by Yuen Wong. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1990. / Chapter Chapter 1 --- Introduction --- p.1 / Chapter Chapter 2 --- Basic Results --- p.5 / Chapter §2.1 --- General Results --- p.5 / Chapter §2.2 --- Williamson's Method --- p.8 / Chapter Chapter 3 --- Algorithms And Subroutines --- p.15 / Chapter §3.1 --- Introduction --- p.15 / Chapter §3.2 --- Increasing Determinant Method --- p.15 / Chapter §3.3 --- Williamson's Method - Direct Computation --- p.21 / Chapter §3.4 --- Williamson's Method - Increasing Determinant --- p.26 / Chapter Chapter 4 --- Comparisons And Recommendations On Algorithms --- p.32 / Chapter §4.1 --- Introduction --- p.32 / Chapter §4.2 --- Comparisons And Recommendations On IMPROV(N) --- p.32 / Chapter §4.3 --- Comparisons And Recommendations On GENHA(N) --- p.34 / Chapter §4.4 --- Comparisons And Recommendations On VTID(N) --- p.35 / Chapter §4.5 --- Summary --- p.37 / Chapter Chapter 5 --- Applications Of Hadamard Matrices --- p.38 / Chapter §5.1 --- Hadamard Matrices And Balanced Incomplete Block Designs' --- p.38 / Chapter §5.2 --- Hadamard Matrices And Optimal Weighing Designs --- p.43 / Chapter Chapter 6 --- Conclusion --- p.51 / References --- p.52 / Appendices --- p.53

Tsang, Chi Shing Sidney. / "October 2010." / Thesis (M.Phil.)--Chinese University of Hong Kong, 2010. / Includes bibliographical references (leaves 58-59). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 1.1 --- The main results --- p.6 / Chapter 1.2 --- Structure of the thesis --- p.8 / Chapter 2 --- The Upper Lyapunov Exponent --- p.10 / Chapter 2.1 --- Notation --- p.10 / Chapter 2.2 --- The upper Lyapunov exponent --- p.11 / Chapter 2.3 --- Cocycles --- p.12 / Chapter 2.4 --- The Theorem of Furstenberg and Kesten --- p.14 / Chapter 3 --- Contraction Properties --- p.19 / Chapter 3.1 --- Two basic lemmas --- p.20 / Chapter 3.2 --- Contracting sets --- p.25 / Chapter 3.3 --- Strong irreducibility --- p.29 / Chapter 3.4 --- A key property --- p.30 / Chapter 3.5 --- Contracting action on P(Rd) and converges in direction --- p.36 / Chapter 3.6 --- Lyapunov exponents --- p.39 / Chapter 3.7 --- Comparison of the top Lyapunov exponents and Fursten- berg's theorem --- p.43 / Chapter 4 --- Analytic Dependence of Lyapunov Exponents on The Probabilities --- p.48 / Chapter 4.1 --- Continuity and analyticity properties for i.i.d. products --- p.49 / Chapter 4.2 --- The proof of the main result --- p.50 / Chapter 5 --- The Expression of The Upper Lyapunov Exponent in Complex Functions --- p.54 / Chapter 5.1 --- The set-up --- p.54 / Chapter 5.2 --- The main result --- p.56 / Bibliography --- p.58

Let P(ε) = P0 + A(ε) be a stochasticity preserving analytic perturbation of a stochastic matrix P0. We characterize the hybrid Cesaro limit lim 1 N(ε) Pk(ε), ε↓0 N(ε) ∑ where N(ε) ↑ ∞ as ε ↓ 0, when P0 has eigenvalues on the unit circle in the complex plane other than 1.

By first looking at the orthonormal basis: Γ = {∑i 4 ibi ∈{0, 1}, finite sums} and the related orthonormal basis 5Γ = {5∑i 4i bi : bi ∈ {0, 1}, finite sums} we find several interesting relationships with the unitary matrix Uα,β arising from the operator U: Γ → 5Γ. Further, we investigate the relationships between U and the operators So : Γ → 4Γ defined by Soe4γ where eγ = e2ΠiΓ and S1: Γ → 4Γ+1 defined by S1eγ = e4γ+1. Most intriguing, we found that when taking powers of the aforementioned Uα,β matrix that although there are infinitely many 1's occurring in the entries of Uα,β only one such 1 occurs in the subsequent higher powers Ukα,β. This means that there are infinitely many γ ∈ Γ ∩ 5Γ, but only one such γ in the intersection Γ and 5kΓ, for k ≥ 2.

vii, 188 leaves : ill. ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.)--University of Adelaide, Dept. of Statistics, 1976

Nonnegative matrices have a myriad of applications in the biological, social, and physical genres. Of particular importance are the primitive matrices. A nonnegative matrix, M, is primitive exactly when there is a positive integer, k, such that M[superscript k] has only positive entries; that is, all the entries in M[superscript k] are strictly greater than zero. This method of determining if a matrix is primitive uses matrix multiplication and so would require time ���(n[superscipt ��]) where ��>2.3 even if fast matrix multiplication were used. Our goal is to find a much faster algorithm. This can be achieved by viewing a nonnegative matrix, M, as the adjacency matrix for a graph, G(M). The matrix, M, is primitive if and only if G(M) is strongly connected and the greatest common divisor of the cycle lengths in G(M) is 1. We devised an algorithm based in breadth-first search which finds a set of cycle lengths whose gcd is the same as that of G(M). This algorithm has runtime O(e) where e is the number of nonzero entries in M and therefore equivalent to the number of edges in G(M). A proof is given shown the runtime of O(n + e) along with some empirical evidence that supports this finding. / Graduation date: 2003

Computing eigenpairs of a matrix corresponding to a specific geometry in the complex plane is an important topic in real time signal processing, pattern recognition, spectral analysis, systems theory, radar, sonar, and geophysics. We have studied the matrix sign and matrix sector function iterations to extract the eigenpairs belonging to various geometries without resorting to computationally expensive eigenanalysis methods. We propose a parallelization of an existing matrix sign function algorithm, which was implemented on a Meiko CS-2 multiprocessor. We obtain a fast and stable algorithm for computing the matrix sector functions using Halley's generalized iteration formula for solving nonlinear equations. We propose a parallel iterative algorithm to compute the principal nth root of a positive definite matrix using Gauss-Legendre integration formula. Furthermore computing functions of square matrices is also an important topic in linear algebra, engineering, and applied mathematics. A parallelization of Parlett's algorithm for computing arbitrary functions of upper triangular matrices is introduced. We propose a block-recursive and a parallel algorithm for fast and efficient computation of functions of triangular matrices. The parallel complexity and cache efficiency of these algorithms for computers with two levels of memory are also analyzed. / Graduation date: 1996

Thesis (M.S.)--Oregon State University, 1968. / Typescript (photocopy). Includes bibliographical references (p. 33). Also available on the World Wide Web.

Thesis (M.S.E.)--University of Alabama in Huntsville, 1968. / Typescript (photocopy).

Consider a simple model of an age-structured population with two age-classes and stochastically varying survival rate of young. Let $m_{1,y},m_{2,t}$ be birth rates per capital and $P_{1,t}$ be a survival rate. egin{eqnarray} left( egin{array}{clr} N_{1,t+1}N_{2,t+1} end{array} ight) = left( egin{array}{clr} m_{1,t+1} & m_{2,t+1} P_{1,t+1} & 0 end{array} ight) left( egin{array}{clr} N_{1,t}N_{2,t} end{array} ight) end{eqnarray} we want to study the large term behavior of $(N_{1,t},N_{2,t})$ the age-structured population through the theory of random matrix product.