Cyclic Hypomonotonicity,Cyclic Submonotonicity,and Integration

Rockafellar has shown that the subdifferentials of convex functions are always cyclically monotone operators. Moreover, maximal cyclically monotone operators are necessarily operators of this type, since one can construct explicitly a convex function, which turns out to be unique up to a constant, whose subdifferential gives back the operator. This result is a cornerstone in convex analysis and relates tightly convexity and monotonicity. In this paper, we establish analogous robust results that relate weak convexity notions to corresponding notions of weak monotonicity, provided one deals with locally Lipschitz functions and locally bounded operators. In particular, the subdifferentials of locally Lipschitz functions that are directionally hypomonotone [respectively, directionally submonotone] enjoy also an additional cyclic strengthening of this notion and in fact are maximal under this new property. Moreover, every maximal cyclically hypomonotone [respectively, maximal cyclically submonotone] operator is always the Clarke subdifferential of some directionally weakly convex [respectively, directionally approximately convex] locally Lipschitz function, unique up to a constant, which in finite dimentions is a lower C² function [respectively, a lower C¹ function].     

Harnack inequality for some classes of Markov processes

In this paper we establish a Harnack inequality for nonnegative harmonic functions of some classes of Markov processes with jumps. Mathematics Subject Classification (2000): Primary 60J45, 60J75, Secondary 60J25.This work was completed while the authors were in the Research in Pairs program at the Mathematisches Forschungsinstitut Oberwolfach. We thank the Institute for the hospitality.The research of this author is supported in part by NSF Grant DMS-9803240.The research of this author is supported in part by MZT grant 0037107 of the Republic of Croatia.     

Low‐gain adaptive stabilization of semilinear second‐order hyperbolic systems

In this paper low‐gain adaptive stabilization of undamped semilinear second‐order hyperbolic systems is considered in the case where the input and output operators are collocated. The linearized systems have an infinite number of poles and zeros on the imaginary axis. The adaptive stabilizer is constructed by a low‐gain adaptive velocity feedback. The closed‐loop system is governed by a non‐linear evolution equation. First, the well‐posedness of the closed‐loop system is shown. Next, an energy‐like function and a multiplier function are introduced and the exponential stability of the closed‐loop system is analysed. Some examples are given to illustrate the theory. Copyright © 2004 John Wiley & Sons, Ltd.     

On the Zeros of Orthogonal Polynomials: The Elliptic Case

Constructive Approximation - Let E = [–1, α] \cup [β, 1], –1 &;lt; α &;lt; β &;lt; 1, and let (pn) be orthogonal on E with respect to the weight function...     

多小波子空间上的单小波表示

本文在较弱的条件下,建立了2重多小波子空间与单小波子空间的关系.即由2重多小波构造出单小波.一方面,这种单小波的平移伸缩与2重多小波的平移伸缩生成的子空间是完全相同的;另一方面,它具有插值性.因此通过构造出的单小波建立了多小波子空间上的Shannon型采样定理.     

ON A SET OF h-HARMONIC HOMOGENEOUS POLYNOMIALS

1　IntroductionLet Sd- 1 bethe unitsphere of Rd,Πdthe space of allpolynomialsofd variables,and x(i)= (x1 ,… ,xi) (i=1 ,… ,d) .In this paper,we are concerned with the homogeneous polyno-mials orthogonal with the weightfunction h(x(d) ) =x2 k11 …x2 kdd on Sd- 1 ,where ki≥ 0 (i=1 ,… ,d) .Connected with the weightfunction h(x(d) ) are the following differential-differenceoperators introduced by Charles F. Dunkl in [3 ] ,Dip(x(d) ) = p(x(d) ) xi+ di=1kip(x(d) ) -p(x1 ,… ,-xi,… ,xd)xi,Δ…     

New inequalities from classical Sturm theorems

Inequalities satisfied by the zeros of the solutions of second-order hypergeometric equations are derived through a systematic use of Liouville transformations together with the application of classical Sturm theorems. This systematic study allows us to improve previously known inequalities and to extend their range of validity as well as to discover inequalities which appear to be new. Among other properties obtained, Szegő's bounds on the zeros of Jacobi polynomials for , are completed with results for the rest of parameter values, Grosjean's inequality (J. Approx. Theory 50 (1987) 84) on the zeros of Legendre polynomials is shown to be valid for Jacobi polynomials with |β|1, bounds on ratios of consecutive zeros of Gauss and confluent hypergeometric functions are derived as well as an inequality involving the geometric mean of zeros of Bessel functions.     

非空交定理及其应用

本文得到了广义区间空间中几个参数型非空交定理，并用此结果证明了一些新型极大极小定理，截口定理，重合定理，以及变分不等式解的存在性定理。本文中的结果包含（１，３，５－１２，１４－１６）中相应结果。     

A Method for Establishing Optimality Conditions for a Nonsmooth Vector-Valued Minimax Problem

Using a method of uniform approximations, necessary and sufficient conditions for a nonsmooth constrained vector-valued minimax problem are established in terms of Mordukhovich subdifferentials.     

Integral Representations,Differentiability Properties and Limits at Infinity for Beppo Levi Functions

The first aim in the present paper is to give an integral representation for Beppo Levi functions on R ⁿ. Our integral representation is an extension of Sobolev's integral representation given for infinitely differentiable functions with compact support. As applications, continuity and differentiability properties of Beppo Levi functions are studied.Our second aim in this paper is to study the existence of limits at infinity for Beppo Levi functions. We also consider the existence of fine-type limits at infinity with respect to Bessel capacities, which yields the radial limit result at infinity.