Letters to the Editor
The Mathematical Intelligencer encourages comments about the material in this issue. Letters to the editor should be sent to the editor-in-chief, Chandler Davis.
Pygmies and their Shadows
In my review of Indiscrete Thoughts, by Gian-Carlo Rota [4], I noted that the last sentence of the book, "When pyg mies cast such long shadows, it must be very late in the day," was an adap tation of Erwin Chargaffs dictum [1, p.641] "That in our days such pygmies throw such giant shadows only shows how late in the day it has become." I am grateful to Professor Kurt Bret terbauer of Technische Universitat Wien for pointing out that Chargaffs formulation is itself based on a well known saying attributed to the Viennese satirist and critic Karl Kraus: "Wenn die Sonne der Kultur tief steht, werfen selbst Zwerge lange Schatten" ("When the sun of culture is low, even pygmies cast long shadows"); cf. [3, p.421]. Since Chargaff cites Kraus in his autobiography as having been "the deepest influence on my formative years" and "truly my only teacher" [2, p.l4], there can be no doubt that he was familiar with Kraus's mot. It will not have escaped the careful reader that Kraus's formulation does
Taking
the
not say quite the same thing as those of Chargaff and Rota; indeed, the lat ter provide a kind of incomplete or de fective converse to the former.The de fect in question was pointed out already in my review ("Can I be the only one to have noticed that shadows are just as long in the morning-or are we all late sleepers?").
REFERENCES
1 . Erwin Chargaff, Preface to a Grammar of Biology,
Science
1 72 (1 971 ), 637-642.
2. Erwin Chargaff, Heraclitean Fire, Rockefeller Univ. Press, New York, 1 978.
3. Johannes John,
Rec/ams Zitaten Lexikon,
Stuttgart, 1 992. 4.
The Mathematical lntelligencer 21
(1 999),
no.2, 72-74.
Lawrence Zalcman Department of Mathematics and Computer Sc ience
Bar-llan University
52900 Ramat-Gan Israel e-mail:
[email protected] Easy Way
...till, demanding proof, And seeking it in everything, I lost All feeling of conviction, and, in fine, Sick, wearied out with contrarieties, Yielded up moral questions in despair, And for my future studies, as the sole Employment of the enquiring faculty, Tum'd towards mathematics, and their clear And solid evidence ... William Wordsworth,
The Prelude
© 2000 SPRINGER-VERLAG NEW YORK, VOLUME 22, NUMBER 2. 2000
3
SAMUEL S. HOLLAND, JR.
My Years as a Fu -Time Industria Mathematician
•
~
ndustrial mathematics has been getting more attention lately; witness the articles [2,6,15,16]. Much of this renewed attention and interest surely derives from the em ployment concerns of our graduate students who have been facing a dismal acade mic job market. Many, many years ago, from
as an industrial mathematician.* While my work since that time has been solely in "pure" mathematics in an academic setting, nonetheless many of the memories from those early days are still fresh with me. So it seemed to me that I might still usefully contribute to the ongoing discussion by putting on record a couple of my own experiences from those days long gone by. And by putting forth some of my own personal conclusions drawn therefrom, about industrial mathematics itself, and about the training of industrial mathematicians. I shall describe two projects in which I was involved, one at the beginning of my time with Technical Operations, Inc. (Tech/Ops), then a fledgling Massachusetts firm, and one at the end of my service with that company. Project One
Project One, a study of the penetration of neutrons in air, was supported in part by the United States Air Force un der a contract monitored by the Director, Research Directorate, Air Force Special Weapons Center (AFSWC), Kirtland Air Force Base, New Mexico. This work was done in 1955. Dr. Paul I. Richards was the primary investigator
on the project, and I was the other half of the team, hav ing just joined the company (Tech/Ops). Testing of nuclear weapons was still underway during this "cold war" period, and the Air Force wanted to know what neutron flux to ex pect from a nuclear weapon explosion in air, not only as a function of the horizontal distance from the explosion, but also as a function of neutron energy. They gave us three months, and they wanted numbers. At this point it might be well to point out to the aspir ing industrial mathematician the difference between work ing in a small start-up company, like Tech/Ops in 1955, where the entire company could go to lunch together, and a large multinational firm like AT&T. A small company is, and must be, very customer-oriented.There is no place to hide-each scientific staff person is usually directly in volved in some sort of commitment to a customer, and needs to get the job done-i.e., fulfill the contract to the customer's satisfaction.There is usually very little time to pursue ancillary mathematical questions that come up, however interesting they might be. In contrast, a large firm can internally finance research, and can allot more time for
"Actually 1 954-1 959 full-time, and 1 960--1 965 full-time summers but part-time otherwise.
4
THE MATHEMATICAL INTELLIGENCER © 2000 SPRINGER-VERLAG NEW YORK
1954 through 1965, I made my living
basic studies that have the potential to enhance the firm's corporate expertise [2]. So, in essence, AFSWC had a contract with Paul Richards and me; we were the "team." We began by settling on a sim plified model that we could realistically hope to analyze, yet would capture the essential features of the actual situation. Our model: an isotropic, monoenergetic point neutron source in an infinite, constant-density air medium. The physical process is this: a neutron, emitted at en ergy E0 by the source, proceeds unimpeded until it strikes an air molecule, either nitrogen (N2), or oxygen (02). In this collision three things can happen: (1) the neutron can scatter elastically-a "billiard ball" collision, (2) the neu tron can scatter inelastically, leaving behind some of its en ergy in the scattering molecule, or (3) the neutron can be absorbed by the struck molecule-it disappears. In the first two cases the neutron loses energy and changes direction, then proceeds to the next collision. The probability of a collision of any one of these three types depends on the energy of the impacting neutron, and is generally known only numerically, where it is known at all. One has analyzed this steady-state process completely when one knows the neutron flux N(r, n, E), the number of neutrons per second per unit energy E crossing a unit area orthogonal to the unit vector n, at distance r from the source. The flux N(r, 0, E) satisfies, and is determined by, the trans port equation. Thus we have modeled this industrial prob lem, as is very often the case, with a differential equation the transport equation. Actually, in this case, an integra differential equation of some considerable complexity. And now, having set up the mathematical model, our job is to use it to provide the information desired by the customer. The industrial mathematician frequently tackles problems in two stages: first, set up a mathematical model for the process under study (which we have just done); second, solve (in some fashion) the mathematical model. As we have just finished with the first stage of this particular project, we have reached a natural point for the question: What lessons are here for the training of today's industrial mathematician? As regards this training, I would first like to look at it apart from its mathematical component; then at the end of this section give my suggestions for the mathematical preparation of industrial mathematicians. First we must note that this specific kind of problem penetration and diffusion of radiation in matter-no longer has any contemporary interest. The day of nuclear testing, and the days of intense interest in nuclear reactors, are over. That particular study, so prominent 40 years ago, has gone out of fashion. Yet I think lessons can be drawn from our project that are still relevant today. Indeed, 40 years from now many of today's hot topics will probably have given way to new, difficult-to-imagine subjects. One would hope to formulate advice that shall remain valid even then. To set up our mathematical model, we needed to know the language and the basic aspects of physics, including atomic physics. I believe that basic instruction in physics and chemistry is a must for the aspiring industrial mathe matician-indeed for any serious student of science.
Further, today's industrial mathematician may have to know the language of molecular biology, or be conversant with the fundamentals of modem electronic circuitry. And as science advances, the training program must keep pace. The confidence of an industrial client will rise in propor tion to the consultant's familiarity with his teirninology and underlying concepts. But, however broad this training in the basic sciences, it seems impossible to have a program that covers all pos sibilities. Industrial mathematics includes an enormous va riety of areas, and requires many different skills. Here is a sample of the variety of jobs that flowed through Tech/Ops while I was there-I've listed the principals in parentheses: Penetration of neutrons in air (Paul Richards and me-this is Project One); Analysis of the method of constructing raised topographic maps from aerial stereo-photographic data (me); Mathematical explanation of "shock waves on the highway" (Paul Richards-these are traffic jams that persist long after the accident has been cleared); Fortran code for Monte Carlo calculations (Dom Raso); Is it pos sible to construct an engine that bums boron? (Everett Reed); Analysis of the exploding-foil hypervelocity gun (me-this is Project Two). And this tiny sample is taken from memory of work done by a small firm 40 years ago. One generalization that may hold up: industrial mathemat ics necessitates continuous on-the-job training. This is part of the fun of industrial mathematics, and part of the chal lenge. Often a given project will take a period of intense effort considerably longer than the three months Paul and I put into the AFSWC contract. Once a project is fmished, then it is on to another totally different one. Variety is the spice of industrial mathematics. Getting back now to our neutron penetration problem, having settled on the transport equation to model the process, Paul Richards and I had three months to use this equation to provide the Air Force with their report con taining quantitative predictions. The Air Force wanted N(r, E) = fN(r, 0, E)dO, the neutron flux integrated over all solid angles. This tells them the number of neutrons with energies between E and E + dE per unit area striking a small spherical target at distance r from the source. The transport equation, a linear but very complicated integra differential equation, is basically just a conservation equa tion-it counts neutrons at a particular energy E entering and leaving an infinitesimal volume in space. So advanced calculus training prepares one to understand this equation. But getting numerical answers from it is another matter, especially in our case where the input data, the various "cross-sections," were known only as rapidly varying nu merical functions of the neutron energy.Furthermore, the angular distributions of scattered neutrons were also com plicated and generally numerically given. The transport equation, in the early 1950's, was the fo cus of a great deal of attention, occasioned by generous government funding. Thus, when Paul and I began to work, we had a substantial high-quality literature to consult. The work of Lewis V. Spencer was crucial [17,18]. Spencer solved the transport equation this way: He noted that it was
VOLUME 22, NUMBER 2, 2000
5
