ASU Electronic Theses and Dissertations
This collection includes most of the ASU Theses and Dissertations from 2011 to present. ASU Theses and Dissertations are available in downloadable PDF format; however, a small percentage of items are under embargo. Information about the dissertations/theses includes degree information, committee members, an abstract, supporting data or media.
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Description
This dissertation will cover two topics. For the first, let $K$ be a number field. A $K$-derived polynomial $f(x) \in K[x]$ is a polynomial that
factors into linear factors over $K$, as do all of its derivatives. Such a polynomial
is said to be {\it proper} if
its roots are distinct. An unresolved question in the literature is
whether or not there exists a proper $\Q$-derived polynomial of degree 4. Some examples
are known of proper $K$-derived quartics for a quadratic number field $K$, although other
than $\Q(\sqrt{3})$, these fields have quite large discriminant. (The second known field
is $\Q(\sqrt{3441})$.) I will describe a search for quadratic fields $K$
over which there exist proper $K$-derived quartics. The search finds examples for
$K=\Q(\sqrt{D})$ with $D=...,-95,-41,-19,21,31,89,...$.\\
For the second topic, by Krasner's lemma there exist a finite number of degree $n$ extensions of $\Q_p$. Jones and Roberts have developed a database recording invariants of $p$-adic extensions for low degree $n$. I will contribute data to this database by computing the Galois slope content, inertia subgroup, and Galois mean slope for a variety of wildly ramified extensions of composite degree using the idea of \emph{global splitting models}.
factors into linear factors over $K$, as do all of its derivatives. Such a polynomial
is said to be {\it proper} if
its roots are distinct. An unresolved question in the literature is
whether or not there exists a proper $\Q$-derived polynomial of degree 4. Some examples
are known of proper $K$-derived quartics for a quadratic number field $K$, although other
than $\Q(\sqrt{3})$, these fields have quite large discriminant. (The second known field
is $\Q(\sqrt{3441})$.) I will describe a search for quadratic fields $K$
over which there exist proper $K$-derived quartics. The search finds examples for
$K=\Q(\sqrt{D})$ with $D=...,-95,-41,-19,21,31,89,...$.\\
For the second topic, by Krasner's lemma there exist a finite number of degree $n$ extensions of $\Q_p$. Jones and Roberts have developed a database recording invariants of $p$-adic extensions for low degree $n$. I will contribute data to this database by computing the Galois slope content, inertia subgroup, and Galois mean slope for a variety of wildly ramified extensions of composite degree using the idea of \emph{global splitting models}.
ContributorsCarrillo, Benjamin (Author) / Jones, John (Thesis advisor) / Bremner, Andrew (Thesis advisor) / Childress, Nancy (Committee member) / Fishel, Susanna (Committee member) / Kaliszewski, Steven (Committee member) / Arizona State University (Publisher)
Created2019
Description
A leading crisis in the United States is the opioid use disorder (OUD) epidemic. Opioid overdose deaths have been increasing, with over 100,000 deaths due to overdose from April 2020 to April 2021. This dissertation presents two mathematical models to address illicit OUD (IOUD), treatment, and recovery within an epidemiological framework. In the first model, individuals remain in the recovery class unless they relapse. Due to the limited availability of specialty treatment facilities for individuals with OUD, a saturation treat- ment function was incorporated. The second model is an extension of the first, where a casual user class and its corresponding specialty treatment class were added. Using U.S. population data, the data was scaled to a population of 200,000 to find parameter estimates. While the first model used the heroin-only dataset, the second model used both the heroin and all-illicit opioids datasets. Backward bifurcation was found in the first IOUD model for realistic parameter values. Additionally, bistability was observed in the second IOUD model with the heroin-only dataset. This result implies that it would be beneficial to increase the availability of treatment. An alarming effect was discovered about the high overdose death rate: by 2038, the disease-free equilibrium would be the only stable equilibrium. This consequence is concerning because although the goal is for the epidemic to end, it would be preferable to end it through treatment rather than overdose. The IOUD model with a casual user class, its sensitivity results, and the comparison of parameters for both datasets, showed the importance of not overlooking the influence that casual users have in driving the all-illicit opioid epidemic. Casual users stay in the casual user class longer and are not going to treatment as quickly as the users of the heroin epidemic. Another result was that the users of the all-illicit opioids were going to the recovered class by means other than specialty treatment. However, the relapse rates for those individuals were much more significant than in the heroin-only epidemic. The results above from analyzing these models may inform health and policy officials, leading to more effective treatment options and prevention efforts.
