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Nonlinear dispersive equations model nonlinear waves in a wide range of physical and mathematics contexts. They reinforce or dissipate effects of linear dispersion and nonlinear interactions, and thus, may be of a focusing or defocusing nature. The nonlinear Schrödinger equation or NLS is an example of such equations. It appears

Nonlinear dispersive equations model nonlinear waves in a wide range of physical and mathematics contexts. They reinforce or dissipate effects of linear dispersion and nonlinear interactions, and thus, may be of a focusing or defocusing nature. The nonlinear Schrödinger equation or NLS is an example of such equations. It appears as a model in hydrodynamics, nonlinear optics, quantum condensates, heat pulses in solids and various other nonlinear instability phenomena. In mathematics, one of the interests is to look at the wave interaction: waves propagation with different speeds and/or different directions produces either small perturbations comparable with linear behavior, or creates solitary waves, or even leads to singular solutions. This dissertation studies the global behavior of finite energy solutions to the $d$-dimensional focusing NLS equation, $i partial _t u+Delta u+ |u|^{p-1}u=0, $ with initial data $u_0in H^1,; x in Rn$; the nonlinearity power $p$ and the dimension $d$ are chosen so that the scaling index $s=frac{d}{2}-frac{2}{p-1}$ is between 0 and 1, thus, the NLS is mass-supercritical $(s>0)$ and energy-subcritical $(s<1).$ For solutions with $ME[u_0]<1$ ($ME[u_0]$ stands for an invariant and conserved quantity in terms of the mass and energy of $u_0$), a sharp threshold for scattering and blowup is given. Namely, if the renormalized gradient $g_u$ of a solution $u$ to NLS is initially less than 1, i.e., $g_u(0)<1,$ then the solution exists globally in time and scatters in $H^1$ (approaches some linear Schr"odinger evolution as $ttopminfty$); if the renormalized gradient $g_u(0)>1,$ then the solution exhibits a blowup behavior, that is, either a finite time blowup occurs, or there is a divergence of $H^1$ norm in infinite time. This work generalizes the results for the 3d cubic NLS obtained in a series of papers by Holmer-Roudenko and Duyckaerts-Holmer-Roudenko with the key ingredients, the concentration compactness and localized variance, developed in the context of the energy-critical NLS and Nonlinear Wave equations by Kenig and Merle. One of the difficulties is fractional powers of nonlinearities which are overcome by considering Besov-Strichartz estimates and various fractional differentiation rules.
ContributorsGuevara, Cristi Darley (Author) / Roudenko, Svetlana (Thesis advisor) / Castillo_Chavez, Carlos (Committee member) / Jones, Donald (Committee member) / Mahalov, Alex (Committee member) / Suslov, Sergei (Committee member) / Arizona State University (Publisher)
Created2011
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Description
The theory of geometric quantum mechanics describes a quantum system as a Hamiltonian dynamical system, with a projective Hilbert space regarded as the phase space. This thesis extends the theory by including some aspects of the symplectic topology of the quantum phase space. It is shown that the quantum mechanical

The theory of geometric quantum mechanics describes a quantum system as a Hamiltonian dynamical system, with a projective Hilbert space regarded as the phase space. This thesis extends the theory by including some aspects of the symplectic topology of the quantum phase space. It is shown that the quantum mechanical uncertainty principle is a special case of an inequality from J-holomorphic map theory, that is, J-holomorphic curves minimize the difference between the quantum covariance matrix determinant and a symplectic area. An immediate consequence is that a minimal determinant is a topological invariant, within a fixed homology class of the curve. Various choices of quantum operators are studied with reference to the implications of the J-holomorphic condition. The mean curvature vector field and Maslov class are calculated for a lagrangian torus of an integrable quantum system. The mean curvature one-form is simply related to the canonical connection which determines the geometric phases and polarization linear response. Adiabatic deformations of a quantum system are analyzed in terms of vector bundle classifying maps and related to the mean curvature flow of quantum states. The dielectric response function for a periodic solid is calculated to be the curvature of a connection on a vector bundle.
ContributorsSanborn, Barbara (Author) / Suslov, Sergei K (Thesis advisor) / Suslov, Sergei (Committee member) / Spielberg, John (Committee member) / Quigg, John (Committee member) / Menéndez, Jose (Committee member) / Jones, Donald (Committee member) / Arizona State University (Publisher)
Created2011
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Description
This dissertation is intended to tie together a body of work which utilizes a variety of methods to study applied mathematical models involving heterogeneity often omitted with classical modeling techniques. I posit three cogent classifications of heterogeneity: physiological, behavioral, and local (specifically connectivity in this work). I consider physiological heterogeneity

