Title: Homotopic structures in algebra, geometry and mathematical physics Author: Eva Černohorská Department: Mathematical Institute of Charles University Supervisor: RNDr. Martin Markl, DrSc., Institute of Mathematics of the Academy of Sciences of the Czech Republic, Mathematical Institute of Charles University Abstract: The aim of this thesis was to generalize the result that associative algebras on finite dimensional vector spaces can be described using differentials on free algebras. This result is limited by the duality of vector spaces. If we assume that the underlying space has a linear topology, then we can use the duality between discrete and linearly compact (profinite) vector spaces. To generalize the notion of an algebra, we need to recall the completed tensor product on linear vector spaces. Since this topics does not seem to be sufficiently covered by the literature, this thesis could serve also as a comprehensive text on linear vector spaces and their completed tensor products. We prove that also A∞ structures on linearly compact vector spaces could be represented by differentials on a free algebra. Keywords: Strongly homotopy associative algebra, linear topological vector space, Pontryagin duality, completed tensor product, differential

Les propriétés d'électrons en deux dimensions (2D) soulèvent des questions fondamentales qui ont été largement explorées au moyen des techniques théoriques de la matière condensée. L'extension de modèles classiques tel le modèle de Hubbard en 2D, en incluant par exemple plusieurs bandes électroniques, ore la possibilité d'accéder à des phénomèmes plus complexes, comme l'interaction du transport électronique et du magnétisme observé dans les composés de fermions lourds. Ces modèles sont en lien direct avec la question de couches minces couplées, les hétérostructures, qui sont depuis peu l'objet d'intenses recherches et orent la possibilité d'intéressantes applications. Dans ce contexte, nous étudions numériquement diérents syst èmes au moyen de la méthode du Monte Carlo Quantique du Déterminant. Tout d'abord, l'eet de la corrélation électronique dans un isolant de bande est évaluée, montrant en particulier l'absence d'une phase métallique intermédiaire. Un deuxième système est composé de deux bandes électroniques couplées, dans lequel l'eet de la largeur de bande de la partie corrélée est exploré de façon systématique. Finalement, nous étudions une interface métal-isolant, qui présente une phase intermédiaire surprenante lorsque le couplage à l'interface est ajusté. / The properties of electrons in two dimensions (2D) raise fundamental questions that have been extensively explored by condensed matter theory. Extending standard frameworks such as the 2D Hubbard model by accounting for more than one electronic band oers the opportunity to access more complex phenomena, such as the interplay between transport and magnetism found in heavy-fermions materials. Such models are directly connected to the problem of coupled layers in complex materials known as heterostructures, which have been widely studied and synthesized in recent years, and are expected to lead to important applications. In that context, we study numerically several systems, by mean of the Determinant Quantum Monte Carlo Method (DQMC). We rst analyze the eect of electronic correlation in a band insulator, showing in particular the absence of an intermediate metallic phase. A second system consists of two coupled bands that modelize a heavy fermion model, in which the role of the bandwidth of the correlated band is systematically investigated. Finally, we consider the case of a metal-insulator interface, unveiling an intriguing intermediate phase as the interfacial coupling is tuned.

Cette thèse traite de la solution numérique de deux types de problèmes stochastiques. Premièrement, nous nous intéressons aux EDS fortement oscillantes, c'est-à-dire, les systèmes composés de variables ergodiques évoluant rapidement par rapport aux autres. Nous proposons un algorithme basé sur des résultats d'homogénéisation. Il est défini par un schéma d'Euler appliqué aux variables lentes couplé avec un estimateur à pas décroissant pour approcher la limite ergodique des variables rapides. Nous prouvons la convergence forte de l'algorithme et montrons que son erreur normalisée satisfait un résultat du type théorème limite centrale généralisé. Nous proposons également une version extrapolée de l'algorithme ayant une meilleure complexité asymptotique en satisfaisant les mêmes propriétés que la version originale. Ensuite, nous étudions la solution des EDS de type McKean-Vlasov (EDSPR-MKV) associées à la solution de certains problèmes de contrôle sous un environnement formé d'un grand nombre de particules ayant des interactions du type champ-moyen. D'abord, nous présentons un nouvel algorithme, basé sur la méthode de cubature sur l'espace de Wiener, pour approcher faiblement la solution d'une EDS du type McKean-Vlasov. Il est déterministe et peut être paramétré pour atteindre tout ordre de convergence souhaité. Puis, en utilisant ce nouvel algorithme, nous construisons deux schémas pour résoudre les EDSPR-MKV découplées et nous montrons que ces schémas ont des convergences d'ordres un et deux. Enfin, nous considérons le problème de réduction de la complexité de la méthode présentée tout en respectant la vitesse de convergence énoncée. / This Ph.D. thesis deals with the numerical solution of two types of stochastic problems. First, we investigate the numeric solution to strongly oscillating SDEs, i.e. systems in which some ergodic state variables evolve quickly with respect to the remaining ones. We propose an algorithm that uses homogenization results and consists of an Euler scheme for the slow scale variables coupled with a decreasing step estimator for the ergodic averages of the fast variables. We prove the strong convergence of the algorithm as well as a generalized central limit theorem result for the normalized error distribution. In addition, we propose an extrapolated version applicable under stronger regularity assumptions and which satisfies the same properties of the original algorithm with lower asymptotic complexity. Then, we treat the problem of solving decoupled Forward Backward Stochastic Differential equations of McKean-Vlasov type (MKV-FBSDE) which appear in some stochastic control problems in an environment of a large number of particles with mean field interactions. As a first step, we propose a new algorithm, based on the cubature method on Wiener spaces, to weakly approach the solution of a McKean-Vlasov SDE. It is deterministic and can be parametrized to obtain any given order of convergence. Using this first forward approximation algorithm, we construct two procedures to solve the decoupled MKV-FBSDE and show that they converge with orders one and two under appropriate regularity conditions. Finally, we consider the problem of reducing the complexity of the presented method while preserving the presented convergence rates.

