Šis darbas yra skirtas šiuolaikinės atomo trikdžių teorijos matematinio aparato, paremto efektinių operatorių formalizmu, plėtojimui. Darbe nuosekliai ir sistemingai, pradedant nuo pačių bendriausių principų, nagrinėjami Foko erdvės apribojimo į redukavimo grupių neredukuotinus poerdvius metodai bei pateikiama neredukuotinų tenzorinių operatorių, charakterizuojančių fizikines ir efektines sąveikas, klasifikacija bendrais ir tam tikrais atskirais atvejais. Gautos išraiškos ir iš jų išplaukiančios išvados yra grindžiamos matematine kalba. Dauguma esminių rezultatų yra suformuluoti teoremų pavidalu. Disertaciją sudaro 101 puslapis, 5 skyriai, 4 priedai, 40 lentelių ir 9 paveikslėliai. Pagrindiniai rezultatai, pateikti disertacijoje, yra publikuoti fizikos ir matematikos mokslų žurnaluose. / The principal goals of the thesis are subjected to general methods and forms of effective operators by the nowadays demands of theoretical application of many-body perturbation theory to atomic physics. The present theoretical research follows up step by step by systematic observation of various possibilities to restrict the Fock space operators to their irreducible subspaces and the classification of irreducible tensor operators which represent the physical as well as the effective interactions. To ground the results of the thesis, the symbolic preparation of obtained expressions is strictly proved mathematically. Most of the main results are listed in theorems. The doctoral dissertation contains 101 pages, 5 sections, 4 appendices, 40 tables and 9 figures. The main results described in the present dissertation have been published in journals of physical and mathematical sciences.

The principal goals of the thesis are subjected to general methods and forms of effective operators by the nowadays demands of theoretical application of many-body perturbation theory to atomic physics. The present theoretical research follows up step by step by systematic observation of various possibilities to restrict the Fock space operators to their irreducible subspaces and the classification of irreducible tensor operators which represent the physical as well as the effective interactions. To ground the results of the thesis, the symbolic preparation of obtained expressions is strictly proved mathematically. Most of the main results are listed in theorems. The doctoral dissertation contains 101 pages, 5 sections, 4 appendices, 40 tables and 9 figures. The main results described in the present dissertation have been published in journals of physical and mathematical sciences. / Šis darbas yra skirtas šiuolaikinės atomo trikdžių teorijos matematinio aparato, paremto efektinių operatorių formalizmu, plėtojimui. Darbe nuosekliai ir sistemingai, pradedant nuo pačių bendriausių principų, nagrinėjami Foko erdvės apribojimo į redukavimo grupių neredukuotinus poerdvius metodai bei pateikiama neredukuotinų tenzorinių operatorių, charakterizuojančių fizikines ir efektines sąveikas, klasifikacija bendrais ir tam tikrais atskirais atvejais. Gautos išraiškos ir iš jų išplaukiančios išvados yra grindžiamos matematine kalba. Dauguma esminių rezultatų yra suformuluoti teoremų pavidalu. Disertaciją sudaro 101 puslapis, 5 skyriai, 4 priedai, 40 lentelių ir 9 paveikslėliai. Pagrindiniai rezultatai, pateikti disertacijoje, yra publikuoti fizikos ir matematikos mokslų žurnaluose.

All procedures in the literature for decomposing symmetric second rank (stress) tensor and symmetric fourth rank (elastic constant) tensor are elaborated and compared which have many engineering and scientific applications for anisotropic materials. The decomposition methods for symmetric second rank tensors are orthonormal tensor basis method, complex variable representation and spectral method. For symmetric fourth rank (elastic constant) tensor, there are four mainly decomposition methods namely as, orthonormal tensor basis, irreducible, harmonic decomposition and spectral. Those are applied to anisotropic materials possessing various symmetry classes which are isotropic, cubic, transversely isotropic, tetragonal, trigonal and orthorhombic. For isotropic materials, an expression for the elastic constant tensor different than the traditionally known form is given. Some misprints found in the literature are corrected. For comparison purposes, numerical examples of each decomposition process are presented for the materials possessing different symmetry classes. Some applications of these decomposition methods are given. Besides, norm and norm ratio concepts are introduced to measure and compare the anisotropy degree for various materials with the same or di&curren / erent symmetries. For these materials,norm and norm ratios are calculated. It is suggested that the norm of a tensor may be used as a criterion for comparing the overall e&curren / ect of the properties of anisotropic materials and the norm ratios may be used as a criterion to represent the anisotropy degree of the properties of materials. Finally, comparison of all methods are done in order to determine similarities and differences between them. As a result of this comparison process, it is proposed that the spectral method is a non-linear decomposition method which yields non-linear orthogonal decomposed parts. For symmetric second rank and fourth rank tensors, this case is a significant innovation in decomposition procedures in the literature.

