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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)

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.
ContributorsElledge, Shawn Michael (Author) / Childress, Nancy (Thesis advisor) / Bremner, Andrew (Committee member) / Fishel, Susanna (Committee member) / Jones, John (Committee member) / Paupert, Julien (Committee member) / Arizona State University (Publisher)
Created2013
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DescriptionReprising the work of Kolpakov and Martelli, a manifold is constructed by face pairings of a four dimensional polytope, the 24-cell. The resulting geometry is a single cusped hyperbolic 4-manifold of finite volume. A short discussion of its geometry and underlying topology is included.
ContributorsAbram, Christopher (Author) / Paupert, Julien (Thesis advisor) / Kawski, Mattias (Committee member) / Kotschwar, Brett (Committee member) / Arizona State University (Publisher)
Created2014
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Description
The uncrossing partially ordered set $P_n$ is defined on the set of matchings on $2n$ points on a circle represented with wires. The order relation is $\tau'\leq \tau$ in $P_n$ if and only if $\tau'$ is obtained by resolving a crossing of $\tau$. %This partial order has been studied by

The uncrossing partially ordered set $P_n$ is defined on the set of matchings on $2n$ points on a circle represented with wires. The order relation is $\tau'\leq \tau$ in $P_n$ if and only if $\tau'$ is obtained by resolving a crossing of $\tau$. %This partial order has been studied by Alman-Lian-Tran, Huang-Wen-Xie, Kenyon, and Lam. %The posets $P_n$ emerged from studies of circular planar electrical networks. Circular planar electrical networks are finite weighted undirected graphs embedded into a disk, with boundary vertices and interior vertices. By Curtis-Ingerman-Morrow and de Verdi\`ere-Gitler-Vertigan, the electrical networks can be encoded with response matrices. By Lam the space of response matrices for electrical networks has a cell structure, and this cell structure can be described by the uncrossing partial orders. %Lam proves that the posets can be identified with dual Bruhat order on affine permutations of type $(n,2n)$. Using this identification, Lam proves the poset $\hat{P}_n$, the uncrossing poset $P_n$ with a unique minimum element $\hat{0}$ adjoined, is Eulerian. This thesis consists of two sets of results: (1) flag enumeration in intervals in the uncrossing poset $P_n$ and (2) cyclic sieving phenomenon on the set $P_n$.

I identify elements in $P_n$ with affine permutations of type $(0,2n)$. %This identification enables us to explicitly describe the elements in $P_n$ with the elements in $\mathcal{MP}_n$.

Using this identification, I adapt a technique in Reading for finding recursions for the cd-indices of intervals in Bruhat order of Coxeter groups to the uncrossing poset $P_n$. As a result, I produce recursions for the cd-indices of intervals in the uncrossing poset $P_n$. I also obtain a recursion for the ab-indices of intervals in the poset $\hat{P}_n$, the poset $P_n$ with a unique minimum $\hat0$ adjoined. %We define an induced subposet $\mathcal{MP}_n$ of the affine permutations under Bruhat order.

Reiner-Stanton-White defined the cyclic sieving phenomenon (CSP) associated to a finite cyclic group action on a finite set and a polynomial. Sagan observed the CSP on the set of non-crossing matchings with the $q$-Catalan polynomial. Bowling-Liang presented similar results on the set of $k$-crossing matchings for $1\leq k \leq 3$. In this dissertation, I focus on the set of all matchings on $[2n]:=\{1,2,\dots,2n\}$. I find the number of matchings fixed by $\frac{2\pi}{d}$ rotations for $d|2n$. I then find the polynomial $X_n(q)$ such that the set of matchings together with $X_n(q)$ and the cyclic group of order $2n$ exhibits the CSP.
ContributorsKim, Younghwan (Author) / Fishel, Susanna (Thesis advisor) / Bremner, Andrew (Committee member) / Czygrinow, Andrzej (Committee member) / Kierstead, Henry (Committee member) / Paupert, Julien (Committee member) / Arizona State University (Publisher)
Created2018
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Description
In the 1980's, Gromov and Piatetski-Shapiro introduced a technique called "hybridization'' which allowed them to produce non-arithmetic hyperbolic lattices from two non-commensurable arithmetic lattices. It has been asked whether an analogous hybridization technique exists for complex hyperbolic lattices, because certain geometric obstructions make it unclear how to adapt this technique.

