Segui questo link per vedere altri tipi di pubblicazioni sul tema: Tiling (mathematics).
Tesi sul tema "Tiling (mathematics)"
Cita una fonte nei formati APA, MLA, Chicago, Harvard e in molti altri stili
Vedi i top-50 saggi (tesi di laurea o di dottorato) per l'attività di ricerca sul tema "Tiling (mathematics)".
Accanto a ogni fonte nell'elenco di riferimenti c'è un pulsante "Aggiungi alla bibliografia". Premilo e genereremo automaticamente la citazione bibliografica dell'opera scelta nello stile citazionale di cui hai bisogno: APA, MLA, Harvard, Chicago, Vancouver ecc.
Puoi anche scaricare il testo completo della pubblicazione scientifica nel formato .pdf e leggere online l'abstract (il sommario) dell'opera se è presente nei metadati.
Vedi le tesi di molte aree scientifiche e compila una bibliografia corretta.
1
Gupta, David (David Gregg) 1969. "Some tiling moves explored." Thesis, Massachusetts Institute of Technology, 1998. http://hdl.handle.net/1721.1/50012.
Testo completo
2
Pachter, Lior 1973. "Domino tiling, gene recognition and mice." Thesis, Massachusetts Institute of Technology, 1999. http://hdl.handle.net/1721.1/85308.
Testo completo
3
Persons, Michael Joseph. "Methods for creating corner colored Wang tiles." Pullman, Wash. : Washington State University, 2010. http://www.dissertations.wsu.edu/Thesis/Spring2010/m_persons_041210.pdf.
Testo completoAbstract (sommario):
Thesis (M.S. in computer science)--Washington State University, May 2010.<br>Title from PDF title page (viewed on May 18, 2010). "School of Engineering and Computer Science." Includes bibliographical references (p. 104-106).
4
Hillman, Chris. "Sturmian dynamical systems /." Thesis, Connect to this title online; UW restricted, 1998. http://hdl.handle.net/1773/5806.
Testo completo
5
Son, Younghwan. "Some results on joint ergodicity, sets of recurrence and substitution and tiling systems." The Ohio State University, 2013. http://rave.ohiolink.edu/etdc/view?acc_num=osu1369046202.
Testo completo
6
Yu, Xiaojiang Gabardo Jean-Pierre. "Wavelet sets, integral self-affine tiles and nonuniform multiresolution analyses." *McMaster only, 2005.
Cerca il testo completo
7
Nielsen, Mark J. "Tilings of topological vector spaces /." Thesis, Connect to this title online; UW restricted, 1990. http://hdl.handle.net/1773/5763.
Testo completo
8
Savinien, Jean P. X. "Cohomology and K-theory of aperiodic tilings." Diss., Atlanta, Ga. : Georgia Institute of Technology, 2008. http://hdl.handle.net/1853/24732.
Testo completoAbstract (sommario):
Thesis (Ph.D.)--Mathematics, Georgia Institute of Technology, 2008.<br>Committee Chair: Prof. Jean Bellissard; Committee Member: Prof. Claude Schochet; Committee Member: Prof. Michael Loss; Committee Member: Prof. Stavros Garoufalidis; Committee Member: Prof. Thang Le.
9
Praggastis, Brenda L. "Markov partitions for hyperbolic toral automorphisms /." Thesis, Connect to this title online; UW restricted, 1992. http://hdl.handle.net/1773/5773.
Testo completo
10
Laperriere, Christiane. "Positive cocycles for minimal Zd-actions on a cantor set resulting from cut and project schemes: The octogonal tiling." Thesis, University of Ottawa (Canada), 2009. http://hdl.handle.net/10393/28269.
Testo completoAbstract (sommario):
We study the cut and projection method, which is a way to construct tilings. This construction leads to a minimal Zd -action on the Cantor set. In this thesis, we will focus our attention on two examples that we will describe in full details. the Fibonacci tiling on R and the octogonal tiling on R2 . For the octogonal tiling, we find small strictly positive cocycles for the minimal action on specific cones of Z2 .
11
Karpenko, Daria. "Self-assembly of Self-similar Structures by Active Tiles." Scholar Commons, 2012. http://scholarcommons.usf.edu/etd/4095.
Testo completoAbstract (sommario):
The natural capacity of DNA for molecular self-assembly has already been exploited to create DNA based tiles which can self-assemble into nano-scale arrays and carry out nano-scale computation. Thus far, however, all such self-assembly has been passive, in the sense that the binding capacities of a tile are never altered throughout the assembly. The idea of active tiles, tiles that can send signals to each other and activate latent binding sites, has been proposed but never incorporated into a formal model. Here, I present an extension of the existent abstract tile assembly model by defining an active tile assembly and give a detailed example of an aperiodic set of active tiles which hierarchically produces a self-similar L-shape tiling. This yields a technique utilizing active tiles for the assembly of aperiodic self-similar shapes.
12
Lightcap, Andrew. "Minimum Degree Conditions for Tilings in Graphs and Hypergraphs." Digital Archive @ GSU, 2011. http://digitalarchive.gsu.edu/math_theses/111.
Testo completoAbstract (sommario):
We consider tiling problems for graphs and hypergraphs. For two graphs and , an -tiling of is a subgraph of consisting of only vertex disjoint copies of . By using the absorbing method, we give a short proof that in a balanced tripartite graph , if every vertex is adjacent to of the vertices in each of the other vertex partitions, then has a -tiling. Previously, Magyar and Martin [11] proved the same result (without ) by using the Regularity Lemma.
In a 3-uniform hypergraph , let denote the minimum number of edges that contain for all pairs of vertices. We show that if , there exists a -tiling that misses at most vertices of . On the other hand, we show that there exist hypergraphs such that and does not have a perfect -tiling. These extend the results of Pikhurko [12] on -tilings.
13
Dietert, Grant. "An integer linear programming formulation for tiling large rectangles using 4 x 6 and 5 x 7 tiles /." Online version of thesis, 2010. http://hdl.handle.net/1850/12275.
Testo completo
14
Colle, Cleber Fernando 1985. "Propriedades dinâmicas e ergódicas de shifts multidimensionais." [s.n.], 2011. http://repositorio.unicamp.br/jspui/handle/REPOSIP/307026.
