Proceedings of the Third Asian Mathematical Conference 2000 : University of the Philippines, Diliman, Philippines, 23-27 October 2000

THIRD ASIAN MATHEMATICAL CONFERENCE 2 0 0 0

This page is intentionally left blank

Proceedings of the

THIRD ASIAN MATHEMATICAL CONFERENCE 2 0 0 0 University of the Philippines Diliman, Philippines

23-27 October 2000

Editors

Toshikazu Sunada Tohoku University, Sendai, Japan

Polly W Sy University of the Philippines, Diliman, Philippines

Yang Lo The Chinese Academy of Sciences, Beijing, China

V|S% World Scientific wl

New Jersey'London 'Singapore* Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

PROCEEDINGS OF THE THIRD ASIAN MATHEMATICAL CONFERENCE 2000 Copyright © 2002 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-02-4947-0

Printed in Singapore by Fulsland Offset Printing

PREFACE The Third Asian Mathematical Conference (AMC2000) was held on 23-27 October 2000 at the University of the Philippines, Diliman Campus. This event was organized by the Southeast Asian Mathematical Society in collaboration with many mathematical societies in Asia. This Conference covered the main fields of mathematics; namely, algebra and number theory, analysis, geometry and topology, applied mathematics, and mathematics education. During the 5-day conference, there were 8 plenary lectures, 47 invited lectures and overlOO short communications. Most of the plenary and invited addresses were delivered by mathematicians from Asia. There were about 500 participants in the Conference. This proceedings volume is a collection of papers presented in the Conference. The AMC2000 has achieved its main purposes of providing a forum for Asian mathematicians to meet and discuss mathematical problems on a wide range of topics and of supporting the objectives of the World Mathematical Year 2000. Moreover, it has fostered friendship among citizens of the participating countries and contributed to the development of mathematics research in the host country as well as participating countries. The first Asian Mathematical Conference (AMC) was held in the University of Hong Kong in 1990. The Second was held in the Suranaree University of Technology, Thailand in 1995 and this Conference was the third in the series. Because of the success of these three AMC conferences, it has been decided that the AMC will be held once every four years. During the third AMC, Singapore has agreed to host the fourth AMC in the year 2004. The Southeast Asian Mathematical Society (SEAMS) is a society formed in 1972 whose members are mathematicians mostly from Southeast Asian countries including Brunei, Hong Kong, Indonesia, Malaysia, the Philippines, Singapore, Thailand and Vietnam. Its primary objectives are (1) To promote the advancement of mathematical sciences in Southeast Asia

v

VI

(2) To facilitate the exchange of information about the current research work and teaching methods not only among mathematicians in Southeast Asia but also with mathematicians from outside the region. (3) To encourage and foster friendly collaboration with the members of mathematical and scientific institutions. Since 1972, the promotion of mathematical research has been done through the sponsorship of mathematical conferences, symposia, meetings of mathematicians, lecture tours and other academic and social activities among its members. The facilitation of the exchange of research information has been realized through the publication of the Southeast Asian Bulletin of Mathematics starting in 1977. We believe that the SEAMS will continue to play its important role of promoting cross cultural and international links for mathematicians in the region as well as the whole of Asia. On behalf of the AMC2000 Steering Committee and AMC2000 Scientific Committee, we would like to express our gratitude and appreciation to all sponsors, donors and friends for their help and support for the Conference. We would like to extend our thanks to all plenary speakers, invited speakers and guests for having kindly accepted our invitation and to all participants for their enthusiastic response. Finally, we would like to acknowledge and express our thanks for the help and support of the faculty, staff, alumni and friends of the Department of Mathematics, University of the Philippines, Diliman Campus in the preparation for and during the Conference.

Toshikazu Sunada Polly W. Sy Yang Lo November 2001

ACKNOWLEDGMENTS The Conference Organizers would like to acknowledge and thank the support given by the following Sponsors and Donors: SPONSORS: International Mathematical Union Abdus Salam International Center for Theoretical Physics UP Diliman Mathematics Foundation, Inc. Ellen Francisco-Fajardo Foundation, Inc. Tan Yan Kee Foundation, Inc Philamlife Foundation, Inc. National Natural Science Foundation of China Chinese Mathematical Society Indonesian Mathematical Society Korean Mathematical Society Mathematical Society of Japan Mathematical Society of the Philippines Mathematical Society of Thailand Singapore Mathematical Society Vietnam Mathematical Society Southeast Asian Mathematical Society Ateneo de Manila University De La Salle University AMA Education System Philippine Social Security System Philippine Health Insurance Corporation Philippine Government Service Insurance System UCPB General Insurance Company, Inc. Aegon Life Insurance (Philippines), Inc. First Guarantee Life Assurance Company, Inc. The Insular Life Assurance Company, Limited CAP Family of Companies Nippon Life Philippines Actuarial Advisers, Inc. CMG Life Insurance Company, Inc. VII

VIII

Aloha Mental Arithmetic Philippines, Inc. Canon Information Technologies Philippines, Inc. First Metro Investment Corporation Medrano Law Office Nestle (Philippines) Pilipinas Shell Petroleum Corporation SoftTech Advantage, Inc. Q.C.D. Ventures Corporation LandBank of the Philippines Anchor Bank

Donors ( Friends and Alumni of the Department) Joaquin B. Bamba Annalizza A. Bernardo Rizaldy T. Capulong Eleonora Y. Cinco Melchor V. Ciubal Peter Godfrey C. Darvin Coralyn G. del Rosario Ronald L. Evangelista Arnold P. Frias Lih Ko-Wei Hiroko and Mitsuo Morimoto Allan R. Santos Judy Frances Aquino See Edgar B. Solilapsi Toshikazu Sunada Horacio T. Templo Kenji Ueno Anonymous Donors

Special Thanks to: HIS Sounds Philippines (Dr. Romulo G. Pizana, Musical Director/Conductor) The University of the Philippines Filipiniana Dance Group (Mr. Van Manalo, Artistic Director)

AMC2000 Committees The AMC2000 Scientific Committee : Yang Lo, The Chinese Academy of Sciences, Beijing (Chairman) Dinh Dung, Hanoi Institute of Mathematics, Hanoi Kim Do Han, Seoul National University, Seoul Lee Seng Luan, National University of Singapore Liu Fon-Che, Academia Sinica, Taipei Ma Zhi Ming, The Chinese Academy of Sciences, Beijing Masaki Maruyama, Kyoto University, Kyoto Milagros P. Navarro, University of the Philippines, Diliman M.S. Raghunathan, Tata Institute of Fundamental Research Vladimir Romanov, Sobolev Institute of Mathematics Novosibirsk, Koptyuga Aner Shalev, The Hebrew University of Jerusalem Toshikazu Sunada, Tohoku University, Sendai The AMC2000 Steering Committee: Polly W. Sy, University of the Philippines, Diliman (Chairperson) Do Long Van, Hanoi Institute of Mathematics, Hanoi Li Daqian, Fudan University, Shanghai Lih Ko-Wei, Academia Sinica, Taipei Ann Chi Kim, Dolsan Mathematical Institute, Pusan Koh KheeMeng, National University of Singapore Mitsuo Morimoto, International Christian University, Tokyo Bienvenido F. Nebres, Ateneo de Manila University Kazuo Okamoto, Tokyo University, Tokyo Shum Kar-Ping, The Chinese University of Hong Kong Kenji Ueno, Kyoto University, Kyoto The AMC2000 International Organizing Committee : Lorna I. Paredes, University of the Philippines, Diliman (Chairperson) Eisa Al-Said, King Saud University, Riyadh Lorna S. Almocera, University of the Philippines, Cebu Yolando B. Beronque, De La Salle University, Manila Sergio R. Canoy, Mindanao State University-IIigan Institute of Technology Wanida Hemakul, Chulalongkorn University, Bangkok Wenlin Li, The Chinese Academy of Sciences, Beijing Ling San, National University of Singapore Maslina, Universiti Kebangsaan Malaysia, Selangor Sri Wahyuni, Gadjah Mada University, Yogyakarta Teofina Rapanut, University of the Philippines, Baguio Walter Roth, Universiti Brunei Darussalam IX

The AMC2000 Local Organizing Committee : Polly W. Sy (Chairperson) Luz R. Nochefranca (Secretary) Maritina T. Castillo Ricardo D.H. Del Rosario Jose Maria L. Escaner IV Augusto Y. Hermosilla Ricardo Enrico C. Namay II Caridad M. Natividad Nenita C. Ocampo Lorna I. Paredes Romulo G. Pizana Dakila A. Reyes Ivy C. Suan Evelyn L. Tan Ma. Vivien V. Visaya ASSISTING OFFICERS & STAFF of the Philippine Social Security System Chairman & Pres. Carlos A. Arellano EVP Leopoldo S. Veroy EVP Horacio T. Templo SVP Maria M. Laurel SVP Edgar B. Solilapsi VP Marissu Bugante VP Alfredo S. Villasanta AVP Rizaldy T. Capulong Amando P. Montano Matias C. Defensor Ronaldo W. Recio Jesse Revecho Jannet V. Villacorta Arnold Frias Melchor Vergel L. Ciubal Nemesiano Mogello

ASSISTING FACULTY and STAFF of the Department of Mathematics, University of the Philippines, Diliman Ma. Nerissa M. Abara Tresio P. Acuzat Julius Caesar C. Agapito Ma. Sonia J. Andres Christine Abegail A. Antonio Carlene Perpetua P. Arceo Joel N. Bangalan Julius M. Basilla Noemi R. Barcial Angela Faith P. Bernardo Rowena Alma L. Betty Carolina H. Briones Rex R. Briones Grace D. Cabading Chona M. Caguintuan Adora A. Calaor Ronel J. Calugay Eloisa S. Cantuba Kristine Joy E. Carpio Don G. Castronuevo Tom N. Chu Filipinas L. Cristobal Laarni dela Cruz Angelito M. delos Reyes Nenita A. delos Santos Danilo C. Dematera Carolina M. Dimasayao Mace I. Eclevia Jeopista C. Espina Rene P. Felix Boogin C. Gamboa Michelle P. Guevara Nelson C. Gomintong Ma. Celeste F. Ignacio Lagrimas D. Iloreta

