Zero sets and uniqueness sets with one cluster point for the Dirichlet space

Zero sets and uniqueness sets of the Dirichlet space are not completely characterized yet. There are sufficient conditions for zero sets and necessary conditions for uniqueness sets and there is a gap in between. Our goal is to take some preliminary steps to fill this gap.     

Asymptotically stable sets for semi-homeomorphisms

Necessary and sufficient conditions for semi-homeomorphisms to have Lyapunov stable sets and asymptotically stable sets are presented. Omega-limit sets for such functions are studied, and the dynamics of related discontinuous functions are investigated.     

Global asymptotic stability analysis by the localization method of invariant compact sets

We study the asymptotic stability and the global asymptotic stability of equilibria of autonomous systems of differential equations. We prove necessary and sufficient conditions for the global asymptotic stability of an equilibrium in terms of invariant compact sets and positively invariant sets. To verify these conditions, we use some results of the localization method for invariant compact sets of autonomous systems. These results are related to finding sets that contain all invariant compact sets of the system (localizing sets) and to the behavior of trajectories of the system with respect to localizing sets. We consider an example of a system whose equilibrium belongs to the critical case.     

Viability Theorems for Higher-Order Differential Inclusions

The aim of this paper is to establish both necessary and sufficient conditions for the existence of viable solutions of a higher-order differential inclusion. Higher-order tangent sets are introduced and studied to get such conditions. These tangent sets are analysed in the case of the viable set defined by equality or inequality constraints and viability conditions are rewritten in terms of these constraints.     

Covering maps of spaces of compact subsets

Mappings generated by covering maps of metric spaces are considered in the spaces of compact subsets of metric spaces. Sufficient conditions for the covering property of maps of spaces of compact sets and sufficient conditions for the existence of coincidence sets of maps of metric spaces are obtained.     

Efficient open domination in Cayley graphs

Efficient open dominating sets in bipartite Cayley graphs are characterized in terms of covering projections. Necessary and sufficient conditions for the existence of efficient open dominating sets in certain circulant Harary graphs are given. Chains of efficient dominating sets, and of efficient open dominating sets, in families of circulant graphs are described as an application.     

L—Fuzzy集分解定理

随着隶属函数真值域的拓广,原来关于Zadch-Fuzzy集的分解定理II和III对于L－Fuzzy集不再成立,尽管已有一些它们的关于L＿Fuzzy集的改进形式,但因条件较强,失去了原来的许多优越性,本文从格论入手,首先引入格中元素的强上集和L－Fuzzy集准截集的两个新概念,并讨论了它们的部分性质,进而借助它们给出了L－Fuzzy集分解定理的两个新形式。     

Characterizations of the strongly unique best approximations

The purpose of this paper is two folded. First, we present some results on strongly Kolmogorov sets, some of which parallel those for Kolmogorov sets. Secondly, we give two conditions which are sufficient for an element of a strongly Kolmogorov set to be a strongly unique best approximation. Then these conditions are shown to be necessary if additional conditions are imposed on either the norm or the set which we approximate from.     

L-可定义集与L-粗糙概念格

对基于剩余格L的L-广形式背景引入了L-可定义集概念,研究L-可定义集的性质及其与L-粗糙概念之间的关系.培出L-广可定义集的刻画;得到常值L-案均为L-可定义集的若干等价条件;证明全体L-可定义集在L-集包含序下是相应L-幂集格及L-粗糙概念格的子完备格;给出全体L-可定义集恰为全体L-粗糙概念外延的一个充要条件.     

Convergence to diffusions with regular boundaries

Earlier results on weak convergence to diffusion processes [8] are generalized to cases where the limiting diffusions may have regular boundaries. The boundaries may be adhesive or reflecting, and in each case we give two different sets of conditions for convergence. It is shown that these conditions are necessary and sufficient for convergence in the same sense as the conditions in [8]. We also extend our results to cases where the coefficients of the diffusions have simple discontinuities, in particular we thereby answer an open question by Keilson and Wellner [9]. Finally we formulate alternative sets of conditions for convergence, with these new sets being more convenient for instance when the sequence under investigation consists of pure jump Markov processes in continuous time.     

Ω-Invariance in control systems with bounded controls

The concepts of weakly Ω-invariant sets and strictly weakly Ω-invariant sets in control systems with bounded controls are defined and analyzed. Computable conditions for weak Ω-invariance are derived, and the question of existence of rest points and stationary points in weakly Ω-invariant sets is considered. For linear dynamics, properties of weakly Ω-invariant sets are studied, and questions of constrained reachability are investigated.     

