The structure of singly-periodic minimal surfaces

The research described in this paper was supported by research grant DE-FG02-86ER250125 of the Applied Mathematical Science subprogram of Office of Energy Research, U.S. Department of Energy, and National Science Foundation grants DMS-8611574 and DMS-8802858     

The strong halfspace theorem for minimal surfaces

The research in this paper was supported by research grants DE-FG02-86ER250125 of the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy and National Science Foundation, Division of Mathematical Sciences research grants DMS-8503350 and DMS-8611574     

The maximum principle at infinity for minimal surfaces in flat three manifolds

The research described in this paper was supported by research grant DE-FG02-86ER250125 of the Applied Mathematical Science subprogram of Office of Energy Research, U.S. Department of Energy, and National Science Foundation grant DMS-8611574.     

Minimal surfaces bounded by convex curves in parallel planes

The research described in this paper was supported by research grant DE-FG02-86ER250125 of the Applied Mathematical Science subprogram of Office of Energy Research, U.S. Department of Energy, and National Science Foundation grant DMS-8900285.     

Mixed hyperbolic-elliptic systems in self-similar flows

From the observation that self-similar solutions of conservation laws in two space dimensions change type, it follows that for systems of more than two equations, such as the equations of gas dynamics, the reduced systems will be of mixed hyperbolic-elliptic type, in some regions of space. In this paper, we derive mixed systems for the isentropic and adiabatic equations of compressible gas dynamics. We show that the mixed systems which arise exhibit complicated nonlinear dependence. In a prototype system, the nonlinear wave system, this behavior is much simplified, and we outline the solution to some typical Riemann problems.Dedicated to Constantine Dafermos on his 60^th birthdayResearch supported by the National Science Foundation, grant DMS-9970310.Research supported by the Department of Energy, grant DE-FG-03-94-ER25222 and by the National Science Foundation, grant DMS-9973475 (POWRE).Research supported by the Department of Energy, grant DE-FG-03-94-ER25222 and by the National Science Foundation, grant DMS-0103823.     

Dines—Fourier—Motzkin quantifier elimination and an application of corresponding transfer principles over ordered fields

A constructive procedure using Dines—Fourier—Motzkin elimination is given for eliminating quantifiers in a linear first order formula over ordered fields. An ensuing transfer principle is illustrated by showing that a locally one-to-one affine map is globally one-to-one and onto all over ordered fields.This research is based on work supported in part by the National Science Foundation under Grant DMS-86-03232, by the Department of Energy grant DE-FG03-87ER25028 and by the United States-Israel Binational Science Foundation Grant 85-00295.     

Spectral conditioning and pseudospectral growth

Using the language of pseudospectra, we study the behavior of matrix eigenvalues under two scales of matrix perturbation. First, we relate Lidskii’s analysis of small perturbations to a recent result of Karow on the growth rate of pseudospectra. Then, considering larger perturbations, we follow recent work of Alam and Bora in characterizing the distance from a given matrix to the set of matrices with multiple eigenvalues in terms of the number of connected components of pseudospectra. J. V. Burke’s research was supported in part by National Science Foundation Grant DMS-0505712. A. S. Lewis’s research was supported in part by National Science Foundation Grant DMS-0504032. M. L. Overton’s research was supported in part by National Science Foundation Grant DMS-0412049.     

Q-Learning Algorithms with Random Truncation Bounds and Applications to Effective Parallel Computing

Motivated by an important problem of load balancing in parallel computing, this paper examines a modified algorithm to enhance Q-learning methods, especially in asynchronous recursive procedures for self-adaptive load distribution at run-time. Unlike the existing projection method that utilizes a fixed region, our algorithm employs a sequence of growing truncation bounds to ensure the boundedness of the iterates. Convergence and rates of convergence of the proposed algorithm are established. This class of algorithms has broad applications in signal processing, learning, financial engineering, and other related fields. G. Yin’s research was supported in part by the National Science Foundation under Grants DMS-0603287 and DMS-0624849 and in part by the National Security Agency under Grant MSPF-068-029. C.Z. Xu’s research was supported in part by the National Science Foundation under Grants CCF-0611750, DMS-0624849, CNS-0702488, and CRI-0708232. L.Y. Wang’s research was supported in part by the National Science Foundation under Grants ECS-0329597 and DMS-0624849 and by the Michigan Economic Development Council.     

A functional limit theorem for waves reflected by a random medium

We introduce a class of distribution-valued stochastic processes that arise in the study of pulse reflection from random media and we analyze their asymptotic properties when they are scaled in a natural way.This research was supported by the National Science Foundation under Grants DMS-8201895 and DMS-90003227.     

The empirical behavior of sampling methods for stochastic programming

We investigate the quality of solutions obtained from sample-average approximations to two-stage stochastic linear programs with recourse. We use a recently developed software tool executing on a computational grid to solve many large instances of these problems, allowing us to obtain high-quality solutions and to verify optimality and near-optimality of the computed solutions in various ways. Research supported by the Mathematical, Information, and Computational Sciences Division subprogram of the Office of Advanced Scientific Computing Research, U.S. Department of Energy, under Contract W-31-109-Eng-38, and by the National Science Foundation under Grant 9726385. Research supported by the Mathematical, Information, and Computational Sciences Division subprogram of the Office of Advanced Scientific Computing Research, U.S. Department of Energy, under Contract W-31-109-Eng-38, and by the National Science Foundation under Grant DMS-0073770. Research supported by the Mathematical, Information, and Computational Sciences Division subprogram of the Office of Advanced Scientific Computing Research, U.S. Department of Energy, under Contract W-31-109-Eng-38, and by the National Science Foundation under Grants 9726385 and 0082065.     

