O objetivo deste trabalho é promover um cotejamento entre as idéias estéticas e científicas de Goethe compreendidas entre o período do Sturm und Drang e o primeiro Classicismo (Frühklassik). Através dessa comparação é delineada a importância que a inter-relação entre arte e ciência desempenhou no desenvolvimento dos conceitos goethianos relativos a essas duas áreas de sua produção. / Nature is the starting point of Goethe in relation to science and art He feels united to the spirit of the creator from the moment he is involved with the natural and the perception of the presence of the divine in nature. When Goethe discovered his creative and intellectual capacities, he used them to create new forms and to further understand the world. Nature in its constant process of transformation gives rise to the creation of an art under the sign of metamorphosis, whose laws Goethe presents as Polarity (Polariãt) and Intensification (Steigerung). The art presented by Goethe is not a copy of nature as a simple formal element, but the reflection of the representation of nature\'s own laws. Thus both science and art integrate with each other and, consequently, they become interdependent. The art of Goethe is constructed from the dialogue between the world and the use of its capacity to create elaborating new forms; therefore, respecting the same laws which originate the natural world. This study presents the formation of goethian\'s organic poiesis created by the dialogue between art and science.

Este trabalho trata da ficção histórica de Goethe em duas fases de sua obra. Em primeiro lugar se ocupa do drama pioneiro do Sturm und Drang, Götz von Berlichingen (1773), para então se voltar às Revolutionsdichtungen conjunto de obras produzidas entre 1791 e 1803, período auge do Classicismo de Weimar, em que o autor emitiu juízos sobre diferentes fases da Revolução Francesa. Ao optar por tal tema, visou-se resgatar Goethe como um importante participante do debate histórico-filosófico que começa com os iluministas e culmina no pensamento político da Restauração. Por esse motivo a pesquisa se iniciou com uma busca dos antecedentes do drama histórico do Sturm und Drang, voltando a Gottsched e Lessing. Geralmente o desenvolvimento da ficção histórica do século XVIII é tomado como passivo em relação aos avanços da filosofia ilustrada, como se uma consciência histórica propriamente moderna houvesse aflorado em certos escritores a partir das leituras que fizeram de novas teorias da época. Contra tal suposição, defendeu-se que há na beletrística alemã da virada do século algumas formulações acerca da relação entre indivíduo e processo histórico que, antes de tudo, desafiaram o progressismo dos iluministas tardios, para os quais o advento da Revolução Francesa marcava o próximo passo no aperfeiçoamento da humanidade. Acompanhar o modo como a história foi tratada nas obras do Sturm und Drang até as do Classicismo de Weimar permite-nos, ademais, constatar a radical atualização da missão da cultura literária, que deveria então lidar com as emergências do presente. A literatura almejada naquele contexto deveria funcionar como uma contracorrente do senso comum e um veículo intelectual autônomo, capaz de erguer-se além das paixões partidárias e posicionar corretivos para as contradições contemporâneas. Tais corretivos traziam o diferencial de serem desvinculados das ciências, religião, cultura política ou sabedoria popular a nova arte deveria, assim, ser tomada como autônoma na medida em que podia cindir radicalmente com a cultura do presente em crise, tornando possível o surgimento de um ideal renovado de humanidade e de vida conjunta que a própria Revolução Francesa não foi capaz de concretizar. Aqui reside o que há de mais polêmico e moderno nas chamadas Revolutionsdichtungen e no Classicismo de Weimar como um todo. / The following research deals with Goethe\'s historical fiction in two distinct phases of his career. First off, it handles with the pioneering Sturm und Drang drama, Götz von Berlichingen (1773), and then with the so-called Revolutionsdichtungen a group of works produced between 1791 and 1803, the heyday of the Weimar Classicism, in which the author made judgments about different aspects of the French Revolution. In choosing such a theme, I expect to highlight the importance of Goethe in the historical-philosophical debate which ranges from the works of Enlightenment intellectuals to the political thinking of the Restauration. Thus, the research sets out with a quest for the predecessors of the historical drama of the Sturm Drang, coming across to figures like Gottsched and Lessing. The development of the historical fiction from the 18th century is commonly taken for granted as somewhat passive towards the advances of the Enlightenment philosophy, as if a properly modern historical consciousness had flourished in certain writers after the readings they made from contemporary theories. Against such a presupposition, I claim that in the German belletristic from the turning of the century one can find interesting formulations about the relation between individual and historical process which, above all, cast doubt on the progressism of late Enlightenment philosophers, to whom the advent of the French Revolution marked the next step toward an evolving mankind. Moreover, following the tracks of how history was portrayed from the Sturm und Drang to the Weimar Classicism allows us to verify a radical reformulation of the mission of literary culture, which thenceforth should respond to the emergencies of the present. The literature of that context should work against the common sense, being thus an autonomous cultural medium able to reach beyond the partisan leanings and offer correctives to contemporary contradictions. Such correctives would have the advantage of being independent of the sciences, religion, political culture or popular wisdom a new art therefore should be called autonomous inasmuch as it could split with the culture of a problematic present, allowing for the emergence of a renewed ideal of humanity and communitary life that the French Revolution itself was incapable of materializing. Therein lies what is most polemical and modern in the so-called Revolutionsdichtungen and in the Weimar Classicism as a whole.

