In this paper we will study the 1p spaces. We will begin with definitions and different examples of 1p spaces. In particular, we will prove Holder's and Minkowski's inequalities for 1p sequence.

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Orientadora: Profa. Dra. Mariana Rodrigues da Silveira / Dissertação (mestrado) - Universidade Federal do ABC, Programa de Pós-Graduação em Matemática , Santo André, 2018.

Le polytope des m-sous-espaces est défini comme l'enveloppe convexe des vecteurs caractéristiques de tous les sous-espaces de dimension m d'un espace affin fini. Le cas particulier du polytope des hyperplans a été étudié par Maurras (1993) et Anglada et Maurras (2003), qui ont obtenu une description complète des facettes. Le polytope général des m-sous-espaces que nous considérons possède une structure plus complexe, notamment concernant les facettes. Néanmoins, nous établissons dans cette thèse plusieurs familles de facettes. Nous caractérisons également complètement le groupe des automorphismes du polytope ainsi que l'adjacence des sommets du polytope des m-sous-espaces. Un tangle est un ensemble d'hyperplans d'un espace affin contenant un hyperplan par classe d'hyperplans parallèles. Anglada et Maurras ont montré que les tangles définissent des facettes du polytope des hyperplans et que toutes les facettes de ce polytope proviennent de tangles. Nous tentons d'établir une généralisation de ce résultat. Nous élaborons une classification des tangles en familles pour de petites dimensions d'espaces affins. / Doctorat en sciences, Spécialisation mathématiques / info:eu-repo/semantics/nonPublished

Goldman [Gol86] introduced a Lie algebra structure on the free vector space generated by the free homotopy classes of oriented closed curves in any orientable surface F . This Lie bracket is known as the Goldman bracket and the Lie algebra is known as the Goldman Lie algebra. In this dissertation, we compute the center of the Goldman Lie algebra for any hyperbolic surface of finite type. We use hyperbolic geometry and geometric group theory to prove our theorems. We show that for any hyperbolic surface of finite type, the center of the Goldman Lie algebra is generated by closed curves which are either homotopically trivial or homotopic to boundary components or punctures. We use these results to identify the quotient of the Goldman Lie algebra of a non-closed surface by its center as a sub-algebra of the first Hochschild cohomology of the fundamental group. Using hyperbolic geometry, we prove a special case of a theorem of Chas [Cha10], namely, the geometric intersection number between two simple closed geodesics is the same as the number of terms (counted with multiplicity) in the Goldman bracket between them. We also construct infinitely many pairs of length equivalent curves in any hyperbolic surface F of finite type. Our construction shows that given a self- intersecting geodesic x of F and any self-intersection point P of x, we get a sequence of such pairs.

In dieser Arbeit beweisen wir für Optimierungsprobleme in L0-Moduln relevante Resultate und untersuchen Anwendungen für die Darstellung von Präferenzen. Im ersten Kapitel geht es um quasikonkave, monotone und lokale Funktionen von einem L0-Modul X nach L0, die wir robust darstellen. Im zweiten Kapitel entwickeln wir das Ekeland’sche Variationsprinzip für L0-Moduln, die eine L0-Metrik besitzen. Wir beweisen eine L0 -Variante einer Verallgemeinerung des Ekeland’schen Theorems. Der Beweis des Brouwerschen Fixpunktsatzes für Funktionen, die auf (L0)^d definiert sind, wird in Kapitel 3 behandelt. Wir definieren das Konzept des Simplexes in (L0)^d und beweisen, dass jede lokale, folgenstetige Funktion darauf einen Fixpunkt besitzt. Dies nutzen wir, um den Fixpunktsatz auch für Funktionen auf beliebigen abgeschlossenen, L0 -konvexen Mengen zu zeigen. Eine allgemeinere Struktur als L0 ist die bedingte Menge. Im vierten Kapitel behandeln wir bedingte topologische Vektorräume. Wir führen das Konzept der Dualität für bedingte Mengen ein und beweisen Theoreme der Funktionalanalysis darauf, unter anderem das Theorem von Banach-Alaoglu und Krein-Šmulian. Im fünften Kapitel widmen wir uns der Darstellung mit wandernden konvexen Mengen. Wir zeigen danach, wie die Transitivität für diese Darstellungsform beschrieben werden kann. Abschließend modellieren wir die Eigenschaft, dass die Transitivität einer Relation nur für ähnliche Elemente gesichert ist und diskutieren Arten der Darstellung solcher Relationen. / In this thesis, we prove results relevant for optimization problems in L0-modules and study applications to order theory. The first part deals with the notion of an Assessment Index (AI). For an L0 -module X an AI is a quasiconcave, monotone and local function mapping to L0. We prove a robust representation of these AIs. In the second chapter of this thesis, we develop Ekeland’s variational principle for L0-modules allowing for an L0-metric. We prove an L0-Version of a generalization of Ekeland’s theorem. A further application of L0 -theory is examined in the third chapter of this thesis, namely an extension of the Brouwer fixed point theorem to functions on (L0)^d . We define a conditional simplex, which is a simplex with respect to L0 , and prove that every local, sequentially continuous function has a fixed point. We extend the fixed point theorem to arbitrary closed, L0-convex sets. A more general structure than L0 -modules is the concept of conditional sets. In the fourth chapter of the thesis, we study conditional topological vector spaces. We examine the concept of duality for conditional sets and prove results of functional analysis: among others, the Banach-Alaoglu and the Krein-Šmulian theorem. Any L0 -module being a conditional set allows to apply all results to L0 -theory. In the fifth chapter, we discuss the property of transitivity of relations and its connection to certain forms of representations. After a survey of common representations of preferences, we attend to relations induced by moving convex sets which are relations of the form that x is preferred to y if and only if x − y is in a convex set depending on y. We examine in which cases such a representation is transitive. Finally, we exhibit nontransitivity due to dissimilarity of the compared object and discuss representations for relations of that type.

