Em uma detalhada revisão nós obtemos a lei do semi-círculo para a densidade de estados no ensemble gaussiano de Wigner. Também falamos sobre a analogia eletrostática de Dyson, enxergando os autovalores como cargas que se repelem no círculo unitário, mostrando que nesse caso a densidade de estados é uniforme. Em um contexto mais geral nós obtemos a lei do semicírculo, provando o teorema de Glivenko-Cantelli para variáveis fortemente correlacionadas usando um método combinatorial de contagem de trajetos, o que nos dá subsídios para falar em estabilidade da lei do semi-círculo. Também, nesta dissertação nós estudamos as funções de correlação nos ensembles gaussiano e circular, mostrando que sob um adequado reescalamento elas são idênticas. Outros ensembles nesta dissertação foram investigados usando o Método de Gram para o caso em que os autovalores são limitados em um intervalo. Computamos a densidade de estados para cada um desses ensembles. Mais precisamente no ensemble de Chebychev, os resultados foram obtidos analiticamente e nesse ensemble além da densidade de estados, também traçamos grá…cos da função de correlação truncada. / In a detailed review we obtain a semi-circle law for the density of states in theWigner’s Gaussian Ensemble. Also we talk about Dyson’s Analogy, seeing the eigenvalues like charges that repulse themselves in the unitary circle, showing that this case the density of states is uniform. In a more general context we obtain the semi-circle law, proving the Glivenko-Cantelli Theorem to strongly correlated variables, using a combinatorial method of Paths' Counting. Thus we are showing the stability of the semi-circle Law. Also, in this dissertation we study the correlation functions in the Gaussian and Circular ensembles showing that using the Gram's Method in the case that eigenvalues are limited in a interval. In these ensembles we computed the density of states. More precisely, in a Chebychev ensemble the results were obtained analytically. In this ensemble, we also obtain graphics of the truncated correlation function.

Produire et faire interagir entre eux des photons optiques de façon contrôlée sont deux conditions nécessaires au développement de communications quantiques à longue distance, et plus généralement au traitement quantique d’information codée sur des photons. Cette thèse présente une étude expérimentale de solutions possibles a ces deux problèmes, en utilisant la conversion des photons en excitations collectives (polaritons) dans un nuage d’atomes froids, placé dans le mode d’une cavité optique de faible finesse (~100). Dans un premier temps, des polaritons entre états atomiques fondamentaux sont utilisés pour « mettre en mémoire » une excitation unique dans le nuage. Celle-ci est ensuite convertie efficacement en un photon unique, dont le champ est analysé par tomographie homodyne. La fonction de Wigner de l’état à un photon est reconstruite a partir des données expérimentales, et présente des valeurs négatives, démontrant que les degrés de liberté de ce photon (mode spatio-temporel et état quantique) sont complètement contrôlés. Dans un second temps, les photons sont couplés à des polaritons impliquant des états de Rydberg. Les fortes interactions dipolaires entre ces derniers se traduisent par des non-linéarités optiques dispersives très importantes, qui sont caractérisées dans un régime d’excitation classique. Ces non-linéarités peuvent être amplifiées jusqu’à ce qu’un seul photon suffise à modifier totalement la réponse du système, permettant en principe de générer des interactions effectives entre photons. / Controllably producing optical photons and making them interact are two key requirements for the development of long-distance quantum communications, and more generally for photonic quantum information processing. This thesis presents experimental studies on possible solutions to these two problems, using the conversion of the photons into collective excitations (polaritons) in a cold atomic cloud, inside the mode of a low-finesse optical cavity (~100). Firstly, ground-state polaritons are used to store a single excitation in the cloud memory. This polariton is then efficiently converted into a single photon, whose field is characterized via homodyne tomography. The single photon state’s Wigner function is reconstructed from the experimental data and exhibits negative values, demonstrating that the photon’s degrees of freedom (spatio-temporal mode and quantum state) are well controlled. Secondly, photons can be coupled to polaritons involving Rydberg states. The strong dipolar interactions between these give rise to very strong optical dispersive nonlinearities, that are characterized in a classical excitation regime. These nonlinearities can be amplified until a single photon is enough to modify the entire system’s response, allowing in principle for the generation of effective photon-photon interactions.

