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Description
Advances in experimental techniques have allowed for investigation of molecular dynamics at ever smaller temporal and spatial scales. There is currently a varied and growing body of literature which demonstrates the phenomenon of \emph{anomalous diffusion} in physics, engineering, and biology. In particular many diffusive type processes in the cell have

Advances in experimental techniques have allowed for investigation of molecular dynamics at ever smaller temporal and spatial scales. There is currently a varied and growing body of literature which demonstrates the phenomenon of \emph{anomalous diffusion} in physics, engineering, and biology. In particular many diffusive type processes in the cell have been observed to follow a power law $\left \propto t^\alpha$ scaling of the mean square displacement of a particle. This contrasts with the expected linear behavior of particles undergoing normal diffusion. \emph{Anomalous sub-diffusion} ($\alpha<1$) has been attributed to factors such as cytoplasmic crowding of macromolecules, and trap-like structures in the subcellular environment non-linearly slowing the diffusion of molecules. Compared to normal diffusion, signaling molecules in these constrained spaces can be more concentrated at the source, and more diffuse at longer distances, potentially effecting the signalling dynamics. As diffusion at the cellular scale is a fundamental mechanism of cellular signaling and additionally is an implicit underlying mathematical assumption of many canonical models, a closer look at models of anomalous diffusion is warranted. Approaches in the literature include derivations of fractional differential diffusion equations (FDE) and continuous time random walks (CTRW). However these approaches are typically based on \emph{ad-hoc} assumptions on time- and space- jump distributions. We apply recent developments in asymptotic techniques on collisional kinetic equations to develop a FDE model of sub-diffusion due to trapping regions and investigate the nature of the space/time probability distributions assosiated with trapping regions. This approach both contrasts and compliments the stochastic CTRW approach by positing more physically realistic underlying assumptions on the motion of particles and their interactions with trapping regions, and additionally allowing varying assumptions to be applied individually to the traps and particle kinetics.
ContributorsHoleva, Thomas Matthew (Author) / Ringhofer, Christian (Thesis advisor) / Baer, Steve (Thesis advisor) / Crook, Sharon (Committee member) / Gardner, Carl (Committee member) / Taylor, Jesse (Committee member) / Arizona State University (Publisher)
Created2014
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Description
Modern measurement schemes for linear dynamical systems are typically designed so that different sensors can be scheduled to be used at each time step. To determine which sensors to use, various metrics have been suggested. One possible such metric is the observability of the system. Observability is a binary condition

Modern measurement schemes for linear dynamical systems are typically designed so that different sensors can be scheduled to be used at each time step. To determine which sensors to use, various metrics have been suggested. One possible such metric is the observability of the system. Observability is a binary condition determining whether a finite number of measurements suffice to recover the initial state. However to employ observability for sensor scheduling, the binary definition needs to be expanded so that one can measure how observable a system is with a particular measurement scheme, i.e. one needs a metric of observability. Most methods utilizing an observability metric are about sensor selection and not for sensor scheduling. In this dissertation we present a new approach to utilize the observability for sensor scheduling by employing the condition number of the observability matrix as the metric and using column subset selection to create an algorithm to choose which sensors to use at each time step. To this end we use a rank revealing QR factorization algorithm to select sensors. Several numerical experiments are used to demonstrate the performance of the proposed scheme.
ContributorsIlkturk, Utku (Author) / Gelb, Anne (Thesis advisor) / Platte, Rodrigo (Thesis advisor) / Cochran, Douglas (Committee member) / Renaut, Rosemary (Committee member) / Armbruster, Dieter (Committee member) / Arizona State University (Publisher)
Created2015
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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

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
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Description
Swarms of animals, fish, birds, locusts etc. are a common occurrence but their coherence and method of organization poses a major question for mathematics and biology.The Vicsek and the Attraction-Repulsion are two models that have been proposed to explain the emergence of collective motion. A major issue

Swarms of animals, fish, birds, locusts etc. are a common occurrence but their coherence and method of organization poses a major question for mathematics and biology.The Vicsek and the Attraction-Repulsion are two models that have been proposed to explain the emergence of collective motion. A major issue for the Vicsek Model is that its particles are not attracted to each other, leaving the swarm with alignment in velocity but without spatial coherence. Restricting the particles to a bounded domain generates global spatial coherence of swarms while maintaining velocity alignment. While individual particles are specularly reflected at the boundary, the swarm as a whole is not. As a result, new dynamical swarming solutions are found.

