Thesis (Ph. D.)--University of Wisconsin, 1966. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

The research contained in this thesis can be divided into two related, but distinct parts. The rst chapter deals with block Toeplitz operators de ned by rational matrix function symbols on discrete sequence spaces. Here we study sequences of operators that converge to the inverses of these Toeplitz operators via an invertibility result involving a special representation of the symbol of these block Toeplitz operators. The second part focuses on a special class of matrices generated by banded Toeplitz matrices, i.e., Toeplitz matrices with a nite amount of non-zero diagonals. The spectral theory of banded Toeplitz matrices is well developed, and applied to solve questions regarding the behaviour of the singular values of Toeplitz-generated matrices. In particular, we use the behaviour of the singular values to deduce bounds for the growth of the norm of the inverse of Toeplitz-generated matrices. In chapter 2, we use a special state-space representation of a rational matrix function on the unit circle to de ne a block Toeplitz operator on a discrete sequence space. A discrete Riccati equation can be associated with this representation which can be used to prove an invertibility theorem for these Toeplitz operators. Explicit formulas for the inverse of the Toeplitz operators are also derived that we use to de ne a sequence of operators that converge in norm to the inverse of the Toeplitz operator. The rate of this convergence, as well as that of a related Riccati di erence equation is also studied. We conclude with an algorithm for the inversion of the nite sections of block Toeplitz operators. Chapter 3 contains the main research contribution of this thesis. Here we derive sharp growth rates for the norms of the inverses of Toeplitz-generated matrices. These results are achieved by employing powerful theory related to the Avram-Parter theorem that describes the distribution of the singular values of banded Toeplitz matrices. The investigation is then extended to include the behaviour of the extreme and general singular values of Toeplitz-generated matrices. We conclude with Chapter 4, which sets out to answer a very speci c question regarding the singular vectors of a particular subclass of Toeplitz-generated matrices. The entries of each singular vector seems to be a permutation (up to sign) of the same set of real numbers. To arrive at an explanation for this phenomenon, explicit formulas are derived for the singular values of the banded Toeplitz matrices that serve as generators for the matrices in question. Some abstract algebra is also employed together with some results from the previous chapter to describe the permutation phenomenon. Explicit formulas are also shown to exist for the inverses of these particular Toeplitz-generated matrices as well as algorithms to calculate the norms and norms of the inverses. Finally, some additional results are compiled in an appendix. / PhD (Mathematics), North-West University, Potchefstroom Campus, 2015

The research contained in this thesis can be divided into two related, but distinct parts. The rst chapter deals with block Toeplitz operators de ned by rational matrix function symbols on discrete sequence spaces. Here we study sequences of operators that converge to the inverses of these Toeplitz operators via an invertibility result involving a special representation of the symbol of these block Toeplitz operators. The second part focuses on a special class of matrices generated by banded Toeplitz matrices, i.e., Toeplitz matrices with a nite amount of non-zero diagonals. The spectral theory of banded Toeplitz matrices is well developed, and applied to solve questions regarding the behaviour of the singular values of Toeplitz-generated matrices. In particular, we use the behaviour of the singular values to deduce bounds for the growth of the norm of the inverse of Toeplitz-generated matrices. In chapter 2, we use a special state-space representation of a rational matrix function on the unit circle to de ne a block Toeplitz operator on a discrete sequence space. A discrete Riccati equation can be associated with this representation which can be used to prove an invertibility theorem for these Toeplitz operators. Explicit formulas for the inverse of the Toeplitz operators are also derived that we use to de ne a sequence of operators that converge in norm to the inverse of the Toeplitz operator. The rate of this convergence, as well as that of a related Riccati di erence equation is also studied. We conclude with an algorithm for the inversion of the nite sections of block Toeplitz operators. Chapter 3 contains the main research contribution of this thesis. Here we derive sharp growth rates for the norms of the inverses of Toeplitz-generated matrices. These results are achieved by employing powerful theory related to the Avram-Parter theorem that describes the distribution of the singular values of banded Toeplitz matrices. The investigation is then extended to include the behaviour of the extreme and general singular values of Toeplitz-generated matrices. We conclude with Chapter 4, which sets out to answer a very speci c question regarding the singular vectors of a particular subclass of Toeplitz-generated matrices. The entries of each singular vector seems to be a permutation (up to sign) of the same set of real numbers. To arrive at an explanation for this phenomenon, explicit formulas are derived for the singular values of the banded Toeplitz matrices that serve as generators for the matrices in question. Some abstract algebra is also employed together with some results from the previous chapter to describe the permutation phenomenon. Explicit formulas are also shown to exist for the inverses of these particular Toeplitz-generated matrices as well as algorithms to calculate the norms and norms of the inverses. Finally, some additional results are compiled in an appendix. / PhD (Mathematics), North-West University, Potchefstroom Campus, 2015

We consider on the interval [0; a], rstly fourth-order di erential operators with eigenvalue parameter dependent boundary conditions and secondly a sixth-order di erential operator with eigenvalue parameter dependent boundary conditions. We associate to each of these problems a quadratic operator pencil with self-adjoint operators. We investigate the spectral proprieties of these problems, the location of the eigenvalues and we explicitly derive the rst four terms of the eigenvalue asymptotics.

