abstract: Distributed inference has applications in a wide range of fields such as source localization, target detection, environment monitoring, and healthcare. In this dissertation, distributed inference schemes which use bounded transmit power are considered. The performance of the proposed schemes are studied for a variety of inference problems. In the first part of the dissertation, a distributed detection scheme where the sensors transmit with constant modulus signals over a Gaussian multiple access channel is considered. The deflection coefficient of the proposed scheme is shown to depend on the characteristic function of the sensing noise, and the error exponent for the system is derived using large deviation theory. Optimization of the deflection coefficient and error exponent are considered with respect to a transmission phase parameter for a variety of sensing noise distributions including impulsive ones. The proposed scheme is also favorably compared with existing amplify-and-forward (AF) and detect-and-forward (DF) schemes. The effect of fading is shown to be detrimental to the detection performance and simulations are provided to corroborate the analytical results. The second part of the dissertation studies a distributed inference scheme which uses bounded transmission functions over a Gaussian multiple access channel. The conditions on the transmission functions under which consistent estimation and reliable detection are possible is characterized. For the distributed estimation problem, an estimation scheme that uses bounded transmission functions is proved to be strongly consistent provided that the variance of the noise samples are bounded and that the transmission function is one-to-one. The proposed estimation scheme is compared with the amplify and forward technique and its robustness to impulsive sensing noise distributions is highlighted. It is also shown that bounded transmissions suffer from inconsistent estimates if the sensing noise variance goes to infinity. For the distributed detection problem, similar results are obtained by studying the deflection coefficient. Simulations corroborate our analytical results. In the third part of this dissertation, the problem of estimating the average of samples distributed at the nodes of a sensor network is considered. A distributed average consensus algorithm in which every sensor transmits with bounded peak power is proposed. In the presence of communication noise, it is shown that the nodes reach consensus asymptotically to a finite random variable whose expectation is the desired sample average of the initial observations with a variance that depends on the step size of the algorithm and the variance of the communication noise. The asymptotic performance is characterized by deriving the asymptotic covariance matrix using results from stochastic approximation theory. It is shown that using bounded transmissions results in slower convergence compared to the linear consensus algorithm based on the Laplacian heuristic. Simulations corroborate our analytical findings. Finally, a robust distributed average consensus algorithm in which every sensor performs a nonlinear processing at the receiver is proposed. It is shown that non-linearity at the receiver nodes makes the algorithm robust to a wide range of channel noise distributions including the impulsive ones. It is shown that the nodes reach consensus asymptotically and similar results are obtained as in the case of transmit non-linearity. Simulations corroborate our analytical findings and highlight the robustness of the proposed algorithm. / Dissertation/Thesis / Ph.D. Electrical Engineering 2013

The study of conformal symmetry is motivated through an example in statistical mechanics and then rigorously developed in quantum field theories in general spatial dimensions. In particular, primary fields are introduced as the fundamental objects of such theories and then studied in the formalism of radial quantization. The implications of conformal invariance on the functional form of correlation functions are studied in detail. Conformal blocks are defined and various approaches to their analytical and numerical calculation are presented with a special emphasis on the one-dimensional case. Building on these preliminaries, a modern formulation of the conformal bootstrap program and its various extensions are discussed. Examples are given in which bounds on the scaling dimensions in a one-dimensional theory are derived numerically. Using these results I motivate the technique of using the extremal functional bootstrap which I then develop in more detail. Many technical details are discussed and examples shown. After a brief discussion of conformal field theories with a boundary I apply numerical methods to find constraints on the spectrum of the 3D Ising model. Another application is presented in which I study the 4-point function on the boundary of a particular theory in Anti-de-Sitter space in order to approximate the mass spectrum of the theory. / O estudo da simetria conforme é motivado através de um exemplo em mecânica estatística e em seguida rigorosamente desenvolvido em teorias de campos quânticos em dimensões espaciais gerais. Em particular, os campos primários são introduzidos como os objetos fundamentais de tais teorias e então estudados através do formalismo de quantização radial. As implicações da invariância conforme na forma funcional das funções de correlação são estudadas em detalhe. Blocos conformes são definidos e várias abordagens para seu cálculo analítico e numérico são apresentadas com uma ênfase especial no caso unidimensional. Com base nessas preliminares, uma formulação moderna do programa de bootstrap conforme e suas várias extensões são discutidas. Exemplos são dados em que limites nas dimensões de escala em uma teoria unidimensional são derivados numericamente. Usando esses resultados, motivei a técnica de usar o bootstrap funcional extremo, que depois desenvolvo em mais detalhes. Diversos detalhes técnicos são discutidos e exemplos são apresentados. Após uma breve discussão das teorias de campo conformes com fronteiras, eu aplico métodos numéricos para encontrar restrições no espectro do modelo de Ising em 3D. Outra aplicação é apresentada em que eu estudo a função de 4 pontos na fronteira de uma teoria particular no espaço Anti-de-Sitter, a fim de aproximar o espectro de massa da teoria.

