Neste trabalho estudaremos a geração do semigrupos por operadores elípticos em dois espaços. Em primeiro lugar estudaremos a geração de semigrupo no espaço \'L POT.2\' (\'OMEGA\') por operadores elípticos de ordem 2m com \'OMEGA\' suficientemente regular. Mais precisamente, se \'OMEGA\' é um domínio limitado com \'PARTIAL OMEGA\' de classe \'C POT. 2m,\' L (x;D) = \'SIGMA\' / [\'alpha\'] \'< ou =\' \'a IND. alpha\' (x) \'D POT. alpha\' é um operador diferencial elíptico de ordem 2m, com \'a IND. alpha\' \'PERTENCE\' \' \'C POT.j\' (\'OMEGA\'), j = max {0, [\'alpha\'] - m}, e A : D(A) \'ESTÁ CONTIDO\' EM \'L POT. 2 (\'OMEGA\') \'SETA\' \' L POT. 2 (\'OMEGA\') é o operador linear dado por D(A) = \'H POT. 2m\' (\'OMEGA\') \'H POT. m INF. 0\' (\'OMEGA\'), (Au)(x) = L (x;D)u; então -A gera um \'C IND. 0\'-semigrupo holomorfo em \'L POT.2\' (\'OMEGA\'). ). Em segundo lugar estudaremos a geração de semigrupo em \'C IND. 0\'(\'OMEGA\") = ) = {u \'PERTENCE A\' C (\'OMEGA\' \'BARRA\") : u[\'PARTIAL omega\' = 0} por operadores elípticos de ordem 2 com \'OMEGA\' satisfazendo uma propriedade geométrica. Mais precisamente, se \'OMEGA\' ESTA CONTIDO EM\' \'R POT. n\' (n \'> ou =\' 2) é um domínio limitado que satisfaz a condição de cone exterior uniforme, L é o operador Lu := - \\\\SIGMA SUP n INF. i,j = 1\' \'a IND. ij \'D IND. ij u + \'\\SIGMA SUP. n IND. j=1 \'b IND. j\' u + cu com coeficientes reais \'a IND. ij\' , \'b IND. j\' , c que satisfazem \'b IND. j \' \'PERTENCE A\' \'L POT. INFTY\' (\'OMEGA\') , j = 1, ..., n, c \'PERTENCE A \' \'L POT> INFTY\' (OMEGA), c \'> ou =\' 0, \'a IND. ij\' \'PERTECE A\' C(\' OMEGA BARRA)\' \' INTERSECCAO\' \'L POT. INFTY\' (OMEGA),e \'A IND. 0\' é parte de L em \'C IND. 0\' (\"OMEGA\'), isto é, D(\'A IND. 0\') = {u \'PERTENCE A\' \'C IND. 0\' (\'OMEGA\') \'INTERSECÇÂO\' \'W POT. 2, n INF. loc\' (\'OMEGA\') : Lu \'PERTENCE A\' \'C IND. 0\' (\'OMEGA\')\' \'A IND. 0\' u = Lu, então -\'A IND. 0\' gera um \'C IND. 0-semigrupo holomorfo limitado em \'C IND. 0\' (\'OMEGA\') / In this work we study the generation of semigroups by elliptic operators in two spaces. Firstly we study the generation of semigroup in the space \'L POT. 2\' (OMEGA) for elliptic operators of order 2m with \'OMEGA\' regular domain. More precisely, if \'OMEGA\' is a bounded domain with \\PARTIAL OMEGA\' \'IT BELONGS\' \'C POT. 2m\', L (x, D) = \\ sigma INF.ALPHA \'> or =\' 2m, \'a IND. alpha\' ( x) \'D POT alpha\' is an elliptic differential operator of order 2m, with \'a IND. alpha\' \' \'IT BELONGS\' \'C POT. j\' (OMEGA), j = max , and A : D (A) \'THIS CONTAINED\' \'L POT. 2\' (OMEGA) \'ARROW\' \'L POT. 2\' (OMEGA) is linear operator given or D(A) = \'H POT. 2m\' (OMEGA) \'INTERSECTION\' \'H POT. m INF. 0 (OMEGA) (Au) (x) = L (x,D) u then -A generates a holomorphic \'C IND. 0\'-semigroup in \'L POT. 2\'.(OMEGA). Secondly we study the generation of semigroup in \'C IND. 0\' (OMEGA) = {u \'IT BELONGS\' (c INF. O\' (OMEGA BAR) : \'u [IND. \\partial omega\' = 0} for elliptic operators of second order with \'OMEGA\' satisfying a geometric property. That is, if \'OMEGA\' \'IT BELONGS\' \'R POT. n\' (n > or = 2) is a bounded domain that satisfies the uniform exterior cone condition, L is the elliptic operator given by Lu : = - \\SIGMA SUP. n INF. i,j = 1\' \'a IND. i, j\' \'D IND. ij \' u + \\SIGMA SUP n INF. j=1\' \'b IND j D IND j\' u + cu with real coefficients \'a IND. ij, \'b IND. j\' , c satisfying \'b ind. j\' \'IT BELONGS\' \' L POT. INFTY\' (omega), j = 1, ..., n, c \'it belongs\' \'L POT. INFTY\' (OMEGA), \'c > or =\' 0, \'\'a IND. ij \'IT BELONGS\' C (OMNEGA BAR) \'INTERSECTION\' (OMEGA), and \'A IND. 0\' is part of L in \'C IND. 0\'(OMEGA), that is, D (\'A IND. 0\') = {u \'IT BELONGS\' \'C IND. 0\' (OMEGA) INTERSECTION \'W POT. 2, n IND. loc (OMEGA)} \'A IND. 0u\' = Lu, then - \'A IND. 0\' generates a bounded holomorphic \'C IND. 0\'-semigroup on \'C IND. 0\' (OMEGA)

