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• The Global ETD Search service is a free service for researchers to find electronic theses and dissertations. This service is provided by the Networked Digital Library of Theses and Dissertations.
Our metadata is collected from universities around the world. If you manage a university/consortium/country archive and want to be added, details can be found on the NDLTD website.
1

#### A Tiling Approach to Chebyshev Polynomials

Walton, Daniel 01 May 2007 (has links)
We present a combinatorial interpretation of Chebyshev polynomials. The nth Chebyshev polynomial of the first kind, Tn(x), counts the sum of all weights of n-tilings using light and dark squares of weight x and dominoes of weight −1, and the first tile, if a square must be light. If we relax the condition that the first square must be light, the sum of all weights is the nth Chebyshev polynomial of the second kind, Un(x). In this paper we prove many of the beautiful Chebyshev identities using the tiling interpretation.
2

#### Discrete Riemann Maps and the Parabolicity of Tilings

Repp, Andrew S. 14 May 1998 (has links)
The classical Riemann Mapping Theorem has many discrete analogues. One of these, the Finite Riemann Mapping Theorem of Cannon, Floyd, Parry, and others, describes finite tilings of quadrilaterals and annuli. It relates to several combinatorial moduli, similar in nature to the classical modulus. The first chapter surveys some of these discrete analogues. The next chapter considers appropriate extensions to infinite tilings of half-open quadrilaterals and annuli. In this chapter we prove some results about combinatorial moduli for such tilings. The final chapter considers triangulations of open topological disks. It has been shown that one can classify such triangulations as either parabolic or hyperbolic, depending on whether an associated combinatorial modulus is infinite or finite. We obtain a criterion for parabolicity in terms of the degrees of vertices that lie within a specified distance of a given base vertex. / Ph. D.
3

#### Actions of Finite Groups on Substitution Tilings and Their Associated C*-algebras

Starling, Charles B 01 February 2012 (has links)
The goal of this thesis is to examine the actions of finite symmetry groups on aperiodic tilings. To an aperiodic tiling with finite local complexity arising from a primitive substitution rule one can associate a metric space, transformation groupoids, and C*-algebras. Finite symmetry groups of the tiling act on each of these objects and we investigate appropriate constructions on each, namely the orbit space, semidirect product groupoids, and crossed product C*-algebras respectively. Of particular interest are the crossed product C*-algebras; we derive important structure results about them and compute their K-theory.
4

#### Actions of Finite Groups on Substitution Tilings and Their Associated C*-algebras

Starling, Charles B 01 February 2012 (has links)
The goal of this thesis is to examine the actions of finite symmetry groups on aperiodic tilings. To an aperiodic tiling with finite local complexity arising from a primitive substitution rule one can associate a metric space, transformation groupoids, and C*-algebras. Finite symmetry groups of the tiling act on each of these objects and we investigate appropriate constructions on each, namely the orbit space, semidirect product groupoids, and crossed product C*-algebras respectively. Of particular interest are the crossed product C*-algebras; we derive important structure results about them and compute their K-theory.
5

#### A minimal subsystem of the Kari-Culik tilings

Siefken, Jason 13 August 2015 (has links)
The Kari-Culik tilings are formed from a set of 13 Wang tiles that tile the plane only aperiodically. They are the smallest known set of Wang tiles to do so and are not as well understood as other examples of aperiodic Wang tiles. We show that a certain subset of the Kari-Culik tilings, namely those whose rows can be interpreted as Sturmian sequences (rotation sequences), is minimal with respect to the Z^2 action of translation. We give a characterization of this space as a skew product as well as explicit bounds on the waiting time between occurrences of m × n configurations. / Graduate / 0405
6

#### Actions of Finite Groups on Substitution Tilings and Their Associated C*-algebras

Starling, Charles B 01 February 2012 (has links)
The goal of this thesis is to examine the actions of finite symmetry groups on aperiodic tilings. To an aperiodic tiling with finite local complexity arising from a primitive substitution rule one can associate a metric space, transformation groupoids, and C*-algebras. Finite symmetry groups of the tiling act on each of these objects and we investigate appropriate constructions on each, namely the orbit space, semidirect product groupoids, and crossed product C*-algebras respectively. Of particular interest are the crossed product C*-algebras; we derive important structure results about them and compute their K-theory.
7

8

#### Actions of Finite Groups on Substitution Tilings and Their Associated C*-algebras

Starling, Charles B January 2012 (has links)
The goal of this thesis is to examine the actions of finite symmetry groups on aperiodic tilings. To an aperiodic tiling with finite local complexity arising from a primitive substitution rule one can associate a metric space, transformation groupoids, and C*-algebras. Finite symmetry groups of the tiling act on each of these objects and we investigate appropriate constructions on each, namely the orbit space, semidirect product groupoids, and crossed product C*-algebras respectively. Of particular interest are the crossed product C*-algebras; we derive important structure results about them and compute their K-theory.
9