In the 1980's, Gromov and Piatetski-Shapiro introduced a technique called "hybridization'' which allowed them to produce non-arithmetic hyperbolic lattices from two non-commensurable arithmetic lattices. It has been asked whether an analogous hybridization technique exists for complex hyperbolic lattices, because certain geometric obstructions make it unclear how to adapt this technique. This thesis explores one possible construction (originally due to Hunt) in depth and uses it to produce arithmetic lattices, non-arithmetic lattices, and thin subgroups in SU(2,1).
ContributorsWells, Joseph (Author) / Paupert, Julien (Thesis advisor) / Kotschwar, Brett (Committee member) / Childress, Nancy (Committee member) / Fishel, Susanna (Committee member) / Kawski, Matthias (Committee member) / Arizona State University (Publisher)
Created2019
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Description
A semi-implicit, fourth-order time-filtered leapfrog numerical scheme is investigated for accuracy and stability, and applied to several test cases, including one-dimensional advection and diffusion, the anelastic equations to simulate the Kelvin-Helmholtz instability, and the global shallow water spectral model to simulate the nonlinear evolution of twin tropical cyclones. The leapfrog

A semi-implicit, fourth-order time-filtered leapfrog numerical scheme is investigated for accuracy and stability, and applied to several test cases, including one-dimensional advection and diffusion, the anelastic equations to simulate the Kelvin-Helmholtz instability, and the global shallow water spectral model to simulate the nonlinear evolution of twin tropical cyclones. The leapfrog scheme leads to computational modes in the solutions to highly nonlinear systems, and time-filters are often used to damp these modes. The proposed filter damps the computational modes without appreciably degrading the physical mode. Its performance in these metrics is superior to the second-order time-filtered leapfrog scheme developed by Robert and Asselin.
Created2016-05
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Description
Honey bees (Apis mellifera) are responsible for pollinating nearly 80\% of all pollinated plants, meaning humans depend on honey bees to pollinate many staple crops. The success or failure of a colony is vital to global food production. There are various complex factors that can contribute to a colony's failure,

Honey bees (Apis mellifera) are responsible for pollinating nearly 80\% of all pollinated plants, meaning humans depend on honey bees to pollinate many staple crops. The success or failure of a colony is vital to global food production. There are various complex factors that can contribute to a colony's failure, including pesticides. Neonicotoids are a popular pesticide that have been used in recent times. In this study we concern ourselves with pesticides and its impact on honey bee colonies. Previous investigations that we draw significant inspiration from include Khoury et Al's \emph{A Quantitative Model of Honey Bee Colony Population Dynamics}, Henry et Al's \emph{A Common Pesticide Decreases Foraging Success and Survival in Honey Bees}, and Brown's \emph{ Mathematical Models of Honey Bee Populations: Rapid Population Decline}. In this project we extend a mathematical model to investigate the impact of pesticides on a honey bee colony, with birth rates and death rates being dependent on pesticides, and we see how these death rates influence the growth of a colony. Our studies have found an equilibrium point that depends on pesticides. Trace amounts of pesticide are detrimental as they not only affect death rates, but birth rates as well.
ContributorsSalinas, Armando (Author) / Vaz, Paul (Thesis director) / Jones, Donald (Committee member) / School of Mathematical and Statistical Sciences (Contributor) / School of International Letters and Cultures (Contributor) / Barrett, The Honors College (Contributor)
Created2016-05
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Description
A Guide to Financial Mathematics is a comprehensive and easy-to-use study guide for students studying for the one of the first actuarial exams, Exam FM. While there are many resources available to students to study for these exams, this study is free to the students and offers an approach to