Em uma detalhada revisão nós obtemos a lei do semi-círculo para a densidade de estados no ensemble gaussiano de Wigner. Também falamos sobre a analogia eletrostática de Dyson, enxergando os autovalores como cargas que se repelem no círculo unitário, mostrando que nesse caso a densidade de estados é uniforme. Em um contexto mais geral nós obtemos a lei do semicírculo, provando o teorema de Glivenko-Cantelli para variáveis fortemente correlacionadas usando um método combinatorial de contagem de trajetos, o que nos dá subsídios para falar em estabilidade da lei do semi-círculo. Também, nesta dissertação nós estudamos as funções de correlação nos ensembles gaussiano e circular, mostrando que sob um adequado reescalamento elas são idênticas. Outros ensembles nesta dissertação foram investigados usando o Método de Gram para o caso em que os autovalores são limitados em um intervalo. Computamos a densidade de estados para cada um desses ensembles. Mais precisamente no ensemble de Chebychev, os resultados foram obtidos analiticamente e nesse ensemble além da densidade de estados, também traçamos grá…cos da função de correlação truncada. / In a detailed review we obtain a semi-circle law for the density of states in theWigner’s Gaussian Ensemble. Also we talk about Dyson’s Analogy, seeing the eigenvalues like charges that repulse themselves in the unitary circle, showing that this case the density of states is uniform. In a more general context we obtain the semi-circle law, proving the Glivenko-Cantelli Theorem to strongly correlated variables, using a combinatorial method of Paths' Counting. Thus we are showing the stability of the semi-circle Law. Also, in this dissertation we study the correlation functions in the Gaussian and Circular ensembles showing that using the Gram's Method in the case that eigenvalues are limited in a interval. In these ensembles we computed the density of states. More precisely, in a Chebychev ensemble the results were obtained analytically. In this ensemble, we also obtain graphics of the truncated correlation function.

《紙・驚》與《鼓・箏》 : 褐紫紅色 : 自《彩露》 : 為京腔女中音、中國大鼓和古箏而寫的雙樂章作品 = "Paper. Jolt" and "Gu. Zheng", op. 23 : Maroon : from the Dews of Color : two-movement piece for Chinese operatic mezzo-soprano, Chinese big drum, and Gu-Zheng (2008) -- Carnival, op. 24 : for symphonic band : coral : from the Dews of Color (2009) -- Celadon, op. 26 : from the Dews of Color : composed for clarinet in B-flat, Gu Zheng, and Percussion (suspended cymbal and vibraphone) = 青瓷色: 自《彩露》: 為降B單簧管, 古箏與敲擊(弔鈸及鋼片琴)而寫 -- Prelude, passacaglia and fugue, op. 25 : for percussion quartet. / "Zhi, jing" yu "Gu, zheng" : he zi hong se : zi "Cai lu" : wei jing qiang nü zhong yin, Zhongguo da gu he gu zheng er xie de shuang yue zhang zuo pin = "Paper, Jolt" and "Gu, zheng", op. 23 : Maroon : from the Dews of Color : two-movement piece for Chinese operatic mezzo-soprano, Chinese big drum, and Gu-zheng (2008) -- Carnival, op. 24 : for symphonic band : coral : from the Dews of Color (2009) -- Celadon, op. 26 : from the Dews of Color : composed for clarinet in B-flat, Gu zheng, and Percussion (suspended cymbal and vibraphone) = Qing ci se : zi "Cai lu": wei xiang B dan huang guan, gu zheng yu qiao ji (diao ba ji gang pian qin) er xie -- Prelude, passacaglia and fugue, op. 25 : for percussion quartet. / Lee, Kar Tai. / Durations: 8 min.; 4 min.; 4 min.; 10 min. / Thesis (M.Mus.)--Chinese University of Hong Kong, 2009. / Abstracts in English and Chinese. / Cover Page --- p.i / Abstract --- p.ii / Acknowledgements --- p.iv / Table of Contents --- p.iv / """Paper. Jolt “ and “ Gu . Zheng""《´紙Ø驚》與《´鼓Ø箏》" --- p.1 / Carnival《嘉年華》 --- p.18 / Celdaon《青瓷》 --- p.35 / "Prelude, Passacaglia and Fugue《前奏、巴沙加牙和賦格曲》" --- p.48