equivalent to an infinite linked system of Volterra integral equations in the moments, fr nN(r, E)dr, of the distribution function. These integral equations admit numerical solu tions so that the first few moments can be calculated. Spencer then devised techniques to reconstruct N with reasonable accuracy from a knowledge of its first few mo ments. His method was tailor-made for our problem. We applied it successfully, and gave the Air Force their num bers in the allotted time [7,8,9]. In this pioneering outstanding work of L. V. Spencer, an exemplary model of industrial mathematics in my opinion, one finds the following topics: series expansions in polyno mials orthogonal with respect to a given weight function; the Fourier-Laplace transform and its inverse-particularly the relation between the strength of a singularity of the trans form and the asymptotic behavior of the original function; Gaussian quadrature; and Bessel and other special functions. These topics are representative of a general area, that one might call classical analysis, that should certainly be part of our graduate training in industrial mathematics. This training in classical analysis can be general. To han dle the variety of special applications that can arise in prac tice, one may consult reference books and on-line sources. On special functions, there is the classic work by Magnus, Oberhettinger, and Soni [13] and the three volumes of the Bateman Manuscript Project [4]. The Comprehensive Hand book [1] edited by Abramowitz and Stegun is currently be ing revised and put on the Web [12]. On integral tables and integral transforms, there are the two volumes [5], also of the Bateman Manuscript Project. The book [3] by Campbell and Foster has an extensive table of Fourier integrals. As for numerical tables and graphs of the special functions, they are rapidly becoming, like me, a relic of the past. Modern software packages such as Mathematica have built-in subroutines for most of the special functions, like Bessel functions, that will print out tables and graphs with a few touches of the keys. And these software packages enable one to use the special functions in programmed cal culations just like the sine and cosine. Beyond classical analysis, there are no other graduate courses that had a direct bearing on my work in industry. But then, I worked only briefly in one small company. Recommendations from other industrial mathematicians would be welcome here. In addition to course work, I be lieve that any industrial mathematics graduate program should provide the opportunity for its students to spend a year or two in industry. One can give all the academic courses in industrial mathematics that one likes, there is no substitute for being there. As for the undergraduate program in mathematics, I would recommend broad training at this stage, rather than a premature specialization. This from my own academic experience with undergraduate mathematics majors who really need to see various kinds of mathematics before they decide in which direction they would like to tilt. Summer work in industry is valuable, both financially and academ ically, for any undergraduate mathematics major. Apart from the course listings in mathematics, my own
6
THE MATHEMATICAL INTELLIGENCER
experience-on both sides of the aisle-has left me with the conviction that the manner in which the mathematics is conveyed is at least as important as what is taught. I have come to the earnest belief that, in the teaching of mathematics, understanding should come before rigor; that motivation, geometric meaning, and physical connections (where applicable), even numerical experimentation, should occupy just as prominent part of the presentation as proofs. This especially at the undergraduate level for students tilting toward applicable mathematics, but ap propriate to some extent for all students at all levels. In my university life, I put considerable effort into implementing this philosophy through the writing of my book [10]. (This solely at the undergraduate level, as most of my graduate teaching centered around my own research in algebra.) My book deals with orthogonal function expansions, the various classical ordinary and partial differential equations, and Dirac's delta function, among other things. My indus trial experience together with my subsequent experience in the undergraduate classroom combined in my mind to shape my presentation of these subjects according to the above-mentioned philosophy. For example, in treating Fourier series and other or thogonal function expansions, I had the students compute a few partial sums of the expansion and graph the result against the function. For Legendre polynomials, I explained how the Legendre partial sum is a global approximation while the Taylor polynomial is local-again having them il lustrate this by numerical tables and graphs. All this nu merical work is easy with currently available hardware and software. As for convergence of these series, many text books spend a great deal of time discussing various hy potheses that will guarantee pointwise, uniform, or ab solute convergence. In our work on the transport equation, Paul Richards and I took for granted that any such series "represented" its function. In this we were following all those we consulted and all those whose work we read. Not wishing to be this cavalier in class lest my academic col leagues suffer from shock, I took a cue from a remark I heard Professor George Mackey make (in connection with his work on quantum mechanics), namely that L2 conver gence is more appropriate for physical applications be cause a physical measurement is an average. So in my book I used L2 convergence exclusively. Testing for L2 conver gence amounts to determining whether or not the integral of Jt12 is finite or infinite, so falls under the calculus topic of "improper integrals". (Lebesgue measurability is irrele vant when dealing with functions that arise in practice.) In dealing with the wave equation, the heat equation, and Schrodinger's equation I took pains to derive them in detail from basic physical principles. (This was a special challenge for Schrodinger's equation.) I did these deriva tions to emphasize the physical connections, and to illus trate how one captures the essence of a complex situation in a mathematical model. Especially valuable training for an aspiring industrial mathematician, and, I think, valuable also for any student of mathematics no matter what his or her eventual specialty.
Dirac's delta function figures prominently in our work in the transport equation. And we used it, as did all others in this area, as ajunction. While the work of Sobolev, Schwartz, Lighthill, and others demonstrates that Dirac's formalism has a rigorous foundation, yet, when it comes to practical cal culations, no alternative formalism comes anywhere near Dirac's in elegance and simplicity. So, in my book, I explained the delta function just as Dirac explained it, and used it just as he used it. I feel that such training provides the students with a very useful tool. And, when properly explained, it does not corrupt their mathematical education in any way. The course based on my book ran for many years here at UMass. Taught not only by me, but also by Professors Richard Ellis, H. T. Ku, and Peter Norman. But, one se mester before I retired, the course was dropped. I know of no other university that developed a similar course. The lack of enthusiasm for my book exceeded my wildest ex pectations. No matter-while I may have overreached to some extent, I know that, basically, I am right. Project Two
block. The gun is fired by discharging a large capacitor through the foil. The explosion of the lower half of the foil is totally contained since it is between the separating insu lator and the lucite block. The upper half of the exploding foil impinges on the Mylar sheet, blowing a circular piece through the 0.32 em hole in the top of the sandwich. This punched out Mylar disk is the projectile of the gun A typi cal firing sequence uses a !-microfarad capacitor bank charged to 100 kilovolts, which thus contains 5000 joules of stored energy. The firing destroys the gun. As I mentioned, the standardized gun described above was the end product of a three-year experimental program. This program included not only trial-and-error evaluation of various gun designs, but included as well development of complex optical and electronic ultra-high-speed diag nostic systems to measure the various physical parameters, such as energy deposition rate and particle velocity. Now, with the testing phase of the project winding down, and with the more-or-less standardized guns being produced in quantity, Tech/Ops, and the Contractor, sought a theory for the gun a theory that might predict some of the observed phenomena-especially particle velocity and might suggest means for improving the performance of the gun, means more economical than cut-and-try. Standard ballistic theories do not apply to this gun for many reasons: (1) the energy is deposited in the breech electrically rather than chemically, (2) the rate of energy deposition is very great, on the order of 500 joules per mi crosecond, (3) the time scale is very short-the whole fir ing sequence is over in a few microseconds, (4) the pro jectile mass is of the order of milligrams, comparable to the mass of the driver gas, and (5) the projectile velocity is in the centimeter-per-microsecond range. A theoretical analysis needs to incorporate these conspicuous features. There is no "typical" industrial mathematics problem. But the problem which I have just described in some consider able detail does exemplify certain aphorisms that I have al luded to earlier. Industrial mathematics is fun. It is exciting. The industrial mathematician sees new things, has a wide va riety of experiences. Can be led into a room, wait while a ca pacitor is charging, see a brilliant flash and a tremendous crack, then be led away with the entreaty, "We need a theory for that." Industrial mathematics differs from academic pure mathematics which relies primarily on self-motivation, and sometimes suffers from lack of such motivation. In industrial mathematics, the problem is here and the time is now. My theory of the exploding-foil gun was based on a num ber of simplifying assumptions: (1) thermal equilibrium is maintained in the breech, (2) energy losses to the breech, projectile friction losses, and blow-by losses are negligible, (3) the pressure and temperature of the breech gas are functions only of time, not of position, (4) the cumulative energy in the breech is a linear function of time Cexperi mentally observed), and (5) the breech is filled with a per fect, monatomic, non-ionizing gas. Let x(t) denote the distance the projectile has moved in time t, x(O) x' (0) = 0. If V0 is the initial breech volume, A the area of the breech (same as the area of the projec.
,
Project Two, done ten years later, in 1964, dealt with the ex ploding-foil hypervelocity gun This project was done under contract between Tech/Ops and the Air Force Materials Laboratory at Wright-Patterson Air Force Base, Ohio. Its pur pose is well described in the introduction to the final report: .
. . . The goal of the over-all investigation was to de velop a system that could accelerate milligram-size particles to velocities in excess of 30 kmlsec. With such a capability, the effects of micrometeoroid impacts on materials could be studied in the laboratory as a requisite first step in the development of protective de vices for space vehicles and missiles subject to dam age and destruction by solid particles moving at high velocities in space. The "system" to generate these high-velocity particles was the exploding-foil hypervelocity gun. The experimen tal program to develop this gun had begun in 1961, and was already at a mature stage in 1964 when I was asked to con tribute a theoretical analysis. So there was an intense learn ing period for me as I was brought up to speed on this com plex experimental program. The exploding-foil hypervelocity gun is a sandwich. The ham is an insulator sheet 0.080 in thick The lower piece of bread is a 1-cm-thick lucite block, and the upper piece a 0.161-cm Fiberglass plate. This upper piece has a 0.32-cm hole in its center. Between the ham and the upper piece of bread is interposed a thin (0.0254 em) sheet of Dupont Mylar. Flat copper strips, fed in through the side of the sandwich on either side of the ham, are joined at the center by an alu minum foil loop (the ham has to be penetrated to complete the loop). Hence, looking down on the sandwich from the top one sees, through the 0.32-cm hole in the top Fiberglass plate, first the Mylar sheet, next the top half of the aluminum foil loop, then the separating insulator sheet (the ham), then the bottom half of the foil loop, and fmally the backup Lucite
=
VOLUME 22, NUMBER 2, 2000
7
tile), then L V0/A has the dimension of length, and y(t) 1 + x(t)IL is dimensionless. Combining the equation of state of a perlect gas, the formula for the internal energy in a perlect monatomic gas, Newton's law F ma, and conservation of energy, one gets the following nonlinear second-order differential equation for y: =
=
=
�:� (��r = at
3y
,
+
Here a =
+
y(O)
=
1,
y'(O)
=
0.
(1)
�)L2 , where ,\is the constant energy dep-
(m + 3
+
s(z ' )2
=
1
g' z(O)
=
1,
z'(O)
=
'
(2)
gration is inherently more accurate than numerical dif ferentiation. Make the substitution w z� in (2), then in =
tegrate once to get =
w
+
Neutron fu l x spectra in air,
J. Appl. Phys. 27 (1 956), 1 042-1050. [9] S. S. Holland, Jr . ,
Neutron penetration in infinite media; calcula
J. Appl. Phys. 29 (1 958),
tion by semi-asymptotic methods,
[1 0] S. S. Holland, Jr.,
Applied analysis by the Hilbert space method,
[1 1] S. S. Holland, Jr. ,
The exploding-foil hypervelocity gun,
1 964,
preprint. [1 2] http://math.nist.gov/DigitaiMathlib/ [13] W. Magnus, F. Oberhettinger, and R. P. Soni,
Formulas and the
orems for the special functions of mathematical physics,
3rd enl.
ed., Springer-Verlag, 1 966. [1 4] R. W. O'Neil, S. S. Holland, Jr., T. Holland, V. E. Scherrer, and H. Stevens,
1
IS
where z(s) = y(t), and primes denote differentiation with respect to s. Equation (2) would seem to require a numer ical solution. To cover the time period of interest, about 3 microseconds, we need information over the range 0 :5 s :5 105. Hence any numerical solution method needs both sta bility and convenience. At this point there comes into play an important rule of numerical methods: numerical inte
3sw'
[8] S. S. Holland, Jr. and P. I . Richards,
AFSWC-TR-55-27 (Unpublished).
Dekker, 1 990.
