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No description available.

Computer simulation of a control system is a valuable tool in design or performance evaluation. This is especially true when non-linear elements cannot be ignored and must be included within the model. A general purpose block diagram oriented simulation program will be developed which can utilize continuous, discrete and non-linear building blocks. The software tool will be demonstrated by means of an example.

<p>The aim of this thesis is to provide mathematical tools for an alternative to the mainstream study of macroeconomics with a focus on debt-driven dynamics.</p> <p>We start with a survey of the literature on formalizations of Minsky's Financial Instability Hypothesis in the context of stock-flow consistent models.</p> <p>We then study a family of macro-economical models that date back to the Goodwin model. In particular, we propose a stochastic extension where noise is introduced in the productivity. Besides proving existence and uniqueness of solutions, we show that orbits must loop around a specific point indefinitely.</p> <p>Subsequently, we analyze the Keen model, where private debt is introduced. We demonstrate that there are two key equilibrium points, intuitively denoted good and bad equilibria. Analytical stability analysis is followed by numerical study of the basin of attraction of the good equilibrium.</p> <p>Assuming low interest rate levels, we derive an approximate solution through perturbation techniques, which can be solved analytically. The zero order solution, in particular, is shown to converge to a limit cycle. The first order solution, on the other hand, is shown to explode, rendering its use dubious for long term assessments.</p> <p>Alternatively, we propose an extension of the Keen model that addresses the immediate completion time of investment projects. Using distributed time delays, we verify the existence of the key equilibrium points, good and bad, followed by their stability analysis. Through bifurcation theory, we verify the existence of limit cycles for certain mean completion times, which are absent in the original Keen model.</p> <p>Finally, we examine the Keen model under government intervention, where we introduce a general form for the government policy. Besides performing stability analysis, we prove several results concerning the persistence of both profits and employment. In economical terms, we demonstrate that when the government is responsive enough, total economic meltdowns are avoidable.</p> / Doctor of Philosophy (PhD)

Diffusion is defined as the movement of a substance down a concentration gradient. The physics of diffusion is well described by Fick’s law. A density-dependent diffusion process is a one in which the diffusion coefficient is a function of the localized density of the diffusing substance. In my thesis I analyze the density-dependent diffusion behavior of two independent processes of biological interest. The first is tumor growth and invasion. The second is single-file diffusion, which in turn has a considerable biological significance, since it has been recently used to model transitions of proteins through DNA molecules. Tumor invasion of normal tissue is a complex process, involving cell migration and proliferation. It is useful to mathematically model tumor invasion in an attempt to find a common, underlying mechanism by which primary and metastatic cancers invade and destroy normal tissue. In our approach we make no assumptions about the process of carcinogenesis, that is, we are not modeling the genetic changes which result in transformation, nor do we seek to understand the causes of these changes. Similarly, we do not attempt to model the large-scale morphological features of tumors such as central necrosis. Rather, we concentrate on the microscopic-scale population interactions occurring at the tumor-host interface, reasoning that these processes strongly influence the clinically-significant manifestations of invasive cancer. We analyze a reaction-diffusion model due to Gawlinski, Gatenby, and others of the acid-mediated tumor invasion mechanism that incorporates the ion concentration as a reaction factor affecting the tumor growth and invasion process. It also adds density-dependent diffusion parameters to the reaction terms yielding independent reaction-diffusion equations for the normal, tumor, and acid populations. For reasonable biological parameters we study the fixed-points of the partial differential equations central of the model and their stability. The fixed-points determine the balance reached by the normal, tumor, and acid populations. As for the second application we present on density-dependent diffusion processes, we consider a model for single-file diffusion that is relevant in a variety of biological processes, for example ion transport in channels. The model is of mathematical and physical as well as biological interest because it exhibits an anomalously-slow tracer diffusion fundamentally different from diffusion without the single-file restriction. We carry out extensive computer simulations to study the role of particle adhesion and space availability (hard-core exclusion) in the model. Both tracer (tagged-particle) and bulk or collective diffusion are considered. Tracer diffusion focuses on the diffusion of the individual particles relative to their starting points, whereas bulk diffusion focuses on the diffusion of the particle distribution as a whole. The nature of the diffusion depends strongly on the initial particle distribution, and both homogeneous and inhomogeneous (for example Gaussian) distributions are considered. For all these models a density-dependent diffusion behavior is confirmed by studying the time evolutions of the moments and widths of these distributions. / Physics

The mechanisms of acceleration and transport of collisionless plasma in the presence of electromagnetic turbulence (EMT) still remains not fully understood. The particle-EMT interaction can be modelled as the interaction of the particle with a particular wave in the presence of random noise. It has been shown that in such a model the acceleration of the charged particles can be almost free. This effect is known as resonance, which can be explained by the so-called “surfatron” mechanism. We have conducted several numerical simulations for the models with and without the presence of EMT. The turbulence has been modeled as small random fluctuations on the background magnetic field. Particles dynamics consist of two regimes of motion: (i) almost free (Larmor) rotation and (ii) captured (resonance) propagation, which are given by two different sets of invariants. We have determined the necessary conditions for capture and release from resonance for the model without fluctuations, as well as the intrinsic structure of the initial conditions domain for particles in order to be captured. We observed a difference in the orders of magnitude of the dispersion of adiabatic invariant due to the effects of the added fluctuations at the resonance. These results are important to describe the mixing of the different energy levels in the presence of EMT. To understand the impact of the EMT on the system dynamics, we have performed statistical analysis of the effects that different characteristics of the random fluctuations have on the system. The particles' energy gain can be viewed as a random walk over the energy levels, which can be described in terms of the diffusion partial differential equation for the probability distribution function. This problem can be reverse-engineered to understand the nature and structure of the EMT, knowing beforehand the energy distribution of a set of particles. / Mechanical Engineering

One goal of systems biology is to predict and modify the behavior of biological networks by accurately monitoring and modeling their responses to certain types of perturbations. The construction of mathematical models based on observation of these responses, referred to as reverse engineering, is an important step in elucidating the structure and dynamics of such networks. Continuous models, described by systems of differential equations, have been used to reverse engineer biochemical networks. Of increasing interest is the use of discrete models, which may provide a conceptual description of the network. In this dissertation we introduce a discrete modeling approach, rooted in computational algebra, to reverse-engineer networks from experimental time series data. The algebraic method uses algorithmic tools, including Groebner-basis techniques, to build the set of all discrete models that fit time series data and to select minimal models from this set. The models used in this work are discrete-time finite dynamical systems, which, when defined over a finite field, are described by systems of polynomial functions. We present novel reverse-engineering algorithms for discrete models, where each algorithm is suitable for different amounts and types of data. We demonstrate the effectiveness of the algorithms on simulated networks and conclude with a description of an ongoing project to reverse-engineer a real gene regulatory network in yeast. / Ph. D.