Induced graphs are used to describe the structure of a graph, one such type of induced graph that has been studied are long paths. In this thesis we show a way to represent such graphs in terms of an array with two colors and a labeled graph. Using this representation and the techniques of Polya counting we will then be able to get upper and lower bounds for graphs containing a long path as an induced subgraph. In particular, if we let P(n,k) be the number of graphs on n+k vertices which contains P_n, a path on n vertices, as an induced subgraph then using our upper and lower bounds for P(n,k) we will show that for any fixed value of k that P(n,k)~2^(nk+k_C_2)/(2k!).

Neste trabalho abordamos dois exemplos de equações de difusão não locais do tipo Neumann: o problema linear homogêneo e um semilinear com termo de reação representado pela função f(u) = u|u|^(p-1). Em ambos os casos, apresentamos condições de existência e unicidade de soluções e analisamos seu comportamento em relação ao tempo. Estudamos uma discretização para o problema linear e a utilizamos para realizar simulações numéricas nas quais podemos verificar algumas das propriedades demonstradas. Também simulamos o problema semilinear observando o comportamento de suas soluções mesmo em casos em que as hipóteses dos teoremas apresentados não são todas satisfeitas. / In this work we approach two examples of nonlocal diffusion equations of Neumann type: the homogeneous linear problem and a semilinear with a reaction term represented by the function f(u) = u|u|^(p-1). In both cases, we present conditions of existence and uniqueness of solutions and we analyze their behavior with respect to time. We study a discretization to the linear problem and use it to perform numerical experiments in order to illustrate some of the demonstrated properties. We also simulate the semilinear problem observing the behavior of its solutions even in cases where the hypothesis of the presented theorems are not all satisfied.

Cette thèse porte sur l’analyse mathématique de différents modèles de la digestion anaérobie. Dans la première partie, nous étudions un modèle à quatre étapes avec dégradation enzymatique du substrat (matière organique) qui peut être sous forme solide. Nous étudions l’effet de l’hydrolyse sur le comportement du processus de la digestion anaérobie et de la production du biogaz (méthane et hydrogène). Nous considèrons, dans un premier modèle, que l’hydrolyse se fait d’une manière enzymatique, alors que dans un second, nous supposons qu’elle est réalisée par un compartiment microbien. Les modèles considérés incluent l’inhibition de croissance des bactéries acétogènes, méthanogènes hydrogénétrophes et acétoclastes par plu- sieurs substrats. Pour étudier l’effet de ces inhibitions en présence de l’étape de l’hydrolyse, nous étudions dans un premier temps un modèle sans inhibition. Nous déterminons les équilibres et nous donnons des conditions nécessaires et suffisantes pour leur stabilité. L’existence et la stabilité des équilibres sont illustrées avec des diagrammes opératoires. Nous montrons que le modèle avec hydrolyse enzymatique change la production du méthane et d’hydrogène. En outre, l’introduction du com- partiment hydrolytique microbien donne de nouveaux équilibres et affecte les régions de stabilité. Nous prouvons que la production de biogaz est maximale en un seul point d’équilibre selon les paramètres opératoires et nous déterminons le taux maxi- mal de biogaz produit, dans chaque cas. Dans la deuxième partie, nous nous sommes intéressés à un modèle à deux étapes décrivant les phases de l’acétogénèse et de la méthanogénèse hydrogénotrophe. Le modèle représente une relation de syntrophie entre deux espèces microbiennes (les bactéries acétogènes et méthanogènes hydro- génotrophes), avec deux substrats à l’entrée (l’acide gras volatile et l’hydrogène), incluant les termes de mortalité et l’inhibition de croissance des bactéries acéto- gènes par un excès d’hydrogène dans le système. L’analyse de l’existence et de la stabilité des équilibres du modèle donne naissance à un nouvel équilibre qui peut être stable selon les paramètres opératoires du système. En utilisant les diagrammes opératoires, on remarque que, quelle que soit la région de l’espace considérée, il existe un seul équilibre localement exponentiellement stable. Cette étude est géné- ralisée dans le cas où la croissance des bactéries méthanogènes hydrogénotrophes est inhibée. Ce modèle donne naissance à deux équilibres strictement positifs et une bistabilité. Nous illustrons, en utilisant les diagrammes opératoires l’effet de cette inhibition sur la réduction des régions de coexistence et l’émergence de régions de bistabilité. / This PhD thesis focuses on the mathematical analysis of different anaerobic digestion (AD) models. In a first part, we study a 4-step model with enzymatic degradation of the substrate (organic matter) that can partly be under a solid form. We investigate the effects of hydrolysis on the behavior of the AD process and the production of biogas (namely, the methane and the hydrogen). We consider, in a first model, that the microbial enzymatic activity is constant, then we take into consideration an explicit hydrolytic microbial compartment for the substrate biodegradation. The considered models include the inhibition of acetogens, hydroge- notrophic methanogens and acetoclastic methanogens growth bacteria. To examine the effects of these inhibitions in presence of a hydrolysis step, we first study an inhibition-free model. We determine the steady states and give sufficient and neces- sary conditions for their stability. The existence and stability of the steady states are illustrated by operating diagrams. We prove that modeling the hydrolysis phase by a constant enzymatic activity affects the production of methane and hydrogen. Furthermore, introducing the hydrolytic microbial compartment yields new steady states and affects the stability regions. We prove that the biogas production occurs at only one of the steady states according to the operating parameters and state variables and we determine the maximal rate of biogas produced, in each case. In the second part, we are interested in a reduced and simplified model of the AD pro- cess. We focus on the acetogenesis and hydrogenetrophic methanogenesis phases. The model describes a syntrophic relationship between two microbial species (the acetogenic bacteria and the hydrogenetrophic methanogenic bacteria) with two in- put substrates (the fatty acids and the hydrogen) including both decay terms and inhibition of the acetogenic bacteria growth by an excess of hydrogen in the sys- tem. The existence and stability analysis of the steady states of the model points out the existence of a new equilibrium point which can be stable according to the operating parameters of the system. By means of operating diagrams, we show that, whatever the region of space considered, there exists only one locally exponentially stable steady state. This study is generalized to the case where the growth of the hydrogenetrophic methanogens bacteria is inhibited. This model exhibits a rich be- havior with the existence of two positive steady states and bistability. We illustrate by means of operating diagrams the effect of this inhibition on the reduction of the coexistence region and the emergence of a bistability region.

