Volume 29, Number 2, February 1982 Notices of the American Mathematical Society 0444854398, 044486329X

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CALENDAR OF AMS MEETINGS THIS CALENDAR lists all meetings which have been approved by the Council prior to the date this issue of the Notices was sent to press. The summer and annual meetings are joint meetings of the Mathematical Association of America and the Ameri· can Mathematical Society. The meeting dates which fall rather far in the future are subject to change; this is particularly true of meetings to which no numbers have yet been assigned. Programs of the meetings will appear in the issues indicated below. First and second announcements of the meetings will have appeared in earlier issues. ABSTRACTS OF PAPERS presented at a meeting of the Society are published in the journal Abstracts of papers presented to the American Mathematical Society in the issue corresponding to that of the Notices which contains the program of the meet· ing. Abstracts should be submitted on special forms which are available in many departments of mathematics and from the office of the Society in Providence. Abstracts of papers to be presented at the meeting must be received at the headquarters of the Society in Providence, Rhode Island, on or before the deadline given below for the meeting. Note that the deadline for ab· stracts submitted for consideration for presentation at special sessions is usually three weeks earlier than that specified below. For additional information consult the meeting announcement and the list of organizers of special sessions. MEETING NUMBER

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DATE

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Notices

of the American Mathematical Society Volume 29, Number 2, February 1982

EDITORIAL COMMITTEE Ralph P. Boas, Ed Dubinsky Richard ]. Griego, Susan Montgomery Mary Ellen Rudin, Bertram Walsh Everett Pitcher {Chairman) MANAGING EDITOR Lincoln K. Durst ASSOCIATE EDITORS Hans Samelson, Queries Ronald L. Graham, Special Articles SUBSCRIPTION ORDERS Subscription for Vol. 29 (1982): $36 list, $18 member. The subscription price for members is included in the annual dues. Subscriptions and orders for AMS publications should be addressed to the American Mathematical Society, P. 0. Box 1571, Annex Station, Providence, Rl 02901. All orders must be prepaid. ORDERS FOR AMS BOOKS AND INQUIRIES ABOUT SALES, SUBSCRIP· TIONS, AND DUES may be made by calling Carol-Ann Blackwood at 800-556-7774 (toll free in U.S.) between 8:00a.m. and 4:15p.m. eastern time, Monday through Friday. See page 17. CHANGE OF ADDRESS. To avoid interruption in service please send address changes four to six weeks in advance. It is essential to include the member code which appears on the address label with all correspondence regarding subscriptions. INFORMATION ABOUT ADVERTISING in the Notices may be obtained from Virginia Biber at 401-272-9500. CORRESPONDENCE, including changes of address should be sent to American Mathematical Society, P.O. Box 6248, Providence, Rl 02940. Second class postage paid at Providence, Rl, and additional mailing offices. Copyright © 1982 by the American Mathematical Society. Printed in the United States of America.

130 The van der Waerden Conjecture: Two Soviet Solutions, j. C. Lagarias 134 1982 Cole Prizes in Number Theory 136 AMS Nomination Procedure is Vulnerable to "Truncation of Preferences," Steven j. Brams 139 1980 CBMS Survey, Major Findings 144 25th AMS Survey (Second Report), Employment of New Doctorates, Faculty Mobility, Enrollments, Class Size 150 Queries 151 Letters to the Editor 152 International Congress, Warsaw 164 Report to Members 166 1982 AMS Elections (Nominations by Petition) 168 News and Announcements 170 New AMS Publications 175 Future Meetings of the Society Bryn Mawr, March 76, 175; Madison, April 76, 190; Bellingham, june 78, 193; AMS Summer Research Institute, 194; AMS-SIAM Summer Seminar, 194; Topics: 1984 Symposium, 195; Invited Speakers, 195; Special Sessions, 195 196 Special Meetings 202 Miscellaneous Personal Items, 202; Deaths, 202; Backlog, 202; Assistantships and Fellowships (Supplement), 204 210 AMS Reports & Communications Recent Appointments, 210; Reports of Meetings: Amherst, 210, Austin, 211, Santa Barbara, 212, Cincinnati, 212; Election Results of 1981, 213, Council for 1982, 213 215 Advertisements

