Showing papers in "Bulletin of the American Mathematical Society in 1983"

the uncertainty principle

Abstract: On considere l'existence et la regularite des solutions d'equations aux derivees partielles, la construction de solutions fondamentales explicites et les valeurs propres d'operateurs de Schrodinger

Fixed point theory and nonlinear problems

Abstract: Introduction. Among the most original and far-reaching of the contributions made by Henri Poincare to mathematics was his introduction of the use of topological or "qualitative" methods in the study of nonlinear problems in analysis. His starting point was the study of the differential equations of celestial mechanics, and in particular of their periodic solutions. His work on this topic began with his thesis in 1879, and was developed in detail in his great three-volume work, Methodes nouvelles de la mechanique celeste, which appeared in the early 1890s and summarized his many memoirs of the intervening period. It continued until his memoir shortly before his death in 1912 in which he put forward the unproved fixed point result usually referred to as "Poincare's last geometric theorem". The ideas introduced by Poincare include the use of fixed point theorems, the continuation method, and the general concept of global analysis. The writer's acquaintance with Poincare's influence came through contact with Solomon Lefschetz and Marston Morse, both of whom were very explicit as to the role of Poincare as an initiator in this direction of mathematical development. In 1934, in the Foreword to his Colloquium volume on The calculus of variations in the large. Morse had put this forward very forcefully in the first paragraph:

The point of pointless topology

Abstract: Introduction. A celebrated reviewer once described a certain paper (in a phrase which never actually saw publication in Mathematical Reviews) as being concerned with the study of "valueless measures on pointless spaces". This article contains nothing about measures, valueless or otherwise; but I hope that by giving a historical survey of the subject known as "pointless topology" (i.e. the study of topology where open-set lattices are taken as the primitive notion) I shall succeed in convincing the reader that it does after all have some point to it. However, it is curious that the point (as I see it) is one which has emerged only relatively recently, after a substantial period during which the theory of pointless spaces has been developed without any very definite goal in view. I am sure there is a moral here; but I am not sure whether it shows that "pointless" abstraction for its own sake is a good thing (because it might one day turn out to be useful) or a bad thing (because it tends to obscure whatever point there might be in a subject). That much I shall leave for the reader to decide. This article is in the nature of a trailer for my book Stone spaces [35], and detailed proofs of (almost) all the results stated here will be found in the book (together with a much fuller bibliography than can be accommodated in this article). However, I should make it plain that I do not claim personal credit for more than a small proportion of these results, and that my own understanding of the nature of pointless topology has been enriched by my contacts with a number of other mathematicians, amongst whom I should particularly mention Bernhard Banaschewski, Michael Fourman, Martin Hyland, John Isbell, Andre Joyal and Myles Tierney. I should also mention the work of Bill Lawvere, particularly as reported in [41], on the nature of continuous variation and the conceptual relation between constant and variable quantities, which has had a profound influence on the developments which I wish to describe; but such questions as these will not be explicitly considered in the present article.

$Q$ valued functions minimizing Dirichlet's integral and the regularity of area minimizing rectifiable currents up to codimension two

Abstract: We announce several results of an extensive study [A] of the size of singular sets in oriented m dimensional surfaces which are area minimizing in m + I dimensional Riemannian manifolds. Our principal result is that the Hausdorff dimension of such a singular set does not exceed ra — 2. Examples show this is the best possible such general estimate when / > 2, i.e., when branching singularities «are possible. The general existence of such surfaces of least area is well known in a variety of settings [F, 5.1.6]. In order to obtain estimates on branching of area minimizing surfaces we were led to use Taylor's expansion in terms of first derivatives at 0 to approximate the nonparametric area integrand by Dirichlet's integrand. Accordingly, we study branched coverings of regions in R m which are graphs of multiple valued functions minimizing the integral of Dirichlet's integrand. As a central estimate in our analysis of area minimizing surfaces we show that the Hausdorff dimension of the branch set of such a minimizing covering does not exceed m — 2. To state several results in more detail we use the terminology of [F]. Suppose that A is a bounded open subset of R m with smooth boundary, and let k, /, ra, n, Q be positive integers with k > 3, I < n, and m > 2.

