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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 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
In an effort to address the lack of literature in on-campus active travel, this study aims to investigate the following primary questions:<br/>• What are the modes that students use to travel on campus?<br/>• What are the motivations that underlie the mode choice of students on campus?<br/>My first stage of research involved a series of qualitative investigations. I held one-on-one virtual interviews with students in which I asked them questions about the mode they use and why they feel that their chosen mode works best for them. These interviews served two functions. First, they provided me with insight into the various motivations underlying student mode choice. Second, they provided me with an indication of what explanatory variables should be included in a model of mode choice on campus.<br/>The first half of the research project informed a quantitative survey that was released via the Honors Digest to attract student respondents. Data was gathered on travel behavior as well as relevant explanatory variables.<br/>My analysis involved developing a logit model to predict student mode choice on campus and presenting the model estimation in conjunction with a discussion of student travel motivations based on the qualitative interviews. I use this information to make a recommendation on how campus infrastructure could be modified to better support the needs of the student population.
ContributorsMirtich, Laura Christine (Author) / Salon, Deborah (Thesis director) / Fang, Kevin (Committee member) / School of Public Affairs (Contributor) / School of Life Sciences (Contributor) / Barrett, The Honors College (Contributor)
Created2021-05
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
Defines the concept of the arcology as conceived by architect Paolo Soleri. Arcology combines "architecture" and "ecology" and explores a visionary notion of a self-contained urban community that has agricultural, commercial, and residential facilities under one roof. Two real-world examples of these projects are explored: Arcosanti, AZ and Masdar City, Abu Dhabi, UAE. Key aspects of the arcology that could be applied to an existing urban fabric are identified, such as urban design fostering social interaction, reduction of automobile dependency, and a development pattern that combats sprawl. Through interviews with local representatives, a holistic approach to applying arcology concepts to the Phoenix Metro Area is devised.
ContributorsSpencer, Sarah Anne (Author) / Manuel-Navarrete, David (Thesis director) / Salon, Deborah (Committee member) / Barrett, The Honors College (Contributor) / School of Geographical Sciences and Urban Planning (Contributor) / School of Sustainability (Contributor)
Created2015-05
Description
The contemporary architectural pedagogy is far removed from its ancestry: the classical Beaux-Arts and polytechnic schools of the 19th century and the Bauhaus and Vkhutemas models of the modern period. Today, the "digital" has invaded the academy and shapes pedagogical practices, epistemologies, and ontologies within it, and this invasion is reflected in teaching practices, principles, and tools. Much of this digital integration goes unremarked and may not even be explicitly taught. In this qualitative research project, interviews with 18 leading architecture lecturers, professors, and deans from programs across the United States were conducted. These interviews focused on advanced practices of digital architecture, such as the use of digital tools, and how these practices are viewed. These interviews yielded a wealth of information about the uses (and abuses) of advanced digital technologies within the architectural academy, and the results were analyzed using the methods of phenomenology and grounded theory. Most schools use digital technologies to some extent, although this extent varies greatly. While some schools have abandoned hand-drawing and other hand-based craft almost entirely, others have retained traditional techniques and use digital technologies sparingly. Reasons for using digital design processes include industry pressure as well as the increased ability to solve problems and the speed with which they could be solved. Despite the prevalence of digital design, most programs did not teach related design software explicitly, if at all, instead requiring students (especially graduate students) to learn to use them outside the design studio. Some of the problems with digital design identified in the interviews include social problems such as alienation as well as issues like understanding scale and embodiment of skill.