Dans cette thèse nous nous intéressons aux propriétés qualitatives et quantitatives des solutions de quelques équations d'ondes en milieux dispersifs ou dispersifs-dissipatifs. Dans une première partie, nous étudions le problème de Cauchy associé aux équations de Benjamin-Ono généralisées. A l'aide de transformées de jauge, combinées avec des outils d'analyse harmonique, nous prouvons des résultats concernant le caractère localement bien posé pour des données initiales de régularité minimale dans l'échelle des espaces de Sobolev. Dans une seconde partie, nous étudions le problème de Cauchy pour des versions dissipatives des équations de Benjamin-Ono et de Korteweg-de Vries. Nous mettons en évidence l'influence des effets dissipatifs sur ces équations en donnant des résultats optimaux sur leur caractère bien ou mal posé. Ceux-ci sont obtenus en travaillant dans des espaces de type Bourgain adaptés à la partie dispersive-dissipative. Pour finir nous étudions le comportement asymptotique des solutions des équations de KdV dissipatives, lorsque celles-ci existent pour tout temps, en calculant explicitement les premiers termes du développement asymptotique dans de nombreux espaces de Sobolev / This thesis deals with the qualitative and quantitative properties of solutions to some wave equations in dispersive or dispersive-dissipative media. In the first part, we study the Cauchy problem for the generalized Benjamin-Ono equations. By means of gauge transforms combined with some harmonic analysis tools, we prove some local well-posedness results for initial data with minimal regularity in Sobolev spaces. In the second part, we study the Cauchy problem for some dissipative versions of the Benjamin-Ono and Korteweg-de Vries equations. We show the influence of the dissipative effects and prove sharp well and ill-posedness results. This is obtained by working in suitable Bourgain's spaces, adapted to the dispersive-dissipative part of the equation. Finally, we study the asymptotic behavior of solutions to the dissipative KdV equations. We explicitly compute the first terms of the asymptotic expansion in Sobolev spaces