Testo completoAbstract (sommario):
Orientador: Eduardo Garibaldi<br>Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica<br>Made available in DSpace on 2018-08-19T00:20:51Z (GMT). No. of bitstreams: 1
Colle_CleberFernando_M.pdf: 1068657 bytes, checksum: 78c9700800b05194ffcf66838581b081 (MD5)
Previous issue date: 2011<br>Resumo: Focaremos sobre aspectos dinâmicos e ergódicos de shifts multidimensionais, atentando especialmente para suas relações com estados fundamentais e quase-cristais em reticulados. Por exemplo, em mecânica estatística, dado um potencial invariante por translação, seus estados fundamentais são medidas de probabilidade invariantes por translação suportadas no conjunto de suas configurações fundamentais, isto é, das configurações com energia específica mínima. Estados fundamentais são naturalmente associados com o bordo de certos polítopos convexos dimensionalmente finitos. Esse bordo se torna drasticamente diferente se a dimensão do modelo em questão passa de d = 1 para d > 1, pois no caso multidimensional existe shift de tipo finito unicamente ergódico sem configurações periódicas<br>Abstract: We will focus on dynamic and ergodic aspects of multidimensional shifts, with particular care to their relations with ground states and quasicrystals in lattices. For example, in statistical mechanics, given a translation-invariant potential, its ground states are translation-invariant probability measures supported on the set of its ground configurations, i.e., of configurations with minimal specific energy. Ground states are naturally associated with the boundary of certain finite-dimensional convex polytopes. This boundary becomes drastically different if the dimension of the model in question changes from d = 1 to d > 1, because in the multidimensional case there exists uniquely ergodic shift of finite type with no periodic configurations<br>Mestrado<br>Matematica<br>Mestre em Matemática
15
Leskow, Lucila Helena Allan 1972. "Tesselações hiperbólicas aplicadas a codificação de geodésicas e códigos de fonte." [s.n.], 2011. http://repositorio.unicamp.br/jspui/handle/REPOSIP/261081.
Testo completoAbstract (sommario):
Orientador: Reginaldo Palazzo Junior<br>Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Elétrica e de Computação<br>Made available in DSpace on 2018-08-18T16:51:18Z (GMT). No. of bitstreams: 1
Leskow_LucilaHelenaAllan_D.pdf: 2583405 bytes, checksum: 3161d9deabaa60a8965a9e3d20ff36aa (MD5)
Previous issue date: 2011<br>Resumo: Neste trabalho apresentamos como contribuição um novo conjunto de tesselações do plano hiperbólico construídas a partir de uma tesselação bem conhecida, a tesselação de Farey. Nestas tesselações a região de Dirichlet é formada por polígonos hiperbólicos de n lados, com n > 3. Explorando as características dessas tesselações, apresentamos alguns tipos possíveis de aplicações. Inicialmente, estudando a relação existente entre a teoria das frações contínuas e a tesselação de Farey, propomos um novo método de codificação de geodésicas. A inovação deste método está no fato de ser possível realizar a codificação de uma geodésica pertencente a PSL(2,Z) em qualquer uma das tesselações ou seja, para qualquer valor de n com n > 3. Neste método mostramos como é possível associar as sequências cortantes de uma geodésica em cada tesselação à decomposição em frações contínuas do ponto atrator desta. Ainda explorando as características dessas novas tesselações, propomos dois tipos de aplicação em teoria de codificação de fontes discretas. Desenvolvendo dois novos códigos para compactação de fontes (um código de árvore e um código de bloco), estes dois métodos podem ser vistos como a generalização dos métodos de Elias e Tunstall para o caso hiperbólico<br>Abstract: In this work we present as contribution a new set of tessellations of the hyperbolic plane, built from a well known tessellation, the Farey tessellation. In this set of tessellations the Dirichlet region is made of hyperbolic polygons with n sides where n > 3. While studying these tessellations and theirs properties, we found some possible applications. In the first one, while exploring the relationship between the continued fractions theory and the Farey tessellation we propose a new method for coding geodesics. Using this method, it is possible to obtain a relationship between the cutting sequence of a geodesic belonging to PSL(2,Z) in each tessellation and the continued fraction decomposition of its attractor point. Exploring the characteristics of these tessellations we also propose two types of applications regarding the discrete memoryless source coding theory, a fixed-to-variable code and a variable length-to-fixed code. These methods can be seen as a generalized version of the Elias and Tunstall methods for the hyperbolic case<br>Doutorado<br>Telecomunicações e Telemática<br>Doutor em Engenharia Elétrica
16
Bérard, Caroline. "Modèles à variables latentes pour des données issues de tiling arrays. Applications aux expériences de ChIP-chip et de transcriptome." Phd thesis, AgroParisTech, 2011. http://tel.archives-ouvertes.fr/tel-00656841.
Testo completoAbstract (sommario):
Les puces tiling arrays sont des puces à haute densité permettant l'exploration des génomes à grande échelle. Elles sont impliquées dans l'étude de l'expression des gènes et de la détection de nouveaux transcrits grâce aux expériences de transcriptome, ainsi que dans l'étude des mécanismes de régulation de l'expression des gènes grâce aux expériences de ChIP-chip. Dans l'objectif d'analyser des données de ChIP-chip et de transcriptome, nous proposons une modélisation fondée sur les modèles à variables latentes, en particulier les modèles de Markov cachés, qui sont des méthodes usuelles de classification non-supervisée. Les caractéristiques biologiques du signal issu des puces tiling arrays telles que la dépendance spatiale des observations le long du génome et l'annotation structurale sont intégrées dans la modélisation. D'autre part, les modèles sont adaptés en fonction de la question biologique et une modélisation est proposée pour chaque type d'expériences. Nous proposons un mélange de régressions pour la comparaison de deux échantillons dont l'un peut être considéré comme un échantillon de référence (ChIP-chip), ainsi qu'un modèle gaussien bidimensionnel avec des contraintes sur la matrice de variance lorsque les deux échantillons jouent des rôles symétriques (transcriptome). Enfin, une modélisation semi-paramétrique autorisant des distributions plus flexibles pour la loi d'émission est envisagée. Dans un objectif de classification, nous proposons un contrôle de faux-positifs dans le cas d'une classification à deux groupes et pour des observations indépendantes. Puis, nous nous intéressons à la classification d'un ensemble d'observations constituant une région d'intérêt, telle que les gènes. Les différents modèles sont illustrés sur des jeux de données réelles de ChIP-chip et de transcriptome issus d'une puce NimbleGen couvrant le génome entier d'Arabidopsis thaliana.
