Relevant bibliographies by topics / Polytopes

Contents

  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers
  6. Reports

Academic literature on the topic 'Polytopes'

Author: Grafiati
Published: 4 June 2021
Last updated: 25 January 2025

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Journal articles on the topic "Polytopes"

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Schulte, Egon, and Asia Ivić Weiss. "Free Extensions of Chiral Polytopes." Canadian Journal of Mathematics 47, no. 3 (June 1, 1995): 641–54. http://dx.doi.org/10.4153/cjm-1995-033-7.

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AbstractAbstract polytopes are discrete geometric structures which generalize the classical notion of a convex polytope. Chiral polytopes are those abstract polytopes which have maximal symmetry by rotation, in contrast to the abstract regular polytopes which have maximal symmetry by reflection. Chirality is a fascinating phenomenon which does not occur in the classical theory. The paper proves the following general extension result for chiral polytopes. If 𝒦 is a chiral polytope with regular facets 𝓕 then among all chiral polytopes with facets 𝒦 there is a universal such polytope 𝓟, whose group is a certain amalgamated product of the groups of 𝒦 and 𝓕. Finite extensions are also discussed.
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Ramanath, Rohan, S. Sathiya Keerthi, Yao Pan, Konstantin Salomatin, and Kinjal Basu. "Efficient Vertex-Oriented Polytopic Projection for Web-Scale Applications." Proceedings of the AAAI Conference on Artificial Intelligence 36, no. 4 (June 28, 2022): 3821–29. http://dx.doi.org/10.1609/aaai.v36i4.20297.

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We consider applications involving a large set of instances of projecting points to polytopes. We develop an intuition guided by theoretical and empirical analysis to show that when these instances follow certain structures, a large majority of the projections lie on vertices of the polytopes. To do these projections efficiently we derive a vertex-oriented incremental algorithm to project a point onto any arbitrary polytope, as well as give specific algorithms to cater to simplex projection and polytopes where the unit box is cut by planes. Such settings are especially useful in web-scale applications such as optimal matching or allocation problems. Several such problems in internet marketplaces (e-commerce, ride-sharing, food delivery, professional services, advertising, etc.), can be formulated as Linear Programs (LP) with such polytope constraints that require a projection step in the overall optimization process. We show that in some of the very recent works, the polytopic projection is the most expensive step and our efficient projection algorithms help in gaining massive improvements in performance.
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Fujita, Naoki. "Newton–Okounkov polytopes of flag varieties and marked chain-order polytopes." Transactions of the American Mathematical Society, Series B 10, no. 15 (April 4, 2023): 452–81. http://dx.doi.org/10.1090/btran/142.

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Marked chain-order polytopes are convex polytopes constructed from a marked poset. They give a discrete family relating a marked order polytope with a marked chain polytope. In this paper, we consider the Gelfand–Tsetlin poset of type A A , and realize the associated marked chain-order polytopes as Newton–Okounkov bodies of the flag variety. Our realization connects previous realizations of Gelfand–Tsetlin polytopes and Feigin–Fourier–Littelmann–Vinberg polytopes as Newton–Okounkov bodies in a uniform way. As an application, we prove that the flag variety degenerates into the irreducible normal projective toric variety corresponding to a marked chain-order polytope. We also construct a specific basis of an irreducible highest weight representation. The basis is naturally parametrized by the set of lattice points in a marked chain-order polytope.
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Hibi, Takayuki, and Nan Li. "Unimodular Equivalence of Order and Chain Polytopes." MATHEMATICA SCANDINAVICA 118, no. 1 (March 7, 2016): 5. http://dx.doi.org/10.7146/math.scand.a-23291.