In his study of spectra of f-rings via pointfree topology, Banaschewski [6] considers lattices of l-ideals, radical l-ideals, and saturated l-ideals of a given f-ring A. In each case he shows that the lattice of each of these kinds of ideals is a coherent frame. This means that it is compact, generated by its compact elements, and the meet of any two compact elements is compact. This will form the basis of our main goal to show that the lattice-ordered rings studied in [6] are coherent frames. We conclude the dissertation by revisiting the d-elements of Mart nez and Zenk [30], and characterise them analogously to d-ideals in commutative rings. We extend these characterisa-tions to algebraic frames with FIP. Of necessity, this will require that we reappraise a great deal of Banaschewski's work on pointfree spectra, and that of Mart nez and Zenk on algebraic frames. / Mathematical Sciences / M. Sc. (Mathematics)

[pt] Um operador definido em um espaço de Hilbert é uniformemente estável se ele converge na topologia da norma para o operador nulo. O problema de Estabilidade Multiplicativa investiga quais são as classes de operadores que estabilizam uniformemente o operador original por uma perturbação multiplicativa. Neste trabalho colocamos este problema no contexto de produto tensorial e investigamos quais as classes que estabilizam multiplicativamente Contrações Fortemente Estáveis sob uma perturbação compacta. Em particular, apresentamos uma solução para o Problema de Estabilidade Multiplicativa para Contrações Fortemente Estáveis. / [en] An operator on a Hilbert space is uniformly stable if it converges to the null operator on the norm topology. The Multiplicative Stabilization Problem investigates which operators classes uniformly stabilize de original operator under multiplicative perturbation. This work consider the previous problem under the tensor product framework and investigates which operators classes multiplicative stabilize Strongly Stable Contraction under compact perturbations. We have established a solution to the Multiplicative Stabilization Problem for Strongly Stable Contractions.

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The research field of graded ring theory is a rich area of mathematics with many connections to e.g. the field of operator algebras. In the last 15 years, algebraists and operator algebraists have defined algebraic analogues of important operator algebras. Some of those analogues are rings that come equipped with a group grading. We want to reach a better understanding of the graded structure of those analogue rings. Among group graded rings, the strongly graded rings stand out as being especially well-behaved. The development of the general theory of strongly graded rings was initiated by Dade in the 1980s and since then numerous structural results have been established for strongly graded rings.  In this thesis, we study the class of epsilon-strongly graded rings which was recently introduced by Nystedt, Öinert and Pinedo. This class is a natural generalization of the well-studied class of unital strongly graded rings. Our aim is to lay the foundation for a general theory of epsilon-strongly graded rings generalizing the theory of strongly graded rings. This thesis is based on three articles. The first two articles mainly concern structural properties of epsilon-strongly graded rings. In the first article, we investigate a functorial construction called the induced quotient group grading. In the second article, using results from the first article, we generalize the Hilbert Basis Theorem for strongly graded rings to epsilon-strongly graded rings and apply it to Leavitt path algebras.  In the third article, we study the graded structure of algebraic Cuntz-Pimsner rings. In particular, we obtain a partial classification of unital strongly, epsilon-strongly and nearly epsilon-strongly graded Cuntz-Pimsner rings up to graded isomorphism.