The influence of elasticity on analysis: The classic heritage

Abstract: Etude des domaines suivants: Le catenaire: solutions doubles, et le calcul variationnel; l'elastique: fonctions elliptiques, analyse qualitative, solutions multiples, geometrie differentielle des courbes gauches; systemes en petites vibrations: nombres propres et modes simples, equations differentielles lineaires, fonctions de Bessel, polynomes de Laguerre, transformees integrales, oscillations resonantes; equations aux derives partielles: concept de «solution», discontinuites, la definition «moderne» d'une fonction, base fonctionnelle, singularites, solutions generalisees; plaques: la courbure «gaussienne»; theorie tridimensionnelle: deformation des regions, deformation et rotation locale, tenseurs, nombres et vecteurs propres, decomposition polaire, resolution spectrale, isotropie et hemitropie, groupes; torsion d'un cylindre: solution principale d'un systeme d'equations differentielles et son etat d'approximation d'autres solutions

Strictly pseudoconvex domains in $C^n$

Abstract: The theory which we have developed so far will be expanded in two ways: 1° In this chapter we shall use the Cauchy-Fantappie calculus to transform the Bochner-Martinelli kernel into a kernel adapted to strictly pseudoconvex domains in ℂ n and prove a number of finiteness and vanishing theorems for these domains; 2° in the next chapter we shall pass from ℂ n to arbitrary Hermitian manifolds and establish the analogue of the BMK formula in that context, obtaining, along the way, the Hodge decomposition theorem on compact manifolds. Both lines of thought will merge in the subsequent chapters and lead to a solution of the Levi problem and finiteness theorems for strictly pseudoconvex manifolds, as well as to a solution of the \\(\\overline \\partial \\)-Neumann problem (to be formulated later).

Cascades of period-doubling bifurcations: A prerequisite for horseshoes

Abstract: On montre qu'une infinite de cascades de dedoublements de periodes doivent apparaitre dans le processus de formation de fer a cheval. Chacune de ces cascades n'a pas besoin d'evoluer de facon reguliere ou monotone, mais elle doit contenir des points periodiques attractifs de toutes les periodes k, 2k, 4k, 8k, ... ∀k

Instantons, double wells and large deviations

Abstract: We find the leading asymptotics of the exponentially small splitting of the two lowest eigenvalues of — ^A -f \\ 2 V in the limit as X —• oo where V is a nonnegative potential with two zeros. In this note, we consider eigenvalues of a Schrödinger operator — | A + \\ 2 y in the limit X —> oo (which is quasiclassical since up to a factor of ti, —hA + V has this form with h = X 1 going to zero). The method can deal with eigenvalues other than the lowest, with multiple minima (even manifolds of minima) or degenerate minima. In this note, for simplicity we discuss only the two lowest eigenvalues, E0(\\) and £i(X), with corresponding normalized eigenvectors Ho(X), Qi(X). More general situations and detailed proofs will appear elsewhere [13]. We will suppose the following about V. (i) V is C, (ii) V(x) > 0 for all x and ^rnL\\x\\^00V(x) > 0, (iii) V vanishes at exactly two points a and b and at these points dV/dXidXj is strictly positive definite. Under these circumstances, one can prove (see e.g. [12]) that ipo is concentrated as X —• oo near the points a, b and that Ei(\\)/\\ has a finite nonzero limit. We want also to suppose that for all e small and for y = a and for y = b (1) Urn f \\ipo{\\x)\\dx>0. One case where (1) holds is when there is a symmetry of order 2 (such as reflection) which leaves V and —A invariant, so that the limit is \\ for y = a or b. Under these circumstances, one expects that the splitting of E\\ and EQ will be governed by tunneling and one goal here is to obtain multidimensional Received by the editors October 21, 1982 and, in revised form, November 29, 1982. 1980 Mathematics Subject Classification. Primary 35P15, 81H99; Secondary 60J65. Research partially supported by USNSF under grant MCS-81-20833. © 1983 American Mathematical Society 0273-0979/82/0000-1204/$01.75 323 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

The influence of elasticity on analysis: Modern developments

Abstract: On examine plusieurs domaines de l'analyse auxquels l'elasticite a apporte une contribution importante: questions de connectivite de la theorie globale des bifurcations; particularites de l'analyse non lineaire; calcul des variations; convergence faible; inegalites variationnelles; autres contributions

