The uniform closure of non-dense rational spaces on the unit interval

Let P_n denote the set of all algebraic polynomials of degree at most n with real coefficients. Associated with a set of poles a₁,a₂,…,a_n R[-1,1] we define the rational function spaces Associated with a set of poles a₁,a₂,… R[-1,1], we define the rational function spacesIt is an interesting problem to characterize sets a₁,a₂,… R[-1,1] for which P(a₁,a₂,…) is not dense in C[-1,1], where C[-1,1] denotes the space of all continuous functions equipped with the uniform norm on [-1,1]. Akhieser showed that the density of P(a₁,a₂,…) is characterized by the divergence of the series .In this paper, we show that the so-called Clarkson–Erdős–Schwartz phenomenon occurs in the non-dense case. Namely, if P(a₁,a₂,…) is not dense in C[-1,1], then it is “very much not so”. More precisely, we prove the following result.Theorem Let a₁,a₂,… R[-1,1]. Suppose P(a₁,a₂,…) is not dense in C[-1,1], that is,Then every function in the uniform closure of P(a₁,a₂,…) in C[-1,1] can be extended analytically throughout the set C -1,1,a₁,a₂,… .     

Robust H ∞ Filtering for a Class of Nonlinear Discrete-Time Markovian Jump Systems

This paper considers the problem of the robust H filtering for a class of nonlinear discrete-time Markovian jump systems with real time-varying norm-bounded parameter uncertainty. For each mode, the nonlinearity is assumed to satisfy the global Lipschitz conditions and appears in both the state and measured output equations. The problem that we address is the design of a nonlinear filter which ensures robust stochastic stability and a prescribed H performance level of the filtering error system for all admissible uncertainties. A sufficient condition for the solvability of this problem is obtained in terms of a set of linear matrix inequalities; an explicit expression of a desired nonlinear H filter is also given. Finally, an example is provided to demonstrate the effectiveness of the proposed approach.     

Infinite Dimensional Isoperimetric Inequalities in Product Spaces with the Supremum Distance

We study the isoperimetric problem for product probability measures with respect to the uniform enlargement. We construct several examples of measures for which the isoperimetric function of coincides with the one of the infinite product . This completes earlier works by Bobkov and Houdré.     

Talagrand＇s T2-transportation Inequality w.r.t, a Uniform Metric for Diffusions

Using the method of Girsanov transformation, we establish the Talagrand‘s T2-inequality for diffusion on the path space C([0, N], R^d) with respect to a uniform metric, with the constant independent of N. This improves the known results for the L2-metric.     

Korn's inequalities for piecewise vector fields

Korn's inequalities for piecewise vector fields are established. They can be applied to classical nonconforming finite element methods, mortar methods and discontinuous Galerkin methods.

     

The length of harmonic forms on a compact Riemannian manifold

We study -dimensional Riemannian manifolds with harmonic forms of constant length and first Betti number equal to showing that they are -step nilmanifolds with some special metrics. We also characterize, in terms of properties on the product of harmonic forms, the left-invariant metrics among them. This allows us to clarify the case of equality in the stable isosytolic inequalities in that setting. We also discuss other values of the Betti number.

     

An Open Question on Cyclic Relaxation

The problem discussed in this note is highly interesting. It is related to several dual iterative methods, such as the methods proposed by Kaczmarz, Hildreth, Agmon, Cryer, Mangasarian, Herman, Lent, Censor, and others. Cast as row-action methods these algorithms have been proved as useful tools for solving large convex feasibility problems that arise in medical image reconstruction from projections, in inverse problems in radiation therapy, and in linear programming.The question that we want to answer is how these algorithms behave when the feasible region is empty. It is shown that under certain conditions the primal sequence still converges while the dual sequence {y _k } obeys the rule y _k =u _k +k v, where {u _k } is a converging sequence and v is a fixed vector that satisfies A ^T v=0,v0,and,b ^T v>0. It is conjectured that these properties hold whenever the feasible region is empty. However, the validity of this claim remains an open question.This revised version was published online in October 2005 with corrections to the Cover Date.     

Extending the Balas-Yu bounds on the number of maximal independent sets in graphs to hypergraphs and lattices

A result of Balas and Yu (1989) states that the number of maximal independent sets of a graph G is at most ^p+1, where is the number of pairs of vertices in G at distance 2, and p is the cardinality of a maximum induced matching in G. In this paper, we give an analogue of this result for hypergraphs and, more generally, for subsets of vectors in the product of n lattices =₁××_n, where the notion of an induced matching in G is replaced by a certain binary tree each internal node of which is mapped into . We show that our bounds may be nearly sharp for arbitrarily large hypergraphs and lattices. As an application, we prove that the number of maximal infeasible vectors x=₁××_n for a system of polymatroid inequalities does not exceed max{Q,^logt/c(2Q,)}, where is the number of minimal feasible vectors for the system, , , and c(,) is the unique positive root of the equation 2^c(^c/log–1)=1. This bound is nearly sharp for the Boolean case ={0,1}ⁿ, and it allows for the efficient generation of all minimal feasible sets to a given system of polymatroid inequalities with quasi-polynomially bounded right-hand sides . This research was supported by the National Science Foundation (Grant IIS-0118635), and by the Office of Naval Research (Grant N00014-92-J-1375). The second and third authors are also grateful for the partial support by DIMACS, the National Science Foundation's Center for Discrete Mathematics and Theoretical Computer Science.Mathematics Subject Classification (2000):20E28, 20G40, 20C20     

Vector Equilibrium Problems Under Asymptotic Analysis

Given a closed convex set K in Rⁿ; a vector function F:K×K R^m; a closed convex (not necessarily pointed) cone P(x) in ^m with non-empty interior, PP(x) Ø, various existence results to the problemfind xK such that F(x,y)- int P(x) y K under P(x)-convexity/lower semicontinuity of F(x,) and pseudomonotonicity on F, are established. Moreover, under a stronger pseudomonotonicity assumption on F (which reduces to the previous one in case m=1), some characterizations of the non-emptiness of the solution set are given. Also, several alternative necessary and/or sufficient conditions for the solution set to be non-empty and compact are presented. However, the solution set fails to be convex in general. A sufficient condition to the solution set to be a singleton is also stated. The classical case P(x)=^m ₊ is specially discussed by assuming semi-strict quasiconvexity. The results are then applied to vector variational inequalities and minimization problems. Our approach is based upon the computing of certain cones containing particular recession directions of K and F.