ContributorsCole, Sandra (Author) / Wirkus, Stephen (Thesis advisor) / Gardner, Carl (Committee member) / Lanchier, Nicolas (Committee member) / Camacho, Erika (Committee member) / Fricks, John (Committee member) / Arizona State University (Publisher)
Created2022
DescriptionCantor sets are totally disconnected, compact, metrizable, and contain no isolated points. All Cantor sets are homeomorphic to each other, but the addition of the metric yields new properties which can be detected by their correspondence with the boundaries of infinite rooted trees.
ContributorsAmes, Robert (Author) / Spielberg, John (Thesis advisor) / Kaliszewski, Steven (Committee member) / Quigg, John (Committee member) / Arizona State University (Publisher)
Created2022
Description
Iwasawa theory is a branch of number theory that studies the behavior of certain objects associated to a $\mathbb{Z}_p$-extension. We will focus our attention to the cyclotomic $\mathbb{Z}_p$-extensions of imaginary quadratic fields for varying primes p, and will give some conditions for when the corresponding lambda-invariants are greater than 1.
ContributorsStokes, Christopher Mathewson (Author) / Childress, Nancy (Thesis advisor) / Sprung, Florian (Committee member) / Montaño, Johnathan (Committee member) / Paupert, Julian (Committee member) / Kaliszewski, Steven (Committee member) / Arizona State University (Publisher)
Created2023
Description
The main part of this work establishes existence, uniqueness and regularity properties of measure-valued solutions of a nonlinear hyperbolic conservation law with non-local velocities. Major challenges stem from in- and out-fluxes containing nonzero pure-point parts which cause discontinuities of the velocities. This part is preceded, and motivated, by an extended study which proves that an associated optimal control problem has no optimal $L^1$-solutions that are supported on short time intervals.
The hyperbolic conservation law considered here is a well-established model for a highly re-entrant semiconductor manufacturing system. Prior work established well-posedness for $L^1$-controls and states, and existence of optimal solutions for $L^2$-controls, states, and control objectives. The results on measure-valued solutions presented here reduce to the existing literature in the case of initial state and in-flux being absolutely continuous measures. The surprising well-posedness (in the face of measures containing nonzero pure-point part and discontinuous velocities) is directly related to characteristic features of the model that capture the highly re-entrant nature of the semiconductor manufacturing system.
More specifically, the optimal control problem is to minimize an $L^1$-functional that measures the mismatch between actual and desired accumulated out-flux. The focus is on the transition between equilibria with eventually zero backlog. In the case of a step up to a larger equilibrium, the in-flux not only needs to increase to match the higher desired out-flux, but also needs to increase the mass in the factory and to make up for the backlog caused by an inverse response of the system. The optimality results obtained confirm the heuristic inference that the optimal solution should be an impulsive in-flux, but this is no longer in the space of $L^1$-controls.
The need for impulsive controls motivates the change of the setting from $L^1$-controls and states to controls and states that are Borel measures. The key strategy is to temporarily abandon the Eulerian point of view and first construct Lagrangian solutions. The final section proposes a notion of weak measure-valued solutions and proves existence and uniqueness of such.
In the case of the in-flux containing nonzero pure-point part, the weak solution cannot depend continuously on the time with respect to any norm. However, using semi-norms that are related to the flat norm, a weaker form of continuity of solutions with respect to time is proven. It is conjectured that also a similar weak continuous dependence on initial data holds with respect to a variant of the flat norm.
The hyperbolic conservation law considered here is a well-established model for a highly re-entrant semiconductor manufacturing system. Prior work established well-posedness for $L^1$-controls and states, and existence of optimal solutions for $L^2$-controls, states, and control objectives. The results on measure-valued solutions presented here reduce to the existing literature in the case of initial state and in-flux being absolutely continuous measures. The surprising well-posedness (in the face of measures containing nonzero pure-point part and discontinuous velocities) is directly related to characteristic features of the model that capture the highly re-entrant nature of the semiconductor manufacturing system.
More specifically, the optimal control problem is to minimize an $L^1$-functional that measures the mismatch between actual and desired accumulated out-flux. The focus is on the transition between equilibria with eventually zero backlog. In the case of a step up to a larger equilibrium, the in-flux not only needs to increase to match the higher desired out-flux, but also needs to increase the mass in the factory and to make up for the backlog caused by an inverse response of the system. The optimality results obtained confirm the heuristic inference that the optimal solution should be an impulsive in-flux, but this is no longer in the space of $L^1$-controls.