This dissertation is intended to tie together a body of work which utilizes a variety of methods to study applied mathematical models involving heterogeneity often omitted with classical modeling techniques. I posit three cogent classifications of heterogeneity: physiological, behavioral, and local (specifically connectivity in this work). I consider physiological heterogeneity using the method of transport equations to study heterogeneous susceptibility to diseases in open populations (those with births and deaths). I then present three separate models of behavioral heterogeneity. An SIS/SAS model of gonorrhea transmission in a population of highly active men-who-have-sex-with-men (MSM) is presented to study the impact of safe behavior (prevention and self-awareness) on the prevalence of this endemic disease. Behavior is modeled in this examples via static parameters describing consistent condom use and frequency of STD testing. In an example of behavioral heterogeneity, in the absence of underlying dynamics, I present a generalization to ``test theory without an answer key" (also known as cultural consensus modeling or CCM). CCM is commonly used to study the distribution of cultural knowledge within a population. The generalized framework presented allows for selecting the best model among various extensions of CCM: multiple subcultures, estimating the degree to which individuals guess yes, and making competence homogenous in the population. This permits model selection based on the principle of information criteria. The third behaviorally heterogeneous model studies adaptive behavioral response based on epidemiological-economic theory within an $SIR$ epidemic setting. Theorems used to analyze the stability of such models with a generalized, non-linear incidence structure are adapted and applied to the case of standard incidence and adaptive incidence. As an example of study in spatial heterogeneity I provide an explicit solution to a generalization of the continuous time approximation of the Albert-Barabasi scale-free network algorithm. The solution is found by recursively solving the differential equations via integrating factors, identifying a pattern for the coefficients and then proving this observed pattern is consistent using induction. An application to disease dynamics on such evolving structures is then studied.
ContributorsMorin, Benjamin (Author) / Castillo-Chavez, Carlos (Thesis advisor) / Hiebeler, David (Thesis advisor) / Hruschka, Daniel (Committee member) / Suslov, Sergei (Committee member) / Arizona State University (Publisher)
Created2012
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Description
The current trend of interconnected devices, or the internet of things (IOT) has led to the popularization of single board computers (SBC). This is primarily due to their form-factor and low price. This has led to unique networks of devices that can have unstable network connections and minimal processing power.

The current trend of interconnected devices, or the internet of things (IOT) has led to the popularization of single board computers (SBC). This is primarily due to their form-factor and low price. This has led to unique networks of devices that can have unstable network connections and minimal processing power. Many parallel program- ming libraries are intended for use in high performance computing (HPC) clusters. Unlike the IOT environment described, HPC clusters will in general look to obtain very consistent network speeds and topologies. There are a significant number of software choices that make up what is referred to as the HPC stack or parallel processing stack. My thesis focused on building an HPC stack that would run on the SCB computer name the Raspberry Pi. The intention in making this Raspberry Pi cluster is to research performance of MPI implementations in an IOT environment, which had an impact on the design choices of the cluster. This thesis is a compilation of my research efforts in creating this cluster as well as an evaluation of the software that was chosen to create the parallel processing stack.
ContributorsO'Meara, Braedon Richard (Author) / Meuth, Ryan (Thesis director) / Dasgupta, Partha (Committee member) / Computer Science and Engineering Program (Contributor) / Barrett, The Honors College (Contributor)
Created2018-05
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Description
This thesis discusses three recent optimization problems that seek to reduce disease spread on arbitrary graphs by deleting edges, and it discusses three approximation algorithms developed for these problems. Important definitions are presented including the Linear Threshold and Triggering Set models and the set function properties of submodularity and monotonicity.

