No description available.

No description available.

No description available.

No description available.

No description available.

Part I: The Theory of Brownian Vacancy Driven Walk We analyze the lattice walk performed by a tagged member of an infinite 'sea' of particles filling a d-dimensional lattice, in the presence of a single vacancy. The vacancy is allowed to be occupied with probability 1/2d by any of its 2d nearest neighbors, so that it executes a Brownian walk. Particle-particle exchange is forbidden; the only interaction between them being hard core exclusion. Thus, the tagged particle, differing from the others only by its tag, moves only when it exchanges places with the hole. In this sense, it is a random walk "driven" by the Brownian vacancy. The probability distributions for its displacement and for the number of steps taken, after n-steps of the vacancy, are derived. Neither is a Gaussian! We also show that the only nontrivial dimension where the walk is recurrent is d=2. As an application, we compute the expected energy shift caused by a Brownian vacancy in a model for an extreme anisotropic binary alloy. In the last chapter we present a Monte-Carlo study and a mean-field analysis for interface erosion caused by mobile vacancies. Part II: One-Dimensional Periodic Hopping Models with Broken Translational Invariance.Case of a Mobile Directional Impurity We study a random walk on a one-dimensional periodic lattice with arbitrary hopping rates. Further, the lattice contains a single mobile, directional impurity (defect bond), across which the rate is fixed at another arbitrary value. Due to the defect, translational invariance is broken, even if all other rates are identical. The structure of Master equations lead naturally to the introduction of a new entity, associated with the walker-impurity pair which we call the quasi-walker. Analytic solution for the distributions in the steady state limit is obtained. The velocities and diffusion constants for both the random walker and impurity are given, being simply related to that of the quasi-particle through physically meaningful equations. As an application, we extend the Duke-Rubinstein reputation model of gel electrophoresis to include polymers with impurities and give the exact distribution of the steady state. / Ph. D.

The Internet has a loosely hierarchical structure. At the top of the hierarchy are the backbones, also called Internet Access Providers (hereafter IAPs). The second layer of the hierarchy is comprised of Internet Service Providers (hereafter ISPs). At the bottom of the hierarchy are the end users, consumers, who browse the web, and websites. To provide access to the whole Internet, the providers must interconnect with each other and share their network infrastructure. Two main forms of interconnection have emerged — peering under which the providers carry each other's traffic without any payments and transit under which the downstream provider pays the upstream provider a certain settlement payment for carrying its traffic. This dissertation develops three game theoretical models to describe the interconnection agreements among the providers, and analysis of those models from two alternative modeling perspectives: a purely non-cooperative game and a network perspective. There are two original contributions of the dissertation. First, we model the formation of peering/transit contracts explicitly as a decision variable in a non-cooperative game, while the current literature does not employ such modeling techniques. Second, we apply network analysis to examine interconnection decisions of the providers, which yields much realistic results. Chapter 1 provides a brief description of the Internet history, architecture and infrastructure as well as the economic literature. In Chapter 2 we develop a model, in which IAPs decide on private peering agreements, comparing the benefits of private peering relative to being connected only through National Access Points (hereafter NAPs). The model is formulated as a multistage game. Private peering agreements reduce congestion in the Internet, and so improve the quality of IAPs. The results show that even though the profits are lower with private peerings, due to large investments, the network where all the providers privately peer is the stable network. Chapter 3 discusses the interconnection arrangements among ISPs. Intra-backbone peering refers to peering between ISPs connected to the same backbone, whereas inter-backbone peering refers to peering between ISPs connected to different backbones. We formulate the model as a two-stage game. Peering affects profits through two channels - reduction of backbone congestion and ability to send traffic circumventing congested backbones. The relative magnitude of these factors helps or hinders peering. In Chapter 4 we develop a game theoretic model to examine how providers decide who they want to peer with and who has to pay transit. There is no regulation with regard to interconnection policies of providers, though there is a general convention that the providers peer if they perceive equal benefits from peering, and have transit arrangements otherwise. The model discusses a set of conditions, which determine the formation of peering and transit agreements. We argue that market forces determine the terms of interconnection, and there is no need for regulation to encourage peering. Moreover, Pareto optimum is achieved under the transit arrangements. / Ph. D.

