This thesis is concerned with the solution of large systems of linear algebraic equations in which the matrix of coefficients is sparse. Such systems occur in the numerical solution of elliptic partial differential equations by finite-difference methods. By applying some well-known iterative methods, usually used to solve linear PDE systems, the thesis investigates their applicability to solve a set of four mildly nonlinear test problems. In Chapter 4 we study the basic iterative methods and semiiterative methods for linear systems. In particular, we derive and apply the CS, SOR, SSOR methods and the SSOR method extrapolated by the Chebyshev acceleration strategy. In Chapter 5, three ways of accelerating the SOR method are described together with the applications to the test problems. Also the Newton-SOR method and the SOR-Newton method are derived and applied to the same problems. In Chapter 6, the Alternating Directions Implicit methods are described. Two versions are studied in detail, namely, the Peaceman-Rachford and the Douglas-Rachford methods. They have been applied to the test problems for cycles of 1, 2 and 3 parameters. In Chapter 7, the conjugate gradients method and the conjugate gradient acceleration procedure are described together with some preconditioning techniques. Also an approximate LU-decomposition algorithm (ALUBOT algorithm) is given and then applied in conjunction with the Picard and Newton methods. Chapter 8 contains the final conclusions.

by To Tze-ming. / Bibliography: leaves 92-93 / Thesis (M.Ph.)--Chinese University of Hong Kong, 1985

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The purpose of this thesis is to determine the ordinary and p-modular irreducible characters and the characters of the principal indecomposable modules of the group SL(2,q), q=pⁿ, for odd p. The decomposition matrix and the Cartan matrix for SL(2,q) are also given. / Science, Faculty of / Mathematics, Department of / Graduate

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A generalization of a theorem of N.R. 0'Brian, zeroes of holomorphic vector fields and the Grothendieck residue, Bull. London Math. Soc, 7 (1975) is given. The theorem of Riemann-Roch and Hirzebruch for V-equivariant holomorphic vector bundles is obtained, via holomorphic vector fields, in the case all zeroes of the holomorphic vector field V are isolated. The Bruhat decomposition of G/B is obtained from the G -action on G/B . It is shown that a theorem of A. Bialynicki-Birula, Some theorems on actions of algebraic groups, Ann. of Math. 98, 480-497 (1973) is the generalization of the Bruhat decomposition on G/B , which was a conjecture of B. Iversen. The existence of a G -action on G/P with only one fixed a point is proved, where G is a connected linear algebraic group defined over an algebraically closed field k of characteristic zero and P is a parabolic subgroup of G . The following is obtained P = N[sub G](Pu) = {geG: Adg(Pu) = Pu} where G is a connected linear algebraic group, P is a parabolic subgroup of G and P^ is the tangent space of the set of unipotent elements of P at the identity. An elementary proof of P = N[sub G](P) = {geG: gPg ⁻¹=P} is given, where G is a connected linear algebraic group and P is a parabolic subgroup of G . / Science, Faculty of / Mathematics, Department of / Graduate

One of the fundamental differences between automorphic representations of classical groups like GL(n) and their metaplectic covers is that in the latter case the space of Whittaker functionals usually has a dimension bigger than one. Gelbart and Piatetski-Shapiro called the metaplectic representations, which possess a unique Whittaker model, distinguished and classified them for the double cover of the group GL(2). Later Patterson and Piatetski-Shapiro used a converse theorem to list the distinguished representations for the degree three cover of GL(3). In their milestone paper on general metaplectic covers of GL(n) Kazhdan and Patterson construct examples of non-cuspidal distinguished representations, which come as residues of metaplectic Eisenstein series. These are generalizations of the classical Jacobi theta functions. Despite some impressive local results to date, cuspidal distinguished representations are not classified or even constructed outside rank 1 and 2. In her thesis Wang makes some progress toward the classification in rank 3. In this dissertation we construct the distinguished representations for the degree four metaplectic cover of GL(4), applying a classical converse theorem like Patterson and Piatetski-Shapiro in the case of rank 2. We obtain the necessary local properties of the Rankin-Selberg convolutions at the ramified places and finish the proof of the construction of cuspidal distinguished representations in rank 3.

The character table of a finite group is a very powerful tool to study the groups and to prove many results. Any finite group is either simple or has a normal subgroup and hence will be of extension type. The classification of finite simple groups, more recent work in group theory, has been completed in 1985. Researchers turned to look at the maximal subgroups and automorphism groups of simple groups. The character tables of all the maximal subgroups of the sporadic simple groups are known, except for some maximal subgroups of the Monster M and the Baby Monster B. There are several well-developed methods for calculating the character tables of group extensions and in particular when the kernel of the extension is an elementary abelian group. Character tables of finite groups can be constructed using various theoretical and computational techniques. In this thesis we study the method developed by Bernd Fischer and known nowadays as the theory of Clifford-Fischer matrices. This method derives its fundamentals from the Clifford theory. Let G = N·G, where N C G and G/N = G, be a group extension. For each conjugacy class [gi]G, we construct a non-singular square matrix Fi, called a Fischer matrix. Once we have all the Fischer matrices together with the character tables (ordinary or projective) and fusions of the inertia factor groups into G, the character table of G is then can be constructed easily. In this thesis we apply the coset analysis technique (this is a method to find the conjugacy classes of group extensions) together with theory of Clifford-Fischer matrices to calculate the ordinary character tables of seven groups of extensions type, in which four are non-split and three are split extensions. These groups are of the forms: 21+8 + ·A9, 37:Sp(6, 2), 26·Sp(6, 2), 25·GL(5, 2), 210:(U5(2):2), 21+6 − :((31+2:8):2) and 22n·Sp(2n, 2) and 28·Sp(8, 2). In addition we give some general results on the non-split group 22n·Sp(2n, 2). / Thesis (Ph.D.)-University of KwaZulu-Natal, Pietermaritzburg, 2012.

Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, August 2000. / Includes bibliographical references. Also available on the Internet.

Thesis (Ph. D.)--University of Alberta, 2010. / Title from pdf file main screen (viewed on June 24, 2010). A thesis submitted to the Faculty of Graduate Studies and Research in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics, [Department of] Mathematical and Statistical Sciences, University of Alberta. Includes bibliographical references.