Apresentamos neste trabalho uma definição de complexidade na categoria derivada de complexos (limitados superiormente) de módulos sobre uma k-álgebra de dimensão finita. Um dos resultados que conseguimos foi uma relação entre a complexidade de objetos indecomponíveis e a noção de dimensão global forte. Mais especificamente, mostramos que a existência de um objeto indecomponível na categoria derivada limitada superiormente com complexidade não nula é condição suficiente para que a respectiva álgebra tenha dimensão global forte infinita. Também investigamos se existe uma relação entre as dimensões global e global forte da classe das álgebras shod (Coelho e Lanzilotta, 2009). Fomos motivados pela caracterização da classe das álgebras quase inclinadas (Happel, Reiten e Smalo, 1996) em termos da sua dimensão global forte, dada por D. Happel e D. Zacharia (2008), e pelo fato das álgebras shod serem uma generalização das álgebras quase inclinadas. Nossa conclusão foi que não existe, em geral, uma caracterização das álgebras shod em termos de sua dimensão global forte. Isto é, mostramos que para cada inteiro d > 2 existe uma álgebra shod estrita cuja dimensão global forte é igual a d. / We introduce in this thesis a definition of complexity in the derived category of bounded above complexes of modules over a finite dimensional k-algebra. One of our result shows a relationship between the complexity of indecomposable objects and the notion of strong global dimension. More specifically, we prove that the existence of an indecomposable object in the category derived bounded above whose complexity is not zero is a sufficient condition for corresponding algebra being of infinite strong global dimension. We also investigate the existence of a relationship between the global dimension and the strong global dimension of shod algebras (Coelho and Lanzilotta, 1999). Our motivation came from characterization of quasitilted algebras (Happel, Reiten and Smalo, 1996) by its strong global dimension, given by D. Happel and D. Zacharia (2008), and from the fact that shod algebras are a generalization of quasitilted algebras. Our conclusion was that there is not in general a characterization of shod algebras in terms of its strong global dimension. This conclusion comes from the fact that we showed that for each integer d > 2 there exists a strictly shod algebra whose strong global dimension is d.

O problema de segmentação de sequências tem o objetivo de particionar uma sequência ou um conjunto delas em um número finito de segmentos distintos tão homogêneos quanto possível. Neste trabalho consideramos o problema de segmentação de um conjunto de sequências aleatórias, com valores em um alfabeto $\\mathcal$ finito, em um número finito de blocos independentes. Supomos ainda que temos $m$ sequências independentes de tamanho $n$, construídas pela concatenação de $s$ segmentos de comprimento $l^{*}_j$, sendo que cada bloco é obtido a partir da distribuição $\\p _j$ em $\\mathcal^{l^{*}_j}, \\; j=1,\\cdots, s$. Além disso denotamos os verdadeiros pontos de corte pelo vetor ${{\\bf k}}^{*}=(k^{*}_1,\\cdots,k^{*}_)$, com $k^{*}_i=\\sum _{j=1}^l^{*}_j$, $i=1,\\cdots, s-1$, esses pontos representam a mudança de segmento. Propomos usar o critério da máxima verossimilhança penalizada para inferir simultaneamente o número de pontos de corte e a posição de cada um desses pontos. Também apresentamos um algoritmo para segmentação de sequências e realizamos algumas simulações para mostrar seu funcionamento e sua velocidade de convergência. Nosso principal resultado é a demonstração da consistência forte do estimador dos pontos de corte quando o $m$ tende ao infinito. / The sequence segmentation problem aims to partition a sequence or a set of sequences into a finite number of segments as homogeneous as possible. In this work we consider the problem of segmenting a set of random sequences with values in a finite alphabet $\\mathcal$ into a finite number of independent blocks. We suppose also that we have $m$ independent sequences of length $n$, constructed by the concatenation of $s$ segments of length $l^{*}_j$ and each block is obtained from the distribution $\\p _j$ over $\\mathcal^{l^{*}_j}, \\; j=1,\\cdots, s$. Besides we denote the real cut points by the vector ${{\\bf k}}^{*}=(k^{*}_1,\\cdots,k^{*}_)$, with $k^{*}_i=\\sum _{j=1}^l^{*}_j$, $i=1,\\cdots, s-1$, these points represent the change of segment. We propose to use a penalized maximum likelihood criterion to infer simultaneously the number of cut points and the position of each one those points. We also present a algorithm to sequence segmentation and we present some simulations to show how it works and its convergence speed. Our principal result is the proof of strong consistency of this estimators when $m$ grows to infinity.

