A note on the relaxation time of two Markov chains on rooted phylogenetic tree spaces

Phylogenetic trees are commonly used to model the evolutionary relationships among a collection of biological species. Over the past fifteen years, the convergence properties for Markov chains defined on phylogenetic trees have been studied, yielding results about the time required for such chains to converge to their stationary distributions. In this work we derive an upper bound on the relaxation time of two Markov chains on rooted binary trees: one defined by nearest neighbor interchanges (NNI) and the other defined by subtree prune and regraft (SPR) moves.     

Multilinear forms and measures of dependence between random variables

Based on an idea of Rosenblatt, the methods of interpolation theory are used to establish moment inequalities and equivalence relations for measures of dependence between two or more families of random variables. A couple of “interpolation” theorems proved here appear to be new.     

Equivalent measures of dependence

The maximal correlation between a pair of σ-fields $\text{A}$ and $\text{B}$ becomes arbitrarily small as sup{|P(A ? B) ? P(A) P(B)|/[P(A) P(B)]^1/2, A ∈ $\text{A}$, B ∈ $\text{B}$, P(A) > 0, P(B) > 0} becomes sufficiently small.     

A note on continuous dependence of solutions of dynamic equations on time scales

We give a precise formulation and a proof as constructive as possible of the widely accepted claim that solutions of a dynamic equation depend continuously on the base time scale. Our approach to this problem is via Euler polygons which opens possibilities for development of numerical analysis of dynamic equations on time scales.     

A note on convex dependence of solutions of IVPs relative to initial values

We generalize a result of Lakshmikantham, Shahzad and Walter on convex dependence of the solutions of IVPs with respect to the initial value to general ordered Banach spaces.     

A framework for positive dependence

This paper presents, for bivariate distributions, a unified framework for studying and relating three basic concepts of positive dependence. These three concepts are positive dependence orderings, positive dependence properties and measures of positive dependence. The latter two concepts are formally defined and their properties discussed. Interrelationships among these three concepts are given, and numerous examples are presented.Supported by the National Science Foundation under Grant DMS-8301361.Supported by the Air Force Office of Scientific Research under Contract 84-0113. Reproduction in whole or part is permitted for any purpose of the United States Government.     

A numerically stable algorithm for two server queue models

In this paper, we consider a queueing system in which there are two exponential servers, each having his own queue, and arriving customers will join the shorter queue. Based on the results given in Flatto and McKean, we rewrite the formula for the probability that there are exactlyk customers in each queue, wherek = 0, 1,…. This enables us to present an algorithm for computing these probabilities and then to find the joint distribution of the queue lengths in the system. A program and numerical examples are given.     

On generalized measures of entropy and dependence

Various authors have studied extensions of Shannon’s entropy but their inferential properties and applications in applied sciences have not invited proper attention from researchers. In the present paper we explore the motivation and implication of using various classes of the generalized entropies and conditional entropies. We evaluate β-class and (α, β)-class entropies for multivariate normal density function. We also obtain the measures of dependence in terms of the classes of generalized entropies.      

A note on the dependence on parameters for a nonlinear system via monotonicity theory

Using monotonicity theory, we investigate the continuous dependence on parameters for the discrete BVPs, which can be written in a form of a nonlinear system.     

A note on multifractal packing dimension of measures

The relations between the multifractal packing dimension of Borel probability measures and the asymptotic behavior of the function φ*(x) = lim sup r→0 logν(B(x,r)) - qlogμ(B(x,r)) logr are discussed and some applications are given.     

A Local Limit Theorem for a Family of Non-Reversible Markov Chains

By proving a local limit theorem for higher-order transitions, we determine the time required for necklace chains to be close to stationarity. Because necklace chains, built by arranging identical smaller Markov chains around a directed cycle, are not reversible, have little symmetry, do not have uniform stationary distributions, and can be nearly periodic, prior general bounds on rates of convergence of Markov chains either do not apply or give poor bounds. Necklace chains can serve as test cases for future techniques for bounding rates of convergence.     

A new perspective on implementation by voting trees

Voting trees describe an iterative procedure for selecting a single vertex from a tournament. They provide a very general abstract model of decision‐making among a group of individuals, and it has therefore been studied which voting rules have a tree that implements them, i.e., chooses according to the rule for every tournament. While partial results concerning implementable rules and necessary conditions for implementability have been obtained over the past 40 years, a complete characterization of voting rules implementable by trees has proven surprisingly hard to find. A prominent rule that cannot be implemented by trees is the Copeland rule, which singles out vertices with maximum degree. In this paper, we suggest a new angle of attack and re‐examine the implementability of the Copeland solution using paradigms and techniques that are at the core of theoretical computer science. We study the extent to which voting trees can approximate the maximum degree in a tournament, and give upper and lower bounds on the worst‐case ratio between the degree of the vertex chosen by a tree and the maximum degree, both for the deterministic model concerned with a single fixed tree, and for randomizations over arbitrary sets of trees. Our main positive result is a randomization over surjective trees of polynomial size that provides an approximation ratio of at least 1/2. The proof is based on a connection between a randomization over caterpillar trees and a rapidly mixing Markov chain. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 39, 59–82, 2011     

A note on measures of nonconvexity

We define the Hausdorff measure of nonconvexity β(C) of a nonempty bounded subset C of a Banach space X as the Hausdorff distance of C to the family of all the nonempty convex bounded subsets of X. We compare the measure β with the Eisenfeld-Lakshmikantham measure of nonconvexity α and prove that the two measures are equivalent (β≤α≤2β), but in general they are different.     

A note on two-person zero-sum communicating stochastic games

For undiscounted two-person zero-sum communicating stochastic games with finite state and action spaces, a solution procedure is proposed that exploits the communication property, i.e., working with irreducible games over restricted strategy spaces. The proposed procedure gives the value of the communicating game with an arbitrarily small error when the value is independent of the initial state.     

A note on the effects of downstream efficiency on upstream pricing

The study of the effects of downstream entry on upstream pricing has revealed the counter-intuitive result of the supplier’s pricing policy being invariant to a new downstream entry under an isoelastic inverse demand function. We show that this counter-intuitive result is reversed when a new downstream entry affects downstream efficiency. We show that in the presence of increased post-entry downstream efficiency the supplier increases the wholesale price by taking advantage of the increased retail efficiency and competition. We also investigate the applicability of our results under other types of inverse demand functions.