Motivated by experiments on titanium alloys of Petrinic et al. (2006), which show the formation of cracks through the growth and coalescence of voids in ductile fracture, we consider the problem of formulating a variational model in nonlinear elasticity compatible both with cavitation and the appearance of discontinuities across two-dimensional surfaces. As in the model for cavitation of Müller and Spector (1995) we address this problem, which is connected to the sequential weak continuity of the determinant of the deformation gradient in spaces of functions having low regularity, by means of adding an appropriate surface energy term to the elastic energy. Based upon considerations of invertibility, we derive an expression for the surface energy that admits a physical and a geometrical interpretation, and that allows for the formulation of a model with better analytical properties. We obtain, in particular, important regularity results for the inverses of deformations, as well as the weak continuity of the determinants and the existence of minimizers. We show, further, that the creation of surface can be modeled by carefully analyzing the jump set of the inverses, and we point out some connections between the analysis of cavitation and fracture, the theory of SBV functions, and the theory of Cartesian currents of Giaquinta, Modica, and Soucek. In addition to the above, we extend previous work of Sivaloganathan, Spector and Tilakraj (2006) on the approximation of minimizers for the problem of cavitation with a constraint in the number of flaw points, and present some numerical results for this problem.

Dans cette thèse nous proposons de nouveaux modèles paramétriques en traitement du signal et de l'image, fondés sur la décomposition de Wold des processus stochastiques. Les approches de modélisation font appel à l'analyse fonctionnelle et harmonique, l'analyse par ondelettes, ainsi qu'à la théorie des champs stochastiques. Le premier chapitre a un caractère introductif théorique et précise les éléments de base concernant le contexte de la prédiction linéaire des processus stochastiques stationnaires et la décomposition Wold, dans le cas 1-D et multi-D. On montre comment les différentes parties de la décomposition sont obtenues à partir de l'hypothèse de stationnarité, via la représentation du processus comme l'orbite d'un certain opérateur unitaire, l'isomorphisme canonique de Kolmogorov et les conséquences sur la prédiction linéaire du théorème de Szégö et de ses extensions multidimensionnelles. Le deuxième chapitre traite une approche de factorisation spectrale de la densité spectrale de puissance qu'on utilisera pour l'identification des modèles de type Moyenne Ajustée (MA), Autorégressif (AR) et ARMA. On utilise la représentation par le noyau reproduisant de Poisson d'une fonction extérieure pour construire un algorithme d'estimation d'un modèle MA avec une densité spectrale de puissance donnée. Cette méthode d'estimation est présentée dans le cadre de deux applications: - Dans la simulation de canaux sans fil de type Rayleigh (cas 1-D). - Dans le cadre d'une approche de décomposition de Wold des images texturées (cas 2-D). Dans le troisième chapitre nous abordons la représentation et la compression hybride d'images. Nous proposons une approche de compression d'images qui utilise conjointement : - les modèles issus de la décomposition de Wold pour la représentation des régions dites texturées de l'image; - une approche fondée sur les ondelettes pour le codage de la partie "cartoon" (ou non-texturée) de l' image. Dans ce cadre, nous proposons une nouvelle approche pour la décomposition d'une image dans une partie texturée et une partie non-texturée fondée sur la régularité locale. Chaque partie est ensuite codée à l'aide de sa représentation particulière. / In this thesis we propose new parametric models in signal and image processing based on the Wold decomposition of stationary stochastic processes. These models rely upon several theoretical results from functional and harmonic analysis, wavelet analysis and the theory of stochastic fields, The first chapter presents the theoretical background of the linear prediction for stationary processes and of the Wold decomposition theorems in 1-D and n-D. It is shown how the different parts of the decomposition are obtained and represented, by the means of the unitary orbit representation of stationary processes, the Kolmogorov canonical model and Szego-type extensions. The second chapter deals with a spectral factorisation approach of the power spectral density used for the parameter estimation of Moving Avergage (MA), AutoRegressif (AR) and ARMA models. The method uses the Poisson integral representation in Hardy spaces in order to estimate an outer transfer function from its power spectral density. - Simulators for Rayleigh fading channels (1-D). - A scheme for the Wold decomposition for texture images (2-D). In the third chapter we deal with hybrid models for image representation and compression. We propose a compression scheme which jointly uses, on one hand, Wold models for textured regions of the image, and on the other hand a wavelet-based approach for coding the 'cartoon' (or non-textured) part of the image. In this context, we propose a new algorithm for the decomposing images in a textured part and a non-textured part. The separate parts are then coded with the appropriate representation.

