Browsing by Author "Dangelmayr, Gerhard, committee member"
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Item Open Access Continued exploration of nearly continuous Kakutani equivalence(Colorado State University. Libraries, 2013) Springer, Bethany Diane, author; Shipman, Patrick, advisor; del Junco, Andres, advisor; Eykholt, Richard, committee member; Dangelmayr, Gerhard, committee member; Pries, Rachel, committee memberNearly continuous dynamical systems, a relatively new field of study, blends together topological dynamics and measurable dynamics/ergodic theory by asking that properties hold modulo sets both meager and of measure zero. In the measure theoretic category, two dynamical systems (X, T) and (Y, S) are called Kakutani equivalent if there exists measurable subsets A subset of X and B subset of Y such that the induced transformations TA and SB are measurably conjugate. We say that a set A subset of X is nearly clopen if it is clopen in the relative topologyof a dense Gδ subset of full measure. Nearly continuous Kakutani equivalence refines the measure-theoretic notion by requiring the sets A and B to be nearly clopen and TA and SB to be nearly continuously conjugate. If A and B have the same measure, then we say that the systems are nearly continuously evenly Kakutani equivalent. All irrational rotations of the circle and all odometers belong to the same equivalence class for nearly continuous even Kakutani equivalence. For our first main result, we prove that if A and B are nearly clopen subsets of the same measure of X and Y respectively, and if ϕ is a nearly continuous conjugacy between TA and SB, then ϕ extends to a nearly continuous orbit equivalence between T and S. We also prove that if A subset of X and B subset of Y are nearly clopen sets such that the measure of A is larger than the measure of B, and if T is a nearly uniquely ergodic transformation and TA is nearly continuously conjugate to SB, then there exists B' subset of Y such that X is nearly continuously conjugate to SB'. We then introduce the natural topological analog of rank one transformations, called strongly rank one transformations, and show that all strongly rank one transformations are nearly continuously evenly Kakutani equivalent to the class containing all adding machines. Our main result proves that all minimal isometries of compact metric spaces are nearly continuously evenly Kakutani equivalent to the binary odometer.Item Open Access Discrete-time topological dynamics, complex Hadamard matrices, and oblique-incidence ion bombardment(Colorado State University. Libraries, 2014) Motta, Francis Charles, author; Shipman, Patrick D., advisor; Dangelmayr, Gerhard, committee member; Peterson, Chris, committee member; Bradley, Mark, committee memberThe topics covered in this dissertation are not unified under a single mathematical discipline. However, the questions posed and the partial solutions to problems of interest were heavily influenced by ideas from dynamical systems, mathematical experimentation, and simulation. Thus, the chapters in this document are unified by a common flavor which bridges several mathematical and scientific disciplines. The first chapter introduces a new notion of orbit density applicable to discrete-time dynamical systems on a topological phase space, called the linear limit density of an orbit. For a fixed discrete-time dynamical system, Φ(χ) : M → M defined on a bounded metric space, we introduce a function E : {γχ : χ ∈ Mg} → R∪{∞} on the orbits of Φ, γχ ≐ {Φt(χ) : t ∈ N}, and interpret E(γχ) as a measure of the orbit's approach to density; the so-called linear limit density (LLD) of an orbit. We first study the family of dynamical systems Rθ : [0; 1) → [0; 1)(θ ∈ (0; 1)) defined by Rθ(χ) = (χ + θ) mod 1. Utilizing a formula derived from the Three-Distance theorem, we compute the exact value of E({RtΦ(χ) : t ∈ N}, χ ∈ [0; 1)), where Φ = √5 – 1) /2. We further compute E({Rtθ(χ) : t ∈ N}; χ ∈ [0, 1)) for a class of irrational rotation angles θ = [j, j,…] with period-1 continued fraction expansions and discuss how this measure distinguishes the topologically transitive behavior of different choices of θ. We then expand our focus to a much broader class of orientation-preserving homeomorphisms of the circle and extend a result of R. Graham and J.H. van Lint about optimal irrational rotations. Finally, we consider the LLD of orbits of the Bernoulli shift map acting on sequences defined over a finite alphabet and prove bounds for a class of sequences built by recursive extension of de Bruijn sequences. To compute approximations of E(γχ) for orbits of the Bernoulli shift map, we develop an efficient algorithm which determines a point in the set of all words of a fixed length over a finite alphabet whose distance to a distinguished subset is maximal. Chapter two represents a departure from a dynamical systems problem by instead exploring the structure of the space of complex Hadamard matrices and mutually unbiased bases (MUBs) of complex Hilbert space. Although the problem is not intrinsically dynamical, our mechanisms for experimentation and exploration include an algorithm which can be viewed as a discrete-time dynamical system as well as a gradient system of ordinary differential equations (ODEs) whose fixed points are dephased complex Hadamards. We use our discrete system to produce numerical evidence which supports existing conjectures regarding complex Hadamards and mutually unbiased bases, including that the maximal size of a set of 6 x 6 MUBs is four. By applying center-manifold theory to