The Attraction-Repulsion Model set with a long-range attraction and short-range repulsion interaction potential typically stabilizes to a well-studied flock steady state solution. The particles for a flock remain spatially coherent but have no spatial bound and explore all space. A bounded domain with specularly reflecting walls traps the particles within a specific region. A fundamental refraction law for a swarm impacting on a planar boundary is derived. The swarm reflection varies from specular for a swarm dominated by

kinetic energy to inelastic for a swarm dominated by potential energy. Inelastic collisions lead to alignment with the wall and to damped pulsating oscillations of the swarm. The fundamental refraction law provides a one-dimensional iterative map that allows for a prediction and analysis of the trajectory of the center of mass of a flock in a channel and a square domain.

The extension of the wall collisions to a scattering experiment is conducted by setting two identical flocks to collide. The two particle dynamics is studied analytically and shows a transition from scattering: diverging flocks to bound states in the form of oscillations or parallel motions. Numerical studies of collisions of flocks show the same transition where the bound states become either a single translating flock or a rotating (mill).
ContributorsThatcher, Andrea (Author) / Armbruster, Hans (Thesis advisor) / Motsch, Sebastien (Committee member) / Ringhofer, Christian (Committee member) / Platte, Rodrigo (Committee member) / Gardner, Carl (Committee member) / Arizona State University (Publisher)
Created2015
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Description
Inverse problems model real world phenomena from data, where the data are often noisy and models contain errors. This leads to instabilities, multiple solution vectors and thus ill-posedness. To solve ill-posed inverse problems, regularization is typically used as a penalty function to induce stability and allow for the incorporation of

Inverse problems model real world phenomena from data, where the data are often noisy and models contain errors. This leads to instabilities, multiple solution vectors and thus ill-posedness. To solve ill-posed inverse problems, regularization is typically used as a penalty function to induce stability and allow for the incorporation of a priori information about the desired solution. In this thesis, high order regularization techniques are developed for image and function reconstruction from noisy or misleading data. Specifically the incorporation of the Polynomial Annihilation operator allows for the accurate exploitation of the sparse representation of each function in the edge domain.

This dissertation tackles three main problems through the development of novel reconstruction techniques: (i) reconstructing one and two dimensional functions from multiple measurement vectors using variance based joint sparsity when a subset of the measurements contain false and/or misleading information, (ii) approximating discontinuous solutions to hyperbolic partial differential equations by enhancing typical solvers with l1 regularization, and (iii) reducing model assumptions in synthetic aperture radar image formation, specifically for the purpose of speckle reduction and phase error correction. While the common thread tying these problems together is the use of high order regularization, the defining characteristics of each of these problems create unique challenges.

Fast and robust numerical algorithms are also developed so that these problems can be solved efficiently without requiring fine tuning of parameters. Indeed, the numerical experiments presented in this dissertation strongly suggest that the new methodology provides more accurate and robust solutions to a variety of ill-posed inverse problems.
ContributorsScarnati, Theresa (Author) / Gelb, Anne (Thesis advisor) / Platte, Rodrigo (Thesis advisor) / Cochran, Douglas (Committee member) / Gardner, Carl (Committee member) / Sanders, Toby (Committee member) / Arizona State University (Publisher)
Created2018
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Description
Predicting resistant prostate cancer is critical for lowering medical costs and improving the quality of life of advanced prostate cancer patients. I formulate, compare, and analyze two mathematical models that aim to forecast future levels of prostate-specific antigen (PSA). I accomplish these tasks by employing clinical data of locally advanced

Predicting resistant prostate cancer is critical for lowering medical costs and improving the quality of life of advanced prostate cancer patients. I formulate, compare, and analyze two mathematical models that aim to forecast future levels of prostate-specific antigen (PSA). I accomplish these tasks by employing clinical data of locally advanced prostate cancer patients undergoing androgen deprivation therapy (ADT). I demonstrate that the inverse problem of parameter estimation might be too complicated and simply relying on data fitting can give incorrect conclusions, since there is a large error in parameter values estimated and parameters might be unidentifiable. I provide confidence intervals to give estimate forecasts using data assimilation via an ensemble Kalman Filter. Using the ensemble Kalman Filter, I perform dual estimation of parameters and state variables to test the prediction accuracy of the models. Finally, I present a novel model with time delay and a delay-dependent parameter. I provide a geometric stability result to study the behavior of this model and show that the inclusion of time delay may improve the accuracy of predictions. Also, I demonstrate with clinical data that the inclusion of the delay-dependent parameter facilitates the identification and estimation of parameters.
ContributorsBaez, Javier (Author) / Kuang, Yang (Thesis advisor) / Kostelich, Eric (Committee member) / Crook, Sharon (Committee member) / Gardner, Carl (Committee member) / Nagy, John (Committee member) / Arizona State University (Publisher)
Created2017
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Description
Need-based transfers (NBTs) are a form of risk-pooling in which binary welfare exchanges

occur to preserve the viable participation of individuals in an economy, e.g. reciprocal gifting