2
3sz")z
Penetration of neutrons in
air,
827-833.
osition rate (Joules/sec), m is the mass of the projectile, and M the mass of the driver gas. While y(t) is dimensionless, t is not. Introduce the di mensionless variables = at?. Equation (1) then becomes
(2z'
[7] S. S. Holland, Jr. and P. I. Richards,
Effects of hypervelocity impacts o n materials,
Tech/Ops
Report AFML-TR-65-1 4 (Unpublished). [1 5] D. G. Schaeffer, math,
Memoirs from a small-scale course on industrial
Notices Amer. Math. Soc. 43 (1 996), 550-557.
[ 1 6] J. Spanier,
The mathematics clinic: an innovative approach to re
alism within an academic environment,
Amer. Math. Monthly 83
(1 976), 771 -775. [1 7] L. V. Spencer and U. Fano,
Penetration and diffusion of x-rays.
Calculation of spatial distributions by polynomial expansion,
J. Res.
Nat'l. Bur. Stds. 46 (1 951 ), 446-456. [1 8] L. V. Spencer,
Penetration and diffusion of x-rays: Calculation of
spatial distributions by semi-asymptotic methods,
Phys. Rev. 88
(1 952), 793-803.
1
4 8 ds Vw- 1, 27 0
w(O)
=
1,
w ' (O)
=
2 . 27
(3)
AUTHOR
Still nonlinear, and still requiring a numerical solution, but much easier to solve accurately than (2). My numerical procedure to solve (3) was coded in Fortran for me by Peter Flusser, a company expert in pro gramming, and was on an IBM7094 (this was in 1964). The single solution to the dimensionless equation (3) al lows one to compute projectile velocities and other gun pa rameters for any particular gun configuration [11,14]. The maximum difference between theory and experiment was 12 percent, despite all the simplifying assumptions. In in dustrial mathematics, as in life, it sometimes pays to be lucky.
run
SAMUEL S. HOLLAND, JR.
Department of Mathematics and Statistics University of Massachusetts Amherst, MA 01 003-4515
REFERENCES
[1 ] M. Abramowitz and I. A. Stegun, tions,
N. B.S. Appl. Math. Ser. 55, U. S. Gov't. Printing Office, 1 965.
[2] R. Calderbank, in industry,
plications,
D. Van Nostrand, 1 954.
[4] A. Erdelyi (Editor),
Higher transcendental functions,
3 vols. ,
Tables of integral transforms,
2 vols. , McGraw
Hill, 1 954. [6] A. Friedman and F. Santosa,
Graduate studies in industrial math
Notices Amer. Math. Soc. 43 (1 996), 564-568.
THE MATHEMATICAL INTELLIGENCER
Samuel S. and his
v
Holland, Jr., recei ed his
Ph.D. in Mathematics under
in 1 961 . His Bachelor's thesis
Review. Mu ch
of Professor
McGraw-Hill, 1 953. [5] A. Erdelyi (Editor),
ematics,
e-mail:
[email protected] A personal perspective on mathematics research
Notices Amer. Math. Soc. 43 (1 996), 569-57 1 .
[3] G . A. Campbell and R . M . Foster, Fourier integrals for practical ap
8
USA
Handbook of mathematical func
of his
life between
B.S. in Physics
in 1950,
Lynn Loomis at Harvard
was published
in
Physical
then and his present status
Emeritus is under considerati o n
in this a rticle .
He
enjoys his wife Mary of 41 years, his children and g randch ild ,
,
his friends and colleagues, the ocean, downhill skiing, choco
late,
scotch whiskey
and a good cigar.
RICHARD KAYE
Minesweeper NP-comp ete NP-completeness
Many programming problems require the design of an al gorithm which has a "yes" or "no" output for each input. For example, the problem of testing a whole number for primality requires an algorithm which answers "yes" if the input number x is prime, and "no" otherwise. In trying to devise an algorithm to solve a given problem, one aspect of obvious practical importance is the time it takes to run. Since a typical algorithm may take more time on some inputs than others, the running time of an algorithm is usually regarded as a function of the input. For technical reasons, it is convenient to consider the way this function varies with the number of symbols required to write the in put. (This number of symbols is usually denoted by n.) For example, for the input 17, our algorithm may require this number to be written in binary (as 10001), so here n = 5. Different algorithms for the same problem may run in different amounts of time, due perhaps to the different cod ing methods used or to different theoretical bases for the algorithms. However, it may be that for a particular prob lem, all valid algorithms can be shown to take at least a certain amount of time, due to the inherent difficulties in the problem being solved. Complexity theory aims to study the inherent difficulties of problems, rather than the time or memory resources used by any particular algorithm or program. It is certainly possible to find problems that can only be solved on a computer using a huge amount of time. It is also possible to fmd sensible-sounding problems that can not be solved on a computer at all! However, there are two classes of problems that are of greatest interest for com plexity-theorists. The first of these classes is the collection, P, of Poly nomial-time computable problems. These are the prob-
IS
lems that can be solved on a normal computer and within an amount of time of order n, or n2, or n3, or n4, (As be fore, n is the number of symbols required to write down the input to the problem. Note in particular that the run ning time of such a program is bounded by a polynomial in the length of the input, not the input itself.) Of course, for a rigorous treatment of the subject, a pre cise definition of the mathematical model of computer we are using and what constitutes the running time of the com puter, must be given. For the purposes of this article I will be less precise, but give here the two main points. Firstly, our computers will have an unlimited amount of memory that is to say that they always have enough memory to com plete the computation in hand. This does not seem partic ularly restrictive, as any terminating computation can only use a fmite amount of memory anyway, and for most al gorithms considered here, the amount of memory required for any particular computation can be estimated fairly ac curately in advance. Secondly, the time taken by the com puter is the number of steps required, where a single step can only process a single character's worth of information and a "character" comes from a fixed alphabet. (Characters could be single bits, or bytes, or 32-bit words, or symbols from some other finite set, provided this finite set is spec ified in advance.) To give an illustrative example, observe that arbitrary natural numbers can be represented on such computers (as sequences of binary digits, for example) and two such numbers can be multiplied together, but the time taken to multiply these numbers will not be a single step it will instead be a function of the length of the numbers, for the computer can only process the numbers character by-character. A large amount of heuristic evidence exists supporting the thesis that the notion of a polynomial-time computable . • • •
© 2000 SPRINGER-VERLAG NEW YORK, VOLUME 22, NUMBER 2, 2000
9
problem is independent of the particular computer model used. That is,
if a problem
is solved in polynomial time on
This algorithm is based on the property that a num
one computer then the algorithm used can be transferred
yx
nomial time there. There is also strong evidence that sug
It is recursive in the sense that it calls itself with
gests that the complexity class P consists of precisely those problems that are soluble
smaller values.
in practice on an ordinary com
1. On input x, if x = 2 answer "yes," and if x = 1 an
puter. Problems not in P may be theoretically soluble, but
swer "no." Otherwise go to the next step.
only with impractical running times even on the very fastest
2. Guess
computer.
3. Guess a prime factorisation
Nondeterministic Polynomial-time computable problems,
and
NP. These are problems that can be solved in polynomial
a 1 a2 . . . an of x
1
run the algorithm recursively to check that
each
time as before, but on a special "enhanced" computer able
-
ai is prime.
4. Verify that 2 c x - 1)/a; =I= 1 mod x for each prime fac tor ai of x- 1. If any of these fail, answer "no;"
to perform "nondeterministic" algorithms. The reason for the interest in NP is that this class contains a great many
otherwise answer "yes."
problems of significant practical importance that are not known to be soluble by an ordinary polynomial-time algo
y and verify that y x - 1 = 1 mod x. (If this
fails, answer "no" and stop.)
The second class of problems of interest is the class of
rithm, including some very well-known problems such as
x > 2 is prime if and only if there is y such that - 1 = 1 mod x and y'l =I= 1 mod x for all q < x - 1.
ber
to a different kind of computer and will also run in poly
Figure 1. Pratt's nondeterministic algorithm for primality.
that of the "travelling salesman." To defme NP, we just need to explain the idea of a non
deterministic
prime numbers, for example, it is not immediately obvious
algorithm. These algorithms are like ordi
how one might show that the set of primes (the comple
nary ("deterministic") ones except that there is an extra
ment of the set of composites) is recognizable in polyno
kind of instruction allowed which instructs the computer
mial time by a nondeterministic algorithm. The problem
to guess a number. The computer performing this instruc
here is to guess something that shows the input x is prime,
tion is assumed to have the very special ability always to
and then to verify our guess quickly, but what should we
make a correct guess if one is available, and it is this as
guess? In fact, there is just such a "certificate of primality,"
pect of nondeterminism that is difficult to implement in
as was first observed by Pratt1 (see Figure
1).
Needless to say, no "nondeterminism chip" has been de
practice! Having made a guess, the nondeterministic algo rithm is required to verify that the guess was indeed a cor
veloped to use in real computers (though some believe that
rect one, because only by doing this can it determine
quantum mechanics implies that something rather like non
whether a correct guess was possible at all.
determinism might be built into a usable device).
As already mentioned, the class NP of Nondeterministic
For example, it is easy to use nondeterminism to tell if a whole number input x is composite (i.e., not prime). The
can be solved in polynomial time on a nondeterministic
yz = x then the machine has
machine. It is generally believed that nondeterminism re
verified that the guess was correct, so may answer yes, the number x is composite. If
problems is the class of problems that
and
machine should guess two whole numbers compute their product, yz. If
y, z > 1
Polynomial-time
yz * x
then the machine may
ally does introduce problems that were not already in P, and also that there are NP problems whose complement
safely answer no, as in this case it is allowed to assume
does not lie in NP, but here lies the main problem. To date,
that no better guess was available, i.e., that x really is
no one has managed to fmd an NP problem and prove it is
prime. Since a single multiplication can be carried out
not in P. The famous "P = NP" question is whether there
rather quickly, this nondeterministic machine will decide
is such a problem. This is one of the most important open
if a number is composite very rapidly without any lengthy
problems in mathematics-perhaps even
search over all the possible factors.
tant open problem. It has the same status as Fermat's last
A nondeterministic machine is not allowed to guess the
the most impor
theorem before Wiles's solution, with a long history (going
answer ("yes" or "no") to the problem and output that, be
back well before computers). The majority of mathemati
cause the machine would not have verified this guess. The
cians believe that P and NP really are different (though sev
special power of these machines lies in the fact that it is
eral well-respected mathematicians consider it quite plau
not necessary to verify that any particular guess was
correct
(because only correct guesses are chosen
if
in
they
are available). It is only required to verify that a guess is
correct.
Because of the different nature of these "yes" and
sible that P
= NP), but no one has a proof. Every
mathematician dreams of solving a problem like this, and a huge number have tried, but no one has succeeded. The difficulty of proving that P * NP is not due to lack
"no" answers, it is not always true that the complement of
of examples of interesting problems in NP. In fact, mathe
a problem solvable using nondeterminism is as easy to
maticians now have a huge list of problems-including the
solve nondeterministically. In the case of composite and
travelling salesman and many others of practical interest-
VR. Pratt, "Every prime has a succinct certificate," SIAM J. Comput. 4 ( 1 9 75), 21 4-220.