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Order polytope and chain polytope are two polytopes that arise naturally from a finite partially ordered set. These polytopes have been deeply studied from viewpoints of both combinatorics and commutative algebra. Even though these polytopes possess remarkable combinatorial and algebraic resemblance, they seem to be rarely unimodularly equivalent. In the present paper, we prove the following simple and elegant result: the order polytope and chain polytope for a poset are unimodularly equivallent if and only if that poset avoid the 5-element "X" shape subposet. We also explore a few equivalent statements of the main result.
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Raza, Hassan, Sakander Hayat, and Xiang-Feng Pan. "Binary Locating-Dominating Sets in Rotationally-Symmetric Convex Polytopes." Symmetry 10, no. 12 (December 6, 2018): 727. http://dx.doi.org/10.3390/sym10120727.

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A convex polytope or simply polytope is the convex hull of a finite set of points in Euclidean space R d . Graphs of convex polytopes emerge from geometric structures of convex polytopes by preserving the adjacency-incidence relation between vertices. In this paper, we study the problem of binary locating-dominating number for the graphs of convex polytopes which are symmetric rotationally. We provide an integer linear programming (ILP) formulation for the binary locating-dominating problem of graphs. We have determined the exact values of the binary locating-dominating number for two infinite families of convex polytopes. The exact values of the binary locating-dominating number are obtained for two rotationally-symmetric convex polytopes families. Moreover, certain upper bounds are determined for other three infinite families of convex polytopes. By using the ILP formulation, we show tightness in the obtained upper bounds.
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DI ROCCO, SANDRA. "PROJECTIVE DUALITY OF TORIC MANIFOLDS AND DEFECT POLYTOPES." Proceedings of the London Mathematical Society 93, no. 1 (June 9, 2006): 85–104. http://dx.doi.org/10.1017/s0024611505015686.

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Non-singular toric embeddings with dual defect are classified. The associated polytopes, called defect polytopes, are proven to be the class of Delzant integral polytopes for which a combinatorial invariant vanishes. The structure of a defect polytope is described.
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Hibi, Takayuki, and Nan Li. "Cutting Convex Polytopes by Hyperplanes." Mathematics 7, no. 5 (April 26, 2019): 381. http://dx.doi.org/10.3390/math7050381.

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Cutting a polytope is a very natural way to produce new classes of interesting polytopes. Moreover, it has been very enlightening to explore which algebraic and combinatorial properties of the original polytope are hereditary to its subpolytopes obtained by a cut. In this work, we devote our attention to all the separating hyperplanes for some given polytope (integral and convex) and study the existence and classification of such hyperplanes. We prove the existence of separating hyperplanes for the order and chain polytopes for any finite posets that are not a single chain, and prove there are no such hyperplanes for any Birkhoff polytopes. Moreover, we give a complete separating hyperplane classification for the unit cube and its subpolytopes obtained by one cut, together with some partial classification results for order and chain polytopes.
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Cunningham, Gabe, Mark Mixer, and Gordon Williams. "Reflexible covers of prisms." Contributions to Discrete Mathematics 19, no. 3 (September 23, 2024): 196–208. http://dx.doi.org/10.55016/ojs/cdm.v19i3.74917.

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The Tomotope provided the first well understood example of an abstract 4-polytope whose connection (monodromy) group was not a string C-group, and which also did not have a unique minimal regular cover. Conversely, we know that if the connection group of a polytope is a string C-group (if the polytope is C-connected), then the polytope will have a unique minimal regular cover. Since the discovery of the Tomotope, an active area of investigation has been determining which abstract $d$-polytopes are C-connected and the ways various constructions for abstract polytopes result in polytopes that do or do not possess unique minimal regular covers. In the current work we show that the prism over every abstract polyhedron is C-connected, or equivalently, that it has a unique minimal regular cover. We also describe a conjecture positing a general condition on the C-connectedness of prisms over polytopes that is independent of rank.
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Gubeladze, Joseph. "Affine-compact functors." Advances in Geometry 19, no. 4 (October 25, 2019): 487–504. http://dx.doi.org/10.1515/advgeom-2019-0004.

