In this thesis we discuss different types of regularity for distributions which appear in the theory of pseudo-differential operators and partial differential equations. Partial differential equations often appear in science and technology. For example the Schrödinger equation can be used to describe the change in time of quantum states of physical systems. Pseudo-differential operators can be used to solve partial differential equations.  They are also appropriate to use when modeling different types of problems within physics and engineering. For example, there is a natural connection between pseudo-differential operators and stationary and non-stationary filters in signal processing. Furthermore, the correspondence between symbols and operators when passing from classical mechanics to quantum mechanics essentially agrees with symbols and operators in the Weyl calculus of pseudo-differential operators. In this thesis we concentrate on investigating how regularity properties for solutions of partial differential equations are affected under the mapping of pseudo-differential operators, and in particular of the free time-dependent Schrödinger operators. The solution of the free time-dependent Schrödinger equation can be expressed as a pseudo-differential operator, with non-smooth symbol, acting on the initial condition. We generalize a result about non-tangential convergence, which was obtained by Sjögren and Sjölin (1989) for the free time-dependent Schrödinger equation. Another way to describe regularity for a distribution is to use wave-front sets. They do not only describe where the singularities are, but also the directions in which these singularities appear. The first types of wave-front sets (analytical wave-front sets) were introduced by Sato (1969, 1970). Later on Hörmander introduced ``classical'' wave-front sets (with respect to smoothness) and showed results in the context of pseudo-differential operators with smooth symbols, cf. Hörmander (1985). In this thesis we consider wave-front sets with respect to Fourier Banach function spaces. Roughly speaking, we take B as a Banach space, which is invariant under translations and embedded between the space of Schwartz functions and the space of temperated distributions. Then we say that the wave-front set of a distribution contains all points (x0, ξ0) such that no localization of the distribution at x0, belongs to FB in the direction ξ0. We prove that pseudo-differential operators with smooth symbols shrink the wave-front set and we obtain opposite embeddings by using sets of characteristic points of the operator symbols. / I denna avhandling diskuterar vi olika typer av regularitet för distributioner som uppkommer i teorin för pseudodifferentialoperatorer och partiella differentialekvationer. Partiella differentialekvationer förekommer inom naturvetenskap och teknik. Exempelvis kan Schrödingerekvationen användas för att beskriva förändringen med tiden av kvanttillstånd i fysikaliska system. Pseudodifferentialoperatorer kan användas för att lösa partiella differential\-ekvationer. De användas också för att modellera olika typer av problem inom fysik och teknik. Det finns till exempel en naturlig koppling mellan pseudodifferentialoperatorer och stationära och icke-stationära filter i signalbehandling. Vidare gäller att relationen mellan symboler och operatorer vid övergången från klassisk mekanik till kvantmekanik i huvudsak överensstämmer med symboler och operatorer inom Weylkalkylen för pseudodifferentialoperatorer. I den här avhandlingen koncentrerar vi oss på att undersöka hur regularitetsegenskaper för lösningar till partiella differentialekvationer påverkas under verkan av pseudodifferentialoperatorer, och speciellt för de fria tidsberoende Schrödingeroperatorerna. Lösningen av den fria tidsberoende Schrödingerekvationen kan uttryckas som en pseudodifferentialoperator, med icke-slät symbol, verkande på begynnelsevillkoret. Vi generaliserar ett resultat om icke-tangentiell konvergens av Sjögren och Sjölin (1989) för den fria tidsberoende Schrödingerekvationen. Ett annat sätt att beskriva regularitet hos en distribution är med hjälp av vågfrontsmängder. De beskriver inte bara var singulariteterna finns, utan också i vilka riktningar dessa singulariteter förekommer. De första typerna av vågfrontsmängder (analytiska vågfrontsmängder) introducerades av Sato (1969, 1970). Senare introducerade Hörmander ''klassiska'' vågfrontsmängder (med avseende på släthet) och visade resultat för verkan av pseudodifferentialoperatorer med släta symboler, se  Hörmander (1985). I denna avhandling betraktar vi vågfrontsmängder med avseende på Fourier Banach funktionsrum. Detta kan ses som att vi låter B vara ett Banachrum, som är invariant under translationer och är inbäddat mellan rummet av Schwartzfunktioner och rummet av tempererade distributioner. Vågfrontsmängden av en distribution innehåller alla punkter (x0, ξ0) så att ingen lokalisering av distributionen kring x0, tillhör FB i riktningen ξ0. Vi visar att pseudodifferentialoperatorer med släta symboler krymper vågfrontsmängden och vi får motsatta inbäddningar med hjälp mängder av karakteristiska punkter till operatorernas symboler.

