3阶不可约符号模式的精细惯量极小临界集

Let S be a nonempty, proper subset of all possible refined inertias of real matrices of order n. The set S is a critical set of refined inertias for irreducible sign patterns of order n,if for each n × n irreducible sign pattern A, the condition S ? ri(A) is sufficient for A to be refined inertially arbitrary. If no proper subset of S is a critical set of refined inertias, then S is a minimal critical set of refined inertias for irreducible sign patterns of order n.All minimal critical sets of refined inertias for full sign patterns of order 3 have been identified in [Wei GAO, Zhongshan LI, Lihua ZHANG, The minimal critical sets of refined inertias for 3×3 full sign patterns, Linear Algebra Appl. 458(2014), 183–196]. In this paper, the minimal critical sets of refined inertias for irreducible sign patterns of order 3 are identified.     

Topological Sorting of Finitely Near Sets

This paper introduces a natural approach in the evaluation of the nearness of sets in topological spaces. The objective is to classify levels of nearness of sets relative to each given set. The main result is a proximity measure of nearness for disjoint sets in an extremally disconnected topological space. Another result is that if A, B are nonempty semi-open sets such that \(A\ \delta \ B\), then \((\text{int }A)\ \delta \ (\text{int }B)\).     

Best proximity point results in non-Archimedean fuzzy metric spaces

We consider the problem of finding a best proximity point which achieves the minimum distance between two nonempty sets in a non-Archimedean fuzzy metric space. First we prove the existence and uniqueness of the best proximity point by using different contractive conditions, then we present some examples to support our best proximity point theorems.     

On minimal sets and strictly weaker topologies

In [1], [2] Farrag characterized the stirictly weaker principal topologies than any given principal topology on a nonempty set by using the minimal open sets which are defined by Steiner [3]. This paper mainly generalizes this result by using the minimal sets, which are defined in the paper with respect to the given topology τ on a nonempty set.     

Remarks on Minimal Sets for Cyclic Mappings in Uniformly Convex Banach Spaces

In this article, we prove that every nonempty and convex pair of subsets of uniformly convex in every direction Banach spaces has the proximal normal structure and then we present a best proximity point theorem for cyclic relatively nonexpansive mappings in such spaces. We also study the structure of minimal sets of cyclic relatively nonexpansive mappings and obtain the existence results of best proximity points for cyclic mappings using some new geometric notions on minimal sets. Finally, we prove a best proximity point theorem for a new class of cyclic contraction-type mappings in the setting of uniformly convex Banach spaces and so, we improve the main conclusions of Eldred and Veeramani.     

On the nonemptiness and compactness of the solution sets for vector variational inequalities

In this article, we investigate conditions for nonemptiness and compactness of the sets of solutions of pseudomonotone vector variational inequalities by using the concept of asymptotical cones. We show that a pseudomonotone vector variational inequality has a nonempty and compact solution set provided that it is strictly feasible. We also obtain some necessary conditions for the set of solutions of a pseudomonotone vector variational inequality to be nonempty and compact.     

Slice-continuous sets in reflexive Banach spaces: convex constrained optimization and strict convex separation

The concept of continuous set has been used in finite dimension by Gale and Klee and recently by Auslender and Coutat. Here, we introduce the notion of slice-continuous set in a reflexive Banach space and we show that the class of such sets can be viewed as a subclass of the class of continuous sets. Further, we prove that every nonconstant real-valued convex and continuous function, which has a global minima, attains its infimum on every nonempty convex and closed subset of a reflexive Banach space if and only if its nonempty level sets are slice-continuous. Thereafter, we provide a new separation property for closed convex sets, in terms of slice-continuity, and conclude this article by comments.     

The satisfactory partition problem

The SATISFACTORY PARTITION problem consists in deciding if a given graph has a partition of its vertex set into two nonempty parts such that each vertex has at least as many neighbors in its part as in the other part. This problem was introduced by Gerber and Kobler [Partitioning a graph to satisfy all vertices, Technical report, Swiss Federal Institute of Technology, Lausanne, 1998; Algorithmic approach to the satisfactory graph partitioning problem, European J. Oper. Res. 125 (2000) 283-291] and further studied by other authors but its complexity remained open until now. We prove in this paper that SATISFACTORY PARTITION, as well as a variant where the parts are required to be of the same cardinality, are NP-complete. However, for graphs with maximum degree at most 4 the problem is polynomially solvable. We also study generalizations and variants of this problem where a partition into k nonempty parts (k?3) is requested.     

Weighting factor results in vector optimization

This paper deals with the study, in a convex vector optimization problem, of the set of efficient solutions and the set of properly efficient solutions, the latter being obtained by a weighting factor technique. Relationships between these two sets are discussed; they are shown to be nonempty when the objective functions have no common direction of recession and to be closed and equal when, moreover, the objective functions are locally polyhedral. An example is provided where the set of efficient solutions is not included in the closure of the nonempty set of properly efficient solutions.The author wishes to thank the unknown referee for the helpful comments that improved the quality of this paper.     

On the Chebyshev Center and the Nonemptiness of the Intersection of Nested Sets

Mathematical Notes - We show that if every bounded set in a Banach space has a Chebyshev center, then the intersection of nested closed bounded sets in this space is nonempty in the case of a...     