Neste trabalho estudaremos a geração do semigrupos por operadores elípticos em dois espaços. Em primeiro lugar estudaremos a geração de semigrupo no espaço \'L POT.2\' (\'OMEGA\') por operadores elípticos de ordem 2m com \'OMEGA\' suficientemente regular. Mais precisamente, se \'OMEGA\' é um domínio limitado com \'PARTIAL OMEGA\' de classe \'C POT. 2m,\' L (x;D) = \'SIGMA\' / [\'alpha\'] \'< ou =\' \'a IND. alpha\' (x) \'D POT. alpha\' é um operador diferencial elíptico de ordem 2m, com \'a IND. alpha\' \'PERTENCE\' \' \'C POT.j\' (\'OMEGA\'), j = max {0, [\'alpha\'] - m}, e A : D(A) \'ESTÁ CONTIDO\' EM \'L POT. 2 (\'OMEGA\') \'SETA\' \' L POT. 2 (\'OMEGA\') é o operador linear dado por D(A) = \'H POT. 2m\' (\'OMEGA\') \'H POT. m INF. 0\' (\'OMEGA\'), (Au)(x) = L (x;D)u; então -A gera um \'C IND. 0\'-semigrupo holomorfo em \'L POT.2\' (\'OMEGA\'). ). Em segundo lugar estudaremos a geração de semigrupo em \'C IND. 0\'(\'OMEGA\") = ) = {u \'PERTENCE A\' C (\'OMEGA\' \'BARRA\") : u[\'PARTIAL omega\' = 0} por operadores elípticos de ordem 2 com \'OMEGA\' satisfazendo uma propriedade geométrica. Mais precisamente, se \'OMEGA\' ESTA CONTIDO EM\' \'R POT. n\' (n \'> ou =\' 2) é um domínio limitado que satisfaz a condição de cone exterior uniforme, L é o operador Lu := - \\\\SIGMA SUP n INF. i,j = 1\' \'a IND. ij \'D IND. ij u + \'\\SIGMA SUP. n IND. j=1 \'b IND. j\' u + cu com coeficientes reais \'a IND. ij\' , \'b IND. j\' , c que satisfazem \'b IND. j \' \'PERTENCE A\' \'L POT. INFTY\' (\'OMEGA\') , j = 1, ..., n, c \'PERTENCE A \' \'L POT> INFTY\' (OMEGA), c \'> ou =\' 0, \'a IND. ij\' \'PERTECE A\' C(\' OMEGA BARRA)\' \' INTERSECCAO\' \'L POT. INFTY\' (OMEGA),e \'A IND. 0\' é parte de L em \'C IND. 0\' (\"OMEGA\'), isto é, D(\'A IND. 0\') = {u \'PERTENCE A\' \'C IND. 0\' (\'OMEGA\') \'INTERSECÇÂO\' \'W POT. 2, n INF. loc\' (\'OMEGA\') : Lu \'PERTENCE A\' \'C IND. 0\' (\'OMEGA\')\' \'A IND. 0\' u = Lu, então -\'A IND. 0\' gera um \'C IND. 0-semigrupo holomorfo limitado em \'C IND. 0\' (\'OMEGA\') / In this work we study the generation of semigroups by elliptic operators in two spaces. Firstly we study the generation of semigroup in the space \'L POT. 2\' (OMEGA) for elliptic operators of order 2m with \'OMEGA\' regular domain. More precisely, if \'OMEGA\' is a bounded domain with \\PARTIAL OMEGA\' \'IT BELONGS\' \'C POT. 2m\', L (x, D) = \\ sigma INF.ALPHA \'> or =\' 2m, \'a IND. alpha\' ( x) \'D POT alpha\' is an elliptic differential operator of order 2m, with \'a IND. alpha\' \' \'IT BELONGS\' \'C POT. j\' (OMEGA), j = max , and A : D (A) \'THIS CONTAINED\' \'L POT. 2\' (OMEGA) \'ARROW\' \'L POT. 2\' (OMEGA) is linear operator given or D(A) = \'H POT. 2m\' (OMEGA) \'INTERSECTION\' \'H POT. m INF. 0 (OMEGA) (Au) (x) = L (x,D) u then -A generates a holomorphic \'C IND. 0\'-semigroup in \'L POT. 2\'.(OMEGA). Secondly we study the generation of semigroup in \'C IND. 0\' (OMEGA) = {u \'IT BELONGS\' (c INF. O\' (OMEGA BAR) : \'u [IND. \\partial omega\' = 0} for elliptic operators of second order with \'OMEGA\' satisfying a geometric property. That is, if \'OMEGA\' \'IT BELONGS\' \'R POT. n\' (n > or = 2) is a bounded domain that satisfies the uniform exterior cone condition, L is the elliptic operator given by Lu : = - \\SIGMA SUP. n INF. i,j = 1\' \'a IND. i, j\' \'D IND. ij \' u + \\SIGMA SUP n INF. j=1\' \'b IND j D IND j\' u + cu with real coefficients \'a IND. ij, \'b IND. j\' , c satisfying \'b ind. j\' \'IT BELONGS\' \' L POT. INFTY\' (omega), j = 1, ..., n, c \'it belongs\' \'L POT. INFTY\' (OMEGA), \'c > or =\' 0, \'\'a IND. ij \'IT BELONGS\' C (OMNEGA BAR) \'INTERSECTION\' (OMEGA), and \'A IND. 0\' is part of L in \'C IND. 0\'(OMEGA), that is, D (\'A IND. 0\') = {u \'IT BELONGS\' \'C IND. 0\' (OMEGA) INTERSECTION \'W POT. 2, n IND. loc (OMEGA)} \'A IND. 0u\' = Lu, then - \'A IND. 0\' generates a bounded holomorphic \'C IND. 0\'-semigroup on \'C IND. 0\' (OMEGA)