Made available in DSpace on 2014-12-17T15:47:44Z (GMT). No. of bitstreams: 1 RaulBP.pdf: 1811907 bytes, checksum: 007d54350318472b95b8e06144b749a5 (MD5) Previous issue date: 2007-07-27 / Although some individual techniques of supervised Machine Learning (ML), also known as classifiers, or algorithms of classification, to supply solutions that, most of the time, are considered efficient, have experimental results gotten with the use of large sets of pattern and/or that they have a expressive amount of irrelevant data or incomplete characteristic, that show a decrease in the efficiency of the precision of these techniques. In other words, such techniques can t do an recognition of patterns of an efficient form in complex problems. With the intention to get better performance and efficiency of these ML techniques, were thought about the idea to using some types of LM algorithms work jointly, thus origin to the term Multi-Classifier System (MCS). The MCS s presents, as component, different of LM algorithms, called of base classifiers, and realized a combination of results gotten for these algorithms to reach the final result. So that the MCS has a better performance that the base classifiers, the results gotten for each base classifier must present an certain diversity, in other words, a difference between the results gotten for each classifier that compose the system. It can be said that it does not make signification to have MCS s whose base classifiers have identical answers to the sames patterns. Although the MCS s present better results that the individually systems, has always the search to improve the results gotten for this type of system. Aim at this improvement and a better consistency in the results, as well as a larger diversity of the classifiers of a MCS, comes being recently searched methodologies that present as characteristic the use of weights, or confidence values. These weights can describe the importance that certain classifier supplied when associating with each pattern to a determined class. These weights still are used, in associate with the exits of the classifiers, during the process of recognition (use) of the MCS s. Exist different ways of calculating these weights and can be divided in two categories: the static weights and the dynamic weights. The first category of weights is characterizes for not having the modification of its values during the classification process, different it occurs with the second category, where the values suffers modifications during the classification process. In this work an analysis will be made to verify if the use of the weights, statics as much as dynamics, they can increase the perfomance of the MCS s in comparison with the individually systems. Moreover, will be made an analysis in the diversity gotten for the MCS s, for this mode verify if it has some relation between the use of the weights in the MCS s with different levels of diversity / Apesar de algumas t?cnicas individuais de Aprendizado de M?quina (AM) supervisionado, tamb?mconhecidos como classificadores, ou algoritmos de classifica??o, fornecerem solu??es que, na maioria das vezes, s?o consideradas eficientes, h? resultados experimentais obtidos com a utiliza??o de grandes conjuntos de padr?es e/ou que apresentam uma quantidade expressiva de dados incompletos ou caracter?sticas irrelevantes, que mostram uma queda na efic?cia da precis?o dessas t?cnicas. Ou seja, tais t?cnicas n?o conseguem realizar um reconhecimento de padr?es de uma forma eficiente em problemas complexos. Com o intuito de obter um melhor desempenho e efic?cia dessas t?cnicas de AM, pensouse na id?ia de fazer com que v?rios tipos de algoritmos de AM consigam trabalhar conjuntamente, dando assim origem ao termo Sistema Multi-Classificador (SMC). Os SMC s apresentam, como componentes, diferentes algoritmos de AM, chamados de classificadores base, e realizam uma combina??o dos resultados obtidos por estes algoritmos para atingir o resultado final. Para que o SMC tenha um desempenho melhor que os classificadores base, os resultados obtidos por cada classificador base devem apresentar uma determinada diversidade, ou seja, uma diferen?a entre os resultados obtidos por cada classificador que comp?em o sistema. Pode-se dizer que n?o faz sentido ter SMC s cujos classificadores base possuam respostas id?nticas aos