Neste trabalho estudamos a equação de ondas do tipo p-Laplaciano \'u IND. tt\' - \'DELTA\' IND.p u + \'(- \'DELTA\' POT. alpha\' u IND. t\' = \' [u] POT.q - 2 u, definida num domínio limitado limitado do \'R POT. n\', com 2 \' > ou = \' p < q e 0 < \' alpha\' < 1. Utilizando o método de Faedo-Galerkin provamos a existência de soluções fracas globais para dados iniciais pequenos. Para essas soluções estudamos também o decaimento polinomial da energia associada. A questão da não existência de soluções globais é considerada para o caso em que a energia inicial do sistema é negativa / In this work we study the p-Laplacian wave equation \'u IND. tt\' - \' DELTA\' IND p u + \'(- \'DELTA\' POT. \'alpha\' \' u IND. t\' = \'[u] POT. q - 2 u, defined in a bounded domain of \'R POT n\', with 2 \'> or =\' p < q and 0 < \' alpha\' < 1. By using the Faedo-Galerkin method we prove the existence of weak global solutions for small initial data. We also study the polynomial decay of the associate energy. The blow-up of solutions in finite time is considered for negative initial energy

Estamos interessados em estudar o comportamento assintótico das soluções de uma classe de Equações Diferenciais Funcionais (EDF) lineares e autônomas do tipo neutro, onde os coeficientes, na parte não neutra, são funções periódicas de período comum w! e os retardamentos são múltiplos de w. Para isto, utilizamo-nos da teoria espectral de operadores aplicada ao chamado operador monodrômico \'PI\' : C \'SETA\' C, cuja ação é evoluir um dado estado um passo de tamanho w. Calculamos o resolvente deste operador, donde inferimos todas as propriedades espectrais que nos permitem determinar o comportamento assintótico das soluções. Mostramos a importância de se determinar autovalores dominantes para a obtenção das estimativas, e mostramos resultados neste sentido. Estudamos em detalhe três exemplos que ilustram a teoria e demonstram sua aplicabilidade / We are interested in the study of the asymptotic behavior of the solutions of a class of linear autonomous Functional Differential Equations (FDE) of neutral type, where the coeficients of the non neutral part are periodic functions with common period w and the time delays are multiples of w. We employ the spectral theory for linear operators applied to the so called monodromic operator \'PI\' : C \'ARROW\'! C, whose action is to evolve a given state one step of size w. We compute the resolvent of this operator, from where we infer the spectral properties that allows us to determine the asymptotic behavior of the solutions. We show the importance to determine whether an eigenvalue is dominant, in order to obtain the estimates for the correspondet solution, and we show results in this direction. Finally we study in detail three examples that illustrate the theory and demonstrate its applicability

Cette thèse traite de l'étude des équations de Navier-Stokes stationnaires incompressibles et, plus précisément, le comportement quand x→∞ de ses solutions. On étudie la situation dans différents types de domaines non bornés en supposant une condition de nullité à l'infini. On regarde d'abord la dimension 3, dans lequel on sait que si le terme de force décroît très vite à l'infini, le comportement asymptotique est donné par les solutions de Landau, qui sont homogènes de degré -1. On généralise donc ce résultat à des termes de force petits dont le comportement asymptotique est donné par un terme avec l'homogénéité correspondante, c'est-à-dire de degré -3. Pour cela, on trouve une condition nécessaire et suffisante qui est que la partie homogène du terme de force soit de moyenne nulle sur la sphère. Pour finir, on généralise ce résultat au cas d'un domaine extérieur. Dans le cas d'un demi-espace, on va plus loin en montrant que si le terme de force décroit assez à l'infini on obtient des solutions décroissant comme 1/|x|2 à l'infini et on trouve une expression explicite du terme dominant. On peut aussi montrer le même type de résultat que dans l'espace entier avec un terme de force en 1/|x|3 mais la condition de moyenne nulle sur la sphère disparaıt. Dans l'étude de la dimension 2 dans le plan tout entier, on se rend compte que les choses sont plus compliquées. D'abord, pour les solutions homogènes, on arrive à trouver les conditions pour que, si le terme de force est suffisamment petit, on obtienne l'existence de solution qui forment alors une famille à deux paramètres. Mais en leur imposant la restriction d'avoir un flux nul sur le cercle unité, on obtient une famille avec un paramètre seulement. Enfin on étudie les solutions non homogènes, mais pour cela on doit supposer certaines conditions de symétrie sur les données. On trouve alors, pour des termes de force décroissant très vite à l'infini, des solutions en 1/|x|3 et on obtient une formule explicite pour le terme principal de leur développement asymptotique. Ce résultat se généralise aussi au cas d'un domaine extérieur et pour finir, dans ce cadre symétrique, on trouve un résultat analogue au cas de la dimension 3 pour des termes de force qui décroissent en 1/|x|3 à l'infini / This thesis deals with the steady incompressible Navier-Stokes equations, more precisely with the asymptotic behavior of its solutions when |x| → ∞. We consider several types of unbounded domains and we assume that the velocity vanishes at infinity. We first look at the three dimensional case, for which we know that if the forcing term decays fast enough at infinity, the asymptotic behavior of the solutions is given by the Landau solutions that are homogeneous of degree -1. We generalize this result to small forcing terms whose asymptotic behavior at infinity is homogeneous of degree -3. To obtain solutions with an asymptotic behavior at infinity homogeneous of degree -1 we find a necessary and sufficient condition on the forcing : the homogeneous part of the forcing term must have zero mean over the unit sphere. Finally, we generalize this result to the case of an exterior domain. In the case of a half space, we prove that if the forcing term decays sufficiently fast at infinity, then we obtain solutions that decay as 1/|x|2 at infinity and we find an explicit formula for the dominant term in the expansion at infinity of the solution. We can also prove the same type of result as in the full space with forcing terms decaying like 1/|x|3 but the condition of zero mean over the sphere is not required any more. The case of the dimension two is much more difficult. We study first homogeneous solutions and find a family indexed on two real parameters. Imposing the restriction of having zero flux through the unit circle, we get a family of solutions with only one parameter. Finally we deal with non homogeneous solutions, but to do this we need to assume some symmetry conditions on the data. If the forcing term is small and decays sufficiently fast at infinity, we find solutions that decay like 1/|x|3 at infinity and we also obtain an explicit formula for the main term in their asymptotic expansion. We generalize this result to the case of an exterior domain and we also obtain, again under symmetry assumptions, an analogous result to the three dimensional case for forcing terms that decay like 1/|x|3 at infinity

