Very accurate methods, based on boundary integral techniques, are developed for the study of multiple, interacting fluid interfaces in an Eulerian fluid. These methods are applied to the evolution of a thin, periodic layer of constant vorticity embedded in irrotational fluid. Numerical regularity experiments are conducted and suggest that the interfaces of the layer develop a curvature singularity in infinite time. This is to be contrasted with the more singular vorticity distribution of a vortex sheet developing such a singularity in a finite time.

Student Number : 9804032J - PhD thesis - School of Mathematics - Faculty of Science / The focus of this thesis is the spectral structure of second order self-adjoint differential operators on graphs. Various function spaces on graphs are defined and we define, in terms of both differential systems and the afore noted function spaces, boundary value problems on graphs. A boundary value problem on a graph is shown to be spectrally equivalent to a system with separated boundary conditions. An example is provided to illustrate the fact that, for Sturm-Liouville operators on graphs, self-adjointness does not necessarily imply regularity. We also show that since the differential operators considered are self-adjoint the algebraic and geometric eigenvalue multiplicities are equal. Asymptotic bounds for the eigenvalues are found using matrix Pr¨ufer angle methods. Techniques common in the area of elliptic partial differential equations are used to give a variational formulation for boundary value problems on graphs. This enables us to formulate an analogue of Dirichlet-Neumann bracketing for boundary value problems on graphs as well as to establish a min-max principle. This eigenvalue bracketing gives rise to eigenvalue asymptotics and consequently eigenfunction asymptotics. Asymptotic approximations to the Green’s functions of Sturm-Liouville boundary value problems on graphs are obtained. These approximations are used to study the regularized trace of the differential operators associated with these boundary value problems. Inverse spectral problems for Sturm-Liouville boundary value problems on graphs resembling those considered in Halberg and Kramer, A generalization of the trace concept, Duke Math. J. 27 (1960), 607-617, for Sturm-Liouville problems, and Pielichowski, An inverse spectral problem for linear elliptic differential operators, Universitatis Iagellonicae Acta Mathematica XXVII (1988), 239-246, for elliptic boundary value problems, are solved. Boundary estimates for solutions of non-homogeneous boundary value problems on graphs are given. In particular, bounds for the norms of the boundary values of solutions to the non-homogeneous boundary value problem in terms of the norm of the non-homogeneity are obtained and the eigenparameter dependence of these bounds is studied. Inverse nodal problems on graphs are then considered. Eigenfunction and eigenvalue asymptotic approximations are used to provide an asymptotic expression for the spacing of nodal points on each edge of the graph from which the uniqueness of the potential, for given nodal data, is deduced. An explicit formula for the potential in terms of the nodal points and eigenvalues is given.

M.Sc., Faculty of Science, University of the Witwatersrand, 2011

by Leung Tin Chi. / Bibliography: leaf 32 / Thesis (M.Phil.)--Chinese University of Hong Kong, 1982

Lam, Hiu Fung. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 58-60). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts also in Chinese.

by Ma Ka Chun. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2004. / Includes bibliographical references (leaves 50-52). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.7 / Chapter 2 --- Queueing Networks --- p.9 / Chapter 2.1 --- One-queue Networks --- p.9 / Chapter 2.2 --- Two-queue Free Networks --- p.12 / Chapter 2.3 --- Two-queue Overflow Networks --- p.13 / Chapter 2.4 --- Networks with Batch Arrivals --- p.14 / Chapter 3 --- ODE Solvers --- p.16 / Chapter 3.1 --- The Initial Value Methods --- p.16 / Chapter 3.1.1 --- The Linear System of Ordinary Differential Equations --- p.16 / Chapter 3.1.2 --- Euler's Method --- p.17 / Chapter 3.1.3 --- Runge-Kutta Methods --- p.17 / Chapter 3.1.4 --- The Stability of the IVMs --- p.19 / Chapter 3.1.5 --- Applications in Queueing Networks --- p.20 / Chapter 3.2 --- The Boundary Value Methods --- p.20 / Chapter 3.2.1 --- The Generalized Backward Differentiation For- mulae --- p.21 / Chapter 3.2.2 --- An example --- p.24 / Chapter 4 --- The Linear Equation Solver --- p.26 / Chapter 4.1 --- Iterative Methods --- p.26 / Chapter 4.1.1 --- The Jacobi method --- p.27 / Chapter 4.1.2 --- The Gauss-Seidel Method --- p.28 / Chapter 4.1.3 --- Other Iterative Methods --- p.29 / Chapter 4.1.4 --- Preconditioning --- p.29 / Chapter 4.2 --- The Multigrid Method --- p.30 / Chapter 4.2.1 --- Iterative Refinement --- p.30 / Chapter 4.2.2 --- Restriction and Prolongation --- p.30 / Chapter 4.2.3 --- The Geometric Multigrid Method --- p.33 / Chapter 4.2.4 --- The Algebraic Multigrid Method --- p.38 / Chapter 4.2.5 --- Higher Dimensional Cases --- p.38 / Chapter 4.2.6 --- Applications in Queueing Networks --- p.38 / Chapter 5 --- Numerical Experiments --- p.41 / Chapter 6 --- Concluding Remarks --- p.49 / Bibliography --- p.50

