In this dissertation, we study the behavior of microscopic organisms utilizing lateral and spiral bending waves to swim in a fluid. More specifically, spermatozoa encounter different fluid environments filled with mucus, cells, hormones, and other large proteins. These networks of proteins and cells are assumed to be stationary and of low volume fraction. They act as friction, possibly preventing or enhancing forward progression of the swimmers. The flow in the medium is described as a viscous fluid with a resistance term known as a Brinkman fluid. It depends on the Darcy permeability parameter affecting the swimming patterns of the flagella. To further understand these effects we study the asymptotic swimming speeds of an infinite-length swimmer propagating planar or spiral bending waves in a Brinkman fluid. We find that, up to the second order expansion, the swimming speeds are enhanced as the resistance increases. The work to maintain the planar bending and the torque exerted on the fluid are also examined. The Stokes limits of the swimming speeds, the work and the torque are recovered as resistance goes to zero. The analytical solutions are compared with numerical results of finite-length swimmers obtained from the method of Regularized Brinkmanlets (MRB). The study gives insight on the effects of the permeability, the length and the radius of the cylinder on the performance of the swimmers. In addition, we develop a grid-free numerical method to study the bend and twist of an elastic rod immersed in a Brinkman fluid. The rod is discretized using a Kirchhoff Rod (KR) model. The linear and angular velocity of the rod are derived using the MRB. The method is validated through a couple of benchmark examples including the dynamics of an elastic rod, and the planar bending of a flagellum in a Brinkman fluid. The studies show how the permeability and stiffness coefficients affect the waveforms, the energy, and the swimming speeds of the swimmers. Also, the beating pattern of the spermatozoa flagellum depends on the intracellular concentrations of calcium ([Ca2+]). An increase of [Ca2+] is linked to hyperactivated motility. This is characterized by highly asymmetrical beating, which allows spermatozoa to reach the oocyte (egg) or navigate along the female reproductive tract. Here, we couple the [Ca2+] to the bending model of a swimmer in a Brinkman fluid. This computational framework is used to understand how internal flagellar [Ca2+] and fluid resistance in a Brinkman fluid alter swimming trajectories and flagellar bending.

In this thesis, the Method of Moments has been applied to calculate capacitance between two arbitrary 3D metal conductors or a capacitance matrix for a 3D multi-conductor system. Capacitance extraction has found extensive use for systems involving sets of long par- allel transmission lines in multi-dielectric environment as well as integrated circuit package including three-dimensional conductors located on parallel planes. This paper starts by reviewing fundamental aspects of transient electro-magnetics followed by the governing dif- ferential and integral equations to motivate the application of numerical methods as Method of Moments(MoM), Finite Element Method(FEM), etc. Among these numerical tools, the surface-based integral-equation methodology - MoM is ideally suited to address the prob- lem. It leads to a well-conditioned system with reduced size, as compared to volumetric methods. In this dissertation, the MoM Surface Integral Equation (SIE)-based modeling approach is developed to realize electrostatic capacitance extraction for 3D geometry. MAT- LAB is employed to validate its e?ciency and e?ectiveness along with design of a friendly GUI. As a base example, a parallel-plate capacitor is considered. We evaluate the accu- racy of the method by comparison with FEM simulations as well as the corresponding quasi-analytical solution. We apply this method to the parallel-plate square capacitor and demonstrate how far could the undergraduate result 0C = A ? "=d' be from reality. For the completion of the solver, the same method is applied to the calculation of line capacitance for two- and multi-conductor 2D transmission lines.

This paper introduces a method to determine the pressure in a fixed thickness, smooth, periodic domain; namely a lead-over-pleat cartridge filter. Finding the pressure within the domain requires the numerical solution of Laplace's equation, the first step of which is approximating, by interpolation, the curved portions of the filter to a circle in the xy plane.A conformal map is then applied to the filter, transforming the region into a rectangle in the uv plane. A finite difference method is introduced to numerically solve Laplace's equation in the rectangular domain. There are currently methods in existence to solve partial differential equations on non- regular domains. In a method employed by Monchmeyer and Muller, a scheme is used to transform from cartesian to spherical polar coordinates. Monchmeyer and Muller stress that for non-linear domains, extrapolation of existing cartesian difference schemes may produce incorrect solutions, and therefore, a volume centered discretization is used. A difference scheme is then derived that relies on mean values. This method has second order accuracy.(Rosenfeld,Moshe, Kwak, Dochan, 1989) The method introduced in this paper is based on a 7-point stencil which takes into account the unequal spacing of the points. From all neighboring pairs, a linear system of equations is constructed, which takes into account the periodic domain.This method is solved by standard iterative methods. The solution is then mapped back to the original domain, with second order accuracy. The method is then tested to obtain a solution to a domain which satisfies $y=sin(x)$ at the center, a shape similar to that of a lead-over-pleat cartridge filter. As a result, a model for the pressure distribution within the filter is obtained.