The arithmetic of Fermat curves

This research was supported in part by National Science Foundation grant DMS-9002095     

The principal pivoting method revisited

The principal pivoting method (PPM) for the linear complementarity problem (LCP) is shown to be applicable to the class of LCPs involving the newly identified class of sufficient matrices.Research partially supported by the National Science Foundation grant DMS-8800589, U.S. Department of Energy grant DE-FG03-87ER25028 and Office of Naval Research grant N00014-89-J-1659.Dedicated to George B. Dantzig on the occasion of his 75th birthday.     

Characterizations of spaces related to metasymplectic spaces

This research was partially supported by the National Science Foundation under Grant Number DMS-8691779.     

Nonlocal superconductivity

The research of D. Brandon was partially supported by the National Science Foundation through grant # DMS-9296011 and by the Army Research Office and the National Science Foundation through the Center for Nonlinear Analysis.     

The central limit theorem and the law of iterated logarithm for empirical processes under local conditions

Summary A CLT and a LIL are proved under weak-L ₂ Gaussian bracketing conditions (weaker than the usual ones). These results have wide applicability and in particular provide an improvement of the Jain-Marcus central limit theorem for C(S)-valued random variables.The research of these authors was carried out while visiting Texas A & M UniversityThe research of these authors has been totally or partially supported by the Danish Natural Science Research Council and by the National Science Foundation Grants numbers DMS-83-18610 and DMS-83-01367     

A Markov-modulated M/G/1 queue II: Busy period and time for buffer overflow

We consider an M/G/1 queue where the arrival and service processes are modulated by a two state Markov chain. We assume that the arrival rate, service time density and the rates at which the Markov chain switches its state, are functions of the total unfinished work (buffer content) in the queue. We compute asymptotic approximations to performance measures such as the mean residual busy period, mean length of a busy period, and the mean time to reach capacity.This research was supported in part by NSF Grants DMS-84-06110, DMS-85-01535 and DMS-86-20267, and grants from the U.S. Israel Binational Science Foundation and the Israel Academy of Sciences.     

Pseudo-monotone complementarity problems in Hilbert space

In this paper, some existence results for a nonlinear complementarity problem involving a pseudo-monotone mapping over an arbitrary closed convex cone in a real Hilbert space are established. In particular, some known existence results for a nonlinear complementarity problem in a finite-dimensional Hilbert space are generalized to an infinite-dimensional real Hilbert space. Applications to a class of nonlinear complementarity problems and the study of the post-critical equilibrium state of a thin elastic plate subjected to unilateral conditions are given.This research was partially supported by the National Science Foundation Grant DMS-89-13089, Department of Energy Grant DE-FG03-87-ER-25028, and Office of Naval Research Grant N00014-89-J-1659. The authors would like to express their sincere thanks to Professor S. Schaible, School of Administration, University of California, Riverside, for his helpful suggestions and comments. They also thank the referees for their comments and suggestions that improved this paper substantially.     

Deforming subdivisions

Subdivisions are continuously deformed subject to three local conditions and shown to retain the global property of being a subdivision. Cell maps, which are introduced, enable elementary arguments.This research is supported in part by the National Science Foundation Grant DMS-8902662.     

On the updating scheme in a class of collinear scaling algorithms for sparse minimization

Sorensen (Ref. 1) has proposed a class of algorithms for sparse unconstrained minimization where the sparsity pattern of the Cholesky factors of the Hessian is known. His updates at each iteration depend on the choice of a vector, and in Ref. 1 the question of choosing this vector is essentially left open. In this note, we propose a variational problem whose solution may be used to choose this vector. The major part of the computation of a solution to this variational problem is similar to the computation of a trust-region step in unconstrained minimization. Therefore, well-developed techniques available for the latter problem can be used to compute this vector and to perform the updating.This research was supported by NSF Grant DMS-8414460 and by DOE Grant DE-FG06-85ER25007, awarded to Washington State University, and by the Applied Mathematical Sciences Subprogram of the US Department of Energy under Contract W-31-109-Eng-38 while the first author was visiting the Mathematics and Computer Science Division of Argonne National Laboratory.     

Consolidation rates for two interacting systems in the plane

Summary This paper is a sequel of a paper of Cox and Griffeath “diffusive clustering in the two dimensional voter model”. We continue our study of the voter model and coalescing random walks on the two dimensional integer lattice. Some exact asymptotics concerning the rate of clustering in the former process and the coalescence rate of the latter are derived. We use these results to prove a limit law, announced in that earlier paper, concerning the size of the largest square centered at the origin which is of solid color at a large time t. Partially supported by the National Science Foundation under Grant DMS-831080 Partially supported by the National Science Foundation under Grant DMS-841317 Partially supported by the National Science Foundation under Grant DMS-830549