Tridiagonal Matrices And Boundary Conditions, 2016 Portland State University

#### Tridiagonal Matrices And Boundary Conditions, J. J. P. Veerman, David K. Hammond

*Mathematics and Statistics Faculty Publications and Presentations*

We describe the spectra of certain tridiagonal matrices arising from differential equations commonly used for modeling flocking behavior. In particular we consider systems resulting from allowing an arbitrary boundary condition for the end of a one-dimensional flock. We apply our results to demonstrate how asymptotic stability for consensus and flocking systems depends on the imposed boundary condition.

Signal Velocity In Oscillator Arrays, 2016 Portland State University

#### Signal Velocity In Oscillator Arrays, Carlos E. Cantos, David K. Hammond, J. J. P. Veerman

*Mathematics and Statistics Faculty Publications and Presentations*

We investigate a system of coupled oscillators on the circle, which arises from a simple model for behavior of large numbers of autonomous vehicles. The model considers asymmetric, linear, decentralized dynamics, where the acceleration of each vehicle depends on the relative positions and velocities between itself and a set of local neighbors. We first derive necessary and sufficient conditions for asymptotic stability, then derive expressions for the phase velocity of propagation of disturbances in velocity through this system. We show that the high frequencies exhibit damping, which implies existence of well-defined signal velocities c+>0 and c−f(x−c ...

A Mechanical Investigation Of Second Order Homogeneous Dynamic Equations On A Time Scale, 2016 Marshall University

#### A Mechanical Investigation Of Second Order Homogeneous Dynamic Equations On A Time Scale, Jacob E. Fischer

*Theses, Dissertations and Capstones*

This thesis covers the basic aspects of time scale calculus, a branch of mathematics combining the theories of differential equations and difference equations. Using the properties of time scale calculus we analyze a second order homogeneous dynamic equation with constant coefficients, in particular, y ∆∆ − 1 6 y ∆ + 1 8 y = 0. Following the analysis, this problem will be graphically evaluated using Marshall University’s Differential Analyzer, affectionately named Art. A differential analyzer is a machine that mechanically integrates by way of related rates of rotating rods. The process for making the jump between intervals on a time scale will be ...

Topological Data Analysis For Systems Of Coupled Oscillators, 2016 Harvey Mudd College

#### Topological Data Analysis For Systems Of Coupled Oscillators, Alec Dunton

*HMC Senior Theses*

Coupled oscillators, such as groups of fireflies or clusters of neurons, are found throughout nature and are frequently modeled in the applied mathematics literature. Earlier work by Kuramoto, Strogatz, and others has led to a deep understanding of the emergent behavior of systems of such oscillators using traditional dynamical systems methods. In this project we outline the application of techniques from topological data analysis to understanding the dynamics of systems of coupled oscillators. This includes the examination of partitions, partial synchronization, and attractors. By looking for clustering in a data space consisting of the phase change of oscillators over a ...

Postural Responses To Perturbations Of The Vestibular System During Walking In Healthy Young And Older Adults, 2015 University of Nebraska Medical Center

#### Postural Responses To Perturbations Of The Vestibular System During Walking In Healthy Young And Older Adults, Jung Hung Chien

*Theses & Dissertations*

It has been shown that approximate one-third of US adults aged 40 years and older (69 million US citizens) have some type of vestibular problems. These declining abilities of the vestibular system affect quality of life. Difficulties in performing daily activities (dressing, bathing, getting in and out of the bed and etc.) have been highly correlated to loss of balance due to vestibular disorders. The exact number of people affected by vestibular disorders is still difficult to quantify. This might be because symptoms are difficult to describe and differences exist in the qualifying criteria within and across studies. Thus, it ...

Gait Transition Dynamics Are Modulated By Experimental Protocol, 2015 University of Connecticut - Storrs

#### Gait Transition Dynamics Are Modulated By Experimental Protocol, Mohammad Abdolvahab, Jason Gordon

*Mohammad Abdolvahab*

No abstract provided.

