A topological space X is said to be completely regular if whenever F is a nonempty closed set and x ∈ X \F , there is a continuous function f : X → [0, 1] such that f(x) = 0 and f(F ) = {1}. A completely regular space need not be Hausdorff. For example, ifX is any set with more than one point, then the trivial topology, in which the only closed sets are ∅ and X, is vacuously completely regular,...

Let $mathbb{F}$ be an algebraically closed field of prime characteristic $p>2$ and $(g, [p])$ a finite-dimensional restricted Lie superalgebra over $mathbb{F}$. It is showed that anyfinite-dimensional indecomposable $g$-module is a module for a finite-dimensional quotient of the universal enveloping superalgebra of $g$. These quotient superalgebras are called the generalized reduced enveloping ...

For the bosonic string on the torus we compute boundary states describing branes with not trivial homology class in presence of constant closed and open background. It turns out that boundary states with non trivial open background generically require the introduction of non physical “twisted” closed sectors, that only F and not F = F +B determines the geometric embedding for Dp branes with p <...

We show that in a Banach space X every closed convex subset is strongly proximinal if and only if the dual norm is strongly sub differentiable and for each norm one functional f in the dual space X∗, JX(f) the set of norm one elements in X where f attains its norm is compact. As a consequence, it is observed that if the dual norm is strongly sub differentiable then every closed convex subset of...

The aim of this note is to offer hyperstability results for linear functional equations of the form f(s) + f(t) = 1 n n ∑ i=1 f(sφi(t)) (s, t ∈ S), where S is a semigroup and where φ1, . . . , φn : S → S are pairwise distinct automorphisms of S such that the set {φ1, . . . , φn} is a group equipped with the composition as the group operation. The main results state that if f satisfies a stabili...

An edge-labeling f of a graph G is an injection from E(G) to the set of integers. The edge-bandwidth of G is B 0 (G) = min f fB 0 (f)g, where B 0 (f) is the maximum diierence between labels of incident edges of G. The m-theta graph (l 1 ; : : : ; l m g is the graph consisting of m pairwise internally disjoint paths with common endpoints and lengths l 1 l m. We determine the edge-bandwidth of al...

Recommended by Sehie Park We obtain new characterizations of open maps in terms of closures, of closed maps in terms of interiors, and of continuous maps in terms of interiors. Further open closed onto maps f : X→Y are described in terms of images under f of certain closed open sets in X. Continuity of onto maps is also characterized in terms of saturated sets.

We consider bounds on the size of families F of subsets of a v-set subject to restrictions modulo a prime p on the cardinalities of the pairwise intersections.We improve the known bound when F is allowed to contain sets of different sizes, but only in a special case. We show that if the bound for uniform families F holds with equality, then F is the set of blocks of what we call a p-ary t-desig...

A classic theorem in combinatorial design theory is Fisher’s inequality, which states that a family F of subsets of [n] with all pairwise intersections of size λ can have at most n non-empty sets. One may weaken the condition by requiring that for every set in F , all but at most k of its pairwise intersections have size λ. We call such families k-almost λ-Fisher. Vu was the first to study the ...

A Room t-cube is standardized if it also satisfies (iv) the ith diagonal cell contains the pair of symbols {co, i}. There are several other equivalent forms that are taken by Room t-cubes, one in terms of graphs and one in terms of Latin squares. The first of these forms is as a set of orthogonal one-factorizations of K,,. Let G be a graph with an even number of vertices. A one-factor in G is a...