of cattle among East African herders or food sharing among vampire bats. With the

broad goal of better understanding the mathematics of such binary welfare and

Need-based transfers (NBTs) are a form of risk-pooling in which binary welfare exchanges

occur to preserve the viable participation of individuals in an economy, e.g. reciprocal gifting

of cattle among East African herders or food sharing among vampire bats. With the

broad goal of better understanding the mathematics of such binary welfare and risk pooling,

agent-based simulations are conducted to explore socially optimal transfer policies

and sharing network structures, kinetic exchange models that utilize tools from the kinetic

theory of gas dynamics are utilized to characterize the wealth distribution of an NBT economy,

and a variant of repeated prisoner’s dilemma is analyzed to determine whether and

why individuals would participate in such a system of reciprocal altruism.

From agent-based simulation and kinetic exchange models, it is found that regressive

NBT wealth redistribution acts as a cutting stock optimization heuristic that most efficiently

matches deficits to surpluses to improve short-term survival; however, progressive

redistribution leads to a wealth distribution that is more stable in volatile environments and

therefore is optimal for long-term survival. Homogeneous sharing networks with low variance

in degree are found to be ideal for maintaining community viability as the burden and

benefit of NBTs is equally shared. Also, phrasing NBTs as a survivor’s dilemma reveals

parameter regions where the repeated game becomes equivalent to a stag hunt or harmony

game, and thus where cooperation is evolutionarily stable.
ContributorsKayser, Kirk (Author) / Armbruster, Dieter (Thesis advisor) / Lampert, Adam (Committee member) / Ringhofer, Christian (Committee member) / Motsch, Sebastien (Committee member) / Gardner, Carl (Committee member) / Arizona State University (Publisher)
Created2018
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Description
The most advanced social insects, the eusocial insects, form often large societies in which there is reproductive division of labor, queens and workers, have overlapping generations, and cooperative brood care where daughter workers remain in the nest with their queen mother and care for their siblings. The eusocial insects

The most advanced social insects, the eusocial insects, form often large societies in which there is reproductive division of labor, queens and workers, have overlapping generations, and cooperative brood care where daughter workers remain in the nest with their queen mother and care for their siblings. The eusocial insects are composed of representative species of bees and wasps, and all species of ants and termites. Much is known about their organizational structure, but remains to be discovered.

The success of social insects is dependent upon cooperative behavior and adaptive strategies shaped by natural selection that respond to internal or external conditions. The objective of my research was to investigate specific mechanisms that have helped shaped the structure of division of labor observed in social insect colonies, including age polyethism and nutrition, and phenomena known to increase colony survival such as egg cannibalism. I developed various Ordinary Differential Equation (ODE) models in which I applied dynamical, bifurcation, and sensitivity analysis to carefully study and visualize biological outcomes in social organisms to answer questions regarding the conditions under which a colony can survive. First, I investigated how the population and evolutionary dynamics of egg cannibalism and division of labor can promote colony survival. I then introduced a model of social conflict behavior to study the inclusion of different response functions that explore the benefits of cannibalistic behavior and how it contributes to age polyethism, the change in behavior of workers as they age, and its biological relevance. Finally, I introduced a model to investigate the importance of pollen nutritional status in a honeybee colony, how it affects population growth and influences division of labor within the worker caste. My results first reveal that both cannibalism and division of labor are adaptive strategies that increase the size of the worker population, and therefore, the persistence of the colony. I show the importance of food collection, consumption, and processing rates to promote good colony nutrition leading to the coexistence of brood and adult workers. Lastly, I show how taking into account seasonality for pollen collection improves the prediction of long term consequences.
ContributorsRodríguez Messan, Marisabel (Author) / Kang, Yun (Thesis advisor) / Castillo-Chavez, Carlos (Thesis advisor) / Kuang, Yang (Committee member) / Page Jr., Robert E (Committee member) / Gardner, Carl (Committee member) / Arizona State University (Publisher)
Created2018
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Description
Rabies is an infectious viral disease. It is usually fatal if a victim reaches the rabid stage, which starts after the appearance of disease symptoms. The disease virus attacks the central nervous system, and then it migrates from peripheral nerves to the spinal cord and brain. At the time when

Rabies is an infectious viral disease. It is usually fatal if a victim reaches the rabid stage, which starts after the appearance of disease symptoms. The disease virus attacks the central nervous system, and then it migrates from peripheral nerves to the spinal cord and brain. At the time when the rabies virus reaches the brain, the incubation period is over and the symptoms of clinical disease appear on the victim. From the brain, the virus travels via nerves to the salivary glands and saliva.