Submitted by Arianne Vellasco Gomes (ariannevellasco@gmail.com) on 2018-06-15T18:52:22Z No. of bitstreams: 1 Arianne_Vellasco_Gomes_TESE_POSMAT_2018.pdf: 4211125 bytes, checksum: 16221f3149817fbc6e4db2f2026f2f14 (MD5) / Approved for entry into archive by Lucilene Cordeiro da Silva Messias null (lubiblio@bauru.unesp.br) on 2018-06-18T17:39:32Z (GMT) No. of bitstreams: 1 gomes_av_dr_bauru.pdf: 3510911 bytes, checksum: 2abe98b4f93107bb6dc267a184ebef70 (MD5) / Made available in DSpace on 2018-06-18T17:39:32Z (GMT). No. of bitstreams: 1 gomes_av_dr_bauru.pdf: 3510911 bytes, checksum: 2abe98b4f93107bb6dc267a184ebef70 (MD5) Previous issue date: 2018-04-17 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Tópicos fundamentais da estrutura eletrônica de materiais cristalinos, são investigados de forma generalizada mediante o Cálculo Fracionário. São calculadas as bandas de energia, as funções de Bloch e as funções de Wannier, para a equação de Schrödinger fracionária com derivada de Riesz. É apresentado um estudo detalhado do caráter não local desse tipo de derivada fracionária. Resolve-se a equação de Schrödinger fracionária para o modelo de Kronig-Penney e estuda-se os efeitos da ordem da derivada e da intensidade do potencial. Verificou-se que, ao passar da derivada de segunda ordem para derivadas fracionárias, o comportamento assintótico das funções de Wannier muda apreciavelmente. Elas perdem o decaimento exponencial, e exibem um decaimento acentuado em forma de potência. Fórmulas simples foram dadas para as caudas das funções de Wannier. A banda de energia mais baixa mostrou-se estar relacionada ao estado ligado de um único poço quântico. Sua função de onda também apresentou decaimento em lei de potência. As bandas de energia superiores mudam de comportamento em função da intensidade do potencial. No caso inteiro, a largura de cada uma dessas bandas diminui. No caso fracionário, diminui inicialmente e depois volta a aumentar, aproximando-se de um valor infinito à medida que a intensidade do potencial tende ao infinito. O grau de localização das funções de Wannier, expresso pelo desvio padrão da posição, mostra um comportamento similar ao da largura das bandas de energia. Além dos cristais perfeitos a Ciência de Materiais estuda cristais com defeito. Os defeitos são responsáveis por muitas propriedades de interesse tecnológico e podem induzir estados localizados. Neste trabalho, calculado o estado localizado de menor energia no modelo de Kronig-Penney fracionário com defeito, mediante método das transformadas de Fourier e das funções de Wannier. Verificou-se que este estado também decai em forma de lei de potência. / Basics topics on the electronic structure of crystalline materials are investigated in a generalized fashion through Fractional Calculus. The energy bands, the Bloch and Wannier functions for the fractional Schr odinger equation with Riesz derivative are calculated. The non-locality of the Riesz fractional derivative is analyzed. The fractional Schr odinger equation is solved for the Kronig-Penney model and the e ects of the derivative order and the potential intensity are studied. It was shown that moving from the integer to the fractional order strongly a ects the asymptotic behavior of the Wannier functions. They lose the exponential decay, gaining a strong power-law decay. Simple formulas have been given for the tails of the Wannier functions. A close relatim between the lowest energy band and the bound state of a single quantum well was found. The wavefunction of the latter decays as a power law. Higher energy bands change their behavior as the periodic potential gets stronger. In the integer case, the width of each one of those bands decreases. In the fractional case, it initially decreases and then increases. The width approaching a nite value as the strength tends to in nity. The degree of localization of the Wannier functions, as expressed by the position standard deviation, behaves similarly to the width of the energy bands. In addition to perfect crystals, Materials Science studies defective crystals. Defects are responsible for many properties of technological interest and can induce localized states. In this work, the localized state of lowest energy in the fractional Kronig-Penney model with defect is calculated through of the Fourier transform method and the Wannier functions. It was shown that is decays as a power law.

