In this thesis we consider in detail the following two fundamental problems for semigroup presentations: 1. Given a semigroup find a presentation defining it. 2. Given a presentation describe the semigroup defined by it. We also establish two links between these two approaches: semigroup constructions and computational methods. After an introduction to semigroup presentations in Chapter 3, in Chapters 4 and 5 we consider the first of the two approaches. The semigroups we examine in these two chapters include completely O-simple semigroups, transformation semigroups, matrix semigroups and various endomorphism semigroups. In Chapter 6 we find presentations for the following semi group constructions: wreath product, Bruck-Reilly extension, Schiitzenberger product, strong semilattices of monoids, Rees matrix semigroups, ideal extensions and subsemigroups. We investigate in more detail presentations for subsemigroups in Chapters 7 and 10, where we prove a number of Reidemeister-Schreier type results for semigroups. In Chapter 9 we examine the connection between the semi group and the group defined by the same presentation. The general results from Chapters 6, 7, 9 and 10 are applied in Chapters 8, 11, 12 and 13 to subsemigroups of free semigroups, Fibonacci semigroups, semigroups defined by Coxeter type presentations and one relator products of cyclic groups. Finally, in Chapter 14 we describe the Todd-Coxeter enumeration procedure and introduce three modifications of this procedure.

This thesis is concerned with extensions and refinements of the Katznelson-Tzafriri theorem, a cornerstone of the asymptotic theory of operator semigroups which recently has received renewed interest in the context of damped wave equations. The thesis comprises three main parts. The key results in the first part are a version of the Katznelson-Tzafriri theorem for bounded C_0-semigroups in which a certain function appearing in the original statement of the result is allowed more generally to be a bounded Borel measure, and bounds on the rate of decay in an important special case. The second part deals with the discrete version of the Katznelson-Tzafriri theorem and establishes upper and lower bounds on the rate of decay in this setting too. In an important special case these general bounds are then shown to be optimal for general Banach spaces but not on Hilbert space. The third main part, finally, turns to general operator semigroups. It contains a version of the Katznelson-Tzafriri theorem in the Hilbert space setting which relaxes the main assumption of the original result. Various applications and extensions of this general result are also presented.

Demonstramos a Lei de Weyl sobre o comportamento assintótico dos autovalores do operador Laplaciano com condições de contorno de Dirichlet em domínios limitados e suaves com o auxílio do núcleo do calor. Para isso, fazemos um estudo dos operadores não-limitados, semigrupos e da transformada de Fourier. Por fim, expomos alguns resultados posteriores motivados pelo artigo de Mark Kac \"Can one hear the shape of a drum?\". / We prove the Weyl Law on the asymptotic behavior of eigenvalues of the Laplace operator with Dirichlet boundary conditions in smooth bounded domains with the help of the heat kernel. To that end, we study unbounded operators, semigroups and the Fourier transform. Lastly, we mention some further results motivated by Mark Kac\'s article \"Can one hear the shape of a drum?\".

Nesta dissertação estudamos uma versão do Teorema de Trotter-Kato que estabelece uma equivalência entre a continuidade, relativamente a um parâmetro, de operadores resolvente e a continuidade dos semigrupos lineares associados. Os operadores ilimitados envolvidos (geradores de semigrupos analíticos) estão definidos em espaços que variam com o parâmetro e isto nos leva a ter que comparar elementos de espaços de Banach diferentes. Este resultado é aplicado a um problema de Neumann em um domínio fino com fronteira altamente oscilante e que se degenera a um intervalo quando o parâmetro varia. Nesta aplicação, utilizamos o método das múltiplas escalas (comum em teoria de homogeneização) para obter formalmente o problema limite (veja [17]) e, em seguida, provamos a convergência compacta dos operadores resolventes utilizando as funções teste oscilantes de Tartar [15], [16] (veja também Cioranescu e Saint Jean Paulin [12]), obtidas através de um problema auxiliar, juntamente com operadores de extensão / In this work we study a version of Trotter-Katos Theorem that establishes an equivalence between the continuity, with respect to a parameter, of the resolvent operators and the continuity of the associated linear semigroups. The unbounded operators involved (generators of analytic semigroups) are defined spaces that vary with the parameter leading us to introduce methods to compare vectors in different Banach spaces. We apply this theorem to an elliptic boundary value problem with Neumann boundary condition in a highly oscillating thin domain that degenerates to a line segment as the parameter varies. In this application we use the multiple scale method (frequently used in the homogenization theory) to obtain, formally, the limiting problem (see [17]) and, in the sequel, we prove the compact convergence of resolvent operators using the oscillating test functions of Tartar [15] (see also [16] and Cioranescu and Saint Jean Paulin [12]) defined with the aid of an auxiliary problem as well as extension operators

