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Dissertations / Theses on the topic 'Polytopes'
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1
Dinh, Thi Ngoc. "Ordinary polytopes." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1999. http://www.collectionscanada.ca/obj/s4/f2/dsk1/tape9/PQDD_0026/NQ49553.pdf.
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Wu, Lei. "Random inscribed polytopes." Connect to a 24 p. preview or request complete full text in PDF format. Access restricted to UC campuses, 2006. http://wwwlib.umi.com/cr/ucsd/fullcit?p3210646.
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Thesis (Ph. D.)--University of California, San Diego, 2006.Title from first page of PDF file (viewed June 7, 2006). Available via ProQuest Digital Dissertations. Vita. Includes bibliographical references (p. 60-65).
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Lundman, Anders. "Classifying Lattice Polytopes." Licentiate thesis, KTH, Matematik (Avd.), 2013. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-134707.
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This thesis consists of two papers in toric geometry. In Paper A we provide a complete classification up to isomorphism of all smooth convex lattice 3- polytopes with at most 16 lattice points. There exist in total 103 different polytopes meeting these criteria. Of these, 99 are strict Cayley polytopes and the remaining four are obtained as inverse stellar subdivisions of such polytopes. We derive a classification, up to isomorphism, of all complete embeddings of smooth toric threefolds in PN where N ≤ 15. Again we have in total 103 such embeddings. Of these, 99 are projective bundles embedded in PN and the remaining four are blow-ups of such toric threefolds. In Paper B we show that a complete smooth toric embedding X ↪ PN having maximal k-th osculating dimension, but not maximal (k + 1)-th osculating dimension, at every point is associated to a Cayley polytope of order k. This result generalises an earlier characterisation by David Perkinson. In addition we prove that the above assumptions are equivalent to requiring that the Seshadri constant is exactly k at every point of X, generalising a result of Atsushi Ito.Denna Licentiatuppsats utgörs av två vetenskapliga artiklar inom torisk geometri. I Paper A ger vi en komplett klassificering, upp till isomorfi, av alla 3-dimensionella glatta konvexa gitter polytoper som innehåller högst 16 gitter punkter. Totalt utgörs klassificeringen av 103 olika polytoper. Av dessa 103 polytoper är 99 stycken strikta Cayley polytoper och resterande fyra är inversa stjärnuppdelningar av Cayley polytoper. Från detta resultat härleder vi en klassificering av alla fullständiga inbäddningar av glatta toriska trefalder i PN för N ≤ 15. Återigen får vi 103 sådana inbäddningar. Av dessa är 99 projektiva fiberknippen inbäddade i PN och resterande fyra är uppblåsningar av dito. I Paper B visar vi att en fullstädig glatt torisk imbäddning X ↪ PN som i varje punkt är sådan att, det k:te oskulerande rummet har maximal dimension, men det (k + 1):a oskulerande rummet ej är av maximal dimension, är associerad till en Cayley polytop av grad k. Detta resultat generaliserar en tidigare känd klassificering av David Perkinson. Vidare visar vi att ovanstående antaganden är ekvivalenta med att anta att Seshadrikonstanten är exakt k för varje punkt på X, vilket generaliserar en tidigare klassificering av Atsushi Ito.
QC 20131129
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Solhjem, Sara Louise. "Sign Matrix Polytopes." Diss., North Dakota State University, 2018. https://hdl.handle.net/10365/28767.
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Motivated by the study of polytopes formed as the convex hull of permutation matrices and alternating sign matrices, several new families of polytopes are defined as convex hulls of sign matrices, which are certain {0,1,-1}--matrices in bijection with semistandard Young tableaux. This bijection is refined to include standard Young tableau of certain shapes. One such shape is counted by the Catalan numbers, and the convex hull of these standard Young tableaux form a Catalan polytope. This Catalan polytope is shown to be integrally equivalent to the order polytope of the triangular poset: therefore the Ehrhart polynomial and volume can be combinatorial interpreted. Various properties of all of these polytope families are investigated, including their inequality descriptions, vertices, facets, and face lattices, as well as connections to alternating sign matrix polytopes, and transportation polytopes.
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Dobbins, Michael Gene. "Representations of Polytopes." Diss., Temple University Libraries, 2011. http://cdm16002.contentdm.oclc.org/cdm/ref/collection/p245801coll10/id/141523.
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MathematicsPh.D.
Here we investigate a variety of ways to represent polytopes and related objects. We define a class of posets, which includes all abstract polytopes, giving a unique representative among posets having a particular labeled flag graph and characterize the labeled flag graphs of abstract polytopes. We show that determining the realizability of an abstract polytope is equivalent to solving a low rank matrix completion problem. For any given polytope, we provide a new construction for the known result that there is a combinatorial polytope with a specified ridge that is always projectively equivalent to the given polytope, and we show how this makes a naturally arising subclass of intractable problems tractable. We give necessary and sufficient conditions for realizing a polytope's interval poset, which is the polytopal analog of a poset's Hasse diagram. We then provide a counter example to the general realizablity of a polytope's interval poset.
Temple University--Theses
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Christophe, Jean. "Le polytope des sous-espaces d'un espace affin fini." Doctoral thesis, Universite Libre de Bruxelles, 2006. http://hdl.handle.net/2013/ULB-DIPOT:oai:dipot.ulb.ac.be:2013/210798.
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Le polytope des m-sous-espaces est défini comme l'enveloppe convexe des vecteurs caractéristiques de tous les sous-espaces de dimension m d'un espace affin fini. Le cas particulier du polytope des hyperplans a été étudié par Maurras (1993) et Anglada et Maurras (2003), qui ont obtenu une description complète des facettes. Le polytope général des m-sous-espaces que nous considérons possède une structure plus complexe, notamment concernant les facettes. Néanmoins, nous établissons dans cette thèse plusieurs familles de facettes. Nous caractérisons également complètement le groupe des automorphismes du polytope ainsi que l'adjacence des sommets du polytope des m-sous-espaces. Un tangle est un ensemble d'hyperplans d'un espace affin contenant un hyperplan par classe d'hyperplans parallèles. Anglada et Maurras ont montré que les tangles définissent des facettes du polytope des hyperplans et que toutes les facettes de ce polytope proviennent de tangles. Nous tentons d'établir une généralisation de ce résultat. Nous élaborons une classification des tangles en familles pour de petites dimensions d'espaces affins.Doctorat en sciences, Spécialisation mathématiques
info:eu-repo/semantics/nonPublished
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Schwartz, Alexander. "Constructions of cubical polytopes." [S.l.] : [s.n.], 2004. http://deposit.ddb.de/cgi-bin/dokserv?idn=970075154.
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Finbow, Wendy. "On stability of polytopes." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1998. http://www.collectionscanada.ca/obj/s4/f2/dsk2/tape15/PQDD_0025/MQ36437.pdf.
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Chapoton, Frédéric. "Opérades, polytopes et bigèbres." Paris 6, 2000. http://www.theses.fr/2000PA066098.
