Two new integral transforms and their applications

This thesis is in two parts. In Part I the independent variable θ in the trigonometric form of Legendre's equation is extended to the range ( -∞, ∞). The associated spectral representation is an infinite integral transform whose kernel is the analytic continuation of the associated Legendre function of the second kind into the complex θ-plane. This new transform is applied to the problems of waves on a spherical shell, heat flow on a spherical shell, and the gravitational potential of a sphere. In each case the resulting alternative representation of the solution is more suited to direct physical interpretation than the standard forms.

In Part II separation of variables is applied to the initial-value problem of the propagation of acoustic waves in an underwater sound channel. The Epstein symmetric profile is taken to describe the variation of sound with depth. The spectral representation associated with the separated depth equation is found to contain an integral and a series. A point source is assumed to be located in the channel. The nature of the disturbance at a point in the vicinity of the channel far removed from the source is investigated.

Part I. On the breaking of nonlinear dispersive waves. Part II. Variational principles in continuum mechanics

A model equation for water waves has been suggested by Whitham to study, qualitatively at least, the different kinds of breaking. This is an integro-differential equation which combines a typical nonlinear convection term with an integral for the dispersive effects and is of independent mathematical interest. For an approximate kernel of the form e^(-b|x|) it is shown first that solitary waves have a maximum height with sharp crests and secondly that waves which are sufficiently asymmetric break into "bores." The second part applies to a wide class of bounded kernels, but the kernel giving the correct dispersion effects of water waves has a square root singularity and the present argument does not go through. Nevertheless the possibility of the two kinds of breaking in such integro-differential equations is demonstrated.

Difficulties arise in finding variational principles for continuum mechanics problems in the Eulerian (field) description. The reason is found to be that continuum equations in the original field variables lack a mathematical "self-adjointness" property which is necessary for Euler equations. This is a feature of the Eulerian description and occurs in non-dissipative problems which have variational principles for their Lagrangian description. To overcome this difficulty a "potential representation" approach is used which consists of transforming to new (Eulerian) variables whose equations are self-adjoint. The transformations to the velocity potential or stream function in fluids or the scaler and vector potentials in electromagnetism often lead to variational principles in this way. As yet no general procedure is available for finding suitable transformations. Existing variational principles for the inviscid fluid equations in the Eulerian description are reviewed and some ideas on the form of the appropriate transformations and Lagrangians for fluid problems are obtained. These ideas are developed in a series of examples which include finding variational principles for Rossby waves and for the internal waves of a stratified fluid.

Nonlinear dynamic transfer functions for certain retinal neuronal systems

The applicability of the white-noise method to the identification of a nonlinear system is investigated. Subsequently, the method is applied to certain vertebrate retinal neuronal systems and nonlinear, dynamic transfer functions are derived which describe quantitatively the information transformations starting with the light-pattern stimulus and culminating in the ganglion response which constitutes the visually-derived input to the brain. The retina of the catfish, Ictalurus punctatus, is used for the experiments.

The Wiener formulation of the white-noise theory is shown to be impractical and difficult to apply to a physical system. A different formulation based on crosscorrelation techniques is shown to be applicable to a wide range of physical systems provided certain considerations are taken into account. These considerations include the time-invariancy of the system, an optimum choice of the white-noise input bandwidth, nonlinearities that allow a representation in terms of a small number of characterizing kernels, the memory of the system and the temporal length of the characterizing experiment. Error analysis of the kernel estimates is made taking into account various sources of error such as noise at the input and output, bandwidth of white-noise input and the truncation of the gaussian by the apparatus.

Nonlinear transfer functions are obtained, as sets of kernels, for several neuronal systems: Light → Receptors, Light → Horizontal, Horizontal → Ganglion, Light → Ganglion and Light → ERG. The derived models can predict, with reasonable accuracy, the system response to any input. Comparison of model and physical system performance showed close agreement for a great number of tests, the most stringent of which is comparison of their responses to a white-noise input. Other tests include step and sine responses and power spectra.