The van der Waerden Conjecture: Two Soviet Solutions by J. C. Lagarias The problem section of the Jahresbericht der Deutschen Mathematiker-Vereinigung publishes both solved and unsolved problems. Several of the unsolved problems have led to an extensive literature. Among these are a problem of A. Scholz 1 on the minimum chain of multiplications needed, to c~culate xn, starting with x [4, 16], Kneser s conJecture (9] (recently proved by Lovasz (12]) and the van der Waerden conjecture (19]. B. L. van der Waerden stated his problem in 1926, as follows. If A is an n X n matrix with entries Uij (i = 1, ... , n; j = 1, ... , n) then the permanent of A, denoted per(A), is defined by per(A) =

I:

conjecture since by Stirling's formula n!/nn is approximately v'2rn e-n. H. Mine gave a detailed history of work on the van der Waerden conjecture (14]. At first glance it appears reasonable to suppose that the van der Waerden conjecture is true, based on the truth of the conjecture for small values of n and the simple heuristic that the minimum of a symmetric function would itself likely be symmetric. The answer to a problem proposed by H. S. Shapiro (17] in 1954 casts some doubt on this heuristic. Shapiro's problem is that of minimizing the function (

al,.,.(l)a2,.,.(2)" • ·an,.,.(n)>

)

gnXl, •.. ,Xn =

C7ESn

where Sn denotes the symmetric group on n symbols. Such a matrix A is said to be doubly stochastic if all its entries are nonnegative real numbers and all its row and column sums are 1. The van der Waerden conjecture asserts that for any doubly stochastic matrix A,

(1)

per(A)

~

per(A)

X2

subject to the constraints that all Xi ~ 0, Xi + X;+ 1 > 0 (to avoid zero denominators). This function is invariant under the transitive permutation group generated by the n-cycle (12 3 · · ·n). The symmetric solution

n!/nn.

X1

The doubly stochastic matrix Jn having all entries equal to 1/n has per(Jn) = n!/nn, showing (1) cannot be improved. A strong version of the ~onjecture asserts that equality holds in (1) only If A=Jn. The van der Waerden conjecture attained the status of a notorious unsolved problem as a consequence of many unsuccessful attempts to The first published results on the prove it. problem appeared in the 1959 paper of Marcus and Newman (13]. They showed Jn was a local minimum of the permanent function in the set of all doubly-stochastic matrices, derived properties of doubly-stochastic matrices that minimize the permanent, and proved the conjecture for n = 3. This was the beginning of a flood of papers by many authors, who proved the conjecture for n = 4, n = 5, and for many special subclasses of doubly-stochastic matrices. In another direction T. Bang (2] and, independently, S. Friedland (8] recently showed that (2)

X1

---+---+··· X3 + X4 X2 + X3 + Xn-1 +~ X1 + X2 Xn + X1

=

X2 = ··· = Xn

gives gn(Xb ..• , Xn) the value n/2, and this is the minimum of gn for all n < 10. However for all n ~ 14 the minimum is Strictly less than n/2, i.e. symmetry is broken. (For a survey of these results, see Mitrinovic (15, pp. 132 fi].) The van der Waerden conjecture has been independently settled by two Soviet mathematicians, D. I. Falikman, at the Scientific Research Institute for Automated Systems of Planning and Management in Construction in the Ukrainian SSR and G. P. Egorychev, at the L. V. Kirensky Institute of Physics, Siberian Branch of the Academy of Sciences of the USSR in Krasnoyarsk. Falikman apparently found his proof first, since he submitted his paper (6] to Mat. Zametki in May J. C. LAGARIAS received his Ph.D. in 1974 from the Massachusetts Institute of Technology; he was a student of Harold M. Stark. Since that time he has been at Bell Laboratories where he is a Member of Technical Staff in the Mathematical Studies Department. In 1978-1979 he was a visiting assistant professor at the University of Maryland. His research interests include number theory, theoretical computer science, discrete mathematics and optimization problems.

~ e-n

for all doubly-stochastic matrices A. This result is not much weaker than the van der Waerden 1 Scholz' problem actually was first proposed in the 19th century.