10
THE MATHEMATICAL INTELLIGENCER
�
m
T F
A
F T
boolean circuit is a circuit built of
with inputs that may be true put
(T)
A
A
A
� � �� A
B
T T F F
T F T F
A
VE T T T F
A
B
AI\B
A
B
A + B
T T F F
T F T F
T F F F
T T F F
T F T F
F T T F
the familiar logic gates such as AND
or false
(F).
(/\),
OR
(V), XOR ( + ), and NOT (--.), each p2, ... , Pn and an out
A circuit will have several inputs labelled p 1,
q. The problem SAT is
Given a boolean circuit C, is there some combination of true/false values for the inputs of C so that the output of C is true? There are algorithms to answer this question, but none running in polynomial time is known. The obvious algorithm (to check all possible combinations of the inputs of C) takes too long, as there are 2n combinations for n inputs. SAT is NP-complete. Figure 2. The NP-complete problem SAT. which are in NP and for which we have a proof that if P =F
Although there are a great many NP-complete problems of
NP, then the problem is
not in P. A problem, A, is typically shown to be of this type by proving that it is NP-complete,
practical importance, no one has found one which may be
i.e., that every other NP problem, B, can be solved by a de
lieved that no such exist. Turning a necessity into a virtue,
terministic polynomial-time program which converts its in
many people have attempted to design cryptosystems so
put, x, for the problem B to an input,j(x), for the problem
that a potential codebreaker would have to solve an NP
solved by a polynomial-time algorithm, and it is widely be
A, with the property that the answer to problem B for in
complete problem in order to break the code-taking too
put x is the same as the answer to problemA for inputj(x).
much time even on the fastest computer. Either way, an
If there is a polynomial-time computable functionj(x) with
answer to the P = NP question would have significant prac
these properties, we say the problem B reduces
tical importance.
to the prob
lem A. Loosely speaking, a problem B reduces to a prob lem A,
if A
"includes" all instances of B as special cases,
and the NP-problem A is NP-complete if it "includes" (in this sense)
all other NP-problems.
The Minesweeper Game Many of the ideas mentioned above may be illustrated ef fectively with a game many readers will be familiar with.
To see the importance of this, consider a problem B in
Minesweeper comes
with Microsoft's Windows operating
NP, and suppose also that we are given an NP-complete
system. 5 In it, the player is presented with an initially blank
problem,A. Then there is a polynomial-time computer pro
grid. Underneath each square there may be a mine, and the
gram that converts each instance, x, of the problem B to
object of the game is to locate all these mines without be
an instance,J(x), of the problemA. But if our NP-complete
ing blown up. You select a square to be revealed; if it is a
problem A is actually in P, the problem A for j(x) can be
mine you are blown up (and the game is over), but with
solved in polynomial time by a deterministic algorithm,
luck, perhaps it isn't. In this second case, when the square
0 to 8, which is the num
hence B also can be solved in deterministic polynomial
is revealed you see a number from
time, because the answers for A on inputj(x) and B on in
put x are the same.2 This also applies to any other C in NP
ber of mines in the eight immediately neighbouring squares. Figure
(with a different functionf(x) of course), so if A is in P,
The numbered squares are the squares that have been re
then every problem in NP will be in P, i.e., P = NP. CooJ. 0;
(8)
Thus, for a fixed a, if K and V are given independently, i.e., there is no additional information on any connection be tween them, then the function s('T) is determined by these two parameters. However, if a relation K(V) is known, then s(T) is determined by the single parameter V. On physical grounds, K is always a function of V (the average diame ter dav depends on the injection rate), so the independence of K and V should be interpreted as a failure as yet to de scribe a function K(V). Of course, one should have physi cal sense when choosing independent values of K and V (see references in [5]). In (8), the empirical coefficient a from (5) is a parameter which belongs to the interval (1, 2). By our assumptions and from physical considerations, s is a positive-valued monotonically increasing function of time 'T, 'T E [0, oo) . Hence y, which is known as the diesel para meter function, is expected to be a positive-valued mono tonically increasing function of x, x E [0, oo) . Inverse problems: an example. In contrast to the usual direct problem of calculating the diesel fuel spray pa rameters based on the given conditions of injection, we call a problem an inverse problem, if we are required to de termine some conditions of the injection (or, more gener ally, some relationships among the injection conditions) on the basis of certain known (desirable) parameters of the fuel spray. Presumably the desired fuel spray parameters are those that produce the most effective fuel spray for the considered diesel system (see [ 1 1 ] , [5] , and their refer ences). The mathematical aspects of diesel spray inverse prob lems seem to be more challenging than those of direct prob lems. Note too that one direct problem gives rise to sev eral inverse problems, depending on the desired qualitative and quantitative features of the spray. With the idealized model as outlined above, we consider the following (now-settled) inverse problem: Determine the conditions of injection, i.e., the values of K and V in (7), from two points s 1 and s2 at the respective times 'T1 and T2 ( 'T1 < 72) on the spray development curve s = s( 'T). Necessary and sufficient conditions were found, namely
( 'T2 )113 T}
s S1
72 2
0,
z
E E.
One can use this criterion and the Herglotz formula to show that the function F(z) is a normalized starlike function if and only if it has the following exponential-integral repre sentation:
mean a nondecreasing function defined on an interval, say
[a , b ].
Each integral
fg f df.L below is a Stieltjes integral.
i) A complex-valued function F defined on ( -oo,
ates the forms �lk=l CjckF (xj
numbers
X1, . . . , Xn (n
) gener
oo
- xk) for every choice ofn real 1, 2, . . . ). Of particular interest in
harmonic analysis are the positive-definite forms, which are =
positive for any nontrivial ch . . .
, Cn· Positive-definite func
tions are those functions which generate positive-definite forms for all n and all choices of the x s. In other words,
�lk =l CjckF(xj - xk) > 0
}
except when all the c s are zero.
Bochner's theorem (e.g., [ 19, Chapter
6])
}
states that F is
F(x) f�oo df.L < oo.
positive-definite and continuous if and only if
f-'='oc exp(ixt)dJ.L(t),
for a positive measure f.L,
=
A
similar theorem holds for locally compact Abelian groups. For 27T-periodic functions, F is positive-definite and contin n uous if and only if F(8) = ��= oo anei fl with an � 0, where � the an's are not all zero and �"' an < oo [ 19, Chapter 7].
ii) Now we tum to geometric function theory. Let E de note the open unit disk {z : jz j < 1 }, and let 91' be the class of all functions p(z), p(O) =
where f.L is a probability measure on
[0, 27T] [4,
Chapter
8].
iii) Finally, we consider completely monotonic functions.
An infmitely differentiable function g is called completely
monotonic on an interval I if ( - 1)ngCnl(x) � 0 on I (n = 0, 1, 2, . . . ). We say that a function g is strictly completely monotonic on an interval I if ( - l)ngCnl(x) > 0 on I for all n. A typical completely monotonic function on (0, oo) is exp( tx) , t � O. lt is clear that positive linear combinations -
of such functions are also completely monotonic. A theo
rem of S. N. Bernstein (see
[ 1]
[3, Chapter 1 3] , [27, Chapter 4],
g is completely monotonic on and
for this and related results) states that a function
(0, oo) if and only if there is [0, oo) such that
a positive measure f.L supported in
g(x)
=
r
exp( -xt) dJ.L(t).
Another useful fact is that if g(x)
exp( -u(x)) and
u'(x)
then g is com
1, that are analytic in E, and such that for z E E, 0t{p(z) } > 0. According to the Herglotz theorem (e.g., [4, Chapter 7]), a function p(z) belongs to
pletely monotonic on I. This follows from the rule for
the class 91' if and only if
di Bruno's formula in
p(z)
=
0 L27T
(1
+ zeit)J( l - zeit) dJ.L(t),
where f.L is a probability measure on
[0, 27T), i.e., f57T df.L
=
cations. We mention two of them, having in mind some deeper sign-regularity conditions and exponential-integral representations.
A function F(z), F(O) = F'(O) - 1 0, is said to be typ ically real in E if it is analytic in E and if for every nonreal =
E E, sign(2J{F(z)})
46
=
sign(2J{z}). For example, a function
THE MATHEMATICAL INTELLIGENCER
differentiating composite functions and induction (see
[25,
Chapter
2]).
Laplace transforms of nonnegative functions
Herglotz representation formula, has many useful appli
z
I,
Originally, completely monotonic functions arose as
and f.L is positive. This representation, known as the
1
=
is completely monotonic on an interval
[27], [15]. Now [16], [14]).
they appear in many areas of mathematics (e.g.,
An instance of complete monotonicity in probability theory variable X with distribution measure f.L is called infinitely
is the concept of infinite divisibility. A non-negative random
divisible if and only if for every positive integer n there are independent non-negative random variablesX1,
• . .
, Xn. each
having the same distribution f.Ln, whose sum is X, that is,
Loo 0
exp( -xt)d J.Ln (t) =
[L"" 0
exp( -xt)dJ.L(t)
]1/n.
It turns out that a probability measure is infinitely divisi ble if and only if the Laplace transform fo exp(- xt)dt-t(t) = exp( -u(x)), where u(O) = 0 and u ' is completely mono tonic on (0, oo), i.e.,
r
exp ( -xt)dt-t(t) = exp
{f
}
[exp( -xt) - 1 ]t-l d v(t) ,
where v is a positive measure. This characterization as an exponential-integral representation is beneficial in estab lishing the complete monotonicity of several probability distributions (like the student t-distribution and the F-dis tribution) using the theory of special functions, for the Laplace transforms of probability measures are usually special functions, and u ' , being the logarithmic derivative of a Laplace transform, is a quotient of two special func tions. For information see [ 17], where the approach uses integral representations. In fact some of the work on com plete monotonicity of quotients of special functions in [ 1 7] led naturally to certain probability distributions which were later found to be hitting-time distributions for Brownian motion [24]. The complete monotonicity of log arithmic derivatives of special functions turned out to be a problem of independent interest; advances along this di rection are exemplified in Hartman's works [ 12], [13].
Now we provide a bridge from the diesel parameter func tion y with which we began to complete monotonicity. For a given a, 1 :::::; a < oo, let [8] xy ' (x) y(x)
,
x > 0,
and
h(O) = 1,
It follows that for a = oo, h(x) = 1 + f' (x) -
�
f(x) X
,
f (: ) - ].
a h
x > 0,
, a
and
1
h(O) = 1,
( 1 1)
(12)
where f(x) satisfies the nonlinear initial-value problem f"
=
(!') -f - ef; 2
x > 0; f(O) = f' (O) = 0.
