Browsing by Author "Shipman, Patrick, advisor"
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Item Open Access Bifurcation of semialgebraic maps(Colorado State University. Libraries, 2014) Drendel, Jesse William, author; Bates, Daniel, advisor; Shipman, Patrick, advisor; Tavener, Simon, committee member; Antolin, Michael, committee memberA semi-algebraic map is a function from a space to itself whose domain and graph are unions of solutions to systems of polynomial equations and inequalities. Thus it is a very general object with many applications, some from population genetics. The isoclines of such a map are semi-algebraic sets, which enjoy many striking properties, the most consequential of which here is that there is an algorithm to compute a "cylindrical decomposition" adapted to any finite family of semi-algebraic sets. The main subject of this paper is that a cylindrical decomposition adapted to the isoclines of a semi-algebraic map partitions parameter space into a tree which isolates bifurcations.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 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 Expected distances on homogeneous manifolds and notes on pattern formation(Colorado State University. Libraries, 2023) Balch, Brenden, author; Shipman, Patrick, advisor; Bradley, Mark, committee member; Shonkwiler, Clay, committee member; Peterson, Chris, committee member; Chen, Hua, committee memberFlag manifolds are generalizations of projective spaces and other Grassmannians: they parametrize flags, which are nested sequences of subspaces in a given vector space. These are important objects in algebraic and differential geometry, but are also increasingly being used in data science, where many types of data are properly understood as subspaces rather than vectors. In Chapter 1 of this dissertation, we discuss partially oriented flag manifolds, which parametrize flags in which some of the subspaces may be endowed with an orientation. We compute the expected distance between random points on some low-dimensional examples, which we view as a statistical baseline against which to compare the distances between particular partially oriented flags coming from geometry or data. Lens spaces are a family of manifolds that have been a source of many interesting phenomena in topology and differential geometry. Their concrete construction, as quotients of odd-dimensional spheres by a free linear action of a finite cyclic group, allows a deeper analysis of their structure. In Chapter 2, we consider the problem of moments for the distance function between randomly selected pairs of points on homogeneous three-dimensional lens spaces. We give a derivation of a recursion relation for the moments, a formula for the kth moment, and a formula for the moment generating function, as well as an explicit formula for the volume of balls of all radii in these lens spaces. Motivated by previous results showing that the addition of a linear dispersive term to the two-dimensional Kuramoto-Sivashinsky equation has a dramatic effect on the pattern formation, we study the Swift-Hohenberg equation with an added linear dispersive term, the dispersive Swift-Hohenberg equation (DSHE) in Chapter 3. The DSHE produces stripe patterns with spatially extended defects that we call seams. A seam is defined to be a dislocation that is smeared out along a line segment that is obliquely oriented relative to an axis of reflectional symmetry. In contrast to the dispersive Kuramoto-Sivashinsky equation, the DSHE has a narrow band of unstable wavelengths close to an instability threshold. This allows for analytical progress to be made. We show that the amplitude equation for the DSHE close to threshold is a special case of the anisotropic complex Ginzburg-Landau equation (ACGLE) and that seams in the DSHE correspond to spiral waves in the ACGLE. Seam defects and the corresponding spiral waves tend to organize themselves into chains, and we obtain formulas for the velocity of the spiral wave cores and for the spacing between them. In the limit of strong dispersion, a perturbative analysis yields a relationship between the amplitude and wavelength of a stripe pattern and its propagation velocity. Numerical integrations of the ACGLE and the DSHE confirm these analytical results. Chapter 4 explores the measurement and characterization of order in non-equilibrium pattern forming systems. The study focuses on the use of topological measures of order, via persistent homology and the Wasserstein metric. We investigate the quantification of order with respect to ideal lattice patterns and demonstrate the effectiveness of the introduced measures of order in analyzing imperfect three-dimensional patterns and their time evolution. The paper provides valuable insights into the complex pattern formation and contributes to the understanding of order in three dimensions.