Dry powder inhalers (DPIs) are a common therapeutic modality for lung diseases such as asthma, but they are also used to treat systemic diseases such as diabetes. Advantages of DPIs include their portable design and low manufacturing costs. Another advantage of DPIs is their breath activation, which makes them popular among patients. In a passive DPI drug is only released when the patient inhales. When the patient inhales, air flows through the device. The flow of air entrains a dry powder formulation inside the device and carries it to the lung. Currently, no DPI exists which can deliver drug independent of the patient to the desired target site in the lung. This is because drug release depends on the patient’s inhalation manoeuvre. To maximize the effect of the treatment it is necessary to optimize DPIs to achieve drug delivery that (A) is independent of the inhalation manoeuvre and (B) is targeted to the correct site in the lung. Therefore, this thesis aims to apply numerical and experimental methods to optimize DPIs systematically. First, two clinically justifiable cost functions have been developed corresponding to the DPI design objectives (A) and (B). An Eulerian-Eulerian (EE) computational fluid dynamics (CFD) approach has then been used to optimize a DPI entrainment geometry. Three different optimized entrainment geometries have been found corresponding to three different therapeutic applications. Second, the CFD approach has been validated experimentally. This is the first experimental study to validate an EE CFD approach for DPI modelling. Third, a personalized medicine approach to DPI design has been proposed. The development of this approach makes it possible to achieve the design objectives for patients with highly different lung functions. Finally, an adaptive DPI with a variable bypass element has been developed. This DPI achieves design objectives (A) and (B) for patients with highly different lung functions with a single device. In contrast to the personalized medicine approach, there is no need to select the optimal amount of bypass, since the device adapts automatically.

<div>This PhD thesis is an important development in the theories, methods, and applications of eigenvalue algorithms for structured matrices. Though eigenvalue problems have been well-studied, the class of matrices that admit very fast (near-linear time) algorithms was quite small until very recently. We developed and implemented a generalization of the famous symmetric tridiagonal divide-and-conquer algorithm to a much larger class of rank structured matrices (symmetric hierarchically semisperable, or HSS) that appear frequently in applications. Altogether, this thesis makes valuable contributions to three different major areas of scientific computing: algorithmic development, numerical analysis, and applications. In addition to the previously stated divide-and-conquer algorithm, we generalize to larger classes of eigenvalue problems and provide several key new low-rank update algorithms. A major contribution the analysis of the structured eigenvalue problem. In addition to standard perturbation analysis, we elucidate some subtle and previously under-examined issues in structured matrix eigenvalue problems such as subspace contributions and secular equation conditioning. Finally, several applications are studied.</div>

Hung Kin Ting. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2004. / Includes bibliographical references (leaves 126-129). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Inverse Problems --- p.1 / Chapter 1.2 --- Applications of Inverse Problems --- p.2 / Chapter 1.3 --- Least-squares Solutions --- p.4 / Chapter 1.4 --- Discrete Systems --- p.4 / Chapter 1.5 --- "Discretization, Regularization and Regularization Pa- rameters" --- p.5 / Chapter 1.6 --- Outline of the Thesis --- p.6 / Chapter 2 --- Some Basic Concepts and Mathematical Tools --- p.8 / Chapter 2.1 --- Singular Value Decomposition (SVD) --- p.8 / Chapter 2.2 --- Generalized Singular Value Decomposition (GSVD) --- p.13 / Chapter 2.3 --- White Noises --- p.16 / Chapter 3 --- Regularized Solutions --- p.18 / Chapter 3.1 --- Derivation of Regularized Solutions --- p.18 / Chapter 3.2 --- Discrete Picard Condition --- p.20 / Chapter 3.3 --- Relationship between Discrete Picard Condition and Regularized Solution --- p.21 / Chapter 3.4 --- Checking for the Discrete Picard Condition --- p.22 / Chapter 4 --- Different Discretization Approaches --- p.23 / Chapter 4.1 --- Problem 1 - Volterra Integral Equation of the First Kind --- p.25 / Chapter 4.2 --- Examples of Problem 1 --- p.30 / Chapter 4.3 --- Problem 2 - Fredholm Integral Equation of the First Kind --- p.49 / Chapter 4.4 --- Examples of Problem 2 --- p.53 / Chapter 4.5 --- Conclusion --- p.57 / Chapter 5 --- Effect of Different Kinds of Observation Data and Differential Operators on Accuracy --- p.59 / Chapter 5.1 --- Pointwise Observation Data --- p.60 / Chapter 5.2 --- Pointwise Observation Data of Heat Fluxes at the Boundary --- p.69 / Chapter 5.3 --- Observation Data with Heat Fluxes --- p.80 / Chapter 5.4 --- Conclusion --- p.89 / Chapter 6 --- L-curve --- p.90 / Chapter 6.1 --- Properties of L-curve --- p.93 / Chapter 6.2 --- L-curve in Log-Log Scale --- p.100 / Chapter 6.3 --- Disadvantages of the L-curve Method --- p.100 / Chapter 7 --- Algorithms of Finding the Corner of L-curve --- p.105 / Chapter 7.1 --- Cubic Spline Curve Fitting --- p.105 / Chapter 7.2 --- Conic Section Fitting --- p.106 / Chapter 7.3 --- Triangle Method --- p.109 / Chapter 8 --- Implementation of the L-curve Method --- p.111 / Chapter 8.1 --- Our Algorithm --- p.111 / Chapter 8.2 --- Numerical Experiments --- p.112 / Chapter 8.3 --- Conclusion --- p.124 / Bibliography --- p.126

