This thesis consists of four papers, all concerning random iteration of isometries. The papers are: I. Ambroladze A, Ådahl M, Random iteration of isometries in unbounded metric spaces. Nonlinearity 16 (2003) 1107-1117. II. Ådahl M, Random iteration of isometries controlled by a Markov chain. Manuscript. III. Ådahl M, Melbourne I, Nicol M, Random iteration of Euclidean isometries. Nonlinearity 16 (2003) 977-987. IV. Johansson A, Ådahl M, Recurrence of a perturbed random walk and an iterated function system depending on a parameter. Manuscript. In the first paper we consider an iterated function system consisting of isometries on an unbounded metric space. Under suitable conditions it is proved that the random orbit {Zn} ∞n=0, of the iterations corresponding to an initial point Z0, “escapes to infinity" in the sense that P(Zn Є K) → 0, as n → ∞ for every bounded set K. As an application we prove the corresponding result in the Euclidean and hyperbolic spaces under the condition that the isometries do not have a common fixed point. In the second paper we let a Markov chain control the random orbit of an iterated function system of isometries on an unbounded metric space. We prove under necessary conditions that the random orbit \escapes to infinity" and we also give a simple geometric description of these conditions in the Euclidean and hyperbolic spaces. The results generalises the results of Paper I. In the third paper we consider the statistical behaviour of the reversed random orbit corresponding to an iterated function system consisting of a finite number of Euclidean isometries of <b>R</b>n. We give a new proof of the central limit theorem and weak invariance principles, and we obtain the law of the iterated logarithm. Our results generalise immediately to Markov chains. Our proofs are based on dynamical systems theory rather than a purely probabilistic approach. In the fourth paper we obtain a suficient condition for the recurrence of a perturbed (one-sided) random walk on the real line. We apply this result to the study of an iterated function system depending on a parameter and defined on the open unit disk in the complex plane.

In this thesis, we study two problems. The first one is the renorming problem in Banach lattices. We state the problem and give some known results related to it. Then we pass to construct a positive doubly power bounded operator with a nonpositive inverse on an infinite dimensional AL-space which generalizes the result of [10]. The second problem is related to the mean ergodicity of positive operators on KBspaces. We prove that any positive power bounded operator T in a KB-space E which satisfies lim n!1 dist1 n n&amp / #8722 / 1 Xk=0 Tkx, [&amp / #8722 / g, g] + BE= 0 (8x 2 E, kxk 1), () where BE is the unit ball of E, g 2 E+, and 0 &lt / 1, is mean ergodic and its fixed space Fix(T) is finite dimensional. This generalizes the main result of [12]. Moreover, under the assumption that E is a -Dedekind complete Banach lattice, we prove that if, for any positive power bounded operator T, the condition () implies that T is mean ergodic then E is a KB-space.

<p>In compressive sensing, the fundamental problem is to reconstruct sparse signal from its nonadaptive insufficient linear measurement. Besides sparse signal reconstruction algorithms, measurement matrix or measurement dictionary plays an important part in sparse signal recovery. Orthogonal Matching Pursuit (OMP) algorithm, which is widely used in compressive sensing, is especially affected by measurement dictionary. Measurement dictionary with small restricted isometry constant or coherence could improve the performance of OMP algorithm. Based on measurement dictionary, sensing dictionary can be constructed and can be incorporated into OMP algorithm. In this thesis, two methods are proposed to design sensing dictionary. In the first method, sensing dictionary design problem is formulated as a linear programming problem. The solution is unique and can be obtained by standard linear programming method such as primal-dual interior point method. The major drawback of linear programming based method is its high computational complexity. The second method is termed sensing dictionary designing algorithm. In this algorithm, each atom of sensing dictionary is designed independently to reduce the maximal magnitude of its inner product with measurement dictionary. Compared with linear programming based method, the proposed sensing dictionary design algorithm is of low computational complexity and the performance is similar. Simulation results indicate that both of linear programming based method and the proposed sensing dictionary designing algorithm can design sensing dictionary with small mutual coherence and cumulative coherence. When the designed sensing dictionary is applied to OMP algorithm, the performance of OMP algorithm improves.</p> / Master of Science in Electrical and Computer Engineering (MSECE)

