Thesis (MSc)--Stellenbosch University, 2011. / ENGLISH ABSTRACT: For solid tumours to grow and metastise, they need to pass through two distinct stages: the avascular growth phase in which the tumour remains in a limited diffusion size and the vascular growth phase where the invasion may take place. In order to accomplish the transition from the former to the latter growth phase, a solid tumour may secrete a substance known as tumour angiogenesis factor (TAF) into the surrounding tissues to stimulate its own blood vessels. Once the tumour has its own blood supply, it can invade other parts of the body destroying healthy tissues organs by secreting the matrix degrading enzymes (MDE). During the invasion, the adhesion both cell-cell and cell-matrix play an extremely important role. In this work, we review some mathematical models dealing with various stages of development of solid tumours and the resulting reaction diffusion equations are solved using the Crank-Nicolson finite differences scheme. We also present a system of reaction-diffusion-taxis partial differential equations, with nonlocal (integral) terms describing the interactions between cancer cells and the host tissue. We then investigate the local and global existence of the solution of the previous model using the semigroup method and Sobolev embeddings. / AFRIKAANSE OPSOMMING: Daar is twee afsonderlike fases nodig vir soliede kanker gewasse om te groei en kwaadaardig te word: die avaskulêre groeifase waarin die gewas tot ’n sekere diffusie grootte beperk word en die vaskulêre groei fase waar die indringing plaasvind. Ten einde die oorgang tussen die twee fases te bewerkstellig, skei die soliede gewas ân stof in die omliggende weefsel af wat bekend staan as âtumor angiogenese factorâ (TAF). Dit stimuleer die vorming van die gewas se eie bloedvate. Wanneer die gewas sy eie bloedtoevoer het, kan dit ander dele van die liggaam indring en gesonde orgaanweefsel vernietig deur die afskeiding van die âmatrix degrading enzymesâ (MDE). Gedurende hierdie proses speel die sel-sel en sel-matriks interaksies ân belangrike rol. In hierdie werk het ons ân paar wiskundige modelle vergelyk wat die verskillende stadiums van die ontwikkeling van soliede gewasse beskryf. Die gevolglike diffusiereaksie vergelykings is opgelos deur gebruik te maak van die âCrank-Nicolson finite differences schemeâ. Ons bied ook ’n stelsel van âreaction-diffusion-taxisâ, met nie-lokale (integrale) terme wat die interaksies tussen kankerselle en die gasheerweefsel beskryf. Ons stel dan ondersoek in na die lokale en globale bestaan van die oplossing van die vorige model, met behulp van die semi-groep metode en die Sobolev ingebeddings.

Continuité des *-représentations et opérateurs de Hankel Cette thèse est comporte deux parties indépendantes. Dans le première partie de ce travail, nous établissons une condition nécessaire et suffisante pour qu'une *-représentation d'un *-semi-groupe abélien topologique S est continu à l'identité e de S. Les résultats sont obtenus moyennant un théorème de représentation intégrale par rapport à une mesure portée par les semi caractères continus. Nous donnons ensuite diverses applications de ces résultats. La deuxième partie de cette thèse traite les opérateurs de Hankel de symboles anti-méromorphes sur les couronne. Dans un premier lieu on met en place le cadre de la théorie générale des opérateurs de Hankel associée à un espace de Hilbert de fonctions holomorphes A^2(µ) de carré intégrable par rapport à une mesure admettant des moments d'indice relatif. Ensuite, nous montrons que l'espace des polynômes de Laurent est dense dans A^2(µ) cela nous permet de définir de façon claire les opérateurs de Hankel et étudier leurs propriétés spectrales. En particulier, pour de nombreux exemples, nous établissons des conditions nécessaires et suffisantes, en termes des moments, garantissant la continuité, la compacité et l'appartenance aux classes de Schatten de ces opérateurs de Hankel. / Continuity of *-representation and Hankel operators This thesis consists of two independent parts. In the first part of this work, we establish a necessary and sufficient condition for a *-representation a *-semigroup abelian topological S is continuous at the identity e of S. The results are obtained by means of a theorem of integral representation with respect to a measure supported by continuous semi characters. We then give several applications of these results. The second part of this thesis deals with Hankel operators anti-meromorphic symbols on an annulus. In the first place we put in place the framework of the general theory of Hankel operators associated with a Hilbert space of holomorphic functions A^2(μ) of square integrable with respect to a measure admitting relative index times. Next, we show that the space of Laurent polynomials is dense in A ^ 2 ( μ ) it allows us to clearly define the Hankel operators and study their spectral properties. In particular, many examples, we establish necessary and sufficient conditions, in terms of time, ensuring continuity compactness and Schatten classes of membership of the Hankel operators.