17
Michel, Jean-Luc. "Polysimplices in euclidean spaces and the enumeration of domino tilings of rectangles." Doctoral thesis, Universite Libre de Bruxelles, 2011. http://hdl.handle.net/2013/ULB-DIPOT:oai:dipot.ulb.ac.be:2013/209922.
Testo completoAbstract (sommario):
Nous étudions, dans la première partie de notre thèse, les polysimplexes d’un espace euclidien de dimension quelconque, c’est-à-dire les objets consistant en une juxtaposition de simplexes réguliers (de tétraèdres si la dimension est 3) accolés le long de leurs faces. Nous étudions principalement le groupe des symétries de ces polysimplexes. Nous présentons une façon de représenter un polysimplexe à l’aide d’un diagramme. Ceci fournit une classification complète des polysimplexes à similitude près. De plus, le groupe des symétries se déduit du groupe des automorphismes du diagramme. Il découle en particulier de notre étude qu’en dimension supérieure à 2, une telle structure ne possède jamais deux faces parallèles et ne contient jamais de circuit fermé de simplexes.<p><p>Dans la seconde partie de notre thèse, nous abordons un problème classique de combinatoire :l’énumération des pavages d’un rectangle mxn à l’aide de dominos. Klarner et Pollack ont montré qu’en fixant m la suite obtenue vérifie une relation de récurrence linéaire à coefficients constants. Nous établissons une nouvelle méthode nous permettant d’obtenir la fonction génératrice correspondante et la calculons pour m <= 16, alors qu’elle n’était connue que pour m <= 10.<p><br>Doctorat en Sciences<br>info:eu-repo/semantics/nonPublished
18
Bush, Albert. "Two Problems on Bipartite Graphs." Digital Archive @ GSU, 2009. http://digitalarchive.gsu.edu/math_theses/72.
Testo completoAbstract (sommario):
Erdos proved the well-known result that every graph has a spanning, bipartite subgraph such that every vertex has degree at least half of its original degree. Bollobas and Scott conjectured that one can get a slightly weaker result if we require the subgraph to be not only spanning and bipartite, but also balanced. We prove this conjecture for graphs of maximum degree 3.
The majority of the paper however, will focus on graph tiling. Graph tiling (or sometimes referred to as graph packing) is where, given a graph H, we find a spanning subgraph of some larger graph G that consists entirely of disjoint copies of H. With the Regularity Lemma and the Blow-up Lemma as our main tools, we prove an asymptotic minimum degree condition for an arbitrary bipartite graph G to be tiled by another arbitrary bipartite graph H. This proves a conjecture of Zhao and also implies an asymptotic version of a result of Kuhn and Osthus for bipartite graphs.
19
Stucky, Eric. "An Exposition of Kasteleyn's Solution of the Dimer Model." Scholarship @ Claremont, 2015. http://scholarship.claremont.edu/hmc_theses/89.
Testo completoAbstract (sommario):
In 1961, P. W. Kasteleyn provided a baffling-looking solution to an apparently simple tiling problem: how many ways are there to tile a rectangular region with dominos? We examine his proof, simplifying and clarifying it into this nearly self-contained work.
20
Karpenko, Daria. "Active Tile Self-assembly and Simulations of Computational Systems." Scholar Commons, 2015. https://scholarcommons.usf.edu/etd/5519.
Testo completoAbstract (sommario):
Algorithmic self-assembly has been an active area of research at the intersection of computer science, chemistry, and mathematics for almost two decades now, motivated by the natural self-assembly mechanism found in DNA and driven by the desire for precise control of nanoscale material manufacture and for the development of nanocomputing and nanorobotics. At the theoretical core of this research is the Abstract Tile Assembly Model (aTAM), the original abstract model of DNA tile self-assembly. Recent advancements in DNA nanotechnology have been made in developing strand displacement mechanisms that could allow DNA tiles to modify themselves during the assembly process by opening or closing certain binding sites, introducing new dynamics into tile self-assembly.
We focus on one way of incorporating such signaling mechanisms for binding site activation and deactivation into the theoretical model of tile self-assembly by extending the aTAM to create the Active aTAM. We give appropriate definitions first for incorporating activation signals and then for incorporating deactivation signals and tile detachment into the aTAM. We then give a comparison of Active aTAM to related models, such as the STAM, and take a look at some theoretical results.
The goal of the work presented here is to define and demonstrate the power of the Active aTAM with and without deactivation. To this end, we provide four constructions of temperature 1 (also known as "non-cooperative") active tile assembly systems that can simulate other computational systems. The first construction concerns the simulation of an arbitrary temperature 2 (also known as "cooperative") standard aTAM system in the sense of producing equivalent structures with a scaling factor of 2 in each dimension; the second construction generates the time history of a given 1D cellular automaton. The third and fourth constructions make use of tile detachment in order to dynamically simulate arbitrary 1D and 2D cellular automata with assemblies that record only the current state updates and not the entire computational history of the specified automaton.
21
Alves, Alessandro Ferreira. "Análise dos emparelhamentos de arestas de polígonos hiperbólicos para a construção de constelações de sinais geometricamente uniformes." [s.n.], 2011. http://repositorio.unicamp.br/jspui/handle/REPOSIP/261080.