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 Modeling of atmospherically important vapor-to-particle reactions(Colorado State University. Libraries, 2014) Hashmi, Bahaudin, author; Shipman, Patrick, advisor; Liu, Jiangguo, advisor; Thompson, Stephen, committee memberLiesegang ring formation is a special type of chemical pattern formation in which a spatial order is formed by density fluctuations of a weakly soluble salt. The Vapor-to-Particle nucleation process that is believed to produce these Liesegang rings is theorized to be the cause of mini-tornadoes and mini-hurricanes developed in a lab. In this thesis, we develop a one-dimensional finite element scheme for modeling laboratory experiments in which ammonia and hydrogen chloride vapor sources are presented to either end of the tubes. In these experiments, a reaction zone develops and propagates along the tube. Both numerical simulations and the laboratory experiments find an increasing amplitude of oscillations at the reaction front.Item Open Access Nonlinear dynamics and machine learning classification of plant pigment patterns(Colorado State University. Libraries, 2023) Wong Dolloff, Kaylee, author; Shipman, Patrick, advisor; Mueller, 6Jennifer, committee member; von Fischer, Joe, committee memberPlants exhibit a variety of vibrant colors that are both beautiful and functional. They owe their reds, purples, and blues to a class of pigments called anthocyanins. Many plants possess spatial variation in their anthocyanin concentration and color, which manifest as diverse patterns on their leaves and flowers. Flower patterns can influence interactions with pollinators, who may have innate preferences for certain patterns and can learn to distinguish between them. Recent work has identified the genes and proteins involved in activation and inhibition of anthocyanin synthesis in some species of Mimulus and showed that their dynamics can be described with a two-component diffusion model. In this thesis, we combine numerical simulations of this model with machine-learning algorithms to classify patterns based on a parameter value that influences the pattern spot size and density. A key challenge is to successfully classify using 2-dimensional spot data, which would permit the classification of real petal data from photos. Our approach makes use of the Voronoi mountain function to construct a 3-dimensional surface from the 2-dimensional data. Classification is very successful with simulated data, and it produces plausible results for real Mimulus petals.Item Open Access Nonlinear dynamics of plant pigmentation(Colorado State University. Libraries, 2022) Hsu, Wei-Yu, author; Shipman, Patrick, advisor; Mueller, Jennifer, committee member; Bradley, Richard, committee member; Finke, Richard, committee memberRed, blue, and purple colors in plants are primarily due to plant pigments called anthocyanins. In a plant cell, an equilibrium is established between anionic and cationic forms of anthocyanins as well electrically neutral colorless forms called hemiketals. In typical cellular pH ranges, the colorless hemiketal would be expected to be the dominant form. Why then, do plants, in fact, display colors? We propose that this is part due to self association and intermolecular association of the colored forms of anthocyanins. We develop a series of models for the interconversion of the colorless and colored forms of anthocyanins, including zwitterionic species and extend these models to include association of the colored species. Analysis of these models leads us to suggest and implement experiments in which the total concentration changes over time, either slowly or quickly compared to the kinetics. Coupling these models to a system of partial differential equations for in vivo anthocyanin synthesis (a modification of the Gierer-Meinhardt activator-inhibitor model), we simulate and analyze a variety of colorful spotted patterns in plant flowers. These studies are aided by a linear stability analysis and nonlinear analysis of the modified Gierer-Meinhardt model. The extended model that we propose is a first model to analyze the effects of association in pattern formation. Association may occur with various geometries which have an effect on the absorbance spectrum. Based on the Beer–Lambert law and our evaporative experiments, we develop methods of deconvoluting absorbance spectra of anthocyanin solutions into absorbance spectra of monomers, dimers and trimers, thus providing clues into the geometry of the smallest associated particles. Finally, we propose a novel geometric method of probing association by observing the changing shape of evaporating solution droplets. The associated mathematical model involves solving the highly nonlinear mean-curvature equation with nonconstant mean curvature (surface tension), and we present new solutions making use of the hodograph transform.Item Open Access Nucleation and growth: modeling the NH3 - HCL reaction(Colorado State University. Libraries, 2012) Shinn, Jaime M., author; Shipman, Patrick, advisor; Liu, James, committee member; Mueller, Jennifer, committee member; Thompson, Stephen, committee memberOne of the trademarks of a Liesegang ring system is the exhibition of a moving reaction front to form a periodic precipitation pattern. This phenomenon has been studied by both chemists and mathematicians. The periodic patterns produced have developed an interest from a mathematical perspective, while the theory and mechanism behind these patterns has created