Matching Items (625)
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
Factory production is stochastic in nature with time varying input and output processes that are non-stationary stochastic processes. Hence, the principle quantities of interest are random variables. Typical modeling of such behavior involves numerical simulation and statistical analysis. A deterministic closure model leading to a second order model for the product density and product speed has previously been proposed. The resulting partial differential equations (PDE) are compared to discrete event simulations (DES) that simulate factory production as a time dependent M/M/1 queuing system. Three fundamental scenarios for the time dependent influx are studied: An instant step up/down of the mean arrival rate; an exponential step up/down of the mean arrival rate; and periodic variation of the mean arrival rate. It is shown that the second order model, in general, yields significant improvement over current first order models. Specifically, the agreement between the DES and the PDE for the step up and for periodic forcing that is not too rapid is very good. Adding diffusion to the PDE further improves the agreement. The analysis also points to fundamental open issues regarding the deterministic modeling of low signal-to-noise ratio for some stochastic processes and the possibility of resonance in deterministic models that is not present in the original stochastic process.
ContributorsWienke, Matthew (Author) / Armbruster, Dieter (Thesis advisor) / Jones, Donald (Committee member) / Platte, Rodrigo (Committee member) / Gardner, Carl (Committee member) / Ringhofer, Christian (Committee member) / Arizona State University (Publisher)
Created2015
Description
Presented is a study on the chemotaxis reaction process and its relation with flow topology. The effect of coherent structures in turbulent flows is characterized by studying nutrient uptake and the advantage that is received from motile bacteria over other non-motile bacteria. Variability is found to be dependent on the initial location of scalar impurity and can be tied to Lagrangian coherent structures through recent advances in the identification of finite-time transport barriers. Advantage is relatively small for initial nutrient found within high stretching regions of the flow, and nutrient within elliptic structures provide the greatest advantage for motile species. How the flow field and the relevant flow topology lead to such a relation is analyzed.
ContributorsJones, Kimberly (Author) / Tang, Wenbo (Thesis advisor) / Kang, Yun (Committee member) / Jones, Donald (Committee member) / Arizona State University (Publisher)
Created2015
Description
A continuously and stably stratified fluid contained in a square cavity subjected to harmonic body forcing is studied numerically by solving the Navier-Stokes equations under the Boussinesq approximation. Complex dynamics are observed near the onset of instability of the basic state, which is a flow configuration that is always an exact analytical solution of the governing equations. The instability of the basic state to perturbations is first studied with linear stability analysis (Floquet analysis), revealing a multitude of intersecting synchronous and subharmonic resonance tongues in parameter space. A modal reduction method for determining the locus of basic state instability is also shown, greatly simplifying the computational overhead normally required by a Floquet study. Then, a study of the nonlinear governing equations determines the criticality of the basic state's instability, and ultimately characterizes the dynamics of the lowest order spatial mode by the three discovered codimension-two bifurcation points within the resonance tongue. The rich dynamics include a homoclinic doubling cascade that resembles the logistic map and a multitude of gluing bifurcations.
The numerical techniques and methodologies are first demonstrated on a homogeneous fluid contained within a three-dimensional lid-driven cavity. The edge state technique and linear stability analysis through Arnoldi iteration are used to resolve the complex dynamics of the canonical shear-driven benchmark problem. The techniques here lead to a dynamical description of an instability mechanism, and the work serves as a basis for the remainder of the dissertation.
The numerical techniques and methodologies are first demonstrated on a homogeneous fluid contained within a three-dimensional lid-driven cavity. The edge state technique and linear stability analysis through Arnoldi iteration are used to resolve the complex dynamics of the canonical shear-driven benchmark problem. The techniques here lead to a dynamical description of an instability mechanism, and the work serves as a basis for the remainder of the dissertation.
ContributorsYalim, Jason (Author) / Welfert, Bruno D. (Thesis advisor) / Lopez, Juan M. (Thesis advisor) / Jones, Donald (Committee member) / Tang, Wenbo (Committee member) / Platte, Rodrigo (Committee member) / Arizona State University (Publisher)
Created2019
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. 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
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. 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
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 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
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 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
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 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.
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