Let Fg 1, g> 1, be the mapping class group consisting of all isotopy classes of base-point and orientation preserving homeomorphisms of a closed, oriented surface F of genus g. Let )~(~1) be its Euler characteristic in the sense of Wall, that is Z(F~I)= [Fgl: F] l z(E/F), where F is any torsion free subgroup of finite index in F~ 1 and E is a contractible space on which F acts freely and proper...

An open (resp., closed) subset A of a topological space (X, T ) is called F-open F-closed) set if cl(A)\A A\int(A)) finite set. In this work, we study the main properties these definitions and examine relationships between F-closed sets with other kinds such as regularly open, closed, sets. Then, establish some operators F-interior, F-closure, F-derived...etc., using At end introduce F-continuo...

The complexity (quasi-metric) space was introduced in [23] to study complexity analysis of programs. Recently, it was introduced in [22] the dual complexity (quasi-metric) space, as a subspace of the function space [0,+∞)ω. Several quasi-metric properties of the complexity space were obtained via the analysis of its dual. We here show that the structure of a quasi-normed semilinear space provid...

The familiar Intermediate Value Theorem of elementary calculus says that if a real valued function f is continuous on the interval [a, b] ⊆ R then it takes each value between f(a) and f(b). As our next result shows, the critical fact is that the domain of f , the interval [a, b], is a connected space, for the theorem generalizes to real-valued functions on any connected space. The Intermediate ...

Let X be a compact metric space and continuous map f:X→X which defines discrete dynamical system (X,f). The f induces two natural maps, namely f¯:K(X)→K(X) on the hyperspace K(X) of non-empty subspaces Zadeh’s extension f^:F(X)→F(X) F(X) normal fuzzy set. In this work, we analyze interaction some orbit tracing properties, specification shadowing properties (X,f) its induced systems (K(X),f¯) (F...

We present some sufficient conditions for the existence of a continuous multi-utility representation of a preorder on a topological space (X, τ) (i.e., for the existence of a family F of continuous increasing realvalued functions on the preordered topological space (X, τ, ) such that, for every pair of elements (x, y) ∈ X×X, x y ⇔ f(x) ≤ f(y) ∀f ∈ F). Such conditions are based on the concept of...

In this paper we introduce the notion of an F-metric, as a function valued distance mapping, on a set X and we investigate the theory of F-metric spaces. We show that every metric space may be viewed as an F-metric space and every F-metric space (X, δ) can be regarded as a topological space (X, τδ). In addition, we prove that the category of the so-called extended Fmetric spaces properly contai...

We consider a basic problem in the general data streaming model, namely, to estimate a vector f ∈ Z that is arbitrarily updated (i.e., incremented or decremented) coordinatewise. The estimate f̂ ∈ Z must satisfy ‖f̂ − f‖∞ ≤ ǫ‖f‖1, that is, ∀i (|f̂i − fi| ≤ ǫ‖f‖1). It is known to have Õ(ǫ) randomized space upper bound [6], Ω(ǫ log(ǫn)) space lower bound [4] and deterministic space upper bound of Ω̃(...

max(u,v)∈E |f(u)− f(v)| if p =∞. If G is connected, then the only functions f satisfying ||f ||E,p = 0 are constant functions, so || · ||E,p is a norm on each linear space of functions on VG which does not contain constants. Usually we shall consider the subspace in the space of all functions on VG given by ∑ v∈V f(v)dv = 0. The obtained normed space will be called a Sobolev space on G and will...