Testo completoAbstract (sommario):
Orientador: Reginaldo Palazzo Junior<br>Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Elétrica e de Computação<br>Made available in DSpace on 2018-08-19T09:31:01Z (GMT). No. of bitstreams: 1
Alves_AlessandroFerreira_D.pdf: 1080224 bytes, checksum: 0748952c3176e9548151bec7e6d9c71d (MD5)
Previous issue date: 2011<br>Resumo: Para projetarmos um sistema de comunicação digital em espaços hiperbólicos é necessário estabelecer um procedimento sistemático de construção de reticulados como elemento base para a construção de constelações de sinais. De outra forma, em codificação de canal é de fundamental importância a caracterização das estruturas algébrica e geométrica associadas a canais discretos sem memória. Neste trabalho, apresentamos a caracterização geométrica de superfícies a partir dos possíveis emparelhamentos das arestas do polígono fundamental hiperbólico com 3 ? n ? 8 lados associado 'a superfície. Esse tratamento geométrico apresenta propriedades importantes na determinação dos reticulados hiperbólicos a serem utilizados no processo de construção de constelações de sinais, a partir de grupos fuchsianos aritméticos e da superfície de Riemann associada. Além disso, apresentamos como exemplo o desenvolvimento algébrico para a determinação dos geradores do grupo fuchsiano 'gama'8 associado ao polígono hiperbólico 'P IND. 8'<br>Abstract: In order to design a digital communication system in hyperbolic spaces is necessary to establish a systematic procedure of constructing lattices as the basic element for the construction of the signal constellations. On the other hand, in channel coding is of fundamental importance to characterize the geometric and algebraic structures associated with discrete memoryless channels. In this work, we present a geometric characterization of surfaces from the edges of the possible pairings of fundamental hyperbolic polygon with 3 ? n ? 8 sides associated with the surface. This treatment has geometric properties important in determining the hyperbolic lattices to be used in the construction of sets of signals derived from arithmetic Fuchsian groups and the associated Riemann surface<br>Doutorado<br>Telecomunicações e Telemática<br>Doutor em Engenharia Elétrica
22
Bennett, Robert. "Fibonomial Tilings and Other Up-Down Tilings." Scholarship @ Claremont, 2016. https://scholarship.claremont.edu/hmc_theses/84.
Testo completoAbstract (sommario):
The Fibonomial coefficients are a generalization of the binomial coefficients with a rather nice combinatorial interpretation. While the ordinary binomial coefficients count lattice paths in a grid, the Fibonomial coefficients count the number of ways to draw a lattice path in a grid and then Fibonacci-tile the regions above and below the path in a particular way. We may forgo a literal tiling interpretation and, instead of the Fibonacci numbers, use an arbitrary function to count the number of ways to "tile" the regions of the grid delineated by the lattice path. When the function is a combinatorial sequence such as the Lucas numbers or the q-numbers, the total number of "tilings" is some multiple of a generalized binomial coefficient corresponding to the sequence chosen.
23
Spreafico, Elen Viviani Pereira 1986. "Novas identidades envolvendo os números de Fibonacci, Lucas e Jacobsthal via ladrilhamentos." [s.n.], 2014. http://repositorio.unicamp.br/jspui/handle/REPOSIP/307509.
Testo completoAbstract (sommario):
Orientador: José Plínio de Oliveira Santos<br>Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica<br>Made available in DSpace on 2018-08-26T02:14:38Z (GMT). No. of bitstreams: 1
Spreafico_ElenVivianiPereira_D.pdf: 1192138 bytes, checksum: 2b12cd351b94a0f2f7ec24fc172305c9 (MD5)
Previous issue date: 2014<br>Resumo: Neste trabalho, colaboramos com provas combinatórias que utilizam a contagem e a q-contagem de elementos em conjuntos de ladrilhamentos com restrições. Na primeira parte do trabalho utilizamos os ladrilhamentos para demonstrar algumas identidades da teoria das partições, dentre elas, o Teorema dos Números Triangulares e o Teorema q-análogo da Série q-Binomial. Na segunda parte do trabalho apresentamos interpretações combinatórias, via ladrilhamento, para algumas identidades envolvendo os números de Jacobsthal e os números generalizados de Jacobsthal . Na terceira parte do trabalho são dadas novas identidades envolvendo os números q-análogos de Jacobsthal e encontramos generalizações para essas novas identidades. Por fim, definimos duas novas sequências: números de Fibonacci generalizados e números de Lucas generalizados e, utilizando ladrilhamentos, estabelecemos e demonstramos novas identidades envolvendo esses números<br>Abstract: In this work we present combinatorial proofs by making use of tilings. In the first part we use tilings to prove some identities on Partitions Theory, including Triangular Numbers' Theorem and q-analogue of q-Binomial Theorem. In the second part we present combinatorial interpretations, using tilings, for some identities involving Jacobsthal numbers and generalized Jacobsthal numbers. Next we find new identities involving an q-analogue of Jacobsthal numbers and generalizations for these new identities. Finally, we define two new sequences: generalized Fibonacci numbers and generalized Lucas numbers, and using tilings, we prove new identities involving these numbers<br>Doutorado<br>Matematica Aplicada<br>Doutora em Matemática Aplicada
24
Nieser, Matthias [Verfasser]. "Parameterization and tiling of polyhedral surfaces / Matthias Nieser." Berlin : Freie Universität Berlin, 2012. http://d-nb.info/1027308481/34.
Testo completo
25
Lenngren, Nils. "k-uniform tilings by regular polygons." Thesis, Uppsala universitet, Algebra, geometri och logik, 2009. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-159395.
Testo completo
26
Korn, Michael Robert 1978. "Geometric and algebraic properties of polyomino tilings." Thesis, Massachusetts Institute of Technology, 2004. http://hdl.handle.net/1721.1/16628.
Testo completoAbstract (sommario):
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2004.<br>Includes bibliographical references (p. 165-167).<br>This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.<br>In this thesis we study tilings of regions on the square grid by polyominoes. A polyomino is any connected shape formed from a union of grid cells, and a tiling of a region is a collection of polyominoes lying in the region such that each square is covered exactly once. In particular, we focus on two main themes: local connectivity and tile invariants. Given a set of tiles T and a finite set L of local replacement moves, we say that a region [Delta] has local connectivity with respect to T and L if it is possible to convert any tiling of [Delta] into any other by means of these moves. If R is a set of regions (such as the set of all simply connected regions), then we say there is a local move property for T and R if there exists a finite set of moves L such that every r in R has local connectivity with respect to T and L. We use height function techniques to prove local move properties for several new tile sets. In addition, we provide explicit counterexamples to show the absence of a local move property for a number of tile sets where local move properties were conjectured to hold. We also provide several new results concerning tile invariants. If we let ai(t) denote the number of occurrences of the tile ti in a tiling t of a region [Delta], then a tile invariant is a linear combination of the ai's whose value depends only on t and not on r.<br>(cont.) We modify the boundary-word technique of Conway and Lagarias to prove tile invariants for several new sets of tiles and provide specific examples to show that the invariants we obtain are the best possible. In addition, we prove some new enumerative results, relating certain tiling problems to Baxter permutations, the Tutte polynomial, and alternating-sign matrices.<br>by Michael Robert Korn.<br>Ph.D.