We shall refer to the conjecture above as the DFS (diesel fuel spray) conjecture. We believe that although h may not be completely monotonic on any interval [0, 77) for a < a0, its conjectured complete monotonicity for a ::o:: a0 seems to spill over to smaller values of a, producing regular sign be havior of the function h and its derivatives. Implications of the DFS Conjecture
A proof of the DFS conjecture would give new information about the nature of the initial-value problem (8) and the model in [ 10] and [9]. Indeed the complete monotonicity of the function h(x) on an interval [0, 7]), 7J = 77(a), a0 :::::; a < oo, will bring the exponential-integral structure to the study of the fuel-spray penetration length, s(T). This struc ture will provide a companion (or possibly an alternative) to the differential equation approach. More precisely, let If h is completely monotonic on (0, 77) then g is completely monotonic on (0, oc), and Bernstein's theorem supplies a representation
(10)
where y is defined by (8). Note that a is now allowed to go beyond the "diesel interval," i.e., the domain (1, 2) spec ified by the diesel spray model [10]. In the asymptotic case (a = oo) we define [8] h(x, oo) = 1 +
and that max[a o,""l 77(a) is attained for some a close to a0.
g(A) = h(7](1 - e-A)), A E [0, oo).
The Logarithmic Derivative of the Diesel Parameter Function and a Related Conjecture
h(x) = h(x, a) =
putations of the function h and its derivatives for various finite values of a, as well as for a = oo (see [8], [23], and below) support the following modified version of the con jecture in [8] . For some a0 ::0:: 3/2 and each a E [ a0, ooj, thefunction_h, defined by (8) and (10) - (13), is strictly completely mo notonic on an interval [0, 77(a)). It is likely that a0 = 3/2
(13)
We can think of h as a family of functions parametrized by a. It tums out that for many values of a E [1, oo], the function h(x) (Figure 3) exhibits certain features of com plete monotonicity on some (maximal) interval [0, 77(a)). In reality, the lifetime of a diesel spray is very short, and thus x-defined by (7)-belongs to quite a restricted seg ment. It would be nice if this were a subset of [0, 77(a)) for "diesel" values of a. It will also be of interest to determine whether or not 77(a) attains its global maximum just on the diesel interval, and to find this maximum. Extensive com-
where I-ta is a positive measure supported in [0, oo) . Since h(O) = 1 , it must be the case that Jo dt-ta 1 . Consequently, the truth of the DFS conjecture, together with equations (14), (10) and (7), would give the following representation of the fuel-spray penetration length s( T): =
s(T)
=
VT exp
{l"'IJaT 0
L0
(1 - x)t - 1 X
]
dx
}
dt-ta(t) , T< a'
1
(15)
where a = a(a) = _KVa - lf7J(a). The representation ( 15) is a characterization of physi cal and geometric properties of diesel sprays converted, via a mathematical model [ 10], to an analytic expression displaying a sign regularity property comparable to those of the examples in i)-iii). It involves an empirical coeffi cient a and a probability measure I-ta depending on a, nei ther of which is known explicitly. Nevertheless this repre sentation, and its possible generalization to evaporated fuel sprays which are generated by given time-dependent in jection pressures, would be useful. In particular this would be important if it is required to solve inverse problems con cerning fuel spray (see above). The question of the exis tence of a solution to a typical inverse problem is closely connected with monotonicity and other qualitative prop-
VOLUME 22, NUMBER 2, 2000
47
h
k------r---,�--�--�� X
250 0 Figure 3a. Graph of the function h(K) in the case
a =
1.
500
750
1000
h
0
250
Figure 3b. Graph of the function h(K) in the case
a =
3/2.
500
750
1000
X
h
0
250
Figure 3c. Graph of the function h(K) in the case
48
THE MATHEMATICAL INTELLIGENCER
a = co .
500
750
1000
X
erties involving various combinations of the function s(r) and its derivatives. It turns out that the representation (15), if proven, will be a convenient tool to answer such a ques tion (see [5] and references there). Our belief that (15) is valid is borne out by computer experiments.
12
• • • • • • •
Numerical Analysis
A computer "confirmation" of the DFS cof\iecture has been based upon a parametric coefficient approach, using both the Taylor coefficient properties of differential equation so lutions and the proper equations with certain auxiliary pa rameters. The detailed analysis will appear elsewhere. Here we discuss some results derived through the Taylor poly nomial approximation. n Let {F}n denote the coefficient of x in the Taylor series expansion of a function F(x) about x = 0. For a given a, a 2:: 1, one obtains the sequence {h}n, n 2:: 1-using equa tions (10) and (8) and triple recursion if a is finite, or equa tions (12) and (13) and double recursion if a = oo [8]. Let n n(a) be the smallest natural n if any such that ( - 1) {h}n < 0. Calculations suggest that there exists a finite n(a) for each a < 3/2. In other words the function h is not completely mo notonic on any interval [0, TJ) if a < 3/2 (Figure 4). For a 2:: 3/2, we take into account some properties of n the n-sequences {h}n (x )Ck) with suitable values of x > 0 and k = 0, 1, 2, . . . (n up to 50,000 in our experiments), and use the Taylor polynomial approximation of the func tion h(xp). Here p is a positive parameter which is close to lim supn..... llhlnl-1/n if the degree of the related Taylor poly nomial is large. This allows us to demonstrate that many successive derivatives of h alternate on a suitably chosen interval [0, TJ), which is contained in the interval of con vergence of the Taylor series expansion of h around the origin (Figures 5 and 6 provide some examples). In addi tion one can use differential equation (8) and some auxil iary parameters to see whether or not for a finite a a num ber of successive derivatives of h alternate on larger intervals. Equations (8) and (13) themselves allow us eas ily to compute at least the function h for values of x on a large interval, showing that h(x) is positive valued and de creases for various a (Figure 3). It turns out that when a = oo the possible interval of complete monotonicity [0, TJ(a)) is a proper subset of the interval of convergence above. Of course, this case of the DFS cof\iecture takes us well beyond the realm of engi neering. However, the asymptotic DFS cof\iecture looks simpler (with its reduced recursion) and might be handled (due to the exponential composition in (13)) by an expo nentiation approach (e.g., Milin's approach [22]). Further more, if for a = oo, it is shown that h is strictly completely monotonic on some interval [0, TJ), then one would expect that for any ij E (0, TJ) and for sufficiently large values of a, h is strictly completely monotonic on [0, ij/a). It is also possible that this property can be extended to certain smaller values of a. The simplest case of numerical examples is when a = oo and parameter p equals 1. The details are too long to be included here and will appear in a future work. Figure 6
:
I
9
p i s used t o denote the probability measure determined b y independent tosses of a coin which, o n each toss, comes u p heads with proba
bility p; and, if r is some event based on such a coin-tossing game, then i>p(f) is the probability of r. Thus, in the formula which follows, the left-hand side should be
= E1 , X2 = e2, . . . , Xn = •n · " 5We will use IEP to denote expectation values which are computed relative to i>p. Recall that if X is a random variable, then its expectation value relative to the proba read "the probability that X1
bility measure i> is nothing more or less than the integral
66
f X di>.
In particular, if X takes on only a countable number of values, then IE [X]
Tl--I E MATHEMATICAL INTELLIGENCER © 2000 SPRINGER-VERLAG NEW YORK
= lx x !>(X = x).
Similarly, ifXm = Xm - 1Ep [Xm] = Xm - p and Sn = !-ih= 1 Xm, then
IEp [Sn]
=
0
and
IEp [S�]
n
=
I
1Ep [Xm Xm •].
m ,m ' = l
where the sum runs over all m = (m 1 , m2, m3, m4) with 1 ::::; mi ::::; n for i E { 1 , 2, 3, 4). Notice that, either by direct computation or by the mean-0 property combined with in dependence,
1Ep [Xm1
But � 1Ep [XmXm •l = 1Ep [X�] = q2p + p2q = pq (p + q) = pq, m ' � 1Ep [XmXm ·l q2p2 - 2(pq)2
• • •
Xm4]
*
=
and so
1Ep [Sn]
=
0
-
IEp [S�]
and
=
+ p2q2 n
=
npq = np(I - P) ::::; 4·
0,
+
(np)Z
=
npq + n2p2.
Of course, what accounts for the difference is the mean-0 property of the Xm's which, together with independence, leads to the vanishing of the off-diagonal terms in the ex pansion of 1Ep [S�]. Because, for any R > 0,6
Hence, at most 4!n2 terms are non-zero, and each of them is no larger than IEp [XI] ::::; pq (1 - pq) ::::; fo. In other words,
(2.2) Now, by repeating the argument with which we passed from (1.1) to (1.3) via (1.2), we see that
IP'p
(l �n - p i 2:: R) ::::; 2n;R4 .
IEp [ eABn]
= JIn 1Ep [eAXm]
= (peAq
+
qe-AP)n for all A E IR,
one can check first that
1Ep (eASn] ::::; en/3pA2 for some {3p E (0, 1], then that
(1.1) yields
( 1.2)
IP'p
(: - p 2: R) V IP'p (: - p ::::; -R) ::::; exp [ -n(AR
As an application,
(I: - p l 2: R)
= IP'p(!Sn - np ! 2:: nR)
= IP'pclsnl
2:: nR) ::::;
and finally, after taking
::;_2 ::::; �2 . 4
IP'P
(1: - p l 2: e) =
is called the weak
0
for each
e
A = 2�,
+ {3pA2)]
for
A > 0,
that
C L3)
That is, with probability at most (4nR2r 1 will n - 1sn (which is the average number of heads) differ from p by at least R. The qualitative conclusion that
J�
(2.3)
Obviously, the preceding success makes one suspect that one can do better than n- 2 too, and, indeed, one can. In fact, from
1Ep [S�] 2:: 1Ep [S�, 1Snl 2: R] 2:: R2 1P'p (ISnl 2: R) ,
IP'p
(2. 1 )
of { 1, 2, 3, 4),
mlT4"
(1.1)
Something slightly subtle has happened here. Because s� involves n2 terms, a priori one would expect 1Ep [S�J to grow quadratically in n. Indeed, this is the case when S� is replaced by S�:
1Ep [S�] = 1Ep [(Sn + np)Z] = IEp [S�] + np iEp [Sn]
a
=
mlTj = m u2 & mu3
m = m' m
=
0 unless, for some permutation
However, for our purposes here, (2.3) will suffice. Namely, what we are seeking is the replacement of the weak law (cf. (1.4)) by the strong law of large numbers. 7 That is, we want to show that
> 0 (1.4)
(
Sn IP'P lim n-'J>oo n
law of large numbers.
= p)
(2.4)
= 1.
To this end, observe that, from (2.3), we have that A Small but Important Refinement
One should ask whether the estimate in (1.3) is shmp. In par ticular, is n - 1 the true rate at which the left hand side tends to 0? In the hope that it will shed light on this question, we consider the expected value of s� instead of SJ. Clearly,
1Ep (S�] =
I IEp [Xm1 m
• • •
Xm4],
n m) ( l �n - p l 2: n - 1 n ::::; � IP'p (l� - p l 2: n - 1 ) n m
IP'p
18 for some
2::
18
9
00
::::; - I n -312 � 0 as m � oo. 2 n= m
6The notation [[X, A] means that the expectation of the random variable X is being computed over the set A. Equivalently, [[X, A] = fA X diP'. 7The sense in which the strong law is stronger than the weak law is a little subtle and probably requires an appreciation of the difference between a/most everywhere convergence and convergence in measure. See, for example, § 3.3 in [81].