Este trabalho introduz uma nova discretização da banda de condução no método de Grupo de Renormalização Numérico. Com essa técnica, a susceptibilidade magnética do modelo de Anderson de duas impurezas, no limite Kondo, e calculada. Como ilustração, a densidade espectral do modelo também é calculada. A nova técnica baseia-se na simetria de paridade do modelo para discretizar diferentemente à banda de condução associada a cada paridade. Sua extensão ao modelo de rede é indicada. A técnica reduz o tempo computacional e permite usar maiores valores do parâmetro de discretização do que no método tradicional. Para um mesmo tempo de cálculo, nossos resultados são muito mais precisos do que os encontrados na literatura. A susceptibilidade é calculada na aproximação de acoplamento independente da energia. Uma interação de troca, tipo RKKY, é somado ao Hamiltoniano do modelo. Para acoplamento ferromagnético, obtém-se efeito Kondo de dois estágios. O estado fundamental é singleto com defasagem de PI/2 na banda de condução. Para acoplamento antiferromagnético fraco, um efeito Kondo é obtido. Para fortes acoplamentos antiferromagnéticos, o estado fundamental e singleto sem defasagens. Um ponto fixo instável é observado separando as regiões de estado fundamental Kondo e antiferromagnético. Nesse ponto a susceptibilidade é nula e a defasagem é indefinida. / This work introduces an extension of the Numerical Renormalization Group approach to compute thermodynamically properties of impurities in metals, based on a novel logarithmic discretization of the conduction band. On the basis of the new method, the thermal dependence of the magnetic susceptibility for the Kondo limit of the two-impurity Anderson model is computed. As another illustration, the impurity spectral density for the same model is calculated analytically in the weakly correlated regime. The new approach takes advantage of the parity-inversion symmetry of the model to discretize differently the odd and the even conduction channels (for Ni impurities, the conduction band could likewise be divided into Ni channels, each of which would be discretized in a different way). The resulting mesh describes better the continuum of the conduction states than the mesh in the standard Numerical Renormalization Group method; as a consequence, the new procedure is substantially less expensive when computing any given thermodynamical property with a given accuracy, thus we are able to compute the temperature dependence of the magnetic susceptibility with a small fraction of the effort involved in the recently reported computation of the ground state properties for the two impurity Kondo model. As in previous Renormalization Group work, the model Hamiltonian is diagonalized within the energy-independent coupling approximation. One well-known shortcoming of this approximation is its inability to generate antiferromagnetic RKKY couplings between the impurities; to compensate, again following previous work; we have added to the Hamiltonian an artificial exchange coupling Io. For weak antiferromagnetic or ferromagnetic couplings, the effective magnetic moment of the impurities decreases with temperature, and as in the one-impurity Kondo effect, the ground-state conduction band is phase shifted by PI /2. For strong ferromagnetic coupling, the Kondo effect takes place in two stages, one for each conduction channel. For strong antiferromagnetic coupling, the magnetic moment also decreases, rapidly, with temperature, but the ground state conduction-band phase shift is zero. The regions of zero and PI /2 ground-state phase shifts are separated by an unstable fixed point. At this point, the magnetic susceptibility vanishes.

