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.

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.

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 ...

Discrete Nonlinear Planar Systems And Applications To Biological Population Models, 2015 Virginia Commonwealth University

#### Discrete Nonlinear Planar Systems And Applications To Biological Population Models, Shushan Lazaryan, Nika Lazaryan, Nika Lazaryan

*Theses and Dissertations*

We study planar systems of difference equations and applications to biological models of species populations. Central to the analysis of this study is the idea of *folding* - the method of transforming systems of difference equations into higher order scalar difference equations. Two classes of second order equations are studied: quadratic fractional and exponential.

We investigate the boundedness and persistence of solutions, the global stability of the positive fixed point and the occurrence of periodic solutions of the quadratic rational equations. These results are applied to a class of linear/rational systems that can be transformed into a quadratic fractional equation ...

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 ...

Geometric Limits Of Julia Sets Of Maps Z^N + Exp(2Πiθ) As N → ∞, 2015 Butler University

#### Geometric Limits Of Julia Sets Of Maps Z^N + Exp(2Πiθ) As N → ∞, Scott R. Kaschner, Reaper Romero, David Simmons

*Scholarship and Professional Work - LAS*

We show that the geometric limit as n → ∞ of the Julia sets J(P_{n,c}) for the maps P_{n,c}(z) = z^{n} + c does not exist for almost every c on the unit circle. Furthermore, we show that there is always a subsequence along which the limit does exist and equals the unit circle.

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.

Zespół Energii Odnawialnej I Zrównoważonego Rozwoju (Eozr), 2014 Wroclaw University of Technology

#### Zespół Energii Odnawialnej I Zrównoważonego Rozwoju (Eozr), Wojciech M. Budzianowski

*Wojciech Budzianowski*

No abstract provided.

Statistics Of The Island-Around-Island Hierarchy In Hamiltonian Phase Space, 2014 Technion-Israel Institute of Technology

#### Statistics Of The Island-Around-Island Hierarchy In Hamiltonian Phase Space, Or Alus, Shmuel Fishman, James Meiss

*Applied Mathematics Faculty Contributions*

The phase space of a typical Hamiltonian system contains both chaotic and regular orbits, mixed in a complex, fractal pattern. One oft-studied phenomenon is the algebraic decay of correlations and recurrence time distributions. For area-preserving maps, this has been attributed to the stickiness of boundary circles, which separate chaotic and regular components. Though such dynamics has been extensively studied, a full understanding depends on many fine details that typically are beyond experimental and numerical resolution. This calls for a statistical approach, the subject of the present work. We calculate the statistics of the boundary circle winding numbers, contrasting the distribution ...

Transients In The Synchronization Of Oscillator Arrays, 2014 Portland State University

#### Transients In The Synchronization Of Oscillator Arrays, Carlos E. Cantos, J. J. P. Veerman

*Mathematics and Statistics Faculty Publications and Presentations*

The purpose of this note is threefold. First we state a few conjectures that allow us to rigorously derive a theory which is asymptotic in N (the number of agents) that describes transients in large arrays of (identical) linear damped harmonic oscillators in R with completely decentralized nearest neighbor interaction. We then use the theory to establish that in a certain range of the parameters transients grow linearly in the number of agents (and faster outside that range). Finally, in the regime where this linear growth occurs we give the constant of proportionality as a function of the signal velocities ...

Exact Tests For Singular Network Data, 2014 Portland State University

#### Exact Tests For Singular Network Data, Ian H. Dinwoodie, Kruti Pandya

*Mathematics and Statistics Faculty Publications and Presentations*

We propose methodology for exact statistical tests of hypotheses for models of network dynamics. The methodology formulates Markovian exponential families, then uses sequential importance sampling to compute expectations within basins of attraction and within level sets of a sufficient statistic for an over-dispersion model. Comparisons of hypotheses can be done conditional on basins of attraction. Examples are presented.