Mathematics Commons^™

Existence And Comparison Of Smallest Eigenvalues For A Fractional Boundary-Value Problem, Paul W. Eloe, Jeffrey T. Neugebauer

Mathematics Faculty Publications

The theory of u0-positive operators with respect to a cone in a Banach space is applied to the fractional linear differential equations

(see PDF)

0 < t < 1, with each satisfying the boundary conditions u(0) = u(1) = 0. The existence of smallest positive eigenvalues is established, and a comparison theorem for smallest positive eigenvalues is obtained.

Data Stories And Pictures; Discovering Lessons And Principles For Statistics And For Life (Abstract), Rafe Donahue

Kenneth C. Schraut Memorial Lectures

Our world is full of data and of analyses of these data. Why are we doing to our data what we are doing to our data?

On Free Stochastic Processes And Their Derivatives, Daniel Alpay, Palle Jorgensen, Guy Salomon

Mathematics, Physics, and Computer Science Faculty Articles and Research

We study a family of free stochastic processes whose covariance kernels K may be derived as a transform of a tempered measure σ. These processes arise, for example, in consideration non-commutative analysis involving free probability. Hence our use of semi-circle distributions, as opposed to Gaussians. In this setting we find an orthonormal bases in the corresponding noncommutative L2 of sample-space. We define a stochastic integral for our family of free processes.

Dispersive And Dissipative Errors In The Dpg Method With Scaled Norms For Helmholtz Equation, Jay Gopalakrishnan, Ignacio Muga, Nicole Olivares

Mathematics and Statistics Faculty Publications and Presentations

This paper studies the discontinuous Petrov--Galerkin (DPG) method, where the test space is normed by a modified graph norm. The modification scales one of the terms in the graph norm by an arbitrary positive scaling parameter. The main finding is that as the parameter approaches zero, better results are obtained, under some circumstances, when the method is applied to the Helmholtz equation. The main tool used is a dispersion analysis on the multiple interacting stencils that form the DPG method. The analysis shows that the discrete wavenumbers of the method are complex, explaining the numerically observed artificial dissipation in the …

Convergence Rates Of The Dpg Method With Reduced Test Space Degree, Timaeus Bouma, Jay Gopalakrishnan, Ammar Harb

Mathematics and Statistics Faculty Publications and Presentations

This paper presents a duality theorem of the Aubin-Nitsche type for discontinuous Petrov Galerkin (DPG) methods. This explains the numerically observed higher convergence rates in weaker norms. Considering the specific example of the mild-weak (or primal) DPG method for the Laplace equation, two further results are obtained. First, the DPG method continues to be solvable even when the test space degree is reduced, provided it is odd. Second, a non-conforming method of analysis is developed to explain the numerically observed convergence rates for a test space of reduced degree

Strong Bounded Solutions For Nonlinear Parabolic Systems, Nsoki Mavinga, M. N. Nkashama

Mathematics & Statistics Faculty Works

In this article we study the existence of strong bounded solutions for nonlinear parabolic systems on a domain which is bounded in space and unbounded in time (namely the entire real line). We use nonlinear iteration arguments combined with some a priori estimates to derive the existence results. We also provide conditions under which we have a positive solution. Some examples are given to illustrate the results.

Hamiltonian Approach To The Modeling Of Internal Geophysical Waves With Vorticity, Alan Compelli

Articles

We examine a simplified model of internal geophysical waves in a rotational 2-dimensional water-wave system, under the influence of Coriolis forces and with gravitationally induced waves. The system consists of a lower medium, bound underneath by an impermeable flat bed, and an upper lid. The 2 media have a free common interface. Both media have constant density and constant (non-zero) vorticity. By examining the governing equations of the system we calculate the Hamiltonian of the system in terms of its conjugate variables and perform a variable transformation to show that it has canonical Hamiltonian structure. We then linearize the system, …

Symmetry And Reductions Of Integrable Dynamical Systems: Peakon And The Toda Chain Systems, Vladimir Gerdjikov, Rossen Ivanov, Gaetano Vilasi

Articles

We are analyzing several types of dynamical systems which are both integrable and important for physical applications. The first type are the so-called peakon systems that appear in the singular solutions of the Camassa-Holm equation describing special types of water waves. The second type are Toda chain systems, that describe molecule interactions. Their complexifications model soliton interactions in the adiabatic approximation. We analyze the algebraic aspects of the Toda chains and describe their real Hamiltonian forms.