Catherine N. Laniog Glolibeth T. Lapay Roosevelt Y. Loto Neddie T. Lozano Benjie L. Luces Frances P'Ann I. Macabuhay Kimberly May E. Magsajo Rhodora D. Monte Augusto Morales J r. Manuel I. Musngi Ann Marie E. Natividad Fidel R. Nemenzo Cherryl N. Ortega Raul O. Padilla Gilbert M. Parrocha Ma. Garnet J. Porciuncula Tricia Caroline M. Quijano Reiko C. Rabe Cherry Rose A. Ramos Crispin R. Ranee Reynaldo Romualdo M. Rey Josephine F. Reyes Gaudella O. Ruiz Virgina G. Saban Gil Sadorra Remedios D. Santos Maranatha Consuelo L. Serrano Anna Marie S. Tiro Natividad A. Valencia Michelle C. Vallejos Cecille V. Verzola Ma. Redina M. Victoria Jesie G. Villanueva Jimmy V. Viloria

Other ASSISTING FACULTY Members : Jaime D.L. Caro, Dept. of Computer Science, University of the Philippines,Diliman Felix P. Muga II, Dept. of Mathematics, Ateneo de Manila University Andrew Vizcarra, Dept. of Mathematics, Ateneo de Manila University XI

This page is intentionally left blank

CONTENTS

Preface

v

Acknowledgments

vii

AMC2000 Committees

ix

Remarks on the Solutions of Apu + |u| 9 _ 2 u = 0 Julius Caesar C. Agapito, Lorna I. Paredes, Reynaldo M. Rey and Polly W. Sy

1

On One-Phase Stefan Problems for Sublinear Heat Equations Toyohiko Aiki, Hitoshi Imai, Naoyuki Ishimura and Yoshio Yamada

6

Hopf Bifurcation in a Chemostat-Related Model Lorna S. Almocera and Polly W. Sy

12

Model-Theoretic Properties of Banach Spaces S. Baratella and S. A. Ng

17

A Vertex Partitioning of a Graph Sergio R. Canoy, Jr., Esperanza B. Arugay and Esamel M. Paluga

38

Construction of Lagrangian and Non-Lagrangian IP Loops Alexander S. Carrascal

48

On Extensions of Differential Operators in L2[a, oo) Maritina T. Castillo

56

Generation of Nafil Loops of Small Order Raoul E. Cawagas

63

XIII

XIV

Contact Topology and CR Geometry in Three Dimensions Jih-Hsin Cheng

70

Upper and Lower Non-Absolute Integrals Cheng Lu Pien and Chew Tuan Seng

81

Constructing Graphs That are Isomorphic to their Clique Graphs G. L. Chia

89

On Barnette's Conjecture and CBP Graphs with Given Number of Hamilton Cycles G. L. Chia and Siew-Hui Ong

94

Multiplier Problems Christian Datry and Gilbert Muraz

112

The Henstock Integral and Measures on Infinite Dimensional Spaces Elvira P. de Lara-Tuprio and Chew Tuan Seng

123

Stability of a Quadrature Method for the Biharmonic Dirichlet Problem V. D. Didenko Shadows of Codes and Lattices Steven T. Dougherty and Patrick Sole Non-Linear N-Term Approximations of Smooth Functions Using Wavelet Decompositions Dinh Dung

131

139

153

Double Series Expansions and their Convergence Domains Keiko Fujita

175

Empirical Pseudorandom Number Generators Kim Gargar

181

Parameter Estimation on a Smart Material Structure: Numerical Computations with Implicit Filtering Optimization N. C. Gomintong and R. C. H. del Rosario

189

XV

On a Parametrization of Skew-Symmetrically Perturbed Gaussian Diffusions Augusto Y. Hermosilla On Non-Ambiguous Biichi V-Automata Phan Trung Huy and Do Long Van The Consequence of Controlled Densed Theorem of Henstock-Kurzweil Integral in n-Dimensional Euclidean Space Christiana Rini Indrati and Soeparna Darmawijaya Local Stability Analysis for a Model of AIDS Transmission Maria Evelyn V. Jacinto An Analysis of Conjunctive Model Reduction Algorithms for Solving Linear Programming Problems V. C. Jacob, T. R. Natesan, V. Rhymend Uthariaraj and P. Narayanasamy

212

224

234

244

254

Weak Convergence of Injective Tensor Products of Vector Measures Jun Kawabe

267

Large Gradients of Solutions to Nonlinear Parabolic Problems M. M. Lavrentiev, Jr.

280

Timetabling Highly Constrained System via Genetic Algorithms Ho Sung C. Lee and Augusto Y. Hermosilla

301

A Deceptive Game with Modified Information K. T. Lee

318

Duadic Codes San Ling

324

The Theory of Binary Nonlinearization and its Applications to Soliton Equations Wen-Xiu Ma

337

XVI

Trigonometric Identities and Geometrical Inequalities for Links and Knots Alexander D. Mednykh Generalized Functions on the Sphere and Heat Functions Mitsuo Morimoto

352

369

Essential Spectrum and Nullities Preserved Under an Admissible Perturbation Marie Redina L. Mumpar-Victoria

375

An Extrapolation Technique for a Singularly Perturbed Problem on a Shishkin Mesh M. C. Natividad and M. Stynes

383

Decomposition-Metaheuristic Method Applied to a Capacitated Facility Location Problem Marrick C. Neri and Augusto Y. Hermosilla

389

Homogeneous Extensions of Shifts with Holder Continuous Skewing Functions Mohd. Salmi Md. Noorani

404

The Abelian Index of Finite Groups Kenneth K. Nwabueze, Norhayati Hamzah and Saiful A. Husain

420

Dirichlet Forms and Symmetric Markovian Semigroups on Von Neumann Algebras Yong Moon Park

427

Symbols for Semilinear Wave Equation with Conormal Cauchy Datas Singular at a Point A. Piriou

444

Triangle Graphs with Maximum Degree At Most 3 Romulo G. Pizana and Rolando E. Ramos

451

XVII

Algebraic Geometry and Physics Shi-shyr Roan

455

Wave Blocking in Parallel Coupled Nerve Fibers M. Rodrigo, H. Ikeda and M. Mimura

483

Nullities of Differential Expressions with Logarithmically Decaying Coefficients in Weighted Space M. P. Roque

488

Problems Concerning the Deficiency Indices of Singular Self-Adjoint Ordinary Differential Operators Bernd Schultze

495

Attainability of a Two Characteristic Manpower Structure Including Demotion A. Srinivasan and P. Mariappan

502

Expected Time for Recruitment in Manpower Planning Associated with Correlated Renewal Sequences — A Shock Model Approach A. Srinivasan and P. Mariappan

511

An Orthogonally Additive Functional on a Space of Banach Lattice Valued Functions Supama and Soeparna Darmawijaya

519

Bounds on Doubling Rates of Investment Capital Due to Incorrect Distributions Choon Peng Tan

532

Tau Ergodic Coefficients Generated by the C-Norm Choon Peng Tan and Pek Wai Liew

541

Finding the Span of Graphs Using Small Circle Unit Representations Joselito A. Uy

552

On the Regularity of a Trust Region-CG Algorithm for Nonlinear Ill-Posed Inverse Problems Yan-Fei Wang and Ya-Xiang Yuan

562

XVIII

Electronic Information and Publications in EMIS — An Offer Supervised by EMS Bernd Wegner Mathematical Modeling of Electromagnetic Sounding for a Conductive 3-D Circular Cylinder Body Embedded in a Conducting Half-Space S. Yooyuanyong and N. Chumchob (Z)-Green's Relations and Perfect rpp Semigroups K. P. Shum, X. J. Guo and X. M. Ren

581

590

604

REMARKS ON THE SOLUTIONS OF APU + \U\Q~2U = 0 JULIUS CAESAR C. AGAPITO*, LORNA I. PAREDES+, REYNALDO M. REY AND POLLY W. SY Department of Mathematics, College of Science, University of the Philippines, Diliman, Quezon City, Philippines In this paper, the positive solution of the equation A p u + |u| 9 _ 2 « = 0, where p ^ q, and its behavior relative to a certain limiting conditions on p and q shall be discussed.

For p, q £ ( 1 , oo), Otani[3] studied t h e following one dimensional equation ^

f Apu + \u\«-2u = 0 \u(0)=u(l)=0

'

on

(0,1)

(1) (2)

(\ux\p~2ux)x.

where A p w =

A function u is said t o be a solution satisfies (1) in t h e distribution sense.

of (Ep'q)

if u € W 0 ' p ( 0 , 1 ) and u

In Otani's paper, he proved t h e following two theorems. T h e o r e m 1 [3] Suppose t h a t p, q > 1 and p / (Ep'q), then u satisfies t h e following: i) u 6 Ca{[0,1]) Z{u) = {x£

n C([0,1]]\Z(«)), [0,1] | ux{x)

= 0 },

q. If u is a solution of

where

2—v a = m i n { < ——y > + 1 , < q > }

and _ J oo if r is an even integer — | m i n { n \ n > r, n nonnegative integer} otherwise. ii) p^-\ux(x)\p

+ ±\ux(x)\q

= constant for all x G [0,1]

hi) lim4_>0+ Ux(t) = lim t _ > 1 -[—u x (i)] 'MACARTHUR AND JOSEFINA DELOS REYES PROFESSORIAL AWARDEE +PCI BANK FOUNDATION PROFESSORIAL CHAIR AWARDEE 1

CHAIR

2

iv) IKIIL(o,i) = IHlL(o,i)

=

£^+£\hmt^o+Mt)\p-

Theorem 2 [3] Suppose that p, q > 1 and p ^ q. Then (Ep'q) has a unique positive solution. Furthermore, for the functional R defined by R(v) -

M

L ^ !KI|Z,P(0,1)

we have R(up,q) = sup{R(v)\ v G W 0 1,p (0,1) denotes the unique positive solution of (Ep'q).

and v ^ 0}, where upoo '

= 2^=1

— — |a; — —| ^2 2 J

for all

x € [0,1].

3

Corollary 3.3 Let a(q) be a function defined on (1, oo) such that 1 < qa(q) ^ q for all q G (l,oo) and limg^.oo a(q) =b. If p — qa(q), then lim u„ g(x) = 2*=* , - - \x - -I , ?->oo ' ^2 2 )

for all

x e [0,11.

Idogawa[2] showed that, Theorem 4

Let Dp,q = ma,xxe[0ti]UPtq(x).

Then

i) for fixed q > 1, we have lim DPiQ = 2 ^ P->I+

ii) for fixed p > 1, we have

lim DPig =

2^(^—)

And in addition to this, we have Proposition 5 For some do > 1, let a(p) be a function defined on (1, l + (5o] such that p 7^ a(p) > 1 for all p € (1,1 + So] a n d limp_>1+ a(p) = a > 1. If q = a(p), then, for any x € (0, | ] lim Up,, (a;) = 2°-*. Proof:

From Theorem 3, we have, for any x € [0, | ]

Let q — a(p) and lim p _n+ a(p) = a > 1. For any fixed 5 6 [0,1), we have

lim fpq(s)= P-+1+ '

lim / ( l - * 9 ) ~ * d * p-n+ 7 0

= f (l-t a )- 1 di.1+ fp,q(s)

= oo if and only if s = 1.