130

1979, while Egorychev distributed a preprint of his paper [5] in late 1980. Falikman proves the van der Waerden conjecture, while Egorychev obtains in addition the stronger result that Jn is the unique permanent-minimizing doubly stochastic matrix. Although the two proofs differ in detail, they both use the same new idea, which is stated as the Main Lemma below. Egorychev's proof was the first to become known in the West. In late 1980 he sent preprints to a number of Western mathematicians, including J. H. van Lint at Eindhoven University of Technology, Ira Gessel and Richard Stanley at M.I.T., and later to Donald Knuth at Stanford. At M.I.T., Ira Gessel, who reads Russian, realized that Egorychev's preprint announced a proof of van der Waerden's conjecture. He prepared a translation in order that the proof could be verified by other mathematicians. At the same time rumors rapidly spread to other universities that the van der Waerden conjecture had been solved. Unlike most such rumors, these turned out to be true. Egorychev's proof was self-contained and elementary, except at one point (the key point!) and was easily verified. That one point was a reference to the Alexandrov inequalities for mixed discriminants of quadratic forms, which was used to prove a permanent inequality called the Main Lemma below. The Main Lemma is in fact a special case of the Alexandrov inequalities, as can be seen after observing that the permanent of a matrix A is the mixed discriminant of a certain set of diagonal quadratic forms. Alexandrov [1, IV] proved his inequalities for mixed discriminants in 1938, in the process of giving his second proof of the better-known Alexandrov-Fenchel inequalities for mixed volumes of convex bodies. His proof is not easily accessible in English. (For a statement of the Alexandrov inequalities and a sketch of Alexandrov's proof see Busemann [3, pp. 51 fl].) After reading Egorychev's preprint, J. H. van Lint and D. Knuth both derived short direct proofs of the Main Lemma by simplifying Alexandrov's proof. They described these in detailed accounts of Egorychev's proof [10, 20]. In July 1981 Falikman's paper appeared in the latest issue of Mat. Zametki. It came as a surprise to most Soviet mathematicians as well as to those in the West. Falikman's paper is short (7 pages) and completely self-contained. He directly proves a special case of the Main Lemma sufficient to obtain his result. We now describe the main ideas in Egorychev's and Falikman's proofs. Both proofs owe something to the 1959 paper of Marcus and Newman. To describe their results, we introduce some notation. Let A[iiJ"] denote the (n -1) X (n- 1) matrix obtained by deleting row i and column J. from the n X n matrix A. The set of all doublystochastic matrices On is a compact, convex subset of the (n -1)2-dimensional vector space V of all n X n matrices with all row and column sums 1. The interior int(On) is the set of all doubly

stochastic matrices A with all aii > 0, while the boundary bd(On) consists of all doubly-stochastic matrices A with some aii = 0. Marcus and Newman first proved there were restrictions on the pattern of zeroes of any matrix A in bd(On) that is a global minimum of the permanent, e.g. that A is indecomposable. 2 Next they examined the necessary conditions for a minimum using Lagrange multipliers, and for indecomposable matrices A they derived the conditions:

(3)

aij

;F. 0 ::::} per A[ilj] = per A.

Next they introduced an "averaging" operation. 3 Write A = [at, ... , an] where ai denotes the ith row of A. Suppose A is permanent-minimizing and that all aii > 0. Then one has

(4)

per [!(at+ a2), !(at + a2), a3, ... , an] n

= i L(ali +

a2i)(per A[lli] +per A[2li])

i=l = per(A), using (3) and the row-stochasticity property. Hence A{l) = [!(a1 + a2), !(at+ a2), a3, ... , an], a matrix derived from A by averaging the first two rows, is also a permanent-minimizing matrix with all aii > 0. By repeating this argument, averaging over different rows, one obtains a sequence A{ 1),A( 2),A(3), ... of permanentminimizing matrices with A(n) -+ Jn. By continuity one concludes that per A;::=: perJn. Marcus and Newman went on to prove that Jn was a strict local minimum of the permanent, and that if A ;F. Jn then one can arrange that all Ak ;F. Jn. This contradicts the local minimality of per(Jn) and proves that Jn is the unique permanentminimizing matrix not on the boundary bd(On)· Marcus and Newman could not apply their averaging argument to matrices on bd(On) because The they could not prove (4) in this case. necessary conditions for a local minimum of per(A) give some more information beyond (3), which however is not sufficient to directly prove (4). London [11] showed that these conditions imply that

(5)

aij =

0 ::::} per A[ilj] ;::=: per A

provided A is a fully indecomposable matrix. Egorychev uses London's result in his proof. The key result that Egorychev supplied is the following. 2 An n X n matrix M is decomposable if there exist permutation matrices P and Q such that the matrix P MQ has an r X s block of zeroes in its upper right comer with r s ~ n. Otherwise it is indecomposable.

