This thesis aims to model the uniaxial deformation of a class of materials consisting of microscopic spherical shells embedded in a rubber matrix. These shells are assumed to buckle as the stress on the material increases. To motivate the analysis we consider the paradigm problem of the debonding of a distribution of cylindrical inclusions in an elastic material undergoing antiplane shear, with bonded and debonded inclusions playing the role of unbuckled and buckled shells respectively. We begin the modelling of the microsphere-containing material by considering the buckling of an isolated embedded shell inclusion with a uniaxial stress field at infinity, using Koiter's theory of shallow shells. The resulting energy functional is solved as an eigenvalue problem by the Rayleigh-Ritz method. Subsequently, we analyse the buckling criterion asymptotically in the limit as the thickness ratio tends to zero by analogy with the WKB analysis of a beam on a variable-stiffness substrate. To model the shell after buckling we consider the simplified case of an embedded shell with a crack around its equator. The system is solved by expressing the displacements in the shell and matrix as series of Love stress functions, with the resulting infinite system of equations solved numerically with the aid of a convergence acceleration method. Finally we consider a composite material consisting of a homogenised dilute distribution of buckled and unbuckled shells, with the proportion of each type of shell dependent on the stress applied to the material, according to an asymptotic formula relating the size of the inclusions and the critical buckling stress that was obtained previously.

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Symbolic dynamics, and in particular β-expansions, are a ubiquitous tool in studying more complicated dynamical systems. Applications include number theory, fractals, information theory, and data storage. In this thesis we will explore the basics of dynamical systems with a special focus on topological dynamics. We then examine symbolic dynamics and β-transformations through the lens of sequence spaces. We discuss observations from recent literature about how matching (the property that the itinerary of 0 and 1 coincide after some number of iterations) is linked to when Tβ,⍺ generates a subshift of finite type. We prove the set of ⍺ in the parameter space for which Tβ,⍺ exhibits matching is symmetric and analyze some examples where the symmetry is both apparent and useful in finding a dense set of ⍺ for which Tβ,⍺ generates a subshift of finite type.

The behavior of many systems of practical interest in communications and other areas is well modeled by a single server exponential queueing system in which the arrival and service rates are dependent upon the state of a Markov chain, the dynamics of which are independent of the queue length. Formal solution to such models based on Neuts's matrix geometric approach have appeared frequently in the literature. A major problem in using the matrix geometric approach is the computation of the rate matrix, which requires the solution of a matrix polynomial. In particular, computational times appear to be unpredictable and excessive for many problems of practical interest. Alternative techniques which employ eigenanalysis have been developed. These techniques are polynomially bounded and yield results very quickly compared to iterative routines. On the other hand, the class of systems to which the eigenanalysis based techniques apply have been somewhat restricted. In this thesis, we modify the eigenanalysis approach initially presented in order to remove some of these restrictions. / Master of Science

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Thesis (MSc)--Stellenbosch University, 2003 / ENGLISH ABSTRACT: A tower of global function fields :F = (FI, F2' ... ) is an infinite tower of separable extensions of algebraic function fields of one variable such that the constituent function fields have the same (finite) field of constants and the genus of these tend to infinity. A study can be made of the asymptotic behaviour of the ratio of the number of places of degree one over the genus of FJWq as i tends to infinity. A tower is called asymptotically good if this limit is a positive number. The well-known Drinfeld- Vladut bound provides a general upper bound for this limit. In practise, asymptotically good towers are rare. While the first examples were non-explicit, we focus on explicit towers of function fields, that is towers where equations recursively defining the extensions Fi+d F; are known. It is known that if the field of constants of the tower has square cardinality, it is possible to attain the Drinfeld- Vladut upper bound for this limit, even in the explicit case. If the field of constants does not have square cardinality, it is unknown how close the limit of the tower can come to this upper bound. In this thesis, we will develop the theory required to construct and analyse the asymptotic behaviour of explicit towers of function fields. Various towers will be exhibited, and general families of explicit formulae for which the splitting behaviour and growth of the genus can be computed in a tower will be discussed. When the necessary theory has been developed, we will focus on the case of towers over fields of non-square cardinality and the open problem of how good the asymptotic behaviour of the tower can be under these circumstances. / AFRIKAANSE OPSOMMING: 'n Toring van globale funksieliggame F = (FI, F2' ... ) is 'n oneindige toring van skeibare uitbreidings van algebraïese funksieliggame van een veranderlike sodat die samestellende funksieliggame dieselfde (eindige) konstante liggaam het en die genus streef na oneindig. 'n Studie kan gemaak word van die asimptotiese gedrag van die verhouding van die aantal plekke van graad een gedeel deur die genus van Fi/F q soos i streef na oneindig. 'n Toring word asimptoties goed genoem as hierdie limiet 'n positiewe getal is. Die bekende Drinfeld- Vladut grens verskaf 'n algemene bogrens vir hierdie limiet. In praktyk is asimptoties goeie torings skaars. Terwyl die eerste voorbeelde nie eksplisiet was nie, fokus ons op eksplisiete torings, dit is torings waar die vergelykings wat rekursief die uitbreidings Fi+d F; bepaal bekend is. Dit is bekend dat as die kardinaliteit van die konstante liggaam van die toring 'n volkome vierkant is, dit moontlik is om die Drinfeld- Vladut bogrens vir die limiet te behaal, selfs in die eksplisiete geval. As die konstante liggaam nie 'n kwadratiese kardinaliteit het nie, is dit onbekend hoe naby die limiet van die toring aan hierdie bogrens kan kom. In hierdie tesis salons die teorie ontwikkel wat benodig word om eksplisiete torings van funksieliggame te konstrueer, en hulle asimptotiese gedrag te analiseer. Verskeie torings sal aangebied word en algemene families van eksplisiete formules waarvoor die splitsingsgedrag en groei van die genus in 'n toring bereken kan word, sal bespreek word. Wanneer die nodige teorie ontwikkel is, salons fokus op die geval van torings oor liggame waarvan die kardinaliteit nie 'n volkome vierkant is nie, en op die oop probleem aangaande hoe goed die asimptotiese gedrag van 'n toring onder hierdie omstandighede kan wees.

