In this thesis we present theoretical and numerical results for a feedback control problem defined by a thermal fluid. The problem is motivated by recent interest in designing and controlling energy efficient building systems. In particular, we show that it is possible to locally exponentially stabilize the nonlinear Boussinesq Equations by applying Neumann/Robin type boundary control on a bounded and connected domain. The feedback controller is obtained by solving a Linear Quadratic Regulator problem for the linearized Boussinesq equations. Applying classical results for semilinear equations where the linear term generates an analytic semigroup, we establish that this Riccati-based optimal boundary feedback control provides a local stabilizing controller for the full nonlinear Boussinesq equations. In addition, we present a finite element Galerkin approximation. Finally, we provide numerical results based on standard Taylor-Hood elements to illustrate the theory. / Ph. D.

Let Ω be the Fraïssé limit of a class of relational structures. We seek to answer the following semigroup theoretic question about Ω. What are the group H-classes, i.e. the maximal subgroups, of End(Ω)? Fraïssé limits for which we answer this question include the random graph R, the random directed graph D, the random tournament T, the random bipartite graph B, Henson's graphs G[subscript n] (for n greater or equal to 3) and the total order Q. The maximal subgroups of End(Ω) are closely connected to the automorphism groups of the relational structures induced by the images of idempotents from End(Ω). It has been shown that the relational structure induced by the image of an idempotent from End(Ω) is algebraically closed. Accordingly, we investigate which groups can be realised as the automorphism group of an algebraically closed relational structure in order to determine the maximal subgroups of End(Ω) in each case. In particular, we show that if Γ is a countable graph and Ω = R,D,B, then there exist 2[superscript aleph-naught] maximal subgroups of End(Ω) which are isomorphic to Aut(Γ). Additionally, we provide a complete description of the subsets of Q which are the image of an idempotent from End(Q). We call these subsets retracts of Q and show that if Ω is a total order and f is an embedding of Ω into Q such that im f is a retract of Q, then there exist 2[superscript aleph-naught] maximal subgroups of End(Q) isomorphic to Aut(Ω). We also show that any countable maximal subgroup of End(Q) must be isomorphic to Zⁿ for some natural number n. As a consequence of the methods developed, we are also able to show that when Ω = R,D,B,Q there exist 2[superscript aleph-naught] regular D-classes of End(Ω) and when Ω = R,D,B there exist 2[superscript aleph-naught] J-classes of End(Ω). Additionally we show that if Ω = R,D then all regular D-classes contain 2[superscript aleph-naught] group H-classes. On the other hand, we show that when Ω = B,Q there exist regular D-classes which contain countably many group H-classes.

Apresentamos duas construções para unidades de uma ordem em uma classe de álgebras de quatérnios que é anel de divisão: as unidades de Pell e as unidades de Gauss. Classificamos os anéis de inteiros de extensões quadráticas racionais, $R$, cujo grupo de unidades $\\U (R G)$ é hiperbólico para um certo grupo $G$ fixado. Também classificamos os semigrupos finitos $S$, tal que, para a álgebra unitária $\\Q S$ e para toda $\\Z$-ordem $\\Gamma$ de $\\Q S$, o grupo de unidades $\\U (\\Gamma)$ é hiperbólico. Nesse mesmo contexto, classificamos os {\\it RA}-loops $L$ cujo loop de unidades $\\U (\\Z L)$ não contém um subgrupo abeliano livre de posto dois. / For a given division algebra of a quaternion algebra, we construct and define two types of units of its $\\Z$-orders: Pell units and Gauss units. Also, for the quadratic imaginary extensions over the racionals and some fixed group $G$, we classify the algebraic integral rings for which the unit group ring is a hyperbolic group. We also classify the finite semigroups $S$, for which all integral orders $\\Gamma$ of $\\Q S$ have hyperbolic unit group $\\U(\\Gamma)$. We conclude with the classification of the $RA$-loops $L$ for which the unit loop of its integral loop ring does not contain a free abelian subgroup of rank two.

Apresentamos duas construções para unidades de uma ordem em uma classe de álgebras de quatérnios que é anel de divisão: as unidades de Pell e as unidades de Gauss. Classificamos os anéis de inteiros de extensões quadráticas racionais, $R$, cujo grupo de unidades $\\U (R G)$ é hiperbólico para um certo grupo $G$ fixado. Também classificamos os semigrupos finitos $S$, tal que, para a álgebra unitária $\\Q S$ e para toda $\\Z$-ordem $\\Gamma$ de $\\Q S$, o grupo de unidades $\\U (\\Gamma)$ é hiperbólico. Nesse mesmo contexto, classificamos os {\\it RA}-loops $L$ cujo loop de unidades $\\U (\\Z L)$ não contém um subgrupo abeliano livre de posto dois. / For a given division algebra of a quaternion algebra, we construct and define two types of units of its $\\Z$-orders: Pell units and Gauss units. Also, for the quadratic imaginary extensions over the racionals and some fixed group $G$, we classify the algebraic integral rings for which the unit group ring is a hyperbolic group. We also classify the finite semigroups $S$, for which all integral orders $\\Gamma$ of $\\Q S$ have hyperbolic unit group $\\U(\\Gamma)$. We conclude with the classification of the $RA$-loops $L$ for which the unit loop of its integral loop ring does not contain a free abelian subgroup of rank two.