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Για ένα, μάλλον, μακρύ διάστημα, (1950-1975) οι Μαθηματικοί ασχολήθηκαν με την εμφύτευση μιας αντιμεταθετικής τοπολογικής ημιομάδας σε ομάδα. Είναι γνωστό ότι για ημιομάδα S έχουμε αλγεβρική εμφύτευση στο σχέση ισοδυναμίας = , όπου στοιχεία της καινούργιας ομάδας). Το νέο στοιχείο είναι ότι ενώ η συνθήκη εμφύτευσης αναφέρεται σε Ομοιόμορφο χώρο, έχει εισαχθεί ο Η- μι-Ομοιόμορφος χώρος. Οι διαφορές μεταξύ των δύο χώρων είναι τεράστιες και ακριβώς, εκεί έγκειται η δημιουργικότητα της νέας δομής. Έτσι, η πρώτη θεώρηση για τη διατριβή είναι η προσπάθεια επιστημόνων να βρούνε συνθήκες, ώστε να μπορεί μια τοπολογική αντιμεταθετική ημιομάδα ( S,.,τ) (με τη συνήθη έννοια των . και τ ) να εμφυτεύεται στη δομή η γνωστή ισοδυ- ναμία ad=bc αν ). Τα έξη πρώτα εδάφια είναι εισαγωγικά. Στη συνέχεια εκθέτουμε όλη τη μεθοδο- λογία του θέματος / --

The theory of semigroups of linear operators forms an integral part of Functional Analysis with substantial applications to many fields of the natural sciences. In this study we are concerned with the application to equations of mathematical physics. The theory of semigroups of bounded linear operators is closely related to the solvability of evolution equations in Banach spaces that model time dependent processes in nature. Well-posed evolution problems give rise to a semigroup of bounded linear operators. However, in many important and interesting cases the problem is ill-posed making it inaccessible to the classical semigroup theory. One way of dealing with this problem is to generalize the theory of semigroups. In this thesis we give an outline of the theory of two such generalizations, namely, C-regularized semigroups and B-bounded semigroups, with the inter-relations between them and show a number of applications to ill-posed problems. / Thesis (Ph.D.)-University of Natal, Durban, 2001.

Thesis (M.S.)--Brigham Young University. Dept. of Mathematics, 2006. / Includes bibliographical references (p. 61-62).

We study an additive perturbation theorem for substochastic semigroups which is known as Kato's Theorem. There are two previously-known generalisations of Kato's Theorem, namely for abstract state spaces and for KB-spaces. We prove a version of Kato's Theorem for a class of spaces which encompasses both, namely ordered Banach spaces with generating cone and monotone norm. We also study a property of the perturbed semigroup in Kato's Theorem known as honesty of the semigroup. We add a few results to the fairly extensive existing theory of honesty for Kato's Theorem for abstract state spaces. In light of our new generalisation of Kato's Theorem to ordered Banach spaces with monotone norm, we investigate generalising the theory of honesty to these spaces as well. The results for the general case are less complete as many of the results for the case of abstract state spaces depend on the additive norm structure of the space. We also consider some new applications of honesty theory in abstract state spaces. We begin by applying honesty theory to the study of the heat equation on graphs. We prove that honesty of the heat semigroup coincides with a concept known as stochastic completeness of the graph which has been studied independently of honesty. We then look at the application of honesty theory to quantum dynamical semigroups. We show that honesty is the natural generalisation of the concept of conservativity of quantum dynamical semigroups. Conservative quantum dynamical semigroups are known to have certain "nice" properties. We show that similar properties hold for honest semigroups using honesty theory results. Finally, we consider a form of boundary perturbations in the context of transport semigroups. There exists an analogous theory of honesty for this set-up. We formulate a general result from which honesty results of both Kato's Theorem and transport semigroups can be derived.

This thesis deals with reachability and freeness problems for systems with linear dynamics, including hybrid systems and matrix semigroups. Hybrid systems are a type of dynamical system that exhibit both continuous and discrete dynamic behaviour. Thus they are particularly useful in modelling practical real world systems which can both flow (continuous behaviour) and jump (discrete behaviour). Decision questions for matrix semigroups have attracted a great deal of attention in both the Mathematics and Theoretical Computer Science communities. They can also be used to model applications with only discrete components. For a computational model, the reachability problem asks whether we can reach a target point starting from an initial point, which is a natural question both in theoretical study and for real-world applications. By studying this problem and its variations, we shall prove in a formal mathematical sense that many problems are intractable or even unsolvable. Thus we know when such a problem appears in other areas like Biology, Physics or Chemistry, either the problem itself needs to be simplified, or it should by studied by approximation. In this thesis we concentrate on a specific hybrid system model, called an HPCD, and its variations. The objective of studying this model is twofold: to obtain the most expressive system for which reachability is algorithmically solvable and to explore the simplest system for which it is impossible to solve. For the solvable sub-cases, we shall also study whether reachability is in some sense easy or hard by determining which complexity classes the problem belongs to, such as P, NP(-hard) and PSPACE(-hard). Some undecidable results for matrix semigroups are also shown, which both strengthen our knowledge of the structure of matrix semigroups, and lead to some undecidability results for other models.