Thesis (Ph.D.)--University of the Witwatersrand, Faculty of Science, 1998. / A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Doctor of Philosophy. Johannesburg 1998 / Let A = (aij) be an m x n matrix. There is a natural way to associate a poset PA with A. A jump in a linear extension of PA is a pair of consecutive elements which are incomparable in Pa. The jump number of A is the minimum number of jumps in any linear extension of PA. The maximum jump number over a class of n x n matrices of zeros and ones with constant row and column sum k, M (n, k), has been investigated in Chapter 2 and 3. Chapter 2 deals with extremization problems concerning M (n ,k). In Chapter 3, we obtain the exact values for M (11,k). M(n,Q), M (n,n-3) and M(n,n-4). The concept of frequency hyperrectangle generalizes the concept of latin square. In Chapter 4 we derive a bound for the maximum number of mutually orthogonal frequency hyperrectangles. Chapter 5 gives two algorithms to construct mutually orthogonal frequency hyperrectangles. Chapter 6 is devoted to some enumerative results about Carlitz compositions (compositions with different adjacent parts).

by Li King Hung. / Thesis submitted in: July, 1998. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (leaves 54-56). / Abstract also in Chinese. / Chapter 1 --- Description of the Number Field Sieve --- p.6 / Chapter 1.1 --- Introduction --- p.6 / Chapter 1.2 --- Outline of the algorithm --- p.8 / Chapter 2 --- Algebraic knowledge --- p.17 / Chapter 2.1 --- Factorization of an ideal over the class of ideals --- p.17 / Chapter 2.2 --- Existence of the square roots of an element in the ring of algebraic integers --- p.26 / Chapter 3 --- Run time analysis and Practical result --- p.31 / Chapter 3.1 --- Relation between sieving over X and finding the linear dependencies --- p.32 / Chapter 3.2 --- Relation between the size of a factor base and finding the linear dependencies --- p.34 / Chapter 3.3 --- Practical consideration --- p.36 / Chapter 4 --- Improvement of the algorithm --- p.38 / Chapter 4.1 --- Quadratic characters --- p.38 / Chapter 4.2 --- Finding the square root --- p.40 / Chapter 4.3 --- Solving the linear system of equation --- p.42 / Chapter 4.4 --- Reusing the computation --- p.47 / Chapter 4.5 --- Using more general purpose data --- p.50 / Chapter 4.6 --- Examples --- p.51 / Chapter 4.6.1 --- "A 18-digit example,761260375069630873" --- p.52 / Chapter 4.6.2 --- "A 23-digit example, 16504377514594481520559" --- p.52 / Bibliography

We give an explicit formula for the arithmetic intersection number of CM cycles on Lubin-Tate spaces for all levels. We prove our formula by formulating the intersection number on the infinite level. Our CM cycles are constructed by choosing two separable quadratic extensions K1, K2/F of non-Archimedean local fields F . Our formula works for all cases, K1 and K2 can be either the same or different, ramify or unramified. As applications, this formula translate the linear Arithmetic Fundamental Lemma (linear AFL) into a comparison of integrals. This formula can also be used to recover Gross and Keating’s result on lifting endomorphism of formal modules.

Thesis (Ph. D.)--University of Texas at Austin, 2006. / Vita. Includes bibliographical references.

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

Let $F$ be a non-Archimedean local field and $G$ be the general linear group $\mathrm{GL}_n$ over $F$. Bushnell and Henniart described the essentially tame local Langlands correspondence of $G(F)$ using rectifiers, which are certain characters defined on tamely ramified elliptic maximal tori of $G(F)$. They obtained such result by studying the automorphic induction character identity. We relate this formula to the spectral transfer character identity, based on the theory of twisted endoscopy of Kottwitz, Langlands, and Shelstad. In this article, we establish the following two main results. (i) To show that the automorphic induction character identity is equal to the spectral transfer character identity when both are normalized by the same Whittaker data. (ii) To express the essentially tame local Langlands correspondence using admissible embeddings constructed by Langlands-Shelstad $\chi$-data and to relate Bushnell-Henniart's rectifiers to certain transfer factors.

It is known that almost all numbers are transcendental in the sense of Lebesgue measure. However there is no simple rule to separate transcendental numbers from algebraic numbers. Today research in this direction is about establishing new transcendence criteria for new families of transcendental numbers. By applying a recent refinement of Subspace Theorem, Boris Adamczewski and Yann Bugeaud determined new transcendence criteria for real numbers which we shall present in this thesis. Published only three years ago, their articles explore combinatorial, algorithmic and dynamic approaches in discussing the notion of complexity of both continued fraction and b-adic expansions of a certain class of real numbers. The condition on the expansions are those of being stammering and non-eventually periodic. Taking together these articles give a well-structured picture of the interrelationships between sequence characteristics of expansion (i.e. complexity, periodicity, type of generator) and algebraic characteristics of number itself (i.e. class, transcendency).

Thesis (Ph. D.)--Rutgers University, 2009. / "Graduate Program in Mathematics." Includes bibliographical references (p. 57-59).

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