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Neste trabalho estudaremos a existência de soluções fracas, semi-clássicas e clássicas, conceitos introduzidos no texto para uma classe de sistemas integro-diferenciais do tipo neutro com retardamento não limitado modelados na forma d/dt D(t, xt) = AD(t, xt) + ∫t0 B(t - s)D(s, xs)ds + g(t, xt), t ∈ (0, a), x0 = φ ∈ B, d/dt (x(t) + F(t, xt)) = Ax(t) + ∫t0 B(t - s)x(s)ds + G(t, xt), t ∈ (0, a), x0 = φ ∈ B, onde A é um operador linear fechado densamente definido em um espaço de Banach X, cada B(t) : D(B(t)) ⊂ X → X, t ≥ 0 é um operador linear fechado, a história xt : (-∞, 0] → X, xt(θ) = x(t + θ), pertence a um espaço de fase abstrato B definido axiomaticamente e D, F, g, G : [0, a] × B → X são funções apropriadas. Para obter alguns de nossos resultados, estudamos a existência e propriedades qualitativas de uma família resolvente de operadores lineares limitados (R(t))t≥0, para o sistema integro-diferencial d/dt (x(t) + ∫t0 N(t - s)x(s)ds) = Ax(t) + ∫t0 B(t - s)x(s) ds, t ∈ (0, a), x(0) = x0, onde (N(t)) t≥0 é uma família de operadores lineares limitados em X. Mencionamos que este tipo de sistemas aparece no estudo da condução de calor em materiais com memória amortecida. / In this work we study the existence of mild, semi-classical and classical solution, concepts introduced be later for a class of abstract neutral functional integrodifferential systems with unbounded delay in the form d/dt D(t, xt) = AD(t, xt) + ∫t0 B(t - s)D(s, xs)ds + g(t, xt), t ∈ (0, a), x0 = φ ∈ B, d/dt (x(t) + F(t, xt)) = Ax(t) + ∫t0 B(t - s)x(s)ds + G(t, xt), t ∈ (0, a), x0 = φ ∈ B, where A : D(A) ⊂ X → X is a closed linear densely defined operator in a Banach space X, each B(t) : D(B(t)) ⊂ X → X, is a closed linear operator, the history xt : (-∞, 0] → X, xt(θ) = x(t + θ), belongs to some abstract phase space B defined axiomatically and D, F, g :[0, a] × B → X are appropriate functions. To establish some of our results, we studied the existence and qualitative properties of a resolvent of bounded linear operators (R(t))t≥0, for a system in the form d/dt (x(t) + ∫t0 N(t - s)x(s)ds) = Ax(t) + ∫t0 B(t - s)x(s) ds, t ∈ (0, a), x(0) = x0, where (N(t)) t≥0 is a family of bounded linear operators on X. We mention that this class of system arise in the study of heat conduction in material with fading memory.

The axiomatic theory of ` Zelazko defines a variety of general spectra where specified axioms are satisfied. However, there arise a number of spectra, usually defined for a single element of a Banach algebra, that are not covered by the axiomatic theory of ` Zelazko. V. Kordula and V. M¨uller addressed this issue and created the theory of regularities. Their unique idea was to describe the underlying set of elements on which the spectrum is defined. The axioms of a regularity provide important consequences. We prove that the set of Koliha-Drazin invertible elements, which includes the Drazin invertible elements, forms a regularity. The properties of the spectrum corresponding to a regularity are also investigated. / Mathematical Sciences / M. Sc. (Mathematics)

The axiomatic theory of ` Zelazko defines a variety of general spectra where specified axioms are satisfied. However, there arise a number of spectra, usually defined for a single element of a Banach algebra, that are not covered by the axiomatic theory of ` Zelazko. V. Kordula and V. M¨uller addressed this issue and created the theory of regularities. Their unique idea was to describe the underlying set of elements on which the spectrum is defined. The axioms of a regularity provide important consequences. We prove that the set of Koliha-Drazin invertible elements, which includes the Drazin invertible elements, forms a regularity. The properties of the spectrum corresponding to a regularity are also investigated. / Mathematical Sciences / M. Sc. (Mathematics)