This thesis discusses three recent optimization problems that seek to reduce disease spread on arbitrary graphs by deleting edges, and it discusses three approximation algorithms developed for these problems. Important definitions are presented including the Linear Threshold and Triggering Set models and the set function properties of submodularity and monotonicity. Also, important results regarding the Linear Threshold model and computation of the influence function are presented along with proof sketches. The three main problems are formally presented, and NP-hardness results along with proof sketches are presented where applicable. The first problem seeks to reduce spread of infection over the Linear Threshold process by making use of an efficient tree data structure. The second problem seeks to reduce the spread of infection over the Linear Threshold process while preserving the PageRank distribution of the input graph. The third problem seeks to minimize the spectral radius of the input graph. The algorithms designed for these problems are described in writing and with pseudocode, and their approximation bounds are stated along with time complexities. Discussion of these algorithms considers how these algorithms could see real-world use. Challenges and the ways in which these algorithms do or do not overcome them are noted. Two related works, one which presents an edge-deletion disease spread reduction problem over a deterministic threshold process and the other which considers a graph modification problem aimed at minimizing worst-case disease spread, are compared with the three main works to provide interesting perspectives. Furthermore, a new problem is proposed that could avoid some issues faced by the three main problems described, and directions for future work are suggested.
ContributorsStanton, Andrew Warren (Author) / Richa, Andrea (Thesis director) / Czygrinow, Andrzej (Committee member) / Computer Science and Engineering Program (Contributor) / Barrett, The Honors College (Contributor)
Created2018-05
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Description
In the last few years, billion-dollar companies like Yahoo and Equifax have had data breaches causing millions of people’s personal information to be leaked online. Other billion-dollar companies like Google and Facebook have gotten in trouble for abusing people’s personal information for financial gain as well. In this new age

In the last few years, billion-dollar companies like Yahoo and Equifax have had data breaches causing millions of people’s personal information to be leaked online. Other billion-dollar companies like Google and Facebook have gotten in trouble for abusing people’s personal information for financial gain as well. In this new age of technology where everything is being digitalized and stored online, people all over the world are concerned about what is happening to their personal information and how they can trust it is being kept safe. This paper describes, first, the importance of protecting user data, second, one easy tool that companies and developers can use to help ensure that their user’s information (credit card information specifically) is kept safe, how to implement that tool, and finally, future work and research that needs to be done. The solution I propose is a software tool that will keep credit card data secured. It is only a small step towards achieving a completely secure data anonymized system, but when implemented correctly, it can reduce the risk of credit card data from being exposed to the public. The software tool is a script that can scan every viable file in any given system, server, or other file-structured Linux system and detect if there any visible credit card numbers that should be hidden.
ContributorsPappas, Alexander (Author) / Zhao, Ming (Thesis director) / Kuznetsov, Eugene (Committee member) / Computer Science and Engineering Program (Contributor) / Barrett, The Honors College (Contributor)
Created2020-05
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Description
Political polarization is the coalescence of political parties -- and the individuals of which parties are composed -- around opposing ends of the ideological spectrum. Political parties in the United States have always been divided, however, in recent years this division has only intensified. Recently, polarization has also wound its

