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.

The purpose of the following essay is to study and explain how young adults use their social networks when it comes to finding a job. This study was conducted using nine semi-structured interviews with young adults with some form of employment. We used Granovetter’s theory about strong and weak ties, Bourdieu’s capital theory, Lin’s social reources theory and Putnam’s reasoning about generalized reciprocity. The result shows that social capital is a decisive factor in how young individuals use their social contacts. The volume of social capital is determined by economic resources, social background and the size of the social network. Less resourceful groups have a tendency to take advantage of the strong ties, i.e. family and relatives, often when looking for their first job. The weak ties, i.e. acquaintances, are used later in life when the individual has greater resources and networks. Individuals from the privileged groups may not always be able to work on their parents’ company as a first job, because they might require certain qualifications to do so. However, these job positions become possible for the young individuals from resourceful groups when they accomplish a relevant education. That's when they get the chance to use the resources that are embedded in their strong ties.

The main topic of this Thesis is a new simplified proof of the Sunyach theorem that provides suffici- ent conditions for existence and uniqueness of an invariant measure for a Markov kernel on a complete separable metric space equipped with its Borel σ-algebra. Weak convergence of measures following from Sunyach's theorem is strengthened to convergence in the total variation norm provided that the Markov kernel is strong Feller. Furthermore, sufficient conditions for geometric ergodicity are stated. Another topic treated is the strong Feller property: its characterization by absolute measurability and uniform integrability and derivation of some other sufficient conditions.

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.

As correções perturbativas da QCD aos decaimentos hadrônicos do tau são obtidas a partir da expansão da função de Adler. Acredita-se que esta série é assintótica e melhor entendida quando sua transformada de Borel é considerada. Usamos o método matemático dos Aproximantes de Padé para reconstruir a transformada de Borel da série e extrair informação sobre as correções de ordens mais altas bem como os pólos devidos aos renôrmalons associados com a divergência da série. Primeiramente, testamos o método no limite large-β0 da QCD, onde a série perturbativa é conhecida em todas as ordens. Neste limite observamos que a variação de esquema de renormalização do acoplamento forte, αs, pode ser útil para a construção de aproximantes que convergem mais rapidamente. Aplicamos o método na QCD completa para obtermos previsões sobre as principais características da série. Em QCD a estrutura analítica da transformada de Borel da função de Adler torna as aproximações com Padés menos eficientes, o que se reflete em incertezas maiores. Chegamos ao resultado de 570 ± 285 para o coeficiente do termo α5s. Devido ao fato de a série prevista pelos aproximantes apresentar comportamento divergente de sinal não-alternado, há uma indicação de que singularidades do tipo infra-vermelho contribuem mais para os coeficientes da série em ordens intermediárias. Além disso, apesar de os resultados para a soma de Borel da função δ(0) serem compatíveis com as duas prescrições mais usadas para fixar a escala de renormalização em decaimentos do tau, o Padé apresenta uma leve preferência pela prescrição de ordem fixa (ou FOPT). / Perturbative QCD corrections to hadronic tau decays are obtained from the expansion of the Adler function. This series is believed to be asymptotic and is better understood when its Borel transform is considered. We use the mathematical method of Padé approximants to reconstruct the Borel transformed series and extract information about higher order corrections as well as renormalon poles associated with the divergence of the series. First, the method is tested in the large-β0 limit of QCD, where the perturbative series is known to all orders. In this limit, we observe that the renormalization scheme variation of the strong coupling, αs, can be useful in constructing approximants that converge faster. We apply the method in complete QCD to obtain predictions about the main characteristics of the series. In QCD, the analytical structure of the Borel transform of the Adler function makes the approximations with Padés less efficient, which is reflected in larger uncertainties. We obtain the result 570 ± 285 for the coefficient of the term α5s. The fixed sign nature of the series predicted by the PAs indicates that there is an indication that infrared singularities contribute more to the coefficients of the series in intermediate orders. In addition, although the results for the Borel sum of the function δ(0) are compatible with the two most frequently used prescriptions for setting the renormalization scale in tau decays, Padé approximants show a slight preference for fixed order prescription (or FOPT).

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.

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.