A note on the finite convergence of alternating projections

We establish sufficient conditions for finite convergence of the alternating projections method for two non-intersecting and potentially nonconvex sets. Our results are based on a generalization of the concept of intrinsic transversality, which until now has been restricted to sets with nonempty intersection. In the special case of a polyhedron and closed half space, our sufficient conditions define the minimum distance between the two sets that is required for alternating projections to converge in a single iteration.     

Asymptotic stability analysis of autonomous systems by applying the method of localization of compact invariant sets

The asymptotic stability and global asymptotic stability of equilibria in autonomous systems of differential equations are analyzed. Conditions for asymptotic stability and global asymptotic stability in terms of compact invariant sets and positively invariant sets are proved. The functional method of localization of compact invariant sets is proposed for verifying the fulfillment of these conditions. Illustrative examples are given.     

Optimality Conditions for Isolated Local Minimum of Nonsmooth Multiobjective Problems

This article is devoted to the study of Fritz John and strong Kuhn-Tucker necessary conditions for properly efficient solutions, efficient solutions and isolated efficient solutions of a nonsmooth multiobjective optimization problem involving inequality and equality constraints and a set constraints in terms of the lower Hadamard directional derivative. Sufficient conditions for the existence of such solutions are also provided where the involved functions have pseudoconvex sublevel sets. Our results are based on the concept of pseudoconvex sublevel sets. The functions with pseudoconvex sublevel sets are a class of generalized convex functions that include quasiconvex functions.     

具强制条件的向量均衡问题

研究锥伪单调、锥拟凸和上锥连续映射在某种强制性条件下的向量均衡问题解集的特征,建立强制性条件与向量均衡问题解集的关系,得到对偶向量均衡问题局部解集含于向量均衡问题解集的性质和向量均衡问题解集的非空性条件,给出在锥伪单调、锥拟凸和上锥连续映射条件下向量均衡问题解集的非空有界性与强制性条件的等价性．     

Solution continuity in variational conditions

We present some results about Lipschitzian behavior of solutions to variational conditions when the sets over which the conditions are posed, as well as the functions appearing in them, may vary. These results rely on calmness and inner semicontinuity, and we describe some conditions under which those conditions hold, especially when the sets involved in the variational conditions are convex and polyhedral. We then apply the results to find error bounds for solutions of a strongly monotone variational inequality in which both the constraining polyhedral multifunction and the monotone operator are perturbed.      

A separation theorem for nonlinear inverse images of convex sets

This paper offers first- and higher-order necessary conditions for the local disjointness of a finite system of sets that are nonlinear inverse images of convex sets. The proof is based on the characterizations of α-admissible and α-tangent variations to nonlinear inverse images of convex sets and a necessary condition for the local disjointness in terms of these variations. As an application, the results are used to obtain first- and higher-order necessary conditions of optimality in constrained optimization problems.     

An extension of building sets and relative difference sets

Building sets are a successful tool for constructing semi‐regular divisible difference sets and, in particular, semi‐regular relative difference sets. In this paper, we present an extension theorem for building sets under simple conditions. Some of the semi‐regular relative difference sets obtained using the extension theorem are new in the sense that their ambient groups have smaller ranks than previously known. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 50–57, 2000     

On the Complexity of Minimizing Quasicyclic Boolean Functions

We investigate the Boolean functions that combine various properties: the extremal values of complexity characteristics ofminimization, the inapplicability of local methods for reducing the complexity of the exhaustion, and the impossibility to efficiently use sufficient minimality conditions. Some quasicyclic functions are constructed that possess the properties of cyclic and zone functions, the dominance of vertex sets, and the validity of sufficient minimality conditions based on independent families of sets. For such functions, we obtain the exponential lower bounds for the extent and special sets and also a twice exponential lower bound for the number of shortest and minimal complexes of faces with distinct sets of proper vertices.     

KKT conditions for weak compact convex sets,theorems of the alternative,and optimality conditions

We present several equivalent conditions for the Karush–Kuhn–Tucker conditions for weak^? compact convex sets. Using them, we extend several existing theorems of the alternative in terms of weak^? compact convex sets. Such extensions allow us to express the KKT conditions and hence necessary optimality conditions for more general nonsmooth optimization problems with inequality and equality constraints. Furthermore, several new equivalent optimality conditions for optimization problems with inequality constraints are obtained.