Investigações da dinâmica crítica em modelos de magnetismo, para tempos curtos, têm aparecido com grande freqüência na literatura. Essa técnica foi descoberta por Li, Schülke e Zheng que, inspirados em trabalhos anteriores de Huse e Janssen et al., mostraram que generalizações de grandezas como a magnetização e o cumulante de Binder exibem comportamento universal já no início da simulação. O estudo da criticalidade em tempos curtos proporciona um caminho alternativo para a estimativa do expoente z, além de permitir o cálculo de um novo expoente dinâmico θ, associado ao comportamento anômalo da magnetização. Da mesma forma, simulações dependentes do tempo tornaram-se ferramenta útil para estudar transições de fase em autômatos celulares e modelos de spin. Em particular, as melhores estimativas para o expoente z do Ising bidimensional foram obtidas por meio da técnica de propagação de danos, introduzida por Kauffman no estudo de autômatos e mais tarde generalizada para modelos de spin. Na primeira parte deste trabalho utilizamos o método Monte Carlo em tempos curtos para investigar o modelo de Baxter-Wu, definido em uma rede bidimensional triangular com variáveis do tipo Ising, acopladas por interações de três corpos. Obtivemos os expoentes críticos dinâmicos z e θ além dos índices críticos estáticos ß e Nû. Os resultados não corroboram aqueles recentemente obtidos por Santos e Figueiredo para o expoente z. Na segunda parte do trabalho, investigamos a propagação de danos no modelo de Ising unidimensional submetido a duas dinâmicas propostas por Hinrichsen e Domany (HD). Em particular, nós estudamos o efeito da atualização síncrona (paralela) e assíncrona (dinâmica contínua) sobre o espalhamento do dano. Mostramos que o dano não se propaga quando a segunda dinâmica é implementada de forma assíncrona. Também mostramos que as regras para atualização do dano produzidas por essa dinâmica, quando a temperatura vai a infinito e um certo parâmetro Lambda é igual a zero, são equivalentes àquelas do bem conhecido autômato celular (modelo A) de Grassberger. / Short-time simulations have been used with great frequency in the literature. That technique was discovered by Li, Shülke and Zheng that, inspired in previous works by Huse and Janssen et al., showed that generalizations of quantities like magnetization and the Binder´s cumulant exhibit universal behavior in the beginning of the simulation (early time behavior). The study of criticality in short-times provides an alternative way to estimate the dynamic critical exponent z, besides allowing the calculation of a new dynamic exponent θ, associated to the anomalous behavior of the magnetization. In the same way, time-dependent simulations became a useful tool to study phase transitions in cellular automata and also for spin models. In fact, the best estimates for the exponent z of the two-dimensional Ising model were obtained through the technique of damage spreading, introduced by Kauffman in the study of cellular automata, later widespread for spin models. In the first part of this work we used short-time Monte Carlo simulations to investigate the Baxter-Wu model, defined in a triangular lattice whose variables are Ising-like coupled by triplet interactions. We have obtained estimates for the dynamic critical exponents z and θ besides static exponents ß e Nû. Our results do not corroborate recent estimates by Santos and Figueiredo for the critical exponent z. In the second part of this work, we investigated the damage spreading in the one-dimensional Ising model under two dynamics introduced by Hinrichsen and Domany (HD). In particular, we study the effects of synchronous (parallel) and asynchronous (continuous dynamics) updating on the spreading properties. We showed that the damage does not spread when the second dynamic is implemented in an asynchronous way. We found that the rules for updating the damage produced by this dynamic, as the temperature goes to infinity and a certain parameter Lambda is zero, are equivalent to those of Grassberger’s well-known model A cellular automaton.