Cette thèse traite de la notion d’observabilité états-entrées (ISO) dans les systèmes de réseaux linéaires. On cherche des caractérisations graphiques en utilisant la notion de structure (resp. s-structural) ISO. Tout d’abord nous nous concentrons sur les systèmes de réseaux linéaires invariant en temps, représentés par des graphes fixes et fournissons des caractérisations pour structure ISO fortes. Par la suite, nous nous intéressons aux systèmes de réseaux linéaires variant en temps, dans lesquels nous nous concentrons d’abord sur le cas particulier des graphes fixes (la structure du graphe reste fixe mais les poids sur les arêtes peuvent varier à la dynamique qui varie dans le temps). Nous montrons que, sous des hypothèses appropriées sur la structure des matrices de commande, d’observation et d’action, l’ISO d’un système équivaut à l’observabilité d’un sous-système défini de manière appropriée. Par la suite, nous exploitons cette équivalence pour obtenir des caractérisations graphiques de l’ISO structurelle (resp. S-structurelles).Ensuite, pour la configuration du LTV, nous considérons le cas plus général des graphes variant en temps tout en ne faisant aucune hypothèse sur la structure des matrices du système. Nous introduisons deux descriptions appropriées de la collection complète de graphes, nommés graphes dynamiques et graphes dynamiques bipartites. Deux caractérisations équivalentes de l’ISO structurelle sont ensuite présentées en termes d’existence d’une liaison et d’une correspondance de taille appropriée, respectivement, dans le graphe dynamique et dans le graphe dynamique bipartite. Pour les ISO fortement structurelles, nous fournissons une condition suffisante et une condition nécessaire, concernant à la fois à l’existence d’une correspondance restreinte de taille appropriée dans le graphe dynamique bipartite et dans un sous-graphe de celle-ci. Lorsqu’il n’y a pas d’action directe de la commande sur les mesures, les deux conditions peuvent être fusionnées pour donner lieu à une condition nécessaire et suffisante.Enfin, nous présentons un algorithme récursif sans biais qui estime simultanément les états et les commandes. Nous nous concentrons sur les systèmes à retard-l inversibles avec retard intrinsèque l > 1, où la reconstruction de la commande est possible uniquement en utilisant des sorties allant jusqu’à un nombre "l" de temps plus tard. En montrant une équivalence avec un système de descripteurs, nous présentons des conditions dans lesquelles le filtre variant en temps converge vers un filtre stationnaire stable, impliquant la solution d’une équation algébrique à temps discret de Riccati. / This thesis deals with the notion of Input and State Observability (ISO) in linear network systems. One seeks graphical characterizations using the notion of structural (resp. s-structural) ISO. We first focus on linear time-invariant network systems ,represented by fixed graphs, and provide characterizations for strong structural ISO. Thereafter, we turn our attention to linear time-varying network systems wherein we first narrow our attention to the particular case of fixed graphs (i.e., the structure of the graph remains fixed; the weights along the edges are allowed to vary, thereby giving rise to time-varying dynamics). We show that, under suitable assumptions on the structure of input, output and feedthrough matrices, ISO of a system is equivalent to observability of a suitably defined subsystem. Subsequently, we exploit this equivalence to obtain graphical characterizations of structural (resp. s-structural) ISO.Thereafter, for the LTV setting, we consider the more general case of time-varying graphs and furthermore make no assumptions on the structure of system matrices. We introduce two suitable descriptions of the whole collection of graphs, which are named as dynamic graph and dynamic bipartite graph. Two equivalent characterizations of structural ISO are then stated in terms of existence of a linking and a matching of suitable size in the dynamic graph and in the dynamic bipartite graph, respectively. For strongly structural ISO, we provide a sufficient condition and a necessary condition, both concerning the existence of a uniquely restricted matching of suitable size in the dynamic bipartite graph and in a subgraph of it. When there is no direct feedthrough of the input on the measurements, the two conditions can be merged to give rise to a necessary and sufficient condition.Finally, we present an unbiased recursive algorithm that simultaneously estimates states and inputs. We focus on delay-$ell$ left invertible systems with intrinsic delay $ellgeq 1$, where the input reconstruction is possible only by using outputs up to $ell$ time steps later in the future. By showing an equivalence with a descriptor system, we state conditions under which the time-varying filter converges to a stationary stable filter, involving the solution of a discrete-time algebraic Riccati equation.