our gradient system, we introduce a novel method to analyze the structure of Hadamards near a fixed matrix. In addition to formalizing this technique, we apply it to prove that a particular 9 x 9 Hadamard does not belong to a continuous family of inequivalent matrices, despite having a positive defect. This is the first known example of this type. The third chapter explores the phenomenon of pattern formation in dynamical systems by considering a model of off-normal incidence ion bombardment (OIIB) of a binary material. We extend the Bradley-Shipman theory of normal-incidence ion bombardment of a binary material by analyzing a system of partial differential equations that models the off-normal incidence ion bombardment of a binary material by coupling surface topography and composition. In this chapter we perform linear and non-linear analysis of the equations modeling the interaction between surface height and composition and derive a system of ODEs which govern the time-evolution of the unstable modes, allowing us to identify parameter ranges which lead to patterns of interest. In particular, we demonstrate that an unusual "dots-on-ripples" topography can emerge for nonzero angles of ion incidence θ. In such a pattern, nanodots arranged in a hexagonal array sit atop a ripple topography. We find that if dots-on-ripples are supplanted by surface ripples as θ or the ion energy are varied, the transition is continuous.Item Open Access Elevation heterogeneity and the spread of white-nose syndrome in bats(Colorado State University. Libraries, 2018) Read, Catherine E., author; Shipman, Patrick, advisor; Dangelmayr, Gerhard, committee member; Tulanowski, Elizabeth, committee memberWhite-nose syndrome (WNS) has been decimating bat populations throughout North America since its discovery in New York during the winter of 2006-2007. The fungus responsible for the disease, Pseudogymnoascus destructans, has since been confirmed as present in Washington, over 3,700 km from the epicenter. In 2012, a stochastic discrete-time dynamical system for WNS spread was developed on a spatially structured network and used to predict the spread of this wildlife epidemic. The model uses a variable for distance and two environmental variables (cave density and winter duration) to generate spread probabilities between counties of the contiguous United States. However, predictions from the 2012 model missed several recently infected counties due to the use of a cave density variable. Major cave formations are both less frequent and poorly documented in the western U.S. Furthermore, cave density may not serve as an accurate proxy for bat hibernacula across the country considering the use of crevice and cavity roosts in rock substrates west of the Great Plains. A Terrain Ruggedness Index (TRI) can thus be calculated from elevation data and used in place of cave density to quantify elevation heterogeneity and represent crevice-dwelling bat populations. Incorporating TRI into the network spread model would generate more accurate WNS presence predictions and aid in more effective management efforts to contain the spread of this deadly bat disease.Item Open Access Mechanism-enabled population balances and the effects of anisotropies in the complex Ginzburg-Landau equation(Colorado State University. Libraries, 2019) Handwerk, Derek, author; Shipman, Patrick, advisor; Dangelmayr, Gerhard, committee member; Oprea, Iuliana, committee member; Finke, Richard, committee memberThis paper considers two problems. The first is a chemical modeling problem which makes use of ordinary differential equations to discover a minimum mechanism capable of matching experimental data in various metal nanoparticle nucleation and growth systems. This research has led to the concept of mechanism-enabled population balance modeling (ME-PBM). This is defined as the use of experimentally established nucleation mechanisms of particle formation to create more rigorous population balance models. ME-PBM achieves the goal of connecting reliable experimental mechanisms with the understanding and control of particle-size distributions. The ME-PBM approach uncovered a new and important 3-step mechanism that provides the best fits to experimentally measured particle-size distributions (PSDs). The three steps of this mechanism are slow, continuous nucleation and two surface growth steps. The importance of the two growth steps is that large particles are allowed to grow more slowly than small particles. This finding of large grow more slowly than small is a paradigm-shift away from the notion of needing nucleation to stop, such as in LaMer burst nucleation, in order to achieve narrow PSDs. The second is a study of the effects of anisotropy on the dynamics of spatially extended systems through the use of the anisotropic Ginzburg-Landau equation (ACGLE) and its associated phase diffusion equations. The anisotropy leads to different types of solutions not seen in the isotropic equation, due to the ability of waves to simultaneously be stable and unstable, including transient spiral defects together with phase chaotic ripples. We create a phase diagram for initial conditions representing both the longwave k = 0 case, and for wavevectors near the circle |k| = μ using the average L² energy.Item Open Access Patterns in dynamics(Colorado State University. Libraries, 2012) Motta, Francis Charles, author; Shipman, Patrick D., advisor; Bradley, R. Mark, committee member; Cavalieri, Renzo, committee member; Dangelmayr, Gerhard, committee memberIn this paper we introduce and explore the idea of