Doctor of Philosophy / Department of Physics / Daniel Rolles / Modern laser sources can produce bursts of light that surpass even the fastest molecular vibrations. With durations this short even moderate pulse energies generate peak powers exceeding the average power output of the entire globe. When focused, this can result in an ultrafast electric field greater than the Coulomb potential that binds electrons to nuclei. This strong electric field strips electrons away from atoms in a process known as strong-field ionization. The first experimental realization of photoionization with intense laser pulses occurred only a few years after the invention of the laser. Yet, despite decades of intensive investigation, open questions remain. At the same time, the knowledge gained has led to the creation of multiple exciting fields such as attoscience, femtochemistry, and ultrafast nano-photonics. In this thesis I present my work to advance the fundamental understanding of intense, ultrafast light-matter interactions as well as efforts to expand the technological capabilities of ultrafast light sources and measurement techniques. This includes the photoionization pro- cess of atoms and nanoparticles subject to intense, mid-infrared laser fields. The resulting photoelectron emission is measured, with high precision, in a velocity map imaging spec- trometer. Other parts of this thesis detail my work on the generation and characterization of non-Gaussian optical pulses. Femtosecond Bessel beams are used to drive and study high harmonic generation with the ultimate goal of creating a compact, high-flux XUV source. Further studies include few-cycle pulses and the carrier-envelope phase, specifically methods of locking and tagging the carrier-envelope phase. A single-shot, all optical tagging method is developed and directly compared to the standard tagging method, the carrier-envelope phase meter. Finally, both experimental and computational studies are presented investigating the ultrafast thermal response cycle of nanowires undergoing femtosecond heating.

As correções perturbativas da QCD aos decaimentos hadrônicos do tau são obtidas a partir da expansão da função de Adler. Acredita-se que esta série é assintótica e melhor entendida quando sua transformada de Borel é considerada. Usamos o método matemático dos Aproximantes de Padé para reconstruir a transformada de Borel da série e extrair informação sobre as correções de ordens mais altas bem como os pólos devidos aos renôrmalons associados com a divergência da série. Primeiramente, testamos o método no limite large-β0 da QCD, onde a série perturbativa é conhecida em todas as ordens. Neste limite observamos que a variação de esquema de renormalização do acoplamento forte, αs, pode ser útil para a construção de aproximantes que convergem mais rapidamente. Aplicamos o método na QCD completa para obtermos previsões sobre as principais características da série. Em QCD a estrutura analítica da transformada de Borel da função de Adler torna as aproximações com Padés menos eficientes, o que se reflete em incertezas maiores. Chegamos ao resultado de 570 ± 285 para o coeficiente do termo α5s. Devido ao fato de a série prevista pelos aproximantes apresentar comportamento divergente de sinal não-alternado, há uma indicação de que singularidades do tipo infra-vermelho contribuem mais para os coeficientes da série em ordens intermediárias. Além disso, apesar de os resultados para a soma de Borel da função δ(0) serem compatíveis com as duas prescrições mais usadas para fixar a escala de renormalização em decaimentos do tau, o Padé apresenta uma leve preferência pela prescrição de ordem fixa (ou FOPT). / Perturbative QCD corrections to hadronic tau decays are obtained from the expansion of the Adler function. This series is believed to be asymptotic and is better understood when its Borel transform is considered. We use the mathematical method of Padé approximants to reconstruct the Borel transformed series and extract information about higher order corrections as well as renormalon poles associated with the divergence of the series. First, the method is tested in the large-β0 limit of QCD, where the perturbative series is known to all orders. In this limit, we observe that the renormalization scheme variation of the strong coupling, αs, can be useful in constructing approximants that converge faster. We apply the method in complete QCD to obtain predictions about the main characteristics of the series. In QCD, the analytical structure of the Borel transform of the Adler function makes the approximations with Padés less efficient, which is reflected in larger uncertainties. We obtain the result 570 ± 285 for the coefficient of the term α5s. The fixed sign nature of the series predicted by the PAs indicates that there is an indication that infrared singularities contribute more to the coefficients of the series in intermediate orders. In addition, although the results for the Borel sum of the function δ(0) are compatible with the two most frequently used prescriptions for setting the renormalization scale in tau decays, Padé approximants show a slight preference for fixed order prescription (or FOPT).