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Cette these comporte deux parties independantes sur des themes voisins. La premiere partie est formee par plusieurs extensions successives d'un article de loday et ronco dans lequel est definie une bigebre y des arbres binaires plans. On definit une bigebre filtree sur les arbres plans qui generalise y. On definit ensuite une bigebre differentielle graduee qui est une extension, distincte de la precedente, de y. Enfin, on interprete ces deux constructions en termes d'operades en construisant une operade filtree et une operade differentielle graduee sur les arbres plans. La seconde partie est un ensemble de travaux sur diverses operades. On introduit l'operade perm des digebres commutatives, ce qui nous permet de retrouver les operades leib des algebres de leibniz et dias des digebres. On decrit ensuite l'operade prelie des algebres pre-lie, on montre qu'elle est la duale quadratique de perm et que ces deux operades sont de koszul. Enfin on demontre l'existence d'une equivalence de categories entre les algebres braces (sur l'operade brace) et certaines algebres dendriformes (sur l'operade dend) munies d'un coproduit coassociatif.
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Philippe, Eva. "Geometric realizations using regular subdivisions : construction of many polytopes, sweep polytopes, s-permutahedra." Electronic Thesis or Diss., Sorbonne université, 2024. http://www.theses.fr/2024SORUS079.
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Cette thèse concerne trois problèmes de réalisations géométriques de structures combinatoires par des polytopes et des subdivisions polyédrales. Un polytope est l'enveloppe convexe d'un ensemble fini de points dans un espace euclidien R^d. Il est muni d'une structure combinatoire donnée par ses faces. Une subdivision est une collection de polytopes dont les faces s'intersectent correctement et dont l'union est convexe. Elle est régulière si elle peut être obtenue en prenant les faces inférieures d'un relèvement de ses sommets dans une dimension de plus.Nous présentons d'abord une nouvelle construction géométrique d'un grand nombre de polytopes combinatoirement distincts, de dimension et nombre de sommets fixés. Cette construction consiste à montrer que certains polytopes admettent un grand nombre de triangulations régulières. Elle nous permet d'améliorer la meilleure borne inférieure connue sur le nombre de types combinatoires de polytopes.Nous étudions ensuite les projections du permutoèdre, nommées polytopes de balayage (sweep polytopes) parce qu'elles modélisent les manières d'ordonner une configuration de points fixée en balayant l'espace par des hyperplans dans une direction constante. Nous introduisons également et étudions une abstraction combinatoire de ces structures : les matroïdes orientés de balayage, qui généralisent en dimension supérieure à 2 la théorie des suites admissibles de Goodman et Pollack.Enfin, nous proposons des réalisations géométriques de l'ordre s-faible, une structure combinatoire qui généralise l'ordre faible sur les permutations, paramétrée par un vecteur s à coordonnées entières strictement positives. En particulier, nous résolvons une conjecture de Ceballos et Pons en montrant que le s-permutoèdre peut être réalisé comme le graphe d'un complexe polytopal qui est une subdivision du permutoèdreThis thesis concerns three problems of geometric realizations of combinatorial structures via polytopes and polyhedral subdivisions. A polytope is the convex hull of a finite set of points in a Euclidean space R^d. It is endowed with a combinatorial structure coming from its faces. A subdivision is a collection of polytopes whose faces intersect properly and such that their union is convex. It is regular if it can be obtained by taking the lower faces of a lifting of its vertices in one dimension higher.We first present a new geometric construction of many combinatorially different polytopes of fixed dimension and number of vertices. This construction relies on showing that certain polytopes admit many regular triangulations. It allows us to improve the best known lower bound on the number of combinatorial types of polytopes.We then study the projections of permutahedra, that we call sweep polytopes because they model the possible orderings of a fixed point configuration by hyperplanes that sweep the space in a constant direction. We also introduce and study a combinatorial abstraction of these structures: the sweep oriented matroids, that generalize Goodman and Pollack's theory of allowable sequences to dimensions higher than 2.Finally, we provide geometric realizations of the s-weak order, a combinatorial structure that generalizes the weak order on permutations, parameterized by a vector s with positive integer coordinates. In particular, we answer Ceballos and Pons conjecture that the s-weak order can be realized as the edge-graph of a polytopal complex that is moreover a subdivision of a permutahedron
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Manoussakis, Georgios Oreste. "Algorithmes combinatoires et Optimisation." Thesis, Université Paris-Saclay (ComUE), 2017. http://www.theses.fr/2017SACLS517.
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Nous introduisons d'abord la classe des graphes $k$-dégénérés qui est souvent utilisée pour modéliser des grands graphes épars issus du monde réel. Nous proposons de nouveaux algorithmes d'énumération pour ces graphes. En particulier, nous construisons un algorithme énumérant tous les cycles simples de tailles fixés dans ces graphes, en temps optimal.Nous proposons aussi un algorithme dont la complexité dépend de la taille de la solution pour le problème d'énumération des cliques maximales de ces graphes. Dans un second temps nous considérons les graphes en tant que systèmes distribués et nous nous intéressons à des questions liées à la notion de couplage lorsqu’aucune supposition n’est faite sur l'état initial du système, qui peut donc être correct ou incorrect. Dans ce cadre nous proposons un algorithme retournant une deux tiers approximation du couplage maximum.Nous proposons aussi un algorithme retournant un couplage maximal quand les communications sont restreintes de telle manière à simuler le paradigme du passage de message. Le troisième objet d'étude n'est pas directement lié à l'algorithmique de graphe, bien que quelques techniques classiques de ce domaine soient utilisées pour obtenir certains de nos résultats.Nous introduisons et étudions certaines familles de polytopes, appelées Zonotopes Primitifs, qui peuvent être décrits comme la somme de Minkowski de vecteurs primitifs. Nous prouvons certaines propriétés combinatoires de ces polytopes et illustrons la connexion avec le plus grand diamètre possible de l'enveloppe convexe de points à coordonnées entières à valeurs dans$[k]$, en dimension $d$. Dans un second temps,nous étudions des paramètres de petites instances de Zonotopes Primitifs, tels que leur nombre de sommets, entre autresWe start by studying the class of $k$-degenerate graphs which are often used to model sparse real-world graphs. We focus one numeration questions for these graphs. That is,we try and provide algorithms which must output, without duplication, all the occurrences of some input subgraph. We investigate the questions of finding all cycles of some givensize and all maximal cliques in the graph. Ourtwo contributions are a worst-case output sizeoptimal algorithm for fixed-size cycleenumeration and an output sensitive algorithmfor maximal clique enumeration for this restricted class of graphs. In a second part weconsider graphs in a distributed manner. Weinvestigate questions related to finding matchings of the network, when no assumptionis made on the initial state of the system. Thesealgorithms are often referred to as selfstabilizing.Our two main contributions are analgorithm returning an approximation of themaximum matching and a new algorithm formaximal matching when communication simulates message passing. Finally, weintroduce and investigate some special families of polytopes, namely primitive zonotopes,which can be described as the Minkowski sumof short primitive vectors. We highlight connections with the largest possible diameter ofthe convex hull of a set of points in dimension d whose coordinates are integers between 0 and k.Our main contributions are new lower bounds for this diameter question as well as descriptions of small instances of these objects
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Hooker, Kevin J. "Hypergraphs and integer programming polytopes /." Search for this dissertation online, 2005. http://wwwlib.umi.com/cr/ksu/main.