VOLUME 22, NUMBER 2 , 2000
67
because
Hence, given any E E (0, 1), we can arrange that
1Pp
(I � - I
)
1Pp(Xn = 1 for all n > m) =
for all n 2:: m 2:: 1 - E
118 P :=;; n-
lim IPp(Xn = 1 for all m < n :=;;
by simply taking m sufficiently large; and so, with proba bility arbitrarily close to 1, n 1sn p as n � oo.
- �
M----::,.c.c
I
m=n+l
Of course, this is not a perfect encoding procedure be cause, although the Xm's uniquely determine Y, Y does not always uniquely determine the Xm's. The problem comes from those t E [0, 1] for which there is an n 2:: 0 such that 2nt is an integer. In order to remove the ambiguity caused by such t's, we adopt the convention that the mth coeffi cient Em(t) in the dyadic expansion of t E [0, 1 ] should be determined so that
n
m 0 :S t - I 2- Em(t) < 2-n
for all n 2:: 1 .
m= 1
(3.2)
Every t E [0, 1] then completely determines {Em(t) : m 2:: 1 } � {0, 1 }. Indeed, the Em(t)'s are generated inductively by the rules: E1 (t) = En+ 1 (t)
- {1
_
{1
0
if 0 :S t < 2- 1 :S X < 1
Em(t) < 2 -n- 1 m 1 2- Em(t) < 2 -n.
E1 = 0
E1 = 1
)[ I
[ E1 = 0, E2 = O)[E1 = 0, E2 = 1)[E1 = 1, E2 = O)[E1 = 1, E2 = 1) I
0
! 2
4
1
The importance of these considerations for us is that they lead to the conclusion that
(
)
1Pp Xn = En(Y) for all n 2:: 1 = 1.
(3.3)
To see this, notice that
n
I m
=1
m 2- xm =
oo
I
m =n+ 1
IPP (there exists n 2:: 1 such that Xm :S
68
I
m= 1
=
1 for all m 2:: n
1Pp(Xn = 1 for all n > m) = 0,
THE MATHEMATICAL INTELLIGENCER
(Y)
=
Sn n
n
�
P
)=1
0.
)
1,
(3.4)
where (cf. (3.2))
n In(t) ""' I Em(t) for t E [0, 1]. m= l
(3.5)
To understand why such a statement might be interesting, notice that
1Pp(2 - nm :=;; Y :=;; 2 - n (m + 1)) = p"in(2 - nm)qn - InC2 -nm) , n 2:: 1 and 0 :=;; m < 2n. (3.6) Hence, if 0 :=;; a :=;; b < 1, then
=
(3. 7) 2nLn(b) "in(2 -nm)qn - In(2 -nm) , I p n->oo m n = 2 Ln(a) lim
and R (t) n
""' Ln(t) + 2 -n,
(3.8)
are, respectively, the closest nth-order dyadic points to the left and right of t E [0, 1]. In general, the limit in (3.7) is es sentially impossible to compute. However, when a = b, the sum degenerates to a single term which, because max{p, q} < 1, tends exponentially fast to 0 as n � oo. Hence, (3.9)
Another case when the computation is possible is when the coin being tossed is fair, and therefore p = t. In this case, all the terms in the sum are equal to 2 -n. Thus, since the number of terms lies between 2n(b - a) and 2n(b a) + 1, we conclude that IP 112 Ca < y :=;; b) = IP 11z(a :=;; y :=;; b) = b - a, 0 :=;; a :=;; b :=;; 1.
2 - mxm :=;; 2-n
(3. 10)
Equivalently,
with equality only if Xm = 1 for all m 2:: n + 1. But 00
for all n 2:: 1
1Pp(Y = x) = 0 for all x E [0, 1] and p E (0, 1).
ETC.
y-
m 2- xm < 2-n
n
) 1
2
( In
Ln(t) ""' I 2- mEm(t) m= 1
More graphically, [ 0
=
where
m
if 0 :S t - !, �= 1 2 if 2 - n - 1 :S t - !, � =
0
IPP
and
if 2 - 1
M__,x
and this is equivalent to (3.3). As an immediate consequence of (3.3) combined with the strong law of large numbers (2.4), we know that
(3. 1)
m= 1
M) = lim pM
=
00
Y = I 2 - mxm E [0, 1].
+
Hence,
Another Representation of Coin-Tossing
The next step in our program entails our encoding the out comes of an infinite coin-tossing game as a real number. Namely, we want to think of our {0, 1 }-valued random vari ables {Xm : m 2:: 1 } as being the coefficients in the dyadic expansion of the random number
m
+
1)
p = t � Y is a uniform random variable on [0, 1].
(3. 1 1)
Now a uniform random variable is a reasonable model for a point which is chosen at random from [0, 1] . So
when p = t (3.4) says that the ratio of O's to 1 's in the dyadic expansion of a typical x E [0, 1] is 1. More pre cisely, if a random x is chosen uniformly from [0, 1], then ��n (X) � t. This observation was made around the tum
of the century by E. Borel and is discussed beautifully and in detail by M. Kac [K]. More recently, G. Goodman [G] discussed some interesting variations on the theme of Borel and Kac. A Problem in Measure Theory
Motivated by these considerations, one should want to de velop some feeling for how many numbers x E [0, 1] are non-random. In particular, if (cf. (3.5)) .1p
=
{t E [0,
�n(t) p},
.
1] : hm
n--. co
--
n
=
(4.1)
how large is the complement .1�t2 of J1v2? A detailed an swer to this question is quite difficult (cf. [DR] for an in teresting discussion from an entirely different point of view). Nonetheless, a few qualitative statements can be made without much effort. I begin by showing that .1�t2 must be reasonably large, in the sense that it must contain an uncountable number of points. Indeed, given any p E (0, 1)\HJ, .1p \:: .1�t2 . We will lrnow that .1�t2 is uncountable as soon as we show, for ex ample, that .11/3 is uncountable. But suppose we could count the points in .1v3, and let (xnl'l be an enumeration of them. Then, (3.4) and (3.9) with p = i would lead us to the contradiction co
1
=
!Pvs(Y E .11/3)
=
I IPv3(Y = Xn) = 0. n= 1
We conclude that .1�t2 is uncountable. Of course, exactly the same reasoning shows that .1p is uncountable for each p E (0, 1), and obviously .1P is disjoint from .1p' when p =I= p'. Knowing that .1�!2 ;;;;? Upnt2 .1p and .1p's are mutu ally disjoint sets each of which is uncountable, one might start believing that .1�t2 is reasonably large. In spite of the evidence to the contrary just given, we lrnow that .1�t2 must also be quite small. Indeed, we lrnow that, with probability 1, a uniform random variable misses .1'it2 entirely. In the terminology of Lebesgue's measure the ory, .1�!2 is a set of measure 0, and sets of measure 0 are characterized by the property that they can be covered by a countable number of intervals the sum of whose lengths is arbitrarily small. That is, for each E > 0, there exists a sequence (In l'l of open intervals such that 00
00
J1'i12 \:: U In and yet I !In! < n= 1 n =1
E
where !In ! is the length of In.
In summary, although .11!2 is not countable, it can nonethe less be covered by a countable number of intervals the sum of whose lengths is as small as you like. Some Exotic Increasing Functions
For 0 < p < given by
1, consider the function Fp : [0, 1] � [0, 1] Fp(x)
=
!Pp(Y ::::; x),
(5.1)
where Y is the random variable in (3.1). Clearly, Fp is non decreasing, Fp(O) = 0, and Fp(1) 1. It is easy to see from (3.7) that =
X < y � Fp(X) < Fp(y).
(5.2)
That is, Fp is strictly increasing. In addition, because, by
(3.9), lim Fp(Y) = 1Pp(Y ::;; x) = !Pp(Y < x) + !Pp(Y = x) = 1Pp(Y < x) = lim Fp(y),
y '>.x
y /' x
we lrnow that Fp is continuous. In other words, we can say that, for each p E (0, 1 ), Fp is a strictly increasing, contin uous function with Fp(O) 0 and Fp(l) 1. When p t, we can say much more: from (3.10), Fv2(x) = x for all x E [0, 1]. However, when p =I= t, the function Fp is somewhat mysterious. It is difficult to pic ture its graph. For example, suppose that we examine its arclength. Recall that, for any continuous F : [0, 1] � IR, the arclength Arc(F) of its graph can be computed by taking the limit8 =
=
=
Arc(F)
=
2n - 1 I Y(2 n)2 n---"'oo m=O lim
+
(F((m
+
1)2 n) - F(m2-n)?,
which may or may not be +oo. For any non-decreasing F, 2 then, because ::;; Va + b2 ::;; a + b for all non-negative a and b,
a_:;;
F(1) - F(O) =
1 � \12 ::::; Arc(F)
::;;
2.
By direct computation (or the Pythagorean Theorem), the lower bound is achieved when F = Flf2. On the other hand, it is hard to imagine how a continuous F could achieve the up per bound. It would seem that the graph of such an F would, at every point, have to be going horizontally or vertically, thereby ruling out the possibility that the function being graphed is continuous. For this reason, it is interesting that
p E (0, 1) \ ( 112 } � Arc(Fp)
=
2.
(5.3)
To see (5.3), one can9 proceed as follows. We are fmd ing the limit of
zn - 1 Ln = I Y4 n m=O
+
2 (Fp((m + 1)2 n) - Fp(m2 - n)) .
8The existence of the limit is assured by the fact that the expression on the right is non-decreasing as n increases. To see this, interpret the mth summand in the nth sum as the length of the 2-dimensional vector whose components are 2-n and F((m + 1 )2-n) - F(m2-n), and note that this vector is the sum of the 2mth and (2m + 1 )st vectors in the (n + 1 )st sum. Thus, the asserted monotonicity is just the triangle inequality for the lengths of vectors in the plane. 9-fhe elegant argument which follows was suggested to me by Alex Perlin. For those who know a little more classical analysis, I have provided an appendix in which it is shown that the graph of any continuous, non-decreasing F on [0, 1] has arclength equal to 1 + F(1 ) - F(O) if and only if F is Lebesgue-singular.
VOLUME 22, NUMBER 2, 2000
69
Set
An(R) = {0 :::;; m < zn : Fp((m + 1)2-n) - Fp(m2 -n) > 2-n RJ Bn(R) = {0 :::;; m < 2n : Fp((m + 1)2-n) - Fp(m2 -n) :::;; 2-n RJ. Obviously, for any R
Ln 2::
I
mEAn(R)
>
0,
Y4 n + (Fp((m + 1)2-n) - Fp(m2- n))2
+ R- 1 )(1
Hence, since10 x 2: R � (1
Ln 2:: (1 + R- 1)- 1
I
mEAn(R)
(
2:: (1 + R- 1 )- 1 1
+
(2-n
+
+
mEAn(R)
mEBn(R)
.x-2) 112 2:: (1 + x),
Fp((m + 1)2 - n)
- Fp(m2- n))
I
The First Derivative of Fp
I
+
I
+
mEBn(R) (Fp((m + 1)2- n)
and clearly this completes the proof that Arc(Fp) 2:: 2. Because we already know that the opposite inequality must hold, (5.3) is now verified.