In the theory of holomorphic functions of one complex variable it is often useful to study subharmonic functions. The subharmonic can be described using the Laplace operator. When one studies holomorphic functions of several complex variables one should study the plurisubharmonic functions instead. Here the complex Monge--Ampère operator has a role similar to that of the Laplace operator in the theory of subharmonic functions. The complex Monge--Ampère operator is nonlinear and therefore it is not as well understood as the Laplace operator. We consider two types of boundary value problems for the complex Monge--Ampere equation in certain pseudoconvex domains. In this thesis the right-hand side in the Monge--Ampère equation will always be smooth, strictly positive and meet a monotonicity condition. The first type of boundary value problem we consider is a Dirichlet problem where we look for plurisubharmonic solutions which are zero on the boundary of the domain. We show that this problem has a unique smooth solution if the domain has a smooth bounded plurisubharmonic exhaustion function which is globally Lipschitz and has Monge--Ampère mass larger than one everywhere. We obtain some results on which domains have such a bounded exhaustion function. The second type of boundary value problem we consider is a boundary blow-up problem where we look for plurisubharmonic solutions which tend to infinity at the boundary of the domain. Here we also assume that the right-hand side in the Monge--Ampère equation satisfies a growth condition. We study this problem in strongly pseudoconvex domains with smooth boundary and show that it has solutions which are Hölder continuous with arbitrary Hölder exponent α, 0 ≤ α < 1. We also show a uniqueness result. A result on the growth of the solutions is also proved. This result is used to describe the boundary behavior of the Bergman kernel.

This thesis is focused on extracting meaningful statistics for the characterization and classification of biological, medical, and financial data and contains four chapters. The first chapter contains theoretical background on scaling and wavelets, which supports the work in chapters two and three. In the second chapter, we outline a methodology for representing sequences of DNA nucleotides as numeric matrices in order to analytically investigate important structural characteristics of DNA. This methodology involves assigning unit vectors to nucleotides, placing the vectors into columns of a matrix, and accumulating across the rows of this matrix. Transcribing the DNA in this way allows us to compute the 2-D wavelet transformation and assess regularity characteristics of the sequence via the slope of the wavelet spectra. In addition to computing a global slope measure for a sequence, we can apply our methodology for overlapping sections of nucleotides to obtain an evolutionary slope. In the third chapter, we describe various ways wavelet-based scaling may be used for cancer diagnostics. There were nearly half of a million new cases of ovarian, breast, and lung cancer in the United States last year. Breast and lung cancer have highest prevalence, while ovarian cancer has the lowest survival rate of the three. Early detection is critical for all of these diseases, but substantial obstacles to early detection exist in each case. In this work, we use wavelet-based scaling on metabolic data and radiography images in order to produce meaningful features to be used in classifying cases and controls. Computer-aided detection (CAD) algorithms for detecting lung and breast cancer often focus on select features in an image and make a priori assumptions about the nature of a nodule or a mass. In contrast, our approach to analyzing breast and lung images captures information contained in the background tissue of images as well as information about specific features and makes no such a priori assumptions. In the fourth chapter, we investigate the value of social media data in building commercial default and activity credit models. We use random forest modeling, which has been shown in many instances to achieve better predictive accuracy than logistic regression in modeling credit data. This result is of interest, as some entities are beginning to build credit scores based on this type of publicly available online data alone. Our work has shown that the addition of social media data does not provide any improvement in model accuracy over the bureau only models. However, the social media data on its own does have some limited predictive power.