This research investigates existing relationships between the three apparently unrelated subjects: Markov process, Semigroups and Partial difierential equations. Markov processes define semigroups through their transition functions. Conversely particular semigroups determine transition functions and can be regarded as Markov processes. We have exploited these relationships to study some Markov chains. The infnitesimal generator of a Feller semigroup on the closure of a bounded domain of Rn; (n ^ 2), is an integro-diferential operator in the interior of the domain and verifes a boundary condition. The existence of a Feller semigroup defined by a diferential operator and a boundary condition is due to the existence of solution of a bounded value problem. From this result other existence suficient conditions on the existence of Feller semigroups have been obtained and we have applied some of them to construct Feller semigroups on the unity disk of R2. / Decision Sciences / M. Sc. (Operations Research)

In this thesis we prove the following: The semigroup which is a disjoint union of two or three copies of a group is a Clifford semigroup, Rees matrix semigroup or a combination between a Rees matrix semigroup and a group. Furthermore, the semigroup which is a disjoint union of finitely many copies of a finitely presented (residually finite) group is finitely presented (residually finite) semigroup. The constructions of the semigroup which is a disjoint union of two copies of the free monogenic semigroup are parallel to the constructions of the semigroup which is a disjoint union of two copies of a group, i.e. such a semigroup is Clifford (strong semilattice of groups) or Rees matrix semigroup. However, the semigroup which is a disjoint union of three copies of the free monogenic semigroup is not just a strong semillatice of semigroups, Rees matrix semigroup or combination between a Rees matrix semigroup and a semigroup, but there are two more semigroups which do not arise from the constructions of the semigroup which is a disjoint union of three copies of a group. We also classify semigroups which are disjoint unions of two or three copies of the free monogenic semigroup. There are three types of semigroups which are unions of two copies of the free monogenic semigroup and nine types of semigroups which are unions of three copies of the free monogenic semigroup. For each type of such semigroups we exhibit a presentation defining semigroups of this type. The semigroup which is a disjoint union of finitely many copies of the free monogenic semigroup is finitely presented, residually finite, hopfian, has soluble word problem and has soluble subsemigroup membership problem.

Submitted by Maria Cristina (library@lncc.br) on 2017-04-12T18:15:16Z No. of bitstreams: 1 thesis.pdf: 855206 bytes, checksum: d10e31928389222e4952d705ce5da5ac (MD5) / Approved for entry into archive by Maria Cristina (library@lncc.br) on 2017-04-12T18:15:30Z (GMT) No. of bitstreams: 1 thesis.pdf: 855206 bytes, checksum: d10e31928389222e4952d705ce5da5ac (MD5) / Made available in DSpace on 2017-04-12T18:15:41Z (GMT). No. of bitstreams: 1 thesis.pdf: 855206 bytes, checksum: d10e31928389222e4952d705ce5da5ac (MD5) Previous issue date: 2016-07-21 / Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) / We studied the uniform stabilization of a class of Timoshenko systems with tip load at the free end of the beam. Our main result is to prove that the semigroup associated to this model is not exponentially stable. Moreover, we prove that the semigroup decays polynomially to zero. When the damping mechanism is e ective only on the boundary of the rotational angle, the solution also decays polynomially with rate depending on the coe cients of the problem. / Estudamos a estabilização uniforme para uma classe de sistemas de Timoshenko com carga pontual na extremidade livre da viga. Nosso principal resultado é provar que o semigrupo associado com este modelo não é exponencialmente estável. Além disso, provamos que o semigrupo decai polinomialmente. Quando a dissipação é eficaz apenas sobre o limite do ângulo de rotação, a solução também decai polinomialmente com taxa de decaimento dependendo dos coeficientes do problema.

