The study of knot invariants is a large and active area of research in the field of knot theory. In the early 1990s, Russian mathematican Victor Vassiliev developed a series of numerical knot invariants, now known as Vassiliev invariants. These invariants have sparked a great deal of interest in the mathematical community, and it is conjectured that, together, they formulate a complete knot invariant. The computation of these invariants is largely algebraic, and unfortunately the values do not appear to describe any intrinsic properties of the knot. In this thesis, a geometric interpretation of the second Vassiliev invariant is provided by examining occurrances of five distinct points on the knot that lie on a common circle in the ambient space. This process is then extended to include an analysis of six-point cocircularities of knots as well. / Graduate

Thèse--Paris. / Bibliography: p. 110-112.

Thèse--Paris. / Bibliography: p. 110-112.

Thesis--Princeton University, 1966. / Includes bibliographical references.

Typescript. Includes vita and abstract. Includes bibliographical references (leaves 163-164) Also available online in Scholars' Bank; and in ProQuest, free to University of Oregon users.

Let n > 1. We prove a homological stability theorem for the diffeomorphism groups of (4n+1)-dimensional manifolds, with respect to forming the connected sum with (2n-1)-connected, (4n+1)-dimensional manifolds that are stably parallelizable. Our techniques involve the study of the action of the diffeomorphism group of a manifold M on the linking form associated to the homology groups of M. In order to study this action we construct a geometric model for the linking form using the intersections of embedded and immersed Z/k-manifolds. In addition to our main homological stability theorem, we prove several results regarding disjunction for embeddings and immersions of Z/k-manifolds that could be of independent interest.

Orientador: Carlos Eduardo Duran Fernandez / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-14T00:21:52Z (GMT). No. of bitstreams: 1 Speranca_LlohamDallagnol_M.pdf: 1442574 bytes, checksum: eec573204734790dd464f4ba0a79d6c5 (MD5) Previous issue date: 2009 / Resumo: Apresentaremos neste trabalho alguns dos modelos clássicos em geometria e topologia diferencial para algumas variedades diferenciáveis com o mesmo tipo homotópico de uma esfera. Em seguida apresentaremos construções mais recentes dos mesmos e algumas de suas propriedades. / Abstract: We show in this work some of the classical models in geometry and diferential topology for some diferentiable manifolds with the same homotopy type of the sphere. We follow with an exposition of recent work and some of its properties. / Mestrado / Topologia / Mestre em Matemática

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Desirability Functions continue to attract attention of scientists and researchers working in the area of multi-response optimization. There are many versions of such functions, differing mainly in formulations of individual and overall desirability functions. Derringer and Suich&rsquo / s desirability functions being used throughout this thesis are still the most preferred ones in practice and many other versions are derived from these. On the other hand, they have a drawback of containing nondifferentiable points and, hence, being nonsmooth. Current approaches to their optimization, which are based on derivative-free search techniques and modification of the functions by higher-degree polynomials, need to be diversified considering opportunities offered by modern nonlinear (global) optimization techniques and related softwares. A first motivation of this work is to develop a new efficient solution strategy for the maximization of overall desirability functions which comes out to be a nonsmooth composite constrained optimization problem by nonsmooth optimization methods. We observe that individual desirability functions used in practical computations are of mintype, a subclass of continuous selection functions. To reveal the mechanism that gives rise to a variation in the piecewise structure of desirability functions used in practice, we concentrate on a component-wise and generically piecewise min-type functions and, later on, max-type functions. It is our second motivation to analyze the structural and topological properties of desirability functions via piecewise max-type functions. In this thesis, we introduce adjusted desirability functions based on a reformulation of the individual desirability functions by a binary integer variable in order to deal with their piecewise definition. We define a constraint on the binary variable to obtain a continuous optimization problem of a nonlinear objective function including nondifferentiable points with the constraints of bounds for factors and responses. After describing the adjusted desirability functions on two well-known problems from the literature, we implement modified subgradient algorithm (MSG) in GAMS incorporating to CONOPT solver of GAMS software for solving the corresponding optimization problems. Moreover, BARON solver of GAMS is used to solve these optimization problems including adjusted desirability functions. Numerical applications with BARON show that this is a more efficient alternative solution strategy than the current desirability maximization approaches. We apply negative logarithm to the desirability functions and consider the properties of the resulting functions when they include more than one nondifferentiable point. With this approach we reveal the structure of the functions and employ the piecewise max-type functions as generalized desirability functions (GDFs). We introduce a suitable finite partitioning procedure of the individual functions over their compact and connected interval that yield our so-called GDFs. Hence, we construct GDFs with piecewise max-type functions which have efficient structural and topological properties. We present the structural stability, optimality and constraint qualification properties of GDFs using that of max-type functions. As a by-product of our GDF study, we develop a new method called two-stage (bilevel) approach for multi-objective optimization problems, based on a separation of the parameters: in y-space (optimization) and in x-space (representation). This approach is about calculating the factor variables corresponding to the ideal solutions of each individual functions in y, and then finding a set of compromised solutions in x by considering the convex hull of the ideal factors. This is an early attempt of a new multi-objective optimization method. Our first results show that global optimum of the overall problem may not be an element of the set of compromised solution. The overall problem in both x and y is extended to a new refined (disjunctive) generalized semi-infinite problem, herewith analyzing the stability and robustness properties of the objective function. In this course, we introduce the so-called robust optimization of desirability functions for the cases when response models contain uncertainty. Throughout this thesis, we give several modifications and extensions of the optimization problem of overall desirability functions.

Made available in DSpace on 2014-11-10T11:09:53Z (GMT). No. of bitstreams: 0 Previous issue date: 2014-08-04Bitstream added on 2014-11-10T11:57:47Z : No. of bitstreams: 1 000788978.pdf: 408960 bytes, checksum: 1cfcc35354bc50d787e5fb17deef1134 (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Estudamos a dimensão de Hausdorff de certos conjuntos invariantes obtidos como repulsores de aplicações com buraco. Neste trabalho relacionamos a dimensão de Hausdorff do repulsor ¤ com o volume do buraco H, em particular mostramos que a dimensão de Hausdorff de ¤ pode ser tomada arbitrariamente próxima da dimensão do ambiente desde que o volume de H seja suficientemente pequeno. Como aplicação dos resultados, mostramos que a dimensão de Hausdorff dos repulsores de uma família desdobrando uma bifurcaçãoo de Hopf varia continuamente no parâmetro de bifurcação / We study the Hausdorff dimension of certain invariant sets obtained as repeller of maps with hole. In this work we relate the Hausdorff dimension of the repeller ¤ with the volume of the hole H, in particular one show that the Hausdorff, dimension of ¤ can be made arbitrarily close to the dimension of ambient provided that the volume of H is suficiently small. As an application, we show that the Hausdorff dimension of repellers of the a family unfolding a Hof bifurcation varies continuously at the bifurcation parameter