We use toric symmetry and blowups to study relationships in the Gromov-Witten theories of $\mathbb{P}^3$ and $\mathbb{P}^1\!\times\!\mathbb{P}^1\!\times\!\mathbb{P}^1$. These two spaces are birationally equivalent via the common blowup space, the permutohedral variety. We prove an equivalence of certain invariants on blowups at only points of $\mathbb{P}^3$ and $\mathbb{P}^1\!\times\!\mathbb{P}^1\!\times\!\mathbb{P}^1$ by showing that these invariants descend from the blowup. Further, the permutohedral variety has nontrivial automorphisms of its cohomology coming from toric symmetry. These symmetries can be forced to descend to the blowups at just points of $\mathbb{P}^3$ and $\mathbb{P}^1\!\times\!\mathbb{P}^1\!\times\!\mathbb{P}^1$. Enumerative consequences are discussed.

The quadratic traveling salesman problem (QTSP) is an extension of the (classical) Traveling Salesman Problem (TSP) where the costs depend on each two nodes that are traversed in succession, i. e., on the edges in the symmetric (STSP) and on the arcs in the asymmetric case (ATSP). The QTSP is motivated by an application in bioinformatics. It can be used in the solution of certain Permuted Markov models that are set up for the recognition of transcription factor binding sites and of splice sites in gene regulation. Important special cases are the Angular-Metric TSP used in robotics and the TSP with Reload Costs used in the planning of telecommunication and transport networks. The SQTSP and the AQTSP can be formulated as integer optimization problems over the polytope associated with the STSP resp. ATSP together with a quadratic cost function. We study the polytopes arising from a linearization of the respective quadratic integer programming formulations. Based on the proof of the dimension of the polytopes using the so called direct method we can prove the facetness of several valid inequalities. These facets and valid inequalities can be divided into three large groups. Some are related to the Boolean quadric polytope. Furthermore we introduce the conflicting edges/arc inequalities that forbid certain configurations of edges and 2-edges resp. of arcs and 2-arcs. Finally, we strengthen valid inequalities of STSP and ATSP in order to get stronger inequalities in the quadratic case. We present two general lifting approaches. One is applicable to all inequalities with nonnegative coefficients and the second allows to strengthen clique tree inequalities. Applying these approaches to the subtour elimination constraints leads to facets in most cases, but in general facetness is not preserved. In addition, the complexity of the separation problems for some of the facet classes is studied. Finally, we present some computational results using a branch-and-cut framework, which is improved by some of the newly derived cutting planes. The tested instances from biology could be solved surprisingly well. Instances with up to 100 nodes could be solved in less than 700 seconds improving the results in the literature by several orders of magnitude. For most of the randomly generated instances using some additional separators allowed to reduce the root gaps and the numbers of nodes in the branch-and-cut tree significantly, often even the running times.

Many important decisions must be made without full information. For example, a woman may need to make a treatment decision regarding breast cancer without full knowledge of important uncertainties, such as how well she might respond to treatment. In the financial domain, in the wake of the housing crisis, the government may need to monitor the credit market and decide whether to intervene. A key input in this case would be a model to describe the chance that one person (or company) will default given that others have defaulted. However, such a model requires addressing the lack of knowledge regarding the correlation between groups or individuals. How to model and make decisions in cases where only partial information is available is a significant challenge. In the past, researchers have made arbitrary assumptions regarding the missing information. In this research, we developed a modeling procedure that can be used to analyze many possible scenarios subject to strict conditions. Specifically, we developed a new Monte Carlo simulation procedure to create a collection of joint probability distributions, all of which match whatever information we have. Using this collection of distributions, we analyzed the accuracy of different approximations such as maximum entropy or copula-models. In addition, we proposed several new approximations that outperform previous methods. The objective of this research is four-fold. First, provide a new framework for approximation models. In particular, we presented four new models to approximate joint probability distributions based on geometric attributes and compared their performance to existing methods. Second, develop a new joint distribution simulation procedure (JDSIM) to sample joint distributions from the set of all possible distributions that match available information. This procedure can then be applied to different scenarios to analyze the sensitivity of a decision or to test the accuracy of an approximation method. Third, test the accuracy of seven approximation methods under a variety of circumstances. Specifically, we addressed the following questions within the context of multivariate discrete distributions: Are there new approximations that should be considered? Which approximation is the most accurate, according to different measures? How accurate are the approximations as the number of random variables increases? How accurate are they as we change the underlying dependence structure? How does accuracy improve as we add lower-order assessments? What are the implications of these findings for decision analysis practice and research? While the above questions are easy to pose, they are challenging to answer. For Decision Analysis, the answers open a new avenue to address partial information, which bing us to the last contribution. Fourth, propose a new approach to decision making with partial information. The exploration of old and new approximations and the capability of creating large collections of joint distributions that match expert assessments provide new tools that extend the field of decision analysis. In particular, we presented two sample cases that illustrate the scope of this work and its impact on uncertain decision making. / text

