The structure of symplectic (canonical) transformations on manifolds with conical singularities is established. The operators associated with these transformations are defined in the weight spaces and their properties investigated.

Green operators on manifolds with edges are known to be an ingredient of parametrices of elliptic (edge-degenerate) operators. They play a similar role as corresponding operators in boundary value problems. Close to edge singularities the Green operators have a very complex asymptotic behaviour. We give a new characterisation of Green edge symbols in terms of kernels with discrete and continuous asymptotics in the axial variable of local model cones.

Option is an asymmetric contract between two parties with future payoff derived from the price of underlying asset. Methods of pricing di erent types of options under more or less general assumptions have been extensively studied since the Nobel price winning works of Black and Scholes [1] and Merton [12] were published in 1973. A new way of pricing options with the use of Mellin transforms have been recently introduced by Panini and Srivastav [15] in 2004. This thesis offers a brief introduction to option pricing with Mellin transforms and a revision of some of the recent research in this field.

Magistro darbe yra gaunamas Z1(s) analizinis pratęsimas į pusplokštumę ir įverčiai. / In the master work, we prove that the function Z1(s) is analytically continuable to the halfplane and estimate.

Darbe pateikta funkcijų tyrimo apžvalga, svarbiausi žinomi rezultatai, suformuluota problema. Pagrindinė ribinė teorema įrodoma, taikant tikimybinius metodus, analizinių funkcijų savybes, aproksimavimo absoliučiai konvertuojančiu integralu principą. / The main limit theorem is proved using probabilistic methods, the analytical functions of the properties.

Many boundary value problems of mathematical physics are modelled by elliptic differential operators L in a given domain Ω . An effective method for treating such problems is the method of layer potentials, whose essence resides in reducing matters to solving a boundary integral equation. This, in turn, requires inverting a singular integral operator, naturally associated with L and Ω, on appropriate function spaces on ƌΩ. When the operator L is of second order and the domain Ω is Lipschitz (i.e., Ω is locally the upper-graph of a Lipschitz function) the fundamental work of B. Dahlberg, C. Kenig, D. Jerison, E. Fabes, N. Rivière, G. Verchota, R. Brown, and many others, has opened the door for the development of a far-reaching theory in this setting, even though several very difficult questions still remain unanswered. In this dissertation, the goal is to solve a number of open questions regarding spectral properties of singular integral operators associated with second and higher-order elliptic boundary value problems in non-smooth domains. Among other spectral results, we establish symmetry properties of harmonic classical double layer potentials associated with the Laplacian in the class of Lipschitz domains in R2. An array of useful tools and techniques from Harmonic Analysis, Partial Differential Equations play a key role in our approach, and these are discussed as preliminary material in the thesis: --Mellin Transforms and Fourier Analysis; --Calderón-Zygmund Theory in Uniformly Rectifiable Domains; -- Boundary Integral Methods. Chapter four deals with proving invertibility properties of singular integral operators naturally associated with the mixed (Zaremba) problem for the Laplacian and the Lamé system in infinite sectors in two dimensions, when considering their action on the Lebesgue scale of p integrable functions, for 1 < p < ∞. Concretely, we consider the case in which a Dirichlet boundary condition is imposed on one ray of the sector, and a Neumann boundary condition is imposed on the other ray. In this geometric context, using Mellin transform techniques, we identify the set of critical integrability indexes p for which the invertibility of these operators fails. Furthermore, for the case of the Laplacian we establish an explicit characterization of the Lp spectrum of these operators for each p є (1,∞), as well as well-posedness results for the mixed problem. In chapter five, we study spectral properties of layer potentials associated with the biharmonic equation in infinite quadrants in two dimensions. A number of difficulties have to be dealt with, the most significant being the more complex nature of the singular integrals arising in this 4-th order setting (manifesting itself on the Mellin side by integral kernels exhibiting Mellin symbols involving hyper-geometric functions). Finally, chapter six, deals with spectral issues in Lipschitz domains in two dimensions. Here we are able to prove the symmetry of the spectra of the double layer potentials associated with the Laplacian. This is in essence a two-dimensional phenomenon, as known examples show the failure of symmetry in higher dimensions. / Mathematics