The aim of this work is to establish precisely what are the functorial connections between the abstract theories of quadratic forms, as well as, to create a short and introductory path from the classic theory to the abstract ones. There is a large amount of literature developed about classic and abstract theories but does note relate them ``geographically\'\'. In this perspective, we discuss the fundamental aspects of the classic and reduced theory of quadratic forms, and sum up the theories of Quaternionic Structures, Cordes Schemes, Abstract Witt Rings, Abstract Ordering Spaces, Special Groups, Abstract Real Spectra and Real Semigroups in a functorial picture, inserting the new aspects involve the recent theory of Multirings and Multifields. / O principal objetivo deste trabalho é estabelecer precisamente quais são as conexões funtoriais entre as teorias abstratas de formas quadráticas, criando uma via introdutória entre a teoria clássica e as abstratas durante este processo. Há uma gama de literatura desenvolvida tanto na teoria clássica quanto nas abstratas, mas nenhuma intercalando-as ``geograficamente\'\'. Nesta perspectiva, discutiremos os aspectos fundamentais da teoria clássica e reduzida de formas quadráticas, encapsulando as teorias das Estruturas Quaterniônicas, Esquemas de Cordes, Anéis de Witt Abstratos, Espaços de Ordens Abstratos, Grupos Especiais, Espectro Real Abstratos e Semigrupos Reais em um quadro funtorial, inserindo os novos elementos envolvendo a teoria recente dos Multi-anéis e Multi-corpos.

This work presents several results on curves of Hurwitz type, defined over a finite field. In 1961, Tallini investigated plane irreducible curves of minimum degree containing all points of the projective plane PG(2,q) over a finite field of order q. We prove that such curves are Fq3(q2+q+1)-projectively equivalent to the Hurwitz curve of degree q+2, and compute some of itsWeierstrass points. In addition, we prove that when q is prime the curve is ordinary, that is, the p-rank equals the genus of the curve. We also compute the automorphism group of such curve and show that some of the quotient curves, arising from some special cyclic automorphism groups, are still curves of Hurwitz type. Furthermore, we solve the problem of explicitly describing the set of all Weierstrass pure gaps supported by two or three special points on Hurwitz curves. Finally, we use the latter characterization to construct Goppa codes with good parameters, some of which are current records in the Mint table. / Este trabalho apresenta vários resultados em curvas do tipo Hurwitz, definidas sobre um corpo finito. Em 1961, Tallini investigou curvas planas irredutíveis de grau mínimo contendo todos os pontos do plano projetivo PG(2,q) sobre um corpo finito de ordem q. Provamos que tais curvas são Fq3(q2+q+1)-projetivamente equivalentes à curva de Hurwitz de grau q+2, e calculamos alguns de seus pontos de Weierstrass. Em adição, provamos que, quando q é primo, a curva é ordinária, isto é, o p-rank é igual ao gênero da curva. Também calculamos o grupo de automorfismos desta curva e mostramos que algumas das curvas quocientes, construídas a partir de certos grupos cíclicos de automorfismos, são ainda curvas do tipo Hurwitz. Além disso, solucionamos o problema de descrever explicitamente o conjunto de todos os gaps puros de Weierstrass suportados por dois ou três pontos especiais em curvas de Hurwitz. Finalmente, usamos tal caracterização para construir códigos de Goppa com bons parâmetros, sendo alguns deles recordes na tabela Mint.

