Showing papers in "Bulletin of the American Mathematical Society in 1966"

Cohomology and deformations in graded lie algebras

Abstract: : The theories of deformations of associative algebras, Lie algebras, and of representations and homomorphisms of these all show a striking similarity to the theory of deformations of complex analytic manifolds The common context of all these turns out to be the problem of deforming one single element x of degree one in a graded Lie algebra subject to the condition (x,x) = 0 The first part of this report gives general properties of graded Lie algebras over fields of characteristic not 2; the second part deals with the case of characteristic 2 The third and fourth parts give general theorems on deformability for the respective cases when the base field is the real or complex numbers (analytic methods), and general algebraically closed fields (algebraic methods) It is shown that deformability is to a large extent determined by certain cohomology groups and mappings of these (Author)

A setting for global analysis

Abstract: Introduction. The primary aim of this report is to present a broad outline for a coherent geometric theory of certain aspects of nonlinear functional analysis. Its setting requires the calculus in topological vector spaces, differential geometry of infinite dimensional manifolds, and the algebraic and differential topology of function spaces. For the most part the developments are of quite recent origin, and at present the theory is in a fluid state (its growth depending strongly on its concrete applications). The beginnings of the subject may be traced to the work of Fréchet, Gâteaux, and Vol terra; we refer to the text [73] of P. Levy for an exposition of some early applications (especially in the calculus of variations and integrable differential systems)—and ask pardon for not presenting any historical perspective in the present survey. About ten years ago it was formally recognized [29] that many of the function spaces which arise in global geometric mathematics possess a natural infinite dimensional differentiable manifold structure. Tha t was not a great surprise; for (1) Many of the most interesting manifolds of differential geometry are well known to have representations as function spaces of rigid maps. (E.g., Riemannian manifolds arise as the configuration spaces of dynamical systems, their cotangent bundles are interpreted as phase spaces, and their Riemannian metrics in terms of kinetic energy.) (2) Much of the language of the classical treatment of the calculus of variations—and the penetrating viewpoint and methods of M. Morse—is that of a function space differential geometry. (E.g., the Euler-Lagrange operator of a variational problem has an interpretation as a gradient vector field, whose trajectories are lines of steepest descent.) (3) Certain eigenvalue problems in integral and differential equations have interpretations in terms of Lagrange's method of multipliers, involving differential geometric ideas in infinite dimensions (e.g., focal point theory, and geometric consequences of the inverse

Rational points in Henselian discrete valuation rings

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Hilbert space is homeomorphic to the countable infinite product of lines

Abstract: As a consequence of this theorem it is possible to investigate topological properties of l2 as topological properties of s. In turn s is a \"natural\" subset of the Hilbert cube (the countable infinite product of closed intervals) which facilitates the study of 5. In 1928 in [5, pp. 94-96] Fréchet raised the general question as to which linear topological spaces were homeomorphic to each other. Specifically he asked whether l2 (called 0) was homeomorphic to 5 (called Eu). In 1932 in [2, p. 233], Banach stated that Mazur had shown that 5 was not homeomorphic to Z2. Subsequently it was understood that the question was still open. The topological classification of complete linear metric spaces initiated by Fréchet has been the subject of considerable research activity with noteworthy contributions by Bessaga, Kadee, Klee and Pelczynski among others. See the bibliography in [3], Particular attention has been given to Fréchet spaces: locally convex complete linear metric spaces. With Theorem I of this paper and recent profound results of Kadec and of Bessaga and Pelczynski, the topological classification of separable infinite-dimensional Fréchet spaces is now complete. All such spaces are homeomorphic to each other. The results leading to this theorem are the following. In a paper to be published in Dokl. Akad. Nauk SSSR, Kadec gives a proof of the theorem \"All separable infinite-dimensional Banach Spaces are homeomorphic.\" Earlier in [4] and in [3, Theorem 9.2], Bessaga and Pelczynski have shown \"Under the conjecture that all separable in-

On nonseparable reflexive Banach spaces

Abstract: The purpose of this paper is to show that certain known results concerning separable spaces hold also for nonseparable reflexive Banach spaces. Our main result (Theorem 1) proves a special case of a conjecture of H. H. Corson and the author [ l] while the corollary proves some conjectures of V. Klee (see for example [2]). In order to state Theorem 1 we introduce the following notation : Let T be a set; by c0(T) we denote the Banach space of scalar valued functions ƒ on T, such that {7; | / ( Y ) | > e } is finite for every e > 0 , with the sup norm.