Diffuse optical tomography (DOT) is an imaging technique that utilizes near-infrared (NIR) light to probe biological tissue and ultimately recover the optical parameters of the tissue. Broadly, the process for image reconstruction in DOT involves three parts: (1) the detected measurements, (2) the modeling of the medium being imaged, and (3) the algorithm that incorporates (1) and (2) to finally estimate the optical properties of the medium. These processes have long been established in the DOT field but are also known to suffer drawbacks. The measurements themselves tend to be susceptible to experimental noise that could degrade reconstructed image quality. Furthermore, depending on the DOT configuration being utilized, the total number of measurements per capture can get very large and add additional computational burden to the reconstruction algorithms. DOT algorithms are reliant on accurate modeling of the medium, which includes solving a light propagation model and/or generating a so-called sensitivity matrix. This process tends to be complex and computationally intensive and, furthermore, does not take into account real system characteristics and fluctuations. Similarly, the inverse algorithms typically utilized in DOT also often take on a high computational volume and complexity, leading to long reconstruction times, and have limited accuracy depending on the measurements, forward model, and experimental system. The purpose of this dissertation is to address and develop computational methods, especially incorporating deep learning, to improve each of these components. First, I evaluated several time-domain data features involving the Mellin and Laplace transforms to incorporate measurements that were robust to noise and sensitive at depth for reconstruction. Furthermore, I developed a method to find the optimal values to use for different imaging depths and scenarios. Second, I developed a neural network that can directly learn the forward problem and sensitivity matrix for simulated and experimental measurements, which allows the computational forward model to adapt to the system's characteristics. Finally, I employed learning-based approaches based on the previous results to solve the inverse problem to recover the optical parameters in a high-speed manner. Each of these components were validated and tested with numerical simulations, phantom experiments, and a variety of in vivo data. Altogether, the results presented in this dissertation depict how these computational approaches lead to an improvement in DOT reconstruction quality, speed, and versatility. It is the ultimate hope that these methods, algorithms, and frameworks developed as a part of this dissertation can be directly used on future data to further validate the research presented here and to further validate DOT as a valuable imaging tool across many applications.

[pt] Ao mesmo tempo em que a autonomia de robôs pessoais e domésticos aumenta, cresce a necessidade de interação dos mesmos com o ambiente. A interação mais básica de um robô com o ambiente é feita pela percepção deste e sua navegação. Para uma série de aplicações não é prático prover modelos geométricos válidos do ambiente a um robô antes de seu uso. O robô necessita, então, criar estes modelos enquanto se movimenta e percebe o meio em que está inserido através de sensores. Ao mesmo tempo é necessário minimizar a complexidade requerida quanto a hardware e sensores utilizados. No presente trabalho, um algoritmo iterativo baseado em entropia é proposto para planejar uma estratégia de exploração visual, permitindo a construção eficaz de um modelo em grafo do ambiente. O algoritmo se baseia na determinação da informação presente em sub-regiões de uma imagem panorâmica 2-D da localização atual do robô obtida com uma câmera fixa sobre o mesmo. Utilizando a métrica de entropia baseada na Teoria da Informação de Shannon, o algoritmo determina nós potenciais para os quais deve se prosseguir a exploração. Através de procedimento de Visual Tracking, em conjunto com a técnica SIFT (Scale Invariant Feature Transform), o algoritmo auxilia a navegação do robô para cada nó novo, onde o processo é repetido. Um procedimento baseado em transformações invariáveis a determinadas variações espaciais (desenvolvidas a partir de Fourier e Mellin) é utilizado para auxiliar o processo de guiar o robô para nós já conhecidos. Também é proposto um método baseado na técnica SIFT. Os processos relativos à obtenção de imagens, avaliação, criação do grafo, e prosseguimento dos passos citados continua até que o robô tenha mapeado o ambiente com nível pré-especificado de detalhes. O conjunto de nós e imagens obtidos são combinados de modo a se criar um modelo em grafo do ambiente. Seguindo os caminhos, nó a nó, um robô pode navegar pelo ambiente já explorado. O método é particularmente adequado para ambientes planos. As componentes do algoritmo proposto foram desenvolvidas e testadas no presente trabalho. Resultados experimentais mostrando a eficácia dos métodos propostos são apresentados. / [en] As the autonomy of personal service robotic systems increases so has their need to interact with their environment. The most basic interaction a robotic agent may have with its environment is to sense and navigate through it. For many applications it is not usually practical to provide robots in advance with valid geometric models of their environment. The robot will need to create these models by moving around and sensing the environment, while minimizing the complexity of the required sensing hardware. This work proposes an entropy-based iterative algorithm to plan the robot´s visual exploration strategy, enabling it to most efficiently build a graph model of its environment. The algorithm is based on determining the information present in sub-regions of a 2- D panoramic image of the environment from the robot´s current location using a single camera fixed on the mobile robot. Using a metric based on Shannon s information theory, the algorithm determines potential locations of nodes from which to further image the environment. Using a Visual Tracking process based on SIFT (Scale Invariant Feature Transform), the algorithm helps navigate the robot to each new node, where the imaging process is repeated. An invariant transform (based on Fourier and Mellin) and tracking process is used to guide the robot back to a previous node. Also, an SIFT based method is proposed to accomplish such task. This imaging, evaluation, branching and retracing its steps continues until the robot has mapped the environment to a pre-specified level of detail. The set of nodes and the images taken at each node are combined into a graph to model the environment. By tracing its path from node to node, a service robot can navigate around its environment. This method is particularly well suited for flat-floored environments. The components of the proposed algorithm were developed and tested. Experimental results show the effectiveness of the proposed methods.