27
Knizel, Alisa. "Random tilings : gap probabilities, local and global asymptotics." Thesis, Massachusetts Institute of Technology, 2017. http://hdl.handle.net/1721.1/112900.
Testo completoAbstract (sommario):
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2017.<br>Cataloged from PDF version of thesis.<br>Includes bibliographical references (pages 119-125).<br>In the thesis we explore and develop two different approaches to the study of random tiling models. First, we consider tilings of a hexagon by rombi, viewed as 3D random stepped surfaces with a measure proportional to q-volume. Such model is closely related to q-Hahn orthogonal polynomial ensembles, and we use this connection to obtain results about the local behavior of this model. In terms of the q-Hahn orthogonal polynomial ensemble, our goal is to show that the one-interval gap probability function can be expressed through a solution of the asymmetric q-Painleve V equation. The case of the q-Hahn ensemble we consider is the most general case of the orthogonal polynomial ensembles that have been studied in this context. Our approach is based on the analysis of q-connections on P1 with a particular singularity structure. It requires a new derivation of a q-difference equation of Sakai's hierarchy [75] of type A(1)/2. We also calculate its Lax pair. Following [7], we introduce the notion of the [tau]-function of a q-connection and its isomonodromy transformations. We show that the gap probability function of the q-Hahn ensemble can be viewed as the [tau]-function for an associated q-connection and its isomonodromy transformations. Second, in collaboration with Alexey Bufetov we consider asymptotics of a domino tiling model on a class of domains which we call rectangular Aztec diamonds. We prove the Law of Large Numbers for the corresponding height functions and provide explicit formulas for the limit. For a special class of examples, an explicit parametrization of the frozen boundary is given. It turns out to be an algebraic curve with very special properties. Moreover, we establish the convergence of the fluctuations of the height functions to the Gaussian Free Field in appropriate coordinates. Our main tool is a recently developed moment method for discrete particle systems.<br>by Alisa Knizel.<br>Ph. D.
28
Linnell, Joshua Randall. "Multilattice Tilings and Coverings." BYU ScholarsArchive, 2021. https://scholarsarchive.byu.edu/etd/8911.
Testo completoAbstract (sommario):
Let L be a discrete subgroup of \mathbb{R}^n under addition. Let D be a finite set of points including the origin. These two sets will define a multilattice of \mathbb{R}^n. We explore how to generate a periodic covering of the space \mathbb{R}^n based on L and $D$. Additionally, we explore the problem of covering when we restrict ourselves to covering \mathbb{R}^n using only dilations of the right regular simplex in our covering. We show that using a set D= {0,d} to define our multilattice the minimum covering density is 5-\sqrt{13}. Furthermore, we show that when we allow for an arbitrary number of displacements, we may get arbitrarily close to a covering density of 1.
29
Bugarin, Enrico Paolo [Verfasser]. "On quotient cohomology of substitution tiling spaces / Enrico Paolo Bugarin." Bielefeld : Universitätsbibliothek Bielefeld, 2014. http://d-nb.info/1053467478/34.
Testo completo
30
Moustafa, Haïja. "Gap-labeling des pavages de type pinwheel." Phd thesis, Université Blaise Pascal - Clermont-Ferrand II, 2009. http://tel.archives-ouvertes.fr/tel-00509886.
Testo completoAbstract (sommario):
Dans cette thèse, nous montrons que le groupe de K-théorie $K_0$ de la $C^*$-algèbre associée aux pavages de type pinwheel est isomorphe à la somme de $\ZZ \oplus \ZZ^6$ et d'un groupe cohomologique $H$.\\ Cette $C^*$-algèbre est de plus munie d'une trace qui induit une application linéaire sur ce groupe de $K$-théorie.\\ Nous calculons explicitement l'image, sous cette application, du sommant $\ZZ \oplus \ZZ^6$, montrant que l'image de $\ZZ$ est nulle et que l'image de $\ZZ^6$ est contenue dans le module de fréquences des patchs du pavage de type pinwheel.\\ Nous montrons également que l'on peut appliquer le théorème de l'indice mesuré dû à A. Connes pour relier l'image de $H$ à une formule cohomologique plus calculable.\\ Pour l'étude de cette partie cohomologique, nous adaptons la cohomologie PV, introduite par J. Savinien et J. Bellissard, au cas des pavages de type pinwheel pour montrer que le groupe de cohomologie de \v{C}ech de dimension maximale de ces pavages est isomorphe au groupe des coinvariants entiers de la transversale canonique associée à ces pavages.\\ Ce résultat nous permet alors de prouver la conjecture du gap-labeling fait par J. Bellissard, dans le cas particulier des pavages de type pinwheel.\\ Nous terminons cette étude par un calcul explicite, montrant que le gap-labeling (ou module de fréquences des patchs) est donné par $\frac{1}{264}\ZZ \left [ \frac{1}{5} \right ]$.
31
Mohammed, Soud. "Tilting and Relative Theories in Subcategories." Doctoral thesis, Norwegian University of Science and Technology, Faculty of Information Technology, Mathematics and Electrical Engineering, 2008. http://urn.kb.se/resolve?urn=urn:nbn:no:ntnu:diva-1825.
Testo completoAbstract (sommario):
<p>We show that, over an artin algebra, the tilting functor preserves (co)tilting modules in the subcategories associated to the functor. We also give a sufficient condition for the category of modules of finite projective dimension over an artin algebra to be contravariantly finite in the category of all finitely generated modules over the artin algebra. This is a sufficient condition for the finitistic dimension of the artin algebra to be finite [3].</p><p>We also develop relative theory and in certain subcategories of the module category over an artin algebra in the sense of [10,11]. We use the theory to generalize the main result of [26]</p>
32
Ricke, Charlotte [Verfasser]. "On tau-Tilting Theory and Perpendicular Categories / Charlotte Ricke." Bonn : Universitäts- und Landesbibliothek Bonn, 2016. http://d-nb.info/1122193823/34.
Testo completo
33
Manimala, Binoy James. "Mathematical modelling for the evaluation of a tiltwing aircraft." Thesis, Glasgow Caledonian University, 1999. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.287856.
