Das „Strongly Constrained and Appropriately Normed“ (SCAN) Austausch-Korrelations-Funktional gehört zur Familie der meta-GGA (generalized gradient approximation) Funktionale. Es gibt aber auch Nachteile Zum einen leiden SCAN Rechnungen oft unter numerischen Instabilitäten, wodurch sehr viele Iteration zum Erreichen von Selbst-Konsistenz benötigt werden. Zum anderen leidet SCAN unter dem von GGA Methoden bekannten Selbstwechselwirkung-Fehler. Im ersten Teil der Arbeit habe ich die numerischen Stabilitätsprobleme in SCAN Rechnungen im Rahmen der numerischen Realraum-Integrationsroutinen im Code FHI-aims untersucht. Diese Analyse zeigt, dass die genannte Probleme durch Anwendung von standardisierten Dichte-Mischalgorithmen für die kinetische Energiedichte abgemildert werden können. Dadurch wird auch in SCAN-Rechnungen eine schnelle und stabile Konvergenz zur selbstkonsistenten Lösung ermöglicht. Im zweiten Teil der Arbeit habe ich untersucht, in welchem Rahmen sich der Selbstwechselwirkung-Fehler in SCAN mittels des von Perdew und Zunger vorgeschlagenen Selbstinteraktionskorrekturalgorithmus (PZ-SIC) verringern lässt. Es wurden aber auch Optimierungen für die PZ-SIC Methode entwickelt. Inspiriert von den ursprünglichen Argumenten in der PZ-SIC-Methode und anderen lokalisierten Methoden, wird in dieser Arbeit eine neuartige Randbedingung (orbital density constraint) vorgeschlagen, die sicherstellt, dass die PZ-SIC Orbitale während des Selbstkonsistenzzyklus lokalisiert bleiben. Dies mildert die Anfangswertabhängigkeit deutlich ab und hilft dabei, in die korrekte selbst-konsistente Lösung mit minimaler Energie zu konvergieren, unabhängig davon ob reelle oder komplexe SIC Orbitale verwendet werden. Die in dieser Arbeit getägtigen Entwicklungen und Untersuchungen sind Wegbereiter dafür, in Zukunft mit SIC-SCAN Rechnungen deutlich genauere ab initio Rechnungen mit nur gering höherem Rechenaufwand durchführen zu können. / The state-of-the-art “Strongly Constrained and Appropriately Normed” (SCAN) functional pertains to the family of meta-generalized-gradient approximation (meta-GGA) exchange-correlation functionals. Nonetheless, SCAN suffers from some well-documented deficiencies. In the first part of this thesis, I revisited the known numerical instability problems of the SCAN functional in the context of the numerical, real-space integration framework used in the FHI-aims code. This analysis revealed that applying standard density-mixing algorithms to the kinetic energy density attenuates and largely cures these numerical issues. By this means, SCAN calculations converge towards the self-consistent solution as fast and as efficiently as lower-order GGA calculations. In the second part of the thesis, I investigated strategies to alleviate the self-interaction error in SCAN calculations by using the self-interaction correction algorithm proposed by Perdew and Zunger (PZ-SIC). Inspired by the original arguments in PZ-SIC and other localized methods, I introduced a mathematical constraint, i.e., the orbital density constraint, that forces the orbitals to retain their localization throughout the self-consistency cycle. In turn, this alleviates the multiple-solutions problem and facilitates the convergence towards the correct, lowest-energy solution both for complex and real SIC orbitals. The developments and investigations performed in this thesis pave the road towards a more wide-spread use of SIC-SCAN calculations in the future, allowing more accurate predictions within only moderate increases of computational cost.

