Fundamental Lemma Research Articles

Exact categories and duality

rings and whose values again were modules over a ring. It will be shown in this paper that the theory may be generalized to functors defined on abstract categories, and whose values are again in such abstract categories. An abstract treatment such as this has several advantages. We list a few: (1) The dualities of the type Kernel - cokernel Projective - injective Z(A) -Z'(A), originally suggested by MacLane [4], may now be formulated as explicit mathematical theorems. (2) In treating derived functors, it suffices to consider left derived functors of a covariant functor of several variables; all other types needed may then be obtained by a dualization process. (3) Further applications of the theory of derived functors are bound to show that the consideration of modules over a ring A will be insufficient. Rings with additional structure such as grading, differentiation, topology, etc., will have to be considered. With the theory developed abstractly, these generalizations are readily available. The paper is divided into four parts. Part I deals with basic definitions, duality, and fundamental lemmas. We make no attempt to prove or even state many of the necessary trivia which are used throughout. Part II is rather short, due to the facts that most of the results follow

Hilbert space methods in the theory of harmonic integrals

The theory of harmonic integrals was created by Hodge [15], and the theorem which bears his name is the central result of the subject. Kodaira [17] and-independently-de Rham and Bidal [1] used the generalized harmonic operator A in their treatments of the theory. A was also used by Milgram and Rosenbloom [19] in their study of harmonic integrals with the heat equation. It is our purpose to develop the properties of A from the point of view of Hilbert space theory, thus arriving at Hodge's theorem without the use of a generalized integral equation theory. In addition, in ?2 we study A on a class of open manifolds-those with negligible boundary; these include all complete manifolds. ?3 contains a proof of the fundamental differentiability lemma. We wish to express our appreciation to Professor M. H. Stone, who suggested this topic to us. He provided us with the proof (in ?2) that I+ 5*+dd* is self-adjoint, and he suggested the application of Rellich's results to the proof of the complete continuity of the Green's operator. 1. Compact manifolds; Hodge's theorem. We assume familiarity with the basic concepts of differential forms on Riemannian manifolds. Expositions of

On Bounds of Polynomials in Hyperspheres and Frechet-Michal Derivatives in Real and Complex Normed Linear Spaces

Introduction: One of the main difficulties-by no means the only onein the development of a theory of analytic functions in normed linear spaces stems from the fact that the modulus of a homogeneous polynomial and that of its polar are not always the same. We give here the theorenm that asserts that the modulus of a homogeneous polynomial and its polar are always equal in complex' Banach spaces. This has many important imi lications including the great simplifications brought about in the proofs of several furndamental theorems. The situation in real2 Banach spaces is quite different as we shall show. We base our proofs on a few fundamental lemmas in normed linear spaces: one generalizes Bernstein's theorem3 on the bounds and first derivative of S, a polynomial in the complex plane and two others generalize theorems of the brothers, A. Markoff4 and V. Markoff5 on the first and higher derivatives of polynomials of a real variable. These lemmas have an interest in themselves and have many applications not discussed in this paper. To present our results as briefly as possible, we assume familiarity with the Frechet differential calculus and analytic function theory in both real and complex Banach spaces. Theorems in Real Banach Spaces. The result of A. Markoff states that if pn(x) is a polynomial of degree n in a real variable x and if lPn(x)I < 1 in the interval xI < 1, then the derivative pn(x) satisfies the inequality lpA(x)l < nZ in |x| < 1. The following lelmma generalizes Markoff's result. Lemma 1. Let n

SIMPLIFIED DECISION FUNCTIONS

central theorems. Our result can also be regarded as a generalization of Neyman & Pearson's fundamental lemma (1933), and it thus forms a bridge from their work to the later work of Wald and Lindley. We arrive at our result by means of two lemmas, one geometrical and one statistical. The geometrical lemma is a purely mathematical result, and its proof would not have been given in this paper had it been longer or more complicated than it is. The reader who wishes to concentrate on the statistical argument can omit the section devoted to this lemma. The statistical lemma, on the other hand, is of some interest in itself as giving a direct frequency interpretation of a likelihood ratio. Our result on decision functions is essentially a slight extension of that first derived by Lindley, but for simplicity we have restricted ourselves to the case where only two decisions are possible. In this respect our results could be extended at the expense of tedious but inessential complication. Our methods of proof, too, have been derived from those of Lindley, by omission of these complications, by use of our geometrical lemma, and by simplification caused by our adoption from the beginning of the notion of a randomized decision function. Lindley went to some trouble to avoid randomized decisions as much as possible (they cannot be altogether avoided), in what now seems to the present writer a misguided endeavour. If we confuse the decision problem with the inference problem, randomization appears as a thing to be avoided. It is unreasonable that the conclusion to be drawn from a set of data should depend on the totally irrelevant throw of a coin; but it is not unreasonable that the action to be taken on the basis of certain facts should

On the Theory of Unbiased Tests of Simple Statistical Hypotheses Specifying the Values of Two or More Parameters

Unbiased critical regions of type D for testing simple hypotheses specifying the values of several parameters are defined and their properties studied. These regions constitute a natural generalization of the Neyman-Pearson regions of type A for testing simple hypotheses specifying the value of one parameter. A theorem is obtained which plays the role of the Neyman-Pearson fundamental lemma in the type A case. Illustrative examples of type D regions are given.