A Posteriori Eigenvalue Error Estimation For The Schrödinger Operator With The Inverse Square Potential, 2015 Wayne State University

#### A Posteriori Eigenvalue Error Estimation For The Schrödinger Operator With The Inverse Square Potential, Hengguang Li, Jeffrey S. Ovall

*Mathematics and Statistics Faculty Publications and Presentations*

We develop an a posteriori error estimate of hierarchical type for Dirichlet eigenvalue problems of the form (−∆ + (c/r) 2 )ψ = λψ on bounded domains Ω, where* *r is the distance to the origin, which is assumed to be in Ω. This error estimate is proven to be asymptotically identical to the eigenvalue approximation error on a family of geometrically-graded meshes. Numerical experiments demonstrate this asymptotic exactness in practice.

Phase Dynamics Of Locset Control Methodology, 2015 University of Tennessee - Knoxville

#### Phase Dynamics Of Locset Control Methodology, Brendan Neschke

*Masters Theses*

Single-mode fiber amplifiers produce diffraction-limited beams very efficiently. Maximum beam intensity requires that an array of these amplifiers have their beams coherently combined at the target. Optical path differences and noise adversely affect beam quality. An existing closed loop phase control methodology, called the locking of optical coherence by single-detector electronic-frequency tagging (LOCSET), corrects phase errors in real time by electronically detecting path length differences and sending signals to lithium niobate phase adjusters. Broadening the line-width using “jitter” of the input signal can increase the output power of an individual amplifier by suppressing nonlinearity. The system dynamics of LOCSET are ...

Investigating Synergy: Mathematical Models For The Coupled Dynamics Of Hiv And Hsv-2 And Other Endemic Diseases, 2015 Purdue University

#### Investigating Synergy: Mathematical Models For The Coupled Dynamics Of Hiv And Hsv-2 And Other Endemic Diseases, Christina M Alvey

*Open Access Dissertations*

This dissertation presents epidemiological models that investigate synergy: synergy between HIV and HSV-2 or between humans and mosquitoes in a malaria study. Each of the three coupled disease models addresses different epidemiological questions with regard to gender or disease structure in the context of sexually-transmitted diseases (STDs), while the malaria model focuses on age-structure of the human population. ^ Mounting evidence indicates that HSV-2 infection may increase susceptibility to HIV infection and that co-infection may increase infectiousness. Accordingly, antiviral treatment of people with HSV-2 may mitigate the incidence of HIV in populations where both pathogens occur. To better understand the epidemiological ...

Thirty Years Of Turnstiles And Transport, 2015 University of Colorado Boulder

#### Thirty Years Of Turnstiles And Transport, James Meiss

*Applied Mathematics Faculty Contributions*

To characterize transport in a deterministic dynamical system is to compute exit time distributions from regions or transition time distributions between regions in phase space. This paper surveys the considerable progress on this problem over the past thirty years. Primary measures of transport for volume-preserving maps include the exiting and incoming fluxes to a region. For area-preserving maps, transport is impeded by curves formed from invariant manifolds that form partial barriers, e.g., stable and unstable manifolds bounding a resonance zone or cantori, the remnants of destroyed invariant tori. When the map is exact volume preserving, a Lagrangian differential form ...

Open And Dense Topological Transitivity Of Extensions By Non-Compact Fiber Of Hyperbolic Systems: A Review, 2015 West Chester University of Pennsylvania

#### Open And Dense Topological Transitivity Of Extensions By Non-Compact Fiber Of Hyperbolic Systems: A Review, Viorel Nitica, Andrei Török

*Mathematics*

Currently, there is great renewed interest in proving the topological transitivity of various classes of continuous dynamical systems. Even though this is one of the most basic dynamical properties that can be investigated, the tools used by various authors are quite diverse and are strongly related to the class of dynamical systems under consideration. The goal of this review article is to present the state of the art for the class of Hölder extensions of hyperbolic systems with non-compact connected Lie group fiber. The hyperbolic systems we consider are mostly discrete time. In particular, we address the stability and genericity ...