The need for impulsive controls motivates the change of the setting from $L^1$-controls and states to controls and states that are Borel measures. The key strategy is to temporarily abandon the Eulerian point of view and first construct Lagrangian solutions. The final section proposes a notion of weak measure-valued solutions and proves existence and uniqueness of such.
In the case of the in-flux containing nonzero pure-point part, the weak solution cannot depend continuously on the time with respect to any norm. However, using semi-norms that are related to the flat norm, a weaker form of continuity of solutions with respect to time is proven. It is conjectured that also a similar weak continuous dependence on initial data holds with respect to a variant of the flat norm.
ContributorsGong, Xiaoqian, Ph.D (Author) / Kawski, Matthias (Thesis advisor) / Kaliszewski, Steven (Committee member) / Motsch, Sebastien (Committee member) / Smith, Hal (Committee member) / Thieme, Horst (Committee member) / Arizona State University (Publisher)
Created2019
Description
This dissertation consists of three papers about opinion dynamics. The first paper is in collaboration with Prof. Lanchier while the other two papers are individual works.
Two models are introduced and studied analytically: the Deffuant model and the Hegselmann-Krause~(HK) model.
The main difference between the two models is that the Deffuant dynamics consists of pairwise interactions whereas the HK dynamics consists of group interactions.
Translated into graph, each vertex stands for an agent in both models.
In the Deffuant model, two graphs are combined: the social graph and the opinion graph.
The social graph is assumed to be a general finite connected graph where each edge is interpreted as a social link, such as a friendship relationship, between two agents.
At each time step, two social neighbors are randomly selected and interact if and only if their opinion distance does not exceed some confidence threshold, which results in the neighbors' opinions getting closer to each other.
The main result about the Deffuant model is the derivation of a positive lower bound for the probability of consensus that is independent of the size and topology of the social graph but depends on the confidence threshold, the choice of the opinion space and the initial distribution. For the HK model, agent~$i$ updates its opinion~$x_i$ by taking the average opinion of its neighbors, defined as the set of agents with opinion at most~$\epsilon$ apart from~$x_i$. Here,~$\epsilon > 0$ is a confidence threshold.
There are two types of HK models: the synchronous and the asynchronous HK models.
In the former, all the agents update their opinion simultaneously at each time step, whereas in the latter, only one agent is selected uniformly at random to update its opinion at each time step.
The mixed model is a variant of the HK model in which each agent can choose its degree of stubbornness and mix its opinion with the average opinion of its neighbors.
The main results of this dissertation about HK models show conditions under which the asymptotic stability holds or a consensus can be achieved, and give a positive lower bound for the probability of consensus and, in the one-dimensional case, an upper bound for the probability of consensus.
I demonstrate the bounds for the probability of consensus on a unit cube and a unit interval.
ContributorsLi, Hsin-Lun (Author) / Lanchier, Nicolas (Thesis advisor) / Camacho, Erika (Committee member) / Czygrinow, Andrzej (Committee member) / Fishel, Susanna (Committee member) / Motsch, Sebastien (Committee member) / Arizona State University (Publisher)
Created2021
Description
This thesis explores several questions concerning the preservation of geometric structure under the Ricci flow, an evolution equation for Riemannian metrics. Within the class of complete solutions with bounded curvature, short-time existence and uniqueness of solutions guarantee that symmetries and many other geometric features are preserved along the flow. However, much less is known about the analytic and geometric properties of solutions of potentially unbounded curvature. The first part of this thesis contains a proof that the full holonomy group is preserved, up to isomorphism, forward and backward in time. The argument reduces the problem to the preservation of reduced holonomy via an analysis of the equation satisfied by parallel translation around a loop with respect to the evolving metric. The subsequent chapter examines solutions satisfying a certain instantaneous, but nonuniform, curvature bound, and shows that when such solutions split as a product initially, they will continue to split for all time. This problem is encoded as one of uniqueness for an auxiliary system constructed from a family of time-dependent, orthogonal distributions of the tangent bundle. The final section presents some details of an ongoing project concerning the uniqueness of asymptotically product gradient shrinking Ricci solitons, including the construction of a certain system of mixed differential inequalities which measures the extent to which such a soliton fails to split.
ContributorsCook, Mary (Author) / Kotschwar, Brett (Thesis advisor) / Paupert, Julien (Committee member) / Kawski, Matthias (Committee member) / Kaliszewski, Steven (Committee member) / Fishel, Susanna (Committee member) / Arizona State University (Publisher)
Created2021