Since, by definition, B(^, 1 - ^) = function, then 1 1 lim B(-, 1 - - ) = lim

r(\+1_f)

T(i) q

w 1

1N

, where T(-) is the gamma 1 • lim T(l - - )

r(i) = ^ • 0 0 = 00.

Therefore, for any x £ (0, | ] , we have limp_).1+ y - B ( i 1 - ^) = oo, and hence, lim f-K^B(-,l--))

P->I+

q

q

p

= l.

Note that [lB(I,l-l)]A[«fcil]^ = ( | ) ^ ( | ) ^ ( F ( ^ | = T y ) - (r(l - i ) ) 5 ^ ( p - l)i±F. Since | is not an integer, we see that

(r(i-l))^( p -i)^ = ( ( ( s i i i ; ) r ( i ) ) P ( p - i ) ) ^ , From p-1 lim p-n+(sin | ) P

1 n'

it follows easily that lim [I*(!,i L ->i+ g V P

- 1 ) ] A [ f c ! ) ] ^ = i. p

Hence, for each x 6 (0, | ] , we have limp_>.1+ up,g(a;) = 2»=T. Proposition 6 For some SQ > 1, let ,9(9) be a function defined on (1,1 + 5o] such that q ^ /3{q) > 1 for all q £ (1,1 + 1. If P — P(Q), t n e n f ° r a n y ^ € (0, §], lim «„,,(«) = 2 ^ r ( ^ ) ( l - (1 - 2a;) A ) . Proof:

Since p = j3(q) and limg_>1+ /3(g) = b > 1, we have for each s £ [0,1]

5

lim fp,q(s)=

/

(l-t)^dt

Jo

«->!+

(-^j)(l-(l-s)b-^)=:9(s).

=

Now if y = g(s) where y 6 [0, ^ ) , then g-^y) thus lim / - i ( y ) =

= 1 - (1 -

(^1))I^T,

and

i-(i-(*zi)y)^r.

Hence, for each x 6 (0, | ] , Theorem 3 yields, limg_).1+ uP),(2;)

= 2^(^l)(l-(l-2x)T^). This proves the proposition. Acknowledgment: This research is supported by the Natural Sciences Research Institute of the University of the Philippines, Diliman. References [1] Agapito, J . C . C , L.I. Paredes, R.M. Rey, P.W. Sy, A Note on the Positive Solution of Apu + \u\q~2u = 0, Matimyas Matematika 22(1999) ppl-8. [2] Idogawa, T., Lecture Notes [3] Otani, M., "On Certain Second Order Ordinary Differential Equations Associated with Sovolev-Poincare-type Inequalities", Nonlinear Analysis, Theory, Method and Applications, 8 (11) 1984, 1225-1270.

ON ONE-PHASE STEFAN PROBLEMS FOR SUBLINEAR HEAT EQUATIONS TOYOHIKO AIKI Department of Mathematics, Faculty of Education, Gifu University, 501-1193, Japan.

Gifu

HITOSHIIMAI Department of Applied Physics and Mathematics, Faculty of Engineering, University of Tokushima, Tokushima 770-8506, Japan. NAOYUKIISHIMURA Hitotsubashi University, Kunitachi, Japan.

Department of Mathematics,

Tokyo 186-8601,

YOSHIO YAMADA Department of Mathematical Sciences, Waseda University, Ohkubo, Tokyo 169-8555, Japan.

Shinjyuku-ku,

We are interested in the one-phase Stefan problem for sublinear heat equations with Dirichlet boundary condition. We survey our recent establishments concerning the unique solvability and the asymptotic behavior of solutions.

1

Introduction

This article surveys our recent studies on the one-phase Stefan problems for sublinear heat equations with Dirichlet boundary condition 5 ' 6 : Find a free boundary x = l(t) > 0 on [0, T] with T > 0 and a function u = u(t,x) on Q{T;l(t)) := {(t,x) |0 < t < T,0 < x < l{t)}, which satisfy

ut = uxx+u1-a

in

u(t, 0) = 0

for 0 < t < T

u{t,l(t))

ior0 0 v{0) = v{k) = 0.

in 0 < x < k

The uniqueness property of (3) leads us to the corollary. Corollary 1 Let v(x;li) denote the solution of (3). Then there holds v(x;l) =

l2/av(x/l;l).

Now we show our main theorem concerning the asymptotic profile.

(

'

9 Theorem 3 Suppose 0 < a < 1. Then any solution {u(t,x),l(t)} as t -4 oo on the whole R+ — {x > 0}; that is, there holds l{t) —> oo

grows up

os t —• oo,

-1

and eventually at every x € J? " one ftas u(£, x) -> oo

as t -¥ oo.

In the above, it is certain that u(t,x) is meaningless initially for any x > lo; however, since l(t) -> oo as t -> oo, u(t,a;) is justified for every x > l0 if t > tx > 0 where t^ depends on x. Our theorem asserts that u(t, x) grows up as t -¥ oo, which should be understood only after u(t,x) is defined. We provide a sketch of proof. Let {u(t, x), l(t)} be a solution to (1), which exists for any T > 0. We introduce a functional F2 associated with (1), which is similar to F\: r«t)

F2[u}(t) := J

/1

i^ux(t,x)2

1

- ^—U(t,x)2-a)

F2 serves as a Lyapunov functional. The proof proceeds along several steps. In the sequel constants which may differ from line to line. Step 1. ||u(t,-)IU- 0.

Step 2. \\u(t, -)|ji°° cannot tend to zero as t —¥ 00. Step 3. l(t) < C(a,u0)(l

+ t( 3 + a )/( 2 a )) for all large t.

Step 4. l(t) cannot converge to a finite lx as t —> 00. Step 5. \\u(t, -)lli°° -> 00 as t -> 00. Step 6. Conclusion with establishing a pointwise estimate. 4

Numerical Justifications

We have analysed one-phase Stefan problems for sublinear heat equations with Dirichlet boundary condition. Here we exhibit numerical computation with the hope of strengthening our current exposition. The details of the numerical procedure and experiments for other types of nonlinearity are reported in our previous paper 14 .

10

Figure 1 simulates the free boundary problem (1) with a = 1/2. The initial datum are u0(x) = Ax2(l -x)2 (A = 0.0001 and A = 1000) and l0 = 1. In both values of A, the growing-up behavior is apparently observed. Figure 2, on the other hand, illustrates the fixed boundary problem (2) with a = 1/2 and /i = 1. The initial data is similar as in Figure 1. Whatever large or small the initial function is, it converges to a nonzero stationary point of F\ defined in §2; this agrees well with analytical results depicted in Theorem 2.

Figure 1: Free boundary problem with a = 0.5. (a) A = 0.0001, u(t,x) t + o o . (b) A = 1000, u(t, x) t +oo.

Figure 2: Fixed boundary problem with a = 0.5. (a) A = 0.001. (b) ,4 = 1000. References 1. J. Aguirre and M. Escobedo, Ann. Fac. Sci. Toulouse 8, 175 (1986-1987). 2. T. Aiki, Nonl. Anal. T. M. A. 26, 707 (1996). 3. T. Aiki and H. Imai in Free boundary problems, theory and applications, eds. M. Niezgodka and P. Strelecki, (Pitman Research Notes Math. 363, Longman, Essex, 1996). 4. T. Aiki and H. Imai, Ann. Mat. Pura Appl. (IV) 175, 327 (1998).

11

5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

T. Aiki, H. Imai, N. Ishimura and Y. Yamada, preprint. T. Aiki, H. Imai, N. Ishimura and Y. Yamada, preprint. T. Aiki and N. Kenmochi, Bull. Fac. Edu., Chiba Univ. 39, 15 (1991). T. Cazenave, F. Dickstein and M. Escobedo, Rend. Mat. Appl. (VII) 19, 211 (1999). A. Fasano and M. Primicerio, J. Math. Anal. Appl. 72, 247 (1979). A. Friedman, Variational Principles and Free-Boundary Problems, (R.E. Krieger, Florida, 1988). H. Fujita, J. Fac. Sci. Univ. Tokyo, Sect. 113, 109 (1966). H. Fujita and S. Watanabe, Comm. Pure Appl. Math. 2 1 , 631 (1968). V.A. Galaktionov, J. Hulshof and J.L. Vazquez, J. Math. Pures Appl. 76, 563 (1994). H. Imai, N. Ishimura and T. Aiki in Free Boundary Problems: Theory and Applications II, ed. N. Kenmochi, (Gakuto, Tokyo, 2000).

H O P F BIFURCATION IN A CHEMOSTAT-RELATED MODEL LORNA S. ALMOCERA University

Department

of the Philippines in the Visayas, Cebu Lahug, Cebu City, Philippines E-mail: [email protected]

P O L L Y W . SY of Mathematics, University of the Philippines, Quezon City, Philippines E-mail: [email protected]

College

Diliman

In this paper, we investigate an autonomous nonlinear system of ordinary differential equations that model competition in a chemostat with an inhibitor. We will illustrate the simplicity of using Hopf's theorem in establishing the existence of periodic solutions bifurcating from the positive equilibrium point.

1

Introduction

The chemostat is a piece of laboratory device frequently used for studying competition between microorganisms. It is a laboratory realization of a simple lake and plays a role in theoretical ecology , in wastewater treatment problems5 and is used as a model for the study of recombinant problems in genetically altered organisms 6 . In its commercial form, the chemostat is used as a model for the manufacture of products by using genetically altered organisms. In this paper, we will investigate the model which is a combination of the model of competition between plasmid-bearing and plasmid-free organisms in a chemostat proposed by Staphanopoulis and Lapidus ° and the model of a chemostat with an inhibitor investigated by Lenski and Hattingh 3 . The equations of the model take the form ii = z i ( ( l - g)/i(l - xi - x 2 ) - 1) x 2 = x 2 ( e _ w 7 2 ( l - xi - x 2 ) - 1) + qxifi(l px\ l-p-6 K +p where Xi{0) > 0,

- xi - x 2 )

(i\ v ;

p(0) > 0,

fi(S) = —*—-, 0 < xx + x2 < 1,0 < q < 1. a, + b We shall restrict system (1) to a positively invariant region r = {(xi,x 2 ,p) : 0 < xi +x2 < 1, 0 < p < 1}. The variables xi and x2 denote the concentration of the competing organisms and the variable p represents the concentration of the inhibitor. The definitions 12

13 of the constants m 1 , m 2 , o i , a 2 , 6, fi and K are given in Hsu et al2. T h e equilibrium points of system (1) are obtained by setting the righthand side of system (1) to zero. A positive equilibrium point E* = (x\,xl,p*) exists if (1-9)^T>1 **d ^ ^ l < L or ( l - 9 ) ^ i > 1 and 1>X*2>\1 where the constants AJ and A2 are denned by the following relations:

/l( Ai)

=-=£L = J o i + Aj;

1- q

,

/2(A5)

=^

a 2 + A2

= eM.