Sandpiles, Spanning Trees, And Plane Duality, 2015 Gettysburg College

#### Sandpiles, Spanning Trees, And Plane Duality, Melody Chan, Darren B. Glass, Matthew Macauley, David Perkinson, Caryn Werner, Qiaoyu Yang

*Math Faculty Publications*

Let G be a connected, loopless multigraph. The sandpile group of G is a finite abelian group associated to G whose order is equal to the number of spanning trees in G. Holroyd et al. used a dynamical process on graphs called rotor-routing to define a simply transitive action of the sandpile group of G on its set of spanning trees. Their definition depends on two pieces of auxiliary data: a choice of a ribbon graph structure on G, and a choice of a root vertex. Chan, Church, and Grochow showed that if G is a planar ribbon graph, it ...

My Finite Field, 2015 Idaho State University

#### My Finite Field, Matthew Schroeder

*Journal of Humanistic Mathematics*

A love poem written in the language of mathematics.

Four Tails Problems For Dynamical Collapse Theories, 2015 Chapman University

#### Four Tails Problems For Dynamical Collapse Theories, Kelvin J. Mcqueen

*Philosophy Faculty Articles and Research*

The primary quantum mechanical equation of motion entails that measurements typically do not have determinate outcomes, but result in superpositions of all possible outcomes. Dynamical collapse theories (e.g. GRW) supplement this equation with a stochastic Gaussian collapse function, intended to collapse the superposition of outcomes into one outcome. But the Gaussian collapses are imperfect in a way that leaves the superpositions intact. This is the tails problem. There are several ways of making this problem more precise. But many authors dismiss the problem without considering the more severe formulations. Here I distinguish four distinct tails problems. The first (bare ...

Control, Stability, And Qualitative Theory Of Dynamical Systems 2014, 2015 Eastern Mediterranean University

#### Control, Stability, And Qualitative Theory Of Dynamical Systems 2014, Nazim I. Mahmudov, Mark A. Mckibben, Sakthivel Rathinasamy, Yong Ren

*Mathematics*

No abstract provided.

Superstable Manifolds Of Invariant Circles And Codimension-One Böttcher Functions, 2015 Butler University

#### Superstable Manifolds Of Invariant Circles And Codimension-One Böttcher Functions, Scott R. Kaschner, Roland K.W. Roeder

*Scholarship and Professional Work - LAS*

Let f:X ⇢ X be a dominant meromorphic self-map, where X is a compact, connected complex manifold of dimension n>1. Suppose that there is an embedded copy of P1 that is invariant under f, with f holomorphic and transversally superattracting with degree a in some neighborhood. Suppose that f restricted to this line is given by z↦zb, with resulting invariant circle S. We prove that if a≥b, then the local stable manifold Wsloc(S) is real analytic. In fact, we state and prove a suitable localized version that can be useful in wider contexts. We then show ...

Rational Maps Of Cp^2 With No Invariant Foliation, 2015 Butler University

#### Rational Maps Of Cp^2 With No Invariant Foliation, Scott R. Kaschner, Rodrigo A. Perez, Roland K.W. Roeder

*Scholarship and Professional Work - LAS*

We present simple examples of rational maps of the complex projective plane with equal first and second dynamical degrees and no invariant foliation.

Discrete Dynamical Systems In Multiple Target And Alternate Selex, 2015 Iowa State University

#### Discrete Dynamical Systems In Multiple Target And Alternate Selex, Howard A. Levine, Yeon-Jung Seo

*Mathematics Publications*

Dynamical systems are often used to model biochemical and biological processes. In Seo et al. (2010, 2014) we studied two mathematical models of the iterative biochemical procedure known as SELEX (Systematic Evolution of Ligands by EXponential Enrichment): multiple target SELEX and alternate SELEX. It is the purpose of this paper to revisit the mathematics of these processes in the language of dynamical systems on compact manifolds but for a dynamical system on a manifold with compact closure. From the experimentalist's point of view, multiple target SELEX provides a way of obtaining the best binding ligands to a pool of ...

Inżynieria Chemiczna Ćw., 2015 Wroclaw University of Technology

Tematyka Prac Doktorskich, 2015 Wroclaw University of Technology

#### Tematyka Prac Doktorskich, Wojciech M. Budzianowski

*Wojciech Budzianowski*

No abstract provided.