In 2003, Johannes Kellendonk and Ian Putnam introduced pattern equivariant cohomology for tilings. In 2006, Betseygail Rand defined a type of pattern equivariant cohomology that incorporates rotational symmetry, using representation of the rotation group. In this doctoral thesis we study the relationship between these two types of pattern equivariant cohomology, showing exactly how to calculate one from the other in the case in which the rotation group is a finitely generated abelian group of free rank 1. We apply our result by calculating the cohomology of the pinwheel tiling.

La motivación principal de esta tesis es el desarrollo de nuevas técnicas de compilación dirigidas a conseguir mayor rendimiento encódigos numéricos complejos que definen es pacios de iteraciones no rectangulares. En particular, nos centramos en la trasformación de "loop tiling" (también conocida como "blocking") y nuestro propósito es mejorar la transformación de loop tiling cuando se aplica a códigos numéricos complejos. Nuestro objetivo es conseguir, a través de la transformación de loop tiling, los mismos o mejores rendimientos que las librerías numéricas proporcionadas por el fabricante que están optimizadas manualmente.En la tesis se muestra que la razón principal por la que los compiladores comerciales actuales consiguen bajos rendimiento en este tipo de aplicaciones es que no son capaces de aplicar loop tiling a nivel de registros. En su lugar, para mejorar la localidad de los datos y el ILP, los compiladores actuales usan y combinan otras transformaciones que no explotan el nivel de registros tan bien como loop tiling. Previamente no se ha considerado aplicar loop tiling a nivel de registro porque en códigos numéricos complejos no es trivial debido a la naturaleza irregular de los espacios de iteraciones. La primera contribución de esta tesis es un algoritmo general de loop tiling a nivel de registros que es aplicable a cualquier tipo de espacio de iteraciones y no sólo a los espacios rectangulares. Nuestro método incluye una heurística muy sencilla para decidir los parámetros de los cortes a nivel de registros. A primera vista parece que loop tiling a nivel de registros (a partir de ahora, register tiling) se tiene que aplicar de tal manera que el bucle que ofrece más reuso temporal de los datos no debe de ser partido. De esta manera maximizamos la reutilización de los registros y minimizamos el número total de load/stores ejecutados. Sin embargo, mostraremos que en espacios de iteraciones no rectangulares, si solamente tenemos en cuenta las direcciones del reuso y no la forma del espacio de iteraciones, los códigos pueden sufrir una degradación en rendimiento. Nuestra segunda contribución es la propuesta de una heurística muy sencilla que determina los parámetros del tiling a nivel de registros considerando no sólo el reuso temporal sino también la forma del espacio de iteraciones. Además, la heurística es suficientemente sencilla para que pueda ser implementada en un compilador comercial.Sin embargo, para conseguir rendimientos similares que códigos optimizados a mano, no es suficiente con aplicar loop tiling a nivel de registros. Con las arquitecturas de hoy en día que disponen de jerarquías de memoria complejas y múltiples procesadores, es necesario que el compilador aplique loop tiling en cuatro o más niveles (paralelismo, cache L2, cache L1 y registros) para conseguir altos rendimientos. Por lo tanto, en las arquitecturas actuales es crucial tener un algoritmo eficiente para aplicar loop tiling en varios niveles de la jerarquía de memoria (tiling multinivel). Además, como mostramos en esta tesis, la transformación de tiling multinivel siempre tendrá que incluir el nivel de registro porque este es el nivel de la jerarquía de memoria que ofrece mayor rendimiento cuando es tratado correctamente.Cuando tiling multinivel incluye el nivel de registros, es necesario que los límites de los bucles sean exactos y que no haya límites redundantes. La razón es que la complejidad y la cantidad de código que se genera con nuestra técnica de register tiling depende polinómicamente del número de límites de los bucles.Sin embargo, hasta ahora, el problema de calcular límites exactos y eliminar límites redundantes es que todas las técnicas conocidas son muy caras en términos de tiempo de compilación y, por ello, difícil de integrar en un compilador comercial. La tercera contribución de esta tesis es una nueva implementación de tiling multinivel que calcula límites exactos y es mucho menos costosa que técnicas tradicionales. Mostraremos que la complejidad de nuestra implementación es proporcional a la complejidad de aplicar una permutación de bucles en el código original (antes de aplicar loop tiling), mientras que las técnicas tradicionales tienen complejidades más altas. Además, nuestra implementación genera menos límites redundantes y permite eliminar los límites redundantes que quedan a menor coste. En conjunto, la eficiencia de nuestra implementación hace posible que pueda ser implementada dentro de un compilador comercial sin tener que preocuparse por los tiempos de compilación.La última parte de esta tesis está dedicada al estudio del rendimiento de tiling multinivel. Se muestran los efectos de tiling en los diferentes niveles de memoria y presentamos datos que comparan los beneficios de tiling a nivel de registros, tiling a nivel de cache y tiling a los dos niveles, cache y registros, simultáneamente. Finalmente, comparamos el rendimiento de códigos optimizados automáticamente con códigos optimizados