2-n
- Fp(m2 -n))
}
This is already grounds for regarding the graphs of the func tions Fp when p =/= t as quite strange indeed. Namely, (5.3) indicates that, at essentially any point, the tangent to the graph of Fp must be either horizontal or vertical. In this section I will provide further evidence that this picture is correct. To be more precise, recall the sets D.p introduced in (4. 1). What we are going to do here is check that
p E (0, 1) \ /tJ � F;(x) = where
F'(x) p
But
I
mEAn (R)
(Fp((m + 1)2- n) - Fp(m2- n)) =
1-
I
mEBn(R)
(Fp((m + 1)2- n) - Fp(m2-n)).
Therefore, we now know that, for every R > 0,
Ln 2:: (1 + R- 1)- 1 X 2-
(
I
mEB,(R)
(Fp((m
+
)
At the same time, by (3.3) and (3.6),
=
=
(
.
mEBn(R)
of X E [0, 1) for which pf.Cx)qn -t,(x) :::;; 2-nR. Thus we will know that
(
)
To this end, we first note that
>
2p log 2p + 2q log 2q 2
Because p
(5.4)
THE MATHEMATICAL INTELLIGENCER
Fp(X
=/=
{
(6.2)
>
+
{
0 if X E /j.l/2 h) - Fp(X) = oo if x E D.p h
·
Hence, since ;In(X) � i,
(
(: Yn(x) -n/2) k X) = n [ Pp ( n� - i) �] log
0,
= ( vpqr (�y
t, pq < i and therefore Pp == 2 Vpq < 1.
log (2vpq)n
and x E (0, 2) � x log x E IR is a strictly convex function which vanishes at x = 1. Finally, by combining (2.4) with (5.4), we arrive at wsquare both sides to check this.
_
Fp(Rn(x)) - Fp(Ln(X)) < Rn(X)) l l p(Ln (X) < y = 2n flu-Rn(X) - Ln(X) = 2npln(X)qn - ln(x) ncx) - n/2 2 .
(5.4) is equivalent to
p log 2p + q log 2q =
h
0 if X E !1 11 - oo if X E D.p 2 .
In other words, everything comes down to proving (6.2). Now letp E (0, 1) \ {112} and x E 11.1!2 be given. Referring to the notation introduced in (3.8) and using (3.6), we have that
0,
p E (0, 1) \ { 1/2} � (2p)P(2q)q > 1 . This is because
Fp(X) - Fp(X - h)
Indeed, if we knew (6.2) for every p E (0, 1) \ { 1/2 }, then, because
l�
)
1Pp ((2p)n -1Sn(2q) I -n-1Snr :::;; R � 0.
h)::._ x_+---' _ FE.:(:_ .._ Fl!..::c. (x:.f_ .) h
it would follow that
where to obtain the first equality we have taken advantage of the fact that U [m2- n, (m+ 1)2-n) is precisely the set
70
h�
1Pp ((2pr-lSn(2q) 1 - n-1Sn)n :::;; R ,
as soon as we show that, for arbitrarily large R
•
(6. 1)
D.q = { 1 - X : x E D.p} and Fq(X) = 1 - Fp(1 - x),
(Fp((m + 1)2-n) - Fp(m2-n)) mEB (R) I
n 1Pp(JJBnqn -Sn :::;; 2 -n R)
hm0
h---.
if X E D.v2 if x E D.p '
is the derivative of Fp at x. (Of course, part of the asser tion is that this limit exists at the indicated points.) To prove (6. 1), it suffices to check it for the left deriv ative. That is, it is enough to show that r
1)2-n) - Fp(m2-n)) .
==
{Ooo
+
log
�
- oo,
which means that lim
n---.oo
Fp(Rn(x)) - Fp(Ln(X)) = 0 exponentially fast. (6.3) Rn(X) - Ln(X)
Although this computation does not provide a definitive proof, it strongly indicates that (6.2) holds at each x E ll 11z. To complete the proof, choose n(x) to be the first n > 1 such that In (X) 2= 2, and, for n 2= n(x), let Mn(x) be the largest 1 ::; m ::; n for which In(x) - Im(x) = 1. Observe that, since n 2:: n(x) => LMn(x)(X) ::; X - 2-n < X ::; RMn(x)(X),
- - 1 < h ::; 2-n
FP (x) - FP (x - h) h (X)) Fp(R - Fp(LMn(x) (X)) Mn(x) ::; 2n -Mn(x) + 1 ; RMn(x) (X) - LMn(x) (X)
n 2:: n(x) and 2
n
=>
from this and (6.3), that it suffices to show
n(X) lim M
= 1. (*) n What would it mean for (*) to fail? There would have to be an r < 1 and there would have to be infinitely many val ues of n for which heads at the (n + 1)st toss (En+t(x) = 1) was preceded by a run of more than (1 - r)n tails (ek(x) = 0 for rn ::; k ::; n, so that Ik(x) remains constant over the run). But then for k at the beginning of the run,
n-.cc
Ik(X)
_
k
I (X) In(X) 2:: n
n
n
2::
2 -n =>
_
r
I=
FP(x) - Fp (x - h) h
2::
{X
E
[(2p)P(2q)q (�rn( ) n-pqNn(x)ln- 1r � +oo n - 1 In (x) - p � 0, X I
as n � oo. Because one can proceed, just as in the demonstration of (*) above, to verify that n - 1Nn(x) - 1 � 0. Finally, as we saw in (5.4), (2p)P(2q)q > 1, and so the desired conclusion should now be evident. In summary, the first part of (6. 1) shows that, when p =F 1/2, F�(x) = 0 when x E .:1 112 (what we called in (3. 1 1) be ing chosen "at random"). On the other hand, to compen sate for the first part and enable Fp to be strictly increas ing, F;(x) = oo when x E llp. Both sets are dense.
.
F(x
�-t((a, b]) = F(b) - F(a)
+
h) - F(x) h
=
0
}
0
for all
0 ::; a < b ::;
1,
then F being Lebesgue-singular is equivalent to f.t being sin gular to Lebesgue measure.l 1 The purpose of this appen dix is to prove that Arc(F) = 2
if and only if F is Lebesgue-singular. (A. 1)
I begin by introducing a little notation. Namely, set = F((m + 1)2- n) - F(m2- n) for n 2:: 0 and 0 ::; m < n 2 . By definition,
dm,n
Arc(F) = lim Ln n--->oo where Ln
2n - l
=
I
m �o
Y4 - n + ll;,,n,
and, as already pointed out, Arc(F) lies between 2. Next note that
2 - Ln
=
2 -n
2n - l
V2 and
((1 + 2ndm,n) - Y1 + (2n dm,n?).
I
m �o
But, for any non-negative
a,
so we now know that
Equivalently, iffn : [0, 1) � [0, oo) is defined so thatfn (X) 2ndm,n when m2-n :S X < (m + 1 )2- n, then
2 - Ln 2
2npln(x)q_NnCx) - ln(X)
[0, 1 ] : k�
[0, 1 ]
One says that F is Lebesgue-singular if the set I has Lebesgue measure 1 . Alternatively, if f.t denotes the Borel measure on [0, 1] determined by
1 .
Thus, it is enough to prove that
=
Given a continuous, non-decreasing function F on with F(O) = 0 and F(1) = 1, set
' .!. ,
In order for n-1In(x) to approach 1/2, as is required by our assumption that x E .:1 112, the left-hand side would have to approach 0 (the Cauchy criterion) and the right-hand side to approach a non-zero limit: contradiction. Therefore (*) must hold. Hence, we have now proved (6.2), and therefore the first half of (6. 1), for every x E ll112· To complete the proof of (6. 1), again letp E (0, 1) \ { 112 } be given, but now suppose that x E llp. For each n 2:: 1, let Nn(X) be the smallest m > n such that Im(X) - In(X) = 1, and observe that, since x - 2 -n :S Ln(X) < LNn(x> (x) :S x,
2- n + 1 > h
Appendix
::; f
[0,1)
1
fn(X) + fn (X)
dx ::; 2 - Ln.
=
(A.2)
In order to complete our program starting from (A.2), we need to invoke some measure theory. In the first place, an immediate consequence of (A.2) is the conclusion that
Ln � 2 ¢::::::> fn � 0 in Lebesgue measure.
(A.3)
Secondly, we need to know that fn � 0 in Lebesgue mea sure if and only if the measure f.t is singular to Lebesgue measure. This second fact, which is also the basis for the comment in footnote 1 1 , is a corollary of a variant (Theorem 5.2.26 in [81]) of Lebesgue's Differentiation Theorem. Namely, fn converges Lebesgue almost every where to the Radon-Nikodym derivative of the absolutely continuous part of the measure f-t. Thus.fn � 0 in Lebesgue
1 1 1n fact, one can show that the preceding set l is assigned measure 0 by the absolutely continuous part of
!"·
VOLUME 22, NUMBER 2, 2000
71
measure if and only if the absolutely continuous part of P vanishes; and, knowing this, one gets (A. l) as an immedi ate consequence of (A.3).
AUTHO R
ACKNOWLEDGMENTS
The author acknowledges support from NFS grant DMS 9625782. He is also grateful to G. Goodman for providing him with most of his bibliographic information. REFERENCES
[B]
Billingsley, P.,
The singular function of bold play,
American
Scientist 71 (1 983), 392-397. [DR] de Rham, G . ,
Sur certaines equations fonctionnelles ,
I'Ouvrage DANIEL W. STROOCK
publie a !'occasion de son centenaire par !'Ecole polytechnique
Department
de I'Universite de Lausanne, pp. 95-97. [D] [G]
Durrett, R.,
Brownian Motion
and Martingales in Analysis,
Goodman, G.,
Theory, Carus Math. Monograph Series #12, J. Wiley, NY, 1 959. [RN] Riesz, F. & Sz.-Nagy, B . ,
Functional Analysis,
translated from the
2nd French edition, Frederick Ungar, New York, 1 955.
[81 ] Stroock, D.,
A Concise Introduction to the Theory of Integration,
3rd Edition, Birkhauser, Boston, 1 999. [S2]
-- ,
Probability Theory, an Analytic View,
e-mail:
[email protected] American Math.
Kac, M., Statistical Independence in Probability, Analysis and Number
Cambridge U. Press,
021 39-4307
USA
Statistical independence and normal numbers, an
Monthly (1 999), 1 1 2-1 26. [K]
Cambridge, MA
Wadsworth, Belmont, CA. 1 984.
aftermath to Mark Kac 's Carus monograph,
of Mathematics MIT
Daniel Stroock got his doctorate
in
1 966 at Rockefeller
University, with Mark Kac. Since 1 984 he has been a Professor at
MIT. He was the 1 997 Colloquium
Lecturer
of the American
Mathematical Society. In addition to his public persona as a leader in probability theory, he is sometimes to be found riding
horses in Colorado. (Well, actually, you can not find him, for he has
not
disclosed where in Colorado
he
does his riding.)
Cambridge, UK and NY, USA, 1 991 .