A semigroup is simply a set with an associative binary operation; computational semigroup theory is the branch of mathematics concerned with developing techniques for computing with semigroups, as well as investigating semigroups with the help of computers. This thesis explores both sides of computational semigroup theory, across several topics, especially in the finite case. The central focus of this thesis is computing and describing maximal subsemigroups of finite semigroups. A maximal subsemigroup of a semigroup is a proper subsemigroup that is contained in no other proper subsemigroup. We present novel and useful algorithms for computing the maximal subsemigroups of an arbitrary finite semigroup, building on the paper of Graham, Graham, and Rhodes from 1968. In certain cases, the algorithms reduce to computing maximal subgroups of finite groups, and analysing graphs that capture information about the regular I-classes of a semigroup. We use the framework underpinning these algorithms to describe the maximal subsemigroups of many families of finite transformation and diagram monoids. This reproduces and greatly extends a large amount of existing work in the literature, and allows us to easily see the common features between these maximal subsemigroups. This thesis is also concerned with direct products of semigroups, and with a special class of semigroups known as Rees 0-matrix semigroups. We extend known results concerning the generating sets of direct products of semigroups; in doing so, we propose techniques for computing relatively small generating sets for certain kinds of direct products. Additionally, we characterise several features of Rees 0-matrix semigroups in terms of their underlying semigroups and matrices, such as their Green's relations and generating sets, and whether they are inverse. In doing so, we suggest new methods for computing Rees 0-matrix semigroups.

