Solving the Social Choice problem under equality constraints

Suppose that a number of equally qualified agents want to choose collectively an element from a set of alternatives defined by equality constraints. Each agent may well prefer a different element, and the social choice problem consists in deciding whether it is possible to design a rule to aggregate all the agents’ preferences into a social choice in an egalitarian way. In this paper we obtain criteria that solve this problem in terms of conditions that are explicitly computable from the constraints. As a theoretical consequence, we show that the only way to avoid running into a social choice paradox consists in designing (if possible) the set of alternatives satisfying certain optimality condition on the constraints, that is, in the natural way from the point of view of economics

On the fine structure of the global attractor of a uniformly persistent flow

We study the internal structure of the global attractor of a uniformly persistent flow. We show that the restriction of the flow to the global attractor has duality properties which can be expressed in terms of certain attractor-repeller decompositions. We also study some natural Morse decompositions of the flow and calculate their Morse equations. These equations provide necessary and sufficient conditions for the existence of attractors with the shape of S-1 or such that their suspension has spherical shape

Combinatorial 2d vector field topology extraction and simplification

Summary: This paper investigates a combinatorial approach to vector field topology. The theoretical basis is given by Robin Forman’s work on a combinatorial Morse theory for dynamical systems defined on general simplicial complexes. We formulate Forman’s theory in a graph theoretic setting and provide a simple algorithm for the construction and topological simplification of combinatorial vector fields on 2D manifolds. Given a combinatorial vector field we are able to extract its topological skeleton including all periodic orbits. Due to the solid theoretical foundation we know that the resulting structure is always topologically consistent. We explore the applicability and limitations of this combinatorial approach with several examples and determine its robustness with respect to noise.