An analogue of the Riemannian structure of a manifold is created for an ultrametric Cantor set using the techniques of Noncommutative Geometry. In particular, a spectral triple is created that can recover much of the fractal geometry of the original Cantor set. It is shown that this spectral triple can recover the metric, the upper box dimension, and in certain cases the Hausdorff measure. The analogy with Riemannian geometry is then taken further and an analogue of the Laplace-Beltrami operator is created for an ultrametric Cantor set. The Laplacian then allows to create an analogue of Brownian motion generated by this Laplacian. All these tools are then applied to the triadic Cantor set. Other examples of ultrametric Cantor sets are then presented: attractors of self-similar iterated function systems, attractors of cookie cutter systems, and the transversal of an aperiodic, repetitive Delone set of finite type. In particular, the example of the transversal of the Fibonacci tiling is studied.

[Truncated abstract] This thesis is concerned with geometrically-defined curves that can be used for interpolation in Riemannian or, more generally, semi-Riemannian manifolds. As in much of the existing literature on such curves, emphasis is placed on manifolds which are important in computer graphics and engineering applications, namely the unit 3-sphere S3 and the closely related rotation group SO(3), as well as other Lie groups and spheres of arbitrary dimension. All geometrically-defined curves investigated in the thesis are either higher order variational curves, namely critical points of cost functionals depending on (covariant) derivatives of order greater than 1, or defined by geometrical algorithms, namely generalisations to manifolds of algorithms from the field of computer aided geometric design. Such curves are needed, especially in the aforementioned applications, since interpolation methods based on applying techniques of classical approximation theory in coordinate charts often produce unnatural interpolants. However, mathematical properties of higher order variational curves and curves defined by geometrical algorithms are in need of substantial further investigation: higher order variational curves are solutions of complicated nonlinear differential equations whose properties are not well-understood; it is usually unclear how to impose endpoint derivative conditions on, or smoothly piece together, curves defined by geometrical algorithms. This thesis addresses these difficulties for several classes of curves. ... The geometrical algorithms investigated in this thesis are generalisations of the de Casteljau and Cox-de Boor algorithms, which define, respectively, polynomial B'ezier and piecewise-polynomial B-spline curves by dividing, in certain ratios and for a finite number of iterations, piecewise-linear control polygons corresponding to finite sequences of control points. We show how the control points of curves produced by the generalised de Casteljau algorithm in an (almost) arbitrary connected finite-dimensional Riemannian manifold M should be chosen in order to impose desired endpoint velocities and (covariant) accelerations and, thereby, piece the curves together in a C2 fashion. A special case of the latter construction simplifies when M is a symmetric space. For the generalised Cox-de Boor algorithm, we analyse in detail the failure of a fundamental property of B-spline curves, namely C2 continuity at (certain) knots, to carry over to M.

Thesis (Doctoral)--Universität Bonn, 1990. / Includes bibliographical references.

Thesis (Ph.D.)--Michigan State University. Dept. of Mathematics, 2008. / Title from PDF t.p. (viewed on July 24, 2009) Includes bibliographical references (p. 75-78). Also issued in print.

Orientador: Diego Sebastian Ledesma / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-18T07:05:24Z (GMT). No. of bitstreams: 1 LuqueJusto_Claudia_M.pdf: 965819 bytes, checksum: 0b70ed73c13a9ad324422e792409e76d (MD5) Previous issue date: 2011 / Resumo: Neste trabalho estudamos o movimento Browniano, sobre uma variedade Riemanniana munida de métricas que variam com respeito ao tempo. Tratamos brevemente os conceitos de semimartingale, equações diferenciáveis estocásticas e processos de difusão sobre variedades diferenciáveis. Apresentamos a construção clássica do movimento Browniano sobre uma variedade Riemanniana (M, g). Finalmente, munindo à variedade com uma família de métricas {g(t)} t ? [0,T] que variam com respeito ao tempo, damos duas construções do movimento Browniano sobre a variedade Riemanniana (M, g(t)), para cada t ? [0, T] (denotamos a este processo como o g(t)-movimento Browniano). Consideramos o fluxo de curvatura média sobre uma hipersuperfície compacta, e damos uma estimativa para o tempo de explosão de um processo definido a partir do g(t)-movimento Browniano. Definimos o transporte paralelo amortiguado ao longo do g(t)-movimento Browniano e damos condições para que este seja de fato uma isometria. / Abstract: We study the Brownian motion on a Riemannian manifold equipped with a family of metrics that vary with respect to time. We treat brief the concepts of semimartingale, stochastic differential equations and diffusion processes on manifolds. We present the classical construction of Brownian motion on an Riemannian manifold (M, g). Finally, equipping the variety with a family of metrics {g(t)}t?[0,T] that vary with respect to time, we give two constructions of Brownian motion on the Riemannian manifold (M, g(t)) for each t ? [0, T] (we denote this process as the g(t)-Brownian motion). We consider the mean curvature flow on a compact hypersurface, and give an estimate for the time of explosion of the g(t)-Brownian motion. We define the Damped parallel transport along of the g(t)-Brownian motion and we give conditions so that in fact is an isometry / Mestrado / Matematica / Mestre em Matemática

