‎let $x$ be a real normed  space, then  $c(subseteq x)$  is  functionally  convex  (briefly, $f$-convex), if  $t(c)subseteq bbb r $ is  convex for all bounded linear transformations $tin b(x,r)$; and $k(subseteq x)$  is  functionally   closed (briefly, $f$-closed), if  $t(k)subseteq bbb r $ is  closed  for all bounded linear transformations $tin b(x,r)$. we improve the    krein-milman theorem  ...

in this article, we have shown, for the add-point monad t, thepartial morphism category set*is isomorphic to the kleisli category sett. alsowe have proved that the category, sett, of t-algebras is isomorphic to thecategory set of pointed sets. finally we have established commutative squaresinvolving these categories.

Rough set theory was proposed by Zdziss law Pawlak 24, 25] in the early 1980's. Since then we have witnessed a systematic, worldwide growth of interest in rough set theory and its applications. Rough set theory deals with the analysis of classiicatory properties of data tables. Data represented in the tables can be acquired from measurements or from human experts; although in principle it must ...

In the past few years, multi-objective optimization (MOO) algorithms have been extensively applied in several fields including engineering design problems. A major reason is the advancement of evolutionary multi-objective optimization (EMO) algorithms that are able to find a set of non-dominated points spread on the respective Pareto-optimal front in a single simulation. Besides just finding a ...

In this paper, we discuss the approximability of the capacitated b-edge dominating set problem, which generalizes the edge dominating set problem by introducing capacities and demands on the edges. We present an approximation algorithm for this problem and show that it achieves a factor of 8/3 for general graphs and a factor of 2 for bipartite graphs. Moreover, we discuss the relationships of t...

Adaptive B-spline Knots Selection Using Multi-Resolution Basis Set Yuan Yuan, Nan Chen, Shiyu Zhou a Department of Industrial and Systems Engineering, the University of Wisconsin – Madison b Department of Industrial & Systems Engineering, National University of Singapore c Corresponding author. Email: [email protected] ABSTRACT B-splines are commonly used in the Computer Aided Design (CAD) an...

UML-B is a graphical formal modelling notation which is based on UML and relies on Event-B and its verification tools. In this paper, we propose annealing and introduce subtyping rules as well-known refactoring rules which can improve and assist the derivation of object-oriented design from an abstract specification written in UML-B. We prove that the proposed annealing rules are behavior prese...