Este trabalho se propõe a estudar o comportamento assintótico dos sistemas dinâmicos autônomos respaldado na Teoria das Dimensões. Mais precisamente, vamos compreender de que maneira nos é útil limitar a dimensão fractal do atrator global de um semigrupo a fim de estudar a dinâmica em dimensão finita, sem que se perca informações sobre a dinâmica ao fazê-lo. Para tanto, o Teorema de Mañé tem um papel decisivo junto às propriedades da dimensão de Hausdorff e a da dimensão fractal; nos permitindo encontrar uma projeção cuja restrição ao atrator é injetora sobre um espaço de dimensão finita. Constatamos ainda que esta abordagem por projeções se aplica largamente a semigrupos originados de equações diferenciais em espaços de Banach de dimensão infinita. / In this work, we study the asymptotic behavior of autonomous dynamical systems supported on the Dimension Theory. More precisely, we understand how fractal dimension finiteness of the global attractor of a semigroup can be used to study the dynamics in finite dimension, without losing information on the dynamics in doing so. For this purpose, the Mañés Theorem plays a decisive role considering the Hausdorff dimension properties and the fractal dimension; thanks to which we managed to find a projection whose restriction to the attractor is an injective application over a finite dimensional space. Besides, we also acknowledge that this projections approach is largely applied to semigroups arrising from differential equations in infinite dimensional Banach spaces.

Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / Seja C uma curva algÃbrica nÃo-singular, irredutÃvel e nÃo-hiperelÃptica sobre um corpo algebricamente fechado K. Neste trabalho trataremos de um resultado geomÃtrico para uma tal curva C. Este resultado à apresentado no teorema 3.0.2 e nos diz que os divisores interseÃÃo de uma curva C mergulhada canonicamente com seus espaÃos osculadores em um ponto P, nÃo considerando a interseÃÃo em P, podem somente mudar em dimensÃes dada pelo semigrupo de Weierstrass de C em P. Sob uma razoÃvel hipÃtese geomÃtrica, obteremos base monomial para os espaÃos vetoriais das diferenciais regulares de ordem superior (teorema 4.0.3). Em seguida, na proposiÃÃo 15, daremos uma condiÃÃo sobre os semigrupos de Weierstrass de C em P de modo que esta hipÃtese geomÃtrica seja verdadeira. Finalmente, daremos exemplos de semigrupos numÃricos satisfazendo tal condiÃÃo. / Let C a non-singular algebraic curve, irredutible and non-hipereliptic over a closed algebrically field K. In this work we to deal of a result geometric to such curve. This result to be introduced in the theorem three and say us that the intersection divisors of a curve C canonically embedded with its osculating spaces at a point P, not considering the intersection at P, can vary only in dimensions given by the Weierstrass semigroup of the curve C at P. Under a reasonable geometrical hypothesis, we to obtain monomial basis for the spaces of higher-order regular differentials (theorem four). Afterwards, in the proposition fifteen,to going a condition on the Weierstrass semigroup of curve C at P in order for this geometrical hipothesis to be true. Finally, we will give examples ofWeierstrass semigroups satisfying such condition.

An e-pseudovariety is a class of finite regular semigroups closed under the taking of homomorphic images, regular subsemigroups and finite direct products. Chapter One consists of a survey of those results from algebraic semigroup theory, universal algebra and lattice theory which are used in the following two chapters. In Chapter Two, a theory of generalised existence varieties is developed. A generalised existence variety is a class of regular semigroups closed under the taking of homomorphic images, regular subsemigroups, finite direct products and arbitrary powers. Equivalently, a generalised e-variety is the union of a directed family of existence varieties. It is demonstrated that a class of finite regular semigroups is an e-pseudovariety if and only if the class consists only of the finite members of some generalised existence variety. The relationship between certain lattices of e-pseudovarieties and generalised existence varieties is explored and a usefu l complete surjective lattice homomorphism is found. A study of complete congruences on lattices of existence varieties and e-pseudovarieties forms Chapter Three. In particular it is shown that a certain meet congruence, whose description is relatively simple, can be extended to yield a complete congruence on a lattice of e-pseudovarieties of finite regular semigroups. Ultimately, theorems describing the method of construction of all complete congruences of lattices of e-pseudovarieties whose members are finite E-solid or locally inverse regular semigroups are proved.