Universidade Federal do Cearà / Neste trabalho estudaremos uma generalizaÃÃo do algoritmo de De Casteljau, que à um procedimento recursivo para construÃÃo de curvas de Bezier em espaÃos euclidianos, para grupos de Lie e esferas S2, com Ãnfase nas curvas de Bezier de grau 3. / In this paper we study a generalization of the De Casteljau algorithm, which is a recursive procedure for constructing Bezier curves in euclidean space, for Lie groups and spheres S2, with emphasis on Bezier curves of degree 3.

Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / Estudamos folheaÃÃes de formas espaciais por hipersuperfÃcies completas, sob certas condiÃÃes sobre as suas curvaturas mÃdias de ordem superior. Em particular, no espaÃo euclidiano obtemos um Teorema tipo-Bernstein para grÃficos cujas curvaturas mÃdia e escalar nÃo mudam de sinal (podendo ser nÃo constantes). NÃs tambÃm estabelecemos a nÃo existÃncia de folheaÃÃes da esfera padrÃo cujas folhas sÃo completas e tÃm curvatura escalar constante, alargando assim um teorema de Barbosa, Kenmotsu e Oshikiri. Para o caso mais geral de folheaÃÃes r-mÃnimas do espaÃo euclidiano, possivelmente com um conjunto singular, somos capazes de invocar um teorema de D. Ferus para dar condiÃÃes sob as quais as folhas nÃo-singulares sÃo folheadas por hiperplanos. / We study foliations of space forms by complete hypersurfaces, under some mild conditions on its higher order mean curvatures. In particular, in Euclidean space we obtain a Bernstein-type theorem for graphs whose mean and scalar curvature do not change sign but may otherwise be nonconstant. We also establish the nonexistence of foliations of the standard sphere whose leaves are complete and have constant scalar curvature, thus extending a theorem of Barbosa, Kenmotsu and Oshikiri. For the more general case of r-minimal foliations of the Euclidean space, possibly with a singular set, we are able to invoke a theorem of Ferus to give conditions under which the nonsigular leaves are foliated by hyperplanes.

Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / Neste trabalho, definimos as transformaÃÃes de Newton e provamos algumas propriedades relacionadas a elas. Fizemos um estudo sobre operador elÃptico e usamos isso para provar que dadas algumas condiÃÃes para a curvatura seccional de uma variedade riemanniana M, conseguimos majorar a funÃÃo altura (em modulo) de um grÃfico vertical compacto imerso em MxR. / In this paper, we define the transformations of Newton and prove some properties related to them. We did a study on elliptic operator and use it to prove that given some conditions for the sectional curvature of a riemannian manifold M,able function of increasing height (in modulus) of a graph vertical compact immersed in MXR.

CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / Neste trabalho, demonstramos teoremas estabelecendo condiÃÃes suficientes para a existÃncia de imersÃes isomÃtricas com curvatura extrÃnseca prescrita em grupos de Lie nilpotentes e solÃveis. Obtemos assim uma generalizaÃÃo do Teorema Fundamental da Teoria de Subvariedades em Rn e, em particular, obtemos resultados de imersÃo em todos os grupos tipo-Heisenberg e em todos os espaÃos de Damek-Ricci. / In this paper, we prove theorems establishing sufficient conditions to existence for isometric immersions with prescribed extrinsic curvature in two-step nilpotent Lie groups and solvmanifolds. We obtain a generalization of the Fundamental Theorem of Submanifold Theory in Rn and, in particular, we one has immersion results in the generally Heisenberg type groups and Damek-Ricci spaces.

Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / O objetivo dessa dissertaÃÃo à apresentar uma fÃrmula para o operador Lr(g) = div(Pr gradiente g) de uma nova funÃÃo suporte g, definida sobre uma hipersuperfÃcie M n em uma forma espacial Riemanniana Mc n+1, bem como mostrar que uma hipersuperfÃcie diferenciÃvel estrelada compacta Σn, com segunda funÃÃo simÃtrica S2 constante positiva na esfera Euclidiana S n+1, deve ser uma esfera geodÃsica Sn (p). Isso generaliza um resultado obtido por Jellett [9] em 1853 para tais tipos de superfÃcies no espaÃo Euclidiano R3. / The purpose of this dissertation is to desire a formula for the operator Lr(g) = div(Pr gradient g) of a new support function g, defined over a hypersurface Mn in a Riemannian space form Mc n +1, and to show that a compact smooth starshaped hypersurface Σn in the Euclidean sphere Sn+1,whose second symmetric function S2 is positive and constant must be a geodesic sphere Sn (p). This generalizes a result obtained by Jellett [9] in 1853 for such surfaces in Euclidean space R3.