Neste trabalho, tivemos como objetivo estudar a existência de soluções fracas não triviais para o problema elíptico com não linearidade crítica { - Δu = λu + u2* - 1+ + g(x, u+) + f(x); em Ω u = 0; sobre ∂ Ω , (P) onde Ω é um domínio limitado com fronteira suave em ℝN, com N ≥ 3, 2* = 2N / (N - 2) é o expoente crítico de Sobolev, u+ = max(u; 0), g ∈ C(Ω̄ x ℝ, ℝ+), λ > λ1, λ ∉ σ (- Δ) e f ∈ Lr> (Ω), com r > N. Com o intuito de observar as mudanças que ocorrem do caso subcrítico para o crítico e as diferentes técnicas variacionais para a resolução de problemas elípticos, estudamos, inicialmente, um problema um pouco mais antigo que (P), que, por sua vez, motivou seu estudo. Tal problema é { - Δu = λ u + up+ +f; em Ω u = 0; sobre ∂ Ω(P\') onde consideramos o caso subcrítico, ou seja, quando p ∈ (1; 2* - 1). Com o auxílio do TEOREMA DE ENLACE verificamos que tanto (P) quanto (P\') têm pelo menos duas soluções fracas não triviais. / In this work, we aimed to study the existence of nontrivial weak solutions for the elliptic problem with critical non-linearity { - Δu = λu + u2* - 1+ + g(x, u+) + f(x); in Ω u = 0; on ∂ Ω , (P) where Ω is a bounded domain with smooth boundary in ℝN, with N ≥ 3, 2* = 2N / N -2 is the critical Sobolev exponent, u+ = max(u; 0), g ∈ C(Ω̄ x ℝ, ℝ+), λ > λ1, λ ∉ σ (- Δ) and f ∈ Lr (Ω), with r > N. In order to observe different variational techniques for solving elliptic problems, we studied initially a problem a little older than (P), which, in turn, led to its study. This problem is { - Δu = λ u + up+ +f; inΩ u = 0; on ∂ Ω(P\') where we consider the subcritical case, that is, when p ∈ (1, 2* - 1). With the aid of the LINKING THEOREM we see that both (P) and (P\') have at least two nontrivial weak solutions.

Neste trabalho estudamos loops de Bol 2-nilpotentes e de expoente 2. Além disso, mostramos que o ideal de aumento de uma álgebra de loop, de um loop finito p-nilpotente em característica p > 0, é nilpotente. Com este resultado conseguimos caracterizar os elementos inversíveis da álgebra de loop de um loop 2-nilpotente sobre um corpo de dois elementos. Provamos também que loops de Bol finitos 2-nilpotentes e de expoente 2 podem ser mergulhados em um loop de Bol à direita de elementos inversíveis de uma álgebra alternativa à direita, sobre um corpo de característica dois. / In this work we study 2-nilpotent Bol loops of exponent 2. Besides, we prove that the augmentation ideal of a loop algebra, of a finite p-nilpotent loop in characteristic p > 0, is nilpotent. With this result we characterized the invertible elements of the loop algebra of a 2-nilpotent loop over a field with two elements. We also proof that 2-nilpotent Bol loops of exponent 2 may be embedded into a right Bol loop of invertible elements of a right alternative algebra, over a field of characteristic 2.