Many functional traits are revealed by these models. Some are: (a) the receptor and horizontal cell systems are nearly linear (small signal) with certain "small" nonlinearities, and become faster (latency-wise and frequency-response-wise) at higher intensity levels, (b) all ganglion systems are nonlinear (half-wave rectification), (c) the receptive field center to ganglion system is slower (latency-wise and frequency-response-wise) than the periphery to ganglion system, (d) the lateral (eccentric) ganglion systems are just as fast (latency and frequency response) as the concentric ones, (e) (bipolar response) = (input from receptors) - (input from horizontal cell), (f) receptive field center and periphery exert an antagonistic influence on the ganglion response, (g) implications about the origin of ERG, and many others.

An analytical solution is obtained for the spatial distribution of potential in the S-space, which fits very well experimental data. Different synaptic mechanisms of excitation for the external and internal horizontal cells are implied.

Projections in a normed linear space and a generalization of the pseudo-inverse

The concept of a "projection function" in a finite-dimensional real or complex normed linear space H (the function P_M which carries every element into the closest element of a given subspace M) is set forth and examined.

If dim M = dim H - 1, then P_M is linear. If P_N is linear for all k-dimensional subspaces N, where 1 ≤ k < dim M, then P_M is linear.

The projective bound Q, defined to be the supremum of the operator norm of P_M for all subspaces, is in the range 1 ≤ Q < 2, and these limits are the best possible. For norms with Q = 1, P_M is always linear, and a characterization of those norms is given.

If H also has an inner product (defined independently of the norm), so that a dual norm can be defined, then when P_M is linear its adjoint P_M^H is the projection on (kernel P_M)^⊥ by the dual norm. The projective bounds of a norm and its dual are equal.

The notion of a pseudo-inverse F⁺ of a linear transformation F is extended to non-Euclidean norms. The distance from F to the set of linear transformations G of lower rank (in the sense of the operator norm ∥F - G∥) is c/∥F⁺∥, where c = 1 if the range of F fills its space, and 1 ≤ c < Q otherwise. The norms on both domain and range spaces have Q = 1 if and only if (F⁺)⁺ = F for every F. This condition is also sufficient to prove that we have (F⁺)^H = (F^H)⁺, where the latter pseudo-inverse is taken using dual norms.

In all results, the real and complex cases are handled in a completely parallel fashion.

Kinetic theory of normal quantum fluids

Close to equilibrium, a normal Bose or Fermi fluid can be described by an exact kinetic equation whose kernel is nonlocal in space and time. The general expression derived for the kernel is evaluated to second order in the interparticle potential. The result is a wavevector- and frequency-dependent generalization of the linear Uehling-Uhlenbeck kernel with the Born approximation cross section.

The theory is formulated in terms of second-quantized phase space operators whose equilibrium averages are the n-particle Wigner distribution functions. Convenient expressions for the commutators and anticommutators of the phase space operators are obtained. The two-particle equilibrium distribution function is analyzed in terms of momentum-dependent quantum generalizations of the classical pair distribution function h(k) and direct correlation function c(k). The kinetic equation is presented as the equation of motion of a two -particle correlation function, the phase space density-density anticommutator, and is derived by a formal closure of the quantum BBGKY hierarchy. An alternative derivation using a projection operator is also given. It is shown that the method used for approximating the kernel by a second order expansion preserves all the sum rules to the same order, and that the second-order kernel satisfies the appropriate positivity and symmetry conditions.

Neutron slowing down with inelastic scattering

The emphasis in reactor physics research has shifted toward investigations of fast reactors. The effects of high energy neutron processes have thus become fundamental to our understanding, and one of the most important of these processes is nuclear inelastic scattering. In this research we include inelastic scattering as a primary energy transfer mechanism, and study the resultant neutron energy spectrum in an infinite medium. We assume that the moderator material has a high mass number, so that in a laboratory coordinate system the energy loss of an inelastically scattered neutron may be taken as discrete. It is then consistent to treat elastic scattering with an age theory expansion. Mathematically these assumptions lead to balance equations of the differential-difference type.

The steady state problem is explored first by way of Laplace transformation of the energy variable. We then develop another steady state technique, valid for multiple inelastic level excitations, which depends on the level structure satisfying a physically reasonable constraint. In all cases the solutions we generate are compared with results obtained by modeling inelastic scattering with a separable, evaporative kernel.