Political polarization is the coalescence of political parties -- and the individuals of which parties are composed -- around opposing ends of the ideological spectrum. Political parties in the United States have always been divided, however, in recent years this division has only intensified. Recently, polarization has also wound its way to the Supreme Court and the nomination processes of justices to the Court. This paper examines how prevalent polarization in the Supreme Court nomination process has become by looking specifically at the failed nomination of Judge Merrick Garland and the confirmations of now-Justices Neil Gorsuch and Brett Kavanaugh. This is accomplished by comparing the ideologies and qualifications of the three most recent nominees to those of previous nominees, as well as analysing the ideological composition of the Senate at the times of the individual nominations.
ContributorsJoss, Jacob (Author) / Hoekstra, Valerie (Thesis director) / Critchlow, Donald (Committee member) / Computer Science and Engineering Program (Contributor) / School of Politics and Global Studies (Contributor) / Barrett, The Honors College (Contributor)
Created2020-05
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Description
The original version of Helix, the one I pitched when first deciding to make a video game
for my thesis, is an action-platformer, with the intent of metroidvania-style progression
and an interconnected world map.

The current version of Helix is a turn based role-playing game, with the intent of roguelike
gameplay and a dark

The original version of Helix, the one I pitched when first deciding to make a video game
for my thesis, is an action-platformer, with the intent of metroidvania-style progression
and an interconnected world map.

The current version of Helix is a turn based role-playing game, with the intent of roguelike
gameplay and a dark fantasy theme. We will first be exploring the challenges that came
with programming my own game - not quite from scratch, but also without a prebuilt
engine - then transition into game design and how Helix has evolved from its original form
to what we see today.
ContributorsDiscipulo, Isaiah K (Author) / Meuth, Ryan (Thesis director) / Kobayashi, Yoshihiro (Committee member) / School of Mathematical and Statistical Sciences (Contributor) / Computer Science and Engineering Program (Contributor) / Barrett, The Honors College (Contributor)
Created2020-05
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Description
RecyclePlus is an iOS mobile application that allows users to be knowledgeable in the realms of sustainability. It gives encourages users to be environmental responsible by providing them access to recycling information. In particular, it allows users to search up certain materials and learn about its recyclability and how to

RecyclePlus is an iOS mobile application that allows users to be knowledgeable in the realms of sustainability. It gives encourages users to be environmental responsible by providing them access to recycling information. In particular, it allows users to search up certain materials and learn about its recyclability and how to properly dispose of the material. Some searches will show locations of facilities near users that collect certain materials and dispose of the materials properly. This is a full stack software project that explores open source software and APIs, UI/UX design, and iOS development.
ContributorsTran, Nikki (Author) / Ganesh, Tirupalavanam (Thesis director) / Meuth, Ryan (Committee member) / Watts College of Public Service & Community Solut (Contributor) / Department of Information Systems (Contributor) / Computer Science and Engineering Program (Contributor) / Barrett, The Honors College (Contributor)
Created2020-05
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Description
At present, the vast majority of human subjects with neurological disease are still diagnosed through in-person assessments and qualitative analysis of patient data. In this paper, we propose to use Topological Data Analysis (TDA) together with machine learning tools to automate the process of Parkinson’s disease classification and severity assessment.

At present, the vast majority of human subjects with neurological disease are still diagnosed through in-person assessments and qualitative analysis of patient data. In this paper, we propose to use Topological Data Analysis (TDA) together with machine learning tools to automate the process of Parkinson’s disease classification and severity assessment. An automated, stable, and accurate method to evaluate Parkinson’s would be significant in streamlining diagnoses of patients and providing families more time for corrective measures. We propose a methodology which incorporates TDA into analyzing Parkinson’s disease postural shifts data through the representation of persistence images. Studying the topology of a system has proven to be invariant to small changes in data and has been shown to perform well in discrimination tasks. The contributions of the paper are twofold. We propose a method to 1) classify healthy patients from those afflicted by disease and 2) diagnose the severity of disease. We explore the use of the proposed method in an application involving a Parkinson’s disease dataset comprised of healthy-elderly, healthy-young and Parkinson’s disease patients.
ContributorsRahman, Farhan Nadir (Co-author) / Nawar, Afra (Co-author) / Turaga, Pavan (Thesis director) / Krishnamurthi, Narayanan (Committee member) / Electrical Engineering Program (Contributor) / Computer Science and Engineering Program (Contributor) / Barrett, The Honors College (Contributor)
Created2020-05