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Abstract Several well known polytopal constructions are examined from the functorial point of view. A naive analogy between the Billera–Sturmfels fiber polytope and the abelian kernel is disproved by an infinite explicit series of polytopes. A correct functorial formula is provided in terms of an affine-compact substitute of the abelian kernel. The dual cokernel object is almost always the natural affine projection. The Mond–Smith–van Straten space of sandwiched simplices, useful in stochastic factorizations, leads to a different kind of affine-compact functors and new challenges in polytope theory.
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Okay, Cihan, Ho Yiu Chung, and Selman Ipek. "Mermin polytopes quantum computation and foundations." Quantum Information & Computation 23, no. 9&10 (July 2023): 733–82. http://dx.doi.org/10.26421/qic23.9-10-2.

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Mermin square scenario provides a simple proof for state-independent contextuality. In this paper, we study polytopes $\MP_\beta$ obtained from the Mermin scenario, parametrized by a function $\beta$ on the set of contexts. Up to combinatorial isomorphism, there are two types of polytopes $\MP_0$ and $\MP_1$ depending on the parity of $\beta$. Our main result is the classification of the vertices of these two polytopes. In addition, we describe the graph associated with the polytopes. All the vertices of $\MP_0$ turn out to be deterministic. This result provides a new topological proof of a celebrated result of Fine characterizing noncontextual distributions on the CHSH scenario. $\MP_1$ can be seen as a nonlocal toy version of $\Lambda$-polytopes, a class of polytopes introduced for the simulation of universal quantum computation. In the $2$-qubit case, we provide a decomposition of the $\Lambda$-polytope using $\MP_1$, whose vertices are classified, and the nonsignaling polytope of the $(2,3,2)$ Bell scenario, whose vertices are well-known.
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Dissertations / Theses on the topic "Polytopes"

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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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Mathematics
Ph.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 permutoèdre
This 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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Books on the topic "Polytopes"

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Grünbaum, Branko. Convex Polytopes. Edited by Volker Kaibel, Victor Klee, and Günter M. Ziegler. New York, NY: Springer New York, 2003. http://dx.doi.org/10.1007/978-1-4613-0019-9.

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Volker, Kaibel, Klee Victor, and Ziegler Günter M, eds. Convex polytopes. 2nd ed. New York: Springer, 2003.

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Arthanari, Tirukkattuppalli Subramanyam. Pedigree Polytopes. Singapore: Springer Nature Singapore, 2023. http://dx.doi.org/10.1007/978-981-19-9952-9.

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Ziegler, Günter M. Lectures on Polytopes. New York, NY: Springer New York, 1995. http://dx.doi.org/10.1007/978-1-4613-8431-1.

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Coxeter, H. S. M. Regular complex polytopes. 2nd ed. Cambridge [England]: Cambridge University Press, 1991.

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Cunningham, Gabriel, Mark Mixer, and Egon Schulte, eds. Polytopes and Discrete Geometry. Providence, Rhode Island: American Mathematical Society, 2021. http://dx.doi.org/10.1090/conm/764.

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Kasprzyk, Alexander M., and Benjamin Nill, eds. Interactions with Lattice Polytopes. Cham: Springer International Publishing, 2022. http://dx.doi.org/10.1007/978-3-030-98327-7.

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Kalai, Gil, and Günter M. Ziegler, eds. Polytopes — Combinatorics and Computation. Basel: Birkhäuser Basel, 2000. http://dx.doi.org/10.1007/978-3-0348-8438-9.

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Richter-Gebert, Jürgen. Realization Spaces of Polytopes. Berlin, Heidelberg: Springer Berlin Heidelberg, 1996. http://dx.doi.org/10.1007/bfb0093761.

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Carman, B. J. Matroids and their polytopes. Manchester: UMIST, 1995.

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Book chapters on the topic "Polytopes"

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Grünbaum, Branko. "Polytopes." In Convex Polytopes, 35–60. New York, NY: Springer New York, 2003. http://dx.doi.org/10.1007/978-1-4613-0019-9_3.

