Neste trabalho, introduzimos os conceitos de controlabilidade e de observabilidade para equações diferenciais ordinárias generalizadas, apresentamos resultados inéditos sobre condições suficientes e necessárias para controlabilidade e para observabilidade para estas equações e também apresentaremos uma aplicação. Utilizando teoremas de correspondência entre equações diferenciais ordinárias generalizadas e outras equações diferenciais, traduzimos os resultados obtidos para os casos particulares de controlabilidade e observabilidade para equações diferenciais em medida e equações diferencias com impulsos. O fato de trabalharmos no ambiente das equações diferenciais ordinárias generalizadas permitiu que os resultados obtidos pudessem envolver funções com muitas descontinuidades e muito oscilantes, ou seja, de variação ilimitada. Os resultados novos apresentados aqui estão contidos no artigo [21] que se encontra em fase final de redação e será submetido à publicação em breve. / In this work, we introduce concepts of controllability and observability for generalized ordinary differential equations, we present new results on necessary and sufficient conditions for controllability and observability for these equations and we also present an application. Using theorems of correspondence between generalized ordinary differential equations and other differential equations, we translate the results obtained for the particular cases of controllability and observability for measure differential equations and differential equations with impulses. The fact that we work in the framework of generalized ordinary differential equations allows us to obtain results where the functions involved can have many discontinuities and be highly oscillating, that is, of unbounded variation. The new results presented here are contained in the preprint [21] which is under final revision and will soon be submitted for publication.

Neste trabalho, estabelecemos a teoria de dicotomias para equações diferenciais ordinárias generalizadas, introduzindo os conceitos de dicotomias para essas equações generalizadas, estudando as suas propriedades e propondo resultados novos. Investigamos condições para a existência de soluções limitadas e condições para a existência de dicotomia exponencial. Utilizando teoremas de correspondência entre equações diferenciais ordinárias generalizadas e outras equações, traduzimos os resultados obtidos para os casos particulares de dicotomias para equações diferenciais em medida e para equações diferenciais com impulsos. O fato de trabalharmos no ambiente das equações diferenciais ordinárias generalizadas faz com que os resultados obtidos para os casos particulares possam envolver funções com muitas descontinuidades e de variação ilimitada. / In this work we establish the theory of dichotomies for generalized ordinary dierential equations, introducing the concepts of dichotomies for these equations, studying their properties and proposing new results. We investigate conditions of existence of exponential dichotomies and bounded solutions. Using correspondence theorems between generalized ordinary dierential equations and other equations, we translate the obtained results to the particular cases of dichotomies for measure dierential equations and for impulsive dierential equations. The fact that we work in the framework of generalized ordinary dierential equations allows us to obtain results for the particular cases where the functions involved can have many discontinuities and be of unbounded variation.

Neste trabalho, estabelecemos a teoria de dicotomias para equações diferenciais ordinárias generalizadas, introduzindo os conceitos de dicotomias para essas equações generalizadas, estudando as suas propriedades e propondo resultados novos. Investigamos condições para a existência de soluções limitadas e condições para a existência de dicotomia exponencial. Utilizando teoremas de correspondência entre equações diferenciais ordinárias generalizadas e outras equações, traduzimos os resultados obtidos para os casos particulares de dicotomias para equações diferenciais em medida e para equações diferenciais com impulsos. O fato de trabalharmos no ambiente das equações diferenciais ordinárias generalizadas faz com que os resultados obtidos para os casos particulares possam envolver funções com muitas descontinuidades e de variação ilimitada. / In this work we establish the theory of dichotomies for generalized ordinary dierential equations, introducing the concepts of dichotomies for these equations, studying their properties and proposing new results. We investigate conditions of existence of exponential dichotomies and bounded solutions. Using correspondence theorems between generalized ordinary dierential equations and other equations, we translate the obtained results to the particular cases of dichotomies for measure dierential equations and for impulsive dierential equations. The fact that we work in the framework of generalized ordinary dierential equations allows us to obtain results for the particular cases where the functions involved can have many discontinuities and be of unbounded variation.

