for student positions in interdisciplinary areas of science and engineering. Traineeships focus on multidisciplinary and intersectoral research opportunities and prepare future faculty in effective teaching methods, applications of advanced educational technologies, and student mentoring techniques.

1998

$67.7

$69

Teacher Professional Continuum

The program addresses critical issues and needs regarding the recruitment, preparation, induction, retention, and lifelong development of kindergarten through grade 12 STEM teachers. Its goals are to improve the quality and coherence of teacher learning experiences across the continuum through research that informs teaching practice and the development of innovative resources for the professional development of kindergarten through grade 12 STEM teachers.

2004

$61.5

$60.2

Research Experiences for Undergraduates

This program supports active participation by undergraduate students in research projects in any of the areas of research funded by the National Science Foundation. The program seeks to involve students in meaningful ways in all kinds of research—whether disciplinary, interdisciplinary, or educational in focus—linked to the efforts of individual investigators, research groups, centers, and national facilities. Particular emphasis is given to the recruitment of women, minorities, and persons with disabilities.

1987

$51.7

$51.1

Graduate Teaching Fellows in This program supports fellowships and associated training that K-12 Education enable graduate students in NSF-supported STEM disciplines to acquire additional skills that will broadly prepare them for professional and scientific careers. Through interactions with teachers, graduate students can improve communication and teaching skills while enriching STEM instruction in kindergarten through grade 12 schools. This program also provides institutions of higher education with an opportunity to make a permanent change in their graduate programs by including partnerships with schools in a manner that will mutually benefit faculties and students.

1999

$49.8

$49.9

Advanced Technological Education (ATE)

With an emphasis on 2-year colleges, the ATE program focuses on the education of technicians for the high-technology fields that drive our nation’s economy. The program involves partnerships between academic institutions and employers to promote improvement in the education of science and engineering technicians at the undergraduate and secondary school levels. The ATE program supports curriculum development, professional development of college faculty and secondary school teachers, career pathways to 2-year colleges from secondary schools and from 2-year colleges to 4-year institutions, and other activities. The program also invites proposals focusing on applied research relating to technician education.

1994

$45.9

$45.1

Course, Curriculum, and Laboratory Improvement

This program emphasizes projects that build on prior work and contribute to the knowledge base of undergraduate STEM education research and practice. In addition, projects should contribute to building a community of scholars who work in related areas of undergraduate education.

1999

$40.7

$40.6

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225

Table 22. Continued Funding (in millions of dollars)

Program

Description

Research on Learning and Education

a

First year

2004

2005

The program seeks to capitalize on important developments across a wide range of fields related to human learning and to STEM education. It supports research across a continuum that includes (1) the biological basis of human learning; (2) behavioral, cognitive, affective, and social aspects of STEM learning; (3) STEM learning in formal and informal educational settings; (4) STEM policy research; and (5) the diffusion of STEM innovations.

2000

$39.4

$38.2

Computer Science, Engineering, and Mathematics Scholarships

This program supports scholarships for academically talented, financially needy students, enabling them to enter the hightechnology workforce following completion of an associate, baccalaureate, or graduate-level degree in computer science, computer technology, engineering, engineering technology, or mathematics. Academic institutions apply for awards to support scholarship activities and are responsible for selecting scholarship recipients, reporting demographic information about student scholars, and managing the project at the institution.

1999

$33.9

$75

Louis Stokes Alliances for Minority Participation

The program is aimed at increasing the quality and quantity of students successfully completing STEM baccalaureate degree programs and increasing the number of students interested in, academically qualified for, and matriculated into programs of graduate study. It also supports sustained and comprehensive approaches that facilitate achievement of the long-term goal of increasing the number of students who earn doctorates in STEM, particularly those from populations underrepresented in STEM fields.

1991

$33.3

$35

Centers for Learning and Teaching

The program focuses on the advanced preparation of STEM educators, as well as the establishment of meaningful partnerships among education stakeholders, especially Ph.D.-granting institutions, school systems, and informal education performers. Its goals are to renew and diversify the cadre of leaders in STEM education; to increase the number of kindergarten through undergraduate educators capable of delivering high-quality STEM instruction and assessment; and to conduct research into STEM education issues of national import, such as the nature of learning, teaching strategies, and reform policies and outcomes.

2000

$30.8

$28.4

Instructional Materials Development

This program contains three components. It supports (1) the creation and substantial revision of comprehensive curricula and supplemental materials that are research-based, enhance classroom instruction, and reflect standards for science, mathematics, and technology education developed by professional organizations; (2) the creation of tools for assessing student learning that are tied to nationally developed standards and reflect the most current thinking on how students learn mathematics and science; and (3) research for development of this program and projects.

1983

$29.3

$28.5

Program

Description

Science, Technology, Engineering, and Mathematics Talent Expansion Program

The program seeks to increase the number of students (U.S. citizens or permanent residents) receiving associate or baccalaureate degrees in established or emerging fields within STEM. Type 1 proposals that provide for full implementation efforts at academic institutions are solicited. Type 2 proposals that support educational research projects on associate or baccalaureate degree attainment in STEM are also solicited.

Historically Black Colleges and Universities (HBCU) Undergraduate Program

Interagency Education Research Initiative

Funding (in millions of dollars)

First year

a

2004

2005

2002

$25

$25.3

This program provides awards to enhance the quality of STEM instructional and outreach programs at HBCUs as a means to broaden participation in the nation’s STEM workforce. Project strategies include curriculum enhancement, faculty professional development, undergraduate research, academic enrichment, infusion of technology to enhance STEM instruction, collaborations with research institutions and industry, and other activities that meet institutional needs.

1998

$23.8

$25.2

This is a collaborative effort with the U.S. Department of Education. The goal is to support scientific research that investigates the effectiveness of educational interventions in reading, mathematics, and the sciences as they are implemented in varied school settings with diverse student populations.

1999

$23.6

$13.8

Information Technology The program is designed to increase the opportunities for students Experiences for Students and and teachers to learn about, experience, and use information Teachers technologies within the context of STEM, including information technology courses. It is in direct response to the concern about shortages of technology workers in the United States. It has two components: (1) youth-based projects with strong emphasis on career and educational paths and (2) comprehensive projects for students and teachers.

2003

$20.9

$25

Enhancing the Mathematical Sciences Workforce in the 21st Century

The long-range goal of this program is to increase the number of U.S. citizens, nationals, and permanent residents who are well prepared in the mathematical sciences and who pursue careers in the mathematical sciences and in other NSF-supported disciplines.

2004

$20.6

$20.7

Centers of Research Excellence in Science and Technology

This program makes resources available to significantly enhance the research capabilities of minority-serving institutions through the establishment of centers that effectively integrate education and research. It promotes the development of new knowledge, enhancements of the research productivity of individual faculty, and an expanded diverse student presence in STEM disciplines.

1987

$19.8

$15.9

ADVANCE: Increasing the Participation and Advancement of Women in Academic Science and Engineering Careers

The program goal is to increase the representation and advancement of women in academic science and engineering careers, thereby contributing to the development of a more diverse science and engineering workforce. Members of underrepresented minority groups and individuals with disabilities are especially encouraged to apply.

2001

$19.4

$19.8

226

United States Government Accountability Office Table 22. Continued Funding (in millions of dollars)a

Program

Description

First year

Federal Cyber Service: Scholarship for Service

2004

2005

This program seeks to increase the number of qualified students entering the fields of information assurance and computer security and to increase the capacity of the United States’ higher education enterprise to continue to produce professionals in these fields to meet the needs of our increasingly technological society. The program has two tracks: provides funds to colleges and universities to (1) award scholarships to students to pursue academic programs in the information assurance and computer security fields for the final 2 years of undergraduate study, or for 2 years of master’s-level study, or for the final 2 years of Ph.D.-level study, and (2) improve the quality and increase the production of information assurance and computer security professionals.

2001

$15.8

$14.1

Alliances for Graduate Education and the Professoriate

This program is intended to increase significantly the number of domestic students receiving doctoral degrees in STEM, with special emphasis on those population groups underrepresented in these fields. The program is interested in increasing the number of minorities who will enter the professoriate in these disciplines. Specific objectives are to develop (1) and implement innovative models for recruiting, mentoring, and retaining minority students in STEM doctoral programs, and (2) effective strategies for identifying and supporting underrepresented minorities who want to pursue academic careers.

1998

$15.3

$14.8

Research on Gender in Science and Engineering

The program seeks to broaden the participation of girls and women in all fields of STEM education by supporting research, dissemination of research, and extension services in education that will lead to a larger and more diverse domestic science and engineering workforce. Typical projects will contribute to the knowledge base addressing gender-related differences in learning and in the educational experiences that affect student interest, performance, and choice of careers, and how pedagogical approaches and teaching styles, curriculum, student services, and institutional culture contribute to causing or closing gender gaps that persist in certain fields.

1993

$10

$9.8

Tribal Colleges and Universities Program

This program provides awards to enhance the quality of STEM instructional and outreach programs, with special attention to the use of information technologies at Tribal colleges and universities, Alaskan Native-serving institutions, and Native Hawaiian-serving institutions. Support is available for the implementation of comprehensive institutional approaches to strengthen STEM teaching and learning in ways that improve access to, retention within, and graduation from STEM programs, particularly those that have a strong technological foundation. Through this program, assistance is provided to eligible institutions in their efforts to bridge the digital divide and prepare students for careers in information technology, science, mathematics, and engineering fields.

2001

$10

$9.8

Program

Description

Funding (in millions of dollars)

a

First year

2004

2005

1998

$32.5

$32.5

Department of Transportation University Transportation Centers Program (UTC)

The UTC program’s mission is to advance U.S. technology and expertise in the many disciplines comprising transportation through the mechanisms of education, research, and technology transfer at university-based centers of excellence. The UTC program’s goals include (1) developing a multidisciplinary program of coursework and experiential learning that reinforces the transportation theme of the center; (2) increasing the numbers of students, faculty, and staff who are attracted to and substantially involved in the undergraduate, graduate, and professional programs of the center; and (3) having students, faculty, and staff who reflect the growing diversity of the U.S. workforce and are substantially involved in the undergraduate, graduate, and professional programs of the center. Source: GAO survey responses from 13 federal agencies.

a The dollar amounts for fiscal years 2004 and 2005 contain actual and estimated program funding levels.

Appendix IV: Data on Students and Graduates in STEM Fields Table 23 provides estimates for the numbers of students in science, technology, engineering, and mathematics (STEM) fields by education level for the 1995-1996 and 2003-2004 academic years. Tables 24 and 25 provide additional information regarding students in STEM fields by gender for the 1995-1996 and 2003-2004 academic years. Table 26 provides

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additional information regarding graduates in STEM fields by gender for the 1994-1995 and 2002-2003 academic years. Appendix V contains confidence intervals for these estimates. Table 23. Estimated Numbers of Students in STEM Fields by Education Level for Academic Years 1995-1996 and 2003-2004 Education level/STEM field

Academic year 1995-1996

Academic year 2003-2004

Percentage change

Bachelor’s level Total 2,218,510 2,876,721 30 Agricultural sciences 101,885 87,025 b Biological sciences 407,336 351,595 -14 Computer sciences 261,139 456,303 75 Engineering 363,504 422,230 16 Mathematics 57,133 64,307 b Physical sciences 107,832 129,207 b Psychology 309,810 409,827 32 Social sciences 536,487 825,495 54 Technology 73,384 130,733 78 Master’s level Total 321,293 403,200 25 Agricultural sciences a 12,977 a Biological sciences 34,701 19,467 -44 Computer sciences 49,071 58,939 b Engineering 66,296 90,234 b Mathematics a 12,531 a Physical sciences a 22,008 a Psychology 30,008 31,918 b Social sciences 82,177 144,895 76 Technology a 10,231 a Doctoral level Total 217,395 198,504 b Agricultural sciences a 5,983 a Biological sciences a 33,884 a Computer sciences a 9,196 a Engineering 32,181 35,687 b Mathematics a 9,412 a Physical sciences 38,058 24,973 b Psychology 30,291 33,994 b Social sciences 54,092 42,464 b Technology a 2,912 a Source: GAO calculations based upon NPSAS data. Note: Enrollment totals differ from those cited in table 9 because table 9 includes students enrolled in certificate, associate’s, other undergraduate, first-professional degree, and post-bachelor’s or postmaster’s certificate programs. a Sample sizes are insufficient to accurately produce estimates. b Changes between academic years 1995-1996 and 2003-2004 are not statistically significant at the 95-percent confidence level. See table 30 for significance of percentage changes.

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United States Government Accountability Office

Table 24. Estimated Percentages of Students by Gender and STEM Field for Academic Years 1995-1996 and 2003-2004 Male Female Agricultural Percent: 1995-1996 Percent: 2003-2004 Percent: 1995-1996 Percent: 2003-2004 sciences Total 58 55 42 45 Bachelor’s 56 54 44 46 Master’s a a a a Doctorate a 61 a 39 Biological sciences Total 46 42 54 58 Bachelor’s 45 42 55 58 Master’s a 26 a 74 Doctorate a 50 a 50 Computer sciences Total 67 76 33 24 Bachelor’s 69 77 31 23 Master’s a 69 a 31 Doctorate a 72 a 28 Engineering Total 83 83 17 17 Bachelor’s 83 83 17 17 Master’s a 81 a 19 Doctorate a 78 a 22 Mathematics Total 62 55 38 45 Bachelor’s 57 54 43 46 Master’s a a a a Doctorate a 68 a 32 Physical sciences Total 62 56 38 44 Bachelor’s 56 53 44 47 Master’s a a a a Doctorate a 68 a 32 Psychology Total 26 26 74 74 Bachelor’s 26 26 74 74 Master’s a 21 a 79 Doctorate a 30 a 70 Social sciences Total 54 41 46 59 Bachelor’s 52 42 48 58 Master’s 51 35 49 65 Doctorate 83 46 17 54 Technology Total 89 81 11 19 Bachelor’s 88 81 12 19 Master’s a a a a Doctorate a a a a Source: GAO calculations based upon NPSAS data. a Sample sizes are insufficient to accurately produce estimates.

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Table 25. Estimated Number of Women Students and Percentage Change by Education Level and STEM Field for Academic Years 1995-1996 and 2003-2004 Number of women students Education level/STEM field

1995-1996

2003-2004

Percentage change in women students b b b b b b +33 +84 +184 a a a a a a b +133 a a a a a a a a +143 a

Bachelor’s level Agricultural sciences 44,444 39,702 Biological sciences 222,323 203,038 Computer sciences 82,013 104,824 Engineering 59,985 70,353 Mathematics 24,597 29,791 Physical sciences 47,421 60,203 Psychology 229,772 304,712 Social sciences 258,023 475,544 Technology 8,871 25,227 Master’s level Agricultural sciences a a Biological sciences a 14,415 Computer sciences a 18,000 Engineering a 17,042 Mathematics a 5,562 Physical sciences a 8,497 Psychology 23,857 25,342 Social sciences 40,395 94,169 Technology a 1,280 Doctoral level Agricultural sciences a 2,353 Biological sciences a 17,074 Computer sciences a 2,556 Engineering a 7,868 Mathematics a 3,042 Physical sciences a 8,105 Psychology a 23,843 Social sciences 9,440 22,931 Technology a 692 Source: GAO calculations based upon NPSAS data. a Sample sizes are insufficient to accurately produce estimates. b Changes between academic years 1995-1996 and 2003-2004 are not statistically significant at the95-percent confidence level. See table 29 for confidence intervals.

Table 26. Comparisons in the Percentage of STEM Graduates by Field and Gender for Academic Years 1994-1995 and 2002-2003 Percentage Percentage Percentage Percentage graduates, men, graduates, men, graduates, women graduates, women 1994-1995 2002-2003 1994-1995 2002-2003

STEM Degree/field Bachelor’s degree Biological/agricultural sciences Earth, atmospheric, and ocean sciences Engineering Mathematics and computer sciences Physical sciences Psychology Social sciences

50 66

40 58

50 34

60 42

83 65 64 27 50

80 69 58 22 45

17 35 36 73 50

20 31 42 78 55

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United States Government Accountability Office Table 26. Continued Percentage Percentage Percentage Percentage graduates, men, graduates, men, graduates, women graduates, women 1994-1995 2002-2003 1994-1995 2002-2003

Technology 90 Master’s degree Biological/agricultural sciences 52 Earth, atmospheric, and ocean 70 sciences Engineering 84 Mathematics and computer sciences 70 Physical sciences 70 Psychology 28 Social sciences 51 Technology 81 Doctoral degree Biological/agricultural sciences 63 Earth, atmospheric, and ocean 78 sciences Engineering 88 80 Mathematics and computer sciences Physical sciences 76 Psychology 38 Social sciences 62 Technology 89 Source: GAO calculations based upon IPEDS data.

88

10

12

45 59

48 30

55 41

79 66 64 23 45 73

16 30 30 72 49 19

21 34 36 77 55 27

57 72

37 22

43 28

83 76 73 31 55 100

12 20 24 62 38 11

17 24 27 69 45 0

Appendix V: Confidence Intervals for Estimates of Students at the Bachelor’s, Master’s, and Doctoral Levels Because the National Postsecondary Student Aid Study (NPSAS) sample is a probability sample of students, the sample is only one of a large number of samples that might have been Table 27. Estimated Changes in the Numbers and Percentages of Students in the STEM and Non-STEM Fields across All Education Levels, Academic Years 1995-1996 and 2003-2004 (95 percent confidence intervals) Lower and upper bounds of 95 percent confidence interval

STEM field

Non-STEM field

Lower bound: number of students: 1995-1996

3,941,589

14,885,171

Upper bound: number of students: 1995-1996

4,323,159

15,601,065

Lower bound: percentage of students: 1995-1996

20

78

Upper bound: percentage of students: 1995-1996

22

80

Lower bound: number of students: 2003-2004

4,911,850

16,740,049

Upper bound: number of students: 2003-2004

5,082,515

17,025,326

Lower bound: percentage of students: 2003-2004

22

77

Upper bound: percentage of students: 2003-2004

23

78

Lower bound: percentage change: 1995/96-2003/04

15

8

Upper bound: percentage change: 1995/96-2003/04

26.9

13.5

Source: GAO calculations based upon 1995-1996 and 2003-2004 NPSAS data. Note: The totals for STEM and non-STEM enrollments include students in addition to the bachelor’s, master’s, and doctorate education levels. These totals also include students enrolled in certificate, associate’s, other undergraduate, first-professional degree, and post-bachelor’s or post-master’s certificate programs. The percentage changes between the 1995-1996 and 2003-2004 academic years for STEM and non-STEM students are statistically significant.

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drawn. Since each sample could have provided different estimates, confidence in the precision of the particular sample’s results is expressed as a 95-percent confidence interval (for example, plus or minus 4 percentage points). This is the interval that would contain the actual population value for 95 percent of the samples that could have been drawn. As a result, we are 95 percent confident that each of the confidence intervals in this report will include the true values in the study population. The upper and lower bounds of the 95 percent confidence intervals for each estimate relied on in this report are presented in the following tables. Table 28. Numbers of Students by Education Level in all STEM Fields for Academic Years 1995-1996 and 2003-2004 (95 percent confidence intervals) Total

Agricultural Sciences

Biological Sciences

Computer Sciences

Engineering

Mathematics

Physical Sciences

Psychology

Social Sciences

Technology

Total

Bachelors

Masters

Doctorate

Lower bound: Number of Students: 1995-1996

2,633,867

2,114,316

271,208

171,824

Upper bound: Number of Students: 1995-1996

2,880,529

2,322,704

377,821

271,230

Lower bound: Number of Students: 2003-2004

3,411,004

2,819,206

366,141

185,230

Upper bound: Number of Students: 2003-2004

3,545,844

2,934,236

442,938

212,471

Lower bound: Number of Students: 1995-1996

93,346

78,241

a

a a

Upper bound: Number of Students: 1995-1996

151,132

130,144

a

Lower bound: Number of Students: 2003-2004

93,543

76,472

7,296

4,661

Upper bound: Number of Students: 2003-2004

119,613

98,590

21,202

7,553

Lower bound: Number of Students: 1995-1996

416,315

360,553

18,883

a

Upper bound: Number of Students: 1995-1996

524,615

454,119

57,066

a

Lower bound: Number of Students: 2003-2004

383,277

330,834

13,728

30,401

Upper bound: Number of Students: 2003-2004

427,502

372,355

26,694

37,367

Lower bound: Number of Students: 1995-1996

275,804

224,616

31,634

a

Upper bound: Number of Students: 1995-1996

363,084

297,662

71,242

a

Lower bound: Number of Students: 2003-2004

495,359

428,927

47,669

7,427

Upper bound: Number of Students: 2003-2004

554,747

483,679

70,210

11,243

Lower bound: Number of Students: 1995-1996

411,868

321,464

45,912

16,620

Upper bound: Number of Students: 1995-1996

516,391

405,544

90,768

54,155

Lower bound: Number of Students: 2003-2004

514,794

400,252

63,632

32,113

Upper bound: Number of Students: 2003-2004

583,058

444,208

116,835

39,261

Lower bound: Number of Students: 1995-1996

68,083

42,910

a

a a

Upper bound: Number of Students: 1995-1996

119,165

74,456

a

Lower bound: Number of Students: 2003-2004

75,705

55,314

7,869

7,687

Upper bound: Number of Students: 2003-2004

97,848

74,318

18,867

11,392

Lower bound: Number of Students: 1995-1996

139,416

87,966

a

21,279

Upper bound: Number of Students: 1995-1996

214,274

130,658

a

60,546

Lower bound: Number of Students: 2003-2004

160,895

116,479

14,944

22,043

Upper bound: Number of Students: 2003-2004

192,534

142,894

31,092

27,903

Lower bound: Number of Students: 1995-1996

327,359

271,188

17,600

16,929

Upper bound: Number of Students: 1995-1996

416,804

348,432

47,037

48,601

Lower bound: Number of Students: 2003-2004

449,858

385,660

24,218

27,846

Upper bound: Number of Students: 2003-2004

502,696

433,995

41,116

40,142

Lower bound: Number of Students: 1995-1996

608,199

478,659

60,792

33,489

Upper bound: Number of Students: 1995-1996

742,107

594,315

103,562

79,414

Lower bound: Number of Students: 2003-2004

974,279

791,462

125,457

38,291

Total

Bachelors

Masters

Doctorate

Upper bound: Number of Students: 2003-2004

1,052,506

859,527

164,333

46,636

Lower bound: Number of Students: 1995-1996

63,910

57,446

a

a

Upper bound: Number of Students: 1995-1996

104,308

92,251

a

a

Lower bound: Number of Students: 2003-2004

130,347

118,492

5,556

1,814

Upper bound: Number of Students: 2003-2004

158,418

143,848

17,158

4,421

Source: GAO calculations based upon 1995-1996 and 2003-2004 NPSAS data. a Sample sizes are insufficient to accurately produce estimates.

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United States Government Accountability Office

Table 29. Estimated Numbers and Percentage Changes in Women Students in STEM Fields, Academic Years 1995-1996 and2003-2004 (95 percent confidence intervals)

Source: GAO calculations based upon NPSAS data. a Sample sizes are insufficient to accurately produce estimates.

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Table 30. Estimated Percentage Changes in Bachelor’s, Master’s, and Doctoral Students in STEM Fields, Academic Years 1995-1996 and 2003-2004 (95 percent confidence intervals)

Lower and upper bounds of 95 percent confidence interval STEM fields

Percentage change in academic years 1995-1996 and 2003-2004

Total

Bachelor’s

Master’s

Doctoral

Agricultural sciences

Lower bound: percentage change

-34.8

-38.7

a

a

11.9

9.5

a

a

no

no

a

a

Lower bound: percentage change

-24.4

-24.8

-79.6

a

Upper bound: percentage change

-2.6

-2.5

-8.3

a

Upper bound: percentage change Statistically significant Biological sciences

yes

yes

yes

a

Lower bound: percentage change

41.1

48.1

-34.8

a

Upper bound: percentage change

89.5

101.3

75

a

Statistically significant

yes

yes

no

a

Lower bound: percentage change

3.5

1.4

-27.5

-55.4

Upper bound: percentage change

33.8

30.9

99.7

77.2

yes

yes

no

no

Lower bound: percentage change

-33.5

-21.8

a

a

Upper bound: percentage change

23

46.9

a

a a

Statistically significant Computer sciences

Engineering

Statistically significant Mathematics

no

no

a

Lower bound: percentage change

-21.7

-6.6

a

-70.2

Upper bound percentage change

24.4

46.3

a

1.4

no

no

a

no

Lower bound: percentage change

11.7

14

-51.2

-48.8

Upper bound: percentage change

45.4

50.5

63.9

73.3

yes

yes

no

no

Lower bound: percentage change

34.6

36.1

24.7

-59.3

Upper bound: percentage change

66.5

71.6

127.9

16.3

yes

yes

yes

no

Lower bound: percentage change

30

33.4

a

a

Upper bound: percentage change

119.6

122.9

a

a

yes

yes

a

a

Lower bound: percentage change

20

23.1

1.8

-29.5

Upper bound: percentage change

32.3

36.3

49.2

12.1

yes

yes

yes

no

Statistically significant Physical sciences

Statistically significant Psychology

Statistically significant Social sciences

Statistically significant Technology

Statistically significant

Total

Statistically significant

Source: GAO calculations based upon 1995-1996 and 2003-2004 NPSAS data. a Sample sizes are insufficient to accurately produce estimates.

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United States Government Accountability Office

Table 31. Estimates of STEM Students by Gender and Field for Academic Years 19951996 and 2003-2004 (95 percent confidence intervals)

Source: GAO calculations based upon 1995-1996 and 2003-2004 NPSAS data. a Sample sizes are insufficient to accurately produce estimates.

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Table 32. Estimates of Students for Selected Racial or Ethnic Groups in STEM Fields for All Education Levels and Fields for the Academic Years 1995-1996 and 2002-2003 (95 percent confidence intervals) Lower bound: number of students, academic year, 1995-1996

Race or ethnicity African American Hispanic

285,3

Asian/Pacific Islander

Upper bound: number of students, academic year, 1995-1996

Lower bound: number of students, academic year, 2003-2004

Upper bound: number of students, academic year, 2003-2004

303,832

416,502

577,854

639,114

81

446,621

461,738

515,423 367,377

247,347

330,541

322,738

Native American

11,464

28,103

30,064

47,694

Other/multiple minorities

17,708

44,434

150,264

183,174

Source: GAO Calculations based upon 1995-1996 and 2003-2004 NPSAS data.

Table 33. Estimates of International Students in STEM Fields by Education Levels for Academic Years 1995-1996 and 2003-2004 (95 percent confidence intervals)

Education level Total Bachelor’s

Lower bound: number of students, 1995-1996

Upper bound: number of students, 1995-1996

Lower bound: number of students, 2003-2004

2

142,192

154,466

186,322

12

102

4

47,684

125,950

154,911

155

523

80,81 20,25

Upper bound: number of students, 2003-2004

Lower bound: Upper bound: percentage percentage change change

Master’s

23,06

3

64,587

16,359

29,899

-76

-13

Doctoral

20,52

5

59,861

5,168

10,735

-90

-68

Source: GAO calculations based upon 1995-1996 and 2003-2004 NPSAS data.

Appendix VI: Confidence Intervals for Estimates of STEM Employment by Gender, Race or Ethnicity, and Wages and Salaries The current population survey (CPS) was used to obtain estimates about employees and wages and salaries in science, technology, engineering, and mathematics (STEM) fields. Because the current population survey (CPS) is a probability sample based on random selections, the sample is only one of a large number of samples that might have been drawn. Since each sample could have provided different estimates, confidence in the precision of the particular sample’s results is expressed as a 95 percent confidence interval (e.g., plus or minus 4 percentage points). This is the interval that would contain the actual population value for 95 percent of the samples that could have been drawn. As a result, we are 95 percent confident that each of the confidence intervals in this report will include the true values in the study population. We use the CPS general variance methodology to estimate this sampling error and report it as confidence intervals. Percentage estimates we produce from the CPS

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United States Government Accountability Office

data have 95 percent confidence intervals of plus or minus 6 percentage points or less. Estimates other than percentages have 95 percent confidence intervals of no more than plus or minus 10 percent of the estimate itself, unless otherwise noted. Consistent with the CPS documentation guidelines, we do not produce estimates based on the March supplement data for populations of less than 75,000. Table 34. Estimated Total Number of Employees by STEM Field between Calendar Years 1994 and 2003

STEM fields Science

Lower bound: calendar year 1994

Upper bound: calendar year 1994

Lower bound: calendar year 2003

Upper bound: calendar year 2003

Statistically significant

2,349,605

2,656,451

2,874,347

3,143,071

yes

Technology

1,285,

321

1,515,671

1,379,375

1,568,189

no

Engineering

1,668,

514

1,929,240

1,638,355

1,843,427

no

2,520,858

2,773,146

yes

Mathematics/ computer sciences

1,369,047

1,606,

395

Table 35. Estimated Numbers of Employees in STEM Fields by Gender for Calendar Years 1994 and 2003

STEM fields Science

Lower Lower Upper Upper Lower Upper Lower Upper bound: bound: bound: bound: bound: bound: bound: bound: calendar calendar calendar calendar calendar calendar calendar calendar year year year year year year year year 1994, 1994, 2003, 2003, Statistically 1994, 1994, 2003, 2003, Statistically women women women women significant men men men men significant 925,548

no

Technology

1,594,527 1,827,685 2,031,124 2,327,390 385,433

505,329

357,805

489,899

yes no

941,960 1,157,900

no

Engineering

107,109

174,669

126,947

210,407

no 1,538,198 1,777,778 1,440,510 1,703,920

no

Mathematics/ computer sciences

372,953

491,053

610,649

779,525

yes

708,673

875,171

863,785 1,046,445

733,358

959,765 1,151,681 1,805,505 2,098,325

yes

Table 36. Estimated Changes in STEM Employment by Gender for Calendar Years 1994 and 2003

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Table 37. Estimated Percentages of STEM Employees for Selected Racial or Ethnic Groups for Calendar Years 1994 and 2003

Table 38. Estimated Changes in Median Annual Wages and Salaries in the STEM Fields for Calendar Years 1994 and 2003

Appendix VII: Comments from the Department of Commerce

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Appendix VIII: Comments from the Department of Health and Human Services

239

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Appendix IX: Comments from the National Science Foundation

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241

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Appendix X: Comments from the National Science and Technology Council

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Acknowledgments Cornelia M. Ashby, Carolyn M. Taylor, Assistant Director; Tim Hall, Analyst in Charge; Mark Ward; Dorian Herring; Patricia Bundy; Paula Bonin; Scott Heacock; Wilfred Holloway; Lise Levie; John Mingus; Mark Ramage; James Rebbe; and Monica Wolford made key contributions to this report.

Bibliography Congressional Research Service, Foreign Students in the United States: Policies and Legislation, RL31146, January 24, 2003, Washington, D.C. Congressional Research Service, Immigration: Legislative Issues on Nonimmigrant Professional Specialty (H-1B) Workers, RL30498, May 5, 2005, Washington, D.C. Congressional Research Service, Monitoring Foreign Students in the United States: The Student and Exchange Visitor Information System (SEVIS), RL32188, October 20, 2004, Washington, D.C. Congressional Research Service, Science, Engineering, and Mathematics Education: Status and Issues, 98-871 STM, April 27, 2004, Washington, D.C. Council on Competitiveness, Innovate America, December 2004, Washington, D.C. Council of Graduate Schools, NDEA 21: A Renewed Commitment to Graduate Education, June 2005, Washington, D.C. Institute of International Education, Open Doors: Report on International Educational Exchange, 2004, New York. Jackson, Shirley Ann, The Quiet Crisis: Falling Short in Producing American Scientific and Technical Talent, Building Engineering and Science Talent, September 2002, San Diego, California. NAFSA: Association of International Educators, In America’s Interest: Welcoming International Students, Report of the Strategic Task Force on International Student Access, January 14, 2003, Washington, D.C. NAFSA: Association of International Educators, Toward an International Education Policy for the United States: International Education in an Age of Globalism and Terrorism, May 2003, Washington, D.C. National Center for Education Statistics, Qualifications of the Public School Teacher Workforce: Prevalence of Out-of-Field Teaching 1987-88 to 1999-2000, May 2002, revised August 2004, Washington, D.C. National Science Foundation, The Science and Engineering Workforce Realizing America’s Potential, National Science Board, August 14, 2003, Arlington, Virginia. National Science Foundation, Science and Engineering Indicators, 2004, Volume 1, National Science Board, January 15, 2004, Arlington, Virginia. Report of the Congressional Commission on the Advancement of Women and Minorities in Science, Engineering and Technology Development. Land of Plenty: Diversity as America’s Competitive Edge in Science, Engineering, and Technology, September 2000. A Report to the Nation from the National Commission on Mathematics and Science Teaching for the 21st Century, Before It’s Too Late, September 27, 2000.

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Seymour, Elaine, and Nancy M. Hewitt, Talking about Leaving: Why Undergraduates Leave the Sciences, Westview Press, 1997, Boulder, Colorado. The National Academies, Policy Implications of International Graduate Students and Postdoctoral Scholars in the United States, 2005, Washington, D.C. U.S. Department of Education, National Center for Education Statistics, Institute of Education Sciences, The Nation’s Report Card, NAEP 2004: Trends in Academic Progress, July 2005, Washington, D.C. U.S. Department of Education, The Secretary’s Third Annual Report on Teacher Quality, Office of Postsecondary Education, 2004, Washington, D.C. U.S. Department of Homeland Security, 2003 Yearbook of Immigration Statistics, Office of Immigration Statistics, September 2004, Washington, D.C.

References [1] [2] [3] [4]

[5] [6] [7]

[8]

For the purposes of this report, we will use the term “agency” when referring to any of the 13 federal departments and agencies that responded to our survey. Core subjects include English, reading or language arts, mathematics, science, foreign languages, civics and government, economics, arts, history, and geography. Other federal programs that are not specifically designed to attract students to STEM education and occupations, such as Pell Grants, may provide financial assistance to students who obtain degrees in STEM fields. There are several types of visas that authorize people to study and work in the United States. F, or student, visas, are for study at 2- and 4-year colleges and universities and other academic institutions; the exchange visitor, or J, visas are for people who will be participating in a cultural exchange program; and M visas are for nonacademic study at institutions, such as vocational and technical schools. In addition, H-1B visas allow noncitizens to work in the United States. GAO, Border Security: Streamlined Visas Mantis Program Has Lowered Burden on Foreign Science Students and Scholars, but Further Refinements Needed, GAO-05-198 (Washington, D.C.: Feb. 18, 2005). Congressional Research Service, Science, Engineering, and Mathematics Education: Status and Issues, 98-871 STM, April 27, 2004, Washington, D.C. A specialty occupation is defined as one that requires the application of a body of highly specialized knowledge, and the attainment of at least a bachelor’s degree (or its equivalent), and the possession of a license or other credential to practice the occupation if required. GAO asked agencies to include STEM and related education programs with one or more of the following as a primary objective: (1) attract and prepare students at any education level to pursue coursework in STEM areas, (2) attract students to pursue degrees (2-year degrees through post doctoral) in STEM fields, (3) provide growth and research opportunities for college and graduate students in STEM fields, such as working with researchers and/or conducting research to further their education, (4) attract graduates to pursue careers in STEM fields, (5) improve teacher (pre-service, inservice, and postsecondary) education in STEM areas, and (6) improve or expand the capacity of institutions to promote or foster STEM fields.

Higher Education: Federal Science, Technology, Engineering… [9] [10] [11] [12] [13] [14] [15]

[16] [17]

[18] [19]

[20] [21] [22] [23] [24] [25] [26] [27] [28] [29]

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The program funding levels, as provided by agency officials, contain both actual and estimated amounts for fiscal year 2004. Six survey respondents did not include the date the program was initially funded. Fiscal year 2005 funding levels were not available for all of the 207 STEM education programs. Three survey respondents did not identify the program goals. One survey respondent did not identify the type of assistance supported by the program. Two survey respondents did not identify the group targeted by the program. Lawful permanent residents, also commonly referred to as immigrants, are legally accorded the privilege of residing permanently in the United States. They may be issued immigrant visas by the Department of State overseas or adjusted to permanent resident status by the Department of Homeland Security in the United States. GAO, Program Evaluation: An Evaluation Culture and Collaborative Partnerships Help Build Agency Capacity,G AO-03-454 (Washington, D.C.: May 2, 2003). In 2004, we reported on women’s participation in federally funded science programs. Among other issues, this report discussed priorities pertaining to compliance with provisions of Title IX of the Education Amendments of 1972. For additional information, see GAO, Gender Issues: Women’s Participation in the Sciences Has Increased, but Agencies Need to Do More to Ensure Compliance with Title IX, GAO04-639, (Washington, D.C.: July 22, 2004). Institute of International Education, Open Doors: Report on International Educational Exchange, 2004, New York. On the basis of March 2004 CPS estimates, the Pew Hispanic Research Center reported that over 10 million unauthorized immigrants resided in the United States and that people of Hispanic and Latino origin constituted a significant portion of these unauthorized immigrants. Other minorities include Asian/Pacific Islanders and American Indian or Alaska Native. GAO, H-1B Foreign Workers: Better Controls Needed to Help Employers and Protect Workers, GAO/HEHS-00-157 (Washington, D.C.: Sept. 7, 2000). GAO, H-1B Foreign Workers: Better Tracking Needed to Help Determine H-1B Program’s Effects on U.S. Workforce, GAO-03-883 (Washington, D.C.: Sept. 10, 2003). National Science Foundation, Science and Engineering Indicators, 2004, Volume 1, National Science Board, January 15, 2004. National Center for Education Statistics, Qualifications of the Public School Teacher Workforce: Prevalence of Out-of-Field Teaching 1987-88 to 1999-2000, May 2002, revised August 2004,Washington, D.C. U.S. Department of Education, The Secretary’s Third Annual Report on Teacher Quality, Office of Postsecondary Education, 2004, Washington, D.C. U.S. Department of Education, National Center for Education Statistics, Institute of Education Sciences, The Nation’s Report Card, NAEP 2004: Trends in Academic Progress, July 2005, Washington, D.C. 27 Seymour, Elaine, and Nancy M. Hewitt, Talking about Leaving: Why Undergraduates Leave the Sciences, Westview Press, 1997, Boulder, Colorado. Seymour and Hewitt. GAO-04-639.

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[30] Report of the Congressional Commission on the Advancement of Women and Minorities in Science, Engineering and Technology Development, Land of Plenty: Diversity as America’s Competitive Edge in Science, Engineering, and Technology, September 2000. [31] NAFSA: Association of International Educators, In America’s Interest: Welcoming International Students, Report of the Strategic Task Force on International Student Access, January 14, 2003, Washington, D.C. [32] GAO, Border Security: Visa Process Should Be Strengthened as an Antiterrorism Tool, GAO-03-132NI (Washington, D.C.: Oct. 21, 2002). [33] GAO, Border Security: New Policies and Increased Interagency Coordination Needed to Improve Visa Process, GAO-03-1013T (Washington, D. C.: July 15, 2003). [34] GAO, Border Security: Improvements Needed to Reduce Time Taken to Adjudicate Visas for Science Students and Scholars, GAO-04-371 (Washington, D.C.: Feb. 25, 2004). [35] GAO-05-198. [36] GAO, Homeland Security: Performance of Information System to Monitor Foreign Students and Exchange Visitors Has Improved, but Issues Remain, GAO-04-690 (Washington, D.C.: June 18, 2004). [37] GAO, Homeland Security: Performance of Foreign Student and Exchange Visitor Information System Continues to Improve, but Issues Remain, GAO-05-440T (Washington, D.C.: March 17, 2005). [38] Report of the Congressional Commission on the Advancement of Women and Minorities in Science, Engineering and Technology Development, Land of Plenty: Diversity as America’s Competitive Edge in Science, Engineering, and Technology, September 2000.

Appendix I [1] [2] [3]

See Technical Paper 63RV:Current Population Survey—Design and Methodology, issued Mar. 2002. Electronic version available at http://www.censusgov/prod/ 2002pubs/tp63rv.pdf. See Technical Paper 63RV, page 11-4. Seymour, Elaine, and Nancy M. Hewitt, Talking about Leaving: Why Undergraduates Leave the Sciences, Westview Press, 1997, Boulder, Colorado.

In: Advances in Mathematics Research, Volume 8 Editor: Albert R. Baswell, pp. 247-275

ISBN: 978-1-60456-454-9 © 2009 Nova Science Publishers, Inc.

Chapter 8

SCIENCE, TECHNOLOGY, ENGINEERING, AND MATHEMATICS (STEM) EDUCATION ISSUES * AND LEGISLATIVE OPTIONS Jeffrey J. Kuenzi, Christine M. Matthew and Bonnie F. Mangan

Abstract There is growing concern that the United States is not preparing a sufficient number of students, teachers, and practitioners in the areas of science, technology, engineering, and mathematics (STEM). A large majority of secondary school students fail to reach proficiency in math and science, and many are taught by teachers lacking adequate subject matter knowledge. When compared to other nations, the math and science achievement of U.S. pupils and the rate of STEM degree attainment appear inconsistent with a nation considered the world leader in scientific innovation. In a recent international assessment of 15-year-old students, the U.S. ranked 28th in math literacy and 24th in science literacy. Moreover, the U.S. ranks 20th among all nations in the proportion of 24-year-olds who earn degrees in natural science or engineering. A recent study by the Government Accountability Office found that 207 distinct federal STEM education programs were appropriated nearly $3 billion in FY2004. Nearly threequarters of those funds and nearly half of the STEM programs were in two agencies — the National Institutes of Health and the National Science Foundation. Still, the study concluded that these programs are highly decentralized and require better coordination. Several pieces of legislation have been introduced in the 109th Congress that address U.S. economic competitiveness in general and support STEM education in particular. These proposals are designed to improve output from the STEM educational pipeline at all levels, and are drawn from several recommendations offered by the scientific and business communities.

*

Excerpted from CRS Report RL33434 dated July 26, 2006.

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Jeffrey J. Kuenzi, Christine M. Matthew and Bonnie F. Mangan The objective of this report is to provide a useful context for these legislative proposals. To achieve this, the report first presents data on the state of STEM education and then examines the federal role in promoting STEM education. The report concludes with a discussion of selected legislative options currently being considered to improve STEM education. The report will be updated as significant legislative actions occur.

Introduction There is growing concern that the United States is not preparing a sufficient number of students, teachers, and professionals in the areas of science, technology, engineering, and mathematics (STEM).1 Although the most recent National Assessment of Educational Progress (NAEP) results show improvement in U.S. pupils’ knowledge of math and science, the large majority of students still fail to reach adequate levels of proficiency. When compared to other nations, the achievement of U.S. pupils appears inconsistent with the nation’s role as a world leader in scientific innovation. For example, among the 40 countries participating in the 2003 Program for International Student Assessment (PISA), the U.S. ranked 28th in math literacy and 24th in science literacy. Some attribute poor student performance to an inadequate supply of qualified teachers. This appears to be the case with respect to subject-matter knowledge: many U.S. math and science teachers lack an undergraduate major or minor in those fields — as many as half of those teaching in middle school math. Indeed, post-secondary degrees in math and physical science have steadily decreased in recent decades as a proportion of all STEM degrees awarded. While degrees in some STEM fields (particularly biology and computer science) have increased in recent decades, the overall proportion of STEM degrees awarded in the United States has historically remained at about 17% of all postsecondary degrees awarded. Meanwhile, many other nations have seen rapid growth in postsecondary educational attainment — with particularly high growth in the number of STEM degrees awarded. According to the National Science Foundation, the United States currently ranks 20th among all nations in the proportion of 24-year-olds who earn degrees in natural science or engineering. Once a leader in STEM education, the United States is now far behind many countries on several measures. What has been the federal role in promoting STEM education? A recent study by the Government Accountability Office (GAO) found that 207 distinct federal STEM education programs were appropriated nearly $3 billion in FY2004.2 Nearly three-quarters of those funds supported 99 programs in two agencies — the National Institutes of Health (NIH) and the National Science Foundation (NSF). Most of the 207 programs had multiple goals, 1

In 2005 and early 2006, at least six major reports were released by highly respected U.S. academic, scientific, and business organizations on the need to improve science and mathematics education: The Education Commission of the States, Keeping America Competitive: Five Strategies To Improve Mathematics and Science Education, July 2005; The Association of American Universities, National Defense Education and Innovation Initiative, Meeting America’s Economic and Security Challenges in the 21st Century, January 2006; The National Academy of Sciences, Committee on Science, Engineering, and Public Policy, Rising Above the Gathering Storm: Energizing and Employing America for a Brighter Economic Future, February 2006; The National Summit on Competitiveness, Statement of the National Summit on Competitiveness: Investing in U.S. Innovation, December 2005; The Business Roundtable, Tapping America’s Potential: The Education for Innovation Initiative, July 2005; the Center for Strategic and International Studies, Waiting for Sputnik, 2005. 2 U.S. Government Accountability Office, Federal Science, Technology, Engineering, and Mathematics Programs and Related Trends, GAO-06-114, Oct. 2005.

