In this paper, the local function, weak semi-local function and closure are compared with each other according to inclusion relation. We define a new operator by using investigate its properties. Thanks operator, we obtain two topologies which finer than some previously defined topologies.

In what follows, the symbol X stands for a real Banach space with norm ‖ · ‖ and (normalized) duality mapping J. Moreover, “continuous” means “strongly continuous” and the symbol “→” (“⇀”) means strong (weak) convergence. The symbol R (R+) stands for the set (−∞,∞) ([0,∞)) and the symbols ∂D, intD, D denote the strong boundary, interior and closure of the set D, respectively. An operator T : X ...

The lack of double negation and de Morgan properties makes fuzzy logic unsymmetrical. This is the reason why fuzzy versions of notions like closure operator or Galois connection deserve attention for both antiotone and isotone cases, since they are not dual. This paper offers them attention, comming to the following conclusions: some kind of hardly describable ”local preduality” still makes pos...

Recently, El-Naschie has shown that the notion of fuzzy topology may be relevant to quantum paretical physics in connection with string theory and E-infinity space time theory. In this paper, we study the concepts of r-fuzzy semi-I-open, r-fuzzy pre-I-open, r-fuzzy α-I-open and r-fuzzy β-I-open sets, which is properly placed between r-fuzzy openness and r-fuzzy α-I-openness (r-fuzzy pre-I-openn...

We introduce fuzzy closure systems and fuzzy closure operators as extensions of closure systems and closure operators. We study relationships between fuzzy closure systems and fuzzy closure spaces. In particular, two families F (S) and F (C) of fuzzy closure systems and fuzzy closure operators on X are complete lattice isomorphic.

This paper presents a new genetic operator called adaptive operator to improve local portions of chromosomes. This new operator is implemented into PseudoBacterial Genetic Algorithm (PBGA). The PBGA was proposed by the authors as a new approach combining a genetic algorithm (GA) with a local improvement mechanism inspired by a process in bacterial genetics. The PBGA was applied for the acquisit...

Rough set theory is an excellent mathematical tool for the analysis of a vague description actions in decision problems. Now, this paper by considering notion equality algebra, lower and upper approximations are introduced some properties them given. Moreover, it proved that define interior operator closure operator, respectively. Also, using D-lower D-upper approximation, conditions nonempty s...

Let R be an excellent local domain of positive characteristic with residue field k and let R+ be its absolute integral closure. If Tor1 (R +, k) vanishes, then R is weakly F-regular. If R has at most an isolated singularity or has dimension at most two, then R is regular.