Thesis (M. Phil.)--University of Hong Kong, 1994. / Includes bibliographical references (leaves 40-42).

Thesis (M.Phil.)--City University of Hong Kong, 2005. / "Submitted to Department of Computer Engineering and Information Technology in partial fulfillment of the requirements for the degree of Master of Philosophy" Includes bibliographical references (leaves 109-111)

Inaug.-Diss.--Bonn. / Extra t.p. with thesis statement inserted. Bibliography: p. 54.

Thesis (Ph. D.)--University of Alberta, 2010. / Title from pdf file main screen (viewed on June 18, 2010). A thesis submitted to the Faculty of Graduate Studies and Research in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics, Department of Mathematical and Statistical Sciences, University of Alberta. Includes bibliographical references.

This thesis consists of an introduction and four research papers related to free boundary problems and systems of fully non-linear elliptic equations. Paper A and Paper B prove optimal regularity of solutions to general elliptic and parabolic free boundary problems, where the operators are fully non-linear and convex. Furthermore, it is proven that the free boundary is continuously differentiable around so called "thick" points, and that the free boundary touches the fixed boundary tangentially in two dimensions. Paper C analyzes singular points of solutions to perturbations of the unstable obstacle problem, in three dimensions. Blow-up limits are characterized and shown to be unique. The free boundary is proven to lie close to the zero-level set of the corresponding blow-up limit. Finally, the structure of the singular set is analyzed. Paper D discusses an idea on how existence and uniqueness theorems concerning quasi-monotone fully non-linear elliptic systems can be extended to systems that are not quasi-monotone. / <p>QC 20151210</p>

The thesis introduces a new version of an elliptic-blending low-Reynolds-number eddy-viscosity Reynolds-averaged Navier Stokes model. It is a model intended to be implemented in an industrial solver. It will be argued that there is still room for such a simple model, though eddy-viscosity models must rely on developments specificallymade for higher order formulations. It is the aim of the v2-f model to integrate elements of Reynolds-stress modelling developments into a simpler formulation, but the former paradoxically suffers from numerical stiffness, which kept it out of reachof industry researchers everyday simulations. The v2-f formulation endeavours to reproduce the near-wall asymptotic behaviour of the turbulent quantities, as sounder alternative to empirical damping functions, and the required near-wall balance of small terms represents a numerical challenge. The present work first provides a comprehensive review of v2-f developments proposed over the past twenty years, and the different remedies for the numericalstiffness linked to the original formulation. The review focuses on ten v2-f variants, proposed between 1991 and 2006, whose behaviour is compared in some fundamental flows: the channel flow for five different Reynolds numbers, the asymptotic case of the logarithmic layer at infinite Reynolds number and the case of a flow with homogeneous sheared turbulence. Based on the conclusions of the review, the thesis proposes new developments. Firstly, the derivation of a new model, namely the φ - α model, is introduced. It relies on the resolution of two non-dimensional variables: φ represents the wall-normal anisotropy and α is a wall-proximity sensor. It is argued that only this formulation can address the numerical problems already mentioned without altering the predictions. Secondly, additional upgrades of the φ - α model are proposed to correct the dissipation rate equation. The aim is to improve the model behaviour in some specific regions of a boundary layer, by isolating some viscous terms and by improving the representation of turbulent transport at the edge of a boundary layer. Final developments are combined in a new model, the BL-v2/k model. The φ - α and BL-v2/k models are then validated for a set of two pressure induced separated flows and two buoyant flows, and beneficial effects of the proposed developments on the predictions are demonstrated. The numerical properties of the convergency of the BL-v2/k model are also reported at the end of this work.

For nearly three centuries mathematicians have been interested in polygons which simultaneously circumscribe and inscribe quadrics. They have shown in many contexts (real, complex, non-euclidean, higher dimensional, etc.) that such polygons may be ``rotated'' while maintaining their circum-inscribed quality. Of particular interest has been conditions on the quadrics which guarantee the existence of such polygons. In 1854 Arthur Cayley provided conditions for closure general to polygons of any size in the complex projective plane. We show that under suitable circumstances the curve, defined by Cayley's conditions, on a fibration of Jacobians over the space of families of quadrics is a reducible curve, particularly in genus two. We may infer additional information about points of finite order on the Jacobians based on the component of the reducible curve in which they lie. Using this information we are able to accomplish two tasks. First we provide sufficient closure conditions for Poncelet's Great Theorem in which each vertex of the polygon lies on a distinct quadric. Next, for a polygon circum-inscribed in quadrics in ℙ^3, we provide additional sufficient conditions for closure beyond what mathematicians had previously believed to be necessary and sufficient.

In the thesis we study the elliptic curves and its use in cryptography. Elliptic curvesencompasses a vast area of mathematics. Elliptic curves have basics in group theory andnumber theory. The points on elliptic curve forms a group under the operation of addition.We study the structure of this group. We describe Hasse’s theorem to estimate the numberof points on the curve. We also discuss that the elliptic curve group may or may not becyclic over finite fields. Elliptic curves have applications in cryptography, we describe theapplication of elliptic curves for discrete logarithm problem and ElGamal cryptosystem.

In the thesis we study the elliptic curves and its use in cryptography. Elliptic curvesencompasses a vast area of mathematics. Elliptic curves have basics in group theory andnumber theory. The points on elliptic curve forms a group under the operation of addition.We study the structure of this group. We describe Hasse’s theorem to estimate the numberof points on the curve. We also discuss that the elliptic curve group may or may not becyclic over finite fields. Elliptic curves have applications in cryptography, we describe theapplication of elliptic curves for discrete logarithm problem and ElGamal cryptosystem.

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