Hsu et al studied system (1) and focused on the asymptotic behavior of the solutions to the system as a function of the parameters of the organisms x\ and x 2 and the alteration of the parameters by the inhibitor p. In this paper we will investigate the qualitative behavior of the system near the positive equilibrium point E*. We will apply Hopf's theorem 4 in establishing the existence of periodic solutions to system (1). This theorem establishes the existence of periodic solutions bifurcating from an equilibrium point x* for a system such as x = f(x, a) if (a) f(x*, ac) = 0 and (b) the Jacobian matrix of / a b o u t x* has exactly two complex conjugate eigenvalues u(a) ± v(a) whose real part u(a) vanishes at a = ac while ^ ^ 0 at a = ac. This paper is organized as follows. In section 2, we investigate the conditions for the stability of E*. Section 3 discusses the existence of periodic solutions of the Hopf type. 2

S t a b i l i t y of t h e P o s i t i v e E q u i l i b r i u m

Let E* = {x\,X2,p*) be the positive equilibrium point of system (1). Jacobian matrix evaluated at this equilibrium point is given by 'nn "12 0 "21 "22 "23 I ri3i 0 ri33

(2)

where _ "11 -

(l-q)mx(l-x'-xl) , (l-(j)miais* _ -x"(l-q)m1ai L (ai + l-xl-x*y > n 1 2 - (ai + l-x{-x'2y ai + l-x{-x*2 — m.2CL2e~^p x2 i grai(1—x* — x2) qmiaix* 7121 ~~~ (ai+l-xl-x'Y2 + ai + l - z j - x ; ; ~~ ( o j + l - x j - x * ) 2 — e~lip ni2(l — x\—x2) -i x2m2a,2e~'J'p _ qmiaix* i "22 — a3 + l - * I - a : 2 ~~ ~ (as + l - x j - x * ) * (ai+l-xj-x*)* -Me"t'p*m2(l-x*-x*) -6p* -, SKx\

"23=

,•, + ! - , ; - , 1

' .

n

31=K+^>

The

"33 = - 1 - (K+p*)2 '

14 At the equilibrium point E* the characteristic equation of the Jacobian matrix J takes t h e form P{\) = A3 + C i A 2 + C 2 A + C 3 = 0 (3) where C\ = - ( n n + "22 + n 3 3 ) C2 = n n ( n 2 2 - n 2 i ) + n 3 3 ( n n + 7 1 2 2 ) (4) C3 = - " l l « 3 3 ( « 2 2 - 2 l ) - "12"23"31With 0 < g < 1, 0 < x i + x 2 < l and with the rest of the constants positive, we conclude t h a t C\ > 0, C 2 > 0, and C 3 > 0. L e m m a 1. ( R o u t h - H u r w i t z s t a b i l i t y c r i t e r i o n ) All roots of (3) have negative real part if and only if C\ > 0, C 3 > 0, C i C 2 — C 3 > 0. Equation (3) has one negative root and a pair of complex roots with a positive real part if and only if d > 0, C3 > 0, C i C 2 - C 3 < 0. L e m m a 2. T h e positive equilibrium point E* is stable if C i C 2 - C3 > 0 and unstable if C i C 2 — C3 < 0. 3

E x i s t e n c e of P e r i o d i c S o l u t i o n s of t h e H o p f T y p e

By substituting the entries of (2) to (4), we can easily see t h a t C\, C 2 , and C3 are functions oi K. After some algebraic computations, C i C 2 —C3 = 0 reduces to A(K) = A4K4+A3Ks + A2K2 + A1K + A0 = 0 (5) where AA = ( " l i + "22)(("11 + n 2 2) - " n ( " 2 2 - ra2i) - 1), A3 = 4p*A4 + Sx^Aoo — 0, C\Ci d{Cl

— CZ\K=K'

%-C3)\K=K.

i = 1,2,3 = 0

7^0

then Hopf bifurcation from E* occurs at K = K*. As the value of K passes through K*, the periodic solution of the Hopf type of system (1) appears which surrounds E*. Here, K* is the Hopf bifurcation value satisfying A(K*) = 0. The existence of periodic solutions of the Hopf type, although a local property, demostrates the case of oscillatory coexistence of the two organisms x\ and a;2 m the presence of an inhibitor p. References 1. 2. 3. 4. 5.

6. 7. 8.

G. D'ans et al, IEEE Trans. AC 62, 341 (1971). S.B. Hsu et al, J.Math. Biol. 34(2), 225 (1995). R.E. Lenski and S. Hattingh in J.Theor. Bio. 122, 83(1986). J.E. Marsden and M. McCracken in The Hopf bifurcation and its applications, (Springer, New York, 1976). E.B. Pike and C.R. Curds in The microbial ecology of activated sludge process, eds. G. Skykes and F. A. Skinner (Academic Press, New York, 1971). G. Stephanopoulis and G. Lapidus, Chem. Engr. Science 43, 49(1988). P. Waltman, Rockey Mt. Jour, of Math. 20, 777(1990). P. Waltman in Competition models in population biology (Philadelphia: Society for Industrial and Applied Mathematics, 1983).

M O D E L - T H E O R E T I C PROPERTIES OF B A N A C H SPACES" S. B a r a t e l l a Dipartimento di Matematica - Universitd di Trento via Sommarive 14, 38050 Povo (TN) Italy E-mail: [email protected] S.-A. N g b School of Mathematics - University of Natal Pietermaritzburg 3209, South Africa E-mail: [email protected] We survey some results on Banach spaces and related structures, viewed as firstorder structures in the model-theoretic setting. We study some geometric properties defined by the neocompact formulas: back-and-forth, density, quantifier elimination, winning strategy. Various kinds of saturation (compactness) are considered. Included also are applications to definable mappings, fixed point theorem and the Banach-Mazur problem on characterizing Hilbert spaces.

1

Introduction

In recent years, the study of Banach spaces has been notably influenced by methods and techniques from other part of mathematics, such as group theory, combinatorics, set theory and mathematical logic. To name two well-known deep results, Gowers dichotomy theorems (Gowers 14 and also Benyamini and Lindenstrauss 7 ) and the construction of Tsirelson space (see Benyamini and Lindenstrauss 7 ) are basically applications of combinatorics and set-theory. Meanwhile application of mathematical logic is less well-known - except the Banach ultraproduct construction, i.e. nonstandard hull constructions (see Dacunha-Castelle and Krivine 11 and Luxemburg 23 ), whose foundation actually lies in the model theory of mathematical logic. In a typical use of Banach ultraproduct, one is concerned with explicit classical properties of the original space and no commitment is made to the investigation of general properties of similar structures. We believe that the latter should not be overlooked. The present short survey paper is aimed at beginning research students and professional researchers who are interested in learning model-theoretic "Mathematics Subject Classification Primary: 03C10. Secondary: 03H05 03C65 46B08 46B04. Keywords: Model theory, Banach space structure, quantifier elimination, neocompact, backand-forth, Banach-Mazur problem. ^Supported by South African NRF 2039556

17

18

properties of Banach spaces and their applications. This is also a good place for those seeking new research problems. Apparently, the present subject is still in its infancy, only the future will tell whether the direction we take here is the correct one. The choice of topics is neither complete nor balanced, in fact it only reflects those results (both published previously and new ones) that we are interested in. We made little reference to the earlier important investigation by Henson and Iovino on the same subject from a different approach, but we recommend the interested readers to Iovino 21 . In the §2, we first consider Banach spaces as a more general class of structures in an appropriate first order logic. Then we consider properties in the neocompact language (Definition 2) which is roughly an approximate higher order language. This language was first introduced by Fajardo and Keisler 12 in abstract settings, but we apply it in formulating concrete geometric properties of Banach spaces. For example, in Theorem 6 we consider the backand-forth property and its relevance to quantifier elimination — a well-known topic in classical model theory. In finite dimensional spaces, back-and-forth is the same as transitivity. But they are different in general spaces. In fact, from model-theoretic point of view, back-and-forth is a more natural notion. Another natural property is density. These properties can be used to describe the shape of the unit sphere, and hence the geometry of the Banach space. We prove a fixed point result from it (Proposition 14). Saturation is a prominent notion in model theory. In our context, there are several natural definitions of saturation (Definition 3). They can be taken as notions of compactness for Banach spaces. Saturation is responsible for the richness of Banach ultraproducts (nonstandard hulls). We give some examples of other usages such as quantifier elimination and back-and-forth. Hilbert spaces satisfy quantifier elimination (Theorem 10). It is quite possible that it already characterizes Hilbert spaces. (See Question 11.) In general, neocompact definable mappings may be hard to describe. But under quantifier elimination, they could only be the trivial maps (Theorem 17). This is yet further evidence that quantifier elimination represents the good behaviour of Hilbert spaces. For the remainder of this paper, we consider game-theoretic type properties such as winning strategy (Definition 19) and its relation to the open problem of Banach-Mazur: Is every separable transitive Banach space a Hilbert space? Winning strategy is a weaker notion than transitivity (Proposition 20). It is unusual in being a self-dual notion in reflexive spaces (Theorem 23). In Theorem 21, we give the equivalence of reflexivity and Radon-Nikodym property under some weak back-and-forth assumption. We end our paper by a list of questions related to the Banach-Mazur problem.

19 For all the unexplained notions in functional analysis, see Beauzamy 6 , Benyamini and Lindenstrauss 7 or any classical book on functional analysis. For the model theory notions see Chang and Keisler 10 or Hodges 20 .

