A steepest descent method is constructed for the general setting of a linear differential equation paired with uniqueness-inducing conditions which might yield a generally overdetermined system. The method differs from traditional steepest descent methods by considering the conditions when defining the corresponding Sobolev space. The descent method converges to the unique solution to the differential equation so that change in condition values is minimal. The system has a solution if and only if the first iteration of steepest descent satisfies the system. The finite analogue of the descent method is applied to example problems involving finite difference equations. The well-posed problems include a singular ordinary differential equation and Laplace’s equation, each paired with respective Dirichlet-type conditions. The overdetermined problems include a first-order nonsingular ordinary differential equation with Dirichlet-type conditions and the wave equation with both Dirichlet and Neumann conditions. The method is applied in an investigation of the Tricomi equation, a long-studied equation which acts as a prototype of mixed partial differential equations and has application in transonic flow. The Tricomi equation has been studied for at least ninety years, yet necessary and sufficient conditions for existence and uniqueness of solutions on an arbitrary mixed domain remain unknown. The domains of interest are rectangular mixed domains. A new type of conditions is introduced. Ladder conditions take the uncommon approach of specifying information on the interior of a mixed domain. Specifically, function values are specified on the parabolic portion of a mixed domain. The remaining conditions are specified on the boundary. A conjecture is posed and states that ladder conditions are necessary and sufficient for existence and uniqueness of a solution to the Tricomi equation. Numerical experiments, produced by application of the descent method, provide strong evidence in support of the conjecture. Ladder conditions allow for a continuous deformation from Dirichlet conditions to initial-boundary value conditions. Such a deformation is applied to a class of Tricomi-type equations which transition from degenerate elliptic to degenerate hyperbolic. A conjecture is posed and states that each problem is uniquely solvable and the solutions vary continuously as the differential equation and corresponding conditions vary continuously. If the conjecture holds true, the result will provide a method of unifying elliptic Dirichlet problems and hyperbolic initial-boundary value problem. Numerical evidence in support of the conjecture is presented.

Um dos mais investigados problemas na teoria qualitativa dos sistemas dinâmicos no plano é o XVI problema de Hilbert que trata dos ciclos limites. Mais precisamente, a segunda parte do referido problema questiona sobre o número máximo de ciclos limites de um sistema diferencial polinomial plano de grau n. Por ciclo limite entendemos uma órbita fechada isolada no conjunto de todas as órbitas periódicas de um sistema diferencial plano.Uma maneira clássica de obter um ciclo limite é perturbando um sistema com uma singularidade do tipo centro. Nesta dissertação apresentamos dois métodos utilizados para a análise do número de ciclos limites que bifurcam de um centro, a saber o método das integrais abelianas e o método do averaging / One of the most investigated problems in the qualitative theory of dynamical systems in the plane is the XVI Hilberts problem which deals with limit cycles. More precisely, the second part of the problem asks about the maximum number of limit cycles of a polynomial differential system of degree n. A limit cycle is a single closed orbit on the set of all periodic orbits of a differential planar system. A classic way to obtain a limit cycle is perturbing a system with a singularity of center type.In this work we discuss about two methods used to investigate the number of limit cycles which bifurcate from a center; they are known as Abelian integrals and averaging theory