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Fiorini, Samuel. "Polyhedral combinatorics of order polytopes." Doctoral thesis, Universite Libre de Bruxelles, 2001. http://hdl.handle.net/2013/ULB-DIPOT:oai:dipot.ulb.ac.be:2013/211629.
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Knight, Vincent. "Alternating sign matrices and polytopes." Thesis, Cardiff University, 2009. http://orca.cf.ac.uk/54880/.
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This thesis deals with two types of mathematical objects: alternating sign matrices and polytopes. Alternating sign matrices were first defined in 1982 by Mills, Robbins and Rumsey. Since then, alternating sign matrices have led to some very captivating research (with multiple open problems still standing), an outline of which is presented in the opening chapter of this thesis. Convex polytopes are extremely relevant when considering enumerations of certain classes of integer valued matrices. An overview of the relevant properties of convex polytopes is presented, before a connection is made between polytopes and alternating sign matrices: the alternating sign matrix polytope. The vertex set of this new polytope is given, as well as a generalization of standard alternating sign matrices to give higher spin alternating sign matrices. From a result of Ehrhart a result concerning the enumeration of these matrices is obtained, namely, that for fixed size and variable line sum the enumeration is given by a particular polynomial. In Chapter 4, we give results concerning the symmetry classes of the alternating sign matrix polytope and in Chapter 3 we study symmetry classes of the Birkhoff polytope. For this classical polytope we give some new results. In the penultimate chapter, another polytope is defined that is a valid solution set of the transportation problem and for which a particular set of parameters gives the alternating sign matrix polytope. Importantly the transportation polytope is a subset of this new polytope.
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Showers, Patrick J. "Abstract Polytopes from Nested Posets." University of Akron / OhioLINK, 2013. http://rave.ohiolink.edu/etdc/view?acc_num=akron1386028871.
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Meyer, Marie. "Polytopes Associated to Graph Laplacians." UKnowledge, 2018. https://uknowledge.uky.edu/math_etds/54.
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Graphs provide interesting ways to generate families of lattice polytopes. In particular, one can use matrices encoding the information of a finite graph to define vertices of a polytope. This dissertation initiates the study of the Laplacian simplex, PG, obtained from a finite graph G by taking the convex hull of the columns of the Laplacian matrix for G. The Laplacian simplex is extended through the use of a parallel construction with a finite digraph D to obtain the Laplacian polytope, PD.
Basic properties of both families of simplices, PG and PD, are established using techniques from Ehrhart theory. Motivated by a well-known conjecture in the field, our investigation focuses on reflexivity, the integer decomposition property, and unimodality of Ehrhart h*-vectors of these polytopes. A systematic investigation of PG for trees, cycles, and complete graphs is provided, which is enhanced by an investigation of PD for cyclic digraphs. We form intriguing connections with other families of simplices and produce G and D such that the h*-vectors of PG and PD exhibit extremal behavior.
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Foschi, Alessandro. "Variétés magnifiques et polytopes moment." Grenoble 1, 1998. http://www.theses.fr/1998GRE10153.
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Soit g un groupe algebrique semi-simple complexe. Les quinze dernieres annees ont connu un developpement substantiel de la theorie des g-varietes spheriques, avec les travaux de brion, luna, pauer, knop. Divers invariants combinatoires algebriques ont ete introduits pour etudier leur geometrie et leur classification. D'un autre cote, les geometres symplectiques ont associe des invariants combinatoires aux operations hamiltoniennes, comme par exemple le polytope moment. L'objectif de ce travail est d'etudier certains liens entre ces deux points de vue, algebrique et symplectique, en mettant l'accent sur le cas particulier des varietes magnifiques (celles-ci jouent un role clef dans la theorie des varietes spheriques). Le premier chapitre ne contient que des rappels. Dans le deuxieme, nous avons rassemble nos resultats. En particulier, par des criteres combinatoires, nous avons particularise au cas magnifique, en les precisant, certains resultats generaux de brion sur les varietes spheriques (concernant les fibres en droites et les espaces de leur sections globales). Nous avons abouti ainsi a une description tout a fait explicite des polytopes moment associes aux differents fibres en droites amples sur une variete magnifique ; description qu'ensuite nous illustrons par de nombreux exemples. Enfin, dans le troisieme chapitre nous construisons une operation hamiltonienne pour su(3, c) qui possede quatre structures complexes differentes munies d'une action de sl(3, c).
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Sjöberg, Hannah [Verfasser]. "On Face Vector Sets and on Alcoved Polytopes : Two Studies on Convex Polytopes / Hannah Sjöberg." Berlin : Freie Universität Berlin, 2020. http://d-nb.info/1219904775/34.
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Paffenholz, Andreas. "Constructions for posets, lattices, and polytopes." [S.l.] : [s.n.], 2005. http://deposit.ddb.de/cgi-bin/dokserv?idn=975678299.
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Müller, Irene. "Corner cuts and corner cut polytopes." Zurich : [ETH Zurich, Department of Mathematics], 2001. http://e-collection.ethbib.ethz.ch/show?type=dipl&nr=26.
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Pfeifle, Julian. "Extremal constructions for polytopes and spheres." [S.l.] : [s.n.], 2003. http://deposit.ddb.de/cgi-bin/dokserv?idn=96753285X.
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Qi, Weinan. "On Resampling Schemes for Uniform Polytopes." Thesis, Université d'Ottawa / University of Ottawa, 2017. http://hdl.handle.net/10393/36057.
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The convex hull of a sample is used to approximate the support of the underlying
distribution. This approximation has many practical implications in real life. For
example, approximating the boundary of a finite set is used by many authors in environmental studies and medical research. To approximate the functionals of convex hulls, asymptotic theory plays a crucial role. Unfortunately, the asymptotic results are mostly very complicated. To address this complication, we suggest a consistent bootstrapping scheme for certain cases. Our resampling technique is used for both semi-parametric and non-parametric cases. Let X1,X2,...,Xn be a sequence of i.i.d. random points uniformly distributed on an unknown convex set. Our bootstrapping scheme relies on resampling uniformly from the convex hull of X1,X2,...,Xn. In this thesis, we study the asymptotic consistency of certain functionals of convex hulls. In particular, we apply our bootstrapping technique to the Hausdorff distance between the actual convex set and its estimator. We also provide a conjecture for the application of our bootstrapping scheme to Gaussian polytopes. Moreover, some other relevant consistency results for the regular bootstrap are developed.
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Moustrou, Philippe. "Geometric distance graphs, lattices and polytopes." Thesis, Bordeaux, 2017. http://www.theses.fr/2017BORD0802/document.
