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Dissertations / Theses on the topic 'Simplicial complexes and polytopes'

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Published: 25 May 2024

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1

Cartier, Noémie. "Lattice properties of acyclic pipe dreams." Electronic Thesis or Diss., université Paris-Saclay, 2023. http://www.theses.fr/2023UPASG065.

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Cette thèse s'inscrit dans le domaine de la combinatoire algébrique. Certains algorithmes de tri peuvent être décrits par des diagrammes appelés réseaux de tri, et l'exécution de ces algorithmes sur des permutations se traduit alors par des arrangements de courbes sur ces réseaux. Ces arrangements donnent des modèles pour des structures combinatoires classiques : par exemple, le treillis de Tamari, dont les relations de couverture sont les rotations sur les arbres binaires, et qui est un quotient bien connu de l'ordre faible sur les permutations. Les complexes de sous-mots généralisent les rés
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2

Jonsson, Jakob. "Simplicial Complexes of Graphs." Doctoral thesis, Stockholm, 2005. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-202.

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3

Jonsson, Jakob. "Simplicial complexes of graphs /." Berlin [u.a.] : Springer, 2008. http://dx.doi.org/10.1007/978-3-540-75858-7.

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4

Mirmohades, Djalal. "Simplicial Structure on Complexes." Licentiate thesis, Uppsala universitet, Algebra och geometri, 2014. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-221410.

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5

Zhang, Zhihan. "Random walk on simplicial complexes." Electronic Thesis or Diss., université Paris-Saclay, 2020. http://www.theses.fr/2020UPASM010.

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La notion de laplacien d’un graphe peut être généralisée aux complexes simpliciaux et aux hypergraphes. Cette notion contient des informations sur la topologie de ces structures. Dans la première partie de cette thèse,nous définissons une nouvelle chaîne de Markov sur les complexes simpliciaux. Pour un degré donné k de simplexes, l’espace d’états n’est pas les k-simplexes comme dans les articles précédents sur ce sujet mais plutôt l’ensemble des k-chaines ou k-co-chaines. Ce nouveau cadre est la généralisation naturelle sur les chaînes de Markov canoniques sur des graphes. Nous montrons que le
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6

Zuffi, Lorenzo. "Simplicial Complexes From Graphs Toward Graph Persistence." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2017. http://amslaurea.unibo.it/13519/.

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Persistent homology is a branch of computational topology which uses geometry and topology for shape description and analysis. This dissertation is an introductory study to link persistent homology and graph theory, the connection being represented by various methods to build simplicial complexes from a graph. The methods we consider are the complex of cliques, of independent sets, of neighbours, of enclaveless sets and complexes from acyclic subgraphs, each revealing several properties of the underlying graph. Moreover, we apply the core ideas of persistence theory in the new context of graph
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7

Petersson, Anna. "Enumeration of spanning trees in simplicial complexes." Licentiate thesis, Uppsala universitet, Matematiska institutionen, 2009. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-138976.

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8

Egan, Sarah. "Nash equilibria in games and simplicial complexes." Thesis, University of Bath, 2008. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.500758.

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Nash's Theorem is a famous and widely used result in non-cooperative game theory which can be applied to games where each player's mixed strategy payoff function is defined as an expectation. Current proofs of this Theorem neither justify why this constraint is necessary or satisfactorily identifies its origins. In this Thesis we change this and prove Nash's Theorem for abstract games where, in particular, the payoff functions can be replaced by total orders. The result of this is a combinatoric proof of Nash's Theorem. We also construct a generalised simplicial complex model and demonstrate a
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9

Hetyei, Gábor. "Simplicial and cubical complexes : anologies and differences." Thesis, Massachusetts Institute of Technology, 1994. http://hdl.handle.net/1721.1/32610.

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10

Perkins, Simon. "Field D* pathfinding in weighted simplicial complexes." Doctoral thesis, University of Cape Town, 2013. http://hdl.handle.net/11427/6433.