Neste trabalho estudamos relações entre expoente de Lyapunov e continuidade absoluta da folheação central para difeomorfismos parcialmente hiperbólicos conservativos de \'T POT. 3\'. Sobre tal tema, provamos que tipicamente (\'C POT. 1\' aberto e \'C POT. 2\' denso) os difeomorfismos parcialmente hiperbólicos, conservativos de classe \'C POT. 2\' , do toro \'T POT. 3\', apresentam folheação central não absolutamente contínua. Desta maneira, respondemos positivamente uma pergunta proposta em [20]. Também neste trabalho, estudamos entropia topológica para Sistema Iterado de Funções. Neste contexto, damos uma nova demonstração para uma conjectura proposta em [14] e provada primeiramente em [15]. Apresentamos um método geométrico que nos permite calcular entropia para transformações de \'S POT. 1\', como em [15]. Além de disso o método apresentado se verifica para casos mais gerais, como por exemplo: transformações não comutativas / In this work we study relations between Lyapunov exponents, absolute continuity of center foliation for conservative partially hyperbolic diffeomorphisms of \'T POT. 3\'. About this theme, (on a \'C POT. 1\' open and \'C POT. 2\'dense set) of conservative partially hyperbolic \'C POT. 2\' diffeomorphisms of the 3-torus presents non absolutely continuous center foliation. So, we answer positively a question proposed in [20]. Also in this work, we study topological entropy for Iterated Functions Systems. In this setting, we give a proof for a conjecture proposed in [14] and firstly proved in [15]. We present a geometrical method that allows us to calcule the entropy for transformations of \'S POT. 1\', like in [15]. Furthermore this method holds for more general cases, for example: non commutative transformations

Sabe-se que, na literatura, existem muitos modelos para se fazer previsão para séries temporais financeiras. Sabe-se também que não há um modelo perfeito e que os mais utilizados atualmente são os modelos de redes neurais recorrentes e os da família GARCH. Referências internacionais apontam que existe uma técnica de medição de uma janela temporal para se identificar o tipo de comportamento existente em uma série temporal; tal técnica é conhecida como Expoente de Hurst. É uma medida que qualifica a série como persistente ou anti-persistente. Este trabalho analisou se o Expoente de Hurst, interfere na qualidade das previsões feitas com o modelo de redes neurais recorrentes com e sem o uso do filtro de ondaletas, utilizando os preços diários das principais commodities, ações negociadas no mercado e a taxa de câmbio. no período de janeiro de 1998 a dezembro de 2010. Com a pesquisa observa-se, na maioria dos casos, há uma possível melhora na qualidade das previsões para as séries antipersistentes. / It is known that there are a lot of models to forecast financial time series. It is known, also, that there is not a perfect model and the most used nowadays are the Recurrent Neural Network models and those from the GARCH family. International references point to a technique of measurement using windowing in order to identify the kind of behavior that is present in time series. This technique is known as Hurst Exponent. It is a measure that qualifies the time series as persistent or anti-persistent. This work analyzed if the Hurst Exponent interferes in the quality of the forecasts made with the Neural Network models with and without the wavelet filter, using the main commodities, stock prices, Ibovespa index and the Dollar/Real exchange rate in the period ranging from January 1998 to December 2010. The initial conclusions concerning the models worked out are positives.

