Ph.D. (Mathematics) / Our objective in this thesis is to investigate two fundamental questions concerning Riesz operators de ned on a Banach space. Recall that Riesz operators are generalizations of compact operators in the sense that Riesz operators have the same spectral properties as compact operators. However, they do not possess the same algebraic properties as compact operators. Our rst question that we investigate is: When is a Riesz operator a nite rank operator? This question is motivated from the fact that if a compact operator de ned on a Banach space has closed range, then it is a nite rank operator. Also, Ghahramani proved that a compact homomorphism de ned on a C -algebra is a nite rank operator, see . Martin Mathieu, in his paper, generalized the result of Ghahramani by proving that a weakly compact homomorphism de ned on a C -algebra is a nite rank operator...

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The purpose of this paper is to present the results of a study of linear spaces with special emphasis of linear transformations, norms, and inner products.

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by Yan Shing. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (leaves 78-80). / Abstract also in Chinese. / Chapter 1 --- Admissible Points of Convex Sets --- p.7 / Chapter 1.1 --- Introduction and Notations --- p.7 / Chapter 1.2 --- The Main Result --- p.7 / Chapter 1.2.1 --- The Proof of Theoreml.2.1 --- p.8 / Chapter 1.3 --- An Application --- p.10 / Chapter 2 --- A Generalization on The Theorems of Admissible Points --- p.12 / Chapter 2.1 --- Introduction and Notations --- p.12 / Chapter 2.2 --- Fundamental Lemmas --- p.14 / Chapter 2.3 --- The Main Result --- p.16 / Chapter 3 --- Introduction to Variational Inequalities --- p.21 / Chapter 3.1 --- Variational Inequalities in Finite Dimensional Space --- p.21 / Chapter 3.2 --- Problems Which Relate to Variational Inequalities --- p.25 / Chapter 3.3 --- Some Variations on Variational Inequality --- p.28 / Chapter 3.4 --- The Vector Variational Inequality Problem and Its Relation with The Vector Optimization Problem --- p.29 / Chapter 3.5 --- Variational Inequalities in Hilbert Space --- p.31 / Chapter 4 --- Vector Variational Inequalities --- p.36 / Chapter 4.1 --- Preliminaries --- p.36 / Chapter 4.2 --- Notations --- p.37 / Chapter 4.3 --- Existence Results of Vector Variational Inequality --- p.38 / Chapter 5 --- The Generalized Quasi-Variational Inequalities --- p.44 / Chapter 5.1 --- Introduction --- p.44 / Chapter 5.2 --- Properties of The Class F0 --- p.46 / Chapter 5.3 --- Main Theorem --- p.53 / Chapter 5.4 --- Remarks --- p.58 / Chapter 6 --- A set-valued open mapping theorem and related re- sults --- p.61 / Chapter 6.1 --- Introduction and Notations --- p.61 / Chapter 6.2 --- An Open Mapping Theorem --- p.62 / Chapter 6.3 --- Main Result --- p.63 / Chapter 6.4 --- An Application on Ordered Normed Spaces --- p.66 / Chapter 6.5 --- An Application on Open Decomposition --- p.70 / Chapter 6.6 --- An Application on Continuous Mappings from Order- infrabarreled Spaces --- p.72 / Bibliography

Dado E um espaço vetorial topológico e T um operador linear contínuo em E, diremos que T é hipercíclico se, para algum elemento x pertencente a E, a órbita de x sob T, Orb(x,T)={x, Tx, T^2 x,...}, for densa em E. Nosso objetivo será apresentar alguns resultados sobre hiperciclicidade e observar como alguns espaços comportam-se diante dessa classe de operadores. \\\\ / Let E be a topological vector space and T a continuous linear operator on E. We say that T is hypercyclic if, for some x in E, the orbit of x on T, Orb(x,T)={x, Tx, T^2 x,...}, is dense in E. Our aim will be to study some results about hypercyclicity and to observe how some spaces behave regarding this class of operators.

Dado E um espaço vetorial topológico e T um operador linear contínuo em E, diremos que T é hipercíclico se, para algum elemento x pertencente a E, a órbita de x sob T, Orb(x,T)={x, Tx, T^2 x,...}, for densa em E. Nosso objetivo será apresentar alguns resultados sobre hiperciclicidade e observar como alguns espaços comportam-se diante dessa classe de operadores. \\\\ / Let E be a topological vector space and T a continuous linear operator on E. We say that T is hypercyclic if, for some x in E, the orbit of x on T, Orb(x,T)={x, Tx, T^2 x,...}, is dense in E. Our aim will be to study some results about hypercyclicity and to observe how some spaces behave regarding this class of operators.

