25,820 research outputs found
The large scale geometry of strongly aperiodic subshifts of finite type
A subshift on a group G is a closed, G-invariant subset of A^G, for some
finite set A. It is said to be a subshift of finite type (SFT) if it is defined
by a finite collection of 'forbidden patterns', to be strongly aperiodic if all
point stabilizers are trivial, and weakly aperiodic if all point stabilizers
are infinite index in G. We show that groups with at least 2 ends have a
strongly aperiodic SFT, and that having such an SFT is a QI invariant for
finitely presented torsion free groups. We show that a finitely presented
torsion free group with no weakly aperiodic SFT must be QI-rigid. The domino
problem on G asks whether the SFT specified by a given set of forbidden
patterns is empty. We show that decidability of the domino problem is a QI
invariant.Comment: 23 pages, 6 figures. The proof of the main theorem has been
simplified and some new corollaries deduce
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
Forcing a sparse minor
This paper addresses the following question for a given graph : what is
the minimum number such that every graph with average degree at least
contains as a minor? Due to connections with Hadwiger's Conjecture,
this question has been studied in depth when is a complete graph. Kostochka
and Thomason independently proved that . More generally,
Myers and Thomason determined when has a super-linear number of
edges. We focus on the case when has a linear number of edges. Our main
result, which complements the result of Myers and Thomason, states that if
has vertices and average degree at least some absolute constant, then
. Furthermore, motivated by the case when
has small average degree, we prove that if has vertices and edges,
then (where the coefficient of 1 in the 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.
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