Spherical means in spaces of constant curvature

Summary Extension of theorems ofF. Riesz on subharmonic functions to spaces of constant curvature by the use of hyperbolic partial differential equations. To Enrico Bompiani on his scientific Jubiles This research was supported in part by the United States Air Force through the Air Force Office of Scientific Research of the Air Research and Development Command under ontract No. AF 49 (638)–228.     

On extreme points inl 1

It is proved that every bounded closed and convex subset ofl ₁ is the closed convex hull of its extreme points. The research reported in this document has been sponsored by the Air Force Office of Scientific Research under Grant AF EOAR 66-18 through the European Office of Aerospace Research (OAR) United States Air Force.     

Stochastic evolution equations driven by nuclear-space-valued martingales

This paper presents a theory of stochastic evolution equations for nuclear-space-valued processes and provides a unified treatment of several examples from the field of applications. (C ₀ , 1) reversed evolution systems on countably Hilbertian nuclear spaces are also investigated.Research supported by the Air Force Office of Scientific Research Grant No. F49620 85 C 0144. The research of V. Perez-Abreu was also supported by CONACYT Grants PCEXCNA-040651 and PCMTCNA-750220 (Mexico).     

Theorems in the additive theory of numbers

Summary This paper extends some earlier results on difference sets andB ₂ sequences bySinger, Bose, Erd?s andTuran, andChowla. This research was supported in part by the United States Air Force through the Air Force Office of Scientific Research of the Air Research and Development Command, under Contract No. AF 49 (638)-213. Reproduction in whole or part is permitted for any purpose of the United States Government.     

Problems and methods in partial differential equations

Summary Finite part and logarithmic part of some divergent integrals with applications to theCauchy problem. A Giovanni Sansone nel suo 70^mo compleanno. The research reported in this document has been sponsored in part by the Air Force Office of Scientific Research of the Air Research and Development Command, United States Air Force, through its European Office under Contract AF 61 (056)-86.     

Geometric convergence of rational approximations toe −z in infinite sectors

Summary In this paper, we study the geometric convergence of rational approximations toe ^–z in infinite sectors symmetric about the positive real axis.Research supported in part by the Air Force Office of Scientific Research under Grant AFOSR-74-2688, and by the University of South Florida Research CouncilResearch supported in part by the Air Force Office of Scientific Research under Grant AFOSR-74-2729, and by the Atomic Energy Commission under Grant AT (11-1)-2075     

On series in linear topological spaces

The main result is that in every complete locally-bounded linear topological space there exist series which are unconditionally yet not absolutely convergent. Relations between absolute, unconditional and metric convergence of series are studied. The research reported in this document has been sponsored in part by the Air Force Office of Scientific Research, OAR through the European Office, Aerospace Research, United States Air Force.     

On the sharpness of theorems concerning zero-free regions for certain sequences of polynomials

Summary In this paper, we establish the sharpness of a theorem concerning zero-free parabolic regions for certain sequences of polynomials satisfying a three-term recurrence relation. Similarly, we establish the sharpness of a zero-free sectorial region for certain sequences of Padé approximants toe ^z .Research supported in part by the Air Force Office of Scientific Research under Grant AFOSR-74-2688Research supported in part by the Air Force Office of Scientific Research under Grant AFOSR-74-2729, and by the Energy Research and Development Administration (ERDA) under Grant E(11-1)-2075     

On complemented subspaces ofm

It is proved that an infinite dimensional subspace ofm is complemented inm if and only if it is isomorphic tom. The research reported in this document has been sponsored by the Air Force Office of Scientific Research under Grant AF EOAR 66-18, through the European Office of Aerospace Research (OAR) United States Air Force.     

The treatment of nonhomogeneous Dirichlet boundary conditions by thep-version of the finite element method

Summary The paper addresses the problem of the implementation of nonhomogeneous essential Dirichlet type boundary conditions in thep-version of the finite element method.Partially supported by the Office of Naval Research under Grant N-00014-85-K-0169Research partially supported by the Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant Number AFOSR 85-0322     

Geometric convergence to e−z by rational functions with real poles

Summary In this paper, we show that there exists a sequence of rational functions of the formR _n(z)=p_n–1(z)/(1+z/n)ⁿ,n=1, 2, ..., with degp _n–1n–1, which converges geometrically toe ^–z in the uniform norm on [0, +), as well as on some infinite sector symmetric about the positive real axis. We also discuss the usefulness of such rational functions in approximating the solutions of heat-conduction type problems.Research supported in part by the Air Force Office of Scientific Research under Grant AFOSR-74-2688, and by the University of South Florida Research Council.Research supported in part by the Air Force Office of Scientific Research under Grant AFOSR-74-2729, and by the Energy Research and Development Administration (ERDA) under Grant E(11-1)-2075.     