Arithmetic characterizations of Sidon sets

Abstract: Let G be any discrete Abelian group. We give several arithmetic characterizations of Sidon sets in G. In particular, we show that a set A is a Sidon set iff there is a number 6 > 0 such that any finite subset A of A contains a subset B Q A with |B| > 6\\A\\ which is quasiindependent, i.e. such that the only relation of the form ]C\\eB e x ^ = '̂ with e\\ equal to + 1 or 0, is the trivial one. Let G be a compact Abelian group and let G be the dual group. For any ƒ in L2(G), we denote by ƒ the Fourier transform of ƒ. A subset A of G is called a Sidon set if there is a constant K with the following property: all the trigonometric polynomials ƒ, such that ƒ is supported by A, satisfy Ei/(7)i<^imic(G). We will denote by 5(A) the smallest constant K with this property. In the theory of Sidon sets (cf. e.g. [2]), there has always been considerable interest in the relations between this analytical definition and the arithmetic properties of the set A (in particular, in the case G = T and A c Z). The aim of this note is to announce several arithmetic characterizations of Sidon sets. Let us make more precise what we mean here by \"arithmetic\". We will denote by RA the set of relations (with coefficients in {—1,0,1}) satisfied by A, i.e. the set of all finitely supported families (e\\)\\eA in {—1,0,1} such that EXGA € X X = 0 By an \"arithmetic\" characterization is usually meant one which depends only on the set RAIn [1], Drury proved that such a characterization exists, but he could not produce any explicit one. Precisely, he proved the following: let A and A' be two sets for which there is a bijection : A' —• A such that the map : RA -> RA', defined by 0((ex)\\eA) = (e^,(x'))\\'€A', is also a bijection. Then, A is a Sidon set iff the same is true for A'. In other words, the property of \"being a Sidon set\" is determined by RAWe give below several explicit arithmetic characterizations, from which the preceding result of Drury follows as a corollary. To state our results, we will need some notation and terminology. We will denote by IA the set of all finitely supported families (e\\)xeA in {—1,0,1}. For any 7 in G, we will denote by H(^,A) the number of ways to write 7 as Received by the editors July 14, 1982. 1980 Mathematics Subject Classification. Primary 43A46, 42A55; Secondary 41A46, 41A65.

Explicit relaxation of a variational problem in optimal design

Abstract: for vector-valued functions u on Lipschitz domains Q c R. The right side of (1) is the relaxation of the left, cf. [1]. Each infimum is over u G if(Q;R), lsupp Vu denotes the characteristic function of the support of Vu, and \\Vu\\ = E(

Physical space-time and nonrealizable ${\text{CR}}$-structures

Abstract: Space-time views leading up to Einstein's general relativity are described in relation to some of Poincare's early ideas on the subject. The basic geometry of twistor theory is introduced as it arises both from Minkowski space-time and the more general curved Einstein models. It is shown how this provides a CR-structure (this being, in essence, another of Poincare's poioneering concepts) in a natural way. Nonrealizable CR-structure can arise, and an example is presented, due to C. D. Hill, G. A. J. Sparling and the author, of a complex manifold-with-boundary which cannot be extended as a complex manifold beyond its C/sup infinity/ boundary.

Entropies and factorizations of topological Markov shifts

Abstract: On presente une caracterisation des valeurs possibles pour l'entropie topologique de decalages de Markov. Si λ est un nombre de Penon, alors il y a une matrice non-negative aperiodique entiere dont le rayon spectral est λ

Hook young diagrams, combinatorics and representations of Lie superalgebras

Abstract: Let V be a finite-dimensional F-vector space, char(F) = 0. Schur introduced the action of the symmetric group Sn on V® n and was then able to determine the representation theory of the general linear group GL{V) [9]. His work was later completed by H. Weyl [10]. This work connects that representation theory with combinatorics via standard and semistandard Young tableaux and via the Schur functions (cf. [6]). Many of the objects in this theory are parametrized by the Young diagrams in a strip. In this work we introduce a slightly more general permutation action of Sn on V® and then describe how most of the above theory generalizes. The main feature here is that most of the generalized objects are parametrized by the partitions inside a hook. The action. Let k,l > 0, k + l > 0, T and U disjoint vector spaces, dimT = /c, dim U = Z, and V = T © U. We define a new right action of Sn on V® , i.e., a map \\j)\\ Sn —• Endj?0^® ), based on Schur's original action and on the functions ƒ/: Sn —• {±1} [5] as follows. Choose bases ti,...,tk G T, tii,...,t6j G U. These induce a basis of V®. Let vx ® • • • <8> vn G V® , vi,...,vn G {ti,...,Uf} be such a basis element, let ƒ = {i\\vi G U} and let aeSn. Then (vi <8• • • <8> vn)#T) = / / ( ( T X V ^ I ) ® ® ! ; ^ ) ) . DEr Extend \\j)(a) to all of V® by linearity: ^((r) G End(V). As usual, we now extend ip to F^n, then check that i\\)\\ FSn —• End(V® ) is an (associative) algebra homomorphism. T/ie question. It is well known that FSn = SxePar(n)®-^» where Par(n) denotes the set of partitions of n and where each I\\ is a simple algebra. It follows that for some T = T(fc,/;n) Ç Par(n), ^ ( F S n ^ S x e r ^ x » and the basic question here is to describe T. Letting B(k, l;ri) be the centralizer of ip(FSn) in EndF(V^ ), the decomposition of V® into irreducible ip(FSn) or S(fc, /; n) modules, will be given by the classical theory of Schur. The answer\\ which extends a theorem of Weyl is