A polynomial is a product of distinct cyclotomic polynomials if and only if it is a divisor over [Special characters omitted.] [x ] of xn - 1 for some positive integer n. In this thesis, we will examine two natural questions concerning the divisors...
Access control is a core component of any information-security strategy. Researchers have spent tremendous energy over the past forty years defining abstract access-control models and proving various properties about them. However, surprisingly...
We study the bijective combinatorics of reduced words. These are fundamental objects in the study of Coxeter groups. We restrict our focus to reduced words of permutations and signed permutations. Our results can all be situated within the context...
In this thesis, we consider several problems relating to cyclic subgroups of the group [mathematical equation]. Each element of [mathematical equation] has a unique representative in one of the two intervals [mathematical equation] and...
In this thesis, we characterize and enumerate the permutations which are realized by the orbits of signed shifts, a family of discrete dynamical systems on words. The permutations, which are called patterns of the signed shifts, are given by the...
In Chapter 2 we look at the distribution of permutation statistics in the context of pattern-avoiding permutations. The first part of this chapter deals with a recursively defined bijection of Robertson [37] between 123- and 132-avoiding...
The Euler '-function and Carmichael -function are extremely important in modern number theory, and much work has been devoted to studying the distribution and arithmetic properties of the values of each function. One interesting unresolved question...
This thesis explores practical and theoretical aspects of several privacy-providing technologies, including tools for anonymous web-browsing, verifiable electronic voting schemes, and private information retrieval from databases. State-of-art...
Polynomials. Finite fields (Algebra). Algebraic functions. Number theory.
The ring of univariate polynomials over a finite field shares many foundational arithmetic properties with the ring of rational integers. This similarity makes it possible for many problems in elementary number theory to be translated 'through the...
Coherent states. Quantum field theory. Nonlinear theories.
In this thesis we study the properties of time-dependent, nontopological configurations and their effect on the macroscopic properties of a system described by a nonlinear field theory. These structures seem to be ubiquitous in relativistic field...
In this thesis we look at several problems that lie in the intersection between combinatorial and multiplicative number theory. A common theme of many of these problems are estimates for and properties of the smooth numbers, those integers not...
'Orthogonal modular forms are algebraic modular forms arising from lattices in quadratic spaces. In this thesis, we define orthogonal modular forms, establish their basic properties, and then apply them to a case of ternary quadratic spaces to...
'The control of waves using periodic structures is crucial for modern optical, electromagnetic and acoustic devices such as diffraction gratings, filters, photonic crystals, solar cells, sensors, and absorbers. We present a high-order accurate...
In this thesis we develop a theory of Fourier analysis and fast Fourier transforms (FFTs) for finite inverse semigroups. Our results generalize results in the theory of Fourier analysis for finite groups. There is a general method for generating...
Partially ordered sets and permutations are combinatorial structures having vast applications in theoretical computer science. In this thesis, we study various computational and algorithmic problems related to these structures. The first chapter of...
The ionosphere is the primary source for heavy ions which are ubiquitous in the terrestrial magnetosphere. Low-altitude energization in the auroral ionosphere results in bulk heating and transverse acceleration of ions, which begin to upwell and/or...
We start by introducing avoidance coupling of Markov chains, with an overview of existing results. We then introduce and motivate a new notion, uniform coupling. We show that the only Markovian avoidance coupling on a cycle is of this type, and...
This dissertation explores the spatial and temporal distributions of impurities in natural ice. Using polarized light, ion chromatography (IC), synchrotron x-ray topography (SXT), scanning electron microscopy (SEM), and energy-dispersive...
Siegel domains. Modular groups. Hecke algebras. Forms
In the 1960s Satake proved the existence of an isomorphism between the local Hecke algebra and the ring of polynomials invariant under the action of the signed permutation group W n (the Weyl group associated to Sp n over a local field)....
Let K be the function field over a finite field of odd order, and let H be a definite quaternion algebra over K. If Î‘ is an order of level M in H , we define theta series for each ideal I of Î‘ using the reduced norm on H. Using harmonic analysis...
We begin with a presentation of the current state of Poljak and Turzik's conjecture that a matroid is sticky if and only if it is modular as described in [5] and [1]. We show that all graphic, cographic, and regular matroids must satisfy the...
Ocean circulation -- Georges Bank -- Mathematical models. ; Submarine topography -- Georges Bank.
A two-dimensional continuously-stratified model is derived for the study of instability properties of oceanic density fronts in the presence of sloping bottom topography. While the model retains several traditional assumptions, such as uniformity...
CD antigens. T cells -- Receptors. Receptor-ligand complexes. Transplantation immunology. Immunological tolerance -- Molecular aspects. Antigens
Voltage-dependent sodium channels (Na v ) are critical determinants of the ability of a neuron to generate and propagate action potentials. While a large family of Na v isoforms has been identified, the impact of specific isoforms on the electrical...