Dans cette thèse, nous étudions la minimalité et la propriété chaotique de systèmes dynamiques p-adiques. Nous étudions aussi des propriétés multifractales des séries de Davenport avec translation de phases. Dans la première partie, nous commençons par l'étude des systèmes dynamiques affines sur Zp. Nous trouvons une condition nécessaire et suffisante pour qu'un tel système soit minimal. En outre, nous exhibons toutes ses composantes strictement ergodiques si le système n'est pas minimal. De plus, nous étudions aussi les systèmes monômes sur le groupe 1+pZp. Ensuite nous étudions les polynômes localement dilatants et transitifs. Pour un tel polynôme, limité sur son ensemble de Julia, nous prouvons qu'il est conjugué à un sous-shift de type fini. Dans la deuxième partie, nous étudions les séries de Davenport avec translation de phases. Après avoir calculé le saut d'une telle série à chaque point, nous trouvons l'ensemble des points discontinus et obtenons une condition nécessaire et suffisante pour qu'une série de Davenport avec translation de phases soit continue sur R. La convergence ponctuelle de la série est aussi étudiée. Ensuite, nous estimons la borne inférieure de l'exposant hölderien de la série de Davenport avec de phase rationnelle et la borne supérieure du spectre de la singularité / In this thesis, we study the minimality and the chaotic property of p-adic dynamical systems and some multifractal properties of phase translated Davenport series. In the first part, we begin with the study of affine dynamical systems on Zp. We find a necessary and sufficient condition for such a system to be minimal. Furthermore, all its strictly ergodic components are exhibited when it is not minimal. In addition, we study monomial systems on the group 1 + pZp. Then transitive locally expanding polynomial systems are studied. It is proved that such a polynomial system, restricted to its Julia set, is conjugate to a subshift of finite type. In the second part, we study phase translated Davenport series. After having calculated the jump of the series at each point, we characterize the set of discontinuous points and get a sufficient and necessary condition for the series to be continuous on R. Furthermore, the pointwise convergence of the series is studied. Then we estimate the lower bound of the Hölder-exponent of rational translated Davenport series and get an upper bound estimation on the spectrum of singularity. The lower bound of the Hölder-exponent are also discussed for some irrational translated series

The key objective of this thesis is the outline of an alternative capital market modeling framework, the Volatility Asset Pricing Model, VAPM, inspired by the innovative dual approach of Mandelbrot and Hudson using the method based on synthesis of two seemingly antagonistic factors -- the volatility of market prices and their serial dependence determining the capital markets' dynamics. The pilot tests of this model in various periods using the market index as well as a portfolio of selected securities delivered generally satisfactory results. Firstly, the work delivers a brief recapitulation regarding the concepts of a consumer/investor choice under general conditions of hypothetical certainty. Secondly, this outline is then followed by a description of the "classical" methodologies in the risky environment of uncertainty, with assessment of their corresponding key models, i.e. the CAPM, SIM, MIM, APTM, etc., notwithstanding results of the related testing approaches. Thirdly, this assessment is based on evaluation of the underlying doctrine of Efficient Market Hypothesis in relation to the so called Random Walk Model. Fourthly, in this context the work also offers a brief exposure to a few selected tests of these contraversial concepts. Fifthly, the main points of conteporary approaches such as the Fractal Dimension and the Hurst Exponent in the dynamic framework of information entropy are subsequently described as the theoretical tools leading to development of the abovementioned model VAPM. The major contribution of this thesis is considered its attempt to apply the abovementioned concepts in practice, with the intention to possibly inspire a further analytical research.