+

3 The averaging argument described below differs in detail from that used in the Marcus and Newman paper; the idea is the same.

131

where x, y are constants depending on E, and y is nonnegative. He then proves the Main Lemma in the special case that per (a1, ... , an) = 0. He uses the Main Lemma to prove that Jn is the only solution to (9) as follows. Suppose (9) holds with a 1 ::;6- a 2. Let t = a 1 - a2, and define s = (s1, ... , sn) by Si = a1,ia2,i· Then the Main Lemma gives

MAIN LEMMA. Let a2, as, ... , an be 1 X n row vectors with nonnegative entries. Then

(6)

per(a1, a1, as, ... , an) per(a2, a2, as, ... , an) ~ [per(a1,a2, ... ,an)] 2 •

If a2, as, ... , an have only positive components, then equality holds in (6) if and only if a 1 = >-a 2 for some real A. There is another way to state the Main Lemma which may be enlightening. Observe that if as, ... , an are held fixed we can regard per(x, y, as, ... , an) as a symmetric bilinear form in the variables x and y. The Main Lemma is equivalent to the assertion that whenever as, ... , an have positive entries this bilinear form is not degenerate and has exactly one positive eigenvalue, i.e. it has signature (1, n -1) [20]. Using the Main Lemma, Egorychev shows that any doubly-stochastic matrix A that satisfies

per(t,t,as, ... ,an)per(s,s,as, ... ,an) ::; [per(t,s,as, ... ,ant· However the conditions (9) imply after a calculation that per(t, s, a3, ... , an) = 0,

n (a1,i- a2,i) 2

per(t, t, as, ... , an) = y

per(s, s, as, ... , an)

must have

(7)

1 ~ i, j

per A[ilj] = per A,

~

n.

Egorychev uses this to show that the existence of a permanent-minimizing matrix on bd(On) implies that there also exists a permanent-minimizing matrix A in int(On) with A ::;6- Jn, a contradiction that proves that Jn is the unique minimum of the permanent function on On. Falikman's proof studies the function

f(A) = per(A)+ E(

II

aij) -

1

1-ols of vector and tensor analysis, the extension of these concepts into abstract fu~~tiO':J spaces (functional analysis), and the umflcat1on of these subjects with the variational calculus and associated methods of numerical (1-09349-1) approximation. approx. 464 pp. Feb.1982 $42.50

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stochastic control by Functional Analysis Methods by ALAIN BENSOUSSAN STUDIES IN MATHEMATICS AND ITS APPLICATIONS, Volume 11 1982 xvi + 410 pages Price: US $58.25/Dfl. 125.00 ISBN 0-444-86329-X This book is an advanced text on stochastic control. It is self-contained and presents most of the results on stochastic control, relying on functional analysis methods and the theory of partial differential equations. Chapter I presents the elements of Stochastic Calculus and Stochastic Differential Equations, Chapter II presents the theory of partial differential equations and Chapter Ill discusses the Martingale problem - thus the first three chapters are devoted to background.

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APPLICATIONS OF CENTRE MAN I FOLD THEORY Jack Carr 1981 /142 pp./ paper $14.00 Applied Mathematical Sciences, Volume 35 ISBN 0-387-905774

BASIC THEORY OF ALGEBRAIC GROUPS AND LIE ALGEBRAS Gerhard P. Hochschild 1981 /267 pp./ cloth $32.00 Graduate Texts in Mathematics ISBN 0-387-90541-3

APPLIED FUNCTIONAL ANALYSIS Second Edition A. V. Balakrishnan 1981 /313 pp./ cloth $34.00 Applications of Mathematics ISBN 0-387-90527-8

Lynn Arthur Steen 1981 /250 pp./12 illus./ cloth $18.00 ISBN 0-387-90564-2

Edited by Raoul Bott 1981 /882 pp./22 illus. I cloth $36.00 ISBN 0-387-905324

QUANTUM PHYSICS A Functional Integral Point of View James Glimm and Arthur Jaffe 1981 /417 pp./43 illus. paper $16.80; cloth $28.00 ISBN 0-387-90562-0 (paper) 0-387-90551.0

THE NON-EUCLIDEAN HYPERBOLIC PLANE Its Structure and Consistency Paul Kelly and Gordon Mathews 1981 /333 pp./201 illus./ paper $24.00 ISBN 0-387-90552-9

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