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Description
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
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 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
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 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
Description
Clean water for drinking, food preparation, and bathing is essential for astronaut health and safety during long duration habitation of the International Space Station (ISS), including future missions to Mars. Despite stringent water treatment and recycling efforts on the ISS, it is impossible to completely prevent microbial contamination of onboard water supplies. In this work, we used a spaceflight analogue culture system to better understand how the microgravity environment can influence the pathogenesis-related characteristics of Burkholderia cepacia complex (Bcc), an opportunistic pathogen previously recovered from the ISS water system. The results of the present study suggest that there may be important differences in how this pathogen can respond and adapt to spaceflight and other low fluid shear environments encountered during their natural life cycles. Future studies are aimed at understanding the underlying mechanisms responsible for these phenotypes.
ContributorsKang, Bianca Younseon (Author) / Nickerson, Cheryl (Thesis director) / Barrila, Jennifer (Committee member) / Ott, Mark (Committee member) / School of Life Sciences (Contributor) / Barrett, The Honors College (Contributor)
Created2016-05
Description
In the traditional setting of quantum mechanics, the Hamiltonian operator does not depend on time. While some Schrödinger equations with time-dependent Hamiltonians have been solved, explicitly solvable cases are typically scarce. This thesis is a collection of papers in which this first author along with Suslov, Suazo, and Lopez, has worked on solving a series of Schrödinger equations with a time-dependent quadratic Hamiltonian that has applications in problems of quantum electrodynamics, lasers, quantum devices such as quantum dots, and external varying fields. In particular the author discusses a new completely integrable case of the time-dependent Schrödinger equation in R^n with variable coefficients for a modified oscillator, which is dual with respect to the time inversion to a model of the quantum oscillator considered by Meiler, Cordero-Soto, and Suslov. A second pair of dual Hamiltonians is found in the momentum representation. Our examples show that in mathematical physics and quantum mechanics a change in the direction of time may require a total change of the system dynamics in order to return the system back to its original quantum state. The author also considers several models of the damped oscillators in nonrelativistic quantum mechanics in a framework of a general approach to the dynamics of the time-dependent Schrödinger equation with variable quadratic Hamiltonians. The Green functions are explicitly found in terms of elementary functions and the corresponding gauge transformations are discussed. The factorization technique is applied to the case of a shifted harmonic oscillator. The time-evolution of the expectation values of the energy related operators is determined for two models of the quantum damped oscillators under consideration. The classical equations of motion for the damped oscillations are derived for the corresponding expectation values of the position operator. Finally, the author constructs integrals of motion for several models of the quantum damped oscillators in a framework of a general approach to the time-dependent Schrödinger equation with variable quadratic Hamiltonians. An extension of the Lewis-Riesenfeld dynamical invariant is given. The time-evolution of the expectation values of the energy related positive operators is determined for the oscillators under consideration. A proof of uniqueness of the corresponding Cauchy initial value problem is discussed as an application.
ContributorsCordero-Soto, Ricardo J (Author) / Suslov, Sergei (Thesis advisor) / Castillo-Chavez, Carlos (Thesis advisor) / Engman, Martin (Committee member) / Herrera-Valdez, Marco (Committee member) / Arizona State University (Publisher)
Created2011
Description
In translating the Russian-language math paper “On Applications of Graph Theory for the Description and Analysis of Information Flow Diagrams in Control Systems” (V.L. Epstein, 1964) we endeavor not only to analyze the specific applications of this paper’s analysis of adjacency matrices to current topics of interest in graph theory, but also to demonstrate in general the value of translating foreign language papers.
ContributorsSaarinen, Emma Catherine (Author) / Inozemtseva, Julia (Thesis director) / Suslov, Sergei (Committee member) / School of Mathematical and Statistical Sciences (Contributor) / Barrett, The Honors College (Contributor)
Created2020-12
Description
Mathematical epidemiology, one of the oldest and richest areas in mathematical biology, has significantly enhanced our understanding of how pathogens emerge, evolve, and spread. Classical epidemiological models, the standard for predicting and managing the spread of infectious disease, assume that contacts between susceptible and infectious individuals depend on their relative frequency in the population. The behavioral factors that underpin contact rates are not generally addressed. There is, however, an emerging a class of models that addresses the feedbacks between infectious disease dynamics and the behavioral decisions driving host contact. Referred to as “economic epidemiology” or “epidemiological economics,” the approach explores the determinants of decisions about the number and type of contacts made by individuals, using insights and methods from economics. We show how the approach has the potential both to improve predictions of the course of infectious disease, and to support development of novel approaches to infectious disease management.