Testo completo
34
Ahmed, Bako. "Limit Shapes for qVolume Tilings of a Large Hexagon." Thesis, KTH, Matematik (Avd.), 2020. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-280758.
Testo completoAbstract (sommario):
Lozenges are polygons constructed by gluing two equilateral triangles along an edge. We can fit lozenge pieces together to form larger polygons and given an appropriate polygon we can tile it with lozenges. Lozenge tilings of the semi-regular hexagon with sides A,B,C can be viewed as the 2D picture of a stack of cubes in a A x B x C box. In this project we investigate the typical shape of a tiling as the sides A,B,C of the box grow uniformly to infinity and we consider two cases: The uniform case where all tilings occur with equal probability and the q^Volume case where the probability of a tiling is proportional to the volume taken up by the corresponding stack of cubes. To investigate lozenge tilings we transform it into a question on families of non-intersecting paths on a corresponding graph representing the hexagon. Using the Lindström–Gessel–Viennot theorem we can define the probability of a non-intersecting path crossing a particular point in the hexagon both for the uniform and the $q$-Volume case. In each case this probability function is connected to either the Hahn or the $q$-Hahn orthogonal polynomials. The orthogonal polynomials depend on the sides of the hexagon and so we consider the asymptotic behaviour of the polynomials as the sides grow to infinity using a result due to Kuijlaars and Van Assche. This determines the density of non-intersecting paths through every point in the hexagon, which we calculate, and a ``Arctic curve" result which shows that the six corners of the hexagon are (with probability one) tiled with just one type of lozenge.<br>"Lozenger" är polygoner konstruerade genom att limma två liksidiga trianglar längs en kant. Vi kan montera lozengstycken ihop för att bilda större polygoner och med en lämplig polygon kan vi lozengplatta den. Lozengplattor av den semi-liksidiga hexagonen med sidorna A, B, C kan ses som 2D-bilden av en stapel kuber i en A x B x C-box. I det här projektet undersöker vi den typiska formen på en platta när sidorna A, B, C på rutan växer till oändlighet och vi tar an två fall: Det likformiga fallet där alla plattor sker med samma sannolikhet och q ^ Volymfallet då sannolikheten för en platta är proportionell mot volymen som tas upp av motsvarande kubstapel. För att undersöka plattor förvandlar vi det till en fråga om samlingar av icke-korsande vägar på en motsvarande graf som representerar hexagonen. Med hjälp av satsen Lindström – Gessel – Viennot kan vi definiera sannolikheten för att en icke-korsande väg går genom en viss punkt i hexagonen både för det enhetliga och $ q $ -volymfallet. I båda fallen är dessa sannolikhetsfunktioner relaterade till Hahn eller $ q $ -Hahn ortogonala polynomer. Dessa ortogonala polynom beror på hexagonens sidor så vi betraktar polynomens asymptotiska beteende när sidorna växer till oändlighet genom ett resultat från Kuijlaars och Van Assche. Detta bestämmer densiteten för de icke-korsande vägarna genom varje punkt i det hexagon vi beräknar. Detta bestämmer också också en '' arktisk kurva '' som visar att hexagonens sex hörn är (med sannolikhet ett) plattade med bara en typ av lozeng.
35
Razafindrazaka, Faniry H. [Verfasser]. "Quad Layout Generation and Symmetric Tilings of Closed Surfaces / Faniry H. Razafindrazaka." Berlin : Freie Universität Berlin, 2016. http://d-nb.info/1084634538/34.
Testo completo
36
Nordenstam, Eric. "Interlaced particles in tilings and random matrices." Doctoral thesis, Stockholm : Matematik, Kungliga Tekniska högskolan, 2009. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-9834.
Testo completo
37
Marks, Frederik [Verfasser], and Steffen [Akademischer Betreuer] König. "Interactions between universal localisations, ring epimorphisms and tilting modules / Frederik Marks. Betreuer: Steffen König." Stuttgart : Universitätsbibliothek der Universität Stuttgart, 2015. http://d-nb.info/1068810939/34.
Testo completo
38
Jakobi, Tobias [Verfasser], Michael [Akademischer Betreuer] Baake, and Friedrich [Akademischer Betreuer] Götze. "Radial projection statistics: a different angle on tilings / Tobias Jakobi ; Michael Baake, Friedrich Götze." Bielefeld : Universitätsbibliothek Bielefeld, 2017. http://d-nb.info/1137624329/34.
Testo completo
39
Kolbe, Benedikt Maximilian [Verfasser], Myfanwy [Akademischer Betreuer] Evans, John [Akademischer Betreuer] Sullivan, Myfanwy [Gutachter] Evans, John [Gutachter] Sullivan, and Jean-Marc [Gutachter] Schlenker. "Structures in three-dimensional Euclidean space from hyperbolic tilings / Benedikt Maximilian Kolbe ; Gutachter: Myfanwy Evans, John Sullivan, Jean-Marc Schlenker ; Myfanwy Evans, John Sullivan." Berlin : Technische Universität Berlin, 2020. http://d-nb.info/1217326049/34.
Testo completo
40
Whitaker, erica j. "Congruence and Noncongruence Subgroups of Γ(2) via Graphs on Surfaces". The Ohio State University, 2011. http://rave.ohiolink.edu/etdc/view?acc_num=osu1321205804.
Testo completo
41
Hlushchanka, Mikhail [Verfasser], Dierk [Akademischer Betreuer] [Gutachter] Schleicher, Keivan [Gutachter] Mallahi-Karai, Mario [Gutachter] Bonk, Volodymyr [Gutachter] Nekrashevych, Kevin M. [Gutachter] Pilgrim, Dylan [Gutachter] Thurston, and Daniel [Gutachter] Meyer. "Invariant graphs, tilings, and iterated monodromy groups / Mikhail Hlushchanka ; Gutachter: Dierk Schleicher, Keivan Mallahi-Karai, Mario Bonk, Volodymyr Nekrashevych, Kevin M. Pilgrim, Dylan Thurston, Daniel Meyer ; Betreuer: Dierk Schleicher." Bremen : IRC-Library, Information Resource Center der Jacobs University Bremen, 2017. http://d-nb.info/1141379759/34.