On von Karman's equations and the buckling of a thin elastic plate

Abstract: The object of this note is to demonstrate the applicability of the methods of nonlinear functional analysis in the investigation of a complex physical problem. In 1910 T. von Karman [9] introduced a system of 2 fourth order elliptic quasilinear partial differential equations which can be used to describe the large deflections and stresses produced in a thin elastic plate subjected to compressive forces along its edge. The most interesting phenomenon associated with this nonlinear situation is the appearance of \"buckling,\" i.e. the plate may deflect out of its plane when these forces reach a certain magnitude. Mathematically this circumstance is expressed by the multiplicity of solutions of the boundary value problem associated with von Karman's equations. With the aid of the modern theory of linear elliptic partial differential equations together with functional analysis on a suitably chosen Hilbert function space, we are able to use the structural pattern of the nonlinearity implicit in Karman s equations to obtain a qualitative nonuniqueness theory for this problem. Among the previous studies of buckling of plates are those of Friedrichs and Stoker [5] and Keller, Keller and Reiss [ó], who study only radially symmetric solutions of circular plates. Numerical studies for rectangular plates have been given by Bauer and Reiss [2] among others. Karmand equations for general domains have been studied by Fife [4] and Morosov [8] in other connections. The authors are grateful to Professors S. Agmon and W. Littman for helpful suggestions. This research was partially supported by the National Science Foundation Grant No. GP-3904 and the Air Force Office of Scientific Research Grant No. 883-65.

On the spectrum of general second order operators

Abstract: This result has been extended to other self ad joint problems for second order operators. See [2], [3], and [ó]. The purpose of this note is to show that the same technique locates the spectrum of a nonself adjoint problem in a half-plane. Such a result is of interest in investigating stability, where one needs to know whether there is any spectrum in the half-plane Re X ^ 0. In a bounded domain D we consider the differential equation

A Lefschetz fixed point formula for elliptic differential operators

Abstract: Introduction. The classical Lefschetz fixed point formula expresses, under suitable circumstances, the number of fixed points of a continuous map ƒ : X-+X in terms of the transformation induced by ƒ on the cohomology of X. If X is not just a topological space but has some further structure, and if this structure is preserved by ƒ, one would expect to be able to refine the Lefschetz formula and to say more about the nature of the fixed points. The purpose of this note is to present such a refinement (Theorem 1) when X is a compact differentiable manifold endowed with an elliptic differential operator (or more generally an elliptic complex). Taking essentially the classical operators of complex and Riemannian geometry we obtain a number of important special cases (Theorems 2,3) . The first of these was conjectured to us by Shimura and was proved by Eichler for dimension one.

Representations of complex semisimple Lie groups and Lie algebras

Abstract: 1. Notation. The object of this note is to announce some results on representations of complex semisimple Lie groups and Lie algebras. © is a semisimple Lie algebra over C, the field of complex numbers. ®, considered over i?, the field of real numbers, is denoted by ® 0. ^ is a Cartan subalgebra of ®, W, the Weyl group of (®, I)). We use the standard terminology in the theory of semisimple Lie algebras (Jacobson [3] and Harish-Chandra [2(a)], [2(b)], [2(c)]). P0 is a positive system of roots, fixed once for all and £0= {#i, • • • , ou}, the associated fundamental system, n= ]C«ep 0 ®~a; tt, considered as a Lie algebra over JR, is denoted by tio. i)o = X)« R'Ha. Fix a square root ( —1)1/2 of — 1 in C. io is a compact form of ® containing ( —1)1/2 i)0. ®o = io+i)o+tto is an Iwasawa decomposition of ®o and G = K-A+-N the corresponding decomposition of G. c(X-*Xc) is the conjugation of ® corresponding to the compact form io. Let ® denote the Lie algebra ®X® over C, and let i:X-*(X°,X)

