Over the past 30 years the study of free boundary problems has stimulated much work. However, there exists a widely occurring, but little studied subclass of free boundary problems in which the free boundary has dimension two fewer than that of the underlying space rather than the more commonly studied case of one less. These problems are called `codimension-two' free boundary problems. In Chapter 1 the typical geometries required for such problems, the main mathematical techniques and the methodology used are discussed. Then, in Chapter 2, the techniques required to solve them are demonstrated using the particular example of the water entry problem. Further results for the water entry problem are then derived including an analysis of the relatively poorly understood water exit problem. In Chapter 3 a review is given of some classical contact and crack problems in solid mechanics. The inclusion of a cohesive zone in a dynamic type-III crack problem is considered. The Muskhelishvili potential method is presented and used to solve both a contact and crack problem. This enables the solution of a type-I crack problem relating to an ink delivery system to be found. In Chapter 4 a problem posed by car windscreen forming is addressed. A local solution near a corner is analysed to explain when and how point forces occur at the corners of the frame on which the simply supported windscreen rests. Then the full problem is solved numerically for different types of boundary condition. Chapters 5 and 6 deal with several sintering problems in viscous flow highlighting the value of the methodology introduced in Chapter 1. It will be shown how the Muskhelishvili potential method also carries over to Stokes flow problems. The difficulties of matching to an inner as opposed to an outer region are investigated. Last two interface problems between immiscible liquids are considered which show how the solution procedure is adapted when the field equation in the thin region is non-trivial. In the final chapter results are summarised, open problems listed and conclusions drawn.

La dystrophie myotonique de type I est la pathologie neuromusculaire la plus répandue chez l'adulte. Elle est caractérisée par une atteinte multisystémique plus ou moins prononcée en fonction de l'extension des répétitions CTG, mutation à l'origine de l'atteinte. Le muscle squelettique est particulièrement touché avec un phénomène de myotonie ainsi qu'une atrophie. Les myoblastes, à l'origine de la formation des muscles et de leur régénération potentielle, présentent, chez les patients DM1, une capacité proliférative limitée par rapport à des myoblastes issus d'individus sains. C'est l'activation précoce de la voie p16 qui est à l'origine de cette sénescence prématurée des cellules DM1. Les mécanismes conduisant à ce phénotype sont néanmoins inconnus. Au cours de cette étude, nous avons tenté de décrypter une partie de ces mécanismes et notamment les liens potentiels entre les expansions CTG, la sensibilité au stress oxydatif et l'activation précoce de la voie p16 conduisant à la sénescence des myoblastes DM1

We consider the motion of a Brownian particle trapped in an arbitrary bounded two or three-dimensional domain, whose boundary is reflecting except for a small absorbing window through which the particle can escape. We use the method of matched asymptotic expansions to calculate the mean first passage time, defined as the time taken for the Brownian particle to escape from the domain through the absorbing window. This is known as the narrow escape problem. Since the mean escape time diverges as the window shrinks, the calculation is a singular perturbation problem. We extend our results to include N absorbing windows of varying length in two dimensions and varying radius in three dimensions. We present findings in two dimensions for the unit disk, unit square and ellipse and in three dimensions for the unit sphere. The narrow escape problem has various applications in many fields including finance, biology, and statistical mechanics.

Thesis (M. Sc.)--University of Adelaide, Dept of Physics, 1986. / Includes bibliographical references (leaves 103-105).

Thesis (Ph. D.)--University of Sydney, 2002. / Title from title screen (viewed Apr. 28, 2008). Submitted in fulfilment of the requirements for the degree of Doctor of Philosophy to the School of Mathematics and Statistics, Faculty of Science. Includes bibliography. Also available in print form.

We study certain confluences of equations with two Fuchsian singularities which produce an irregular singularity of Poincaré rank one. We demonstrate a method to understand how to pass from solutions with power-like behavior which are analytic in neighbourhoods to solutions with exponential behavior which are analytic in sectors and have divergent asymptotic behavior. We explicitly calculate the Stokes' matrices of the confluent system in terms of the monodromy data, specifically the connection matrices, of the original system around the merging singularities. The confluence of Gauss' hypergeometric equation gives an excellent opportunity to show our approach with a concrete example. We explicitly show how the Stokes' data arise in the confluences of the isomonodromic deformation problems for the Painlevé equations PVI to PV and PV to PIII(D6).

Cette thèse est consacrée à l'étude de l'influence d'une fine couche hétérogène sur le comportement élastique linéaire d'une structure tridimensionnelle.Deux types d'hétérogénéités sont pris en compte : des cavités et des inclusions élastiques. Une étude complémentaire, dans le cas d'inclusions de grande rigidité, a été réalisée en considérant un problème de conduction thermique.Une analyse formelle par la méthode des développements asymptotiques raccordés conduit à un problème d'interface qui caractérise le comportement macroscopique de la structure. Le comportement microscopique de la couche est lui déterminé sur une cellule de base. Le modèle asymptotique obtenu est ensuite implémenté dans un code éléments finis. Une étude numérique permet de valider les résultats de l'analyse asymptotique. / This thesis is devoted to the study of the influence of a thin heterogeneous layeron the linear elastic behavior of a three-dimensional structure. Two types of heterogeneties are considered : cavities and elastic inclusions. For inclusions of high rigidty a further study was performed in the case of a heat conduction problem.A formal analysis using the matched asymptotic expansions method leads to an interface problem which characterizes the macroscopic behavior of the structure. The microscopic behavior of the layer is determined in a basic cell.The asymptotic model obtained is then implemented in a finite element software.A numerical study is used to validate the results of the asymptotic analysis.

This paper deals with the joint and conditional distributions concerning the maximum of random walk paths and the number of times this maximum is achieved. This joint distribution was studied first by Dwass [1967]. Based on his result, the correlation and some conditional moments are derived. The main contributions are however asymptotic expansions concerning the conditional distribution and conditional moments. (author's abstract) / Series: Forschungsberichte / Institut für Statistik

We consider the motion of a Brownian particle trapped in an arbitrary bounded two or three-dimensional domain, whose boundary is reflecting except for a small absorbing window through which the particle can escape. We use the method of matched asymptotic expansions to calculate the mean first passage time, defined as the time taken for the Brownian particle to escape from the domain through the absorbing window. This is known as the narrow escape problem. Since the mean escape time diverges as the window shrinks, the calculation is a singular perturbation problem. We extend our results to include N absorbing windows of varying length in two dimensions and varying radius in three dimensions. We present findings in two dimensions for the unit disk, unit square and ellipse and in three dimensions for the unit sphere. The narrow escape problem has various applications in many fields including finance, biology, and statistical mechanics. / Science, Faculty of / Mathematics, Department of / Graduate

This work motivates the study of open dynamical systems corresponding to the doubling map. In particular, the dynamical properties of the attractor of the doubling map when a symmetric, centred open interval is removed are studied. Using the arithmetical properties of the binary expansion of the points on the boundary of the removed interval, we study properties such as topological transitivity, the specification property and intrinsic ergodicity. The properties of the function that associates to each hole $(a,b)$ the topological entropy of the attractor of the considered dynamical system are also shown. For these purposes, a subshift corresponding to an element of the lexicographic world is associated to each attractor and the mentioned properties are studied symbolically.