模糊粗糙近似算子的拓扑性质

考虑论域上一二元关系所决定的模糊粗糙近似算子的拓扑性质,证明了任一自反二元关系可以决定一模糊拓扑.并且,当二元关系自反对称时,该模糊拓扑中的元是开集当且仅当它是闭集;当二元关系自反传递时,该模糊拓扑的闭包与内部算子恰为模糊粗糙上、下近似算子.     

由子基生成的内部算子和闭包算子

本文研究粗糙集与拓扑空间的关系,统一地使用拓扑空间中的集合关于子基的内部和闭包来研究粗糙集理论和覆盖广义粗糙集理论中的下近似集和上近似集,以及由它们导出的关于子基的开集,导集,闭集,边界．研究这两个概念及由它们导出的相关概念的性质不仅对于粗糙集理论,而且对于拓扑学本身都有重要的理论和实际应用意义．     

Rodabaugh α-闭包的推广

对LF拓扑空间的分明集及模糊格L中的元素α,Rodabaugh提出了α-闭包的概念.本文对此做了推广，对LF拓扑空间中的任一LF集，定义了HFα-闭包．并顺便引入了HFα-闭集的概念.文中讨论了这两个概念的基本性质。     

决策信息系统中的拓扑学方法

在决策信息系统中引入拓扑结构,借助拓扑学的基本概念(拓扑、内部和闭包等)研究决策问题,用它们刻画决策信息系统中的一些重要概念(决策协调集、决策约简集、下近似协调集、下近似约简集、上近似协调集、上近似约简集),并利用它们把这些重要概念推广到最一般的情况,建立起相应的约简理论.     

覆盖空间及粗糙集与拓扑的统一

引入覆盖空间,定义了其邻域、内部、闭包、测度等概念,研究了它们的性质.得出了粗糙集近似空间和拓扑空间都是具体覆盖空间的重要结论,从而用覆盖空间统一了粗糙集和拓扑.利用覆盖空间,得到了粗糙集和拓扑中更深刻的性质,从算子论和集合论的角度丰富和深化了粗糙集与拓扑的内容.     

近似空间的拓扑性质及其应用

近似空间(U,R)的全体可定义集构成X上的一个拓扑.本文在不要求论域U是有限的前提下探讨近似空间上这个拓扑的局部性质和可数性质,以及拓扑空间可近似化的充要条件及公理化体系,并寻找它们在粗糙集理论中的应用.     

由上近似数诱导的拟阵及其特征

拟阵理论与粗糙集理论之间有很多相似之处,近年来,探讨这二者之间的联系成为一个研究热点.首先利用基于等价关系的上近似数诱导了一系列拟阵结构,然后刻画了这类拟阵的独立集、基集、秩函数以及闭包等,讨论了与其它拟阵之间的一些联系.     

从边界运算出发建立拓扑空间

拓扑空间是现代数学中的一个重要的基本概念 .在集合上建立拓扑空间的方法很多 ,通常用开集公理来刻划 ,也可以选取点的邻域系 ,闭集 ,集合的闭包和内部等作为拓扑的原始概念 .本文选取集合的边界作为原始概念 ,在集合上建立拓扑空间     

覆盖下近似算子的拓扑性质

在不限制U为有限论域的情况下,研究了覆盖下近似算子XL和CL的拓扑性质。证明了覆盖下近似算子XL是内部算子,而且由XL生成的拓扑TXL为包含由覆盖C本身作为子基生成的拓扑TC的最小Alexandrov拓扑。特别地,当U为有限论域时,TXL=TC.然而,覆盖下近似算子CL不是内部算子。当覆盖C为某拓扑的基时,CL是内部算子,且此时由CL生成的拓扑TCL与TC是同一个拓扑。若进一步要求U为有限论域,则TCL=TXL=TC,进而CL=XL.     

定向闭包空间的若干性质

拓扑空间X称为定向闭包空间(简称DC空间),若它是T₀的，且其任一既约闭集都是某定向子集的闭包，此处X赋予特殊化序。本文讨论了定向闭包空间的一些基本性质，证明了偏序集赋予Alexandroof拓扑所得空间和其Smyth幂空间都是DC空间；偏序集赋予上拓扑是quasisober空间当且仅当它是DC空间；DC性对开子空间遗传，但对饱和子空间一般不遗传；对T₀空间X,其Smyth幂空间是DC空间一般不蕴含X是DC空间。     

Approximate Optimality and Approximate Duality for Quasi Approximate Solutions in Robust Convex Semidefinite Programs

In this paper, we study quasi approximate solutions for a convex semidefinite programming problem in the face of data uncertainty. Using the robust optimization approach (worst-case approach), approximate optimality conditions and approximate duality theorems for quasi approximate solutions in robust convex semidefinite programming problems are explored under the robust characteristic cone constraint qualification. Moreover, some examples are given to illustrate the obtained results.     

Approximate Duality

We extend the Lagrangian duality theory for convex optimization problems to incorporate approximate solutions. In particular, we generalize well-known relationships between minimizers of a convex optimization problem, maximizers of its Lagrangian dual, saddle points of the Lagrangian, Kuhn–Tucker vectors, and Kuhn–Tucker conditions to incorporate approximate versions. As an application, we show how the theory can be used for convex quadratic programming and then apply the results to support vector machines from learning theory.     

Approximate Inclusion-Exclusion

The Inclusion-Exclusion formula expresses the size of a union of a family of sets in terms of the sizes of intersections of all subfamilies. This paper considers approximating the size of the union when intersection sizes are known for only some of the subfamilies, or when these quantities are given to within some error, or both.In particular, we consider the case when allk-wise intersections are given for everykK. It turns out that the answer changes in a significant way aroundK=n: ifKO(n) then any approximation may err by a factor of (n/K ²), while ifK (n) it is shown how to approximate the size of the union to within a multiplicative factor of .When the sizes of all intersections are only given approximately, good bounds are derived on how well the size of the union may be approximated. Several applications for Boolean function are mentioned in conclusion.Partially supported by NSF 865727-CCR and ARO DALL03-86-K-017. Part of this work was done in U.C. Berkeley, supported by NSF CCR-8411954.     

Approximate Homomorphisms

which ``almost everywhere' looks like an ultrafilter has to be close to some fixed ultrafilter. Received: August 20, 1997     

Approximate Hermite quasi-interpolation

In this article, we derive approximate quasi-interpolants when the values of a function u and of some of its derivatives are prescribed at the points of a uniform grid. As a byproduct of these formulas we obtain very simple approximants, which provide high-order approximations for solutions to elliptic differential equations with constant coefficients.     

Approximate Indexed Lists

Let the position of a list element in a list be the number of elements preceding it plus one. Anindexed listsupports the following operations on a list: Insert; delete; return the position of an element; and return the element at a certain position. The order in which the elements appear in the list is completely determined by where the insertions take place; we do not require the presence of any keys that induce the ordering.We considerapproximate indexed listsand show that a tiny relaxation in precision of the query operations allows a considerable improvement in time complexity. The new data structure has applications in two other problems; namely,list labelingandsubset rank.