We develop a numerical method for solving singular differential equations and demonstrate the method on a variety of singular problems including first order ordinary differential equations, second order ordinary differential equations which have variational principles, and one partial differential equation.

The method of steepest descent is used to minimize typical functionals from elasticity.

This paper is a study of continue and related metric spaces, Chapter I is an introductory chapter. Irreducible continua and noncut points are the main topics in Chapter II. The third chapter begins with a few results on locally connected spaces. These results are then used to prove results in locally connected continua. Decomposable and indecomposable continua are dealt with in Chapter IV. Totally disconnected metric spaces are studied in the beginning of Chapter V. Then we see that every compact metric space is a continuous image of the Cantor set. A continuous map from the Cantor set onto [0,1] is constructed. Also, a continuous map from [0,1] onto [0,1]x[0,1] is built, Then an order preserving homeomorphism is constructed from a metric arc onto [0,1],

This paper is a study of the relation between smoothness of the norm on a normed linear space and the property that every Chebyshev subset is convex. Every normed linear space of finite dimension, having a smooth norm, has the property that every Chebyshev subset is convex. In the second chapter two properties of the norm, uniform Gateaux differentiability and uniform Frechet differentiability where the latter implies the former, are given and are shown to be equivalent to smoothness of the norm in spaces of finite dimension. In the third chapter it is shown that every reflexive normed linear space having a uniformly Gateaux differentiable norm has the property that every weakly closed Chebyshev subset, with non-empty weak interior that is norm-wise dense in the subset, is convex.

Primary objective of the thesis is proof of the statement that if for ∈ ℕ a ≥ 1 a bilipschitz mapping belongs to +1, loc ∩ ,∞ loc then also its inverse −1 belongs to +1, loc . We prove a similar statement also for spaces loc . For this purpose we construct a new ordering of -th partial derivatives to generalized Jacobian matrix. Thanks to this matrix we are able to differentiate matrices in an applicable way. Generalized Jacobian matrix is projected so that there still holds the Chain rule and, in some way, also rules for matrices product differentiation. 1

Doctor of Philosophy / Department of Mathematics / Diego Maldonado / Originally introduced in 1961 by Carl Gustav Axel Harnack [36] in the context of harmonic functions in R[superscript]2, the so-called Harnack inequality has since been established for solutions to a wide variety of different partial differential equations (PDEs) by mathematicians at different times of its historical development. Among them, Moser's iterative scheme [47-49] and Krylov-Safonov's probabilistic method [43, 44] stand out as pioneering theories, both in terms of their originality and their impact on the study of regularity of solutions to PDEs. Caffarelli's work [12] in 1989 greatly simplified Krylov-Safonov's theory and established Harnack's inequality in the context of fully non-linear elliptic PDEs. In this scenario, Caffarelli and Gutierrez's study of the linearized Monge-Ampere equation [15, 16] in 2002-2003 served as a motivation for axiomatizations of Krylov-Safonov-Caffarelli theory [3, 25, 57]. The main work in this dissertation is a new axiomatization of Krylov-Safonov-Caffarelli theory. Our axiomatic approach to Harnack's inequality in spaces of homogeneous type has some distinctive features. It sheds more light onto the role of the so-called critical density property, a property which is at the heart of the techniques developed by Krylov and Safonov. Our structural assumptions become more natural, and thus, our theory better suited, in the context of variational PDEs. We base our method on the theory of Muckenhoupt's A[subscript]p weights. The dissertation also gives an application of our axiomatic approach to Harnack's inequality in the context of infinite graphs. We provide an alternate proof of Harnack's inequality for harmonic functions on graphs originally proved in [21].

Thesis is submitted in partial fulfilment for the degree of Master of Architecture (Professional) to the Faculty of Engineering and the Built Environment, School of Architecture and Planning at the University of the Witwatersrand, Johannesburg, 2017 / ABSTRACT The aim of this dissertation is to explore alternative ways of looking at architecture through the use of theory, the type of theory, alternative building programme development, representing architecture and how architecture is implemented. By doing so, an intervention is designed to expose various social truths, stimulating self reflection and adding value to the Alexandra-Sandton corridor context. This research project utilises the spirit of carnivals as subversive and radical events to change a community’s behaviour. This dissertation explores Bakhtin’s theory of the “carnivalesque”. This theory was used as the theoretical framework based on four characteristics. Throughout the research process these are used to analyse site context and create an intervention. The four carnivalesque characteristics are: - Usurping of hierarchies; - Pushing taboos; - Unusual connections; - Eccentric behaviour. The social inequalities along the Alexandra-Sandton corridor are broken down into every day activities and juxtaposed to amplify and expose hidden rules that we have come to accept in Johannesburg. The intervention acts as a commentary on the future connection of the two contrasting communities for spectacle and self reflection, transcending the everyday experience into a surreal playground through virtual reality and other means. Architecturally, Social Fiction has three main design strands 1. Theoretical exploration; 2. Architecture as emotional stimulus; 3. Virtual reality as fictional representation. Social Fiction is a project that bridges architecture, politics, socioeconomics and philosophy, using the medium of virtual reality and comic book fantasy as an open and accessible way, challenging the traditional plan, section elevation as a means of communication. / GR2017