這篇論文的目的是討論下述半線性橢圓方程的穩定解。 / -Δu = f (u) in Ω, / 這裡Ω是R{U+207F}中的光滑區域。對於非線性項f (u) = / [附圖]. / 我們得到了關於穩定解的Liouville-type結果。對於帶有更一般非線性項的半線性橢圓方程，在二到四維的情況下，我們得到一個關於穩定解的先驗估計。 / 最後，在維數很大的情形下，我們具體的給出一個關於以下雙調和方程的指數P的上界P，使得當P滿足 <1 P < Pc ‘下述雙調和方程不存在穩定解。 / Δ²u=u{U+1D56}, u>0 in R{U+207F} / The main aim of this thesis is to review recent results on stable solutions to the following semilinear elliptic equation / -Δu = f (u) in Ω, / where Ω ⊆ R{U+207F}.For the special case f (u) = / [With mathematic formula]. / For the general nonlinearity f in a bounded domain, we obtain a priori estimate for the stable solution when 2 ≤ n ≤ 4 / Finally, we give a explicit bound on the exponent for the nonexistence of stable solutions to the following biharmonic problem in large dimensions. / Δ²u=u{U+1D56}, u>0 in R{U+207F} / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Yang, Wen. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 66-69). / Abstracts also in Chinese. / Chapter 1 --- Introduction --- p.5 / Chapter 2 --- On the Lane-Emden equation --- p.12 / Chapter 2.1 --- On stable solutions to the Lane-Emden equation --- p.12 / Chapter 2.2 --- On nite Morse index solutions to the Lane-Emden equation --- p.19 / Chapter 3 --- On general semilinear elliptic equation --- p.28 / Chapter 4 --- On the biharmonic equation --- p.44 / Chapter 4.1 --- The rst part of the proof of Theorem 1.6 --- p.44 / Chapter 4.2 --- The second part of the proof of Theorem 1.6 --- p.54 / Bibliography --- p.66

Lee, Chak Shing. / "August 2011." / Thesis (M.Phil.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (leaves 60-62). / Abstracts in English and Chinese. / Chapter 1 --- Model Problems --- p.1 / Chapter 1.1 --- Introduction --- p.1 / Chapter 1.2 --- The curl-curl operator --- p.2 / Chapter 1.3 --- The convection-diffusion equation --- p.6 / Chapter 2 --- Staggered DG method for the Curl-Curl operator --- p.8 / Chapter 2.1 --- Introduction --- p.8 / Chapter 2.2 --- Discontinuous Galerkin discretization --- p.8 / Chapter 2.3 --- Stability for aligned fields --- p.14 / Chapter 2.4 --- Error estimates --- p.17 / Chapter 2.5 --- Numerical experiments --- p.21 / Chapter 2.6 --- Concluding Remarks --- p.32 / Chapter 3 --- Staggered DG method for the convection-diffusion equation --- p.33 / Chapter 3.1 --- Introduction --- p.33 / Chapter 3.2 --- Method description --- p.33 / Chapter 3.3 --- Preservation of physical structures --- p.38 / Chapter 3.4 --- Stability and convergence --- p.42 / Chapter 3.4.1 --- Static problem --- p.42 / Chapter 3.4.2 --- Time-dependent problem --- p.46 / Chapter 3.5 --- Fully discrete scheme --- p.49 / Chapter 3.6 --- Numerical examples --- p.55 / Chapter 3.6.1 --- The static problem --- p.55 / Chapter 3.6.2 --- Time dependent problem --- p.56 / Chapter 3.7 --- Concluding Remark --- p.59 / Bibliography --- p.60

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