A fundamental lemma from the theory of holomorphic functions on an algebraic variety

A fundamental lemma from the theory of holomorphic functions on an algebraic variety

The differential invariants of a two-index tensor

Riemannian geometry, based upon a metric form ds = gijdxdx', gives us the curvature tensor R)u as the sole basic differential invariant of the space, and of the symmetric tensor gy. The general tensor g^ can be broken up into the sum of two irreducible components, namely the symmetric and antisymmetric portions defined respectively by 2giij)=gij+gji and 2g[ij]=gij--gji. The latter disappears in constructing ds; but the general differential invariants of gij must necessarily be composed of those derivable from g(ij) (the curvature tensor above), from gun, and a group of mixed invariants dependent upon both. I t is proposed to investigate the general problem by use of a well known and easily proved fundamental lemma of the calculus of variations : The Euler equations derived from a variational principle are tensor-invariant under the group of transformations which leaves the original integral invariant. Actually the equations as directly obtained state that a certain covariant vector vanishes. Given the tensor ga{x • • • x) we first introduce two (implicit) absolute parameters u, v, and construct the variational problem

A note on the du Bois-Reymond equations in the calculus of variations

satisfies the du Bois-Reymond form of the Euler equations. The principal purpose of the present note is to give an alternate proof of these results of McShane and Tonelli. In addition to being simpler in detail than the previous proofs, the differentiability theorems of §§2 and 3 involve weaker hypotheses than the corresponding theorems of McShane and Tonelli, both in regard to conditions on the integrand function and in regard to the class of arcs considered. Basically, the present proof is intimately related to the proof of the fundamental lemma as given by Bliss [l, pp. 20-21].

On the Extension of the Pflastersatz

In this paper I prove some new Pflastersätze for r−dimensional sets in the n−dimensional Euclidean space Rn belonging to one and the same family, in which Lebesgue's fundamental lemma represents a “0-dimensional” case out of r + 1 possible cases of dimensions 0, 1, 2, …, r.

An elementary proof of a fundamental lemma concerning the limit of a sum

An elementary proof of a fundamental lemma concerning the limit of a sum

Construction of automorphic Galois representations, II

Recent developments in the theory of the stable trace formula, especially the proof by Laumon and Ngo of the fundamental lemma for unitary groups, has revived Langlands’ strategy for constructing Galois representations in the `-adic cohomology of Shimura varieties. I will describe the kinds of representations one can expect to obtain in this way, or more generally as `-adic limits of representations appearing in the cohomology of Shimura varieties attached to unitary groups. I will also report on the current state of the collective project of the Paris automorphic forms seminar to carry out the construction following the prescription outlined by Langlands thirty years ago.

II. Second memoir on plane stigmatics

Let there be two groups of points upon a plane, termed, for distinction, indices and stigmata respectively, bearing such relations to each other that any one index determines the position of n stigmata, and any one stigma determines the position of m indices. The theory of these relations between indices and stigmata constitutes plane stigmatics . Each related pair of index X and stigma Y constitutes a stigmatic point , henceforth written “the s. point ( xy )." The straight lines joining any index with each of its corresponding stigmata are termed ordinates . If, when the index moves upon a straight line, the ordinate remains parallel to some other straight line, the relation between index and stigma is that expressed by the relation between abscissa and ordinate in the coordinate geometry of Descartes. When only one index corresponds to one stigma and conversely, and both indices and stigmata lie always on one and the same straight line, or the indices upon one and the stigmata upon another, the relations between indices and stigmata are those between homologous points in the homographic geometry of Chasles. The general expression of the stigmatic relation is obtained by a generalization of Chasles’s fundamental lemma in his theory of characteristics ( Comptes Rendus , June 27, 1864, vol. lviii. p. 1175), clinants being substituted for scalars. It results that in certain forms of the law of coordination , which “ coordinates ” the stigmata with the indices, there may be solitary indices which have no corresponding stigmata, and solitary stigmata which have no corresponding indices, and also double points in which the index coincides with its stigma (76). The particular case in which one index corresponds to one stigma and conversely, and no solitary index or stigma occurs, is termed a stigmatic line (henceforth written “s. line”), because the Cartesian case is that of a Cartesian straight line in ordinary coordinate geometry, but in the general s. line the figures described by index and stigma may be any directly similar plane figures (77). The investigation of this particular case occupies almost the whole of the Introductory Memoir . When one index corresponds to one stigma and conversely, but there is one solitary index and one solitary stigma, we have s. homography , provided the solitary index is distinct from the solitary stigma (79), and s. involution when the solitary index coincides with the solitary stigma (78), so called because they generalize the relations treated of under these names by Chasles.