Codensity monads are ubiquitous, as are various different notions of compactness and finiteness. Two such examples of "compact" spaces are compact Hausdorff Spaces and Linearly Compact Vector Spaces. Compact Hausdorff Spaces are the algebras of the codensity monad induced by the inclusion of finite sets in the category of sets. Similarly linearly compact vector spaces are the algebras of the codensity monad induced by the inclusion of finite dimensional vector spaces in the category of vector spaces. So in these two examples the notions of finiteness, compactness and codensity are intertwined. In this thesis we generalise these results. To do this we generalise the notion of ultrafilter, and follow the intuition of the compact Hausdorff case. We give definitions of general notions of "finiteness" and "compactness" and show that the algebras for the codensity monad induced by the "finite" objects are exactly the "compact" objects.

O presente trabalho realiza o estudo e implementação de um filtro ativo de potência paralelo (FAPP) aplicado em sistemas trifásicos a quatro fios utilizando a topologia de um inversor de tensão four-legs, visando a redução do conteúdo harmônico gerado por cargas não lineares e a compensação de reativos. A geração das correntes de referência de compensação é obtida através da estratégia de compensação baseada no sistema de eixos de referência síncrona (SRF) a qual utiliza conceitos de controle vetorial. O sincronismo do sistema SRF é realizado através de um circuito q-PLL (q-Phase Locked Loop) o qual é baseado no conceito da teoria da potência instantânea imaginária (teoria pq). A análise matemática da topologia four-legs é desenvolvida a fim de se obter um modelo linear em espaço de estados que represente o sistema físico para os sistemas de coordenadas abc, αβ0 e dq0. O método de controle proposto é implementado em eixos dq0, através de três controladores do tipo PI (Proporcional-Integral), os quais são projetados utilizando os conceitos de margem de estabilidade. Um estudo detalhado da modulação Space Vector aplicada a sistemas trifásicos four-legs é apresentado. Resultados de simulações são apresentados para validar o modelo do filtro proposto bem como a técnica de controle aplicada. Por fim, os resultados experimentais obtidos são avaliados considerando a a recomendação IEEE 519-1992 para mostrar a capacidade do FAPP na supressão de correntes harmônicas e compensação de potência reativa. / This work deals with the study and analysis of a parallel active power filter (APF) applied to three-phase four-wire systems using a four-leg inverter, aiming the suppresion of the harmonic content of non-linear loads and reactive power compensation. The generation of the compensation current references is obtained by means of the strategy based on the synchronous reference frame (SRF) system, which utilizes the concepts of vector control. The timing of the SRF system is performed through a q-PLL (q-Phase Locked Loop) circuit that is based on the imaginary instantaneous power theory. Mathematical analysis is developed in order to obtain a linear model in state space that represents the physical system in the coordinate systems abc, αβ0 and dq0. The proposed control method is implemented in dq0 axes through three Proportional-Integral (PI) controllers, which are designed using the concepts of stability margin. A detailed study of Space Vector modulation applied to three-phase four-leg inverter is presented. Simulation results are presented to validate the model of the APF and the control technique adopted. Finally, experimental results are obtained and evaluated considering the recommendation IEEE 519-1992 to show the capability of the parallel APF of current harmonic suppression and reactive power comensation.

Grothendieck published an extraordinary paper entitled ”Resume de la theorie metrique des pro¬duits tensoriels topologiques” in 1953. The main result of this paper is the inequality which is commonly known as Grothendieck Inequality. Following Kirivine, in this article, we give the proof of Grothendieck Inequality. We refor¬mulate it in different forms. We also investigate the famous Grothendieck constant KG. The Grothendieck constant was achieved by taking supremum over a special class of matrices. But our attempt will be to investigate it, considering a smaller class of matrices, namely only the positive definite matrices in this class. Actually we want to use it to get a counterexample of Matsaev’s conjecture, which was proved to be right by Von Neumann in some specific cases. In chapter 1, we shall state and prove the Grothendieck Inequality. In chapter 2, we shall introduce tensor product of vector spaces and different tensor norms. In chapter 3, we shall formulate Grothendieck Inequality in different forms and use the notion of tensor norms for its equivalent formation .In the last chapteri.ein chapter4we shall investigate on the Grothendieck constant.