Este trabalho estuda o modelo de Falicov e Kimball com duas impurezas. O modelo consiste de um metal com duas impurezas separadas por uma distância R, cada uma das quais é representada por um único nível eletrônico. Um acoplamento V permite transferência de carga entre cada impureza e a banda de condução do metal. Além disso, cada impureza introduz um potencial espalhador G cuja intensidade depende da ocupação do seu nível, assim simulando a interação eletrostática entre um buraco na impureza e os elétrons de condução. Esta dissertação diagonaliza o Hamiltoniano do modelo pelo método do grupo de renormalização numérico. Dá-se atenção à possível equivalência entre este modelo (desprovido de spin) e o modelo de Kondo para duas impurezas. Discute-se em particular essa equivalência para R=0 e para R= INFINITO. Para R finito, apenas um primeiro passo na direção de se estabelecer a equivalência é dado: obtém-se uma expressão analítica para a taxa de transição eletrônica entre os níveis das impurezas e a banda de condução. / In this work, the two-impurity Falicov-Kimball model is studied. The model consists of a metal containing two impurities separated by a distance R, each represented by a single electronic level. A coupling V allows charge transfer between each impurity and the conduction band. In addition, each impurity scatters the conduction electrons with a localized potential G whose intensity depends on the occupancy of the impurity level; this mimics the Coulomb attraction between na impurity hole and the conduction band. This dissertation diagonalizes the model Hamiltonian with the numerical renormalization group method. In two special limits, R=0 and R=INFINITO, the equivalence between the (spinless) Falicov-Kimball model and the two-impurity Kondo model is discussed. For other impurity separations, only a first step torwards establishing that equivalence is taken: na analytical expression for the electronic transition rate between the impurity levels and the conduction states is obtained.

No presente estudo, amostras policristalinas ricas em Ta e com estequiometrias Ta1-xZrx; x < 0.15; foram preparadas através da mistura apropriada dos elementos metálicos, os quais foram fundidos em forno a arco elétrico sobre uma placa de cobre refrigerada a água e sob atmosfera de argônio de alta pureza. Os padrões de difração de raios-X das ligas, como fundidas (as cast) e tratadas termicamente a 850 °C por 24 h, revelaram a ocorrência de uma estrutura cristalina cúbica de corpo centrada bcc, tipo W, e parâmetros de rede que aumentam suavemente com o aumento do teor de Zr nas ligas. Medidas de susceptibilidade magnética dc, conduzidas nas condições de resfriamento da amostra em campo zero (ZFC) e do resfriamento com o campo magnético aplicado (FC), indicaram que supercondutividade volumétrica é observada abaixo de ~ 5.8, 6.9, 7.0 K em amostras com x = 0.05, 0.08, e 0.10, respectivamente. Essas temperaturas críticas supercondutoras são bastante superiores àquela observada no Ta elementar ~ 4.45 K. Medidas de resistividade elétrica na presença de campos magnéticos aplicados de até 9 T confirmaram a temperatura crítica supercondutora das amostras estudadas. O campo crítico superior Hc2 e o comprimento de coerência E foram estimados a partir dos dados de magnetorresistência. Os valores estimados de Hc2 foram de ~ 0.46, 1.78, 3.85 e 3.97 T, resultando em valores de E ~ 26.0, 13.6, 9.2 e 9.1 nm para as ligas as cast com x = 0.00, 0.05, 0.08 e 0.10, respectivamente. A partir dos dados experimentais do calor específico Cp das ligas, magnitudes estimadas do salto em Cp nas vizinhanças das transições supercondutoras indicaram valores maiores que o previsto pela teoria BCS. Utilizando as equações analíticas derivadas da teoria do acoplamento forte da supercondutividade foi então proposto que o aumento da temperatura de transição supercondutora nas ligas devido a substituição parcial do Ta por Zr está intimamente relacionado ao aumento do acoplamento elétron-fônon, visto que a densidade de estados eletrônicos no nível de Fermi foi estimada ser essencialmente constante através da série Ta1-xZrx com x < 0.10. / In the present study, polycrystalline samples of Ta-rich binary alloys with stoichiometry Ta1-xZrx; x < 0.15; were prepared by mixing appropriate amounts of the metallic elements which were arc-melted on a water-cooled hearth under high-purity argon atmosphere. The X-ray