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

Orientador: Vanderlei Minori Horita / Banca: Ali Messaoudi / Banca: Marcio Ricardo Alves Gouveia / Banca: Thiago Aparecido Catalan / Banca: Nivaldo Costa Muniz / Resumo: 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 / Abstract: 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 / Doutor

Orientadores: Alcibiades Rigas, Carlos Eduardo Duran Fernandez / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-20T13:56:12Z (GMT). No. of bitstreams: 1 Speranca_LlohamDallagnol_D.pdf: 994466 bytes, checksum: 472919d7eec0f563b673a0307450dc49 (MD5) Previous issue date: 2012 / Resumo: Nesse trabalho estudaremos a geometria e a topologia de algumas variedades homeomorfas, porém não difeomorfas, à esfera padrão Sn, chamadas esferas exóticas. Realizaremos duas dessas variedades como quocientes isométricos de fibrados principais com métricas de conexão sobre esferas de curvatura constante. Através disso, apresentaremos simetrias desses espaços e exemplos explícitos de difeomorfismos não isotópicos a identidade, usando-os para o cálculo de grupos de homotopia equivariante. Como mais uma aplicação dessa construção, provaremos que, se uma esfera homotópica de dimensão 15 é realizável como um fibrado linear sobre S8, então a mesma esfera é realizável como um fibrado linear sobre a esfera exótica de dimensão 8 com as mesmas funções de transição. No ultimo capítulo lidaremos com a geometria de fibrados induzidos, deduzindo uma condição necessária sobre a função indutora para que a métrica da conexão induzida tenha curvatura seccional não-negativa / Abstract: In this work we study the geometry and topology of manifolds homemorphic, but not diffeomorphic, to the standard sphere Sn, the so called exotic spheres. We realize two of these manifolds as isometric quotients of principal bundles with connection metrics over the constant curved sphere. Through this, we present symmetries in these spaces and explicit examples of diffeomorphisms not isotopic to the identity, using them for the calculation of equivariant homotopy groups. As another application, we prove that, if a homotopy 15-sphere is realizeble as the total space of a linear bundle over the standard 8-sphere, then, it is realizeble as the total space of a linear bundle over the exotic 8-sphere with the same transition maps. In the last chapter we deal with the geometry of pull-back bundles, deducing a necessary condition on the pull-back map for nonnegative curvature of the induced connection metric / Doutorado / Matematica / Doutor em Matemática

Indiana University-Purdue University Indianapolis (IUPUI) / In 1970, Serge Novikov made a statement which is now called, "The Novikov Conjecture" and is considered to be one of the major open problems in topology. This statement was motivated by the endeavor to understand manifolds of arbitrary dimensions by relating the surgery map with the homology of the fundamental group of the manifold, which becomes diffi cult for manifolds of dimension greater than two. The Novikov Conjecture is interesting because it comes up in problems in many different branches of mathematics like algebra, analysis, K-theory, differential geometry, operator algebras and representation theory. Yu later proved the Novikov Conjecture holds for all closed manifolds with discrete fundamental groups that are coarsely embeddable into a Hilbert space. The class of groups that are uniformly embeddable into Hilbert Spaces includes groups of Property A which were introduced by Yu. In fact, Property A is generally a property of metric spaces and is stable under quasi-isometry. In this thesis, a new version of Yu's Property A in the case of locally compact groups is introduced. This new notion of Property A coincides with Yu's Property A in the case of discrete groups, but is different in the case of general locally compact groups. In particular, Gromov's locally compact hyperbolic groups is of Property A.

Indiana University-Purdue University Indianapolis (IUPUI) / Let f:X\rightarrow X be a dominant meromorphic self-map, where X is a compact, connected complex manifold of dimension n > 1. Suppose there is an embedded copy of \mathbb P^1 that is invariant under f, with f holomorphic and transversally superattracting with degree a in some neighborhood. Suppose also that f restricted to this line is given by z\rightarrow z^b, with resulting invariant circle S. We prove that if a ≥ b, then the local stable manifold W^s_loc(S) is real analytic. In fact, we state and prove a suitable localized version that can be useful in wider contexts. We then show that the condition a ≥ b cannot be relaxed without adding additional hypotheses by resenting two examples with a < b for which W^s_loc(S) is not real analytic in the neighborhood of any point.

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[pt] Esse trabalho tem como objetivo reunir os teoremas topológicos de ponto fixo clássicos e seus corolários, além de teoremas de ponto fixo provenientes da teoria do grau e algumas importantes aplicações desses teoremas a variadas áreas - desde as clássicas aplicações à teoria de EDOs e EDPs à uma aplicação à teoria dos jogos. Um exemplo é o Teorema do Ponto Fixo de Schauder-Tychonoff, para aplicações compactas em convexos de espaços localmente convexos, do qual segue como corolário que todo compacto convexo de um espaço vetorial normado (não necessariamente de dimensão finita) possui a propriedade do ponto fixo. No que se refere à teoria dos jogos em particular, foi deduzido o Teorema de Nash, que determina condições sobre as quais certos jogos possuem equilíbrios nos seus espaços das estratégias. Toda a topologia geral necessária nas demonstrações foi desenvolvida extensiva e detalhadamente a partir de topologia elementar, seguindo algumas das referências bibliográficas. O Teorema de Extensão de Dugundji - uma extensão do Teorema de Extensão de Tietze a fechados de espaços métricos sobre espaços localmente convexos -, por exemplo, é demonstrado com detalhes e usado diversas vezes ao longo da dissertação. / [en] The goal of the present work is to gather the classical fixed-point theorems and their corollaries, as well as other fixed-point theorems arising from degree theory, and some important applications to diverse fields - from the classical applications to ODEs and PDEs to an application to the game theory. An example is the Schauder-Tychonoff Fixed-Point Theorem, 1 concerning compact mappings in convex subsets of locally convex spaces, from which it follows as a corollary that every compact convex subset of a normed vector space is a fixed-point space. In regard to game theory in particular, we obtained Nash s theorem, 2 which ascertains conditions over which certain games have equilibria in their strategy spaces. All general topology necessary in the proofs was developed extensively and in details from a basic topology starting point, following some of the bibliographic references. Dugundji s Extension Theorem 3 - an extension of Tietze s Extension Theorem 4 for closed subsets of metric spaces into locally convex spaces-, for instance, is obtained with detais and used throughout the dissertation.