Cette thèse est une contribution à l’étude de problèmes de contrôle optimal dont le caractère non linéaire se traduit par la présence, dans les équations d’état, d’un terme bilinéaire relativement à l’état et au contrôle. Malgré les difficultés liées à la non linéarité, nous obtenons des propriétés spécifiques au cas bilinéaire. L’introduction générale constitue la première partie. La seconde partie est consacrée à l’étude des équations d’état ; ce sont des équations aux dérivées partielles d’évolution. Nous établissons des estimations a priori sur les solutions à partir des inégalités de Willett et Wong et nous démontrons que les équations d’états sont bien posées. Dans le cas où les contrôles subissent une contrainte liée aux états, ces estimations permettent de déduire l’existence de solutions dans le cadre des inclusions différentielles. Les troisième et quatrième parties de ce mémoire sont dévolues à la démonstration de l’existence de contrôles optimaux, puis à l’analyse de la sensibilité relative à une perturbation qui intervient de façon additive dans l’équation d’état. Le caractère bilinéaire permet de vérifier des conditions suffisantes d’optimalité du second ordre. Nous fournissons sur des exemples, une formule explicite des dérivées directionnelles de la fonction valeur optimale / This thesis is devoted to the analysis of nonlinear optimal control problems governed by an evolution state equation involving a term which is bilinear in state and control. The difficulties due to nonlinearity remain, but bilinearity adds a lot of structure to the control problem under consideration. In Section 2, by using Willet and Wong inequalities we establish a priori estimates for the solutions of the state equation. These estimates allow us to prove that the state equation is well posed in the sense of Hadamard. In the case of a feedback constraint on the control, the state equation becomes a differential inclusion. Under mild assumptions, such a differential inclusion is solvable. In Section 3, we prove the existence of solutions to the optimal control problem. Section 4 is devoted to the sensitivity analysis of the optimal control problem. We obtain a formula for the directional derivative of the optimal value function. This general formula is worked out in detail for particular examples

Qualitative properties of solutions to equations of fluid mechanics Mgr. Jakub Tichý Supervisor: doc. Mgr. Petr Kaplický, Ph.D. Department: Department of Mathematical Analysis Abstract This thesis is devoted to the boundary regularity of weak solutions to the system of nonlinear partial differential equations describing incompressible flows of a certain class of generalized Newtonian fluids in bounded domains. Equations of motion and continuity equation are complemented with perfect slip boundary conditions. For stationary generalized Stokes system in Rn with growth condi- tion described by N-function Φ the existence of the second derivatives of velocity and their regularity up to the boundary are shown. For the same system of equa- tions integrability of velocity gradients is proven. Lq estimates are obtained also for classical evolutionary Stokes system via interpolation-extrapolation scales. Hölder continuity of velocity gradients and pressure is shown for evolutionary generalized Navier-Stokes equations in R2 . Keywords Generalized Stokes and Navier - Stokes equations, incompressible fluids, perfect slip boundary conditions, regularity up to the boundary

Cette thèse s’intéresse à diverses composantes du traitement et de l’analyse d’images par méthodes non locales. Ces méthodes sont basées sur la redondance d’information présente dans d’autres images, et utilisent des algorithmes de recherche de correspondance, généralement basés sur l’utilisation patchs, pour extraire et transférer de l’information depuis ces images d’exemples. Ces approches, largement utilisées par la communauté de vision par ordinateur, sont souvent limitées par le temps de calcul de l’algorithme de recherche, appliqué à chaque pixel, et par la nécessité d’effectuer un prétraitement ou un apprentissage pour utiliser de grandes bases de données.Pour pallier ces limites, nous proposons plusieurs méthodes générales, sans apprentissage,rapides, et qui peuvent être facilement adaptées à diverses applications de traitement et d’analyse d’images naturelles ou médicales. Nous introduisons un algorithme de recherche de correspondances permettant d’extraire rapidement des patchs d’une grande bibliothèque d’images 3D, que nous appliquons à la segmentation d’images médicales. Pour utiliser de façon similaire aux patchs,des présegmentations en superpixels réduisant le nombre d’éléments de l’image,nous présentons une nouvelle structure de voisinage de superpixels. Ce nouveau descripteur permet d’utiliser efficacement les superpixels dans des approches non locales. Nous proposons également une méthode de décomposition régulière et précise en superpixels. Nous montrons comment évaluer cette régularité de façon robuste, et que celle-ci est nécessaire pour obtenir de bonnes performances de recherche de correspondances basées sur les superpixels. / This thesis focuses on several aspects of image analysis and processing with non local methods. These methods are based on the redundancy of information that occurs in other images, and use matching algorithms, that are usually patch-based, to extract and transfer information from the example data. These approaches are widely used by the computer vision community, and are generally limited by the computational time of the matching algorithm, applied at the pixel scale, and by the necessity to perform preprocessing or learning steps to use large databases. To address these issues, we propose several general methods, without learning, fast, and that can be easily applied to different image analysis and processing applications on natural and medical images. We introduce a matching algorithm that enables to quickly extract patches from a large library of 3D images, that we apply to medical image segmentation. To use a presegmentation into superpixels that reduces the number of image elements, in a way that is similar to patches, we present a new superpixel neighborhood structure. This novel descriptor enables to efficiently use superpixels in non local approaches. We also introduce an accurate and regular superpixel decomposition method. We show how to evaluate this regularity in a robust manner, and that this property is necessary to obtain good superpixel-based matching performances.