Ein wichtiges Gebiet der Finanzmathematik ist die Bestimmung von Auszahlungsprofilen, die den erwarteten Nutzen eines Agenten unter einer Budgetrestriktion maximieren. Wir charakterisieren optimale Auszahlungsprofile für einen Agenten, der unsicher ist in Bezug auf das genaue Marktmodell. Der hier benutzte Dualitätsansatz führt zu einem Minimierungsproblem für bestimmte konvexe Funktionale über zwei Mengen von Wahrscheinlichkeitsmaßen, das wir zunächst lösen müssen. Schließlich führen wir noch eine zweite Restriktion ein, die das Risiko beschränkt, das der Agent eingehen darf. Wir gehen dabei wie folgt vor: Kapitel 1. Wir betrachten das Problem, die f-Divergenz f(P|Q) über zwei Mengen von Wahrscheinlichkeitsmaßen zu minimieren, wobei f eine konvexe Funktion ist. Wir zeigen, dass unter der Bedingung "f( undendlich ) / undendlich = undendlich" Minimierer existieren, falls die erste Menge abgeschlossen und die zweite schwach kompakt ist. Außerdem zeigen wir, dass unter der Bedingung "f( undendlich ) / undendlich = 0" ein Minimierer in einer erweiterten Klasse von Martingalmaßen existiert, falls die zweite Menge schwach kompakt ist. Kapitel 2. Die Existenzresultate aus dem ersten Kapitel implizieren die Existenz eines Auszahlungsprofils, das das robuste Nutzenfunktional inf E_Q[u(X)] über eine Menge von finanzierbaren Auszahlungen maximiert, wobei das Infimum über eine Menge von Modellmaßen betrachtet wird. Die entscheidende Idee besteht darin, die minimierenden Maße aus dem ersten Kapitel als gewisse "worst-case"-Maße zu identifizieren. Kapitel 3. Schließlich fordern wir, dass das Risiko der Auszahlungsprofile beschränkt ist. Wir lösen das robuste Problem in einem unvollständigen Marktmodell für Nutzenfunktionen, die nur auf der positiven Halbachse definiert sind. In einem Beispiel vergleichen wir das optimale Auszahlungsprofil unter der Risikorestriktion mit den optimalen Auszahlungen ohne eine solche Restriktion und unter einer Value-at-Risk-Nebenbedingung. / Finding payoff profiles that maximize the expected utility of an agent under some budget constraint is a key issue in financial mathematics. We characterize optimal contingent claims for an agent who is uncertain about the market model. The dual approach that we use leads to a minimization problem for a certain convex functional over two sets of measures, which we first have to solve. Finally, we incorporate a second constraint that limits the risk that the agent is allowed to take. We proceed as follows: Chapter 1. Given a convex function f, we consider the problem of minimizing the f-divergence f(P|Q) over these two sets of measures. We show that, if the first set is closed and the second set is weakly compact, a minimizer exists if f( infinity ) / infinity = infinity. Furthermore, we show that if the second set of measures is weakly compact and f( infinifty ) / infinity = 0, then there is a minimizer in a class of extended martingale measures. Chapter 2. The existence results in Chapter 1 lead to the existence of a contingent claim which maximizes the robust utility functional inf E_Q[u(X)] over some set of affordable contingent claims, where the infimum is taken over a set of subjective or modell measures. The key idea is to identify the minimizing measures from the first chapter as certain worst case measures. Chapter 3. Finally, we require the risk of the contingent claims to be bounded. We solve the robust problem in an incomplete market for a utility function that is only defined on the positive halfline. In an example we compare the optimal claim under this risk constraint with the optimal claims without a risk constraint and under a value-at-risk constraint.