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provided multiple types of assistance, and were targeted at multiple groups. The study concluded that these programs are highly decentralized and could benefit from stronger coordination, while noting that the creation of the National Science and Technology Council in 1993 was a step in the right direction.3 Several pieces of legislation have been introduced in the 109th Congress that would support STEM education in the United States. Many of the proposals in these bills have been influenced by the recommendations of several reports recently issued by the scientific, business, and policy-making communities. Of particular influence has been a report issued by the National Academy of Sciences (NAS), Rising Above the Gathering Storm: Energizing and Employing America for a Brighter Economic Future — also known as the “Augustine” report. Many of the recommendations appearing in the NAS report are also contained in the Administration’s American Competitiveness Initiative.4 Among the report’s many recommendations, five are targeted at improving STEM education. These five recommendations seek to increase the supply of new STEM teachers, improve the skills of current STEM teachers, enlarge the pre-collegiate pipeline, increase postsecondary degree attainment, and enhance support for graduate and early-career research. The purpose of this report is to put these legislative proposals into a useful context. The first section analyzes data from various sources to build a more thorough understanding of the status of STEM education in the United States. The second section looks at the federal role in promoting STEM education, providing a broad overview of nearly all of the programs in federal agencies and a detailed look at a few selected programs. Finally, the third section discusses legislative options currently being considered to improve STEM education. This discussion focuses primarily on the proposals that have seen congressional action to date.

STEM Education in the United States Elementary and Secondary Education Assessments of Math and Science Knowledge National-level assessment of U.S. students’ knowledge of math and science is a relatively recent phenomenon, and assessments in other countries that provide for international comparisons are even more recent. Yet the limited information available thus far is beginning to reveal results that concern many individuals interested in the U.S. educational system and the economy’s future competitiveness. The most recent assessments show improvement in U.S. pupils’ knowledge of math and science; however, the large majority still fail to reach adequate levels of proficiency. Moreover, when compared to other nations, the achievement of U.S. students is seen by many as inconsistent with the nation’s role as a world leader in scientific innovation. 3

4

These points were reiterated by Cornelia M. Ashby, Director of GAO’s Education, Workforce, and Income Security Team. Her testimony can be found at [http://edworkforce. house.gov/hearings/109th/ fc/competitiveness050306/wl5306.htm], as well as on the GAO website at [http://www.gao.gov/ new.items/d06702t.pdf]. Office of Science and Technology Policy, Domestic Policy Council, American Competitiveness Initiative — Leading the World In Innovation, Feb. 2006.

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The National Assessment of Educational Progress (NAEP) is the only nationally representative, continuing assessment of elementary and secondary students’ math and science knowledge. Since 1969, NAEP has assessed students from both public and nonpublic schools at grades 4, 8, and 12. Students’ performance on the assessment is measured on a 0500 scale, and beginning in 1990 has been reported in terms of the percentages of students attaining three achievement levels: basic, proficient, and advanced.5 Proficient is the level identified by the National Assessment Governing Board as the degree of academic achievement that all students should reach, and “represents solid academic performance. Students reaching this level have demonstrated competency over challenging subject matter.” In contrast, the board states that “Basic denotes partial mastery of the knowledge and skills that are fundamental for proficient work at a given grade.”6 The most recent NAEP administration occurred in 2005. Figure 1 displays the available results from the NAEP math tests administered between 1990 and 2005. Although the proportion of 4th and 8th grade students achieving the proficient level or above has been increasing each year, overall math performance has been quite low. The percentage performing at the basic level has not improved in 15 years. About two in five students continue to achieve only partial mastery of math. In 2005, only about one-third of 4th and 8th grade students performed at the proficient level in math — 36% and 30%, respectively.7 The remainder of students — approximately 20% of 4th graders and just over 30% of 8th graders — scored below the basic level. For 12th grade students, the most recently published NAEP results are from the 2000 assessments.8 Only 17% of 12th grade students performed at the proficient or higher level on the math assessment that year.9 This figure was only slightly higher than the previous two assessments in 1996 (16%) and 1992 (15%), but was significantly higher, in statistical terms, than the 12% reported proficient in 1990. Progress aside, it appears that very few students graduate from U.S. high schools with math skills considered adequate. More than half of all 12th grade students performed below even the basic level in each assessment year except 1996. Similarly low levels of achievement have been found with regard to knowledge of science. Less than one-third of 4th and 8th grade students and less than one-fifth of 12th grade students score at or above proficient in science. In 2005, the percentage of 4th, 8th, and 12th grade students scoring proficient or above was 29%, 29%, and 18%, respectively; compared to 27%, 30%, and 18% in 2000 and 28%, 29%, and 21% in 1996.10

5

For more information on NAEP and other assessments, see CRS Report RL31407, Educational Testing: Implementation of ESEA Title I-A Requirements Under the No Child Left Behind Act, by Wayne C. Riddle. 6 The National Assessment Governing Board is an independent, bipartisan group created by Congress in 1988 to set policy for the NAEP. More information on the board and NAEP achievement levels can be found at [http://www.nagb.org/]. 7 U.S. Department of Education, National Center for Education Statistics, The Nation’s Report Card: Mathematics 2005, (NCES 2006-453), Oct. 2005, p. 3. 8 The reporting delay for the 2005 grade 12 math assessments is due, in part, to substantial changes made in the assessment framework, and will not include comparisons to results from previous years. 9 U.S. Department of Education, National Center for Education Statistics, The Nation’s Report Card: Mathematics 2000 (NCES 2001-517) Aug. 2001, Figure B. 10 U.S. Department of Education, National Center for Education Statistics, The Nation’s Report Card: Science 2005 (NCES 2006-466) May 2006, Figures 4, 14, and 24.

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Source: U.S. Department of Education, National Center for Education Statistics, The Nation’s Report Card, various years.

Figure 1. Percentages of Students Scoring Basic and Proficient in Math, Selected Years: 1990-2005.

U.S. Students Compared to Students in Other Nations Another relatively recent development in the area of academic assessment has been the effort by a number of nations to produce reliable cross-national comparison data.11 The Trends in International Mathematics and Science Study (TIMSS) assesses achievement in these subjects at grades 4 and 8 among students in several countries around the world. TIMSS has been administered to 4th grade students on two occasions (1995 and 2003) and to 8th grade students on three occasions (1995, 1999, and 2003). In the latest administration, 25 countries participated in assessments of their 4th grade students, and 45 countries participated in assessments of their 8th grade students. Unlike NAEP, TIMSS results are reported only in terms of numerical scores, not achievement levels.

11

More information on the development of this assessment can be found in archived CRS Report 86-683, Comparison of the Achievement of American Elementary and Secondary Pupils with Those Abroad — The Examinations Sponsored by the International Association for the Evaluation of Educational Achievement (IEA), by Wayne C. Riddle (available on request).

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U.S. 4th grade pupils outscored the international average on the most recent TIMSS assessment.12 The international average score for all countries participating in the 2003 4th grade TIMSS was 495 in math and 489 in science.13 The average score for U.S. students was 518 in math and 536 in science. U.S. 4th grade students outscored students in 13 of the 24 countries participating in the math assessment in 2003. In science, U.S. students outperformed students in 16 of the 24 countries. Among the 10 Organization for Economic Co-operation and Development (OECD) member states participating in the 2003 TIMSS, U.S. 4th grade students ranked fourth in math and tied for second in science. U.S. 8th grade pupils also outscored the international average. Among 8th grade students, the international average on the 2003 TIMSS was 466 in math and 473 in science. The average score for U.S. students was 504 in math and 527 in science. Among the 44 countries participating in the 8th grade assessments in 2003, U.S. students outscored students in 25 countries in math and 32 countries in science. Twelve OECD countries participated in the 8th grade TIMSS in 2003 — five outscored the United States in math and three outscored the United States in science. TIMSS previously assessed students at grade 4 in 1995 and grade 8 in 1995 and 1999. Although there was no measurable difference between U.S. 4th graders’ average scores in 1995 and 2003, the standing of the United States declined relative to that of the 14 other countries participating in both math and science assessments. In math, U.S. 4th graders outperformed students in nine of these countries in 1995, on average, compared to six countries in 2003. In science, U.S. 4th graders outperformed students in 13 of these countries in 1995, on average, compared to eight countries in 2003. Among 8th graders, U.S. scores increased on both the math and science assessments between 1995 and 2003. The increase in scores translated into a higher ranking of the United States relative to other countries. In math, 12 of the 21 participating countries outscored U.S. 8th graders in 1995, while seven did so in 2003. In science, 15 of the 21 participating countries outscored U.S. 8th graders in 1995, while 10 did so in 2003. Table 1 displays the 2003 TIMSS math and science scores of 4th and 8th grade students by country (scores in bold are higher than the U.S. score). The Program for International Student Assessment (PISA) is an OECD-developed effort to measure, among other things, mathematical and scientific literacy among students 15 years of age — i.e., roughly at the end of their compulsory education.14 In 2003, U.S. students scored an average of 483 on math literacy —behind 23 of the 29 OECD member states that participated and behind four of the 11 non-OECD countries. The average U.S. student scored 491 on science literacy —behind 19 of the 29 OECD countries and behind three of the 11 non-OECD countries. Table 2 displays the 2003 PISA scores on math and science literacy by country (scores in bold are higher than the U.S. score). 12

Performance on the 1995 TIMSS assessment was normalized on a scale in which the average was set at 500 and the standard deviation at 100. Each country was weighted so that its students contributed equally to the mean and standard deviation of the scale. To provide trend estimates, subsequent TIMSS assessments are pegged to the 1995 average. 13 All the TIMSS results in this report were taken from, Patrick Gonzales, Juan Carlos Guzmán, Lisette Partelow, Erin Pahlke, Leslie Jocelyn, David Kastberg, and Trevor Williams, Highlights From the Trends in International Mathematics and Science Study (TIMSS) 2003 (NCES 2005 — 005), Dec. 2004. 14 Like the TIMSS, PISA results are normalized on a scale with 500 as the average score, and results are not reported in terms of achievement levels. In 2003, PISA assessments were administered in just over 40 countries.

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Table 1. TIMSS Scores by Grade and Country/Jurisdiction, 2003

International average United States United Kingdom Tunisia Sweden South Africa Slovenia Slovak Republic Singapore Serbia Scotland Saudi Arabia Russian Federation Romania Philippines Palestinian National Authority Norway New Zealand Netherlands Morocco Moldova, Republic of Malaysia Macedonia, Republic of Lithuania Lebanon Latvia Korea, Republic of Jordan Japan Italy Israel Iran, Islamic Republic of Indonesia Hungary Hong Kong SAR Ghana Estonia Egypt Cyprus Chinese Taipei Chile Bulgaria Botswana Belgium-Flemish Bahrain Australia Armenia

4th Math 495 518 531 339 — — 479 — 594 — 490 — 532 — 358 — 451 493 540 347 504 — — 534 — 536 — — 565 503 — 389 — 529 575 — — — 510 564 — — — 551 — 499 456

Grade Science 489 536 540 314 — — 490 — 565 — 502 — 526 — 332 — 466 520 525 304 496 — — 512 — 532 — — 543 516 — 414 — 530 542 — — — 480 551 — — — 518 — 521 437

8th Math 466 504 — 410 499 264 493 508 605 477 498 332 508 475 378 390 461 494 536 387 460 508 435 502 433 508 589 424 570 484 496 411 411 529 586 276 531 406 459 585 387 476 366 537 401 505 478

Grade Science 473 527 — 404 524 244 520 517 578 468 512 398 514 470 377 435 494 520 536 396 472 510 449 519 393 512 558 475 552 491 488 453 420 543 556 255 552 421 441 571 413 479 365 516 438 527 461

Source: U.S. Department of Education, National Center for Education Statistics, Highlights From the Trends in International Mathematics and Science Study (TIMSS) 2003, NCES 2005-005, Dec. 2004.

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OECD average United States Turkey Switzerland Sweden Spain Slovak Republic Portugal Poland Norway New Zealand Netherlands Mexico Luxembourg Korea, Republic of Japan Italy Ireland Iceland Hungary Greece Germany France Finland Denmark Czech Republic Canada Belgium Austria Australia Non-OECD Countries Uruguay United Kingdom Tunisia Thailand Serbia and Montenegro Russian Federation Macao SAR Liechtenstein Latvia Indonesia Hong Kong SAR

Math 500 483 423 527 509 485 498 466 490 495 524 538 385 493 542 534 466 503 515 490 445 503 511 544 514 517 533 529 506 524

Science 500 491 434 513 506 487 495 468 498 484 521 524 405 483 538 548 487 505 495 503 481 502 511 548 475 523 519 509 491 525

422 508 359 417 437 468 527 536 483 360 550

438 518 385 429 436 489 525 525 489 395 540

Source: U.S. Department of Education, National Center for Education Statistics, International Outcomes of Learning in Mathematics Literacy and Problem Solving, NCES 2005-003, Dec. 2004.

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Math and Science Teacher Quality Many observers look to the nation’s teaching force as a source of national shortcomings in student math and science achievement. A recent review of the research on teacher quality conducted over the last 20 years revealed that, among those who teach math and science, having a major in the subject taught has a significant positive impact on student achievement.15 Unfortunately, many U.S. math and science teachers lack this credential. The Schools and Staffing Survey (SASS) is the only nationally representative survey that collects detailed data on teachers’ preparation and subject assignments.16 The most recent administration of the survey for which public data are available took place during the 19992000 school year. That year, there were just under 3 million teachers in U.S. schools, about evenly split between the elementary and secondary levels. Among the nation’s 1.4 million public secondary school teachers, 13.7% reported math as their main teaching assignment and 11.4% reported science as their main teaching assignment.17 Nearly all public secondary school math and science teachers held at least a baccalaureate degree (99.7%), and most had some form of state teaching certification (86.2%) at the time of the survey.18 However, many of those who taught middle school (classified as grades 5-8) math and science lacked an undergraduate or graduate major or minor in the subject they taught. Among middle-school teachers, 51.5% of those who taught math and 40.0% of those who taught science did not have a major or minor in these subjects. By contrast, few of those who taught high school (classified as grades 9-12) math or science lacked an undergraduate or graduate major or minor in that subject. Among high school teachers, 14.5% of those who taught math and 11.2% of those who taught science did not have a major or minor in these subjects.19. Table 3 displays these statistics for teachers in eight subject areas. Table 3. Percentage of Middle and High School Teachers Lacking a Major or Minor in Subject Taught, 1999-2000. English Foreign language Mathematics Science Social science ESL/bilingual education Arts and music Physical/health education

Middle School 44.8% 27.2% 51.5% 40.0% 29.6% 57.6% 6.8% 12.6%

High School 13.3% 28.3% 14.5% 11.2% 10.5% 59.4% 6.1% 9.5%

Source: U.S. Department of Education, National Center for Education Statistics, Qualifications of the Public School Teacher Workforce: Prevalence of Out-of-Field Teaching 1987-88 to 1999-2000, NCES 2002603, May 2002.

15

Michael B. Allen, Eight Questions on Teacher Preparation: What Does the Research Say?, Education Commission of the States, July 2003. 16 The sample is drawn from the Department of Education Common Core of Data, which contains virtually every school in the country. 17 U.S. Department of Education, Digest of Education Statistics, 2004, NCES 2005-025, Oct. 2005, Table 67. 18 CRS analysis of Schools and Staffing Survey data, Mar. 29, 2006. 19 U.S. Department of Education, Qualifications of the Public School Teacher Workforce, May 2002, Tables B-11 and B-12.

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Given the link between teachers’ undergraduate majors and student achievement in math and science, these data appear to comport with some of the NAEP findings discussed earlier. Recall that those assessments revealed that only about one-third of 4th and 8th grade students performed at the proficient or higher level in math and science. On the other hand, at the high school level, the data seem to diverge. While four-fifths of math and science teachers at this level have a major in the subject, only two-fifths of high school students scored proficient or above on the NAEP in those subjects.

Postsecondary Education STEM Degrees Awarded in the United States The number of students attaining STEM postsecondary degrees in the U.S. more than doubled between 1960 and 2000; however, as a proportion of degrees in all fields, STEM degree awards have stagnated during this period.20 In the 2002-2003 academic year, more than 2.5 million degrees were awarded by postsecondary institutions in the United States.21 That year, just under 16% (399,465) of all degrees were conferred in STEM fields; all STEM degrees comprised 14.6% of associate degrees, 16.7% of baccalaureate degrees, 12.9% of master’s degrees, and 34.8% of doctoral degrees.22 Table 4 displays the distribution of degrees granted by academic level and field of study. At the associate and baccalaureate levels, the number of STEM degrees awarded was roughly equivalent to the number awarded in business. In 2002-2003, 92,640 associate degrees and 224,911 baccalaureate degrees were awarded in STEM fields, compared to 102,157 and 293,545, respectively, in business. However, nearly twice as many master’s degrees were granted in business (127,545) as in STEM (65,897), and an even larger number of master’s degrees were awarded in education (147,448). At the doctoral level, STEM plays a larger role. Doctoral degrees awarded in STEM fields account for more than one-third of all degrees awarded at this level. Education is the only field in which more doctoral degrees (6,835) were awarded than in the largest three STEM fields — biology, engineering, and the physical sciences (5,003, 5,333, and 3,858, respectively). Specialization within STEM fields also varies by academic level. Engineering was among the most common STEM specialties at all levels of study in 2002-2003. Biology was a common specialization at the baccalaureate and doctoral levels, but not at the master’s level. Computer science was common at all but the doctoral level. Physical sciences was a common specialization only at the doctoral level.

20

Through various “completions” surveys of postsecondary institutions administered annually since 1960, ED enumerates the number of degrees earned in each field during the previous academic year. 21 U.S. Department of Education, National Center for Education Statistics, Digest of Education Statistics, 2004, NCES 2005-025, Oct. 2005, Table 169. 22 Includes Ph.D., Ed.D., and comparable degrees at the doctoral level, but excludes first-professional degrees, such as M.D., D.D.S., and law degrees.

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Table 4. Degrees Conferred by Degree-Granting Institutions by Academic Level and Field of Study, 2002-2003. All fields STEM fields, total STEM, percentage of all fields Biological and biomedical sciences Computer and information sciences Engineering and engineering technologies Mathematics and statistics Physical sciences and science technologies

Associate 632,912 92,640 14.6% 1,496 46,089 42,133 732 2,190

Baccalaureate Master’s 1,348,503 512,645 224,911 65,897 16.7% 12.9% 60,072 6,990 57,439 19,503 76,967 30,669 12,493 17,940

3,626 5,109

Doctoral 46,024 16,017 34.8% 5,003 816 5,333

Total 2,540,084 399,465 15.7% 73,561 123,847 155,102

1,007 3,858

17,858 29,097

Non-STEM fields, total 540,272 1,123,592 446,748 30,007 2,140,619 Business 102,157 293,545 127,545 1,251 524,498 Education 11,199 105,790 147,448 6,835 271,272 English language and literature/letters 896 53,670 7,413 1,246 63,225 Foreign languages and area studies 1,176 23,530 4,558 1,228 30,492 Liberal arts and sciences, general 216,814 40,221 3,312 78 260,425 studies, and humanities Philosophy, theology, and religious 804 18,270 6,677 1,983 27,734 studies/vocations Psychology 1,784 78,613 17,123 4,831 102,351 Social sciences 5,422 115,488 12,109 2,989 136,008 History 316 27,730 2,525 861 31,432 Other 199,704 366,735 118,038 8,705 693,182 Source: U.S. Department of Education, National Center for Education Statistics, Digest of Education Statistics, 2004, NCES2005-025, Oct. 2005, Table 249-252.

Figure 2 displays the trends in STEM degrees awarded over the last three decades (excluding associate degrees). The solid line represents the number of STEM degrees awarded as a proportion of the total number of degrees awarded in all fields of study. The flat line indicates that the ratio of STEM degrees to all degrees awarded has historically hovered at around 17%. The bars represent the number of degrees awarded in each STEM sub-field as a proportion of all STEM degrees awarded. The top two segments of each bar reveal a consistent decline, since 1970, in the number of degrees awarded in math and the physical sciences. The bottom segment of each bar shows a history of fluctuation in the number of degrees awarded in biology over the last 30 years. The middle two segments in the figure represent the proportion of degrees awarded in engineering and computer science. The figure reveals a steady decline in the proportion of STEM degrees awarded in engineering since 1980, and a steady increase in computer science degrees (except for a contraction that occurred in the late 1980s following a rapid expansion in the early 1980s).

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Figure 2. STEM Degrees Awarded, 1970-2003.

U.S. Degrees Awarded to Foreign Students The increased presence of foreign students in graduate science and engineering programs and in the scientific workforce has been and continues to be of concern to some in the scientific community. Enrollment of U.S. citizens in graduate science and engineering programs has not kept pace with that of foreign students in these programs. According to the National Science Foundation (NSF) Survey of Earned Doctorates, foreign students earned one-third of all doctoral degrees awarded in 2003. Doctoral degrees awarded to foreign students were concentrated in STEM fields. The NSF reports that foreign students earned “more than half of those [awarded] in engineering, 44% of those in mathematics and computer science, and 35% of those in the physical sciences.”23 Many of these degree recipients remain in the United States to work. The same NSF report indicates that 53% of those who earned a doctorate in 1993 remained in the U.S. as of 1997, and 61% of the 1998 cohort were still working in the United States in 2003. In addition to the number of foreign students in graduate science and engineering programs, a significant number of university faculty in the scientific disciplines are foreign, and foreign doctorates are employed in large numbers by industry.24 23

National Science Board, Science and Engineering Indicators, 2006, (NSB 06-1). Arlington, VA: National Science Foundation, Jan. 2006, p. O-15. 24 For more information on issues related to foreign students and foreign technical workers, see the following: CRS Report 97-746, Foreign Science and Engineering Presence in U.S. Institutions and the Labor Force, by Christine M. Matthews; CRS Report RL31973, Programs Funded by the H-1B Visa Education and Training Fee and Labor Market Conditions for Information Technology (IT) Workers, by Linda Levine; and CRS Report RL30498, Immigration: Legislative Issues on Nonimmigrant Professional Specialty (H-1B) Workers, by Ruth Ellen Wasem.

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International Postsecondary Educational Attainment The United States has one of the highest rates of postsecondary educational attainment in the world. In 2003, the most recent academic year for which international data are available, 38% of the U.S. population aged 25-64 held a postsecondary degree — 9% at the tertiary-type B (vocational level) and 29% at the tertiary-type A (university level) or above. The OECD compiled comparison data from 30 OECD member states and 13 other nations. Three countries (Canada, Israel, and the Russian Federation) had larger shares at the two tertiary levels combined; however, all three had lower rates at the tertiary-type A level. At the tertiary-type A level, only one country (Norway) had a rate as high as the United States. The average for OECD member states was 16% at tertiary-type A and 8% at tertiary-type B.25 China and India were not included in the OECD data. Reliable information on postsecondary educational attainment is very difficult to obtain for these countries. The World Bank estimates that, in 1998, tertiary enrollment of the population between 18 and 24 years old was 6% in China and 8% in India, up from 1.7% and 5.2%, respectively, in 1980.26 Based on measures constructed by faculty at the Center for International Development (CID), the National Science Foundation (NSF) has generated an estimate of the distribution of the world’s population that possesses a tertiary education.27 The NSF estimates that the number of people in the world who had a tertiary education more than doubled from 73 million in 1980 to 194 million in 2000. Moreover, the two fastest-growing countries were China and India. China housed 5.4% of the world’s tertiary degree holders in 1980, and India had 4.1%; by 2000, the share in these countries was 10.5% and 7.7%, respectively. Indeed, as Figure 3 indicates, China and India were the only countries to substantially increase their share of the world’s tertiary degree-holders during that period.

International Comparisons in STEM Education The NSF has compiled data for many countries on the share of first university degrees awarded in STEM fields.28 According to these data, the United States has one of the lowest rates of STEM to non-STEM degree production in the world. In 2002, STEM degrees accounted for 16.8% of all first university degrees awarded in the United States (the same NCES figure reported at the outset of this section). The international average for the ratio of STEM to non-STEM degrees was 26.4% in 2002. Table 5 displays the field of first university degrees for regions and countries that award more than 200,000 university degrees annually. 25

Organization for Economic Co-operation and Development, Education at a Glance, OECD Indicators 2005, Paris, France, Sept. 2005. The OECD compiles annual data from national labor force surveys on educational attainment for the 30 OECD member countries, as well as 13 non-OECD countries that participate in the World Education Indicators (WEI) program. More information on sources and methods can be found at [http://www.oecd.org/ dataoecd/36/39/35324864.pdf]. 26 The World Bank, Constructing Knowledge Societies: new challenges for tertiary education, Washington, D.C., October 2002. Available at [http://siteresources.worldbank .org/EDUCATION/Resources/2782001099079877269/547664-1099079956815/ ConstructingKnowledgeSocieties.pdf]. 27 Unlike the OECD data, which are based on labor-force surveys of households and individuals, the CID data are based on the United Nations Educational, Scientific and Cultural Organization (UNESCO) census and survey data of the entire population. Documentation describing methodology as well as data files for the CID data is available at [http://www.cid.harvard.edu/ciddata/ciddata.html].

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Source: National Science Foundation, Science and Engineering Indicators, 2006, Volume 1, Arlington, VA, NSB 06-01, Jan. 2006.

Figure 3. Population 15 Years Old or Older With Tertiary Education by Country, 1980 and 2000.

28

First university degrees are those designated Level 5A by the International Standard Classification of Education (ISCED 97), and usually require less than five years to complete. More information on this classification and the ISCED is available at [http://www.unesco. org/education/information/nfsunesco/doc/isced_1997.htm].

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Among these nations, only Brazil awards a smaller share (15.5%) of STEM degrees than the United States. By contrast, the world leaders in the proportion of STEM degrees awarded are Japan (64.0%) and China (52.1%). Although the U.S. ranks near the bottom in the proportion of STEM degrees, it ranks third (behind Japan and China) in the absolute number of STEM degrees awarded. Table 5. Field of First University Degree, by Selected Region and Country, 2002 or the Most Recent Year Available Region/Country All regions Asia China India Japan South Korea Middle East Europe France Spain United Kingdom Central/Eastern Europe Russia North/Central America Mexico United States South America Brazil

All fields 9,057,193 3,224,593 929,598 750,000 548,897 239,793 445,488 2,682,448 309,009 211,979 282,380 1,176,898 554,814 1,827,226 321,799 1,305,730 543,805 395,988

STEM Fields 2,395,238 1,073,369 484,704 176,036 351,299 97,307 104,974 713,274 83,984 55,418 72,810 319,188 183,729 341,526 80,315 219,175 96,724 61,281

Percent STEM 26.4% 33.3% 52.1% 23.5% 64.0% 40.6% 23.6% 26.6% 27.2% 26.1% 25.8% 27.1% 33.1% 18.7% 25.0% 16.8% 17.8% 15.5%

Source: National Science Foundation, Science and Engineering Indicators, 2006, Volume 1, Arlington, VA, NSB 06-01, January 2006, Table 2-37.

Federal Programs that Promote STEM Education Government Accountability Office Study According to a 2005 Government Accountability Office (GAO) survey of 13 federal civilian agencies, in FY2004 there were 207 federal education programs designed to increase the number of students studying in STEM fields and/or improve the quality of STEM education.29 About $2.8 billion was appropriated for these programs that year, and about 71% ($2 billion) of those funds supported 99 programs in two agencies. In 2004, the National Institutes of Health (NIH) received $998 million that funded 51 programs, and the National Science Foundation (NSF) received $997 million that funded 48 programs. Seven of the 13 29

U.S. Government Accountability Office, Federal Science, Technology, Engineering, and Mathematics Programs and Related Trends, GAO-06-114, Oct. 2005. The GAO study does not include programs in the Department of Defense because the department decided not to participate. Other programs were omitted from the report for various reasons; typically because they did not meet the GAO criteria for a STEM-related educational program (according to an Apr. 26, 2006 conversation with the report’s lead author, Tim Hall).

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agencies had more than five STEM-related education programs. In addition to the NIH and NSF, only three other agencies received more than $100 million for STEM-related education programs. In FY2004, the National Aeronautics and Space Administration (NASA) received $231 million that funded five programs, the U.S. Department of Education (ED) received $221 million that funded four programs, and the Environmental Protection Agency (EPA) received $121 million that funded 21 programs. The GAO study found that most of the 207 programs had multiple goals, provided multiple types of assistance, and were targeted at multiple groups. The analysis identified six major program goals, four main types of assistance, and 11 target groups. The findings revealed that federal STEM education programs are heavily geared toward attracting college graduates into pursuing careers in STEM fields by providing financial assistance at the graduate and postdoctoral levels. Moreover, improving K-12 teacher education in STEM areas was the least frequent of the major goals, improving infrastructure was the least frequent of the main types of assistance, and elementary and secondary students were the least frequent group targeted by federal STEM education programs.30 The major goals of these programs were found by GAO to be the following (the number of programs with this goal is shown in parentheses): • • • • • •

attract and prepare students at all educational levels to pursue coursework in STEM areas (114), attract students to pursue STEM postsecondary degrees (two-year through Ph.D.) and postdoctoral appointments (137), provide growth and research opportunities for college and graduate students in STEM fields (103), attract graduates to pursue careers in STEM fields (131), improve teacher education in STEM areas (73), and improve or expand the capacity of institutions to promote STEM fields (90).

The four main types of assistance provided by these programs were as follows (the number of programs providing this service is shown in parentheses): • • • •

financial support for students or scholars (131), institutional support to improve educational quality (76), support for teacher and faculty development (84), and institutional physical infrastructure support (27).

The 11 target groups served by these programs were the following (the number of programs targeting them is shown in parentheses): • • • 30

elementary school students (28), middle school students (34), high school students (53),

Attrition rates among college students majoring in STEM fields combined with the growth of foreign students in U.S. graduate STEM programs suggest that pre-college STEM education may be a major source of the nation’s difficulty in this area.

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two-year college students (58), four-year college students (96), graduate students (100), postdoctoral scholars (70), elementary school teachers (39), secondary school teachers (50), college faculty or instructional staff (79), and institutions (82).

Description of Selected Federal STEM Programs The 2005 GAO report did not discuss federal STEM programs in detail (a very brief description of programs funded at $10 million or more is contained in Appendix III of the report). This section describes the kinds of activities the largest of these programs support, and how they operate at the federal, state, and/or local levels.31.

NIH National Research Service Awards The NIH was appropriated $998 million in FY2004 in support of its 51 STEM educational programs. Nearly two-thirds ($653 million) of those funds went to three programs under the National Research Service Awards (NRSA), first funded in 1975.32 Most of these funds ($547 million) went to one program, the NRSA Institutional Research Training Grants, which provides pre- and postdoctoral fellowships in health-related fields. An additional $73 million went to NRSA Individual Postdoctoral Fellowship Grants and $34 million went to NRSA Predoctoral Fellowship Grants. The Training Grants are awarded to institutions to develop or enhance research training opportunities for individuals, selected by the institution, who are training for careers in specified areas of interest to the institution or principal investigator. The Fellowship Grants are awarded directly to individuals from various organizations within the NIH (e.g., the National Institute on Aging) to support the particular research interests of the individual receiving the award. NRSA grant applicants must be U.S. citizens or nationals, or permanent resident aliens of the United States — individuals on temporary or student visas are not eligible. Predoctoral trainees must have received a baccalaureate degree by the starting date of their appointment, and must be training at the postbaccalaureate level and be enrolled in a program leading to a Ph.D. in science or in an equivalent research doctoral degree program. Health-profession students who wish to interrupt their studies for a year or more to engage in full-time research training before completing their professional degrees are also eligible. Postdoctoral trainees must have received, as of the beginning date of their appointment, a Ph.D., M.D., or comparable doctoral degree from an accredited domestic or foreign institution. Institutional grants are made for a five-year period. Trainee appointments are normally made in 12-month increments, although short-term (two- to three-month) awards are available. No individual 31

Additional program descriptions are available in the CRS congressional distribution memorandum, Federally Sponsored Programs for K-12 Science, Mathematics, and Technology Education, by Bonnie F. Mangan, available upon request. 32 More information on the NRSA program is available at [http://grants.nih.gov/ training/nrsa.htm].

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trainee may receive more than five years of aggregate NRSA support at the predoctoral level or three years of support at the postdoctoral level, including any combination of support from institutional training grants and individual fellowship awards. The annual stipend for predoctoral trainees in 2005 was about $12,000, and the postdoctoral stipend was between $20,000 and $32,000 (depending on years of experience). In FY2004, Training Grants were awarded to 293 institutions in all but six states. A total of 2,356 grants were awarded, which funded nearly 9,000 predoctoral fellowships and nearly 5,500 postdoctoral fellowships. The Fellowship Grant programs supported around 2,500 preand postdoctoral students in 2004. The large majority of the Training Grants were awarded through the National Institute of General Medical Sciences.

NSF Graduate Research Fellowships The largest of the NSF STEM education programs — the Graduate Research Fellowships ($97 million in FY2005) — is also one of the longest-running federal STEM programs (enacted in 1952). The purpose of this program is to increase the size and diversity of the U.S. workforce in science and engineering. The program provides three years of support to approximately 1,000 graduate students annually in STEM disciplines who are pursuing research-based master’s and doctoral degrees, with additional focus on women in engineering and computer and information sciences. In 2006, 907 awards were given to graduate students studying in nine major fields at 150 instituions. Applicants must be U.S. citizens or nationals, or permanent resident aliens of the United States; must have completed no more than twelve months of full-time graduate study at the time of their application; and must be pursuing an advanced degree in a STEM field supported by the National Science Foundation.33 The fellows’ affiliated institution receives a $40,500 award — $30,000 for a 12-month stipend and $10,500 for an annual cost-ofeducation allowance. These awards are for a maximum of three years and usable over a fiveyear period, and provide a one-time $1,000 International Research Travel Allowance. All discipline-based review panels, made up of professors, researchers, and others respected in their fields, convene for three days each year to read and evaluate applications in their areas of expertise. In 2005, there were 29 such panels made up of more than 500 experts.

NSF Mathematics and Science Partnerships The Mathematics and Science Partnerships program was the NSF’s second-largest program in FY2005 ($79 million in FY2005) and was the agency’s largest program in FY2004 ($139 million). Since its inception in 2002, this program has awarded grants that support four types of projects (the number of awards is shown in parentheses): • •

33

Comprehensive Partnership projects (12) to implement change in mathematics and science education across the K-12 continuum; Targeted Partnership projects (28) to improve K-12 student achievement in a narrower grade range or disciplinary focus in mathematics and/or science;

A list of NSF-supported fields of study can be found at [http://www.nsf.gov/pubs/2005/nsf05601/ nsf05601.htm#study].

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Institute Partnership projects (8) to focus on improving middle and high school mathematics and science through the development of school-based intellectual leaders and master teachers; and Research, Evaluation and Technical Assistance projects (22) to build research, evaluation, and infrastructure capacity for the MSP.

One of the Comprehensive Partnership projects is between the Baltimore County Public Schools (BCPS) and the University of Maryland, Baltimore County (UMBC). The two main goals of the UMBC-BCPS STEM Partnership are to (1) facilitate the implementation, testing, refinement, and dissemination of promising practices for improving STEM student achievement, and (2) improve teacher quality and retention in selected high-need elementary, middle, and high schools in Baltimore County Public Schools. Centered on creating and evaluating performance-based pre-service (internship) teacher education programs and sustainable professional development programs for teachers and administrators, the project is designed to increase K-12 student achievement in STEM areas by increasing teacher and administrator knowledge. Ongoing assessments of student work and the differentiation of instruction based upon these assessments serve to evaluate and refine instruction, curricula and assessments, professional development programs, administrative leadership strategies, and directions for overall school improvement in STEM areas. UMBC and BCPS collaboration is facilitated by the creation of the Center for Excellence in STEM Education, where UMBC faculty and BCPS teachers and administrators develop projects to serve the needs of the BCPS district and the university. At the center, faculty and teachers work together to simultaneously improve the university’s STEM and teacher education departments and the teaching and learning culture in the BCPS. One of the Targeted Partnership grants supports the Promoting Reflective Inquiry in Mathematics Education Partnership, which includes Black Hills State University, Technology and Innovations in Education (TIE) of the Black Hills Special Services Cooperative, and the Rapid City School District in South Dakota. The overall goal of the partnership is aimed at improving achievement in mathematics for all students in Rapid City schools, with a particular goal of reducing the achievement gap between Native American and non-Native American students. The project seeks to improve the professional capacity and sustain the quality of K-12 in-service teachers of mathematics in the Rapid City School District, and student teachers of mathematics from Black Hills State University in order to provide effective, inquiry-based mathematics instruction. Objectives include reducing the number of high school students taking non-college preparatory mathematics, increasing the number of students taking upper level mathematics, and increasing student performance on college entrance exams. To accomplish these goals, the project provides 100 hours of professional development in combination with content-based workshops at the district level, and buildingbased activities involving modeling of effective lessons, peer mentoring and coaching, and lesson study. Mathematics education and discipline faculty from Black Hills State University are involved in district-wide professional development activities. A cadre of building-based Mathematics Lead Teachers convenes learning teams composed of mathematics teachers, mathematics student teachers, school counselors, and building administrators to identify key issues in mathematics curriculum and instruction.

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NSF Research Experiences for Undergraduates The Research Experiences for Undergraduates (REU) program is the largest of the NSF STEM education programs that supports active research participation by undergraduate students ($51 million in FY2005). REU projects involve students in research through two avenues. REU Sites are based on independent proposals to initiate and conduct projects that engage a number of students in research. REU Supplements are requested for ongoing NSFfunded research projects or are included as a component of proposals for new or renewal NSF grants or cooperative agreements. REU projects may be based in a single discipline or academic department, or on interdisciplinary or multi-department research opportunities with a coherent intellectual theme. Undergraduate student participants in either Sites or Supplements must be citizens or permanent residents of the United States or its possessions. Students apply directly to REU Sites (rather that to the NSF) to participate in the program. One of the grantees under this program is the REU Site in Microbiology at the University of Iowa. The goals of this project are to (1) recruit and select bright students, including women, individuals with diverse backgrounds with respect to geographic origin and ethnicity, and students from non-Ph.D.-granting institutions where research possibilities are limited; (2) involve students in basic, experimental research in microbiology; (3) expose students to a broad range of bioscience research; (4) develop each student’s critical-thinking skills; and (5) develop each student’s ability to record, analyze, and present scientific information. The student participants are integrated into faculty research programs and expected to perform like beginning graduate students. Informal faculty-student discussions and weekly seminars supplement laboratory research. Weekly informal lunches, two picnics, and a banquet facilitate social and scientific interactions. At the end of each summer’s program, the students prepare oral presentations to be given at a Summer Program Symposium. Each student also prepares a written research report under the guidance of a mentor.

NASA Minority University Research Education Program Nearly half of the funds ($106 million of $231 million) appropriated for NASA’s STEM education programs in FY2004 went to the Minority University Research Education Program (MUREP). MUREP supports grants to expand and advance NASA’s scientific and technological base through collaborative efforts with Historically Black Colleges and Universities (HBCUs) and other minority universities, including Hispanic-serving institutions and tribal colleges and universities. The program provides (1) K-12 awards to build and support successful pathways for students to progress to the next level of mathematics and science through a college preparatory curriculum and enrollment in college; (2) highereducation awards to improve the rate at which underrepresented minorities are awarded degrees in STEM disciplines; and (3) partnership awards to higher-education institutions and school districts that improve K-12 STEM teaching. One of the partnership programs, the Minority University Mathematics, Science and Technology Awards for Teacher and Curriculum Enhancement Program, supports collaborative efforts between universities and school districts to increase the number and percentage of state-certified STEM teachers in schools with high percentages of disadvantaged students. Grant awards range from $50,000 to $200,000 annually for each of three years of support, for a total of up to $600,000. A longstanding grant funded under this

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program involves a partnership between Florida International University and Miami-Dade County Public Schools. Students from three middle schools and one high school attend mathematics, science, and technology classes for half of each day at the university and spend the other half of the day in their home school. University faculty, graduate students, and preservice secondary mathematics teachers work with district teachers in providing the at-risk students with standards-based curriculum and instruction.

ED Mathematics and Science Partnerships Three-quarters of the STEM program funds in the Department of Education ($149 million of $221 million) in FY2004 went to the Mathematics and Science Partnership (MSP) program. The MSP is intended to increase the academic achievement of students in mathematics and science by enhancing the content knowledge and teaching skills of classroom teachers. These partnerships — between state education agencies, high-need school districts, and STEM faculty in institutions of higher education — are supported by state-administered formula grants and carried out in collaboration with the NSF-MSP program. Partnerships must use their grants for one or more of several specific activities. Among them are the following: • • • •

• • •

professional development to improve math and science teachers’ subject knowledge; activities to promote strong teaching skills among these teachers and teacher educators; math and science summer workshops or institutes with academic-year followup; recruitment of math, science, and engineering majors to teaching jobs through signing and performance incentives, stipends for alternative certification, and scholarships for advanced course work; development and redesign of more rigorous, standards-aligned math and science curricula; distance-learning programs for math and science teachers; and opportunities for math and science teachers to have contact with working mathematicians, scientists, and engineers.

A review of projects funded in 2004 revealed that most grantees focus on math (as opposed to science) instruction in middle schools, and provide professional development to roughly 46 teachers over a period of about 21 months.34 The survey found that most projects link content to state standards, and that algebra, geometry, and problem-solving are the top three math topics addressed by professional development activities. Most projects administer content knowledge tests to teachers, conduct observations, and make pre-and post-test comparisons. About half of the projects develop their own tests for teachers, and most rely on state tests of academic achievement to measure student knowledge.

34

Analysts at the Brookings Institution conducted a survey of 266 winning MSP projects from 41 states. Results of the survey are available at [http://www.ed.gov/programs/mathsci/ proposalreview.doc].

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Proposals to Improve STEM Education Several pieces of legislation have been introduced in the 109th Congress with the purpose of improving STEM education in the United States. Many of the proposals in these bills have been influenced by the recommendations of several reports recently issued by leading academic, scientific, and business organizations (mentioned in the introduction of this report).35 These recommendations, particularly those from the business community, are not limited to the educational system. This report does not discuss these non-educational policy recommendations (e.g., immigration policies that affect the supply of foreign workers to fill U.S. demand in STEM occupations). The concluding section of this report discusses STEM education policy recommendations in detail, as well as selected pieces of legislation that have been introduced in this area.

Recommendations by the Scientific Community The recommendations to improve federal STEM policy concern every aspect of the educational pipeline. All of the recent reports issuing STEM education policy recommendations focus on five areas: improving elementary and secondary preparation in math and science, recruiting new elementary and secondary math and science teachers, retooling current math and science teachers, increasing the number of undergraduate STEM degrees awarded, and supporting graduate and early-career research. As mentioned at the outset of this report, one report that has been of particular influence in the STEM debate is from the National Academy of Sciences (NAS) — Rising Above the Gathering Storm. This influence is perhaps due to the clear targets and concrete programs laid out in the report. The NAS report’s five recommendations to improve STEM education follow. • • • • •

quadruple middle- and high-school math and science course-taking by 2010, recruit 10,000 new math and science teachers per year, strengthen the skills of 250,000 current math and science teachers, increase the number of STEM baccalaureate degrees awarded, and support graduate and early-career research in STEM fields.

To enlarge the pipeline of future STEM degree recipients, NAS sets a goal of quadrupling the number of middle and high school students taking Advanced Placement (AP) or International Baccalaureate (IB) math or science courses, from the current 1.1 million to 4.5 million by 2010. NAS further sets a goal of increasing the number of students who pass either the AP or IB tests to 700,000 by 2010. To enlarge the pipeline, NAS also supports the 35

The Education Commission of the States, Keeping America Competitive: Five Strategies To Improve Mathematics and Science Education, July 2005; The Association of American Universities, National Defense Education and Innovation Initiative, Meeting America’s Economic and Security Challenges in the 21st Century, Jan. 2006; The National Academy of Sciences, Committee on Science, Engineering, and Public Policy, Rising Above the Gathering Storm: Energizing and Employing America for a Brighter Economic Future, Feb. 2006; The National Summit on Competitiveness, Statement of the National Summit on Competitiveness: Investing in U.S. Innovation, Dec. 2005; The Business Roundtable, Tapping America’s Potential: The Education for Innovation Initiative, July 2005; The Center for Strategic and International Studies, Waiting for Sputnik, 2005.

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expansion of programs such as statewide specialty high schools for STEM immersion and inquiry-based learning through laboratory experience, summer internships, and other research opportunities. To recruit 10,000 new STEM teachers, NAS advocates the creation of a competitive grant program to award merit-based scholarships to obtain a four-year STEM degree in conjunction with certification as a K-12 mathematics or science teacher. These $10,000 to $20,000 awards could be used only for educational expenses and would require a five-year service commitment. An additional $10,000 annual bonus would be awarded to participating teachers in underserved schools in inner cities and rural areas. In further support of this scholarship program, NAS recommends that five-year, $1 million matching grants be awarded to postsecondary institutions to encourage the creation of programs that integrate the obtainment of a STEM bachelor’s degree with teacher certification. NAS proposes four approaches to achieving the goal of strengthening the skills of 250,000 current STEM teachers. First, NAS proposes that matching grants be awarded to support the establishment of state and regional summer institutes for STEM teachers modeled after the Merck Institute for Science Education. Second, NAS proposes that additional grants go to postsecondary institutions that support STEM master’s degree programs for current STEM teachers (with or without STEM bachelor’s degrees) modeled after the University of Pennsylvania Science Teachers Institute. Third, NAS proposes that programs be created to train current teachers to provide AP, IB, and pre-AP or pre-IB instruction modeled after the Advanced Placement Initiative and the Laying the Foundation programs. Fourth, NAS proposes the creation of a national panel to collect, evaluate, and develop rigorous K-12 STEM curricula modeled after Project Lead the Way. To increase STEM bachelor’s degree attainment, NAS proposes providing 25,000 new scholarships each year. These Undergraduate Scholar Awards in Science, Technology, Engineering, and Mathematics (USA-STEM) would be distributed to each state in proportion with its population, and awarded to students based on competitive national exams. The $20,000 scholarships could only go to U.S. citizens, and could only be used for the payment of tuition and fees in pursuit of a STEM degree at a U.S. postsecondary institution. To increase graduate study in areas of national need, including STEM, NAS proposes the creation of 5,000 new fellowships each year to U.S. citizens pursuing doctoral degrees. The fellowships would be administered by the National Science Foundation, which would also draw on the advice of several federal agencies in determining the areas of need. An annual stipend of $30,000 would be accompanied by an additional $20,000 annually to cover the cost of tuition and fees. These fellowships would also be portable, so that students could choose to study at a particular institution without the influence of faculty research grants.