interest from a chemist's point of view. Many mathematical models have been proposed, and much interest has been invested in studying the mechanism behind these Liesegang ring systems. In particular, we will consider the NH3-HCl system, a gas-phase system in which the two gases (NH3 and HCl) diffuse into a tube and meet to form a solid precipitate. The reaction front then moves down the tube, forming a Liesegang banding pattern along the way. In this thesis, we derive a model for this system and examine some results of the model, which contribute to the theory and mechanism behind the NH3-HCl system. We predict the position of the first and last Liesegang band formed, and we examine the effect of the tube length of our system. Front velocity data from the model has also been obtained and is shown to correlate well with experimental data. We also note that the width of the heterogeneous nucleation zone increases as the concentration ratio of NH3 to HCl decreases, and we discuss the effect that water vapor has on the system.Item Open Access Persistent homology of the logistic map: an exploration of chaos(Colorado State University. Libraries, 2014) Neville, Rachel, author; Shipman, Patrick, advisor; Peterson, Chris, committee member; Thompson, Stephen, committee memberGiven a discrete sampling of points, how can one reconstruct the underlying geometric object? Further, the question arises how can one discern between noise and sampling distortion and important topological features. Algebraic and topological techniques used computationally can prove to be powerful and currently unconventional tools to understand the "shape" of data. In recent years, persistent homology has been explored as a computational way to capture information regarding the longevity of topological features of discrete data sets. In this project, the persistent homology of functions is explored specifically as a way of examining features of functions. Persistent homology tracks the longevity of connected components of level sets in a persistence diagram. By connecting points generated by a discrete time dynamical system with line segments, this data can be viewed as a (piece-wise linear) function, persistent homology is used to track features of the data. This provides a novel and useful tool for computationally examining dynamical systems. The logistic map is one of the simplest examples of a nonlinear map that displays periodic behavior for some parameter values, but for others, displays chaotic behavior. When the persistence diagram is generated for an orbit of the logistic map, all of the points surprisingly lie approximately on a line. This is not true for a general sequence. This pattern arises not only after stability has been reached in the periodic case, but also as points approach stability for parameters in the periodic regime but also perhaps more surprisingly, for parameter values that lie in the chaotic regime as well. In fact, the slope of this line is fairly similar as the parameter values are varied. This arises from the order in which the points pair to form the persistence diagram and a scaling factor seen in the periodic regime of a class of maps (including the logistic map). It is interesting that the effects of this scaling are still seen in the chaotic regime. This pattern not only arises for the logistic map, but for other unimodal maps and other higher dimensional systems that are "close" to these maps such as the Lorenz system.Item Open Access The mathematical modeling and analysis of nonlocal ecological invasions and savanna population dynamics(Colorado State University. Libraries, 2013) Strickland, William Christopher, author; Dangelmayr, Gerhard, advisor; Shipman, Patrick, advisor; Zhou, Yongcheng, committee member; Brown, Cynthia, committee memberThe main focus of this dissertation is the development and analysis of two new mathematical models that individually address major open problems in ecology. The first challenge is to characterize and model the processes that result in a savanna ecosystem as a stable state between grassland and forest, and the second involves modeling the non-local spread of a biological invader over heterogeneous terrain while incorporating the influence of a mass transportation network on the system. Both models utilize and compare work done in other, often more opaque, modeling paradigms to better develop transparent and application-ready solutions which can be easily adapted and inform ecological work done in the field. Savanna is defined by the coexistence of trees and grass in seasonally dry areas of the tropics and sub-tropics, but there is no consensus as to why savanna occurs as a stable state between tropical grassland and forest. To understand the dynamics behind the tree-grass relationship, we begin by reviewing and analyzing approaches in currently available savanna models. Next, we develop a mathematical model for savanna water resource dynamics based on FLAMES, an Australian process-based software model created to capture the effects of seasonal rainfall and fire disturbance on savanna tree stands. As a mathematically explicit dynamical system represented by coupled differential equations, the