A mathematical model is developed for the spread of rabies in a spatially distributed fox population to model the spread of the rabies epizootic through middle Europe that occurred in the second half of the 20th century. The model considers both territorial and wandering rabid foxes and includes a latent period for the infection. Since the model assumes these two kinds of rabid foxes, it is a system of both partial differential and integral equations (with integration

over space and, occasionally, also over time). To study the spreading speeds of the rabies epidemic, the model is reduced to a scalar Volterra-Hammerstein integral equation, and space-time Laplace transform of the integral equation is used to derive implicit formulas for the spreading speed. The spreading speeds are discussed and implicit formulas are given for latent periods of fixed length, exponentially distributed length, Gamma distributed length, and log-normally distributed length. A number of analytic and numerical results are shown pertaining to the spreading speeds.

Further, a numerical algorithm is described for the simulation

of the spread of rabies in a spatially distributed fox population on a bounded domain with Dirichlet boundary conditions. I propose the following methods for the numerical approximation of solutions. The partial differential and integral equations are discretized in the space variable by central differences of second order and by

the composite trapezoidal rule. Next, the ordinary or delay differential equations that are obtained this way are discretized in time by explicit

continuous Runge-Kutta methods of fourth order for ordinary and delay differential systems. My particular interest

is in how the partition of rabid foxes into

territorial and diffusing rabid foxes influences

the spreading speed, a question that can be answered by purely analytic means only for small basic reproduction numbers. I will restrict the numerical analysis

to latent periods of fixed length and to exponentially

distributed latent periods.

The results of the numerical calculations

are compared for latent periods

of fixed and exponentially distributed length

and for various proportions of territorial

and wandering rabid foxes.

The speeds of spread observed in the

simulations are compared

to spreading speeds obtained by numerically solving the analytic formulas

and to observed speeds of epizootic frontlines

in the European rabies outbreak 1940 to 1980.
ContributorsAlanazi, Khalaf Matar (Author) / Thieme, Horst R. (Thesis advisor) / Jackiewicz, Zdzislaw (Committee member) / Baer, Steven (Committee member) / Gardner, Carl (Committee member) / Kuang, Yang (Committee member) / Smith, Hal (Committee member) / Arizona State University (Publisher)
Created2018
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Description
Chapter 1 introduces some key elements of important topics such as; quantum mechanics,

representation theory of the Lorentz and Poincare groups, and a review of some basic rela- ´

tivistic wave equations that will play an important role in the work to follow. In Chapter 2,

a complex covariant form of the classical

Chapter 1 introduces some key elements of important topics such as; quantum mechanics,

representation theory of the Lorentz and Poincare groups, and a review of some basic rela- ´

tivistic wave equations that will play an important role in the work to follow. In Chapter 2,

a complex covariant form of the classical Maxwell’s equations in a moving medium or at

rest is introduced. In addition, a compact, Lorentz invariant, form of the energy-momentum

tensor is derived. In chapter 3, the concept of photon helicity is critically analyzed and its

connection with the Pauli-Lubanski vector from the viewpoint of the complex electromag- ´

netic field, E+ iH. To this end, a complex covariant form of Maxwell’s equations is used.

Chapter 4 analyzes basic relativistic wave equations for the classical fields, such as Dirac’s

equation, Weyl’s two-component equation for massless neutrinos and the Proca, Maxwell

and Fierz-Pauli equations, from the viewpoint of the Pauli-Lubanski vector and the Casimir ´

operators of the Poincare group. A connection between the spin of a particle/field and ´

consistency of the corresponding overdetermined system is emphasized in the massless

case. Chapter 5 focuses on the so-called generalized quantum harmonic oscillator, which

is a Schrodinger equation with a time-varying quadratic Hamiltonian operator. The time ¨

evolution of exact wave functions of the generalized harmonic oscillators is determined

in terms of the solutions of certain Ermakov and Riccati-type systems. In addition, it is

shown that the classical Arnold transform is naturally connected with Ehrenfest’s theorem

for generalized harmonic oscillators. In Chapter 6, as an example of the usefulness of the

methods introduced in Chapter 5 a model for the quantization of an electromagnetic field

in a variable media is analyzed. The concept of quantization of an electromagnetic field

in factorizable media is discussed via the Caldirola-Kanai Hamiltonian. A single mode

of radiation for this model is used to find time-dependent photon amplitudes in relation

to Fock states. A multi-parameter family of the squeezed states, photon statistics, and the

uncertainty relation, are explicitly given in terms of the Ermakov-type system.
ContributorsLanfear, Nathan A (Author) / Suslov, Sergei (Thesis advisor) / Kotschwar, Brett (Thesis advisor) / Platte, Rodrigo (Committee member) / Matyushov, Dmitry (Committee member) / Kuiper, Hendrik (Committee member) / Gardner, Carl (Committee member) / Arizona State University (Publisher)
Created2016