Optimal lower bounds for cubature error on the sphere

We show that the worst-case cubature error E(Q_m;H^s) of an m-point cubature rule Q_m for functions in the unit ball of the Sobolev space H^s=H^s(S²),s>1, has the lower bound , where the constant c_s is independent of Q_m and m. This lower bound result is optimal, since we have established in previous work that there exist sequences of cubature rules for which with a constant independent of n. The method of proof is constructive: given the cubature rule Q_m, we construct explicitly a ‘bad’ function f_mH^s, which is a function for which Q_mf_m=0 and . The construction uses results about packings of spherical caps on the sphere.     

KKM property and fixed point theorems in metric spaces

In this paper, first part, we establish some fixed point theorems for a k-set contraction map on the nearly-subadmissible subsets of a complete metric space. The second part, we deduce a generalization of the approximate fixed point theorem for the lower semicontinuous mappings on a metric space.     

On the Lower Tail Probabilities of Some Random Sequences in <Emphasis Type="Italic">l</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

We investigate the behaviour of the logarithmic small deviation probability of a sequence (σ _n θ _n ) in l _p , 0<p≤∞, where (θ _n ) are i.i.d. random variables and (σ _n ) is a decreasing sequence of positive numbers. In particular, the example σ _n ∼n ^−μ (1+log n)^−ν is studied thoroughly. Contrary to the existing results in the literature, the rate function and the small deviation constant are expressed expli- citly in the present treatment. The restrictions on the distribution of θ ₁ are kept to an absolute minimum. In particular, the usual variance assumption is removed. As an example, the results are applied to stable and Gamma-distributed random variables.     

集值映射的Nash平衡点集的通有稳定性

本文讨论了集值映射的Nash平衡点的存在及平衡点集的通有稳定性,得到大多数的集值映射的Nash平衡点集是稳定的。     

模糊关系性质的模糊T-近似算子刻画

在定义一般模糊关系下的模糊T-近似的基础上，用模糊T-近似算子来刻画模糊关系的自反性、对称性、T-传递性等性质，并讨论了模糊T-粗糙集的一些性质。     

On the Existence of Solutions of Quasi-equilibrium Problems with Constraints

The quasi-equilibrium problems with constraints are formulated and some sufficient conditions on the existence of their solutions are shown. As special cases, we obtain several results on the existence of solutions of some vector quasivariational inequality and vector optimization problems. An application of the obtained results is given to show the existence of solutions of quasi-optimization problems with constraints.     

On the worst case complexity of potential reduction algorithms for linear programming

There are several classes of interior point algorithms that solve linear programming problems in O(√nL) iterations. Among them, several potential reduction algorithms combine both theoretical (O(√nL) iterations) and practical efficiency as they allow the flexibility of line searches in the potential function, and thus can lead to practical implementations. It is a significant open question whether interior point algorithms can lead to better complexity bounds. In the present paper we give some negative answers to this question for the class of potential reduction algorithms. We show that, even if we allow line searches in the potential function, and even for problems that have network structure, the bound O(√nL) is tight for several potential reduction algorithms, i.e., there is a class of examples with network structure, in which the algorithms need at least Θ(√nL) iterations to find an optimal solution. The research of the author was partially supported by a Presidential Young Investigator Award DDM-9158118 with matching funds from Draper Laboratory.     

An efficient solver for weighted Max-SAT

We present a new branch and bound algorithm for weighted Max-SAT, called Lazy which incorporates original data structures and inference rules, as well as a lower bound of better quality. We provide experimental evidence that our solver is very competitive and outperforms some of the best performing Max-SAT and weighted Max-SAT solvers on a wide range of instances.     

On fuzzy multi-valued functions part 2: Some topological properties

In this paper new topological concepts connected with fuzzy multi-valued functions theory-as presented in the first part of this work-are introduced and their properties are investigated. As will be shown in future papers these concepts and properties are indispensable in connection with the analysis of fuzzy economic systems.     

Orientor field equations in Banach spaces

This paper focuses on the interplay between seminormality requirements and convergence hypotheses on trajectories in the lower closure theorems for orientor field equations. With the use of a weak form of seminormality, called the intermediate property (Q*), we obtain lower closure theorems (and thereby closure theorems) for orientor field equations in Banach spaces, under the assumption of strong convergence of some coordinates of the trajectories, while only weak convergence is assumed in the others. In the Euclidean case, this requirement of property (Q*) is further reduced to mere Kuratowski property (K), under the usual growth-type conditions. Finally, in the appendix, property (Q*) is investigated in detail.This research has been partially supported by Research Project AFOSR-71-2122 at the University of Michigan, Ann Arbor, Michigan.