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De Concini, Corrado, and Claudio Procesi. "Polytopes." In Topics in Hyperplane Arrangements, Polytopes and Box-Splines, 3–23. New York, NY: Springer New York, 2010. http://dx.doi.org/10.1007/978-0-387-78963-7_1.

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Buchstaber, Victor, and Taras Panov. "Polytopes." In University Lecture Series, 7–20. Providence, Rhode Island: American Mathematical Society, 2002. http://dx.doi.org/10.1090/ulect/024/02.

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Gelfand, Israel M., Mikhail M. Kapranov, and Andrei V. Zelevinsky. "Newton Polytopes and Chow Polytopes." In Discriminants, Resultants, and Multidimensional Determinants, 193–213. Boston, MA: Birkhäuser Boston, 1994. http://dx.doi.org/10.1007/978-0-8176-4771-1_7.

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Grünbaum, Branko. "3-Polytopes." In Convex Polytopes, 263–328. New York, NY: Springer New York, 2003. http://dx.doi.org/10.1007/978-1-4613-0019-9_13.

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Grünbaum, Branko. "Neighborly Polytopes." In Convex Polytopes, 136–45. New York, NY: Springer New York, 2003. http://dx.doi.org/10.1007/978-1-4613-0019-9_7.

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Grünbaum, Branko. "Notation and Prerequisites." In Convex Polytopes, 1–9. New York, NY: Springer New York, 2003. http://dx.doi.org/10.1007/978-1-4613-0019-9_1.

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Grünbaum, Branko. "Extremal Problems Concerning Numbers of Faces." In Convex Polytopes, 192–222. New York, NY: Springer New York, 2003. http://dx.doi.org/10.1007/978-1-4613-0019-9_10.

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Grünbaum, Branko. "Properties of Boundary Complexes." In Convex Polytopes, 223–50. New York, NY: Springer New York, 2003. http://dx.doi.org/10.1007/978-1-4613-0019-9_11.

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Grünbaum, Branko. "k-Equivalence of Polytopes." In Convex Polytopes, 251–62. New York, NY: Springer New York, 2003. http://dx.doi.org/10.1007/978-1-4613-0019-9_12.

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Conference papers on the topic "Polytopes"

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Ke, Yekun, Xiaoyu Li, Zhao Song, and Tianyi Zhou. "Faster Sampling Algorithms for Polytopes with Small Treewidth." In 2024 IEEE International Conference on Big Data (BigData), 44–53. IEEE, 2024. https://doi.org/10.1109/bigdata62323.2024.10825010.

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Devanathan, Srikanth, and Karthik Ramani. "Creating Polytope Representation of Design Spaces for Visual Exploration Using Consistency Technique." In ASME 2009 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2009. http://dx.doi.org/10.1115/detc2009-86887.

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A polytope-based representation is presented to approximate the feasible space of a design concept that is described mathematically using constraints. A method for constructing such design spaces is also introduced. Constraints include equality and inequality relationships between design variables and performance parameters. The design space is represented as a finite set of (at most) 3-dimensional (possibly non-convex) polytopes, i.e., points, intervals, polygons (both open and closed) and polyhedra (both open and closed). These polytopes approximate the locally connected design space around an initial feasible point. The algorithm for constructing the design space is developed by adapting consistency algorithm for polytope representations.
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O'Donnell, Ryan, Rocco A. Servedio, and Li-Yang Tan. "Fooling polytopes." In STOC '19: 51st Annual ACM SIGACT Symposium on the Theory of Computing. New York, NY, USA: ACM, 2019. http://dx.doi.org/10.1145/3313276.3316321.

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Arroyave-Tobón, Santiago, Denis Teissandier, and Vincent Delos. "Tolerance Analysis With Polytopes in HV-Description." In ASME 2016 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2016. http://dx.doi.org/10.1115/detc2016-59027.

