Probability Distributions on Partially Ordered Sets and Network Interdiction Games

This article poses the following problem: Does there exist a probability distribution over subsets of a finite partially ordered set (poset), such that a set of constraints involving marginal probabilities of the poset’s elements and maximal chains is satisfied? We present a combinatorial algorithm to positively resolve this question. The algorithm can be implemented in polynomial time in the special case where maximal chain probabilities are affine functions of their elements. This existence problem is relevant for the equilibrium characterization of a generic strategic interdiction game on a capacitated flow network. The game involves a routing entity that sends its flow through the network while facing path transportation costs and an interdictor who simultaneously interdicts one or more edges while facing edge interdiction costs. Using our existence result on posets and strict complementary slackness in linear programming, we show that the Nash equilibria of this game can be fully described using primal and dual solutions of a minimum-cost circulation problem. Our analysis provides a new characterization of the critical components in the interdiction game. It also leads to a polynomial-time approach for equilibrium computation.

Orbital shadowing property on chain transitive sets for generic diffeomorphisms

Let f : M → M be a diffeomorphism on a closed smooth n(≥ 2) dimensional manifold M. We show that C1 generically, if a diffeomorphism f has the orbital shadowing property on locally maximal chain transitive sets which admits a dominated splitting then it is hyperbolic.

Flip Posets of Bruhat Intervals

In this paper we introduce a way of partitioning the paths of shortest lengths in the Bruhat graph $B(u,v)$ of a Bruhat interval $[u,v]$ into rank posets $P_{i}$ in a way that each $P_{i}$ has a unique maximal chain that is rising under a reflection order. In the case where each $P_{i}$ has rank three, the construction yields a combinatorial description of some terms of the complete $\textbf{cd}$-index as a sum of ordinary $\textbf{cd}$-indices of Eulerian posets obtained from each of the $P_{i}$.

On the structure of random hypergraphs

Let Hn be a countable random n-uniform hypergraph for n > 2, and P(Hn) = {f[Hn] : f : Hn ? Hn is an embedding}. We prove that a linear order L is isomorphic to the maximal chain in the partial order ?P(Hn)?{?},?? if and only if L is isomorphic to the order type of a compact set of reals whose minimal element is nonisolated.

A special chain theorem in the set of intermediate rings

Let [Formula: see text] be an extension of integral domains, and let [Formula: see text] be the integral closure of [Formula: see text] in [Formula: see text]. The main purpose of this paper is to study [Formula: see text], the set of intermediate rings between [Formula: see text] and [Formula: see text]. As a main tool, we establish an explicit description of any intermediate ring in terms of localizations of [Formula: see text] (or [Formula: see text]). This study effectively enables us to characterize the minimal extensions in [Formula: see text]. We also prove a special chain theorem concerning the length of an arbitrary maximal chain in [Formula: see text].