La première partie de la thèse réunit des résultats classiques sur l’espace de Hardy, les espaces modèles et l’espace de Bergman. Puis sur cette base, nous exposons des travaux relatifs aux opérateurs de Toeplitz, en particulier, nous présentons la description du spectre et du spectre essentiel de ces opérateurs sur l’espace de Hardy et de Bergman. La première partie de notre recherche tire son inspiration de deux faits établis pour un opérateur de Toeplitz T. Premièrement, sur l’espace de Hardy, la norme de T, la norme essentielle de T et la norme infinie du symbole de T sont égales. Nous étudions ce cas d’égalité sur l’espace de Bergman pour les opérateurs de Toeplitz à symbole quasihomogène et radial. Deuxièmement, sur l’espace de hardy, le spectre et le spectre essentiel sont fortement liés à l’image du symbole de T. Nous étudions le cas d’égalité entre le spectre et l’image essentielle du symbole pour les symboles quasihomogènes et radials. Pour répondre à ces deux questions, nous utilisons la transformée de Berezin, les coefficients de Mellin et la moyenne du symbole. La dernière partie de la thèse s’interesse au théorème de Szegö qui donne un lien entre les valeurs propres d’une suite de matrices de Toeplitz de taille n, et le symbole de cette suite de matrice. Nous donnons un résultat du même type sur l’espace de Bergman pour les symboles harmoniques sur le disque et continus sur le cercle. Enfin, nous étudions une généralisation de ce théorème en compressant l’opérateur de Toeplitz sur une suite d’espaces modèles de dimension finie. / This thesis deals with the spectral properties of the Toeplitz operators in relation to their associated symbol. In the first part, we give some classical results about Hardy space, model spaces and Bergman space. Afterwards, we expose some results about Toeplitz operator on the Hardy space. In particular, we discuss their spectrum and essential spectrum. Our work is inspired from two facts which have been proved on the Hardy space. First, considering a Toeplitz operator T, the norm, essential norm, spectral radius of T and the supremum of its symbol are equal. Secondly, on the Hardy space, spectrum, essential spectrum and essential range are strongly related. We answer the question of the equality between the norms, the spectral radius and the supremum of the symbol and between spectrum and essential range on the Bergman space. We look at these two properties on the Bergman space when the symbol is radial or quasihomogeneous. We answer these questions using the Berezin transform, the Mellin coefficients and the mean value of the symbol. The last part deals with the classical Szegö theorem which underline a link between the eigenvalues of a Toeplitz matrix sequence and its symbol. We give a result of the same type on Bergman space considering harmonic symbol wich have a continuous extension. We give a generalization, considering the sequence of the compressions of a Toeplitz operator on a sequence of model spaces.

Dans cette thèse on étudie la régularité de problèmes elliptiques (Laplace, Helmholtz) ou paraboliques (équation de la chaleur) avec donnée mesure dans divers cadres géométriques. Ainsi, on considère pour les seconds membres des masses de Dirac en un point, sur une ligne infinie, semi-infinie ou finie, et également sur une courbe régulière. Les solutions de ces problèmes étant singulières sur la fracture (modélisée par la masse de Dirac dans le second membre), on étudie la régularité dans des espaces de Sobolev avec poids. Dans le cas d'une fracture droite, on utilise une technique classique qui consiste à appliquer une transformée de Fourier ou de Mellin à l'équation de Laplace. Ceci nous amène à étudier l'équation de Helmholtz en 2D. Pour ce dernier, on montre des estimations uniformes qui permettent ensuite de prendre la transformée inverse et d'obtenir le résultat de régularité attendu. De même, la transformée de Laplace transforme l'équation de la chaleur dans la même équation de Helmholtz en 2D. Dans le cas d'une fracture courbe régulière, grâce aux résultats de [D'angelo:2012], en utilisant un argument de localisation et un recouvrement dyadique, on obtient une régularité améliorée de la solution toujours dans les espaces de Sobolev avec poids. / In this thesis, we study the regularity of elliptic problems (Laplace, Helmholtz) or parabolic problems (heat equation) with measure data in different geometric frames. Thus, we consider for the second members, Dirac masses at a point, on a line, on a half-line, or on a bounded segment, and also on a regular curve.  As the solutions of these problems are singular on the fracture (modeled by Dirac mass in the second member), we study their regularity in weighted Sobolev spaces.   In the case of a straight fracture, using Fourier or Mellin technique reduces the problem in dimension three to a Helmholtz problem in dimension two. For the latter, we prove uniform estimates, which are then used to apply the inverse transform and to obtain the expected regularity result. Similarly, the Laplace transformation transforms the heat equation into the same Helmholtz equation in 2D.  In the case of a smooth curve fracture, thanks to the results of [D'angelo:2012], using a localization argument and a dyadic recovery we get an improved smoothness of the solution always in weighted Sobolev spaces.