A bounded operator T on a complex separable Hilbert space is said to be homogeneous if '(T ) is unitarily equivalent to T for all ' in M•ob, where M•ob is the M•obius group. A complete description of all homogeneous weighted shifts was obtained by Bagchi and Misra. The first examples of irreducible bi-lateral homogeneous 2-shifts were given by Koranyi. We describe all irreducible homogeneous 2-shifts up to unitary equivalence completing the list of homogeneous 2-shifts of Koranyi. After completing the list of all irreducible homogeneous 2-shifts, we show that every homogeneous operator whose associated representation is a direct sum of three copies of a Complementary series representation, is reducible. Moreover, we show that such an operator is either a direct sum of three bi-lateral weighted shifts, each of which is a homogeneous operator or a direct sum of a homogeneous bi-lateral weighted shift and an irreducible bi-lateral 2-shift. It is known that the characteristic function T of a homogeneous contraction T with an associated representation is of the form T (a) = L( a) T (0) R( a); where L and R are projective representations of the M•obius group M•ob with a common multiplier. We give another proof of the \product formula". We point out that the defect operators of a homogeneous contraction in B2(D) are not always quasi-invertible (recall that an operator T is said to be quasi-invertible if T is injective and ran(T ) is dense). We prove that when the defect operators of a homogeneous contraction in B2(D) are not quasi-invertible, the projective representations L and R are unitarily equivalent to the holomorphic Discrete series representations D+ 1 and D++3, respectively. Also, we prove that, when the defect operators of a homogeneous contraction in B2(D) are quasi-invertible, the two representations L and R are unitarily equivalent to certain known pairs of representations D 1; 2 and D +1; 1 ; respectively. These are described explicitly. Let G be either (i) the direct product of n-copies of the bi-holomorphic automorphism group of the disc or (ii) the bi-holomorphic automorphism group of the polydisc Dn: A commuting tuple of bounded operators T = (T1; T2; : : : ; Tn) is said to be homogeneous with respect to G if the joint spectrum of T lies in Dn and '(T); defined using the usual functional calculus, is unitarily equivalent to T for all ' 2 G: We show that a commuting tuple T in the Cowen-Douglas class of rank 1 is homogeneous with respect to G if and only if it is unitarily equivalent to the tuple of the multiplication operators on either the reproducing kernel Hilbert space with reproducing kernel n 1 i=1 (1 ziwi) i or Q n i i n; are positive real numbers, according asQG is as in (i) or 1 ; where ; i, 1 i i=1 (1 z w ) (ii). Finally, we show that a commuting tuple (T1; T2; : : : ; Tn) in the Cowen-Douglas class of rank 2 is homogeneous with respect to M•obn if and only if it is unitarily equivalent to the tuple of the multiplication operators on the reproducing kernel Hilbert space whose reproducing kernel is a product of n 1 rank one kernels and a rank two kernel. We also show that there is no irreducible tuple of operators in B2(Dn), which is homogeneous with respect to the group Aut(Dn):

Submitted by ANA KARLA PEREIRA RODRIGUES (anakarla_@hotmail.com) on 2017-08-17T12:22:57Z No. of bitstreams: 1 arquivototal.pdf: 697305 bytes, checksum: 0b28f8f8c4f8b4509047eb441817be7c (MD5) / Made available in DSpace on 2017-08-17T12:22:57Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 697305 bytes, checksum: 0b28f8f8c4f8b4509047eb441817be7c (MD5) Previous issue date: 2016-03-09 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work, we present an explicit geometric characterization for the space of quadratcs form vanishing precisely on a twisted cubic. We show that the set of degenerate quadrics lying on a net of quadrics containing a twisted cubic is described by a curve whose equation is given by the square of an irreducible conic. Conversely, if is a net of quadrics whosw intersection with the set of degenerate quadrics is a curve given by the square of an irreducible conic, we furnish conditions under which the cammon zero locus of turns out to be a twisted cubic. It is enough to require that does not contain a pair of planes. / Neste trabalho, apresentamos uma caracteriza c~ao geom etrica expl cita para o espa co das formas quadr aticas que se anulam precisamente sobre uma c ubica reversa. Mostramos que o conjunto das qu adricas degeneradas pertencentes a uma rede de qu adricas que cont em a c ubica reversa e descrita por uma curva cuja equa c~ao e dada pelo quadrado de uma c^onica irredut vel. Rec procamente, se e uma rede de qu adricas cuja interse c~ao com o conjunto das qu adricas n~ao degeneradas e uma curva dada pelo quadrado de uma c^onica irredut vel, fornecemos condi c~oes sob as quais o lugar dos zeros comuns de seja uma c ubica reversa. E su ciente que n~ao contenha um par de plano.

Made available in DSpace on 2015-03-26T13:45:35Z (GMT). No. of bitstreams: 1 texto completo.pdf: 1037099 bytes, checksum: 73ef343c6773880a845ee58f11f86a80 (MD5) Previous issue date: 2013-01-18 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / This dissertation is devoted to the study of stable maps from closed surfaces to the sphere, from a global viewpoint. Associated of such maps domain, we study grafs with integers positive weight in the vertices as invariants, based in the Hacon, Mendes and Romero and works. And associated the image of these maps, we study the minimal contour based in the Kamenosono-Yamamoto work. / Essa dissertação é dedicada ao estudo de aplicações estáveis de superfícies fechadas na esfera, do ponto de vista global. Associado ao domínio de tais aplicações, estudamos grafos com pesos inteiros positivos nos vértices como invariantes, baseado nos trabalhos e de Hacon, Mendes e Romero. E associado à imagem dessas aplicações, estudamos o contorno minimal baseado no trabalho de Kamenosono-Yamamoto.

