Showing papers in "Pacific Journal of Mathematics in 1976"

On the inverse function theorem.

Abstract: This short set of notes includes a complete proof of the Inverse Function Theorem. There will be more notes later covering smooth manifolds, immersions, and submer-sions. Our goal is to prove the following theorem:

Divided rings and going-down

Abstract: Going-down rings are characterized in terms of the AVdomains of Akiba. As a result, the internal characterization of going-down rings which has recently been established by McAdam in the integrally closed case is generalized to the root-closed case, but is shown to fail in general.

Convex and concave functions of singular values of matrix sums.

Abstract: when / is an increasing concave function of a nonnegative real variable, with /(0) = 0. This inequality is of some interest as in previously published work a convexity (rather than concavity) hypothesis has usually been necessary to establish results of the general type of (1). See, for example, Gohberg and Krein [3], page 49, or Marcus and Mine [4], pages 103 and 116. In this paper we shall uncover the algebraic foundation of (1) by giving a short proof of a generalization. Our proof, which is simpler and more direct than the proof of (1) given by RotfeΓd, will be based on an interesting matrix valued triangle inequality, a special case of which was given by RotfeΓd. We note that the methods used by RotfeΓd are very much adapted to the inequality (1) that he wished to prove, and do not appear capable of proving the extensions of his results to be presented below.

On Liouville’s theory of elementary functions

Abstract: l For the convenience of the reader, we provide in this section a succinct and somewhat simplified treatment of the necessary parts of Ax's paper [1], Let k~-+K be a fixed homomorphism of commutative rings. (Thus K is a λ -algebra. In all our applications K will be a field and k a subfield, but we may as well begin with the extra generality.) If M is a iΓ-module, by a k-derivation of K into M is meant a klinear map D: K—+M such that D(xy) = x(Dy) + y(Dx) for all x, y eK. In such a situation we have Dx = nx~Dx for all x e K and all positive integers n; taking x = 1, n = 2, we get Dl = 0, and hence D vanishes on the image of k in K. The ^-derivations of K into M form a if-submodule Derfe (K, M) of Homfc (K, M). A derivation on (or of) the ring K is simply a Z-derivation of K into K (that is, we take k = Z, K = M). A derivation on an integral domain extends to a unique derivation on its field of quotients, by means of the equation D(x/y) = (yDx — xDy)/y.

Insertion of a continuous function

Abstract: Necessary and sufficient conditions in terms of lower cut sets are given for the insertion of a continuous function between two comparable real valued functions with a certain pair of a general class of properties. The class of properties is defined by being preserved when added to a continuous function and by being possessed by any constant function.

Stochastic convex programming: singular multipliers and extended duality singular multipliers and duality

Abstract: A two-stage stochastic programming problem with recourse is studied here in terms of an extended Lagrangian function which allows certain multipliers to be elements of a dual space (i?00)*, rather than an ϊ£λ space. Such multipliers can be decomposed into an i^-component and a "singular" component. The generalization makes it possible to characterize solutions to the problem in terms of a saddle-point, if the problem is strictly feasible. The Kuhn-Tucker conditions for the basic duality framework are modified to admit singular multipliers. It is shown that the optimal multiplier vectors in the extended dual problem are, in at least one broad case, ideal limits of maximizing sequences of multiplier vectors in the basic dual problem.

The Krull intersection theorem. II.

Abstract: 1. Introduction. This paper is a continuation of [1]. In §2 we show that for a nonminimal principal prime (p), /= Π* =1(p)" is a prime ideal and pJ = /. An example is given to show that the condition that (p) be nonminimal is necessary. We also consider the question of when a prime ideal minimal over a principal ideal has rank one. Of particular interest is the example of a domain D with a doubly generated ideal / such that Π;=1 Γϊl Π; βl Γ. In §3 we prove that Π: =1 ΓA = lΓ\™=ιΓA for any finitely generated module A over a valuation ring. In §4 we consider certain converses to the usual Krull Intersection Theorem for Noetherian rings. It is shown that for (R, M) a quasi-local ring whose maximal ideal M is finitely generated, many classical results for local rings are actually equivalent to the ring R being Noetherian.