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Un graphe métrique G(X;D) est un graphe dont l’ensemble des sommets est l’ensemble X des points d’un espace métrique (X; d), et dont les arêtes relient les paires fx; yg de sommets telles que d(x; y) 2 D. Dans cette thèse, nous considérons deux problèmes qui peuvent être interprétés comme des problèmes de graphes métriques dans Rn. Premièrement, nous nous intéressons au célèbre problème d’empilements de sphères, relié au graphe métrique G(Rn; ]0; 2r[) pour un rayon de sphère r donné. Récemment, Venkatesh a amélioré d’un facteur log log n la meilleure borne inférieure connue pour un empilement de sphères donné par un réseau, pour une suite infinie de dimensions n. Ici nous prouvons une version effective de ce résultat, dans le sens où l’on exhibe, pour la même suite de dimensions, des familles finies de réseaux qui contiennent un réseaux dont la densité atteint la borne de Venkatesh. Notre construction met en jeu des codes construits sur des corps cyclotomiques, relevés en réseaux grâce à un analogue de la Construction A. Nous prouvons aussi un résultat similaire pour des familles de réseaux symplectiques. Deuxièmement, nous considérons le graphe distance-unité G associé à une norme k_k. Le nombre m1 (Rn; k _ k) est défini comme le supremum des densités réalisées par les stables de G. Si la boule unité associée à k _ k pave Rn par translation, alors il est aisé de voir que m1 (Rn; k _ k) > 1 2n . C. Bachoc et S. Robins ont conjecturé qu’il y a égalité. On montre que cette conjecture est vraie pour n = 2 ainsi que pour des régions de Voronoï de plusieurs types de réseaux en dimension supérieure, ceci en se ramenant à la résolution de problèmes d’empilement dans des graphes discretsA distance graph G(X;D) is a graph whose set of vertices is the set of points X of a metric space (X; d), and whose edges connect the pairs fx; yg such that d(x; y) 2 D. In this thesis, we consider two problems that may be interpreted in terms of distance graphs in Rn. First, we study the famous sphere packing problem, in relation with thedistance graph G(Rn; (0; 2r)) for a given sphere radius r. Recently, Venkatesh improved the best known lower bound for lattice sphere packings by a factor log log n for infinitely many dimensions n. We prove an effective version of this result, in the sense that we exhibit, for the same set of dimensions, finite families of lattices containing a lattice reaching this bound. Our construction uses codes over cyclotomic fields, lifted to lattices via Construction A. We also prove a similar result for families of symplectic lattices. Second, we consider the unit distance graph G associated with a norm k _ k. The number m1 (Rn; k _ k) is defined as the supremum of the densities achieved by independent sets in G. If the unit ball corresponding with k _ k tiles Rn by translation, then it is easy to see that m1 (Rn; k _ k) > 1 2n . C. Bachoc and S. Robins conjectured that the equality always holds. We show that this conjecture is true for n = 2 and for several Voronoï cells of lattices in higher dimensions, by solving packing problems in discrete graphs
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Gottcheiner, Alain. "Constructions et taxonomies de polytopes combinatoires." Doctoral thesis, Universite Libre de Bruxelles, 2002. http://hdl.handle.net/2013/ULB-DIPOT:oai:dipot.ulb.ac.be:2013/211469.
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Kulas, Katja [Verfasser]. "Combinatorics of Tropical Polytopes / Katja Kulas." München : Verlag Dr. Hut, 2013. http://d-nb.info/1037286804/34.
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Loiskekoski, Lauri [Verfasser]. "Separators of simple polytopes / Lauri Loiskekoski." Berlin : Freie Universität Berlin, 2018. http://d-nb.info/1155420810/34.
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Paterson, Harry David. "On the combinatorics of convex polytopes." Thesis, University College London (University of London), 2004. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.412282.
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Kasprzyk, A. "Toric Fano varieties and convex polytopes." Thesis, University of Bath, 2006. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.428355.
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In this thesis we study toric Fano varieties. Toric varieties are a particular class of algebraic variety which can be described in terms of combinatorial data. Toric Fano varieties correspond to certain convex lattice polytopes whose boundary lattice points are dictated by the singularities involved. Terminal toric Fano varieties correspond to convex lattice polytopes which contain only the origin as an internal lattice point, and whose boundary lattice points are precisely the vertices of the polytope. The situation is similar for canonical toric Fano varieties, with the exception that the condition on boundary lattice points is relaxed. We call these polytopes terminal (or canonical) Fano polytopes. The heart of this thesis is the development of an approach to classifying Fano polytopes, and hence the associated varieties. This is achieved by ordering the polytopes with respect to inclusion. There exists a finite collection of polytopes which are minimal with respect to this ordering. It is then possible to “grow” these minimal polytopes in order to obtain a complete classification. Critical to this method is the ability to find the minimal polytopes. Their description is inductive, requiring an understanding of the lower-dimensional minimal polytopes. A generalisation of weighted projective space plays a crucial role – the associated simplices form the building blocks of the minimal polytopes. A significant part of this thesis is dedicated to attempting to understand these building blocks. A classification of all toric Fano threefolds with at worst terminal singularities is given. The three-dimensional minimal canonical polytopes are also found, making a complete classification possible.
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Mészáros, Karola. "Root polytopes, triangulations, and subdivision algebras." Thesis, Massachusetts Institute of Technology, 2010. http://hdl.handle.net/1721.1/60199.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2010.Cataloged from PDF version of thesis.
Includes bibliographical references (p. 99-100).
In this thesis a geometric way to understand the relations of certain noncommutative quadratic algebras defined by Anatol N. Kirillov is developed. These algebras are closely related to the Fomin-Kirillov algebra, which was introduced in the hopes of unraveling the main outstanding problem of modern Schubert calculus, that of finding a combinatorial interpretation for the structure constants of Schubert polynomials. Using a geometric understanding of the relations of Kirillov's algebras in terms of subdivisions of root polytopes, several conjectures of Kirillov about the reduced forms of monomials in the algebras are proved and generalized. Other than a way of understanding Kirillov's algebras, this polytope approach also yields new results about root polytopes, such as explicit triangulations and formulas for their volumes and Ehrhart polynomials. Using the polytope technique an explicit combinatorial description of the reduced forms of monomials is also given. Inspired by Kirillov's algebras, the relations of which can be interpreted as subdivisions of root polytopes, commutative subdivision algebras are defined, whose relations encode a variety of possible subdivisions, and which provide a systematic way of obtaining subdivisions and triangulations.
by Karola Mészáros.
Ph.D.
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Macchia, Marco. "Two level polytopes :geometry and optimization." Doctoral thesis, Universite Libre de Bruxelles, 2018. http://hdl.handle.net/2013/ULB-DIPOT:oai:dipot.ulb.ac.be:2013/276475.