Tubular reactors cover a large class of processes in chemical and biochemical engineering. They are typically reactors in which the medium is not homogeneous (like fixed-bed reactors, packed-bed reactors, fluidized-bed reactors,...) and possibly involve diferent phases (liquid/solid/gas). The dynamics of nonisothermal axial dispersion or plug flow tubular reactors are described by semilinear partial differential equations (PDE's) derived from mass and energy balances. The main source of nonlinearities in such dynamics is concentrated in the kinetics terms of the model equations. Like tubular reactors many physical phenomena are modelled by partial differential equations (PDE's). Such systems are called distributed parameter systems. Control problems of these systems can be formulated in state-space form in a way analogous to those of lumped parameter systems (those described by ordinary differential equations) if one introduces a suitable infinite-dimensional state-space and suitable operators instead of the usual matrices. This thesis deals with the synthesis of optimal control laws with a view to regulate the temperature and the reactant concentration of a nonisothermal plug flow reactor model. Several tools of linear and semilinear infinite-dimensional system theory are extended and/or developed, and applied to this model. On the one hand, the concept of asymptotic stability is studied for a class of infinite-dimensional semilinear Banach state- space systems. Asymptotic stability criteria are established, which are based on the concept of strictly m-dissipative operator. This theory is applied to a nonisothermal plug flow reactor. On the other hand, the concept of optimal Linear-Quadratic (LQ) feedback is studied for class of infinite-dimensional linear systems. This theory is applied to a linearized plug flow reactor model in order to design an LQ optimal feedback controller. Then the resulting nonlinear closed-loop system performances are analyzed. Finally this control design strategy is extended to a large class of first-order hyperbolic PDE's systems.

In this paper we establish the regularity, exponential stability of global (weak) solutions and existence of uniform compact attractors of semiprocesses, which are generated by the global solutions, of a two-parameter family of operators for the nonlinear 1-d non-autonomous viscoelasticity. We employ the properties of the analytic semigroup to show the compactness for the semiprocess generated by the global solutions.

An n-tuple of operators V=(V_1,…,V_n) acting on a Hilbert space H is said to be isometric if the corresponding row operator is an isometry. A free semigroup algebra is the weakly closed algebra generated by an isometric n-tuple V. The structure of a free semigroup algebra contains a great deal of information about V. Thus it is natural to study this algebra in order to study V. A free semigroup algebra is said to be analytic if it is isomorphic to the noncommutative analytic Toeplitz algebra, which is a higher-dimensional generalization of the classical algebra of bounded analytic functions on the complex unit disk. This notion of analyticity is of central importance in the general theory of free semigroup algebras. A vector x in H is said to be wandering for an isometric n-tuple V if the set of words in the entries of V map x to an orthonormal set. As in the classical case, the analytic structure of the noncommutative analytic Toeplitz algebra is determined by the existence of wandering vectors for the generators of the algebra. In the first part of this thesis, we prove the following dichotomy: either an isometric n-tuple V has a wandering vector, or the free semigroup algebra it generates is a von Neumann algebra. This implies the existence of wandering vectors for every analytic free semigroup algebra. As a consequence, it follows that every free semigroup algebra is reflexive, in the sense that it is completely determined by its invariant subspace lattice. In the second part of this thesis we prove a decomposition for an isometric tuple of operators which generalizes the classical Lebesgue-von Neumann-Wold decomposition of an isometry into the direct sum of a unilateral shift, an absolutely continuous unitary and a singular unitary. The key result is an operator-algebraic characterization of an absolutely continuous isometric tuple in terms of analyticity. We show that, as in the classical case, this decomposition determines the weakly closed algebra and the von Neumann algebra generated by the tuple.