Submitted by Cristiane Chim (cristiane.chim@ucpel.edu.br) on 2017-02-14T11:02:21Z No. of bitstreams: 1 JulianoRB.pdf: 1568199 bytes, checksum: d4d56c5973ce16deee6885df8b5cc27f (MD5) / Made available in DSpace on 2017-02-14T11:02:21Z (GMT). No. of bitstreams: 1 JulianoRB.pdf: 1568199 bytes, checksum: d4d56c5973ce16deee6885df8b5cc27f (MD5) Previous issue date: 2016-12-14 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES# / #2075167498588264571# / #600 / Acoustic echo is a phenomenon caused by the reflection of sound when it is perceived by its emission source with some delay. It appears in full duplex communication systems in a detrimental manner to the interaction between the speakers, reducing the intelligibility of exchanged information. Because it is a very unpleasant effect, acoustic echo attenuation or canceling procedures, if possible, become necessary procedures to mitigate such phenomenon. Several techniques have been reported in the literature in order to cancel acoustic echo, which includes the application of structures based on adaptive filtering. Typically, strategies where the adaptive filtering is used, it is assumed that the path traveled by the acoustic echo can be modeled as a linear time-invariant system. However, because the elements that form such a path, such as amplifiers and speakers, present non-linear characteristics, linear models fail to adequately represent the whole path traveled by the acoustic echo. In recent approaches it has been assumed that the echo response should obey non-linear and non-stationary models, whereas in acoustic environments the signal to be modeled can be modified by the movement of people and objects. Thus, the appropriate response representation model becomes essential for satisfactory adaptive system’s performance on echo identification and cancellation. This work aims to propose a new technique in the nonlinear and time variant estimation from the application of the Kalman filtering theory, which presents a good answer in the identification of non-stationary systems, in a Hilbert Space defined by Reproductive Kernel, where linearity can be treated linearly, using as an example of application the cancellation of acoustic echo / O eco acústico é um fenômeno causado pela reflexão do som quando este é percebido por sua fonte emissora com algum atraso, agindo, nos sistemas de comunicação full duplex de forma prejudicial na interação entre os interlocutores, ocasionando a redução da inteligibilidade das informações trocadas. Dessa forma, por se tratar de um efeito bastante desagradável, faz-se necessária a aplicação de procedimentos cuja finalidade seja atenuar, ou se possível, eliminar tal fenômeno. Diversas técnicas têm sido apresentadas na literatura com o objetivo de cancelar o eco acústico, onde se destaca a aplicação de estruturas baseadas na filtragem adaptativa. Normalmente, em estratégias onde a filtragem adaptativa é utilizada, assume-se que o caminho percorrido pelo eco acústico pode ser modelado como um sistema linear e invariante no tempo. Entretanto, em virtude do conjunto de elementos que formam tal caminho, tais como os amplificadores e os alto-falantes, apresentarem características não lineares, os modelos lineares não conseguem representar de forma adequada todo o caminho percorrido pelo eco acústico. Considerando que em ambientes acústicos o sinal a ser modelado pode ser modificado pela circulação de pessoas e movimentação de objetos, em recentes abordagens, têm-se assumido que a resposta do eco deve obedecer a modelos não lineares e não estacionários. Dessa forma, o modelo adequado de representação da resposta se torna fundamental para que o sistema adaptativo utilizado na identificação e cancelamento do eco tenha um desempenho satisfatório. Este trabalho propõe uma nova técnica na estimação não linear e variante no tempo a partir da aplicação da teoria de filtragem de Kalman, que apresenta uma boa resposta na identificação de sistemas não estacionários, em um Espaço de Hilbert definido por Kernel Reprodutivo, onde a não linearidade poderá ser tratada de forma linear, utilizando como exemplo de aplicação o cancelamento do eco acústico. Palavras

The broad topic of this dissertation is the study of algebraic structure arising from polyhedral geometric objects. There are three distinct topics covered over three main chapters. However, each of these topics are further linked by a connection to the Eulerian polynomials. Chapter 2 studies Euler-Mahonian identities arising from both the symmetric group and generalized permutation groups. Specifically, we study the algebraic structure of unit cube semigroup algebra using Gröbner basis methods to acquire these identities. Moreover, this serves as a bridge between previous methods involving polyhedral geometry and triangulations with descent bases methods arising in representation theory. In Chapter 3, the aim is to characterize Hilbert basis elements of certain 𝒔-lecture hall cones. In particular, the main focus is the classification of the Hilbert bases for the 1 mod 𝑘 cones and the 𝓁-sequence cones, both of which generalize a previous known result. Additionally, there is much broader characterization of Hilbert bases in dimension ≤ 4 for 𝒖-generated Gorenstein lecture hall cones. Finally, Chapter 4 focuses on certain algebraic and geometric properties of 𝒔-lecture hall polytopes. This consists of partial classification results for the Gorenstein property, the integer-decomposition property, and the existence of regular, unimodular triangulations.