manualmente (librerías numéricas que ofrecen los fabricantes) sobre dos arquitecturas diferentes (ALPHA 21164 and MIPS R10000) para concluir que actualmente la tecnología de los compiladores hace posible que códigos numéricos complejos consigan el mismo rendimiento que códigos optimizados manualmente. / The main motivation of this thesis is to develop new compilation techniques that address the lack of performance of complex numerical codes consisting of loop nests defining non-rectangular iteration spaces. Specifically, we focus on the loop tiling transformation (also known as blocking) and our purpose is the improvement of the loop tiling transformation when dealing with complex numerical codes. Our goal is to achieve via the loop tiling transformation the same or better performance as hand-optimized vendor-supplied numerical libraries. We will observe that the main reason why current commercial compilers perform poorly when dealing with this type of codes is that they do not apply tiling for the register level. Instead, to enhance locality at this level and to improve ILP, they use/combine other transformations that do not exploit the register level as well as loop tiling. Tiling for the register level has not generally been considered because, in complex numerical codes, it is far from being trivial due to the irregular nature of the iteration space. Our first contribution in this thesis will be a general compiler algorithm to perform tiling at the register level that handles arbitrary iteration space shapes and not only simple rectangular shapes.Our method includes a very simple heuristic to make the tile decisions for the register level. At first sight, register tiling should be performed so that whichever loop carries the most temporal reuse is not tiled. This way, register reuse is maximized and the number of load/store instructions executed is minimized. However, we will show that, for complex loop nests, if we only consider reuse directions and do not take into account the iteration space shape, the tiled loop nest can suffer performance degradation. Our second contribution will be a proposal of a very simple heuristic to determine the tiling parameters for the register level, that considers not only temporal reuse, but also the iteration space shape. Moreover, the heuristic is simple enough to be suitable for automatic implementation by compilers.However, to be able to achieve similar performance to hand-optimized codes, it is not enough by tiling only for the register level. With today's architectures having complex memory hierarchies and multiple processors, it is quite common that the compiler has to perform tiling at four or more levels (parallelism, L2-cache, L1-cache and registers) in order to achieve high performance. Therefore, in today's architectures it is crucial to have an efficient algorithm that can perform multilevel tiling at multiple levels of the memory hierarchy. Moreover, as we will see in this thesis, multilevel tiling should always include the register level, as this is the memory hierarchy level that yields most performance when properly tiled.When multilevel tiling includes the register level, it is critical to compute exact loop bounds and to avoid the generation of redundant bounds. The reason is that the complexity and the amount of code generated by our register tiling technique both depend polynomially on the number of loop bounds. However, to date, the drawback of generating exact loop bounds and eliminating redundant bounds has been that all techniques known were extremely expensive in terms of compilation time and, thus, difficult to integrate in a production compiler. Our third contribution in this thesis will be a new implementation of multilevel tiling that computes exact loop bounds at a much lower complexity than traditional techniques. In fact, we will show that the complexity of our implementation is proportional to the complexity of performing a loop permutation in the original loop nest (before tiling), while traditional techniques have much larger complexities. Moreover, our implementation generates less redundant bounds in the multilevel tiled code and allows removing the remaining redundant bounds at a lower cost. Overall, the efficiency of our implementation makes it possible to integrate multilevel tiling including the register level in a production compiler without having to worry about compilation time.The last part of this thesis is dedicated to studying the performance of multilevel tiling. We will discuss the effects of tiling for different memory levels and present quantitative data comparing the benefits of tiling only for the register level, tiling only for the cache level and tiling for both levels simultaneously. Finally, we will compare automatically-optimized codes against hand-optimized vendor-supplied numerical libraries, on two different architectures (ALPHA 21164 and MIPS R10000), to conclude that compiler technology can make it possible for complex numerical codes to achieve the same performance as hand-optimized codes on modern microprocessors.

In this paper we study the hierarchical structure of the 2-d polyominoes. We introduce a new infinite family of polyominoes which we prove tiles a strip. We discuss applications of algebra to tiling. We discuss the algorithmic decidability of tiling the infinite plane Z x Z given a finite set of polyominoes. We will then discuss tiling with rectangles. We will then get some new, and some analogous results concerning the possible hierarchical structure for the 3-d polycubes.

