Extension of Simons' inequality

We prove the following extended version of Simons' inequality and present its applications. Let be a set and be a subset of . Let be a subset of a Hausdorff topological vector space which is invariant under infinite convex combinations. Let be a bounded function such that the functions are convex for all and whenever 0$">, and Let be a sequence in . Assume that, for every , there exists satisfying . Then

If , then the set in the above inequality can be replaced by .

     

Vector Variational Inequalities for Nondifferentiable Convex Vector Optimization Problems

In this paper, we consider a nondifferentiable convex vector optimization problem (VP), and formulate several kinds of vector variational inequalities with subdifferentials. Here we examine relations among solution sets of such vector variational inequalities and (VP). Mathematics Subject classification (2000). 90C25, 90C29, 65K10 This work was supported by the Brain Korea 21Project in 2003. The authors wish to express their appreciation to the anonymous referee for giving valuable comments.     

Concatenating Indicators of Flats for Designing Cryptographic Functions

Boolean functions with good cryptographic characteristics are needed for the design of robust pseudo-random generators for stream ciphers and of S-boxes for block ciphers. Very few general constructions of such cryptographic Boolean functions are known. The main ones correspond to concatenating affine or quadratic functions. We introduce a general construction corresponding to the concatenation of indicators of flats. We show that the functions it permits to design can present very good cryptographic characteristics.     

A Formula for <Emphasis Type="Italic">N</Emphasis>-Row Macdonald Polynomials

We derive a formula for the n-row Macdonald polynomials with the coefficients presented both combinatorically and in terms of very-well-poised hypergeometric series.     

Saddle Points and Pareto Points in Multiple Objective Programming

In this paper relationships between Pareto points and saddle points are studied in convex and nonconvex multiple objective programming. The analysis is based on partitioning the index sets of objectives and constraints and splitting the original problem into subproblems having a special structure. The results are based on scalarizations of multiple objective programs and related linear and augmented Lagrangian functions. In the nonconvex case, a saddle point characterization of Pareto points is possible under assumptions that guarantee existence of Pareto points and stability conditions of single objective problems. Essentially, these conditions are not stronger than those in analogous results for single objective programming.This research was partially supported by ONR Grant N00014-97-1-784AMS Subject Classification: 90C29, 90C26     

Computation of efficient compromise arcs in Convex Quadratic Multicriteria Optimization

Given a Convex Quadratic Multicriteria Optimization Problem, we show the stability of the Domination Problem. By modifying Benson’s single parametric method, which is based on the Domination Problem, we are able to show the existence of an efficient compromise arc connecting any two efficient points. Moreover, we deduce an algorithm which realizes the modification in polynomial time.Part of the article is taken from the author’s doctoral dissertation at the University of Eichstätt-Ingolstadt     

On the \Gamma-limit of the Mumford-Shah functional

We study by means of -convergence the asymptotics of the rescaled Mumford-Shah functional

Boundary behavior of Gleason''''s problem in hyperbolic harmonic Bergman spaces

Necessary and sufficient conditions are obtained for the boundedness of Berezin transformation on Lebesgue space Lp(B, dVβ) in the real unit ball B in Rn. As an application, we prove that Gleason type problem is solvable in hyperbolic harmonic Bergman spaces. Furthermore we investigate the boundary behavior of the solutions of Gleason type problem.     

On the finite sum representations of the Lauricella functions <Emphasis Type="Italic">F</Emphasis><Subscript>D</Subscript>

By using divided differences, we derive two different ways of representing the Lauricella function of n variables F_D⁽ⁿ⁾(a,b₁,b₂,...,b_n;c;x₁,x₂,...,x_n) as a finite sum, for b₁,b₂,...,b_n positive integers, and a,c both positive integers or both positive rational numbers with c–a a positive integer. AMS subject classification 33D45, 40B05, 40C99Jieqing Tan: Research supported by the National Natural Science Foundation of China under Grant No. 10171026 and Anhui Provincial Natural Science Foundation under Grant No. 03046102.Ping Zhou: Corresponding author. Research supported by NSERC of Canada.     

Positive definite dot product kernels in learning theory

In the classical support vector machines, linear polynomials corresponding to the reproducing kernel K(x,y)=xy are used. In many models of learning theory, polynomial kernels K(x,y)=_l=0^Na_l(xy)^l generating polynomials of degree N, and dot product kernels K(x,y)=_l=0⁺a_l(xy)^l are involved. For corresponding learning algorithms, properties of these kernels need to be understood. In this paper, we consider their positive definiteness. A necessary and sufficient condition for the dot product kernel K to be positive definite is given. Generally, we present a characterization of a function f:RR such that the matrix [f(xⁱx^j)]_i,j=1^m is positive semi-definite for any x¹,x²,...,x^mRⁿ, n2. Supported by CERG Grant No. CityU 1144/01P and City University of Hong Kong Grant No. 7001342.AMS subject classification 42A82, 41A05