Legislation in the 109th Congress Several bills containing STEM education-related proposals have been introduced in the 109th Congress, and have also seen additional legislative action. Some of these bills have already been passed by Congress and signed into law by the President. The National Aeronautics and Space Administration Authorization Act of 2005 (P.L. 109-155) directed the Administrator to develop, expand, and evaluate educational outreach programs in science and space that serve elementary and secondary schools. The National Defense Authorization Act of 2006 (P.L.

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109-163) made permanent the Science, Mathematics and Research for Transformation pilot program initiated by the Defense Act of 2005 (P.L. 108-375) to address deficiencies of scientists and engineers in the national security workforce. The Deficit Reduction Act of 2005 (P.L. 109-171) established the Academic Competitiveness Grants and the National Science and Mathematics Access to Retain Talent Grants programs, which supplement Pell Grants for students studying mathematics, technology, engineering, critical foreign languages, and physical, life, and computer sciences. The act also established the Academic Competitiveness Council, chaired by the Secretary of Education and charged with identifying and evaluating all federal STEM programs, and recommending reforms to improve program integration and coordination. Additional bills that have been introduced in the 109th Congress that would make substantial changes or additions to current federal STEM education policy include two intended to reauthorize the Higher Education Act (HEA), as well as several pieces of socalled “competitiveness” legislation. On February 28, 2006, the Senate Committee on Health, Education, Labor, and Pensions reported S. 1614, the Higher Education Amendments of 2005 (S.Rept. 109-218). On March 30, 2006, the House passed H.R. 609, the College Access and Opportunity Act of 2005. On April 24, 2006, the Senate Committee on Energy and Natural Resources reported S. 2197, Protecting America’s Competitive Edge Through Energy Act of 2006 (S.Rept. 109-249). A companion bill, S. 2198, Protecting America’s Competitive Edge Through Education and Research Act of 2006, has been the subject of two hearings (February 28, 2006 and March 1, 2006). Another bill that would make substantial additions to federal STEM education policy is S. 2109, the National Innovation Act. On June 22, 2006, the House Committee on Science reported two bills — H.R. 5358, the Science and Mathematics Education for Competitiveness Act, and H.R. 5356, the Early Career Research Act.

Secondary School Math and Science Preparation S. 2197 would provide experiential-based learning opportunities for students by establishing a summer internship program for middle school and secondary school students at the National Laboratories funded by the Department of Energy (DOE). Language in the bill requires that 40% of the participants be from low-income families. The bill also requires that the participants be from schools where teachers are teaching “out-of-field,” hold temporary certification, or have a high turnover rate. For this purpose, S. 2197 would authorize appropriations of $50 million for each of five fiscal years — FY2007 through FY2011. S. 2109 would increase support for science education through the NSF. The bill would authorize the following amounts for expansion of science, mathematics, engineering, and technology talent under the NSF Authorization Act of 2002 (P.L. 107-368): FY2007, $35 million; FY2008, $50 million; FY2009, $100 million; and FY2010, $150 million. S. 2109 would also promote innovation-based experiential learning. This bill would allow NSF to award grants to local education agencies (LEAs) for implementation of innovation-based experiential learning. A total of 500 elementary or middle schools and 500 secondary schools would participate. Funding would total $10 million in FY2007 and $20 million each for FY2008 and FY2009.

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Recruiting and Retaining New STEM Teachers Provisions in S. 2198 would direct the NSF to provide two types of support for future science and mathematics teachers. One such award would be four-year fellowships in the amount of $10,000 annually to individuals who complete a baccalaureate degree in science, engineering, or mathematics, with concurrent teacher certification. A requirement would be that these individuals teach as full-time mathematics, science, or elementary school teachers in highneed elementary and secondary schools. Additional support for teacher recruitment would be through scholarships for science and mathematics teachers. The NSF Director would award merit-based scholarships of up to $20,000 per year for not more than four years to students majoring in science, mathematics, and engineering education who pursue concurrent teacher certification to assist students in paying their college education expenses. S. 2198 would also authorize institutional grants to provide an integrated course of study in mathematics, science, engineering, or teacher education that leads to a baccalaureate degree in the STEM disciplines with concurrent teacher certification. The awards would total $1 million per year for a period of five years. Matching funds in predetermined amounts would be required from non-federal sources — not less than 25% of the amount for the first year, not less than 35% for the second year, and not less than 50% of the amount of the grant award for each succeeding year of the grant. S. 2198 would provide further institutional grants to develop part-time, three-year master’s degree programs in science and mathematics education for teacher enhancement. Eligible participants must collaborate with a teacher preparation program of an institution of higher education. The competitively awarded grants are not to exceed $1 million. Priority would be given to applicants who consult with LEAs, use online technology, and develop innovative efforts directed at reducing shortages of science and mathematics teachers in lowincome urban or rural areas. H.R. 609 and S. 1614 would expand and extend the current loan forgiveness program for STEM teachers. Currently, HEA Title IV, Section 428J and 465, as amended by the Taxpayer-Teacher Protection Act of 2004 (P.L. 108-409), provides a higher maximum debt relief for qualified math and science secondary school teachers; up to $17,500 in loan forgiveness compared to $5,000 for other eligible teachers. However, only teachers who were new borrowers between October 1, 1998 and October 1, 2005 are eligible. Both S. 1614 and H.R. 609 would extend eligibility for loan forgiveness of up to $17,500 to qualified math and science secondary school teachers who were new borrowers after October 1, 1998; i.e., they would extend eligibility beyond the October 1, 2005 limit set by P.L. 108-409.36 H.R. 609 would establish the Mathematics and Science Incentive Program to provide eligible math and science teachers relief from interest payments on student loans in return for working in high-need schools. Participating teachers would serve at least five years and could not receive more than $5,000 in total relief. S. 1614 would allow state and partnership grantees under the HEA Title II, Teacher Enhancement Grant program to provide scholarships to students who later teach in the areas of math or science. H.R. 609 would create an Adjunct Teacher Corps that would award grants to LEAs or other educational organizations (private or public) to recruit professionals with math and 36

More information on teacher loan forgiveness can be found in CRS Report RL32516, Student Loan Forgiveness Programs, by Gail McCallion.

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science skills to serve as adjunct teachers. Grants could be used to develop outreach programs, fund signing bonuses, and compensate outside entities for the costs associated with allowing employees to serve. Grantees would have to match the federal funds received dollarfor-dollar.

Upgrading the STEM Skills of Current Teachers Another type of award provided in S. 2198 would be in the form of a fellowship for five years, in the amount of $10,000 annually, to teachers who have successfully completed a master’s degree in science or mathematics education, and who undertake increased responsibilities such as teacher mentoring and other leadership activities. S. 2198 would also direct the Secretary of Education to award grants to nonprofit entities that partner with local school districts for the training of teachers who will lead Advanced Placement or International Baccalaureate (AP-IB) and pre-AP-IB programs in science and mathematics. The grantees must demonstrate an ability to serve not fewer than 10,000 children from lowincome families. S. 2198 would also support a National Clearinghouse on Mathematics and Science Teaching Materials. The Secretary of Education would establish a national panel, after consultation with the National Academies, to collect proven effective K-12 mathematics and science teaching materials, and create a clearinghouse of such materials for dissemination to states and school districts. The bill would authorize the appropriation of $20 million for each fiscal year — FY2007 through FY2011. S. 2197 would provide for assistance to speciality schools for science and mathematics. The bill would require that funds and staff of the National Laboratories be made available to assist in teaching courses at statewide speciality schools with comprehensive programs in science, mathematics, and engineering. Both S. 1614 and H.R. 609 would target the HEA, Title II, Teacher Enhancement Grant program to math and science teachers. Under the current program, partnership grantees may provide professional development to improve teachers’ content knowledge; however, no particular subject areas are specified. Both bills would support efforts by partnership grantees to increase the number of math and science teachers and to provide opportunities for clinical experience in the areas of math and science. Both bills would also allow state grantees to support bonus pay for math and science teachers. H.R. 5358 would rename the current NSF Math and Science Partnership program that supports teacher training partnerships between LEAs and either IHEs or eligible nonprofit organizations. Grantees would operate teacher institutes that provide intensive content instruction in science and mathematics, as well as induction programs for new teachers, professional development, and training in the use of technology and laboratory equipment. Grants would be awarded for a period of five years at between $75,000 and $2 million per year. The bill would authorize appropriations of $50 million for each of the fiscal years 2007 through 2011.

Increase STEM Baccalaureate Degree Attainment S. 2198 would provide support for a Future American Scientist Scholarships program. This program would provide 25,000 new competitive merit-based undergraduate scholarships to

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students who are U.S. citizens. The scholarships would be awarded in the amount of $20,000 per year. The bill would authorize an appropriation of $375 million for FY2007; $750 million for FY2008; $1.125 billion for FY2009; and $1.5 billion annually for FY2010 through FY2013. S. 1614 would create a new Mathematics and Science Scholars Program that would award $1,000 for each of up to two years to eligible undergraduate students majoring in mathematics, science, technology, or engineering. To be eligible, students would have to complete a “rigorous secondary school curriculum in mathematics and science” as determined by the state. In determining priority for the scholarship winners, the governor of the state may take into consideration (1) the student’s regional or geographic location, (2) whether the student attended school in a high-need area, (3) attended a low-performing school, or (4) is a member of a group under-represented in STEM fields. H.R. 609 would amend the current Robert C. Byrd Honors Scholarship Program to focus these awards on students pursuing a major in studies leading to a baccalaureate, master’s, or doctoral degree in mathematics, engineering, or the physical, life, or computer sciences. These scholarships would be for a period of up to five years, and would be equal to the students’ unmet financial need (i.e., the cost of attendance minus any non-loan aid). At least 50% of the cost of these scholarships would have to come from non-federal funds. H.R. 5358 would amend the current NSF Robert Noyce Scholarship Program, which provides scholarships to undergraduates majoring in STEM fields. Students would be eligible for a $10,000 scholarship annually (up from the current $7,500), and would be required to teach math or science in a high-need school. Recipients must serve two years as a teacher for every year they received a scholarship, with a maximum of four years of service. The bill would also make special considerations for students and working professionals attending parttime. The bill authorizes $50 million for this program in FY2007, $70 million in FY2008, and $90 million for FY2009 through FY2011. H.R. 5358 would establish a Talent Expansion Program, as well as Centers for Undergraduate Education in Science, Mathematics, and Engineering, which would award grants to IHEs intended to improve and expand course-taking in STEM fields. The former program would be authorized at $40 million for FY2007, $45 million for FY2008, and $50 million for FY2009 through FY2011. The latter program would be authorized at $4 million for FY2007, $10 million for FY2008 through FY2011.

Graduate Research and Early-Career Scholarship S. 2198 would provide funding for graduate research fellowships in the critical fields of science, mathematics, and engineering. The bill would establish a fellowship program to provide tuition and financial support for eligible students pursuing master’s and doctoral degrees in science, mathematics, and engineering, and other areas of national need. S. 2198 would authorize appropriations of $225 million for FY2007, $450 million for FY2008, and $675 million for FY2009 through FY2013. Both S. 1614 and H.R. 609 would amend the Graduate Assistance in Areas of National Need program (HEA, Title VII, Section 711) to encourage study in STEM fields. The current program provides grants to institutions of higher education that award graduate degrees in areas of national need. This program also funds fellowships to graduate students pursuing doctoral degrees in areas of national need at eligible institutions. The Secretary of Education,

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in consultation with federal agencies and nonprofit organizations, is given the authority to determine the areas of national need. S. 1614 would require the Secretary to consult with specific federal and nonprofit agencies and organizations (including NAS) in determining the areas of national need. H.R. 609 would require that in determining the areas of national need, the Secretary shall prioritize “math and science teachers, special education teachers, and teachers who provide instruction for limited English proficient individuals.” S. 2109 proposes expanded graduate fellowship and graduate traineeship programs in the NSF. It would authorize the appropriation of $34 million each year for FY2007 through FY2011 for expansion of the Graduate Research Fellowship Program in NSF. Such funding would allow for an additional 250 fellowships to be awarded to U.S. citizens each year during the five-year period. The total number of fellowships to be awarded is 1,250. The bill also provides $57 million each year for a period of five years for the expansion of the Integrative Graduate Education and Research Traineeship program in NSF. The support would provide an additional 250 grants per year to U.S. citizens, for a total of 1,250 awards. S. 2109 would direct the Secretary of the Department of Defense (DOD) to utilize appropriations for expansion of the Science, Mathematics, and Research for Transformation Defense Scholarship Program (SMART). The bill would provide support for an additional 160 doctoral degrees and 60 master’s degrees for FY2007 through FY2011. The Secretary of Defense would be charged with expanding by 200 the number of participants in the National Defense Science and Engineering Graduate Fellowship program each fiscal year from FY2007 through FY2011. S. 2109 calls for the establishment of a program to award traineeships to undergraduate and graduate students pursuing studies in areas important to DOD in science, mathematics, and engineering. The selected programs should expose students to multidisciplinary studies, innovation-oriented studies, and academic, private sector, or government laboratories and research. Awardees would be required to work for DOD for 10 years following the completion of the degree program. The bill would authorize appropriations in the amount of $11.1 million each year for FY2007 through FY2011. S. 2109 would establish a clearinghouse of successful professional science master’s degree programs, and make the program elements available to colleges and universities. Grants would be awarded for pilot programs at four-year institutions that foster improvement of professional science master’s degree programs. The bill directs that preference be given to institutions that obtain two-thirds of their funding from non-federal support. A maximum of 200 grants would be awarded to four-year institutions for one three-year term, with renewal possible for a maximum of two additional years. Language included in the bill states that performance benchmarks be developed prior to the beginning of the program. The amount of $20 million would be made available for evaluation and reporting of the pilot program. S. 2197 would authorize research grants for early-career scientists and engineers. The participants must have completed their degrees no more than 10 years before the awarding of the grants. Not less than 65 grants per year would be awarded to outstanding early-career researchers to support other researchers in the DOE National Laboratories, and federally funded research and development centers. The grants would be awarded for a period of five years in duration, and at a level of $100,000 for each year of the grant period. The bill would authorize the appropriation of $6.5 million in FY2007, $13 million in FY2008, $19.5 million in FY2009, $26 million in FY2010, and $32.5 million in FY2011. H.R. 5356 would authorize two new programs (one carried out by the NSF Director and the other by the DOE Under Secretary of Science) that would award grants to scientists and

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engineers at the early stages of their careers at IHEs and other research institutions. Five-year, merit-based grants are to be awarded at a minimum of $80,000 per year to support innovative projects that integrate research and education.

Federal Program Coordination Under S. 2198, the Director of the Office of Science and Technology Policy would establish a standing committee on education in mathematics, science, and engineering. The responsibility of the committee would include the development of national goals for support of the disciplines by the federal government. Language in the bill stipulates that there should be public comment on the national goals. In addition, plans should be periodically reviewed and updated by the Director. S. 2197 calls for the creation of a position of Director of Mathematics, Science and Engineering Education, and charges that the Director administer and oversee programs at the DOE. In conjunction with the NAS, the Director would assess, after five years, the performance of science and mathematics programs at DOE. The bill would establish a Mathematics, Science, and Engineering Education fund, using 0.3% of funding made available to DOE for research, education, demonstration, and commercial application. H.R. 609 would establish Mathematics and Science Education Coordinating Council Grants to assist states in coordinating math- and science-related activities supported by the Elementary and Secondary Education Act Title II, Part B, Mathematics and Science Partnerships Program, and the HEA Title II, Teacher Quality Enhancement Program.

In: Advances in Mathematics Research, Volume 8 ISBN 978-1-60456-454-9 c 2009 Nova Science Publishers, Inc. Editor: Albert R. Baswell, pp. 277-294

Chapter 9

O N C OMPUTATIONAL M ODELS FOR THE H YPERSPACE Salvador Romaguera∗ Instituto Universitario de Matem´atica Pura y Aplicada, Universidad Polit´ecnica de Valencia, 46022 Valencia, Spain.

Abstract Let BX be the continuous poset of formal balls of a metric space (X, d) endowed by the weightable quasi-metric qd induced by d. We show that the continuous poset B(CX) of formal balls of the space CX of nonempty closed bounded subsets of X endowed by the quasi-metric qHd induced by the Hausdorff metric Hd on CX is isometric to a sup-closed subspace of the space C(BX) of nonempty sup-closed bounded subsets of BX endowed with the Hausdorff quasi-metric Hqd . We also show that the quasi-metric space (B(CX), qHd ) is bicomplete if and only if the metric space (X, d) is complete. Several consequences are derived. In particular, our approach provides an interesting class of weightable quasi-metric spaces for which weightability of the Hausdorff quasi-metric holds on certain paradigmatic subspaces. Moreover, some properties from topological algebra are discussed; for instance, we prove that if (X, d) is a metric monoid (respectively, a metric cone), then (B(CX), qHd ) is a quasi-metric monoid (respectively, (B(Cc X), qHd ) is a quasi-metric cone, where by Cc (X) we denote the family of all convex members of C(X)).

MSC (2000): 06B35, 54B20, 54E35, 54E50, 54F05, 54H12 Keywords: computational model, metric space, hyperspace, Hausdorff metric, formal ball, domain, complete, weightable quasi-metric, partial metric, quasi-metric monoid.

1.

Introduction

What is a computational model? Following Edalat and S¨underhauf [8, Introduction], and informally speaking, by a computational model we could mean a mathematical structure ∗

E-mail address: [email protected] The author thanks the support of the Spanish Ministry of Education and Science, and FEDER, under grant MTM2006-14925-C02-01.

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which is constructed in an easy way and serves useful in performing certain computations on the structure. Motivated by the fact that both (continuous) domains and metric spaces are the basic mathematical structures in computer science, some authors have formalized the idea of “computational model” by connecting the domain theory with the theory of metric spaces and related topological spaces (see [11, 19, 24, 31, etc]). On the other hand, it is well known that the Hausdorff metric constitutes an efficient tool in several branches of Mathematics and Computer Science, such as convex analysis ( [4, 27]), fractals ( [9]), image processing ( [16,33,35]) , denotational semantics of programming languages ( [1–3]), asymptotic complexity of algorithms ( [28]), etc. It then suggests the natural problem of establishing connections between computational models for hyperspaces of a metric space (X, d), endowed with the Hausdorff metric, and the corresponding hyperspaces of computational models for (X, d). Here we investigate this problem. In particular, we study the case that the computational model is constructed from the continuous poset BX of formal balls of (X, d) as discussed by Edalat and Heckmann in [7] and by Heckmann in [14]; more precisely, we prove that the continuous poset B(CX) of formal balls of the hyperspace CX of nonempty closed bounded subsets of X endowed by the weightable quasi-metric qHd induced by the Hausdorff metric Hd on CX, is isometric to a sup-closed subspace of the hyperspace C(BX) of nonempty sup-closed bounded subsets of BX endowed with the Hausdorff quasi-metric Hqd , where qd denotes the weightable quasi-metric on BX induced by the metric d. We also prove that the quasi-metric space (B(CX), qHd ) is bicomplete if and only if the metric space (X, d) is complete. Several consequences are derived. In particular, it follows that the Hausdorff quasi-metric Hqd is weightable on the subspace of C(BX) which is isometric to (B(CX), qHd ). Moreover, some properties from topological algebra are explored; for instance, we show that if (X, d) is a metric monoid, then (B(CX), qHd ) is a quasi-metric monoid. The paper is organized as follows. In Section 2 we recall the basic concepts and results from domain theory and from the theory of metric spaces and their generalizations, respectively, that will be used later on. In Sections 3 and 4 we prove our main results and, finally, we present our conclusion in the light of the obtained results.

2.

Background

In the following the letters N and R+ will denote the set of natural numbers and the set of nonnegative real numbers, respectively. Our basic reference for Domain Theory is [13]. Let us recall that a partial order (or simply an order) on a nonempty set D is a binary relation ≤ on D such that for each x, y, z ∈ D : (i) x ≤ x (reflexivity); (ii) if x ≤ y and y ≤ z, then x ≤ z (transitivity); (iii) if x ≤ y and y ≤ x, then x = y (antisymmetry). A partially ordered set, or poset for short, is a nonempty set D equipped with a partial order ≤; it will be denoted by (D, ≤) or simply by D if no confusion arises. A subset A of a poset D is directed provided that it is nonempty and every finite subset of A has an upper bound in A. The least upper bound of a subset A of D is denoted by sup A if it exists.

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A poset D is said to be directed complete, and is called a dcpo, if every directed subset of D has a least upper bound. A point x in D is called maximal if condition x ≤ y implies x = y. The set of all maximal points of D will be denoted by max D. Let D be a poset and x, y ∈ D; we say that x is way below y, in symbols x ≪ y, if for each directed subset A of D for which sup A exists, the relation y ≤ sup A implies the existence of some z ∈ A with x ≤ z. A poset D is called continuous if for each x ∈ D, the set x ⇓= {u ∈ D : u ≪ x} is directed and x = sup(x ⇓). A continuous poset which is also a dcpo is called a continuous domain or, simply, a domain. A subset B of a poset D is a basis for D if for each x ∈ D, the set xB ⇓= {u ∈ B : u ≪ x} is directed and x = sup(xB ⇓). Recall that a poset has a basis if and only if it is continuous. Therefore, a dcpo has a basis if and only if it is a domain. A dcpo having a countable basis is said to be an ω-continuous domain [7]. The Scott topology σ(D) of a dcpo (D, ≤) is constructed as follows [13, Chapter II]: A subset U of D is open with respect to σ(D) provided that: (i) U =↑ U, where ↑ U = {y ∈ D : x ≤ y for some x ∈ U }; (ii) for each directed subset A of D with sup D ∈ U, it follows that D ∩ U 6= ∅. If (D, ≤) is a domain, then the sets x ⇑, x ∈ D, form an open base for σ(D), where x ⇑= {y ∈ D : x ≪ y} (see [13, Proposition II-1.10]). Moreover, σ(D) has a countable base if and only if (D, ≤) is an ω-continuous domain. In case that (D, ≤) is a continuous poset it is possible to show yet that the sets x ⇑, x ∈ D, form an open base for a topology on D, which is also called the Scott topology of (D, ≤) and it is also denoted by σ(D) (see, for instance, [7, p. 58]). If A is a subset of D, we denote by σ(D)|A the restriction of σ(D) to A. Our basic reference for general topology is [10] and for quasi-metric spaces they are [12] and [21]. By a quasi-metric on a set X we mean a function q : X × X → R+ such that for all x, y, z ∈ X : (i) x = y ⇔ q(x, y) = q(y, x) = 0; (ii) q(x, z) ≤ q(x, y) + q(y, z). A quasi-metric space is a pair (X, q) such that X is a set and q is a quasi-metric on X. Each quasi-metric q on X induces a T0 topology τq on X which has as a base the family of open balls {Bq (x, r) : x ∈ X, ε > 0}, where Bq (x, ε) = {y ∈ X : q(x, y) < ε} for all x ∈ X and ε > 0. Given a quasi-metric q on X, then the function q −1 defined on X × X by q −1 (x, y) = q(y, x), is also a quasi-metric on X, called the conjugate of q, and the function q s defined on X × X by q s (x, y) = max{q(x, y), q −1 (x, y)} is a metric on X. A subset A of a quasi-metric space (X, q) is called sup-closed if A is closed in the metric space (X, q s ). A quasi-metric space (X, q) is said to be bicomplete if (X, q s ) is a complete metric space. In this case, we say that q is a bicomplete quasi-metric on X. The notion of a partial metric space, and its equivalent weightable quasi-metric space, was introduced by Matthews in [25] as a part of the study of denotational semantics of dataflow networks.

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Let us recall that a partial metric [25] on a set X is a function p : X × X → R+ such that for all x, y, z ∈ X : (i) x = y ⇔ p(x, x) = p(x, y) = p(y, y); (ii) p(x, x) ≤ p(x, y); (iii) p(x, y) = p(y, x); (iv) p(x, z) ≤ p(x, y) + p(y, z) − p(y, y). A partial metric space is a pair (X, p) such that X is a set and p is a partial metric on X. Each partial metric p on X induces a T0 -topology τp on X which has as a base the family of open p-balls {Bp (x, ε) : x ∈ X, ε > 0}, where Bp (x, ε) = {y ∈ X : p(x, y) < ε + p(x, x)} for all x ∈ X and ε > 0. A quasi-metric space (X, q) is called weightable if there exists a function w : X → R+ such that for all x, y ∈ X, q(x, y) + w(x) = q(y, x) + w(y). The function w is said to be a weighting function for (X, q) and the quasi-metric q is weightable by the function w. The relationship between partial metric spaces and weightable quasi-metric spaces is given in the following result. Theorem A [25]. a) Let (X, p) be a partial metric space. Then, the function q(p) : X × X → R+ defined by q(p)(x, y) = p(x, y) − p(x, x) for all x, y ∈ X, is a weightable quasi-metric on X with weighting function w given by w(x) = p(x, x) for all x ∈ X. Furthermore τp = τq(p) . b) Conversely, if (X, q) is a weightable quasi-metric space with weighting function w, then the function p(q) : X × X → R+ defined by p(q)(x, y) = q(x, y) + w(x) is a partial metric on X. Furthermore τq = τp(q) . Note that if p is a partial metric on a set X, we have p(q(p)) = p. According to [25, Definition 5.2], a sequence (xn )n in a partial metric space (X, p) is called a Cauchy sequence if there exists limn,m p(xn , xm ). A partial metric space (X, p) is said to be complete [25, Definition 5.3] if every Cauchy sequence (xn )n in X converges, with respect to τp , to a point x ∈ X such that p(x, x) = limn,m p(xn , xm ). The relationship between complete partial metric spaces and bicomplete weightable quasi-metric spaces is given in the following result. Theorem B [26]. A partial metric space (X, p) is complete if and only if (X, q(p)) is a bicomplete quasi-metric space. The following are easy but useful examples of posets. Example 1. a) It is well known that if (X, q) is a quasi-metric space, then the binary relation ≤q defined on X by x ≤q y ⇔ q(x, y) = 0, is a partial order on X. Hence (X, ≤q ) is a poset. b) Similarly, if (X, p) is a partial metric space, then the binary relation ≤p defined on X by x ≤p y ⇔ p(x, y) = p(x, x), is a partial order on X. Hence (X, ≤p ) is a poset [14, 25, 30, etc]. Therefore, if (X, p) is a partial metric space, one has ≤p =≤q(p) . We conclude this section by recalling the construction of the Hausdorff (quasi-)metric of a (quasi-)metric space.

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Recall that given a metric space (X, d), we denote by CX the family of all nonempty closed bounded subsets of X. For each pair A, B ∈ CX put Hd (A, B) = max{Hd+ (A, B), Hd− (A, B)}, where Hd+ (A, B) = supb∈B d(A, b) and Hd− (A, B) = supa∈A d(a, B). Then Hd is a metric on CX which is called the Hausdorff metric of (X, d) (see, for instance, [4]), and we refer to (CX, Hd ) as an hyperspace ( [15, p. 125]) . This notion admits a natural extension to the setting of quasi-metric spaces (compare [5, 22]). Let (X, q) be a quasi-metric space. Denote by CX the family of all nonempty closed bounded subsets of the metric space (X, q s ). For each pair A, B ∈ CX put Hq (A, B) = max{Hq+ (A, B), Hq− (A, B)}, where Hq+ (A, B) = supb∈B q(A, b) and Hq− (A, B) = supa∈A q(a, B). Then Hq is a quasi-metric on CX which is called the Hausdorff quasi-metric of (X, q), and we refer to (CX, Hq ) as an hyperspace.

3.

The Weightable Quasi-metric Space (B(CX), qHd )

In this section we shall prove some results of the structure of the quasi-metric space (B(CX), qHd ) which were announced in Section 1. First, we give formal notions of a computational model and of a quantifiable computational model, respectively. Definition 1 (compare [11, 19, 24]). A computational model for a topological space (X, τ ) is a pair (D, φ) such that D is a domain and φ is an homeomorphism from (X, τ ) onto (max D, σ(D)|max D ). Definition 2. A quantifiable computational model for a metric space (X, d) is a pair (D, φ) such that: (i) D is a domain equipped with a bicomplete quasi-metric q satisfying: (i1 ) the topology induced by q coincides with the Scott topology inherited from D; (i2 ) the partial order ≤q induced by q coincides with the partial order of D. (ii) φ is an isometry from (X, d) to (D, q) with max D = φ(X). In the last decade several authors ( [6–8, 11, 14, 19, 20, 23, 24, 31, 32, 34, etc]) have constructed computational models for various topological structures. In particular, Lawson proved in [23] that a metrizable space X is a Polish space if and only if there is an ωcontinuous domain D such that: (i) (X, τ ) is homeomorphic to (max D, σ(D)|max D ), and (ii) σ(D)|max D coincides with the Lawson topology on max D. (Let us recall that a Polish space is a separable metrizable topological space that admits a compatible complete metric, and that a topological space is said to be separable if it has a countable dense subset). Therefore, each Polish space has a computational model. In [7], Edalat and Heckmann gave a more direct and explicit construction of an ωcontinuous domain for any Polish space satisfying conditions (i) and (ii) above, with the help of the notion of a formal ball. In fact, they proved, among other results that each complete metric space, has a computational model. Later on, Heckmann ( [14]) improved this result by essentially showing that each complete metric space has a quantifiable computational model.

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Next we recall the main results from [7] and [14] because our approach is based on them. Given a metric space (X, d), define BX := {(x, r) : x ∈ X, r ∈ R+ }. Then, each pair (x, r) ∈ BX is called a formal ball in (X, d). Denote by ⊑d the binary relation on BX given by (x, r) ⊑d (y, s) ⇐⇒ d(x, y) ≤ r − s. Then (BX, ⊑d ) is a poset. Furthermore (BX, ⊑d ) is a domain if and only if the metric space (X, d) is complete. It is also proved in [7] that for each metric space (X, d), (BX, ⊑d ) is a continuous poset such that max BX = {(x, 0) : x ∈ X}. Moreover, the mapping φ : X → BX given by φ(x) = (x, 0) for all x ∈ X, is an homeomorphism from (X, τd ) onto (max BX, σ(BX)|max BX ). In [14] Heckmann essentially proved, among others, the following crucial result: Given a (complete) metric space (X, d), the function pd defined on BX × BX by pd ((x, r), (y, s)) = max{d(x, y), |r − s|} + r + s, is a (complete) partial metric on BX. Furthermore, the topology induced by pd coincides with the Scott topology on BX, and the partial order ≤pd induced by pd coincides with ⊑d on BX. The weightable quasi-metric q(pd ) induced by pd will be simply denoted by qd , and, by Theorem A, we have qd ((x, r), (y, s)) = max{d(x, y), |r − s|} + s − r, for all (x, r), (y, s) ∈ BX. We will refer to qd as the weightable quasi-metric induced by d. Remark 1. Taking into account the preceding observations and Theorem B, we deduce that if (X, d) is a complete metric space, then the pair (BX, φ) is a quantifiable computational model for (X, d), where BX is equipped with the bicomplete quasi-metric qd induced by d, and φ : X → BX is given by φ(x) = (x, 0) for all x ∈ X. From the above constructions we also deduce that for a metric space (X, d), the pair (B(CX), qHd ) is a weightable quasi-metric space where B(CX) is the continuous poset of formal balls of the metric hyperspace (CX, Hd ) and qHd is the weightable quasi-metric on B(CX) induced by Hd , i.e., qHd ((A, r), (B, s)) = max{Hd (A, B), |r − s|} + s − r, for all (A, r), (B, s) ∈ B(CX). On the other hand, we can construct the quasi-metric hyperspace (C(BX), Hqd ), where C(BX) is the family of all nonempty subsets of BX that are closed and bounded in the metric space (BX, (qd )s ).

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In the light of these facts, it suggests the natural question of obtaining the precise relationship between the weightable quasi-metric space (B(CX), qHd ) and the quasi-metric hyperspace (C(BX), Hqd ). In Theorem 1 below we answer this question. Let us recall that an isometry from a quasi-metric space (X, q) to a quasi-metric space (Y, q ′ ) is a mapping f : X → Y such that q ′ (f (x), f (y)) = q(x, y) for all x, y ∈ X. (Note that every isometry is one-to-one.) The quasi-metric spaces (X, q) and (Y, q ′ ) are said to be isometric if there is an isometry from (X, q) onto (Y, q ′ ). Theorem 1. Let (X, d) be a metric space. Then (B(CX), qHd ) is isometric to a supclosed subspace of (C(BX), Hqd ), via the isometry Φ : (B(CX), qHd ) → (C(BX), Hqd ) given by Φ((A, r)) = A × {r}, for all A ∈ CX and r ∈ R+ . Proof. We first show that Φ is well-defined. Indeed, let A ∈ CX and r ∈ R+ , and let (y, s) ∈ BX such that (y, s) belongs to the closure of A × {r} in (BX, (qd )s ). Then, there is a sequence (an )n in A such that lim(qd )s ((y, s), (an , r)) = 0. n

Hence limn {max{d(y, an ), |r − s|}} = 0, so y ∈ A and r = s. Thus A × {r} ∈ C(BX) and consequently Φ is well-defined. Now let (A, r), (B, s) ∈ B(CX). We easily deduce the following relations inf a∈A {max{d(a, b), |r − s|} + s − r} = max{inf a∈A d(a, b), |r − s|} + s − r, and supb∈B {max{d(A, b), |r − s|} + s − r} = max{supb∈B d(A, b), |r − s|} + s − r. Therefore Hq+d (Φ((A, r)), Φ((B, s))) = sup qd ((A × {r}), (b, s)) b∈B

= sup{ inf qd ((a, r), (b, s))} b∈B a∈A = sup inf {max{d(a, b), |r − s|} + s − r} b∈B a∈A = sup max{ inf d(a, b), |r − s|} + s − r b∈B

a∈A

= sup {max{d(A, b), |r − s|} + s − r} b∈B

= max{sup d(A, b), |r − s|} + s − r =

b∈B max{Hd+ (A, B), |r

− s|} + s − r.

Similarly, we show that Hq−d (Φ((A, r)), Φ((B, s))) = max{Hd− (A, B), |r − s|} + s − r.

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Hence Hqd (Φ((A, r)), Φ((B, s))) = max max{Hd+ (A, B), |r − s|} + s − r, max{Hd− (A, B), |r − s|} + s − r = max max{Hd+ (A, B), Hd− (A, B)}, |r − s| + s − r

= max{Hd (A, B), |r − s|} + s − r = qHd ((A, r), (B, s)).

Consequently Φ is an isometry from (B(CX), qHd ) to (C(BX), Hqd ). Finally, we show that the set Φ(B(CX)) is closed in the metric space (C(BX), (Hqd )s ). Let U ∈ C(BX) and let ((An , rn ))n be a sequence in B(CX) such that lim(Hqd )s (U, An × {rn }) = 0. n

Then for each (x, sx ) ∈ U there are sequences (an )n , (bn )n , with an , bn ∈ An for all n ∈ N, such that lim {{max{d(x, an ), |sx − rn |} + rn − sx } = 0,

and

n

lim {{max{d(x, bn ), |sx − rn |} + sx − rn } = 0. n

Therefore limn rn = sx (and also limn d(x, an ) = limn d(x, bn ) = 0). Thus, if we put s = limn rn and B = {x ∈ X : (x, s) ∈ U }, we have that U = B × {s}. Since U is closed in the metric space (BX, (qd )s ), it easily follows that B is closed in (X, d). Thus U ∈ Φ(B(CX)), and hence Φ(B(CX)) is closed in (C(BX), (Hqd )s ). This concludes the proof. The preceding result is interesting because from the weightable quasi-metric space (BX, qd ) one obtains a nice subspace of the quasi-metric hyperspace (C(BX), Hqd ), namely (Φ(B(CX)), Hqd ), which is also weightable. The following example shows that, nevertheless, (C(BX), Hqd ) is not weightable, in general. Example 2. Let X = {a} and let d be the trivial metric on X, i.e., d(a, a) = 0. We show that (C(BX), Hqd ) is not weightable. Assume the contrary. Then, there exists a weighting function W for (C(BX), Hqd ). We have the following relations qd ((a, 0), (a, 1)) = max{d(a, a), 1} + 1 = 2,

and

qd ((a, 1), (a, 0)) = max{d(a, a), 1} − 1 = 0. Then Hqd ({(a, 0)}, {(a, 1)}) = qd ((a, 0), (a, 1)) = 2, and Hqd ({(a, 1)}, {(a, 0)}) = qd ((a, 1), (a, 0)) = 0. Hence (1)

2 + W ({(a, 0)}) = 0 + W ({(a, 1)}).

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On the other hand, if B = {(a, 0), (a, 1)}, we obtain Hqd ({(a, 0)}, B) = max{Hq+d ({(a, 0)}, B), Hq−d ({(a, 0)}, B)} = max sup qd ({(a, 0)}, b), qd ({(a, 0)}, B) b∈B

= 2, and Hqd (B, {(a, 0)}) = max{Hq+d (B, {(a, 0)}), Hq−d (B, {(a, 0)})} = max qd (B, {(a, 0)}), sup qd (b, {(a, 0)}) b∈B

= 0. Hence (2)

2 + W ({(a, 0}) = 0 + W (B).

Moreover, we have Hqd ({(a, 1)}, B) = max{Hq+d ({(a, 1)}, B), Hq−d ({(a, 1)}, B)} = max sup qd ({(a, 1)}, b), qd ({(a, 1)}, B) b∈B

= 0, and Hqd (B, {(a, 1)}) = max{Hq+d (B, {(a, 1)}), Hq−d (B, {(a, 1)})} = max qd (B, {(a, 1)}), sup qd (b, {(a, 1)}) b∈B

= 2. Hence (3)

0 + W ({(a, 1}) = 2 + W (B).

From (1) and (2) it follows that W ({(a, 1)}) = W (B), which contradicts relation (3). We conclude that (C(BX), Hqd ) is not weightable. Theorem 2. For a metric space (X, d) the following are equivalent. (1) (X, d) is complete. (2) (B(CX), qHd ) is weightable and bicomplete. (3) (Φ(B(CX)), Hqd ) is weightable and bicomplete. Proof. (1) =⇒ (2). If (X, d) is complete, then the hyperspace (CX, Hd ) is complete ( [4, Theorem 3.2.4]). So, by Remark 1, the weightable quasi-metric space (B(CX), qHd ) is bicomplete.

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(2) =⇒ (3). It is an obvious consequence of Theorem 1. (3) =⇒ (1). Let (xn )n be a Cauchy sequence in (X, d). Then ({xn } × {0})n is a Cauchy sequence in (Φ(B(CX)), (Hqd )s ). Let A ∈ CX and r ∈ R+ such that limn (Hqd )s (A × {r}, {xn } × {0}) = 0. (Of course, r = 0). Choose a ∈ A. Clearly limn d(a, xn ) = 0. We conclude that (X, d) is complete. (Note that, in particular, A = {a}). Observe that if (X, d) is a metric space, then for each A, B ∈ CX and r, s ∈ R+ we have Hqd (A × {r}, B × {s}) = 0 ⇔ qHd ((A, r), (B, s)) = 0 ⇔ Hd (A, B) ≤ r − s, and hence A × {r} ≤Hqd B × {s} ⇔ (A, r) ⊑Hd (B, s). Consequently, from the fact that (B(CX), ⊑Hd ) is a continuous poset, it follows that (Φ(B(CX)), ≤Hqd ) is a continuous poset, where ≤Hqd is the partial order induced by Hqd (see Example 1 a)). Moreover, if (X, d) is complete, then (B(CX), ⊑Hd ) is a domain, and, thus, (Φ(B(CX)), ≤Hqd ) is also a domain. So, by Theorem 2, (Φ(B(CX)), φ) is a quantifiable computational model for (CX, Hd ), where φ(A) = A × {0} for all A ∈ CX.

4.

Properties from Topological Algebra

In this section we obtain some properties of (B(CX), qHd ) from topological algebra. Actually, we shall do this approach in a more general setting where the space BX of formal balls is replaced by a product space X × Y with X and Y be metric monoids or metric cones, respectively. In this direction, our first result generalizes Heckmann’s construction of the partial metric pd given in Section 3. Proposition 1. Let (X, d) and (Y, e) be metric spaces. Fix u0 ∈ Y. Then the function pd,e : (X × Y ) × (X × Y ) → R+ given by pd,e ((x, u), (y, v)) = max{d(x, y), e(u, v)} + e(u0 , u) + e(u0 , v), for all x, y ∈ X and u, v ∈ Y, is a partial metric on X × Y. Proof. Let x, y, z ∈ X and u, v, w ∈ Y. Then pd,e ((x, u), (y, v)) = pd,e ((x, u), (x, u)) = pd,e ((y, v), (y, v)) ⇐⇒ max{d(x, y), e(u, v)} + e(u0 , u) + e(u0 , v) = 2e(u0 , u) = 2e(u0 , v) ⇐⇒ (x, u) = (y, v).

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Moreover pd,e ((x, u), (x, u)) = 2e(u0 , u) ≤ e(u0 , u) + e(u0 , v) + e(v, u) ≤ pd,e ((x, u), (y, v)), and, obviously, pd,e ((x, u), (y, v)) = pd,e ((y, v), (x, u)). Finally pd,e ((x, u), (y, v)) = max{d(x, y), e(u, v)} + e(u0 , u) + e(u0 , v) ≤ max{d(x, z), e(u, w)} + max{d(z, y), e(w, v)} + e(u0 , u) + e(u0 , v) = pd,e ((x, u), (z, w)) + pd,e ((z, w), (y, v)) − pd,e ((z, w), (z, w)). We conclude that (X × Y, pd,e ) is a partial metric space. Note that if (Y, e) is the set R+ endowed with the Euclidean metric and put u0 = 0, then pd,e is exactly the partial metric pd on BX constructed in Section 3. Remark 2. The partial metric pd,e of Theorem 3 satisfies pd,e ((x, u), (x, u)) = 2e(u0 , u), for all x ∈ X and u ∈ Y. Hence, the weightable quasi-metric q(pd,e ) induced by pd,e , has the function w given by w((x, u)) = 2e(u0 , u), for all x ∈ X and u ∈ Y , as a weighting function. Furthermore, we have q(pd,e )((x, u), (y, v)) = max{d(x, y), e(u, v)} + e(u0 , v) − e(u0 , u), and hence

q(pd,e ) + (q(pd,e ))−1 2

((x, u), (y, v)) = max{d(x, y), e(u, v)},

for all x, y ∈ X and u, v ∈ Y. Therefore, the metric (q(pd,e )) + (q(pd,e ))−1 )/2 coincides with the product metric d × e on X × Y. The following result generalizes Theorem 2 to this context. Proposition 2. Let (X,d) and (Y,e) be complete metric spaces. Then (X × Y, q(pd,e )) is a bicomplete quasi-metric space. Proof. By Remark 2 it follows that (X × Y, (q(pd,e )) + (q(pd,e ))−1 )/2) is a complete metric space, and thus (X, (q(pd,e ))s ) is a complete metric space. We conclude that (X, q(pd,e )) is bicomplete.