Dans un premier temps, nous étudions la continuité d'une <br />représentation $\theta$ du groupe topologique $G$ dans une algèbre de Banach $A$ en fonction du comportement de $\limsup_{u \rightarrow 1}\| \theta(u)-I \|$, où $1$ désigne l'élément unité de $G$ et $I$ celui de $A$. Nous obtenons aussi des résultats de continuité automatique pour une large catégorie de représentations de groupes. <br /><br />Nous étudions ensuite, dans des cas concrets le spectre de l'opérateur $S_M: E/M \rightarrow E/M$ défini par $S(f+M)=Sf +M$, c'est-à-dire la compression de $S$ à $E/M$ où $E$ est un espace de Banach, $S:E \rightarrow E$ un opérateur borné et $M$ un sous-espace vectoriel fermé invariant par $S$, c'est-à-dire vérifiant $S(M) \subset M$. D'abord nous nous plaçons dans des espaces de Banach $E$ de fonctions analytiques sur le disque unité pour lesquels le shift usuel $S:z \mapsto zf$ et le shift arrière $T: f \mapsto \frac{f-f(0)}{z}$ ont leur spectre égal au cercle unité et vérifient la condition de non-quasianalyticité. Nous montrons que si $f \in M$ admet une extension analytique à $\D \cup D(\zeta,r)$, avec $|\zeta|=1$, $f(\zeta)\neq 0$, alors $\zeta \notin Spec(S_M)$. Nous appliquons ce résultat à l'espace de Hardy pondéré $H_{\sigma_{\alpha}}(\D)$, avec $\sigma_{\alpha}(n)=e^{-n^{\alpha}}$, $n \geq 0$, $\alpha \in (\frac{1}{2},1)$.<br /><br />Enfin nous étudions une situation quasianalytique, celle des espaces $l^2(w,\Z)$ à poids "$\log$-impairs". Soit $L$ un arc fermé non vide du cercle unité; nous montrons que la construction de Y.Domar de sous-espaces invariants par translations pour les espaces $l^2(w,\Z)$ vérifiant une condition naturelle de régularité, permet d'obtenir des sous-espaces $M_L$ tels que $Spec (S_{M_L})=L$, où $S: (u_n)_{n \in \Z} \mapsto (u_{n-1})_{n \in \Z}$ désigne le shift bilatéral usuel sur $l^2(w,\Z)$.

Nous proposons une dérivation théorético-informationnelle de la théorie quantique via une axiomatique informationnelle. La 1ère partie de la thèse est consacrée aux fondements philosophiques de cette approche. Elle est présentée dans un cadre épistémologique sous la forme d'une boucle entre descriptions théoriques. La 2ème partie est consacrée à la dérivation du formalisme de la théorie quantique. Nous posons un système d'axiomes et nous analysons le double rôle de l'observateur. A l'aide des techniques de la logique quantique, nous établissons les théorèmes montrant les étapes de la reconstruction du formalisme de la théorie quantique. Dans la 3ème partie, nous introduisons la théorie des C*-algèbres et nous proposons de cette dernière une interprétation théorético-informationnelle. Nous concluons par une liste de problèmes ouverts dans l'approche informationnelle, y compris ceux relevant des sciences cognitives, de la théorie de la décision et des technologies de l'information.

Nous nous intéressons à un aspect quantitatif du dix-septième problème de Hilbert : construire une famille de polynômes en deux variables, à coefficients réels, de degré 8 en l'une des deux variables qui sont positifs mais ne sont pas somme de trois carrés de fractions rationnelles.<br /><br />Comme expliqué par Huisman et Mahé, un polynôme donné P en deux variables à coefficients réels, totalement positif, unitaire, sans facteur carré et de degré multiple de 4 en l'une des variables est une somme de trois carrés de fractions rationnelles si et seulement si la jacobienne d'une certaine courbe hyperelliptique (associée à P) possède un point ”antineutre”.<br /><br />Grâce à ce critère, et en suivant une méthode de Cassels, Ellison et Pfister, nous résolvons notre problème : à l'aide d'une 2-descente, nous montrons que la jacobienne associée à un certain polynôme positif est de rang de Mordell-Weil nul, puis nous vérifions que cette jacobienne n'a aucun point de torsion antineutre.