Cette étude intéresse la partie savoyarde de la " zone subbriançonnaise " comprise entre Arc et Isère et correspond globalement aux massifs du Perron des Encombres et de la Grande Moendaz, regroupés sous le nom de " Nappe du Pas du Roc " Cette nappe, faite de 3 écailles tectoniques qui sont, de l'Est à l'Ouest, l'unité de la Croix des Têtes, l'unité de Saint-Martin-de-la-Porte et l'unité de la Grande Moendaz, est limitée : - À l'Est, par l'accident gypseux frontal du Briançonnais externe (Zone Houillère) - À l'Ouest, par un cisaillement redressé qui la fait reposer sur des unités composites à flysch priabonien correspondant anciennement aux " Ecailles externes " subbriançonnaises et à la zone ultradauphinoise ou du flysch des Aiguilles d'Arves et qui appartiennent, en réalité, à des unités avancées à substratum anteflysch d'affinités briançonnaise. La nappe du Pas du Roc ne comprend que des terrains allant du trias supérieur au Malm inférieur ; la série malm supérieur à nummulitique, qui ici a disparu par érosions et/ou diverticulations, est cependant conservée au Sud de l'Arc dans la haute vallée de la Valloirette. Les études stratigraphiques, tectoniques et bibliographiques de ce mémoire permettent la reconstitution géodynamique suivante : - Du Trias supérieur au Lias inférieur, le Subbriançonnais appartient à une plate-forme lagunaire à nérétique se développant du Delphino-hélvétique au Briançonnais. - A partir du Lias moyen, concomitamment à l'ouverture téthysienne par fracturation, le Subbriançonnais apparaît comme une marge faillée entre le sillon delphino-helvétiques à vocontien et le haut-fond briançonnais et provençal en position méridionale. Plus précisément, dans le secteur entre Arc et Isère, il est possible de distinguer du Lias moyen à l'Oxfordien deux domaines, le domaine externe de la Grande Moendaz aux sédiments pélagiques et argileux faisant transition au sillon delphino-helvétique interne et le domaine interne du Perron des Encombres, en position haute et aux sédiments à dominante carbonatée se raccordant au Briançonnais. L'instabilité tectonique de cette marge se manifeste, en particulier au Callovo-Oxfordien, par la présence de brèches sur le domaine interne, d'olistilites sur le talus et de turbidites donnant naissance à un véritable flysch " oxfordien " dont les éléments lithiques se fondent progressivement dans la sédimentation pélagique du sillon externe. - L'histoire postérieure est reconstituée à partir des secteurs environnants (Sud de l'Arc, klippe de Sulens, Chablais,...) Toujours en position de marge, le Subbriançonnais correspond à un domaine de mer ouverte à sédimentation pélagique affectée par des apports détritiques épisodiques internes et se terminant par le " flysch noir " de l'Eocène inférieur à moyen. Au cours du Crétacé supérieur et du Paléogène, cette marge découpée par de grands coulissages sénestres, est dilacérée en blocs qui remontent avec le Briançonnais vers le Nord, mettant donc en place finalement, selon une transversale ouest-est, les zones isopiques apparentes suivantes delphino-helvétique, subbriançonnaise à lignes de faciès cependant en biais (NE-SW), briançonnaise. A ces mouvements transcurrents correspondant à une contrainte compressive générale N-S sont associées des érosions, des structures de transpression et l'individualisation et le fonctionnement d'un bassin de flysch Priabonien-Oligocène inférieur qui cachète définitivement la tectonique décrochante antérieure. De l'Oligocène au Miocène, se mettent en place les nappes à vergence globale ouest. Cette tectonique alpine est polyphasée. Dans le temps, se succèdent, en particulier : - une phase généralisée de charriage des zones isopiques vers l'Ouest, le Briançonnais externe dépassant le Subbriançonnais ; - une phase de cisaillement responsable des écaillages les plus visibles et du pincement des unités briançonnaises avancées, au front de la nappe du Pas du Roc ; - une phase de blocage avec des manifestations de rétrodéversements. Postérieurement à cette technique nappée E-W, une reprise des contraintes N-S, donnerait naissance à des coulissages senestres méridiens. Ceux-ci sont peu exprimés entre Arc et Isère. Enfin, les bombements de Belledonne et de la zone Houillère donnent à la région son cachet structural actuel.

