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Designing optical multi-band networks : polyhedral analysis and algorithms / Conception de réseaux optiques multi-bandes : Analyse polyédrale et algorithmesBenhamiche, Amal 12 December 2013 (has links)
Dans cette thèse, on s'intéresse à deux problèmes de conception de réseaux, utilisant la technologie OFDM multi-bandes. Le premier problème concerne la conception d'un réseau mono-couche avec contraintes spécifiques. Nous donnons une formulation en PLNE pour ce problème et étudions le polyèdre associé à sa restriction sur un arc. Nous introduisons deux familles d'inégalités valides définissant des facettes et développons un algorithme de coupes et branchements pour le problème. Nous étudions la variante multicouche du problème précédent et proposons plusieurs PLNE pour le modéliser. Nous identifions plusieurs familles de facettes et discutons des problèmes de séparation associés. Nous développons un algorithme de coupes et branchements utilisant l'ensemble des contraintes identifiées. Enfin, une formulation compacte et deux formulations basées sur des chemins sont proposées pour le problème. Nous présentons deux algorithmes de génération de colonnes et branchements pour le problème. / In this thesis we consider two capacitated network design (CND) problems, using OFDM multi-band technology. The first problem is related to single-layer network design with specific requirements. We give an ILP formulation for this problem and study the polyhedra associated with its arc-set restriction. We describe two families of facet defining inequalities. We devise a Branch-and-Cut algorithm for the problem. Next, we investigate the multilayer version of CND using OFDM technology. We propose several ILP formulations and study the polyhedron associated with the first (cut) formulation. We identify several classes of facets and discuss the related separation problem. We devise a Branch-and-Cut algorithm embedding valid inequalities of both single-layer and multilayer problems. The second formulation is compact, and holds a polynomial number of constraints and variables. Two further path formulations are given which yield two efficient Branch-and-Price algorithms for the problem.
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The classification and dynamics of the momentum polytopes of the SU(3) action on points in the complex projective plane with an application to point vorticesShaddad, Amna January 2018 (has links)
We have fully classified the momentum polytopes of the SU(3) action on CP(2)xCP(2) and CP(2)xCP(2) xCP(2), both actions with weighted symplectic forms, and their corresponding transition momentum polytopes. For CP(2)xCP(2) the momentum polytopes are distinct line segments. The action on CP(2)xCP(2) xCP(2), has 9 different momentum polytopes. The vertices of the momentum polytopes of the SU(3) action on CP(2)xCP(2) xCP(2), fall into two categories: definite and indefinite vertices. The reduced space corresponding to momentum map image values at definite vertices is isomorphic to the 2-sphere. We show that these results can be applied to assess the dynamics by introducing and computing the space of allowed velocity vectors for the different configurations of two-vortex systems.
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Phylogenetic Inference Using a Discrete-Integer Linear Programming ModelSands, William Alvah January 2017 (has links)
No description available.
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Randomized integer convex hullHong Ngoc, Binh 12 February 2021 (has links)
The thesis deals with stochastic and algebraic aspects of the integer convex hull. In the first part, the intrinsic volumes of the randomized integer convex hull are investigated. In particular, we obtained an exact asymptotic order of the expected intrinsic volumes difference in a smooth convex body and a tight inequality for the expected mean width difference. In the algebraic part, an exact formula for the Bhattacharya function of complete primary monomial ideas in two variables is given. As a consequence, we derive an effective characterization for complete monomial ideals in two variables.
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Implicit representation of inscribed volumesSahbaei, Parto 01 May 2017 (has links)
We present an implicit approach for constructing smooth isolated or interconnected 3-D inscribed volumes which can be employed for volumetric modeling of various kinds of spongy or porous structures, such as volcanic rocks, pumice stones, Cancellus bones *, liquid or dry foam, radiolarians, cheese, and other similar materials. The inscribed volumes can be represented in their normal or positive forms to model natural pebbles or pearls, or in their inverted or negative forms to be used in porous structures, but regardless of their types, their smoothness and sizes are controlled by the user without losing the consistency of the shapes. We introduce two techniques for blending and creating interconnections between these inscribed volumes to achieve a great flexibility to adapt our approach to different types of porous structures, whether they are regular or irregular. We begin with a set of convex polytopes such as 3-D Voronoi diagram cells and compute inscribed volumes bounded by the cells. The cells can be irregular in shape, scale, and topology, and this irregularity transfers to the inscribed volumes, producing natural-looking spongy structures. Describing the inscribed volumes with implicit functions gives us a freedom to exploit
volumetric surface combinations and deformations operations effortlessly / Graduate
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Uma contribui??o ao estudo das categorias internas e de sua prolifera??o em redes ARTMAPAlves, Robinson Luis de Souza 05 November 2012 (has links)
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Previous issue date: 2012-11-05 / ART networks present some advantages: online learning; convergence in a few epochs of training;
incremental learning, etc. Even though, some problems exist, such as: categories proliferation,
sensitivity to the presentation order of training patterns, the choice of a good vigilance parameter, etc.