In this work, we present our investigations into two different systems, both far from equilibrium. We first present the relaxation and ordering processes in magnetic skyrmion systems. This is followed by a study of the behavior of many species interacting on a spatially heterogeneous lattice. Magnetic skyrmions have been a subject of great interest in recent years. They have been proposed to be at the heart of next-generation computing and information storage devices. One interesting feature of magnetic skyrmions is the presence of the non-dissipative Magnus force. The Magnus force causes the skyrmions to be deflected from their direction of motion. In this work, we examine the effect the strength of this Magnus force has on the late-time ordering behavior of magnetic skyrmions. We show that the late-time ordering also shows enhanced relaxation with an increase in the Magnus force. We also studied the behavior of magnetic skyrmions when confined to a narrow channel. We show that, like before, the Magnus force helps the system order faster while experiencing a constant drive. Interestingly, when the drive was periodic, the Magnus force inhibited the relaxation in the system. Interacting populations have been a topic of scientific interest since the late eighteenth century. We studied the effect of spatial heterogeneity on a two-dimensional lattice. Using cyclic predator-prey interaction schemes, we numerically simulated the effect of asymmetric predation rates inside "habitats." We show that, due to the non-linearity of the system, the species that had a chance to escape predation did not see the largest benefit from this change. Instead, the predator of this prey saw the largest benefit from this change. The material on skyrmion systems is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Division of Materials Science and Engineering under Award Number DE-SC0002308. The population dynamics research was sponsored by the Army Research Office and was accomplished under Grant No. W911NF17-1-0156. / Doctor of Philosophy / In this work, we present our investigations into two different systems. Both of these systems are considered to be not in equilibrium. We first present is the behavior of magnetic skyrmions as the system settles into an arranged state. This is followed by a study of the behavior of multiple species interacting on a lattice where different parts of the lattice have different rules of interaction. Magnetic skyrmions are small defects that occur in specific types of magnetic materials. They have been proposed to be useful in next-generation computing devices. Similar to a curve-ball in baseball, but due to a different physical phenomenon, magnetic skyrmions follow curved paths when pushed. This effect, known as a Magnus force, causes the magnetic skyrmions to settle faster into a position relative to the other magnetic skyrmions in the system. We show that this effect also occurs when the magnetic skyrmions are being pushed through a narrow channel. If the push is periodically started and stopped, the Magnus force instead slows down the ability for magnetic skyrmions to settle into a position relative to the other magnetic skyrmions. Interacting populations have been a topic of scientific interest since the late eighteenth century. We studied the effect of changing the rules of species interaction based on where on a two-dimensional lattice the interaction occurred. Using cyclic predator-prey interaction schemes, we numerically simulated the effect of asymmetric predation rates inside "habitats." We showed that, due to the complex interaction scheme present in the system, the species that had a chance to escape predation did not see the largest benefit from this change. Instead, the predator of this prey saw the largest benefit from this change. The material on skyrmion systems is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Division of Materials Science and Engineering under Award Number DE-SC0002308. The population dynamics research was sponsored by the Army Research Office and was accomplished under Grant No. W911NF17-1-0156.