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A (convex) polytope P is said to be 2-level if every hyperplane H that is facet-defining for P has a parallel hyperplane H' that contains all the vertices of P which are not contained in H.Two level polytopes appear in different areas of mathematics, in particular in contexts related to discrete geometry and optimization. We study the problem of enumerating all combinatorial types of 2-level polytopes of a fixed dimension d. We describe the first algorithm to achieve this. We ran it to produce the complete database for d <= 8. Our results show that the number of combinatorial types of 2-level d-polytopes is surprisingly small for low dimensions d.We provide an upper bound for the number of combinatorially inequivalent 2-level d-polytopes. We phrase this counting problem in terms of counting some objects called 2-level configurations, that capture the class of "maximal" rank d 0/1-matrices, including (maximal) slack matrices of 2-level cones and 2-level polytopes. We provide a proof that the number of d-dimensional 2-level configurations coming from cones and polytopes, up to linear equivalence, is at most 2^{O(d^2 log d)}.Finally, we prove that the extension complexity of every stable set polytope of a bipartite graph with n nodes is O(n^2 log n) and that there exists an infinite class of bipartite graphs such that, for every n-node graph in this class, its stable set polytope has extension complexity equal to Omega(n log n).Doctorat en Sciences
info:eu-repo/semantics/nonPublished
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Ferroni, Rivetti Luis <1993>. "The Ehrhart Theory of Matroid Polytopes." Doctoral thesis, Alma Mater Studiorum - Università di Bologna, 2021. http://amsdottorato.unibo.it/9945/1/FerroniRivetti_Luis_tesi.pdf.
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In this thesis we investigate the Ehrhart polynomials of the basis polytope and the independence polytope of a matroid. We disprove conjectures by De Loera, Haws and Koppe (2007) and by Castillo and Liu (2015) asserting the positivity of the coefficients of these polynomials. Through the way we prove several positive results such as the Ehrhart positivity of hypersimplices and matroids of rank 2. Also, we provide an explicit formula for the Ehrhart polynomial of the basis polytope and the independence polytope of sparse paving matroids, a class that is conjecturally predominant.
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Brunel, Victor Emmanuel. "Non parametric estimation of convex bodies and convex polytopes." Thesis, Paris 6, 2014. http://www.theses.fr/2014PA066146/document.
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Dans ce travail, nous nous intéressons à l'estimation d'ensembles convexes dans l'espace Euclidien $\R^d$, en nous penchant sur deux modèles. Dans le premier modèle, nous avons à notre disposition un échantillon de $n$ points aléatoires, indépendants et de même loi, uniforme sur un ensemble convexe inconnu. Le second modèle est un modèle additif de régression, avec bruit sous-gaussien, et dont la fonction de régression est l'indicatrice d'Euler d'un ensemble convexe ici aussi inconnu. Dans le premier modèle, notre objectif est de construire un estimateur du support de la densité des observations, qui soit optimal au sens minimax. Dans le second modèle, l'objectif est double. Il s'agit de construire un estimateur du support de la fonction de régression, ainsi que de décider si le support en question est non vide, c'est-à-dire si la fonction de régression est effectivement non nulle, ou si le signal observé n'est que du bruit. Dans ces deux modèles, nous nous intéressons plus particulièrement au cas où l'ensemble inconnu est un polytope convexe, dont le nombre de sommets est connu. Si ce nombre est inconnu, nous montrons qu'une procédure adaptative permet de construire un estimateur atteignant la même vitesse asymptotique que dans le cas précédent. Enfin, nous démontrons que ce même estimateur pallie à l'erreur de spécification du modèle, consistant à penser à tort que l'ensemble convexe inconnu est un polytope. Nous démontrons une inégalité de déviation pour le volume de l'enveloppe convexe des observations dans le premier modèle. Nous montrons aussi que cette inégalité implique des bornes optimales sur les moments du volume manquant de cette enveloppe convexe, ainsi que sur les moments du nombre de ses sommets. Enfin, dans le cas unidimensionnel, pour le second modèle, nous donnons la taille asymptotique minimale que doit faire l'ensemble inconnu afin de pouvoir être détecté, et nous proposons une règle de décision, permettant un test consistant du caractère non vide de cet ensembleIn this thesis, we are interested in statistical inference on convex bodies in the Euclidean space $\R^d$. Two models are investigated. The first one consists of the observation of $n$ independent random points, with common uniform distribution on an unknown convex body. The second one is a regression model, with additive subgaussian noise, where the regression function is the indicator function of an unknown convex body. In the first model, our goal is to estimate the unknown support of the common uniform density of the observed points. In the second model, we aim either to estimate the support of the regression function, or to detect whether this support is nonempty, i.e., the regression function is nonzero. In both models, we investigate the cases when the unknown set is a convex polytope, and when we know the number of vertices. If this number is not known, we propose an adaptive method which allows us to obtain a statistical procedure performing asymptotically as well as in the case of perfect knowledge of that number. In addition, this procedure allows misspecification, i.e., provides an estimator of the unknown set, which is optimal in a minimax sense, even if the unknown set is not polytopal, in the contrary to what may have been thought. We prove a universal deviation inequality for the volume of the convex hull of the observations in the first model. We show that this inequality allows one to derive tight bounds on the moments of the missing volume of this convex hull, as well as on the moments of the number of its vertices. In the one-dimensional case, in the second model, we compute the asymptotic minimal size of the unknown set so that it can be detected by some statistical procedure, and we propose a decision rule which allows consistent testing of whether of that set is empty
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Taylor, Clifford T. "Deletion-Induced Triangulations." UKnowledge, 2015. http://uknowledge.uky.edu/math_etds/24.
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Let d > 0 be a fixed integer and let A ⊆ ℝd be a collection of n ≥ d + 2 points which we lift into ℝd+1. Further let k be an integer satisfying 0 ≤ k ≤ n-(d+2) and assign to each k-subset of the points of A a (regular) triangulation obtained by deleting the specified k-subset and projecting down the lower hull of the convex hull of the resulting lifting. Next, for each triangulation we form the characteristic vector defined by Gelfand, Kapranov, and Zelevinsky by assigning to each vertex the sum of the volumes of all adjacent simplices. We then form a vector for the lifting, which we call the k-compound GKZ-vector, by summing all the characteristic vectors. Lastly, we construct a polytope Σk(A) ⊆ ℝ|A| by taking the convex hull of all obtainable k-compound GKZ-vectors by various liftings of A, and note that $\Sigma_0(\A)$ is the well-studied secondary polytope corresponding to A. We will see that by varying k, we obtain a family of polytopes with interesting properties relating to Minkowski sums, Gale transforms, and Lawrence constructions, with the member of the family with maximal k corresponding to a zonotope studied by Billera, Fillamen, and Sturmfels. We will also discuss the case k = d = 1, in which we can provide a combinatorial description of the vertices allowing us to better understand the graph of the polytope and to obtain formulas for the numbers of vertices and edges present.
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Dall, Aaron Matthew. "Matroids : h-vectors, zonotopes, and Lawrence polytopes." Doctoral thesis, Universitat Politècnica de Catalunya, 2015. http://hdl.handle.net/10803/286280.
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The main objects of study in this thesis are matroids. In particular we are interested in three particular classes matroids: regular matroids, arithmetic matroids, and internally perfect matroids. Of these families, regular matroids are the oldest and most well-known. In contrast, arithmetic matroids are relatively new structures that simultaneously capture combinatorial and geometric invariants of rational vector configurations. We introduce the class of internally perfect matroids in order to use the structure of the internal order of such a matroid to prove Stanley's conjecture that (under a certain assumption) any h-vector of a matroid is a pure O-sequence in this case.