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Let us recall that a monoid is a semigroup (X, ∗) having neutral element. If (X, ∗) and (Y, ⋆) are (commutative) monoids, then (X × Y, ∗⋆)) is a (commutative) monoid, where (x, u) ∗ ⋆(y, v) = (x ∗ y, u ⋆ v) for all x, y ∈ X, u, v ∈ Y. By a (quasi-)metric monoid we mean a triple (X, ∗, d) such that (X, ∗) is a monoid and d is a (quasi-)metric on X such that d(x ∗ a, y ∗ b) ≤ d(x, y) + d(a, b) for all x, y, a, b ∈ X, or equivalently, d(x ∗ z, y ∗ z) ≤ d(x, y) and d(z ∗ x, z ∗ y) ≤ d(x, y) for all x, y, z ∈ X (see, for instance, [30]). According to [30, Definition 1], by a partial metric monoid we mean a triple (X, ∗, p) such that (X, ∗) is a monoid and p is a partial metric on X such that p(x ∗ a, y ∗ b) ≤ p(x, y) + p(a, b) for all x, y, a, b ∈ X. Lemma 1. Let (X, ∗, d) and (Y, ⋆, e) be metric monoids. Then (X × Y, ∗⋆, pd,e ) is a partial metric monoid whenever the point u0 of Proposition 1 is the neutral element of (Y, ⋆). Proof. First note that if u0 is the neutral element of (Y, ⋆), we have e(u0 , u ⋆ v) ≤ e(u0 , u) + e(u0 , v) for all u, v ∈ Y. Now let x, y, a, b ∈ X and u, v, s, t ∈ Y. Then, by Proposition 1, pd,e ((x, u) ∗ ⋆(y, v), (a, s) ∗ ⋆(b, t)) = pd,e ((x ∗ y, u ⋆ v), (a ∗ b, s ⋆ t)) = max {d(x ∗ y, a ∗ b), e(u ⋆ v, s ⋆ t)} + e(u0 , u ⋆ v) + e(u0 , s ⋆ t) ≤ max {d(x, a) + d(y, b), e(u, s) + e(v, t)} + e(u0 , u) + e(u0 , v) + e(u0 , s) + e(u0 , t) ≤ max {d(x, a), e(u, s)} + max {d(y, b), e(v, t)} + e(u0 , u) + e(u0 , v) +e(u0 , s) + e(u0 , t) = pd,e ((x, u), (a, s)) + pd,e ((y, v), (b, t)). We have shown that (X × Y, ∗⋆, pd,e ) is a partial metric monoid. Lemma 2. Let (X, ∗, d) and (Y, ⋆, e) be metric monoids. Then (X × Y, ∗⋆, q(pd,e )) is a quasi-metric monoid whenever the point u0 of Proposition 1 is the neutral element of (Y, ⋆) and e(u0 , u ⋆ v) = e(u0 , u) + e(u0 , v) for all u, v ∈ Y . Proof. Let x, y, a, b ∈ X and u, v, s, t ∈ Y. From Theorem A, Lemma 1 and Remark 2, we deduce the following relations q(pd,e )((x, u) ∗ ⋆(y, v), (a, s) ∗ ⋆(b, t)) = q(pd,e )((x ∗ y, u ⋆ v), (a ∗ b, s ⋆ t)) = pd,e ((x ∗ y, u ⋆ v), (a ∗ b, s ⋆ t)) − pd,e ((x ∗ y, u ⋆ v), (x ∗ y, u ⋆ v)) ≤ pd,e ((x, u), (a, s)) + pd,e ((y, v), (b, t)) − 2e(u0 , u ⋆ v) = pd,e ((x, u), (a, s)) + pd,e ((y, v), (b, t)) − 2e(u0 , u) − 2e(u0 , v) = q(pd,e )((x, u), (a, s)) + q(pd,e )((y, v), (b, t)). We have shown that (X × Y, ∗⋆, q(pd,e )) is a quasi-metric monoid.

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If we denote by + the usual addition on R+ , then the monoid (R+ , +), endowed with the Euclidean metric, has the structure of a metric monoid. From this fact and Lemmas 1 and 2, we immediately deduce the following result. Proposition 3. Let (X, ∗, d) be a metric monoid. Then (BX, ∗+, pd ) is a partial metric monoid and (BX, ∗+, qd ) is a quasi-metric monoid. Following [17], a cone (a semilinear space in [29]) is a triple (X, ∗, ◦) such that (X, ∗) is a commutative monoid with neutral element 0 and ◦ is a function from R+ × X to X which satisfies for all r, s ∈ R+ and x, y ∈ X : (i) r◦(s◦x) = (rs)◦x; (ii) (r+s)◦x = r◦x∗s◦x; (iii) r ◦ (x ∗ y) = r ◦ x ∗ r ◦ y; (iv) 1 ◦ x = x; (v) 0 ◦ x = 0 (see [18] for related structures). If (X, ∗, ◦) and (Y, ⋆, ·) are cones, then (X × Y, ∗⋆, ◦·) is a cone, where we define r ◦ ·(x, u) = (r ◦ x, r · u) for all x ∈ X, u ∈ Y and r ∈ R+ . By a (quasi-)metric cone we mean a quadruple (X, ∗, ◦, d) such that (X, ∗, ◦) is a cone and d is a (quasi-)metric on X for which d(x∗a, y∗b) ≤ d(x, y)+d(a, b) and d(r◦x, r◦y) ≤ rd(x, y) for all x, y, a, b ∈ X and r ∈ R+ . By a partial metric cone we mean a quadruple (X, ∗, ◦, p) such that (X, ∗, ◦) is a cone and p is a partial metric on X for which p(x ∗ a, y ∗ b) ≤ p(x, y) + p(a, b) and p(r ◦ x, r ◦ y) ≤ rp(x, y) for all x, y, a, b ∈ X and r ∈ R+ . Lemma 3. Let (X, ∗, ◦, d) and (Y, ⋆, ·, e) be metric cones. Then (X × Y, ∗⋆, ◦·, pd,e ) is a partial metric cone whenever the point u0 of Proposition 1 is the neutral element of (Y, ⋆, ·). Proof. By virtue of Lemma 1 we only need to prove that for each x, y ∈ X, u, v ∈ Y and r ∈ R+ , it follows that pd,e (r ◦ ·(x, u), r ◦ ·(y, v)) ≤ rpd,e ((x, u), (y, v)). In fact, pd,e (r ◦ ·(x, u), r ◦ ·(y, v)) = pd,e ((r ◦ x, r · u), (r ◦ y, r · v)) = max {d(r ◦ x, r ◦ y), e(r · u, r · v)} + e(u0 , r · u) + e(u0 , r · v) ≤ max {rd(x, y), re(u, v)} + re(u0 , u) + re(u0 , v) = r [max {d(x, y), e(u, v)} + e(u0 , u) + e(u0 , v)] = rpd,e ((x, u), (y, v)). We have shown that (X × Y, ∗⋆, ◦·, pd,e ) is a partial metric cone. Lemma 4. Let (X, ∗, ◦, d) and (Y, ⋆, ·, e) be metric cones. If u0 is the neutral element of (Y, ⋆, ·). Then (X × Y, ∗⋆, ◦·, q(pd,e )) is a quasi-metric cone whenever the point u0 of Proposition 1 is the neutral element of (Y, ⋆, ·) and e(u0 , u ⋆ v) = e(u0 , u) + e(u0 , v), e(u0 , r · u) = re(u0 , u), for all u, v ∈ Y and r ∈ R+ . Proof. By virtue of Lemma 2 we only need to prove that for each x, y ∈ X, u, v ∈ Y and r ∈ R+ , it follows that q(pd,e )(r ◦ ·(x, u), r ◦ ·(y, v)) ≤ rq(pd,e )((x, u), (y, v)). In fact,

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from Theorem A, Lemma 3 and Remark 2, we obtain q(pd,e )(r ◦ ·(x, u), r ◦ ·(y, v)) = q(pd,e )((r ◦ x, r · u), (r ◦ y, r · v)) = pd,e ((r ◦ x, r · u), (r ◦ y, r · v)) − 2e(u0 , r · u) ≤ rpd,e ((x, u), (y, v)) + re(u0 , v) − 2re(u0 , u) = rq(pd,e )((x, u), (y, v)). We have shown that (X × Y, ∗⋆, ·◦, q(pd,e )) is a quasi-metric cone. If we denote by · the usual multiplication on R+ , then the cone (R+ , +, ·), endowed with the Euclidean metric, has the structure of a metric cone. From this fact and Lemmas 3 and 4, we immediately deduce the following result. Proposition 4. Let (X, ∗, ◦, d) be a metric cone. Then (BX, ∗+, ◦·, pd ) is a partial metric cone and (BX, ∗+, ◦·, qd ) is a quasi-metric cone. In order to apply Propositions 3 and 4 to B(CX) and B(Cc X), respectively, where by Cc X we denote the family of all nonempty closed bounded convex subsets of a metric cone (X, ∗, ◦, d), we need to establish some properties of the structure of CX and Cc X (related results may be found, for instance, in [4, p. 89 and 91] and in [27]). Let (X, ∗, d) be a metric monoid with neutral element 0. For each A, B ∈ CX, define A ∗ B = {a ∗ b : a ∈ A, b ∈ B}, and A ⊛ B = A ∗ B. Then, we have the following. Lemma 5. (CX, ⊛, Hd ) is a metric monoid. Proof. Let A, B, C ∈ CX. Obviously A ⊛ B is a nonempty closed subset of (X, d). Moreover, it is bounded because A and B are bounded and (X, ∗, d) is a metric monoid. Thus A ⊛ B ∈ CX. Furthermore (A ⊛ B) ⊛ C = (A ⊛ B) ∗ C = A ∗ B ∗ C = A ∗ B ∗ C = A ∗ B ∗ C = A ∗ (B ⊛ C) = A ⊛ (B ⊛ C). Since for each A ∈ CX, A ⊛ {0} = A, it follows that (CX, ⊛) is a monoid. If, in addition, (X, ∗) is commutative then it is obvious that (CX, ⊛) is also commutative. Next we show that Hd+ (A ⊛ C, B ⊛ C) ≤ Hd+ (A, B). Indeed, choose an arbitrary ε > 0. Let x ∈ B ⊛ C. Then, there exist b ∈ B and c ∈ C such that d(b ∗ c, x) < ε. Now let a ∈ A such that d(a, b) < ε + d(A, b). Thus d(A ⊛ C, x) ≤ d(A ∗ C, x) ≤ d(a ∗ c, x) ≤ d(a ∗ c, b ∗ c) + d(b ∗ c, x) < 2ε + d(A, b) ≤ 2ε + Hd+ (A, B). Hence Hd+ (A ⊛ C, B ⊛ C) ≤ Hd+ (A, B). Similarly we show the following inequalities: Hd+ (C ⊛ A, C ⊛ B) ≤ Hd+ (A, B), Hq− (A ⊛ C, B ⊛ C) ≤ Hq− (A, B) and Hq− (C ⊛ A, C ⊛ B) ≤ Hq− (A, B). We conclude that (CX, ⊛, Hd ) is a metric monoid.

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Theorem 3. Let (X, ∗, d) be a metric monoid. Then (B(CX), ⊛+, pHd ) is a partial metric monoid and (B(CX), ⊛+, qHd ) is a quasi-metric monoid. Proof. Since, by Lemma 5, (CX, ⊛, Hd ) is a metric monoid, it follows from Proposition 3 that (B(CX), ⊛+, pHd ) is a partial metric monoid and (B(CX), ⊛+, qHd ) is a quasi-metric monoid. If (X, ∗, ◦, d) is a metric cone and for each A, B ∈ Cc X and r ∈ R+ we define A ⊛ B = A ∗ B, and r ◦ A = {r ◦ a : a ∈ A}, we have the following. Lemma 6. (Cc X, ⊛, ◦, Hd ) is a metric cone. Proof. We first show that (Cc X, ⊛, ◦) is a cone. It is easily seen that for each A, B ∈ Cc X and r ∈ R+ , A ⊛ B ∈ Cc X and r ◦ A ∈ Cc X, so, by virtue of Lemma 5, we only need to prove that for each A, B ∈ Cc X and r, s ∈ R+ . the conditions (i)-(v) of the notion of a cone given above, hold. In fact, it is clear that r ◦ (s ◦ A) = (rs) ◦ A. Moreover r ◦ (A ⊛ B) = r ◦ A ∗ B = r ◦ (A ∗ B) = r ◦ A ∗ r ◦ B = r ◦ A ⊛ r ◦ B. On the other hand, by convexity of A, we obtain (r + s) ◦ A = r ◦ A ∗ s ◦ A, so (r + s) ◦ A ⊆ r ◦ A ⊛ s ◦ A. Now let x ∈ r ◦ A ⊛ s ◦ A. Then there exist two sequences (an )n , (bn )n , in A such that limn d(x, r ◦ an ∗ s ◦ bn ) = 0. Therefore 1 1 lim d ◦ x, ◦ (r ◦ an ∗ s ◦ bn ) = 0. n r+s r+s 1 ◦ x ∈ A, so x ∈ (r + s) ◦ A. We conclude Since A is closed convex, we deduce that r+s that (r + s) ◦ A = r ◦ A ⊛ s ◦ A. Finally, it is obvious that 1 ◦ A = A and 0 ◦ A = 0. Hence (Cc X, ⊛, ◦) is a cone. Now from the fact that for each A, B ∈ Cc X and r ∈ R+ , we have

d(r ◦ A, r ◦ b) ≤ rd(A, b),

and d(r ◦ a, r ◦ B) ≤ r(d(a, B),

for all a ∈ A, b ∈ B, it follows that Hd (r ◦ A, r ◦ B) ≤ rHd (A, B). By Lemma 5 we conclude that (Cc X, ⊛, ◦, Hd ) is a metric cone. Theorem 4. Let (X, ∗, ◦, d) be a metric cone. Then (B(CX), ⊛+, ◦·, pHd ) is a partial metric cone and (B(CX), ⊛+, ◦·, qHd ) is a quasi-metric cone. Proof. Since, by Lemma 6, (CX, ⊛, ◦, Hd ) is a metric cone, it follows from Proposition 4 that (B(CX), ⊛+, ◦·, pHd ) is a partial metric cone and (B(CX), ⊛+, ◦·, qHd ) is a quasimetric cone.

5.

Conclusion

It is known that for a metric space (X, d), the continuous poset BX of formal balls can be endowed with a weightable quasi-metric qd whose induced topology coincides with

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the Scott topology. We have analyzed the continuous poset B(CX) of the hyperspace (CX, Hd ). We have proved that the weightable quasi-metric space (B(CX), qHd ) is isometric to a sup-dense subspace of the hyperspace (C(BX), Hqd ). If (X, d) is complete, then this subspace provides a quantifiable computational model for (CX, Hd ). Some properties from topological algebra have been also discussed. In particular, if (X, d) is a metric monoid, then (B(CX), qHd ) admits a structure of quasi-metric monoid, and if (X, d) is a metric cone, then (B(Cc X), qHd ) admits a structure of quasi-metric cone.

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[14] R. Heckmann. Approximation of metric spaces by partial metric spaces, Appl. Categ. Struct. 1999, vol. 7, 71-83. [15] N.R. Howes. Modern Analysis and Topology, Universitext, Springer-Verlag, New York, 1995. [16] D.P. Huttenlocher; G.A. Klanderman; W.J. Rucklidge. Comparing images using the Hausdorff distance, IEEE Trans. Pattern Anal. Mach. Intelligence 1993, vol. 15, 850863. [17] K. Keimel; W. Roth. Ordered Cones and Approximation, Lecture Notes Math. vol. 1517, Springer, Berlin, 1992. [18] R. Kopperman. All topologies come from generalized metrics, Amer. Math. Monthly 1988, vol. 95, 89-97. [19] R. Kopperman; H.P. K¨unzi; P. Waszkiewicz. Bounded complete models of topological spaces, Topology Appl. 2004, vol. 139, 285-297. [20] M. Kr¨otzsch. Generalized ultrametric spaces in quantitative domain theory, Theoret. Comput. Sci. 2006, vol. 368, 30-49. [21] H.P.A. K¨unzi. Nonsymmetric distances and their associated topologies: About the origins of basic ideas in the area of asymmetric topology, In: Handbook of the History of General Topology, C.E. Aull, R. Lowen (Eds.), Kluwer, Drodrecht, 2001, Vol. 3, 853-968. [22] H.P.A. K¨unzi; C. Ryser. The Bourbaki quasi-uniformity, Topology Proc. 1995, vol. 20, 161-183. [23] J. Lawson. Spaces of maximal points, Math. Struct. Comput. Sci. 1997, vol. 7, 543556. [24] K. Martin. Domain theoretical models of topological spaces, Electronic Notes Theoret. Comput. Sci. 1998, vol. 13, 173-181. [25] S.G. Matthews. Partial metric topology, Ann. New York Acad. Sci. 1994, vol. 728, 183-197. [26] S. Oltra; S. Romaguera; E.A. S´anchez-P´erez. Bicompleting weightable quasi-metric spaces and partial metric spaces, Rend. Circolo Mat. Palermo 2002, vol. 51, 151-162. [27] J. Rodr´ıguez-L´opez; S. Romaguera. Closedness of bounded convex sets of asymmetric normed linear spaces and the Hausdorff quasi-metric, Bull. Belgian Math. Soc. 2006, vol. 13, 551-562. [28] J. Rodr´ıguez-L´opez; S. Romaguera; O. Valero. Asymptotic complexity of algorithms via the nonsymmetric Hausdorff distance, Computing Letters 2006, vol. 2, 155-161. [29] S. Romaguera; M. Sanchis. Semi-Lipschitz functions and best approximation in quasimetric spaces, J. Approx. Theory 2000, vol. 103, 292-301.

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[30] S. Romaguera; M. Schellekens. Partial metric monoids and semivaluation spaces, Topology Appl. 2005, vol. 153, 948-962. [31] J.J.M.M. Rutten. Weighted colimits and formal balls in generalized metric spaces, Topology Appl. 1998, vol. 89, 179-202. [32] M.P. Schellekens. A characterization of partial metrizability. Domains are quantifiable, Theoret. Comput. Sci. 2003, vol. 305, 409-432. [33] B. Sendov. Hausdorff distance and image processing, Russian Math. Surveys 2004, vol. 59, 319-328. [34] P. Waszkiewicz. Partial metrisability of continuous posets, Math. Struct. Comput. Sci. 2006, vol. 16, 359-372. [35] C. Zhao; W. Shi; Y. Deng. A new Hausdorff distance for image matching, Pattern Recogn. Letters 2005, vol. 26, 581-586.

In: Advances in Mathematics Research, Volume 8 ISBN 978-1-60456-454-9 c 2009 Nova Science Publishers, Inc. Editor: Albert R. Baswell, pp. 295-353

Chapter 10

P ERIODIC -T YPE S OLUTIONS OF D IFFERENTIAL I NCLUSIONS Jan Andres∗ Dept. of Math. Analysis, Fac. of Science, Palack´y University, Tomkova 40, 779 00 Olomouc–Hejˇc´ın, Czech Republic

1.

Introduction

Periodicity is one of the most influental phenomena for everybody’s life. However, its pure form occurs in nature rather rarely. For instance, periodic structure in three independent directions (lattice symmetry) which is typical for crystals is never purely periodic in a mathematical (ideal) sense. Moreover, in the early 80’s, certain materials were found with diffraction patterns as those for crystals, but with other symmetries that are not commensurate with lattice symmetry. These new substances, called quasicrystals, have much to do with (aperiodic) Penrose tiling and quasi-periodicity (or even with almost-periodicity; cf. [Me1], [Me2]). Regular time repetitions, like clock measurements, are similarly always periodic only with some accuracy. That is why mathematical models deal also with various sorts of (generalized) periodicity which can be rather far from its original meaning (cf. e.g. [A4], [AKZ], [Fe], [JMe], [LF], [LLY], [LY], [Ru]). Periodic dynamics (in a pure or generalized sense) are usually described by solutions of differential or difference equations or inclusions. In order to extend the notion of a dynamical system in an appropriate way, the “terminus technicus” periodic process was introduced (cf. e.g. [AG1] and the references therein). Although many monographs have been already written about free and forced linear as well as nonlinear oscillations (see e.g. [H1], [Md], [Mi]), especially the theory of almost-periodic oscillations is far from to be built in a satisfactory level (cf. [AG1], [BMC], [HM], [J3], [Mw2], [SY]). Moreover, there are many sorts of almost-periodicity and some of them are a bit curious (cf. [ABG], [Be], [Bs], [Le]). It is interesting that the creator of the classical theory of almost-periodic functions, Harald Bohr [Bh], came to (uniformly) almost-periodic functions not in order to extend the notion of a ∗

E-mail address: 6198959214).

[email protected]

Supported by the Council of Czech Government (MSM

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periodic function, but because of his investigation of Dirichlet series. There is still a lot to be also said about quasi-periodic (cf. [BC1], [BC2], [CZ], [Lv], [Or], [SM]), anti-periodic (cf. [AAP1], [AAP2], [AF], [AP], [FMT]) and derivo-periodic (cf. [A2], [A3], [AGMP], [Mw1]) oscillations. Hence, the main purpose of our contribution is two-fold: (i) rather than a complete account or a systematic study, we would like to indicate a flavour of the theory of periodictype oscillations, and (ii) to present some of our own results for periodic-type solutions of differential equations and inclusions. For (i), we preferably selected in Section 4. (Primer of periodic-type oscillations) the related results (including ours) which are easy for formulation while, for (ii), some technicalities had to be involved in Section 5. in order to derive sufficiently general criteria of the effective solvability of given actual problems. Results are, nevertheless, sketched in a form that is convenient for exposition and not necessarily in the greatest generality possible. Our objective is so to give the reader an overall idea of what the standard theory is like as well as to include enough information about its most recent progress. Formally, the focus of the object is simply the determination of the readable text for a wider audience with some parts to yield also a profit for the specialists. Although Section 3. (Periodic-type maps, multivalued maps and their selections) contains a lot of a new material, eventually of an independent interest, its choice was tendentiously determined by the needs of the main Sections 4. and 5.. Not to break the context, we also recall in Preliminaries several useful facts about multivalued maps and their selections. In Concluding remarks, possible generalizations, extensions and improvements are only indicated.

2.

Preliminaries

In the entire text, by a multivalued map ϕ : X ⊸ Y , we mean the one with at least nonempty values, i.e. ϕ : X → 2Y \ {0}. It is convenient to recall the definitions of multivalued maps which will be under consideration and their basic properties. In particular, the existence of appropriate single-valued selections will be of our interest here. For more details, see [AG1], [HP], [Ry]. D EFINITION 2.1 A map ϕ : X ⊸ Y , where X, Y are metric spaces, is said to be upper semicontinuous (u.s.c.) if, for every open U ⊂ Y , the set {x ∈ X | ϕ(x) ⊂ U } is open in X. It is said to be lower semicontinuous (l.s.c.) if, for every open U ⊂ Y , the set {x ∈ X | ϕ(x) ∩ U 6= ∅} is open in X. If it is both u.s.c. and l.s.c., then it is called continuous. Obviously, in the single-valued case, if f : X → Y is u.s.c. or l.s.c., then it is continuous. Moreover, the compact-valued map ϕ : X ⊸ Y is continuous if and only if it is Hausdorff-continuous, i.e. continuous w.r.t. the metric d in X and the Hausdorff-metric dH in {B ⊂ Y | B is nonempty and bounded}, where dH (A, B) := inf{ε > 0 | A ⊂ Oε (B) and B ⊂ Oε (A)} and Oε (B) := {x ∈ X | ∃y ∈ B : d(x, y) < ε}. Every u.s.c. map ϕ : X ⊸ Y with closed values has a closed graph Γϕ , but not vice versa. Nevertheless, if the graph Γϕ of a compact map ϕ : X ⊸ Y is closed, then ϕ is u.s.c.

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The following proposition will be important for the existence of single-valued continuous selections (see, e.g. [HP, Corollary 1.4.8]). P ROPOSITION 2.1 Let X be a paracompact (e.g. metric) space and Y be a Banach space. If ϕ : X ⊸ Y is l.s.c. with convex closed values, then given [xi , yi ] ∈ Γϕ , i = 1, . . . , n, there exist a single-valued continuous selection f ⊂ ϕ of ϕ such that f (xi ) = yi , i = 1, . . . , n. We say that ϕ : X ⊸ Y is Lipschitzian or Lipschitz-continuous with constant L (w.r.t. the metric d in X and the Hausdorff-metric dH in {B ⊂ Y | B is nonempty and bounded}) if dH (ϕ(x1 ), ϕ(x2 )) ≤ Ld(x1 , x2 ), for all x1 , x2 ∈ X. For metric X and Banach Y , Lipschitz-continuous ϕ : X ⊸ Y with bounded, closed, convex values admits a Lipschitz-selection f ⊂ ϕ if and only if Y is finite-dimensional.√The Lipschitz constant of f is not necessarily the same as for ϕ. For Y = Rn , it is n( 12 3+ 5 1)L, where L is a constant of ϕ. For more details, see [HP, pp. 101–105]. Besides semicontinuous maps, measurable and semi-Carath´eodory maps will be also of importance. Hence, assume that Y = (Y, d) is a separable metric space and (Ω, U, ν) is a measurable space, i.e. a set Ω equipped with σ-algebra U of subsets and a countably additive measure ν on U. A typical example is when Ω is a bounded domain in Rn , equipped with the Lebesque measure. D EFINITION 2.2 A map ϕ : Ω ⊸ Y is called strongly measurable if there exists a sequence of step multivalued maps ϕn : Ω ⊸ Y such that dH (ϕn (ω), ϕ(ω)) → 0, for almost all (a.a.) ω ∈ Ω, as n → ∞. In the single-valued case, one can simply replace multivalued step maps by single-valued step maps and dH (ϕn (ω), ϕ(ω)) by d (ϕn (ω), ϕ(ω)). A map ϕ : Ω ⊸ Y is called measurable if {ω ∈ Ω | ϕ(ω) ⊂ V } ∈ U, for each open V ⊂Y. A map ϕ : Ω ⊸ Y is called weakly measurable if {ω ∈ Ω | ϕ(ω) ⊂ V } ∈ U, for each closed V ⊂ Y . Obviously, if ϕ is strongly measurable, then it is measurable and if ϕ is measurable, then it is also weakly measurable. If ϕ has compact values, then the notions of measurability and weak measurability coincide. In separable Banach spaces Y , the notions of strong measurability and measurability coincide for multivalued maps with compact values as well as for single-valued maps (see [KOZ, Theorem 1.3.1 on pp. 45–49]). If Y is a not necessarily separable Banach space, then a strongly measurable map ϕ : Ω ⊸ Y with compact values has a single-valued strongly measurable selection (see e.g. [De, Proposition 3.4(b) on pp. 25– 26]). Furthermore, if Y is a separable complete space, then every measurable ϕ : Ω ⊸ Y with closed values has, according to the following Kuratowski–Ryll-Nardzewski theorem (see e.g. [AG1, Theorem 3.49 in Chapter I.3]), a single-valued measurable selection. P ROPOSITION 2.2 (Kuratowski–Ryll-Nardzewski) Let Ω be as above and Y be a separable complete space. Then every measurable map ϕ : Ω ⊸ Y with closed values has a singlevalued measurable selection. Now, let Ω = [0, a] be equipped with the Lebesque measure and X, Y be Banach. D EFINITION 2.3 A map ϕ : [0, a] × X ⊸ Y with nonempty, compact and convex values is called u-Carath´eodory (resp. l-Carath´eodory, resp. Carath´eodory) if it satisfies (i) t ⊸ ϕ(t, x) is strongly measurable, for every x ∈ X,

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(ii) x ⊸ ϕ(t, x) is u.s.c. (resp. l.s.c., resp. continuous), for almost all t ∈ [0, a], (iii) kykY ≤ r(t)(1 + kxkX ), for every (t, x) ∈ [0, a] × X, y ∈ ϕ(t, x), where r : [0, a] → [0, ∞) is an integrable function. For our needs (semi-) Carath´eodory maps will be employed only in Euclidean spaces. Moreover, for X = Rm and Y = Rn , one can state the following proposition. P ROPOSITION 2.3 (i) Carath´eodory maps are product-measurable (measurable as the whole (t, x) ⊸ ϕ(t, x)), i.e. w.r.t. minimal σ-algebra U[0,a] ⊗ B(Rm ), generated by U[0,a] × B(Rm ), where U[0,a] is the σ-algebra of subsets of [0, a], and B(Rm ) stands for the Borel sets of Rm , (ii) they possess a single-valued Carath´eodory selection f ⊂ ϕ. It need not be so for u-Carath´eodory or l-Carath´eodory maps. Nevertheless, for uCarath´eodory maps, we have at least (again X = Rm and Y = Rn ). P ROPOSITION 2.4 u-Carath´eodory maps (in the sense of Definition 2.3) are weakly superpositionally measurable, i.e. the composition ϕ(t, q(t)) admits, for every q ∈ C([0, a], Rm ), a single-valued measurable selection. If they are still product-measurable, then they are also superpositionally measurable, i.e. the composition ϕ(t, q(t)) is measurable, for every q ∈ C([0, a], Rm ). R EMARK 2.1 If X, Y are separable Banach spaces and ϕ : X ⊸ Y is a Carath´eodory mapping, then ϕ is also superpositionally measurable, i.e. ϕ(t, q(t)) is measurable, for every q ∈ C([0, a], X) (see [KOZ, Theorem 1.3.4 on p. 56]). Under the same assumptions, Proposition 2.3 can be appropriately generalized (see [KOZ, Proposition 7.9 on p. 229 and Proposition 7.23 on pp. 234–235]). If ϕ : X ⊸ Y is only u-Carath´eodory and X, Y are (not necessarily separable) Banach spaces, then ϕ is weakly superpositionally measurable, i.e. ϕ(t, q(t)) admits a single-valued measurable selection, for every q ∈ C([0, a], X) (see e.g. [De, Proposition 3.5 on pp. 26– 27] or [KOZ, Theorem 1.3.5 on pp. 57–58]). For l-Carath´eodory maps, single-valued Carath´eodory selections can be guaranteed, under suitable restrictions (cf. [Ry] and the references therein). Nevertheless, since lCarath´eodory maps will not be employed in the sequel, the related statements are omitted here.

3. 3.1.

Periodic-Type Maps, Multivalued Maps and Their Selections Periodic Maps

Function f , defined on Rn and having values in an arbitrary set S, is called periodic if there exists a nonzero vector (0 6=) ω ∈ Rn such that f (x + ω) = f (x), for all x ∈ Rn . Any ω ∈ Rn satisfying this equality is called a period of f . The set P of all periods of f generates a subgroup in Rn which (by the hypothesis) does not degenerate to 0 (cf. [Bo]). Sometimes (e.g. when f is not defined on the whole Rn ), we can restrict ourselves to a subset X ⊂ Rn , e.g. for X = Rn+ . The equality f (x + ω) ≡ f (x) then means f (x + ω) = f (x), for all x ∈ X, where f (x) and f (x + ω) are defined. If f : Rn ⊸ S or f : X ⊸ S

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is multivalued with nonempty values, i.e. f : Rn → 2S \ {0} or f : X → 2S \ {0}, then by a periodic (multivalued) function f , we often mean only that f (x) ⊂ f (x + ω), provided each component of ω is positive, for all x, where f (x) and f (x + ω) are defined. If f : Rn → H is a continuous map from Rn into a Hausdorff topological space H, then the group P of its periods is closed. If V ⊂ P is the biggest vector subspace of P , then f is constant on any class (mod V ). Thus, f can be defined in this way by its restriction on a subspace which is complementary to V . That is why it is enough to consider continuous periodic functions with a discrete group of periods P . If rank P = p, then f is called pperiodic1 and any system of p vectors generating P is called the main system of periods of f . For two different main systems of periods, one can be obtained from the other by a linear transformation with integer coefficients whose determinant has an absolute value equal to 1. If P is a closed subgroup of the group Rn and π : Rn → Rn /P is a canonical projection, then g → g ◦π is a bijective mapping of all functions from Rn /P into S whose periods are contained in P . If S is a topological space, then for continuity of g it is necessary and sufficient g ◦ π to be continuous. For more details, concerning the algebraic aspects of periodic maps, see e.g. [Bo]. If S is a vector space and f : Rn → S is single-valued, then the basic equality f (x + ω) = f (x) can be equivalently rewritten into f (x + ω) − f (x) = 0. If S = (S, d) is a metric space, then we can still write d(f (x), f (x + ω)) = 0. On the other hand, if S = (S, d0 ) is only pseudometric, i.e. if d0 (f (x), f (x + ω)) = 0 does not necessarily imply that f (x) = f (x + ω), then the set {f : Rn → S | d0 (f (x), f (x + ω)) = 0} can be rather far from the set of periodic functions. For multivalued maps f : Rn ⊸ S, the situation is even more delicate. If A, B ⊂ S are subsets of a vector space, then defining A − B := {a − b | a ∈ A, b ∈ B}, the equality f (x + ω) − f (x) = 0 can never be satisfied, provided f is not single-valued. If S = (S, d) is a metric space, then dist(f (x), f (x + ω)) = inf[d(y1 , y2 ) | y1 ∈ f (x), y2 ∈ f (x + ω)] = 0, for the set distance, can imply rather curious possibilities, but including the case f (x) ⊂ f (x + ω). On the other hand, the equality f (x) = f (x + ω) can be expressed equivalently by means of the Hausdorff metric dH (for its definition, see Preliminaries) as dH (f (x), f (x+ω)) = 0 while, for the inclusion f (x) ⊂ f (x+ω), we have not equivalently that dH (f (x), f (x + ω)) ≥ 0. Moreover, if f : Rn ⊸ B, where B is a Banach space, is a periodic lower semicontinuous (l.s.c.) map with convex closed values, then it admits a single-valued continuous periodic selection f0 ⊂ f with the same period (see Proposition 2.1). If f : Rn ⊸ S, 1

By a p-periodic map f , we usually mean in a completely different way that f (x + p) ≡ f (x), where p ∈ Rn is the minimal period. The different meaning can be, however, easily recognized from the context. If the minimal period is 0, the function f is constant.

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where S is a separable complete space, is a measurable periodic map with closed values, then according to the well-known Kuratowski–Ryll–Nardzewski theorem (cf. Proposition 2.2) f admits a single-valued measurable periodic selection f0 ⊂ f with the same period. If f : R → Rn is (almost everywhere) differentiable and ω-periodic, then f˙ is also ω-periodic. Denoting therefore, as usual, CP (R, Rn ) := {f ∈ C (R, Rn ) | f (x) ≡ f (x + ω)}, C (k)P (R, Rn ) := {f ∈ C (k) (R, Rn ) | f (x) ≡ f (x + ω)}, we have C (k)P (R, Rn ) = {f ∈ C (k) (R, Rn ) | f (j) (x) ≡ f (j) (x + ω), j = 0, . . . , k}. If we endow CP (R, Rn ) by the norm kf k1 := maxx∈[0,ω] |f (x)| and C (k)P (R, Rn ) by Pk (j) (x)|, then the spaces (CP (R, Rn ), k · k ) and the norm kf k2 := 1 j=0 maxx∈[0,ω] |f (C (k)P (R, Rn ), k · k2 ) are Banach. Similarly, denoting AC (k−1)P (R, Rn ) := {f ∈ AC (k−1) (R, Rn ) | f (x) ≡ f (x + ω), } where AC (k−1) (R, Rn ) is a space of functions f : R → Rn whose (k − 1)th derivatives f (k−1) (x) are absolutely continuous, and endowing it by the norm kf k3 :=

k−1 X j=0

max |f

x∈[0,ω]

(j)

(x)| +

Z

ω

|f (k) (x)| dx,

0

the space (AC (k−1)P (R, Rn ), k · k3 ) is Banach as well, etc. The basic problem of the Fourier harmonic analysis consists in representation of (continuous) periodic functions by means of series of trigonometric functions of the form ̺1 cos T x resp. ̺2 sin T x. More precisely, let f : R → R be a function such that f (x + ω) ≡ f (x) and assume that the following Dirichlet conditions are satisfied: (i) f is single-valued and finite, for every x ∈ [0, ω] (observe that this condition already follows from the notation f : R → R), (ii) f admits in [0, ω] at most a finite number of (finite) discontinuities, (iii) f admits in [0, ω] a finite number of local extremal points. Then f can be uniquely expressed by means of the Fourier series as follows: ∞

X 1 f (x) = A0 + (An cos nT x + Bn sin nT x), 2 n=1

where T =

2π ω ,

2 An = ω

Z

0

ω

f (x) cos nT x dx,

2 Bn = ω

Z

ω

f (x) sin nT x dx.

0

This series is convergent in each point x = x0 ∈ [0, ω] to the value 1 [f (x0 + 0) + f (x0 − 0)]. 2

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In fact, an ω-periodic f can be equivalently expressed by means of the series consisting only of trigonometric functions of one type, namely ∞ p X 1 Bn 2 2 f (x) = A0 + An + Bn cos nT x − arctan 2 An n=1

resp. f (x) =

∞ p X 1 Bn 1 A2n + Bn2 sin nT x + π − arctan , A0 + 2 2 An n=1

where the Fourier coefficients An , Bn are as above. If an ω-periodic f is additionally even or odd, then simply ∞

X 1 f (x) = A0 + An cos nT x 2 n=1

or f (x) =

∞ X

Bn sin nT x,

n=1

respectively. For more details concerning trigonometric series and the Fourier analysis, see e.g. [Z1, Z2]. For abstract harmonic analysis, see e.g. the monograph in two volumes [HR1, HR2]. For the classification and approximation of periodic functions, see [St].

3.2.

Anti-periodic Maps

Function f , defined on Rn and having values in a vector space V (unlike for periodic maps, we must also consider function −f which requires values in a vector space) is called antiperiodic if there exists a nonzero vector (0 6=) ω ∈ Rn such that f (x + ω) = −f (x), for all x ∈ Rn . Any ω ∈ Rn satisfying this equality is called an anti-period of f . Obviously, every anti-periodic function with anti-period ω ∈ Rn is 2ω-periodic, but not vice versa. The class of anti-periodic functions can be therefore regarded as a special subclass of periodic functions. As for periodic functions, if not otherwise stated, by an ω-anti-periodic function, we shall mean the one with the minimal anti-period ω ∈ Rn . A generalization appears in the context of Bloch waves and the Floquet theory (studied in Section 4.2. below), where solutions of given differential equations are typically of the form x(t + T ) ≡ ̺x(t), where ̺ = e(p/q)πi ; p, q ∈ Z, because, for ̺ = 1 (p = 0), we obtain a T -periodic solution, while for ̺ = −1 (p = q 6= 0), we obtain a T -anti-periodic solution. Functions of this form are therefore sometimes called periodic in the sense of Bloch. Another generalization might appear in this context, namely functions satisfying the functional equality f (x + ω) = α(x)f (x), where α is a suitable function. These maps are sometimes also rather incorrectly called as quasi-periodic. We saw in Section 3.1. that odd ω-periodic functions f can be expressed in the form f (x) =

∞ X

n=1

Bn sin nT x,

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where T =

2π ω

and Bn =

2 ω

Z

ω

f (x) sin nT x dx.

0

Hence, if the graph of an ω-anti-periodic function f (x + c) is, for some c = c0 ∈ Rn , still symmetric w.r.t. the origin 0, then the function f (x + c0 ) is an odd ω-anti-periodic as well as odd 2ω-periodic by which f can be expressed in the form ∞ X

f (x) =

Bn sin

n=1

where 1 Bn = ω

Z

2ω+c0

c0

π (x − c0 ), ω

π f (x) sin n x dx. ω

Otherwise, the related Fourier series includes also the terms An cos n ωπ x, as for general 2ω-periodic functions, in Section 3.1.. If V is a vector space and f : Rn → V is single-valued, then the basic equality f (x + ω) = −f (x) can be rewritten into f (x + ω) + f (x) = 0. If V = (V, d) is a metric vector space, then we can still write d(−f (x), f (x + ω)) = 0. On the other hand, if V = (V, d0 ) is only a pseudometric vector space, i.e. if d0 (−f (x), f (x + ω)) = 0 does not necessarily imply that f (x + ω) = −f (x), then the set {f : Rn → V | d0 (−f (x), f (x + ω)) = 0} can be rather far from the set of anti-periodic functions. For multivalued maps f : Rn ⊸ V , the situation is again more delicate. The equality f (x) + f (x + ω) = 0 can never be satisfied, provided f is not single-valued. On the other hand, the equality f (x+ω) = −f (x) can be expressed equivalently by means of the Hausdorff metric dH (for its definition, see Preliminaries) as dH (−f (x), f (x + ω)) = 0. By anti-periodic (multivalued) functions, we can only mean, similarly as in Section 3.1., that −f (x) ⊂ f (x + ω), provided each component of ω ∈ Rn is positive, for all x, where f (x) and f (x + ω) are defined. This implies that dist(−f (x), f (x + ω)) = inf[d(y1 , y2 ) | y1 ∈ −f (x), y2 ∈ f (x + ω)] = 0, but the reverse implication does not obviously hold. We also have dH (−f (x), f (x + ω)) ≥ 0, but the reverse implication means something completely different. If f : Rn ⊸ B, where B is a Banach space, is an anti-periodic lower semicontinuous (l.s.c.) map with convex closed values, then it admits a single-valued continuous antiperiodic selection f0 ⊂ f with the same anti-period (see Proposition 2.1). If f : Rn ⊸ B, where B is a separable Banach space, is a measurable anti-periodic map with closed values then, according to the well-known Kuratowski–Ryll–Nardzewski theorem (cf. Proposition 2.2), f admits a single-valued measurable anti-periodic selection f0 ⊂ f with the same anti-period. If f : R → Rn is (almost everywhere) differentiable and ω-anti-periodic, then f˙ is also ω-anti-periodic. Denoting therefore CAP (R, Rn ) := {f ∈ C (R, Rn ) | −f (x) ≡ f (x + ω)}, C (k)AP (R, Rn ) := {f ∈ C (k) (R, Rn ) | −f (x) ≡ f (x + ω)}, AC (k−1)AP (R, Rn ) := {f ∈ AC (k−1) (R, Rn ) | −f (x) ≡ f (x + ω)},

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the spaces (CAP (R, Rn ), k · k1 ), (C (k)AP (R, Rn ), k · k2 ), (AC (k−1)AP (R, Rn ), k · k3 ) are Banach, provided the norms k · k1 , k · k2 , k · k3 are defined as in Section 3.1..

3.3.

Quasi-periodic Maps

D EFINITION 3.1 A continuous function f : R → Rn is said to be k-period quasiperiodic if it takes the form f (x) = f (ω1 x, ω2 x, . . . , ωk x), where f is 1-periodic separately in its k arguments, i.e. f (x1 , . . . , xj , . . . , xk ) ≡ f (x1 , . . . , xj+1 , . . . , xk ), for every j ∈ {1, . . . , k}, and ω1 , ω2 , . . . , ωk are incommensurate frequencies, namely if n1 ω1 + n2 ω2 + · · · + nk ωk = 0 holds only when all the integers nj ∈ Z, j = 1, . . . , k, are zero, i.e. n1 = n2 = . . . = nk = 0. R EMARK 3.1 As for periodic functions, Definition 3.1 can be easily extended in a natural way to k-period quasi-periodic maps f : Rn → S, where S is an arbitrary set. For our needs, it is however enough to be restricted to real vector functions as in Definition 3.1. A more general class of (not necessarily continuous) quasi-periodic functions was studied in [BP]. √ E XAMPLE 3.1 Function f (x) = sin(2πx) + sin(2 2πx) is obviously 2-period quasi√ √ ω1 2 1 √ periodic with frequencies ω1 = 1 and ω2 = 2, because ω2 = 2 = 2 6∈ Q. R EMARK 3.2 Of course, 1-period quasi-periodic functions are periodic in the usual sense. Furthermore, it is well-known that quasi-periodic functions are related to the problem of invariant tori and to the celebrated KAM theory (cf. e.g. [BHS]). Every (continuous) quasi-periodic function is well-known (cf. e.g. [NB, p. 232]) to be uniformly almost-periodic (for the definition of a uniform almost-periodicity, see the following section). The reverse statement does not hold in general. Nevertheless, we know necessary and sufficient additional conditions under which a uniformly almost-periodic function becomes quasi-periodic. For Nakajima’s proof of the following theorem, see e.g. [Yo, pp. 30–34]. T HEOREM 3.1 Function f : R → Rn is k-period quasi-periodic if and only if it is uniformly almost-periodic and its module (spectrum) has a finite integer basis, namely f (x) =

X m

m1 mk am exp 2πix + ··· + , ω1 ωk

where ω1 , . . . , ωk are some reals and m = (m1 , . . . , mk ), for m1 , m2 , . . . , mk ∈ Z. In [MP], the authors found necessary and sufficient conditions under which a function f : R → R can be represented as a sum of n periodic functions. Defining, for given constants h0 , . . . , hn > 0, the differences ∆(h0 )f (x) := f (x + h0 ) − f (x) and ∆(h0 , h1 , . . . , hn )f (x) := ∆(h0 , . . . , hn−1 )(∆(hn ))f (x), their theorem reads as follows.

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T HEOREM 3.2 ([MP]) A function f : R → R is the sum of finitely many periodic functions if and only if there exist positive numbers h0 , . . . , hn−1 such that hi /hj ∈ Q, for i 6= j, and ∆(h0 , . . . , hn−1 )f = 0. Recalling that a function f : R → R is Darboux if, for an arbitrary interval I ⊂ R, the image f (I) is connected, the authors of [NW] specified Theorem 3.2 for Darboux summands as follows. T HEOREM 3.3 ([NW]) Assume, for given constants h0 , . . . , hn−1 > 0, that hi /hj ∈ Q, for i 6= j. A function f : R → R is the sum of n ∈ N Darboux functions f0 , . . . , fn−1 of periods h0 , . . . , hn−1 , respectively, if and only if ∆(h0 , . . . , hn−1 )f = 0. Recalling that a function f : R → R is Marczewski measurable if, for each perfect (i.e. closed with no isolated points) set P ⊂ R, there exists a perfect subset Q ⊂ P such that the restriction f |Q is continuous, the authors of [NW] still specified Theorem 3.2 for Marczewski measurable summands as follows. T HEOREM 3.4 ([NW]) Assume, for given constants h0 , . . . , hn−1 > 0, that hi /hj ∈ Q, for i 6= j. A function f : R → R is the sum of n ∈ N Marczewski measurable periodic functions f0 , . . . , fn−1 of periods h0 , . . . , hn−1 , respectively, if and only if f is Marczewski measurable and ∆(h0 , . . . , hn−1 )f = 0. R EMARK 3.3 As was proved in [NW], although the identity map f (x) = x cannot be represented as the sum of a finite number of Lebesgue measurable functions or periodic functions with the Baire property (i.e. their domain is R \ M , where M ⊂ R is a meager (first Bair category) set and f −1 (W ) ∩ R differs from an open set by a meager set in R, for every open subset W of R, by which they are, according to the Kuratowski theorem, continuous on R \ M ), it can be written as the sum of two periodic Marczewski measurable functions. Finite sums of periodic functions form the most typical subclass of quasi-periodic functions. If the summands are multivalued lower semicontinuous periodic maps with convex closed values or measurable periodic maps with closed values then, as pointed out in Section 3.1., there exist single-valued periodic continuous or measurable selections, respectively, forming the single-valued summands. Moreover, it has again meaning to consider such multivalued summands only on a subset X ⊂ R, e.g. for X = [0, ∞), and with the inclusion periodicity property, namely fj (x) ⊂ fj (x + hj ), j = 0, . . . , n − 1, for all x ∈ X. Unfortunately, quasi-periodic functions are well-known to form a vector space which is not closed w.r.t. the uniform convergence on R (cf. e.g. [F3]). Since the Banach space of uniformly almost-periodic functions has not this handicap, that is perhaps why it is preferebly studied. Since (apart from the special case of periodic summands) we shall not deal in this paper with quasi-periodic solutions, let us finally mention its relationship to partial differential equations, as explained e.g. in [OT]. Consider the equation x˙ = f (ω1 t, . . . , ωk t, x), (3.1) where f : R2 → R is continuous and f (·, x) is a (continuous) k-periodic quasi-periodic function (i.e. ω1 , . . . , ωk are linearly independent reals over the rationals Q), for every x ∈ R. Then x(t) = x(ω1 t, . . . , ωk t), where x(t1 . . . , tk ) is 1-periodic in each tj , j = 1, . . . , k,

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is a quasi-periodic solution of (3.1) if and only if x(t) satisfies the partial differential equation, as its equation of characteristics, ω1

∂x ∂x + · · · + ωk = f (t1 , . . . , tk , x), ∂t1 ∂tk

(3.2)

in a distributional sense. In [OT], the authors did not consider (3.2) in a distributional sense, but in a classical setting, for which they interpreted the operator on the left-hand side of (3.2) as the directional derivative. This approach lead them to quasi-periodic solutions of (3.1) in a weaker sense (close to almost automorphic solutions; cf. [NG], [SY]), but still with significant properties, allowing them to preserve a version of the Massera transformation theorem, studied in Section 4.3. below.

