Topological approximations of multisets

Rough set theory is a powerful mathematical tool for dealing with inexact, uncertain or vague information. The core concept of rough set theory are information systems and approximation operators of approximation spaces. In this paper, we define and investigate three types of lower and upper multiset approximations of any multiset. These types based on the multiset base of multiset topology induced by a multiset relation. Moreover, the relationships between generalized rough msets and mset topologies are given. In addition, an illustrative example is given to illustrate the relationships between different types of generalized definitions of rough multiset approximations.     

A rigorous definition of Robustness Analysis

Robustness Analysis (RA) is a problem structuring method for flexible planning. This paper presents a revised definition of RA which is based on earlier presentations of RA as well as new results derived from empirical research. This paper first identifies the basic concepts that are necessary for a rigorous definition of RA and relationships between these concepts. On this foundation, this paper constructs new definitions of robustness score and discusses their uses.     

On unified concepts of detectability and observability for continuous-time stochastic systems

This paper is concerned with detectability and observability of continuous-time stochastic linear systems. By adopting the idea used in defining these two concepts for time-varying systems and Markovian jump linear systems that have been studied in the literature, corresponding definitions for continuous-time stochastic linear systems are proposed. These two definitions are not only able to unify some recent definitions on these two concept reported in the literature, but also allow us to propose an efficient rank criterion to test observability of continuous-time stochastic linear systems. It seems that this rank criterion is quite analogous to the rank criterion for deterministic linear systems. With the help of these two concepts and the new criteria, the stochastic Lyapunov equation is revisited and some recent published work on this equation are generalized. Numerical examples are given to illustrate the effectiveness of the proposed approach.     

Classification of stability-like concepts and their study using vector Lyapunov functions

In the first section, stability-like definitions for ordinary differential equations are derived from a general qualitative concept. It is shown that the classical definitions of stability in the sense of Lyapunov, and their extensions can easily be deduced from this general formulation. A classification of all the definitions which may be derived is proposed.The second section contains the main results of this paper. It deals with the “comparison method” based upon one of T. Wazewski's theorems on differential inequalities. Several authors have used this method in order to investigate stability-like properties. We display the structure of this method, in order to state and prove some general comparison principles. These apply to the class of concepts considered earlier.In the last section some new results about stability and attractivity of sets are obtained as examples for the comparison principles. A theorem on stability in tube-like domains is proved in order to emphasize the generality and the flexibility of the comparison method.     

Cosine transforms over fields of characteristic 2

In this paper, we introduce cosine transforms over fields of characteristic 2. Our approach complements previous definitions of finite field trigonometric transforms, which only hold for fields whose characteristic is an odd prime. Besides introducing some new concepts related to trigonometry in finite fields, we discuss the eigenstructure and other important properties of the proposed transforms.     

微分积分定义的几种新途径对比分析

微分与积分的常规定义具有复杂性且两者之间缺乏直观的联系.对于项武义教授、张景中院士与林群院士所给出的新定义进行一一阐述,然后从概念引入、微分与积分之间的联系、定义的应用效果以及微积分基本定理的所指这四个方面进行比较分析.对微积分阐释途径的多元化研究,有助于教改的推进与完善.     

基于区间数度量的区间值模糊集合的贴近度和模糊度的关系

给出了基于区间数度量的区间值模糊集合的贴近度和模糊度的概念,详细研究了区间值模糊集合的贴近度和模糊度之间的关系,并基于公理化定义,证明了它们二者之间的相互转化关系,最后,给出了若干公式来计算区间值模糊集合的贴近度和模糊度。     

一类非光滑多目标半无限规划的最优性条件

利用K-方向导数,给出了一类存在性更为广泛的广义凸函数.即广义一致K-（F,α,ρ,d）-I型凸函数,进而讨论了涉及这些新广义凸性的一类多目标半无限规划的最优性条件。     

直观模糊集的熵、距离测度、相似测度及它们的关系

熵、距离测度和相似测度是模糊集理论中的三个重要概念.首先系统地给出了直观模糊集的熵、距离测度和相似测度的公理化定义,并讨论了它们之间的一些基本关系.然后在距离测度公理化定义的基础上产生了一些新的直观模糊集的熵公式.     

Stability and Shapley value for an n-persons fuzzy game

The aim of the paper is to explain new concepts of solutions for n-persons fuzzy games. Precisely, it contains new definitions for ‘core’ and ‘Shapley value’ in the case of the n-persons fuzzy games. The basic mathematical results contained in the paper are these which assert the consistency of the ‘core’ and of the ‘Shapley value’. It is proved that the core (defined in the paper) is consistent for any n-persons fuzzy game and that the Shapley values exists and it is unique for any fuzzy game with proportional values.     