A nonconstructive proof of Gentzen’s Hauptsatz for second order predicate logic

Abstract: Takeuti [3] showed that the consistency of analysis (i.e. second order number theory) is finitistically implied by the Hauptsatz for second order logic» i.e. by the proposition that every theorem of this system is derivable without cut. We will prove that, conversely, the Hauptsatz for this system follows from a certain generalization of the consistency of analysis; namely from: I. Every countable set of relations among natural numbers is included in an o)-model. An co-model is a collection of relations among natural numbers which is closed under the second order comprehension axiom. Henkin [l ] has shown that a second order formula is derivable with the cut rule if and only if it is valid in all (countable) co-models. When the given set of relations consists only of the successor relation, I asserts the consistency of analysis. The formalism for second order predicate logic which we will use is obtained from the system of predicate logic of finite order given in Schutte [2] by dropping all expressions and bound variables of types other than 0 (individuals), 1 (propositions) and (0, 0, • • • , 0) (relations among individuals). Thus, expressions of type 0 are built up from constants and free variables of type 0 using function constants. The expressions of type (0, • • • , 0) are constants, free variables and expressions Xx? • • • x£4(x?, • • • , x£), where A (a°u • • • , a£) is a wff (expression of type 1). The logical symbols other than X are —*, v and V. The notation and terminology of [2] will be assumed. In particular, the notions of strict derivation and partial valuation will be the same as in [2], except that they refer to the second order logic and not the full system of [2], and that we require of a partial valuation that whenever VxA(x) is true (t), then so is A{a) for some free

Structure of categories

Abstract: Introduction. This paper sets out to develop a structure theory of categories and carries it, not very far, but far enough for some applications. We need a new definition of complete (coinciding with old definitions [2], [8] for well-powered co-well-powered categories). The new definition is needed even to construct images of mappings. With it, we can show that every completion of a small category, and also every primitive category of algebras, is a retract of any category in which it is fully embedded. Such categories are called injective] strictly stronger injectiveness properties are rather trivial. By a completion of & is meant a complete category in which Cfc is fully embedded so that no complete full proper subcategory contains it. The results stated come from the regular completion theory concerning complete extensions of GL, regularly represented in Cat((£*, U), and the statements given are simply the main applications of two theorems to the effect that complete categories satisfying certain boundedness conditions are injective. Apparatus is set up, bu t not developed, for a general completion theory and finer tests for injectiveness. Precise statements of results cannot well be given before we establish the set-theoretic foundation (§1). The new clause in the definition of completeness requires every intersection of extremal subobjects [8] to be representable, and the dual. Then every completion of a small category 0, is well-powered, no intermediate full subcategory is left complete, and the embedding preserves all limits that may exist in Ô (and dually). Unfortunately, retraction preserves completeness only in the weaker sense of [8] ; and, since not every category has a completion in the same Grothendieck universe, and injectiveness is defined relative to a universe, I can prove that an injective category is complete only in the still weaker sense of Freyd [2]. In any of these senses, up to an equivalence of categories, a left complete full subcategory of a complete category is both left closed in its right closure and right closed in its left closure. Accordingly, one would hope, from Freyd's theorems on existence of adjoints, to retract by a reflector and a coreflector. This question is pursued for

Certain Hilbert spaces of entire functions

Abstract: 1. Introduction. The research reported on in the present note was motivated by the following Proposition (F), due to Ernest Fischer ([5], see also [4] for an earlier version; actually Fischer proved a more general result, but the special case suffices as a point of departure for our discussion) : (F) Let P denote a homogeneous polynomial in si, • • • , z k with complex coefficients. Then every polynomial in Si, • • • , z k has a unique representation QP+R where