Submitted by ANA KARLA PEREIRA RODRIGUES (anakarla_@hotmail.com) on 2017-08-17T14:00:23Z No. of bitstreams: 1 arquivototal.pdf: 1661057 bytes, checksum: 913a7f671e2e028b60d14a02274f932a (MD5) / Made available in DSpace on 2017-08-17T14:00:23Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 1661057 bytes, checksum: 913a7f671e2e028b60d14a02274f932a (MD5) Previous issue date: 2015-02-27 / In this work we investigate the concept of topological vector spaces and their properties. In the rst chapter we present two sections of basic results and in the other sections we present a more general study of such spaces. In the second chapter we restrict ourselves to the real scalar eld and we study, in the context of locally convex spaces, the Hahn-Banach and Banach-Alaoglu theorems. We also build the weak, weak-star, of bounded convergence and of pointwise convergence topologies. Finally we investigate the Theorem of Banach-Steinhauss, the Open Mapping Theorem and the Closed Graph Theorem. / Neste trabalho, estudamos o conceito de espa cos vetoriais topol ogicos e suas propriedades. No primeiro cap tulo, apresentamos duas se c~oes de resultados b asicos e, nas demais se c~oes, apresentamos um estudo sobre tais espa cos de forma mais ampla. No segundo cap tulo, restringimo-nos ao corpo dos reais e fazemos um estudo sobre os espa cos localmente convexos, o Teorema de Hahn-Banach, o Teorema de Banach- Alaoglu, constru mos as topologias fraca, fraca-estrela, da converg^encia limitada e da converg^encia pontual. Por ultimo, estudamos o Teorema da Limita c~ao Uniforme, o Teorema do Gr a co Fechado e o da Aplica c~ao Aberta no contexto mais geral dos espa cos de Fr echet.

This thesis deals with mining of relevant information from documents and automatic splitting of multiple documents merged together. Moreover, it describes the design and implementation of software for data mining from documents and for automatic splitting of multiple documents. Methods for acquiring textual data from scanned documents, named entity recognition, document clustering, their supportive algorithms and metrics for automatic splitting of documents are described in this thesis. Furthermore, an algorithm of implemented software is explained and tools and techniques used by this software are described. Lastly, the success rate of the implemented software is evaluated. In conclusion, possible extensions and further development of this thesis are discussed at the end.

Compressed sensing has been proposed as a computationally efficient method to estimate the finite-dimensional signals. The idea is to develop an undersampling operator that can sample the large but finite-dimensional sparse signals with a rate much below the required Nyquist rate. In other words, considering the sparsity level of the signal, the compressed sensing samples the signal with a rate proportional to the amount of information hidden in the signal. In this dissertation, first, we employ compressed sensing for physical layer signal processing of directional millimeter-wave communication. Second, we go through the theoretical aspect of compressed sensing by running a comprehensive theoretical analysis of compressed sensing to address two main unsolved problems, (1) continuous-extension compressed sensing in locally convex space and (2) computing the optimum subspace and its dimension using the idea of equivalent topologies using Köthe sequence. In the first part of this thesis, we employ compressed sensing to address various problems in directional millimeter-wave communication. In particular, we are focusing on stochastic characteristics of the underlying channel to characterize, detect, estimate, and track angular parameters of doubly directional millimeter-wave communication. For this purpose, we employ compressed sensing in combination with other stochastic methods such as Correlation Matrix Distance (CMD), spectral overlap, autoregressive process, and Fuzzy entropy to (1) study the (non) stationary behavior of the channel and (2) estimate and track channel parameters. This class of applications is finite-dimensional signals. Compressed sensing demonstrates great capability in sampling finite-dimensional signals. Nevertheless, it does not show the same performance sampling the semi-infinite and infinite-dimensional signals. The second part of the thesis is more theoretical works on compressed sensing toward application. In chapter 4, we leverage the group Fourier theory and the stochastical nature of the directional communication to introduce families of the linear and quadratic family of displacement operators that track the join-distribution signals by mapping the old coordinates to the predicted new coordinates. We have shown that the continuous linear time-variant millimeter-wave channel can be represented as the product of channel Wigner distribution and doubly directional channel. We notice that the localization operators in the given model are non-associative structures. The structure of the linear and quadratic localization operator considering group and quasi-group are studied thoroughly. In the last two chapters, we propose continuous compressed sensing to address infinite-dimensional signals and apply the developed methods to a variety of applications. In chapter 5, we extend Hilbert-Schmidt integral operator to the Compressed Sensing Hilbert-Schmidt integral operator through the Kolmogorov conditional extension theorem. Two solutions for the Compressed Sensing Hilbert Schmidt integral operator have been proposed, (1) through Mercer's theorem and (2) through Green's theorem. We call the solution space the Compressed Sensing Karhunen-Loéve Expansion (CS-KLE) because of its deep relation to the conventional Karhunen-Loéve Expansion (KLE). The closed relation between CS-KLE and KLE is studied in the Hilbert space, with some additional structures inherited from the Banach space. We examine CS-KLE through a variety of finite-dimensional and infinite-dimensional compressible vector spaces. Chapter 6 proposes a theoretical framework to study the uniform convergence of a compressible vector space by formulating the compressed sensing in locally convex Hausdorff space, also known as Fréchet space. We examine the existence of an optimum subspace comprehensively and propose a method to compute the optimum subspace of both finite-dimensional and infinite-dimensional compressible topological vector spaces. To the author's best knowledge, we are the first group that proposes continuous compressed sensing that does not require any information about the local infinite-dimensional fluctuations of the signal.