Le feedback auditif se définit comme un signal auditif qui contient de l'information sur un mouvement. Il a été montré que le feedback auditif peut guider le mouvement en temps réel, mais son influence sur l'apprentissage moteur est moins clair. Cette thèse a pour but d'examiner l'influence du feedback auditif sur l'apprentissage moteur, en se focalisant sur le contrôle temporel des mouvements. Premièrement, nous étudions l'apprentissage moteur chez les non-musiciens sains et montrons qu'ils bénéficient de l'information temporelle contenue dans le feedback auditif et qu'ils sont sensibles aux distorsions de cette information temporelle. Deuxièmement, nous appliquons ces connaissances à la réhabilitation de patients cérébro-lésés. Nous trouvons que ces patients améliorent leurs capacités de mouvement mais ne dépendent pas de la correspondance temporelle entre le mouvement et le son. Paradoxalement, ces patients ont même bénéficié des distorsions temporelles dans le feedback. Troisièmement, nous étudions les experts musicaux, car ils ont établi des liens particulièrement forts entre leur mouvement et le son. Nous développons de nouveaux outils d'analyse qui nous permettent de séparer les déviations temporelles en variation systématique et non-systématique. Le résultat principal est que ces experts sont devenus largement indépendants du feedback auditif. La proposition centrale de cette thèse est que le feedback auditif joue un rôle dans l'apprentissage moteur de la régularité, mais la façon dont le cerveau l'utilise dépend de la population étudiée. Ces résultats donnent une nouvelle perspective sur l'intégration audio-motrice et contribuent au développement de nouvelles approches pour l'apprentissage de la musique et la réhabilitation / Auditory feedback is an auditory signal that contains information about performed movement. Music performance is an excellent candidate to study its influence on motor actions, since the auditory result is the explicit goal of the movement. Indeed, auditory feedback can guide online motor actions, but its influence on motor learning has been investigated less. This thesis investigates the influence of auditory feedback in motor learning, focusing particularly on how we learn temporal control over movements. First, we investigate motor learning in non-musicians, finding that they benefit from temporal information supplied by the auditory signal and are sensitive to distortions of this temporal information. Second, we turn to stroke patients that are re-learning motor actions in a rehabilitation setting. Patients improved their movement capacities but did not depend on the time-locking between movements and the resulting auditory feedback. Surprisingly, they appear to benefit from distortions in feedback. Third, we investigate musical experts, who arguably have established strong links between movement and auditory feedback. We develop a novel analysis framework that allows us to segment timing into systematic and non-systematic variability. Our finding is that these experts have become largely independent of the auditory feedback. The main claim defended in this thesis is that auditory feedback can and does play a role in motor learning of regularity, but the way in which it is used varies qualitatively between different populations. These findings provide new insights into auditory-motor integration and are relevant for developing new perspectives on the role of music in training and rehabilitation settings

Einst als Hilfssatz für Szemerédis Theorem entwickelt, hat sich das Regularitätslemma in den vergangenen drei Jahrzehnten als eines der wichtigsten Werkzeuge der Graphentheorie etabliert. Im Wesentlichen hat das Lemma zum Inhalt, dass dichte Graphen durch eine konstante Anzahl quasizufälliger, bipartiter Graphen approximiert werden können, wodurch zwischen deterministischen und zufälligen Graphen eine Brücke geschlagen wird. Da letztere viel einfacher zu handhaben sind, stellt diese Verbindung oftmals eine wertvolle Zusatzinformation dar. Vom Regularitätslemma ausgehend gliedert sich die vorliegende Arbeit in zwei Teile. Mit Fragestellungen der Extremalen Hypergraphentheorie beschäftigt sich der erste Teil der Arbeit. Es