Estamos interessados em estudar o comportamento assintótico das soluções de uma classe de Equações Diferenciais Funcionais (EDF) lineares e autônomas do tipo neutro, onde os coeficientes, na parte não neutra, são funções periódicas de período comum w! e os retardamentos são múltiplos de w. Para isto, utilizamo-nos da teoria espectral de operadores aplicada ao chamado operador monodrômico \'PI\' : C \'SETA\' C, cuja ação é evoluir um dado estado um passo de tamanho w. Calculamos o resolvente deste operador, donde inferimos todas as propriedades espectrais que nos permitem determinar o comportamento assintótico das soluções. Mostramos a importância de se determinar autovalores dominantes para a obtenção das estimativas, e mostramos resultados neste sentido. Estudamos em detalhe três exemplos que ilustram a teoria e demonstram sua aplicabilidade / We are interested in the study of the asymptotic behavior of the solutions of a class of linear autonomous Functional Differential Equations (FDE) of neutral type, where the coeficients of the non neutral part are periodic functions with common period w and the time delays are multiples of w. We employ the spectral theory for linear operators applied to the so called monodromic operator \'PI\' : C \'ARROW\'! C, whose action is to evolve a given state one step of size w. We compute the resolvent of this operator, from where we infer the spectral properties that allows us to determine the asymptotic behavior of the solutions. We show the importance to determine whether an eigenvalue is dominant, in order to obtain the estimates for the correspondet solution, and we show results in this direction. Finally we study in detail three examples that illustrate the theory and demonstrate its applicability

Neste trabalho estudamos a equação de ondas do tipo p-Laplaciano \'u IND. tt\' - \'DELTA\' IND.p u + \'(- \'DELTA\' POT. alpha\' u IND. t\' = \' [u] POT.q - 2 u, definida num domínio limitado limitado do \'R POT. n\', com 2 \' > ou = \' p < q e 0 < \' alpha\' < 1. Utilizando o método de Faedo-Galerkin provamos a existência de soluções fracas globais para dados iniciais pequenos. Para essas soluções estudamos também o decaimento polinomial da energia associada. A questão da não existência de soluções globais é considerada para o caso em que a energia inicial do sistema é negativa / In this work we study the p-Laplacian wave equation \'u IND. tt\' - \' DELTA\' IND p u + \'(- \'DELTA\' POT. \'alpha\' \' u IND. t\' = \'[u] POT. q - 2 u, defined in a bounded domain of \'R POT n\', with 2 \'> or =\' p < q and 0 < \' alpha\' < 1. By using the Faedo-Galerkin method we prove the existence of weak global solutions for small initial data. We also study the polynomial decay of the associate energy. The blow-up of solutions in finite time is considered for negative initial energy