2

Preliminaries and general results

From the point of view of first order logic, we can view a Banach space M over the real numbers as a structure of the form A4 = < M , + , 0 , ( r _ ) r e Q , | | ||>, where - r_ is the operation of multiplication by the rational scalar r; - {M. + . 0 . (r_) r6 Q) is the underlying vector space structure over the rational numbers; - J] j| is the norm. In applications, a Banach space may be equipped with additional functions, for instance a family of seminorms, lattice operations in a Banach lattice, multiplication operation in a Banach algebra, inner product in a Hilbert space... So, in general, we deal with a structure JV=<M(/i)i6j>, where M. is a Banach space and (/i)ie/ is a countable family of functions from finite powers of M. to M. or to R that are uniformly continuous on each bounded set. We will say that M is a Banach space structure. A detailed description of these structures (in a slightly more general setting) can be found in Iovino 21 . The language for A/ is the same one adopted by Baratella and Ng 2 (see also Iovino 21 ). Here we recall that we work in the language of vector spaces over the rationals, augmented by symbols for the rational bounds of the norm. Furthermore, for each vector-valued /*, we simply add the symbol / , and, for each real-valued fa, we include function symbols for rational bounds of values of/,;. In the pure language (i.e. the language adequate for a structure like M above), the typical term t{x\,... ,xn) has the form X^o 7 "* 3 '*' w n e r e the Xi's are variables and the TVS are rationals. Among the simplest formulas that we can write in the above language are those of the form r < \\t(xi,... ,xn)\\ < s, for some rational numbers r,s. For example, a unit vector is expressed by 1 < ||i|| < 1. But in order to express irrational norm such as \/2, we are forced to use an infinitary

20

formula such as /\(r < ||x|| < s), where the conjunction is taken over all r, s € Q, r < y/2 < s. Consequently, it is natural to work with a family of innnitary expressions called neocompact formulas and with its subfamily of finitary positive bounded formulas (see Baratella and Ng 2 ). Neocompact formulas (briefly: nc) have been introduced in an abstract framework by Keisler 22 . Positive bounded formulas (briefly: pb) are due to Henson 17 . We recall now the recursive definition of pb formula. (As a technical convenience, we assume that in all the formulas all the variables are bound in norm to the closed unit ball.) When we refer to the the norm of a tuple, we mean the supremum norm. Definition 1 Let C be the language for some Banach space structure. C-terms are constructed as in the first order case, by using function symbols for vectorvalued functions only. We define the class of pb formulas over £ as follows: 1. If t is a term and r is a rational number, then \\t\\ < r

and

\\t\\ > r

are pb formulas.

2. If fi is an m-ary real-valued function, t\,... ,tm rational number, then fi{h,...

,tm) < r

and

fi{t\,...,tm)>r

are terms and r is a

are pb formulas.

3. If 9 and ip are pb formulas, then 9\/ tp and 8 A tp are pb formulas. 4- If ip is a pb formula and 9 is a quantifier free pb formula, then 3yip(x,y)

and

~iy{9{y) —-> tp(S-y))

are pb formulas.

The formulas without quantifiers are called quantifier free pb (briefly: qfpb). We also recall the definition of nc formula. Definition 2

1. Atomic neocompact formulas have the form

where each <j>i is qfpb and x is a finite list of variables. 2. If <j> and xp are nc, then V ip is nc.

21 3. If {4>i}i ip(y,x))

is nc.

Let M be a Banach space structure and let a £ Mn. T h e (qf)pb type of a in M. is t h e set of (quantifier free) pb formulas without parameters t h a t are t r u e of am. M. We will use the notation tpM(a) for the qfpb t y p e and we will drop reference to t h e structure, when there is no ambiguity. An n-type is a maximal consistent set of qfpb formulas in n free variables. We call the Banach space structures in t h e pure language simply Banach spaces. A Hilbert space is a Banach space whose norm is given by an inner product. Notice t h a t in a Hilbert space, the inner product does not appear explicitly as a function of the structure. Indeed it is a remark in Baratella and Ng 3 t h a t if M. is a Hilbert space in the pure language and M+ is t h e expansion of A4 t o a language with a symbol for t h e inner product t h e n for any t y p e p(x) in t h e pure language, the set q(x) of qfpb formulas in t h e expanded language t h a t are implied by p(x) is a type, and the two types define the same set in the corresponding structures. By BM w e denote the closed unit ball of M (we simply write B when it is safe). We use SM (or <S) for the unit sphere. We are interested in Banach space structures being "sufficiently rich" in t h e sense made precise by the definition below: D e f i n i t i o n 3 We say that a Banach space structure A4 is pb-saturated if every finitely satisfiable set in M of pb formulas in finitely many free variables and finitely many parameters is satisfiable in M. Similarly for qfpb- and ncsaturaiion. We say that M is Vqfpb-saturated (Vnc-saturated) if it is saturated for sets made of qfpb and one universal pb (nc,) in finitely many free variables and parameters. When we refer to saturation for sets of formulas without parameters, we make the index 0 follow the appropriate kind of formulas (so p b 0 saturation means saturation for sets of pb formulas without parameters). It is easy to show t h a t finite dimensional Banach space structure and Hilbert spaces are nc-saturated (see Baratella and N g 2 for the details).

22

Keeping in mind that in every formula all free variables are bound to the closed unit ball, we formulate the following Definition 4 (Formulas are assumed to be parameter-free.) Let M be a Banach space structure in the language C We say that M. has Quantifier Elimination (briefly: QE) if, for every n, every subset of Bn definable in Ai by a nc formula in C is also definable by an atomic nc in C, i.e. it is 11° over the qfpb formulas. Keisler 22 has proved quantifier elimination theorems in a very general setting. As a consequence we have that, in Banaoh space structures satisfying certain conditions, every set defined by a nc also admits a simpler definition by means of a countable conjunction of qfpb. The relevant conditions are singled out in Definition 4 . 1 2 , where equivalent formulations of Keisler's properties in the setting of Banach space structures are provided. In particular, we recall the following: Definition 5 Let M be a Banach space structure. 1. Ai is dense if, for each rn, any two m-tuples which satisfy the same qfpb formulas in M. also satisfy the same existential formulas in M.. (Recall that an existential formula is obtained by existential quantification from a qfpb formula.) 2. M. has the back-and-forth property (briefly: BF) if, whenever tp(oj) = tp(a2) and b\ G Bm for some m, then there is 62 such that tp(ai,6i) = tp(a 2 ,6 2 ). 3. A sequence tp(a,) of types converges to a type p (notation: tp(5i) —> p) if all qfpb 8 that appear infinitely often in the sequence are also in p. (See the topology defined by Keisler22 on the set of types and Definition 4- M. has the strong open mapping property (briefly: SOM) if for every a € Mn, for any sequence tp(a») such that tp(a») —> tp(a) and b £ Bm, for some rn, there is a sequence c~i € Mn. such that tp(a^) = tp(c,;) and tp(5j,6) —> tp(a,6). We are interested in Banach space structures satisfying the above properties because of the following (see Baratella and Ng 2 ):

23

T h e o r e m 6 For a Vnco-saturated Banach space structure M the following are equivalent: 1. M has BF; 2. M has BF and SOM; 3. M has QE.

The next result shows how BF and SOM are related to density and to various degrees of saturation. L e m m a 7 Let M be a qfpb0-saturated Banach space structure. Then in M. 1. BF if and only if qfpb-saturation and density; 2. QE if and only ij'Vnco-saturation, qfpb-saturation and density.

In the sequel we deal with nonstandard hulls of Banach space structures. (Banach ultraproducts form a special subclass of such spaces.) First we briefly recall the nonstandard hull construction. We recall that we always work in nonstandard universes that are at least a>i-saturated. Now fix a nonstandard universe. Let M be an internal Banach space structure. Let || || be the norm and let fin(M) be the R-vector subspace of norm-finite elements of M. Let Mo denote the Banach space part of M, i.e. the reduct to the pure language. Recall that the nonstandard hull Ma of Mo is, as R-vector space, the quotient of fin(.M) modulo the R-vector subspace of elements having infinitesimal norm. We denote by [a] the image of a tuple a of elements of fin(.M) under the canonical homomorphism n : fin(M) —> M and let ° : fin(*R) —• R be the standard part map. The norm of [o] given by °||a|| turns Mo into a Banach space (topological completeness is ensured by saturation, even if we start from an internal normed space only). Furthermore, if the extra functions in M are S'-continuous and 5-bounded on bounded sets, then as in Keisler 22 , if we include also the standard part (defined in the natural way) of extra functions from M, then we get a Banach space structure (still denoted by M): the nonstandard hull of the Banach space structure M • If M is an ordinary Banach space structure (like those introduced at the beginning of this section), we write M for *M and, as customary, we say that M is the nonstandard hull of the Banach space structure M-

24

There is a canonical isometric embedding of an ordinary Banach space structure M into its nonstandard hull M. We always identify elements of M and their images under such an embedding. The following is a straightforward corollary of Lemma 7. Corollary 8 The nonstandard hull of an internal Banach space structure has QE if and only if it is dense and \/nco-saturated. In Baratella and Ng 2 it is proved that if M is is an ordinary Banach space structure that is q/pfr-saturated and if its nonstandard hull has QE then M also has QE. They also proved that QE is invariant with respect to different nonstandard hull constructions. Theorem 9 Let M and M be nonstandard hulls of a Banach space structure M. Then __ M has QE & M has QE. Furthermore, in Baratella and Ng 2 QE is compared with a notion of quantifier elimination introduced by Henson (briefly: QEH). QEH is related to the approximate satisfiability of Henson 17 , which is weaker than first order satisfiability. QE and QEH, although different, agree on nonstandard hulls of internal Banach spaces that are Vnco-saturated. In an arbitrary Banach space structure, for any rational A > 1 we define the relation of X-equivalence on tuples a = ( a i , . . . , an) and b = (b\,..., bn) as follows:

a~A6

A-11| Y^aiai II ^ \\Y1aibi I' - AH Yla,<Xl I'

foralltteC n

!

In the pure language it is not difficult to show that the properties of BF and SOM are equivalent to the following: BF: Given \\a,i\\ = 1, i < n + 1 and ||6j|| = 1, i < n, if (ai,...,an) ~i (&i,... ,bn) then there is bn+\ such that {a\.... ,an+\) ~ i (&i, • • •,&n+i)SOM: For any n-tuples a, (a fc ) fc<w andm-tuple 6, if || E"=i a i fl fcj|| -> IIZ"=i a J a j!l for any a € Q n , then there are n-tuples {c~k)k] £ L p ([0,1] A ) such that [9] = [ip ° ^A}Note also that X C Lp([0,l]"2) as a subspace via [9] H [ J O 7rWl). In particular, as a subspace, Z C Y ~ L p ([0, l]" 2 ) © y . We denote by G(Y) the group of isometric isomorphisms of Y onto itself. Again by Greim et al.15, we have that G(Y) acts transitively on the unit sphere of Y. In order to prove that Z has WS let ai,&i € Z (denoting both functions and equivalence classes) with a\ ~ i bx and let 1\ € G(V) be such that Ti(ai) = Assume now inductively that ( a i , . . . , o n ) ~ j (&i,..., fcn) and I \ , . . . , Tn in G(Y) are constructed with aj, h € Z and Tj(aj) = h, i < j < n. Write a* = aj © o^ and 6, = 6j © b}, where a j , &j € X c Lp([0,1}U2) and a?, bf e y. Now let a n + 1 G Z C y be given and let b = Tn(an+i). Regard b e Lp([0,1]W2) © y, i.e. we can decompose b = c© d, where c € £ P ([0, l]" 2 ) and d £Y. Let i c w 2 be a countable set of coordinate indices on which c depends on. Let B C wi be a countable set of coordinate indices on which a } , . . . , a\ and b\,... ,b\ depend on and let p : 0)2 —> 0)2 be a bijection that fixes 5 pointwise and p(A) C o»i. Let p : [0, l]" 2 —> [0, l]" 2 be the corresponding bijection.