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Includes abstract.<br>Includes bibliographical references.<br>The development of algorithms to efficiently determine an optimal path through a complex environment is a continuing area of research within Computer Science. When such environments can be represented as a graph, established graph search algorithms, such as Dijkstra’s shortest path and A*, can be used. However, many environments are constructed from a set of regions that do not conform to a discrete graph. The Weighted Region Problem was proposed to address the problem of finding the shortest path through a set of such regions, weig
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11

Newman, J. Andrew. "Torsion in Homology of Random Simplicial Complexes." The Ohio State University, 2018. http://rave.ohiolink.edu/etdc/view?acc_num=osu1531499208297615.

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12

Kahle, Matthew. "Topology of random simplicial complexes and phase transitions for homology /." Thesis, Connect to this title online; UW restricted, 2007. http://hdl.handle.net/1773/5809.

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13

Zagrodny, Christopher Michael. "Algebraic Concepts in the Study of Graphs and Simplicial Complexes." Digital Archive @ GSU, 2006. http://digitalarchive.gsu.edu/math_theses/7.

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This paper presents a survey of concepts in commutative algebra that have applications to topology and graph theory. The primary algebraic focus will be on Stanley-Reisner rings, classes of polynomial rings that can describe simplicial complexes. Stanley-Reisner rings are defined via square-free monomial ideals. The paper will present many aspects of the theory of these ideals and discuss how they relate to important constructions in commutative algebra, such as finite generation of ideals, graded rings and modules, localization and associated primes, primary decomposition of ideals and Hilber
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14

Zhu, Xueyun. "Vlist and Ering: compact data structures for simplicial 2-complexes." Thesis, Georgia Institute of Technology, 2013. http://hdl.handle.net/1853/50389.

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Various data structures have been proposed for representing the connectivity of manifold triangle meshes. For example, the Extended Corner Table (ECT) stores V+6T references, where V and T respectively denote the vertex and triangle counts. ECT supports Random Access and Traversal (RAT) operators at Constant Amortized Time (CAT) cost. We propose two novel variations of ECT that also support RAT operations at CAT cost, but can be used to represent and process Simplicial 2-Complexes (S2Cs), which may represent star-connecting, non-orientable, and non-manifold triangulations along with dangling
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15

Abramchuk, Yauheniya [Verfasser], and Volker [Gutachter] Kaibel. "Undominated complexes of cut polytopes / Yauheniya Abramchuk ; Gutachter: Volker Kaibel." Magdeburg : Universitätsbibliothek Otto-von-Guericke-Universität, 2018. http://d-nb.info/121996543X/34.

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16

Muhammad, Abubakr. "Graphs, Simplicial Complexes and Beyond: Topological Tools for Multi-agent Coordination." Diss., Available online, Georgia Institute of Technology, 2005, 2005. http://etd.gatech.edu/theses/available/etd-11152005-171405/.

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Thesis (Ph. D.)--Electrical and Computer Engineering, Georgia Institute of Technology, 2006.<br>Symington, Margaret, Committee Member ; Howard, Ayanna, Committee Member ; Tannenbaum, Allen, Committee Member ; Verriest, Erik, Committee Member ; Egerstedt, Magnus, Committee Chair. Vita. Includes bibliographical references.
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17

Weigel, Christian Jens [Verfasser]. "Gated chamber complexes, simplicial arrangements and Coxeter groups / Christian Jens Weigel." Gießen : Universitätsbibliothek, 2015. http://d-nb.info/1075144884/34.

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18

Rauße, Christian [Verfasser], and Christoph [Akademischer Betreuer] Böhm. "Simplicial complexes of compact homogeneous spaces / Christian Rauße ; Betreuer: Christoph Böhm." Münster : Universitäts- und Landesbibliothek Münster, 2017. http://d-nb.info/1142528421/34.

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19

Chebbi, Yassin. "Laplacien discret d'un 2-complexe simplicial." Thesis, Nantes, 2018. http://www.theses.fr/2018NANT4028/document.