3.4.

Almost-periodic Maps

The theory of almost-periodic (a.p.) functions was created by H. Bohr in the Twenties, but it was restricted to the class of uniformly continuous functions. Let us, therefore, consider it firstly as a subspace of the space C (R, R) of all continuous functions, defined on R and with the values in R. Let us recall that a set X ⊂ R is said to be relatively dense (r.d.) if there exists a number l > 0 s.t. every interval [a, a + l] contains at least one point of X. D EFINITION 3.2 (Bohr-type definition) A function f ∈ C (R, R) is said to be uniformly almost-periodic (u.a.p.) if, for every ǫ > 0, there corresponds a r.d. set {τ }ǫ s.t. sup |f (x + τ ) − f (x)| < ǫ x∈R

∀τ ∈ {τ }ǫ .

Each number τ ∈ {τ }ǫ is called an ǫ-uniformly almost-period (or a uniformly ǫtranslation number) of f . The class Cap of u.a.p. functions has the following important properties (see e.g. [Bh, Bs]): • Every u.a.p. function is uniformly continuous. • Every u.a.p. function is uniformly bounded. • If a sequence of u.a.p. functions fn converges uniformly in R to a function f , then f is u.a.p., too. In other words, the set of u.a.p. functions is closed w.r.t. the uniform convergence. Since it is a closed subset of the Banach space Cb := C ∩L∞ (i.e. the space of bounded continuous functions, endowed with the sup-norm), it is Banach, too. It is easy to show that the space is even a commutative Banach algebra, w.r.t. the usual product of functions. D EFINITION 3.3 (normality or Bochner-type definition) A function f ∈ C (R, R) is called uniformly normal if, for every sequence {hi } of real numbers, there corresponds a subsequence {hni } s.t. the sequence of functions {f (x + hni )} is uniformly convergent. The numbers hi are called translation numbers and the functions f hi (x) := f (x + hi ) are called translates.

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In other words, f is uniformly normal if the set of translates is precompact in Cb , i.e. if it contains a fundamental (Cauchy) subsequence. Obviously, in complete spaces, it is equivalent to say that the set of translates is precompact or relatively compact (i.e. the closure is compact). Since every trigonometric polynomial P (x) =

n X

ak eiλk x

(ak ∈ R; λk ∈ R)

k=1

is well-known to be u.a.p., every function f , obtained as the limit of a uniformly convergent sequence of trigonometric polynomials, is u.a.p. Thus, it is natural to introduce the third definition of a.p. (continuous) functions. D EFINITION 3.4 (approximation) We call Cap (R, R) the (Banach) space obtained as the closure of the space P(R, R) of all trigonometric polynomials in the space Cb , endowed with the sup-norm. R EMARK 3.4 Equivalently (cf. [Bh], [Bs]), function f belongs to Cap (R, R) if, for any ǫ > 0, there exists a trigonometric polynomial Tǫ s.t. sup |f (x) − Tǫ (x)| < ǫ . x∈R

It is easy to show that Cap , like C , is invariant under translations, that is Cap contains, together with f , the functions f t (x) := f (x + t) ∀t ∈ R. The Definitions 3.2, 3.3 and 3.4, are equivalent (see e.g. [Bh], [Bs]): T HEOREM 3.5 A continuous function f is u.a.p. if and only if it is uniformly normal and if and only if it belongs to Cap (R, R). For every function f , we will call by the mean value of f the number 1 T →∞ 2T

M [f ] = lim

Z

T

f (x) dx.

−T

The mean value of every u.a.p. function f exists and (cf. [Bh], [Bs]) 1 (a) M [f ] = lim T →∞ T

Z

1 (b) M [f ] = lim T →∞ 2T

T

0

Z

1 f (x) dx = lim T →∞ T

Z

0

f (x) dx,

−T

a+T

f (x) dx;

(3.3)

uniformly w.r.t. a ∈ R.

a−T

R EMARK 3.5 Every even function satisfies (3.3), while necessary condition for an odd function to be u.a.p. is that M [f ] = 0. Furthermore, since, for every u.a.p. function f and for every real number λ, the function f (x)e−iλx is a u.a.p. function, the number a(λ, f ) := M [f (x)e−iλx ] always exists.

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307

For every u.a.p. function f , there always exists at most a countable infinite set of values λ (called the Bohr–Fourier exponents or frequencies) for which a(λ, f ) 6= 0 (see e.g. [Bh], [Bs]). The numbers a(λ, f ) are called the Bohr–Fourier coefficients and the set σ(f ) := {λn | a(λn , f ) 6= 0} is called the spectrumP of f . The formal series n a(λn , f )e−iλx is called the Bohr–Fourier series of f and we write X f (x) ∼ a(λn , f )e−iλx . n

In [F2], the author found necessary and sufficient conditions in order a uniformly a.p. function f ∈ C (R, R) to be uniformly approximated on R by continuous periodic functions with an arbitrary accuracy. It is so if and only if f has a one-point basis, i.e. that there exists a number r ∈ R such that σ(f ) ⊂ rQ = {rq | q ∈ Q}. If σ(f ) has a finite integer basis, then f is, according to Theorem 3.1, quasi-periodic, and vice versa. Every nonconstant continuous ω-periodic function f has a one-point basis {2π/ω} which, however, need not be integer. On the other hand, there also exist a.p. functions with a one-point basis which are not periodic. In fact, according to [F2], function f can have values in a Banach space. In [F1], uniformly a.p. functions which are (uniform) limits of sequences of continuous periodic functions were considered with a special respect to the structure of their Bohr– Fourier exponents. Let us note that A. S. Besicovitch [Bs] has shown that there are infinitely many trigonometric series convergent, in every finite interval, to any continuous bounded function of a bounded variation and, therefore, that the sum-function of an everywhere convergent trigonometric series is not necessarily uniformly almost-periodic. For more details like the connection between the Bohr–Fourier exponents and almostperiods, see e.g. [ABG] and the references therein. In order to deal with locally Lebesgue integrable almost-periodic functions, let us introduce the following Stepanov, Weyl and Besicovitch (pseudo)norms and distances Z x+L 1 p 1 p p (Stepanov) kf kS := sup |f (t)| dt , DS p (f, g) = kf − gkS p ; L L L x∈R L x (Weyl) kf kW p := lim kf kS p , DW p (f, g) = limL→∞ DS p (f, g); L

L→∞

(Besicovitch) kf kB p

1 := lim sup 2T T →∞

L

Z

T

p

|f (t)| dt

−T

1

p

, DB p (f, g) = kf − gkB p ,

where f, g ∈ Lploc (R, R), p ∈ N and L > 0. It is without any loss of generality to fix L > 0 as L = 1 (cf. [ABG], [Bh], [Bs]). Hence, taking Gp successively as S p (= S1p ) or W p or B p , we can generalize Definitions 3.2, 3.3 and 3.4 as follows. D EFINITION 3.5 (Bohr-type definition) A function f ∈ Lploc (R, R) is said to have the Gpap property if, for every ǫ > 0, there corresponds a r.d. set {τ }ǫ s.t. DGp (f (x + τ ), f (x)) < ǫ;

∀τ ∈ {τ }ǫ .

308

Jan Andres Each number τ ∈ {τ }ǫ is called an ǫ-Gp -almost-period of f .

D EFINITION 3.6 (normality or Bochner-type definition) A function f ∈ Lploc (R, R) is called Gp -normal if the family of functions {f h }, defined as f h (t) = f (t + h), where h ∈ R is an arbitrary number, is Gp -precompact, i.e. if for each sequence {f hi }, we can choose a fundamental subsequence. D EFINITION 3.7 (approximation) We denote by G p (R, R) the space obtained as the closure in BG p := {f ∈ Lploc (R, R) | kf kGp < ∞} of the space P(R, R) of all trigonometric polynomials w.r.t. the Gp -norm. p We can still define, rather curiously, the class e-Wap , e-W p -normal and e-W p of equiWeyl a.p. functions by means of the Stepanov (pseudo)metric as follows.

D EFINITION 3.8 (Bohr-type definition) A function f ∈ Lploc (R, R) is said to be equip almost-periodic in the sense of Weyl (e-Wap ) if, for every ǫ > 0, there corresponds a r.d. set {τ }ǫ and a number L0 = L0 (ǫ) s.t.

Z x+L 1 p 1 p sup |f (t + τ ) − f (t)| dt < ǫ; x∈R L x

∀L ≥ L0 (ǫ).

Each number τ ∈ {τ }ǫ is called an ǫ-equi-Weyl-almost-period (or equi-Weyl ǫtranslation number of f ). D EFINITION 3.9 (equi-W p -normality) A function f ∈ Lploc (R, R) is said to be equi-W p normal if the family of functions {f h } (h is an arbitrary real number) is SLp -precompact, for sufficiently large L, i.e. if for each sequence {f hi }, we can choose an SLp -fundamental subsequence, for a sufficiently large L. D EFINITION 3.10 (approximation) We will denote by equi-W p (R, R) the space obtained as the closure in BS p := {f ∈ Lploc (R, R) | f b ∈ L∞ (R, Lp ([0, 1]))} of the space of all trigonometric polynomials w.r.t. to the SLp -norm, for a sufficiently large L, i.e. for every f ∈ e − W p and for every ǫ > 0, there exist L0 = L0 (ǫ) and a trigonometric polynomial Tǫ s.t. DS p (f, Tǫ ) < ǫ L

∀L ≥ L0 (ǫ).

p Although the classes Sap , S p -normal and S p are, as in Theorem 3.5, equivalent, it is not so for other classes. The hierarchy of almost-periodic function spaces was established in [ABG] in the form of the following Table 1.

Periodic-Type Solutions of Differential Inclusions

309

Table 1. a. periods

normal

Cap

Bohr Stepanov equi-Weyl

⇔

u. normal

⇑ ⇓r

⇑ ⇓r

p Sap

⇔

S p -normal ⇑ ⇓r

p e-Wap

e-W p -normal

⇑ ⇓r

............

⇐ ;

p Wap

Weyl

⇑ ⇓r

Besicovitch

⇑ ⇓r

u.a.p.

⇔

Sp m

⇐ ; ⇐ ;

Wp ⇑ ⇓r

............

B p -normal

e-W p ⇑ ⇓r

............

W p -normal ............

⇐ ;

p Bap

⇑ ⇓r

⇔

⇑ ⇓r

⇑ ⇓r ⇔

approx.

⇐ ;

Bp

For uniformly continuous bounded (ucb) functions (w.r.t. the sup-norm), like solutions of differential equations and inclusions studied in Section 5.4. below, the following relations take place for classes of a.p. functions in Table 1 (cf. [ABG]): p {Cap }ucb = {u-normal}ucb = {u.a.p.}ucb = {Sap }ucb = {S p -normal}ucb = {S p }ucb p = {e- W p }ucb ⊂ {e-Wap }ucb = {e-W p -normal}ucb p = {W p }ucb ⊂ {Wap }ucb = {W p -normal}ucb p ⊂ {Bap }ucb = {B p -normal}ucb ⊃ {B p }ucb .

Modifying the examples in [ABG], one can check that the inclusions in the second and p p p third lines are strict, i.e. that {Wap }ucb 6⊂ {e-Wap }ucb 6⊂ {Sap }ucb . Replacing the Euclidean distances |.−.| in Definitions 3.2, 3.5 and 3.8 by the Hausdorff metric dH (·, ·) defined in Preliminaries, we can extend in a natural correct way the Bohrp p p p type definitions of classes Cap , Sap , e-Wap , Wap , Bap of a.p. functions to those of a.p. multivalued functions with nonempty compact values as follows (cf. [ABG], [AG1]). D EFINITION 3.11 (Bohr-type definition) A (Hausdorff) continuous multivalued function ϕ : R ⊸ R with nonempty compact values is said to be uniformly-almost-periodic (u.a.p.) if, for every ǫ > 0, there corresponds a r.d. set {τ }ǫ s.t. sup dH ϕ(t + τ ), ϕ(t) < ǫ ∀t ∈ {τ }ǫ . t∈R

Furthermore, a measurable multivalued function ϕ : R ⊸ R with nonempty compact p p p p p values is said to have the Gap -property, where Gap is Sap or Wap or Bap , if, for every ǫ > 0, there corresponds a r.d. set {τ }ǫ s.t. ∀t ∈ {τ }ǫ : Z sup x∈R

or

x

x+1

dH

p 1 p ϕ(t + τ ), ϕ(t) dt < ǫ,

Z x+L p 1 p 1 lim sup dH ϕ(t + τ ), ϕ(t) dt < ǫ, L→∞ x∈R L x

310 or

Jan Andres p 1 Z T p 1 lim sup dH ϕ(t + τ ), ϕ(t) dt < ǫ, 2T −T T →∞

respectively. At last, a measurable multivalued function ϕ : R ⊸ R with nonempty compact values p is said to be equi-almost-periodic in the sense of Weyl (e-Wap ) if, for every ǫ > 0, there corresponds a r.d. set {τ }ǫ and a number L0 = L0 (ǫ) s.t. Z x+L p 1 p 1 sup dH ϕ(t + τ ), ϕ(t) dt < ǫ, ∀t ∈ {τ }ǫ , ∀L ≥ L0 (ǫ). x∈R L x Each number τ ∈ {τ }ǫ is called a respective ǫ-almost-period (or a respective ǫtranslation number of ϕ). p p Although multivalued Sap and e-Wap maps ϕ were shown to admit a single-valued p and e-Wap selection f ⊂ ϕ in [DS], [D1] and [D2], a multivalued u.a.p. map need not possess, rather curiously, a single-valued u.a.p. selection, as pointed out in [BVLL]. Moreover, the values of ϕ can be even in a complete metric space.

p Sap

R EMARK 3.6 The replacement of the Euclidean distances |.−.| in Bochner-type definitions 3.3, 3.6 and 3.9 by the Hausdorff metric dH (·, ·) leads also to natural correct definitions of u. normal, S p -normal, e-W p -normal, W p -normal and B p -normal measurable (in case of u. normal: (Hausdorff-) continuous) multivalued functions with nonempty compact values. On the other hand, it is not so for the classes of a.p. functions defined by means of approximation by trigonometric polynomials, because the arbitrary accuracy requirement for these approximations would reduce multivalued maps to single-valued functions, only. It is, therefore, more natural to define, as in [D3], the class of B p -multivalued measurable functions with nonempty compact values, when replacing the Euclidean distances |. − .| by p the Hausdorff metric dH (·, ·), jointly with trigonometric polynomials by Sap -multivalued maps. For B p -multivalued maps ϕ, defined in this way, the existence of a single-valued B p selection f ⊂ ϕ, was proved in [D3]. The values of ϕ can be also in a complete metric space.

3.5.

Derivo-periodic Maps

A function f is said to be derivo-periodic if f˙(t) ≡ f˙(t + ω), for some ω > 0. If f˙ is measurable, then we can obviously assume, without any loss of generality, that f˙(t) = f˙(t + ω), almost everywhere (a.e.). By a function, we will understand here a single-valued one, i.e. f : R → R, or a multivalued one, i.e. f : R ⊸ R (or, equivalently, f : R → 2R \ {0}, but (for the sake of simplicity) always with R as its domain as well as its range. Besides the standard derivative (for a single-valued function), we will also consider the one in the sense of F. S. De Blasi [DB] (for multivalued functions). It is well-known (cf. [Fa, p. 235]) that a continuously differentiable (single-valued) function is derivo-periodic if and only if it takes the form f (t) = f0 (t) + αt, where f0 is periodic and α ∈ R. More precisely, for f ∈ C 1 (R, R), we have: f˙(t) = f˙(t + ω), for some ω > 0 ⇐⇒ f (t) = f0 (t) + αt,

(3.4)

Periodic-Type Solutions of Differential Inclusions

311

where f0 (t) = f0 (t+ω). It follows immediately that a primitive function (an antiderivative) to such f˙ is ω-periodic if and only if it is bounded, i.e. α = 0. Physically, derivo-periodic functions can correspond to a motion with a periodic velocity, a subsynchronous level of performance of the motor or a motion of particles in a sinusoidal potential related to a free-electron laser. For many references concerning derivoperiodic motions and their applications in astronomy, engineering, laser physics, etc., see [AG1]. Particularly, in quantum physics, a “slalom orbit” of an electron beam should be described, in view of the Heisenberg Uncertainty Principle, by means of a multivalued function with a periodic single-valued derivative. This requirement can be satisfied for De Blasi-like differentiable multivalued functions, as pointed out in [ABP], because according to the result of our former Ph.D. student L. J¨utner (cf. also [AG1, Theorem 2.18 in Appendix 2]), a De Blasi-like differentiable function is always a sum of a single-valued continuous function having right-hand side and left-hand side derivatives plus a multivalued constant. Furthermore, it is well-known that if right-hand side or left-hand side derivatives of a single-valued part are continuous, then this single-valued part becomes continuously differentiable, and so satisfies (3.4). On the other hand, a natural question arises, namely how much regular must be f in order (3.4) to be satisfied? It is intuitively clear that this has a lot to do with the Fundamental Theorem of Calculus (the Newton-Leibniz formula), because the simplest proof of (3.4) relies on it (see [Fa, p. 235] or [AG1, p. 657]). In this light, the usage of the Newton integral or the Lebesgue integral allows us to replace the C 1 -class in (3.4) by differentiable or absolutely continuous functions, respectively; of course, satisfying (3.4) almost everywhere, in the latter case. As we will see, relation (3.4) holding a.e. can be verified for the related class of ACG∗ functions defined below. Since a continuous (single-valued) function f with right-hand side or left-hand side derivatives f˙+ or f˙− is known to belong to the ACG∗ -class, the singlevalued part of De Blasi-like differentiable multivalued function satisfies (3.4), a.e. Consequently, a multivalued analogy of (3.4) holds for De Blasi-like differentiable multivalued functions. Hence, let us start with the class of generalized absolutely continuous in the restricted sense (ACG∗ -) functions. Usually, the definitions of AC∗ and ACG∗ -functions are given on a subset of a closed (bounded) interval in R. For our needs, we extend the notion of ACG∗ -functions onto an arbitrary subset of R in the following definition. D EFINITION 3.12 A function f : [a, b] → R is said to be AC∗ on S ⊂ [a, b] if, for every ǫ > 0, there exists δ > 0 such that, for any subpartition P = {[aj , bj ]}sj=1 of [a, b] with aj , bj ∈ S, for every j = 1, . . . , s, s X j=1

|bj − aj | < δ ⇒

s X

ω(f ; [aj , bj ]) < ǫ,

j=1

where ω denotes the oscillation of f on the interval [aj , bj ] : ω(f ; [aj , bj ]) := sup{|f (y) − f (x)| | x, y ∈ [aj , bj ]}. Furthermore, a function f : J → R, J ⊂ R, is said to be ACG∗ on S ⊂ J if f is continuous on S and S is a countable union of sets on which f is AC∗ .

312

Jan Andres

In [ABP], we established the following characterization of a.e. derivo-periodic functions for ACG∗ -functions. T HEOREM 3.6 If f : R → R is ACG∗ on R, then the following conditions are equivalent: (i) f˙(t) = f˙(t + ω), ω > 0, for a.a. t ∈ R, (ii) f˙+ (t) = f˙+ (t + ω) or f˙− (t) = f˙− (t + ω), ω > 0, for a.a. t ∈ R, (iii) f (t) = f0 (t) + αt, for all t ∈ R, where α ∈ R, f0 is ω-periodic, ω > 0, and ACG∗ on R. R EMARK 3.7 The class of functions having derivative everywhere, but on a set with zero (Lebesgue) measure on which have negligible variation, coincides on any closed interval with the class of ACG∗ -functions. It follows from definitions that a function f : R → R is ACG∗ on R if and only if it is ACG∗ on [k, k + 1], for every k ∈ Z. Also, a function f : R → R has derivative everywhere, but on a set with zero (Lebesgue) measure on which have negligible variation if and only if this holds on [k, k + 1], for every k ∈ Z. Therefore, Theorem 3.6 can be equivalently expressed in terms of functions having derivative everywhere, but on a set with zero measure on which have negligible variation. This was done by means of the Kurzweil–Henstock integral in [ABP], where the precise definitions and more details can be found. Now, we shall proceed to De Blasi-like differentiable multivalued functions. The original notion introduced in [DB] reduces in R into the following definition. D EFINITION 3.13 A multivalued function ϕ : R ⊸ R (i.e. ϕ : R → 2R \ {∅}) is said to be De Blasi-like differentiable at t ∈ R if there exists (a single-valued(!); cf. [ABP], [AG1]) mapping Dt ϕ : R → R such that Dt ϕ is positively homogeneous (i.e. Dt ϕ(λs) = λDt ϕ(s), λ ≥ 0, for all s ∈ R) and a number δ > 0 such that dH (ϕ(t + h), ϕ(t) + Dt ϕ(h)) = o(h),

whenever |h| ≤ δ,

where o(h) denotes a nonnegative function such that limh→0 o(h)/|h| = 0, dH (·, ·) stands for the Hausdorff metric (cf. Preliminaries). Dt ϕ is called the differential of ϕ at t. Of course, ϕ is said to be De Blasi-like differentiable on J ⊂ R (or simply, for J = R, De Blasi-like differentiable) if it is so at every point t ∈ R. R EMARK 3.8 According to an important statement in [AG1, Theorem (A2.18)]), we can define equivalently a De Blasi-like differentiable function ϕ : R ⊸ R as a sum of a singlevalued continuous function f : R → R having (standard) right-hand side and left-hand side derivatives plus a bounded interval (i.e., a multivalued constant) {C}, namely ϕ = f +{C}. R EMARK 3.9 It is well-known that a function f : [a, b] → R having right-hand side and left-hand side derivatives, for all t ∈ E ⊂ [a, b], admits derivatives with at most countably many exceptions on E ⊂ [a, b]. Moreover, continuous functions which are differentiable for all, but countably many t ∈ E ⊂ [a, b] are known to be there ACG∗ . Thus, continuous functions f : [a, b] → R having right-hand side and left-hand side derivatives, for all t ∈ E ⊂ [a, b], are there ACG∗ . For more details, see [ABP] and the references therein. Since the single-valued part f of De Blasi-like differentiable multivalued function ϕ : R ⊸ R (see Remark 3.8) is, according to Remark 3.9, ACG∗ on any closed interval, and

Periodic-Type Solutions of Differential Inclusions

313

since (cf. [ABP] or [AG1, Theorem (A2.20)]) Dt ϕ(t) ≡ Dt ϕ(t + ω) ⇐⇒ f˙+ (t) ≡ f˙+ (t + ω) and f˙− (t) ≡ f˙− (t + ω),

(3.5)

we arrive by means of Theorem 3.6 and Remark 3.8 at the following result. T HEOREM 3.7 Let ϕ : R ⊸ R be a De Blasi-like differentiable multivalued function in the sense of Definition 3.13. Then ϕ is derivo-periodic with period ω > 0, i.e. (3.5) holds, if and only if there exist α ∈ R and an ω-periodic continuous function f0 : R → R such that ϕ(t) = [f0 (t) + αt] + {C},

for all t ∈ R,

where {C} is a bounded interval (a multivalued constant). R EMARK 3.10 For a single-valued De Blasi-like differentiable function ϕ = f (i.e. {C} = {0}), Theorem 3.7 represents only a particular case of Theorem 3.6. On the other hand, Theorem 3.7 generalizes the statement that a differentiable (in a usual sense) function f : R → R is derivo-periodic if and only if f (t) = f0 (t) + αt with a (differentiable) periodic function f0 : R → R. In view of the above results, the primitives of periodic derivatives become, under natural assumptions, periodic if and only if they are bounded. According to the well-known Bohl–Bohr theorem (see e.g. [Bs]), the same is true in the class of uniformly continuously differentiable functions w.r.t. (uniformly or Bohr-type) almost-periodic (a.p.) functions. For the definitions of a.p. functions and their properties; see the foregoing section. Thus, a natural question arises, whether or not (?) the almost-periodicity of a (uniformly continuous) derivative implies that the function itself takes the form of a sum of an a.p. function plus some linear part. The following arguments demonstrate that this must be answered negatively in general. Since there are examples (see e.g. [J1], [J2], [JM] or, more recently, [OT]) showing the existence of a.p. functions with a zero mean value whose primitives are unbounded, they cannot be (according to the mentioned Bohl–Bohr theorem) a.p. Moreover, if x˙ 0 (t) is a.p. with a zero mean value, namely ¯˙0 := lim 1 x T →∞ 2T

Z

T

−T

1 x˙ 0 (t) dt = lim T →∞ T

Z

T

x0 (t) = 0, t→∞ t

x˙ 0 (t) dt = lim 0

and such that x0 (t) is (as in the mentioned examples) unbounded, we can conclude that x0 (t) is not a.p., and because of x0 (t) = o(t), it does not contain any linear part. A bit more ¯˙ generally, let x(t) ˙ be a.p. having not necessarily a zero mean value. Then x(t) ˙ = x˙ 0 (t) + x, where again Z 1 T ¯˙ = 0. ¯ x(t) ˙ dt ∈ R and x x˙ := lim T →∞ T 0 Thus, integrating x(t) ˙ from 0 to t, we obtain that ¯˙ + C, x(t) = x0 (t) + xt where C is a suitable real constant. Since x0 (t) need not be a.p. and, as a consequence of x0 (t) = o(t), no linear part can be extracted from it, there is no chance, in general, for such

314

Jan Andres

¯˙ + C and an a.p. function, as claimed an x(t) to be a sum of the present linear function xt above. Since the mentioned example in [JM] concerns in fact quasi-periodic functions with only two basic frequencies, the same is also true for this subclass.

4. 4.1.

Primer of Periodic-Type Oscillations Linear Systems with Constant Coefficients

Consider the linear system x ∈ Rn ,

x˙ + Ax = p(t),

(4.6)

where A is an (n × n)-matrix with (real) constant entries and p : R → Rn is a (measurable) locally Lebesgue integrable n-vector function. It is well-known (see e.g. [Fa, Chapter 2.1]) that system (4.6) possesses a unique ωperiodic solution if and only if σ(A) ∩ 2πiZ/ω = ∅, where σ(A) is a spectrum of A, i.e. the set of eigenvalues of A, provided p(t) ≡ p(t + ω). If p is anti-periodic, p(t) ≡ −p(t + ω), it is also half-periodic, p(t) ≡ p(t + 2ω), but not vice versa. That is why the spectral condition cannot be used for the characterization of existence and uniqueness result about anti-periodic solutions. On the other hand, if σ(A)∩πiZ/ω = ∅, then system (4.6) admits a unique ω-anti-periodic solution, provided p(t) ≡ −p(t + ω). It follows from the Fredholm alternative, which is usually called in the context of ODEs as Conti’s lemma, that system (4.6) has a unique 2qω-periodic solution if and only if the homogeneous system x˙ + Ax = 0,

x ∈ Rn ,

(4.7)

has only the trivial 2qω-periodic solution. Unlike for periodic solutions, the system x˙ = p(t) possesses the only ω-anti-periodic solution, provided p(t) ≡ −p(t + ω). Because of the superposition principle, system x˙ + Ax =

k X

pj (t),

x ∈ Rn ,

(4.8)

j=1

where pj : R → Rn are (measurable) locally Lebesgue integrable n-vector functions such that pj (t) ≡ pj (t+ωj ), where ωj are linearly independent reals over the rationals Q, admits a k-period-quasi-periodic solution, provided σ(A) ∩ 2πiZ/ωj = ∅, for each j = 1, . . . , k. If p : R → Rn is an essentially bounded Stepanov-almost-periodic n-vector function then, according to the well-known Bohr–Neugebauer-type result (cf. e.g. [Ra], and the references therein), every entirely bounded solution of system (4.6), with an arbitrary real matrix A, is uniformly-almost-periodic. Analogous results also hold for almost-periodic solutions in a more general sense (Weyl, Besicovitch); cf. [Ra]. Now, let A be a regular real (n × n)-matrix and f ∈ ACG ∗ (R, Rn ) be ω-derivo-periodic, i.e. p(t) ˙ ≡ p(t ˙ + ω). Then (4.6) admits a unique (smooth) ω-derivoperiodic solution if and only if σ(A) ∩ 2πiZ/ω = ∅ (for more details and the definition of the ACG ∗ -class of generalized absolutely continuous functions (see [ABP] and cf. Section 3.5.). The same is true for a unique, but this time an absolutely continuous (i.e.

Periodic-Type Solutions of Differential Inclusions

315

Carath´eodory) ω-derivo-periodic solution of (4.6) if p takes the form p(t) = p0 (t) + αt, where p0 : R → Rn is, more generally, a (measurable) locally Lebesgue integrable n-vector function such that p0 (t) ≡ p0 (t + ω) and α ∈ Rn is a real n-vector. All the above existence (not uniqueness) results can be extended, on the basis of selection theorems, to linear differential inclusions x˙ + Ax ∈ P (t),

x ∈ Rn ,

(4.9)

resp. x˙ + Ax ∈

k X

Pj (t),

x ∈ Rn .

(4.10)

j=1

Since every measurable multivalued map with closed values possesses, according to the Kuratowski–Ryll-Nardzewski theorem (cf. Proposition 2.2) (single-valued) measurable selections, the same is true for ω-periodic and ω-anti-periodic multivalued maps w.r.t. measurable ω-periodic and ω-anti-periodic selections. That is why the above existence conclusions for ω-periodic and ω-anti-periodic solutions can be extended to (4.9) as well as those for k-period-quasi-periodic solutions to (4.10). Furthermore, since Stepanov-almost-periodic multivalued maps with compact convex values possess (single-valued) Stepanov-almost-periodic selections (cf. Section 3.4.), the analogous extension holds for (4.9) w.r.t. uniformly-almost-periodic multivalued solutions. Since, rather curiously, uniformly-almost-periodic multivalued maps with compact convex values need not possess, according to the observation in [BVLL] (cf. Section 3.4.), (single-valued) uniformly-almost-periodic selections, we cannot obtain in this way smooth uniformly-almost-periodic solutions of (4.9). At last, the similar extension can be done for (smooth) ω-derivo-periodic solutions of (4.9), provided P : R ⊸ Rn is a DeBlasi-like differentiable multivalued map whose derivative is ω-periodic (for more details, see [ABP]). The same is true for (4.9) w.r.t. Carath´eodory ω-derivo-periodic solutions, provided P takes the from P (t) = P0 (t) + αt, where P0 is a measurable multivalued map with closed values such that P0 (t) ≡ P0 (t + ω) and α ∈ Rn is a real n-vector.

4.2.

Linear Systems with Time-Variable Coefficients

Now, consider the linear system x˙ + A(t)x = p(t),

x ∈ Rn ,

(4.11)

where A is this time an (n × n)-matrix with (real) time-variable entries which are (measurable) locally Lebesgue integrable and p : R → Rn is again a (measurable) locally Lebesgue integrable n-vector function. We shall discuss the possibility of further extension of results from the foregoing section. It follows from the Floquet theory (see e.g. [Fa, Chapters 2.2 and 2.3], [YS1], [YS2]) that system (4.6) has a unique (harmonic) ω-periodic solution if and only if 1 is not a Floquet (characteristic) multiplier of A, provided A(t) ≡ A(t + ω) and p(t) ≡ p(t + ω). Let us recall that by Floquet multipliers we mean, as usual, eigenvalues of the monodromy

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matrix X(ω), where X(t) = Φ(t) exp(Λt) is the fundamental matrix of solutions to the homogeneous system x˙ + A(t)x = 0, x ∈ Rn , (4.12) with Φ(0) = I, Φ(t) ≡ Φ(t + ω), and Λ is a constant matrix. More generally, system (4.11) has a unique (subharmonic) 2qω-periodic solution if and only if exp( pq πi), where p and q > 0 are integers, is not a characteristic multiplier of A, provided A(t) ≡ A(t + ω) and p(t) ≡ p(t + 2qω). Thus, it has a unique ω-anti-periodic solution if −1 is not a characteristic multiplier of A, provided A(t) ≡ A(t + ω) and p(t) ≡ −p(t + ω). However, since it also has a unique ω-anti-periodic solution, when A(t) ≡ 0 and p(t) ≡ −p(t + ω), the multiplier condition cannot be used for the “only if” part. According to the Massera transformation theorem [Ma] (cf. [Yo, Theorem 15.3]), if system (4.11) admits a bounded solution in the future, then it also has a (harmonic) ωperiodic solution, provided A(t) ≡ A(t + ω) and p(t) ≡ p(t + ω). Unfortunately, the analogous deduction from the above conclusions for k-period-quasiperiodic solutions of the system x˙ + A(t)x =

k X

pj (t),

x ∈ Rn ,

j=1

is impossible, because the period of A should be contradictionally an integer multiple of all linearly independent over Q periods of pj , j = 1, . . . , k. There is also no direct analogy of Bohr–Neugebauer-type or Massera type results for almost-periodic A and p. The appropriate theory for (4.11) with almost-periodic A and p is due to J. Favard (cf. e.g. [Yo, ChapterIII.18]). Nevertheless, if the trivial solution of the homogeneous system (4.12) is uniformly asymptotically stable, then there is a unique (smooth) uniformly-almost-periodic solution of (4.11) which is globally uniformly asymptotically stable, provided A and p are uniformly-almost-periodic (cf. [Yo, Theorem 19.4]). In order system (4.11), where A(t) ≡ A(t + ω), to have an ω-derivo-periodic solution, p should take the special form p(t) = p0 (t) + A(t)αt, where p0 (t) ≡ p0 (t + ω) and α ∈ Rn ˙ is an n-vector. But since then p(t + ω) = p(t) + A(t)αω and p(t ˙ + ω) = p(t) ˙ + A(t)αω, provided A and p are differentiable, p is ω-periodic resp. ω-derivo-periodic if and only if ˙ A(t)α = 0 resp. A(t)α = 0. Therefore, system (4.11), where A is differentiable with A(t) ≡ A(t + ω) and p is ω-derivo-periodic, can have a pure (α 6= 0) ω-derivo-periodic solution only if A(t) ≡ A is a constant matrix. As concerns the extensions of the existence results (sufficiency criteria) to differential inclusions of the form x˙ + A(t)x ∈ P (t), x ∈ Rn , where A is the same as above and P : R ⊸ Rn is at least a measurable multivalued map with closed values, the situation is quite analogous to the foregoing section, because these multivalued extensions rely on the same selection theorems for P . Since a uniformlyalmost-periodic P with compact convex values need not admit, as already mentioned, a uniformly-almost-periodic selection, but only Stepanov-almost-periodic selection, it would be nice to show the existence of a uniformly-almost-periodic solution of (4.11), provided A

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is uniformly-almost-periodic such that the trivial solution of (4.12) is uniformly asymptotically stable and p is an essentially bounded Stepanov-almost-periodic n-vector function. For a constant matrix A(t) ≡ A, it is true.

4.3.

Nonlinear Scalar Equations

Consider the nonlinear scalar equation x˙ = f (t, x),

x ∈ Rn ,

(4.13)

where f : R × R → R is a Carath´eodory function (cf. Preliminaries). J. L. Massera [Ma] (cf. [Yo, Theorem 15.3]) proved, for a continuous f such that f (t, x) ≡ f (t + ω, x), that the existence of a solution of (4.13) which is bounded in the future implies, under the uniqueness assumption, the existence of a (harmonic) ω-periodic solution of (4.13). This Massera’s theorem was improved in [OO1] (cf. [DOO]), for (4.13) with L1 -Carath´eodory function f on [0, ω] × R, by showing that, for the same conclusion, the uniqueness assumption can be omitted. If equation (4.13), where f (t, x) ≡ f (t+ω, x) satisfies the L1 -Carath´eodory conditions on [0, ω]×R, admits an nω-periodic solution with n > 1 then, for every k ∈ N, there exists a kω-periodic solution of (4.13). Moreover, the set of all (subharmonic) k-periodic solutions of (4.13) has a topological dimension at least k as a subset of L∞ (R). This remarkable multiplicity results was obtained in [OO2] (for an alternative proof, see [AFP1], [AFP2]). Under the same assumptions, there still exists an infinite-dimensional subset of L∞ (R) of uniformly-almost-periodic solutions of (4.13) which are not nω-periodic, for any n ∈ N (see [OO1, Theorem 3.6]). If f (·, x) : R → R is quasi-periodic, uniformly w.r.t. x ∈ R, then the boundedness of solutions need not imply the existence of quasi-periodic (and all the worse, uniformlyalmost-periodic) solutions of (4.13). Z. Opial (cf. [Yo, p. 181], [OT]) has constructed f such that f (·, x) is 2-period-quasi-periodic and all solutions of (4.13) are bounded in the future, but none is uniformly-almost-periodic. A. M. Fink and P. O. Frederickson (cf. [Yo, p. 181], [OT]) have modified Opial’s example by showing that even uniform ultimate boundedness (i.e. dissipativity: lim supt→∞ |x(t)| < D, for all solutions x(t) of (4.13), where D is a suitable common constant) is insufficient to imply the existence of a uniformly-almostperiodic (and so, quasi-periodic) solution of (4.13), provided again f (·, x) is 2-period-quasiperiodic, uniformly w.r.t. x ∈ R. Nevertheless, it was proved in [OT] that the existence of a bounded solution of a quasi-periodic in time equation (4.13) implies the existence of a quasi-periodic solution in a certain weaker (close to almost-automorphic) sense.

4.4.

Nonlinear Planar Systems

Consider the nonlinear planar system x˙ = f (t, x),

x ∈ R2 ,

(4.14)

where f : R × R2 → R2 is a continuous function such that f (t, x) ≡ f (t + ω, x). J. L. Massera [Ma] (cf. [Yo, Theorem 15.5]) established a theorem saying that, under the uniqueness assumption, if all solutions of (4.14) exist in the future and at least one of

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them is bounded (on the half-line), then there exists a (harmonic) ω-periodic solution of (4.14). He also gave in [Ma] and example (cf. [Yo, Example 15.1]) showing that the sole existence of a bounded solution (i.e. without assuming the existence of all solutions on the half-line) need not imply ω-periodic solutions. Nevertheless, in particular, Lagrange-stable systems (4.14) imply the existence of harmonics. In this case, the further information about the structure of solutions of (4.14) can be deduced from the result in [Mu]. It follows from slightly improved results of T. Matsuoka (cf. [A1, Theorem 1.2], [AG1, Chapter III.9]) that, under the uniqueness assumption, three (harmonic) ω-periodic solutions of (4.14) imply “generically” the existence of infinitely many (subharmonic) kωperiodic solutions of (4.14), k ∈ N. The “genericity” is however understood in terms of the Artin braid group theory, i.e. with the exception of certain simplest braids, representing the three given harmonics. The natural way how to prove at least three harmonics of (4.14) is by means of the Nielsen fixed point theory. In [A1], [AG1], we constructed in this way an example of system (4.14) possessing at least three harmonics. To combine these results for obtaining infinitely many subharmonics by excluding the related exceptional braids is, however, a difficult task.

4.5.

Nonlinear Systems in Rn

J. L. Massera constructed in [Ma] an example that, for n > 2, there is no analogy of his transformation theorem in [Ma] for planar systems. According to the example of S.-N. Chow in (cf. [Yo, pp. 177–180]), even the existence of a bounded uniformly asymptotically stable solution does not necessarily imply the existence of a harmonic of a Lagrange-stable (i.e. all solutions are bounded) systems in R3 . Consider the nonlinear system x˙ = f (t, x),

x ∈ Rn ,

(4.15)

where f : R × Rn → Rn is a Carath´eodory function (cf. Preliminaries). Paraphrasing the planar result of M. L. Cartwright, T. Yoshizawa asserted (see [Yo, Theorem 15.8] and cf. [H2]) that, for a continuous f such that f (t, x) ≡ f (t + ω, x), dissipative system (4.15) implies, under the uniqueness condition, the existence of a (harmonic) ω-periodic solution. In [AG2], we have slightly improved this result, namely that the uniformly dissipative, not necessarily uniquely solvable, systems (4.15) imply the existence of harmonics, provided f is Carath´eodory and f (t, x) ≡ f (t + ω, x). More generally, (4.15) can be replaced, for the same conclusion, by the inclusion x˙ ∈ F (t, x), where F : R × Rn ⊸ Rn is an upperCarath´eodory multivalued map (cf. Preliminaries) with compact convex values such that F (t, x) ≡ F (t + ω, x). Let us note that, under the assumptions of Yoshizawa’s theorem, uniquely solvable dissipative systems are, according to the result of N. Pavel (cf. [AG1], [Yo]), uniformly dissipative, i.e. ∀D1 > 0 ∃△t > 0 : [t0 ∈ R, |x(t0 )| < D1 , t ≥ t0 + △t] ⇒ |x(t)| ≤ D2 , where D2 > 0 is a common constant, for all solutions x(t) of (4.15).

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On the basis of the statements from the foregoing Sections 4.1. and 4.2., we can already obtain, by means of the standard fixed point theorems, a lot of classical results about periodic-type solutions of nonlinear systems. Let us demonstrate it on two simple examples for anti-periodic and almost-periodic oscillations. Since the homogeneous system x = 0, x ∈ Rn , has only the trivial anti-periodic solutions, there is exactly one ω-anti-periodic solution of the nonhomogeneous system x˙ = p(t), where p : R → Rn is a (measurable) locally Lebesgue integrable n-vector function such that p(t) ≡ −p(t + ω). Therefore, applying the Schauder linearization technique and the standard Schauder fixed point theorem (cf. [A1], [AG1]), one can immediately obtain an ω-anti-periodic solution of the form 1 x(t) = 2

Z

0

f (s, x(s)) ds + −ω

Z

t

f (s, x(s)) ds 0

to (4.15), where f (t, x) ≡ −f (t + ω, x), provided |f (t, x)| ≤ α|x| + β holds, for all (t, x) ∈ [0, ω] × Rn with a sufficiently small constant α. Similarly, since the homogeneous system (4.7), where the real parts of all eigenvalues of A are nonzero, has only the trivial entirely bounded solution, there is exactly one uniformlyalmost-periodic solution of the nonhomogeneous system (4.6), where p : R → Rn is a uniformly-almost-periodic n-vector function. Therefore, applying the Schauder linearization technique and the standard Banach fixed point theorem (cf. [AG1]), one can immediately obtain a unique uniformly-almost-periodic solution of the form Z ∞ G(t − s)f (s, x(s)) ds x(t) = −∞

to the system x˙ + Ax = f (t, x),

(4.16)

where f (·, x) : R → Rn is uniformly-almost-periodic, uniformly w.r.t. x ∈ Rn , provided f (t, ·) : Rn → Rn is Lipschitz-continuous with a sufficiently small Lipschitz constant. This well-known theorem was probably proved for the first time by G. I. Birjuk in 1954 (cf. also [H1]). Because of the mentioned Kuratowski–Ryll-Nardzewski selection theorem (cf. Proposition 2.2), the first example can be directly extended to system x˙ ∈ f (t, x) + P (t), where P : R ⊸ Rn is a (multivalued) bounded, measurable map with closed values such that P (t) ≡ −P (t + ω). On the other hand, since the uniformly-almost-periodic multivalued map P : R → Rn with compact convex values need not possess single-valued uniformlyalmost-periodic selections, but only Stepanov-almost-periodic selections, a multivalued extension of the second example is not so straightforward (see [AG1, Chapter III.103] and cf. Section 5.4. below). Finally, consider again the semilinear system (4.16) and assume that, for given ω > 0 and α ∈ Rn , A is a real (n × n)-matrix such that σ(A) ∩ 2πiZ/ω = ∅ and f takes the special form f (t, x) := f0 (t, x) + (tA + E)α, where f0 : R × Rn → Rn is an n-vector bounded Carath´eodory function such that f0 (t, x + αt) ≡ f0 (t + ω, x + α(t + ω)),

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e.g. f0 (t, x) ≡ f0 (t + ω, x) ≡ f0 (t, x + αω). We will show that, under these assumptions, system (4.16) possesses an ω-derivo-periodic solution. One can readily check that x(t) = x0 (t) + αt is an ω-derivo-periodic solution of (4.16) if and only if x0 (t) is an ω-periodic solution of the functional system x˙ + Ax = f0 (t, x + αt),

x ∈ Rn .

