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Orientador : Ary Orozimbo Chiacchio / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-01T00:45:56Z (GMT). No. of bitstreams: 1 Radin_LuceliaAparecida_M.pdf: 1620497 bytes, checksum: a50467e4398741fd1797e0571c2bef4c (MD5) Previous issue date: 2002 / Resumo: Neste trabalho, abordamos alguns tópicos de Teoria de Aproximação em espaços normados reais tais como a existência e unicidade do elemento de melhor aproximação e a continuidade da projeção métrica. Apresentamos exemplos e contra-exemplos de conjuntos de existência e de conjuntos de Chebyshev. Abordamos também, o problema da conexidade dos conjuntos de Chebyshev em espaços de Hilbert. Além disso, estudamos algumas classes de espaços com propriedades específicas, tais como convexidade uniforme e suavidade, para os quais é possível garantir uma resposta afirmativa para a seguinte questão: todo conjunto de Chebyshev é convexo? / Abstract: In this work we treat some topics of the Approximation Theory in the real normed spaces, such as the existence and uniqueness of the element of best approximation and the continuity of the metric projection. We present examples and counter-examples of sets of existence and Chebyshev sets. We also treat the problem of connectedness of Chebyshev sets in Hilbert spaces. Furthermore, we study some classes of spaces with specific geometrical properties such as uniform convexity and smootheness, for which is possible to guarantee an affirmative answer for the following question: is every set of Chebyshev convex? / Mestrado / Mestre em Matemática

Thesis (M.A.)--Boston University / PLEASE NOTE: Boston University Libraries did not receive an Authorization To Manage form for this thesis or dissertation. It is therefore not openly accessible, though it may be available by request. If you are the author or principal advisor of this work and would like to request open access for it, please contact us at open-help@bu.edu. Thank you. / The problem discussed in this paper is that of finding a best approximation to a given real-valued function f(x) over a continuum by means of finding a best approximation over a discrete set of points. We are also seeking to find a numerical method of finding a best approximation over our discrete set of points. A best approximation is one which minimizes the maximum deviation of our approximation from our given function f(x). We first discuss the concept of linear programming. In this paper we are not so much concerned with the theory behind linear programming as we are with the method to solve a linear programming problem, namely the simplex method. We discuss the simplex method from the point of view of a programmer, noting how results are continually updated until an optimal solution to t he problem is found. The only theoretical aspect of linear programming which we discuss is the notion of duals and the relationship between the solution of a primal and a dual problem. This becomes very important later in the paper when we try to formulate a best a proximat ion problem as a linear programming problem. Next we discuss the theoretical aspects of best approximation over a continuum. We prove existence, uniqueness, and, most important for our purposes, characterization. Our approximating functions are assumed to form a Chebyschev set throughout this paper. Finally we discuss best approximation over a discrete set of points. We first prove that the characterization theorem holds for problems of this type. Now that we have a way to tell whether our approximation is the best that can be obtained, we turn our attention to the relationship between the best approximation problem over a continuum and a discrete set of points. We prove in a quite general context that the best approximation over a discrete set of points converges uniformly to the solution to the problem over the continuum. We then retrace our steps and establish similar results for the particular case of polynomial approximation. After this we try to find out about the rate at which this convergence takes place. In general this question has no answer for its depends on the smoothness of the functions involved; if, however, we assume the fun ctions satisfy a Holder condition we may obtain some bounds on the rate of convergence. Finally, we reformulate the best approximation problem, showing how it can be considered as a linear programming problem which we already have a means of solving. / 2031-01-01

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Spiders weave intricate webs for catching prey, providing shelter and setting mating rituals; arguably the most notable of these creations is the orb web. In this thesis we model the essential vibrations of orb webs by first considering a spider web as a multilinked structure of elastic strings. We then solve the associated eigenvalue problem using a Chebyshev pseudospectral collocation method to discretize the system. This thesis first examines the vibrations of webs with uniform material properties throughout, then investigates the effects of using biologically realistic material parameters for silks within a single web. Understanding how spiders detect and react to the vibrations produced by their webs is of interest for both biological and engineering applications. / Master of Science

