The Faraday rotation spectrum of composites containing magnetite nanoparticles is found to be dependent on the interparticle spacing of the constituent nanoparticles. The composite materials are prepared by combining chemicallysynthesized Fe3O4 (magnetite) nanoparticles (8 nm diameter) and poly(methylmethacrylate) (PMMA). Composites are made containing a range of nanoparticle concentrations. The peak of the main spectral feature depends on nanoparticle concentration; this peak is observed to shift from approximately 470 nm for (dilute composites) to 560 nm (concentrated). A theory is presented based on the dipole approximation which accounts for optical coupling between magnetite particles. Qualitative correlations between theoretical calculations and experimental data suggest the shifts in spectral peak position depend on both interparticle distance and geometrical configuration.

by Lee Siu-keung. / Parallel title in Chinese. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1992. / Includes bibliographical references (leaves 162-164). / Chapter Chapter 1 --- Introduction --- p.5 / Chapter Chapter 2 --- Path Integrals / Chapter 2.1 --- Path´ؤIntegral Approach to Quantum Mechanics --- p.8 / Chapter 2.2 --- Path´ؤIntegral Approach to Statistical Mechanics --- p.14 / Chapter 2.3 --- Variational Principle --- p.18 / Chapter 2.4 --- "Variational Method Proposed by Giachetti and Tognetti, and by Feynman and Kleinert" / Chapter 2.4.1 --- Effective Classical Partition Function --- p.24 / Chapter 2.4.2 --- Particle Distribution Function From Effective Classical Potential --- p.34 / Chapter Chapter 3 --- Systematic Perturbation Corrections to the Variational Approximation Proposed in Section2.4 / Chapter 3.1 --- Formalism / Chapter 3.1.1 --- Free Energy --- p.38 / Chapter 3.1.2 --- Particle Distribution Function --- p.49 / Chapter 3.2 --- Second Order Correction to Free Energy --- p.53 / Chapter 3.3 --- First Order Correction to Particle Distribution Function --- p.60 / Chapter Chapter 4 --- Examples and Results / Chapter 4.1 --- Quartic Anharmonic Oscillator / Chapter 4.1.1 --- "Free Energy, Internal Energy and Specific Heat" --- p.69 / Chapter 4.1.2 --- Particle Distribution Function --- p.87 / Chapter 4.2 --- Symmetric Double-well Potential / Chapter 4.2.1 --- "Free Energy, Internal Energy and Specific Heat" --- p.88 / Chapter 4.2.2 --- Particle Distribution Function --- p.106 / Chapter 4.3 --- Quartic-cubic Anharmonic Potential / Chapter 4.3.1 --- Free Energy --- p.108 / Chapter 4.3.2 --- Particle Distribution Function --- p.115 / Chapter Chapter 5 --- Application to the One-dimensional Ginzburg-Landau Model / Chapter 5.1 --- Introduction --- p.120 / Chapter 5.2 --- Exact Partition Function and Free Energy Per Unit Length --- p.123 / Chapter 5.3 --- Zeroth Order Approximation to Free Energy Per Unit Length --- p.126 / Chapter 5.4 --- Exact Specific Heat --- p.133 / Chapter 5.5 --- Zeroth Order Approximation to Specific Heat --- p.139 / Chapter Chapter 6 --- Conclusion --- p.141 / Chapter Appendix I --- Functional Calculus - Differentiation --- p.145 / Chapter Appendix II --- Evaluation of Feynman Propagator Δf(τ) --- p.147 / Chapter Appendix III --- Vanishing of the First Order Correction-βf1 --- p.150 / Chapter Appendix IV --- Numerical Method for the Energy Eigenvalues and Eigenfunctions of the One-dimensional Schroedinger Equation with ax2 + bx4 Potential --- p.153 / Chapter Appendix V --- Numerical Integrations with imaginary Ω --- p.158 / References --- p.162 / Figures --- p.165