The classification of finite semigroups is difficult even for small orders because of their large number. Most finite semigroups are nilpotent of nilpotency rank 3. Formulae for their number up to isomorphism, and up to isomorphism and anti-isomorphism of any order are the main results in the theoretical part of this thesis. Further studies concern the classification of nilpotent semigroups by rank, leading to a full classification for large ranks. In the computational part, a method to find and enumerate multiplication tables of semigroups and subclasses is presented. The approach combines the advantages of computer algebra and constraint satisfaction, to allow for an efficient and fast search. The problem of avoiding isomorphic and anti-isomorphic semigroups is dealt with by supporting standard methods from constraint satisfaction with structural knowledge about the semigroups under consideration. The approach is adapted to various problems, and realised using the computer algebra system GAP and the constraint solver Minion. New results include the numbers of semigroups of order 9, and of monoids and bands of order 10. Up to isomorphism and anti-isomorphism there are 52,989,400,714,478 semigroups with 9 elements, 52,991,253,973,742 monoids with 10 elements, and 7,033,090 bands with 10 elements. That constraint satisfaction can also be utilised for the analysis of algebraic objects is demonstrated by determining the automorphism groups of all semigroups with 9 elements. A classification of the semigroups of orders 1 to 8 is made available as a data library in form of the GAP package Smallsemi. Beyond the semigroups themselves a large amount of precomputed properties is contained in the library. The package as well as the code used to obtain the enumeration results are available on the attached DVD.

In this thesis we study two different topics, both in the context of semigroup constructions. The first is the investigation of an embedding problem, specifically the problem of whether it is possible to embed any given finitely presentable semigroup into a D-simple finitely presentable semigroup. We consider some well-known semigroup constructions, investigating their properties to determine whether they might prove useful for finding a solution to our problem. We carry out a more detailed study into a more complicated semigroup construction, the Byleen extension, which has been used to solve several other embedding problems. We prove several results regarding the structure of this extension, finding necessary and sufficient conditions for an extension to be D-simple and a very strong necessary condition for an extension to be finitely presentable. The second topic covered in this thesis is relative rank, specifically the sequence obtained by taking the rank of incremental direct powers of a given semigroup modulo the diagonal subsemigroup. We investigate the relative rank sequences of infinite Cartesian products of groups and of semigroups. We characterise all semigroups for which the relative rank sequence of an infinite Cartesian product is finite, and show that if the sequence is finite then it is bounded above by a logarithmic function. We will find sufficient conditions for the relative rank sequence of an infinite Cartesian product to be logarithmic, and sufficient conditions for it to be constant. Chapter 4 ends with the introduction of a new topic, relative presentability, which follows naturally from the topic of relative rank.

Orientador: Fernando Eduardo Torres Orihuela / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-12T08:34:02Z (GMT). No. of bitstreams: 1 VillanuevaZevallos_JuanElmer_D.pdf: 1127069 bytes, checksum: 8ac303abd191b4c264038dcd1ce40be1 (MD5) Previous issue date: 2008 / Resumo: Os principais tópicos aqui considerados são do tipo aritmético. Introduzimos e estudamos semigrupos que generalizam os chamados semigrupos de Arf. Além de seu interesse particular, eles podem ser usados para esclarecer a estrutura de anéis de semigrupos no sentido de Lipman. Também calculamos os valores exatos dos pesos de semigrupos usando o número de lacunas pares. Isto está relacionado ao recobrimento duplo de curvas e tem interesse no estudo de moduli e constelação de curvas. / Abstract: The main topics considered here are of arithmetical type. We introduce and study semigroups that generalize the so-called Arf semigroups. Apart from being interesting by their own, they may be used to clarify the structure of semigroup rings in the sense of Lipman. We also compute the true value of the weights of semigroups by using the number of even gaps. This is related to double covering of curves and is useful to the study of moduli and constellation of curves. / Doutorado / Geometria Algebrica / Doutor em Matemática

Cette thèse est composée de deux parties. Nous étudions dans la première la conjecture de Wilf pour les semi-groupes numériques et la résolvons dans certains cas. Dans la seconde nous considérons une classe de semi-groupes presque arithmétiques et donnons pour ces semi-groupes des formules explicites pour la base d’Apéry, le nombre de Frobenius, et les nombres de pseudo-Frobenius. Nouscaractérisons aussi ceux qui sont symétriques (resp. pseudo-symétriques). / The thesis is made up of two parts. We study in the first part Wilf’s conjecture for numerical semigroups. We give an equivalent form of Wilf’s conjecture in terms of the Apéry set, embedding dimension and multiplicity of a numerical semigroup. We also give an affirmative answer for the conjecture in certain cases. In the second part, we consider a class of almost arithmetic numerical semigroups and give for this class of semigroups explicit formulas for the Apéry set, the Frobenius number, the genus and the pseudo-Frobenius numbers. We also characterize the symmetric (resp. pseudo-symmetric) numerical semigroups for this class of numerical semigroups.