Dans la première partie de cette thèse, on s'intéresse à la convergence de certaines séries aléatoires. On détermine des conditions suffisantes sur la suite $(a_k)_{k\geq 1}$ et sur les variables aléatoires indépendantes $(X_k)_{k\geq 1}$, afin que pour presque tout $\omega\in\Omega$, on ait la convergence uniforme ou pour presque tout $x$ de la série $\sum_{k\geq 1}a_k f(x\cdot(X_1+\cdots+X_k)(\omega))$ pour une certaine classe de fonctions $f$. On trouve, aussi, des conditions suffisantes sur la suite $(a_k)_{k\geq 1}$ et sur les variables aléatoires indépendantes $(X_k)_{k\geq 1}$, afin que pour presque tout $\omega\in\Omega$, on ait la convergence dans $L^2(\mu)$ ou $\mu$-presque partout de la série $\sum_{k\geq 1}a_k T^{(X_1+\cdots+X_k)(\omega)}(g)(x)$ pour certaines classes de fonctions $g\in L^2(\mu)$ et de flot $\{T^{t},t\in G\}$ d'opérateurs de $L^2(\mu)$ dans $L^2(\mu)$ (où $G$ est un semi-groupe de $\mathbb{R}$). La deuxième partie porte sur des propriétés de mesures : quasi-Bernoulli et quasi-Bernoulli faible. On trouve notamment des conditions nécessaires et suffisantes pour qu'une mesure de Markov inhomogène soit quasi-Bernoulli ou quasi-Bernoulli faible. On caractèrise à l'aide de ces conditions les mesures de Bernoulli qui sont quasi-Bernoulli ou quasi-Bernoulli faible. On prouve que si une mesure de Bernoulli n'ayant que des probabilités non nulles est quasi-Bernoulli faible alors elle est quasi-Bernoulli. La dernière partie est consacrée à l'étude du problème du temps de retour uniforme dans les systèmes exponentiellement mélangeant. Il s'agit d'avoir un recouvrement aléatoire de l'espace engendré par un processus exponentiellement mélangeant. Etant donné un recouvrement aléatoire, on obtient une estimation du nombre de recouvrements en fonction de la dimension maximale locale de la mesure

This thesis consists of six papers. In Paper I, we give an algorithm for merging sorted lists of monomials and together with a projection technique, we obtain a new complexity bound for the Buchberger-Möller algorithm and the FGLM algorithm. In Paper II, we discuss four different constructions of vector space bases associated to vanishing ideals of points. We show how to compute normal forms with respect to these bases and give complexity bounds. As an application we drastically improve the computational algebra approach to the reverse engineering of gene regulatory networks. In Paper III, we introduce the concept of multiplication matrices for ideals of projective dimension zero. We discuss various applications and, in particular, we give a new algorithm to compute the variety of an ideal of projective dimension zero. In Paper IV, we consider a subset of projective space over a finite field and give a geometric description of the minimal degree of a non-vanishing form with respect to this subset. We also give bounds on the minimal degree in terms of the cardinality of the subset. In Paper V, we study an associative version of an algorithm constructed to compute the Hilbert series for graded Lie algebras. In the commutative case we use Gotzmann's persistence theorem to show that the algorithm terminates in finite time. In Paper VI, we connect the commutative version of the algorithm in Paper V with the Buchberger algorithm. / At the time of doctoral defence, the following papers were unpublished and had a status as follows: Paper 3: Manuscript. Paper 4: Manuscript. Paper 5: Manuscript. Paper 6: Manuscript

Microwave signal processing in the optical domain is investigated in this thesis. Two signal processors including an all-optical fractional Hilbert transformer and an all-optical microwave differentiator are investigated and experimentally demonstrated. Specifically, the photonic-assisted fractional Hilbert transformer with tunable fractional order is implemented based on a temporal pulse shaping system incorporating a phase modulator. By applying a step function to the phase modulator to introduce a phase jump, a real-time fractional Hilbert transformer with a tunable fractional order is achieved. The microwave bandpass differentiator is implemented based on a finite impulse response (FIR) photonic microwave delay-line filter with nonuniformly-spaced taps. A microwave bandpass differentiator based on a six-tap nonuniformly-spaced photonic microwave delay-line filter with all- positive coefficients is designed, simulated, and experimentally demonstrated. The reconfigurability of the microwave bandpass differentiator is experimentally investigated. The employment of the differentiator to perform differentiation of a bandpass microwave signal is also experimentally demonstrated.