In dieser Arbeit untersuchen wir das Problem der Maximierung bestimmter konkaver Funktionale auf dem Raum der optionalen, zufälligen Maße. Deartige Funktionale treten in der mikroökonomischen Literatur auf, wo ihre Maximierung auf die Bestimmung des optimalen Konsumplans eines ökomischen Agenten hinausläuft. Als Alternative zu den wohlbekannten Methoden der dynamischen Programmierung wird ein neuer Zugang vorgestellt, der es erlaubt, die Struktur der maximierenden Maße in einem über den üblicherweise angenommenen Markovschen Rahmen hinausgehenden, allgemeinen Semimartingalrahmen zu klären. Unser Zugang basiert auf einer unendlichdimensionalen Version des Kuhn-Tucker-Theorems. Die implizierten Bedingungen erster Ordnung erlauben es uns, das Maximierungsproblem auf ein neuartiges Darstellungsproblem für optionale Prozesse zu reduzieren, das damit als ein nicht-Markovsches Substitut für die Hamilton-Jacobi-Bellman Gleichung der dynamischen Programmierung dient. Um dieses Darstellungsproblem im deterministischen Fall zu lösen, führen wir eine zeitinhomogene Verallgemeinerung des Konvexitätsbegriffs ein. Die Lösung im allgemeinen stochastischen Fall ergibt sich über eine enge Beziehung zur Theorie des Gittins-Index der optimalen dynamischen Planung. Unter geeigneten Annahmen gelingt ihre Darstellung in geschlossener Form. Es zeigt sich dabei, daß die maximierenden Maße absolutstetig, diskret und auch singulär sein können, je nach Struktur der dem Problem zugrundeliegenden Stochastik. Im mikroökonomischen Kontext ist es natürlich, daß Problem in einen Gleichgewichtsrahmen einzubetten. Der letzte Teil der Arbeit liefert hierzu ein allgemeines Existenzresultat für ein solches Gleichgewicht. / In this thesis, we study the problem of maximizing certain concave functionals on the space of optional random measures. Such functionals arise in microeconomic theory where their maximization corresponds to finding the optimal consumption plan of some economic agent. As an alternative to the well-known methods of Dynamic Programming, we develop a new approach which allows us to clarify the structure of maximizing measures in a general stochastic setting extending beyond the usually required Markovian framework. Our approach is based on an infinite-dimensional version of the Kuhn-Tucker Theorem. The implied first-order conditions allow us to reduce the maximization problem to a new type of representation problem for optional processes which serves as a non-Markovian substitute for the Hamilton-Jacobi-Bellman equation of Dynamic Programming. In order to solve this representation problem in the deterministic case, we introduce a time-inhomogeneous generalization of convexity. The stochastic case is solved by using an intimate relation to the theory of Gittins-indices in optimal dynamic scheduling. Closed-form solutions are derived under appropriate conditions. Depending on the underlying stochastics, maximizing random measures can be absolutely continuous, discrete, and also singular. In the microeconomic context, it is natural to embed the above maximization problem in an equilibrium framework. In the last part of this thesis, we give a general existence result for such an equilibrium.

In this work, our goal is to prove results on prolongation of solutions, uniform boundedness of solutions, uniform stability as well uniform asymptotic stability (in the classical sense of Lyapunov) for measure differential equations and for dynamic equations on time scales. In order to get our results, we employ the theory of generalized ODEs, since these equations encompass measure differential equations and dynamic equations on time scales. Therefore, to get our results, we start by proving the expected result for abstract generalized ODEs. Then, using the correspondence between the solutions of these equations and the solutions of measure differential equations (see [38]), we extend all the results to these the latter. After that, using the correspondence between the solutions of measure differential equations and the solutions of dynamic equations on time scales (see [21]), we extend all the results to these last equations. Finally, we investigate autonomous generalized ODEs and show that these equations do not enlarge the class of classical autonomous ODEs, even when we consider a more general class of functions as right-hand sides. All the new results presented in this work are contained in papers [16, 17, 18, 19]. / Neste trabalho, nosso objetivo e provar resultados sobre prolongamento de soluções, limitação uniforme de soluções, estabilidade uniforme e estabilidade uniforme assintótica (no sentido clássico de Lyapunov) para equações diferenciais em medida e para equações dinâmicas em escalas temporais. A fim de obter os nossos resultados, empregamos a teoria de EDOs generalizadas, uma vez que estas equações abrangem equações diferenciais em medida e equações dinâmicas em escalas temporais. Portanto, para obter nossos resultados, vamos começar por provar, os resultados que queremos para EDOs generalizadas abstratas. Em seguida, usando a correspondência entre as soluções de EDOs generalizadas e soluções de equações diferenciais em medida (ver [38]), estenderemos os resultados para estas ultimas equações. Depois disso, usando a correspondência entre as soluções de equações diferenciais em medida e as soluções de equações dinâmicas em escalas temporais (ver [21]), estenderemos todos os resultados para estas ultimas equações. Finalmente, investigamos EDOs generalizadas autônomas e mostramos que estas equações não aumentam a classe de EDOs autônomas clássicas, mesmo quando consideramos uma classe mais geral de funções nos lados direitos das equações. Os novos resultados encontrados estão contidos em [16, 17, 18, 19].