The thesis represents a collection of solved problems concerned with covering planar shapes (mostly rectangles with integer sides) by tiles known as polyominoes (e.g., domi- noes, trominoes, tetrominoes, etc.). In most cases, the goal is to find a tiling or to prove that no such tiling exists. In more difficult problems, the task is to deduce conditions for the rectangle to be tileable by specified polyominoes. The last chapter is devoted to calcu- lating the number of all possible tilings of the specified rectangle.

Liu, Xin. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 61-63). / Abstracts in English and Chinese. / Abstract --- p.1 / Acknowledgments --- p.4 / Chapter 0 --- Introduction --- p.5 / Chapter 1 --- Basics of self-affine tiles --- p.8 / Chapter 1.1 --- Self-affine sets --- p.8 / Chapter 1.2 --- Self-affine tiles --- p.11 / Chapter 1.3 --- Structure of tiling sets --- p.15 / Chapter 1.4 --- Integral self-affine tiles --- p.20 / Chapter 1.4.1 --- Lebesgue measure of integral self-affine tile --- p.22 / Chapter 1.4.2 --- Classification of digit set --- p.24 / Chapter 2 --- Connectedness of self-affine tiles --- p.28 / Chapter 2.1 --- Connectedness --- p.28 / Chapter 2.2 --- Disk-likeness --- p.33 / Chapter 3 --- Tiles with rotations and reflections --- p.39 / Chapter 3.1 --- Tiles --- p.39 / Chapter 3.2 --- Integral tiles --- p.54 / Bibliography --- p.60

I have developed a tiling interpretation of the q-binomial coefficients. The aim of this thesis is to apply this combinatorial interpretation to a variety of q-identities to provide straightforward combinatorial proofs. The range of identities I present include q-multinomial identities, alternating sum identities and congruences.

Recent developments in tiling theory, primarily in the study of anisohedral shapes, have been the product of exhaustive computer searches through various classes of polygons. I present a brief background of tiling theory and past work, with particular emphasis on isohedral numbers, aperiodicity, Heesch numbers, criteria to characterize isohedral tilings, and various details that have arisen in past computer searches. I then develop and implement a new ``boundary-based'' technique, characterizing shapes as a sequence of characters representing unit length steps taken from a finite language of directions, to replace the ``area-based'' approaches of past work, which treated the Euclidean plane as a regular lattice of cells manipulated like a bitmap. The new technique allows me to reproduce and verify past results on polyforms (edge-to-edge assemblies of unit squares, regular hexagons, or equilateral triangles) and then generalize to a new class of shapes dubbed polysnakes, which past approaches could not describe. My implementation enumerates polyforms using Redelmeier's recursive generation algorithm, and enumerates polysnakes using a novel approach. The shapes produced by the enumeration are subjected to tests to either determine their isohedral number or prove they are non-tiling. My results include the description of this novel approach to testing tiling properties, a correction to previous descriptions of the criteria for characterizing isohedral tilings, the verification of some previous results on polyforms, and the discovery of two new 4-anisohedral polysnakes.

Recent developments in tiling theory, primarily in the study of anisohedral shapes, have been the product of exhaustive computer searches through various classes of polygons. I present a brief background of tiling theory and past work, with particular emphasis on isohedral numbers, aperiodicity, Heesch numbers, criteria to characterize isohedral tilings, and various details that have arisen in past computer searches. I then develop and implement a new ``boundary-based'' technique, characterizing shapes as a sequence of characters representing unit length steps taken from a finite language of directions, to replace the ``area-based'' approaches of past work, which treated the Euclidean plane as a regular lattice of cells manipulated like a bitmap. The new technique allows me to reproduce and verify past results on polyforms (edge-to-edge assemblies of unit squares, regular hexagons, or equilateral triangles) and then generalize to a new class of shapes dubbed polysnakes, which past approaches could not describe. My implementation enumerates polyforms using Redelmeier's recursive generation algorithm, and enumerates polysnakes using a novel approach. The shapes produced by the enumeration are subjected to tests to either determine their isohedral number or prove they are non-tiling. My results include the description of this novel approach to testing tiling properties, a correction to previous descriptions of the criteria for characterizing isohedral tilings, the verification of some previous results on polyforms, and the discovery of two new 4-anisohedral polysnakes.

Thesis (Ph. D.)--University of Washington, 1990. / Vita. Includes bibliographical references (leaves [126]-128).

Thesis (Ph. D.)--Oregon State University, 2009. / Printout. Includes bibliographical references (leaves 88-89). Also available on the World Wide Web.