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20

Salve, Dias Fabio Augusto. "A study of some morphological operators in simplicial complex spaces." Phd thesis, Université Paris-Est, 2012. http://pastel.archives-ouvertes.fr/pastel-00824751.

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In this work we study the framework of mathematical morphology on simplicial complex spaces. Simplicial complexes are a versatile and widely used structure to represent multidimensional data, such as meshes, that are tridimensional complexes, or graphs, that can be interpreted as bidimensional complexes. Mathematical morphology is one of the most powerful frameworks for image processing, including the processing of digital structures, and is heavily used for many applications. However, mathematical morphology operators on simplicial complex spaces is not a concept fully developped in the liter
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21

Adams-Florou, Spiros. "Homeomorphisms, homotopy equivalences and chain complexes." Thesis, University of Edinburgh, 2012. http://hdl.handle.net/1842/6250.

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This thesis concerns the relationship between bounded and controlled topology and in particular how these can be used to recognise which homotopy equivalences of reasonable topological spaces are homotopic to homeomorphisms. Let f : X → Y be a simplicial map of finite-dimensional locally finite simplicial complexes. Our first result is that f has contractible point inverses if and only if it is an ε- controlled homotopy equivalences for all ε > 0, if and only if f × id : X × R → Y × R is a homotopy equivalence bounded over the open cone O(Y +) of Pedersen and Weibel. The most difficult part, t
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22

Du, Dong. "Contributions to Persistence Theory." The Ohio State University, 2012. http://rave.ohiolink.edu/etdc/view?acc_num=osu1338304358.

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23

Ellis, Robert B. "A Kruskal-Katona theorem for cubical complexes." Thesis, Virginia Tech, 1996. http://hdl.handle.net/10919/45075.

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<p>The optimal number of faces in cubical complexes which lie in cubes refers to the maximum number of faces that can be constructed from a certain number of faces of lower dimension, or the minimum number of faces necessary to construct a certain number of faces of higher dimension. If <i>m</i> is the number of faces of <i>r</i> in a cubical complex, and if s > r(s < r), then the maximum(minimum) number of faces of dimension s that the complex can have is m<sub>(s/r)</sub> +. (m-m<sub>(r/r)</sub>)<sup>(s/r)</sup>, in terms of upper and lower semipowers. The corresponding formula for simpli
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24

Nisse, Mounir. "Sur la géométrie et la topologie des amibes et coamibes des variétés algébriques complexes." Paris 6, 2010. http://www.theses.fr/2010PA066131.

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Un des nouveaux domaine de mathématiques pures appelé “géométrie tropicale” a vu un développement spectaculaire au cours de ces derniéres années. En géométrie énumérative, récemment Gregory Mikhalkin a donné une interprétation des invariants de Gromov-Witten en termes de géométrie tropicale en comptant des chemins entiers dans des polytopes entiers (Théorème de correspondance de Mikhalkin \cite{M2-04}). En utilisant des outils analogues, Andreas Gathmann et Hannah Markwig redécouvrent la formule de Caporaso-Harris pour les courbes complexes planes ainsi que les formules de Kontsevich pour les
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25

Zeckner, Matthew. "TOPOLOGICAL AND COMBINATORIAL PROPERTIES OF NEIGHBORHOOD AND CHESSBOARD COMPLEXES." UKnowledge, 2011. http://uknowledge.uky.edu/gradschool_diss/163.

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This dissertation examines the topological properties of simplicial complexes that arise from two distinct combinatorial objects. In 2003, A. Björner and M. de Longueville proved that the neighborhood complex of the stable Kneser graph SGn,k is homotopy equivalent to a k-sphere. Further, for n = 2 they showed that the neighborhood complex deformation retracts to a subcomplex isomorphic to the associahedron. They went on to ask whether or not, for all n and k, the neighborhood complex of SGn,k contains as a deformation retract the boundary complex of a simplicial polytope. Part one of this diss
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26

Tambour, Jérôme. "Complexes moment-angle et variétés complexes." Phd thesis, Université de Bourgogne, 2010. http://tel.archives-ouvertes.fr/tel-00648247.