Submitted by Angela Maria de Oliveira (amolivei@uepg.br) on 2019-03-14T19:25:14Z No. of bitstreams: 2 license_rdf: 811 bytes, checksum: e39d27027a6cc9cb039ad269a5db8e34 (MD5) Renan Augusto Pontes Ribeiro.pdf: 9570923 bytes, checksum: a291ba63c045a11cb0a642a480367e27 (MD5) / Made available in DSpace on 2019-03-14T19:25:14Z (GMT). No. of bitstreams: 2 license_rdf: 811 bytes, checksum: e39d27027a6cc9cb039ad269a5db8e34 (MD5) Renan Augusto Pontes Ribeiro.pdf: 9570923 bytes, checksum: a291ba63c045a11cb0a642a480367e27 (MD5) Previous issue date: 2019-02-26 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / O desenvolvimento da spintrônica tem motivado a busca por novos materiais multiferróicos devido à multifuncionalidade desses compostos associada ao acoplamento entre diferentes ordens ferróicas em uma estrutura cristalina. No presente estudo, propomos a investigação teórica, baseada na Teoria do Funcional de Densidade, dos materiais ATiO3 (A = Mn, Fe, Ni) na estrutura R3c com objetivo de esclarecer o efeito da substituição do cátion A sobre as propriedades estruturais, magnéticas e eletrônicas, bem como descrever diferentes mecanismos de controle das propriedades multiferróicas baseados em arquiteturas de filmes-finos, morfologia e controle de defeitos intrínsecos. Para uma maior compreensão dos efeitos envolvidos nos materiais ATiO3, diferentes funcionais de troca e correlação foram investigados e o funcional PBE0 apresentou os menores desvios, consequentemente, a melhor representação comparado aos resultados experimentais. Com objetivo de investigar as propriedades conectadas a filmes-finos dos materiais ATiO3, propomos uma metodologia inovadora que permite descrever as deformações uni- e biaxial que se originam na região de interface entre o filme e o substrato. Nesse caso, os resultados obtidos indicam que as distorções estruturais induzem uma transição magnética para o NiTiO3, originando ordenamento ferromagnético a partir de um critério magneto-estrutural associado a deformação dos clusters [MO6] que reproduz satisfatoriamente os resultados experimentais reportados na literatura. De modo análogo, para elucidar a relação entre o magnetismo e a morfologia dos materiais ATiO3, combinamos cálculos de Energia de Superfície, Construção de Wulff e um formalismo avançado para descrever o magnetismo superficial considerando a existência de spins não compensados ao longo dos planos polares (100), (001), (101), (012), (111) e apolares (110). Os resultados indicam que a redução do número de coordenação dos metais A e Ti para os planos (001) e (111) resulta na transferência de carga entre os cátions A2+ e Ti4+, originando espécies Ti3+ magnéticas que aumentam o magnetismo superficial ao longo desses planos. Além disso, esse efeito é capaz de induzir uma alteração do caráter eletrônico para esses materiais, permitindo indicar que a clivagem das superfícies contribui para o controle das propriedades eletrônicas, reduzindo o valor de band-gap ou gerando comportamento meio-metálico. Os mapas morfológicos obtidos indicam que o controle da exposição majoritária do plano (001) para obtenção de discos hexagonais induz um aumento do magnetismo superficial para os materiais ATiO3 em acordo com resultados experimentais, além de predizer diferentes morfologias acessíveis com interessantes propriedades magnéticas. Ademais, o efeito de defeitos intrínsecos como vacâncias de oxigênio no bulk e superfície apolar (110) dos materiais ATiO3 foi investigado indicando que a redução do número de coordenação na região do defeito induz que os elétrons remanescentes sejam localizados, principalmente, nos orbitais 3d vazios dos cátions Ti vizinhos, gerando espécies [TiO5]ꞌ e [TiO4]ꞌ (3d1 ) que possibilitam uma interação ferromagnética nos materiais MnTiO3 e FeTiO3. A combinação