A new method is developed to determine a set of informative and refined interface assertions satisfied by functions that are represented by feed-forward neural networks. Neural networks have often been criticized for their low degree of comprehensibility.It is difficult to have confidence in software components if they have no clear and valid interface description. Precise and understandable interface assertions for a neural network based software component are required for safety critical applications and for theintegration into larger software systems. The interface assertions we are considering are of the form &quote if the input x of the neural network is in a region (alpha symbol) of the input space then the output f(x) of the neural network will be in the region (beta symbol) of the output space &quote and vice versa. We are interested in computing refined interface assertions, which can be viewed as the computation of the strongest pre- and postconditions a feed-forward neural network fulfills. Unions ofpolyhedra (polyhedra are the generalization of convex polygons in higher dimensional spaces) are well suited for describing arbitrary regions of higher dimensional vector spaces. Additionally, polyhedra are closed under affine transformations. Given a feed-forward neural network, our method produces an annotated neural network, where each layer is annotated with a set of valid linear inequality predicates. The main challenges for the computation of these assertions is to compute the solution of a non-linear optimization problem and the projection of a polyhedron onto a lower-dimensional subspace.

Η παρούσα διατριβή αφορά στη μελέτη του συνόρου υπερβολικών δισδιάστατων πολυέδρων. Οι χώροι οι οποίοι μελετώνται κατασκευάζονται κολλώντας υπερβολικά τρίγωνα τα οποία έχουν 2 τουλάχιστον κορυφές στο άπειρο. Οι συγκολλήσεις γίνονται με ισομετρίες κατά μήκος των πλευρών των τριγώνων και οι χώροι οι οποίοι προκύπτουν εφοδιάζονται φυσιολογικά με μία γεωμετρία η οποία έχει ομοιότητες με την γεωμετρία των υπερβολικών πολλαπλοτήτων. Αρχικά μελετάμε τις βασικές ιδιότητες των δισδιάστατων ιδεωδών πολυέδρων και αποδεικνύουμε ότι: «Για κάθε δύο σημεία του συνόρου του καθολικού καλύμματος του χώρου που κατασκευάζουμε, υπάρχει άπειρο πλήθος υποχώρων του συνόρου ομοιομορφικών με το οι οποίοι περιέχουν τα σημεία αυτά». Στη συνέχεια, για μια ειδική κλάση πολυέδρων που κατασκευάζουμε κολλώντας με ισομετρίες κατά μήκος των πλευρών τους πεπερασμένα υπερβολικά τρίγωνα τα οποία έχουν δύο κορυφές στο άπειρο, αποδεικνύουμε επιπλέον ότι: «το σύνορο του καθολικού καλύμματος του χώρου που κατασκευάζουμε είναι τοπικά συνεκτικό κατά τόξα». Τέλος, στην τρίτη ενότητα δίδουμε μια τοπολογική περιγραφή του συνόρου των ιδεωδών πολυέδρων διάστασης 2. / The present work is related to the study of the visual boundary of hyperbolic two dimensional simplicial complexes. We construct (and study) spaces by gluing hyperbolic triangles with at least two vertices at infinity. We glue the triangles by isometries along their sides and we study the derived spaces. In the first chapter it is proved that for every two points in the visual boundary of the universal covering of a two dimensional ideal polyhedron, there is an infinity of paths joining them. In the second chapter, a class of hyperbolic two dimensional complexes X is defined. Is is shown that the limit set of the action of π1(X) on the universal covering of X, is equal to the visual boundary and also that the visual boundary is path connected and locally path connected. Finally, in the third chapter a kind of Sierpinski set is described which is homeomorphic to the visual boundary of certain ideal polyhedra.