The time dependent problem presents some new difficulties. By modeling the elastic scattering cross section in a particular way, we generate solutions to this more interesting problem. We conjecture the method of characteristics may be useful in analyzing time dependent problems with general cross sections. These ideas are briefly explored.

Pulsed neutron experiments in graphite

Detailed pulsed neutron measurements have been performed in graphite assemblies ranging in size from 30.48 cm x 38.10 cm x 38.10 cm to 91.44 cm x 66.67 cm x 66.67 cm. Results of the measurement have been compared to a modeled theoretical computation.

In the first set of experiments, we measured the effective decay constant of the neutron population in ten graphite stacks as a function of time after the source burst. We found the decay to be non-exponential in the six smallest assemblies, while in three larger assemblies the decay was exponential over a significant portion of the total measuring interval. The decay in the largest stack was exponential over the entire ten millisecond measuring interval. The non-exponential decay mode occurred when the effective decay constant exceeded 1600 sec^( -1).

In a second set of experiments, we measured the spatial dependence of the neutron population in four graphite stacks as a function of time after the source pulse. By doing an harmonic analysis of the spatial shape of the neutron distribution, we were able to compute the effective decay constants of the first two spatial modes. In addition, we were able to compute the time dependent effective wave number of neutron distribution in the stacks.

Finally, we used a Laplace transform technique and a simple modeled scattering kernel to solve a diffusion equation for the time and energy dependence of the neutron distribution in the graphite stacks. Comparison of these theoretical results with the results of the first set of experiments indicated that more exact theoretical analysis would be required to adequately describe the experiments.

The implications of our experimental results for the theory of pulsed neutron experiments in polycrystalline media are discussed in the last chapter.

Identificação de áreas sob maior risco para leishmaniose visceral na cidade de Teresina, Piauí, Brasil

O propósito desta Tese foi detectar e caracterizar áreas sob alto risco para leishmaniose visceral (LV) e descrever os padrões de ocorrência e difusão da doença, entre os anos de 1993 a 1996 e 2001 a 2006, em Teresina, Piauí, por meio de métodos estatísticos para análise de dados espaciais, sistemas de informações geográficas e imagens de sensoriamento remoto. Os resultados deste estudo são apresentados na forma de três manuscritos. O primeiro usou análise de dados espaciais para identificar as áreas com maior risco de LV na área urbana de Teresina entre 2001 e 2006. Os resultados utilizando razão de kernels demonstraram que as regiões periféricas da cidade foram mais fortemente afetadas ao longo do período analisado. A análise com indicadores locais de autocorrelação espacial mostrou que, no início do período de estudo, os agregados de alta incidência de LV localizavam-se principalmente na região sul e nordeste da cidade, mas nos anos seguintes os eles apareceram também na região norte da cidade, sugerindo que o padrão de ocorrência de LV não é estático e a doença pode se espalhar ocasionalmente para outras áreas do município. O segundo estudo teve como objetivo caracterizar e predizer territórios de alto risco para ocorrência da LV em Teresina, com base em indicadores socioeconômicos e dados ambientais, obtidos por sensoriamento remoto. Os resultados da classificação orientada a objeto apontam a expansão da área urbana para a periferia da cidade, onde antes havia maior cobertura de vegetação. O modelo desenvolvido foi capaz de discriminar 15 conjuntos de setores censitário (SC) com diferentes probabilidades de conterem SC com alto risco de ocorrência de LV. O subconjunto com maior probabilidade de conter SC com alto risco de LV (92%) englobou SC com percentual de chefes de família alfabetizados menor que a mediana (≤64,2%), com maior área coberta por vegetação densa, com percentual de até 3 moradores por domicílio acima do terceiro quartil (>31,6%). O modelo apresentou, respectivamente, na amostra de treinamento e validação, sensibilidade de 79% e 54%, especificidade de 74% e 71%, acurácia global de 75% e 67% e área sob a curva ROC de 83% e 66%. O terceiro manuscrito teve como objetivo avaliar a aplicabilidade da estratégia de classificação orientada a objeto na busca de possíveis indicadores de cobertura do solo relacionados com a ocorrência da LV em meio urbano. Os índices de acurácia foram altos em ambas as imagens (>90%). Na correlação da incidência da LV com os indicadores ambientais verificou-se correlações positivas com os indicadores Vegetação densa, Vegetação rasteira e Solo exposto e negativa com os indicadores Água, Urbana densa e Urbana verde, todos estatisticamente significantes. Os resultados desta tese revelam que a ocorrência da LV na periferia de Teresina está intensamente relacionada às condições socioeconômicas inadequadas e transformações ambientais decorrentes do processo de expansão urbana, favorecendo a ocorrência do vetor (Lutzomyia longipalpis) nestas regiões.