Dans cette thèse, nous nous intéressons à plusieurs aspects des systèmes de racines des algèbres de Lie simples. Dans un premier temps, nous étudions les coordonnées des vecteurs propres des matrices de Cartan. Nous commençons par généraliser les travaux de physiciens qui ont montré que les masses des particules dans la théorie des champs de Toda affine sont égales aux coordonnées du vecteur propre de Perron -- Frobenius de la matrice de Cartan. Puis nous adoptons une approche différente, puisque nous utilisons des résultats de la théorie des singularités pour calculer les coordonnées des vecteurs propres de certains systèmes de racines. Dans un deuxième temps, en s'inspirant des idées de Givental, nous introduisons les matrices de Cartan q-déformées et étudions leur spectre et leurs vecteurs propres. Puis, nous proposons une q-déformation des équations de Toda et construisons des 1-solitons solutions en adaptant la méthode de Hirota, d'après les travaux de Hollowood. Enfin, notre intérêt se porte sur un ensemble de transformations agissant sur l'ensemble des bases ordonnées de racines comme le groupe de tresses. En particulier, nous étudions les bases distinguées, qui forment l'une des orbites de cette action, et des matrices que nous leur associons. / In this thesis, our goal is to study various aspects of root systems of simple Lie algebras. In the first part, we study the coordinates of the eigenvectors of the Cartan matrices. We start by generalizing the work of physicists who showed that the particle masses of the affine Toda field theory are equal to the coordinates of the Perron -- Frobenius eigenvector of the Cartan matrix. Then, we adopt another approach. Namely, using the ideas coming from the singularity theory, we compute the coordinates of the eigenvectors of some root systems. In the second part, inspired by Givental's ideas, we introduce q-deformations of Cartan matrices and we study their spectrum and their eigenvectors. Then, we propose a q-deformation of Toda's equations et compute 1-solitons solutions, using the Hirota's method and Hollowood's work. Finally, our interest is focused on a set of transformations which induce an action of the braid group on the set of ordered root basis. In particular, we study an orbit for this action, the set of distinguished basis and some associated matrices.

Neste trabalho, introduzimos os conceitos de controlabilidade e de observabilidade para equações diferenciais ordinárias generalizadas, apresentamos resultados inéditos sobre condições suficientes e necessárias para controlabilidade e para observabilidade para estas equações e também apresentaremos uma aplicação. Utilizando teoremas de correspondência entre equações diferenciais ordinárias generalizadas e outras equações diferenciais, traduzimos os resultados obtidos para os casos particulares de controlabilidade e observabilidade para equações diferenciais em medida e equações diferencias com impulsos. O fato de trabalharmos no ambiente das equações diferenciais ordinárias generalizadas permitiu que os resultados obtidos pudessem envolver funções com muitas descontinuidades e muito oscilantes, ou seja, de variação ilimitada. Os resultados novos apresentados aqui estão contidos no artigo [21] que se encontra em fase final de redação e será submetido à publicação em breve. / In this work, we introduce concepts of controllability and observability for generalized ordinary differential equations, we present new results on necessary and sufficient conditions for controllability and observability for these equations and we also present an application. Using theorems of correspondence between generalized ordinary differential equations and other differential equations, we translate the results obtained for the particular cases of controllability and observability for measure differential equations and differential equations with impulses. The fact that we work in the framework of generalized ordinary differential equations allows us to obtain results where the functions involved can have many discontinuities and be highly oscillating, that is, of unbounded variation. The new results presented here are contained in the preprint [21] which is under final revision and will soon be submitted for publication.

Since the introduction of the Riemann integral in the middle of the nineteenth century, integration theory has been subject to significant breakthroughs on a relatively frequent basis. We have now reached a point where integration theory has been thoroughly researched to a point where one has to delve quite deep into a particular subject in order to encounter open conjectures. In education the Riemann integral has for quite some time been the standard integral in elementary analysis courses and as the complexity of these courses incrementally increase the more general Lebesgue integral eventually becomes the standard integral.  Unfortunately, in the transition from the Riemann integral to the Lebesgue integral there are certain topics of pure theoretical interest which to a certain extent are neglected. This is particularly the case for topics regarding the inverse relationship between differential and integral calculus and the integration of exceedingly complicated functions which for example might be of a highly oscillatory nature. From an applied mathematician's point of view, the partial neglection of these topics in the case of highly problematic functions might be justified in the sense that this theory is unnecessary for modeling most problems that appear in nature. From a theoretician's point of view however this negligence is unacceptable. Consequently, there are alternative integrals which give rise to theories which one can use in an attempt to study these aforementioned topics. An example of such an integral is the Henstock–Kurzweil integral, which can be developed in a rather similar manner to that of the Riemann integral.  In this thesis we will develop the Henstock–Kurzweil integral in order to answer some of the questions which to a certain extent are beyond the scope of the Lebesgue integral while using rather basic proof techniques from complex analysis and measure theory. In addition to that we extended various properties of the Lebesgue integral to the Henstock–Kurzweil integral, in particular when it comes to Lebesgue's fundamental theorem of calculus and the basic convergence theorems of the Lebesgue integral.

