Made available in DSpace on 2014-06-11T19:30:22Z (GMT). No. of bitstreams: 0 Previous issue date: 2013-01-28Bitstream added on 2014-06-13T19:25:46Z : No. of bitstreams: 1 marchiori_rm_me_rcla.pdf: 441197 bytes, checksum: dfc9ce6e00b97ad657ecd6859c6787a4 (MD5) / Tudo é número, diria o famoso matemático grego Pitágoras. Os números estão a nossa volta, como o oxigênio que respiramos. Primeiro vieram os naturais, depois os inteiros, os racionais e os incríveis irracionais, que deixaram os pitagóricos tão perplexos a ponto de escondê-los. Números primos, perfeitos e outros vieram. E quando tudo parecia ser real apareceram os imaginários. Que imaginação tem esses matemáticos! Vamos nos aprofundar em um grupo intrigante de números chamados transcendentes e aos números estudados por um matemático francês chamado Liouville / All is number, say the famous Greek mathematician Pythagoras. The numbers are all around us, like the oxygen we breathe. First came the natural, then the integers, the rational and the irrational incredible that left perplexed the Pythagoreans so as to hide them. Prime numbers, perfect and others came. And when everything seemed to be real the imaginary appeared. What have these mathematical imagination! Let's delve in a group of intriguing numbers called transcendental numbers and studied by a French mathematician named Liouville