In typical attempts to control spatio-temporal chaos, spatially extended systems were subjected to protocols that perturbed them as a whole, often overlooking the potential stabilizing interaction between adjacent regions. We have shown that through the application of a time-delayed feedback scheme to a specific localized region of a system periodic patterns can be generated that are distinct from those observed when controlling the whole system. In this thesis, we present the results of two interconnected studies: 1) Spatio-temporal patterns emerging from spatially localized time-delayed feedback perturbations within transient chaotic states of the Gray-Scott reaction-diffusion system 2) Spatio-temporal patterns emerging from spatially localized time-delayed feedback perturbations within chaotic states of the cubic complex Ginzburg-Landau equation We present an investigation of two model systems: the Gray-Scott reaction-diffusion equation and the complex Ginzburg-Landau equation. Specifically we numerically study two models characterized by exhibiting various chaotic regimes. We first consider a comprehensive study of the Gray-Scott model highlighting key details about different parameter space regimes and their relative proximity to the chaotic regime. Through a systematic investigation of the effects of the model control parameters, time-delayed feedback control strength parameters, perturbed region widths, and other quantities, we show that novel patterns can be formed through the appropriate choice of perturbation region and strength. For the second study we use spatially localized time-delayed feedback on the one-dimensional complex Ginzburg-Landau equation and demonstrate, through the numerical integration of the resulting real and imaginary equations, the stabilization of novel periodic patterns within three distinct chaotic regimes. In these studies we have shown that selectively applying a time-delayed feedback scheme to a specific spatially localized region of a chaotic system can bring forth periodic patterns that are distinct from those observed when applying a perturbation to the whole system. Depending on the protocol used, these new patterns can emerge either in the perturbed or the unperturbed region. The mechanism underlying the observed pattern generation is related to the interplay between diffusion across the interfaces separating the different regions and time-delayed feedback. Research was sponsored by the Army Research Office and was accomplished under Grant No. W911NF-17-1-0156. / Doctor of Philosophy / In typical attempts to control spatio-temporal chaos, spatially extended systems were subjected to protocols that perturbed them as a whole, often overlooking the potential stabilizing interaction between adjacent regions. We have shown that through the application of a time-delayed feedback scheme to a specific localized region of a system periodic patterns can be generated that are distinct from those observed when controlling the whole system. We present an investigation of two model systems: the Gray-Scott reaction-diffusion equation and the complex Ginzburg-Landau equation. We first consider a comprehensive study of the Gray-Scott model highlighting key details about different parameter space regimes and their relative proximity to the chaotic regime. Through a systematic investigation of the effects of the model control parameters, time-delayed feedback control strength parameters, perturbed region widths, and other quantities, we show that novel patterns can be formed through the appropriate choice of perturbation region and strength. For the second study we use spatially localized time-delayed feedback on the one-dimensional complex Ginzburg-Landau equation and demonstrate, through the numerical integration of the resulting real and imaginary equations, the stabilization of novel periodic patterns within chaotic regimes. In these studies we have shown that selectively applying a time-delayed feedback scheme to a specific region of a chaotic system can generate periodic patterns that are distinct from those observed when controlling the whole system. Depending on the protocol used, these new patterns can emerge either in the perturbed or the unperturbed region. The mechanism underlying the observed pattern generation is related to the interplay between diffusion across the interfaces separating the different regions and time-delayed feedback. Research was sponsored by the Army Research Office and was accomplished under Grant No. W911NF-17-1-0156.

The morphologic variability of a fossil assemblage is of interest in many paleontological studies. However, many fossil assemblages are time-averaged; that is, many generations of non-contemporaneous organisms are mixed into the same fossil bed. Assemblages of robust mollusk shells deposited in nearshore marine environments are often time-averaged over 100's to 1000's of years. Mixing many generations of a taxon can increase measured morphologic variability over that of a single generation if morphology is changing during the interval of time-averaging. If morphology is changing, time-averaging can also alter observed correlations between morphologic variables, as well as allometric growth patterns. If morphology is static, then time-averaging will not increase variability or otherwise obscure patterns of morphologic variability. Testing the effects of time-averaging on morphology will help determine the reliability of information derived from the fossil record. In this study, morphologic variability was compared between 6 standing crop, living populations of Mercenaria campechiensis (Bivalvia) and two fossil assemblages of M. campechiensis and M. permagna. One fossil sample was collected as a series of superposed units that could be analyzed individually or in aggregate. The x,y coordinates of 13 landmarks and pseudolandmarks were recorded on over 600 valves, and variability was calculated using Least Squares Procrustes Analysis. Once corrections were made for allometry, the variabilities of the samples drawn from single time-averaged fossil beds were indistinguishable from the variabilities of the recent samples. For this data set, the variabilities of the fossil samples could be used without reservation to estimate the variability of the standing crop populations from which they formed. Morphology was quite stable over the 100's to 1000's of years that likely passed as the assemblages accumulated. A small amount of analytical time-averaging of the samples increases variability slightly, but additional analytical time-averaging causes no further increase. Very slight morphologic fluctuations are evident at time spans exceeding 100's to 1000's of years. Lumping geographically separated samples and samples of different species also increases variability. Morphologic stasis is evident in Mercenaria over 100's to 1000's of years, but previous studies have indicated that evolutionary rates over this time frame are typically high. These studies are based on colonization events, however, and are biased towards high rates. Data gathered here and in previous studies suggest that local populations may evolve rapidly at their founding, but that stasis follows this initial burst of change. This model describes a pattern similar to Punctuated Equilibrium at a lower level of the genealogical hierarchy, and is here termed "Punctuated Equilibrium, Jr." This model can be further tested in empirical studies and should aid in determining the causes of species-level evolutionary patterns. / Master of Science