In this paper, we study different types of non-additive set multifunctions (such as: uniformly autocontinuous, null-additive, null-null-additive), presenting relationships among them and some of their properties regarding atoms and pseudo-atoms. We also study non-atomicity and non-pseudo-atomicity of regular null-additive set multifunctions defined on the Baire (Borel respectively) δ-ring of a ...

let c be a field of subsets of a set i. also, let     1 i i be a non-decreasing positivesequence of real numbers such that 1, 1 0 1   i  and    11 i i . in this paper we prove thatbv of all the games of -bounded variation on c is a non-separable and norm dual banach space of thespace of simple games on c . we use this fact to establish the existence of a linear mapping t from...

For a non-empty ground set X, finite or infinite, the set-valuation or set-labeling of a given graph G is an injective function f : V (G) ! P(X) such that the induced edge-function f : E(G) ! P(X) ?? f;g is defined by f (uv) = f(u) f(v) for every uv2E(G), where P(X) is the power set of the set X and is a binary operation on sets. A set-indexer of a graph G is an set-labeling f : V (G) such that...

§1. The basic definitions and the main theorem. 1. Definition. (1) We define addition on 2 as addition modulo 2 on each component, i.e., if x, y, z ∈ 2 and x+ y = z then for every n we have z(n) = x(n) + y(n) (mod 2). (2) For A,B ⊆ 2 and x ∈ 2 we set x + A = {x + y : y ∈ A}, and we define A + B similarly. (3) We denote the Lebesgue measure on 2 with μ. We say that X ⊆ 2 is null-additive if for ...

In this paper we study the asymptotic behavior of stochastic approximation schemes with set-valued drift function and non-additive iterate-dependent Markov noise. We show that a linearly interpolated trajectory of such a recursion is an asymptotic pseudotrajectory for the flow of a limiting differential inclusion obtained by averaging the set-valued drift function of the recursion w.r.t. the st...

This paper proposes an effective algorithm based on the level set method (LSM) to solve shape and topology optimization problems. Since the conventional LSM has several limitations, a binary level set method (BLSM) is used instead. In the BLSM, the level set function can only take 1 and -1 values at convergence. Thus, it is related to phase-field methods. We don’t need to solve the Hamilton-Jac...

Let N0 denote the set of all non-negative integers and X be any subset of X. Also denote the power set of X by P(X). An integer additive set-labeling (IASL) of a graph G is an injective function f : V (G) ! P(X) such that the induced function f+ : E(G) ! P(X) is defined by f+(uv) = f(u) + f(v), where f(u) + f(v) is the sumset of f(u) and f(v). An IASL f is said to be a topological IASL (Top-IAS...