Neste trabalho buscamos caracterizar o espectro de uma classe de operadores bloco--Jacobi limitados definidos em $l^2(\\Lambda,\\mathbb{C}^L)$ ($\\Lambda: \\mathbb{Z}_+\\times\\{0,1,\\ldots,L-1\\}$ representa uma faixa de largura $L\\ge 2$ no semi--plano $\\mathbb{Z}_+^2$) e sujeitos a perturbações esparsas (no sentido que as distâncias entre as ``barreiras\'\' crescem geometricamente à medida que estas se afastam da origem) distribuídas aleatoriamente. Tais operadores são construídos a partir da soma de Kronecker de matrizes de Jacobi $J$, cada qual atuando em uma direção do espaço. Demonstramos, por meio da bloco--diagonalização do operador, que %o estudo de suas principais propriedades espectrais dependem da %se limita à caracterização da ``medida de mistura\'\' $\\frac{1}{L}\\sum_{j=0}^{L-1}\\mu_j$, $\\mu_j$ a medida espectral associada à matriz de Jacobi $J^j=J+2\\cos(2\\pi j/L)I $. Para tanto, buscamos primeiramente caracterizar cada uma das medidas $\\mu_j$, explorando e aperfeiçoando algumas técnicas bastante conhecidas no estudo de operadores esparsos unidimensionais. Demonstramos, por exemplo, que a seqüência de ângulos de Prüfer (variáveis que, juntamente com os raios de Prüfer, parametrizam as soluções da equação de autovalores) é uniformemente distribuída no intervalo $[0,\\pi)$, o %que %resultado que nos permite determinar o comportamento assintótico médio das soluções da equação de autovalores. Tal resultado, aliado às técnicas desenvolvidas por Marchetti \\textit{et. al.} em \\cite{MarWre} e a uma adaptação dos critérios de Last e Simon \\cite{LS} para operadores esparsos, nos permitem demonstrar a existência de uma transição aguda (pontual) entre os espectros singular--contínuo e puramente pontual. Empregamos em seguida os resultados de Jitomirskaya e Last presentes em \\cite{JitLast} e obtemos a dimensão Hausdorff exata associada à medida $\\mu_j$, dada por $\\alpha_j=1+\\frac{4(1-p^2)^2}{p^2(4- (\\lambda-2\\cos(2\\pi j/L))^2)}$ ($\\lambda\\in[-2,2]$), recuperando um resultado análogo obtido por Zlato\\v s em \\cite{Zla}. Por fim, adaptamos tais resultados à situação da medida de mistura associada à matriz bloco--Jacobi, obtendo $\\alpha=\\min_{j\\in\\mathcal{I}(\\lambda)}\\alpha_j$, $\\mathcal{I}(\\lambda):\\{m \\in\\{0,1,\\ldots,L-1\\}:\\lambda\\in[-2+2\\cos(2\\pi j/L),2+2\\cos(2\\pi j/L)]\\}$, como sua dimensão Hausdorff exata. Estudamos modelos idênticos com esparsidades sub e super-geométricas, obtendo na primeira situação um espectro puramente pontual (de dimensão Hausdorff nula) e na segunda um espectro puramente singular--contínuo (de dimensão Hausdorff 1). Finalmente, verificamos a existência de transição entre os espectros puramente pontual e singular--contínuo em um modelo com esparsidade super-geométrica cuja dimensão Hausdorff associada à medida espectral é nula. / In this work we attempt to caracterize the spectrum of a class of limited block--Jacobi operators defined in $l^2(\\Lambda,\\mathbb{C}^L)$ ($\\Lambda: \\mathbb{Z}_+\\times\\{0,1,\\ldots,L-1\\}$ represents a strip of width $L\\ge 2$ on the semi--plane $\\mathbb{Z}_+^2$) subject to a sparse perturbation (which means that the distance between the ``barries\'\' grow geometrically with their distance to the origin) randomly distributed. Such operators are defined as Kronecker sums of unidimensional Jacobi matrices $J$, each one acting in different directions of the space. We prove, by means of a block--diagonalization of the operator, that %the study of its most relevant spectral properties depend on %is related to the caracterization of the ``mixture measure\'\' $\\frac{1}{L}\\sum_{j=0}^{L-1}\\mu_j$, $\\mu_j$ the spectral measure of the Jacobi matrix $J^j=J+2\\cos(2\\pi j/L)I$. For this, we must characterize at first each one of the measures $\\mu_j$, exploiting and improving some well known techniques developed in the study of unidimensional sparse operators. We prove, for instance, that the sequence of Prüfer angles (variables which parametrize the solutions of the eigenvalue equation) are uniform distributed on the interval $[0,\\pi)$, a result which gives us condition to determine the average asymptotic behavior of the solutions of the eigenvalue equation. Such result, in association with the techniques developed by Marchetti \\textit{et. al.} in \\cite{MarWre} and with an adaptation of Last--Simon \\cite{LS} criteria for sparse operator, permit us to prove the existence of a sharp transition between singular continuous and pure point spectra. Following on, we use the results from Jitomirskaya--Last of \\cite{JitLast} and obtain the exact Hausdorff dimension of the measure $\\mu_j$, given by $\\alpha_j=1+\\frac{4(1-p^2)^2}{p^2(4-(\\lambda-2\\cos(2\\pi j/L))^2)}$ ($\\lambda\\in[- 2,2]$), recovering an analogous result due to Zlato\\v s in \\cite{Zla}. At last, we adapt these results to the mixture measure of the block--Jacobi matrix, obtaining $\\alpha=\\min_{j\\in\\mathcal{I}(\\lambda)}\\alpha_j$, $\\mathcal{I}(\\lambda):\\{m \\in\\{0,1,\\ldots,L-1\\}:\\lambda\\in[-2+2\\cos(2\\pi j/L),2+2\\cos(2\\pi j/L)]\\}$, as its exact Hausdorff dimension. We study as well identical models with sub and super geometric sparsities conditions, obtaining a pure point spectrum (with null Hausdorff dimension) in the first case, and a purely singular continuous spectrum (such that its Hausdorff dimension is 1) in the second. Finally, we prove the existence of a transition between pure point and singular continuous spectra in a model with sub--geometric sparsity whose Hausdorff dimension related to the spectral measure is null.