Since, furthermore, f1 (t + ω, x) ≡ f1 (t, x) := f0 (t, x + αt) ≡ f0 (t + ω, x + α(t + ω)), the functional system really admits, by the analogous arguments as above, an ω-periodic solution, and subsequently system (4.16) has a desired ω-derivo-periodic solution, as claimed. In particular, system x˙ + Ax = f (x) + p(t),

x ∈ Rn ,

where A is the same as above, admits an ω-derivo-periodic solution, provided f : Rn → Rn is a continuous n-vector function such that f (x) ≡ f (x + αω) and p : R → Rn takes the special form p(t) = p0 (t) + (tA + E)α, where p0 : R → Rn is an n-vector measurable bounded function such that p0 (t) ≡ p0 (t + ω), because f0 (t + ω, x) ≡ f0 (t, x) := f (x + αt) ≡ f (x + α(t + ω)). Since, according Kuratowski–Ryll-Nardzewski theorem (cf. Proposition 2.2) a measurable, bounded, ω-periodic map P : R ⊸ Rn with closed values possesses a single valued measurable, bounded, ω-periodic selection p0 ⊂ P0 , the same is true for the inclusion x˙ + Ax ∈ f (x) + P (t),

x ∈ Rn ,

where A, f are the same as above and P takes the special form P (t) = P0 (t) + (tA + E)α, provided P0 : R ⊸ Rn is measurable bounded multivalued map with closed values such that P0 (t) ≡ P0 (t + ω). Further extensions are available, e.g. for the inclusion of the form x˙ + Ax ∈ F (x) + P (t),

x ∈ Rn ,

provided F : Rn ⊸ Rn is an l.s.c. map with convex compact values such that F (x) ≡ F (x + αω). On the other hand, since u.s.c. maps F with convex compact values need not possesses continuous selections, the related extension is not so straightforward.

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5.

321

General Theorems for Periodic-Type Solutions

5.1.

Bounded Solutions

Firstly, we are interested in the existence of an entirely bounded solution to the semilinear differential inclusion x˙ + A(t)x ∈ F (t, x),

for a.a. t ∈ R, x ∈ Rn ,

(5.17)

with A ∈ L(Rn ), where L(Rn ) denotes the space of linear continuous transformations in Rn , and a set-valued transformation F . Our assumptions concerning the inclusion (5.17) will be the following: (A1) A : R → L(Rn ) is a (measurable) locally Lebesgue integrable matrix. (A2) Assume that x˙ + A(t) x = 0 (5.18) admits a regular exponential dichotomy (cf. Remark 5.1 below). Denote by G Green’s function for (5.18). (F) Let F : R × Rn ⊸ Rn be a u-Carath´eodory set-valued map (cf. Definition 2.3) such that |F (t, x)| ≤ m(t) + K|x|, for a.a. t ∈ R, x ∈ Rn , (5.19) where K ≥ 0 is a sufficiently small constant and m ∈ L1loc (R) is such that, for a constant M , Z t+1 m(s) ds t ∈ R < M. sup t

The following theorem is a special (finite-dimensional) case of Theorem 5.1 in [A1] (cf. also [AG1, Theorem III.5.33]). T HEOREM 5.1 Under the assumptions (A1), (A2), (F), the semilinear differential inclusion (5.17) admits an entirely bounded solution of the form x(t) =

Z

∞

G(t, s)f (s, x(s)) ds,

f ⊂ F.

−∞

R EMARK 5.1 Condition (A2) is satisfied, provided there exists a projection matrix P (P = P 2 ) and constants k > 0, λ > 0 such that ( |X(t)P X −1 (s)| ≤ k exp(−λ(t − s)), for s ≤ t, (5.20) −1 |X(t)(I − P )X (s)| ≤ k exp(−λ(s − t)), for t ≤ s, where X(t) is the fundamental matrix of (5.18), satisfying X(0) = I, i.e., the unit matrix. If A in (A1) is a piece-wise continuous and periodic, then it is well-known that (5.20) takes place, whenever all the associated Floquet multipliers lie off the unit cycle. If A in (A1) is (continuous and) almost-periodic, then it is enough (see [Pa]) that (5.20) holds only on a half-line [t0 , ∞) or even on a sufficiently long finite interval.

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R EMARK 5.2 If A : R → L(Rn ) is a linear, bounded operator whose spectrum does not intersect the imaginary axis, then the constant K in (F) can be easily taken as K < 1/C(A), where ( Z ∞ e−A(t−s) P− , for t > s, (5.21) |G(t, s)| ds ≤ C(A), G(t, s) = sup e−A(t−s) P+ , for t < s t∈R −∞

and P− , P+ stand for the corresponding spectral projections to the invariant subspaces of A. The existence of bounded solutions can be also obtained in the sequential way by means of the following proposition (for its proof, see e.g. [A1, Proposition 4.5] or [AG1, Proposition III.1.37]).

P ROPOSITION 5.1 Let F : R × Rn ⊸ Rn be a u-Carath´eodory mapping with nonempty, compact and convex values such that |F (t, x)| ≤ α(t) + β(t)|x|, for all (t, x) ∈ R × Rn , where α, β are locally Lebesgue integrable functions in R. Then, for every x0 ∈ Rn , there exists a solution x ∈ ACloc (R, Rn ) of the Cauchy problem ( x˙ ∈ F (t, x), for a.a. t ∈ (−∞, ∞), x ∈ Rn , x(0) = x0 . Let {xm (t)} be a sequence of absolutely continuous functions such that (i) for every m ∈ N, xm ∈ AC ([−m, m], Rn ) is a solution of x˙ ∈ F (t, x),

for a.a. t ∈ [−m, m], x ∈ Rn ,

(ii) sup{|xm (t)| | m ∈ N, t ∈ [−m, m]} := M < ∞ and xm (t) ∈ D ⊂ Rn , for every t ∈ [−m, m], where D is a given closed subdomain of Rn . Then there exists an entirely bounded solution x ∈ ACloc (R, Rn ) of the inclusion x˙ ∈ F (t, x),

for a.a. t ∈ (−∞, ∞),

such that sup |x(t)| ≤ M (< ∞) and x(t) ∈ D, for all t ∈ R. t∈R

Hence, consider still the boundary value problems of the type ( x˙ + A(t)x ∈ F (t, x), for a.a. t ∈ [0, τ ], x ∈ Rn , Lx = Θ,

(5.22)

where (i) A : [0, τ ] → L(Rn ) is a measurable linear operator such that |A(t)| ≤ γ(t), for all t ∈ [0, τ ] and some integrable function γ : [0, τ ] → [0, ∞), (ii) the associated homogeneous problem ( x˙ + A(t)x = 0, for a.a. t ∈ [0, τ ], x ∈ Rn , Lx = 0

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has only the trivial solution, (iii) F : [0, τ ] × Rn ⊸ Rn is a u-Carath´eodory mapping with nonempty, compact and convex values (cf. Definition 2.3), (iv) there are two nonnegative Lebesgue-integrable functions δ1 , δ2 : [0, τ ] → [0, ∞) such that |F (t, x)| ≤ δ1 (t) + δ2 (t)|x|,

for a.a. t ∈ [0, τ ] and all x ∈ Rn ,

where |F (t, x)| = sup{|y| | y ∈ F (t, x)}. P ROPOSITION 5.2 Consider problem (5.22) with (i)–(iv) above and let G : [0, τ ] × Rn × Rn × [0, 1] → Rn be a product-measurable u-Carath´eodory map (cf. Definition 2.3 and Proposition 2.3) such that G(t, c, c, 1) ⊂ F (t, c),

for all (t, c) ∈ [0, τ ] × Rn .

Assume, furthermore, that (v) there exists a (bounded) retract Q of C([0, τ ], Rn ) such that Q \ ∂Q is nonempty (open) and such that G(t, x, q(t), λ) is Lipschitzian in x with a sufficiently small Lipschitz constant (cf. Preliminaries), for a.a. t ∈ [0, τ ] and each (q, λ) ∈ Q × [0, 1], (vi) there exists a Lebesgue integrable function α : [0, τ ] → [0, ∞) such that |G(t, x(t), q(t), λ)| ≤ α(t),

a.e. in [0, τ ],

for any (x, q, λ) ∈ ΓT (i.e. from the graph of T ), where T denotes the set-valued map which assigns, to any (q, λ) ∈ Q × [0, 1], the set of solutions of ( x˙ + A(t)x ∈ G(t, x, q(t), λ), for a.a. t ∈ [0, 1], x ∈ Rn , Lx = Θ, (vii) T (Q × {0}) ⊂ Q holds and the boundary ∂Q of Q is fixed point free w.r.t. T , for every (q, λ) ∈ Q × [0, 1]. Then problem (5.22) has a solution. R EMARK 5.3 Rescaling t in (5.22), the interval [0, τ ] can be obviously replaced in Proposition 5.2 by any compact interval J, e.g. J = [−m, m], m ∈ N. Therefore, the second part of Proposition 5.1 can be still applied for obtaining an entirely bounded solution, as claimed. E XAMPLE 5.1 Consider problem (5.22), on the interval [−m, m], m ∈ N. Assume that the appropriate conditions (i)–(iv) are satisfied. Taking (for a product-measurable F : [−m, m] × Rn ⊸ Rn ) G(t, q(t)) = F (t, q(t)),

for q ∈ Q,

where Q = {µ ∈ C([−m, m], Rn ) | maxt∈[−m,m] |µ(t)| ≤ D} and D > 0 is a sufficiently big constant which will be specified below, we can see that (v) holds trivially. Furthermore, according to (iv), we get |G(t, q(t))| ≤ δ1 (t) + δ2 (t)D,

for a.a. t ∈ [−m, m],

(5.23)

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i.e. (vi) holds as well with α(t) = δ1 (t) + δ2 (t)D. At last, the associated linear problem ( x˙ + A(t)x ∈ F (t, q(t)), for a.a. t ∈ [−m, m], x ∈ Rn , Lx = Θ has, according to the Fredholm alternative (Conti’s lemma), for every q ∈ Q, a nonempty set of solutions of the form Z m T (q) = H(t, s)f (s, q(s)) ds, −m

where H is the related Green function and f ⊂ F is a measurable selection. For more details, see [A1] or [AG1, Chapter III.5]. Therefore, in order to apply Proposition 5.2 for the solvability of (5.22), we only need to show (cf. (vii)) that T (Q) ⊂ Q (and that ∂Q is fixed point free w.r.t. T , for every q ∈ Q, which is, however, not necessary here). Hence, in view of (5.23), we have that Z m H(t, s)f (s, q(s)) ds max |T (q)| = max t∈[−m,m] t∈[−m,m] −m Z m ≤ max |H(t, s)|(δ1 (s) + δ2 (s)D) ds t∈[−m,m] −m Z m Z m δ2 (t) dt , δ1 (t) dt + D = max |H(t, s)| t,s∈[−m,m]

−m

−m

and subsequently the above requirement holds for D≥ provided Z

maxt,s∈[−m,m] |H(t, s)| 1−

Rm

−m δ1 (t) dt Rm , maxt,s∈[−m,m] |H(t, s)| −m δ2 (t) dt

m

δ2 (t) dt < −m

1 maxt,s∈[−m,m] |H(t, s)|

,

m ∈ N.

(5.24)

(5.25)

(Observe that for D strictly bigger than the above quantity, ∂Q becomes fixed point free). The problem (5.22) is solvable on the interval J = [−m, m], for every m ∈ N, and subsequently the inclusion (5.17) admits, according to the second part of Proposition 5.2, an entirely bounded solution x(t) such that supt∈(−∞,∞) |x(t)| ≤ D. If, instead of (5.23), we have |F (t, x)| ≤ δ1 + δ2 D′ ,

for a.a. t ∈ (−∞, ∞) and |x| ≤ D′ ,

where D′ satisfies, for every m ∈ N, the inequality Rm δ max 1 t∈[−m,m] −m |H(t, s)| ds Rm , D′ ≥ 1 − δ2 maxt∈[−m,m] −m |H(t, s)| ds provided

δ2 <

maxt∈[−m,m]

1 Rm

−m |H(t, s)| ds

,

(5.23’)

(5.24’)

(5.25’)

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then problem (5.22) is again solvable on the interval J = [−m, m], for every m ∈ N, and subsequently the inclusion (5.17) admits, according to the second part of Proposition 5.2, an entirely bounded solution x(t) such that supt∈(−∞,∞) |x(t)| ≤ D′ . R EMARK 5.4 Observe that the growth restriction (5.23) with conditions (5.24), (5.25), resp. the growth restriction (5.23’) with conditions (5.24’), (5.25’), are of the same quality as the growth restriction (5.19) with condition (5.21) in Theorem 5.1 . If problem (5.22) takes the special form (m ∈ N) ( x˙ + A(t)x ∈ F (t, x), for a.a. t ∈ [−m, m], x ∈ Rn , x(−m) = M x(m), where M is a regular (n × n)-matrix,

(5.26)

usually called the Floquet problem, then the bound sets approach allows us to say something about the localization of solution of (5.26). Therefore, if the solution values a located, for every m ∈ N, in a given set K ⊂ Rn , the second part of Proposition 5.2 guarantees the existence of an entirely bounded solution of (5.17). The conditions for the solvability of (5.26) were established in [AMT3]. T HEOREM 5.2 Assume that A : R → L(Rn ) is an essentially bounded, locally Lebesgue integrable matrix and F : R × Rn ⊸ Rn is a u-Carath´eodory mapping with nonempty, convex, compact values. Assume, furthermore, that 1) the homogeneous periodic problem ( x˙ + A(t)x = 0, for a.a. t ∈ [−m, m], x ∈ Rn , x(−m) = x(m), has, for every m ∈ N, only the trivial solution, 2) there exists a u-Carath´eodory mapping G : R × Rn × Rn × [0, 1] ⊸ Rn such that, for all (t, x, y) ∈ R × R2n and λ ∈ [0, 1], it follows: G(t, x, y, 0) = G0 (t, x), G(t, y, y, 1) ⊂ F (t, y); |w| ≤ s(t)(1 + |x| + |y| + λ), with w ∈ G(t, x, y, λ) and s ∈ L1loc (R), 3) there exists a bounded, non-empty and open K ⊂ Rn , whose closure is a retract of Rn , and a function V : Rn → R of class C 2 such that V (x) ≤ 0 on K, V (x) = 0 and ∇V (x) 6= 0, for every x ∈ ∂K, and <∇V (x), w> ≤ 0, for all t ∈ R, x ∈ ∂K, λ ∈ (0, 1], w ∈ G(t, x, x, λ) − A(t)x, 4) G(t, ·, y, λ) is Lipschitzian with a sufficiently small Lipschitz constant L, for every (t, y, λ) ∈ R × K × [0, 1], 5) for each m ∈ N, the set of solutions of problem ( x˙ + A(t)x ∈ G0 (t, x), for a.a. t ∈ [−m, m], x ∈ Rn , (5.27) x(m) = x(−m),

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is a subset of C([−m, m], K). Then inclusion (5.22) admits a bounded solution x(t) of the form Z ∞ G(t, s)f (s, x(s)) ds, f ⊂ F, x(t) = −∞

such that x(t) ∈ K, for all t ∈ R. Proof. Consider problem (5.26) with M = I, i.e. the periodic boundary value problem ( x˙ + A(t)x ∈ F (t, x), for a.a. t ∈ [−m, m], x ∈ Rn , (5.28) x(m) = x(−m), whose associated homogeneous problem has, according to 1), only the trivial solution. According to [AMT3, Theorem 1] (cf. Theorem 5.5 in Section 5. below), problem (5.28) admits, for every m ∈ N, a solution xm (t) such that xm (t) ∈ K, for all t ∈ [−m, m], m ∈ N. According to the second part of Proposition 5.1, inclusion (5.17) admits an entirely bounded solution x(t) of the form Z ∞ G(t, s)f (s, x(s)) ds, f ⊂ F x(t) = −∞

(cf. Remark 5.3 and Theorem 5.1) with x(t) ∈ K, for all t ∈ R. R EMARK 5.5 If G in Theorem 5.2 is globally u.s.c., then the bounding function V can be e.g. only locally Lipschitzian and its derivative can be replaced by the Dini derivatives. For more details, see [AMT1] (cf. also [AG1, Chapter III.8 (a)]). R EMARK 5.6 If G in Theorem 5.2 is u-Carath´eodory and the bounding function does not belong to the class C 2 , then the related conclusions must be assumed in some neighbourhood of the boundary ∂K of K. For more details, see [AMT2] (cf. also [AG1, Chapter III.8 (b)]). As an application of a modified, in view of Remark 5.5, Theorem 5.2, let us give the following example. E XAMPLE 5.2 At first, consider the family of periodic problems ( x˙ + A(t)x ∈ F1 (t, x) + F2 (t, x), x(−m) = x(m), m ∈ N,

(5.29)

where x = (x1 , . . . , xn ), A = (aij )i,j=1,...,n : R → L(Rn ) is a bounded continuous matrix, F = F1 + F2 = (f11 , . . . , f1n ) + (f21 , . . . , f2n ), F1 , F2 : R × Rn ⊸ Rn are globally u.s.c. multivalued functions which are bounded in t ∈ (−∞, ∞), for every x ∈ Rn , and linearly bounded in x ∈ Rn , for every t ∈ (−∞, ∞). Let A be such that the homogeneous problem ( x˙ + A(t)x = 0, x(−m) = x(m),

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327

has, for every m ∈ N, only the trivial solution. Assume, furthermore, that there exist positive constants Ri , i = 1, . . . , n, such that ± [aii (t)(±Ri ) − λf1i (t, x(±Ri )) − λf2i (t, x(±Ri ))] >

n X j=1 j6=i

Rj

sup

|aij (t)|,

t∈(−∞,∞)

i = 1, . . . n, t ∈ (−∞, ∞), where x(±Ri ) = (x1 , . . . , xi−1 , ±Ri , xi+1 , . . . , xn ), |xj | ≤ Rj , j 6= i, λ ∈ (0, 1], and that F1 (t, ·) is Lipschitzian with a sufficiently small constant L, for every t ∈ (−∞, ∞) (cf. Preliminaries). In order to apply Theorem 5.1 for the solvability of (5.29), let us still consider the enlarged family of problems ( x˙ + A(t)x ∈ λF1 (t, x) + λF2 (t, q(t)), λ ∈ [0, 1], (5.30) x(m) = x(−m), where q ∈ Q = {e q ∈ C([−m, m], Rn ) | qe(t) ∈ K, for all t ∈ [−m, m]}. Observe that if ξ ∈ ∂K, then ξ = ξ(±Ri ) = (ξ1 , . . . , ξi−1 , ±Ri , ξi+1 , . . . , ξn ), for some i and |ξj | ≤ Rj for all j 6= i. Therefore, let us define for (5.30) the locally Lipschitzian bounding function composed of Vξ (x) = ±xi − Ri , i = 1, . . . , n, where ξ = ξ(±Ri ) ∈ ∂K. One can easily check that, under the above assumptions, we have: ad 3) Vξ (ξ) = 0 and Vξ (x) ≤ 0, for x ∈ K, and (∇Vξ (ξ)(λF1 (t, ξ) + λF2 (t, ξ) − A(t)ξ)) = n X = ±λ[f1i (t, ξ) + f2i (t, ξ)] ∓ aij (t)ξj < 0, j=1

for all t ∈ [−m, m], m ∈ N, where λ ∈ (0, 1]. Since the conditions 1), 2), 4), 5) are satisfied either by hypotheses or trivially, inclusion x˙ + A(t)x ∈ F1 (t, x) + F2 (t, q(t)),

for a.a. t ∈ (−∞, ∞), x ∈ Rn ,

(5.31)

admits, according to Theorem 5.2, a bounded solution x(t) such that x(t) ∈ K, for all t ∈ R. Via anti-periodic problems, we can still obtain the following theorem (cf. [AMT3, Theorem 2]). T HEOREM 5.3 Assume that A(t) ≡ 0 and let conditions 2)–4) in Theorem 5.2 be satisfied with K ⊂ Rn symmetric w.r.t. the origin (observe that condition 1) holds automatically). Let, furthermore, the sets of solutions of problem ( x˙ ∈ G0 (t, x), for a.a. t ∈ [−m, m], (5.32) x(m) = −x(−m), be, for each m ∈ N, a subset of C([−m, m], K). Then inclusion (5.17) admits a bounded solution x(t) such that x(t) ∈ K, for all t ∈ R.

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Proof. Given m ∈ N, let us consider problem (5.26) with M = −I, and A ≡ 0, i.e. the anti-periodic boundary value problem ( x˙ ∈ F (t, x), for a.a. t ∈ [−m, m], x ∈ Rn , (5.33) x(m) = −x(−m), whose associated homogeneous problem has only the trivial solution. In this case, the invariance of ∂K w.r.t. M = −I is equivalent to the symmetry of the origin. According to [AMT3, Theorem 1] (cf. Theorem 5.8 in Section 5.3. below), problem (5.33) admits, for each m ∈ N, a solution xm (·) such that xm (t) ∈ K, for all t ∈ [−m, m]. The conclusion follows by means of the second part of Proposition 5.2. R EMARK 5.7 A typical case occurs when G(t, x, y, λ) = λF (t, y) + (1 − λ)G0 (t, x), where G0 is a u-Carath´eodory multivalued mapping. If, in particular, G0 (t, x) ≡ 0 then, according to condition 1), the set of solutions of (5.32) is a subset of C([−m, m], K) if and only if K is a neighbourhood retract of the origin. E XAMPLE 5.3 Consider the differential inclusion x˙ ∈ F1 (t, x) + F2 (t, x),

for a.a. t ∈ R, x ∈ Rn ,

(5.34)

where F1 , F2 : R × Rn ⊸ Rn are u-Carath´eodory maps and F1 (t, ·) is Lipschitzian, with a sufficiently small Lipschitz constant, for all t ∈ R. Assume, furthermore, the existence of positive constants R such that <x, w> ≤ 0, for all t ∈ R, x ∈ Rn , with |x| = R, and w ∈ F1 (t, x) + F2 (t, x). Take the u-Carath´eodory map G(t, x, y, λ) = λ(F1 (t, x) + F2 (t, y)), and put K as the ball B0R , i.e. K = B0R . Since K is a neighbourhood of the origin and G0 ≡ 0, the set of solutions of (5.32) is, for all m, a subset of C([−m, m], K) (see Remark 5.7). Furthermore, since all assumptions of Theorem 5.2 can be satisfied by means of C 2 -function V (x) = x2 − R2 , the inclusion (5.34) admits, according to Theorem 5.2, a bounded solution x(t) such that x(t) ∈ K, for all t ∈ R.

5.2.

Periodic Solutions

Consider inclusion (5.17), where A(t) ≡ A(t + τ ) and F (t, x) ≡ F (t + τ, x), for some τ > 0. Modifying slightly the proof of Theorem 5.1, we can give immediately the following theorem. T HEOREM 5.4 Let A : [0, τ ] → L(Rn ) be a Lebesgue integrable (n × n)-matrix function whose Floquet multipliers are different from 1 (cf. Section 4.2.). Let F : [0, τ ] × Rn ⊸ Rn be a u-Carath´eodory set-valued map such that |F (t, x)| ≤ m(t) + K|x|,

for a.a. t ∈ [0, τ ], x ∈ Rn ,

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where K ≥ 0 is a sufficiently small constant and m ∈ L1 ([0, τ ]). Then inclusion (5.17) admits a τ -periodic solution x(t) of the form Z τ G(t, s)f (s, x(s)) ds, f ⊂ F. x(t) = 0

R EMARK 5.8 Constant K in Theorem 5.4 can be taken, similarly as in Remark 5.2, as K < 1/C(A), where Z τ max |G(t, s)| ds ≤ C(A). t∈[0,τ ] 0

More generally, as a particular case of Proposition 5.2, we can give immediately the following corollary. C OROLLARY 5.1 Consider problem ( x(t) ˙ + A(t)x(t) ∈ F (t, x(t)), x(0) = x(τ ),

for a.a. t ∈ [0, τ ],

(5.35)

where F (t, x) ≡ F (t + τ, x) satisfies conditions (iii) and (iv) in Proposition 5.2. Let G : [0, τ ] × Rn × Rn × [0, 1] ⊸ Rn be a product-measurable u-Carath´eodory map (cf. Definition 2.3 and Proposition 2.3) such that G(t, c, c, 0) ≡ −A(t)c and G(t, c, c, 1) ⊂ F (t, c),

for all (t, c) ∈ [0, τ ] × Rn .

Assume that A : [0, τ ] → L(Rn ) is an essentially bounded τ -periodic (n × n)-matrix, satisfying condition (i) in Proposition 5.2, whose Floquet multipliers are different from 1. Let, furthermore, conditions (v), (vi) in Proposition 5.2 be satisfied with 0 ∈ Q (instead of T (Q) × {0} ⊂ Q, in condition (vii)), where Lx = x(0) − x(τ ) and Θ = 0. Then inclusion (5.17) admits a τ -periodic solution. Proof. Since 1 is not a Floquet multiplier of A, the homogeneous problem ( x˙ + A(t) = 0, for a.a. t ∈ [0, τ ], x ∈ Rn , x(0) = x(τ )

(5.36)

admits only the trivial solution, by which condition (ii) in Proposition 5.2 holds. Thus, for G(t, c, c, 0) ≡ −A(t)c, all assumptions of Proposition 5.2, where Lx = x(0) − x(τ ) and Θ = 0, are satisfied, because 0 ∈ Q implies that T (Q) × {0} ⊂ Q in condition (vii). The assertion, therefore, follows directly from Proposition 5.2. R EMARK 5.9 Example 5.1 can be easily specified for the periodic problem, when taking Lx = x(0) − x(τ ) and Θ = 0, with the same growth restrictions (5.23), (5.24), (5.25) resp. (5.23’), (5.24’), (5.25’). If Floquet problem (5.26) takes (for M = I) the special form (5.35), then the bound sets approach allows us to say something about the localization of solutions. The following theorem is a special case (for M = I) of [AMT3, Theorem 1].

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T HEOREM 5.5 Let A : [0, τ ] → L(Rn ) and F : [0, τ ] × Rn ⊸ Rn be the same as in Corollary 5.1. Assume that the following conditions are satisfied: 1) there exists a u-Carath´eodory mapping G : [0, τ ] × R2n × [0, 1] ⊸ Rn such that, for all t ∈ [0, τ ], x, y ∈ Rn and λ ∈ [0, 1], it holds G(t, x, y, 0) = G0 (t, x), G(t, y, y, 1) ⊂ F (t, y); |w| ≤ s(t)(1 + |x| + |y| + λ), with w ∈ G(t, x, y, λ) and s ∈ L1 ([0, τ ]), 2) there exists a nonempty, open and bounded set K ⊂ Rn , whose closure is a retract of Rn , and a C 2 -function V : Rn → R such that V (x) ≤ 0 on K, V (x) = 0 and ∇V (x) 6= 0 on ∂K, and <∇V (x), w> ≤ 0, (5.37) for all t ∈ [0, τ ], x ∈ ∂K, λ ∈ (0, 1], and w ∈ G(t, x, x, λ) − A(t)x, where <·, ·> denotes the inner product, 3) G(t, ·, y, λ) is Lipschitzian with a sufficiently small Lipschitz constant L (cf. Preliminaries), for each (t, y, λ) ∈ [0, τ ] × K × [0, 1], 4) for each solution x(t) of problem ( x˙ + A(t)x ∈ G0 (t, x), for a.a. t ∈ [0, τ ], x ∈ Rn , (5.38) x(b) = x(a) it holds x(t) ∈ K, for all t ∈ [0, τ ]. Then inclusion (5.17) admits a τ -periodic solution Z τ G(t, s)f (s, x(s)) ds, x(t) =

f ⊂ F,

0

such that x(t) ∈ K, for all t ∈ (−∞, ∞). R EMARK 5.10 A typical case occurs when G(t, x, y, λ) = λF (t, y) + (1 − λ)G0 (t, x), where G0 is a u-Carath´eodory multivalued mapping. If, in particular, G0 (t, x) ≡ 0 then, because of the assumption that 1 is not a Floquet multiplier of A, homogeneous problem (5.36) has only the trivial solution, by which condition 4) is satisfied if and only if 0 ∈ K. R EMARK 5.11 Remarks 5.5 and 5.6 from the foregoing section hold here fully as well. As an application of a modified, in view of Remark 5.11, Theorem 5.5, let us give the following example. E XAMPLE 5.4 Consider the periodic problem ( x˙ + A(t)x ∈ F1 (t, x) + F2 (t, x), x(0) = x(τ ),

(5.39)

where x = (x1 , . . . , xn ), matrix A = (aij )i,j=1,...,n : [0, τ ] → L(Rn ) is continuous and such that 1 is not its Floquet multiplier, F = F1 + F2 = (f11 , . . . , f1n ) + (f21 , . . . , f2n ), F1 , F2 : [0, τ ] × Rn ⊸ Rn are globally upper semicontinuous multivalued functions which

Periodic-Type Solutions of Differential Inclusions

331

are bounded in t ∈ [0, τ ], for every x ∈ Rn , and linearly bounded in x ∈ Rn , for every t ∈ [0, τ ]. Assume, furthermore, that there exist positive constants Ri , i = 1, . . . , n such that ± [aii (t)(±Ri ) − λf1i (t, x(±Ri )) − λf2i (t, x(±Ri ))] >

n X

Rj max |aij (t)|,

j=1 j6=i

t∈[0,τ ]

i = 1, . . . n, t ∈ [0, τ ], where x(±Ri ) = (x1 , . . . , xi−1 , ±Ri , xi+1 , . . . , xn ), |xj | ≤ Rj , j 6= i, λ ∈ (0, 1], and that F1 (t, ·) is Lipschitzian with a sufficiently small constant L, for every t ∈ [0, τ ] (cf. Preliminaries). In order to apply Theorem 5.5 for the solvability of (5.39), let us still consider the enlarged family of problems ( x˙ + A(t)x ∈ λF1 (t, x) + λF2 (t, q(t)), λ ∈ [0, 1], (5.40) x(0) = x(τ ), where q ∈ Q = {e q ∈ C([0, τ ], Rn ) | qe(t) ∈ K, for all t ∈ [0, τ ]}. Observe that if ξ ∈ ∂K, then ξ = ξ(±Ri ) = (ξ1 , . . . , ξi−1 , ±Ri , ξi+1 , . . . , ξn ), for some i and |ξj | ≤ Rj for all j 6= i. Therefore, let us define for (5.40) the locally Lipschitzian bounding function composed of Vξ (x) = ±xi − Ri , i = 1, . . . , n, where ξ = ξ(±Ri ) ∈ ∂K. One can easily check that, under the above assumptions, we have: ad 2) Vξ (ξ) = 0 and Vξ (x) ≤ 0, for x ∈ K, and (∇Vξ (ξ)(λF1 (t, ξ) + λF2 (t, ξ) − A(t)ξ)) = n X aij (t)ξj < 0, = ±λ[f1i (t, ξ) + f2i (t, ξ)] ∓ j=1

for all t ∈ [0, τ ], where λ ∈ (0, 1]. Since conditions 1), 3) are satisfied by hypotheses and condition 4) follows trivially from the fact that 1 is not a Floquet multiplier of A, inclusion x˙ + A(t)x ∈ F1 (t, x) + F2 (t, q(t)) admits, according to Theorem 5.5, a τ -periodic solution x(t) such that x(t) ∈ K, for all t ∈ R. For the completeness, applying the Nielsen fixed point theory developed in [AG1], we can establish the following slight modification of the multiplicity results in [A1, Theorem 6.1] and [AG1, Theorem III.6.4]. T HEOREM 5.6 Consider boundary value problem (5.17) on the interval J = [0, τ ]. Assume 2 that A : J → Rn is a single-valued essentially bounded, Lebesgue measurable (n × n)matrix such that 1 is not its Floquet multiplier and F : J × R ⊸ Rn is a u-Carath´eodory

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product-measurable mapping with nonempty, compact and convex values (cf. Definition 2.3 and Proposition 2.3) satisfying |F (t, x)| ≤ µ(t)(|x| + 1),

(5.41)

where µ : J → [0, ∞) is a suitable Lebesgue integrable bounded function. Then inclusion (5.17) has at least N = N (r|T (Q) ◦ T (·)) solutions (for the definition of the Nielsen number N , see [AG1, Chapter I.10]), provided there exists a closed connected subset Q of C(J, Rn ) with a finitely generated abelian fundamental group such that (i) T (Q) is bounded, (ii) T : Q ⊸ U is retractible onto Q with a retraction r, where U is an open subset of C(J, Rn ) containing Q, i.e. there is a continuous retraction r : U → Q and p ∈ U \ Q with r(p) = q implies that p 6∈ T (q), (iii) T (Q) ⊂ {x ∈ AC (J, Rn ) | x(0) = x(τ )}, where T (q) denotes the set of (existing) solutions to (5.17). R EMARK 5.12 In the single-valued case, we can obviously assume the unique solvability of the associated linearized problem. Moreover, Q need not then have a finitely generated abelian fundamental group. In the multivalued case, the latter is true, provided Q is compact and T (Q) ⊂ Q. For more details, see [A1] and [AG1, Chapter III.6]. R EMARK 5.13 The Nielsen number N ∈ N ∪ {0} is a homotopy invariant guaranteeing the existence of at least N τ -periodic solutions of inclusion (5.17). Its computation is always a difficult task (cf. [AG1, Chapter III.6]). R EMARK 5.14 We constructed a nontrivial example of a planar (i.e. in R2 ) differential inclusion with at least three τ -periodic solutions in [A1, Theorem 6.2] and [AG1, Theorem III.6.17]. This result can be, however, alternatively obtained by different techniques. Finding the multiplicity criteria which can be obtained by Theorem 5.6, but not alternatively by different techniques, is an open problem. For functional differential equations and inclusions, we have found such criteria in [AF1] and [AF2].

5.3.

Anti-periodic Solutions

Consider inclusion (5.17), where this time A(t) ≡ A(t + τ ) and F (t, x) ≡ −F (t + τ, x), for some τ > 0. Modifying slightly the proof of Theorem 5.1, we can give immediately the following theorem. T HEOREM 5.7 Let either A : [0, τ ] → L(Rn ) be a Lebesgue integrable (n × n)-matrix function whose Floquet multipliers are different from −1 (cf. Section 4.2.) or A(t) ≡ 0. Let F : [0, τ ] × Rn ⊸ Rn be a u-Carath´eodory set-valued map such that |F (t, x)| ≤ m(t) + K|x|,

for a.a. t ∈ [0, τ ], x ∈ Rn ,

where K ≥ 0 is a sufficiently small constant and m ∈ L1 ([0, τ ]). Then inclusion (5.17) admits a τ -anti-periodic solution x(t) of the form Z τ G(t, s)f (s, x(s)) ds, f ⊂ F, x(t) = 0

Periodic-Type Solutions of Differential Inclusions or

1 x(t) = 2

Z

0

f (s, x(s)) ds + −τ

Z

333

t

f (s, x(s)) ds,

f ⊂ F,

0

respectively. R EMARK 5.15 Constant K in Theorem 5.7 can be taken either as K < 1/C(A), where Z τ max |G(t, s)| ds ≤ C(A), t∈[0,τ ] 0

or as K < 1/τ , respectively. More generally, as a particular case of Proposition 5.2, we can give immediately the following corollary. C OROLLARY 5.2 Consider problem ( x˙ + A(t)x ∈ F (t, x), for a.a. t ∈ [0, τ ], x ∈ Rn , x(0) = −x(τ ),

(5.42)

where A : [0, τ ] → L(Rn ) is either an essentially bounded τ -periodic (n × n)-matrix, satisfying condition (i) Proposition 5.2, whose Floquet multipliers are different from −1, or A(t) ≡ 0, and F (t, x) ≡ −F (t+τ, −x) satisfies conditions (iii) and (iv) in Proposition 5.2. Let G : [0, τ ] × Rn × Rn × [0, 1] ⊸ Rn be a product-measurable u-Carath´eodory map (cf. Definition 2.3 and Proposition 2.3) such that G(t, c, c, 0) ≡ −A(t)c and G(t, c, c, 1) ⊂ F (t, c),

for all (t, c) ∈ [0, τ ] × Rn .

Assume that conditions (v), (vi) in Proposition 5.2 hold with 0 ∈ Q (instead of T (Q) × {0} ⊂ Q, in condition (vii)), where Lx = x(0) + x(τ ) and Θ = 0. Then inclusion (5.17) admits a τ -anti-periodic (2τ -periodic) solution. Proof. Since −1 is not a Floquet multiplier of A or A(t) ≡ 0, the homogeneous problem ( x˙ + A(t) = 0, for a.a. t ∈ [0, τ ], x ∈ Rn , (5.43) x(0) = −x(τ ) admits only the trivial solution, by which condition (ii) in Proposition 5.2 holds. Thus, G(t, c, c, 0) ≡ −A(t)c, and all assumptions of Proposition 5.2, where Lx = x(0) − x(τ ) and Θ = 0, are satisfied, because 0 ∈ Q implies that T (Q × {0}) ⊂ Q, in condition (vii). The assertion, therefore, follows directly from Proposition 5.2. R EMARK 5.16 Example 5.1 can be easily specified for the anti-periodic problem, when taking Lx = x(0) + x(τ ) and Θ = 0, either with the same growth restrictions (5.23), (5.24), (5.25) resp. (5.23’), (5.24’), (5.25’), or (when A(t) ≡ 0) with Rτ Z τ 2 0 Rδ1 (t) dt D≥ 2 δ2 (t) dt < , where τ 3 0 3 − 0 δ2 (t) dt resp.

D≥

2 3τ

δ1 , − δ2

where δ2 <

2 . 3τ

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In the case of ODEs, Corollary 5.2 can be still improved, when A(t) ≡ 0, as follows (cf. [A1, Corollary 5.4] and [AG1, Chapter III.5.38]). C OROLLARY 5.3 Consider problem ( x˙ = f (t, x), for a.a. t ∈ [0, τ ], x ∈ Rn , x(0) = −x(τ ), where f (t, x) ≡ −f (t + τ, −x) is a Carath´eodory function. Let g : [0, τ ] × Rn × Rn × [0, 1] → Rn be a Carath´eodory function such that g(t, c, c, 1) = f (t, c),

for all (t, c) ∈ [0, τ ] × Rn .

Assume that (i) there exists a bounded retract Q of C([0, τ ], Rn ) such that Q\∂Q is nonempty (open) and such that g(t, x, q(t), λ) satisfies |g(t, x, q(t), λ) − g(t, y, q(t), λ)| ≤ p(t)|x − y|,

x, y ∈ Rn

for a.a. t ∈ [0, τ ] and each (q, λ) ∈ Q × [0, 1], where p : [0, τ ] → [0, ∞) is a Lebesgue integrable function with Z τ p(t) dt ≤ π, 0

(ii) there exists a Lebesgue integrable function α : [0, τ ] → [0, ∞) such that |g(t, x(t), q(t), λ)| ≤ α(t),

a.e. in [0, τ ],

for any (x, q, λ) ∈ ΓT , where T denotes the set-valued map which assigns, to any (q, λ) ∈ Q × [0, 1], the set of solutions of ( x(t) ˙ = g(t, x(t), q(t), λ), for a.a. t ∈ [0, τ ], x(0) = −x(τ ), (iii) T (Q × {0}) ⊂ Q holds and ∂Q is fixed point free w.r.t. T , for every (q, λ) ∈ Q × [0, 1]. Then the equation x˙ = f (t, x) admits a τ -anti-periodic (2τ -periodic) solution. R EMARK 5.17 As in Corollary 5.2, the requirement T (Q × {0}) ⊂ Q in condition (vii) reduces to {0} ⊂ Q, provided g(t, x, q, λ) = λg(t, x, λ), λ ∈ [0, 1]. If Floquet problem (5.26) takes (for M = −I) the special form (5.42), then the bound sets approach allows us to say something about the localization of solutions. The following theorem is a special case (for M = −I) of [AMT3, Theorem 1]. T HEOREM 5.8 Let A : [0, τ ] → L(Rn ) and F : [0, τ ] × Rn ⊸ Rn be the same as in Corollary 5.2. Assume that the following conditions are satisfied: 1) there exists a u-Carath´eodory mapping G : [0, τ ] × R2n × [0, 1] ⊸ Rn such that, for all t ∈ [0, τ ], x, y ∈ Rn and λ ∈ [0, 1], it holds G(t, x, y, 0) = G0 (t, x), G(t, y, y, 1) ⊂ F (t, y); |w| ≤ s(t)(1 + |x| + |y| + λ),

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with w ∈ G(t, x, y, λ) and s ∈ L1 ([0, τ ]), 2) there exists a nonempty, open and bounded set K ⊂ Rn , whose closure is a retract of Rn , and whose boundary ∂K is symmetric w.r.t. the origin 0 ∈ Rn , and a C 2 -function V : Rn → R such that V (x) ≤ 0 on K, V (x) = 0, ∇V (x) 6= 0 on ∂K, and <∇V (x), w> ≤ 0,

(5.44)

for all t ∈ [0, τ ], x ∈ ∂K, λ ∈ (0, 1], and w ∈ G(t, x, x, λ) − A(t)x, where <·, ·> denotes the inner product, 3) G(t, ·, y, λ) is Lipschitzian with a sufficiently small Lipschitz constant L, for each (t, y, λ) ∈ [0, τ ] × K × [0, 1], 4) for each solution x(t) of problem ( x˙ + A(t)x ∈ G0 (t, x), for a.a. t ∈ [0, τ ], x ∈ Rn , (5.45) x(b) = −x(a), it holds x(t) ∈ K, for all t ∈ [0, τ ]. Then inclusion (5.17) admits a τ -anti-periodic solution Z τ G(t, s)f (s, x(s)) ds, f ⊂ F, x(t) = 0

resp. 1 x(t) = 2

Z

0

f (s, x(s)) ds + −τ

Z

t

f (s, x(s)) ds,

f ⊂ F,

0

such that x(t) ∈ K, for all t ∈ (−∞, ∞). R EMARK 5.18 If A(t) ≡ 0 and F = f resp. G = g is single-valued, then the Lipschitz constant L in condition 3) can be taken, according to condition (i) in Corollary 5.3, as π L R τ < τ or can be replaced by the Lipschitzian function p : [0, τ ] → [0, ∞) such that 0 p(t)dt = π.

R EMARK 5.19 A typical case occurs when G(t, x, y, λ) = λF (t, y) + (1 − λ)G0 (t, x), where G0 is u-Carath´eodory multivalued mapping. If, in particular, G0 (t, x) ≡ 0 then, because of the assumption that either −1 is not a Floquet multiplier of A or A(t) ≡ 0, homogeneous problem (5.43) has only the trivial solution, by which condition 4) is automatically satisfied, namely 0 ∈ K. R EMARK 5.20 Remarks 5.5 and 5.6 from Section 5.1. hold here fully as well. As the first application of Theorem 5.8 (with A(t) ≡ 0), let us give the following example. E XAMPLE 5.5 Consider the differential inclusion ( x˙ ∈ F1 (t, x) + F2 (t, x), F1 (t, x) ≡ −F1 (t + τ, −x), F2 (t, x) ≡ −F2 (t + τ, −x),

(5.46)

where F1 , F2 : R × Rn ⊸ Rn are upper-Carath´eodory maps and F1 (t, ·) is Lipschitzian, with a sufficiently small Lipschitz constant L (in the single-valued case, it is enough, in

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view of Remark 5.18, to take L ≤ πτ ), for all t ∈ R. Assume, furthermore, the existence of a positive constants R such that <x, w> ≤ 0, for all t ∈ R, x ∈ Rn , with |x| = R, and w ∈ F1 (t, x) + F2 (t, x). Take the u-Carath´eodory map G(t, x, y, λ) := λ(F1 (t, x) + F2 (t, y)), and put K as the ball B0R , i.e. K = B0R . Since K is a neighbourhood of the origin and G0 ≡ 0, the set of solutions of (5.43) is, with A(t) ≡ 0, a subset of C([0, τ ], K) (see Remark 5.19). Furthermore, since all assumptions of Theorem 5.8 can be satisfied by means of C 2 -function V (x) = |x|2 − R2 , the inclusion (5.46) admits, according to Theorem 5.8, a τ -anti-periodic solution x(t) such that x(t) ∈ K, for all t ∈ R. As the second application of a modified, in view of Remark 5.20, Theorem 5.8 (with A(t) 6≡ 0), let us give the following example. E XAMPLE 5.6 Consider the anti-periodic problem x˙ + A(t)x ∈ F1 (t, x) + F2 (t, x), F1 (t, x) ≡ −F1 (t + τ, −x), F2 (t, x) ≡ −F2 (t + τ, −x), x(0) = −x(τ ),

(5.47)

where x = (x1 , . . . , xn ), matrix A = (aij )i,j=1,...,n : [0, τ ] → L(Rn ) is continuous with A(t) ≡ A(t + τ ), and such that −1 is not its Floquet multiplier, F = F1 + F2 = (f11 , . . . , f1n ) + (f21 , . . . , f2n ), F1 , F2 : [0, τ ] × Rn ⊸ Rn are globally upper semicontinuous multivalued functions with nonempty, convex, compact values which are bounded in t ∈ [0, τ ], for every x ∈ Rn , and linearly bounded in x ∈ Rn , for all t ∈ [0, τ ]. Assume, furthermore, that there exist positive constants Ri , i = 1, . . . , n, such that ± [aii (t)(±Ri ) − λf1i (t, x(±Ri )) − λf2i (t, x(±Ri ))] >

n X j=1 j6=i

Rj max |aij (t)|, t∈[0,τ ]

i = 1, . . . n, t ∈ (0, τ ), where x(±Ri ) = (x1 , . . . , xi−1 , ±Ri , xi+1 , . . . , xn ), |xj | ≤ Rj , j 6= i, λ ∈ (0, 1], n h i X ±λf1i (0, x(±Ri )) ± λf2i (0, x(±Ri )) ∓ aii (0)(±Ri ) < − |aij (0)|Rj j=1 j6=i

and n h i X ±λf1i (τ, −x(±Ri )) ± λf2i (τ, −x(±Ri )) ± aii (τ )(±Ri ) < − |aij (τ )|Rj , j=1 j6=i

i=1,. . . ,n, where λ ∈ (0, 1], and that F1 (t, ·) is Lipschitzian with a sufficiently small constant L, for every t ∈ [0, τ ] (cf. Preliminaries).