diffraction patterns of the as cast alloys and heat treated ones at 850 °C by 24 h revealed the occurrence of the body-centered cubic crystal structure bcc, type W, and lattice parameters that increase slightly with increasing Zr content. Magnetic susceptibility measurements dc, performed in zero-field cooling ZFC and field cooling FC processes, indicated that bulk superconductivity is observed below ~ 5.8, 6.9, and 7.0 K, in samples with x = 0.05, 0.08, and 0.10, respectively. These superconducting critical temperatures are higher than that of ~ 4.45 K found in elemental Ta. Electrical resistivity measurements under applied magnetic fields to 9 T corroborated the superconducting critical temperatures for the samples studied. The thermodynamic upper critical field Hc2 and the coherence length E were estimated from the magnetoresistance data. The estimated values of Hc2 were 0.46, 1.78, 3.85, and 3.97 T, leading to E 26.0, 13.6, 9.2, and 9.1 nm for the as cast alloys with x = 0.00, 0.05, 0.08, and 0.10, respectively. In addition to this, from the results of heat capacity Cp data, jumps in the vicinity of the superconducting transition were estimated and found to be larger than the one expected from the BCS theory. By using analytic equations derived from the strong coupling theory of superconductivity we argued that the enhancement of Tc in alloying Ta with Zr is due to the increase of the electron-phonon coupling, provided that the density of states in the Fermi level was found to be essentially constant in the series Ta1-xZrx; x < 0.10.

Doctor of Philosophy / Department of Physics / Itzhak Ben-Itzhak / The use of ultrafast lasers allows one to study and even control quantum mechanical systems on their natural timescales. Our aim is to study the fragmentation of small molecules in strong laser fields as a means to gain understanding of molecular dynamics and light-matter interactions. Our research group has utilized fast, positively charged molecular ion beams as targets to study and control fragmentation by strong laser fields. This approach allows for detection of all molecular fragments including neutrals, and a coincidence three-dimensional momentum imaging technique is used to characterize the fragmentation. A natural extension of these types of studies is to expand the types of molecular systems that can be studied, from positively charged molecules to neutral and negatively charged molecules. To that end, the primary technical development of this dissertation involved the generation and use of fast, negatively charged molecular beams. Using fast molecular anion beams as targets allows for the study of fragmentation in which all fragments are neutral. As a demonstration, we employ this capability to study F2- dissociation and photodetachment. The dissociation pathways are identified and used to evaluate the initial vibrational population of the F2- beam. The role of dissociation in photodetachment is also explored, and we find that it competes with other dissociative (F+F) and non-dissociative (F2) photodetachment mechanisms. Also highlighted are studies of fragmentation of LiO-, in which the dissociation into Li+O- fragments provides information about the structure of Li O-, including the bond dissociation energy, which was found to be larger than values based on theory. Studies of the autodetachment lifetimes of Li O- were also performed using a pump-probe technique. Additional experimental advancements have made successful pump-probe studies of the ionization of HD+ and Ar2+ possible. Enhancement in the ionization of dissociating HD+ and Ar2+ was observed at surprisingly large internuclear separation where the fragments are expected to behave like separate atoms. The analysis methods used to quantify this enhancement are also described. Finally, the production of excited Rydberg D* fragments from D2 molecules was studied utilizing a state-selective detection method. The carrier-envelope phase dependence of D* formation was found to depend on the range of excited final states of the atomic fragments. We also measured the excited state population of the D* fragments. Together, the studies presented in this work provide new information about fragmentation of positive, negative, and neutral molecules in strong laser fields, and the experimental developments serve as building blocks for future studies that will lead to a better understanding of molecular dynamics.

