Matching Items (27)
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
We analyzed multiple different models that can be utilized when measuring effects effects of fire and fire behavior in a forest ecosystem. In the thesis we focused on exploring ordinary differential equations, stochastic models, and partial differential equations
ContributorsVo, Sabrina (Author) / Jones, Donald (Thesis director) / Parker, Nathan (Committee member) / School of Mathematical and Statistical Sciences (Contributor) / School of Sustainability (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
Honey bee (Apis mellifera) colonies have experienced substantial losses due to colony collapse disorder (CCD) since the first officially reported cases in 2006. Many factors have been implicated in CCD, including pests, pathogens, malnutrition, and pesticide use, but no correlation has been found between a single factor and the occurrence of CCD. Fungicides have received less research attention compared to insecticides, despite the fact that fungicide application coincides with bloom and the presence of bees. Pristine fungicide is widely used in agriculture and is commonly found as a residue in hives. Several studies have concluded that Pristine can be used without harming bees, but reports of brood loss following Pristine application continue to surface across the country. The primary objectives of this study were to determine whether Pristine causes an aversive gustatory response in bees and whether consumption of an acute dose affects responsiveness to sucrose. An awareness of how foragers interact with contaminated food is useful to understand the likelihood that Pristine is ingested and how that may affect bees' ability to evaluate floral resources. Our results indicated that Pristine has no significant effect on gustatory response or sucrose responsiveness. There was no significant difference between bee responses to Pristine contaminated sucrose and sucrose alone, and no significant effect of Pristine on sucrose responsiveness. These results indicate that honey bees do not have a gustatory aversion to Pristine. A lack of aversion means that honey bees will continue collecting contaminated resources and dispersing them throughout the colony where it can affect brood and clean food stores.
ContributorsMcHugh, Cora Elizabeth (Co-author) / Jernigan, Christopher (Co-author, Committee member) / Burden, Christina (Co-author) / DeGrandi-Hoffman, Gloria (Co-author) / Smith, Brian (Thesis director) / Fewell, Jennifer (Committee member) / Barrett, The Honors College (Contributor) / School of Geographical Sciences and Urban Planning (Contributor) / School of Life Sciences (Contributor) / School of Art (Contributor)
Created2015-05
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, 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
Description
Dividing the plane in half leaves every border point of one region a border point of both regions. Can we divide up the plane into three or more regions such that any point on the boundary of at least one region is on the border of all the regions? In fact, it is possible to design a dynamical system for which the basins of attractions have this Wada property. In certain circumstances, both the Hénon map, a simple system, and the forced damped pendulum, a physical model, produce Wada basins.
ContributorsWhitehurst, Ryan David (Author) / Kostelich, Eric (Thesis director) / Jones, Donald (Committee member) / Armbruster, Dieter (Committee member) / Barrett, The Honors College (Contributor) / School of Mathematical and Statistical Sciences (Contributor) / Department of Chemistry and Biochemistry (Contributor)
Created2013-05
Description
Computer simulations are gaining recognition as educational tools, but in general there is still a line dividing a simulation from a game. Yet as many recent and successful video games heavily involve simulations (SimCity comes to mind), there is not only the growing question of whether games can be used for educational purposes, but also of how a game might qualify as educational. Endemic: The Agent is a project that tries to bridge the gap between educational simulations and educational games. This paper outlines the creation of the project and the characteristics that make it an educational tool, a simulation, and a game.
ContributorsFish, Derek Austin (Author) / Karr, Timothy (Thesis director) / Marcus, Andrew (Committee member) / Jones, Donald (Committee member) / Barrett, The Honors College (Contributor) / School of Mathematical and Statistical Sciences (Contributor) / Department of Physics (Contributor)
Created2013-05
Description
In large part, the great success of eusocial insects is due to efficient division of labor (Duarte et al. 2011; Dornhaus 2008). Within ant colonies, the process of dividing labor is not clearly defined, but it may be key to understanding the productivity and success of these colonies. This study analyzed data from an experiment that was conducted with the goal of examining how finely division of labor is organized in ant colonies. The experiment considered the actions of all ants from three Temnothorax rugatulus colonies. The colonies were each carefully recorded during five distinct emigrations per colony. The experiment produced such a large quantity of data that it was challenging to analyze the results, a major obstacle for many studies of collective behavior. Therefore, I designed a computer program that successfully sorted all of the data and prepared it for an initial statistical analysis that was performed in R. The preliminary results suggest that while most of the ants perform little to no work, there is an overall pattern of elitism; it seems that division of labor in ants is not more finely divided than previously shown. Future studies should provide further analysis of the data and will be useful in forming a more complete understanding of the division of labor within the emigrations of Temnothorax rugatulus colonies.
ContributorsJones, Samantha (Author) / Pratt, Stephen (Thesis director) / Jones, Donald (Committee member) / Shaffer, Zachary (Committee member) / Barrett, The Honors College (Contributor) / W. P. Carey School of Business (Contributor)
Created2012-12
Description
A model has been developed to modify Euler-Bernoulli beam theory for wooden beams, using visible properties of wood knot-defects. Treating knots in a beam as a system of two ellipses that change the local bending stiffness has been shown to improve the fit of a theoretical beam displacement function to edge-line deflection data extracted from digital imagery of experimentally loaded beams. In addition, an Ellipse Logistic Model (ELM) has been proposed, using L1-regularized logistic regression, to predict the impact of a knot on the displacement of a beam. By classifying a knot as severely positive or negative, vs. mildly positive or negative, ELM can classify knots that lead to large changes to beam deflection, while not over-emphasizing knots that may not be a problem. Using ELM with a regression-fit Young's Modulus on three-point bending of Douglass Fir, it is possible estimate the effects a knot will have on the shape of the resulting displacement curve.
ContributorsSexton, Thurston Bryant (Author) / Takahashi, Timothy (Thesis director) / Jones, Donald (Committee member) / Barrett, The Honors College (Contributor) / Mechanical and Aerospace Engineering Program (Contributor) / School of Mathematical and Statistical Sciences (Contributor) / School of Human Evolution and Social Change (Contributor)
Created2015-05
Description
Distinguishing between projectile and blunt force or sharp force trauma can be complicated by processes that result in fragmentation or loss of skeletal features. Postmortem processes that obscure skeletal features may result in the inability to properly determine the mechanism of trauma using morphology alone. The presence of gunshot residue (GSR) is indicative of a gunshot event and can be used to differentiate between etiologies of skeletal trauma. Primer GSR is composed of barium (Ba), antimony (Sb), and lead (Pb), which are vaporized during the firearm discharge and can be deposited in small quantities on surfaces within proximity of a gunshot event. Scanning Electron Microscopes with Energy Dispersive X-Ray Spectroscopy (SEM-EDX) have been used in the past to detect GSR on a variety of surfaces including bone. The purpose of this study is to determine the ability to detect GSR on bone tissue using SEM-EDX following warm-water maceration or decomposition.
ContributorsSweeney, Kaylin (Co-author) / Boyd, Derek A. (Co-author) / Cheek, Kimber G. (Co-author) / Ehlers, Blake (Co-author) / Falsetti, Anthony B. (Co-author) / Langley, Natalie R. (Co-author) / Lasala, AmberCherie (Co-author) / Pittman, Bethany (Co-author) / Sweat, Ken (Thesis director) / Falsetti, Anthony (Committee member) / School of Mathematical and Natural Sciences (Contributor) / Barrett, The Honors College (Contributor)
Created2019-05