This is a literature study, in linear algebra, about positive and nonnegative matrices and their special properties. We say that a matrix or a vector is positive/nonnegative if all of its entries are positive/nonnegative. First, we study some generalities and become acquainted with two types of nonnegative matrices; irreducible and reducible. After exploring their characteristics we investigate and prove the two main theorems of this subject, namely Perron's and Perron-Frobenius' theorem. In short Perron's theorem from 1907 tells us that the spectral radius of a positive matrix is a simple eigenvalue of the matrix and that its eigenvector can be taken to be positive. In 1912, Georg Frobenius generalized Perron's results also to irreducible nonnegative matrices. The two theorems have a wide range of applications in both pure mathematics and practical matters. In real world scenarios, many measurements are nonnegative (length, time, amount, etc.) and so their mathematical formulations often relate to Perron-Frobenius theory. The theory's importance to linear dynamical systems, such as Markov chains, cannot be overstated; it determines when, and to what, an iterative process will converge. This result is in turn the underlying theory for the page-ranking algorithm developed by Google in 1998. We will see examples of all these applications in chapters four and five where we will be particularly interested in different types of Markov chains.  The theory in this thesis can be found in many books. Here, most of the material is gathered from Horn-Johnson [5], Meyer [9] and Shapiro [10]. However, all of the theorems and proofs are formulated in my own way and the examples and illustrations are concocted by myself, unless otherwise noted. / Det här är en litteraturstudie, inom linjär algebra, om positiva och icke-negativa matriser och deras speciella egenskaper. Vi säger att en matris eller en vektor är positiv/icke-negativ om alla dess element är positiva/icke-negativa. Inledningsvis går vi igenom några grundläggande begrepp och bekanta oss med två typer av icke-negativa matriser; irreducibla och reducibla. Efter att vi utforskat deras egenskaper så studerar vi och bevisar ämnets två huvudsatser; Perrons och Perron-Frobenius sats. Kortfattat så säger Perrons sats, från 1907, att spektralradien för en positiv matris är ett simpelt egenvärde till matrisen och att dess egenvektor kan tas positiv. År 1912 så generaliserade Georg Frobenius Perrons resultat till att gälla också för irreducibla icke-negativa matriser.  De två satserna har både många teoretiska och praktiska tillämpningar. Många verkliga scenarios har icke-negativa mått (längd, tid, mängd o.s.v) och därför relaterar dess matematiska formulering till Perron-Frobenius teori. Teorin är betydande även för linjära dynamiska system, såsom Markov-kedjor, eftersom den avgör när, och till vad, en iterativ process konvergerar. Det resultatet är i sin tur den underliggande teorin bakom algoritmen PageRank som utvecklades av Google år 1998. Vi kommer se exempel på alla dessa tillämpningar i kapitel fyra och fem, där vi speciellt intresserar oss för olika typer av Markov-kedjor. Teorin i den här artikeln kan hittas i många böcker. Det mesta av materialet som presenteras här har hämtats från Horn-Johnson [5], Meyer [9] och Shapiro [10]. Däremot är alla satser och bevis formulerade på mitt eget sätt och alla exempel, samt illustrationer, har jag skapat själv, om inget annat sägs.

We show that amongst all trees with a given degree sequence it is a ball where the vertex degrees decrease with increasing distance from its center that maximizes the spectral radius of the graph (i.e., its adjacency matrix). The resulting Perron vector is decreasing on every path starting from the center of this ball. This result it also connected to Faber-Krahn like theorems for Dirichlet matrices on trees. The above result is extended to connected graphs with given degree sequence. Here we give a necessary condition for a graph that has greatest maximum eigenvalue. Moreover, we show that the greatest maximum eigenvalue is monotone on degree sequences with respect to majorization. (author's abstract). Note: There is a more recent version of this paper available: "Graphs with Given Degree Sequence and Maximal Spectral Radius", Research Report Series / Department of Statistics and Mathematics, no. 72. / Series: Research Report Series / Department of Statistics and Mathematics

Some of the theory of nonnegative matrices is first presented. The Perron-Frobenius theorem is highlighted. Some of the important linear algebraic methods of information retrieval are surveyed. Latent Semantic Indexing (LSI), which uses the singular value de-composition is discussed. The Hyper-Text Induced Topic Search (HITS) algorithm is next considered; here the power method for finding dominant eigenvectors is employed. Through the use of a theorem by Sinkohrn and Knopp, a modified HITS method is developed. Lastly, the PageRank algorithm is discussed. Numerical examples and MATLAB programs are also provided.