积分凸性及其应用

该文在Banach空间中通过向量值函数的Bochner积分引进集合与泛函的积分凸性以及集合的积分端点等概念. 文章主要证明有限维凸集、开凸集和闭凸集均是积分凸集，下半连续凸泛函与开凸集上的上半连续凸泛函均是积分凸的, 非空紧集具有积分端点, 对紧凸集来说其积分端点集与端点集一致, 最后给出积分凸性在最优化理论方面的两个应用.     

Well-positioned Closed Convex Sets and Well-positioned Closed Convex Functions

We characterize the class of those closed convex sets which have a barrier cone with a nonempty interior. As a consequence, we describe the set of those proper extended-real-valued functionals for which the domain of their Fenchel conjugate has a nonempty interior. As an application, we study the stability of the solution set of a semi-coercive variational inequality.     

Proximity Function Minimization Using Multiple Bregman Projections, with Applications to Split Feasibility and Kullback–Leibler Distance Minimization

Problems in signal detection and image recovery can sometimes be formulated as a convex feasibility problem (CFP) of finding a vector in the intersection of a finite family of closed convex sets. Algorithms for this purpose typically employ orthogonal or generalized projections onto the individual convex sets. The simultaneous multiprojection algorithm of Censor and Elfving for solving the CFP, in which different generalized projections may be used at the same time, has been shown to converge for the case of nonempty intersection; still open is the question of its convergence when the intersection of the closed convex sets is empty.Motivated by the geometric alternating minimization approach of Csiszár and Tusnády and the product space formulation of Pierra, we derive a new simultaneous multiprojection algorithm that employs generalized projections of Bregman to solve the convex feasibility problem or, in the inconsistent case, to minimize a proximity function that measures the average distance from a point to all convex sets. We assume that the Bregman distances involved are jointly convex, so that the proximity function itself is convex. When the intersection of the convex sets is empty, but the closure of the proximity function has a unique global minimizer, the sequence of iterates converges to this unique minimizer. Special cases of this algorithm include the Expectation Maximization Maximum Likelihood (EMML) method in emission tomography and a new convergence result for an algorithm that solves the split feasibility problem.     

向量互补问题和极小元问题

本文不仅引入了非空集的有效下界的概念,而且引入了新的严格C-伪单调映射,在C-伪单调性的假设下,文内证明了向量互补问题（VCP）的可行集及其相关集合的有效下界和下有效点的存在性。     

Behavior of Trajectories of Time-Invariant Systems

Finitely many embedded localizing sets are constructed for invariant compact sets of a time-invariant differential system. These localizing sets are used to divide the state space into three subsets, the least localizing set and two sets called sets of the first kind and the second kind. We prove that the trajectory passing through a point of the set of the first kind remains in this set and tends to infinity. For a trajectory passing through a point of the set of the second kind, there are three possible types of behavior: it either goes to infinity or, at some finite time, enters the least localizing set, or has a nonempty ω-limit set contained in the intersection of the boundary of one of the constructed localizing sets with the universal section of the corresponding localizing function.     

Galois closed entity sets and k-balls of quasi-ultrametric multi-way dissimilarities

Quasi-ultrametric multi-way dissimilarities and their respective sets of k-balls extend the fundamental bijection in classification between ultrametric pairwise dissimilarities and indexed hierarchies. We show that nonempty Galois closed subsets of a finite entity set coincide with k-balls of some quasi-ultrametric multi-way dissimilarity. This result relates the order theoretic Galois connection based clustering approach to the dissimilarity based one. Moreover, it provides an effective way to specify easy-to-interpret cluster systems, from complex data sets, as well as to derive informative attribute implications.     

Generalized closed sets with respect to an ideal in bitopological spaces

An ideal on a set X is a nonempty collection of subsets of X with heredity property which is also closed under finite unions. The concept of generalized closed sets in bitopological spaces was introduced by Sundaram. In this paper, we introduce and study the concept of generalized closed sets with respect to an ideal in an ideal bitopological space.     

Strong Restricted-Orientation Convexity

Strong restricted-orientation convexity is a generalization of standard convexity. We explore the properties of strongly convex sets in multidimensional Euclidean space and identify major properties of standard convex sets that also hold for strong convexity. We characterize strongly convex flats and halfspaces, and establish the strong convexity of the affine hull of a strongly convex set. We then show that, for every point in the boundary of a strongly convex set, there is a supporting strongly convex hyperplane through it. Finally, we show that a closed set with nonempty interior is strongly convex if and only if it is the intersection of strongly convex halfspaces; we state a condition under which this result extends to sets with empty interior.     

Solution Sets of Interval-Valued Min-S-Norm Fuzzy Relational Equations

This paper introduces the concepts of tolerable solution set, united solution set, and controllable solution set for interval-valued fuzzy relational equations. Given a continuous s-norm, it is shown that each of the three types of the solution sets of interval-valued fuzzy relational equations with a min-s-norm composition, if nonempty, is composed of one minimum solution and a finite number of maximal solutions. Necessary and sufficient conditions for the existence of solutions are given. Computational procedures based on the constructive proofs are proposed to generate the complete solution sets. An example is given to illustrate the proposed procedures. An erratum to this article is available at .