Testo completo
42
Thapper, Johan. "Combinatorial Considerations on Two Models from Statistical Mechanics." Licentiate thesis, Linköping : Matematiska institutionen, Linköpings universitet, 2007. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-10141.
Testo completo
43
Julien, Antoine. "Complexité des pavages apériodiques : calculs et interprétations." Phd thesis, Université Claude Bernard - Lyon I, 2009. http://tel.archives-ouvertes.fr/tel-00466323.
Testo completoAbstract (sommario):
La théorie des pavages apériodiques a connu des développements rapides depuis les années 1980, avec la découvertes d'alliages métalliques cristallisant dans une structure quasi-périodique.Dans cette thèse, on étudie particulièrement deux méthodes de construction de pavages : par coupe et projection, et par substitution. Deux angles d'approche sont développés : l'étude de la fonction de complexité, et l'étude métrique de l'espace de pavages.Dans une première partie, on calcule l'asymptotique de la fonction de complexité pour des pavages coupe et projection, généralisant ainsi des résultats connus en dynamiques symbolique pour la dimension 1. On montre que pour un pavage coupe et projection canonique N sur d sans période, la complexité croît (à des constantes près) comme n à la puissance a, où a est un entier compris entre d et N-d.Ensuite, on se base sur une construction de Pearson et Bellissard qui construisent un triplet spectral sur les ensembles de Cantor ultramétriques. On suit leur construction dans le cas d'ensembles de Cantor auto-similaires. Elle s'applique en particulier aux transversales d'espaces de pavages de substitution.Enfin, on fait le lien entre la distance usuelle sur l'enveloppe d'un pavage et la complexité de ce pavage. Les liens entre complexité et métrique permettent de donner une preuve directe du fait suivant : la complexité des pavages de substitution apériodiques de dimension d croît comme n à la puissance d.La question de liens entre la complexité et la topologie (et pas seulement avec la distance) reste ouverte. Nous apportons cependant des réponses partielles dans cette direction.
44
Front, Mathias. "Émergence et évolution des objets mathématiques en Situation Didactique de Recherche de Problème : le cas des pavages archimédiens du plan." Thesis, Lyon 1, 2015. http://www.theses.fr/2015LYO10250/document.
Testo completoAbstract (sommario):
Étudier l'émergence de savoirs lors de situations didactiques non finalisées par un savoir préfabriqué et pré-pensé nécessite un bouleversement des points de vue, aussi bien épistémologique que didactique. C'est pourquoi, pour l'étude de situations didactiques pour lesquelles le problème est l'essence, nous développons une nouvelle approche historique et repensons des outils pour les analyses didactiques. Nous proposons alors, pour un problème particulier, l'exploration des pavages archimédiens du plan, une enquête historique centrée sur l'activité du savant cherchant et sur l'influence de la relation aux objets dans la recherche. De ce point de vue, l'étude des travaux de Johannes Kepler à la recherche d'une harmonie du monde est particulièrement instructive. Nous proposons également, pour l'analyse des savoirs émergents en situation didactique, une utilisation d'outils liés à la sémiotique qui permet de mettre en évidence la dynamique de l'évolution des objets mathématiques. Nous pouvons finalement conclure quant à la possibilité de construire et mettre en œuvre des ≪ Situations Didactiques de Recherche de Problème ≫ assurant l'engagement du sujet dans la recherche, l'émergence et le développement d'objets mathématiques, la genèse de savoirs. L'étude nous conforte dans la nécessité d'une approche pragmatique des situations et la pertinence d'un regard différent sur les savoirs à l'école<br>The study of the emergence of knowledges in teaching situations not finalized by a prefabricated and pre-thought knowledge requires an upheaval of point of view, epistemological as well as didactic. For the study of learning situations in which the problem is the essence, we develop a new historical approach and we rethink the tools for didactic analyzes. We propose, then, for a particular problem, exploration of Archimedean tilings of the plane, a historical inquiry centered on the activity of the scientist in the process of research and on the influence of the relationship with objects. From this perspective, the study of Johannes Kepler’s work in search of a world harmony is particularly instructive. We also propose, for the analysis of the emerging knowledge in teaching situations, to use tools related to semiotics, which allows to highlight the dynamic of evolution of mathematical objects. We can finally conclude on the opportunity to build and implement “Didactic Situations of Problem Solving”, which ensure the commitment of the subject in the research, the emergence and development of mathematical objects, the genesis of knowledges. The study reinforces the necessity of a pragmatic approach of situations and the relevance of a different look at the knowledge at school
45
Zhang, Tianfang. "Machine learning multicriteria optimization in radiation therapy treatment planning." Thesis, KTH, Matematisk statistik, 2019. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-257509.
