On the rate of convergence in the entropic central limit theorem

We study the rate at which entropy is produced by linear combinations of independent random variables which satisfy a spectral gap condition.Mathematics Subjects Classification (2000):94A17; 60F05Supported in part by the EU Grant HPMT-CT-2000-00037, The Minkowski center for Geometry and the Israel Science Foundation.Supported in part by NSF Grant DMS-9796221.Supported in part by EPSRC Grant GR/R37210.Supported in part by the BSF, Clore Foundation and EU Grant HPMT-CT-2000-00037.     

On Best Approximations from RS-sets in Complex Banach Spaces

The concept of an RS-set in a complex Banach space is introduced and the problem of best approximation from an RS-set in a complex space is investigated. Results consisting of characterizations, uniqueness and strong uniqueness are established,     

Superlinear convergence of affine scaling interior point newton method for linear inequality constrained minimization without strict complementarity

In this paper we extend and improve the classical affine scaling interior-point Newton method for solving nonlinear optimization subject to linear inequality constraints in the absence of the strict complementarity assumption. Introducing a computationally efficient technique and employing an identification function for the definition of the new affine scaling matrix, we propose and analyze a new affine scaling interior-point Newton method which improves the Coleman and Li affine sealing matrix in [2] for solving the linear inequlity constrained optimization. Local superlinear and quadratical convergence of the proposed algorithm is established under the strong second order sufficiency condition without assuming strict complementarity of the solution.     

On self-mapping degrees of <Emphasis Type="Italic">S</Emphasis><Superscript>3</Superscript>-geometry manifolds

In this paper we determined all of the possible self-mapping degrees of the manifolds with S3-geometry, which are supposed to be all 3-manifolds with finite fundamental groups. This is a part of a project to determine all possible self-mapping degrees of all closed orientable 3-manifold in Thurston＇s picture.     

A fluctuation type limit theorem for Jiřina processes with immigration

It is proved by the theory of semigroup that the Ornstein-Uhlenbeck type process with jumps can arise from the fluctuation limit of a sequence of Jirina processes with immigration under suitable moments conditions.     

Finite-termination schemes for solving semi-infinite satisfycing problems

The problem of finding a parameter which satisfies a set of specifications in inequality form is sometimes referred to as the satisfycing problem. We present a family of methods for solving, in a finite number of iterations, satisfycing problems stated in the form of semi-infinite inequalities. These methods range from adaptive uniform discretization methods to outer approximation methods.The research reported herein was sponsored in part by the National Science Foundation Grant ECS-8713334, the Air Force Office of Scientific Research Contract AFOSR-86-0116, and the State of California MICRO Program Grant 532410-19900.     

On reconstruction of distributions by the translated moments of linear statistics

The problem of reconstruction of a multivariate distribution by means of translated moments is considered. Supported in part by the Russian Foundation for Fundamental Research (project code # 95-011-01260) and the International Science Foundation. Proceedings of the XVII Seminar on Stability Problems for Stochastic Models, Kazan, Russia, 1995, Part II.     

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.     

Limiting behavior of bang-bang controls for Sobolev problems

We consider the limiting behavior of optimal bang-bang controls as a family of Sobolev equations formally converges to a wave equation. The weak-starlimit of the sequence of bang-bang controls is an optimal control for the wave equation problem. The associated optimal states converge strongly and, for the optimal time problem, the optimal times converge to the optimal time for the wave equation.This work was supported in part by the National Science Foundation, Grant No. MCS-79-02037.     

Classification of non-rigid families of K3 surfaces and a finiteness theorem of Arakelov type

Supported in part by the National Science Foundation, through Grant DMS-8800355     

An outer approximations algorithm for computer-aided design problems

This paper presents an implementable algorithm of the outer approximations type for solving nonlinear programming problems with functional inequality constraints. The algorithm was motivated by engineering design problems in circuit tolerancing, multivariable control, and shock-resistant structures.This research was sponsored by the National Science Foundation, Grant No. ENG73-08214A01, and the National Science Foundation (RANN), Grant No. ENV76-04264.     

Muirhead-Rado inequality for compact groups

Muirhead’s majorization inequality was extended by Rado to the case of arbitrary permutation groups. We further generalize this inequality to compact groups and their linear representations over the reals. We characterize saturation of the inequality, and describe the saturation condition in detail for the case of actions on Hermitian operators. Supported in part by the National Science Foundation and the Center for the Mathematics of Information.     