In this thesis we analyze lumpability of infinite dimensional dynamical systems. Lumping is a method to project a dynamics by a linear reduction operator onto a smaller state space on which a self-contained dynamical description exists. We consider a well-posed dynamical system defined on a Banach space X and generated by an operator F, together with a linear and bounded map M : X → Y, where Y is another Banach space. The operator M is surjective but not an isomorphism and it represents a reduction of the state space. We investigate whether the variable y = M x also satisfies a well-posed and self-contained dynamics on Y . We work in the context of strongly continuous semigroup theory. We first discuss lumpability of linear systems in Banach spaces. We give conditions for a reduced operator to exist on Y and to describe the evolution of the new variable y . We also study lumpability of nonlinear evolution equations, focusing on dissipative operators, for which some interesting results exist, concerning the existence and uniqueness of solutions, both in the classical sense of smooth solutions and in the weaker sense of strong solutions. We also investigate the regularity properties inherited by the reduced operator from the original operator F . Finally, we describe a particular kind of lumping in the context of C*-algebras. This lumping represents a different interpretation of a restriction operator. We apply this lumping to Feller semigroups, which are important because they can be associated in a unique way to Markov processes. We show that the fundamental properties of Feller semigroups are preserved by this lumping. Using these ideas, we give a short proof of the classical Tietze extension theorem based on C*-algebras and Gelfand theory.

Cette thèse est consacrée à l’enrichissement du modèle mathématique classique des structures intelligentes, en tenant compte des effets thermiques, et à son étude analytique et numérique. Il s'agit typiquement de structures se présentant sous forme de capteurs ou actionneurs, piézoélectriques et/ou magnétostrictifs, dont les propriétés dépendent de la température. On présente d'abord des résultats d'existence et unicité concernant deux problèmes posés sur un domaine tridimensionnel : le problème dynamique et le problème quasi-statique. A partir du problème quasi-statique on déduit un modèle bidimensionnel de plaque grâce à la méthode des développements asymptotiques en considérant quatre types différents de conditions aux limites, chacun visant à modéliser un comportement de type capteur et/ou actionneur. Chacun des quatre problèmes se découple en un problème membranaire et un problème de flexion. Ce dernier est un problème d'évolution qui tient compte d'un effet d'inertie de rotation. On focalise ensuite notre attention sur ce problème et on en présente une étude mathématique et numérique. L'analyse numérique est complétée avec des tests effectués sous l'environnement FreeFEM++. / This thesis is devoted to the enrichment of the usual mathematical model of smart structures, by taking into account thermal effects, and to its mathematical and numerical study. By the expression "smart structures" we refer to structures acting as sensors or actuators, whose properties depend on the temperature. We present at first the results of existence and uniqueness concerning two problems posed on a three-dimensional domain: the dynamic problem and the quasi-static problem. Based on the quasi-static problem, we infer a two-dimensional plate model by means of the asymptotic expansion method by considering four different sets of boundary conditions, each one featuring a sensor-like or an actuator-like behavior. Each of the four problems decouples into a membrane problem and a flexural problem. The latter is an evolution problem that accounts for a rotational inertia effect. Attention is then focused on this problem by presenting a mathematical and numerical study of it. Our numerical analysis is complemented with numerical tests carried out under the FreeFEM++ environment.

In the presented work we study the existence of periodic solution to infinite dimensional stochastic equation with periodic coefficients driven by Cylindrical Wiener process. Used theory of infinite dimensional stochastic equations in Hilbert spaces and Markov processes is summarized in the first two chapters. In the third and last chapter we present the result itself. Necessary technical background mostly from operator theory is encapsulated in the Appendix. The proof of existence of periodic solution of corresponding equation is a combination of arguments by Khasminskii, which ensure under suitable conditions the existence of periodic Markov process, and the results of Da Prato, G¸atatrek and Zabczyk for the existence of invariant measure for homogeneous stochastic equation in Hilbert spaces. At the end we derive sufficient condition for the existence of periodic solution in the language of coefficients using the work of Ichikawa and illustrate the results by the example of Stochastic PDE. The work is written in English.