the concept of algebraic hyperstructures introduced by marty as a generalization of ordinary algebraic structures. in an ordinary algebraic structure, the composition of two elements is an element, while in an algebraic hyperstructure, the composition of two elements is a set. the concept of ?-semihyperrings is a generalization of semirings, a generalization of semihyper rings and a generalizat...

In a standard f -connectivity network design problem, we are given an undirected graph G = (V,E), a cut-requirement function f : 2 → N, and non-negative costs c(e) for all e ∈ E. We are then asked to find a minimum-cost vector x ∈ N such that x(δ(S)) ≥ f(S) for all S ⊆ V . We focus on the class of such problems where f is a proper function. This encodes many well-studied NP-hard problems such a...

The Loewner partial order is deened on the space of Hermitian matrices by A B if A ? B is positive semideenite. Given a strictly increasing function f :(a; b) ! R we deene the partial order f on the set of Hermitian matrices with spectrum contained in (a; b) by A f B if f(A) f(B): We say that the partial orders and f are equivalent on a set S of Hermitian matrices if A B if and only if A f B fo...

and Applied Analysis 3 that the function f : (X, α) → (Y, β) is strongly proximally continuous on A if ∀E ⊂ A, ∀S ⊂ X, EαS 󳨐⇒ f (E) βf (S) . (5) Finally, one says that f is strongly proximally continuous on B if f is strongly proximally continuous on A, for every A ∈ B. We shall use the notation C B (X, Y) to denote the family of the functions from X to Y which are strongly proximally continuou...

A discrete function f defined on Zn is said to be logconcave if f(λx+(1− λ)y) ≥ [f(x)]λ[f(y)]1−λ for x, y, λx + (1− λ)y ∈ Zn. A more restrictive notion is strong unimodality. Following Barndorff-Nielsen (1973) a discrete function p(z), z ∈ Zn is called strongly unimodal if there exists a convex function f(x), x ∈ Rn such that f(x) = − log p(x), if x ∈ Zn. In this paper sufficient conditions are...

We give an overview of the F–product construction and the corresponding nonstandard constructions and show that (in the case of bounded ultrapowers) both constructs are isomorphic (theorem 3.15). From this we also follow a little “classical” corollary (3.14). The nonstandard construction has been investigated in [Wol84] and [Wol], the corresponding classical constructions first appeared in [Der...

We give examples of non-amenable ICC groups Γ with the Haagerup property, weakly amenable with constant Λcb(Γ) = 1, for which we show that the associated II1 factors L(Γ) are strongly solid, i.e. the normalizer of any diffuse amenable subalgebra P ⊂ L(Γ) generates an amenable von Neumann algebra. Nevertheless, for these examples of groups Γ, L(Γ) is not isomorphic to any interpolated free group...