Cette thèse porte sur les techniques d'analyse formelle de systèmes hybrides à dynamiques continues non linéaire. Ses contributions portent sur les algorithmes d'atteignabilité et sur les problèmatiques liées à la representation des ensembles atteignables. This thesis deals with formal analysis of hybrid system with non linear continous dynamic. It contributes to the fields of reachability analysis algorithm and the set representation. / In this thesis, we presented our contributions to the formal analysis of dynamical systems. We focused on the problem of efficiently computing an accurate approximation of the reachable sets under nonlinear dynamics given by differential equations. Our aim was also to design scalable methods which can handle large systems. The first contribution of this thesis concerns the dynamic hybridization technique for a large class of nonlinear systems. We focused on the hybridization domain construction such that the linear interpolation realized in this domain ensures a desired error between the original system trajectories and those computed with the approximated system. We propose a construction method which tends to maximize the domain volume which reduce the number of creation of new domains during the analysis. The second research direction that we followed concerns a subclass of nonlinear dynamical systems which are the polynomial systems. Our results for the reachability analysis of these systems are based on the Bernstein expansion properties. We approximate an initial reachability computation (which requires solving polynomial optimization problems) with an accurate over-approximation (which requires solving linear optimization problems). The last theoretical contribution concerns the reachability analysis of linear systems with polyhedral input which often result from approximation of nonlinear systems. We proposed a technique to refine

Submitted by Maike Costa (maiksebas@gmail.com) on 2016-03-28T12:29:14Z No. of bitstreams: 1 arquivototal.pdf: 800250 bytes, checksum: 42e76ab05ea580b4fd24a3312b9b4212 (MD5) / Made available in DSpace on 2016-03-28T12:29:14Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 800250 bytes, checksum: 42e76ab05ea580b4fd24a3312b9b4212 (MD5) Previous issue date: 2014-08-29 / In this work we present a study of the regular polyhedra, comparing and discussing the concepts and de nitions given in the study of regular polyhedra in textbooks most widely used in Brazilian high schools. We prove the theorem of Euler, we calculate surface areas and volumes of regular polyhedra. Finally, we present some mathematical software that can be used by students and mathematics teachers in the spatial geometry classes as auxiliary material in the teaching and learning of this subject in the classroom. / Neste trabalho apresentamos um estudo sobre os poliedros regulares, comparando e discutindo os conceitos e as de nições que são dadas no estudo dos poliedros regulares nos livros didáticos mais utilizados nas escolas brasileiras de Ensino Médio. Provamos o teorema de Euler, calculamos áreas de superfícies e os volumes dos poliedros regulares. Por m, apresentamos alguns softwares matemáticos que podem ser utilizados pelos alunos e professores de Matemática nas aulas de geometria espacial como material auxiliar no processo de ensino e aprendizagem deste tema em sala de aula.

Submitted by Clebson Anjos (clebson.leandro54@gmail.com) on 2015-05-18T18:12:57Z No. of bitstreams: 1 arquivototal.pdf: 13930019 bytes, checksum: d5e52fb67904848f89fafaf5ec93c06d (MD5) / Approved for entry into archive by Clebson Anjos (clebson.leandro54@gmail.com) on 2015-05-18T18:15:19Z (GMT) No. of bitstreams: 1 arquivototal.pdf: 13930019 bytes, checksum: d5e52fb67904848f89fafaf5ec93c06d (MD5) / Made available in DSpace on 2015-05-18T18:15:19Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 13930019 bytes, checksum: d5e52fb67904848f89fafaf5ec93c06d (MD5) Previous issue date: 2013-09-24 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This dissertation is focused on the Euler's theorem for polyhedra homeomorphic to the sphere. Present statements made by Cauchy, Poincaré and Legendre. As a consequence we show that there are only ve regular convex polyhedra, called polyhedra Plato. / Esta dissertação tem como tema central o Teorema de Euler para poliedros homeomorfos à esfera. Apresentamos demonstrações feitas por Cauchy, Poincaré e Legendre. Como consequência mostramos a existência de apenas cinco poliedros convexos regulares, os chamados poliedros de Platão.