The strict topology and compactness in the space of measures

Abstract: The strict topology ß on C(S), the bounded continuous complex valued functions on the locally compact Hausdorff space S, was first introduced by R. C. Buck [3], [4], [5]. It has also been studied by I. Glicksberg [10], J. Wells [20], and C. Todd [17]. This topology has been used in the study of various problems in spectral synthesis [11], spaces of bounded holomorphic functions [15], and multipliers of Banach algebras [18], [19]. This paper is a detailed account of results announced by the author in [6], [7] on the relationship of C(S)ß with its dual M(S), the bounded Radon measures on S. In particular, we are concerned with the question (posed by Buck) of whether or not C(S)e is a Mackey space and, consequently, with compactness criteria in M(S). The existence and description of the Mackey topology, the strongest topology yielding a given adjoint space, is known, and there are several properties (e.g., metrizable) which imply that a designated topology is the Mackey topology. However, the author knows of no example of a topological vector space with an intrinsically defined topology which is a Mackey space, except by virtue of some formally stronger property (e.g., metrizable, barrelled, bornological). This is not true for C(S)e. In fact, we show that if S is paracompact then every /3-weak* countably compact subset of M(S) is ß-equicontinuous ; consequently, C(S)e is a Mackey space (Theorem 2.6). Also, if S is not compact then C(S)B is not barrelled, bornological, nor metrizable. It can also happen that C(S)e is not a Mackey space, as we show for the case when S is the space of ordinal numbers less than the first uncountable ordinal. In §3 we examine the subspace problem for C(S)e. That is, if C(S)ß is a Mackey space, which subspaces of C(S) are Mackey spaces when furnished with the relative strict topology? We are able to solve this problem when S is the space of positive integers. Also, we show that 77°°, the bounded holomorphic functions on the open unit disk D, is not a Mackey space when endowed with the ß topology—even though C(D)ß is. From these results we prove the existence of a closed subspace N of I1 such that there is no bounded projection of I1 onto 7Y. Finally, (/°°, ß) is a semireflexive Mackey space with a closed subspace which is not a Mackey space.

Counterexample to Euler’s conjecture on sums of like powers

Abstract: 5 as the smallest instance in which four fifth powers sum to a fifth power. This is a counterexample to a conjecture by Euler [l] that at least n nth powers are required to sum to an nth power, n>2.

Functional differential equations close to differential equations

Abstract: when the lag function r(t) is nearly constant for large /, and has also asked for conditions on the function r under which all solutions approach zero as t—» oo. The purpose of this announcement is to initiate a study of various stability and oscillation problems for equations with perturbed lag functions, and to suggest that a modification of the familiar method of conversion to an integral equation can be profitably employed for these problems. In this announcement, we characterize the asymptotic behavior of solutions of Equation (1) in case r(t) tends to zero with a certain order as £-^oo. We go beyond the problem posed by Bellman by establishing the asymptotic equivalence of Equation (1) and an approximating ordinary differential equation. In subsequent work, we shall deal with cases in which r{t) is asymptotically constant, or is oscillatory, extensions to higher order equations, and so on. In particular, we shall deal with the remaining parts of Bellman's Research Problem. The first result obtained is as follows:

Wiener's contributions to generalized harmonic analysis, prediction theory and filter theory

Abstract: 0. Prologue. The strong cohesive forces permeating the work of Norbert Wiener complicate the task of surveying his contributions to specific areas. Where is one to begin and where to end? In the realm of prediction, for instance, Wiener's book [TS] stands out as his first major contribution. But an important part of this book concerns the synthesis of predictors, for which as Kakutani remarked (32) : \"The theory of generalized harmonic analysis developed by the author some 20 years ago is exactly the right tool . . . .\" Now the latter theory, given in the memoir [GHA] of 1930, was itself the culmination of researches begun in 1924, which were motivated by even earlier investigations in the theory of Brownian motion. I t would seem tha t a thorough review of Wiener's work in prediction should start from about the year 1919 when he looked at the Charles River from his office a t M.I .T. and began to wonder whether the Lebesgue integral was the right tool for the analysis of the undulating water surface. Such a review would be beyond the abilities of this writer, even if he were granted the necessary space. In this review we shall first survey those aspects of Wiener's great memoir [GHA] which bear on his later work on prediction and filtering (I). We shall then describe briefly how the mathematical activity of the thirties influenced his thought (II). Next we shall discuss Wiener's general theory of nonlinear prediction (III) . From this we shall turn to his many contributions to linear prediction and filtering theory (IV). Lastly we shall dwell on his theory of filters (V).

