Closure System and Its Semantics

It is well known that topological spaces are axiomatically characterized by the topological closure operator satisfying the Kuratowski Closure Axioms. Equivalently, they can be axiomatized by other set operators encoding primitive semantics of topology, such as interior operator, exterior operator, boundary operator, or derived-set operator (or dually, co-derived-set operator). It is also known that a topological closure operator (and dually, a topological interior operator) can be weakened into generalized closure (interior) systems. What about boundary operator, exterior operator, and derived-set (and co-derived-set) operator in the weakened systems? Our paper completely answers this question by showing that the above six set operators can all be weakened (from their topological counterparts) in an appropriate way such that their inter-relationships remain essentially the same as in topological systems. Moreover, we show that the semantics of an interior point, an exterior point, a boundary point, an accumulation point, a co-accumulation point, an isolated point, a repelling point, etc. with respect to a given set, can be extended to an arbitrary subset system simply by treating the subset system as a base of a generalized interior system (and hence its dual, a generalized closure system). This allows us to extend topological semantics, namely the characterization of points with respect to an arbitrary set, in terms of both its spatial relations (interior, exterior, or boundary) and its dynamic convergence of any sequence (accumulation, co-accumulation, and isolation), to much weakened systems and hence with wider applicability. Examples from the theory of matroid and of Knowledge/Learning Spaces are used as an illustration.

On Some Properties of the Limit Points of (z(n)/n)n

Let (Fn)n≥0 be the sequence of Fibonacci numbers. The order of appearance of an integer n≥1 is defined as z(n):=min{k≥1:n∣Fk}. Let Z′ be the set of all limit points of {z(n)/n:n≥1}. By some theoretical results on the growth of the sequence (z(n)/n)n≥1, we gain a better understanding of the topological structure of the derived set Z′. For instance, {0,1,32,2}⊆Z′⊆[0,2] and Z′ does not have any interior points. A recent result of Trojovská implies the existence of a positive real number t<2 such that Z′∩(t,2) is the empty set. In this paper, we improve this result by proving that (127,2) is the largest subinterval of [0,2] which does not intersect Z′. In addition, we show a connection between the sequence (xn)n, for which z(xn)/xn tends to r>0 (as n→∞), and the number of preimages of r under the map m↦z(m)/m.

On Strongly b − θ -Continuous Mappings in Fuzzifying Topology

In this article, we will define the new notions (e.g., b − θ -neighborhood system of point, b − θ -closure (interior) of a set, and b − θ -closed (open) set) based on fuzzy logic (i.e., fuzzifying topology). Then, we will explain the interesting properties of the above five notions in detail. Several basic results (for instance, Definition 7, Theorem 3 (iii), (v), and (vi), Theorem 5, Theorem 9, and Theorem 4.6) in classical topology are generalized in fuzzy logic. In addition to, we will show that every fuzzifying b − θ -closed set is fuzzifying γ -closed set (by Theorem 3 (vi)). Further, we will study the notion of fuzzifying b − θ -derived set and fuzzifying b − θ -boundary set and discuss several of their fundamental basic relations and properties. Also, we will present a new type of fuzzifying strongly b − θ -continuous mapping between two fuzzifying topological spaces. Finally, several characterizations of fuzzifying strongly b − θ -continuous mapping, fuzzifying strongly b − θ -irresolute mapping, and fuzzifying weakly b − θ -irresolute mapping along with different conditions for their existence are obtained.

Vague Separation

In this paper we are introducing VT1 space, vague haussdorff space (VT2) and then we derive every vague subspace of VT1 space is VT1 and also for VT2. And also we derive the Cartesian product of two vague closed sets is also vague closed set in the vague product topological space X x Y .Finally we define Vague limit point, Vague isolated point, Vague adherent point, Vague perfect, Vague derived set, vague exterior and also derive some theorems on this .

AN ALTERNATIVE APPROACH TO MULTI-GAP SUPERCONDUCTIVITY

We draw attention to a feature suggested by a widely-cited paper by Suhl, Matthias, and Walker in the context of multi-gap superconductivity that seems to have escaped serious attention: interaction parameters in a superconductor characterized by two zero-temperature gaps but a single critical temperature must be temperature-dependent. Guided by this cue, we have presented a plausible scenario for a quantitative explanation of the superconducting properties of MgB 2 via an alternative approach — the approach provided by the recently derived set of generalized-BCS equations. Attention is drawn to earlier work in diverse fields where a similar T-dependent approach has been fruitful.

THE MODAL LOGIC OF STONE SPACES: DIAMOND AS DERIVATIVE

We show that if we interpret modal diamond as the derived set operator of a topological space, then the modal logic of Stone spaces is K4 and the modal logic of weakly scattered Stone spaces is K4G. As a corollary, we obtain that K4 is also the modal logic of compact Hausdorff spaces and K4G is the modal logic of weakly scattered compact Hausdorff spaces.

Sexual Script Theory: An Integrative Exploration of the Possibilities and Limits of Sexual Self-Definition

Sexual Script Theory (SST) and its clinical applications are premised on the notion that the subjective understandings of individuals of their sexuality determine the persons' choices of sexual actions and the qualitative experiencing of those sexual acts. The key elements of SST and key Christian control beliefs about sexuality are described, and then related in an integrative exploration of SST. The limits of an understanding of psychological scripting grounded in an unfettered Constructivism, and the limits of a purely pragmatic understanding of script legitimacy, are each discussed. We develop the pervasive theme of the necessary connectedness of sexual scripting to the broader processes of self-definition, which for the Christian, are to be rooted in a biblically-derived set of categories that connect sexuality to the character of the whole person, to their union with a spouse in marriage, and to the human community (individually and corporately) in its relationship to God.