ContributorsAlqabandy, Hamad (Author) / Brandt, Beverly (Thesis advisor) / Mesch, Claudia (Committee member) / Newton, David (Committee member) / Arizona State University (Publisher)
Created2012
Description
In the middle of the 20th century, juried annuals of Native American painting in art museums were unique opportunities because of their select focus on two-dimensional art as opposed to "craft" objects and their inclusion of artists from across the United States. Their first fifteen years were critical for patronage and widespread acceptance of modern easel painting. Held at the Philbrook Art Center in Tulsa (1946-1979), the Denver Art Museum (1951-1954), and the Museum of New Mexico Art Gallery in Santa Fe (1956-1965), they were significant not only for the accolades and prestige they garnered for award winners, but also for setting standards of quality and style at the time. During the early years of the annuals, the art was changing, some moving away from conventional forms derived from the early art training of the 1920s and 30s in the Southwest and Oklahoma, and incorporating modern themes and styles acquired through expanded opportunities for travel and education. The competitions reinforced and reflected a variety of attitudes about contemporary art which ranged from preserving the authenticity of the traditional style to encouraging experimentation. Ultimately becoming sites of conflict, the museums that hosted annuals contested the directions in which artists were working. Exhibition catalogs, archived documents, and newspaper and magazine articles about the annuals provide details on the exhibits and the changes that occurred over time. The museums' guidelines and motivations, and the statistics on the award winners reveal attitudes toward the art. The institutions' reactions in the face of controversy and their adjustments to the annuals' guidelines impart the compromises each made as they adapted to new trends that occurred in Native American painting over a fifteen year period. This thesis compares the approaches of three museums to their juried annuals and establishes the existence of a variety of attitudes on contemporary Native American painting from 1946-1960. Through this collection of institutional views, the competitions maintained a patronage base for traditional style painting while providing opportunities for experimentation, paving the way for the great variety and artistic progress of Native American painting today.
ContributorsPeters, Stephanie (Author) / Duncan, Kate (Thesis advisor) / Fahlman, Betsy (Thesis advisor) / Mesch, Claudia (Committee member) / Arizona State University (Publisher)
Created2012
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
This project is a critical look at Chicano artist Vincent Valdez's 2002-2004 series Stations. The theoretical framework for this work is the concept of cultural citizenship, which refers to a variety of ways in which marginalized groups of people create, fight for, and retain space, identity, and rights within American society through acts of daily life. This research considers how the ten large-scale charcoal drawings that comprise Stations contribute to the construction and representation of distinct and unique Latino spaces and identities. Valdez establishes space in the sense of belonging and community engagement that his work allows. Within this context, thoughtful attention is paid to the cultural meaning of the artist's subject choices of boxing and religion. This research considers the significance of these subject choices and how the connections between the two create unique spaces of shared experience and consciousness for a viewer of the work. However, the parallels that Valdez draws between the Christ figure and his boxer also allow for a careful examination of the representations and contradictions of contemporary constructions of masculinity that are present in the series. Within this project, the work of Gloria Anzaldúa is critical in understanding and discussing the fluid nature of Chicano identity. This study also considers how in the tradition of Chicana writers, Valdez expresses and affirms identity through autobiographical methods. Further, the artist's use of charcoal to create these large scale drawings is considered for its narrative qualities. This study concludes that Valdez's series Stations is an act of cultural citizenship.
ContributorsStemm Patel, Shannon (Author) / Malagamba-Ansótegui, Amelia (Thesis advisor) / Mesch, Claudia (Committee member) / Sweeney, Gray (Committee member) / Arizona State University (Publisher)
Created2010
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
The Indian princess began as an imposition, a Eurocentric conception based in preconceived notions of cultural structures and gendered power roles - a mixture of noble woman and provocative demure maiden - created by Anglo men to epitomize an idyllic image of otherness and womanhood. This analysis begins by exploring the history of the icon that was first conceived through sixteenth century explorer's tales of exotic queens then traces her progression through the romantic idealizations of the Indian woman Pocahontas. Research then explores how the character, comprised of a mixture of feathers, beads, and buckskin, was implemented into performance, and discusses how her flesh and blood enactment became critical to her survival. Drawing on the theories of contemporary critics, final examination turns to twentieth century perceptions of the Princess and reactions to her by contemporary Native artists whose manipulations of the character opens alternative dialogs about the stereotype to offer reconstructions of her historic discourse.
ContributorsHanawalt, Tammi Jo (Author) / Duncan, Kate (Thesis advisor) / Fahlman, Betsy (Committee member) / Malagamba-Ansótegui, Amelia (Committee member) / Mesch, Claudia (Committee member) / Arizona State University (Publisher)
Created2011