ContributorsPerrings, Charles (Author) / Castillo-Chavez, Carlos (Author) / Chowell-Puente, Gerardo (Author) / Daszak, Peter (Author) / Fenichel, Eli P. (Author) / Finnoff, David (Author) / Horan, Richard D. (Author) / Kilpatrick, A. Marm (Author) / Kinzig, Ann (Author) / Kuminoff, Nicolai (Author) / Levin, Simon (Author) / Morin, Benjamin (Author) / Smith, Katherine F. (Author) / Springborn, Michael (Author) / Simon M. Levin Mathematical, Computational and Modeling Sciences Center (Contributor) / College of Liberal Arts and Sciences (Contributor) / School of Life Sciences (Contributor) / School of Human Evolution and Social Change (Contributor) / W.P. Carey School of Business (Contributor) / Economics (Contributor) / Julie Ann Wrigley Global Institute of Sustainability (Contributor)
Created2015-12-01
Description
Preserving a system’s viability in the presence of diversity erosion is critical if the goal is to sustainably support biodiversity. Reduction in population heterogeneity, whether inter- or intraspecies, may increase population fragility, either decreasing its ability to adapt effectively to environmental changes or facilitating the survival and success of ordinarily rare phenotypes. The latter may result in over-representation of individuals who may participate in resource utilization patterns that can lead to over-exploitation, exhaustion, and, ultimately, collapse of both the resource and the population that depends on it. Here, we aim to identify regimes that can signal whether a consumer–resource system is capable of supporting viable degrees of heterogeneity. The framework used here is an expansion of a previously introduced consumer–resource type system of a population of individuals classified by their resource consumption. Application of the Reduction Theorem to the system enables us to evaluate the health of the system through tracking both the mean value of the parameter of resource (over)consumption, and the population variance, as both change over time. The article concludes with a discussion that highlights applicability of the proposed system to investigation of systems that are affected by particularly devastating overly adapted populations, namely cancerous cells. Potential intervention approaches for system management are discussed in the context of cancer therapies.
ContributorsKareva, Irina (Author) / Morin, Benjamin (Author) / Castillo-Chavez, Carlos (Author) / Simon M. Levin Mathematical, Computational and Modeling Sciences Center (Contributor) / College of Liberal Arts and Sciences (Contributor) / School of Human Evolution and Social Change (Contributor) / Julie Ann Wrigley Global Institute of Sustainability (Contributor)
Created2015-02-01
Description
Background
In the weeks following the first imported case of Ebola in the U. S. on September 29, 2014, coverage of the very limited outbreak dominated the news media, in a manner quite disproportionate to the actual threat to national public health; by the end of October, 2014, there were only four laboratory confirmed cases of Ebola in the entire nation. Public interest in these events was high, as reflected in the millions of Ebola-related Internet searches and tweets performed in the month following the first confirmed case. Use of trending Internet searches and tweets has been proposed in the past for real-time prediction of outbreaks (a field referred to as “digital epidemiology”), but accounting for the biases of public panic has been problematic. In the case of the limited U. S. Ebola outbreak, we know that the Ebola-related searches and tweets originating the U. S. during the outbreak were due only to public interest or panic, providing an unprecedented means to determine how these dynamics affect such data, and how news media may be driving these trends.
Methodology
We examine daily Ebola-related Internet search and Twitter data in the U. S. during the six week period ending Oct 31, 2014. TV news coverage data were obtained from the daily number of Ebola-related news videos appearing on two major news networks. We fit the parameters of a mathematical contagion model to the data to determine if the news coverage was a significant factor in the temporal patterns in Ebola-related Internet and Twitter data.
Conclusions
We find significant evidence of contagion, with each Ebola-related news video inspiring tens of thousands of Ebola-related tweets and Internet searches. Between 65% to 76% of the variance in all samples is described by the news media contagion model.
In the weeks following the first imported case of Ebola in the U. S. on September 29, 2014, coverage of the very limited outbreak dominated the news media, in a manner quite disproportionate to the actual threat to national public health; by the end of October, 2014, there were only four laboratory confirmed cases of Ebola in the entire nation. Public interest in these events was high, as reflected in the millions of Ebola-related Internet searches and tweets performed in the month following the first confirmed case. Use of trending Internet searches and tweets has been proposed in the past for real-time prediction of outbreaks (a field referred to as “digital epidemiology”), but accounting for the biases of public panic has been problematic. In the case of the limited U. S. Ebola outbreak, we know that the Ebola-related searches and tweets originating the U. S. during the outbreak were due only to public interest or panic, providing an unprecedented means to determine how these dynamics affect such data, and how news media may be driving these trends.
Methodology
We examine daily Ebola-related Internet search and Twitter data in the U. S. during the six week period ending Oct 31, 2014. TV news coverage data were obtained from the daily number of Ebola-related news videos appearing on two major news networks. We fit the parameters of a mathematical contagion model to the data to determine if the news coverage was a significant factor in the temporal patterns in Ebola-related Internet and Twitter data.
