A quantum channel is a mapping which sends density matrices to density matrices. The estimation of quantum channels is of great importance to the field of quantum information. In this thesis two topics related to estimation of quantum channels are investigated. The first of these is the upper bound of Sarovar and Milburn (2006) on the Fisher information obtainable by measuring the output of a channel. Two questions raised by Sarovar and Milburn about their bound are answered. A Riemannian metric on the space of quantum states is introduced, related to the construction of the Sarovar and Milburn bound. Its properties are characterized. The second topic investigated is the estimation of unitary channels. The situation is considered in which an experimenter has several non-identical unitary channels that have the same parameter. It is shown that it is possible to improve estimation using the channels together, analogous to the case of identical unitary channels. Also, a new method of phase estimation is given based on a method sketched by Kitaev (1996). Unlike other phase estimation procedures which perform similarly, this procedure requires only very basic experimental resources.

Doctor of Philosophy / Department of Mathematics / Diego Maldonado / Originally introduced in 1961 by Carl Gustav Axel Harnack [36] in the context of harmonic functions in R[superscript]2, the so-called Harnack inequality has since been established for solutions to a wide variety of different partial differential equations (PDEs) by mathematicians at different times of its historical development. Among them, Moser's iterative scheme [47-49] and Krylov-Safonov's probabilistic method [43, 44] stand out as pioneering theories, both in terms of their originality and their impact on the study of regularity of solutions to PDEs. Caffarelli's work [12] in 1989 greatly simplified Krylov-Safonov's theory and established Harnack's inequality in the context of fully non-linear elliptic PDEs. In this scenario, Caffarelli and Gutierrez's study of the linearized Monge-Ampere equation [15, 16] in 2002-2003 served as a motivation for axiomatizations of Krylov-Safonov-Caffarelli theory [3, 25, 57]. The main work in this dissertation is a new axiomatization of Krylov-Safonov-Caffarelli theory. Our axiomatic approach to Harnack's inequality in spaces of homogeneous type has some distinctive features. It sheds more light onto the role of the so-called critical density property, a property which is at the heart of the techniques developed by Krylov and Safonov. Our structural assumptions become more natural, and thus, our theory better suited, in the context of variational PDEs. We base our method on the theory of Muckenhoupt's A[subscript]p weights. The dissertation also gives an application of our axiomatic approach to Harnack's inequality in the context of infinite graphs. We provide an alternate proof of Harnack's inequality for harmonic functions on graphs originally proved in [21].

Neste trabalho apresentaremos o princípio seletivo S1(O;H ) que caracteriza a propriedade da diagonal Gd . Iremos também apresentar um jogo topológico G1(O;H ) induzido por esse princípio seletivo e estudaremos as relações entre esse jogo e a propriedade da diagonal Gd . Além disso, apresentaremos outros jogos topológicos e mostraremos quais são as relações destes com o jogo G1(O;H ). Finalmente, daremos algumas aplicações desses jogos e exemplos / In this work we present a selection principle S1(O;H) that characterizes the Gd -diagonal property. We also present a topological game G1(O;H ) induced by this selection principle and we study the relations between this game and the Gd -diagonal property. Besides that, we present other topological games and we show which are the relations between those topological games and the game G1(O;H ). Finally, we give some applications and examples.

Fractal geometry is a new branch of mathematics. This report presents the tools, methods and theory required to describe this geometry. The power of Iterated Function Systems (IFS) is introduced and applied to produce fractal images or approximate complex estructures found in nature. The focus of this thesis is on how fractal geometry can be used in applications to computer graphics or to model natural objects.