Kwok Chun Ming. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (leaves [389]-393). / Text in English; abstracts in English and Chinese. / Kwok Chun Ming. / Abstract --- p.i / Acknowledgement --- p.iii / Contents --- p.iv / List of Figures --- p.ix / Chapter Chapter 1. --- Introduction --- p.1 / Chapter 1.1 --- Introduction --- p.1 / Chapter 1.2 --- Objective and Methodology --- p.3 / Chapter Chapter 2. --- Theory of the Jaynes-Cummings model --- p.5 / Chapter 2.1 --- Formulation of the Jaynes-Cummings model --- p.5 / Chapter 2.1.1 --- Quantization of the Electromagnetic Field --- p.6 / Chapter 2.1.2 --- Quantization of the Matter Field --- p.11 / Chapter 2.1.3 --- The Interaction between the Radiation and the Matter --- p.13 / Chapter 2.1.4 --- Formulation of the One-quantum JCM --- p.15 / Chapter 2.2 --- Energy Eigenstates and Eigenenergy Spectrum --- p.18 / Chapter 2.3 --- Initial States and Observables --- p.20 / Chapter 2.3.1 --- Initial States --- p.20 / Chapter 2.3.2 --- Field Observables --- p.24 / Chapter 2.3.3 --- Atomic Observables --- p.25 / Chapter 2.4 --- Conclusion --- p.27 / Chapter Chapter 3. --- "Generalized SU(1,1) JCM" --- p.28 / Chapter 3.1 --- "Diagonalization of the SU(1,1) JCM" --- p.28 / Chapter 3.2 --- "SU(1,1) Coherent States and Observables" --- p.32 / Chapter 3.2.1 --- "Realizations of the SU(1,1) JCM" --- p.33 / Chapter 3.2.2 --- "SU(1,1) Coherent States" --- p.33 / Chapter 3.2.3 --- Field Observables --- p.35 / Chapter 3.3 --- Conclusion --- p.36 / Chapter Chapter 4. --- "One-mode, Intensity-dependent JCM" --- p.37 / Chapter 4.1 --- "Properties of the One-mode, Intensity-dependent JCM" --- p.37 / Chapter 4.2 --- Squeezing Effect --- p.40 / Chapter 4.2.1 --- Ordinary Amplitude Squeezing --- p.41 / Chapter 4.2.2 --- "SU(1,1) Squeezing" --- p.44 / Chapter 4.2.3 --- SU(2) Squeezing --- p.47 / Chapter 4.3 --- Atomic Inversion --- p.49 / Chapter 4.4 --- Q-function --- p.52 / Chapter 4.4.1 --- Ordinary Q-function --- p.53 / Chapter 4.4.2 --- "SU(1,1) Q-function" --- p.59 / Chapter 4.5 --- Purity Function --- p.65 / Chapter 4.5.1 --- Field Purity Function --- p.65 / Chapter 4.5.2 --- Atomic Purity Function --- p.68 / Chapter 4.6 --- Asymptotic Behavior of Field Squeezing --- p.70 / Chapter 4.7 --- Conclusion --- p.75 / Chapter Chapter 5. --- "One-mode, Two-quantum JCM" --- p.191 / Chapter 5.1 --- "Properties of the One-mode, Two-quantum JCM" --- p.191 / Chapter 5.2 --- Squeezing --- p.196 / Chapter 5.2.1 --- Ordinary Amplitude Squeezing --- p.197 / Chapter 5.2.2 --- "SU(1,1) squeezing" --- p.202 / Chapter 5.2.3 --- SU(2) squeezing --- p.205 / Chapter 5.3 --- Atomic Inversion --- p.206 / Chapter 5.4 --- Q-function --- p.210 / Chapter 5.4.1 --- Ordinary Q-function --- p.210 / Chapter 5.4.2 --- "SU(1,1) Q-function" --- p.215 / Chapter 5.5 --- Purity Function --- p.217 / Chapter 5.5.1 --- Field Purity Function --- p.217 / Chapter 5.5.2 --- Atomic Purity Function --- p.222 / Chapter 5.6 --- Conclusion --- p.225 / Chapter Chapter 6. --- "Two-mode, Two-quantum JCM" --- p.254 / Chapter 6.1 --- "Properties of the Two-mode, Two-quantum JCM" --- p.254 / Chapter 6.2 --- Squeezing --- p.260 / Chapter 6.2.1 --- Ordinary Amplitude Squeezing --- p.260 / Chapter 6.2.2 --- "SU(1,1) Squeezing" --- p.264 / Chapter 6.2.3 --- SU(2) Squeezing --- p.267 / Chapter 6.3 --- Atomic Inversion --- p.269 / Chapter 6.4 --- Q-function --- p.271 / Chapter 6.4.1 --- "SU(1,1) Q-function" --- p.271 / Chapter 6.5 --- Purity Function --- p.273 / Chapter 6.5.1 --- Field Purity Function --- p.273 / Chapter 6.5.2 --- Atomic Purity Function --- p.275 / Chapter 6.6 --- Conclusion --- p.277 / Chapter Chapter 7. --- "Generalized One-mode, Intensity-dependent JCM" --- p.300 / Chapter 7.1 --- "Diagonalization of the Generalizated One-mode, Intensity-dependent JCM" --- p.301 / Chapter 7.2 --- Energy Eigenstates and Eigenenergy Spectrum --- p.307 / Chapter 7.2.1 --- Energy Eigenstates --- p.307 / Chapter 7.2.2 --- Eigenergy Spectrum --- p.309 / Chapter 7.3 --- Conclusion --- p.310 / Chapter Chapter 8. --- Single Trapped and Laser-irradiated JCM --- p.311 / Chapter 8.1 --- Properties of the One-quantum STLI JCM --- p.311 / Chapter 8.2 --- Squeezing Effect --- p.315 / Chapter 8.2.1 --- Ordinary Amplitude Squeezing --- p.315 / Chapter 8.2.2 --- "SU(1,1) Squeezing" --- p.320 / Chapter 8.2.3 --- SU(2) Squeezing --- p.323 / Chapter 8.3 --- Atomic Inversion --- p.326 / Chapter 8.4 --- Q-function --- p.329 / Chapter 8.4.1 --- Ordinary Q-function --- p.329 / Chapter 8.4.2 --- "SU(1,1) Q-function" --- p.332 / Chapter 8.5 --- Purity Function --- p.334 / Chapter 8.5.1 --- Field Purity Function --- p.335 / Chapter 8.5.2 --- Atomic Purity function --- p.338 / Chapter 8.6 --- Non-classical Effects of the Two-quantum STLI JCM --- p.341 / Chapter 8.7 --- Conclusion --- p.342 / Chapter Chapter 9. --- Conclusion --- p.386 / Bibliography --- p.389

