In this thesis, some characterizations of continua of finite degree are given. It turns out that being of finite degree (by formal definition) can be described by saying there exists an equivalent metric in which Hausdorff linear measure of the continuum is finite. I discuss this result in detail.

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Nesta pesquisa o objetivo principal é avaliar a influência da estimativa para a compreensão do conceito de área de uma superfície plana. / In this research the main objective is to evaluate the influence of the estimation for the understanding of the concept of the area of a flat surface.

Orientador: Ricardo de Sá Teles / Resumo: Esta dissertação discorrerá sobre como as desigualdades são abordadas no ensino básico, com foco no ensino médio. Traremos à tona algumas desigualdades notórias seja por sua importância histórica, elegância ou importância dentro da Matemática. Com auxílio de tópicos de medida e integração deduziremos algumas desigualdades em sua forma mais geral e mostraremos aplicações das mesmas em conjuntos discretos de números reais em problemas diversos. / Abstract: This work will talk about how the inequalities are treat in basic school, with focus in high school. We will address some notorious inequalities, due for its historical importance, elegance or importance inside Mathematics. With topics of measure and integration, we deduce some inequalities in their more general form and we will show its applications in discrete sets of real numbers in several problems. / Mestre

Estudamos o efeito de uma tangência homoclínica na variação da entropia topológica. Provamos que um difeomorfismo com uma tangência homoclínica associada a uma peça básica com máxima entropia é um ponto de variação da entropia na topologia \'C POT. INFINITO\'. Além disso, discutimos o problema variacional na topologia \'C POT.1\' e apresentamos um exemplo de descontinuidade da entropia em dimensão três. Um resultado devido a Newhouse afirma que um difeomorfismo genérico sobre uma superfície com um conjunto homoclínico que contém uma tangência correspondente a um ponto periódico dissipativo, não pode ter medidas de SRB suportadas no conjunto homoclínico. Generalizamos este resultado para dimensões maiores, no caso em que a tangência homoclínica está associada com uma sela seccionalmente dissipativa / We study the effect of a homoclinic tangency in the variation of the topological entropy. We prove that a diffeomorphism with a homoclinic tangency associated to a basic hyperbolic set with maximal entropy is a point of entropy variation in the \'C POT THE INFINITE\' topology. We also discuss variational problem in \'C POT.1\' topology and we show an example of discontinuity of the entropy in dimension three. A result due to Newhouse states that a generic surface diffeomorphism with a homoclinic set containing a tangency associated to a dissipative periodic point, can not have SRB (Sinai-Ruelle-Bowen) measures supported on the homoclinic set. We generalize this result to higher dimensions, in the case where the homoclinic tangency is associated to a sectionally dissipative saddle

There are two main features in this thesis. The first is a study of the role of "tracts" of finite determination on the growth of integral functions. Such a study is of importance, since most of the known results in this field are based on the extreme case where the tracts of finite determination reduce to single lines. The second is the construction of functions, meromorphic and integral, bounded in tracts of positive angular measure, where the tracts are not confined to radial sectors.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1982. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE / Bibliography: leaves 106-109. / by Rae Michael Andrew Shortt. / Ph.D.

Este trabalho está preocupado com o conceito de medida e dimensão de Hausdorff usando ferramentas matemáticas adequadas. Como, frequentemente, é importante e difícil determinar a dimensão Hausdorff 1 de um conjunto e ainda mais difícil de encontrar ou mesmo estimar a sua medida Hausdorff, por auto proteção é usado o conjunto ternário de Cantor. A construção ternária simplifica certas dificuldades técnicas sobre a teoria da dimensão. O conjunto de Cantor é um exemplo interessante de um conjunto magro, perfeito, compacto e não enumerável, cuja medida e dimensão topológica são nulas. A análise de muitas das suas propriedades e consequências interessantes nos campos da teoria dos conjuntos e da topologia nos oferece uma rota direta que leva à medida Hausdorff do conjunto Cantor e sua dimensão fractal que é igual à sua dimensão Hausdorff. Também é calculada a dimensão Hausdorff para alguns fractais clássicos, como o tapete Sierpinski e a curva de flocos de neve von Koch. / This work is concerned with the concept of Hausdorff measure and dimension using suitable mathematical tools. Since it is often important and dificult to determine the Hausdorff dimension2 of a set and even harder to find or even to estimate its Hausdorff measure, by self-protection choices, it is used the ternary Cantor set. The ternary construction reduces technical difficulties about dimension theory. Cantor set is an interesting example of a meager, perfect, compact, uncountable set whose measure and topologic dimension are zero. Analysis of many of its interesting properties and consequences in the fields of set theory and topology provides a direct route that leads to the Hausdorff measure of the Cantor set and its fractal dimension that is equal to its Hausdorff dimension. It is also computed the Hausdorff dimension for some classical fractals such as the Sierpinski carpet and the von Koch snowflake curve.

The Group Questionnaire (GQ) was developed to measure group therapeutic processes-which are linked to successful prediction of patient outcome and therapeutic factors-across three qualitative dimensions (positive bond, positive work, and negative relationship) and three structural dimensions (member-leader, member-member, and member-group). The GQ model has been shown to be valid across 5 settings and 4 countries. As a clinical measure given after each session, length is of particular concern. Although shorter measures are more convenient for clients and therapists to use, fewer items necessarily means less information, a loss of psychometrics, and possible floor and ceiling effects. This study examined the effects of shortening the GQ on its clinical utility and psychometric integrity. Methods. Archival data from 7 previous studies was used, with 2,594 participants in an estimated 455 groups gathered from counseling centers, non-clinical process groups, inpatient psychiatric hospitals, outpatient psychiatric hospitals, and an inpatient state hospital. Participants answered questions from the Group Questionnaire administered during the productive working phase of a group. Analysis. Analysis was done using multilevel structural equation modeling in Mplus to account for the nested nature of groups. Items were selected using clinical judgment and statistical judgment considering inter item correlation and factor loading. Model fit was analyzed in comparison to the standards in the literature and in comparison to the full length GQ. Discussion. The revised 12 item GQ has good model fit and acceptable reliability. Further assessment is needed to determine how the reduction affects clinical utility.

<p>The Laplace equation and the related p-Laplace equation are closely associated with Sobolev spaces. During the last 15 years people have been exploring the possibility of solving partial differential equations in general metric spaces by generalizing the concept of Sobolev spaces. One such generalization is the Newtonian space where one uses upper gradients to compensate for the lack of a derivative.</p><p>All papers on this topic are written for an audience of fellow researchers and people with graduate level mathematical skills. In this thesis we give an introduction to the Newtonian spaces accessible also for senior undergraduate students with only basic knowledge of functional analysis. We also give an introduction to the tools needed to deal with the Newtonian spaces. This includes measure theory and curves in general metric spaces.</p><p>Many of the properties of ordinary Sobolev spaces also apply in the generalized setting of the Newtonian spaces. This thesis includes proofs of the fact that the Newtonian spaces are Banach spaces and that under mild additional assumptions Lipschitz functions are dense there. To make them more accessible, the proofs have been extended with comments and details previously omitted. Examples are given to illustrate new concepts.</p><p>This thesis also includes my own result on the capacity associated with Newtonian spaces. This is the theorem that if a set has p-capacity zero, then the capacity of that set is zero for all smaller values of p.</p>