Distributed Control and Optimization for Communication and Power Systems

We are at the cusp of a historic transformation of both communication system and electricity system. This creates challenges as well as opportunities for the study of networked systems. Problems of these systems typically involve a huge number of end points that require intelligent coordination in a distributed manner. In this thesis, we develop models, theories, and scalable distributed optimization and control algorithms to overcome these challenges.

This thesis focuses on two specific areas: multi-path TCP (Transmission Control Protocol) and electricity distribution system operation and control. Multi-path TCP (MP-TCP) is a TCP extension that allows a single data stream to be split across multiple paths. MP-TCP has the potential to greatly improve reliability as well as efficiency of communication devices. We propose a fluid model for a large class of MP-TCP algorithms and identify design criteria that guarantee the existence, uniqueness, and stability of system equilibrium. We clarify how algorithm parameters impact TCP-friendliness, responsiveness, and window oscillation and demonstrate an inevitable tradeoff among these properties. We discuss the implications of these properties on the behavior of existing algorithms and motivate a new algorithm Balia (balanced linked adaptation) which generalizes existing algorithms and strikes a good balance among TCP-friendliness, responsiveness, and window oscillation. We have implemented Balia in the Linux kernel. We use our prototype to compare the new proposed algorithm Balia with existing MP-TCP algorithms.

Our second focus is on designing computationally efficient algorithms for electricity distribution system operation and control. First, we develop efficient algorithms for feeder reconfiguration in distribution networks. The feeder reconfiguration problem chooses the on/off status of the switches in a distribution network in order to minimize a certain cost such as power loss. It is a mixed integer nonlinear program and hence hard to solve. We propose a heuristic algorithm that is based on the recently developed convex relaxation of the optimal power flow problem. The algorithm is efficient and can successfully computes an optimal configuration on all networks that we have tested. Moreover we prove that the algorithm solves the feeder reconfiguration problem optimally under certain conditions. We also propose a more efficient algorithm and it incurs a loss in optimality of less than 3% on the test networks.

Second, we develop efficient distributed algorithms that solve the optimal power flow (OPF) problem on distribution networks. The OPF problem determines a network operating point that minimizes a certain objective such as generation cost or power loss. Traditionally OPF is solved in a centralized manner. With increasing penetration of volatile renewable energy resources in distribution systems, we need faster and distributed solutions for real-time feedback control. This is difficult because power flow equations are nonlinear and kirchhoff's law is global. We propose solutions for both balanced and unbalanced radial distribution networks. They exploit recent results that suggest solving for a globally optimal solution of OPF over a radial network through a second-order cone program (SOCP) or semi-definite program (SDP) relaxation. Our distributed algorithms are based on the alternating direction method of multiplier (ADMM), but unlike standard ADMM-based distributed OPF algorithms that require solving optimization subproblems using iterative methods, the proposed solutions exploit the problem structure that greatly reduce the computation time. Specifically, for balanced networks, our decomposition allows us to derive closed form solutions for these subproblems and it speeds up the convergence by 1000x times in simulations. For unbalanced networks, the subproblems reduce to either closed form solutions or eigenvalue problems whose size remains constant as the network scales up and computation time is reduced by 100x compared with iterative methods.

