Homogeneous chains, heterogeneous chains, and delay of reinforcement

Continuous-time homogeneous Markov chains

Hosoya polynomials of random benzenoid chains

Let $G$ be a molecular graph with vertex set $V(G)$, $d_G(u, v)$ the topological distance between vertices $u$ and $v$ in $G$. The Hosoya polynomial $H(G, x)$ of $G$ is a polynomial $sumlimits_{{u, v}subseteq V(G)}x^{d_G(u, v)}$ in variable $x$. In this paper, we obtain an explicit analytical expression for the expected value of the Hosoya polynomial of a random benzenoid chain with $n$ hexagon...

Boundary and Entropy of Space Homogeneous Markov Chains

CNRS, Université de Rennes-1 and Technische Universität Graz We study the Poisson boundary (≡ representation of bounded harmonic functions) of Markov operators on discrete state spaces that are invariant under the action of a transitive group of permutations. This automorphism group is locally compact, but not necessarily discrete or unimodular. The main technical tool is the entropy theory whi...

On the tutte polynomial of benzenoid chains

The Tutte polynomial of a graph G, T(G, x,y) is a polynomial in two variables defined for every undirected graph contains information about how the graph is connected. In this paper a simple formula for computing Tutte polynomial of a benzenoid chain is presented.

An efficient Imperialistic Competitive Algorithm for the closed-loop supply chains considering pricing for product, and fleet of heterogeneous vehicles

 This paper investigates a pricing multi-period, closed-loop supply chains (CLSCs) with two echelons of producers and customers. Products are delivered to customers might be defective which are picked up and gathered in the collection center and will be fixed if it is possible and will be returned to the chain. Otherwise, they are sold as waste. This problem is determining price and distributio...