Indiana University-Purdue University Indianapolis (IUPUI) / Associated to any finite simple graph $\Gamma$ is the {\em chromatic polynomial} $\P_\Gamma(q)$ whose complex zeros are called the {\em chromatic zeros} of $\Gamma$. A hierarchical lattice is a sequence of finite simple graphs $\{\Gamma_n\}_{n=0}^\infty$ built recursively using a substitution rule expressed in terms of a generating graph. For each $n$, let $\mu_n$ denote the probability measure that assigns a Dirac measure to each chromatic zero of $\Gamma_n$. Under a mild hypothesis on the generating graph, we prove that the sequence $\mu_n$ converges to some measure $\mu$ as $n$ tends to infinity. We call $\mu$ the {\em limiting measure of chromatic zeros} associated to $\{\Gamma_n\}_{n=0}^\infty$. In the case of the Diamond Hierarchical Lattice we prove that the support of $\mu$ has Hausdorff dimension two. The main techniques used come from holomorphic dynamics and more specifically the theories of activity/bifurcation currents and arithmetic dynamics. We prove a new equidistribution theorem that can be used to relate the chromatic zeros of a hierarchical lattice to the activity current of a particular marked point. We expect that this equidistribution theorem will have several other applications, and describe one such example in statistical mechanics about the Lee-Yang-Fisher zeros for the Cayley Tree.

We address the problem of finding groupoid models for C*-algebras given some prescribed K-theory data. This is a reasonable question because a groupoid model for a C*-algebra reveals much about the structure of the algebra. A great deal of progress towards solving this problem has been made using constructions with inductive limits, subgroupoids, and dynamical systems. This dissertation approaches the question with a more specific methodology in mind, with factor groupoids. In the first part, we develop a portrait of relative K-theory for C*-algebras using the general framework of Banach categories and Banach functors due to Max Karoubi. The purpose of developing such a portrait is to provide a means of analyzing the K-theory of an inclusion of C*-algebras, or more generally of a *-homomorphism between two C*-algebras. Another portrait may be obtained using a mapping cone construction and standard techniques (it is shown that the two presentations are naturally and functorially isomorphic), but for many examples, including the ones considered in the second part, the portrait obtained by Karoubi's construction is more convenient. In the second part, we construct examples of factor groupoids and analyze their C*-algebras. A factor groupoid setup (two groupoids with a surjective groupoid homomorphism between them) induces an inclusion of two C*-algebras, and therefore the portrait of relative K-theory developed in the first part, together with an excision theorem, can be used to elucidate the structure. The factor groupoids are obtained as quotients of AF-groupoids and certain extensions of Cantor minimal systems using iterated function systems. We describe the K-theory in both cases, and in the first case we show that the K-theory of the resulting C*-algebras can be prescribed through the factor groupoids. / Graduate

No description available.

In this Thesis we study two types of mechanical nonholonomic systems, namely systems with linear constraints and lagrangian with a linear term in the velocities, and nonholonomic systems with affine constraints and lagrangian without a linear term in the velocities. For the former type of systems we construct an almost-Poisson bracket using elements related to a riemannian metric induced by the kinetic energy, and we show that under certain conditions gauge momenta exist. For the latter type of systems, we focus on the ones possessing a \emph{Noether symmetry}. To everyone of these systems we associate an equivalent system of the former type, and we exhibit the procedure to relate them and their gauge momentum. As a test case for the theory, we analyze the system of a heavy ball rolling without slipping on a rotating surface of revolution: we elucidate that also in this framework the so-called Routh integrals are related to symmetries, we give conditions for boundedness of the motions. In the particular case the surface of revolution is an inverted cone we characterize the qualitative behavior of the motions.

In this paper, we study various examples of dynamical systems found in nature and extract the necessary concepts to build upon. Then, we develop and propose a new deterministic model for COVID-19 propagation. Our model should serve two purposes. First, we will approximate the infected and deceased individuals after a given time during the pandemic. Then, using a linearized subsystem describing infectious compartments about the disease- free equilibrium (DFE), we will determine the basic reproductive number (R0) by the next-generation matrix method.

No description available.

This report describes the simulation of an inverted pendulum control system. The purpose is to provide an interesting learning process through high resolution color graphics animations in the control of dynamic systems. The software uses the graphic capabilities extensively to make it very user-friendly and highly interactive. A numerical analysis method is used to solve the systems of equations. The animation driven by the results is then displayed on the video terminal. Facilities range from selection of controllers, changing of system parameters, plotting graphs, and hardcopy outputs.

Cognitive behavioral therapy (CBT) has been shown to be an efficacious treatment for youth anxiety, but we do not have a satisfactory understanding of how CBT achieves its beneficial effects. The present study used a dynamical systems framework to model ecological momentary assessment (EMA) data collected via a cellular telephone and to examine patterns of affective variability over time and across CBT and client-centered therapy (CCT) treatments. Dynamical systems are systems that change over time in response to input from the environment and from itself at an earlier time. Associations between pretreatment variables and patterns of affect at pretreatment and over the course of the treatments were also examined. Results revealed significant decreases in affective variability over the course of treatment for participants in the CBT condition, but not for those in the CCT condition. Several variables (i.e., emotion regulation coping related to anger, depressive symptoms, and affiliative temperament) predicted initial affective variability ratings and changes in affective variability over time. Findings provide initial support for the dynamical systems approach to examining changes that occur during treatment. Implications for the examination of mechanisms of change are discussed. / Psychology

We analyze a simplified mathematical model of the complete degradation of monochlorophenol. The model takes form of a system of six ordinary differential equations, the dynamics of which can be reduced to the dynamics of a three-dimensional system on the invariant set. We extend the previous analysis by considering multiple substrate inflow. We also focus on the bifurcations occurring in the system and their biological meaning. / Thesis / Master of Science (MSc)

The objective of this dissertation is to develop theoretical and computational tools for the study of qualitative changes in the dynamics of systems with discontinuities, also known as nonsmooth or hybrid dynamical systems, under parameter variations. Accordingly, this dissertation is divided into two parts. The analytical section of this dissertation discusses mathematical tools for the analysis of hybrid dynamical systems and their application to a series of model examples. Specifically, qualitative changes in the system dynamics from a nonimpacting to an impacting motion, referred to as grazing bifurcations, are studied in oscillators where the discontinuities are caused by impacts. Here, the study emphasizes the formulation of conditions for the persistence of a steady state motion in the immediate vicinity of periodic and quasiperiodic grazing trajectories in an impacting mechanical system. A local analysis based on the discontinuity-mapping approach is employed to derive a normal-form description of the dynamics near a grazing trajectory. Also, the results obtained using the discontinuity-mapping approach and direct numerical integration are found to be in good agreement. It is found that the instabilities caused by the presence of the square-root singularity in the normal-form description affect the grazing bifurcation scenario differently depending on the relative dimensionality of the state space and the steady state motion at the grazing contact. The computational section presents the structure and applications of a software program, TC-HAT, developed to study the bifurcation analysis of hybrid dynamical systems. Here, we present a general boundary value problem (BVP) approach to locate periodic trajectories corresponding to a hybrid dynamical system under parameter variations. A methodology to compute the eigenvalues of periodic trajectories when using the BVP formulation is illustrated using a model example. Finally, bifurcation analysis of four model hybrid dynamical systems is performed using TC-HAT. / Ph. D.