The primary objects of study in this thesis are noncommutative surfaces; that is, noncommutative noetherian domains of GK dimension 2. Frequently these rings will also be singular, in the sense that they have infinite global dimension. Very little is known about singularities of noncommutative rings, particularly those which are not finite over their centre. In this thesis, we are able to give a precise description of the singularities of a few families of examples. In many examples, we lay the foundations of noncommutative singularity theory by giving a precise description of the singularities of the fundamental examples of noncommutative surfaces. We draw comparisons with the fundamental examples of commutative surface singularities, called Kleinian singularities, which arise from the action of a finite subgroup of SL(2; k) acting on a polynomial ring. The main tool we use to study the singularities of noncommutative surfaces is the singularity category, first introduced by Buchweitz in [Buc86]. This takes a (possibly noncommutative) ring R and produces a triangulated category Dsg(R) which provides a measure of "how singular" R is. Roughly speaking, the size of this category reflects how bad the singularity is; in particular, Dsg(R) is trivial if and only if R has finite global dimension. In [CBH98], Crawley-Boevey-Holland introduced a family of noncommutative rings which can be thought of as deformations of the coordinate ring of a Kleinian singularity. We give a precise description of the singularity categories of these deformations, and show that their singularities can be thought of as unions of (commutative) Kleinian singularities. In particular, our results show that deforming a singularity in this setting makes it no worse. Another family of noncommutative surfaces were introduced by Rogalski-Sierra-Stafford in [RSS15b]. The authors showed that these rings share a number of ring-theoretic properties with deformations of type A Kleinian singularities. We apply our techniques to show that the "least singular" example has an A1 singularity, and conjecture that other examples exhibit similar behaviour. In [CKWZ16a], Chan-Kirkman-Walton-Zhang gave a definition for a quantum version of Kleinian singularities. These require the data of a two-dimensional AS regular algebra A and a finite group G acting on A with trivial homological determinant. We extend a number of results in [CBH98] to the setting of quantum Kleinian singularities. More precisely, we show that one can construct deformations of the skew group rings A#G and the invariant rings AG, and then determine some of their ring-theoretic properties. These results allow us to give a precise description of the singularity categories of quantum Kleinian singularities, which often have very different behaviour to their non-quantum analogues.

The non-linear Hartree-Fock (HF) equations are usually solved via the iterative self-consistent field method. However, there is no a priori guarantee of convergence, especially in systems with strong electron correlation where symmetry breaking occurs. This work focuses on closed- shell systems in the HF approximation and the (in)stability of the found solutions, and proposes new deterministic methods for the localization of both symmetry-adapted and broken symmetry solutions. We employ a perturbative method and show how one can always obtain a symmetry-adapted solution of the HF equations. We also determine the radius of convergence, related to the existence of at least one bound state, of the perturbative series. Next, we rederive the matrix of stability and adapt it to spin and orbital symmetry. Calculation of higher energy variations follows, first in terms of spin-orbitals and then orbitals. Motivated by the investigation of the structure of a broken-symmetry solution, we propose a new deterministic method for the localization of a broken-symmetry solution. The general expressions are verified by reformulating the stability conditions for simple cases. The proposed methods are successfully applied to helium-, beryllium- and neon-like systems.

No description available.

The advent of modern computer technologies provided a powerful tool in numerical simulations. One of the most frequently used method for the discretization of the physical domain is Finite element Method (FEM). One of the main problems in a finite element analysis is the adequacy of the finite element mesh. Since the quality of the finite element solution directly depends on the quality of meshes, the additional process to improve the quality of meshes is necessary for reliable finite element approximation. In order to perform quality-assessed numerical simulation, the adaptive finite element strategies have been developed. These strategies integrate the finite element analysis with error estimation and fully automatic mesh modification, user interaction with this process is limited by initial geometry data and possible error tolerance definition. The finite element solution, obtained during adaptive finite element strategy process, approximates quite good different engineering structures. Despite many works in this area the problem of the adequate finite element mesh is not fully solved and additional developments are needed in order to improve adaptive mesh refinement strategy process. The aim and tasks of the work – to obtain methodology for quality assessed discretization to finite elements for complex geometry engineering structures by adaptive finite element strategies. To realize this purpose the following scientific tasks have been dealt: to develop an automatic... [to full text]

The advent of modern computer technologies provided a powerful tool in numerical simulations. One of the most frequently used method for the discretization of the physical domain is Finite element Method (FEM). One of the main problems in a finite element analysis is the adequacy of the finite element mesh. Since the quality of the finite element solution directly depends on the quality of meshes, the additional process to improve the quality of meshes is necessary for reliable finite element approximation. In order to perform quality-assessed numerical simulation, the adaptive finite element strategies have been developed. These strategies integrate the finite element analysis with error estimation and fully automatic mesh modification, user interaction with this process is limited by initial geometry data and possible error tolerance definition. The finite element solution, obtained during adaptive finite element strategy process, approximates quite good different engineering structures. Despite many works in this area the problem of the adequate finite element mesh is not fully solved and additional developments are needed in order to improve adaptive mesh refinement strategy process. The aim and tasks of the work – to obtain methodology for quality assessed discretization to finite elements for complex geometry engineering structures by adaptive finite element strategies. To realize this purpose the following scientific tasks have been dealt: to develop an automatic... [to full text]

Zur Bestimmung singulärer Punkte eines bestimmten Typs muss eine zugehörige reduzierte Funktion und deren Ableitungen bestimmte Bedingungen erfüllen. Dabei ist diese reduzierte Funktion implizit durch ein nichtlineares Gleichungssystem definiert. Man erhält letztendlich ein minimal erweitertes System, das auch Ableitungen der reduzierten Funktion enthält,und den singulären Punkt als reguläre Lösung besitzt. In der vorliegenden Arbeit wird die Technik des automatischen Differenzierens für die Vorwärtsmethode dargestellt, insbesondere wird die Differentiation iterativer Verfahren untersucht. Es wird ein Überblick über die Theorie von singulären Punkten gegeben und das Erkennungsproblem definiert. Ein zweistufiges Verfahren zur Bestimmung singulärer Punkte wird auf Basis der Vorwärtsmethode und des Newton-Verfahrens beschrieben und wurde an verschiedenen Typen von singulären Punkten getestet.