We study the problem of the distribution of certain GL(3) Maass forms, namely, we obtain a Weyl’s law type result that characterizes the distribution of their eigenvalues, and an orthogonality relation for the Fourier coefficients of these Maass forms. The approach relies on a Kuznetsov trace formula on GL(3) and on the inversion formula for the Lebedev-Whittaker transform. The family of Maass forms being studied has zero density in the set of all GL(3) Maass forms and contains all self-dual forms. The self-dual forms on GL(3) can also be realised as symmetric square lifts of GL(2) Maass forms by the work of Gelbart-Jacquet. Furthermore, we also establish an explicit inversion formula for the Lebedev-Whittaker transform, in the nonarchimedean case, with a view to applications.

<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>

Chan Pui Man. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2001. / Includes bibliographical references (leaves 103-105). / Abstracts in English and Chinese. / Chapter Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Main Problems --- p.1 / Chapter 1.2 --- Motivation --- p.5 / Chapter 1.3 --- Present Works --- p.7 / Chapter Chapter 2 --- Estimation of the Eigenvalues in a Wishart Distribution --- p.11 / Chapter 2.1 --- Review of Previous Works --- p.14 / Chapter 2.2 --- Some Useful Statistical and Mathematical Results --- p.17 / Chapter 2.3 --- Improved Estimation of A under Squared Error Loss L1 --- p.23 / Chapter 2.4 --- Simulation Study for the Wishart Distribution under Squared Error Loss --- p.28 / Chapter 2.5 --- Discussions on Wishart Distribution Under Squared Error Loss --- p.32 / Chapter 2.6 --- Improved Estimation of A under the Entropy Loss(det) L2 --- p.33 / Chapter 2.7 --- Simulation Study for the Wishart Distribution Under Entropy Loss L2 --- p.38 / Chapter 2.8 --- Discussions on Wishart Distribution Under Entropy Loss --- p.44 / Chapter Chapter 3 --- Estimation of the Eigenvalues in a Multivariate F Distribution --- p.46 / Chapter 3.1 --- Review of Previous Works --- p.49 / Chapter 3.2 --- Some Useful Statistical and Mathematical Results --- p.50 / Chapter 3.3 --- Improved Estimation of A under the Squared Loss L1 --- p.54 / Chapter 3.4 --- Simulation Study for F Distribution under Squared Error Loss L1 --- p.62 / Chapter 3.5 --- Discussions on F distribution under Squared Error Loss --- p.68 / Chapter 3.6 --- Improved Estimation of A under the Entropy Loss(det) L2 --- p.69 / Chapter 3.7 --- Simulation Study for Multivariate F Distribution under Entropy Loss(det) L2 --- p.76 / Chapter 3.8 --- Discussions on F distribution under Entropy Loss --- p.86 / Chapter Chapter 4 --- Inheritance of Dominance between Eigenvalues Loss Function and Matrix Function --- p.87 / Chapter 4.1 --- Significance of The Problem --- p.87 / Chapter 4.2 --- Inheritance of Dominance between Eigenvalues Estimator and Matrix Estimator under Squared Error Loss --- p.92 / Chapter 4.3 --- Inheritance of Dominance between Eigenvalues Estimator and Matrix Estimator under Entropy Loss --- p.97 / Chapter 4.4 --- Conclusion --- p.102 / BIBLIOGRAPHY --- p.103

In this work we tackle the challenge of designing quantum unitary operators which represent solutions to optimization problems. We start with a novel method which combines an evolutionary algorithm known as an Evolution Strategy (ES) with a method to randomly generate unitary operators. With this new method, a quantum operator is represented for the first time using real-valued vectors and can be "evolved" or designed to meet certain target criteria. This criteria could be the solution to an optimization problem. With the ability to evolve quantum operators, we attempt to evolve various known single and multi-qubit quantum gates as well as quantum oracles. We evolve quantum operators which solve instance problems of a known NP-Hard problem and even attempt to evolve a generalized solution operator. We evolve multiple operators with varying size and investigate their properties through eigenanalysis methods as well as by synthesizing them into quantum logic gates using the quantum compiler Qubiter. We also present a new quantum logic algebra which offers a new way to represent quantum circuits and demonstrate its immediate uses in quantum computing.

My research is concentrated on applications of linear algebra in engineering, including wavelet analysis and structured matrices and (currently) approximation of large dense matrices by a mosaic of low rank blocks. / Singapore-MIT Alliance (SMA)

Thesis (M.S.)--Brigham Young University. Dept. of Physics and Astronomy, 2003. / Includes bibliographical references (p. 73).