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Tilings and model theory
ISBN 978-5-94057-377-7International audienceIn this paper we emphasize the links between model theory and tilings. More precisely, after giving the definitions of what tilings are, we give a natural way to have an interpretation of the tiling rules in first order logics. This opens the way to map some model theoretical properties onto some properties of sets of tilings, or tilings themselves
Consequences of Pure Point Diffraction Spectra for Multiset Substitution Systems
There is a growing body of results in the theory of discrete point sets and
tiling systems giving conditions under which such systems are pure point
diffractive. Here we look at the opposite direction: what can we infer about a
discrete point set or tiling, defined through a primitive substitution system,
given that it is pure point diffractive? Our basic objects are Delone multisets
and tilings, which are self-replicating under a primitive substitution system
of affine mappings with a common expansive map . Our first result gives a
partial answer to a question of Lagarias and Wang: we characterize repetitive
substitution Delone multisets that can be represented by substitution tilings
using a concept of "legal cluster". This allows us to move freely between both
types of objects. Our main result is that for lattice substitution multiset
systems (in arbitrary dimensions) being a regular model set is not only
sufficient for having pure point spectrum--a known fact--but is also necessary.
This completes a circle of equivalences relating pure point dynamical and
diffraction spectra, modular coincidence, and model sets for lattice
substitution systems begun by the first two authors of this paper.Comment: 36 page
Tilings With Very Elastic Tiles
We consider tiles of some fixed size, with an associated weighting on the
shapes of tile, of total mass 1. We study the pressure, , of tilings with
those tiles; the pressure, one over the volume times the logarithm of the
partition function. (The quantity we define as "pressure" could, perhaps
equally harmoniously with physics notation, be called "entropy per volume",
neither nomenclature is "correct".) We let (easy to compute) be the
pressure in the limit of absolute smoothness (the weighting function is
constant). Then as smoothness, suitably defined, increases, converges to
, uniformly in the volume. It is the uniformity requirement that
makes the result non-trivial. This seems like a very basic result in the theory
of pressure of tilings. Though at the same time, perhaps non-glamorous, being
bereft of geometry and not very difficult. The problem arose for us out of
study of a problem in mathematical physics, associated to a model of
ferromagnetism.Comment: 25 page
Brane Tilings and Their Applications
We review recent developments in the theory of brane tilings and
four-dimensional N=1 supersymmetric quiver gauge theories. This review consists
of two parts. In part I, we describe foundations of brane tilings, emphasizing
the physical interpretation of brane tilings as fivebrane systems. In part II,
we discuss application of brane tilings to AdS/CFT correspondence and
homological mirror symmetry. More topics, such as orientifold of brane tilings,
phenomenological model building, similarities with BPS solitons in
supersymmetric gauge theories, are also briefly discussed.
This paper is a revised version of the author's master's thesis submitted to
Department of Physics, Faculty of Science, the University of Tokyo on January
2008, and is based on his several papers: math.AG/0605780, math.AG/0606548,
hep-th/0702049, math.AG/0703267, arXiv:0801.3528 and some works in progress.Comment: 208 pages, 92 figures, based on master's thesis; v2: minor
corrections, to appear in Fortschr. Phy
Statistical mechanics of glass transition in lattice molecule models
Lattice molecule models are proposed in order to study statistical mechanics
of glass transition in finite dimensions. Molecules in the models are
represented by hard Wang tiles and their density is controlled by a chemical
potential. An infinite series of irregular ground states are constructed
theoretically. By defining a glass order parameter as a collection of the
overlap with each ground state, a thermodynamic transition to a glass phase is
found in a stratified Wang tiles model on a cubic lattice.Comment: 18 pages, 8 figure
A Comprehensive Survey of Brane Tilings
An infinite class of gauge theories can be engineered on
the worldvolume of D3-branes probing toric Calabi-Yau 3-folds. This kind of
setup has multiple applications, ranging from the gauge/gravity correspondence
to local model building in string phenomenology. Brane tilings fully encode the
gauge theories on the D3-branes and have substantially simplified their
connection to the probed geometries. The purpose of this paper is to push the
boundaries of computation and to produce as comprehensive a database of brane
tilings as possible. We develop efficient implementations of brane tiling tools
particularly suited for this search. We present the first complete
classification of toric Calabi-Yau 3-folds with toric diagrams up to area 8 and
the corresponding brane tilings. This classification is of interest to both
physicists and mathematicians alike.Comment: 39 pages. Link to Mathematica modules provide
Enumeration of Matchings: Problems and Progress
This document is built around a list of thirty-two problems in enumeration of
matchings, the first twenty of which were presented in a lecture at MSRI in the
fall of 1996. I begin with a capsule history of the topic of enumeration of
matchings. The twenty original problems, with commentary, comprise the bulk of
the article. I give an account of the progress that has been made on these
problems as of this writing, and include pointers to both the printed and
on-line literature; roughly half of the original twenty problems were solved by
participants in the MSRI Workshop on Combinatorics, their students, and others,
between 1996 and 1999. The article concludes with a dozen new open problems.
(Note: This article supersedes math.CO/9801060 and math.CO/9801061.)Comment: 1+37 pages; to appear in "New Perspectives in Geometric
Combinatorics" (ed. by Billera, Bjorner, Green, Simeon, and Stanley),
Mathematical Science Research Institute publication #37, Cambridge University
Press, 199
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