On critical poins for Gaussian vectors with infinitely divisible squares
Year of Dissertation:
2009
This paper is concerned with necessary conditions for infinite divisibility of the Gaussian squares with non-zero means. A Gaussian vector G with zero mean is said to have a critical point α, such that 0≤α≤∞
Non-commutative cryptography: Diffie-Hellman and CCA secure cryptosystems using matrices over group rings and digital signatures
Author:
Charalambos Koupparis
Year of Dissertation:
2012
Advisor:
Delaram Kahrobaei
As computing speed has been following Moore's law without any inclination of tapering out, the need for ever more secure cryptographic protocols is becoming more and more relevant. During the past one and a half decades the field of non-commutative (or on-abelian) group based cryptography has seen a surge in interest.
The Geometry of Lattice-Gauge-Orbit Space
Year of Dissertation:
2011
In this paper, the Riemannian geometry of gauge-orbit space on the lattice with open boundary conditions is explored. It is shown how the metric and inverse metric tensors can be calculated, and further how the Ricci curvature might be calculated. The metric tensor and the inverse metric tensor are calculated for special cases, and some conjectures about the curvature of the space are made, which, if true, would move towards implying a mass gap in the theory.
Resplendent models generated by indiscernibles
Year of Dissertation:
2013
In this thesis I study the question: Which first-order structures are generated by indiscernibles? J. Schmerl showed that if L<\italic> is a finite language, every countable recursively saturated L<\italic>-structure in which a form of coding of finite functions is available is generated by indiscernibles. Further, he showed that such a structure has arbitrarily large extensions which are generated by a set of indiscernibles, resplendent, and L<\italic>_{&infin,&omega}-equivalent to the original structure. Proofs of these theorems are complex and use a combinatorial lemma whose proof in Schmerl's paper has an acknowledged gap. I offer a complete proof of a more direct combinatorial lemma from which Schmerl's theorems follow.
Dual Graphs and Poincaré Series of Valuations
Year of Dissertation:
2012
Valuations on function fields of dimension two have been studied from the perspectives of dual graphs, generating sequences, Poincare series, and the valuative tree, among others. The goal of this dissertation is to greater unify these various approaches. Spivakovsky's dual graphs are used to calculate the Poincare series of non-divisorial valuations. With Galindo's results in the divisorial case already known, the equivalence of Poincare series with dual graphs is shown. A new elementary constructive proof of minimal generating sequences for non-divisorial valuations is given along the way, using only modest prerequisites from number theory. It is fair to say that the proof of minimal generating sequences is the crux of this dissertation, while the results on Poincare series are all corollaries.
Problems in additive number theory
Year of Dissertation:
2011
In the first chapter we obtain the Biro-type upper bound for the smallest period of B in the case when A is a finite multiset of integers and B is a multiset such that A and B are t-complementing multisets of integers. In the second chapter we answer an inverse problem for lattice points proving that if K is a compact subset of R×R such that K+Z×Z=R×R then the integer points of the difference set of K is not contained on the coordinate axes, Z×{0}U{0}×Z. In the third chapter we show that there exist infinite sets A and M of positive integers whose partition function has weakly superpolynomial but not superpolynomial growth. The last chapter deals with the size of a sum of dilates 2·A+k·A. We prove that if k is a power of an odd prime or product of two primes and A a finite set of integers such that |A|>8k^k, then |2· A+k·A|≥ (k+2)|A|-k^2-k+2.
Total Variation of Gaussian Processes and Local Times of Associated Levy Processes
Year of Dissertation:
2009
Results of Taylor and Marcus and Rosen on the total variation of Gaussian processes and local times of associated symmetric stable processes are extended to a large class of symmetric Lévy processes. In this extension, the increments variance of the Gaussian process is generalized to a regularly varying function with index 0<α< 2. The result also includes a generalization of the total variation function.
The Hilbert Projective Metric, Multi-type Branching Processes and Mathematical Biology: a Model of the Evolution of Resistance
Author:
Christopher McCarthy
Year of Dissertation:
2010
Bacteria, viruses, or cancer cells, by means of mutation and replication, are sometimes able to escape the selective pressure exerted by treatment. This is called the development, or evolution, of resistance.
Algebraic Models for the Free Loop Space and Differential Forms of a Manifold
Year of Dissertation:
2011
Advisor:
Mahmoud Zeinalian
Our initial goal is to give a chain level description of the string topology loop product for a large class of spaces. This effort is described in two parts; the first uses Brown's theory of twisting cochains to obtain a model for the free loop space of a manifold and the second constructs a minimal model for the Frobenius algebra of differential forms of a manifold. The first part defines the loop product for closed, oriented manifolds and Poincare Duality spaces. The second part is an attempt to understand the minimal model for the Frobenius algebra of a manifold, with the idea of extending the methods in the first section to define the loop product for open manifolds.
An unstable variant of the Morava Change of Rings theorem for K(n) theory
Year of Dissertation:
2011
We formulate a very general criteria for a base change comonads for Ext