Établir une théorie de gravité quantique qui décrit de manière cohérente les propriétés quantiques de la matière et de l'espace-temps est l'un des défis majeurs de la physique théorique. Malgré plusieurs décennies de recherches, de nombreux problèmes conceptuels et techniques doivent encore être résolus. L'étude de modèles simplifiés donne des idées de résolution. La première partie de la thèse traite de la gravité quantique bidimensionnelle. À deux dimensions, la gravité quantique est beaucoup mieux comprise et de nombreux calculs peuvent être faits exactement. Si la gravité quantique bidimensionnelle a été largement étudiée quand elle est couplée à de la matière conforme, le cas de la matière non-conforme était très peu connu jusque récemment. Nous calculons d'abord l'action gravitationnelle pour un champ scalaire massif sur une surface de Riemann avec bords puis pour un fermion de Majorana massif sur une variété compacte. Ce dernier cas correspond à une CFT perturbée par une perturbation conforme et est d'ordinaire étudié grâce à l'ansatz de DDK, mais les résultats sont différents. Finalement, on calcule le spectre de l'action de Mabuchi dans l'approximation du minisuperespace. La seconde partie étudie les propriétés thermales des trous noirs dans le contexte de la correspondance AdS/CFT. On construit un modèle de mécanique quantique fondé sur les principes holographiques pour simuler la dynamique des trous noirs quantiques. Ce modèle permet d'obtenir des résultats numériques exacts. / Finding a theory of quantum gravity describing in a consistent way the quantum properties of matter and spacetime geometry is one of the greatest challenges of modern theoretical physics. However after several decades of research, many conceptual and technical issues are still to be resolved. Insights on these questions can be given by simplified toy models that allow for exact computations. The first part of the thesis deals with two-dimensional quantum gravity. In two dimensions quantum gravity is much better understood and many computations can be carried out exactly. Whereas two-dimensional quantum gravity coupled to conformal matter has been widely studied and is now well understood, much less was known until recently when matter is non-conformal. First we compute the gravitational action for a massive scalar field on a Riemann surface with boundaries and then for a massive Majorana fermion on a manifold without boundary. The latter case corresponds to a CFT perturbed by a conformal perturbation and is usually tackled through the DDK ansatz, but the results do not seem to match. Finally we give a minisuperspace computation of the spectrum of the Mabuchi action, a functional that appears in the gravitational action for a massive scalar field. In the second part we focus on black hole thermal behaviour which provides a lot of insight of how a theory of quantum gravity should look like. In the context of string theory the AdS/CFT correspondence provides powerful tools for understanding the microscopic origin of black holes thermodynamics. We construct a quantum mechanical toy model based on holographic principles to study the dynamics of quantum black holes.

Les collisions entre atomes ultrafroids et surfaces matérielles sont caractérisées par la réflexion de l'onde de matière atomique sur le potentiel attractif de Casimir-Polder. Cette réflexion quantique est déterminante pour des expériences telles que GBAR, qui mesurera l'accélération d'un atome d'antihydrogène froid chutant vers une plaque de détection. Dans cette thèse, le potentiel de Casimir-Polder est calculé à partir des propriétés de diffusion électromagnétique de l'atome et de la surface. Il s'avère dépendre de la réponse diélectrique, de l'épaisseur et de la densité du milieu. Nous montrons que la réflexion sur ce potentiel est associée à une rupture de l'approximation semiclassique et qu'elle augmente pour des atomes lents et des potentiels faibles. Les transformations de Liouville relient des équations de Schrödinger avec des potentiels différents mais les mêmes amplitudes de diffusion. L'équivalence entre la réflexion quantique sur un puits de potentiel et l'effet tunnel à travers une barrière offre de nouvelles perspectives sur le problème. Nous discutons aussi des effets de la gravité sur le paquet d'onde atomique et de ses conséquences pour les expériences avec des atomes en chute libre. Associée à la réflexion quantique sur un miroir horizontal, la gravité permet de maintenir des particules dans des états à longue durée de vie aux applications prometteuses pour la métrologie. En particulier, nous proposons un système pour améliorer la précision de GBAR en réduisant la dispersion en vitesse des atomes d'antihydrogène. / Collisions between ultracold atoms and material surfaces are characterized by the reflection of the atomic matter wave from the attractive Casimir-Polder potential. This quantum reflection is particularly relevant to experiments such as GBAR, which will determine the gravitational acceleration of a cold antihydrogen atom by timing its fall onto a detection plate. In this thesis, the Casimir-Polder potential is computed from the electromagnetic scattering properties of the atom and surface and it is found to depend notably on the dielectric response, thickness and density of the medium. We show that reflection on this potential is associated with a breakdown of the semiclassical approximation and that it is enhanced for slow atoms and weak potentials. Liouville transformations relate Schrödinger equations with different potential landscapes but identical scattering properties. We gain new insights on the problem of quantum reflection on a potential well by mapping it onto an equivalent problem of tunneling through a wall. We also discuss the effect of gravity on the atomic wavepacket and its implications for free fall experiments with atoms. When combined with quantum reflection from a horizontal mirror, gravity can be used to trap particles in long lived states with promising applications for metrology. In particular, we suggest a scheme to improve the precision of the GBAR experiment by reducing the velocity dispersion of the falling atoms.