Recent experiments have shown the fundamental role that random fluctuations play in many chemical systems in living cells, such as gene regulatory networks. Mathematical models are thus indispensable to describe such systems and to extract relevant biological information from experimental data. Recent decades have seen a considerable amount of modelling effort devoted to this task. However, current methodologies still present outstanding mathematical and computational hurdles. In particular, models which retain the discrete nature of particle numbers incur necessarily severe computational overheads, greatly complicating the tasks of characterising statistically the noise in cells and inferring parameters from data. In this thesis we study analytical approximations and inference methods for stochastic reaction dynamics. The chemical master equation is the accepted description of stochastic chemical reaction networks whenever spatial effects can be ignored. Unfortunately, for most systems no analytic solutions are known and stochastic simulations are computationally expensive, making analytic approximations appealing alternatives. In the case where spatial effects cannot be ignored, such systems are typically modelled by means of stochastic reaction-diffusion processes. As in the non-spatial case an analytic treatment is rarely possible and simulations quickly become infeasible. In particular, the calibration of models to data constitutes a fundamental unsolved problem. In the first part of this thesis we study two approximation methods of the chemical master equation; the chemical Langevin equation and moment closure approximations. The chemical Langevin equation approximates the discrete-valued process described by the chemical master equation by a continuous diffusion process. Despite being frequently used in the literature, it remains unclear how the boundary conditions behave under this transition from discrete to continuous variables. We show that this boundary problem results in the chemical Langevin equation being mathematically ill-defined if defined in real space due to the occurrence of square roots of negative expressions. We show that this problem can be avoided by extending the state space from real to complex variables. We prove that this approach gives rise to real-valued moments and thus admits a probabilistic interpretation. Numerical examples demonstrate better accuracy of the developed complex chemical Langevin equation than various real-valued implementations proposed in the literature. Moment closure approximations aim at directly approximating the moments of a process, rather then its distribution. The chemical master equation gives rise to an infinite system of ordinary differential equations for the moments of a process. Moment closure approximations close this infinite hierarchy of equations by expressing moments above a certain order in terms of lower order moments. This is an ad hoc approximation without any systematic justification, and the question arises if the resulting equations always lead to physically meaningful results. We find that this is indeed not always the case. Rather, moment closure approximations may give rise to diverging time trajectories or otherwise unphysical behaviour, such as negative mean values or unphysical oscillations. They thus fail to admit a probabilistic interpretation in these cases, and care is needed when using them to not draw wrong conclusions. In the second part of this work we consider systems where spatial effects have to be taken into account. In general, such stochastic reaction-diffusion processes are only defined in an algorithmic sense without any analytic description, and it is hence not even conceptually clear how to define likelihoods for experimental data for such processes. Calibration of such models to experimental data thus constitutes a highly non-trivial task. We derive here a novel inference method by establishing a basic relationship between stochastic reaction-diffusion processes and spatio-temporal Cox processes, two classes of models that were considered to be distinct to each other to this date. This novel connection naturally allows to compute approximate likelihoods and thus to perform inference tasks for stochastic reaction-diffusion processes. The accuracy and efficiency of this approach is demonstrated by means of several examples. Overall, this thesis advances the state of the art of modelling methods for stochastic reaction systems. It advances the understanding of several existing methods by elucidating fundamental limitations of these methods, and several novel approximation and inference methods are developed.

