The number of non-solutions to an equation in a group and non-topologizable torsion-free groups

It is shown that, for any pair of cardinals with infinite sum, there exist a group and an equation over this group such that the first cardinal is the number of solutions to this equation and the second cardinal is the number of non-solutions to this equation. A countable torsion-free non-topologizable group is constructed.Comment: 5 pages; minor changes in the introduction and reference

Mathematical Structures in Group Decision-Making on Resource Allocation Distributions.

Optimal decisions on the distribution of finite resources are explicitly structured by mathematical models that specify relevant variables, constraints, and objectives. Here we report analysis and evidence that implicit mathematical structures are also involved in group decision-making on resource allocation distributions under conditions of uncertainty that disallow formal optimization. A group's array of initial distribution preferences automatically sets up a geometric decision space of alternative resource distributions. Weighted averaging mechanisms of interpersonal influence reduce the heterogeneity of the group's initial preferences on a suitable distribution. A model of opinion formation based on weighted averaging predicts a distribution that is a feasible point in the group's implicit initial decision space

New spectral multiplicities for ergodic actions

Let G be a locally compact second countable Abelian group. Given a measure preserving action T of G on a standard probability space, let M(T) denote the set of essential values of the spectral multiplicity function of the Koopman unitary representation of G associated with T. In the case when G is either a discrete countable Abelian group or R^n, n>0, it is shown that the sets of the form {p,q,pq}, {p,q,r,pq,pr,qr,pqr} etc. or any multiplicative (and additive) subsemigroup of N are realizable as M(T) for a weakly mixing G-action T.Comment: 17 page

Optimal Recombination in Genetic Algorithms

This paper surveys results on complexity of the optimal recombination problem (ORP), which consists in finding the best possible offspring as a result of a recombination operator in a genetic algorithm, given two parent solutions. We consider efficient reductions of the ORPs, allowing to establish polynomial solvability or NP-hardness of the ORPs, as well as direct proofs of hardness results

Differential Inequalities in Multi-Agent Coordination and Opinion Dynamics Modeling

Distributed algorithms of multi-agent coordination have attracted substantial attention from the research community; the simplest and most thoroughly studied of them are consensus protocols in the form of differential or difference equations over general time-varying weighted graphs. These graphs are usually characterized algebraically by their associated Laplacian matrices. Network algorithms with similar algebraic graph theoretic structures, called being of Laplacian-type in this paper, also arise in other related multi-agent control problems, such as aggregation and containment control, target surrounding, distributed optimization and modeling of opinion evolution in social groups. In spite of their similarities, each of such algorithms has often been studied using separate mathematical techniques. In this paper, a novel approach is offered, allowing a unified and elegant way to examine many Laplacian-type algorithms for multi-agent coordination. This approach is based on the analysis of some differential or difference inequalities that have to be satisfied by the some "outputs" of the agents (e.g. the distances to the desired set in aggregation problems). Although such inequalities may have many unbounded solutions, under natural graphic connectivity conditions all their bounded solutions converge (and even reach consensus), entailing the convergence of the corresponding distributed algorithms. In the theory of differential equations the absence of bounded non-convergent solutions is referred to as the equation's dichotomy. In this paper, we establish the dichotomy criteria of Laplacian-type differential and difference inequalities and show that these criteria enable one to extend a number of recent results, concerned with Laplacian-type algorithms for multi-agent coordination and modeling opinion formation in social groups.Comment: accepted to Automatic