entre os diferentes mecanismos investigados permitiu estabelecer um guia científico para o estudo teórico de materiais multiferróicos, contribuindo para descrever as potencialidades dos diferentes materiais bem como predizer novos candidatos. / The development of spintronic has motivated the search for new multiferroic materials due to the multifunctionality of these materials that are associated with the coupling of different ferroic orders into a single crystalline structure. In the present study, we propose a theoretical investigation, based on Density Functional Theory, of ATiO3 (A = Mn, Fe, Ni) materials in the R3c structure in order to clarify the effect of A-site cation replacement on the structural, magnetic and electronic properties, as well as to describe a different mechanism to control the multiferroic properties based on thin-film architectures, morphology and point defects. For a more comprehensive overview of the main effects involved on the ATiO3 materials several exchange-correlation functionals were investigated, being the PBE0 the functional with smallest deviations and, consequently, the best representation in comparison to the experimental results. Aiming to describe the main fingerprints related with the creation of ATiO3 thin-films, we propose an innovative methodology that allows to describe the uniaxial and biaxial deformations originated in the interface region between the film and the substrate. In this case, the results indicate that structural distortions induce a magnetic transition for the NiTiO3, originating ferromagnetic ordering from magneto-structural criteria, which is associated to the deformation of the [MO6] clusters that reproduces satisfactorily the experimental results reported in the literature. Similarly, in order to elucidate the relationship between the magnetism and the morphology of the ATiO3 materials, we combined Surface Energy, Wulff Construction, and an advanced formalism to describe surface magnetism by considering the existence of uncompensated spins along the polar planes (100), (001), (101), (012), (111) and non-polar (110). The results indicate that the reduction of the coordination for both A and Ti metals along the (001) and (111) planes induces a charge transfer between the A 2+ and Ti4+ cations, resulting in magnetic Ti3+ species that increase the superficial magnetism along such planes. Moreover, this effect allowed a change in the electronic structure for these materials, allowing to point out that the cleavage of the surfaces contribute to the control of the electronic properties reducing the band-gap value or generating half-metallic behavior. The morphological maps indicated that the control of the major exposure for the (001) surface to obtain hexagonal discsinduces an increase of the superficial magnetism for the ATiO3 materials according to experimental results, besides predicting different accessible morphologies with interesting magnetic properties. In addition, the effect of intrinsic defects such as oxygen vacancies on the bulk and non-polar (110) surface of the ATiO3 materials were investigated, indicating that the reduction of coordination in the defect region induces the localization of the remaining electrons in the empty 3d orbitals of neighboring Ti cations, generating [TiO5]'and [TiO4]' (3d1 ) species that allow a ferromagnetic interaction for MnTiO3 and FeTiO3 materials. The combination of the different mechanisms investigated has allowed to stablish a scientific guide for the theoretical study of multiferroic materials, contributing to describe the potentialities of the different materials as well as to predict new candidates.

Part I - Level crossings and other level functionals.<br />Part II - Some contributions in statistics of extremes and in statistical mechanics.