Digitized by Kansas Correctional Industries

Solving operations research problems with approximation algorithms has been an important topic since approximation algorithm can provide near-optimal solutions to NP-hard problems while achieving computational efficiency. In this thesis, we consider two different problems in the field of optimal control and scheduling theory respectively and develop efficient approximation algorithms for those problems with performance guarantee. Chapter 2 presents approximation algorithms for solving the optimal execution problem for demand-response contract in electricity markets. Demand side participation is essential for achieving real-time energy balance in today's electricity grid. Demand-response contracts, where an electric utility company buys options from consumers to reduce their load in the future, are an important tool to increase demand-side participation. In this chapter, we consider the operational problem of optimally exercising the available contracts over the planning horizon such that the total cost to satisfy the demand is minimized. In particular, we consider the objective of minimizing the sum of the expected ℓ_β-norm of the load deviations from given thresholds and the contract execution costs over the planning horizon. For β=∞, this reduces to minimizing the expected peak load. The peak load provides a good proxy to the total cost of the utility as spikes in electricity prices are observed only in peak load periods. We present a data driven near-optimal algorithm for the contract execution problem. Our algorithm is a sample average approximation (SAA) based dynamic program over a multi-period planning horizon. We provide a sample complexity bound on the number of demand samples required to compute a (1+ε)-approximate policy for any ε>0. Our SAA algorithm is quite general and we show that it can be adapted to quite general demand models including Markovian demands and objective functions. For the special case where the demand in each period is i.i.d., we show that a static solution is optimal for the dynamic problem. We also conduct a numerical study to compare the performance of our SAA based DP algorithm. Our numerical experiments show that we can achieve a (1+ε)-approximation in significantly smaller numbers of samples than what is implied by the theoretical bounds. Moreover, the structure of the approximate policy also shows that it can be well approximated by a simple affine function of the state. In Chapter 3, we study the NP-hard coflow scheduling problem and develop a polynomial-time approximation algorithm for the problem with constant approximation ratio. Communications in datacenter jobs (such as the shuffle operations in MapReduce applications) often involve many parallel flows, which may be processed simultaneously. This highly parallel structure presents new scheduling challenges in optimizing job-level performance objectives in data centers. Chowdhury and Stoica [13] introduced the coflow abstraction to capture these communication patterns, and recently Chowdhury et al. [15] developed effective heuristics to schedule coflows. In this chapter, we consider the problem of efficiently scheduling coflows so as to minimize the total weighted completion time, which has been shown to be strongly NP-hard [15]. Our main result is the first polynomial-time deterministic approximation algorithm for this problem, with an approximation ratio of $64/3$, and a randomized version of the algorithm, with a ratio of 8+16sqrt{2}/3. Our results use techniques from both combinatorial scheduling and matching theory, and rely on a clever grouping of coflows. In Chapter 4, we carry out a comprehensive experimental analysis on a Facebook trace and extensive simulated instances to evaluate the practical performance of several algorithms for coflow scheduling, including our approximation algorithms developed in Chapter 3. Our experiments suggest that simple algorithms provide effective approximations of the optimal, and that the performance of the approximation algorithm of Chapter 3 is relatively robust, near optimal, and always among the best compared with the other algorithms, in both the offline and online settings.

This thesis consists of two unrelated parts. In the first part of this thesis, we give explicit expressions for the Fourier coefficients of Eisenstein series E∗(z, s, χ) twisted by modular symbols ⟨γ, f⟩ in the case where the level of f is prime and equal to the conductor of the Dirichlet character χ. We obtain these expressions by computing the spectral decomposition of an automorphic function closely related to E∗(z, s, χ). We then give applications of these expressions. In particular, we evaluate sums such as Σχ(γ)⟨γ, f⟩, where the sum is over γ ∈ Γ∞\Γ0(N) with c^2 + d^2 < X, with c and d being the lower-left and lower-right entries of γ respectively. This parallels past work of Goldfeld, Petridis, and Risager, and we observe that these sums exhibit different amounts of cancellation than what one might expect. In the second part of this thesis, given an elliptic curve E and a point P in E(R), we investigate the distribution of the points nP as n varies over the integers, giving bounds on the x and y coordinates of nP and determining the natural density of integers n for which nP lies in an arbitrary open subset of {R}^2. Our proofs rely on a connection to classical topics in the theory of Diophantine approximation.