Etant donné trois entiers k, s et a, nous prouvons dans le premier chapitre qu'il existe un graphe k-régulier fini (resp. un graphe k-régulier connexe fini) dont le groupe d'automorphismes a exactement s orbites sur l'ensemble des sommets et a orbites sur l'ensemble des arêtes si et seulement si<p><p>(s,a)=(1,0) si k=0,<p>(s,a)=(1,1) si k=1,<p>s=a>0 si k=2,<p>0< s <= 2a <= 2ks si k>2.<p><p>(resp.<p>(s,a)=(1,0) si k=0,<p>(s,a)=(1,1) si k=1 ou 2,<p>s-1<=a<=(k-1)s+1 et s,a>0 si k>2.)<p><p>Nous étudions les polyèdres convexes de R³ dans le second chapitre. Pour tout polyèdre convexe P, nous notons Isom(P) l'ensemble des isométries de R³ laissant P invariant. Si G est un sous-groupe de Isom(P), le f_G-vecteur de P est le triple d'entiers (s,a,f) tel que G ait exactement s orbites sur l'ensemble sommets de P, a orbites sur l'ensemble des arêtes de P et f orbites sur l'ensemble des faces de P. Remarquons que (s,a,f) est le f_{id}-vecteur (appelé f-vecteur dans la littérature) d'un polyèdre si ce dernier possède exactement s sommets, a arêtes et f faces. Nous généralisons un théorème de Steinitz décrivant tous les f-vecteurs possibles. Pour tout groupe fini G d'isométries de R³, nous déterminons l'ensemble des triples (s,a,f) pour lesquels il existe un polyèdre convexe ayant (s,a,f) comme f_G-vecteur. Ces résultats nous permettent de caractériser les triples (s,a,f) pour lesquels il existe un polyèdre convexe tel que Isom(P) a s orbites sur l'ensemble des sommets, a orbites sur l'ensemble des arêtes et f orbites sur l'ensemble des faces.<p><p>La structure d'incidence I(P) associée à un polyèdre P consiste en la donnée de l'ensemble des sommets de P, l'ensemble des arêtes de P, l'ensemble des faces de P et de l'inclusion entre ces différents éléments (la notion de distance ne se trouve pas dans I(P)). Nous déterminons également l'ensemble des triples d'entiers (s,a,f) pour lesquels il existe une structure d'incidence I(P) associée à un polyèdre P dont le groupe d'automorphismes a exactement s orbites de sommets, a orbites d'arêtes et f orbites de sommets. / Doctorat en sciences, Spécialisation mathématiques / info:eu-repo/semantics/nonPublished

Orientadores: Sandra Augusta Santos, Roberto Andreani / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-14T11:40:11Z (GMT). No. of bitstreams: 1 Shirabayashi_WesleyVagnerInes_D.pdf: 877226 bytes, checksum: a814cd9947431a0aee17517c4cc953f4 (MD5) Previous issue date: 2009 / Resumo: Neste trabalho reformulamos o problema de complementaridade não linear generalizado (GNCP) em cones poliedrais como um sistema não linear com restrição de não negatividade em algumas variáveis, e trabalhamos na resolução de tal reformulação por meio de estratégias de pontos interiores. Em particular, definimos dois algoritmos e provamos a convergência local de tais algoritmos sob hipóteses usuais. O primeiro algoritmo é baseado no método de Newton, e o segundo, no método tensorial de Chebyshev. O algoritmo baseado no método de Chebyshev pode ser visto como um método do tipo preditor-corretor. Tal algoritmo, quando aplicado a problemas em que as funções envolvidas são afins, e com escolhas adequadas dos parâmetros, torna-se o bem conhecido algoritmo preditor-corretor de Mehrotra. Também apresentamos resultados numéricos que ilustram a competitividade de ambas as propostas. / Abstract: In this work we reformulate the generalized nonlinear complementarity problem (GNCP) in polyhedral cones as a nonlinear system with nonnegativity in some variables and propose the resolution of such reformulation through interior-point methods. In particular we define two algorithms and prove the local convergence of these algorithms under standard assumptions. The first algorithm is based on Newton's method and the second, on the Chebyshev's tensorial method. The algorithm based on Chebyshev's method may be considered a predictor-corrector one. Such algorithm, when applied to problems for which the functions are affine, and the parameters are properly chosen, turns into the well-known Mehrotra's predictor corrector algorithm. We also present numerical results that illustrate the competitiveness of both proposals. / Doutorado / Otimização / Doutor em Matemática Aplicada