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This article proposes the use of polytopes in HV-description to solve tolerance analysis problems. Polytopes are defined by a finite set of half-spaces representing geometric, contact or functional specifications. However, the list of the vertices of the poly-topes are useful for computing other operations as Minkowski sums. Then, this paper proposes a truncation algorithm to obtain the V-description of polytopes in ℝn from its H-description. It is detailed how intersections of polytopes can be calculated by means of the truncation algorithm. Minkowski sums as well can be computed using this algorithm making use of the duality property of polytopes. Therefore, a Minkowski sum can be calculated intersecting some half-spaces in the dual space. Finally, the approach based on HV-polytopes is illustrated by the tolerance analysis of a real industrial case using the open source software PolitoCAT and politopix.
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Ng, Louis. "Technically, squares are polytopes." In Summer Workshop on Lattice Polytopes. WORLD SCIENTIFIC, 2019. http://dx.doi.org/10.1142/9789811200489_0019.

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Aronov, Boris, and Tamal K. Dey. "Polytopes in arrangements." In the fifteenth annual symposium. New York, New York, USA: ACM Press, 1999. http://dx.doi.org/10.1145/304893.304963.

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Dobkin, D., H. Edelsbrunner, and C. K. Yap. "Probing convex polytopes." In the eighteenth annual ACM symposium. New York, New York, USA: ACM Press, 1986. http://dx.doi.org/10.1145/12130.12174.

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Boya, Luis J., Oscar J. Garay, Marisa Fernández, Luis Carlos de Andrés, and Luis Ugarte. "Supersymmetry and Polytopes." In SPECIAL METRICS AND SUPERSYMMETRY: Proceedings of the Workshop on Geometry and Physics: Special Metrics and Supersymmetry. AIP, 2009. http://dx.doi.org/10.1063/1.3089204.

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Engström, Alexander, and Florian Kohl. "Lattice polytopes in mathematical physics." In Summer Workshop on Lattice Polytopes. WORLD SCIENTIFIC, 2019. http://dx.doi.org/10.1142/9789811200489_0011.

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Steinmeyer, Johanna. "On the faces of simple polytopes." In Summer Workshop on Lattice Polytopes. WORLD SCIENTIFIC, 2019. http://dx.doi.org/10.1142/9789811200489_0025.

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Reports on the topic "Polytopes"

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Lewis, Daniel, and Andreas Christoforides. Super-Closed Polytopes and Semi-Locally Stable Morphisms with moderation incentives. Web of Open Science, March 2020. http://dx.doi.org/10.37686/emj.v1i1.20.

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Conforti, Michele, Gwrard Cornuwjols, and Klaus Truemper. From Totally Unimodular to Balanced O, +-1 Matrices: A Family of Integer Polytopes,. Fort Belvoir, VA: Defense Technical Information Center, July 1992. http://dx.doi.org/10.21236/ada254552.

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Lawrence, Jim. Polytope volume computation. Gaithersburg, MD: National Institute of Standards and Technology, 1989. http://dx.doi.org/10.6028/nist.ir.89-4123.

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Savov, Svetoslav, and Ivan Popchev. Stability Tests for Discrete-time Polytopic Systems. "Prof. Marin Drinov" Publishing House of Bulgarian Academy of Sciences, September 2018. http://dx.doi.org/10.7546/crabs.2018.09.10.

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Qi, Liqun, and Egon Balas. Linear-Time Separation Algorithms for the Three-Index Assignment Polytope. Fort Belvoir, VA: Defense Technical Information Center, September 1990. http://dx.doi.org/10.21236/ada228854.

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Balas, Egon, and Shu M. Ng. On the Set Covering Polytope. 2. Lifting the Facets with Coefficients in (0,1,2). Revision. Fort Belvoir, VA: Defense Technical Information Center, March 1988. http://dx.doi.org/10.21236/ada196237.

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Carr, Bob. Using the Dimension Reduction Technique to Prove that Clique Trees Define Facets for the Asymmetric Traveling Salesman Polytope. Fort Belvoir, VA: Defense Technical Information Center, September 1994. http://dx.doi.org/10.21236/ada289397.

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