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Le but de cette thèse est d'étendre les résultats de l'article [B-M] sur les relations entre variétés moment-angle et variétés complexes. On s'intéressera ici aux variétés moment-angle issues d'une décomposition simpliciale (et non simplement polytopale) de la sphère. On cherchera ensuite à utiliser la relation entre ces deux types d'objets pour comprendre la topologie de certaines variétés complexes.[B-M] F.Bosio, L.Meersseman, Real quadrics in Cn, complex manifolds and polytopes, Acta Mathematica, 197 (2006), n° 1, 53 -- 127.
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27

Knöppel, Felix Jakob [Verfasser], Ulrich [Akademischer Betreuer] Pinkall, Ulrich [Gutachter] Pinkall, Boris [Gutachter] Springborn, and Johannes [Gutachter] Wallner. "Complex line bundles over simplicial complexes / Felix Jakob Knöppel ; Gutachter: Ulrich Pinkall, Boris Springborn, Johannes Wallner ; Betreuer: Ulrich Pinkall." Berlin : Technische Universität Berlin, 2016. http://d-nb.info/1156013682/34.

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28

Kowalick, Ryan. "Discrete Systolic Inequalities." The Ohio State University, 2013. http://rave.ohiolink.edu/etdc/view?acc_num=osu1384873457.

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29

Maria, Clément. "Algorithmes et structures de données en topologie algorithmique." Thesis, Nice, 2014. http://www.theses.fr/2014NICE4081/document.

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La théorie de l'homologie généralise en dimensions supérieures la notion de connectivité dans les graphes. Étant donné un domaine, décrit par un complexe simplicial, elle définit une famille de groupes qui capturent le nombre de composantes connexes, le nombre de trous, le nombre de cavités et le nombre de motifs équivalents en dimensions supérieures. En pratique, l'homologie permet d'analyser des systèmes de données complexes, interprétés comme des nuages de points dans des espaces métriques. La théorie de l'homologie persistante introduit une notion robuste d'homologie pour l'inférence topol
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30

McDonald, Terry Lynn. "Piecewise polynomial functions on a planar region: boundary constraints and polyhedral subdivisions." Texas A&M University, 2003. http://hdl.handle.net/1969.1/3915.

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Splines are piecewise polynomial functions of a given order of smoothness r on a triangulated region (or polyhedrally subdivided region) of Rd. The set of splines of degree at most k forms a vector space Crk() Moreover, a nice way to study Cr k()is to embed n Rd+1, and form the cone b of with the origin. It turns out that the set of splines on b is a graded module Cr b() over the polynomial ring R[x1; : : : ; xd+1], and the dimension of Cr k() is the dimension o This dissertation follows the works of Billera and Rose, as well as Schenck and Stillman, who each approached the study of splines fr
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31

Criado, Gallart Francisco [Verfasser], Michael [Akademischer Betreuer] Joswig, Leal Francisco [Akademischer Betreuer] Santos, Michael [Gutachter] Joswig, Günter [Gutachter] Rote, and Leal Francisco [Gutachter] Santos. "Tropical bisectors and diameters of simplicial complexes / Francisco Criado Gallart ; Gutachter: Michael Joswig, Günter Rote, Francisco Santos Leal ; Michael Joswig, Francisco Santos Leal." Berlin : Technische Universität Berlin, 2021. http://d-nb.info/1232319600/34.

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32

Ault, Shaun V. "On the Symmetric Homology of Algebras." The Ohio State University, 2008. http://rave.ohiolink.edu/etdc/view?acc_num=osu1218237992.

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33

Guinard, Stéphane. "Reconstruction et généralisation de complexes simpliciaux à partir de scans lidar de scènes urbaines." Thesis, Paris Est, 2020. http://www.theses.fr/2020PESC2012.