Wiener and integration in function spaces

Abstract: 1. Evolution of Mathematics is, by and large, a continuous process and its growth and progress seldom deviate greatly from the natural historical lines. I t is because of this that we tend, in retrospect, to admire most those developments which though born well outside it have grown to join and to enrich the mainstream of our science. I t was the great fortune and the great achievement of Norbert Wiener to initiate such a development when, in the early twenties, he introduced a measure, now justly bearing his name, in the space of continuous functions.

Persistent and invariant formulas relative to theories of higher order

Abstract: The purpose of this department is to provide early announcement of significant new results, with some indications of proof. Although ordinarily a research announcement should be a brief summary of a paper to be published in full elsewhere, papers giving complete proofs of results of exceptional interest are also solicited. Manuscripts more than eight typewritten double spaced pages long will not be considered as acceptable.

On central topological groups

Abstract: Introduction. Let G be a locally compact group and Z its center. We shall be concerned with the class [Z] of all locally compact groups G such that G/Z is compact. In studying [Z]-groups the chief goal we have in mind is to obtain a natural generalization of the theory of compact groups and that of locally compact abelian groups broad enough to be nontrivial and which enables one to extend virtually all the important results pertaining to the aforementioned theories; this is done for the structure and representation theory. (For the latter see \"Representation Theory of Central Topological Groups,\" p. 831 of this Bulletin).

Existence theory for two point boundary value problems

Abstract: for some M è O , a^t^b and all continuously differentiable functions u(t). Let the constants a^ bi satisfy: (5) at ^ 0, bi Ê; 0, i = 0, 1; a0 + b0 > 0. Tfeew a unique solution of (1), (2), (3) exists f or each (a, /3). PROOF. We sketch the proof. The initial value problem u" = ƒ(/, u, u'), a S t S b\ (6) a0u(a) — aiu'(a) = a\ a\Co — a$ci = 1; co^(a) — Ciu' has the unique solution u(s\ t). The problem (1), (2), (3) has as many solutions as there are real roots, s* of (j>(s) = bou(s; b) + b\U(s\ b) = ft Since w(s; t) is continuously differentiable with respect to 5 the derivative %(t)^du(s\ t)/ds satisfies the variational problem [ l ] , *" « PM + S(o«,

A characterization of $Q$-domains

Abstract: Let R be an integral domain with quotient field K. By an overring of R is meant a ring B with RÇ1BÇ.K. R is a Q-domain if every overring of R is a ring of quotients of R with respect to some multiplicative system in R. A P-domain is a Prüfer ring. Q-domains have been investigated by Gilmer and Ohm [3] and by Davis [2]. All (^-domains are P-domains, and a long list of characterizations of P-domains is available in Bourbaki [l, pp. 93-94]. Noetherian Q-domains are characterized in [3] as those Dedekind domains whose ideal class group is a torsion group. The purpose of this paper is to obtain a characterization of general Q-domains (Theorem 5). Let K* denote the set of nonzero elements of K. If x£i£*, we define the numerator ideal of x to be JV(X)= {aÇzR> a — bx, for some &£i?} and the denominator ideal of x to be D(x) = {&£P: bxÇzR}. Since N(x)=Rxr\\R and D(x)=N(l/x), N(x) and D(x) are ideals in P. If P is a prime ideal in P, RP denotes the local ring of R at P.