Testo completoAbstract (sommario):
In radiation therapy treatment planning, recent works have used machine learning based on historically delivered plans to automate the process of producing clinically acceptable plans. Compared to traditional approaches such as repeated weighted-sum optimization or multicriteria optimization (MCO), automated planning methods have, in general, the benefits of low computational times and minimal user interaction, but on the other hand lack the flexibility associated with general-purpose frameworks such as MCO. Machine learning approaches can be especially sensitive to deviations in their dose prediction due to certain properties of the optimization functions usually used for dose mimicking and, moreover, suffer from the fact that there exists no general causality between prediction accuracy and optimized plan quality.In this thesis, we present a means of unifying ideas from machine learning planning methods with the well-established MCO framework. More precisely, given prior knowledge in the form of either a previously optimized plan or a set of historically delivered clinical plans, we are able to automatically generate Pareto optimal plans spanning a dose region corresponding to plans which are achievable as well as clinically acceptable. For the former case, this is achieved by introducing dose--volume constraints; for the latter case, this is achieved by fitting a weighted-data Gaussian mixture model on pre-defined dose statistics using the expectation--maximization algorithm, modifying it with exponential tilting and using specially developed optimization functions to take into account prediction uncertainties.Numerical results for conceptual demonstration are obtained for a prostate cancer case with treatment delivered by a volumetric-modulated arc therapy technique, where it is shown that the methods developed in the thesis are successful in automatically generating Pareto optimal plans of satisfactory quality and diversity, while excluding clinically irrelevant dose regions. For the case of using historical plans as prior knowledge, the computational times are significantly shorter than those typical of conventional MCO.<br>Inom strålterapiplanering har den senaste forskningen använt maskininlärning baserat på historiskt levererade planer för att automatisera den process i vilken kliniskt acceptabla planer produceras. Jämfört med traditionella angreppssätt, såsom upprepad optimering av en viktad målfunktion eller flermålsoptimering (MCO), har automatiska planeringsmetoder generellt sett fördelarna av lägre beräkningstider och minimal användarinteraktion, men saknar däremot flexibiliteten hos allmänna ramverk som exempelvis MCO. Maskininlärningsmetoder kan vara speciellt känsliga för avvikelser i dosprediktionssteget på grund av särskilda egenskaper hos de optimeringsfunktioner som vanligtvis används för att återskapa dosfördelningar, och lider dessutom av problemet att det inte finns något allmängiltigt orsakssamband mellan prediktionsnoggrannhet och kvalitet hos optimerad plan. I detta arbete presenterar vi ett sätt att förena idéer från maskininlärningsbaserade planeringsmetoder med det väletablerade MCO-ramverket. Mer precist kan vi, givet förkunskaper i form av antingen en tidigare optimerad plan eller en uppsättning av historiskt levererade kliniska planer, automatiskt generera Paretooptimala planer som täcker en dosregion motsvarande uppnåeliga såväl som kliniskt acceptabla planer. I det förra fallet görs detta genom att introducera dos--volym-bivillkor; i det senare fallet görs detta genom att anpassa en gaussisk blandningsmodell med viktade data med förväntning--maximering-algoritmen, modifiera den med exponentiell lutning och sedan använda speciellt utvecklade optimeringsfunktioner för att ta hänsyn till prediktionsosäkerheter.Numeriska resultat för konceptuell demonstration erhålls för ett fall av prostatacancer varvid behandlingen levererades med volymetriskt modulerad bågterapi, där det visas att metoderna utvecklade i detta arbete är framgångsrika i att automatiskt generera Paretooptimala planer med tillfredsställande kvalitet och variation medan kliniskt irrelevanta dosregioner utesluts. I fallet då historiska planer används som förkunskap är beräkningstiderna markant kortare än för konventionell MCO.
46
Benali, Abdelkader. "Comportement dynamique des butées hydrodynamiques." Poitiers, 1988. http://www.theses.fr/1988POIT2011.
Testo completo
47
Funke, Frank. "Steuerung der Beinmotorik von Grillen durch ein Paar gravizeptiver Interneurone." Doctoral thesis, [S.l.] : [s.n.], 2004. http://webdoc.sub.gwdg.de/diss/2004/funke/funke.pdf.
Testo completo
48
"Tiling of the integers." 2008. http://library.cuhk.edu.hk/record=b5893703.
Testo completoAbstract (sommario):
Leung, Fung Bun.<br>Thesis (M.Phil.)--Chinese University of Hong Kong, 2008.<br>Includes bibliographical references (leaves 36).<br>Abstracts in English and Chinese.<br>Abstract --- p.i<br>Acknowledgement --- p.iii<br>Chapter 1 --- Introduction --- p.1<br>Chapter 2 --- Tiling of Integers --- p.3<br>Chapter 2.1 --- General Assumptions --- p.3<br>Chapter 2.2 --- Periodicity of Tiling --- p.4<br>Chapter 2.3 --- Cyclotomic Polynomials --- p.5<br>Chapter 2.4 --- Equivalence --- p.10<br>Chapter 3 --- Theorems of Integer Tiles --- p.12<br>Chapter 3.1 --- Main Theorems --- p.12<br>Chapter 3.2 --- Jusifications on Assumptions --- p.12<br>Chapter 3.3 --- Proofs of Main Theorems --- p.14<br>Chapter 3.3.1 --- Proof of Theorem 3.1.1 --- p.14<br>Chapter 3.3.2 --- Proof of Theorem 3.1.2 --- p.15<br>Chapter 3.3.3 --- Proof of Theorem 3.1.3 --- p.15<br>Chapter 4 --- Structure of Integer Tiles --- p.20<br>Chapter 4.1 --- Classification --- p.20<br>Chapter 4.2 --- Structure of Tiles with Cardinality pα --- p.22<br>Chapter 4.3 --- Structure of Tiles with Cardinality pαqβ --- p.23<br>Chapter 4.4 --- More Examples --- p.25<br>Chapter 5 --- Complementing Pairs Mod pqr --- p.26<br>Chapter 5.1 --- Introduction --- p.26<br>Chapter 5.2 --- A Conjecture and Proofs --- p.27<br>Chapter 5.3 --- A Summary of Our Results --- p.29<br>Chapter A --- Number Systems --- p.30<br>Chapter A.l --- Definitions --- p.30<br>Chapter A.2 --- Structure of tiles of Z+ and Nn --- p.31<br>Chapter B --- Notations --- p.35
49
"Self-affine tilings, digit sets classification, and related problems." 2004. http://library.cuhk.edu.hk/record=b5892153.
Testo completoAbstract (sommario):
Cheung Wai-Man.<br>Thesis (M.Phil.)--Chinese University of Hong Kong, 2004.<br>Includes bibliographical references (leaves 62-64).<br>Abstracts in English and Chinese.<br>Abstract --- p.ii<br>Acknowledgement --- p.iv<br>Chapter 1 --- Introduction --- p.1<br>Chapter 2 --- Fundamental Results of Self-Affine Tiles --- p.11<br>Chapter 2.1 --- Construction of self-affine tiles --- p.11<br>Chapter 2.2 --- The requirements of self-affine tiles --- p.14<br>Chapter 2.3 --- Tiling sets of self-affine tiles --- p.22<br>Chapter 3 --- Digit sets of Self-Affine Tiles --- p.42<br>Chapter 3.1 --- Integral self-affine tiles --- p.42<br>Chapter 3.2 --- Classification of digit sets --- p.49<br>Chapter 3.3 --- Weak product-form digit set --- p.52<br>Chapter 3.4 --- Remarks --- p.55<br>Chapter 4 --- Discussion --- p.57<br>Chapter 4.1 --- Concluding remarks --- p.57<br>Chapter 4.2 --- Conjectures --- p.58<br>Bibliography --- p.64
50
Starling, Charles Benjimen. "Computation of the Ruelle-Sullivan map for substitution tilings." 2005. http://hdl.handle.net/1828/778.
Testo completo