Na Capa, Folha de Rosto e Ficha Catalogr?fica da obra constam o t?tulo: "Os Conselhos de Alimenta??o Escolar da Comarca de Diamantina: percep??es e estrutura". / Submitted by Jos? Henrique Henrique (jose.neves@ufvjm.edu.br) on 2018-06-14T18:58:03Z No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) caio_guedes_oliveira.pdf: 3164792 bytes, checksum: 20acd36d2a5df005a431d8b58d2b8787 (MD5) / Approved for entry into archive by Rodrigo Martins Cruz (rodrigo.cruz@ufvjm.edu.br) on 2018-07-18T11:24:43Z (GMT) No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) caio_guedes_oliveira.pdf: 3164792 bytes, checksum: 20acd36d2a5df005a431d8b58d2b8787 (MD5) / Made available in DSpace on 2018-07-18T11:24:43Z (GMT). No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) caio_guedes_oliveira.pdf: 3164792 bytes, checksum: 20acd36d2a5df005a431d8b58d2b8787 (MD5) Previous issue date: 2017 / Esta pesquisa est? inserida na linha de pesquisa da Promo??o da Sa?de, Preven??o e Controle de Doen?as e buscou analisar os 9 (nove) Conselhos de Alimenta??o Escolar (CAEs) da Comarca de Diamantina, MG, com a finalidade de levantar, analisar e desvelar se atendem aos crit?rios m?nimos estabelecidos na legisla??o em vig?ncia, se h? coer?ncia entre suas atribui??es legais e o que ocorre na pr?tica, se disp?em de estrutura m?nima necess?ria para o desempenho das suas atribui??es e quais s?o as percep??es atribu?das pelos conselheiros sobre as atividades de Controle Social exercido perante o Conselho. Trata-se de um estudo transversal retrospectivo, descritivo, do tipo estudo de caso, com abordagem quantitativa e qualitativa, no qual foi feita uma minuciosa an?lise documental nos arquivos dos Conselhos de Alimenta??o Escolar para remontar a forma de atua??o deles, buscando conhecer suas principais qualidades e defeitos. Al?m disso, foram realizadas entrevistas com os conselheiros de alimenta??o escolar para entender quais eram as percep??es que eles tinham sobre as atividades que desempenham, por meio da t?cnica denominada an?lise de conte?do. A an?lise documental demonstrou que h? muitas inconsist?ncias na forma de escolha dos conselheiros e in?meras defici?ncias a serem corrigidas pelos Conselhos, por outro lado, tamb?m foram salientados muitos aspectos positivos dos trabalhos desenvolvidos. A an?lise das atas dos CAEs e das entrevistas com os membros dos Conselhos desvelou as percep??es que eles tinham sobre a forma de constitui?ao do Conselho, as dificuldades vividas para desempenhar o controle social, tais como problemas de infraestrutura, de acesso a informa??es e pouco incentivo a se capacitarem. Por outro lado, demonstraram o empenho para realizar suas atribui??es, zelar pela qualidade da alimenta??o escolar dos seus Munic?pios e pelo uso correto do dinheiro p?blico. Ao final, foi poss?vel constatar que a capacita??o, o est?mulo ao desempenho das fun??es e a participa??o popular s?o fatores que influenciam a atua??o efetiva e eficiente dos conselheiros de alimenta??o escolar, os quais devem ser garantidos e estimulados pelo Poder P?blico, pelos pr?prios conselheiros e pela sociedade. / Disserta??o (Mestrado Profissional) ? Programa de P?s-Gradua??o em Sa?de, Sociedade e Ambiente, Universidade Federal dos Vales do Jequitinhonha e Mucuri, 2017. / This research is included in the research line of Health Promotion, Prevention and Control of Diseases and sought to analyze the 9 (nine) School Feeding Councils of the Region of Diamantina, in the State of Minas Gerais, with the purpose of raising, analyzing and unveiling if they meet the minimum criteria established in the legislation in force, if there is a consistency between the legal attributions of the CAEs and what happens in practice, if they have the minimum structure necessary for the performance of their duties, and what are the perceptions attributed by the counselors on the activities of social control exercised before the Council. This is a cross-sectional, retrospective, descriptive, case-study with a quantitative and qualitative approach, in which a detailed documentary analysis was carried out in the School Feeding Council archives to reassemble their way of acting, seeking to know their main qualities and defects. In addition, interviews were conducted with school feeding counselors to understand what perceptions they had of the activities they perform through the technique called content analysis. The documentary analysis showed that there are many inconsistencies in the choice of counselors and numerous shortcomings to be corrected by the Councils, on the other hand, also highlighted many positive aspects of the work developed. The analysis of the minutes of the CAEs and the interviews with the members of the Councils revealed the perceptions they had about the form of constitution of the council, the difficulties experienced to carry out social control, such as infrastructure problems, access to information and little incentive to qualify. On the other hand, they demonstrated the commitment to carry out their duties, to ensure the quality of school feeding of their Municipalities and the correct use of public money. In the end, it was possible to see that the training, stimulating functions and popular participation are factors that influence the effective and efficient performance of school feeding counselors, which must be guaranteed and encouraged by the Government, by its own counselors and society.