#### Tilings and other combinatorial results

Gruslys, Vytautas January 2018 (has links)
In this dissertation we treat three tiling problems and three problems in combinatorial geometry, extremal graph theory and sparse Ramsey theory. We first consider tilings of $\mathbb{Z}^n$. In this setting a tile $T$ is just a finite subset of $\mathbb{Z}^n$. We say that $T$ tiles $\mathbb{Z}^n$ if the latter set admits a partition into isometric copies of $T$. Chalcraft observed that there exist $T$ that do not tile $\mathbb{Z}^n$ but tile $\mathbb{Z}^{d}$ for some $d > n$. He conjectured that such $d$ exists for any given tile. We prove this conjecture in Chapter 2. In Chapter 3 we prove a conjecture of Lonc, stating that for any poset $P$ of size a power of $2$, if $P$ has a greatest and a least element, then there is a positive integer $k$ such that $[2]^k$ can be partitioned into copies of $P$. The third tiling problem is about vertex-partitions of the hypercube graph $Q_n$. Offner asked: if $G$ is a subgraph of $Q_n$ such $|G|$ is a power of $2$, must $V(Q_d)$, for some $d$, admit a partition into isomorphic copies of $G$? In Chapter 4 we answer this question in the affirmative. We follow up with a question in combinatorial geometry. A line in a planar set $P$ is a maximal collinear subset of $P$. P\'or and Wood considered colourings of finite $P$ without large lines with a bounded number of colours. In particular, they examined whether monochromatic lines always appear in such colourings provided that $|P|$ is large. They conjectured that for all $k,l \ge 2$ there exists an $n \ge 2$ such that if $|P| \ge n$ and $P$ does not contain a line of cardinality larger than $l$, then every colouring of $P$ with $k$ colours produces a monochromatic line. In Chapter 5 we construct arbitrarily large counterexamples for the case $k=l=3$. We follow up with a problem in extremal graph theory. For any graph, we say that a given edge is triangular if it forms a triangle with two other edges. How few triangular edges can there be in a graph with $n$ vertices and $m$ edges? For sufficiently large $n$ we prove a conjecture of F\"uredi and Maleki that gives an exact formula for this minimum. This proof is given in Chapter 6. Finally, Chapter 7 is concerned with degrees of vertices in directed hypergraphs. One way to prescribe an orientation to an $r$-uniform graph $H$ is to assign for each of its edges one of the $r!$ possible orderings of its elements. Then, for any $p$-set of vertices $A$ and any $p$-set of indices $I \subset [r]$, we define the $I$-degree of $A$ to be the number of edges containing vertices $A$ in precisely the positions labelled by $I$. Caro and Hansberg were interested in determining whether a given $r$-uniform hypergraph admits an orientation where every set of $p$ vertices has some $I$-degree equal to $0$. They conjectured that a certain Hall-type condition is sufficient. We show that this is true for $r$ large, but false in general.
10

#### Interlaced particles in tilings and random matrices

Nordenstam, Eric January 2009 (has links)
This thesis consists of three articles all relatedin some way to eigenvalues of random matrices and theirprincipal minors and also to tilings of various planar regions with dominoes or rhombuses.Consider an $N\times N$ matrix $H_N=[h_{ij}]_{i,j=1}^N$ from the Gaussian unitary ensemble (GUE). Denote its principal minors (submatrices in the upper left corner) by $H_n=[h_{ij}]_{i,j=1}^n$ for  $n=1$, \dots, $N$. We show in paper A that  all the $N(N+1)/2$ eigenvaluesof $H_1$, \dots, $H_N$ form a determinantal process on $N$ copies of the real line $\mathbb{R}$. We also show that this distribution arises as a scaling limit in tilings of an Aztec diamond with dominoes.We discuss a corresponding result for rhombus tilings of a hexagonwhich was later proved by Okounkov and Reshtikhin. We give a new proof of that statement in the introductionto this thesis.In paper B we perform a similar analysis for the Anti-symmetric Gaussian unitary ensemble (A-GUE). We show that the positive eigenvalues of an $N\times N$ A-GUE matrix andits principal minors form a determinantal processon $N$ copies of the positive real line $\mathbb{R}^+$.We also show that this distribution of all these eigenvalues appears as a scaling limit of tilings of half a hexagon with rhombuses. In paper C we study the shuffling algorithm for tilings of an Aztec diamond. This leads to the study of an interacting set of interlacedparticles that evolve in time. We conjecture that the diffusion limit of thisprocess is a process studied by Warrenand establish some results in this direction. / QC 20100804

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