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Grâce à leur résolution et à leur accessibilité toujours meilleures, les capteurs LiDAR sont de plus en plus utilisés pour cartographier les villes. En effet, ces capteurs sont capables de réaliser efficacement des acquisitions à haut résolution, qui peuvent ensuite être utilisées pour produire des reconstructions géométriquement détaillées de scènes complexes. Cependant, une telle reconstruction nécessite d’organiser les données avec une structure de données adaptée, comme des nuages de points ou des maillages. Les nuages de points fournissent une représentation compacte des données, mais leu
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34

Bigo, Louis. "Représentations symboliques musicales et calcul spatial." Thesis, Paris Est, 2013. http://www.theses.fr/2013PEST1074/document.

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Représentations symboliques musicales et calcul spatial. La notion d'espace symbolique est fréquemment utilisée en théorie, analyse et composition musicale. La représentation de séquences dans des espaces de hauteurs, comme le Tonnetz, permet de capturer des propriétés mélodiques et harmoniques qui échappent aux systèmes de représentation traditionnels. Nous généralisons cette approche en reformulant d'un point de vue spatial différents problèmes musicaux (reconnaissance de style, transformations mélodiques et harmoniques, classification des séries tous-intervalles, etc.). Les espaces sont for
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35

Bettiol, Enrico. "Column generation methods for quadratic mixed binary programming." Thesis, Paris 13, 2019. http://www.theses.fr/2019PA131073.

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La programmation non linéaire mixte peut modéliser un grand nombre de problèmes réels. Cependant, ces problèmes peuvent contenir de nombreuses variables ou contraintes, il convient donc de proposer des méthodes de décomposition afin de les résoudre efficacement. Parmi ces techniques on peut citer la génération de colonnes et notamment la décomposition de Dantzig-Wolfe. Il s’agit d’une reformulation du problème original, qui permet de générer une séquence de sous-problèmes plus simples, appelés maître etpricing, pour obtenir la valeur optimale. Développée d’abord pour les problèmes linéaires, l
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36

Adimoolam, Santosh Arvind. "A Calculus of Complex Zonotopes for Invariance and Stability Verification of Hybrid Systems." Thesis, Université Grenoble Alpes (ComUE), 2018. http://www.theses.fr/2018GREAM027/document.

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Le calcul des ensembles atteignables est une approche de facto utilisée dans de nombreuses méthodes de vérification formelles pour les systèmes hybrides. Mais le calcul exact de l'ensemble atteignable est un problème insurmontable pour de nombreux types de systèmes hybrides, soit en raison de l'indécidabilité ou de la complexité de calcul élevée. Alternativement, beaucoup de recherches ont été axées sur l'utilisation de représentations d'ensembles qui peuvent être manipulées efficacement pour calculer une surestimation suffisamment précise de l'ensemble atteignable. Les zonotopes sont une repr
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37

Calvez, Philippe. "Modélisation d'agencements énergétiques durables dans les zones urbaines intelligentes : une approche pour la réduction de l’emprise énergétique par les pratiques soutenables." Thesis, Paris 1, 2015. http://www.theses.fr/2015PA010056.

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D’un côté, la transition écologique et les enjeux de développement durable sont de nos jours une réalité que l’on ne peut ignorer compte tenu des impacts négatifs des activités humaines sur leurs environnements. De l’autre côté, une numérisation toujours plus importante de ces environnements entraîne la génération de volumes massifs de traces numériques, qui sont autant d’indices sur le monde dans lequel vivent les acteurs de ces activités. Une difficulté non négligeable existe pour comprendre les tenants et aboutissants faisant que d’une activité à une autre, l’impact sur l’environnement mesu
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Poncio, Carlos Henrique Felicio. "Versões do teorema de Tverberg e aplicações." Universidade Federal de São Carlos, 2016. https://repositorio.ufscar.br/handle/ufscar/8044.