Wang Jin. / "September 2000." / Thesis (Ph.D.)--Chinese University of Hong Kong, 2000. / Includes bibliographical references (p. 93-105). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Mode of access: World Wide Web. / Abstracts in English and Chinese.

Leung, Cheung Yu. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (leaves 85-86). / Abstracts in English and Chinese. / Introduction --- p.6 / Chapter 1 --- Preliminaries --- p.7 / Chapter 1.1 --- Locally compact groups and unitary representations --- p.7 / Chapter 1.2 --- Positive definite functions --- p.10 / Chapter 1.3 --- Affine isometric actions of groups --- p.23 / Chapter 1.4 --- Ultraproducts --- p.29 / Chapter 2 --- Amenability --- p.33 / Chapter 2.1 --- Reiter's property --- p.33 / Chapter 2.2 --- Fφlner's property --- p.41 / Chapter 3 --- Kazhdan's Property (T) --- p.43 / Chapter 3.1 --- Definition and basic properties --- p.43 / Chapter 3.2 --- Property (FH) --- p.51 / Chapter 3.3 --- Spectral criterion for Property (T) --- p.56 / Chapter 3.4 --- Property (T) for SL3(Z) --- p.60 / Chapter 3.5 --- Expanders --- p.72 / Approximation Properties of Groups --- p.5 / Chapter 4 --- Haagerup Property --- p.74 / Chapter 4.1 --- Equivalent formulations of Haagerup Property --- p.74 / Chapter 4.2 --- Trees and wall structures --- p.82 / Bibliography --- p.85

abstract: In the realm of network science, many topics can be abstracted as graph problems, such as routing, connectivity enhancement, resource/frequency allocation and so on. Though most of them are NP-hard to solve, heuristics as well as approximation algorithms are proposed to achieve reasonably good results. Accordingly, this dissertation studies graph related problems encountered in real applications. Two problems studied in this dissertation are derived from wireless network, two more problems studied are under scenarios of FIWI and optical network, one more problem is in Radio- Frequency Identification (RFID) domain and the last problem is inspired by satellite deployment. The objective of most of relay nodes placement problems, is to place the fewest number of relay nodes in the deployment area so that the network, formed by the sensors and the relay nodes, is connected. Under the fixed budget scenario, the expense involved in procuring the minimum number of relay nodes to make the network connected, may exceed the budget. In this dissertation, we study a family of problems whose goal is to design a network with “maximal connectedness” or “minimal disconnectedness”, subject to a fixed budget constraint. Apart from “connectivity”, we also study relay node problem in which degree constraint is considered. The balance of reducing the degree of the network while maximizing communication forms the basis of our d-degree minimum arrangement(d-MA) problem. In this dissertation, we look at several approaches to solving the generalized d-MA problem where we embed a graph onto a subgraph of a given degree. In recent years, considerable research has been conducted on optical and FIWI networks. Utilizing a recently proposed concept “candidate trees” in optical network, this dissertation studies counting problem on complete graphs. Closed form expressions are given for certain cases and a polynomial counting algorithm for general cases is also presented. Routing plays a major role in FiWi networks. Accordingly to a novel path length metric which emphasizes on “heaviest edge”, this dissertation proposes a polynomial algorithm on single path computation. NP-completeness proof as well as approximation algorithm are presented for multi-path routing. Radio-frequency identification (RFID) technology is extensively used at present for identification and tracking of a multitude of objects. In many configurations, simultaneous activation of two readers may cause a “reader collision” when tags are present in the intersection of the sensing ranges of both readers. This dissertation ad- dresses slotted time access for Readers and tries to provide a collision-free scheduling scheme while minimizing total reading time. Finally, this dissertation studies a monitoring problem on the surface of the earth for significant environmental, social/political and extreme events using satellites as sensors. It is assumed that the impact of a significant event spills into neighboring regions and there will be corresponding indicators. Careful deployment of sensors, utilizing “Identifying Codes”, can ensure that even though the number of deployed sensors is fewer than the number of regions, it may be possible to uniquely identify the region where the event has taken place. / Dissertation/Thesis / Doctoral Dissertation Computer Science 2019