32

Let Tn+x e G(Y) be defined as follows: for all a e Y, regarding Tn(a) as an element of Lp([0,1]W2) © Y, with Tn(a) = r ®s, where r G Lp([0, l]" 2 ) and s e "F, define Tn+i(a) = r o p'1 @ s. Define bn+\ = c o p " 1 ©d, Since cop-1 depends on p(A) C wi, we have: bn+i € Z, Tn+i(a,i) = 6, for i < n + 1, and therefore ( a i , . . . , a n + i ) ~ i (6i,...,6n+i). • Along the lines of the proof of Theorem 2.1 in Cabello Sanchez 9 , we have that in a Banach space with BF\ some relevant properties coincide. Theorem 21 For a Banach space M. with BF± the following are equivalent: 1. Ai has the Radon-Nikodym property; 2. .M is reflexive; 3. TW is superreflexive; 4- Ai is uniformly convex. Proof By a known result, every reflexive space has Radon-Nikodym property and that every uniformly convex space is superreflexive. Therefore it remains to prove that (1) => (4). For 0 < e < 1 and x <E M let A(x,e) = i n f { l - A : there exists y e Ai such that ||Aui±y|| < 1 and \\y\\ > e}. It is known that if inf{A(x, e) : x € SM} > 0 for all 0 < e < 1, then .M is uniformly convex (see Cabello Sanchez 8 and Finet 1 3 ). By Radon-Nikodym property there exists x 0 G SM such that A(xo, e) > 0 for all 0 < e < 1 (see Cabello Sanchez 8 ). An easy calculation shows that BF\ implies A(xo,e) = A(x,e) for all x G SM and all 0 < e < 1. Then .M is uniformly convex. • The following gives dichotomy results with respect to smoothness and convexity property. Theorem 22 Let Ai be a Banach space. 1. If Ai has WS then either all points on its unit sphere are smooth or none is smooth. 2. If .M has BF\ then all points on its unit sphere are extreme or none is extreme.

33

Proof 1. For sake of contradiction, suppose u,v are unit vectors, such that u is a smooth but v is not. Let w b e a unit vector witnessing the nonsmoothness for v. (By this we mean that there exist two unit functionals (f> and 8 both norming v but 4>(w) ^ 8(w)). Using WS we can find a closed separable subspace Mo of M including elements u, v, w, and an isometric isomorphism T : Mo —+ Mo such that T(u) = v. By Hahn-Banach theorem, u is a smooth point in Mo and, by including the witness w, we conclude that v is nonsmooth in Mo- We thus get a contradiction since there can be no isometric isomorphism mapping a smooth point to a nonsmooth point. 2. Note that BF\ is equivalent to saying that whenever two vectors of M have the same qfpb type (i.e. the same norm), they also satisfy the same existential pb formulas with one existential quantifier. Now we can prove the dichotomy by noticing that, for some e > 0, either all unit vectors satisfy the existential pb formula 3 l / ( W = ||l/|| = ||2a;-y|| = l A | | a : - y | | > e ) or none does.

• The following shows that, for reflexive Banach spaces, WS is a self-dual property. Theorem 23 Let M be a reflexive Banach space. Then M has WS if and only if the dual space M' has WS. Proof Suppose M has WS. Then, by Theorem 21, it is strictly convex and superreflexive. It follows from the results in Finet 13 (see Definition 7 and Proposition 9) that there are smooth points on the unit sphere of M (Frechet differentiability is stronger than Gateux differentiability). So, by Theorem 22, M is smooth. Recall also that, for a reflexive space, strict convexity is equivalent to smoothness of the dual. We want to prove that, for all unit vectors 4> a n d 8 in M', player 3 has a winning strategy for T\{M' ,4>),{M' .8)}. Let a,b be unit vectors in M norming , 8 respectively. The main idea of the proof is that, during a play of r[(M',4>), (M1,9)}, we keep records of a simultaneous play of T[(A4, a), (M,b)].

Suppose that, for some 4>,9 € (M')n+1 and some a, b € Mn+l such that <po = 4>,9o = 9 and ao = a,bo = b, we already have the following from the first n moves of a play in the game r[(M',(f>o), (M',90)} : • (00, • --An)

~ 1 (#0, •• -,6n);

• ( a 0 ) . . . ,o n ) ~ i (6o,- • - , M ; . (norming) ||a,|| - ||^|| = &( 0 i ) = 1,

||6):|| = \\9%\\ = 9^)

= 1,

i < n.

Let (f>n+i 6 A4' with ||0 n + i|| = 1. We are going to respond by producing 9n+\ e M! such that the above properties extend to some (n + 2)-tuples. By smoothness and renexivity of M!, let a„+i £ JM be the unique vector so that ||a n + i|| = <j)n+i(an+i) = 1. By WS in M, choose bn+i € M so that (ao, • • •, a n +i) ~ i (&o, • • •, bn+i). Now by smoothness and renexivity of M., there is 9n+i e A4' so that ||6 n + i|| = | | 0 n + 1 | | = 5 n + 1 ( 6 n + 1 ) = l. C l a i m : (0,..., i < n + 1;

( iu ) II TH=o oci4>i\\Mo = II E r i ) 1 aii\\M and || E " ^ 1 M i l U o = II Y^=o "AIL for all a € R n + 2 (i.e. the norms are the same whether one views the functionals as functionals on M. or as functionals on the subspace M.§). The details of the construction are provided in Baratella and Ng 5 . The same argument applies to M' to prove that if M' has WS then M has WS. • The following is a consequence of the previous theorem and of the equivalence of WS and transitivity for separable spaces. Corollary 24 Let M be a reflexive, separable Banach space. transitive if and only if M' is transitive.

Then M is

35 Proof We notice that reflexivity and separability of M imply separability of M'. (Indeed, it is known that a separable Banach space has separable dual if and only if the dual has Radon-Nikodym property, and, as already noticed, reflexivity implies the Radon-Nikodym property.) Therefore one can apply the previous theorem. • Remark 25 A stronger form of the previous corollary is proved by Cabello Sanchez^: he shows that transitivity is a self-dual property for reflexive spaces. Recall that two elements in Lp[0,1] are disjoint if they have disjoint supports. For p ^ 2, oo a characterization of disjointness in Lp[0,1] is the following: elements / , g are disjoint if and only if

Ikf + Pg\\p = Hp\\f\\p + W\\f\\pforall

a,peR.

By using such characterization and the remark that WS implies BF\, one can easily prove that Lp[0,1] does not have WS when 1 < p < oo, p / 2. Similarly, the following do not have WS: £p(X), when X is any set with at least two elements and 1 < p < oo, p ^ 2; the space C(X) of bounded real valued continuous functions on a compact Hausdorff topological space X, equipped with supremum norm, when X has at least two isolated points. A Banach space is almost-transitive if the orbit of each unit vector under the action of the group of isometric isomorphisms is dense in the unit sphere. It is an easy remark that the nonstandard hull of an almost-transitive Banach space is transitive. Since L p [0,l], 1 < p < oo, p ^ 2 is almost-transitive but not transitive (see Greim et al. 1 5 ), we have an example showing that WS is not a super-property and it is not preserved under subspace. We finish with a list of open problems. The main one, of course, is the Banach and Mazur problem posed over 60 years ago, which we restated equivalently as (1) Is every separable WS Banach space a Hilbert space? Here are some weakenings of this problem: (2) Is a (separable) WS Banach space reflexive? (3) Is WS a self-dual property for (separable) Banach spaces? I.e. does the statement of Theorem 23 hold without the additional assumption of reflexivity? (4) Is a (separable) transitive Banach space reflexive? Finally, it would be of interest to clarify further the properties in this section: (5) Is WS the same as BF1

1. S. Banach, Theorie des Operations Lineaires, Chelsea Publishing Company, New York, 1955. 2. S. Baratella and S.-A. Ng, Neocompact quantifier elimination in structures based on Banach spaces, to appear in Annals of Pure and Applied Logic. 3. S. Baratella and S.-A. Ng, Applications of neocompact quantifier elimination, in preparation. 4. S. Baratella and S.-A. Ng, Fixed points in the nonstandard hull of a Banach space,Nonlinear Analysis: Theory, Methods and Applications 34, 299-306 (1998). 5. S. Baratella and S.-A. Ng, Playing isometry games in banach spaces, in preparation. 6. B. Beauzamy, Introduction to Banach spaces and their geometry, North-Holland, 1982. 7. Y. Benyamini and J. Lindenstrauss, Geometric nonlinear functional analysis, Vol. 1, American Mathematical Society colloquium publications, 2000. 8. F. Cabello Sanchez, Regards sur le probleme des rotations de Mazur, Extracta Mathematicae 12, 97-116 (1997). 9. F. Cabello Sanchez, Maximal symmetric norms on Banach spaces, Math. Proc. R. Irish Academy 98A, 121-130 (1998). 10. C.C. Chang and H.J. Keisler, Model theory. 3rd edition, NorthHolland, 1990. 11. D. Dacunha-Castelle and J.-L. Krivine, Applications des ultraproduits a l'etude des espaces et des algebres de Banach, Studia Mathematica 4 1 , 315-334 (1972). 12. S. Fajardo and H.J. Keisler, Neometric Spaces, Advances in Mathematics 118, 134-175 (1996). 13. C. Finet, Uniform convexity properties of norms on a superreflexive Banach space, Israel Journal of Mathematics 53, 81-92 (1986). 14. W.T. Gowers, An infinite Ramsey theory and some Banach-space dichotomies, preprint. 15. P. Greim, J.E. Jamison and A. Kaminska, Almost transitivity in some function spaces, Mathematical Proceedings of the Cambridge Philosophical Society 116, 475-488 (1994). 16. P.B. Guerrero and A.R. Palacios, Transitivity of the norm on Banach spaces having a Jordan structure, Manuscripta Mathematica

37

116, 475-488 (1994). 17. C.W. Henson, Nonstandard hulls of Banach spaces, Israel Journal of Mathematics 25, 108-144 (1976). 18. C.W. Henson, Model theory for structures based on Banach spaces, abstract of the talk given at "X Latin American Symposium on Mathematical Logic", in Bulletin of Symbolic Logic 2, 223-224 (1996). 19. C.W. Henson and L.C. Moore Jr., Nonstandard Analysis and the Theory of Banach spaces, in "Nonstandard analysis - recent developments" (A.E. Hurd ed.), Lecture notes in Mathematics 983, Springer, Berlin-New York 1983. 20. W. Hodges, Model theory, Cambridge University Press, 1993. 21. J. Iovino, A quick introduction to Banach space model theory, lecture notes, Carnegie Mellon University 1995 (available at http://sphere.math.utsa.edu/~iovino/). 22. H.J. Keisler, Quantifier elimination for neocompact sets, Journal of Symbolic Logic 63, 1442-1472 (1998). 23. W.A.J. Luxemburg, A general theory of monads, in "Applications of Model Theory to Algebra, Analysis and Probability" (W.A.J. Luxemburg ed.), Holt, Rinehart and Winston, New York, 1969.