La dynamique d'un difféomorphisme d'une variété compacte est essentiellement concentrée sur l'ensemble récurrent par chaînes, qui est partitionné en classes de récurrence par chaînes, disjointes et indécomposables. Le travail de Bonatti et Crovisier [BC] montre que, pour les difféomorphismes C¹-génériques, une classe de récurrence par chaînes ou bien est une classe homocline, ou bien ne contient pas de point périodique. Une classe de récurrence par chaînes sans point périodique est appelée classe apériodique.Il est clair qu'une classe homocline hyperbolique ni contient d'orbite périodique faible ni supporte de mesure non hyperbolique.Cette thèse tente de donner une caractérisation des classes homoclines non hyperboliques en montrant qu'elles contiennent des orbites périodiques faibles ou des mesures ergodiques non hyperboliques. Cette thèse décrit également les décompositions dominées sur les classes apériodiques.Le premier résultat de cette thèse montre que, pour les difféomorphismes C¹-génériques, si les orbites périodiques contenues dans une classe homocline H(p) ont tous leurs exposants de Lyapunov bornés loin de zéro, alors H(p) doit être (uniformément) hyperbolique. Ceci est dans l'esprit des travaux sur la conjecture de stabilité, mais il y a une différence importante lorsque la classe homocline H(p) n'est pas isolée. Par conséquent, nous devons garantir que des orbites périodiques "faibles'', crées par perturbations au voisinage de la classe homocline, sont contenues dans la classe. En ce sens, le problème est de nature "intrinsèque'', et l'argument classique de la conjecture de stabilité est impraticable.Le deuxième résultat de cette thèse prouve une conjecture de Díaz et Gorodetski [DG]: pour les difféomorphismes C¹-génériques, si une classe homocline n'est pas hyperbolique, alors elle porte une mesure ergodique non hyperbolique. C'est un travail en collaboration avec C. Cheng, S. Crovisier, S. Gan et D. Yang. Dans la démonstration, nous devons appliquer une technique introduité dans [DG], et qui améliore la méthode de [GIKN], pour obtenir une mesure ergodique comme limite d'une suite de mesures périodiques.Le troisième résultat de cette thèse énonce que, génériquement, une décomposition dominée non-triviale sur une classe apériodique stable au sens de Lyapunov est en fait une décomposition partiellement hyperbolique. Plus précisément, pour les difféomorphismes C¹-génériques, si une classe apériodique stable au sens de Lyapunov a une décomposition dominée non-triviale Eoplus F, alors, l'un des deux fibrés est hyperbolique: soit E contracté, soit F dilaté.Dans les démonstrations des résultats principaux, nous construisons des perturbations qui ne sont pas obtenues directement à partir des lemmes de connexion classiques. En fait, il faut appliquer le lemme de connexion un grand nombre (et même un nombre infini) de fois. Nous expliquons les méthodes de connexions multiples dans le Chapitre 3. / The dynamics of a diffeomorphism of a compact manifold concentrates essentially on the chain recurrent set, which splits into disjoint indecomposable chain recurrence classes. By the work of Bonatti and Crovisier [BC], for C¹-generic diffeomorphisms, a chain recurrence class either is a homoclinic class or contains no periodic point. A chain recurrence class without a periodic point is called an aperiodic class.Obviously, a hyperbolic homoclinic class can neither contain weak periodic orbit or support non-hyperbolic ergodic measure.This thesis attempts to give a characterization of non-hyperbolic homoclinic classes via weak periodic orbits inside or non-hyperbolic ergodic measures supported on it. Also, this thesis gives a description of the dominated splitting on Lyapunov stable aperiodic classes.The first result of this thesis shows that for C¹-generic diffeomorphisms, if the periodic orbits contained in a homoclinic class H(p) have all their Lyapunov exponents bounded away from 0, then H(p) must be (uniformly) hyperbolic. This is in spirit of the works of the stability conjecture, but with a significant difference that the homoclinic class H(p) is not known isolated in advance. Hence the "weak'' periodic orbits created by perturbations near the homoclinic class have to be guaranteed strictly inside the homoclinic class. In this sense the problem is of an "intrinsic" nature, and the classical argument of the stability conjecture does not pass through.The second result of this thesis proves a conjecture by Díaz and Gorodetski [DG]: for C¹-generic diffeomorphisms, if a homoclinic class is not hyperbolic, then there is a non-hyperbolic ergodic measure supported on it. This is a joint work with C. Cheng, S. Crovisier, S. Gan and D. Yang. In the proof, we have to use a technic introduced in [DG], which developed the method of [GIKN], to get an ergodic measure by taking the limit of a sequence of periodic measures.The third result of this thesis states that, generically, a non-trivial dominated splitting over a Lyapunov stable aperiodic class is in fact a partially hyperbolic splitting. To be precise, for C¹-generic diffeomorphisms, if a Lyapunov stable aperiodic class admits a non-trivial dominated splitting Eoplus F, then one of the two bundles is hyperbolic: either E is contracted or F is expanded.In the proofs of the main results, we construct several perturbations which are not simple applications of the connecting lemmas. In fact, one has to apply the connecting lemma several (even infinitely many) times. We will give the detailed explanations of the multi-connecting processes in Chapter 3.