Uso do espaço do marsupial Caluromys philander (Didelphimorphia, Didelphidae) no Parque Nacional da Serra dos Órgãos - Rio de Janeiro, Brasil

Uso do espaço é um padrão bem estudado em ecologia. Entretanto, formação de área de vida e posição relativa dos abrigos com área de uso são pouco estudados, principalmente para marsupiais didelfídeos. Dentre estes animais, podemos destacar Caluromys philander, devido ao seu baixo registro em armadilhas na Mata Atlântica e características peculiares dentro do grupo, como seu desenvolvimento, longevidade e hábito alimentar. Neste estudo foram investigadas as formações das áreas de uso destes animais através da comparação com seus movimentos diários, e a posição dos seus abrigos dentro das suas áreas de uso. Para isso foram monitorados seis indivíduos de C. philander através de colares rádio transmissores. Estes indivíduos se deslocaram em média 534 153 m por noite. Além disso, apresentaram área diária de 9548 3591 m e área de vida de 2,8 0,4 ha. Noventa e sete por cento das áreas diárias apresentaram sobreposição entre si, com média de 19,4% de sobreposição. Não houve diferença nos locais dos abrigos dos indivíduos monitorados, dentro dos seus Mínimos Polígonos Convexos. Entretanto, estes mesmos abrigos não estiveram localizados nas áreas de maior intensidade de uso. A média de abrigos utilizados por indivíduo foi de 6,3 (3-10) com utilização média de 2,9 (1-17) vezes em cada abrigo, sendo que 48% das vezes os animais só possuíram um registro em cada abrigo, demonstrando baixa fidelidade. Entretanto, dois indivíduos apresentaram diferença de utilização entre seus abrigos, com três destes sendo mais utilizados que os outros. O trabalho sugere que os indivíduos de C. philander monitorados apresentam área de vida propriamente dita (restrita), onde ocorrem as sobreposições entre suas áreas diárias. C. philander utilizam mais de um abrigo em suas vidas, trocando com frequência de abrigos, apesar de alguns destes abrigos poderem ser mais utilizados que outros. Além disso, os abrigos destes animais não são localizados nas suas áreas de maior intensidade de uso.

Evaluation of the nutrient composition of some unconventional feedstuffs

A trial was carried out to evaluate the nutrient composition of some unconventional feedstuffs to ascertain their suitability for incorporation into fish feed. Pawpaw seed, mudskipper meal, water melon seed, almond kernel and groundnut husk were processed and analyzed for their proximate and mineral composition using internationally established procedure. The results obtained showed that the experimental feed stuffs had percentage crude protein that ranged from 7.38% to 55.85%. The highest level (55.85%) was found in mudskipper meal while the lowest (7.38%) was found in groundnut husk. The experimental feed stuffs had appreciable levels of calcium, potassium and phosphorus ranging from 0.29% to 1.87%, 0.17 to 1.26 and 0.06 to 0.97 respectively. Almond kernel can be used to replace soybean meal in aqua feed. Also, mudskipper meal can be used to replace fish meal in order to produce a cheaper fish feed; but it was recommended that feeding trials should be conducted to ascertain the inclusion level at which they can be effectively incorporated. The potentials of some of these feedstuffs in fish feed formulation seem high.

Análise espaço-temporal do uso do habitat pelo boto-cinza (Sotalia Guianensis) na Baía de Guanabara, Rio de Janeiro

Determinar áreas de vida tem sido um tema amplamente discutido em trabalhos que procuram entender a relação da espécie estudada com as características de seu habitat. A Baía de Guanabara abriga uma população residente de botos-cinza (Sotalia guianensis) e o objetivo do presente estudo foi analisar o uso espacial de Sotalia guianensis, na Baía de Guanabara (RJ), entre 2002 e 2012. Um total de 204 dias de coleta foi analisado e 902 pontos selecionados para serem gerados os mapas de distribuição. A baía foi dividida em quatro subáreas e a diferença no esforço entre cada uma não ultrapassou 16%. O método Kernel Density foi utilizado nas análises para estimativa e interpretação do uso do habitat pelos grupos de botos-cinza. A interpretação das áreas de concentração da população também foi feita a partir de células (grids) de 1,5km x 1,5km com posterior aplicação do índice de sobreposição de nicho de Pianka. As profundidades utilizadas por S. guianensis não apresentaram variações significativas ao longo do período de estudo (p = 0,531). As áreas utilizadas durante o período de 2002/2004 foram estimadas em 79,4 km com áreas de concentração de 19,4 km. Os períodos de 2008/2010 e 2010/2012 apresentaram áreas de uso estimadas em um total de 51,4 e 58,9 km, respectivamente e áreas de concentração com 10,8 e 10,4 km, respectivamente. As áreas utilizadas envolveram regiões que se estendem por todo o canal central e região nordeste da Baía de Guanabara, onde também está localizada a Área de Proteção Ambiental de Guapimirim. Apesar disso, a área de vida da população, assim como suas áreas de concentração, diminuiu gradativamente ao longo dos anos, especialmente no entorno da Ilha de Paquetá e centro-sul do canal central. Grupos com mais de 10 indivíduos e grupos na classe ≥ 25% de filhotes em sua composição, evidenciaram reduções de mais de 60% no tamanho das áreas utilizadas. A população de botos-cinza vem decrescendo rapidamente nos últimos anos, além de interagir diariamente com fontes perturbadoras, sendo estas possíveis causas da redução do uso do habitat da Baía de Guanabara. Por esse motivo, os resultados apresentados são de fundamental importância para a conservação desta população já que representam consequências da interação em longo prazo com um ambiente costeiro altamente impactado pela ação antrópica.