Among the problems, the most important is the category proliferation that is probably the most
critical. This problem makes the network create too many categories, consuming resources to store
unnecessarily a large number of categories, impacting negatively or even making the processing time
unfeasible, without contributing to the quality of the representation problem, i. e., in many cases, the
excessive amount of categories generated by ART networks makes the quality of generation inferior
to the one it could reach. Another factor that leads to the category proliferation of ART networks is
the difficulty of approximating regions that have non-rectangular geometry, causing a generalization
inferior to the one obtained by other methods of classification. From the observation of these
problems, three methodologies were proposed, being two of them focused on using a most flexible
geometry than the one used by traditional ART networks, which minimize the problem of categories
proliferation. The third methodology minimizes the problem of the presentation order of training
patterns. To validate these new approaches, many tests were performed, where these results
demonstrate that these new methodologies can improve the quality of generalization for ART
networks / As redes do tipo ART apresentam algumas vantagens: aprendizado online; converg?ncia em poucas
?pocas de treinamento; aprendizado incremental, etc. Contudo, alguns problemas existem:
prolifera??o de categorias, sensibilidade a ordem de apresenta??o dos padr?es, escolha de um bom
valor para o par?metro de vigil?ncia. O mais importante deles ? o problema da prolifera??o de
categorias e ? provavelmente o mais cr?tico. Esse problema faz com que a rede crie v?rias categorias
consumindo recursos (recursos para armazenar uma grande quantidade de categorias desnecess?rias
impactando negativamente ou at? mesmo inviabilizando o tempo de processamento da rede) sem
contribuir para a qualidade da representa??o do problema, ou seja, em muitos casos a quantidade
excessiva de categorias geradas pelas redes ART faz com que a qualidade da generaliza??o da rede
seja inferior. Outro fator que leva a prolifera??o de categorias das redes do tipo ART ? a dificuldade
de aproximar regi?es de classes que tem geometria n?o retangular, ocasionando uma generaliza??o
inferior a outros m?todos de classifica??o. A partir da observa??o desses problemas, foi desenvolvido
esse trabalho que prop?e tr?s metodologias. Duas dessas metodologias utilizam uma geometria mais
flex?vel do que a geometria regular retangular presente nas redes ART tradicionais e minimizam o
problema da prolifera??o de categorias. A terceira metodologia minimiza o problema da ordem de
apresenta??o dos padr?es e a prolifera??o de categorias. Com o objetivo de validar as novas
abordagens, v?rios testes foram realizados. Os resultados obtidos nesses testes demonstram a
viabilidade das metodologias propostas em reduzir o n?mero de categorias e melhorar a qualidade da
generaliza??o. Em muitos desses testes a quantidade m?nima de categorias necess?rias para classificar
corretamente as classes foi atingida ap?s o treinamento, o que demonstra uma significativa melhora
em rela??o aos m?todos j? existentes. Al?m disso, devido a nova geometria das categorias, utilizando
politopos convexos, a qualidade da generaliza??o melhorou em rala??o ao estado da arte
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Geometric distance graphs, lattices and polytopes / Graphes métriques géométriques, réseaux et polytopesMoustrou, Philippe 01 December 2017 (has links)
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 discrets. / A 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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Representation of Monoids and Lattice Structures in the Combinatorics of Weyl Groups / Représentations de monoïdes et structures de treillis en combinatoire des groupes de Weyl.Gay, Joël 25 June 2018 (has links)
La combinatoire algébrique est le champ de recherche qui utilise des méthodes combinatoires et des algorithmes pour étudier les problèmes algébriques, et applique ensuite des outils algébriques à ces problèmes combinatoires. L’un des thèmes centraux de la combinatoire algébrique est l’étude des permutations car elles peuvent être interprétées de bien des manières (en tant que bijections, matrices de permutations, mais aussi mots sur des entiers, ordre totaux sur des entiers, sommets du permutaèdre…). Cette riche diversité de perspectives conduit alors aux généralisations suivantes du groupe symétrique. Sur le plan géométrique, le groupe symétrique engendré par les transpositions élémentaires est l’exemple canonique des groupes de réflexions finis, également appelés groupes de Coxeter. Sur le plan monoïdal, ces même transpositions élémentaires deviennent les opérateurs du tri par bulles et engendrent le monoïde de 0-Hecke, dont l’algèbre est la spécialisation à q=0 de la q-déformation du groupe symétrique introduite par Iwahori. Cette thèse se consacre à deux autres généralisations des permutations. Dans la première partie de cette thèse, nous nous concentrons sur les matrices de permutations partielles, en d’autres termes les placements de tours ne s’attaquant pas deux à deux sur un échiquier carré. Ces placements de tours engendrent le monoïde de placements de tours, une généralisation du groupe symétrique. Dans cette thèse nous introduisons et étudions le 0-monoïde de placements de tours comme une généralisation du monoïde de 0-Hecke. Son algèbre est la dégénérescence