Seja p um primo, considere q = pe com e ≥ 1 inteiro. Dado o polinômio f (x) = x4+ax3+bx2+ cx+d ∈ Fq[x], consideremos o polinômio F(T) = T4 +aT3 +bT2 +cT + d - y ∈ Fq(y)[T], com y = f (x) sobre Fq(y). O objetivo desse trabalho é determinar o número de polinômios f (x) que tem seu grupo de galois associado GF isomorfo a cada subgrupo transitivo (prefixado) de S4. O trabalho foi baseado no artigo: Galois closures of quartic sub-fields of rational function fields, usando equações auxiliares associadas ao polinômio minimal F(T) de graus 3 e 2 (DUMMIT, 1994); bem como uma caraterização das curvas projetivas planas de grau 2 não singulares. Se car(k) ≠ 2, associamos a F(T) sua cúbica resolvente RF(T) e seu discriminante ΔF. Em seguida obtemos condições para GF ≅ C4 (vide Teorema 2.9), que é ocaso fundamental para determinação dos demais casos. Se car(k) = 2, procuramos determinar condições para GRF ≅ A3, associando ao polinômio RF(T) sua quadrática resolvente P(T) (vide a Proposição 2.13). Apos ter homogeneizado P(T), usamos uma das consequências do teorema de Bézout, a saber, uma curva algébrica projetiva plana C de grau 2 é irredutível se, e somente se, C não tem pontos singulares. Nesta dissertação obtemos resultados semelhantes com uma abordagem relativamente diferente daquela usada pelo autor R. Valentini. / Let be p a prime, q = pe whit e ≥ 1 integer. Let a polynomial f (x) = x4+ax3+bx2+cx+d ∈ Fq[x], considering the polynomial F(T)=T4+aT3+bT2+cT +d, with y= f (x) over Fq(y)[T]. The purpose of the current research is to determine the numbers of polynomials f (x) which have its associated Galois group GF, this GF is isomorphic for each transitive subgroup (prefixed) of A4. This project is based on the article: Galois closures of quartic sub-fields of rational function fields, using auxiliary equations associated to the minimal polynomial F(T) of degrees 3 and 2 (DUMMIT, 1994); besides a characterization of non-singular projective plane curves of degree 2 was used. If car(k) ≠ 2, associated to F(T) the resolvent cubic RF(T) and its discriminant ΔF then conditions for GF are obtained as GF ≅ C4 which is the fundamental case for determining the other cases (Theorem 2.9). If car(k) = 2, to find conditions for GRF ≅ A3, associated to the polynomial RF(T) its resolvent quadratic p(T) (Proposition 2.13). Homogenizing p(T), one of the consequences of the Bezout theorem was applied. It is, a projective plane curve C, which grade 2, is irreducible if and only if C is smooth. In the current dissertation, similar results were obtained using a different approach developed by the author R. Valentini.

Une méthode de calcul de forçage optimal a été employée afin d'analyser le caractère amplificateur sélectif de bruit d'écoulements compressibles pariétaux. Une telle approche inclut la prise en compte de croissances non-modales induites par la non-normalité des équations de Navier-Stokes linéarisées. La méthode numérique repose sur le calcul de la matrice résolvante globale et la résolution d'un problème aux valeurs propres associé à un problème d'optimisation. Les densités d'énergie des forçages et réponses optimaux calculés pour une couche limite supersonique ont pu être reliés à la courbe neutre expérimentale obtenue par Laufer et Vrebalovich, à condition de contraindre la localisation du forçage en amont de la branche inférieure. Par la suite, une étude paramétrique en nombre de Mach de la réceptivité 2D d'une interaction choc/couche limite laminaire a permis de caractériser le développement d'instabilités convectives de Kelvin-Helmholtz et Tollmien-Schlichting (TS) à haute fréquence. La réceptivité basse fréquence de ce système a été mise en relation avec la résonance d'un mode global stable. Par ailleurs, une extension de la méthode numérique 2D a été proposée pour le calcul de perturbations 3D. Son application au calcul du forçage optimal d'une couche limite à M=4.5 a permis de mettre en évidence la croissance non-modale 3D de streaks ainsi que le développement d'ondes TS obliques dont la croissance, en régime compressible, est favorisée par rapport à celle des ondes 2D. Cette étude a également permis d'observer la croissance du mode de Mack à plus haute fréquence. / Parietal compressible flows have been studied by means of optimal forcing computations in order to characterize the noise amplifier nature of these flows. This approach is able to take into account the non-modal growth of linear perturbations induced by the non-normality of the linearized Navier-Stokes equations. The numerical strategy is based on the computation of the global resolvent matrix and an eigenvalue problem stemming from an optimization problem. Optimal forcing and response energy densities of a supersonic boundary layer have been linked to the experimental neutral curve obtained by Laufer et Vrebalovich, provided that the forcing localization is constrained upstream from the lower branch. Afterwards, a parametric study with respect to the Mach number of the 2D receptivity of the laminar shock wave/boundary layer interaction flow has allowed to analyze the growth of Kelvin-Helmholtz and Tollmien-Schlichting instabilities (TS) occurring at high frequencies. At low frequencies, the receptivity of the system has been linked to the resonance of a stable global mode. Furthermore, the 2D numerical method has been extended to allow the computation of 3D perturbations. This approach has been applied to a supersonic boundary layer flow at M=4.5 in which the 3D non-modal growth of streaks has been identified, as well as the development of oblique TS waves, whose growth is larger than the one associated to 2D waves in compressible regime. This study has also allowed to detect the growth of Mack mode at higher frequencies.