Firstly, we consider the following critical elliptic Neumann problem --Deltau + muu = uN+2N-2 , u > 0 in O; 6u6n = 0 on ∂O, where O is a smooth bounded domain in RN , N ≥ 7, mu is a large positive number and nu denotes exterior unit normal vector. We show that at a positive nondegenerate local minimum point Q0 of the mean curvature function, for any fixed integer K ≥ 2, there exists a mu K > 0 such that for mu > muK, the above problem has K -- bubble solution umu concentrating at the same point Q 0. Precisely, we show that umu has K local maximum points Qm1,...,Qm K ∈ ∂O with the property that umQmj ∼mN-22 ,Qmj→Q0 , j = 1, ..., K, and mN-3N Q'1 m,...,Q'K m approaches an optimal configuration that minimizes the following functional RQ'1,...,Q 'K=c1 j=1K4Q' j+c2 i≠j1&vbm0;Q' i-Q'j&vbm0;N-2 where Qmi=Qm i,1,...,Qmi,N-1 ,Qmi,N:= Q'i m,Qmi,N , c1, c2 > 0 are two generic constants and ϕ(Q) = Q T GQ with G = (∇ijH(Q0)). / In my thesis, I will address different concentration phenomena for some second order elliptic problems. / Lastly, we consider the problem &egr;2Delta u -- u + uq = 0 in a smooth bounded domain O ⊂ R2 with Neumann boundary condition where &egr; > 0 is a small parameter and q > 1. We prove for some explicit &egr;'s the existence of positive solution u&egr; concentrating at any connected component of ∂O, exponentially small in &egr; at any positive distance from it. / Secondly, we study positive solutions of the equation &egr;2Delta u -- u + uN+2N-2 = 0, where N = 3, 4, 5, and &egr; > 0 is small, with Neumann boundary condition in a smooth bounded domain O ⊂ RN . We prove that, along some sequence {&egr;j} with &egr;j → 0, there exists a solution with an interior bubble at an innermost part of the domain and a boundary layer on the boundary ∂O. / Wang, Liping. / "June 2008." / Adviser: Jun Cheng Wei. / Source: Dissertation Abstracts International, Volume: 70-03, Section: B, page: 1707. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (p. 107-117). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307.

In [DS1,2,3], Denker and Sato initiated a new point of view to study the problem. They identified the Sierpinski gasket as a Martin boundary of some canonical Markov chain and used the associated theory to consider the problem. In this thesis, we will extend their result so as to be applicable to all single-point connected monocyclic post critically finite (m.p.c.f.) self-similar sets. / In the first chapter, we review some basic facts of the self-similar sets and the Martin boundaries, and we prove that every m.p.c.f. self-similar set K is homeomorphic to the quotient space of the symbolic space associated with K, moreover, the homeomorphism is a Lipschitz equivalence for some special m.p.c.f. self-similar sets. / In the second chapter, we first prove that the quotient space of the symbolic space associated with K is homeomorphic to the Martin boundary with respect to the state space associated with K if K is a single-point connected m.p.c.f. self-similar set. Combining this result and the result in the first chapter, we conclude that every single-point connected m.p.c.f. self-similar set can be identified with the Martin boundary of some canonical Markov chain. Then for the 3-level Sierpinski gasket, we prove that there exists a one to one relation between the strongly P-harmonic functions on the 3 state space and K-harmonic functions constructed by Kigami. / In the third chapter, we define a new Markov chain on the pentagasket K which is a single-point connected m.p.c.f. self-similar also. Under the new Markov chain, we prove that K can be identified with the Martin boundary of the new Markov chain and that there exists a one to one relation between the strongly P-harmonic functions and the K-harmonic functions. / One of the fundamental problems in fractal analysis is to construct a Laplacian on fractals. Since fractals, like the Sierpinski gasket and the pentagasket, do not have any smooth structures, it is not possible to construct it from the classical point of view. Hence, until now there is no systematic way to define such a notion on the general class of fractals. / There are two approaches for the problem which have achieved some success in certain special situations. The first one is a probabilistic approach via constructing Brownian motions on self-similar sets. The second approach is an analytical one proposed by Kigami. He approximated the underlying self-similar set K by an increasing sequence of finite sets equipped with the discrete Laplacians Hm in a consistent way. He showed that if K is strongly symmetric, then Hm converge to a Laplacian on K. / by Ju, Hongbing. / "March 2008." / Adviser: Lau Ka Sing. / Source: Dissertation Abstracts International, Volume: 70-03, Section: B, page: 1702. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (p. 91-94). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307.

In chapter 1, we focus on the full potential equation in an infinite nozzle with some decay cross-sections and prove the existence and stability of the transonic shock wave; which is a solution to a free boundary value problem for a quasi-linear mix-typed partial differential equation with the position of shock as a free boundary. To achieve this conclusion, we reduce it to a free boundary value problem for a quasi-linear elliptic equation in an unbounded domain. The crucial step in our analysis is to derive some uniform a priori estimates in such a domain. Then we apply the fixed point theorem to establish the existence of solutions to the full potential equation. / In chapter 2, we study the short time existence of discontinuous shock front solutions of the pressure gradient system which is the Euler system without inertial terms, where the initial data can have shock discontinuities of arbitrary strength which lie on a given smooth initial surface with arbitrary geometry. These shock solutions are constructed via a classical iteration scheme. The key step is to obtain the uniform stability for the related linearized equation by calculating the Lopatinski's determinant, which enables us to modify the technique of Majda and establish the local existence of solutions to the pressure gradient system without the structural constraints as for the full Euler system. / In this thesis we study two kinds of multi-dimensional shock phenomena for the compressible fluid dynamics. / Xie Feng. / "December 2005." / Adviser: Zhou Ping Xin. / Source: Dissertation Abstracts International, Volume: 67-11, Section: B, page: 6446. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2005. / Includes bibliographical references (p. 71-80). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307.

No description available.