Nesta tese foi introduzido o conceito de folheação Finsleriana singular, que generaliza ação Finsleriana, submersão Finsler e folheação Finsleriana. O primeiro resultado desta tese afirma que qualquer folheação Finsleriana singular sobre uma variedade Randers com data (h,W) é folheação Riemanniana singular com respeito a h e W é um campo folheado. Para obter este resultado provou-se um teorema de redução ao slice, que permite relacionar uma folheação Finsleriana singular com uma folheação Finsleriana singular em um espaço de Minkowski. O terceiro resultado garante a equifocalidade para as fibras regulares de uma submersão singular analítica que na parte regular é uma submersão Finsleriana. No transcurso do trabalho verificou-se propriedades relevantes das folheações Finslerianas singulares e a existência de vizinhanças tubulares Finslerianas, uma propriedade básica que não estava escrita na literatura. / In this thesis we introduce the concept of singular Finsler foliation, which generalizes the concepts of Finsler actions, Finsler submersions and Finsler foliations. In the first theorem we show that if F is a singular Finsler foliation on a Randers manifold (M,Z) with Zermelo data (h,W), then F is a singular Riemannian foliation on the Riemannian manifold (M, h). In order to prove this theorem we present in the second theorem a slice reduction that relates local singular Finsler foliations on Finsler manifolds with singular Finsler foliations on Minkowski spaces. Finally in the third result we prove the equifocality of the fibers of an analytic singular submersion that is a Finsler submersion in the regular part. Along the work we stress relevant properties on singular Finsler foliations and we also remark the existence of tubular neigborhoods in Finsler geometry, a basic result that we did not find properly written in the literature.

In this thesis we study certain singular Sturm-Liouville differential expressions from an operator theoretic point of view.In particular we are interested in expressions that involve strongly singular potentials as introduced by Gesztesy and Zinchenko.On the ODE side, analyzing these expressions involves the so-called $m$-functions, often generalized Nevanlinna functions, who encapsulate spectral information of the underlying problem.The aim of the two papers in this thesis is to further understanding on the operator theory side.In the first paper, we use a model for super singular perturbations to describe a family of induced self-adjoint realizations of a perturbed Schr\"o\-din\-ger operator, i.e., with a potential of the form $c/x^2 + q$ where $q$ is a perturbation.Following the unperturbed example of Kurasov and Luger, we find that the so-called $Q$-function appearing in this approach is in good agreement with the above named $m$-function.Furthermore, we show that the operator model can be chosen such that $Q \equiv m$.In the second paper, we present a negative result in this area, namely that the supersingular perturbations model cannot be used for all strongly singular potentials.For a potential with a stronger singularity at the origin, namely $1/x^4$, we discuss the asymptotic behaviour of the Weyl solution at zero.It turns out that this function cannot be regularized appropriately and the operator model breaks down.

Nesta tese foi introduzido o conceito de folheação Finsleriana singular, que generaliza ação Finsleriana, submersão Finsler e folheação Finsleriana. O primeiro resultado desta tese afirma que qualquer folheação Finsleriana singular sobre uma variedade Randers com data (h,W) é folheação Riemanniana singular com respeito a h e W é um campo folheado. Para obter este resultado provou-se um teorema de redução ao slice, que permite relacionar uma folheação Finsleriana singular com uma folheação Finsleriana singular em um espaço de Minkowski. O terceiro resultado garante a equifocalidade para as fibras regulares de uma submersão singular analítica que na parte regular é uma submersão Finsleriana. No transcurso do trabalho verificou-se propriedades relevantes das folheações Finslerianas singulares e a existência de vizinhanças tubulares Finslerianas, uma propriedade básica que não estava escrita na literatura. / In this thesis we introduce the concept of singular Finsler foliation, which generalizes the concepts of Finsler actions, Finsler submersions and Finsler foliations. In the first theorem we show that if F is a singular Finsler foliation on a Randers manifold (M,Z) with Zermelo data (h,W), then F is a singular Riemannian foliation on the Riemannian manifold (M, h). In order to prove this theorem we present in the second theorem a slice reduction that relates local singular Finsler foliations on Finsler manifolds with singular Finsler foliations on Minkowski spaces. Finally in the third result we prove the equifocality of the fibers of an analytic singular submersion that is a Finsler submersion in the regular part. Along the work we stress relevant properties on singular Finsler foliations and we also remark the existence of tubular neigborhoods in Finsler geometry, a basic result that we did not find properly written in the literature.