padr?es apresentados. Apesar dos SMC s apresentarem melhores resultados que os sistemas executados individualmente, h? sempre a busca para melhorar os resultados obtidos por esse tipo de sistema. Visando essa melhora e uma maior consist?ncia nos resultados, assim como uma maior diversidade dos classificadores de um SMC, v?m sendo recentemente pesquisadas metodologias que apresentam como caracter?sticas o uso de pesos, ou valores de con- fian?a. Esses pesos podem descrever a import?ncia que um determinado classificador forneceu ao associar cada padr?o a uma determinada classe. Esses pesos ainda s?o utilizados, em conjunto com as sa?das dos classificadores, durante o processo de reconhecimento (uso) dos SMC s. Existem diferentes maneiras de se calcular esses pesos e podem ser divididas em duas categorias: os pesos est?ticos e os pesos din?micos. A primeira categoria de pesos se caracteriza por n?o haver a modifica??o de seus valores no decorrer do processo de classifica??o, ao contr?rio do que ocorre com a segunda categoria, onde os valores sofrem modifica??es no decorrer do processo de classifica??o. Neste trabalho ser? feito uma an?lise para verificar se o uso dos pesos, tanto est?ticos quanto din?micos, conseguem aumentar o desempenho dos SMC s em compara??o com estes sistemas executados individualmente. Al?m disso, ser? feita uma an?lise na diversidade obtida pelos SMC s, para dessa forma verificar se h? alguma rela??o entre o uso dos pesos nos SMC s com diferentes n?veis de diversidade

Dans ce travail, nous proposons deux nouvelles approches ensemblistes pourl’estimation d’état basées sur la caractérisation explicite des bornes d’erreur d’estimation. Ces approches peuvent être vues comme la combinaison entre un observateur ponctuel et une caractérisation ensembliste de l’erreur d’estimation. L’objectif est de réduire la complexité de leur implémentation, de réduire le temps de calcul en temps réel et d’améliorer la précision et des encadrements des vecteurs d’état.La première approche propose un observateur ensembliste basé sur des ensembles invariants ellipsoïdaux pour des systèmes linéaires à temps-discret et aussi des systèmes à paramètres variables. L’approche proposée fournit un intervalle d’état déterministe qui est construit comme une somme entre le vecteur état estimé du système et les bornes de l’erreur d’estimation. L’avantage de cette approche est qu’elle ne nécessite pas la propagation des ensemble d’état dans le temps.La deuxième approche est une version intervalle de l’observateur d’état de Luenberger, pour les systèmes linéaires incertains à temps-discret, basés sur le calcul d’intervalle et les ensembles invariants. Ici, le problème d’estimation ensembliste est considéré comme un problème d’estimation d’état ponctuel couplé à une caractérisation intervalle de l’erreur d’estimation. / In This work, we propose two main new approaches for the set-membershipstate estimation problem based on explicit characterization of the estimation error bounds. These approaches can be seen as a combination between a punctual observer and a setmembership characterization of the observation error. The objective is to reduce the complexity of the on-line implimentation, reduce the on-line computation time and improve the accuracy of the estimated state enclosure.The first approach is a set-membership observer based on ellipsoidal invariant sets for linear discrete-time systems and also for Linear Parameter Varying systems. The proposed approach provides a deterministic state interval that is build as the sum of the estimated system states and its corresponding estimation error bounds. The important feature of the proposed approach is that does not require propagation of sets.The second approach is an interval version of the Luenberger state observer for uncertain discrete-time linear systems based on interval and invariant set computation. The setmembership state estimation problem is considered as a punctual state estimation issue coupled with an interval characterization of the estimation error.