Conclusions
We find significant evidence of contagion, with each Ebola-related news video inspiring tens of thousands of Ebola-related tweets and Internet searches. Between 65% to 76% of the variance in all samples is described by the news media contagion model.
ContributorsTowers, Sherry (Author) / Afzal, Shehzad (Author) / Bernal, Gilbert (Author) / Bliss, Nadya (Author) / Brown, Shala (Author) / Espinoza, Baltazar (Author) / Jackson, Jasmine (Author) / Judson-Garcia, Julia (Author) / Khan, Maryam (Author) / Lin, Michael (Author) / Mamada, Robert (Author) / Moreno, Victor (Author) / Nazari, Fereshteh (Author) / Okuneye, Kamaldeen (Author) / Ross, Mary (Author) / Rodriguez, Claudia (Author) / Medlock, Jan (Author) / Ebert, David (Author) / Castillo-Chavez, Carlos (Author) / Simon M. Levin Mathematical, Computational and Modeling Sciences Center (Contributor) / College of Liberal Arts and Sciences (Contributor) / School of Human Evolution and Social Change (Contributor) / Mary Lou Fulton Teachers College (Contributor) / Educational Leadership and Innovation (Contributor) / School of Mathematical and Statistical Sciences (Contributor) / Global Security Initiative (Contributor)
Created2015-06-11
Description
Background
Seroepidemiological studies before and after the epidemic wave of H1N1-2009 are useful for estimating population attack rates with a potential to validate early estimates of the reproduction number, R, in modeling studies.
Methodology/Principal Findings
Since the final epidemic size, the proportion of individuals in a population who become infected during an epidemic, is not the result of a binomial sampling process because infection events are not independent of each other, we propose the use of an asymptotic distribution of the final size to compute approximate 95% confidence intervals of the observed final size. This allows the comparison of the observed final sizes against predictions based on the modeling study (R = 1.15, 1.40 and 1.90), which also yields simple formulae for determining sample sizes for future seroepidemiological studies. We examine a total of eleven published seroepidemiological studies of H1N1-2009 that took place after observing the peak incidence in a number of countries. Observed seropositive proportions in six studies appear to be smaller than that predicted from R = 1.40; four of the six studies sampled serum less than one month after the reported peak incidence. The comparison of the observed final sizes against R = 1.15 and 1.90 reveals that all eleven studies appear not to be significantly deviating from the prediction with R = 1.15, but final sizes in nine studies indicate overestimation if the value R = 1.90 is used.
Conclusions
Sample sizes of published seroepidemiological studies were too small to assess the validity of model predictions except when R = 1.90 was used. We recommend the use of the proposed approach in determining the sample size of post-epidemic seroepidemiological studies, calculating the 95% confidence interval of observed final size, and conducting relevant hypothesis testing instead of the use of methods that rely on a binomial proportion.
Seroepidemiological studies before and after the epidemic wave of H1N1-2009 are useful for estimating population attack rates with a potential to validate early estimates of the reproduction number, R, in modeling studies.
Methodology/Principal Findings
Since the final epidemic size, the proportion of individuals in a population who become infected during an epidemic, is not the result of a binomial sampling process because infection events are not independent of each other, we propose the use of an asymptotic distribution of the final size to compute approximate 95% confidence intervals of the observed final size. This allows the comparison of the observed final sizes against predictions based on the modeling study (R = 1.15, 1.40 and 1.90), which also yields simple formulae for determining sample sizes for future seroepidemiological studies. We examine a total of eleven published seroepidemiological studies of H1N1-2009 that took place after observing the peak incidence in a number of countries. Observed seropositive proportions in six studies appear to be smaller than that predicted from R = 1.40; four of the six studies sampled serum less than one month after the reported peak incidence. The comparison of the observed final sizes against R = 1.15 and 1.90 reveals that all eleven studies appear not to be significantly deviating from the prediction with R = 1.15, but final sizes in nine studies indicate overestimation if the value R = 1.90 is used.
Conclusions
Sample sizes of published seroepidemiological studies were too small to assess the validity of model predictions except when R = 1.90 was used. We recommend the use of the proposed approach in determining the sample size of post-epidemic seroepidemiological studies, calculating the 95% confidence interval of observed final size, and conducting relevant hypothesis testing instead of the use of methods that rely on a binomial proportion.
ContributorsNishiura, Hiroshi (Author) / Chowell-Puente, Gerardo (Author) / Castillo-Chavez, Carlos (Author) / Simon M. Levin Mathematical, Computational and Modeling Sciences Center (Contributor) / College of Liberal Arts and Sciences (Contributor) / School of Human Evolution and Social Change (Contributor)
Created2011-03-24