In this dissertation, we explore the approximate diagonalization of unital homomorphisms between C*-algebras. In particular, we prove that unital homomorphisms from commutative C*-algebras into simple separable unital C*-algebras with tracial rank at most one are approximately diagonalizable. This is equivalent to the approximate diagonalization of commuting sets of normal matrices. We also prove limited generalizations of this theorem. Namely, certain injective unital homomorphisms from commutative C*-algebras into simple separable unital C*-algebras with rational tracial rank at most one are shown to be approximately diagonalizable. Also unital injective homomorphisms from AH-algebras with unique tracial state into separable simple unital C*-algebras of tracial rank at most one are proved to be approximately diagonalizable. Counterexamples are provided showing that these results cannot be extended in general. Finally, we prove that for unital homomorphisms between AF-algebras, approximate diagonalization is equivalent to a combinatorial problem involving sections of lattice points in cones.

Orientador: Prof. Dr. Ricardo Suyama / Dissertação (mestrado) - Universidade Federal do ABC, Programa de Pós-Graduação em Engenharia da Informação, 2017. / O metodo estatistico de Independent Component Analysis (ICA) e um dos mais amplamente utilizados para solucionar o problema de Blind Source Separation (BSS) que, junto a outros metodos de processamento de sinais, sao colocados a prova com o aumento do numero das fontes de sinais e amostras disponiveis para processamento, e sao a base de aplicacoes com requisitos de desempenho cada vez maiores. O objetivo deste trabalho e realizar o estudo do metodo ICA e analise dos algoritmos FastICA e Joint Approximate Diagonalization of Eigen-matrices (JADE) implementados em Field-Programmable Gate Array (FPGA) e seu comportamento quando variamos o numero de amostras das misturas e os numeros de iteracoes ou updates. Outros trabalhos de pesquisa ja foram realizados com o objetivo de demonstrar a viabilidade da implementacao de tais algoritmos em FPGA, mas pouco apresentam sobre o metodo utilizado para definir detalhes de implementacao como numero de amostradas utilizados, a razao da representacao numerica escolhida e sobre o thoughtput alcancado. A analise que este trabalho propos realizar, num primeiro momento, passa por demonstrar o comportamento do core dos algoritmos quando implementados utilizando diferentes representacoes numericas de ponto flutuante com precisao simples (32 bits) e ponto fixo com diferentes numeros de amostras e fontes a serem estimadas, por meio de simulacoes. Foi verificada a viabilidade desses serem utilizados para atender aplicacoes que precisam resolver o problema de BSS com boa acuracia, quando comparados com implementacoes dos mesmos algoritmos que se utilizaram de uma representacao numerica de ponto flutuante com precisao dupla (64 bits). Utilizando o Simulink R¿e a biblioteca DSP Builder R¿da Altera R¿para implementar os modelos de cada algoritmo, foi possivel analisar outros aspectos importantes, em busca de demonstrar a possibilidade da utilizacao de tais implementacoes em aplicacoes com requisitos de tempo real, que necessitam de alto desempenho, utilizando FPGA de baixo custo, como: a quantidade de recursos de FPGA necessarios na implementacao de cada algoritmo, principalmente buscando minimizar a utilizacao de blocos DSP, a latencia, e maximizar o throughput de processamento. / Independent Component Analysis (ICA) is one of the most widely used statistical method to solve the problem of Blind Source Separation (BSS), which, along with other signal processing methods, faces new challenges with the increasing the number of signal sources and samples available for processing, being the base of applications with increasing performance requirements. The aim of this work is to study the FastICA and the Joint Approximate Diagonalization of Eigen-matrices (JADE) algorithms and implement them in Field- Programmable Gate Array (FPGA). Other researches have already been carried out with the objective of demonstrating the feasibility of implementing such algorithms in FPGA, but they present little about the methodology used and implementation details such as the number of samples used, why the numerical representation was chosen and the obtained thoughtput. The analysis carried out in this work demonstrates the behavior of the core of the algorithms when implemented using different representations, such as singleprecision floating-point (32 bits) and fixed point with different numbers of samples and sources to be estimated. It was verified these immplementations are able to solve the BSS problem with good accuracy when compared with implementations of the same algorithms that exmploy a double-precision floating-point representation (64 bits). Using the Simulink R¿ and Alterafs R¿ DSP Builder R¿ library to implement the models of each algorithm, it was possible to analyze other important aspects, in order to demonstrate the possibility of using such implementations in applications with real-time requirements that require high performance, using low cost FPGA, such as: the necessary FPGA resources in the implementation of each algorithm, mainly seeking to minimize the use of DSP blocks, latency, and to maximize the processing throughput.

In this work a numerical approach for the determination of the energy spectra and the calculation of thermodynamic properties of magnetic molecules is presented. The work is focused on the treatment of spin systems which exhibit point-group symmetries. Ring-like and archimedean-type structures are discussed as prominent examples. In each case the underlying spin quantum system is modeled by an isotropic Heisenberg Hamiltonian. Its energy spectrum is calculated either by numerical exact diagonalization or by an approximate diagonalization method introduced here. In order to implement full spin-rotational symmetry the numerical approach at hand is based on the use of irreducible tensor operators. Furthermore, it is shown how an unrestricted use of point-group symmetries in combination with the use of irreducible tensor operators leads to a reduction of the dimensionalities as well as to additional information about the physics of the systems. By exemplarily demonstrating how the theoretical foundations of the irreducible tensor operator technique can be realized within small spin systems the technical aspect of this work is covered. These considerations form the basis of the computational realization that was implemented and used in order to get insight into the investigated systems.