Dans cette thèse de doctorat, nous apportons quelques contributions à la stabilité de Lyapunov non-régulière des inclusions différentielles de premier ordre avec opérateurs monotones maximaux, dans un cadre Hilbertien de dimension infini. Nous fournissons des caractérisations explicites, primales et/ou duales, des paires de Lyapunov faibles et fortes, dont les fonctions sont semi-continues inférieurement à valeurs réelles étendues, et associées à des inclusions différentielles dont la partie de droite est gouvernée par des perturbations Lipschitziennes des opérateurs dits Cusco F, ou des opérateurs monotones maximaux A, ou les deux à la fois x(t) ∈ F(x(t}} A(x(t}} t ≥ 0, x(0) ∈ domA. De manière équivalente, nous étudions l'invariance faible et forte des ensembles fermés pour ces inclusions différentielles. Comme dans L'approche classique de Lyapunov à la stabilité des équations différentielles, les résultats présentés dans cette thèse n'utilisent que les données du système différentiel; c'est-à-dire, l'opérateur A et la multifonction F, et donc pas besoin de connaître les solutions, ni les semi-groupes générés par les opérateurs monotones en question. Parce que les paires de Lyapunov sont formées par des fonctions qui sont simplement semi-continues inférieurement, et les ensembles invariants ne sont que ensembles fermés, nous faisons usage dans cette thèse à des outils de l'analyse non-lisse, afin de fournir des critères du premier ordre, utilisant des sous-différentiels généraux et des cônes normaux. Nous fournissons une analyse similaire pour les inclusions différentielles gouvernées par le cône normal proximal à des ensembles prox-réguliers. Notre analyse ci-dessus, nous a permis de présenter ces systèmes prox-réguliers d’apparence plus générale, comme des inclusions différentielles avec opérateurs monotones maximaux. Nous utilisons aussi nos résultats pour étudier la géométrie des opérateurs monotones maximaux, et plus précisément, la caractérisation de la frontière des valeurs de ces opérateurs seulement au moyen des valeurs situées à proximité, distinctes du point de référence. Ce résultat a des applications dans la stabilité des problèmes de la programmation semi-infinie. Nous utilisons également nos résultats sur les paires de Lyapunov et les ensembles invariants pour établir une étude systématique des observateurs de type Luenberger pour des inclusions différentielles avec des cônes normaux à des ensembles prox-réguliers. / In this PhD thesis, we make some contributions to nonsmooth Lyapunov stability of first-order differential inclusions with maximal monotone operators, in the setting of infinite-dimensional Hilbert spaces. We provide primal and dual explicit characterizations for parameterized weak and strong Lyapunov pairs of lower semicontinuous extended-real-valued functions, referred to as a-Lyapunov pairs, associated to differential inclusions with right-hand-sides governed by Lipschitz or Cusco perturbationsF of maximal monotone operators A, x(t) ∈ F(x(t}} A(x(t}} t ≥ 0, x(0) ∈ domA. Equivalently, we study the weak and strong invariance of sets with respect to such differential inclusions. As in the classical Lyapunov approach to the stability of differential equations, the presented results make use of only the data of the differential system; that is, the operator A and the multifunction F, and so no need to know about the solutions, nor the semi-groups generated by the monotone operators. Because our Lyapunov pairs and invariant sets candidates are just lower semicontinuous and closed, respectively, we make use of nonsmooth analysis to provide first-order-like criteria using general subdifferentials and normal cones. We provide similar analysis to non-convex differential inclusions governed by proximal normal cones to prox-regular sets. Our analysis above allowed to prove that such apparently more general systems can be easily coined into our convex setting. We also use our results to study the geometry of maximal monotone operators, and specifically, the characterization of the boundary of the values of such operators by means only of the values at nearby points, which are distinct of the reference point. This result has its application in the stability of semi-infinite programming problems. We also use our results on Lyapunov pairs and invariant sets to provide a systematic study of Luenberger-like observers design for differential inclusions with normal cones to prox-regular sets.