Nous proposons des méthodes simples et efficaces pour manipuler des expressions trigonométriques de la forme $F=\sum_{k} f_k\cos\tfrac{k\pi}{n}, f_k\in Z$ où $d<n$ fixé. Nous utilisons les polynômes unitaires de Chebyshev qui forment une base de $Z[x]$ avec laquelle toutes les opérations arithmétiques peuvent être exécutées aussi rapidement qu'avec la base de monômes, mais également déterminer le signe et une approximation de $F$, calculer le polynôme minimal de $F$. Dans ce cadre nous calculons efficacement le polynôme minimal de $2\cos\frac{\pi}{n}$ et aussi le polynôme cyclotomique $\Phi_n$. Nous appliquons ces méthodes au calcul des diagrammes de nœuds de Chebyshev $C(a,b,c,\varphi) : x=T_a(t), y=T_b(t), z=T_c(t+\varphi)$, ce qui permet de tester si une courbe donnée est un nœud, et aussi lister tous les nœuds de Chebyshev possibles quand un triple $(a,b,c)$ fixé en bonne complexité. / We propose a set of simple and fast algorithms for evaluating and using trigonometric expressions in the form $F=\sum_{k}f_k\cos\frac{k\pi}{n}$, $f_k\in Z$ where $d<n$ fixed. We make use of the monic Chebyshev polynomials as a basis of $Z[x]$. We can perform arithmetic operations (multiplication, division, gcd) on polynomials expressed in a Chebyshev basis (with the same bit-complexity as in the monomial basis), compute the sign of $F$, evaluate it numerically and compute its minimal polynomial in $Q[x]$. We propose simple and efficient algorithms for computing the minimal polynomial of $2\cos\frac{\pi}{n}$ and also the cyclotomic polynomial $\Phi_n$. As an application, we give a method to determine the Chebyshev knot's diagrams $C(a,b,c,\varphi) : x=T_a(t),y=T_b(t), z=T_c(t+\varphi)$ which allows to test if a given curve is a Chebyshev knot, and point out all the possible Chebyshev knots coressponding a fixed triple $(a,b,c)$, all of these computings can be done with a good bit complexity.

In this thesis, we have analyzed one-dimensional fermion-spin interactions in the dilute limit. The two cases we analyze represent different paradigms. For the first part, we look at the existence of spins for all sites as an effective model to describe the rearrangement of core electrons within the dynamic Hubbard model. Within this model, the behavior of electrons and holes will be compared in the presence of fermion-spin coupling and on-site repulsion. It will be shown that in this framework, electrons and holes behave differently and even though electrons experience increased repulsion, holes show attraction for a range of on-site repulsions. The characteristics of the interaction show effective nearest-neighbor attraction though no such term exists within the model. By the analysis of dynamic properties, two regions of interaction are identified. The gradual change from weak to strong coupling of fermions is presented. The effect of introducing on-site repulsion for both ranges of coupling is presented for both the dynamic Hubbard model and electron-hole symmetric version. For the second case involving fermion-spin interaction, we look at the interaction of a fermion with spins existing only for a small portion of the lattice, representing a coupled magnetic layer that an itinerant fermion interacts with through Heisenberg-like spin flip interaction. The interaction represents a spin-flip interaction of a spin current and magnetic layer. This interaction has been extensively studied for its relevance to computer hard drives both experimentally and theoretically. Most theoretical descriptions utilize the semi-classical Landau-Lifshitz-Gilbert (LLG) formalism. However, with recent improvements in experimental methods with very small magnetic layers and very fast real time measurements, quantum effects become more pronounced. We present quantum mechanical results that show considerable modification to spin-flip interaction. We identify a set of conditions that exhibits the existence of an emerging bound state for the spin current both numerically and analytically. The bound state is a quantum mechanical state and cannot be achieved with a classical picture. We present results in a one-dimensional lattice for a spin-1/2 system, and generalize our arguments to higher dimension and spins with S > 1/2.

Thesis (Ph. D.)--West Virginia University, 2002. / Title from document title page. Document formatted into pages; contains xi, 107 p. : ill. (some col.). Includes abstract. Includes bibliographical references (p. 102-107).