Dans de nombreuses applications, nous souhaitons interpoler ou approcher une fonction de plusieurs variables possédant certaines propriétés ou “formes” géométriques, telles que la régularité, la monotonie, la convexité ou la non-négativité. Ces propriétés sont importantes pourdes applications en physique (par exemple, la courbe pression-volume doit avoir une dérivée non négative), aussi bien où le problème de l’interpolation conservant la forme est essentiel dans divers problèmes de l’industrie (par exemple, modélisation automobile, construction de la surface dumasque). Par conséquent, une question importante se pose : comment calculer la meilleure approximation possible à une fonction donnée f lorsque certaines de ses propriétés caractéristiques supplémentaires sont connues ?Cette thèse présente plusieurs nouvelles techniques pour trouver une bonne approximation des fonctions de plusieurs variables par des opérateurs linéaires dont l’erreur d’approximation A( f ) - f garde un signe constant pour toute fonction f satisfaisant une certaine convexité généralisée. Nous nous concentrons dans cette thèse sur la classe des fonctions convexesou fortement convexes. Nous décrirons comment la connaissance a priori de cette information peut être utilisée pour déterminer une bonne majoration de l’erreur pour des fonctions continuellement différentiables avec des gradients Lipschitz continus. Plus précisément, nous montrons que les estimations d’erreur basées sur ces opérateurs sont toujours contrôléespar les constantes de Lipschitz des gradients, le paramètre de la convexité forte ainsi que l’erreur commise associée à l’utilisation de la fonction quadratique. En supposant en plus que la fonction que nous voulons approcher est également fortement convexe, nous établissons de meilleures bornes inférieures et supérieures pour les estimations d’erreur de l’approximation. Lesméthodes de quadrature multidimensionnelle jouent un rôle important, voire fondamental, en analyse numérique. Une analyse satisfaisante des erreurs provenant de l’utilisationdes formules de quadrature multidimensionnelle est bien moins étudiée que dans le cas d’une variable. Nous proposons une méthode d’approximation de l’intégrale d’une fonction réelle donnée à plusieurs variables par des formules de quadrature, qui conduisent à des valeurs approchées par excès (respectivement par défaut) des intégrales des fonctions ayantun certain type de convexité. Nous verrons aussi, comme nous l’avons fait pour l’approximation des fonctions, que pour de telles formules d’intégration, on peut établir un résultat de caractérisation en termes d’estimations d’erreur. En outre, nous avons étudié le problèmede l’approximation d’une intégrale définie d’une fonction donnée quand un certain nombre d’intégrales de cette fonction sur certaines sections hyperplanes d’un l’hyper-rectangle sont seulement disponibles.La motivation derrière ce type de problème est multiple. Il se pose dans de nombreuses applications, en particulier en physique expérimentale et en ingénierie, où les valeurs standards des échantillons discrets des fonctions ne sont pas disponibles, mais où seulement leurs valeurs moyennes sont accessibles. Par exemple, ce type de données apparaît naturellement dans la tomographie par ordinateur avec ses nombreuses applications en médecine, radiologie, géologie, entre autres. / In many applications, we may wish to interpolate or approximate a multivariate function possessing certain geometric properties or “shapes” such as smoothness, monotonicity, convexityor nonnegativity. These properties may be desirable for physical (e.g., a volume-pressure curve should have a nonnegative derivative) or practical reasons where the problem of shape preserving interpolation is important in various problems occurring in industry (e.g., car modelling, construction of mask surface). Hence, an important question arises: How can we compute the best possible approximation to a given function f when some of its additional characteristic properties are known?This thesis presents several new techniques to find a good approximation of multivariate functions by a new kind of linear operators, which approximate from above (or, respectively, from below) all functions having certain generalized convexity. We focus on the class of convex and strongly convex functions. We would wish to use this additional informationin order to get a good approximation of f . We will describe how this additional condition can be used to derive sharp error estimates for continuously differentiable functions with Lipschitz continuous gradients. More precisely we show that the error estimates based on such operators are always controlled by the Lipschitz constants of the gradients, the convexity parameter of the strong convexity and the error associated with using the quadratic function. Assuming, in addition, that the function, we want to approximate, is also strongly convex, we establish sharp upper as well as lower refined bounds for the error estimates.Approximation of integrals of multivariate functions is a notoriously difficult tasks and satisfactory error analysis is far less well studied than in the univariate case. We propose a methodto approximate the integral of a given multivariate function by cubature formulas (numerical integration), which approximate from above (or from below) all functions having a certain type of convexity. We shall also see, as we did for for approximation of functions, that for such integration formulas, we can establish a characterization result in terms of sharp error estimates. Also, we investigated the problem of approximating a definite integral of a given function when a number of integrals of this function over certain hyperplane sections of d-dimensional hyper-rectangle are only available rather than its values at some points.The motivation for this problem is multifold. It arises in many applications, especially in experimental physics and engineering, where the standard discrete sample values fromfunctions are not available, but only their mean values are accessible. For instance, this data type appears naturally in computer tomography with its many applications inmedicine, radiology, geology, amongst others.