Some singular cauchy problems

Summary Existence and uniqueness theorems for some generalizedEuler-Poisson-Darboux equations are proved and growth and convexity properties of the solutions are studied for multiply subharmonic initial values. This research was supported in part by the United States Air Force through the Air Force Office of Scientific Research of the Air Research and Development Command under Contract No. AF 49 (638)-228, at the University of Maryland, and in part by the National Science Foundation, under a fellowship grant.     

Classes of biorthonormal systems

Summary The (A)-summability and mean-convergence of Fourier series in terms of certain biorthonormal systems have been investigated. The problem of completing a set of polynomials to a biothonormal system has been solved in the case of certain sets of type zero (Sheffer’s classification). The research reported in this document has been sponsored in part by the Air Force Office of Scientific Research of the Air Research and Development Command, United States air Force, through its European Office, under Contract N₀ AF 61 (052)-324.     

On the zeros and poles of Padé approximants toe z

Summary In this paper, we study the location of the zeros and poles of general Padé approximats toe ^z. The location of these zeros and poles is useful in the analysis of stability for related numerical methods for solving systems of ordinary differential equations.Research supported in part by the Air Force Office of Scientific Research under Grant AFOSR-74-2688, and by the University of South Fla. Research Council.Research supported in part by the Air Force Office of Scientific Research under Grant AFOSR-74-2729, and by the Atomic Energy Commission under Grant AT(11-1)-2075.     

A priori estimates for truncation error of continued fractionsK(1/b n )

A priori estimates are obtained for the truncation error of continued fractions of the formK(1/b _n ), with complex elementsb _n . The method employed is based on the calculation of bounds for successive diameters of a sequence of nested disks, where then-th approximant of the continued fraction is contained in then-th disk. Numerical examples are given to illustrate useful procedures and typical error estimates for continued fraction expansions of the complex logarithm and the ratio of consecutive Bessel functions.This research was supported by the National Science Foundation under Grant No. GP-9009 and by the United States Air Force through the Air Force Office of Scientific Research under Grant No. AFOSR-70-1888.     

Polynomial dual network simplex algorithms

We show how to use polynomial and strongly polynomial capacity scaling algorithms for the transshipment problem to design a polynomial dual network simplex pivot rule. Our best pivoting strategy leads to an O(m ² logn) bound on the number of pivots, wheren andm denotes the number of nodes and arcs in the input network. If the demands are integral and at mostB, we also give an O(m(m+n logn) min(lognB, m logn))-time implementation of a strategy that requires somewhat more pivots.Research supported by AFOSR-88-0088 through the Air Force Office of Scientific Research, by NSF grant DOM-8921835 and by grants from Prime Computer Corporation and UPS.Research supported by NSF Research Initiation Award CCR-900-8226, by U.S. Army Research Office Grant DAAL-03-91-G-0102, and by ONR Contract N00014-88-K-0166.Research supported in part by a Packard Fellowship, an NSF PYI award, a Sloan Fellowship, and by the National Science Foundation, the Air Force Office of Scientific Research, and the Office of Naval Research, through NSF grant DMS-8920550.     

On the Feynman-Kac formula and its applications to filtering theory

In this article the Feynman-Kac formula is obtained for a Markov process (X _t) whose transition probability function is not stationary. A converse to the Feynman-Kac formula is also obtained. This is used to prove the uniqueness of the solution to a measure-valued equation satisfied by the optimal filter in the white-noise approach to nonlinear filtering theory.Research partially supported by the Air Force Office of Scientific Research Contract No. F49620 85 C 0144 and by the Indian Statistical Institute.     

Notes on Klee’s paper “Polyhedral sections of convex bodies”

In sections 2 and 3 two methods for proving the non existence of certain universal Banach spaces, are presented. In section 4 it is proved that every infinite-dimensional conjugate Banach space has a two-dimensional subspace whose unit cell is not a polygon. The research reported in this document has been sponsored by the Air Force Office of Scientific Research under Grant AF EOAR 66-18, through the European Office of Aerospace Research (OAR) United States Air Force.     

On the zeros and poles of Padé approximants toe z . III

Summary In this paper, we continue our study of the location of the zeros and poles of general Padé approximants toe ^z . We state and prove here new results for the asymptotic location of the normalized zeros and poles for sequences of Padé approximants toe ^z , and for the asymptotic location of the normalized zeros for the associated Padé remainders toe ^z . In so doing, we obtain new results for nontrivial zeros of Whittaker functions, and also generalize earlier results of Szegö and Olver.Research supported in part by the Air Force Office of Scientific Research under Grant AFOSR-74-2688Research supported in part by the Air Force Office of Scientific Research under Grant AFOSR-74-2729, and by the Energy Research and Development Administration (ERDA) under Grant EY-76-S-02-2075     

Remarks on conservative Markov processes

Necessary and sufficient conditions, for a Markov process to be conservative, are studied. As a consequence it is proved: 1. IfP is conservative so isP ^k . 2.P is conservative if and only if the probability of returning toA, from a point inA, is one. The research reported in this document has been sponsored by the Air Force of Scientific Research under Grant AF EOAR 66-18, through the European Office of Aerospace Research (OAR), United States Air Force.