Clear, Simple, Stimulating Undergraduate Texts from the Gelfand School Outreach Program
Trigonometry
I. M. Gelfand, Rutgers University, New Brunswick, NJ & M. Saul, The Bronxville School, Bronxville, NY
This new text in the collec tion of the Gelfand School T�gonometry Outreach Program is written in an engaging style, and approaches the material in a � unique fashion that will moti vate students and teachers alike. All basic topics are cov ered with an emphasis on .. ·� -- -beautiful illustrations and �-------' examples that treat elemen tary trigonometry as an outgrowth of geometry, but stimulate the reader to think of all of mathe matics as a unified subject. The definitions of the trigonometric functions are geometrically moti vated. Geometric relationships are rewritten in trigonometric form and extended. The text then makes a transition to a study of the algebraic and analytic properties of trigonometric functions, in a way that provides a solid foundation for more advanced mathematical discussions.
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72
THE MATHEMATICAL INTELLIGENCER
Algebra
I.M. Gelfand & A. Shen
''The idea behind teaching is to expect students to learn why things are true, rather than have them memorize ways ofsolving a few problems . . . [Thisj same philosophy lies behind the current text. . . A serious yet lively look at algebra. " -The American Mathematical Monthly 1 993, 3rd printing 2000 I 1 60 pp. I Soltcover ISBN 0-81 76-3677-3 I $ 1 9.95
The Method of Coordinates
I.M. Gelfand, E.G. Glagoleva & A.A. Kirillov
"High school students (or teachers) reading
through these two books would learn an enor mous amount ofgood mathematics. More impor tantly, they would also get a glimpse of how mathematics is done. " -The Mathematical lntelligencer 1 990, 3rd printing 1 996 I 84 pp. I Soltcover ISBN 0-81 76-3533-5 I $ 1 9.50
Functions and Graphs
I.M. Gelfand, E.G. Glagoleva & E.E. Shnol
''All through both volumes, one finds a careful
description of the step-by-step thinking process that leads up to the correct definition ofa concept or to an argument that clinches in the proof of a theorem. We are. . . very fortunate thatan account ofthis caliber has finally made it to printedpages. " -The Mathematical lntelligencer 1 990, 5th printing 1 999 I 1 1 0 pp. I Soltcover ISBN 0-81 76-3532-7 1 $ 1 9.95
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Textbook evaluation copies available upon request, caU ext. 669. Prices ore valid in North America on� ond subie 0? If f is an entire function of finite order at most p, meaning that limJzl -+oo [.flz)[e-l z JP+< = 0 for every positive E, then Hadamard's fac torization theorem implies that lim supr___.co m(r)erP+ • = oo for every pos itive E. In other words, m(r) cannot tend to zero too fast. This weak estimate can not be improved in general. For exam ple, the exponential function ez has or r der 1 and m(r) = e - . On the other hand, iff is a nonconstant polynomial, then m(r) tends to infinity like a power of r. The question arises whether an en tire function of sufficiently small order is enough like a polynomial that its min imum modulus must be unbounded. In 1905, A. Wiman confirmed [16] =
74
THE MATHEMATICAL INTELLIGENCER
that the minimum modulus of ev ery nonconstant entire function of order p strictly less than t is in unbounded. Moreover, deed r lim SUPr-+oc m(r)e- P- • = oo when 0 < E < p < t. The cutoff at t is sharp, for the convergent infinite product I4:'= 1 (1 - zln2), which equals (sin 1rVZ)/ ( 1rYz), has order t and m(r) :s r- 112f1T. (See [5, Chap. 3] for more about the minimum modulus of entire functions of small order.) Wiener's dissertation gives a new proof of Wiman's theorem. The proof is elementary in the sense that it uses only arguments about series and prod ucts of real numbers; it avoids using theorems from function theory. Wiener's proof even supplements Wiman's theorem by giving some in formation in the endpoint cases p = 0 and p = i; namely, Wiener shows that if j(z) = II�=l (1 - zlan), where {anl�= l is a sequence of nonzero complex numbers of increasing modu lus, and if limn___.co n21[anf = 0, then lim supr___.oo m(r)r-k = oo for every pos itive k. This result applies to all tran scendental entire functions of order 0 n [e.g., to II�=l (1 - zln )] and to some entire functions of order t [e.g., to II�=2 (1 - zl(n2 log n))]. The second part of Wiener's disser tation is motivated by a theorem of Landau [8] that generalizes Picard's lit tle theorem.
There is a positive junction R such that every polynomial of the form a0 + z + a2z2 + + anzn assumes at least one of the values 0 and 1 in the disk {z: fzl :s R(ao) ). THEOREM 3
(Landau).
· · ·
One might hope that a theorem about polynomials would have an ele mentary proof, which would then yield an elementary proof of Picard's theo rem. Wiener was able to find an ele mentary proof (using RoucM's theo rem, but nothing else from function theory) of Landau's theorem under an additional hypothesis about the loca tion of the zeros of the polynomial; namely, he assumed that the zeros are located within the two equal acute an gles determined by two lines inter secting at the origin. If the radian mea-
sure of the acute angle is t1r - {3, then one can take R(ao) = 28fao f logfaof/ sin {3. (The cases a0 = 0 and a0 = 1 are of no concern, because then the poly nomial takes the value 0 or 1 at the ori gin.) ACKNOWLEDGMENTS
For assistance in this project of iden tifying and tracing F. Wiener, we thank Samuel J. Patterson (Georg August-Universitat Gottingen), Con stance Reid, Heinrich Wefelscheid (Gerhard-Mercator-Universitat Ges amthochschule Duisburg), and Law rence A. Zalcman (Bar Ilan University). We are especially indebted to Profes sor Wefelscheid for locating and send ing us a copy of Wiener's dissertation. We thank Heidemarie Wormann Boas for help with German translation. The authors' research was partially supported by grants from the National Science Foundation. REFERENCES
1 . L. Aizenberg, A. Aytuna, and P. Djakov, "An abstract approach to Bohr's phenom enon,"
Proc. Am. Math. Soc.
(in press).
2. L. Aizenberg, A. Aytuna, and P. Djakov, "Generalization of Bohr's theorem for arbi trary bases in spaces of holomorphic func tions of several variables," 1 998, preprint. 3. L. Aizenberg, "Multidimensional analogues of Bohr's theorem on power series," Am. Math. Soc.
Proc.
(in press).
4. H .P. Boas and D. Khavinson, "Bohr's power series theorem in several variables," Proc. Am. Math.
Soc.
1 25(1 0) (1 997),
2975-2979. 5. R.P. Boas,
Entire Functions,
Academic
Press, New York, 1 954. 6. H. Bohr, "A theorem concerning power se ries,"
Proc. London Math. Soc.
(1 9 1 4), 1 -5.
(2) 1 3
7. G. H. Hardy, J . E. Littlewood, and G. P61ya, Inequalities,
Cambridge University Press,
Cambridge, 1 934; second edition 1 952. 8. E. Landau, " U ber eine Verallgemeinerung des Picardschen Satzes, "
Sitzungsber.
Konig/. Preuss. Akad. Wissensch.
(1 904),
1 1 1 8-1 1 33. 9. E. Landau,
Oarstellung und BegnJndung
einiger neuerer Ergebnisse der Funktionen theorie,
Springer-Verlag, Berlin, 1 91 6, sec
ond edition 1 929, third edition 1 986 edited and supplemented by Dieter Gaier.
A U T H OR S
HAROLD P. BOAS
DMITRY KHAVINSON
Department of Mathematics
Department of Mathematical Sciences
Texas A&M Un iversity
University of Arkansas Fayetteville, AR 72701 USA
College Station, TX 77843-3368 USA e- mail :
[email protected] e- mail :
[email protected] Harold P. Boas received his A.B. and S.M. from Harvard in
Dmitry Khavinson was born in Moscow, Russia. He obtained
Technology in 1 980. After
his Ph. D . from Brown University in 1 983. Since 1 983, he has
his M . S . from Moscow State Pedagogical Institute in 1 978 and
1 976 and his Ph.D. from the Massachusetts Institute of
4 years as a J.F. Ritt Assistant
been teaching at the University of Arkansas in Fayetteville. He
Professor at Columbia University in New York, he joined the
faculty of Texas A&M University in College Station in 1 984. He
has been a visiting
Bergman Prize for their research on function theory in multi
Michigan, University of Alabama, and Universidad de La
space.
Editor for the international journal Complex Variables.
and his collaborator, Emil J. Straube, shared the 1 995 Stefan
Laguna (Tenerife). Since 1 991 , he has served as an Associate
dimensional complex space. Currently, he is lost in cyber
1 0. Deutsche Mathematiker-Vereinigung, Nach
rufe, an index of the obituary notices from
W I N N E R OF T H E 1 99 8 ASSOC IATION OF A M E RICAN I'LJI�I.ISH ERS BEST N EW J'ITL.E IN rvlATH EMATICSI
the DMV Jahresbericht is available on the
JAMES.
KEENER, University of Utah and JAMES SNEYD. University of Michigan
World-Wide Web at the address http:// www. mathematik. uni-bielefeld. de/DMVI
Mathematical Physiology
archiv/nachrufe.html.
1 1 . G. P61ya and G. Szego, Problems and Theorems
Springer-Verlag,
in Analysis,
professor at the Royal Institute of
Technology (Stockholm), the Technion (Haifa), University of
Mathematical Physiology
James Keener james sneyd
Heidelberg, 1 972, Vol. 1 .
1 2 . I. Schur, "Bemerkungen zur Theorie der
beschrankten Bilinearformen mit unendlich vielen Veranderlichen," J. Reine Angew. Math. 1 40 (1 9 1 1 ), 1 -28. 1 3. H.
Weyl,
mit
"Singulare lntegralgleichungen
besonderer BerOck-sichtigung
Fourierschen
lntegraltheorems,"
des Ph.D.
1 4. F. Wiener, "Eiementarer Beweis eines Reihensatzes von Herrn Hilbert," Math. Ann. 68 (1 9 1 0), 361 -366. 1 5. F.W.
Wiener,
"Eiementare Beitrage
zur
neueren Funktionentheorie," Ph.D. thesis, Georg-August-Universitat, Gottingen, 1 91 1 . 1 6. A.
Wiman,
"Sur
une
This book provides an overview of mathematical physiology
thesis, Gottingen, 1 908.
extension
d ' un
theoreme de M. Hadamard, " Ark. Math.
Astron. Fys. 2(1 4) (1 905), 1 -5.
containing a variety of physio logical models.
Physiology
Mathemarical
is divided into two parts: the
first
pan
is a pedagogical presentation of some of the basic theory; the second pan is devmed to an exten
sive discussion of particular physiological
tems. Machemarical Physiology will
�
sys
be of interest
to researchers and graduate students in applied
m athematics interested in physiological prob
lems. a n d to quantitative physiologists wishing to know about current and new mathematical techniques.
I 998/785 PP. . 360 ILUJS. HAROCOVERI$69.95 ISBN 0·387-9838 1 -3 INTERDISCIPLI ARY APPuED MATHEMATICS. VOLUME 8
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