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Submitted by Livia Mello (liviacmello@yahoo.com.br) on 2016-10-05T14:40:49Z No. of bitstreams: 1 DissCHFP.pdf: 1216039 bytes, checksum: e21e062b0283d2bfe6ec436442e824a5 (MD5)<br>Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-10-20T19:23:38Z (GMT) No. of bitstreams: 1 DissCHFP.pdf: 1216039 bytes, checksum: e21e062b0283d2bfe6ec436442e824a5 (MD5)<br>Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-10-20T19:23:43Z (GMT) No. of bitstreams: 1 DissCHFP.pdf: 1216039 bytes, checksum: e21e062b0283d2bfe6ec436442e824a5 (MD5)<br>Made availab
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Buchet, Mickaël. "Topological inference from measures." Thesis, Paris 11, 2014. http://www.theses.fr/2014PA112367/document.

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La quantité de données disponibles n'a jamais été aussi grande. Se poser les bonnes questions, c'est-à-dire des questions qui soient à la fois pertinentes et dont la réponse est accessible est difficile. L'analyse topologique de données tente de contourner le problème en ne posant pas une question trop précise mais en recherchant une structure sous-jacente aux données. Une telle structure est intéressante en soi mais elle peut également guider le questionnement de l'analyste et le diriger vers des questions pertinentes. Un des outils les plus utilisés dans ce domaine est l'homologie persistant
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Benzeghli, Brahim. "Étude explicite de quelques n-champs géométriques." Phd thesis, Université Nice Sophia Antipolis, 2013. http://tel.archives-ouvertes.fr/tel-00868795.

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Dans [PRID], Pridham a montré que tout n-champs d'Artin M admet une présentation en tant que schéma simplicial X. → M, telle que le schéma simplicial X satisfait à certaines propriétés notées par G.Pn,k de [GROTH]. Dans la présentation (...→ X2 → X1 → X0 → M), le schéma X1 représente une carte pour X0 x MX0. Donc, la lissité de X0 → M est équivalente à la lissité des deux projections ә0,ә1 : X1 → X0. Ce sont les deux premières parties de la condition de Grothendieck-Pridham, notées G.P1,0 et G.P1,1. Dans [BENZ12] nous avons introduit un n-champ d'Artin M des éléments de Maurer-Cartan d'une dg-
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41

Brunink, Jan-Marten. "Subdivisions of simplicial complexes." Doctoral thesis, 2021. https://repositorium.ub.uni-osnabrueck.de/handle/urn:nbn:de:gbv:700-202109145342.

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The topic of this thesis are subdivisions of simplicial complexes, in particular we focus on the so-called antiprism triangulation. In the first main part, the real-rootedness of the h-polynomial of the antiprism triangulation of the simplex is proven. Furthermore, we study combinatorial interpretations of several invariants as the h- and local h-vector. In the second part, we show the almost strong Lefschetz property of the antiprism triangulation for every shellable simplicial complex.
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42

Huntemann, Svenja. "Simplicial Complexes of Placement Games." 2013. http://hdl.handle.net/10222/35472.

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Placement games are a subclass of combinatorial games which are played on graphs. In this thesis, we demonstrate that placement games could be considered as games played on simplicial complexes. These complexes are constructed using square-free monomials. We define new classes of placement games and the notion of Doppelgänger. To aid in exploring the simplicial complex of a game, we introduce the bipartite flip and develop tools to compare known bounds on simplicial complexes (such as the Kruskal-Katona bounds) with bounds on game complexes.
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Couto, Maria Inês Gomes da Rocha. "Measuring Distances Between Paving Simplicial Complexes." Master's thesis, 2018. https://hdl.handle.net/10216/114586.

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44

Akinwande, Grace Itunuoluwa. "Limit Theorems for Random Simplicial Complexes." Doctoral thesis, 2020. https://repositorium.ub.uni-osnabrueck.de/handle/urn:nbn:de:gbv:700-202010223623.