Thesis (MSc)--Stellenbosch University, 2015 / ENGLISH ABSTRACT : Two applications of analytic techniques to combinatorial problems with number-theoretic flavours are shown. The first is an application of the real saddle point method to derive second-order asymptotic expansions for the number of solutions to the signum equation of a general class of sequences. The second is an application of more elementary methods to yield asymptotic expansions for the number of partitions of a large integer into powers of an integer b where each part has bounded multiplicity. / AFRIKAANSE OPSOMMING : Ons toon twee toepassings van analitiese tegnieke op kombinatoriese probleme met getalteoretiese geure. Die eerste is ’n toepassing van die reële saalpuntmetode wat tweede-orde asimptotiese uitbreidings vir die aantal oplossings van die ‘signum’ vergelyking vir ’n algemene klas van rye aflewer. Die tweede is ’n toepassing van meer elementêre metodes wat asimptotiese uitbreidings vir die aantal partisies van ’n groot heelgetal in magte van ’n heelgetal b, waar elke deel ’n begrensde meervoudigheid het, aflewer

Geng Di. / Thesis (Ph.D.)--Chinese University of Hong Kong, 1996. / Includes bibliographical references (leaves 116-119). / Acknowledgement --- p.i / Abstract --- p.ii / Introduction --- p.iii / Part I --- p.1 / Chapter 1 --- Critical Dimension of a Semilinear Degenerate Elliptic Equation Involving Critical Sobolev-Hardy Exponent --- p.2 / Chapter 1.1 --- Introduction --- p.2 / Chapter 1.2 --- Non-existence (I) --- p.5 / Chapter 1.3 --- Non-existence (II) --- p.11 / Chapter 1.4 --- Existence --- p.13 / Chapter 1.5 --- Radial Symmetry of Solutions --- p.16 / Appendix A --- p.20 / Appendix B --- p.23 / Chapter 2 --- Critical Dimension of a Hessian Equation Involving Critical Ex- ponent --- p.27 / Chapter 2.1 --- Introduction --- p.27 / Chapter 2.2 --- Preliminary Results --- p.29 / Chapter 2.3 --- Existence Results --- p.32 / Chapter 2.4 --- Non-existence Results --- p.43 / Chapter 3 --- Absence of Critical Dimension for the Subelliptic Laplacian on the Heisenberg Group --- p.48 / Chapter 3.1 --- Introduction and Main Result --- p.48 / Chapter 3.2 --- Proof of the Theorem --- p.49 / Part2 --- p.55 / Chapter 4 --- Asymptotic Behavior for Weighted p-Laplace Equations Involv- ing Critical Growth on the Ball --- p.56 / Chapter 4.1 --- Introduction --- p.56 / Chapter 4.2 --- A Crucial Lemma --- p.59 / Chapter 4.3 --- Proof of the Main Theorems --- p.61 / Chapter 5 --- Asymptotics for a Semilinear Weighted Elliptic Equation In- volving Critical Sobolev-Hardy Exponent --- p.71 / Chapter 5.1 --- Introduction --- p.71 / Chapter 5.2 --- Some Preliminary Results --- p.73 / Chapter 5.3 --- A Crucial Estimate --- p.80 / Chapter 5.4 --- Proof of the Main Theorem --- p.85 / Appendix --- p.88 / Chapter 6 --- Asymptotics for Positive Solutions for a Biharmonic Equation Involving Critical Exponent --- p.93 / Chapter 6.1 --- Introduction --- p.93 / Chapter 6.2 --- Preliminary Results --- p.94 / Chapter 6.3 --- Pohozaev's identity and Green's Function --- p.98 / Chapter 6.4 --- A Crucial Lemma --- p.103 / Chapter 6.5 --- Proof of Main Theorem --- p.112 / Bibliography --- p.115