A V E R T E X P A R T I T I O N I N G OF A G R A P H SERGIO R. CANOY, JR. ESPERANZA B. ARUGAY ESAMEL M. PALUGA Department of Mathematics, MSU-IIT 9200 Iligan City, Philippines Abstract Basically, the tree chromatic number of a graph is the minimum number of subsets the vertex set of a graph can be partitioned so that each subset induces a tree. Among others, this paper gives a characterization of graphs which have tree chromatic number greater than or equal to two. We also give a characterization of graphs which have tree chromatic number exactly equal to two.

1. I n t r o d u c t i o n There have been a great deal of results on studies regarding the partitioning of a vertex set as well as an edge set of a graph such t h a t each element of a partition induces a certain graph theoretic property. This paper considers a partitioning of the vertex set of a graph. Such a kind of partitioning, which was defined and investigated in [5], gives rise to what we call the tree-chromatic number of a graph. Henceforth, every graph G is simple, and V(G) denotes the vertex set of G. 2. Definitions a n d R e s u l t s

38

39

The first two definitions have the greatest roles to play. Definition 2.1 Let G be a simple graph. A family Q, — {Vi, V2,..., Vjt} of nonempty subsets of the vertex set V(G) is a vertex-tree partition of G if (i) 0 is a partition of V(G), that is, ft is a pairwise disjoint family and (ii) each element V$ of ft induces a subgraph of G which is a tree. Definition 2.2 Let G be a simple graph and P(G) the set of all vertex-tree partitions of G. The tree-chromatic number of G, is given by r{G) = min{\Q\ : ft G P(G)}. We first give the following remarks which are easy to verify. Remark 1 r(G) exists for every simple graph G and 1 < r(G). Remark 2 If G is a graph of order \G\, then r(G) < \G\. Remark 3 If G and H are isomorphic graphs, then r(G) = T(H).

Lemma 2.3 G is a tree if and only if T(G) = 1. Proof: Suppose G is a tree. Then ft = {V(G)} is a vertex-tree partition of G. By Definition 2.2, r(G) < 1. Combining this with Remark 1 gives us the desired result. Conversely, suppose that T{G) = 1 and assume that G is not a tree. Then there

40

exists no partition Q of G that contains a single element. This means that r(G) > 2, contrary to our assumption. Thus, G is a tree. The following remark is immediate from the above lemma. Remark 4 For all n > 1, T(Pn) = 1 and for all n > 3, r(Ki,n) = 1, where Pn is the path of order n and K\,n is the star graph. Lemma 2.4 Let G be a graph with exactly two components Gx and G 2 . Then r{G) = r(Gi) + r ( G 2 ) . Proof: First, suppose that r(G) = k and choose a vertex-tree partition Q, of G consisting of k elements. Then Q, = f2i U f22, where Oi Pi fi2 = 0 and fix and f22 are vertex-tree partitions of G\ and G2, respectively. By Definition 2.2, we have r(G!) + T(G2) < Iflxl + |Q 2 | = |Q| = r(G). On the other hand, if f2i and Q2 are vertex-tree partitions of G\ and G2, respectively, with If^l = r(Gi) and |fi 2 | = T{G2), then Q = Qi U f22 is a vertex-tree partition of G. Hence, with Definition 2.2, we have riGr) + r(G2) = \Qt\ + | 0 2 | = |Q| < r(G). The above two inequalities imply that T ( G I ) + r(G 2 ) = T(G).

By induction on the number of components of a graph, we obtain the following extension of Lemma 2.4.

41

Theorem 2.5 If Gi, G2, •••, Gk are the components of a graph G, then T(G) =

Y,T(G1).

Remark 5 There exist a graph G and a subgraph H of G such that r{G) < r(H). To see this, let G be a tree such that E(G) ^ 0 and let H be a disconnected subgraph of G. By Lemma 2.3, 1 = T{G) < T(H).

Lemma 2.6 Let G be a connected graph. If G is unicyclic, then T(G) = 2. Proof: Let C be the cycle in G and v G ^ ( C ) . Since G is not a tree, r(G) > 2. Now the set V(G)\{v} induces a subgraph H of G. Let Gi, G 2 , ••-, G m (m > 1) be the components of H. Then each Gi is a tree. If Gi contains the other vertices of C, then the set X = V(G2) U V(G3) U ... U V(Gm) U {w} induces a tree. It follows that Q = { V ( ( J I ) , X } is a vertex-tree partition of G. By Definition 2.2, r(G) < 2. Thus, r(G) = 2. Lemma 2.7 Let G be a connected graph with exactly n cycles. If these cycles are disjoint, i.e., any two cycles do not share a common vertex, then T(G) > n + 1. Proof: By Lemma 2.6, the assertion holds for n = 1. Suppose that the conclusion holds for n. Assume that G has exactly n + 1 cycles and that these cycles are all disjoint. Let Ci,C2, ...,G n ,G n + i be the disjoint cycles of G. Let Q be a vertex-tree partition of G with T(G) = | 0 | . Without loss of

42

generality, we may assume that Cn+\ is an endcycle, i.e., Cn+i has only one vertex, say v, that is in any shortest path connecting this cycle to another cycle in G. Note that such an endcycle in G exists because the cycles in G are mutually disjoint. Let W be the element of Q having the following property: (i) v ^ W and (ii) W contains at least one vertex of Cn+i- Note that X = V(G)\W induces a subgraph H of G which contains exactly n cycles (the cycles C\, C2, •••, Cn). Naturally, the family Q+ = Q\{W} is a vertex-tree partition of H. Hence by Definition 2.2 and the inductive hypothesis, |Q + | > r(H) > n + 1. Therefore, r(G) = \Q\ > n + 2. This proves the lemma. We now show that the reverse inequality in Lemma 2.8 is also true. Hence, we have the following result. Theorem 2.8 Let G be a connected graph with exactly n cycles. If these cycles are disjoint, then r(G) — n + 1. Proof: We prove this by induction on the number of cycles in G. First, suppose that n = 2. Then T ( G ) > 3 by Lemma 2.7. Let C\ and C 2 be the disjoint cycles of G. Let P be a shortest path connecting C\ to C2. Then every path that connects C\ to C2 has always the path P as a portion (i.e. P is unique). For if there is a path P+ that connects C\ to C 2 and yet does not contain P, then P and P + , together with some vertices of C\ and C2, will form a cycle different from C\ and Ci- This is contrary to our assumption. Now let v G V(Ci) be a vertex of the shortest path connecting C\ and C2. Then V(G)\{v} induces a subgraph H of G. Let Gi,G2,...,Gm(m > 2) be the components of H. One

43

of these components, say G±, contains the other vertices of C\. Hence, Gx is a tree. Let X = V(G2)uV(G3)U...UV(Gm)U{v}. Then X induces a connected unicyclic subgraph K of G. By Lemma 2.6, r{K) = 2. Let SI = {Vi,!^} be a vertex treepartition of K. Then Vt+ = Q U {V(G\)} is vertex-tree partition of G. By Definition 2.2, it follows that T{G) < 3. Thus, T(G) — 3 and the assertion holds for n — 2. Next, suppose that the assertion holds for all A; < n. We show that it also holds for n+1. To this end, let Ci, C2, •••, Cn, Cn+i be the disjoint cycles of G. Then T{G) > n + 2 by Lemma 2.7. Let v G V{Cn+i). Then V(G)\{v} induces a subgraph H of G. Consider the following cases: Case 1. Suppose H is connected. Then it contains exactly n cycles (the cycles Ci, C 2 ,..., Cn). Hence, by assumption, r(H) = n + 1. Let Q, = {Vi, V2,, Vn+i\ be a vertex-tree partition of H. Then Q+ = fiu{{w}} is a vertex-tree partition of G. Hence, r{G) < n + 2. Case £. Suppose if has components Gi, G2, •••, Gm(rn > 2). Then one of these components, say G\, contains the other vertices of Cn+\. Suppose first that G\ does not contain all the n cycles C\, C2,..., Cn. Without loss of generality, assume that Ci,C2, ...,Ck(k < n) are the only cycles contained in Gi. Let X = {v} U V(G2) U ... U V(Gm). Then X induces a connected subgraph K of G. Further, K contains the cycles Cfc+i, Ck+2, •••, Cn. By the inductive hypothesis, r{G\) = k + 1 and T(K) = (n - k) + 1. Let Qx = {Vi, V2,, Vk+1} and fi2 = {Wi, W2,, W(n_fc)+1} be vertex-tree partitions of Gy and K, respectively. Then Q, — Vti U f^ is a vertex-tree partition of G. Hence, r(G) < |fi| = (k + 1) + (n - k) + 1 = n + 2. If Gx contains all the cycles Ci,C2, ...,Cn, then r(Gi) = n + 1. In this

44

case, the subgraph K induced by X is a tree and so,r(H) = 1. Consequently, r(G) < n + 2. In both cases, we see that r(G) < n + 2. Therefore, T(G) = n + 2. This proves the theorem. T h e o r e m 2.9 Let G be a graph with exactly k components. If G has exactly n cyles and these cycles are disjoint, then T(G) = n + k. Proof: Let Gx, G2,..., Gk be the components of G. Then r(G) = E*U 7"(Gi) by Theorem 2.5. For each % e {1,2,, A;}, denote by ?7J the number of cycles in G{. Then, by Theorem 2.8, we have

r(G) = Y.riG,) »=i

=

n + k.