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We consider random simplicial complexes constructed on a Poisson point process within a convex set in a Euclidean space, especially the Vietoris-Rips complex and the Cech complex both of whose 1-skeleton is the Gilbert graph. We investigate at first the Vietoris-Rips complex by considering the volume-power functionals defined by summing powers of the volume of all k-dimensional faces in the complex. The asymptotic behaviour of these functionals is investigated as the intensity of the underlying Poisson point process tends to infinity and the distance parameter goes to zero. This behaviour is o
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Couto, Maria Inês Gomes da Rocha. "Measuring Distances Between Paving Simplicial Complexes." Dissertação, 2018. https://hdl.handle.net/10216/114586.

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46

Steenbergen, John Joseph. "Towards a Spectral Theory for Simplicial Complexes." Diss., 2013. http://hdl.handle.net/10161/8256.

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<p>In this dissertation we study combinatorial Hodge Laplacians on simplicial com-</p><p>plexes using tools generalized from spectral graph theory. Specifically, we consider</p><p>generalizations of graph Cheeger numbers and graph random walks. The results in</p><p>this dissertation can be thought of as the beginnings of a new spectral theory for</p><p>simplicial complexes and a new theory of high-dimensional expansion.</p><p>We first consider new high-dimensional isoperimetric constants. A new Cheeger-</p><p>type inequality is proved, under certain conditions, between an isoperimetric constan
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47

Perkins, Simon James. "Field D* Pathfinding in Weighted Simplicial Complexes." Thesis, 2014. http://pubs.cs.uct.ac.za/archive/00000924/.

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The development of algorithms to efficiently determine an optimal path through a complex environment is a continuing area of research within Computer Science. When such environments can be represented as a graph, established graph search algorithms, such as Dijkstra’s shortest path and A*, can be used. However, many environments are constructed from a set of regions that do not conform to a discrete graph. The Weighted Region Problem was proposed to address the problem of finding the shortest path through a set of such regions, weighted with values representing the cost of traversing the regio
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48

Venturello, Lorenzo. "Combinatorial and algebraic properties of balanced simplicial complexes." Doctoral thesis, 2019. https://repositorium.ub.uni-osnabrueck.de/handle/urn:nbn:de:gbv:700-201911192203.

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Simplicial complexes are mathematical objects whose importance stretches from topology to commutative algebra and combinatorics. In this thesis we focus on the family of balanced simplicial complexes. A d-dimensional simplicial complex is balanced if its 1-skeleton can be properly (d+1)-colored, as in the classical graph theoretic sense. Equivalently, a d-dimensional complex is balanced iff it admits a non-degenerate simplicial projection to the d-simplex. We present results on these complexes from a number of different points of view. After two introductory chapters, we exhibit in chapter 3 a
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Shih, Jen-Chieh, and 施仁傑. "The simplicial complexes and the multiplicity of determinantal rings." Thesis, 2001. http://ndltd.ncl.edu.tw/handle/58212839789821132963.

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碩士<br>國立中正大學<br>數學研究所<br>89<br>Let K be a field and X be a generic m ×n matrix over K. Let R=K[X] and I be the ideal generated by the (r+1) ×(r+1) minors of X. There are several ways to prove the above formula since 1950, however, they are not easy to understand. In this paper, we use the simplicial complexes to simplify our difficulty. We are able to quickly calculate the multiplicity of R.
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50

Crowley, Katherine Dutton. "Discrete Morse theory and the geometry of nonpositively curved simplicial complexes." Thesis, 2001. http://hdl.handle.net/1911/17951.

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Understanding the conditions under which a simplicial complex collapses is a central issue in many problems in topology and combinatorics. Let K be a simplicial complex endowed with the piecewise Euclidean geometry given by declaring edges to have unit length, and satisfying the property that every 2-simplex is a face of at most two 3-simplices in K. Our main theorem is that if |K| is nonpositively curved (in the sense of CAT(0)) then K simplicially collapses to a point. The main tool used in the proof is Forman's discrete Morse theory (see section 2.2), a combinatorial version of the classic
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