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In order to apply Theorem 5.8 for the solvability of (5.47), let us still consider the enlarged family of problems ( x˙ + A(t)x ∈ λF1 (t, x) + λF2 (t, q(t)), x(0) = −x(τ ),

λ ∈ [0, 1],

(5.48)

where q ∈ Q = {e q ∈ C([0, τ ], Rn ) | qe(t) ∈ K, for all t ∈ [0, τ ]}. Observe that if ξ ∈ ∂K, then ξ = ξ(±Ri ) = (ξ1 , . . . , ξi−1 , ±Ri , ξi+1 , . . . , ξn ), for some i and |ξj | ≤ Rj for all j 6= i. Therefore, let us define for (5.48) the locally Lipschitzian bounding function composed of Vξ (x) = ±xi − Ri , i = 1, . . . , n, where ξ = ξ(±Ri ) ∈ ∂K. One can easily check that, under the above assumptions, we have: ad 2) Vξ (ξ) = 0 and Vξ (x) ≤ 0, for x ∈ K, (∇Vξ (ξ)(λF1 (t, ξ) + λF2 (t, ξ) − A(t)ξ)) = = ±λf1i (t, x(±Ri )) ± λf2i (t, x(±Ri )) ∓ aii (t)(±Ri ) ∓ for all t ∈ (0, τ ), where λ ∈ (0, 1],

n X

aij (t)ξj < 0,

j=1 j6=i

(∇Vξ (ξ)(λF1 (0, ξ) + λF2 (0, ξ) − A(0)ξ)) = = ±λf1i (0, x(±Ri )) ± λf2i (0, x(±Ri )) ∓ aii (0)(±Ri ) ∓

n X

aij (0)ξj < 0,

j=1 j6=i

where λ ∈ (0, 1], and

(∇V−ξ (−ξ)(λF1 (τ, −ξ) + λF2 (τ, −ξ) + A(τ )ξ)) = = ∓λf1i (τ, −x(±Ri )) ∓ λf2i (τ, −x(±Ri )) ∓ aii (τ )(±Ri ) ∓

n X

aij (τ )ξj < 0,

j=1 j6=i

where λ ∈ (0, 1].

Since conditions 1), 3) are satisfied by hypotheses and condition 4) follows trivially from fact that −1 is not a Floquet multiplier of A, inclusion x˙ + A(t)x ∈ F1 (t, x) + F2 (t, x), admits, according to Theorem 5.8, a τ -anti-periodic solution x(t) such that x(t) ∈ K, for all t ∈ R. As the third application of a modified, in view of Remark 5.20, Theorem 5.8, let us give the following example.

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E XAMPLE 5.7 Consider the differential inclusion ( x˙ + A(t)x ∈ F1 (t, x) + F2 (t, x), F1 (t, x) ≡ −F1 (t + τ, −x), F2 (t, x) ≡ −F2 (t + τ, −x),

(5.49)

where matrix A = (aij )i,j=1,...,n : [0, τ ] → L(Rn ) is either Lebesgue-integrable essentially bounded function with A(t) ≡ A(t + τ ) such that −1 is not its Floquet multiplier or A(t) ≡ 0, F1 , F2 : [0, τ ] × Rn ⊸ Rn are upper-Carath´eodory multivalued functions with nonempty convex and compact values such that there exist c1 and c2 ∈ L1 ([0, τ ]), satisfying |F1 (t, 0)| ≤ c1 (t), |F2 (t, x)| ≤ c2 (t),

for all t ∈ [0, τ ], for all (t, x) ∈ [0, τ ] × K,

(K is defined below) and F1 (t, · ) is Lipschitzian, with a sufficiently small Lipschitz constant L, for all t ∈ R (⇒ |F1 (t, x)| ≤ L|x| + |F1 (t, 0)| ≤ L|x| + c1 (t) ≤ (L + c1 (t))(1 + |x|), for all (t, x) ∈ [0, τ ] × R). Assume, furthermore, the existence of positive constants ε and Rj , j = 1, . . . , n, such that K = Πnj=1 (−Rj , Rj ), ∂Kj = {ξ ∈ ∂K | ξj = ±Rj } and Q = {q ∈ C([0, τ ]) | q(t) ∈ K, for all t ∈ [0, τ ]}, and take, for all j = 1, . . . , n, ξ ∈ ∂Kj , x ∈ K ∩ Bξε , t ∈ [0, τ ], q ∈ Q, λ ∈ (0, 1], and w ∈ −A(t)x + λ[F1 (t, x) + F2 (t, x)], satisfying (sign ξj · wj ) < 0. Let us consider the family of multivalued functions defined as G(t, x, q, λ) = λ(F1 (t, x) + F2 (t, q)) which, recalling the growth conditions imposed on F1 and F2 and the boundedness of K, satisfy the conditions 1), 3) of Theorem 5.8. Moreover, condition 4) is trivially satisfied, because the only solution of (5.45) with G0 ≡ 0 is x = 0 ∈ int Q = Q \ ∂Q. Taking the locally Lipschitzian bounding function V : Rn → R as composed of Vξj (xj ) = sign ξj · xj − Rj , i = 1, . . . , n, we have Vξ (ξ) = 0 and Vξ (x) ≤ 0, for x ∈ K. Moreover (∇Vξ (x), w) = sign ξj · wj < 0, for ξ ∈ ∂Kj , x ∈ K ∩ Bξε , w ∈ −A(t)x + λ[F1 (t, x) + F2 (t, x)], λ ∈ (0, 1], j = 1, . . . , n. Thus condition 2) holds, too. A modified version, in view of Remark 5.20, Theorem 5.8 therefore implies the existence of τ -anti-periodic solution x(t) of inclusion (5.49) such that x(t) ∈ K, for all t ∈ R.

5.4.

Almost-periodic Solutions

Consider the inclusion x˙ + Ax ∈ F (x) + P (t),

for a.a. t ∈ R, x ∈ Rn ,

(5.50)

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339

where A is an (n×n)-matrix with real constant entries whose eigenvalues have nonzero real parts, F : Rn ⊸ Rn is Lipschitzian mapping with convex, compact values and P : R ⊸ Rn is an essentially bounded measurable map with closed values which will be successively assumed to be almost-periodic in the sense of Stepanov, Weyl and Besicovitch. Since such F possesses a (single-valued) Lipschitzian selection√ f ⊂ F , whose constant is, however, not necessarily the same, but can be taken as Ln(12 3/5 + 1), where L is a constant of F (see Preliminaries), and since such P possesses (single-valued) almostperiodic selections p ⊂ P in the sense of Stepanov, Weyl and Besicovitch, respectively (cf. Section 3.4.), we shall firstly consider the system x˙ + Ax = f (x) + p(t),

for a.a. t ∈ R, x ∈ Rn ,

(5.51)

√ where f : Rn → Rn is a Lipschitzian map with the constant L0 = Ln(12 3/5 + 1) and P : R → Rn is an essentially bounded function which is almost-periodic in a given sense (Stepanov, Weyl, Besicovitch). We already know from Section 5.1. that if L0 is sufficiently small (cf. Example 5.1), then system (5.51) admits an entirely bounded solution x(t) of the form (cf. Remark 5.1)

x(t) =

Z

∞

G(t − s)[f (x(s)) + p(s)] ds, where sup t∈R

−∞

Z

∞

|G(t − s)| ds ≤ −∞

2k . λ

(5.52)

In fact, there is a unique bounded solution of (5.51). If p is Stepanov almost-periodic (p ∈ Sap ) (cf. Section 3.4.), then in order to prove that x ∈ Sap , we need the Bochner transform xb of x, defined as xb (t) := x(t + η),

η ∈ [0, 1], t ∈ R,

and satisfying xb ∈ C(R, L([0, 1])n ), whenever x ∈ Lloc (R, Rn ) which is trivially satisfied by the hypothesis that x(t) is a solution of (5.51). Moreover, n o Sap = x ∈ Lloc (R, Rn ) | xb ∈ Cap (R, L([0, 1])n ) and kxkSap = kxb kBC(R,L([0,1])n ) . For more details, see e.g. [AG1, Appendix 1]. Since x(t + τ ) =

Z

∞

−∞

G(t − s)[f (x(s − t)) + p(s − t)] ds,

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we obtain (cf. (5.52)) kxb (t + τ ) − xb (t)kBC =

Z ∞

b G (t − s)[f (x(s + τ )) − f (x(s)) + p(s + τ ) − p(s)] ds =

−∞ BC Z ∞ b b b b |G(t − s)| kf (x(s + τ )) − f (x(s)) + p (s + τ ) − p (s)kBC ds ≤ sup t∈R Z−∞ ∞ |G(t − s)| ds kf b (x(t + τ )) − f b (x(t)) + pb (t + τ ) − pb (t)kBC ≤ sup t∈R

−∞

2k b ≤ kf (x(t + τ )) − f b (x(t)) + pb (t + τ ) − pb (t)kBC λ i 2k h L0 kxb (t + τ ) − xb (t)kBC + kpb (t + τ ) − pb (t)kBC , ≤ λ

i.e. kxb (t + τ ) − xb (t)kBC ≤ provided L0 <

kpb (t + τ ) − pb (t)kBC , 1/ 2k λ − L0

1 C(A) .

Thus, if kpb (t + τ ) − pb (t)kBC < ε, i.e. if τ is an ε-almost-period of p, then it is also an ε(1/ 2k λ − L0 )-almost-period of x, by which the solution x(t) of (5.51) is almost-periodic in the sense of Stepanov. In fact, since x(t) is also bounded and uniformly continuous, it is uniformly almost-periodic (cf. Section 3.4.). We can give the first theorem for almost-periodic solutions of (5.50). T HEOREM 5.9 Let A, F, P be as above. If the Lipschitz constant L of F still satisfies the λ inequality L ≤ 2kn(12√λ 3/5+1) (⇒ L0 < 2k ), where P is almost-periodic in the sense of Stepanov, then inclusion (5.50) admits a uniformly almost-periodic solution. R EMARK 5.21 For a uniformly almost-periodic (multivalued) map P , we cannot improve Theorem 5.9 in the sense that the solution is classical (smooth), because as pointed out in Section 3.4., P need not possess a uniformly almost-periodic selection. Theorem 5.9 can be slightly improved, on the basis of Theorem 5.2 in Section 5.1., as follows. C OROLLARY 5.4 Let the assumptions of Theorem 5.2 in Section 5.1. be satisfied with A(t) ≡ A, whose eigenvalues have nonzero real parts, and G(t, x, y, λ) = λ[F (y) + P (t)]. λ If F is Lipschitzian with constant L < 2kn(12√ , on the set K, and P is almost3/5+1) periodic in the sense of Stepanov, then inclusion (5.50) admits a uniformly almost-periodic solution x(t) such that x(t) ∈ K, for all t ∈ R. Consider again (5.50), where A, F are as above, but P be this time almostperiodic in the sense of Weyl. More precisely, let P be the sum P = P0 + p1 , where P0 is a multivalued e-Wap -perturbation of a single-valued Wap -function p1 . Since P0 possesses a single-valued e-Wap -selection p0 (cf. Section 3.4.) which is (e-)W -uniformly continuous (cf. [AG1, Appendix 1]), it can be proved by the similar arguments as in [AG1, p. 564] that the sum p = p0 + p1 : R → Rn is also a Wap -function (i.e. p ∈ Wap ). For definitions and more details, see Section 3.4..

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Hence, consider also √ (5.51), where f ⊂ F is a Lipschitzian selection of F with the constant L0 = Ln(12 3/5 + 1) and p ∈ Wap . System (5.51) admits, by the same reasons as above, a (unique) entirely bounded solution x(t) of the form (5.52). Since again Z ∞ G(t − s)[f (x(s − t)) + p(s − t)] ds, x(t + τ ) = −∞

and so Z

a+l

|x(t + τ ) − x(t)| dt = Z a+l Z ∞ dt |G(t − s)|[f (x(s + τ )) − f (x(s)) + p(s + τ ) − p(s)] ds = a

−∞ ∞

a

≤

Z

a+l

dt

a

Z

|G(t − s)| |f (x(s + τ )) − f (x(s)) + p(s + τ ) − p(s)| ds,

−∞

we can proceed similarly as above (this time, without using the Bochner transform, but applying the Fubini theorem; for more details, see [AG1, pp. 548–549]) to obtain Z a+l 2εk 1 |x(t + τ ) − x(t)| dt < , lim sup l→∞ a∈R l λ − 2L0 k a λ provided L0 < 2k , where τ is an ε-almost-periodic of p in the Wap -pseudometric and λ, k are suitable constants related to A (cf. (5.52)). Thus, x(t) is Wap -almost-periodic (i.e. x ∈ Wap ). In fact, since it is bounded and uniformly continuous, it is also Wap -uniformly-continuous, and subsequently W -normal (for more details, see [AG1, Appendix 1] and cf. Section 3.4.). We can, therefore, give the second theorem for almost-periodic solutions of (5.50).

T HEOREM 5.10 Let A, F, P be as above. If the Lipschitz constant L of F still satisfies λ λ ), where λ, k are suitable constants related the inequality L < 2kn(12√ (⇒ L0 < 2k 3/5+1) to A (cf. (5.52)), and P = P0 + p1 , where P0 is a multivalued e-Wap -perturbation of a single-valued Wap -function p1 , then inclusion (5.50) admits a W -normal solution. R EMARK 5.22 W -normality is a bit more than the Wap -property, but a bit less than the e-Wap -property (see Table 1 in Section 3.4.). It is, therefore, a question whether the e-Wap property of P implies the existence of an e-W -normal solution of (5.50). Theorem 5.10 can be slightly improved, on the basis of Theorem 5.2 in Section 5.1., as follows. C OROLLARY 5.5 Let the assumptions of Theorem 5.2 in Section 5.1. be satisfied with A(t) ≡ A, whose eigenvalues have nonzero real parts, and G(t, x, y, λ) = λ[F (y) + P (t)]. λ If F is Lipschitzian with constant L < 2kn(12√ , on the set K, and P = P0 + p1 , 3/5+1) where P0 is a multivalued e-Wap -perturbation of a single-valued Wap -function p1 , then inclusion (5.50) admits a W -normal solution x(t) such that x(t) ∈ K, for all t ∈ R. Finally, consider (5.50), where A, F are as above, but P be this time almost-periodic in the sense of Besicovitch. More precisely, let P be the sum P = P0 + p1 , where P0 is a multivalued B-perturbation of a single-valued Bap -function p1 . Since P0 possesses

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a single-valued B-selection p0 (cf. Section 3.4.) which is Bap -uniformly continuous (cf. [AG1, Appendix 1]), it can be proved by the similar arguments as in [AG1, p. 564] that the sum p = p0 + p1 : R → Rn is also a Bap -function (i.e. p ∈ Bap ). For definitions and more details, see Section 3.4.. Hence, consider also √ (5.51), where f ⊂ F is a Lipschitzian selection of F with the constant L0 = Ln(12 3/5 + 1) and p ∈ Bap . System (5.51) admits, by the same reasons as above, a (unique) entirely bounded solution x(t) of the form (5.52). Since again Z ∞ G(t − s)[f (x(s − t)) + p(s − t)] ds, x(t + τ ) = −∞

and so Z

T

|x(t + τ ) − x(t)| dt = Z T Z ∞ |G(t − s)|[f (x(s + τ )) − f (x(s)) + p(s + τ ) − p(s)] ds dt = −T

−∞ ∞

−T T

≤

Z

−T

dt

Z

|G(t − s)| |f (x(s + τ )) − f (x(s)) + p(s + τ ) − p(s)| ds,

−∞

we can proceed similarly as above (for more details, see [AG1, pp. 550–551]) to obtain Z T 2εk 1 |x(t + τ ) − x(t)| dt < , lim T →∞ 2T λ − 2L0 k −T λ provided L0 < 2k , where τ is an ε-almost-periodic of p in the Bap -pseudometric and λ, k are suitable constants related to A (cf. (5.52)). Thus, x(t) is Bap -almost-periodic (i.e. x ∈ Bap ). In fact, since it is bounded and uniformly continuous, it is also Bap -uniformly-continuous, and subsequently B-normal (for more details, see [AG1, Appendix 1] and cf. Section 3.4.). We can, therefore, give the third theorem for almost-periodic solutions of (5.50).

T HEOREM 5.11 Let A, F, P be as above. If the Lipschitz constant L of F still satisfies the λ inequality L < 2kn(12√λ 3/5+1) (⇒ L0 < 2k ), where λ, k are suitable constants related to A (cf. (5.52)), and P = P0 + p1 , where P0 is a multivalued B-perturbation of a single-valued Bap -function p1 , then inclusion (5.50) admits a B-normal solution. R EMARK 5.23 B-normality is a bit more than the Bap -property, but a bit less than the Bproperty (see Table 1 in Section 3.4.). It is, therefore, a question whether the B-property of P implies the existence of a B-solution of (5.50). Theorem 5.11 can be slightly improved, on the basis of Theorem 5.2 in Section 5.1., as follows. C OROLLARY 5.6 Let the assumptions of Theorem 5.2 in Section 5.1. be satisfied with A(t) ≡ A, whose eigenvalues have nonzero real parts, and G(t, x, y, λ) = λ[F (y) + P (t)]. λ , on the set K, and P = P0 + p1 , If F is Lipschitzian with constant L < 2kn(12√ 3/5+1) where P0 is a multivalued B-perturbation of a single-valued Bap -function p1 , then inclusion (5.50) admits a B-normal solution x(t) such that x(t) ∈ K, for all t ∈ R.

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R EMARK 5.24 In the single-valued case, Lipschitz constant L can be taken as L = L0 < λ 2k . The proved bounded almost-periodic solutions are then unique (in the case of corollaries, on K). R EMARK 5.25 Theorems 5.10 and 5.11 are slight generalizations of the analogous results in [AG1, Chapter III.10], because multivalued perturbations P0 in the sum P = P0 + p1 were there assumed almost-periodic in the sense of Stepanov, everywhere. Corollaries 5.4, 5.5, 5.6 are formally new.

5.5.

Derivo-periodic Solutions

Consider the inclusion x˙ ∈ F (x) + P (t),

for a.a. t ∈ R, x ∈ Rn ,

(5.53)

where x = (x1 , . . . , xn ) and F (x) = (f1 (x), . . . , fn (x)). D EFINITION 5.1 We say that x ∈ ACloc (R, Rn ) is an ω-derivo-periodic solution of (5.53) if x(t) ˙ = x(t ˙ + ω), for a.a. t ∈ R. Assume, we have a suitable notion of a multivalued derivative (see e.g. [AG1, Appendix 2], say D∗ , or multivalued partial derivatives Dx∗j (w.r.t. xj ), j = 1, . . . , n, such that F : Rn ⊸ Rn is αω-D∗ -periodic, α ∈ Rn , ω > 0, i.e. D∗ F (x) ≡ D∗ F (x + αω),

(5.54)

where D∗ F means the Jacobi matrix, namely D∗ F = (D∗ (fi )xj )ni,j=1 . H YPOTHESIS 5.1 The αω-D∗ -periodicity of F , i.e. (5.54), implies that F can be written as F (x) = F0 (x) − Ax,

(5.55)

F0 (x) ≡ F0 (x + αω),

(5.56)

where F0 is αω-periodic, i.e. and A is a suitable (n × n)-matrix. If so, then (5.53) with an αω-D∗ -periodic F would take the following quasi-linear form: x˙ + Ax ∈ F0 (x) + P (t),

(5.57)

where F0 satisfies (5.56). Moreover, x0 (t) is obviously an ω-periodic solution of x˙ + Ax ∈ F0 (x + αt) + P (t),

(5.58)

where F0 satisfies (5.56) and P is ω-periodic, i.e. P (t) ≡ P (t + ω),

(5.59)

if and only if x(t) = x0 (t) + αt is an ω-derivo-periodic solution of x˙ + Ax ∈ F0 (x) + [P (t) + (tA + I)α], where I denotes the unit matrix.

(5.60)

344

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H YPOTHESIS 5.2 [P (t) + (tA + I)α] is ω-D∗ -periodic, i.e. D∗ {P (t + ω) + [(t + ω)A + I]α} ≡ D∗ [P (t) + (tA + I)α] = D∗ P (t) + Aα ≡ D∗ P (t + ω) + Aα

(5.61)

holds, provided (5.59) takes place. More important is, however, the “reverse” formulation: x(t) = x0 (t) + αt is an ωderivo-periodic solution of (5.57) if and only if x0 (t) is an ω-periodic solution of x˙ + Ax ∈ F0 (x + αt) + [P (t) − (tA + I)α],

(5.62)

where F0 satisfies (5.56). H YPOTHESIS 5.3 An ω-D∗ -periodic P implies that P (t) = P0 (t) + (tA + I)α

(5.63)

holds with an ω-periodic P0 , i.e. P0 (t) ≡ P0 (t + ω).

(5.64)

If so, then (5.62) would take the form x˙ + Ax ∈ F0 (x + αt) + P0 (t),

(5.65)

where, in view of (5.56), F0 (x + α(t + ω)) ≡ F0 (x + αt) =: F1 (t, x) ≡ F1 (t + ω, x). For (5.65), we can easily find sufficient conditions for the existence of at least one ωperiodic solution, e.g. if σ(A) ∩ 2πiZ/ω = ∅ (see Sections 4.5. and 4.1.), provided a u.s.c. F0 with nonempty, convex and compact values is αω-periodic, i.e. (5.56), and a measurable, essentially bounded P0 with nonempty, convex and compact values is ω-periodic, i.e. (5.64). Therefore, we can give immediately P ROPOSITION 5.3 If Hypothesis 5.3 is satisfied for a D∗ -differentiable, ω-D∗ -periodic P , i.e. if (5.63) is implied, then (5.65) with the same α can be written in the form of (5.62), and subsequently (5.57) admits an ω-derivo-periodic solution, provided σ(A) ∩ 2πiZ/ω = ∅ and (5.56) takes place for a u.s.c. multivalued function F0 with nonempty, convex and compact values. P ROPOSITION 5.4 Let the assumptions of Proposition 5.3 be satisfied. If Hypothesis 5.1 is still satisfied, for an αω-D∗ -periodic, D∗ -differentiable multivalued function F with nonempty, convex and compact values and with the same A as in (5.57), then even inclusion (5.53) admits an ω-derivo-periodic solution. Although we have to our disposal several different notions of multivalued derivatives (see [AG1, Appendix 2]), to satisfy Hypothesis 5.1 or 5.3, which are assumed in Propositions 5.3 and 5.4, only the usage of a multivalued derivative due to F. S. De Blasi, introduced in Section 3.5., is available.

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345

Moreover, F in Hypothesis 5.1 must have a special form to satisfy (5.55), e.g. such that D∗ (Fi (x))xj = D∗ (Fi (xj ))xj ,

for all i, j = 1, . . . , n.

(5.66)

As pointed out in Remark 3.8 in Section 3.5., in Rn , we can define equivalently a De Blasi-like differentiable function as a sum of single-valued continuous function having right-hand side and left-hand side derivatives plus a multivalued constant. Thus, according to Theorem 3.7 in Section 3.5., a De Blasi-like differentiable multivalued function F with (5.66) takes the form (5.55) with F0 satisfying (5.56) if and only if (5.54) takes place jointly with Fi (αj ω) = Fi (0) − aij αj ,

for all i, j = 1, . . . , n,

(5.67)

where A = (aij )ni,j=1 and α = (α1 , . . . , αn ). Similarly, a De Blasi-like differentiable multivalued function P takes the form (5.63) with P0 satisfying (5.64) if and only if P is ω-D∗ -periodic, i.e. D∗ P (t) ≡ D∗ P (t + ω),

(5.68)

P (ω) = P (0) + (ωA + I)α.

(5.69)

jointly with Hence, we are ready to give the following three theorems, when applying Propositions 5.3 and 5.4. T HEOREM 5.12 Let σ(A) ∩ 2πiZ/ω = ∅ hold for a real (n × n)-matrix A. Assume, furthermore, that F0 : Rn ⊸ Rn is a u.s.c. multivalued function with nonempty, convex and compact values which is αω-periodic, i.e. (5.56), where α ∈ Rn , ω > 0. Let, at last, P : R ⊸ Rn be a De Blasi-like differentiable multivalued function with nonempty, convex and compact values which is ω-derivo-periodic, i.e. (5.68), satisfying (5.69), where I is a unit matrix. Then the inclusion (5.57) admits an ω-derivo-periodic solution in the sense of Definition 5.1. T HEOREM 5.13 Let F : Rn ⊸ Rn be a De Blasi-like differentiable multivalued function with nonempty, convex and compact values which is αω-derivo-periodic, i.e. (5.54), satisfying (5.66) and (5.67), where A is a real (n × n)-matrix such that σ(A) ∩ 2πiZ/ω = ∅, I is a unit matrix and α ∈ Rn , ω > 0. Then the inclusion x˙ ∈ F (x) + [P (t) + (tA + I)α]

(5.70)

admits an ω-derivo-periodic solution in the sense of Definition 5.1, provided an essentially bounded measurable multivalued function P : R ⊸ Rn with nonempty, convex and compact values is ω-periodic, i.e. (5.59). T HEOREM 5.14 Let F : Rn ⊸ Rn be a De Blasi-like continuously differentiable multivalued function with nonempty, convex and compact values which is αω-derivo-periodic, i.e. (5.54), satisfying (5.66) and (5.67), where A is a real (n × n)-matrix such that σ(A) ∩ 2πiZ/ω = ∅, I is a unit matrix and α ∈ Rn , ω > 0. Assume, furthermore, that P : R ⊸ Rn is a De Blasi-like differentiable multivalued function with nonempty, convex and compact values which is ω-derivo-periodic, i.e. (5.68), satisfying (5.69). Then the inclusion (5.53) admits an ω-derivo-periodic solution in the sense of Definition 5.1.

346

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Without applying Propositions 5.3 and 5.4, we can give immediately the following statement. C OROLLARY 5.7 Let A be a real (n × n)-matrix such that σ(A) ∩ 2πiZ/ω = ∅. Assume that F0 : Rn ⊸ Rn is a u.s.c. multivalued map with nonempty, convex and compact values satisfying (5.56) and P : R ⊸ Rn is an essentialy bounded measurable multivalued function with nonempty, convex and compact values which is ω-periodic, i.e. (5.59). Then the inclusion (5.60) admits an ω-derivo-periodic solution in the sense of Definition 5.1. R EMARK 5.26 All the above statements were obtained, via appropriate transformations, by means of the results for periodic solutions in Section 5.2.. It would be much more interesting, but also more delicate, to find sufficient conditions for the existence of ω-derivoperiodic solutions for inclusions of the form (5.57), where F0 satisfies (5.56) and P satisfies (5.59). Such criteria are known, for for pendulum-type equations with suitable instance, 0 −1 constants a, b, where n = 2, A = 0 a , F = (0, −b sin x1 )T , P = (0, p(t))T . For the related references, see e.g. Remarks and comments to [AG1, Chapter III.11].

6.

Concluding Remarks

• Sufficient conditions for the existence of periodic-type solutions of differential systems can be significantly improved for those of higher-order (scalar) differential equations and inclusions (for bounded, periodic and anti-periodic solutions, cf. [AG1, Chapter III.5]; for almost-periodic solutions, cf. [AG1, Chapter III.10], [ABR1], [ABR2]; for derivo-periodic solutions, cf. [A2]). • Periodic-type investigations have been extended to second-order vector differential equations and inclusions (for bounded, periodic and anti-periodic solutions, cf. [AKM]; for anti-periodic solutions, cf. [AAP2]; for quasi-periodic solutions, cf. [BC2]; for almostperiodic solutions, cf. [BMC]). • Appropriate generalizations have been formulated in abstract spaces (for boun-ded, periodic and anti-periodic solutions, cf. [A1], [AMT4], [AG1, Chapter III.5]; for periodic solutions, cf. [HM], [KOZ]; for anti-periodic solutions, cf. [AAP1], [AF], [AP]; for almostperiodic solutions, cf. [AG1, Chapter III.10], [NG], [Za]). • For implicit differential equations, the study of periodic solutions can be transformed to periodic problems for (explicit) differential inclusions (cf. [AG1, Chapter III.11], [FGK], [FK]). • Periodic solutions of random differential equations and inclusions can be investigated in a deterministic way (cf. [AG1, Chapter III.4], [A5]). • Recently, an enormous activity has been spent to the research of difference equations and equations on time scales. It is natural to consider in this frame periodic-type oscillations to (possibly multivalued) discrete models. Some results have been already obtained especially for periodic and almost-periodic solutions of difference equations (cf. e.g. the related papers published in Journal of Difference Equations and Applications). • Obviously, many problems of periodic-type remain open. Although not formulated explicitly, some of them could be easily recognized in our text.

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INDEX 1 1G, 121

4 4G, 121, 133

A Aβ, 8, 315, 316, 317, 321, 322, 323, 324, 325, 326, 327, 328, 329, 330, 331, 332, 333, 334, 335, 336, 337, 338, 340, 341, 342 academic, 91, 149, 153, 159, 180, 182, 191, 192, 193, 194, 201, 202, 204, 205, 208, 210, 211, 221, 222, 223, 224, 225, 226, 227, 229, 230, 233, 235, 244, 248, 250, 251, 256, 259, 266, 267, 268, 274 academic performance, 250 academic success, 202 access, 152, 160, 169, 188, 203, 224, 226 accidental, 124 accuracy, 18, 21, 29, 30, 39, 56, 78, 295, 307, 310 achievement, ix, 151, 153, 158, 182, 205, 221, 224, 225, 247, 248, 249, 250, 251, 252, 255, 265, 267 acoustic, 14 activation, 223 ad hoc, 56 adaptation, 94, 166, 175 adjustment, viii, 162 administration, 215, 250, 251, 255 administrative, 265 administrators, 179, 207, 212, 213, 215, 265 adult, 91 advertisement, 166 advertisements, 171 advertising, 169, 170, 171 aeronautical, 183 aesthetics, 90, 91 affective dimension, 95 Africa, 253

African American, 180, 181, 184, 189, 190, 192, 197, 198, 212, 222, 235 African Americans, 189, 190, 197, 198, 222 Ag, 125, 126, 128, 129, 133 age, 112, 152, 211, 252 agents, 98 aggregation, 163, 165 agricultural, 183, 213, 221, 229, 230 aid, 144, 273 AIDS, 219 Alaska, 184, 198, 216, 222, 245 Alaska Natives, 222 Alaskan Native, 189, 226 Albert Einstein, 217 algorithm, 68 aliens, 263, 264 alternative, 98, 147, 165, 267, 314, 317, 324 alternatives, 164, 170 American Association for the Advancement of Science, 215 American Competitiveness Initiative, 249 American Indian, 184, 189, 198, 245 American Indians, 189 amplitude, 25, 45, 61 AMS, 348 Amsterdam, 347, 349 analysts, 183, 212, 213 angular momentum, 121, 137 anomalous, 349 anthropology, 92 ants, 216 appendix, 179, 182, 185, 191, 192, 194, 198, 207, 211 application, vii, 1, 6, 56, 67, 75, 79, 93, 101, 117, 124, 143, 145, 146, 149, 153, 157, 159, 160, 165, 174, 223, 244, 264, 275, 326, 330, 335, 336, 337, 347 applied research, 224 appropriations, 270, 272, 273, 274 Arabia, 253 Arctic, 221 arithmetic, 126, 145 Arizona, 218 Armenia, 253

356

Index

arousal, 164, 170 Asia, 220, 261 Asian, 184, 192, 198, 235, 245 assessment, ix, 141, 142, 143, 201, 210, 221, 225, 247, 249, 250, 251, 252 assets, 162, 163, 170, 176 assignment, 147, 255 assumptions, vii, 89, 90, 94, 95, 103, 141, 163, 165, 174, 298, 313, 317, 318, 320, 321, 327, 328, 329, 331, 333, 336, 337, 340, 341, 342, 344 astronomy, 311 asymptotic, 278, 349 asymptotically, 316, 317, 318 Athens, 112 atmosphere, 56 atoms, 115, 116, 123, 137 attachment, 143, 209 attitudes, 95, 110, 111, 112 attractiveness, 163, 165, 171, 172, 173, 175 auditing, 180, 209 Australasia, 113 Australia, 109, 112, 114, 202, 253, 254 Australian Research Council, 98 Austria, 254 authenticity, 113 authority, 274 automation, 146 autonomy, 158 availability, 162, 165, 168, 169 awareness, 162, 165, 169, 170

B Bahrain, 253 Banach spaces, 292, 297, 298, 350 barriers, 110 basic research, 223 beginning teachers, 109 behavior, viii, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 174, 175 behaviours, 96 Belgium, 254 beliefs, vii, 89, 94, 95, 96, 98, 99, 103, 106, 110, 112, 113 benchmarks, 274 benefits, 153, 212 benzene, 122, 123 bias, 97 bifurcation, 143 bilingual, 200, 255 binomial distribution, 126 biotechnology, 184 bipartisan, 250 Bohr, 295, 305, 307, 309, 348 bonus, 269, 272 border security, 178 borrowers, 271 borrowing, 162

boson, 118, 119 bosons, viii, 115, 117, 118, 128 Botswana, 253 boundary conditions, 9, 12, 23, 30, 33, 50, 67, 68, 69, 80, 82 boundary value problem, 322, 326, 328, 331, 347, 348 bounded solution, 314, 316, 317, 318, 319, 321, 322, 323, 324, 325, 326, 327, 328, 339, 341, 342 bounds, 211, 230, 231, 233 boys, 152, 202 braids, 318 brain, 101 Brazil, 261 breakdown, 141 broad spectrum, vii Bulgaria, 253 Business Roundtable, 248, 268 buyer, 163

C calculus, 201 Canada, 1, 202, 254, 259 candidates, 222 CAP, 302, 303 capacity, 187, 189, 190, 216, 221, 223, 226, 244, 245, 262, 265 cast, 48 categorization, 182, 210 category a, 147 category d, 162 cation, 222 Cauchy problem, 322 CCR, 219 cell, 6 Census, 182, 211, 215 Census Bureau, 211, 215 Central America, 261 certificate, 191, 227, 230 certification, 182, 200, 255, 267, 269, 270, 271 chaos, 352 characteristic viscosity, 143 charge density, 20, 30 chemicals, 221 children, 105, 182, 200, 208, 221, 222, 272 Chile, 253 China, 202, 259, 261 chiral, 126 chirality, 126 Cincinnati, 217 citizens, 91, 186, 188, 208, 224, 225, 258, 263, 264, 266, 269, 273, 274 citizenship, 188, 208 civilian, 178, 180, 182, 184, 185, 186, 197, 208, 209, 216, 221, 261 classes, 116, 122, 123, 136, 152, 157, 200, 204, 267, 308, 309, 310

Index classical, 116, 295, 305, 319, 340 classification, viii, 115, 117, 149, 211, 212, 260, 301 classroom, vii, 89, 90, 93, 94, 96, 97, 98, 99, 101, 103, 105, 107, 108, 112, 149, 151, 155, 156, 157, 158, 159, 221, 224, 225, 267 classroom practice, 90 classroom teacher, 221, 267 classroom teachers, 221, 267 classrooms, 92, 96, 111, 113, 114, 200, 201, 204, 224 closure, 283, 306, 308, 325, 330, 335 Co, 139, 216, 246, 252, 259 codes, 6, 14, 29, 36, 47, 78, 212, 213 cognition, 95, 96, 101 cognitive, viii, 95, 96, 97, 105, 107, 110, 114, 145, 146, 147, 149, 150, 157, 158, 159, 160, 175, 222, 225 cognitive process, 150 cognitive tool, viii, 145, 149, 158, 160 coherence, viii, 89, 103, 224 cohort, 258 collaboration, 79, 181, 207, 265, 267 college students, 188, 201, 221, 262, 263 colleges, 179, 182, 184, 203, 205, 207, 209, 213, 214, 215, 222, 223, 224, 226, 244, 274 collisions, 5 Colorado, 244, 245, 246 colors, 126 Columbia, 175 Columbia University, 175 combinatorics, viii, 115, 117, 126 combined effect, 20 commercials, 169 commodity, 165 communication, 160, 166, 169, 183, 224 communities, ix, 113, 222, 247, 249 community, 93, 205, 223, 224, 258, 268 comparative advantage, 164 compassion, 107 competence, 105, 107, 110, 164 competency, 250 competition, 163, 172, 175, 202, 203 competitive advantage, ix, 165, 167, 168, 171, 172, 173, 176, 177, 178, 205 competitive markets, 161, 164, 174 competitiveness, 249, 270 complexity, 93, 97, 103, 278, 293 compliance, 121, 245 components, 101, 146, 165, 225 composition, 298 comprehension, 147 compulsory education, 252 computation, 65, 145, 146, 147, 148, 150, 332 computer science, 183, 191, 194, 197, 198, 199, 225, 229, 230, 236, 248, 257, 258, 270, 273, 278 computer systems, 212 computer technology, 184, 225 computing, viii, 145, 152, 159, 161 concrete, 147, 268

357

confidence, vii, 89, 90, 104, 105, 106, 107, 110, 179, 191, 192, 198, 211, 212, 213, 214, 227, 229, 230, 231, 232, 233, 234, 235, 236 confidence interval, 179, 191, 192, 198, 211, 212, 213, 214, 227, 229, 230, 231, 232, 233, 234, 235, 236 confidence intervals, 179, 191, 192, 198, 211, 213, 214, 227, 229, 230, 231, 232, 233, 234, 235, 236 configuration, 47, 117, 120, 122, 123, 147 confinement, 14 confusion, 278 Congress, ix, 178, 209, 247, 249, 250, 268, 269, 270 consensus, 95 conservation, 47, 213 consolidation, 153 constraints, 47, 162, 163, 174 construction, 94, 95, 97, 98, 110, 280, 281, 286 constructivist, 98 consumer choice, viii, 162, 163 consumers, viii, 162, 163, 165, 166, 168, 171, 174 consumption, 162, 163, 166, 168, 175 consumption function, 175 content analysis, 153 continuity, 299 control, 7, 56, 70, 146, 175, 213, 292 convergence, 32, 49, 67, 69, 174, 304, 305 conversion, 146, 147 convex, x, 167, 277, 278, 290, 291, 293, 297, 299, 302, 304, 315, 316, 318, 319, 320, 322, 323, 325, 332, 336, 338, 339, 344, 345, 346 coordination, ix, 159, 203, 215, 247, 249, 270 correlation, 75, 77, 117, 133, 135, 136, 137, 138 correlation coefficient, 77 correlations, 133, 137 costs, 180, 189, 201, 272 Coulomb, 41 Coulomb gauge, 41 coupling, 6, 27, 116, 134, 137 course work, 267 coverage, 164, 172, 173, 175 CPU, 65 CRC, 351, 353 credentials, 200 critical density, 25 CRM, 352 cross-sectional, 165 CRS, 247, 250, 251, 255, 258, 263, 271 crystals, 295 cues, 172 cultural practices, 101 culture, vii, 89, 90, 91, 94, 95, 96, 101, 103, 108, 110, 112, 113, 214, 226 curiosity, 96, 224 Current Population Survey (CPS), 178, 179, 182, 183, 184, 190, 198, 210, 211, 212, 213, 235, 236, 245, 246 curriculum, 92, 93, 94, 96, 100, 106, 108, 111, 146, 201, 204, 221, 222, 223, 224, 225, 226, 265, 266, 267, 273

358

Index

curriculum development, 224 customer preferences, 166 customers, 162, 163, 164, 165, 166, 167, 169, 170, 171, 173, 174, 175 cycles, 123 cyclotron, 14, 48 Cyprus, 253 Czech Republic, 254, 295

D data analysis, 99 data collection, 161 data communication, 212 data set, 210 database, 184, 210 debt, 271 decisions, ix, 90, 91, 95, 162, 164, 166, 170, 171, 176, 177, 179, 181, 199, 200, 201, 202, 203, 205, 207, 208, 214, 215 deduction, 316 deficit, 107 Deficit Reduction Act, 270 deficits, 206 definition, 32, 38, 83, 299, 302, 303, 305, 306, 307, 308, 309, 311, 312, 314, 332 deformation, 62 degenerate, 122, 123, 136, 298 demand, viii, 100, 112, 146, 162, 163, 166, 167, 168, 170, 174, 176, 268 demographic characteristics, 210 Denmark, 254 density, 6, 9, 14, 18, 19, 20, 29, 39, 45, 71, 73 dentistry, 222 Department of Agriculture, 180, 185, 216, 221 Department of Commerce, 185, 216, 237 Department of Defense (DOD), 179, 274 Department of Education, 98, 179, 185, 186, 209, 210, 215, 216, 221, 225, 244, 245, 250, 251, 253, 254, 255, 256, 257, 262, 267 Department of Energy (DOE), 139, 185, 217, 270 Department of Health and Human Services, 181, 185, 206, 218, 239 Department of Homeland Security, 178, 185, 203, 219, 244, 245 Department of State, 184, 245 Department of the Interior, 185, 219 Department of Transportation, 185, 221, 226 derivatives, 60, 61, 300, 311, 312, 313, 326, 343, 344, 345 derived demand, 174 designers, 113 desire, 102, 203 destruction, viii, 141, 142, 143, 144 destruction processes, 143, 144 destructive process, 141 deviation, 92 dichotomy, 321

differential equations, x, 296, 301, 304, 309, 332, 346, 347, 348, 349, 350, 351, 352 differentiation, 265 diffraction, 295 diffusion, 29, 32, 53, 55, 56, 61, 67, 68, 71, 75, 79, 225 diffusion process, 67 digital divide, 226 dimensionality, 6, 78 Dirichlet boundary conditions, 30, 50 Dirichlet condition, 300 disadvantaged students, 266 discipline, vii, 89, 90, 91, 92, 93, 95, 96, 101, 103, 104, 109, 110, 114, 265, 266 discourse, 94, 98 Discovery, 219, 220 dispersion, 29 displacement, 69 disposable income, 163, 165, 168 dissipative system, 318, 348 distribution, 5, 6, 7, 8, 9, 14, 15, 16, 17, 21, 23, 28, 29, 36, 37, 40, 41, 42, 46, 48, 75, 118, 120, 122, 124, 126, 144, 167, 172, 200, 256, 259, 263 distribution function, 5, 6, 7, 8, 9, 14, 16, 17, 21, 23, 28, 29, 40, 41, 42, 46, 48 divergence, 68, 93 diversity, 95, 149, 179, 214, 222, 224, 226, 264 division, 18 doctors, 212 domain-specificity, 104 dominance, 75, 95, 163, 164, 170 draft, 181, 188, 189, 206, 207, 208 drinking, 221 duality, 167 duopoly, 163, 175 duplication, 208 duration, 274 dynamical system, 295

E earth, 200 East Asia, 220 Eastern Europe, 261 ecology, 100 economic competitiveness, ix, 247 economics, 244 ecosystem, 221 Education, 89, 95, 98, 100, 102, 103, 109, 111, 112, 113, 114, 145, 159, 160, 161, 178, 179, 181, 182, 183, 184, 185, 186, 187, 189, 190, 191, 192, 193, 195, 196, 197, 199, 200, 201, 203, 204, 205, 207, 209, 210, 211, 213, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224, 225, 226, 227, 229, 230, 231, 233, 235, 237, 239, 241, 243, 244, 245, 248, 249, 250, 251, 253, 254, 255, 256, 257, 258, 259, 260, 261, 262, 263, 265, 266, 267, 268, 269, 270, 271, 272, 273, 274, 275

