Mixing times of lozenge tiling and card shuffling Markov chains

We show how to combine Fourier analysis with coupling arguments to bound the mixing times of a variety of Markov chains. The mixing time is the number of steps a Markov chain takes to approach its equilibrium distribution. One application is to a class of Markov chains introduced by Luby, Randall, and Sinclair to generate random tilings of regions by lozenges. For an L X L region we bound the mixing time by O(L^4 log L), which improves on the previous bound of O(L^7), and we show the new bound to be essentially tight. In another application we resolve a few questions raised by Diaconis and Saloff-Coste, by lower bounding the mixing time of various card-shuffling Markov chains. Our lower bounds are within a constant factor of their upper bounds. When we use our methods to modify a path-coupling analysis of Bubley and Dyer, we obtain an O(n^3 log n) upper bound on the mixing time of the Karzanov-Khachiyan Markov chain for linear extensions.Comment: 39 pages, 8 figure

Mixing Time of the Rudvalis Shuffle

We extend a technique for lower-bounding the mixing time of card-shuffling Markov chains, and use it to bound the mixing time of the Rudvalis Markov chain, as well as two variants considered by Diaconis and Saloff-Coste. We show that in each case Theta(n^3 log n) shuffles are required for the permutation to randomize, which matches (up to constants) previously known upper bounds. In contrast, for the two variants, the mixing time of an individual card is only Theta(n^2) shuffles.Comment: 9 page

The UNLV Jazz Ensemble II and III

Program listing performers and works performe

Forcing a sparse minor

This paper addresses the following question for a given graph $H$: what is the minimum number $f(H)$ such that every graph with average degree at least $f(H)$ contains $H$ as a minor? Due to connections with Hadwiger's Conjecture, this question has been studied in depth when $H$ is a complete graph. Kostochka and Thomason independently proved that $f(K_t)=ct\sqrt{\ln t}$. More generally, Myers and Thomason determined $f(H)$ when $H$ has a super-linear number of edges. We focus on the case when $H$ has a linear number of edges. Our main result, which complements the result of Myers and Thomason, states that if $H$ has $t$ vertices and average degree $d$ at least some absolute constant, then $f(H)\leq 3.895\sqrt{\ln d}\,t$. Furthermore, motivated by the case when $H$ has small average degree, we prove that if $H$ has $t$ vertices and $q$ edges, then $f(H) \leq t+6.291q$ (where the coefficient of 1 in the $t$ term is best possible)

Learning and Assessment in a Reading Group Format

The purpose of this paper is to outline how a traditional learning format the reading group was used to deliver a third-year political economy module (Critique of Political Economy). We begin by outlining the module delivery which is student-centred and where assessment is via presentations. The presenter/discussant format we use mirrors that at many academic conferences. Thereafter, we consider the nature of the reading material we used (Marx's Capital (1976)) before discussing the criteria for a good text. Finally, on the basis of these experiences we consider problems and issues that emerged in the reading group format. In concluding we argue that the reading group format has much to commend it, though we would suggest it as a complement to, rather than a substitute for, the more traditional lecture/seminar approach.