Cette thèse étudie quelques problèmes d’analyse harmonique sur les groupes quantiques compacts. Elle consiste en trois parties. La première partie présente la théorie Lp élémentaire des transformées de Fourier, les convolutions et les multiplicateurs sur les groupes quantiques compacts, y compris la théorie de Hausdorff-Young et les inégalités de Young.Dans la seconde partie, nous caractérisons les opérateurs de convolution positifs sur un groupe quantique fini qui envoient Lp dans L2, et donnons aussi quelques constructions sur les groupes quantiques compacts infinis. La méthode pour étudier les états non-dégénérés fournit une formule générale pour calculer les états idempotents associés aux images deHopf, qui généralise un travail de Banica, Franz et Skalski. La troisième partie est consacrée à l’étude des ensembles de Sidon, des ensembles _(p) et des notions associées pour les groupes quantiques compacts. Nous établissons différentes caractérisations des ensembles de Sidon, et en particulier nous démontrons que tout ensemble de Sidon est un ensemble de Sidon fort au sens de Picardello. Nous donnons quelques liens entre les ensembles de Sidon, les ensembles _(p) et les lacunarités pour les multiplicateurs de Fourier sur Lp, généralisant un travail de Blendek et Michali˘cek. Nous démontrons aussi l’existence des ensembles de type _(p) pour les systèmes orthogonaux dans les espaces Lp non commutatifs, et déduisons les propriétés correspondantes pour les groupes quantiques compacts. Nous considérons aussi les ensembles de Sidon centraux, et nous prouvons que les groupes quantiques compacts ayant les mêmes règles de fusion et les mêmes fonctions de dimension ont des ensemble de Sidon centraux identiques. Quelques exemples sont aussi étudiés dans cette thèse. Les travaux présentés dans cette thèse se basent sur deux articles de l’auteur. Le premier s’intitule “Lp-improving convolution operators on finite quantum groups” et a été accepté pour publication dans Indiana University Mathematics Journal, et le deuxième est un travail intitulé “Lacunary Fourier series for compact quantum groups” et a été publié en ligne dans Communications in Mathematical Physics. / This thesis studies some problems in the theory of harmonic analysis on compact quantum groups. It consists of three parts. The first part presents some elementary Lp theory of Fourier transforms, convolutions and multipliers on compact quantum groups, including the Hausdorff-Young theory and Young’s inequalities. In the second part, we characterize positive convolution operators on a finite quantum group G which are Lp-improving, and also give some constructions on infinite compact quantum groups. The methods for ondegeneratestates yield a general formula for computing idempotent states associated to Hopf images, which generalizes earlier work of Banica, Franz and Skalski. The third part is devoted to the study of Sidon sets, _(p)-sets and some related notions for compact quantum groups. We establish several different characterizations of Sidon sets, and in particular prove that any Sidon set in a discrete group is a strong Sidon set in the sense of Picardello. We give several relations between Sidon sets, _(p)-sets and lacunarities for Lp-Fourier multipliers, generalizing a previous work by Blendek and Michali˘cek. We also prove the existence of _(p)-sets for orthogonal systems in noncommutative Lp-spaces, and deduce the corresponding properties for compact quantum groups. Central Sidon sets are also discussed, and it turns out that the compact quantum groups with the same fusion rules and the same dimension functions have identical central Sidon sets. Several examples are also included. The thesis is principally based on two works by the author, entitled “Lp-improvingconvolution operators on finite quantum groups” and “Lacunary Fourier series for compact quantum groups”, which have been accepted for publication in Indiana University Mathematics Journal and Communications in Mathematical Physics respectively.