wird zunächst eine Version des Regularitätslemmas Hypergraphen angewandt, um asymptotisch scharfe Schranken für das Auftreten von Hamiltonkreisen in uniformen Hypergraphen mit hohem Minimalgrad herzuleiten. Nachgewiesen werden des Weiteren asymptotisch scharfe Schranken für die Existenz von perfekten und nahezu perfekten Matchings in uniformen Hypergraphen mit hohem Minimalgrad. Im zweiten Teil der Arbeit wird ein neuer, Szemerédis ursprüngliches Konzept generalisierender Regularitätsbegriff eingeführt. Diesbezüglich wird ein Algorithmus vorgestellt, welcher zu einem gegebenen Graphen ohne zu dichte induzierte Subgraphen eine reguläre Partition in polynomieller Zeit berechnet. Als eine Anwendung dieses Resultats wird gezeigt, dass das Problem MAX-CUT für die oben genannte Graphenklasse in polynomieller Zeit bis auf einen multiplikativen Faktor von (1+o(1)) approximierbar ist. Der Untersuchung von Chung, Graham und Wilson zu quasizufälligen Graphen folgend wird ferner der sich aus dem neuen Regularitätskonzept ergebende Begriff der Quasizufälligkeit studiert und in Hinsicht darauf eine Charakterisierung mittels Eigenwertseparation der normalisierten Laplaceschen Matrix angegeben. / Once invented as an auxiliary lemma for Szemerédi''s Theorem the regularity lemma has become one of the most powerful tools in graph theory in the last three decades which has been widely applied in several fields of mathematics and theoretical computer science. Roughly speaking the lemma asserts that dense graphs can be approximated by a constant number of bipartite quasi-random graphs, thus, it narrows the gap between deterministic and random graphs. Since the latter are much easier to handle this information is often very useful. With the regularity lemma as the starting point two roads diverge in this thesis aiming at applications of the concept of regularity on the one hand and clarification of several aspects of this concept on the other. In the first part we deal with questions from extremal hypergraph theory and foremost we will use a generalised version of Szemerédi''s regularity lemma for uniform hypergraphs to prove asymptotically sharp bounds on the minimum degree which ensure the existence of Hamilton cycles in uniform hypergraphs. Moreover, we derive (asymptotically sharp) bounds on minimum degrees of uniform hypergraphs which guarantee the appearance of perfect and nearly perfect matchings. In the second part a novel notion of regularity will be introduced which generalises Szemerédi''s original concept. Concerning this new concept we provide a polynomial time algorithm which computes a regular partition for given graphs without too dense induced subgraphs. As an application we show that for the above mentioned class of graphs the problem MAX-CUT can be approximated within a multiplicative factor of (1+o(1)) in polynomial time. Furthermore, pursuing the line of research of Chung, Graham and Wilson on quasi-random graphs we study the notion of quasi-randomness resulting from the new notion of regularity and concerning this we provide a characterisation in terms of eigenvalue separation of the normalised Laplacian matrix.

Die Aktionspotential (AP) -Aktivität einzelner kortikaler Neuronen kann messbare sensorische Effekte hervorrufen. Es ist jedoch nicht bekannt, wie AP-Sequenzen Parameter und spezifische neuronale Subtypen die hervorgerufenen Sinnesempfindungen beeinflussen. Hier haben wir einen ‘Reverse-Physiology‘ Ansatz angewendet, um die Beziehung zwischen der Aktivität einzelner Neuronen und der Empfindung zu untersuchen. Zunächst wird der Prozess der Nanostimulation, eine von der juxtazellulären Markierungstechnik abgeleiteten Einzelzell-Stimulationsmethode, detailliert beschrieben. Nanostimulation ist einfach anzuwenden und kann auf eine Vielzahl von identifizierbaren Neuronen in narkotisierten und wachen Tieren angewandt werden. Wir beschreiben die Aufnahmetechnik und die elektrische Konfiguration für Nanostimulation. Während eine exakte zeitliche Bestimmung der AP nicht erreicht wurde, konnten Frequenz und Anzahl der AP parametrisch kontrolliert werden. Wir zeigen, dass Nanostimulation auch angewendet werden kann, um sensorische Reaktionen in identifizierbaren