Dans cette thèse, nous nous intéressons à des singularités particulières dans les variétés de caractères. Dans le premier chapitre, on justifie que les caractères de représentations irréductibles d'un groupe fuchsien vers un groupe de Lie complexe semi-simple forment une orbifolde. Le lieu orbifold (i.e. l'ensemble des points dont l'isotropie n'est pas triviale) est constitué des caractères de représentations exceptionnelles. Dans le second chapitre, nous décrivons précisément le lieu orbifold quand le groupe de Lie est le groupe projectif linéaire sur un espace vectoriel complexe dont la dimension est un nombre premier. Dans le troisième et le quatrième chapitre nous cherchons à classifier les groupes d'isotropies possibles à conjugaison près apparaissant quand le groupe de Lie est respectivement un quotient du groupe spécial linéaire pour un espace vectoriel complexe de dimension finie quelconque dans le troisième chapitre et un quotient du groupe de spin complexe dans le quatrième chapitre. / Ln this thesis, we want to understand some singularities in the character variety. ln a first chapter, we justify that the characters of irreducible representations from a Fuchsian group to a complex semi-simple Lie group is an orbifold. The orbifold locus is, then, the characters of bad representations. ln the second chapter, we focus on the case where the Lie group is the projectif linear group over a complex vector space whose dimension is a prime number. ln particular we give an explicit description of this locus. ln the third and fourth chapter, we describe the isotropy groups (i.e. the centralizers of bad subgroups) arising in the cases when the Lie group is a quotient of the special linear group of a complex vector space of finite dimension (third chapter) and when the Lie group is a quotient of a complex spin group in the fourth chapter.

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Soit q ⩾ 2 une puissance première fixe. L’objectif principal de cette thèse est d’étudier le comportement asymptotique de la fonction arithmétique Π_q(n,k) comptant le nombre de polynômes moniques de degré n et ayant exactement k facteurs irréductibles (avec multiplicité) sur le corps fini F_q. Warlimont et Car ont montré que l’objet Π_q(n,k) est approximativement distribué de Poisson lorsque 1 ⩽ k ⩽ A log n pour une constante A > 0. Plus tard, Hwang a étudié la fonction Π_q(n,k) pour la gamme complète 1 ⩽ k ⩽ n. Nous allons d’abord démontrer une formule asymptotique pour Π_q(n,k) en utilisant une technique analytique classique développée par Sathe et Selberg. Nous reproduirons ensuite une version simplifiée du résultat de Hwang en utilisant la formule de Sathe-Selberg dans le champ des fonctions. Nous comparons également nos résultats avec ceux analogues existants dans le cas des entiers, où l’on étudie tous les nombres naturels jusqu’à x avec exactement k facteurs premiers. En particulier, nous montrons que le nombre de polynômes moniques croît à un taux étonnamment plus élevé lorsque k est un peu plus grand que logn que ce que l’on pourrait supposer en examinant le cas des entiers. Pour présenter le travail ci-dessus, nous commençons d’abord par la théorie analytique des nombres de base dans le contexte des polynômes. Nous introduisons ensuite les fonctions arithmétiques clés qui jouent un rôle majeur dans notre thèse et discutons brièvement des résultats bien connus concernant leur distribution d’un point de vue probabiliste. Enfin, pour comprendre les résultats clés, nous donnons une discussion assez détaillée sur l’analogue de champ de fonction de la formule de Sathe-Selberg, un outil récemment développé par Porrit et utilisons ensuite cet outil pour prouver les résultats revendiqués. / Let q ⩾ 2 be a fixed prime power. The main objective of this thesis is to study the asymptotic behaviour of the arithmetic function Π_q(n,k) counting the number of monic polynomials that are of degree n and have exactly k irreducible factors (with multiplicity) over the finite field F_q. Warlimont and Car showed that the object Π_q(n,k) is approximately Poisson distributed when 1 ⩽ k ⩽ A log n for some constant A > 0. Later Hwang studied the function Π_q(n,k) for the full range 1 ⩽ k ⩽ n. We will first prove an asymptotic formula for Π_q(n,k) using a classical analytic technique developed by Sathe and Selberg. We will then reproduce a simplified version of Hwang’s result using the Sathe-Selberg formula in the function field. We also compare our results with the analogous existing ones in the integer case, where one studies all the natural numbers up to x with exactly k prime factors. In particular, we show that the number of monic polynomials grows at a surprisingly higher rate when k is a little larger than logn than what one would speculate from looking at the integer case. To present the above work, we first start with basic analytic number theory in the context of polynomials. We then introduce the key arithmetic functions that play a major role in our thesis and briefly discuss well-known results concerning their distribution from a probabilistic point of view. Finally, to understand the key results, we give a fairly detailed discussion on the function field analogue of the Sathe-Selberg formula, a tool recently developed by Porrit and subsequently use this tool to prove the claimed results.