The following is a quick consequence of Theorem 2.9. Corollary 2.10 Let G b e a graph with exactly k components. If G is unicyclic, then r(G) = k 4-1. Proof: Apply Theorem 2.9 with n — 1. R e m a r k 6 For all n > 3, the cycle graph Cn is connected and unicyclic. Therefore, from Corollary 2.10, T{Cn) = 2. T h e o r e m 2.11 Let G be a graph. Then T(G) > 2 if and only if G is disconnected or else G has at least one cycle.

45

Proof: Suppose that r(G) > 2. If G were disconnected, then we are done. So, suppose that G is connected. If G does not contain a cycle, then it must be a tree. Thus, by Lemma 2.3, T(G) = 1. This gives a contradiction. Hence, G has at least one cycle. Conversely, assume that G is disconnected. Then G is not a tree. Hence, r(G) > 2 by Lemma 2.3. The same argument holds if G has at least one cycle. Theorem 2.12 Let G be a graph. Then T(G) = 2 if and only if G is a forest with exactly two components or else G is connected, has a cycle and a subgraph H which is a tree such that X\V(H), in turn, induces a tree. Proof: Suppose that r(G) = 2 and suppose that G is not a forest with exactly two components. Observe that if G is disconnected with T(G) = 2, then G cannot have more than two components. Thus our assumptions force G to be connected. Further, since r(G) = 2, it follows from Lemma 2.3 that G is not a tree. Consequently, G contains at least one cycle. Now, let 0 = {Vi, V2} be a vertex-tree partition of G and H the subgraph of G induced by Vi. Then H is a tree by Definition 2.1. Again, by Definition 2.1, V(G)\V(H) = V2 induces a tree. Conversely, suppose that G is a forest with exactly two components G\ and G2. From Theorem 2.10, we have r{G) > 2. Note that by definition of a forest, G\ and G2 are both trees. Let V1 = V(GX) and V2 = V(G2). Then Q = {VUV2} is a vertex-tree partition of G. By Definition 2.2, r(G) < 2. Therefore, T(G) = 2. Suppose now that G is connected, has a cycle and a sub-

46

graph H which is tree such that X\V(H) induces a tree. Put Vi = V(H) and V2 = X\VL Then Q = {VUV2} is a vertextree partition of G. By Definition 2.2, r(G) < 2. But since G contains a cycle, it follows from Theorem 2.11 that T(G) > 2. These two inequalities mean that r(G) = 2. This completes the proof of the theorem. Corollary 2.13 Let G be a connected graph with cycles. If the cycles of G have a common non-cutvertex v , then r(G) = 2. Proof: Let Vi = {v} and V2 = V(G)\{v}. Let H be the graph induced by Vi- Since v is a non-cutvertex, its removal yields a connected graph. Hence, H is a connected subgraph of G. Now, if H has a cycle, say C, then C must be a cycle in G. Hence, by assumption, v 6 V"(C). But this is not possible because v is not in V2 = V(i?). Therefore, H is a tree. Since Vi induces the path Pi (which is a tree), the desired result now follows from Theorem 2.12.

References [1] Akiyama, J., Era, H., Gervacio, S., and Watanabe, M., Path Chromatic Number of Graphs, Journal of Graph Theory, Vol. 13, No. 13, 1989, pp. 569-575. [2] Arugay, E. B., On the Path Chromatic Number of Graphs, Ph.D. Thesis, Ateneo de Manila University, Philippines, 1990.

47

[3] Buenavista, R.N., On the Tree Chromatic Index of a Graph, M.S. Thesis, MSU-Iligan Institute of Technology, Philippines, 1996. [4] Harrary, F., Graph Theory, Addison-Wesley Reading, MA, 1969. [5] Mania, E., On the Tree Chromatic Number of Graphs, M.S. Thesis, MSU-Iligan Institute of technology, Philippines, 1997. [6] Toft, B., Graph Colouring Problems Part I, Universitas Othiniensis, Denmark,1987.

C O N S T R U C T I O N OF L A G R A N G I A N A N D N O N - L A G R A N G I A N I P LOOPS ALEXANDER S. CARRASCAL SciTech R&D Center, Polytechnic University of the Philippines E-mail: [email protected] This paper introduces several practical methods for constructing Lagrangian and non-Lagrangian IP loops. The construction of Lagrangian IP loops L involves three known factorable groups Gk, k = 1,2,3, each of order 2m, having a common normal subgroup H of order m such that G^/H is isomorphic to C2, where C2 is the cyclic group of order 2. On the other hand, let Gk, fc = 1,2,3, be groups, each of order m — rs, IT < rs, with a common subgroup H of order r. Then non-Lagrangian IP loops of order (3s — 2)r can be constructed.

1

Introduction

The study of quasigroups and loops is gaining much interest because of their many applications in both pure and applied mathematics. However, not much is known about these interesting structures because their theory is a relatively young discipline. A quasigroup (Q; •) is a non-empty set Q with a binary operation • such that for any two elements a,b S Q, the linear equations a- x = b,

y•a=b

have unique solutions for x and y lying in Q. A loop (L; •) is a quasigroup variety with a unique identity element e such that e • x = x = x • e for all x e L. A quasigroup (loop) (Q; •} is said to possess the left inverse property (LIP) if there exists a bijection J\ : a —• ax on Q such that ax{ax) = x for every x € Q. Similarly, a quasigroup (loop) {Q; •) is said to possess the right inverse property (RIP) if there exists a bijection Jp : a —> ap such that (xa)ap = x for every x £ Q. If a quasigroup (loop) has both LIP and RIP, it is said to have the inverse property (IP). 2

Parastrophs and Complements

Let a be a fixed element in Q. We define the left and right translation maps L(a) : Q —• Q and R(a) :Q-^Q, respectively, by xL(a) — a • x

and 48

xR(a) = x • a

49

for all x € Q. Since the mappings L(a) and -R(o) are bisections, they have inverse maps L{a)~l and R{a)~x. These inverse maps can be exploited to define two binary operations \ and / for the set Q as follows x

\y = yL(x)~1

and

x/y =

xL{y)~x

where x\y = z if and only if x • z = y and x/y = z if and only if z • y = x. These operations are called left division and right division, respectively, and the quasigroups {Q; \ ) and (Q; /) are called conjugates of (Q; •). Using functional notation, one can write F(a, b) =c instead of a- b = c and substitute symbols F _ 1 and _ 1 F for operations / and \ , respectively. Going a step further, one can define three other operations on Q associated with F, as shown in the table below:

(l)F(x,z)=y (2)F~1(y,z)=x (3)-lF(x,y) = z

(4)-\F-i)(y,x)=z (5)(-1F)-1(z,y)=x (6)(-\F-L))-\z,x)=y

The six quasigroups associated with each of these operations are also referred to as parastrophs 1 . It is easy to show that ( F - 1 ) " 1 = F, ~\~1F) = F and ( ^ F - 1 ) ) - 1 = " ' ( ( ^ F ) ^ 1 ) . Moreover, the following pairs of quasigroups: (1) (Q; F " 1 ) and (Q; ( ^ F ) " 1 ) , (2) (Qr'F) and ( Q ; " ^ 1 ) ) , and (3) xa as the right and left a-complement maps, respectively. If Y{a) = *7(a)> we shall call it simply as the a-complement map and denote it by 7(a). 3

Multi-^ Systems and Quasidirect Products

Definition 3 Let {{Q; k) \k = 1, ...,r} be a set ofr quasigroups each of order n with binary operations <j>k. The collection of such quasigroups, denoted by (Q; 3>), where $ = { system2 of order (n;r). Definition 4 Let (P; •) be a quasigroup of order m and let {Q; $) be mvMv system of order (n;m 2 ), where $ = {ij\i,j G P}. The quasidirect

50

product*3 of(P;-) and(Q;$), denoted by (P;-)®(Q;$) or simply by P®Q, is a system (G;o) such that G = {(p,q)\p € P, q € Q} with composition (pi,qh) o (pj,qk) = {vi • Pj,qh4>pUP:jqk) where each <j>Pi p. defines a local operation on Q. 4

Construction of Special Classes of Lagrangian I P Loops

We shall now discuss several methods for constructing special classes of Lagrangian IP loops using parastrophs and their isotopes. 4-1

Construction of Totally Symmetric Loops of the Form L(V, C = {<j>ij\i,j € P} where 01,1 02,1 03,1 04,1

= = = =

° ° ° °

01,2 = ° 02.2 = °

03.2 - - M ^ 1 ) 04.2 = r ^ -

1

) ) -

1

01.3 02.3 03,3 04.3

= ° =~* F = ° = t ^ ) " '

01,4 = ° 02,4 = F = * 03.4 = F-1 04,4 = °

Then (G; o) is a totally symmetric IP loop of order n = 2 m + 2 which we shall denote by L(V,C2m,Q). Proof. Let i,j, k € {2,3,4} and let x,y,z G G where x — (i,qx), y = (i>9»)> a n d z = {kiQz)- It is not difficult to show that the parastroph operations satisfy 0 i;fe 0 ij - = 0 i) j0i )fe = t and 4>i,k4>j,k = 0j,fc0i,fc = itk{qx,qz) = qy, then 0 i ) ) j (g x ,%) = qz and 0j)fe(kqz) = (i, qx) o (j, qy) = (* • J, 9 * 0 ^ ) = {k, qz) = z. Similarly, (xoz)ozf = (xo;z)o.2 = [(i, &,)(*, &)](*, ge) = (i-fc, qx(j>i>kqz)o{k,qz) = {j,qy) o (Kqz) = (j • k, qy4>jikqz) = (i,qx) = a;. Hence (G;o) = L f V . V ^ . Q ) is IP and totally symmetric. 4-2

Construction of IP Loops of the Form L(V,

Dm,Q)

Theorem 6 Suppose (Dm; x) = (C 2 ;o) ® (i>k(qx,qz) = qv, then <j>ij(qx,qy) = qz and <j>jik(qy>qz) = qx. It follows that xA • (x * z) = x * (x • z) = (i,qx) * [(i,iikqz) = (i,qx) * (j,qy) = (i • j , qx<j>itjqy) = (k,qz) = z. Similarly, (x*z)*zf = (x*z)*z = [(i,qx)*(k,qz)]-k(k,qz) = (i-k, qxiikqz)* (k,qz) = (j,qy)*(k,qz) = (j • k, qy(j>jtkqz) = {i,qx) = x. Thus, the system (G; •) = L(V, Dm, Q) has inverse property.

52

4-3

Construction of IP Loops of the Form L(G\, G2, G3)

We shall now present a construction method that utilizes isotopes of the parastrophs of a given quasigroup. Theorem 7 Suppose (Gk;