The thesis is structured as follows. We give all relevant background information in Chapter 1. In Chapter 2 we give a new proof of a generalization of Kirchoff's matrix-tree theorem to regular matroids. After recasting the problem into the world of polyhedral geometry via two zonotopes determined by a regular matroid, we reprove the theorem by showing that the volumes of these zonotopes are equal by providing an explicit bijection between the points in them (up to a set of measure zero). We then generalize to the weighted case, and conclude by using our technique to reprove the the classical matrix-tree theorem by working out the details when the matrices involved have rank-plus-one many rows. This chapter is joint work with Julian Pfeifle.
In Chapter 3 we exploit a well-known connection between the zonotope and Lawrence polytope generated by a fixed integer representation of a rational matroid to prove relations between various polynomials associated to these two polytopes and the underlying matroid. First we prove a relationship between the Ehrhart polynomial of the zonotope and the numerator of the Ehrhart series of the Lawrence polytope. On the level of arithmetic matroids, this relation allows us to view the numerator of the Ehrhart series of the Lawrence polytope as the arithmetic matroid analogue of the usual matroid h-vector of the matroid. After proving the previous result, we use it to give a new interpretation of the coefficients of a certain evaluation of the arithmetic Tutte polynomial. Finally, we give a new proof that the h-vector of the matroid and the numerator of the Ehrhart series of the Lawrence polytope coincide when the matrix representing the matroid is unimodular.
In Chapter 4, we consider a new class of matroids consisting of those matroids whose internal order makes them especially amenable to proving Stanley's conjecture. Stanley's conjecture states that for any matroid there exists a pure order ideal whose O-sequence coincides with the h-vector of the matroid. We give a brief review of known results in Section 4.1 before turning to ordered matroids and the internal order in Section 4.2, where we also define internally perfect bases and matroids. In Section 4.3 we first prove preliminary results about internally perfect bases culminating in Theorem 4.11 in which we show that, under a certain assumption, any internally perfect matroid satisfies Stanley's conjecture. Moreover, we conjecture that the assumption in the previous sentence holds for all internally perfect matroids.El principal objeto de estudio de la presente tesis son las matroides, que generalizan propiedades de matrices a un contexto más combinatorio. Nos interesaremos principalmente por tres clases particulares: matroides regulares, matroides aritméticas, y matroides internamente perfectas. De estas famílias, las matroides regulares son las mejor estudiadas. En cambio, las matroides aritméticas son estructuras relativamente nuevas que capturan simultáneamente invariantes combinatorias y geométricas de configuraciones racionales de vectores. Introducimos en esta tesis la clase de matroides internamente perfectas, que nos permiten usar la estructura del orden interno de dichas matroides para probar, en este caso y suponiendo la veracidad de una afirmación, la conjetura de Stanley que cualquier h-vector de una matroide es una O-secuencia pura. Esta tesis está estructurada de la siguiente forma. En el Capítulo 1 damos los antecedentes relevantes. En el Capítulo 2 ofrecemos una nueva demostración de una generalización del teorema de Kirchhoff. Después reestructuramos el problema en el mundo de la geometría poliédrica a través de dos zonotopos determinados por una matroide regular, demostrando que los volúmenes de estos zonotopos son iguales, y construyendo una biyección explícita entre ellos (fuera de un conjunto de medida cero). Generalizamos entonces al caso de una matroide con pesos. Concluimos mostrando que nuestra técnica pude ser usada para volver a demostrar el teorema clásico de Kirchhoff, puliendo los detalles cuando las matrices tienen corrango igual a uno. Este capítulo es fruto de trabajo conjunto con Julian Pfeifle. En el Capítulo 3 sacamos provecho de una conexión entre el zonotopo y el politopo de Lawrence generado por una representación íntegra (con coeficientes enteros) de una matroide racional para probar relaciones entre varios polinomios asociados con ellos. Primero demostramos una relación entre el polinomio de Ehrhart del zonotopo y el numerador de la serie de Ehrhart del politopo de Lawrence. Al nivel de matroides aritméticas esta relación nos permite ver el numerador de la serie de Ehrhart del politopo de Lawrence como el análogo, para matroides aritméticas, del usual h-vector de la matroide. Después de demostrar el resultado mencionado, lo usamos para ofrecer una nueva interpretación de los coeficientes de una evaluación particular del polinomio aritmético de Tutte. Finalmente mostramos que el h-vector de la matroide y la serie de Ehrhart del politopo de Lawrence coinciden cuando la representación es unimodular. En el Capítulo 4 consideramos una nueva clase de matroides, cuyo orden interno las vuelve especialmente dispuestas para demostrar la conjetura de Stanley. Esta conjetura dice que para cualquier matroide existe un ideal de orden puro cuya O-secuencia coincide con el h-vector de la matroide. Damos un breve repaso de los resultados conocidos en la Sección 4.1 antes de enfocarnos en las matroides ordenadas y el orden interno en la Sección 4.2, donde también definimos las bases y matroides internamente perfectas. En la Sección 4.3 probamos resultados preliminares sobre bases internamente perfectas culminando en el Teorema 4.11, dónde mostramos que, suponiendo la veracidad de cierta afirmación, cualquier matroide perfecta satisface la conjetura de Stanley. Por otra parte, conjeturamos que esta afirmación, en efecto, es válida para todas las matroides internamente perfectas.
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Lopez, Mario A., Shlomo Reisner, and reisner@math haifa ac il. "Linear Time Approximation of 3D Convex Polytopes." ESI preprints, 2001. ftp://ftp.esi.ac.at/pub/Preprints/esi1005.ps.
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Gonska, Bernd [Verfasser]. "Inscribable polytopes via Delaunay triangulations / Bernd Gonska." Berlin : Freie Universität Berlin, 2013. http://d-nb.info/1031104100/34.
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Sturgeon, Stephen. "Boij-Söderberg Decompositions, Cellular Resolutions, and Polytopes." UKnowledge, 2014. http://uknowledge.uky.edu/math_etds/20.
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Boij-Söderberg theory shows that the Betti table of a graded module can be written as a linear combination of pure diagrams with integer coefficients. In chapter 2 using Ferrers hypergraphs and simplicial polytopes, we provide interpretations of these coefficients for ideals with a d-linear resolution, their quotient rings, and for Gorenstein rings whose resolution has essentially at most two linear strands. We also establish a structural result on the decomposition in the case of quasi-Gorenstein modules. These results are published in the Journal of Algebra, see [25].
In chapter 3 we provide some further results about Boij-Söderberg decompositions. We show how truncation of a pure diagram impacts the decomposition. We also prove constructively that every integer multiple of a pure diagram of codimension 2 can be realized as the Betti table of a module.