persistent homology (PH) and discuss several applications of this computational topology tool beyond its intended purpose. In particular we apply persistence to data generated by dynamical systems. The application of persistent homology to the circle map will lead us to rediscover the well-known result about the distribution of points in the orbit of this ergodic system called the Three Distance Theorem. We then apply PH to data extracted from several models of ion bombardment of a solid surface. This will present us with an opportunity to discuss new ways of interpreting PH data by introducing statistics on its output. Using these statistics we will begin to develop a technique to answer questions of interest to physicists about the degree of ordering present in the topography of a solid surface after ion bombardment. Finally we observe some inherent limitations in PH and, through simple examples, develop techniques to improve the technology. Specically we will implement algorithms to iteratively spread points on a real algebraic variety and demonstrate that the methodology works to improve the signals in the output of PH.Item Open Access The effects of environmental flow on the internal dynamics of tropical cyclones(Colorado State University. Libraries, 2012) Williams, Gabriel Jason, author; Schubert, Wayne H., advisor; Dangelmayr, Gerhard, committee member; Maloney, Eric D., committee member; van den Heever, Sue, committee memberThis dissertation focuses on two projects that examine the interaction between the internal dynamics of tropical cyclones and the large-scale environmental flow using a hierarchy of numerical model simulations. Diabatic heating from deep moist convection in the hurricane eyewall produces a towering annular structure of elevated potential vorticity (PV) called a hollow PV tower. For the first project, the three-dimensional rearrangement of hurricane-like hollow PV towers is examined in an idealized framework. For the adiabatic PV tower in the absence of environmental flow, barotropic instability causes air parcels with high PV to be mixed into the eye preferentially at lower levels, where unstable PV wave growth rates are the largest. When the diabatic forcing is included, diabatic PV production accompanies the inward mixing at low levels, and similarly diabatic PV destruction accompanies the outflow at upper-levels. The largest variation in PV is produced when the diabatic forcing is placed within the radius of maximum winds (RMW) due to its ability to efficiently extract kinetic energy from the specified heating source. For the adiabatic PV tower in vertical shear, the initial response of the vortex to the vertical shear is to tilt downshear and rotate cyclonically about the mid-level center. The cyclonic precession of the vortex around the center demonstrates the existence of an azimuthal wavenumber-1 quasimode that prevents the vertical alignment of the vortex. When the effects of diabatic forcing are included, the increase in inertial stability causes the resonant damping of the quasimode to become more efficient, leading to the emission of sheared vortex Rossby waves (VRWs) and vortex alignment. Generally, it is shown that the vortex response to vertical shear depends sensitively on the Rossby deformation radius, Rossby penetration depth, and the vortex beta Rossby number of the vortex. For the second project, we examine the development of shock-like structures in the tropical cyclone boundary layer for a stationary and slowly moving tropical cyclone. Using a two-dimensional slab boundary layer model and a three-dimensional boundary layer model, we show that both boundary layer models approximate the nonlinear viscous Burgers' equation in the tropical cyclone boundary layer. For the stationary tropical cyclone, radial inflow creates a circular shock near the surface while vertical mixing communicates the shock throughout the boundary layer. The peak Ekman pumping occurs at a height of 600 m, which is also the location of maximum turbulent transport, consistent with Hurricane Hugo (1989). For a moving TC, the asymmetry in the frictional drag causes an asymmetry in the boundary layer response. As the translation speed of the TC increases, the nonlinear asymmetric advective interactions amplify, leading to an anticyclonic spiral in the vertical velocity field and pronounced inflow in the right-front quadrant of the storm.Item Open Access Two-step coding theorem in the nearly continuous category(Colorado State University. Libraries, 2013) Salvi, Niketa, author; Shipman, Patrick, advisor; Şahin, Ayşe, advisor; Dangelmayr, Gerhard, committee member; Oprea, Iuliana, committee member; Wang, Haonan, committee memberIn measurable dynamics, one studies the measurable properties of dynamical systems. A recent surge of interest has been to study dynamical systems which have both a measurable and a topological structure. A nearly continuous Z-system consists of a Polish space X with a non-atomic Borel probability measure μ and an ergodic measure-preserving homeomorphism T on X . Let ƒ : X → R be a positive, nearly continuous function bounded away from 0 and ∞. This gives rise to a flow built over T under the function ƒ in the nearly continuous category. Rudolph proved a representation theorem in the 1970's, showing that any measurable flow, where the function ƒ is only assumed to be measure-preserving on a measurable Z-system, can be represented as a flow built under a function where the ceiling function takes only two values. We show that Rudolph's theorem holds in the nearly continuous category.