Il existe aujourd’hui, peut-être plus encore que par le passé, de multiples façons de faire de l’orchestre. Orchestres symphoniques permanents, orchestres dit « par projet », orchestres associatifs, ensembles ayant une démarche historisante, les formes de pratiques orchestrales se répartissent en des catégories complexes, organisées selon leurs constructions juridiques, les modes de recrutements de leurs artistes et bien entendu, leurs projets musicaux. Les conventions ayant lieu au sein de ces structures diffèrent totalement, tant du point de vue musical que social. Cette thèse propose d’observer ces pratiques orchestrales en questionnant les différents modes de construction du collectif à l’œuvre dans les orchestres professionnels. Par-delà cette interrogation, mon but est d’observer comment et dans quelle mesure l’organisation de chaque orchestre peut influencer l’engagement des musiciens (socialement mais aussi musicalement) et la prestation artistique. Cette recherche puise son influence dans les travaux anglo-saxons, en s’intéressant notamment aux interactions entre le cadre organisationnel propre à chaque orchestre et les stratégies qu’y développent les individus. Pour cela je questionne, outre les artistes, ceux qui travaillent à leurs côtés : l’orchestre est envisagé en tant que processus de coopération entre techniciens, administrations et musiciens. L’observation de ces mécanismes met en relief les relations particulières que les orchestres et les individus entretiennent avec l’idée d’un engagement artistique obligatoirement corrélé au risque social. / The orchestra exists in many forms today, perhaps even more so than in the past. Permanent symphony orchestras, so-called "project" orchestras, self-governing orchestras, historically informed performance ensembles, the forms of orchestral practice are divided into complex categories, organized according to their legal status, recruitment procedures and, of course, musical mission. The agreements within these structures are totally different, as much from the musical as the social point of view.This thesis aims to observe these orchestral practices by questioning the various modes of constructing the collective in the professional orchestras. Beyond this interrogation, my purpose is to observe how and to what extent the organization of every orchestra can influence the commitment of the musicians (socially but also musically) and the artistic performance. This research draws its influence from Anglo-Saxon sources, focusing particularly on the interactions between the organizational framework specific to each orchestra and the strategies which the individuals develop therein. For this I question not only the artists but those who work by their side: the orchestra is envisaged as a process of cooperation between technicians, administrations, and musicians.The observation of these mechanisms emphasizes the particular relations which the orchestras and the individuals maintain with the idea of an artistic commitment necessarily correlated with social risk.

In this thesis we present results about some eigenvalue statistics of the beta-Hermite ensembles, both in the classical cases corresponding to beta = 1, 2, 4, that is the Gaussian orthogonal ensemble (consisting of real symmetric matrices), the Gaussian unitary ensemble (consisting of complex Hermitian matrices) and the Gaussian symplectic ensembles (consisting of quaternionic self-dual matrices) respectively. We also look at the less explored general beta-Hermite ensembles (consisting of real tridiagonal symmetric matrices). Specifically we look at the empirical distribution function and two different scalings of the largest eigenvalue. The results we present relating to these statistics are the convergence of the empirical distribution function to the semicircle law, the convergence of the scaled largest eigenvalue to the Tracy-Widom distributions, and with a different scaling, the convergence of the largest eigenvalue to 1. We also use simulations to illustrate these results. For the Gaussian unitary ensemble, we present an expression for its level density. To aid in understanding the Gaussian symplectic ensemble we present properties of the eigenvalues of quaternionic matrices. Finally, we prove a theorem about the symmetry of the order statistic of the eigenvalues of the beta-Hermite ensembles. / I denna kandidatuppsats presenterar vi resultat om några olika egenvärdens-statistikor från beta-Hermite ensemblerna, först i de klassiska fallen då beta = 1, 2, 4, det vill säga den gaussiska ortogonala ensemblen (bestående av reella symmetriska matriser), den gaussiska unitära ensemblen (bestående av komplexa hermitiska matriser) och den gaussiska symplektiska ensemblen (bestående av kvaternioniska själv-duala matriser). Vi tittar även på de mindre undersökta generella beta-Hermite ensemblerna (bestående av reella symmetriska tridiagonala matriser). Specifikt tittar vi på den empiriska fördelningsfunktionen och två olika normeringar av det största egenvärdet. De resultat vi presenterar för dessa statistikor är den empiriska fördelningsfunktionens konvergens mot halvcirkel-fördelningen, det normerade största egenvärdets konvergens mot Tracy-Widom fördelningen, och, med en annan normering, största egenvärdets konvergens mot 1. Vi illustrerar även dessa resultat med hjälp av simuleringar. För den gaussiska unitära ensemblen presenterar vi ett uttryck för dess nivåtäthet. För att underlätta förståelsen av den gaussiska symplektiska ensemblen presenterar vi egenskaper hos egenvärdena av kvaternioniska matriser. Slutligen bevisar vi en sats om symmetrin hos ordningsstatistikan av egenvärdena av beta-Hermite ensemblerna.