AN ABSTRACT OF THE DISSERTATION OFSaleh Aljurbua, for the Doctor of Philosophy degree in APPLIED MATHEMATICS, presented on January 27th, 2021, at Southern Illinois University Carbondale. TITLE: THE EXISTENCE OF SOLUTIONS FOR FRACTIONAL DIFFERENTIAL EQUATIONS FOR ORDER q ∈ (n − 1, n], n ∈ N, WITH ANTIPERIODIC BOUNDARY CONDITIONS MAJOR PROFESSOR: Dr. Mingqing Xiao Differential equations play a major role in natural science, physics and technology. Fractional differential equations (FDE) gained a lot of popularity in the past three decades and they became very important in economics, physics and chemistry. In fact, fractional integrals and derivatives became essential and made a significant contribution in dynamical systems which simulate it. They fill the gaps between the integer-types of integrations and derivatives in the classical settings. This work consists of four Chapters. The first Chapter will be covering background, preliminary and fundamental tools used in our dissertation topic. The second Chapter consists of the existence of solutions for nonlinear fractional differential equations of some specific orders with antiperiodic boundary conditions followed by the main topic which is the existence of solutions for nonlinear fractional differential equations of order q ∈ (n−1, n], n ∈ N with antiperiodic boundary conditions of a continuous function f(t, x(t)). Moreover, definitions, theorems and some lemmas will be provided. v In the third Chapter, we offer some examples to illustrate our approach in the main topic. Finally, the fourth Chapter includes the summary and perspective researches.

In this PhD thesis we study some weighted regular Sturm-Liouville problems in which the weight function takes on both positive and negative signs in an appropriate interval [a,b]. With such  problems there is the possible existence of non-real eigenvalues, unlike in the definite case (i.e. left or right definite) in which only real eigenvalues exist. This PhD thesis consists of five papers (papers A-E) and an introduction to this area, which puts these papers into a more general frame. In paper A we give some precise estimates on the Richardson number for the two turning point case, thereby complementing the work of Jabon and Atkinson from 1984 in an essential way. We also give a corrected version of their result since there seems to be a typographical error in their paper. In paper B we show that the interlacing property, which holds in the one turning point case, does not hold in the two turning point case. The paper consists of a detailed presentation of numerical results of the case in which the weight function is allowed to change its sign twice in the interval (-1, 2). We also present some theoretical results which support the numerical results. Moreover, a number of new open questions are raised. We also observe that the real and imaginary parts of a non-real eigenfunction either have the same number of zeros in the interval (-1,2) or the numbers of zeros differ by two. In paper C, we obtain bounds on real and imaginary parts of non-real eigenvalues of a non-definite Sturm-Liouville problem, with Dirichlet boundary conditions, thus complementing the results obtained in a paper byBehrndt et.al. from 2013 in an essential way. In paper D we obtain a lower bound on the eigenvalue of the smallest modulus associated with a Dirichlet problem in the general case of a regular Sturm-Liouville problem. In paper E we expand upon the basic oscillation theory for general boundary problems of the form   -y''+q(x)y=λw(x)y,  on  I = [a,b], where  q(x) and w(x) are real-valued continuous functions on [a,b] and y is required to satisfy a pair of homogeneous separated boundary conditions at the end-points. Already in 1918 Richardson proved that, in the case of the Dirichlet problem,  if w(x) changes its sign exactly once and the boundary problem is  non-definite, then the zeros of the real and imaginary parts of any non-real eigenfunction interlace. We show that, unfortunately, this result is false in the case of two turning points, thus removing any hope for a general separation theorem for the zeros of the non-real eigenfunctions. Furthermore, we show that when a non-real eigenfunction vanishes inside I, then the absolute value of the difference between the total number of zeros of its real and imaginary parts is exactly 2.

