For some families of one-dimensional locally infinitely divisible Markov processes xet 0≤t≤T with frequent small jumps, large deviation expansions for expectations are proved: as epsilon ↓ 0 Ee expe-1F xe =expe -1Ff0 -Sf0 0≤i≤ s/2Ki˙ei+o&parl0; es/2&parr0; where s is a positive integer, S is the normalized action functional, constants Ki are expressed through derivatives of the smooth functional F, and &phis;0 is the unique maximizer of F -- S The proof of above large deviation expansions relies on asymptotic expansions for expectations of a smooth functional G of stochastic processes etaepsilon = epsilon--1/2(xi epsilon -- &phis;0) : as epsilon ↓ 0 EeGhe =EGh +e1/2EA1Gh +˙˙˙+es/2EAsG h+o&parl0;es/2&parr0; for some Gaussian diffusion eta and suitable differential operators Ai / acase@tulane.edu

In this paper we make use of the the concept of covering spaces for homogeneous continua as pioneered by Rogers and refined by Maciaas. For a continuum X embedded essentially into the product of the circle S1 and the Hilbert cube Q , we examine the structure of the space X˜ which is the preimage of X under a standard covering map p : RxQ→S1xQ . We show that the compactification of components of X˜ must be decomposable, and are aposyndetic whenever X is. We also demonstrate the conditions under which higher order forms of aposyndesis are inherited by the components of X˜ as well. We conclude by examining the continuum component structure of X˜ and develop several theorems that allow us to determine the cardinality of the collection of continuum components / acase@tulane.edu

<p> The exponential nature of Moore's law has inadvertently created huge data storage complexes that are scattered around the world. Data elements are continuously being searched, processed, erased, combined and transferred to other storage units without much regard to power consumption. The need for faster searches and power efficient data processing is becoming a fundamental requirement. Quantum computing may offer an elegant solution to speed and power through the utilization of the natural laws of quantum mechanics. Therefore, minimal cost quantum circuit implementation methodologies are greatly desired. </p><p> This thesis explores the decomposition of group functions and the Walsh spectrum for implementing quantum canonical cascades with minimal cost. Three different methodologies, using group decomposition, are presented and generalized to take advantage of different quantum computing hardware, such as ion traps and quantum dots. Quantum square root of swap gates and fixed angle rotation gates comprise the first two methodologies. The third and final methodology provides further quantum cost reduction by more efficiently utilizing Hilbert spaces through variable angle rotation gates. The thesis then extends the methodology to realize a robust quantum circuit synthesis tool for single and multi-output quantum logic functions.</p>

abstract: The chromatic number $\chi(G)$ of a graph $G=(V,E)$ is the minimum number of colors needed to color $V(G)$ such that no adjacent vertices receive the same color. The coloring number $\col(G)$ of a graph $G$ is the minimum number $k$ such that there exists a linear ordering of $V(G)$ for which each vertex has at most $k-1$ backward neighbors. It is well known that the coloring number is an upper bound for the chromatic number. The weak $r$-coloring number $\wcol_{r}(G)$ is a generalization of the coloring number, and it was first introduced by Kierstead and Yang \cite{77}. The weak $r$-coloring number $\wcol_{r}(G)$ is the minimum integer $k$ such that for some linear ordering $L$ of $V(G)$ each vertex $v$ can reach at most $k-1$ other smaller vertices $u$ (with respect to $L$) with a path of length at most $r$ and $u$ is the smallest vertex in the path. This dissertation proves that $\wcol_{2}(G)\le23$ for every planar graph $G$. The exact distance-$3$ graph $G^{[\natural3]}$ of a graph $G=(V,E)$ is a graph with $V$ as its set of vertices, and $xy\in E(G^{[\natural3]})$ if and only if the distance between $x$ and $y$ in $G$ is $3$. This dissertation improves the best known upper bound of the chromatic number of the exact distance-$3$ graphs $G^{[\natural3]}$ of planar graphs $G$, which is $105$, to $95$. It also improves the best known lower bound, which is $7$, to $9$. A class of graphs is nowhere dense if for every $r\ge 1$ there exists $t\ge 1$ such that no graph in the class contains a topological minor of the complete graph $K_t$ where every edge is subdivided at most $r$ times. This dissertation gives a new characterization of nowhere dense classes using generalized notions of the domination number. / Dissertation/Thesis / Doctoral Dissertation Mathematics 2020

<p>We study first order deformations of the tangent sheaf of resolutions of Calabi-Yau threefolds that are of the form $\CC^3/\ZZ_r$, focusing</p><p> on the cases where the orbifold has an isolated singularity. We prove a lower bound on the number </p><p>of deformations of the tangent bundle for any crepant resolution of this orbifold. We show that this lower bound is achieved when the resolution used is the </p><p>G-Hilbert scheme, and note that this lower bound can be found using a combinatorial count of (0,2)-deformation moduli fields for</p><p>N=(2,2) conformal field theories on the orbifold. We also find that in general this minimum is not achieved, and expect the discrepancy </p><p>to be explained by worldsheet instanton corrections coming from rational curves in the orbifold resolution. We show that </p><p>irreducible toric rational curves will account for some of the discrepancy, but also prove that there must be additional</p><p>worldsheet instanton corrections beyond those from smooth isolated rational curves.</p> / Dissertation