Orientador: Edmea Cassia Baptista / Resumo: O problema de Despacho Econômico com Ponto de Válvula é um importante problema relacionado aos Sistemas Elétricos de Potência, que pode ser formulado como um problema de otimização não linear, não convexo e não diferenciável, o que dificulta sua resolução através de métodos exatos. Pode-se observar na literatura que diversos métodos heurísticos são propostos para a resolução do mesmo, os quais são eficientes e com um baixo custo computacional. Uma das desvantagens desses métodos é o tamanho do espaço de busca para realizer tais testes. Pesquisas realizadas apontam que, na grande maioria das vezes, os pontos ótimos para o problema de Despacho Econômico com Ponto de Válvula se encontram em pontos nos quais a função modular, presente na formulação do problema, possui valor nulo, ou estão na região destes e tais pontos são denominados de Pontos Singulares. Neste trabalho, com o bjetivo de propor um método heurístico com espaço de busca reduzido, é proposto um método de Busca Tabu direcionada a Pontos Singulares, o qual utiliza o método de Busta Tabu para percorrer os pontos nos quais a função modular se anula. O método se mostra eficiente para problemas de DEPV de 3, 13 e 40 geradores, com valores próximos aos valores ótimos obtidos por métodos determinísticos e com baixo custo computacional. / Abstract: The problem of Economic Load Dispatch with Valve Point (EDVP) is an important problem related to Electric Power Systems, that can be formulated as a non-linear, non-convex and non-differentiable optimization problem, that difficults resolution through deterministic methods. We can observe in the literature that many heuristic methods are proposed for the resolution of the same, being efficient with a low computational cost. One of the advantages of this methods is the size of the search space necessary to perform the tests. Researches points out that, in most cases, the optimal points for the Economic Load Dispatch with Valve Point problem are at points where the modular function present in the problem formulation has zero value, or in the region thereof, these points are called Singular Points. In this work is proposed, with the objective to propose a heuristic method with the search space reducted, a Tabu Search Directed to Singular Point Search, which uses he tatbu search method to the points in which the modular function cancels out. The method is efficient for resolution of Economic Load Dispatch with Valve Point problems of 3, 13 and 40 generators unities, with values close to optimal obtained by deterministic methods values and low computational cost. / Mestre

Dado um sistema diferencial no plano, muito se questiona sobre o comportamento de suas soluções. Nas vizinhanças dos pontos singulares existem ferramentas que nos indicam o tipo e a estabilidade estrutural de cada um deles; são as chamadas formas normais. No entanto, o interesse vai mais além do conhecimento local das soluções em cada singularidade. Nesse trabalho apresentamos algumas ferramentas clássicas da teoria qualitativa das equações diferenciais ordinárias empregadas na investigação global dos campos de vetores polinomiais planares e as empregamos na investigação de duas famílias paramétricas de campos quadráticos encontradas no estudo dos campos com hipérboles invariantes. Dentre as ferramentas estudadas destacamos a classificação local das soluções em pontos singulares elementares e semi-elementares e a técnica de compactificação de Poincaré. / Given a planar differential system, many questions are raised about the behavior of their solutions. In the neighborhood of singular points there exist many tools which indicate their type and their structural stability; they are known as normal forms. However, the interest goes beyond the local behavior in the neighborhood of each singularity. In this dissertation we present some classical tools from the qualitative theory of ordinary differential equations which are usually applied to the global investigation of planar polinomial vector fields and we apply them to the investigation of two parametric families of quadratic fields from the study of the vector fields with invariant hyperbolas. Among the studied tools we highlight the local classification of the solutions around elementary and semi-elementary singular points and the technique known as Poincarés compactification.

Dado um sistema diferencial no plano, muito se questiona sobre o comportamento de suas soluções. Nas vizinhanças dos pontos singulares existem ferramentas que nos indicam o tipo e a estabilidade estrutural de cada um deles; são as chamadas formas normais. No entanto, o interesse vai mais além do conhecimento local das soluções em cada singularidade. Nesse trabalho apresentamos algumas ferramentas clássicas da teoria qualitativa das equações diferenciais ordinárias empregadas na investigação global dos campos de vetores polinomiais planares e as empregamos na investigação de duas famílias paramétricas de campos quadráticos encontradas no estudo dos campos com hipérboles invariantes. Dentre as ferramentas estudadas destacamos a classificação local das soluções em pontos singulares elementares e semi-elementares e a técnica de compactificação de Poincaré. / Given a planar differential system, many questions are raised about the behavior of their solutions. In the neighborhood of singular points there exist many tools which indicate their type and their structural stability; they are known as normal forms. However, the interest goes beyond the local behavior in the neighborhood of each singularity. In this dissertation we present some classical tools from the qualitative theory of ordinary differential equations which are usually applied to the global investigation of planar polinomial vector fields and we apply them to the investigation of two parametric families of quadratic fields from the study of the vector fields with invariant hyperbolas. Among the studied tools we highlight the local classification of the solutions around elementary and semi-elementary singular points and the technique known as Poincarés compactification.