Cette thèse comporte deux chapitres principaux. Deux problèmes indépendants de Modélisation Mathématique y sont étudiés. Au chapitre 1, on étudiera le problème de l’existence et de l’unicité des distributions quasi-stationnaires (DQS) pour un mouvement Brownien avec dérive, tué en zéro dans le cas où la frontière d’entrée est l’infini et la frontière de sortie est zéro selon la classification de Feller.Ce travail est lié à l’article pionnier dans ce sujet  par Cattiaux, Collet, Lambert, Martínez, Méléard, San Martín; où certaines conditions suffisantes ont été établies pour prouver l’existence et l’unicité de DQS dans le contexte d’une famille de Modèles de Dynamique des Populations.Dans ce chapitre, nous généralisons les théorèmes les plus importants de ce travail pionnier, la partie technique est basée dans la théorie de Sturm-Liouville sur la demi-droite positive. Au chapitre 2, on étudiera le problème d’obtenir des bornes inférieures optimales sur l’Hamiltonien du Modèle d’Ising avec interactions à longue portée, l’interaction entre deux spins situés à distance d décroissant comme d^(2-a), où a ϵ[0,1).Ce travail est lié à l’article publié en 2005 par Cassandro, Ferrari, Merola, Presutti où les bornes inférieures optimales sont obtenues dans le cas où a est dans [0,(log3/log2)-1) en termes de structures hiérarchiques appelées triangles et contours.Les principaux théorèmes obtenus dans cette thèse peuvent être résumés de la façon suivante:1. Il n’existe pas de borne inférieure optimale pour l’Hamiltonien en termes de triangles pour a dans ϵ[log2/log3,1). 2. Il existe une borne optimale pour l’Hamiltonien en termes de contours pour a dans a ϵ [0,1). / This thesis contains two main Chapters, where we study two independent problems of Mathematical Modelling : In Chapter 1, we study the existence and uniqueness of Quasi Stationary Distributions (QSD) for a drifted Browian Motion killed at zero, when $+infty$ is an entrance Boundary and zero is an exit Boundary according to Feller's classification. The work is related to the previous paper published in 2009 by { Cattiaux, P., Collet, P., Lambert, A., Martínez, S., Méléard, S., San Martín, where some sufficient conditions were provided to prove the existence and uniqueness of QSD in the context of a family of Population Dynamic Models. This work generalizes the most important theorems of this work, since no extra conditions are imposed to get the existence, uniqueness of QSD and the existence of a Yaglom limit. The technical part is based on the Sturm Liouville theory on the half line. In Chapter 2, we study the problem of getting quasi additive bounds on the Hamiltonian for the Long Range Ising Model when the interaction term decays according to d^{2-a}, a ϵ[0,1). This work is based on the previous paper written by Cassandro, Ferrari, Merola, Presutti, where quasi-additive bounds for the Hamiltonian were obtained for a in [0,(log3/log2)-1) in terms of hierarchical structures called triangles and Contours. The main theorems of this work can be summarized as follows: 1 There does not exist a quasi additive bound for the Hamiltonian in terms of triangles when a ϵ [0,(log3/log2)-1), 2. There exists a quasi additive bound for the Hamiltonian in terms of Contours for a in [0,1).

Ce mémoire présente des résultats de régularité pour des problèmes d'équations aux dérivées partielles paraboliques. Dans la première partie nous nous intéressons à des problèmes à frontière libre issus du problème de<br />l'obstacle parabolique à coefficients variables. Nous montrons des résultats de régularité de la solution et de la frontière libre. Cette étude utilise des méthodes d'explosion et des formules de monotonie. La seconde partie est consacrée à l'étude d'un problème issu de la modélisation de l'agrégation en biologie : le système de<br />Keller-Segel. En utilisant une énergie libre, nous montrons l'existence d'une masse critique en deçà de laquelle les solutions existent et au delà de laquelle elles explosent en temps fini. Nous précisons leur comportement asymptotique, dans le cas où les solutions existent en temps long.

Cette thèse est divisée en deux parties. Dans la première partie, on considère le système de réaction-diffusion-advection (Pε), qui est un modèle d'haptotaxie, mécanisme lié à la dissémination de tumeurs cancéreuses. Le résultat principal concerne la convergence de la solution du systeme (Pε) vers la solution d'un problème à frontière libre (P0) qui est bien défini. Dans la seconde partie, on considère une classe générale d'équations elliptiques du type Hénon:−∆u = |x|^{α} f(u) dans Ω ⊂ R^N avec α > -2. On examine deux cas classiques : f(u) = e^u, |u|^{p−1} u et deux autres cas : f(u) = u^{p}_{+} puis f(u) nonlinéarité générale. En étudiant les solutions stables en dehors d'un ensemble compact (en particulier, solutions stables et solutions avec indice de Morse fini) avec différentes méthodes, on obtient des résultats de classification.

In this paper we investigate (d + 1)-point D-optimal designs for d-th degree polynomial regression with weight function w(x) > 0 on the interval [a, b]. Suppose that w'(x)/w(x) is a rational function and the information of whether the optimal support contains the boundary points a and b is available. Then the problem of constructing (d + 1)-point D-optimal designs can be transformed into a differential equation problem leading us to a certain matrix with k auxiliary unknown constants. We characterize the weight functions corresponding to the cases when k= 0 and k= 1. Then, we can solve (d + 1)-point D-optimal designs directly from differential equation (k = 0) or via eigenvalue problems (k = 1). The numerical results show us an interesting relationship between optimal designs and ordered eigenvalues.