This thesis consists of three parts. In the first part, we recall some background theory that will be used throughout the thesis. In the second part, we studied the existence and uniqueness of solutions of the stochastic heat equations with Markovian switching. In the third part, we investigate the properties of solutions, such as Feller property, strong Feller property and stability.

Two applications of Category Theory are considered. The link between them is applications to Physics and more specifically to Entropy. The first research chapter is broader in scope and not explicitly about Physics, although connections to Statistical Mechanics are made towards the end of the chapter. Matroids are abstract structures that describe dependence, and strong maps are certain structure-preserving functions between them with desirable properties. We examine properties of various categories of matroids and strong maps: we compute limits and colimits; we find free and cofree constructions of various subcategories; we examine factorisation structures, including a translation principle from geometric lattices; we find functors with convenient properties to/from vector spaces, multisets of vectors, geometric lattices, and graphs; we determine which widely used operations on matroids are functorial (these include deletion, contraction, series and parallel connection, and a simplification monad); lastly, we find a categorical characterisation of the greedy algorithm. In conclusion, this project determines which aspects of Matroid Theory are most and least conducive to categorical treatment. The purpose of the second research chapter is to provide a categorical framework for generalising and unifying notions of Entropy in various settings, exploiting the fact that Entropy is a monotone subadditive function. A categorical characterisation of Entropy through a category of thermodynamical systems and adiabatic processes is found. A modelling perspective (adiabatic categories) that directly generalises an existing model is compared to an axiomatisation through topological and linear structures (topological weak semimodules), where the latter is based on a categorification of semimodules. Properties of each class of categories are examined; most notably a cancellation property of adiabatic categories generalising an existing result, and an adjunction between the categories of weak semimodules and symmetric monoidal categories. An adjunction between categories of adiabatic categories and topological weak semimodules is found. We examine in which cases each of these classes of categories constitutes a traced monoidal category. Lastly, examples of physical applications are provided. In conclusion, this project uncovers a way of, and makes progress towards, retrieving the statistical formulation of Entropy from simple axioms.

Investigamos o papel da interação forte nas colisões periféricas de íons pesados, contrastando-a com a interação eletromagnética, e o efeito que ela desempenha na excitação de modos coletivos, processos envolvendo a correlação de dois núcleons e processos completamente incoerentes. Explicamos dados experimentais recentes (Brookhaven, E814 Collaboration), separando quantitativa e qualitativamente as contribuições nuclear e eletromagnética, coerentes e incoerentes, comparando-as também com outras experiências similares. / We have compared the role played by strong interactions with the one played e1ectromagnetic interactions in re1ativistic heavy ion collisions. We also analyze its effects of strong interactions in the excitation of collective modes and in the emission of one and two-nucleon correlations. We explain recent experimental data (Brookhaven, E814 Collaboration), separating qualitatively and quantitatively the nuclear and electromagnetic, coherent and incoherent contributions to the one-nucleon emission cross section. We also compare them with results of other similar experiments.