Le présent mémoire traite de la description du LaOFeAs, le premier matériau découvert de la famille des pnictures de fer, par la théorie de la fonctionnelle de la densité (DFT). Plus particulièrement, nous allons exposer l’état actuel de la recherche concernant ce matériau avant d’introduire rapidement la DFT. Ensuite, nous allons regarder comment se comparent les paramètres structuraux que nous allons calculer sous différentes phases par rapport aux résultats expérimentaux et avec les autres calculs DFT dans la littérature. Nous allons aussi étudier en détails la structure électronique du matériau sous ses différentes phases magnétiques et structurales. Nous emploierons donc les outils normalement utilisés pour mieux comprendre la structure électronique : structures de bandes, densités d’états, surfaces de Fermi, nesting au niveau de Fermi. Nous tirerons profit de la théorie des groupes afin de trouver les modes phononiques permis par la symétrie de notre cristal. De plus, nous étudierons le couplage électrons-phonons pour quelques modes. Enfin, nous regarderons l’effet de différentes fonctionnelles sur nos résultats pour voir à quel point ceux-ci sont sensibles à ce choix. Ainsi, nous utiliserons la LDA et la PBE, mais aussi la LDA+U et la PBE+U. / We present DFT calculations of the electronic structure of LaOFeAs, the parent compound of the new family of superconductors, the iron pnictides, in this master thesis. Specifically, we are going to take a look at the present state of the research done on this material before giving a quick introduction to DFT. Then, we will compare the optimized structural parameters as calculated by our DFT code with the experimental data as well as results obtained by other groups. We studied the electronic structure of LaOFeAs using the standard set of tools : band structure, density of state (DOS), fermi surface and fermi surface nesting. We used theoretical methods to determine the allowed phonon modes in this crystal structure. This, in turn, enabled us to explore the electron-phonon coupling in our material for the most important modes. We’ll also discuss the influence different functionals may have for calculating the electronic structure. This will allow us to validate our results. In detail, we will compare results obtained with the following functionals: LDA and PBE, as well as LDA+U and PBE+U.

Ce mémoire est consacré à des résultats d'analyse harmonique réelle dans des cadres géométriques discrets (graphes) ou continus (groupes de Lie).Soit $\Gamma$ un graphe (ensemble de sommets et d'arêtes) muni d'un laplacien discret $\Delta=I-P$, où $P$ est un opérateur de Markov.Sous des hypothèses géométriques convenables sur $\Gamma$, nous montrons la continuité $L^p$ de fonctionnelles de Littlewood-Paley fractionnaires. Nous introduisons des espaces de Hardy $H^1$ de fonctions et de $1$-formes différentielles sur $\Gamma$, dont nous donnons plusieurs caractérisations, en supposant seulement la propriété de doublement pour le volume des boules de $\Gamma$. Nous en déduisons la continuité de la transformée de Riesz sur $H^1$. En supposant de plus des estimations supérieures ponctuelles (gaussiennes ou sous-gaussiennes) sur les itérées du noyau de l'opérateur $P$, nous obtenons aussi la continuité de la transformée de Riesz sur $L^p$ pour $1<p<2$.Nous considérons également l'espace de Besov $B^{p,q}_\alpha(G)$ sur un groupe de Lie unimodulaire $G$ muni d'un sous-laplacien $\Delta$. En utilisant des estimations du noyau de la chaleur associé à $\Delta$, nous donnons plusieurs caractérisations des espaces de Besov, et montrons une propriété d'algèbre pour $B^{p,q}_\alpha(G) \cap L^\infty(G)$, pour $\alpha>0$, $1\leq p\leq+\infty$ et $1\leq q\leq +\infty$. Les résultats sont valables en croissance polynomiale ou exponentielle du volume des boules. / This thesis is devoted to results in real harmonic analysis in discrete (graphs) or continuous (Lie groups) geometric contexts.Let $\Gamma$ be a graph (a set of vertices and edges) equipped with a discrete laplacian $\Delta=I-P$, where $P$ is a Markov operator.Under suitable geometric assumptions on $\Gamma$, we show the $L^p$ boundedness of fractional Littlewood-Paley functionals. We introduce $H^1$ Hardy spaces of functions and of $1$-differential forms on $\Gamma$, giving several characterizations of these spaces, only assuming the doubling property for the volumes of balls in $\Gamma$. As a consequence, we derive the $H^1$ boundedness of the Riesz transform. Assuming furthermore pointwise upper bounds for the kernel (Gaussian of subgaussian upper bounds) on the iterates of the kernel of $P$, we also establish the $L^p$ boundedness of the Riesz transform for $1<p<2$.We also consider the Besov space $B^{p,q}_\alpha(G)$ on a unimodular Lie group $G$ equipped with a sublaplacian $\Delta$.Using estimates of the heat kernel associated with $\Delta$, we give several characterizations of Besov spaces, and show an algebra property for $B^{p,q}_\alpha(G) \cap L^\infty(G)$ for $\alpha>0$, $1\leq p\leq+\infty$ and $1\leq q\leq +\infty$.These results hold for polynomial as well as for exponential volume growth of balls.