à q=0 de la q-déformation du monoïde de placements de tours introduite par Solomon. On étudie par la suite les propriétés monoïdales fondamentales du 0-monoïde de placements de tours (ordres de Green, propriété de treillis du R-ordre, J-trivialité) ce qui nous permet de décrire sa théorie des représentations (modules simples et projectifs, projectivité sur le monoïde de 0-Hecke, restriction et induction le long d’une fonction d’inclusion).Les monoïdes de placements de tours sont en fait l’instance en type A de la famille des monoïdes de Renner, définis comme les complétés des groupes de Weyl (c’est-à-dire les groupes de Coxeter cristallographiques) pour la topologie de Zariski. Dès lors, dans la seconde partie de la thèse nous étendons nos résultats du type A afin de définir les monoïdes de 0-Renner en type B et D et d’en donner une présentation. Ceci nous conduit également à une présentation des monoïdes de Renner en type B et D, corrigeant ainsi une présentation erronée se trouvant dans la littérature depuis une dizaine d’années. Par la suite, nous étudions comme en type A les propriétés monoïdales de ces nouveaux monoïdes de 0-Renner de type B et D : ils restent J-triviaux, mais leur R-ordre n’est plus un treillis. Cela ne nous empêche pas d’étudier leur théorie des représentations, ainsi que la restriction des modules projectifs sur le monoïde de 0-Hecke qui leur est associé. Enfin, la dernière partie de la thèse traite de différentes généralisations des permutations. Dans une récente séries d’articles, Châtel, Pilaud et Pons revisitent la combinatoire algébrique des permutations (ordre faible, algèbre de Hopf de Malvenuto-Reutenauer) en terme de combinatoire sur les ordres partiels sur les entiers. Cette perspective englobe également la combinatoire des quotients de l’ordre faible tels les arbres binaires, les séquences binaires, et de façon plus générale les récents permutarbres de Pilaud et Pons. Nous généralisons alors l’ordre faibles aux éléments des groupes de Weyl. Ceci nous conduit à décrire un ordre sur les sommets des permutaèdres, associaèdres généralisés et cubes dans le même cadre unifié. Ces résultats se basent sur de subtiles propriétés des sommes de racines dans les groupes de Weyl qui s’avèrent ne pas fonctionner pour les groupes de Coxeter qui ne sont pas cristallographiques / Algebraic combinatorics is the research field that uses combinatorial methods and algorithms to study algebraic computation, and applies algebraic tools to combinatorial problems. One of the central topics of algebraic combinatorics is the study of permutations, interpreted in many different ways (as bijections, permutation matrices, words over integers, total orders on integers, vertices of the permutahedron…). This rich diversity of perspectives leads to the following generalizations of the symmetric group. On the geometric side, the symmetric group generated by simple transpositions is the canonical example of finite reflection groups, also called Coxeter groups. On the monoidal side, the simple transpositions become bubble sort operators that generate the 0-Hecke monoid, whose algebra is the specialization at q=0 of Iwahori’s q-deformation of the symmetric group. This thesis deals with two further generalizations of permutations. In the first part of this thesis, we first focus on partial permutations matrices, that is placements of pairwise non attacking rooks on a n by n chessboard, simply called rooks. Rooks generate the rook monoid, a generalization of the symmetric group. In this thesis we introduce and study the 0-Rook monoid, a generalization of the 0-Hecke monoid. Its algebra is a proper degeneracy at q = 0 of the q-deformed rook monoid of Solomon. We study fundamental monoidal properties of the 0-rook monoid (Green orders, lattice property of the R-order, J-triviality) which allow us to describe its representation theory (simple and projective modules, projectivity on the 0-Hecke monoid, restriction and induction along an inclusion map).Rook monoids are actually type A instances of the family of Renner monoids, which are completions of the Weyl groups (crystallographic Coxeter groups) for Zariski’s topology. In the second part of this thesis we extend our type A results to define and give a presentation of 0-Renner monoids in type B and D. This also leads to a presentation of the Renner monoids of type B and D, correcting a misleading presentation that appeared earlier in the litterature. As in type A we study the monoidal properties of the 0-Renner monoids of type B and D : they are still J-trivial but their R-order are not lattices anymore. We study nonetheless their representation theory and the restriction of projective modules over the corresponding 0-Hecke monoids. The third part of this thesis deals with different generalizations of permutations. In a recent series of papers, Châtel, Pilaud and Pons revisit the algebraic combinatorics of permutations (weak order, Malvenuto-Reutenauer Hopf algebra) in terms of the combinatorics of integer posets. This perspective encompasses as well the combinatorics of quotients of the weak order such as binary trees, binary sequences, and more generally the recent permutrees of Pilaud and Pons. We generalize the weak order on the elements of the Weyl groups. This enables us to describe the order on vertices of the permutahedra, generalized associahedra and cubes in the same unified context. These results are based on subtle properties of sums of roots in Weyl groups, and actually fail for non-crystallographic Coxeter groups.