359

Index education reform, 108 educational attainment, 259 educational institutions, 214 educational process, 158 educational programs, ix, 177, 178, 180, 208, 209 educational quality, 188, 262 educational research, 113, 225 educational settings, 225 educational system, 92, 249, 268 educators, ix, 94, 96, 177, 179, 187, 207, 208, 225, 267 Egypt, 253 elasticity, 168 electric field, 8, 9, 13, 14, 15, 16, 18, 19, 20, 22, 24, 25, 28, 29, 36, 38, 39, 45, 46, 78, 79 electromagnetic, 5, 21, 22, 26, 27, 41, 42, 43, 78 electromagnetic fields, 42 electromagnetic wave, 21, 27, 41, 43, 78 electron, 7, 14, 15, 21, 26, 29, 37, 40, 43, 47, 48, 54, 116, 117, 120, 121, 311 electron beam, 311 electronic systems, 117 electrons, viii, 14, 15, 20, 21, 26, 29, 37, 40, 41, 43, 115, 116, 118, 120, 121, 122, 134, 137 Elementary and Secondary Education Act, 275 elementary school, 262, 263, 271 emotion, 95, 96, 101, 114 emotional, 96, 103, 107, 114 emotionality, 111 emotions, 95, 107, 164, 170 employees, ix, 177, 179, 180, 181, 190, 191, 197, 198, 199, 205, 206, 208, 209, 210, 211, 235, 272 employers, 184, 215, 223, 224 employment, 179, 181, 184, 190, 199, 205, 211, 223 empowered, 91, 158 encouragement, 202 enculturation, 94 endocrine, 221 energy, viii, 6, 12, 47, 52, 54, 67, 70, 75, 77, 115, 120, 124 energy transfer, 70 engagement, 101, 102, 103, 107, 109, 112, 224 English as a second language, 200 enrollment, 179, 181, 183, 191, 192, 208, 211, 214, 223, 259, 266 enterprise, 93, 226 enthusiasm, viii, 102, 110, 145, 158, 200 environment, viii, 156, 162, 164, 168, 170, 214, 222 environmental conditions, 147 Environmental Protection Agency, 178, 181, 185, 206, 217, 262 epistemological, 92 epistemology, vii, 89, 90, 96, 103 equality, 157, 298, 299, 301, 302 equilibrium, 6, 12, 32, 33, 48, 50, 62 equipment, 94, 156, 213, 272 ergonomics, 159 ESL, 255 ESR, 124

estimating, 169, 174 Estonia, 253 ethnic groups, 182, 189, 198, 222 ethnicity, 184, 190, 198, 214, 235, 266 Euclidean space, 298 Euler equations, 29, 30, 55 Eulerian, 3, 6, 12, 15, 29, 56, 57, 61, 67, 78, 79, 86 Europe, 261 evolution, 6, 7, 10, 11, 12, 13, 30, 50, 54, 56, 62, 73, 74, 75, 77, 78, 347 execution, 146 exercise, 157, 162 expenditures, 174 expertise, 106, 108, 109, 226, 264 exposure, 110, 201, 223 extrusion, 143, 144

F failure, 146, 147, 150 family, x, 211, 277, 279, 280, 281, 282, 290, 308, 326, 327, 331, 337, 338 FCL, 94 fear, 113 February, 114, 184, 203, 209, 248, 270 federal funds, 205, 272 federal government, viii, 177, 178, 182, 205, 275 feedback, viii, 99, 145, 155, 156, 158 feelings, 95, 105 fees, 269 fermions, viii, 115, 117, 118, 121 filament, 53, 78 finance, 161, 164 financial resources, 206 financial support, 187, 205, 262, 273 Finland, 140, 254 firms, 161, 162, 163, 164, 165, 166, 169, 170, 174 first-time, 193 fixation, 147 flexibility, 147, 149, 158, 159, 160 floating, 14, 15, 20, 37 flow, 6, 7, 30, 33, 47, 56, 61, 67, 101, 112, 292 fluid, vii, 1, 5, 55, 67, 78, 79 focus group, 99, 101, 102, 214 focus groups, 214 focusing, 90, 110, 147, 215, 224 food, 183, 212, 213, 221 foreign language, 93, 270 foreign nation, 222 foreign nationals, 222 Forestry, 183, 213 forgiveness, 271 Fourier, 10, 11, 12, 13, 31, 50, 55, 68, 300, 301, 302, 307, 349 Fourier analysis, 301 fractals, 278, 292 framing, 94, 102, 162, 176 France, 84, 254, 259, 261, 348

360

Index

fullerene, 124 function values, 84 funding, 185, 186, 187, 189, 205, 209, 216, 217, 218, 219, 220, 221, 226, 245, 273, 274, 275 funds, ix, 185, 186, 205, 222, 226, 247, 248, 261, 263, 266, 267, 271, 272, 273

G games, 175 gauge, 41 gender, 194, 214, 226, 227 gender gap, 226 gene, 278, 296, 343, 346 generalization, 91, 124, 301 generalizations, 278, 296, 343, 346 generation, 117, 119, 121 generators, 119, 122 geography, 244 geology, 91, 200 Georgia, 179, 213, 214, 215 Germany, 165, 254 girls, 152, 202, 226 globalization, 165 goals, 113, 180, 182, 185, 187, 189, 202, 209, 224, 225, 226, 245, 248, 262, 265, 266, 275 gold, 116 government, viii, 177, 178, 180, 182, 189, 205, 209, 224, 244, 274, 275 Government Accountability Office (GAO), ix, 177, 178, 180, 182, 184, 185, 186, 187, 188, 190, 191, 192, 193, 194, 195, 196, 198, 200, 202, 204, 206, 207, 208, 210, 212, 213, 214, 215, 216, 218, 220, 221, 222, 224, 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 238, 240, 242, 244, 245, 246, 247, 248, 249, 261, 262, 263 grades, 199, 200, 204, 215, 250, 251, 255 graduate education, 205, 208 graduate students, 187, 190, 215, 221, 222, 223, 224, 244, 262, 263, 264, 266, 267, 273, 274 grants, 221, 222, 263, 264, 265, 266, 267, 269, 270, 271, 272, 273, 274, 275 graph, 296, 302, 323 gravitational field, 57 gravity, 55, 56 Greece, 254 grounding, 104 groups, 98, 100, 117, 122, 123, 124, 135, 137, 152, 180, 181, 182, 185, 187, 188, 189, 198, 205, 207, 212, 222, 224, 225, 226, 249, 262 growth, 12, 33, 34, 50, 54, 70, 111, 114, 164, 175, 187, 193, 244, 248, 262, 325, 329, 333, 338 growth rate, 33 guidance, 152, 202, 266 guidelines, 95, 213, 236

H H1, 295, 319, 350 H2, 318, 350 Hamiltonian, 7, 22, 41, 48, 116, 134 hanging, 112 hardships, 203 harm, 96 harmonics, 318 harmony, 96 Harvard, 114, 175, 176 Hawaii, 216 health, 106, 182, 212, 221, 222, 224, 255 Health and Human Services (HHS), 178, 181, 185, 206, 207, 218, 239 health education, 255 health effects, 221 health services, 222 heart, 96, 103 heat, 47 hedonic, 163 height, 56, 57, 61, 62, 65 Heisenberg, 311 Heisenberg Uncertainty Principle, 311 Helmholtz equation, 55, 56 high school, 100, 181, 199, 200, 201, 204, 250, 255, 256, 262, 265, 267, 268, 269 high tech, 162, 163, 167, 174 high temperature, 47 higher education, 103, 178, 186, 214, 221, 222, 224, 226, 267, 271, 273 Higher Education Act (HEA), 270 Hilbert, 347, 348 Hilbert space, 347, 348 hips, 272 Hispanic, 180, 184, 189, 192, 197, 198, 212, 216, 217, 222, 235, 245 holistic, 147, 175 Homeland Security, 178, 181, 185, 203, 206, 209, 219, 223, 244, 245, 246 homeomorphic, 281 homogeneity, 92, 163, 165, 174 Hong Kong, 145, 146, 152, 253, 254 House, 111, 113, 270 household, 163, 170, 176, 212 households, 211, 259 human, 94, 97, 98, 159, 189, 221, 222, 224, 225 human capital, 224 human subjects, 222 Hungary, 253, 254 hydrodynamics, 29, 30, 55 hyperbolic, vii, 1, 3, 78 hypermedia, 160 hypertext, 160 hypothesis, 298, 339 hysteresis, 147

Index

I ice, 178, 180, 182, 184, 186, 188, 190, 192, 194, 196, 198, 200, 202, 204, 206, 208, 210, 212, 214, 216, 218, 220, 222, 224, 226, 228, 230, 232, 234, 236, 238, 240, 242, 244 icosahedral, 124, 136 id, 80, 179, 202, 204 identity, viii, 89, 95, 105, 107, 108, 110, 111, 112, 114, 116, 304 IEA, 251 Illinois, 179, 213 images, 293 immersion, 269 immigrants, 245 immigration, 268 Immigration and Customs Enforcement, 183 implementation, 157, 178, 222, 225, 226, 265, 270 in situ, 147 incentive, 166, 201 incentives, 205, 267 inclusion, 148, 149, 150, 151, 156, 158, 299, 304, 318, 320, 321, 322, 324, 325, 326, 327, 328, 329, 330, 331, 332, 333, 335, 336, 337, 338, 340, 341, 342, 343, 344, 345, 346 income, 162, 163, 165, 167, 168, 174, 211, 271, 272 income tax, 211 incomes, 174 incompressible, vii, 1, 5, 29, 30, 55, 67, 79 increasing returns, 163 indexing, 213 India, 161, 202, 259, 261 Indian, 185, 209, 218 Indian Health Service, 209, 218 Indiana, 179, 213 Indians, 218 indicators, 175 indices, 14, 21, 124 individual perception, 164 individual students, 155, 156, 222 Indonesia, 253, 254 induction, 97, 224, 272 industry, 189, 211, 225, 258 inequality, 324, 340, 341, 342 inertia, 47 infinite, 116, 307 Information System, 246 Information Technology (IT), 198, 220, 225, 258 infrastructure, 188, 262, 265 inherited, 281 innovation, viii, ix, 108, 109, 165, 166, 175, 177, 178, 247, 248, 249, 268, 270 insight, vii, 89, 90, 91, 96, 99, 109, 162 inspectors, 212 inspiration, 103 instabilities, 6, 29, 70 instability, 6, 8, 12, 33, 56, 62, 70

361

institutions, 178, 179, 180, 186, 187, 205, 210, 211, 214, 221, 222, 223, 224, 225, 226, 244, 256, 262, 263, 264, 266, 267, 269, 273, 274, 275 institutions of higher education, 186, 221, 222, 267, 272, 273, 275 instruction, 92, 93, 94, 95, 102, 113, 149, 155, 159, 160, 200, 204, 224, 225, 265, 267, 269, 272, 274 instruments, 153 insurance, 212 intangible, 98, 163, 165, 166 Integrated Postsecondary Education Data, 178, 179 integration, 6, 9, 29, 48, 50, 55, 61, 65, 94, 111, 142, 155, 157, 158, 159, 161, 164, 202, 270 intellect, 98, 107 intelligence, 96 intensity, 25, 141, 143, 144, 173 intentions, 165 interaction, 6, 21, 29, 40, 47, 78, 79, 97, 100, 124, 156, 202, 224 interactions, 5, 98, 108, 155, 156, 157, 159, 224, 266 Interactive Perimeter Learning Tool, viii, 145, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159 interdependence, viii, 162 interdisciplinary, 205, 224, 266 interface, 106, 150, 151 International Baccalaureate, 268, 272 international students, 178, 180, 181, 182, 190, 192, 193, 194, 202, 203, 205, 206, 207, 208, 213 internship, 223, 265, 270 interpersonal communication, 169 interpretation, 95 interstitial, 4, 32, 84 interval, 198, 211, 212, 231, 235, 304, 305, 307, 311, 312, 313, 321, 323, 324, 325, 331 interview, 99, 105, 153 interviews, 98, 99, 101, 153, 155, 156, 158, 201, 207, 214, 215 intrinsic, viii, 162, 224 intuition, 96 invariants, 48, 55, 57, 60 inversion, 80, 126 investment, 172, 173, 223 ions, vii, viii, 7, 14, 15, 21, 26, 29, 38, 40, 41, 43, 67, 78, 79, 89, 101, 114, 181, 206, 207, 211, 225, 338 IPEDS, 178, 179, 182, 183, 190, 193, 195, 196, 210, 211, 230 Iran, 253 Ireland, 254 Islamic, 253 island, 50 isolation, 202 Israel, 253, 259, 350 Italy, 253, 254 iteration, 31, 32, 55, 56, 59 iterative solution, 37

362

Index

J January, 112, 113, 114, 189, 243, 245, 246, 248, 261 Japan, 85, 253, 254, 261 jobs, 202, 267 Jordan, 253 Josephson junction, 348 judge, 147, 164, 170 judgment, 164, 170 junior high, 188 junior high school, 188 justification, 151, 153

K K-12, 209, 220, 222, 223, 224, 262, 263, 264, 265, 266, 269, 272 Kelvin-Helmholtz instability, 33 kindergarten, 181, 182, 187, 199, 200, 204, 215, 222, 224, 225 kinetic energy, 54, 67 kinetic equations, 78 Korea, 253, 254

L L1, 321, 325, 329, 330, 332, 335, 338 labor, 178, 182, 197, 259 labor force, 178, 182, 197, 259 lack of confidence, 107 Lagrangian, 48, 55, 56 language, vii, 89, 90, 93, 98, 161, 182, 244, 255, 257 laser, 25, 26, 27, 43, 311 Latino, 184, 198, 212, 245 lattice, 295 Latvia, 253, 254 law, 27, 44, 184, 201, 256, 269 laws, 97 lead, 2, 33, 105, 166, 171, 204, 221, 226, 261, 272, 305 leadership, 94, 113, 189, 205, 265, 272 learners, viii, 90, 93, 104, 145, 150 learning, vii, viii, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 101, 103, 104, 105, 106, 107, 108, 109, 110, 111, 113, 114, 145, 146, 147, 149, 150, 151, 152, 153, 155, 156, 157, 158, 159, 160, 163, 164, 166, 170, 180, 221, 224, 225, 226, 265, 269, 270 learning culture, 265 learning environment, 149, 158 learning process, 158, 159, 164 learning task, 152 Lebanon, 253 legislation, ix, 247, 249, 268, 270 legislative, ix, 248, 249, 269 legislative proposals, ix, 248, 249 leisure, 168

lens, 95, 108 lesson plan, 104 life cycle, 165, 166 life span, 162 lifelong learning, 91 lifestyle, 91 lifetime, 94, 162, 165, 175 Likert scale, 153 limitation, 147, 215 limitations, 107, 214 Lincoln, 98 linear, 8, 21, 27, 32, 40, 44, 50, 56, 58, 78, 81, 94, 145, 148, 159, 295, 299, 313, 314, 315, 321, 322, 324 linear function, 314 linkage, 98 links, 223 literacy, ix, 106, 247, 248, 252 Lithuania, 253 loans, 271 localization, 325, 329, 334 location, 158, 273 logistics, 156 London, 84, 111, 112, 114, 161 long-term, viii, 162, 225 Los Angeles, 179, 213, 215 losses, 163 love, 101 low-income, 204, 221, 270 loyalty, 174 Luxembourg, 254

M Macao, 254 Macedonia, 253 macromolecular chains, 142 macromolecules, 141 magnetic, 6, 14, 15, 19, 20, 22, 29, 30, 36, 37, 38, 47, 48, 50, 54, 55, 67, 68, 70, 73, 74, 75, 77, 78, 79 magnetic field, 6, 14, 19, 22, 29, 30, 36, 37, 38, 47, 67, 70 magnetosphere, 47 Malaysia, 253 management, 183, 224 manifold, 176 manifolds, 176 manufacturing, 161, 164 mapping, 163, 212, 213, 282, 283, 298, 299, 312, 322, 323, 325, 328, 330, 332, 334, 335, 339 market, viii, 162, 163, 164, 165, 166, 169, 171, 172, 173, 174, 175, 176, 203 market penetration, 172 market segment, 169, 171 market value, 163, 165 marketing, 162, 164, 165, 166, 170, 171 markets, 165, 172, 176

363

Index Markov, 351 Maryland, 265 masculine image, 94 Massachusetts, 292 Massera type, 316 mastery, 250 mathematicians, 95, 209, 214, 267 mathematics, vii, viii, ix, 89, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 109, 110, 111, 112, 113, 114, 145, 146, 149, 151, 152, 158, 177, 178, 181, 182, 183, 190, 193, 194, 197, 199, 200, 201, 202, 204, 205, 208, 214, 216, 221, 223, 224, 225, 226, 235, 244, 247, 248, 258, 264, 265, 266, 267, 269, 270, 271, 272, 273, 274, 275 mathematics education, 91, 94, 112, 113, 248, 271, 272 Matrices, 139, 140 matrix, 9, 23, 31, 80, 82, 175, 314, 316, 317, 321, 325, 326, 330, 331, 336, 338, 343, 345 measurement, 146, 159 measures, 153, 154, 155, 163, 165, 191, 193, 203, 212, 248, 259, 292 media, 109, 222 median, 199, 212 medicine, 222 Mediterranean, 112 melt, 143, 144 melts, viii, 141, 144 memory, 65 men, 191, 194, 195, 197, 202, 212, 215, 229, 230, 236 mental health, 222 mentor, 202, 222, 266 mentoring, 181, 199, 202, 205, 223, 224, 226, 265, 272 merchandise, 164, 170 Merck, 269 merit-based, 269, 271, 272, 275 metaphors, 103 methodological procedures, 97 metric, ix, x, 174, 277, 278, 279, 280, 281, 282, 283, 284, 285, 286, 287, 288, 289, 290, 291, 292, 293, 294, 296, 297, 299, 302, 308, 309, 310, 312 metric spaces, 278, 280, 286, 287, 292, 293, 294, 296 Mexico, 161, 171, 254, 261 Mexico City, 161 MHD, vii, 1, 47, 67, 68, 75, 79 Middle East, 261 middle schools, 267, 270 mining, 212 Ministry of Education, 93, 277 minorities, 178, 180, 181, 182, 190, 197, 198, 199, 202, 204, 205, 208, 222, 223, 224, 226, 235, 245, 266 minority, 179, 180, 188, 190, 192, 194, 205, 212, 222, 223, 225, 226, 266 minority groups, 212, 225

minority students, 180, 190, 192, 222, 226 mirror, 91, 97 misleading, 206, 207 MIT, 292 ML, 121 modeling, 55, 56, 265 models, 47, 78, 93, 114, 163, 165, 170, 174, 202, 224, 226, 278, 281, 292, 293, 295, 346 Moldova, 253 molecular mass, 141, 142, 144 molecular weight, 142 molecules, viii, 115, 116, 117, 123, 124, 137 momentum, 21, 22, 40, 41, 48, 121, 137 monodromy, 315 monograph, 301 monoids, 286, 288, 294 monotone, 348 Montenegro, 254 Morocco, 253 Moscow, 144, 350, 351, 352 motion, 7, 17, 30, 55, 311 motivation, 92 motives, 164, 170, 175 motors, 351 mouth, 169 movement, 92, 94, 146, 147, 148, 152, 156 MSC, 277 MSP, 224, 265, 267 multidimensional, 55 multidisciplinary, 224, 226, 274 multiplication, 116, 167, 172, 290 multiplicity, 127, 317, 331, 332 multiplier, 315, 316, 329, 330, 331, 333, 335, 336, 337, 338 music, 255 myopic, 166

N nation, ix, 177, 178, 182, 190, 205, 209, 211, 221, 224, 225, 247, 248, 249, 255, 262 national, 180, 182, 190, 203, 205, 221, 224, 225, 255, 259, 269, 270, 272, 273, 274, 275 National Academy of Sciences (NAS), 248, 249, 268, 269, 274, 275 National Aeronautics and Space Administration, 178, 185, 186, 207, 209, 219, 223, 224, 262, 266 National Assessment of Educational Progress, 201, 250 National Center for Education Statistics (NCES), 178, 179, 182, 183, 210, 211, 215, 243, 244, 245, 250, 251, 252, 253, 254, 255, 256, 257, 259 National Defense Authorization Act, 269 National Institute of Standards and Technology, 219 National Institutes of Health, ix, 178, 180, 185, 186, 209, 218, 219, 247, 248, 261, 262, 263

364

Index

National Postsecondary Student Aid Study (NPSAS), 178, 179, 182, 183, 190, 191, 192, 208, 210, 211, 227, 228, 229, 230, 231, 232, 233, 234, 235 National Science and Technology Council, 178, 182, 189, 249 National Science Foundation, ix, 139, 177, 178, 180, 181, 185, 218, 220, 224, 240, 243, 245, 247, 248, 259, 260, 261, 264, 269 national security, 203, 270 Native American, 189, 194, 222, 235, 265 Native Americans, 189, 222 Native Hawaiian, 216, 226 natural, ix, 97, 104, 116, 141, 247, 248, 278, 281, 283, 303, 306, 309, 310, 311, 313, 318, 346 natural science, ix, 97, 247, 248 natural sciences, 97 negativity, 163, 165 negotiation, vii, 89 Netherlands, 253, 254 network, 97, 163 New York, 84, 87, 112, 113, 114, 139, 140, 175, 243, 245, 292, 293, 348, 349, 350, 351 New Zealand, 202, 253, 254 Newton, 4, 311 Nielsen, 318, 331, 332, 347, 350 NMR, 123, 124, 129 No Child Left Behind, 178, 182, 250 noise, 6, 29, 45, 57, 79 non-citizens, 188 nonlinear, viii, 5, 6, 7, 12, 50, 56, 70, 162, 295, 317, 318, 319, 347, 348, 351 nonlinear dynamics, 70 non-linear equations, viii, 162 nonlinear systems, 319, 348 non-Newtonian, 144 normal, 14, 22, 42, 104, 116, 134, 144, 152, 305, 306, 309, 310 normalization, 36, 40 normed linear space, 293 norms, 303, 307 Norway, 253, 254, 259 novelty, 166 nuclear, viii, 115, 117, 118, 121, 123, 124, 127, 128, 129, 132, 133, 184 nuclei, 115, 123, 128, 129 nucleus, 118, 124 nurses, 212, 222 nursing, 222

O obedience, 103 observations, 34, 56, 99, 200, 206, 207, 267, 282 occupational, 212 OECD, 252, 254, 259 Office of Management and Budget, 189 Office of Science and Technology Policy, 189, 249, 275

oligopoly, 166 Oncology, 219 one dimension, 3, 6, 21, 32, 78 online, 271 operator, 6, 116, 167, 172, 305, 322 ophthalmic, 183, 212 optical, 347 optical systems, 347 optimization, viii, 162, 167, 168, 174, 176 oral, 266 oral presentations, 266 orbit, 123, 124, 311 ordinary differential equations, 347 organ, 113 organic, 169 organization, 94, 166, 183, 203, 204 organizations, 164, 180, 203, 204, 205, 207, 209, 211, 215, 222, 225, 248, 263, 268, 271, 272, 274 orientation, 118, 174 oscillation, 34, 311 oscillations, x, 18, 39, 55, 295, 296, 319, 346, 347, 348, 349, 350 outpatient, 222 outreach programs, 225, 226, 269

P Pacific, 189, 192, 198, 220, 222, 235, 245, 349, 350 Pacific Islander, 192, 198, 235, 245 Pacific Islanders, 198, 245 paper, viii, 61, 107, 111, 112, 113, 142, 153, 154, 161, 162, 163, 165, 166, 174, 175, 246, 278, 304 parabolic, 1 parallel computers, 6, 78 parameter, 8, 57, 169 parents, 202 Paris, 259, 349 Parliament, 112 partial differential equations, vii, 1 particles, viii, 7, 9, 17, 41, 47, 115, 117, 118, 119, 120, 121, 129, 311 particulate matter, 221 partition, 118, 119, 126 partnership, 113, 265, 266, 267, 271, 272 partnerships, 214, 222, 223, 224, 225, 267, 272 pathophysiology, 222 pathways, 223, 224, 266 PDAs, 152, 155 pedagogical, vii, viii, 89, 90, 103, 104, 105, 106, 110, 111, 145, 146, 157, 158, 159, 226 pedagogies, 90, 95 pedagogy, vii, 89, 91, 92, 94, 95, 96, 97, 98, 101, 103, 108, 112, 114, 214 pediatric, 222 peer, 265 peers, 202 Pell Grants, 244, 270 pendulum, 351

Index Pennsylvania, 179, 213, 269 pensions, 212 perception, 153, 157, 164, 170 perceptions, 97, 111, 112, 161, 163, 165 performance, 5, 12, 13, 29, 36, 40, 56, 78, 146, 163, 164, 165, 167, 172, 203, 224, 226, 248, 250, 265, 267, 274, 275, 311 performers, 225 periodic, 6, 8, 9, 31, 32, 50, 55, 62, 68, 117, 120, 121, 122, 124, 129, 133, 136, 137, 295, 296, 298, 299, 300, 301, 303, 304, 307, 310, 311, 313, 314, 316, 318, 321, 325, 326, 329, 330, 339, 340, 344, 346, 348, 349, 350, 351, 352 Periodic Table, viii, 115, 117 periodicity, viii, 67, 69, 115, 116, 117, 118, 121, 122, 123, 124, 126, 129, 133, 134, 135, 136, 137, 295, 304, 349, 351 permanent resident, 188, 208, 225, 245, 263, 264, 266 personal, 90, 94, 95, 96, 99, 104, 105, 108, 109, 152 personality, 96, 166 personality traits, 166 persons with disabilities, 189, 224 persuasion, 172 perturbation, 12, 33, 50, 54 perturbations, 343 Petroleum, 212, 213 phase space, 6, 12 Philippines, 253 philosophical, 97 philosophy, 104 physical sciences, 256 physicists, 183, 212 physics, vii, 1, 5, 6, 13, 47, 79, 91, 92, 100, 104, 105, 106, 109, 200, 201, 204, 311 physiology, 222 pilot programs, 274 PISA, 248, 252, 254 planar, 317, 318, 332 planning, 93, 104 plasma, vii, 1, 5, 6, 7, 12, 13, 14, 18, 19, 20, 21, 24, 25, 26, 27, 29, 30, 36, 37, 38, 39, 43, 45, 47, 48, 54, 67, 78, 79 plasma physics, 1, 5, 79 play, 48, 67, 96, 117, 161, 165, 175, 189, 202 pleasure, 96 Poisson, 6, 7, 8, 9, 15, 16, 22, 30, 31, 36, 42, 48 Poisson equation, 7, 8, 9, 15, 16, 22, 30, 31, 36, 42 Poland, 254 polarization, 21, 40, 44, 47, 78 policy makers, 206 pollution, 221 polygons, 147 polymer, 141, 142, 143, 144 polymer destruction, 141 polymer melts, 141 polynomial, 60, 61, 81, 82, 83, 117, 123, 124, 306, 308 polynomials, 61, 119, 124, 306, 308, 310

365

polypropylene, viii, 141, 143, 144 poor, 93, 109, 201, 248 population, 182, 184, 202, 209, 211, 212, 226, 231, 235, 259, 269 population group, 226 portfolios, 176 Portugal, 254 positivism, 97 positivist, 97 postsecondary education, 210, 211, 221, 248, 259 poststructuralism, 114 power, 97, 121, 125, 162, 163 powers, 91, 117, 125 Prandtl, 68 prediction, 55 preference, 151, 153, 157, 158, 163, 169, 274 pressure, 14, 18, 38, 39, 68 prevention, 221 price competition, 163 prices, 165, 167, 168 pricing policies, 163 primary care, 222 primary school, 152 primitives, 313 prior knowledge, 149, 151 priorities, 245 private, 99, 107, 179, 214, 271, 274 proactive, 203 probability, 166, 168, 211, 212, 230, 235 probe, 124 problem solving, 94 problem-solving, 146, 147, 149, 267 problem-solving strategies, 149 procedural knowledge, 146, 148, 150, 151 product attributes, 163, 164, 165, 170 product design, 161, 164 product market, 173 product performance, 167 production, 226, 259 productivity, 225 professional careers, 223 professional development, 100, 107, 110, 114, 224, 225, 265, 267, 272 profit, x, 107, 163, 165, 174, 175, 296 profit margin, 163 profitability, 164, 175 program, 180, 181, 184, 186, 187, 188, 189, 198, 203, 205, 206, 207, 208, 209, 211, 221, 222, 223, 224, 225, 226, 244, 245, 259, 261, 262, 263, 264, 266, 267, 269, 270, 271, 272, 273, 274 programming, 198, 278, 292 programming languages, 292 promote, 159, 181, 187, 205, 224, 244, 262, 267, 270 promote innovation, 270 propagation, 2, 13 property, 7, 304, 307, 341, 342 proportionality, 143 proposition, 297, 298, 322 prosperity, 224

366

Index

protons, 115, 129 pruning, 172 pseudo, 307, 308 psychology, 95, 112, 176, 191 public, 107, 179, 214, 215, 222, 224, 250, 253, 255, 271, 275 public health, 222 public interest, 224 public schools, 215 publishers, 92 pupils, ix, 90, 102, 247, 248, 249, 252

Q qualifications, 109, 215 qualitative differences, 214 qualitative research, 97, 98 quality control, 215 quality of life, 224 quantum, 116, 117, 120, 124, 127, 129, 311, 347 quantum mechanics, 116 quasi-linear, 343 quasiparticles, 12 quasi-periodic, 296, 301, 303, 304, 305, 307, 314, 317, 346, 349, 352 questionnaire, 153, 157, 209, 210, 215 questionnaires, 161, 209, 210

R race, 190, 191, 211, 214 radius, 37, 48, 62 random, 160, 163, 203, 212, 214, 235, 346 random access, 160 range, 62, 149, 158, 187, 206, 207, 225, 264, 266, 310 Rayleigh, 33 reading, 182, 225, 244 real numbers, 278, 305 realism, 98 reality, 97, 98 reasoning, 159 recall, 99, 149, 278, 280, 281, 282, 283, 288, 296, 305, 315 recalling, 280, 338 recognition, 97 recruiting, 222, 226, 268 recursion, 83 reduction, 3, 147 referees, 161 reflection, 99, 104 reflexivity, 278 reforms, 270 regional, 56, 183, 212, 269, 273 registered nurses, 212, 222 regular, viii, 124, 145, 146, 147, 148, 150, 151, 156, 158, 162, 311, 314, 321, 325

reinforcement, 146, 166, 175 reinforcement learning, 166, 175 rejection, 97, 202 relationship, viii, 90, 91, 94, 96, 101, 108, 115, 171, 174, 183, 280, 283, 304 relationships, vii, viii, 89, 93, 103, 115 relativity, 116 reliability, 179, 210 repetitions, 295 representative samples, 215 research, vii, viii, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 100, 103, 104, 107, 110, 111, 112, 113, 114, 139, 143, 145, 149, 151, 158, 159, 160, 165, 174, 180, 181, 182, 183, 186, 187, 188, 189, 190, 204, 205, 207, 212, 221, 222, 223, 224, 225, 226, 244, 249, 255, 262, 263, 265, 266, 268, 269, 273, 274, 275, 346 research and development, 175, 182, 224, 274 researchers, 95, 113, 143, 190, 199, 200, 201, 212, 213, 214, 222, 244, 264, 274 resolution, 6, 7, 56 resources, 104, 105, 107, 149, 158, 162, 181, 189, 190, 204, 205, 206, 209, 222, 224, 225 response time, 165 responsibilities, 272 retail, viii, 162, 163, 164, 166, 168, 170, 171, 176 retention, 163, 165, 202, 214, 221, 222, 224, 226, 265 returns, 164, 211 revenue, 175 Reynolds, 67 Reynolds number, 67 Rhode Island, 350 risk, 176, 221 risk assessment, 221 robotics, 184 robustness, 174 Romania, 253 rotations, 116, 347 routines, 92 rural, 269, 271 rural areas, 269, 271 Russia, 261 Russian, 253, 254, 259, 294, 349, 350, 351, 352

S safety, 212 salaries, 179, 190, 199, 212, 235 salary, 212 sales, 92, 164, 165, 168, 169, 171 sample, 153, 154, 179, 203, 208, 210, 211, 212, 215, 230, 231, 235, 255 sample design, 211, 215 sampling, 179, 209, 213, 235 sampling error, 179, 209, 213 SAR, 253, 254

Index satisfaction, 162, 163, 164, 165, 166, 167, 170, 171, 173, 175, 202 saturation, 12, 34, 54 Saturday, 218 Saudi Arabia, 253 scalar, 317, 346 scattering, 47 scholarship, 103, 113, 225, 269, 273 Scholarship Program, 216, 217, 218, 220, 273, 274 scholarships, 186, 188, 205, 208, 222, 223, 225, 226, 267, 269, 271, 272, 273 school, vii, ix, 89, 90, 91, 92, 93, 95, 98, 99, 100, 103, 108, 109, 111, 112, 113, 114, 152, 158, 181, 182, 184, 188, 190, 199, 200, 201, 203, 204, 205, 215, 221, 222, 223, 224, 225, 244, 247, 248, 250, 255, 256, 262, 263, 265, 266, 267, 268, 269, 270, 271, 272, 273 Schools and Staffing Survey (SASS), 215, 255 science education, 91, 92, 98, 103, 108, 109, 111, 112, 113, 114, 190, 204, 205, 264, 270 science educators, 222 science literacy, ix, 247, 248, 252 science teaching, 95, 107, 113, 272 scientific community, 93 scientific knowledge, 93 scientific method, 92 scientific theory, 91, 97 scientists, 91, 92, 95, 96, 183, 188, 189, 206, 207, 208, 209, 212, 213, 214, 221, 222, 223, 267, 270, 274 scores, 251, 252 search, 171 second language, 200 secondary school students, ix, 247, 270 secondary schools, vii, 89, 103, 112, 113, 114, 224, 269, 270, 271 secondary students, 250, 262 secondary teachers, 187, 200 security, 178, 203, 223, 224, 226, 270 selecting, 108, 143, 181, 225 self-concept, 95 self-efficacy, 110 semantics, 278, 279, 292 semigroup, 288 Senate, 270 sensitivity, 162, 163 separation, 13, 15, 20, 78 September 11, 178, 202, 203 Serbia, 253, 254 series, 92, 211, 296, 300, 301, 302, 307 service quality, 164, 170 services, viii, 161, 162, 163, 164, 170, 171, 174, 175, 176, 222, 226 sex, 211 shape, viii, 94, 105, 118, 119, 120, 145, 146, 147, 148, 150, 151, 152, 154, 155, 156, 158 shaping, 95 shareholder value, 165 shares, 259

367

sharing, 94, 97, 102, 203 shear, 19, 33, 38 Shell, 121 shocks, 2, 173 short run, 166 shortage, 109 short-term, 222, 263 sign, 40, 116, 135, 338 signs, 25 simulation, 24, 29, 71 Singapore, 253 SIS, 109 skills, 93, 94, 109, 110, 146, 147, 200, 221, 224, 249, 250, 266, 267, 268, 269, 272 skin, 48 Slovenia, 253 smoothing, 70, 162 social construct, 111 social phenomena, 97 social sciences, 191 socioeconomic, 214 sociology, 92, 111 software, 149, 183, 212 solar, 47 solar wind, 47 solutions, x, 33, 49, 93, 161, 164, 175, 295, 296, 301, 304, 305, 309, 314, 315, 316, 317, 318, 319, 323, 324, 325, 327, 328, 329, 332, 334, 336, 340, 341, 342, 343, 346, 347, 348, 349, 350, 351, 352 South Africa, 253 South America, 261 South Dakota, 265 South Korea, 261 Soviet Union, 205 space exploration, 224 Spain, 254, 261, 277 spatial, 2, 6, 9, 40, 41, 50, 147, 164 special education, 274 special theory of relativity, 115 specialisation, vii, 89, 90 specialization, 256 species, 48, 117, 124, 128, 129 spectroscopy, viii, 115, 116, 117, 121, 124 spectrum, vii, 25, 26, 27, 43, 44, 45, 125, 303, 307, 314, 322 speed, 2, 116, 166 speed of light, 116 spin, viii, 115, 117, 118, 119, 120, 121, 122, 123, 124, 125, 127, 128, 129, 132, 133, 134, 136, 137, 138 spin-1, 118, 119 sponsor, 222 Sputnik, 248, 268 SRS, 26, 43 stability, 5, 29, 32, 56 stabilize, 174 stages, 146, 151, 275 stakeholder, 215 stakeholders, 225

368

Index

standard deviation, 153, 154, 155, 252 standards, 93, 103, 111, 113, 180, 182, 209, 215, 225, 267 State Department, 203 statistics, 100, 183, 255, 257 steady state, 18, 38 STEM, ix, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 221, 223, 224, 225, 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 244, 245, 247, 248, 249, 251, 253, 255, 256, 257, 258, 259, 261, 262, 263, 264, 265, 266, 267, 268, 269, 270, 271, 272, 273, 275 STEM fields, ix, 177, 178, 179, 180, 181, 182, 183, 184, 187, 188, 190, 191, 192, 193, 194, 195, 197, 198, 199, 200, 201, 202, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 227, 233, 236, 244, 248, 256, 257, 258, 259, 261, 262, 268, 273 STM, 243, 244 strain, 141, 143 strains, 143 strategic, 163, 173, 175, 190 strategies, 105, 107, 111, 147, 149, 150, 159, 161, 162, 164, 169, 170, 171, 173, 174, 175, 182, 189, 224, 225, 226, 265 stress, 29, 38, 42, 53 student achievement, 112, 224, 256, 264, 265 Student and Exchange Visitor Information System, 178, 182, 184, 203, 243 student enrollment, 179, 193, 205, 210 student populations, 224, 225 student retention, 214 student teacher, 265 students, viii, ix, 90, 91, 92, 93, 94, 95, 96, 99, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 177, 178, 179, 180, 181, 182, 184, 187, 188, 190, 191, 192, 193, 194, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 213, 214, 215, 221, 222, 223, 224, 225, 226, 227, 229, 230, 235, 244, 247, 248, 249, 250, 251, 252, 256, 258, 261, 262, 263, 264, 265, 266, 267, 268, 269, 270, 271, 273, 274 subjective, 96, 97, 98, 169 substances, 144, 295 substitution, 80, 125, 142 subtraction, 160 suffering, 102 summer, 223, 224, 266, 267, 269, 270 superposition, 314 supervision, 222 supplemental, 225 supply, 161, 163, 164, 214, 248, 249, 268 supporting institutions, 224 surgery, 56 Sweden, 253, 254 switching, 122

Switzerland, 254 symbols, 98, 117, 121, 124, 279 symmetry, viii, 52, 53, 96, 115, 116, 122, 123, 124, 133, 135, 137, 163, 165, 295, 328 symplectic, 13 synchronous, 351 syndrome, 162 systems, 6, 7, 12, 21, 39, 52, 98, 113, 120, 124, 134, 137, 161, 164, 178, 182, 183, 198, 212, 213, 225, 292, 299, 318, 346, 347, 348, 349, 350, 351

T talent, 270 tangible, 163, 165 target population, 209 targets, 268 taste, 202 Taylor expansion, 33, 42, 59, 79 teacher effectiveness, 204 teacher instruction, 94 teacher preparation, 271 teacher training, 272 teachers, vii, viii, ix, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 107, 108, 109, 110, 111, 112, 113, 114, 145, 151, 152, 153, 155, 156, 157, 158, 182, 188, 199, 200, 202, 204, 215, 222, 224, 225, 247, 248, 249, 255, 256, 263, 265, 266, 267, 268, 269, 270, 271, 272, 274 teaching, vii, viii, 89, 90, 91, 92, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 145, 146, 149, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 182, 201, 207, 221, 222, 224, 225, 226, 248, 255, 265, 266, 267, 270, 272 teaching process, 111, 153 teaching strategies, 225 technician, 224 technicians, 183, 212, 213, 224 technology, ix, 113, 159, 160, 163, 166, 167, 171, 174, 175, 177, 178, 182, 184, 191, 193, 197, 203, 205, 208, 216, 221, 225, 226, 235, 247, 248, 267, 270, 271, 272, 273 technology gap, 175 technology transfer, 184, 203, 226 temperature, 14, 15, 47, 143 temporal, 164 tertiary education, 259 testimony, 249 textbooks, 92, 93 Thailand, 254 theology, 257 theory, vii, viii, x, 1, 5, 92, 93, 94, 96, 97, 98, 115, 117, 122, 160, 162, 163, 164, 167, 173, 176, 278, 292, 293, 295, 296, 301, 303, 305, 315, 316, 318, 331, 347 therapeutic interventions, 222 thermo-mechanical, 142, 143, 144

369

Index thermonuclear, viii, 141, 144 thinking, 90, 92, 93, 104, 105, 111, 225 threat, 106 threatening, 106 TIE, 265 time, 2, 3, 4, 6, 7, 8, 9, 14, 16, 18, 21, 22, 23, 29, 36, 39, 42, 47, 48, 49, 50, 53, 54, 55, 56, 57, 58, 59, 61, 65, 68, 69, 73, 74, 75, 78, 79, 90, 94, 99, 101, 104, 107, 141, 150, 152, 158, 159, 162, 163, 164, 165, 167, 168, 169, 171, 172, 173, 175, 184, 189, 193, 201, 203, 204, 209, 210, 215, 223, 255, 264, 273, 295, 314, 315, 317, 319, 332, 340, 341, 346 tin, 173 tissue, 222 Title I-A, 250 Title IX of the Education Amendments of 1972, 245 tokamak, 13, 47 topological, ix, 47, 277, 278, 281, 286, 292, 293, 299, 317, 349 topological structures, 281 topology, 47, 67, 279, 281, 282, 291, 292, 293 total energy, 54, 74 tourism, 100 tracking, 92 trade, 94 tradition, vii, 89, 91, 93 traffic, 175 trainees, 222, 263, 264 training, viii, 89, 100, 102, 103, 109, 110, 149, 159, 180, 186, 200, 202, 204, 205, 222, 223, 224, 263, 264, 272 training programs, 180, 186, 222 trajectory, 9, 110 trans, 305, 306 transfer, 70, 146, 184, 226 transformation, viii, 89, 299, 305, 316, 318, 321 transformations, 142, 144, 321, 346 transition, 20, 167 transitions, 124 translation, 161, 305 transport, 13, 18, 38 transportation, 182, 226 travel, 203 trend, 137, 162, 252 trial, 162 tribal, 266 tribal colleges, 266 tuition, 201, 222, 269, 273 Tunisia, 253, 254 turbulence, 19, 38, 67 turbulent, 68 Turkey, 254 turnover, 270 tutoring, 79 two-dimensional, 5, 6, 30, 40, 47, 48, 57, 67, 68, 79, 136, 351 two-dimensional space, 5, 30

U U.S. Geological Survey, 219 uncertainty, 162 Uncertainty Principle, 311 undergraduate, 99, 191, 201, 205, 210, 215, 223, 224, 225, 226, 227, 230, 248, 255, 256, 266, 268, 272, 273, 274 undergraduate education, 224 undergraduates, 215, 273 unemployment, 211 UNESCO, 259 unification, viii, 89 uniform, 36, 62, 79, 303, 304, 305, 307, 317 United Kingdom, 253, 254, 261 United Nations, 259 United States, viii, ix, 177, 178, 180, 181, 182, 184, 186, 188, 190, 192, 193, 194, 196, 198, 200, 202, 203, 204, 205, 206, 208, 210, 212, 214, 216, 218, 220, 222, 224, 225, 226, 228, 230, 232, 234, 236, 238, 240, 242, 243, 244, 245, 246, 247, 248, 249, 252, 253, 254, 256, 258, 259, 261, 263, 266, 268 universe, 215 universities, 113, 179, 180, 181, 182, 184, 188, 200, 201, 202, 203, 204, 205, 207, 209, 213, 214, 215, 222, 223, 224, 226, 244, 266, 274 Uruguay, 254 USDA, 180

V vacuum, 23, 42 Valencia, 277 validation, 94 validity, 167 values, 3, 14, 23, 32, 33, 34, 48, 52, 58, 59, 60, 61, 62, 66, 69, 78, 82, 83, 84, 95, 96, 108, 111, 120, 124, 127, 136, 137, 138, 142, 143, 144, 162, 163, 164, 165, 166, 167, 170, 171, 172, 173, 174, 211, 212, 231, 235, 296, 297, 298, 299, 300, 301, 302, 304, 305, 307, 309, 310, 315, 316, 318, 319, 320, 322, 323, 325, 332, 336, 338, 339, 344, 345, 346 variability, viii, 162, 166, 168, 174, 210 variable, 2, 9, 32, 80, 143, 164 variables, 56, 61, 68, 69, 70, 147, 161, 163, 164, 165, 166, 168, 170, 171, 174, 175 variance, 213, 235 variation, 3, 149, 171, 307, 312 vector, 22, 40, 41, 43, 166, 167, 168, 298, 299, 301, 302, 303, 304, 346, 348 vegetables, 163 velocity, 2, 3, 7, 8, 9, 14, 15, 21, 22, 26, 30, 33, 34, 36, 39, 43, 47, 48, 56, 57, 58, 61, 68, 78, 311 veterinary medicine, 222 Victoria, 100, 103, 109, 112 visa, 181, 184, 198, 200, 202, 203, 206, 207, 246, 258

370

Index

visa system, 203 visas, 182, 184, 198, 203, 206, 207, 244, 245, 263 viscosity, 70, 141, 143, 144 vocabulary, 93 vocational, 244, 259 volatility, 163 vortex, 6, 8, 12, 34, 52, 68 vortices, 6, 12, 29, 45, 56, 62, 73

W wages, 179, 190, 199, 212, 235 water, vii, 1, 5, 55, 56, 57, 62, 78, 79, 221 water quality, 221 wave propagation, 2 weather prediction, 55 web, 152 web-based, 150 welfare, 224 well-being, 105 wind, 47 wine, 109 winemaking, 109

winning, 267 women, 178, 179, 180, 181, 182, 190, 191, 192, 194, 195, 197, 199, 202, 204, 205, 212, 214, 215, 224, 225, 226, 229, 230, 236, 245, 264, 266 workers, 184, 212, 213, 223, 225, 258, 268 workforce, 185, 188, 189, 190, 205, 208, 222, 223, 224, 225, 226, 258, 264, 270 working groups, 189 workstation, 65 World Bank, 259 writing, 60

Y yield, x, 5, 116, 133, 137, 296 yogurt, 162

Z zoology, 100

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