Dans cette thèse, nous étudions la contrôlabilité à zéro de quelques systèmes paraboliques continus et semi-discrétisés. Nous considérons tout d'abord des systèmes en cascade d'équations paraboliques de la forme ∂t −(∂xγ∂x +q). La variable spatiale évolue dans un intervalle réel borné et ce système est semi-discrétisé en espace par un schéma aux différences finies. En appliquant la méthode des moments, nous démontrons des résultats de contrôlabilité à zéro et de φ(h) contrôlabilité à zéro, suivant les hypothèses formulées sur le maillage et les fonctions γ et q. Puis nous étendons ces résultats lorsque la variable d'espace évolue dans un domaine cylindrique, la zone de contrôle se situant dans une partie d'une section au bord du cylindre. Ce domaine cylindrique se décompose en un produit de deux espaces. Sur le premier, de dimension 1, nous appliquons les résultats décrits précédemment. Sur le second, nous appliquons la méthode de Lebeau-Robbiano. Cette approche permet à la fois de montrer que le problème discrétisé est φ(h) contrôlable à zéro et de retrouver un résultat de contrôlabilité à zéro sur le système continu. Dans une autre partie, nous nous intéressons au temps minimal de contrôle à zéro de l'équation de Grushin posée sur un domaine rectangulaire dont le domaine de contrôle est une bande verticale. L'étude se ramène à une infinité dénombrable, indexée par le paramètre de Fourier $n$, de problèmes de contrôle à zéro d'équations paraboliques, traitée, ici encore, à l'aide de la méthode des moments. / In this thesis, we study the null controllability of some continous and semi discretized parabolic systems. We first consider cascade systems of parabolic equations of the form ∂t −(∂xγ∂x +q). The space variable belongs to a real and bounded interval and this system is semi-discretized in space by a finite differences scheme. Applying the so called moments method, we prove null controllability and φ(h) null controllability results, depending on the hypotheses on the mesh and on functions γ and q. Then, we extend this results when the space variable belongs to a cylindrical domain which control zone is in a section at the border of the cylinder. This cylindrical domain is decomposed into a product of two spaces. On the first, of dimension 1, we apply the results described previously. On the second, we use the Lebeau-Robbiano's procedure. In this framework, we prove φ(h) null controllability results on the discretized domain as well as null controllability results on the continous problem. In another section, we investigate the computation of minimal time of null controllability of Grushin's equation defined on a rectangular domain which control region is a vertical strip. This problem of control amounts to study a countably infinite family, indexed by the Fourier parameter $n$, of null control problems of parabolic equations, tackled, once again, with the moments method.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / In this dissertation, we study the Sturm Liouvile's theorems for the zeros of the solutions of linear differential equations of second order. These classical theorems are applied to analysis of the monotonicity of functions involving the zeros of classical orthogonal polynomials. in particular, Gegenbauer polynomials. / Neste trabalho estudamos os Teoremas de Sturm Liouville para zeros de soluções de equações diferenciais lineares de segunda ordem. Estes teoremas clássicos são aplicados para análise do crescimento e decrescimento de certas funções que envolvem os zeros de Polinômios Ortogonais Clássicos, como os Polinômios de Gegenbauer. / Mestre em Matemática

