A study of singular integral operators with shift

Nesta tese, consideram-se operadores integrais singulares com a acção extra de um operador de deslocacamento de Carleman e com coeficientes em diferentes classes de funções essencialmente limitadas. Nomeadamente, funções contínuas por troços, funções quase-periódicas e funções possuíndo factorização generalizada. Nos casos dos operadores integrais singulares com deslocamento dado pelo operador de reflexão ou pelo operador de salto no círculo unitário complexo, obtêm-se critérios para a propriedade de Fredholm. Para os coeficientes contínuos, uma fórmula do índice de Fredholm é apresentada. Estes resultados são consequência das relações de equivalência explícitas entre aqueles operadores e alguns operadores adicionais, tais como o operador integral singular, operadores de Toeplitz e operadores de Toeplitz mais Hankel. Além disso, as relações de equivalência permitem-nos obter um critério de invertibilidade e fórmulas para os inversos laterais dos operadores iniciais com coeficientes factorizáveis. Adicionalmente, aplicamos técnicas de análise numérica, tais como métodos de colocação de polinómios, para o estudo da dimensão do núcleo dos dois tipos de operadores integrais singulares com coeficientes contínuos por troços. Esta abordagem permite também a computação do inverso no sentido Moore-Penrose dos operadores principais. Para operadores integrais singulares com operadores de deslocamento do tipo Carleman preservando a orientação e com funções contínuas como coeficientes, são obtidos limites superiores da dimensão do núcleo. Tal é implementado utilizando algumas estimativas e com a ajuda de relações (explícitas) de equivalência entre operadores. Focamos ainda a nossa atenção na resolução e nas soluções de uma classe de equações integrais singulares com deslocamento que não pode ser reduzida a um problema de valor de fronteira binomial. De forma a atingir os objectivos propostos, foram utilizadas projecções complementares e identidades entre operadores. Desta forma, as equações em estudo são associadas a sistemas de equações integrais singulares. Estes sistemas são depois analisados utilizando um problema de valor de fronteira de Riemann. Este procedimento tem como consequência a construção das soluções das equações iniciais a partir das soluções de problemas de valor de fronteira de Riemann. Motivados por uma grande diversidade de aplicações, estendemos a definição de operador integral de Cauchy para espaços de Lebesgue sobre grupos topológicos. Assim, são investigadas as condições de invertibilidade dos operadores integrais neste contexto.

Laplacian eigenvectors and eigenvalues and almost equitable partitions

Relations between Laplacian eigenvectors and eigenvalues and the existence of almost equitable partitions (which are generalizations of equitable partitions) are presented. Furthermore, on the basis of some properties of the adjacency eigenvectors of a graph, a necessary and sufficient condition for the graph to be primitive strongly regular is introduced. © 2006 Elsevier Ltd. All rights reserved.

Regularity of Wiener-Hopf plus Hankel operators

In this thesis we consider Wiener-Hopf-Hankel operators with Fourier symbols in the class of almost periodic, semi-almost periodic and piecewise almost periodic functions. In the first place, we consider Wiener-Hopf-Hankel operators acting between L2 Lebesgue spaces with possibly different Fourier matrix symbols in the Wiener-Hopf and in the Hankel operators. In the second place, we consider these operators with equal Fourier symbols and acting between weighted Lebesgue spaces Lp(R;w), where 1 < p < 1 and w belongs to a subclass of Muckenhoupt weights. In addition, singular integral operators with Carleman shift and almost periodic coefficients are also object of study. The main purpose of this thesis is to obtain regularity properties characterizations of those classes of operators. By regularity properties we mean those that depend on the kernel and cokernel of the operator. The main techniques used are the equivalence relations between operators and the factorization theory. An invertibility characterization for the Wiener-Hopf-Hankel operators with symbols belonging to the Wiener subclass of almost periodic functions APW is obtained, assuming that a particular matrix function admits a numerical range bounded away from zero and based on the values of a certain mean motion. For Wiener-Hopf-Hankel operators acting between L2-spaces and with possibly different AP symbols, criteria for the semi-Fredholm property and for one-sided and both-sided invertibility are obtained and the inverses for all possible cases are exhibited. For such results, a new type of AP factorization is introduced. Singular integral operators with Carleman shift and scalar almost periodic coefficients are also studied. Considering an auxiliar and simpler operator, and using appropriate factorizations, the dimensions of the kernels and cokernels of those operators are obtained. For Wiener-Hopf-Hankel operators with (possibly different) SAP and PAP matrix symbols and acting between L2-spaces, criteria for the Fredholm property are presented as well as the sum of the Fredholm indices of the Wiener-Hopf plus Hankel and Wiener-Hopf minus Hankel operators. By studying dependencies between different matrix Fourier symbols of Wiener-Hopf plus Hankel operators acting between L2-spaces, results about the kernel and cokernel of those operators are derived. For Wiener-Hopf-Hankel operators acting between weighted Lebesgue spaces, Lp(R;w), a study is made considering equal scalar Fourier symbols in the Wiener-Hopf and in the Hankel operators and belonging to the classes of APp;w, SAPp;w and PAPp;w. It is obtained an invertibility characterization for Wiener-Hopf plus Hankel operators with APp;w symbols. In the cases for which the Fourier symbols of the operators belong to SAPp;w and PAPp;w, it is obtained semi-Fredholm criteria for Wiener-Hopf-Hankel operators as well as formulas for the Fredholm indices of those operators.