In chapter 4 we introduce the idea of a c-polar self-dual polytope. We prove that in dimension 2 only the odd n-gons have an embedding which is polar self-dual. We also define the family of Ferrers polytopes. We prove that the Ferrers polytope in dimension d is d-polar self-dual hence establishing a nontrivial example of a polar self-dual polytope in all dimension. Finally we prove that the Ferrers polytope in dimension d supports a cellular resolution of the Stanley-Reisner ring of the (d+3)-gon.
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Wacheux, Christophe. "Semi-toric integrable systems and moment polytopes." Phd thesis, Université Rennes 1, 2013. http://tel.archives-ouvertes.fr/tel-00932926.
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Un système intégrable semi-torique sur une variété symplectique de dimension 2n est un système intégrable dont le flot de n − 1 composantes de l'application moment est 2 -périodique. On obtient donc une action hamiltonienne du tore Tn−1. En outre, on demande que tous les points critiques du système soient non-dégénérés et sans composante hyperbolique. En dimension 4, San V˜u Ngo.c et Álvaro Pelayo ont étendu à ces systèmes semi-toriques les résultats célèbres d'Atiyah, Guillemin, Sternberg et Delzant concernant la classification des systèmes toriques. Dans cette thèse nous proposons une extension de ces résultats en dimension quelconque, à commencer par la dimension 6. Les techniques utilisées relèvent de l'analyse comme de la géométrie symplectique, ainsi que de la théorie de Morse dans des espaces différentiels stratifiés. Nous donnons d'abord une description de l'image de l'application moment d'un point de vue local, en étudiant les asymptotiques des coordonnées actionangle au voisinage d'une singularité foyer-foyer, avec le phénomène de monodromie du feuilletage qui en résulte. Nous passons ensuite à une description plus globale dans la veine des polytopes d'Atiyah, Guillemin et Sternberg. Ces résultats sont basés sur une étude systématique de la stratification donnée par les fibres de l'application moment. Avec ces résultats, nous établissons la connexité des fibres des systèmes intégrables semi-toriques de dimension 6 et indiquons comment nous comptons démontrer ce résultat en dimension quelconque.
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Chan, Clara S. (Clara Sophia). "On shellings and subdivisions of convex polytopes." Thesis, Massachusetts Institute of Technology, 1992. http://hdl.handle.net/1721.1/13245.
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Akhtar, Mohammad Ehtisham. "Mutations of Laurent polynomials and lattice polytopes." Thesis, Imperial College London, 2015. http://hdl.handle.net/10044/1/28115.
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It has been conjectured that Fano manifolds correspond to certain Laurent polynomials under Mirror Symmetry. This correspondence predicts that the regularized quantum period of a Fano manifold coincides with the classical period of a Laurent polynomial mirror. This correspondence is not one-to-one, as many different Laurent polynomials can have the same classical period; it should become one-to-one after imposing the correct equivalence relation on Laurent polynomials. In this thesis we introduce what we believe to be the correct notion of equivalence: this is algebraic mutation of Laurent polynomials. We also consider combinatorial mutation, which is the transformation of lattice polytopes induced by algebraic mutation of Laurent polynomials supported on them. We establish the basic properties of algebraic and combinatorial mutations and give applications to algebraic geometry, most notably to the classification of Fano manifolds up to deformation. Our main focus is on the surface case, where the theory is particularly rich.
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Balletti, Gabriele. "Classifications and volume bounds of lattice polytopes." Licentiate thesis, Stockholms universitet, Matematiska institutionen, 2017. http://urn.kb.se/resolve?urn=urn:nbn:se:su:diva-139823.
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In this licentiate thesis we study relations among invariants of lattice polytopes, with particular focus on bounds for the volume.In the first paper we give an upper bound on the volume vol(P^*) of a polytope P^* dual to a d-dimensional lattice polytope P with exactly one interiorlattice point, in each dimension d. This bound, expressed in terms of the Sylvester sequence, is sharp, and is achieved by the dual to a particular reflexive simplex. Our result implies a sharp upper bound on the volume of a d-dimensional reflexive polytope. In the second paper we classify the three-dimensional lattice polytopes with two lattice points in their strict interior. Up to unimodular equivalence thereare 22,673,449 such polytopes. This classification allows us to verify, for this case only, the sharp conjectural upper bound for the volume of a lattice polytope with interior points, and provides strong evidence for more general new inequalities on the coefficients of the h^*-polynomial in dimension three.
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Royer, Tiago. "Ehrhart theory for real dilates of polytopes." Universidade de São Paulo, 2018. http://www.teses.usp.br/teses/disponiveis/45/45134/tde-31052018-093012/.
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The Ehrhart function L_P(t) of a polytope P is defined to be the number of integer points in the dilated polytope tP. Classical Ehrhart theory is mainly concerned with integer values of t; in this master thesis, we focus on how the Ehrhart function behaves when the parameter t is allowed to be an arbitrary real number. There are three main results concerning this behavior in this thesis. Some rational polytopes (like the unit cube [0, 1]^d) only gain integer points when the dilation parameter t is an integer, so that computing L_P(t) yields the same integer point count than L_P(t). We call them semi-reflexive polytopes. The first result is a characterization of these polytopes in terms of the hyperplanes that bound them. The second result is related to the Ehrhart theorem. In the classical setting, the Ehrhart theorem states that L_P(t) will be a quasipolynomial whenever P is a rational polytope. This is also known to be true with real dilation parameters; we obtained a new proof of this fact starting from the chraracterization mentioned above. The third result is about how the real Ehrhart function behaves with respect to translation in this new setting. It is known that the classical Ehrhart function is invariant under integer translations. This is far from true for the real Ehrhart function: not only there are infinitely many different functions L_{P + w}(t) (for integer w), but under certain conditions the collection of these functions identifies P uniquely.A função de Ehrhart L_P(t) de um politopo P é definida como sendo o número de pontos com coordenadas inteiras no politopo dilatado tP. A teoria de Ehrhart clássica lida principalmente com valores inteiros de t; esta dissertação de mestrado foca em como a função de Ehrhart se comporta quando permitimos que o parâmetro t seja um número real arbitrário. São três os resultados principais desta dissertação a respeito deste comportamento. Alguns politopos racionais (como o cubo unitário [0, 1]^d) apenas ganham pontos inteiros quando o parâmetro de dilatação t é um inteiro, de tal forma que computar L_P(t) devolve a mesma contagem de pontos que L_P(t). Eles são chamados de politopos semi-reflexivos. O primeiro resultado desta dissertação é uma caracterização destes politopos em termos de suas descrições como interseção de semi-espaços. O segundo resultado é relacionado ao teorema de Ehrhart. No contexto clássico, o teorema de Ehrhart afirma que L_P(t) será um quasi-polinômio sempre que P for um politopo racional. Sabe-se que este teorema generaliza para parâmetros reais de dilatação; nesta dissertação é apresentada uma nova demonstração deste fato, baseada na caracterização mencionada acima. O terceiro resultado é sobre como a função real de Ehrhart se comporta com respeito à translação neste novo contexto. Sabe-se que a função de Ehrhart clássica é invariante sob translações por vetores com coordenadas inteiras. Por outro lado, a função real de Ehrhart está bem longe de ser invariante: não só existem infinitas funções L_{P + w}(t) distintas, mas também, sob certas condições, esta coleção de funções identifica P unicamente.