x, 110 p. : ill. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number. / This dissertation examines the existence of the self-intersection local time for a superprocess over a stochastic flow in dimensions d ≤ 3, which through constructive methods, gives a Tanaka like representation. The superprocess over a stochastic flow is a superprocess with dependent spatial motion, and thus Dynkin's proof of existence, which requires multiplicity of the log-Laplace functional, no longer applies. Skoulakis and Adler's method of calculating moments is extended to higher moments, from which existence follows. / Committee in charge: Hao Wang, Co-Chairperson, Mathematics; David Levin, Co-Chairperson, Mathematics; Christopher Sinclair, Member, Mathematics; Huaxin Lin, Member, Mathematics; Van Kolpin, Outside Member, Economics

abstract: Higher-rank graphs, or k-graphs, are higher-dimensional analogues of directed graphs, and as with ordinary directed graphs, there are various C*-algebraic objects that can be associated with them. This thesis adopts a functorial approach to study the relationship between k-graphs and their associated C*-algebras. In particular, two functors are given between appropriate categories of higher-rank graphs and the category of C*-algebras, one for Toeplitz algebras and one for Cuntz-Krieger algebras. Additionally, the Cayley graphs of finitely generated groups are used to define a class of k-graphs, and a functor is then given from a category of finitely generated groups to the category of C*-algebras. Finally, functoriality is investigated for product systems of C*-correspondences associated to k-graphs. Additional results concerning the structural consequences of functoriality, properties of the functors, and combinatorial aspects of k-graphs are also included throughout. / Dissertation/Thesis / Masters Thesis Mathematics 2016

abstract: Since the seminal work of Tur ́an, the forbidden subgraph problem has been among the central questions in extremal graph theory. Let ex(n;F) be the smallest number m such that any graph on n vertices with m edges contains F as a subgraph. Then the forbidden subgraph problem asks to find ex(n; F ) for various graphs F . The question can be further generalized by asking for the extreme values of other graph parameters like minimum degree, maximum degree, or connectivity. We call this type of question a Tura ́n-type problem. In this thesis, we will study Tura ́n-type problems and their variants for graphs and hypergraphs. Chapter 2 contains a Tura ́n-type problem for cycles in dense graphs. The main result in this chapter gives a tight bound for the minimum degree of a graph which guarantees existence of disjoint cycles in the case of dense graphs. This, in particular, answers in the affirmative a question of Faudree, Gould, Jacobson and Magnant in the case of dense graphs. In Chapter 3, similar problems for trees are investigated. Recently, Faudree, Gould, Jacobson and West studied the minimum degree conditions for the existence of certain spanning caterpillars. They proved certain bounds that guarantee existence of spanning caterpillars. The main result in Chapter 3 significantly improves their result and answers one of their questions by proving a tight minimum degree bound for the existence of such structures. Chapter 4 includes another Tur ́an-type problem for loose paths of length three in a 3-graph. As a corollary, an upper bound for the multi-color Ramsey number for the loose path of length three in a 3-graph is achieved. / Dissertation/Thesis / Doctoral Dissertation Mathematics 2018

abstract: In 1959, Iwasawa proved that the size of the $p$-part of the class groups of a $\mathbb{Z}_p$-extension grows as a power of $p$ with exponent ${\mu}p^m+{\lambda}\,m+\nu$ for $m$ sufficiently large. Broadly, I construct conditions to verify if a given $m$ is indeed sufficiently large. More precisely, let $CG_m^i$ (class group) be the $\epsilon_i$-eigenspace component of the $p$-Sylow subgroup of the class group of the field at the $m$-th level in a $\mathbb{Z}_p$-extension; and let $IACG^i_m$ (Iwasawa analytic class group) be ${\mathbb{Z}_p[[T]]/((1+T)^{p^m}-1,f(T,\omega^{1-i}))}$, where $f$ is the associated Iwasawa power series. It is expected that $CG_m^i$ and $IACG^i_m$ be isomorphic, providing us with a powerful connection between algebraic and analytic techniques; however, as of yet, this isomorphism is unestablished in general. I consider the existence and the properties of an exact sequence $$0\longrightarrow\ker{\longrightarrow}CG_m^i{\longrightarrow}IACG_m^i{\longrightarrow}\textrm{coker}\longrightarrow0.$$ In the case of a $\mathbb{Z}_p$-extension where the Main Conjecture is established, there exists a pseudo-isomorphism between the respective inverse limits of $CG_m^i$ and $IACG_m^i$. I consider conditions for when such a pseudo-isomorphism immediately gives the existence of the desired exact sequence, and I also consider work-around methods that preserve cardinality for otherwise. However, I primarily focus on constructing conditions to verify if a given $m$ is sufficiently large that the kernel and cokernel of the above exact sequence have become well-behaved, providing similarity of growth both in the size and in the structure of $CG_m^i$ and $IACG_m^i$; as well as conditions to determine if any such $m$ exists. The primary motivating idea is that if $IACG_m^i$ is relatively easy to work with, and if the relationship between $CG_m^i$ and $IACG_m^i$ is understood; then $CG_m^i$ becomes easier to work with. Moreover, while the motivating framework is stated concretely in terms of the cyclotomic $\mathbb{Z}_p$-extension of $p$-power roots of unity, all results are generally applicable to arbitrary $\mathbb{Z}_p$-extensions as they are developed in terms of Iwasawa-Theory-inspired, yet abstracted, algebraic results on maps between inverse limits. / Dissertation/Thesis / Ph.D. Mathematics 2013

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