Made available in DSpace on 2016-04-27T16:59:06Z (GMT). No. of bitstreams: 1 JuNIOR TEODORO DA SILVA.pdf: 5677889 bytes, checksum: 8cd741014ed99609aade05fe8af085ab (MD5) Previous issue date: 2010-05-21 / This research is inserted in the scope of teaching and learning of Geometry, in particular in the Geometric Transformations with a specific approach in isometric axial symmetry transformation. This proposal led to an investigation about the concepts of this kind of isometry through the use of error in a reconstructive approach from pedagogical strategies with use of technologies. The development occurred in two stages, being the first one done in a sequence of activities in the static environment paper and pencil . The second sequence was made with activities mediated by software of dynamic geometry named Geogebra. The errors that occurred in the static environment were considered for a reconstructive approach in the stage in which the dynamic geometry was used. For such occurrence, the research tried to identify how the pedagogical books of 3rd and 4th stages of Fundamental Teaching received the general reports made by PNLD (National Program of Pedagogical Book) and how some books highlight this subject in these contents. Moreover, it was verified how this subject is handled in the schools according to authors like Almouloud (2007), Catunda et al (1998) and Pavanello (1993), as well as according to the instructions given by National Curricula Parameters. Concerned by the reconstructive approach of error, the research aimed to understand the function of error in Mathematics learning according Brousseau (1986), Almouloud (2007), Perrenoud (2000), Astolfi (1997), Macedo (1997) and Pinto (2000). In the first stage, the learners participated in the development of the sequence with ruler, compass and square method, pointing that they never had used those tools. This fact limited the students to empirical validations. In the second stage, the students showed progress related to construction provided by the resources offered, mainly related to quick corrections, validations and proofs facilitated by resources provided by a pedagogical strategy with use of technologies among which the dynamic geometry Geogebra software / Este trabalho insere-se no âmbito do ensino e aprendizagem da Geometria, em particular as Transformações Geométricas com uma abordagem específica na transformação isométrica Simetria Axial. Esta proposta conduziu a uma investigação sobre os conceitos desse tipo de isometria através do uso do erro numa abordagem reconstrutiva a partir de estratégias pedagógicas com uso de tecnologias. O desenvolvimento ocorreu em duas etapas, sendo a primeira realizada com uma sequência de atividades realizada no ambiente estático papel e lápis e a segunda com uma sequência de atividades construída por intermédio do software de geometria dinâmica Geogebra. Os erros ocorridos no ambiente estático foram considerados para uma abordagem reconstrutiva na etapa que se valeu da geometria dinâmica. Para tal ocorrência, buscou-se identificar como os livros didáticos dos 3º e 4º ciclos recebem pareceres gerais pelo PNLD (Programa Nacional do Livro Didático) e como alguns livros enfatizam esse tema em seus conteúdos. Além disso, verificou-se como o tema é tratado nas escolas de acordo com autores como Almouloud (2004), Catunda (1998) e Pavanello (1993), bem como de acordo com as instruções dadas pelos Parâmetros Curriculares Nacionais. Quanto à abordagem reconstrutiva do erro, buscou-se entender a função do erro na aprendizagem de Matemática segundo Brousseau (1986), Almouloud (2007), Perrenoud (2000), Astolfi (1997), Macedo (1997) e Pinto (2000). Na primeira etapa, os aprendizes participaram no desenvolvimento da sequência com o método da construção com régua, compasso e esquadros, indicando que nunca haviam usado esses instrumentos, limitando-se, assim, às validações empíricas. Na segunda etapa, os sujeitos apresentaram avanços quanto às construções devido aos recursos oferecidos, principalmente em relação à correção imediata e às validações e provas facilitadas pelos recursos oportunizados por uma estratégia pedagógica com uso de tecnologias, dentre as quais, o programa de geometria dinâmica Geogebra

We demonstrate the Splitting Theorem due to Cheeger and Gromoll, which ensures that a complete Riemannian n-manifold which has nonnegative Ricci curvature and a line, can be split isometrically into the Riemannian product of real with a (n-1 )- manifold. / Conselho Nacional de Desenvolvimento Científico e Tecnológico / Demonstramos o Teorema de Decomposição de Cheeger-Gromoll, o qual garante que uma variedade Riemanniana completa ndimensional, com curvatura de Ricci não-negativa, que possui uma linha, pode ser decomposta isometricamente num produto Riemanniano de uma variedade (n-1 )-dimensional com o conjunto dos reais.

