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A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of requirements for the degree of Master of Science. May 23, 2017. / After studying matrix model quantum mechanics for one and two matrices and the eigenvalue dynamics of the one matrix model, evidence is found that there is a sector of the two matrix model that can be reduced to eigenvalue dynamics. This sector is defined by the SU(2) sector of N = 4 super Yang-Mills theory. Evidence is found by doing explicit computations of correlation functions using an interesting generalization of the usual Van der Monde determinant. The observables we study are the BPS operators of the SU(2) sector and include traces of products of both matrices, which are genuine multi matrix observables. These operators are associated to supergravity solutions of string theory. / XL2018

Chan David. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2000. / Includes bibliographical references (leaves [73]-76). / Abstracts in English and Chinese. / Abstract --- p.i / Acknowledgement --- p.ii / Contents --- p.iii / List of Figures --- p.vi / List of Tables --- p.viii / Chapter Chapter 1. --- Introduction --- p.1 / Chapter Chapter 2. --- Relaxation Phenomena --- p.8 / Chapter 2.1 --- Introduction --- p.8 / Chapter 2.2 --- Relaxation Phenomena --- p.8 / Chapter 2.3 --- Ising Model as an Example --- p.9 / Chapter 2.3.1 --- Monte Carlo Simulation --- p.10 / Chapter 2.3.2 --- Transition Matrix Method --- p.11 / Chapter 2.3.3 --- Results and Comparison --- p.13 / Chapter 2.4 --- Conclusion --- p.17 / Chapter Chapter 3. --- Stochastic Processes --- p.18 / Chapter 3.1 --- Introduction --- p.18 / Chapter 3.2 --- Stochastic Processes --- p.18 / Chapter 3.3 --- Markov Processes and Markov Chains --- p.19 / Chapter 3.4 --- Transition Matrix --- p.20 / Chapter 3.5 --- Detailed Balance Condition --- p.21 / Chapter 3.5.1 --- Weaker Balance Condition --- p.22 / Chapter 3.6 --- Ergodicity --- p.22 / Chapter 3.7 --- Conclusion --- p.23 / Chapter Chapter 4. --- Master Equation --- p.24 / Chapter 4.1 --- Introduction --- p.24 / Chapter 4.2 --- Master Equation --- p.24 / Chapter 4.3 --- Properties of the Transition Matrix W --- p.25 / Chapter 4.4 --- Eigenvalue Equation --- p.26 / Chapter 4.5 --- Relaxation Time --- p.28 / Chapter 4.6 --- Properties of Eigenvalues and Eigenvectors --- p.29 / Chapter 4.7 --- Power Method --- p.32 / Chapter 4.7.1 --- Power Method for λx --- p.33 / Chapter 4.7.2 --- Power Method for λx --- p.35 / Chapter 4.7.3 --- Power Method for λ2 of W --- p.38 / Chapter 4.8 --- Conclusion --- p.39 / Chapter Chapter 5. --- Decimation --- p.40 / Chapter 5.1 --- Introduction --- p.40 / Chapter 5.2 --- Method of Decimation --- p.40 / Chapter 5.2.1 --- Exact Formalism --- p.42 / Chapter 5.2.2 --- Iterative Formalism --- p.44 / Chapter 5.2.3 --- Approximate Formalism --- p.46 / Chapter 5.3 --- Decimation as an Effective Master Equation --- p.46 / Chapter 5.4 --- One Dimensional Energy Landscape --- p.47 / Chapter 5.5 --- Conclusion --- p.58 / Chapter Chapter 6. --- Mean Field Theory --- p.59 / Chapter 6.1 --- Introduction --- p.59 / Chapter 6.2 --- Toy Model A --- p.59 / Chapter 6.3 --- Toy Model B --- p.61 / Chapter 6.4 --- Mean Field Theory : a Lower Bound of Relaxation Time --- p.62 / Chapter 6.5 --- Mean Field Theory with the Decimation Formalism --- p.62 / Chapter 6.6 --- Mean Field Theory with Fluctuations --- p.63 / Chapter 6.6.1 --- Fluctuation Phenomena with the Decimation Formalism --- p.67 / Chapter 6.7 --- Conclusion --- p.69 / Chapter Chapter 7. --- Conclusion --- p.70 / Bibliography --- p.73

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Nous établissons une nouvelle majoration optimale pour les plus petites valeurs propres de l'opérateur de Dirac sur une hypersurface compacte de l'espace hyperbolique. / We prove a new upper bound for the smallest eigenvalues of the Dirac operator on a compact hypersurface of the hyperbolic space.