Neuronen selektiv zu inhibieren. Als nächstes haben wir untersucht wie sich die Frequenz und Anzahl der AP sowie die Regelmäßigkeit der Pulsfolge auf die Detektion von Einzelzell-Stimulationen im somatosensorischen Kortex von Ratten auswirken. Für mutmaßlichen erregende regular-spiking Neuronen erhöhte sich die Nachweisbarkeit mit abnehmender Frequenz und Anzahl der AP. Die Stimulation einzelner, mutmaßlichen inhibitorischer und schnell feuernder Neuronen führte zu wesentlich stärkeren sensorischen Effekten, die unabhängig von Frequenz und Anzahl der AP waren. Außerdem fanden wir heraus, dass Unregelmäßigkeiten der Pulsfolge die sensorischen Effekte von putativ erregenden Neuronen stark erhöhten. Diese Unregelmäßigkeiten wurden in durchschnittlich 8% der Durchgänge festgestellt. Unsere Daten deuten darauf hin, dass das es auf Verhaltnisebene eine große Sensivität für kortikale AP und deren zeitlichen Abfolge gibt. / The action potential (AP) activity of single cortical neurons can evoke measurable sensory effects, but it is not known how spiking parameters and specific neuronal subtypes affect the evoked sensations. Here we applied a reverse physiology approach to investigate the relationship between single neuron activity and sensation. First, we provide a detailed description of the procedures involved in nanostimulation, a single-cell stimulation method derived from the juxtacellular labeling technique. Nanostimulation is easy to apply and can be directed to a wide variety of identifiable neurons in anesthetized and awake animals. We describe the recording approach and the parameters of the electric configuration underlying nanostimulation. While exact AP timing has not been achieved, AP frequency and AP number can be parametrically controlled. We demonstrate that nanostimulation can also be used to selectively inhibit sensory responses in identifiable neurons. Next, we examined the effects of AP frequency, AP number and spike train regularity on the detectability of single-cell stimulation in rat somatosensory cortex. For putative excitatory, regular spiking neurons detectability increased with decreasing AP frequencies and decreasing AP numbers. Stimulation of single putative inhibitory, fast spiking neurons led to much larger sensory effects that were not dependent on AP frequency and AP number. In addition, we found that spike train irregularity greatly increased the sensory effects of putative excitatory neurons, with irregular spike trains being detected in on average 8% of trials. Our data suggest that the behaving animal is extremely sensitive to cortical APs and their temporal patterning.

Neste projeto apresentamos algumas direções de pesquisa desenvolvidas no estudo da geometria/topologia da singularidade, no ambiente real e complexo, para funções e aplicações polinomiais. Para isso, utilizaremos as ferramentas da teoria de estratificação, técnicas de decomposição Open book, condições de regularidade no sentido Malgrange, t-regularidade, \'rho\'E-regularidade e trivialidade topológica no infinito / On this project we present some research lines developed in the study of the geometry/ topology of singularity, on the real and complex settings, for functions and polynomial mappings. For this, we use tools from stratification theory, techniques of Open Book decomposition, Malgrange regularity condition, t-regularity condition, \'rho\'E-regularity and topological triviality at infinity

Submitted by CAROLINE DE ARRUDA SIGNORINI null (carolineasignorini@gmail.com) on 2017-03-22T17:30:47Z No. of bitstreams: 1 Dissertação - versão definitiva [22.03.2017].pdf: 1265324 bytes, checksum: cb95983dd59698aa1bb765a4dd7f9866 (MD5) / Approved for entry into archive by Luiz Galeffi (luizgaleffi@gmail.com) on 2017-03-23T13:46:47Z (GMT) No. of bitstreams: 1 signorini_ca_me_sjrp.pdf: 1265324 bytes, checksum: cb95983dd59698aa1bb765a4dd7f9866 (MD5) / Made available in DSpace on 2017-03-23T13:46:47Z (GMT). No. of bitstreams: 1 signorini_ca_me_sjrp.pdf: 1265324 bytes, checksum: cb95983dd59698aa1bb765a4dd7f9866 (MD5) Previous issue date: 2017-03-07 / Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) / Nosso principal propósito neste trabalho é o estudo de condições