Cette thèse se situe à l'intersection de plusieurs domaines mathématiques particulièrement actifs actuellement : l'analyse fonctionnelle, la théorie des opérateurs, la dynamique complexe et la théorie des semigroupes. Nous étudierons ici les semigroupes d'opérateurs de composition sur quelques espaces de Hardy pondérés, notamment l'espace de Hardy du disque et l'espace de Dirichlet. Dans un premier temps, nous allons voir pourquoi se placer à cette intersection est pertinent, en montrant comment utiliser les propriétés des semigroupes pour calculer explicitement les normes de certains opérateurs de composition. Dans un second temps, nous étudierons les propriétés des semigroupes d'opérateurs de compositions qui sont directement accessibles à partir de la seule donnée du générateur infinitésimal du semigroupe, en nous concentrant tout particulièrement sur les notions d'analyticité et de compacité / This thesis takes place at the intersection of several particularly active mathematical areas : functional analysis, operator theory, complex dynamics and theory of semigroups. Here, we study semigroups of composition operators on some weighted Hardy spaces, in particular the Hardy space of the disk and the Dirichlet space. First, we will show why this intersection is relevant for our study, pointing out how to use the properties of semigroups to explicitly compute the norms of some composition operators. Secondly, we will study the properties of semigroups of composition operators that are directly accessible from the only data of the infinitesimal generator, focusing on analyticity and compactness

L’objectif de ce travail de thèse est d’étudier quelques problèmes elliptiques et paraboliques d’écoulement de fluides non Newtoniens incompressibles et non isothermes gouvernés par l’équation aux dérivées partielles de Stokes avec la condition de Tresca sur une partie du bord quand la viscosité dépend à la fois de la température, de la vitesse et du module du tenseur des taux de déformations. Dans le premier chapitre, on a fait une introduction générale. Dans le deuxième chapitre, nous nous intéressons au couplage entre le système de Stokes et l’équation de la chaleur en régime stationnaire. On montre l’existence de la solution de l’inéquation variationnelle décrivant le système de Stokes pour une température donnée quand la viscosité dépend à la fois de la température, de la vitesse et du module du tenseur des taux de déformations en utilisant la méthode de monotonie pour la vitesse et le théorème de De Rham pour la pression. Dans un deuxième temps, on étudie l’existence et l’unicité de la température solution de l’équation de la chaleur avec un terme L1(Ω) au second membre quand la viscosité dépend à la fois de la température, de la vitesse et du module du tenseur des taux de déformations. On montre ensuite l’existence de la solution du problème variationnel couplé avec la viscosité dépend de la température et du module du tenseur des taux de déformations, en utilisant le théorème de point fixe de Schauder. Dans le troisième et le quatrième chapitre, on traite l’existence et l’unicité de la solution du système de Stokes en régime instationnaire quand la viscosité dépend de la température et du module du tenseur des taux de déformations dans les cas p = 2, p > 2 et 6 5 < p < 2 en utilisant la notion des semi-groupes et la méthode de monotonie pour la vitesse et le théorème de De Rham pour la pression. Par contre, lorsque la viscosité dépend de plus de la vitesse on obtient seulement l’existence par le théorème de point fixe de Schauder / The objective of this thesis is to study some elliptic and parabolic problems of the non-Newtonian flow of an incompressible and non isothermal fluid governed by partial differential equation of Stokes with Tresca’s condition on a part of the boundary when the fluid viscosity depends on temperature and also on the modulus of strain rate tensor and the velocity of the fluid. In the first chapter, we did a general introduction. In the second chapter, we consider the coupling between the Stokes systemand the heat equation in steady state. We prove the existence of a solution of the variational inequality describing the Stokes system when the fluid viscosity depends on temperature and also on the modulus of strain rate tensor and the velocity of the fluid of a given temperature by using the monotony methods for the velocity and De Rham’s theorem for the pressure. We study the existence and uniqueness of the temperature solution of the heat equation with L1 (Ω) term to the second member when the fluid viscosity depends on temperature and also on the modulus of strain rate tensor and the velocity of the fluid. We show the existence of a solution of the coupled variational problem when the fluid viscosity depends on temperature and also on the modulus of strain rate tensor by using Schauder fixed point theorem. In the third and the fourth chapter, we treate the existence and uniqueness of a solution of the Stokes system in unsteady state when the fluid viscosity depends only on temperature and on the modulus of strain rate tensor in the cases p = 2, p > 2 and 6 5 < p < 2 by using the notion of semigroup and monotony methods for the velocity and De Rham’s theorem for the pressure. However, when the fluid viscosity depends also on the velocity of the fluid we obtain only the existence by Schauder fixed point theorem

This thesis treats a protocol for anonymous database search (or if one prefer, a protocol for user-private information retrieval), that is based on the use of combinatorial configurations. The protocol is called P2P UPIR. It is proved that the (v,k,1)-balanced incomplete block designs (BIBD) and in particular the finite projective planes are optimal configurations for this protocol. The notion of n-anonymity is applied to the configurations for P2P UPIR protocol and the transversal designs are proved to be n-anonymous configurations for P2P UPIR, with respect to the neighborhood points of the points of the configuration. It is proved that to the configurable tuples one can associate a numerical semigroup. This theorem implies results on existence of combinatorial configurations. The proofs are constructive and can be used as algorithms for finding combinatorial configurations. It is also proved that to the triangle-free configurable tuples one can associate a numerical semigroup. This implies results on existence of triangle-free combinatorial configurations.