A Guide to Financial Mathematics is a comprehensive and easy-to-use study guide for students studying for the one of the first actuarial exams, Exam FM. While there are many resources available to students to study for these exams, this study is free to the students and offers an approach to the material similar to that of which is presented in class at ASU. The guide is available to students and professors in the new Actuarial Science degree program offered by ASU. There are twelve chapters, including financial calculator tips, detailed notes, examples, and practice exercises. Included at the end of the guide is a list of referenced material.
ContributorsDougher, Caroline Marie (Author) / Milovanovic, Jelena (Thesis director) / Boggess, May (Committee member) / Barrett, The Honors College (Contributor) / Department of Information Systems (Contributor) / School of Mathematical and Statistical Sciences (Contributor)
Created2015-05
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Description
Covering subsequences with sets of permutations arises in many applications, including event-sequence testing. Given a set of subsequences to cover, one is often interested in knowing the fewest number of permutations required to cover each subsequence, and in finding an explicit construction of such a set of permutations that has

Covering subsequences with sets of permutations arises in many applications, including event-sequence testing. Given a set of subsequences to cover, one is often interested in knowing the fewest number of permutations required to cover each subsequence, and in finding an explicit construction of such a set of permutations that has size close to or equal to the minimum possible. The construction of such permutation coverings has proven to be computationally difficult. While many examples for permutations of small length have been found, and strong asymptotic behavior is known, there are few explicit constructions for permutations of intermediate lengths. Most of these are generated from scratch using greedy algorithms. We explore a different approach here. Starting with a set of permutations with the desired coverage properties, we compute local changes to individual permutations that retain the total coverage of the set. By choosing these local changes so as to make one permutation less "essential" in maintaining the coverage of the set, our method attempts to make a permutation completely non-essential, so it can be removed without sacrificing total coverage. We develop a post-optimization method to do this and present results on sequence covering arrays and other types of permutation covering problems demonstrating that it is surprisingly effective.
ContributorsMurray, Patrick Charles (Author) / Colbourn, Charles (Thesis director) / Czygrinow, Andrzej (Committee member) / Barrett, The Honors College (Contributor) / School of Mathematical and Statistical Sciences (Contributor) / Department of Physics (Contributor)
Created2014-12
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Description
Deconvolution of noisy data is an ill-posed problem, and requires some form of regularization to stabilize its solution. Tikhonov regularization is the most common method used, but it depends on the choice of a regularization parameter λ which must generally be estimated using one of several common methods. These methods

Deconvolution of noisy data is an ill-posed problem, and requires some form of regularization to stabilize its solution. Tikhonov regularization is the most common method used, but it depends on the choice of a regularization parameter λ which must generally be estimated using one of several common methods. These methods can be computationally intensive, so I consider their behavior when only a portion of the sampled data is used. I show that the results of these methods converge as the sampling resolution increases, and use this to suggest a method of downsampling to estimate λ. I then present numerical results showing that this method can be feasible, and propose future avenues of inquiry.
ContributorsHansen, Jakob Kristian (Author) / Renaut, Rosemary (Thesis director) / Cochran, Douglas (Committee member) / Barrett, The Honors College (Contributor) / School of Music (Contributor) / Economics Program in CLAS (Contributor) / School of Mathematical and Statistical Sciences (Contributor)
Created2015-05
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Description
This paper focuses on the Szemerédi regularity lemma, a result in the field of extremal graph theory. The lemma says that every graph can be partitioned into bounded equal parts such that most edges of the graph span these partitions, and these edges are distributed in a fairly uniform way.

This paper focuses on the Szemerédi regularity lemma, a result in the field of extremal graph theory. The lemma says that every graph can be partitioned into bounded equal parts such that most edges of the graph span these partitions, and these edges are distributed in a fairly uniform way. Definitions and notation will be established, leading to explorations of three proofs of the regularity lemma. These are a version of the original proof, a Pythagoras proof utilizing elemental geometry, and a proof utilizing concepts of spectral graph theory. This paper is intended to supplement the proofs with background information about the concepts utilized. Furthermore, it is the hope that this paper will serve as another resource for students and others to begin study of the regularity lemma.
ContributorsByrne, Michael John (Author) / Czygrinow, Andrzej (Thesis director) / Kierstead, Hal (Committee member) / Barrett, The Honors College (Contributor) / School of Mathematical and Statistical Sciences (Contributor) / Department of Chemistry and Biochemistry (Contributor)
Created2015-05