Orientador: Ricardo Luís Doretto / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Física Gleb Wataghin / Made available in DSpace on 2018-09-03T02:41:35Z (GMT). No. of bitstreams: 1 Leite_LeonardoDaSilvaGarcia_M.pdf: 1468749 bytes, checksum: 2f4e22a34c4a72b7b68eec6673285298 (MD5) Previous issue date: 2017 / Resumo: Nesse trabalho, calculamos a entropia de emaranhamento de um antiferromagneto de Heisenberg dimerizado em uma rede quadrada. Dois padrões de dimerização distintos são considerados: colunar e alternado. Em ambos os casos, focamos na fase de sólidos de singletos (VBS) que é descrita pela representação dos operadores de ligação. Nesse formalismo, o hamiltoniano de spin original é mapeado em um modelo efetivo de bósons interagentes com excitações de tripleto. O hamiltoniano efetivo é estudado na aproximação harmônica e o espectro das excitações elementares e o diagrama de fase dos dois modelos dimerizados são determinados. Consideramos um subsistema unidimensional (cadeia) de comprimento $L$ dentro de uma rede quadrada com condições periódicas de contorno e calculamos a entropia de emaranhamento. Seguimos um procedimento analítico baseado na teoria de ondas de spin modificadas que havia sido desenvolvido originalmente para calcular a entropia de emaranhamento em fases magneticamente ordenadas. Em particular, esse procedimento nos permite considerar subsistemas unidimensionais compostos por até 200 sítios. Combinamos esse procedimento com o formalismo dos operadores de ligação na aproximação harmônica e mostramos que, para os dois modelos de Heisenberg dimerizados, a entropia de emaranhamento da fase VBS obedece uma lei de área. Tanto para a dimerização colunar quanto para a alternada, mostramos que a entropia de emaranhamento aumenta à medida que o sistema se aproxima da transição de fase quântica entre as fases Néel-VBS / Abstract: Nesse trabalho, calculamos a entropia de emaranhamento de um antiferromagneto de Heisenberg dimerizado em uma rede quadrada. Dois padrões de dimerização distintos são considerados: colunar e alternado. Em ambos os casos, focamos na fase de sólidos de singletos (VBS) que é descrita pela representação dos operadores de ligação. Nesse formalismo, o hamiltoniano de spin original é mapeado em um modelo efetivo de bósons interagentes com excitações de tripleto. O hamiltoniano efetivo é estudado na aproximação harmônica e o espectro das excitações elementares e o diagrama de fase dos dois modelos dimerizados são determinados. Consideramos um subsistema unidimensional (cadeia) de comprimento $L$ dentro de uma rede quadrada com condições periódicas de contorno e calculamos a entropia de emaranhamento. Seguimos um procedimento analítico baseado na teoria de ondas de spin modificadas que havia sido desenvolvido originalmente para calcular a entropia de emaranhamento em fases magneticamente ordenadas. Em particular, esse procedimento nos permite considerar subsistemas unidimensionais compostos por até 200 sítios. Combinamos esse procedimento com o formalismo dos operadores de ligação na aproximação harmônica e mostramos que, para os dois modelos de Heisenberg dimerizados, a entropia de emaranhamento da fase VBS obedece uma lei de área. Tanto para a dimerização colunar quanto para a alternada, mostramos que a entropia de emaranhamento aumenta à medida que o sistema se aproxima da transição de fase quântica entre as fases Néel-VBS / Mestrado / Física / Mestre em Física / 1547615/2015 / CAPES

A number of attempts have been made to facilitate agricultural trade liberalisation over the last decades. In spite of these efforts, trade liberalisation of agricultural and food processed goods has been modest. It is argued that this lack of trade liberalisation is explained by the existence of governments that are politically biased in the sense that they place anti-trade policies in order to favour powerful sectors in the economy. While there exists some evidence supporting this argument, it is difficult to assess how these biases influence agricultural trade patterns because existing quantitative modelling approaches do not normally consider simultaneously key aspects that characterise the food industry such as intra-industry trade and the existence of intermediaries in the supply chain with significant market power, among others. The objective of this thesis is to offer an alternative theoretical model that has the potential to accommodate these key aspects and corresponds to an international trade network model that extends the framework developed by Goyal and Joshi (2006). The model was solved by means of simulations and the results revealed that policy biased indeed can prevent trade liberalisation of agricultural and food processed goods. However, other factors that apparently have not been reported so far and that are related to the market power exercised by intermediaries were identified. They correspond to the position of a country in the trade network (i.e. a country occupying a central position in the network is less likely to support trade liberalisation independently of any policy bias), the possibility that global free trade is an unlikely outcome, and the possibility that the world is trapped in an inefficient international trade network. The results also revealed that the adoption of compensatory lump sum payments across countries (i.e. inter-node transfers) or across sectors within a country (i.e. intra-node transfers) could be used a potential tools to achieve global free trade in agriculture as they can compensate losers from trade by gainers achieving, as a consequence, Pareto improving outcomes.