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Lienkaemper, Caitlin. "Toric Ideals, Polytopes, and Convex Neural Codes." Scholarship @ Claremont, 2017. http://scholarship.claremont.edu/hmc_theses/106.
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How does the brain encode the spatial structure of the external world?
A partial answer comes through place cells, hippocampal neurons which
become associated to approximately convex regions of the world known
as their place fields. When an organism is in the place field of some place
cell, that cell will fire at an increased rate. A neural code describes the set
of firing patterns observed in a set of neurons in terms of which subsets
fire together and which do not. If the neurons the code describes are place
cells, then the neural code gives some information about the relationships
between the place fields–for instance, two place fields intersect if and only if
their associated place cells fire together. Since place fields are convex, we are
interested in determining which neural codes can be realized with convex
sets and in finding convex sets which generate a given neural code when
taken as place fields. To this end, we study algebraic invariants associated
to neural codes, such as neural ideals and toric ideals. We work with a
special class of convex codes, known as inductively pierced codes, and seek
to identify these codes through the Gröbner bases of their toric ideals.
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Herrmann, Sven. "Splits and tight spans of convex polytopes." München Verl. Dr. Hut, 2009. http://d-nb.info/993259189/04.
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Holt, Fredrick Baden. "Linear algebra, polytopes, and the Hirsch conjecture /." Thesis, Connect to this title online; UW restricted, 1996. http://hdl.handle.net/1773/5795.
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Codenotti, Giulia [Verfasser]. "Covering properties of lattice polytopes / Giulia Codenotti." Berlin : Freie Universität Berlin, 2020. http://d-nb.info/1205735569/34.
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Margot, François Margot François. "Composition de polytopes combinatoires : une approche par projection /." [S.l.] : [s.n.], 1994. http://library.epfl.ch/theses/?nr=1209.
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PEIXOTO, CAMILLA NERES. "GOSSET POLYTOPES AND THE COXETER GROUPS E(N)." PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO, 2010. http://www.maxwell.vrac.puc-rio.br/Busca_etds.php?strSecao=resultado&nrSeq=16433@1.
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PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIROCONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICO
Um politopo convexo é semiregular se todas as suas faces forem regulares e o grupo de isometrias agir transitivamente sobre os vértices. A classificação dos politopos semiregulares inclui algumas famílias infinitas, algumas exceções em dimensão baixa e uma família, os politopos de Gosset, que está definida para dimensão entre 3 e 8. Certos grupos de isometrias de R(n) gerados por reflexões são chamados grupos de Coxeter. A classificação dos grupos de Coxeter inclui três famílias infinitas, algumas exceções em dimensão menor ou igual a 4 e os grupos excepcionais E(6), E(7) e E(8). O grupo E(n) é o grupo das isometrias do politopo de Gosset em dimensao n. Nesta dissertação construiremos os grupos de Coxeter En, os politopos de Gosset e indicaremos a relação destes objetos com os reticulados e as álgebras de Lie também conhecidos como E(n).
A convex polytope is semiregular if all its faces are regular and the group of isometries acts transitively over vertices. The classification of semiregular polytopes includes a few infinite families, some low dimensional exceptions and a family, the Gosset polytopes, which is defined for dimension 3 to 8. Certain groups of isometries of R(n) generated by reflections are called Coxeter groups. The classification of finite Coxeter groups includes three infinite families, some exceptions in dimension 4 or lower and the exceptional groups E(6), E(7) and E(8). The group En is the group of isometries of the Gosset polytope in dimension n. In this dissertation we construct the Coxeter groups En, the Gosset polytopes and indicate the relationship of these objects with the lattices and Lie algebras which are also known as E(n).
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Gillette, Andrew, and Alexander Rand. "INTERPOLATION ERROR ESTIMATES FOR HARMONIC COORDINATES ON POLYTOPES." EDP SCIENCES S A, 2016. http://hdl.handle.net/10150/621355.
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Interpolation error estimates in terms of geometric quality measures are established for harmonic coordinates on polytopes in two and three dimensions. First we derive interpolation error estimates over convex polygons that depend on the geometric quality of the triangles in the constrained Delaunay triangulation of the polygon. This characterization is sharp in the sense that families of polygons with poor quality triangles in their constrained Delaunay triangulations are shown to produce large error when interpolating a basic quadratic function. Non-convex polygons exhibit a similar limitation: large constrained Delaunay triangles caused by vertices approaching a non-adjacent edge also lead to large interpolation error. While this relationship is generalized to convex polyhedra in three dimensions, the possibility of sliver tetrahedra in the constrained Delaunay triangulation prevent the analogous estimate from sharply reflecting the actual interpolation error. Non-convex polyhedra are shown to be fundamentally different through an example of a family of polyhedra containing vertices which are arbitrarily close to non-adjacent faces yet the interpolation error remains bounded.
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Bederoff, Ericksson Jonas. "Graph properties of DAG associahedra and related polytopes." Thesis, KTH, Matematik (Avd.), 2018. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-228406.
Full textAbstract:
A directed acyclic graph (DAG) can be thought of as encoding a set of conditional independence (CI) relations among random variables. Assuming we sample data from a probability distribution satisfying these CI relations, a fundamental problem in causal inference is to recover the edge-structure of the underlying DAG. An algorithm known as the greedy SP algorithm aims to recover the underlying DAG by walking along edges of a polytope known as the DAG associahedron. Hence, the edge-structure of the DAG associahedron plays an important role in understanding the complexity of the greedy SP algorithm. In this thesis, we study graph properties of the edge-graph of DAG associahedra, related polytopes, and their important subgraphs. The properties considered include diameter, radius and center. Our results on the diameter of DAG associahedra lead to a new causal inference algorithm with improved theoretical consistency guarantees and complexity bounds relative to the greedy SP algorithm.En riktad acyklisk graf kan betraktas som en representant för relationer av betingat oberoende mellan stokastiska variabler. Ett grundläggande problem inom kausal inferens är, givet att vi får data från en sannolikhetsfördelning som uppfyller en mängd relationer av betingat oberoende, att hitta den underliggande graf som representerar dessa relationer. Den giriga SP-algoritmen strävar efter att hitta den underliggande grafen genom att traversera kanter på en polytop kallad DAG-associahedern. Därav spelar kantstrukturen hos DAG-associahedern en stor roll för vår förståelse för den giriga SP-algoritmens komplexitet. I den här uppsatsen studerar vi grafegenskaper hos kantgrafer av DAG-associahedern, relaterade polytoper och deras viktiga delgrafer. Exempel på grafe-genskaper som vi studerar är diameter, radie och center. Våra diameterresultat för DAG-associahedern ger upphov till en ny kausal inferensalgoritm med förbättrade teoretiska på-litlighetsgarantier och komplexitetsbegränsningar jämfört med den giriga SP-algoritmen.