Sectorial antennas for WSN localization system

This work investigates low cost localization systems (LS) based on received signal strength (RSS) and integrated with different types of antennas with main emphasis on sectorial antennas. The last few years have witnessed an outstanding growth in wireless sensor networks (WSN). Among its various possible applications, the localization field became a major area of research. The localization techniques based on RSS are characterized by simplicity and low cost of integration. The integration of LS based on RSS and sectorial antennas (SA) was proven to provide an effective solution for reducing the number of required nodes of the networks and allows the combination of several techniques, such as RSS and angle of arrival (AoA). This PhD thesis focuses on studying techniques, antennas and protocols that best meet the needs of each LS with main focus on low cost systems based on RSS and AoA. Firstly there are studied localization techniques and system that best suit the requirements of the user and the antennas that are most appropriate according to the nature of the signal. In this step it is intended to provide a fundamental understanding of the undertaken work. Then the developed antennas are presented according to the following categories: sectorial and microstrip antennas. Two sectorial antennas are presented: a narrowband antenna operating at 2.4 to 2.5 GHz and a broadband antenna operating at 800MHz-2.4GHz. The low cost printed antennas were designed to operate at 5 GHz, which may be used for vehicular communication. After presenting the various antennas, several prototypes of indoor/outdoor LS are implemented and analyzed. Localization protocols are also proposed, one based on simplicity and low power, and the other on interoperability with different types of antennas and system requirements.

Network virtualisation from an operator perspective

Network virtualisation is seen as a promising approach to overcome the so-called “Internet impasse” and bring innovation back into the Internet, by allowing easier migration towards novel networking approaches as well as the coexistence of complementary network architectures on a shared infrastructure in a commercial context. Recently, the interest from the operators and mainstream industry in network virtualisation has grown quite significantly, as the potential benefits of virtualisation became clearer, both from an economical and an operational point of view. In the beginning, the concept has been mainly a research topic and has been materialized in small-scale testbeds and research network environments. This PhD Thesis aims to provide the network operator with a set of mechanisms and algorithms capable of managing and controlling virtual networks. To this end, we propose a framework that aims to allocate, monitor and control virtual resources in a centralized and efficient manner. In order to analyse the performance of the framework, we performed the implementation and evaluation on a small-scale testbed. To enable the operator to make an efficient allocation, in real-time, and on-demand, of virtual networks onto the substrate network, it is proposed a heuristic algorithm to perform the virtual network mapping. For the network operator to obtain the highest profit of the physical network, it is also proposed a mathematical formulation that aims to maximize the number of allocated virtual networks onto the physical network. Since the power consumption of the physical network is very significant in the operating costs, it is important to make the allocation of virtual networks in fewer physical resources and onto physical resources already active. To address this challenge, we propose a mathematical formulation that aims to minimize the energy consumption of the physical network without affecting the efficiency of the allocation of virtual networks. To minimize fragmentation of the physical network while increasing the revenue of the operator, it is extended the initial formulation to contemplate the re-optimization of previously mapped virtual networks, so that the operator has a better use of its physical infrastructure. It is also necessary to address the migration of virtual networks, either for reasons of load balancing or for reasons of imminent failure of physical resources, without affecting the proper functioning of the virtual network. To this end, we propose a method based on cloning techniques to perform the migration of virtual networks across the physical infrastructure, transparently, and without affecting the virtual network. In order to assess the resilience of virtual networks to physical network failures, while obtaining the optimal solution for the migration of virtual networks in case of imminent failure of physical resources, the mathematical formulation is extended to minimize the number of nodes migrated and the relocation of virtual links. In comparison with our optimization proposals, we found out that existing heuristics for mapping virtual networks have a poor performance. We also found that it is possible to minimize the energy consumption without penalizing the efficient allocation. By applying the re-optimization on the virtual networks, it has been shown that it is possible to obtain more free resources as well as having the physical resources better balanced. Finally, it was shown that virtual networks are quite resilient to failures on the physical network.