In this work, our goal is to prove results on prolongation of solutions, uniform boundedness of solutions, uniform stability as well uniform asymptotic stability (in the classical sense of Lyapunov) for measure differential equations and for dynamic equations on time scales. In order to get our results, we employ the theory of generalized ODEs, since these equations encompass measure differential equations and dynamic equations on time scales. Therefore, to get our results, we start by proving the expected result for abstract generalized ODEs. Then, using the correspondence between the solutions of these equations and the solutions of measure differential equations (see [38]), we extend all the results to these the latter. After that, using the correspondence between the solutions of measure differential equations and the solutions of dynamic equations on time scales (see [21]), we extend all the results to these last equations. Finally, we investigate autonomous generalized ODEs and show that these equations do not enlarge the class of classical autonomous ODEs, even when we consider a more general class of functions as right-hand sides. All the new results presented in this work are contained in papers [16, 17, 18, 19]. / Neste trabalho, nosso objetivo e provar resultados sobre prolongamento de soluções, limitação uniforme de soluções, estabilidade uniforme e estabilidade uniforme assintótica (no sentido clássico de Lyapunov) para equações diferenciais em medida e para equações dinâmicas em escalas temporais. A fim de obter os nossos resultados, empregamos a teoria de EDOs generalizadas, uma vez que estas equações abrangem equações diferenciais em medida e equações dinâmicas em escalas temporais. Portanto, para obter nossos resultados, vamos começar por provar, os resultados que queremos para EDOs generalizadas abstratas. Em seguida, usando a correspondência entre as soluções de EDOs generalizadas e soluções de equações diferenciais em medida (ver [38]), estenderemos os resultados para estas ultimas equações. Depois disso, usando a correspondência entre as soluções de equações diferenciais em medida e as soluções de equações dinâmicas em escalas temporais (ver [21]), estenderemos todos os resultados para estas ultimas equações. Finalmente, investigamos EDOs generalizadas autônomas e mostramos que estas equações não aumentam a classe de EDOs autônomas clássicas, mesmo quando consideramos uma classe mais geral de funções nos lados direitos das equações. Os novos resultados encontrados estão contidos em [16, 17, 18, 19].

Dans ce travail nous avons étudié des réactions chimiques sur la surface de deux métaux :le platine et l'or, en utilisant la microscopie ionique à effet de champ électrique (FIM) et la spectrométrie de masse de désorption par champ pulsé (PFDMS). En complément de ces données expérimentales, nous apportons des résultats obtenus par la théorie de la fonctionnelle de la densité (DFT). La taille et la morphologie de nos échantillons font qu’ils sont de bons modèles de grains de phase active dans un catalyseur réel.<p>\ / Doctorat en sciences, Spécialisation chimie / info:eu-repo/semantics/nonPublished