Seja p um primo, considere q = pe com e ≥ 1 inteiro. Dado o polinômio f (x) = x4+ax3+bx2+ cx+d ∈ Fq[x], consideremos o polinômio F(T) = T4 +aT3 +bT2 +cT + d - y ∈ Fq(y)[T], com y = f (x) sobre Fq(y). O objetivo desse trabalho é determinar o número de polinômios f (x) que tem seu grupo de galois associado GF isomorfo a cada subgrupo transitivo (prefixado) de S4. O trabalho foi baseado no artigo: Galois closures of quartic sub-fields of rational function fields, usando equações auxiliares associadas ao polinômio minimal F(T) de graus 3 e 2 (DUMMIT, 1994); bem como uma caraterização das curvas projetivas planas de grau 2 não singulares. Se car(k) ≠ 2, associamos a F(T) sua cúbica resolvente RF(T) e seu discriminante ΔF. Em seguida obtemos condições para GF ≅ C4 (vide Teorema 2.9), que é ocaso fundamental para determinação dos demais casos. Se car(k) = 2, procuramos determinar condições para GRF ≅ A3, associando ao polinômio RF(T) sua quadrática resolvente P(T) (vide a Proposição 2.13). Apos ter homogeneizado P(T), usamos uma das consequências do teorema de Bézout, a saber, uma curva algébrica projetiva plana C de grau 2 é irredutível se, e somente se, C não tem pontos singulares. Nesta dissertação obtemos resultados semelhantes com uma abordagem relativamente diferente daquela usada pelo autor R. Valentini. / Let be p a prime, q = pe whit e ≥ 1 integer. Let a polynomial f (x) = x4+ax3+bx2+cx+d ∈ Fq[x], considering the polynomial F(T)=T4+aT3+bT2+cT +d, with y= f (x) over Fq(y)[T]. The purpose of the current research is to determine the numbers of polynomials f (x) which have its associated Galois group GF, this GF is isomorphic for each transitive subgroup (prefixed) of A4. This project is based on the article: Galois closures of quartic sub-fields of rational function fields, using auxiliary equations associated to the minimal polynomial F(T) of degrees 3 and 2 (DUMMIT, 1994); besides a characterization of non-singular projective plane curves of degree 2 was used. If car(k) ≠ 2, associated to F(T) the resolvent cubic RF(T) and its discriminant ΔF then conditions for GF are obtained as GF ≅ C4 which is the fundamental case for determining the other cases (Theorem 2.9). If car(k) = 2, to find conditions for GRF ≅ A3, associated to the polynomial RF(T) its resolvent quadratic p(T) (Proposition 2.13). Homogenizing p(T), one of the consequences of the Bezout theorem was applied. It is, a projective plane curve C, which grade 2, is irreducible if and only if C is smooth. In the current dissertation, similar results were obtained using a different approach developed by the author R. Valentini.

The thesis consists of two parts, a theoretical part and an applied part, and in addition an Appendix. Except for a very short chapter in the applied part and the appendix we only present previously unknown results leading to a very concise style. In the theoretical part we study rates of decay for vector-valued functions and semigroups of operators depending on a real and positive variable. Under boundedness assumptions on the function/semigroup itself and under analytic extendability assumptions of its Laplace transform/resolvent across the imaginary axis we provide (almost) sharp rates of decay. Our results improve known results in this very active area of research. In the second part of the thesis we apply our results to specific examples (from the field of PDEs): local energy decay for wave equations on exterior domains, energy decay for damped wave equations on bounded domains and decay for a viscoelastic boundary damping model for sound waves. Many more examples can be found in the vast literature.

Specific non-relativistic quantum mechanical one-dimensional systems, interacting via point interactions, are discussed within the resolvent algebra setting.

Submitted by Irene Nascimento (irene.kessia@ufpe.br) on 2017-03-29T19:13:09Z No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) Tese-Thamires.pdf: 818214 bytes, checksum: 5697cce4e93e09e89c5150c064df333e (MD5) / Made available in DSpace on 2017-03-29T19:13:09Z (GMT). No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) Tese-Thamires.pdf: 818214 bytes, checksum: 5697cce4e93e09e89c5150c064df333e (MD5) Previous issue date: 2016-02-26 / CNPQ / Este trabalho trata da teoria de existência, unicidade, regularidade, continuação e alternativa de Blow-up de solução brandas para Equação de Volterra Fracionarias com condições iniciais locais cujo termo não linear satisfaz certas propriedades localmente Lipschitz. Analisamos também o caso de condições iniciais não locais e não linearidades verificando condições do tipo Caratheodory. Neste caso estudamos as propriedades topológicas do conjunto soluções de tais equações. / his work deals with existence, uniqueness, regularity, continuation and Blow up Alternative of mild solutions for Fractional Volterra Equations with local initial conditions, whose nonlinear terms satisfy some locally Lipschitz properties. Moreover we analyse thecase of nonlocal initial conditions and nonlinearities of Caratheodory type. In this case, we study topological properties of the solution set of such equations.