Bounded controllers for robust exponential convergence

We consider the problem of stabilizing an uncertain system when the norm of the control input is bounded by a prespecified constant. We treat continuous-time dynamical systems whose nominal part is linear and whose uncertain part is norm-bounded by a known affine function of the norm of the system state and the norm of the control input. Given a prespecified rate of convergence and a ball containing the origin of the state space, we present controllers which guarantee that, for all allowable uncertainties and nonlinearities, there is a region of attraction from which all solutions converge to the given ball with the prespecified convergence rate.This research was supported by the National Science Foundation under Grant MSS-90-57079.     

L p -dual Quermassintegral sums

In this paper, we first introduce a concept of L _p -dual Quermassintegral sum function of convex bodies and establish the polar projection Minkowski inequality and the polar projection Aleksandrov-Fenchel inequality for L _p -dual Quermassintegral sums. Moreover, by using Lutwak’s width-integral of index i, we establish the L _p -Brunn-Minkowski inequality for the polar mixed projection bodies. As applications, we prove some interrelated results. This work was partially supported by the National Natural Science Foundation of China (Grant No. 10271071), Zhejiang Provincial Natural Science Foundation of China (Grant No. Y605065) and Foundation of the Education Department of Zhejiang Province of China (Grant No. 20050392)     

Solution differentiability and continuation of Newton's method for variational inequality problems over polyhedral sets

In this paper, we derive some further differentiability properties of solutions to a parametric variational inequality problem defined over a polyhedral set. We discuss how these results can be used to establish the feasibility of continuation of Newton's method for solving the variational problem in question.This work was based on research supported by the National Science Foundation under Grant No. ECS-87-17968.     

Necessary Conditions for Impulsive Nonlinear Optimal Control Problems without a priori Normality Assumptions

First-order and second-order necessary conditions of optimality for an impulsive control problem that remain informative for abnormal control processes are presented and derived. One of the main features of these conditions is that no a priori normality assumptions are required. This feature follows from the fact that these conditions rely on an extremal principle which is proved for an abstract minimization problem with equality constraints, inequality constraints, and constraints given by an inclusion in a convex cone. Two simple examples illustrate the power of the main result.The first author was partially supported by the Russian Foundation for Basic Research Grant 02-01-00334. The second author was partially supported by the Russian Foundation for Basic Research Grant 00-01-00869. The third author was partially supported by Fundacao para a Ciencia e Tecnologia and by INVOTAN Grant.     

On circumscribed cylinders of convex sets

Two inequalities regarding the average size of circumscribed cylinders of convex sets are proved. These inequalities generalize and supplement an inequality which Knothe has proved for the purpose of deriving a converse of a theorem of Archimedes concerning the surface area of circumscribed cylinders of spheres.Supported by National Science Foundation Research Grant DMS 8922399.     

Nonlinear Boundary Stabilization of Nonuniform Timoshenko Beam

In this paper, the stabilization problem of nonuniform Timoshenko beam by some nonlinear boundary feedback controls is considered. By virtue of nonlinear semigroup theory, energy-perturbed approach and exponential multiplier method, it is shown that the vibration of the beam under the proposed control actiondecays exponentially or in negative power of time t as t→∞.     

A proof of the Grothendieck inequality

The fundamental constant of Grothendieck's inequality, defined below, was shown by Grothendieck to be less than sinh π/2=2.301+. We improve the bound slightly, and show that for the positive definite case π/2 suffices. Partially supported by Grant #901-052, Gustavus Adolphus College Research Fund, and a National Science Foundation Summer Traineeship grant. This is based on part of the author's Ph.D. dissertation, University of Minnesota, under Prof. C.A. McCarthy. The author is grateful for Prof. McCarthy's assistance.     

Two extensions of Lubinsky’s universality theorem

We extend some remarkable recent results of Lubinsky and Levin-Lubinsky from [−1, 1] to allow discrete eigenvalues outside σ _ess and to allow σ _ess first to be a finite union of closed intervals and then a fairly general compact set in ℝ (one which is regular for the Dirichlet problem). Supported in part by NSF grant DMS-0140592 and U.S.-Israel Binational Science Foundation (BSF) Grant No. 2002068.