Cette thèse est une contribution au problème de détection de fines structures tubulaires dans une image2-D ou 3-D. Nous avons plus précisément en vue le cas des images angiographiques. Celles-ci étant bruitées, les vaisseaux ne se détachent pas nettement du reste de l’image, la question est donc de segmenter avec précision le réseau sanguin. Le cadre théorique de ce travail est le calcul des variations eten particulier l'énergie de Mumford-Shah. Cependant, ce modèle n'est adapté qu'à la détection de structures volumiques étendues dans toutes les directions de l’image. Le but de ce travail est donc deconstruire une énergie qui favorise les ensembles qui ne sont étendus que dans une seule direction, cequi est le cas de fins tubes. Pour cela, une nouvelle inconnue est introduite, une métrique Riemannienne,qui a pour but la détection de la structure géométrique de l’image. Une nouvelle formulation de l’énergie de Mumford-Shah est donnée avec cette nouvelle métrique. La preuve de l'existence d'une solution au problème de la minimisation de l’énergie est apportée. De plus, une approximation par gamma-convergence est démontrée, ce qui permet ensuite de proposer et de mettre en oeuvre une implémentation numérique. / This thesis is a contribution to the fine tubular structures detection problem in a 2-D or 3-D image. We arespecifically interested in the case of angiographic images. The vessels are surrounded by noise and thenthe question is to segment precisely the blood network. The theoretical framework of our work is thecalculus of variations and we focus on the Mumford-Shah energy. Initially, this model is adapted to thedetection of volumetric structures extended in all directions of the image. The aim of this study is to buildan energy that favors sets which are extended in one direction, which is the case of fine tubes. Then, weintroduce a new unknown, a Riemannian metric, which captures the geometric structure at each point ofthe image and we give a new formulation of the Mumford-Shah energy adapted to this metric. Thecomplete analysis of this model is done: we prove that the associated problem of minimization is wellposed and we introduce an approximation by gamma-convergence more suitable for numerics. Eventually,we propose numerical experimentations adapted to this approximation.

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)

Nous considérons certaines séries de Fourier liées à la théorie des formes modulaires. Nous étudions leurs propriétés analytiques : la dérivabilité, le module de continuité et l'exposant de Hölder. Nous utilisons deux méthodes différentes. La première revient à trouver et itérer une équation fonctionnelle de la fonction étudiée (méthode d'Itatsu) et la deuxième provient de l'analyse en ondelettes (méthode de Jaffard). L'étape essentielle de chacune dépend de la modularité sous-jacente. Nous trouvons que les propriétés analytiques de ces séries aux points irrationnels sont liées aux propriétés diophantiennes de ces points. Ce travail a été motivé par l'étude de la fonction de Riemann. / We consider certain Fourier series which arise from modular or automorphicforms. We study their analytic properties: differentiability, modulus of continuity and theH¨older regularity exponent. We use two different methods. One is based on finding anditerating a functional equation for the function studied (Itatsu’s method), the second onecomes from wavelet analysis (Jaffard’s method). The crucial steps in both of them arebased on the underlined modularity. We find that the analytic properties of these seriesat an irrational x are related to the fine diophantine properties of x, in a very precise way.The work was motivated by the study of the Riemann series.