Submitted by CAMILA TOLIN SANTOS DA SILVA (camilatolin@yahoo.com.br) on 2018-08-30T19:02:42Z No. of bitstreams: 1 Camila Tolin - Dissertação mestrado versão final.pdf: 53772690 bytes, checksum: e820d5f4112aab671ce28d3bd47eab57 (MD5) / Rejected by Elza Mitiko Sato null (elzasato@ibilce.unesp.br), reason: Solicitamos que realize correções na submissão seguindo as orientações abaixo: Problema 01) É necessário que o arquivo contendo a sua dissertação esteja no formato PDF (Portable Document Format) e não esteja protegido . Problema 02) A ficha catalográfica deve ser corrigida pois nela consta: ”Instituto de Biociências, Letras e Ciências Exatas”, Presidente Prudente, o nome do Câmpus de Presidente Prudente é Faculdade de Ciências e Tecnologia. Problema 03) Solicito que corrija também na descrição na natureza da pesquisa na folha de rosto e aprovação: Dissertação apresentada como parte dos requisitos para obtenção do título de Mestre, junto ao programa de Mestrado Profissional em Matemática em Rede Nacional - PROFMAT, da Faculdade de Ciências e Tecnologia da Universidade Estadual Paulista Júlio de Mesquita Filho , Câmpus de Presidente Prudente. Problema 04) O arquivo contém 12(doze) páginas em branco (pags. 04, 06, 08, 10, 12, 14, 24, 42, 52, 82, 98 e 138) as mesmas devem retiradas pois o arquivo não pode conter páginas em branco. e deve ser corrigido também na ficha catalográfica o número de folhas. Lembramos que o arquivo depositado no repositório deve ser igual ao impresso, o rigor com o padrão da Universidade se deve ao fato de que o seu trabalho passará a ser visível mundialmente. Agradecemos a compreensão. on 2018-08-31T13:19:11Z (GMT) / Submitted by CAMILA TOLIN SANTOS DA SILVA (camilatolin@yahoo.com.br) on 2018-09-03T19:10:00Z No. of bitstreams: 1 Camila Tolin - Dissertação mestrado versão final 03-09-2018.pdf: 53765385 bytes, checksum: 97efaf5bb1d8f5cfbffec24611d1e062 (MD5) / Approved for entry into archive by Elza Mitiko Sato null (elzasato@ibilce.unesp.br) on 2018-09-03T19:52:36Z (GMT) No. of bitstreams: 1 silva_cts_me_sjrp.pdf: 9381939 bytes, checksum: 551bd372a749383dd387d1f2049a82bf (MD5) / Made available in DSpace on 2018-09-03T19:52:36Z (GMT). No. of bitstreams: 1 silva_cts_me_sjrp.pdf: 9381939 bytes, checksum: 551bd372a749383dd387d1f2049a82bf (MD5) Previous issue date: 2018-07-31 / A topologia é um ramo da matemática, sutilmente entrelaçado com a Geometria, de aplicação em diversas áreas do conhecimento, cuja conceituação foi apresentada de forma expressiva nas escolas durante as décadas de 60 e 70, com o movimento educacional conhecido como Matemática Moderna. Através das mudanças curriculares, muitos temas abordados no ensino fundamental e médio foram reestruturados dentro de um conjunto de parâmetros para a organização curricular da base nacional comum, os PCN's, que normatizam a base do ensino e orientam que a matemática deve ser apresentada para o desenvolvimento de habilidades inerentes à resolução de problemas, aquisição de linguagem simbólica, modelagem e interpretação de situações cotidianas, argumentação e aplicação em situações da vida real. Portanto, esse trabalho foi elaborado com o objetivo de fornecer suporte para o ensino da topologia no ensino básico, através da compilação de fatos históricos, formalização de definições básicas de caráter introdutório como continuidade, espaços métricos, espaços topológicos, entre outros, apresentação de atividades que poderão ser trabalhadas conjuntamente com o ensino da geometria, que de forma lúdica e intuitiva, ajudarão a alicerçar a base para um futuro aprofundamento desses conceitos, auxiliando no desenvolvimento do pensamento topológico. / Topology is a branch of mathematics, subtly intertwined with geometry, of application in several areas of knowledge, whose conceptualization was presented expressively in schools during the 60s and 70s, with the educational movement known as Modern Mathematics. Through the curricular changes, many topics addressed in elementary and secondary education have been restructured within a set of parameters for the curriculum organization of the common national base, the PCNs, that normalize the base of the teaching and guide that the mathematics must be presented for the development of inherent abilities to solve problems, acquisition of symbolic language, modeling and interpretation of everyday situations, argumentation and application in real life situations. Thus, this work was developed with the purpose of providing support for the teaching of topology in basic education, through the compilation of historical facts, formalization of basic de nitions of introductory character such as continuity, metric spaces, topological spaces, among others, presentation of activities which can be worked together with the teaching geometry, which in a playful and intuitive way, will help to lay the foundation for a future deepening of these concepts, aiding in the development of topological thinking.

Neste trabalho apresentaremos o princípio seletivo S1(O;H ) que caracteriza a propriedade da diagonal Gd . Iremos também apresentar um jogo topológico G1(O;H ) induzido por esse princípio seletivo e estudaremos as relações entre esse jogo e a propriedade da diagonal Gd . Além disso, apresentaremos outros jogos topológicos e mostraremos quais são as relações destes com o jogo G1(O;H ). Finalmente, daremos algumas aplicações desses jogos e exemplos / In this work we present a selection principle S1(O;H) that characterizes the Gd -diagonal property. We also present a topological game G1(O;H ) induced by this selection principle and we study the relations between this game and the Gd -diagonal property. Besides that, we present other topological games and we show which are the relations between those topological games and the game G1(O;H ). Finally, we give some applications and examples.

The relationship between the convergence of a sequence of self mappings of a metric space and their fixed points, known as the stability (or continuity) of fixed points has been of continuing interest and widely studied in fixed point theory. In this thesis we study the stability of common fixed points in a Hausdorff uniform space whose uniformity is generated by a family of pseudometrics, by using some general notations of convergence. These results are then extended to 2-metric spaces due to S. Gähler. In addition, a well-known theorem of T. Suzuki that generalized the Banach Contraction Principle is also extended to 2-metric spaces and applied to obtain a coincidence theorem for a pair of mappings on an arbitrary set with values in a 2-metric space. Further, we prove the existence of coincidence and fixed points of Ćirić type weakly generalized contractions in metric spaces. Subsequently, the above result is utilized to discuss applications to the convergence of modified Mann and Ishikawa iterations in a convex metric space. Finally, we obtain coincidence, fixed and stationary point results for multi-valued and hybrid pairs of mappings on a metric space.

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.

<p>Predmet istraživanja u doktorskoj disertaciji su metode za egzistenciju i konstrukciju nepokretne tačke za jednoznačna i višeznačna preslikavanja kontraktivnog tipa u metričkom i generalizovanim metričkim prostorima (konveksan metrički, fazi metrički i fazi G metrički prostor).</p> / <p>The subject of research in the doctoral dissertation is the methods for the existence and construction of a fixed point for the single and multivalued mappings of a contraction type in metric and in generalized spaces(convex metric spaces, fuzzy metric spaces and fuzzy G-metric spaces).</p>

In this thesis, we discuss three main projects which are related to Polish groups and their actions on standard Borel spaces. In the first part, we show that the complexity of the classification problem of continua is Borel bireducible to a universal orbit equivalence relation induce by a Polish group on a standard Borel space. In the second part, we compare the relative complexity of various types of classification problems concerning subspaces of [0,1]^n for all natural number n. In the last chapter, we give a topological characterization theorem for the class of locally compact two-sided invariant non-Archimedean Polish groups. Using this theorem, we show the non-existence of a universal group and the existence of a surjectively universal group in the class.