PRME AND k-PRIME IDEALS IN Ω-GROUPS

Abstract

In this paper we define two concepts of prime ideals for Ω-groups. The first generalizes the definitions of prime ideal in rings, nearrings, Γ-rings, associative algebras and Lie algebras. The second generalizes a concept defined for groups by ??ukin ([21]). We show that both lead to radicals in the sense of Hoehnke ([10]). Furthermore in the case of rings, Γ-rings, abelian zero-symmetric nearrings and cubic rings these two definitions coincide, thus obtaining a new characterization for the prime ideal. Zero-symmetric Ω-groups are defined analogously to the nearring case and a new characterization in term of ideals is given.     

Areas of attraction for nonautonomous differential equations on finite time intervals

In this article, new concepts of (exponential) attractivity for nonautonomous differential equations on a finite time interval are introduced. Due to nonuniqueness of finite-time objects, areas of attraction are studied rather than solutions which are attractive. A sufficient and necessary condition for the existence of such areas is presented, which makes use of a time-varying Riemannian metric. Several examples illustrate the theoretical results and definitions.     

Ageing concepts: An approach based on quantile function

The notion of ageing plays an important role in reliability analysis and in identifying life distributions. Most of the ageing concepts existing in the literature are described on the basis of measures defined in terms of the distribution function. Recently, the role of quantile functions has also been identified as lifetime models, and reliability functions based on distribution functions were redefined in terms of quantile functions. In the present paper, we redefine some important popular ageing concepts using quantile functions. The uses of new definitions are illustrated by discussing ageing properties of some quantile function models.     

Submanifolds in the Context of Infinite Dimensional Manifolds with Corners

In this paper we introduce new definitions of submanifold and immersion in the context of infinite dimensional manifolds with corners. We show that they are the natural concepts in this context by giving positive answers to the problems of transitivity of submanifolds, inverse image of submanifolds and transversality, the problem of good immersion in quadrants of Banach spaces and the relation between a map being differentiable and its graph being a submanifold.     

关于“L—Fuzzy群同态与同构及其性质”一文的注记

文（１）给出了Ｌ－Ｆｕｚｚｙ群间的同态映射，满同态映射以及Ｌ－Ｆｕｚｚｙ群的Ｌ－正规子群等概念。本文指出以上三种定义的不合理性，并给出它们的合理性定义。     

Decision and forecast horizons,agreeable plans,and the maximum principle for infinite horizon control problems

This paper establishes a link between the concepts of optimality used in economic theory for infinite horizon planning models, and the concepts of decision and forecast horizons used in several areas of Management Science. It is shown that decision and forecast horizons induce an alternate definition of optimality which is stronger than the concept of ‘agreeable plan’ proposed by Hammond. All concepts of optimality share a common property, namely a Principle of Optimality. In an optimal control framework this implies that the maximum principle will be a necessary condition for optimality according to any of these definitions.     

Tensor products and matrix differential calculus

This paper sets forth some of the principal results of the algebra of Kronecker products in a way which relates them directly to the abstract algebra of tensor products. The concepts and the results that are developed in this way are used to analyse three alternative definitions that have been proposed for the derivative of a matrix function Y=Y(X) with respect to its matrix argument X. It is argued that only one of these definitions is viable. The other definitions, which are widely used in econometrics, are not consistent with the classical representation of linear algebra via matrix theory; and they lead to serious practical difficulties that do not arise when the appropriate definition is adopted.     

Classes of perfect graphs

The strong perfect graph conjecture, suggested by Claude Berge in 1960, had a major impact on the development of graph theory over the last 40 years. It has led to the definitions and study of many new classes of graphs for which the strong perfect graph conjecture has been verified. Powerful concepts and methods have been developed to prove the strong perfect graph conjecture for these special cases. In this paper we survey 120 of these classes, list their fundamental algorithmic properties and present all known relations between them.     

Cumulative measures of information and stochastic orders

In this paper we present some known results on cumulative measures of information, study their properties and relate these definitions to concepts of reliability theory. We give some relations of these measures of discrimination with some well-known stochastic orders and with the relative reversed hazard rate order. We investigate also a stochastic comparison among the empirical cumulative measures that can be related to the cumulative measures. Large part of this paper is a survey article; however, in the last section, we define a new measure of discrimination between residual lifetimes and study some of its properties.     

Affine Properties of Convex Equal-Area Polygons

In this paper we discuss some affine properties of convex equal-area polygons, which are convex polygons such that all triangles formed by three consecutive vertices have the same area. Besides being able to approximate closed convex smooth curves almost uniformly with respect to affine length, convex equal-area polygons admit natural definitions of the usual affine differential geometry concepts, like affine normal and affine curvature. These definitions lead to discrete analogous to the six-vertex theorem and an affine isoperimetric inequality. One can also define discrete counterparts of the affine evolute, parallels and the affine distance symmetry set preserving many of the properties valid for smooth curves.