Mapeamento de difusão no reconhecimento e reconstrução de sinais

Em muitas representações de objetos ou sistemas físicos se faz necessário a utilização de técnicas de redução de dimensionalidade que possibilitam a análise dos dados em baixas dimensões, capturando os parâmetros essenciais associados ao problema. No contexto de aprendizagem de máquina esta redução se destina primordialmente à clusterização, reconhecimento e reconstrução de sinais. Esta tese faz uma análise meticulosa destes tópicos e suas conexões que se encontram em verdadeira ebulição na literatura, sendo o mapeamento de difusão o foco principal deste trabalho. Tal método é construído a partir de um grafo onde os vértices são os sinais (dados do problema) e o peso das arestas é estabelecido a partir do núcleo gaussiano da equação do calor. Além disso, um processo de Markov é estabelecido o que permite a visualização do problema em diferentes escalas conforme variação de um determinado parâmetro t: Um outro parâmetro de escala, Є, para o núcleo gaussiano é avaliado com cuidado relacionando-o com a dinâmica de Markov de forma a poder aprender a variedade que eventualmente seja o suporte do dados. Nesta tese é proposto o reconhecimento de imagens digitais envolvendo transformações de rotação e variação de iluminação. Também o problema da reconstrução de sinais é atacado com a proposta de pré-imagem utilizando-se da otimização de uma função custo com um parâmetro regularizador, γ, que leva em conta também o conjunto de dados iniciais.

Precondicionadores baseados na aproximação da inversa.

Neste trabalho de dissertação apresentaremos uma classe de precondicionadores baseados na aproximação esparsa da inversa da matriz de coecientes, para a resolução de sistemas lineares esparsos de grandes portes através de métodos iterativos, mais especificamente métodos de Krylov. Para que um método de Krylov seja eficiente é extremamente necessário o uso de precondicionadores. No contexto atual, onde computadores de arquitetura híbrida são cada vez mais comuns temos uma demanda cada vez maior por precondicionadores paralelizáveis. Os métodos de inversa aproximada que serão descritos possuem aplicação paralela, pois so dependem de uma operação de produto matriz-vetor, que é altamente paralelizável. Além disso, alguns dos métodos também podem ser construídos em paralelo. A ideia principal é apresentar uma alternativa aos tradicionais precondicionadores que utilizam aproximações dos fatores LU, que apesar de robustos são de difícil paralelização.

Gene function prediction from synthetic lethality networks via ranking on demand.

MOTIVATION: Synthetic lethal interactions represent pairs of genes whose individual mutations are not lethal, while the double mutation of both genes does incur lethality. Several studies have shown a correlation between functional similarity of genes and their distances in networks based on synthetic lethal interactions. However, there is a lack of algorithms for predicting gene function from synthetic lethality interaction networks. RESULTS: In this article, we present a novel technique called kernelROD for gene function prediction from synthetic lethal interaction networks based on kernel machines. We apply our novel algorithm to Gene Ontology functional annotation prediction in yeast. Our experiments show that our method leads to improved gene function prediction compared with state-of-the-art competitors and that combining genetic and congruence networks leads to a further improvement in prediction accuracy.