Thesis (M.A.)--Wake Forest University. Dept. of Mathematics, 2008. / Vita. Includes bibliographical references (leaves 48-49)

The objective of the thesis is to investigate the trace formulas and their applications on Hecke eigenvalues, especially on the distribution of Hecke eigenvalues. This thesis is divided into two parts.. In the first part of the thesis, a review is firstly carried out for the equidistribution of Hecke eigenvalues as primes vary and for the expected size of the error term in this equidistribution problem. Then the Kuznetsov trace formula is applied to prove a result on the size of the error term in the asymptotic distribution formula of Hecke eigenvalues. These eigenvalues become equidistributed with respect to the p-adic Plancherel measures as Hecke eigenforms vary. Next, this problem is generalized to Satake parameters of GL2 representations with prescribed supercuspidal local representations. Such a generalization is novel to the case of classical automorphic forms. To achieve this result, a trace formula of Arthur-Selberg type with a couple of key refinements is used. In the second part of the thesis, a density theorem is proved which counts the number of exceptional nontrivial zeros of a family of symmetric power L-functions attached to primitive Maass forms in the critical strip. In addition, a large sieve inequality of Elliott-Montgomery-Vaughan type for primitive Maass forms is established. The density theorem and large sieve inequality have many applications. For instance, they are used to prove statistical results on Hecke eigenvalues of primitive Maass forms and the extreme values of the symmetric power L-functions attached to primitive Maass forms. / published_or_final_version / Mathematics / Doctoral / Doctor of Philosophy

Eigenvalue problems find important applications in engineering electromagnetics. The most studied examples are cavity resonance problems and wave guidance problems, in which the electromagnetic fields are bounded or semi-bounded. Efficient eigenvalue based analysis is developed to investigate both bounded and unbounded fields in a unified manner. Incorporating a locally-conformal finite difference technique, a modal expansion based model order reduction is proposed to elucidate complicated physics of wave-matter interaction, and the fast field approximation provides quick guidance to design and optimization problems of a variety of devices. As the key to modal expansion, modal analysis for interior spectrum needs to be carried out efficiently in large scale applications. A null-space shift algorithm and a fast cosine- and sine- transform based preconditioning scheme are proposed to greatly enhance the performance of a conventional eigensolver. To model multi-junction parallel plate waveguides, both explicit and implicit mode matching schemes are introduced, where eigenanalysis are incorporated. Both schemes intelligently conform to the geometrical characteristics, which are suitable for demonstrating interesting phenomena of zero index materials. Finally, a differential forms inspired discretization scheme is proposed for the edge element based finite element analysis of waveguide problems where the variational expressions involve transverse fields only. Differential forms provide useful insight on the discretization of complicated variational expressions. / published_or_final_version / Electrical and Electronic Engineering / Doctoral / Doctor of Philosophy

published_or_final_version / Mathematics / Master / Master of Philosophy

Two direct algorithms are suggested for the computation of the linear state feedback for multi-input systems such that the resultant closed-loop system matrix has specified eigenvalues. The extra freedom can be used in different ways, for example to decrease some norm of the feedback matrix or to improve the condition of some of the eigenvalues of the closed-loop matrix. The algorithms appear to be more efficient than the techniques already in existence and have several desirable numerical properties. They use unitary transformations for numerical reliability, and are based on ideas from the QR algorithm for solving the eigenproblem. / The above algorithms have been developed jointly with Chris Paige.