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[en] AUTOMATED SYNTHESIS OF OPTIMAL DECISION TREES FOR SMALL COMBINATORIAL OPTIMIZATION PROBLEMS / [pt] SÍNTESE AUTOMATIZADA DE ÁRVORES DE DECISÃO ÓTIMAS PARA PEQUENOS PROBLEMAS DE OTIMIZAÇÃO COMBINATÓRIACLEBER OLIVEIRA DAMASCENO 24 August 2021 (has links)
[pt] A análise de complexidade clássica para problemas NP-difíceis é geralmente
orientada para cenários de pior caso, considerando apenas o comportamento
assintótico. No entanto, existem algoritmos práticos com execução em um tempo razoável para muitos problemas clássicos. Além disso, há evidências que apontam para algoritmos polinomiais no modelo de árvore de decisão linear para resolver esses problemas, embora não muito explorados. Neste trabalho, exploramos esses resultados teóricos anteriores. Mostramos que a solução ótima para problemas combinatórios 0-1 pode ser encontrada reduzindo esses problemas para uma Busca por Vizinho Mais Próximo sobre o conjunto de vértices de Voronoi correspondentes. Utilizamos os hiperplanos que delimitam essas regiões para gerar sistematicamente uma árvore de decisão que repetidamente divide o espaço até que possa separar todas as soluções, garantindo uma resposta ótima. Fazemos experimentos para testar os limites de tamanho para os quais podemos construir essas árvores para os casos do 0-1 knapsack, weighted minimum cut e symmetric traveling salesman. Conseguimos encontrar as árvores desses problemas com tamanhos até 10, 5 e 6, respectivamente. Obtemos também as relações de adjacência completas para os esqueletos dos politopos do knapsack
e do traveling salesman até os tamanhos 10 e 7. Nossa abordagem supera
consistentemente o método de enumeração e os métodos baseline para o weighted
minimum cut e symmetric traveling salesman, fornecendo soluções ótimas em
microssegundos. / [en] Classical complexity analysis for NP-hard problems is usually oriented to
worst-case scenarios, considering only the asymptotic behavior. However, there
are practical algorithms running in a reasonable time for many classic problems. Furthermore, there is evidence pointing towards polynomial algorithms in
the linear decision tree model to solve these problems, although not explored
much. In this work, we explore previous theoretical results. We show that the
optimal solution for 0-1 combinatorial problems can be found by reducing these
problems into a Nearest Neighbor Search over the set of corresponding Voronoi
vertices. We use the hyperplanes delimiting these regions to systematically generate a decision tree that repeatedly splits the space until it can separate all solutions, guaranteeing an optimal answer. We run experiments to test the size limits for which we can build these trees for the cases of the 0-1 knapsack, weighted minimum cut, and symmetric traveling salesman. We manage to find the trees of these problems with sizes up to 10, 5, and 6, respectively. We also obtain the complete adjacency relations for the skeletons of the knapsack and traveling salesman polytopes up to size 10 and 7. Our approach consistently outperforms the enumeration method and the baseline methods for the weighted minimum cut and symmetric traveling salesman, providing optimal solutions within microseconds.
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Okounkov Bodies of Borel Orbit Closures in Wonderful Group CompactificationsMiller, Jason A. 09 July 2014 (has links)
No description available.
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