"It is well known that singularities occur when solving elliptic value problems with non-convex domains or when some part of the data or the coefficients of the PDE are not smooth. Such problems and correspondent singularities often arise in practice, for instance, in fracture mechanics, in the material science with heterogeneities, or when dealing with mixed boundary conditions. A great deal is known about the nature of the singularities, which arise in some of these problems. In this thesis, we consider the scalar transmission problems with straight interfaces and with cross points across coefficients and possibly on a non-convex region ($L$-shaped domain). It is known that only $H^{1+au}$ ($0 < au< 1$) regularity on the solution is obtained and therefore the use of finite element method with the piecewise linear continuous function space does not give optimal accuracy. In this thesis, we introduce a new algorithm which are second order accurate on the (weighted) $L_2$, first order accurate on the (weighted) $H_1$ norm and second order accurate for the Stress Intensive Factor (SIF). The new methods take advantage of Mortar techniques. The main feature of the proposed algorithms is that we use primal singular functions {it without} cutting-off functions. The old algorithms use cutting-off functions as a tool of satisfying boundary conditions. In algorithms proposed in this thesis, use instead Mortar finite element technique to match the boundary and interfaces conditions. In this thesis, we also consider non-matching meshes sizes for different coefficients. We note that a new Mortar Lagrange multiplier is required in order to obtain optimal consistence errors for transmission problems. The proposed algorithms are very appealing over other methods because they are very accurate, do not require complicated numerical quadratures or interpolations, it is simple to design PCGs, and it can be generalized to other PDEs and to higher order methods."

Chiu Ho Man Edward. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2005. / Includes bibliographical references (leaves 36-39). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.4 / Chapter 2 --- Some Preliminary Analysis --- p.11 / Chapter 3 --- An Auxiliary Linear problem --- p.16 / Chapter 4 --- Construction of natural constraint --- p.22 / Chapter 5 --- Energy computation for reduced energy functional --- p.26 / Chapter 6 --- Proof of Theorem 1.1 --- p.29 / Bibliography --- p.36

Ao, Weiwei. / "August 2010." / Thesis (M.Phil.)--Chinese University of Hong Kong, 2010. / Includes bibliographical references (leaves 92-97). / Abstracts in English and Chinese. / Abstract --- p.ii / Acknowledgement --- p.v / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Spikes on Single Line-Segments --- p.12 / Chapter 2.1 --- Ansatz and sketch of the proof --- p.12 / Chapter 2.2 --- Linear theory --- p.15 / Chapter 2.3 --- The non linear projected problem --- p.20 / Chapter 2.4 --- Projection of the error and proof of Theorem 1.0.1 --- p.24 / Chapter 3 --- The triple junction solutions --- p.33 / Chapter 3.1 --- Approximate solutions --- p.33 / Chapter 3.2 --- linear and nonlinear projected problem --- p.35 / Chapter 3.3 --- Error estimates and the proof of theorem 1.0.2 --- p.35 / Chapter 4 --- Layer concentrations in three-dimensional domain --- p.45 / Chapter 4.1 --- Preliminaries and setting up the problem --- p.45 / Chapter 4.1.1 --- A linear model problem --- p.45 / Chapter 4.1.2 --- Setting up the problem in suitable coordinates --- p.53 / Chapter 4.2 --- The gluing procedure --- p.62 / Chapter 4.3 --- The invertibility of L2 --- p.65 / Chapter 4.4 --- Solving the nonlinear projected problem --- p.67 / Chapter 4.5 --- Estimates of the projection against ∇w and Z --- p.72 / Chapter 4.5.1 --- estimates for the projection of the error --- p.73 / Chapter 4.5.2 --- projection of terms involving φ --- p.78 / Chapter 4.5.3 --- projection of errors on the boundary --- p.80 / Chapter 4.6 --- "The system for (f1, f2, e):proof of the theorem" --- p.81

Spherical Harmonics (SPHARM) and Singular Value Decomposition (SVD) utilize the orthogonal relations of its parameters to represent and process images. The process involve mapping of the image from its original parameter domain to a new domain where the processing is performed. This process induces distortion and smoothing is required. The image now mapped to the new parameter domain is descripted using SPHARM and SVD using one at a time. The least significant values for the SPHARM coefficients and singular values of SVD are truncated which induces compression in the reconstructed image keeping the memory allocation in view. In this thesis, we have applied SPHARM and SVD tools to represent and reconstruct an image. The image is first mapped to the unit sphere (a sphere with unit radius). The image gets distorted that is maximum at the north and south poles, for which smoothing is approached by leaving 0.15*π space blank at each pole where no mapping is done. Sampling is performed for the θ and φ parameters and the image is represented using spherical harmonics and its coefficients are calculated. The same is then repeated for the SVD and singular values are computed. Reconstruction is performed using the calculated parameters, but defined over some finite domain, which is done by truncating the SPHARM coefficients and the singular values inducing image compression. Results are formulated for the various truncation choices and analyzed and finally it is concluded that SPHARM is better as compared with SVD as compression tool as there is not much difference in the quality of the reconstructed image with both tools, though SVD seem better quality wise, but with much higher memory allocation than SPHARM.

Pines’ and Gottlieb’s Formulations for the gravitational potential provides expressions for the gravitational potential, U, and its derivatives in a co-ordinate system that produces non-singular values. This report summarizes the origin of the singularities due to the spherical co-ordinate system and a discussion of the methods by which the singularity produced by the conventional representation of the gravitational potential is removed by the implementations described in this report. / text

No description available.

Thesis (M.S.)--Virginia Polytechnic Institute and State University, 1988. / Vita. Abstract. Includes bibliographical references (leaves 154-155). Also available via the Internet.