Neste trabalho, alguns espaços de funções que surgem naturalmente no contexto da topologia geral são estudados. Por meio da noção de bornologias, problemas de Cp-teoria e Ck-teoria são analisados simultaneamente, como a caracterização de certas funções cardinais no espaço das funções contínuas e propriedades relativas a jogos seletivos. A compactificação de Stone- Cech também é estudada, onde a existência de P-pontos no resíduo dos naturais é considerada sob a Hipótese do Contínuo. Com a adição de certas hipóteses sobre pequenos cardinais, alguns resultados obtidos ao longo do texto são utilizados em problemas relacionados com espaços de Michael e espaços de Alster / In this work, some function spaces that naturally arise in the general topology context are studied. By the notion of bornologies, Cp-theory and Ck-theory problems are simultaneously analysed, as the characterization of some cardinal functions in the space of continuous real functions and properties related to selective games. The Stone-Cech compactification is also studied, and the existence of P-points in the remainder of the set of the natural numbers is considered under the Continuum Hypothesis. With the addition of some hypotheses about small cardinals, some results obtained through the text are used in problems related to Michael spaces and Alster spaces

Nesta pesquisa o objetivo é trazer à tona a discussão sobre a acessibilidade universal, no espaço livre projetado. Buscando evidenciar dificuldades encontradas não só por pessoas com deficiências, mas também por toda a sociedade, no uso do território. Focando na sensibilização dos profissionais, especialmente arquitetos paisagistas. Este estudo busca discutir aspectos da concepção de projetos que, embora existam legislações e normas que tratem da questão de forma bastante ampla, a prática da construção da cidade \" real\" não garante os direitos adquiridos legalmente. Os espaços livres de uso público são locais onde se desenvolve a cidadania e têm fundamental importância no processo de inclusão social. Alguns espaços construídos, embora de acordo com a norma (NBR 9050 - Acessibilidade a Edificações, Mobiliário, Espaços e Equipamentos Urbanos), muitas vezes, não atende à pessoa com deficiência (PCD), porque o meio externo não é acessível. / In this research the purpose is to bring to light the discussion about universal accessibility on the designed free space, by trying to make evident the difficulties found not only by the disabled, but also by the entire society. It focuses on the professional, especially architects and landscape architects. This study tries to argue different aspects of the project conception which, even though there are laws and norms that handle the issue in an ample way, the practice of the \"real\" city construction does not guarantee the rights acquired legally. The public free spaces are spaces where the citizenship is developed and are really important in the process of social inclusion. Some of the built spaces sometimes doesn´t consider the disabled, although they follow the law (NBR 9050), because the exterior is not accessible.

Neste trabalho aplicamos técnicas de combinatória infinitária e forcing na teoria dos espaços de Banach, investigando propriedades dos espaços de Banach da forma C(K), formado pelas funções reais contínuas sobre K com a norma do supremo, com poucos operadores, no sentido de que todo operador em C(K) é da forma gI+S, onde I é o operador identidade, g pertence a C(K) e S é fracamente compacto. Enfatizamos as construções onde K é conexo, o que implica que C(K) é indecomponível. Assumindo Axioma Diamante, um axioma combinatório mais forte que a Hipótese do Contínuo, construímos um espaço de Banach C(K) tal que C(L) tem poucos operadores, para todo L subespaço fechado de K. Sob a Hipótese do Contínuo construímos um espaço C(K) indecomponível com poucos operadores tal que K contém $\\beta N$ homeomorficamente. Em ZFC construímos um espaço C(K) com poucos operadores em um sentido estritamente mais fraco. Também mostramos a existência de pelo menos contínuo espaços de Banach C(K) indecomponíveis dois a dois essencialmente incomparáveis. Usando forcing provamos que existe consistentemente um espaço de Banach C(K) de densidade menor que contínuo com poucos operadores e um C(K) indecomponível de densidade menor que contínuo. / In this work we apply techniques of infinitary combinatorics and forcing in Banach spaces theory, investigating the compact topological spaces K such that the Banach space C(K), consisting of the continuous real-valued functions on K with the supremum norm, has few operators, in the sense that all operators on C(K) have the form gI+S, where I is the identity operator, g\\ belongs to C(K) and S is weakly compact. We emphasize the constructions where K is connected, which implies that C(K) is indecomposable. Assuming Diamond Axiom, a combinatoric axiom stronger than the continuum hypothesis, we construct a Banach space C(K) where C(L) has few operators, for every L closed subspace of K. Under continuum hypothesis we construct an indecomposable C(K) with few operators such that K contains $\\beta \\mathbb$ homeomorphically. In ZFC we construct a space C(K) with few operators in a strictly weaker sense. We also show the existence of at least continuum pairwise essentially incomparable indecomposable Banach spaces C(K). Using forcing, we prove that there exists consistently a Banach space C(K) of density smaller than continuum having few operators and an indecomposable C(K) of density smaller than continuum.