3,016 research outputs found
An algorithm to generate exactly once every tiling with lozenges of a domain
We first show that the tilings of a general domain form a lattice which we
then undertake to decompose and generate without any redundance. To this end,
we study extensively the relatively simple case of hexagons and their
deformations. We show that general domains can be broken up into hexagon-like
parts. Finally we give an algorithm to generate exactly once every element in
the lattice of the tilings of a general domain.Comment: 36 pages, 61 figures. To be published in Theoretical Computer Science
(special issue on tilings.
Pattern occurrence statistics and applications to the Ramsey theory of unavoidable patterns
As suggested by Currie, we apply the probabilistic method to problems
regarding pattern avoidance. Using techniques from analytic combinatorics, we
calculate asymptotic pattern occurrence statistics and use them in conjunction
with the probabilistic method to establish new results about the Ramsey theory
of unavoidable patterns in the full word case (both nonabelian sense and
abelian sense) and in the partial word case.
Keywords: Combinatorics on words; Partial words; Unavoidable patterns;
Abelian patterns; Probabilistic method; Analytic combinatorics; Ramsey theory.Comment: 22 pages, 5 figures, presented at SIAM DM 201
Constructing Words with High Distinct Square Densities
Fraenkel and Simpson showed that the number of distinct squares in a word of
length n is bounded from above by 2n, since at most two distinct squares have
their rightmost, or last, occurrence begin at each position. Improvements by
Ilie to and by Deza et al. to 11n/6 rely on the study of
combinatorics of FS-double-squares, when the maximum number of two last
occurrences of squares begin. In this paper, we first study how to maximize
runs of FS-double-squares in the prefix of a word. We show that for a given
positive integer m, the minimum length of a word beginning with m
FS-double-squares, whose lengths are equal, is 7m+3. We construct such a word
and analyze its distinct-square-sequence as well as its
distinct-square-density. We then generalize our construction. We also construct
words with high distinct-square-densities that approach 5/6.Comment: In Proceedings AFL 2017, arXiv:1708.0622
Visibility graphs and deformations of associahedra
The associahedron is a convex polytope whose face poset is based on
nonintersecting diagonals of a convex polygon. In this paper, given an
arbitrary simple polygon P, we construct a polytopal complex analogous to the
associahedron based on convex diagonalizations of P. We describe topological
properties of this complex and provide realizations based on secondary
polytopes. Moreover, using the visibility graph of P, a deformation space of
polygons is created which encapsulates substructures of the associahedron.Comment: 18 pages, 16 figure
Fourier transform on high-dimensional unitary groups with applications to random tilings
A combination of direct and inverse Fourier transforms on the unitary group
identifies normalized characters with probability measures on -tuples
of integers. We develop the version of this correspondence by
matching the asymptotics of partial derivatives at the identity of logarithm of
characters with Law of Large Numbers and Central Limit Theorem for global
behavior of corresponding random -tuples.
As one application we study fluctuations of the height function of random
domino and lozenge tilings of a rich class of domains. In this direction we
prove the Kenyon-Okounkov's conjecture (which predicts asymptotic Gaussianity
and exact form of the covariance) for a family of non-simply connected
polygons.
Another application is a central limit theorem for the quantum random
walk with random initial data.Comment: 66 pages, 10 figure
Fixed-point tile sets and their applications
An aperiodic tile set was first constructed by R. Berger while proving the
undecidability of the domino problem. It turned out that aperiodic tile sets
appear in many topics ranging from logic (the Entscheidungsproblem) to physics
(quasicrystals). We present a new construction of an aperiodic tile set that is
based on Kleene's fixed-point construction instead of geometric arguments. This
construction is similar to J. von Neumann self-reproducing automata; similar
ideas were also used by P. Gacs in the context of error-correcting
computations. This construction it rather flexible, so it can be used in many
ways: we show how it can be used to implement substitution rules, to construct
strongly aperiodic tile sets (any tiling is far from any periodic tiling), to
give a new proof for the undecidability of the domino problem and related
results, characterize effectively closed 1D subshift it terms of 2D shifts of
finite type (improvement of a result by M. Hochman), to construct a tile set
which has only complex tilings, and to construct a "robust" aperiodic tile set
that does not have periodic (or close to periodic) tilings even if we allow
some (sparse enough) tiling errors. For the latter we develop a hierarchical
classification of points in random sets into islands of different ranks.
Finally, we combine and modify our tools to prove our main result: there exists
a tile set such that all tilings have high Kolmogorov complexity even if
(sparse enough) tiling errors are allowed.Comment: v7: updated reference to S.G.Simpson's pape
Tiling with arbitrary tiles
Let be a tile in , meaning a finite subset of
. It may or may not tile , in the sense of
having a partition into copies of . However, we prove that
does tile for some . This resolves a conjecture of
Chalcraft.Comment: 23 pages, 19 figures; slightly update
Singular polynomials of generalized Kasteleyn matrices
Kasteleyn counted the number of domino tilings of a rectangle by considering
a mutation of the adjacency matrix: a Kasteleyn matrix K. In this paper we
present a generalization of Kasteleyn matrices and a combinatorial
interpretation for the coefficients of the characteristic polynomial of KK^\ast
(which we call the singular polynomial), where K is a generalized Kasteleyn
matrix for a planar bipartite graph. We also present a q-version of these ideas
and a few results concerning tilings of special regions such as rectangles.Comment: 15 pages, 4 figure
Interpolation by periods in planar domain
Let be a countably connected domain. To any closed
differential form of degree in with components in
one associates the sequence of its periods around holes in , that is
around bounded connected components of . For which
the collection of such period sequences coincides with ? We
give the answer in terms of metric properties of holes in .Comment: 87 pages, 13 figure
A unified homogenization approach for the Dirichlet problem in Perforated Domains
We revisit the periodic homogenization of Dirichlet problems for the Laplace
operator in perforated domains, and establish a unified proof that works for
different regimes of hole-cell ratios, that is the ratio between the scaling
factor of the holes and that of the periodic cells. The approach is then made
quantitative and it yields correctors and error estimates for vanishing
hole-cell ratios. For positive volume fraction of holes, the approach is just
the standard oscillating test function method; for vanishing volume fraction of
holes, we study asymptotic behaviors of a properly rescaled cell problems and
use them to build oscillating test functions. Our method reveals how the
different regimes are intrinsically connected through the cell problems and the
connection with periodic layer potentials.Comment: 28 page
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