necessárias e suficientes de otimalidade para problemas de Cálculo das Variações no contexto não-suave. Este estudo partirá da formulação básica suave, passando por problemas com restrições Lagrangianas, até o caso em que consideramos Lagrangianas não-suaves e soluções absolutamente contínuas. Neste caminho, abordaremos um importante avanço na teoria de Cálculo das Variações: os resultados de existência e regularidade de soluções. Além das condições necessárias, analisaremos as condições suficientes através de um conceito de convexidade generalizada, o qual denominamos E-pseudoinvexidade. / Our main purpose in this work is the study of necessary and sufficient optimality conditions for Calculus of Variations problems in the nonsmooth context. This study will comprehend the smooth basic formulation, constrained problems (with Lagrangian restrictions), non-smooth Lagrangians and absolutely continuous solutions. Moreover, we will approach an important advance in Calculus of Variations theory: the existence and regularity of solutions. In addition to necessary conditions, we will analyze sufficient conditions through a generalized convexity concept, which we called E-pseudoinvexity. / FAPESP: 2014/24271-6

Let f : \'K POT. \' be a \'C POT. 2\' semi-algebraic mapping for K = R and a polynomial mapping for K = C. It is well-known that f is a locally trivial topological fibration over the complement of the bifurcation set B(f), also called atypical set. In this work, we consider the notion of t-regularity and \'ho E\'-regularity to study the bifurcation set of semi-algebraic mappings f : \'R POT. n\' \'ARROW\' \'R POT. p\' and polynomial mappings f : \'C POT. n\' \'ARROW\' \'C POT. p\'. We show that t-regularity is equivalent to regularity conditions at infinity which have been used by Rabier (1997), Gaffney (1999), Kurdyka, Orro and Simon (2000) and Jelonek (2003) in order to control the asymptotic behaviour of mappings. In addition, we prove that t-regularity implies \'ho E\'-regularity. The \'ho E\'-regularity enables one to define the set of asymptotic non \'ho E\'-regular values S(f) \'This contained\' \' K POT. p\', and the set \'A IND. \'ho E\'\' := f(Singf) U S(f). For \'C POT. 2\' semi-algebraic mappings f : \'R POT. n\' ARROW \' \'R POT. p\' and polynomial mappings f : \'C POT. n\' \'ARROW\' \'C POT. p\', based on a partial Thom stratification at infinity, we rove that S(f) and \'A IND. ho E\' are closed real semi-algebraic sets of dimension at most p - 1 (real dimension at most 2p - 2, for f : \'C POT. n\' \'ARROW\' \'C POT. p\'). Moreover, based on a new fibration theorem at infinity, i.e. holding in the complement of a sufficiently large ball, we obtain B(f) \'this contained\' \'A IND. ho E\'. We study two special classes of polynomial mappings f : \'R POT. n\' \"ARROW\' \'R POT. p\', the class of fair polynomial mappings and the class of Newton non-degenerate polynomial mappings. For fair polynomial mappings, we give an interpretation of t-regularity in terms of integral closure of modules, which is a real counterpart of Gaffney\'s result (1999). For non-degenerate polynomial mappings, we obtain an approximation for B(f) through a set which depends on the Newton polyhedron of f (results like this have been obtained by Némethi and Zaharia (1990) for polynomial functions f : \'C POT. n\' \'ARROW\' C and recently for mixed polynomial functions by Chen and Tibar (2012)). To finish, we discuss some simple consequences of our work: the equivalence t regularity Rabier (equivalently Gaffney, Kuo-KOS, Jelonek) condition for mappings f : X \'ARROW\' \'K POT. p\', where X \'this contained\' \'K POT. n\' is a smooth ane variety; the problem of bijectivity of semi-algebraic mappings; and a formula to compute the Euler characteristic of regular fibres of polynomial mappings f : \'R POT. n\' \'AROOW\' \'R POT. n-1\'. The above results are also extensions of some results obtained, for polynomial functions f : \'K POT. n\' \'ARROW K, by Némethi and Zaharia (1990), Siersma and Tibar (1995), Paunescu and Zaharia (1997), Parusinski (1995) and Tibar (1998). Title: Regularity at infinity and global fibrations of real algebraic maps / Considere f : \'K POT. n\' \"SETA\' \'K