Tez (Yüksek Lisans) - Süleyman Demirel Üniversitesi, Fen Bilimleri Enstitüsü, Matematik Anabilim Dalı, 2007. / Bibliyografya var.

Dans cette thèse, nous nous intéressons à la contrôlabilité interne et à son coût pour une ou plusieurs équations aux dérivées partielles conservatives. ?Dans la première partie, nous introduisons et détaillons deux méthodes permettant d'estimer le coût du contrôle (et par dualité, de la constante d'observabilité) de l'équation des ondes avec potentiel $l^{\infty}$ en dimension un d'espace. La première utilise la propagation des ondes le long des caractéristiques en s'appuyant sur le rôle symétrique de la variable de temps et d'espace. La deuxième méthode repose sur la décomposition spectrale de l'équation des ondes et sur l'utilisation des inégalités d'ingham. L'estimation de la constante d'observabilité se ramène alors à l'étude d'un problème d'optimisation faisant intervenir les vecteurs propres du laplacien-dirichlet avec potentiel. Nous fournissons ensuite des propriétés qualitatives sur le minimiseurs ainsi qu'une estimation du minimum ne dépendant que de la mesure de l'ensemble d'observation. ?Dans la deuxième partie, nous étudions la contrôlabilité de certains systèmes d'équations avec un nombre de contrôles réduits, autrement dit le nombre de contrôles est plus petit que le nombre d'équations. En particulier, nous caractérisons exactement les données initiales qui peuvent être contrôlées pour des systèmes d'équations couplées de type schrödinger et nous énonçons une condition nécessaire et suffisante de type kalman pour des systèmes d'équations des ondes couplées. La preuve repose sur une méthode de contrôle fictif combinée à la résolution algébrique d'un système sous-déterminé et sur certains résultats de régularité. / In this work, we focus on the internal controllability and its cost for some linear partial differential equations. In the first part, we introduce and describe two methods to provide precise estimates of the cost of control (and by duality, of the observability constant) for general one dimensional wave equations with potential. The first one is based on a propagation argument along the characteristics relying on the symmetrical roles of the time and space variables. The second one uses a spectral decomposition of the solution of the wave equation and ingham's inequalities. This relates the estimation of the observability constant to the study of an optimal problem involving dirichlet eigenfunctions of laplacian with potential. We provide some qualitative properties of the minimizers, and also precise bounds on the minimum. In the second part, we are concerned with the controllability of some systems of equations by a reduced number of controls (i.e. the number of controls is less that the number of equations). In particular, in the case of coupled systems of schrödinger equations, we exactly characterize the initial conditions that can be controlled and we give a necessary and sufficient condition of kalman type for the controllability of coupled systems of wave equations. The proof relies on the fictitious control method coupled with the proof of an algebraic solvabilityproperty for some related underdetermined system, as well as on some regularity results.

In this paper we consider the question of uniqueness of positive solutions for Dirichlet problems of the form - Δ u(x)= g(λ,u(x)) in B, u(x) = 0 on ϑB, where A is the Laplace operator, B is the unit ball in RˆN, and A>0. We show that if g(λ,u)=uˆ(N+2)/(N-2) + λ, that is g has "critical growth", then large positive solutions are unique. We also prove uniqueness of large solutions when g(λ,u)=A f(u) with f(0) < 0, f "superlinear" and monotone. We use a number of methods from nonlinear functional analysis such as variational identities, Sturm comparison theorems and methods of order. We also present a regularity result on linear elliptic equation where a coefficient has critical growth.