new model immediately has the advantage of being concise and transparent compared to previous models, yet still robust in its ability to account for different climate and soil characteristics. Through analytical analysis of the model, we show a clear connection between climate and stand structure, with particular emphasis on the length and severity of the dry season. As a result, we can numerically quantify the parameter space of year-by-year stochastic variability in stand structure based on rainfall and fire probabilities. This results in a characterization of savanna existence in the absence of extreme fire suppression based on the availability of water resources in the soil due to climate and ground water retention. One example of the model's success is its ability to predict a savanna environment for Darwin, Australia and a forest environment for Sydney, even though Sydney receives less annual rainfall than Darwin. The majority of this dissertation focuses on modeling the spread of a biological invader in heterogeneous domains, where invasion often takes place non-locally, through nearby human transportation networks. Since early detection and ecological forecasting of invasive species is urgently needed for rapid response, accurately modeling invasions remains a high priority for resource managers. To achieve this goal, we begin by revisiting a particular class of deterministic contact models obtained from a stochastic birth process for invasive organisms. We then derive a deterministic integro-differential equation of a more general contact model and show that the quantity of interest may be interpreted not as population size, but rather as the probability of species occurrence. We then proceed to show how landscape heterogeneity can be included in the model by utilizing the concept of statistical habitat suitability models which condense diverse ecological data into a single statistic. Next, we develop a model for vector-based epidemic transport on a network as represented by a strongly connected, directed graph, and analytically compute the exact optimal control for suppression of the infected graph vectors. Since this model does not require any special assumptions about the underlying spatiotemporal epidemic spread process, it should prove suitable in a variety of application contexts where network based disease vector dynamics need to be understood and properly controlled. We then discuss other methods of control for the special case of the integro-differential model developed previously and explore numerical results of applying this control. Finally, we validate model results for the Bromus tectorum invasion of Rocky Mountain National Park using data collected by ecologists over the past two decades, and illustrate the effect of various controls on this data. A final chapter concerns a problem of cognitive population dynamics, namely vowel pronunciation in natural languages. We begin by developing a structured population approach to modeling changes in vowel systems, taking into account learning patterns and effects such as social trends. Our model treats vowel pronunciation as a continuous variable in vowel space and allows for continuous dependence of vowel pronunciation on time and age of the speaker. The theory of mixtures with continuous diversity provides a framework for the model, which extends the McKendrick-von Foerster equation to populations with age and phonetic structures. Numerical integrations of the model reveal how shifts in vowel pronunciation may occur in jumps or continuously given perturbations such as the influx of an immigrant population.Item Open Access Topological techniques for characterization of patterns in differential equations(Colorado State University. Libraries, 2017) Neville, Rachel A., author; Shipman, Patrick, advisor; Adams, Henry, committee member; Krummel, Amber, committee member; Shonkwiler, Clayton, committee memberComplex data can be challenging to untangle. Recent advances in computing capabilities has allowed for practical application of tools from algebraic topology, which have proven to be useful for qualitative and quantitative analysis of complex data. The primary tool in computational topology is persistent homology. It provides a valuable lens through which to study and characterize complex data arising as orbits of dynamical systems and solutions of PDEs. In some cases, this includes leveraging tools from machine learning to classify data based on topological characteristics. We see a unique pattern arising in the persistence diagram of a class of one-dimensional discrete dynamical systems--even in chaotic parameter regimes, and connect this to the dynamics of the system in Chapter 2. Geometric pattern structure tell us something about the parameters driving the dynamics in the system as is the case for anisotropic Kuramoto-Sivashinsky equation which displays chaotic bubbling. We will see this in Chapters 3 and 4. Defects in pattern-forming systems be detected and tracked and studied to characterize the degree of order in near-hexagonal nanodot structures formed by ion bombardment, which will be developed in Chapter 5.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.