Cette thèse comporte deux parties, axées sur des aspects différents de la géométrie lorentzienne. La première partie porte sur les groupes d’isométries de surfaces lorentziennes globalement hyperboliques spatialement compactes, particulièrement lorsque le groupe exhibe une dynamique non triviale (action non propre). Le groupe d'isométries agit naturellement sur le cercle par difféomorphismes, et les résultats principaux portent sur la classification de ces représentations. Sous une hypothèse sur le bord conforme, on obtient une conjugaison par homéomorphisme avec l'action projective d'un sous-groupe de PSL(2,R) ou de l'un de ses revêtements finis. La différentiabilité de la conjuguante est étudiée, avec des résultats qui garantissent une conjugaison dans le groupe de difféomorphismes du cercle dans certains cas. On donne également des contre-exemples à l'existence d'une conjugaison différentiable, y compris pour des groupes ayant une dynamique riche. Ces constructions s'appuient sur l'étude de flots hyperboliques en dimension trois. Sans l'hypothèse sur le bord conforme, on obtient une semi conjugaison et un isomorphisme de groupes. On construit également des exemples pour lesquels il n'existe pas de conjugaison topologique. La seconde partie de cette thèse étudie un espace-temps vu comme un système dynamique multi-valuée : à un point on associe sont futur causal. Cette approche, déjà présente dans les travaux de Fathi et Siconolfi, permet de concrétiser le lien entre fonctions de Lyapunov en systèmes dynamiques et fonctions temps. Le résultat principal est une version lorentzienne du Théorème de Conley : on peut définir l'ensemble récurrent par chaînes d'un espace-temps, et il existe une fonction continue croissante le long de toute courbe causale orientée vers le futur, strictement croissante si le point de départ de la courbe n'est pas dans l'ensemble récurrent par chaînes. Ces techniques s'adaptent aussi dans un espace-temps stablement causal, ce qui permet de donner une nouvelle preuve d'une partie du Théorème d'Hawking. / This thesis is divided into two parts, dealing with two different aspects of Lorentzian geometry. The first part deals with isometry groups of globally hyperbolic spatially compact Lorentz surfaces, especially when it has a non trivial dynamical behavior (non proper action). The isometry group acts on circle by diffeomorphisms, and the main results of this part concern the classification of these actions. Under a hypothesis on the conformal boundary, we show that they are topologically conjugate to the projective action of a subgroup of PSL(2,R), or one of its finite covers. The differentiability of the conjugacy is studied, with some results giving a differentiable conjugacy under additional hypotheses. We also give counter examples to such a differentiable conjugacy, even for groups with rich dynamics. These constructions use hyperbolic flows on three manifolds. Without the hypothesis on the conformal boundary, we obtain a semi conjugacy and a group isomorphism. We also give examples where a topological conjugacy cannot exist. In the second part of this thesis, we see a spacetime as a multi valued dynamical system: we map a point to its causal future. This point of view was already adopted by Fathi and Siconolfi, and it gives a concrete meaning to the link between Lyapunov functions in dynamical systems and time functions. The main result is a Lorentzian version of Conley's Theorem: we define the chain recurrent set of a spacetime, and construct a continuous function that increases along future directed causal curves outside the chain recurrent set, and that is non decreasing along other future curves. These techniques also apply to the stably causal setting, and we obtain a new proof of a part of Hawking's Theorem.

Spectral methods have become increasingly popular in designing fast algorithms for modern highdimensional datasets. This thesis looks at several problems in which spectral methods play a central role. In some cases, we also show that such procedures have essentially the best performance among all randomised polynomial time algorithms by exhibiting statistical and computational trade-offs in those problems. In the first chapter, we prove a useful variant of the well-known Davis{Kahan theorem, which is a spectral perturbation result that allows us to bound of the distance between population eigenspaces and their sample versions. We then propose a semi-definite programming algorithm for the sparse principal component analysis (PCA) problem, and analyse its theoretical performance using the perturbation bounds we derived earlier. It turns out that the parameter regime in which our estimator is consistent is strictly smaller than the consistency regime of a minimax optimal (yet computationally intractable) estimator. We show through reduction from a well-known hard problem in computational complexity theory that the difference in consistency regimes is unavoidable for any randomised polynomial time estimator, hence revealing subtle statistical and computational trade-offs in this problem. Such computational trade-offs also exist in the problem of restricted isometry certification. Certifiers for restricted isometry properties can be used to construct design matrices for sparse linear regression problems. Similar to the sparse PCA problem, we show that there is also an intrinsic gap between the class of matrices certifiable using unrestricted algorithms and using polynomial time algorithms. Finally, we consider the problem of high-dimensional changepoint estimation, where we estimate the time of change in the mean of a high-dimensional time series with piecewise constant mean structure. Motivated by real world applications, we assume that changes only occur in a sparse subset of all coordinates. We apply a variant of the semi-definite programming algorithm in sparse PCA to aggregate the signals across different coordinates in a near optimal way so as to estimate the changepoint location as accurately as possible. Our statistical procedure shows superior performance compared to existing methods in this problem.

No description available.

The aim of this work is to derive mathematical descriptions of gravitation. Postulating gravitation as a force field, Newton's law of gravitation is heuristically derived by considering linear differential operators invariant under euclidean isometries and by finding the fundamental solution to Helmholtz equation in three dimensions. Thereafter, the theory of differential geometry is introduced, providing a framework for the subsequent review of gravitation as curvature. Lastly, in the light of Einstein's postulates and equivalence principle, Lovelock's proof of uniqueness of Einstein's field equations is presented.

The notion of a fuzzy metric space due to George and Veeramani has many advantages in analysis since many notions and results from classical metric space theory can be extended and generalized to the setting of fuzzy metric spaces, for instance: the notion of completeness, completion of spaces as well as extension of maps. The layout of the dissertation is as follows: Chapter 1 provide the necessary background in the context of metric spaces, while chapter 2 presents some concepts and results from classical metric spaces in the setting of fuzzy metric spaces. In chapter 3 we continue with the study of fuzzy metric spaces, among others we show that: the product of two complete fuzzy metric spaces is also a complete fuzzy metric space. Our main contribution is in chapter 4. We introduce the concept of a standard fuzzy pseudo metric space and present some results on fuzzy metric identification. Furthermore, we discuss some properties of t-nonexpansive maps. / Mathematical Sciences / M. Sc. (Mathematics)