Orientador: Marta Cilene Gadotti / Banca: Suzete Maria Silva Afonso / Banca: Katia Andreia Gonçalves de Azevedo / Resumo: Este texto aborda os principais resultados sobre a Teoria de Sturm-Liouville assim como os pré-requisitos necessários para construí-los, entre eles o Teorema Espectral para Operadores Compactos e a Teoria de Fredholm. Também são apresentados alguns exemplos e uma aplicação envolvendo uma equação diferencial parcial que modela o problema da corda vibrante / Abastract: This research approaches the main results on the Sturm-Liouville Theory, as well the necessary prerequisites for constructing them, including the Spectral Theorem for Compact Operators and Fredholm Theory. It is also presented some examples and an application involving a partial differential equation that models the vibrating string problem / Mestre

Hourya Sinaceur présente dans son ouvrage Corps et Modèles la théorie des caractéristiques de Kronecker comme la fin d’« un cycle d’idées sturmiennes », la situant ainsi dans une histoire de l’algébrisation du théorème de Sturm. Pourtant, cette théorie est souvent aussi présentée comme le point de départ de certains des premiers travaux de topologie de la toute fin du dix-neuvième siècle. Nous souhaitons dans cette thèse rendre compte de la façon dont ces deux histoires se rencontrent au sein de la théorie de Kronecker. Pour cela, nous disposons d’un matériel particulièrement éclairant et jusqu’alors très peu exploité : les manuscrits des cours que Kronecker a donnés à l’Université de Berlin entre 1872 et 1891 sur la Théorie des équations algébriques. Nous commencerons par présenter ces manuscrits, leur contenu et leur contexte de rédaction. Nous nous intéresserons ensuite plus précisément à la reprise du théorème de Sturm par Kronecker, et nous montrerons que la théorie des caractéristiques n’est pas seulement un prolongement algébrique de ce théorème, mais qu’elle se transforme pour fournir certains des outils analytiques que Poincaré utilisera lors de la construction de son Analysis Situs. L’exposition de la théorie des caractéristiques dans ses cours est l’occasion pour Kronecker de reprendre trois des quatre démonstrations que Gauss a données du théorème fondamental de l’algèbre, et nous montrerons comment, dans la pratique de Kronecker, la notion même de racine est interrogée / In her book Corps et Modèles, Hourya Sinaceur presented Kronecker’s theory of characteristics as the end of a “cycle of Sturmian ideas”, making it a step in history of the algebraization of Sturm’s theorem. However, this theory is often also introduced as the starting point of some of the early works of topology of the very end of the nineteenth century. In this PhD thesis, I will describe how these two stories are connected in Kronecker’s theory. To achieve this, I used material which has seldom been discussed before : the manuscripts of the courses that Kronecker gave at the University of Berlin between 1872 and 1891 on the Theory of algebraic equations. I begin with the presentation of these manuscripts, their contents and their writing contexts. I then look more closely at Kronecker’s rework of the theorem of Sturm and show that the theory of characteristics is not only an algebraic extension of this theorem but also that it is transformed in order to provide some of the analytic tools that Poincaré will use for the construction of his Analysis Situs. The exposition of the theory of characteristics in his courses is an opportunity for Kronecker to take up three of the four demonstrations from Gauss of the fundamental theorem of algebra, and I will show how, in Kronecker’s practice, the very notion of root is questioned

We study several unbounded operators with view to extending von Neumann's theory of deficiency indices for single Hermitian operators with dense domain in Hilbert space. If the operators are non-commuting, the problems are difficult, but special cases may be understood with the use representation theory. We will further study the partial derivative operators in the coordinate directions on the L2 space on various covering surfaces of the punctured plane. The operators are defined on the common dense domain of C∞ functions with compact support, and they separately are essentially selfadjoint, but the unique selfadjoint extensions will be non-commuting. This problem is of a geometric flavor, and we study an index formulation for its solution. The applications include the study of vector fields, the theory of Dirichlet problems for second order partial differential operators (PDOs), Sturm-Liouville problems, H.Weyl's limit-point/limit-circle theory, Schrödinger equations, and more.