POT. p\' uma aplicação semi-algébrica de classe \'C POT. 2\' para K = R e uma aplicação polinomial para K = C. Por resultados clássicos, sabe-se que f é uma fibração topologicamente trivial sobre o complementar dos valores de bifurcação B(f), também chamado de valores atípicos. Neste trabalho, consideramos a t-regularidade e a \'ho E\'-regularidade no estudo dos valores de bifurcação de aplicações semi-algébricas f : \'R POT. n\' \'SETA\' \'R POT. p\' de classe \'C POT. 2\' e aplicações polinomiais f : \'C POT. n\' \'SETA\' \'C POT. p\'. Mostramos que t-regularidade é equivalente às condições de regularidade no infinito usadas por Rabier (1997), Gaffney (1999), Kurdyka, Orro e Simon (2000) e Jelonek (2003) no controle do comportamento assintótico de aplicações. Também mostramos que t-regularidade implica \'ho E\'-regularidade. Através da \'ho E\'-regularidade, definimos o conjunto dos valores assintóticos não \'ho E\'- regulares S(f) \'K POT. p\', e o conjunto \'A IND. ho E\' : = f(Singf) U S(f). Para aplicações semialgébricas f : \'R POT. n\' \'SETA\' \'R POT. p\' de classe \'C POT. 2\' e aplicações polinomiais f : \'C POT. \' \'SETA\' \'C POT. p\', baseados na existência de uma estraticação parcial de Thom no infinito, provamos que S(f) e \'A IND. ho E\' são conjuntos semi-algébricos reais de dimensão no máximo p - 1 (dimensão real no máximo 2p 2, para f : \'C POT. \' \'SETA\' \' C POT. p\'). Além disso, baseados em um novo teorema de fibração no infinito, ou seja na existência de fibração no complementar de uma bola de raio suficientemente grande, obtemos que o conjunto de bifurcação B(f) está contido no conjunto \'A IND. ho E\'. Estudamos também duas classes de aplicações polinomiais f : \'R POT. n\' \'SETA\' \'R POT. p\', a classe de aplicações polinomiais fair e a classe de aplicações Newton não degeneradas. Para aplicações polinomiais fair, obtemos uma interpretação da t-regularidade em termos da teoria de fecho integral de módulos, estendendo para o caso real os resultados de Gaffney (1999). Para aplicações não degeneradas, obtemos uma aproximação de B(f) através de um conjunto que depende do poliedro de Newton de f (resultados deste tipo foram obtidos por Némethi e Zaharia (1990) para funções polinomiais f : \'C POT. \' \'SETA\' C e recentemente para funções polinomiais mistas por Chen e Tibar (2012)). No final, discutimos algumas consequências simples do nosso trabalho: a equivalência t-regularidade condição de Rabier (equivalentemente Gaffney, Kuo-KOS, Jelonek) para aplicações f : X \'SETA\' \'K POT. p\', onde X \'está contido\' \'K POT. n\' é uma variedade suave afim; o problema de bijetividade de aplicações semi-algébricas; e uma fórmula para o cálculo da característica de Euler de fibras regulares de aplicações polinomiais f : \'R POT. n\' \'SETA\' \'R POT. n-1\'. Os resultados acima também são extensões de alguns resultados obtidos para funções polinomiais f : \'K POT. n\' \'SETA\' K, por Némethi e Zaharia (1990), Siersma e Tibar (1995), Paunescu e Zaharia (1997), Parusinski (1995) e Tibar (1998). Título: Regularidade no infinito e fibrações globais de aplicações algébricas reais

<p>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.</p><p>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.</p><p>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.</p><p>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).</p><p>In this thesis we consider wave-front sets with respect to Fourier Banach function spaces. Roughly speaking, we take <em>B</em> 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 <em>FB</em> 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.</p> / <p>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.</p><p>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.</p><p>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.</p><p>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).</p><p>I denna avhandling betraktar vi vågfrontsmängder med avseende på Fourier Banach funktionsrum. Detta kan ses som att vi låter <em>B</em> 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 <em>FB</em> 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.</p>

<p>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.</p>