Complex Absorbing Potential Method For Dirac Operators. Clusters Of Resonances, Jimmy Kungsman, Michael Melgaard

Articles

For both nonrelativistic and relativistic Hamiltonians, the Complex Absorbing Potential (CAP) method has been applied extensively to calculate resonances in Physics and Chemistry. We study clusters of resonances for the perturbed Dirac operator near the real axis and, in the semiclassical limit, we establish the CAP method rigorously by showing that resonances are perturbed eigenvalues of the nonselfadjoint CAP Hamiltonian, and vice versa.

Evaporation Screening, Scattering, And Quantum Tunneling Near The Horizons Of Schwarzschild And Reissner-Nordstroem Black Holes, Emil Prodanov

Articles

For the radial motion of massive particles with large angular momenta in Schwarzschild geometry and that of massive charged particles with large
angular momenta or energy in a particular range in Reissner-Nordstroem geometry, there exist classically forbidden regions on the outside of
the respective event horizons which scatter certain in-falling geodesics or screen some of the black holes' evaporation by reflecting the emitted

particles back into the black holes. Quantum tunneling across this forbidden regions is studied.

Culturally Relevant Pedagogy: Secondary Mathematics In An Urban Classroom, Julia Glissmann North

Honors Program Theses

Research and test scores have shown that African-American, Latino, Native American, and other minority students are underachieving in secondary mathematics. This is concerning not only to school personnel – who are under pressure to have students perform well on standardized tests – but also to the future of the country. When teachers adopt a culturally relevant pedagogy, diverse students will have a better opportunity to learn and retain mathematical content. When academic content is taught in a culturally relevant way, students are able to retain the information, improve their performance in school, and become more informed participants in society. Through …

Ill-Posedness Of The Two-Dimensional Keller-Segel Model In Triebel-Lizorkin Spaces, Chao Deng, John Villavert

School of Mathematical & Statistical Sciences Faculty Publications

This article proves the ill-posedness of the Cauchy problem for the two-dimensional Keller–Segel model in Triebel–Lizorkin spaces, for . In particular, it is shown that solutions can develop norm inflation under certain settings in that the solution can become arbitrarily large after an arbitrarily short time even for small initial data.

Building A Predictive Model For Baseball Games, Jordan Robertson Tait

All Graduate Theses, Dissertations, and Other Capstone Projects

In this paper, we will discuss a method of building a predictive model for Major League Baseball Games. We detail the reasoning for pursuing the proposed predictive model in terms of social popularity and the complexity of analyzing individual variables. We apply a coarse-grain outlook inspired by Simon Dedeos' work on Human Social Systems, in particular the open source website Wikipedia [2] by attempting to quantify the influence of winning and losing streaks instead of analyzing individual performance variables. We will discuss initial findings of data collected from the LA Dodgers and Colorado Rockies and apply further statistical analysis to …

Indemics: An Interactive High-Performance Computing Framework For Data Intensive Epidemic Modeling, Keith R. Bisset, Jiangzhuo Chen, Suruchi Deodhar, Xizhou Feng, Yifei Ma, Madhav V. Marathe

Mathematics, Statistics and Computer Science Faculty Research and Publications

We describe the design and prototype implementation of Indemics (_Interactive; Epi_demic; _Simulation;)—a modeling environment utilizing high-performance computing technologies for supporting complex epidemic simulations. Indemics can support policy analysts and epidemiologists interested in planning and control of pandemics. Indemics goes beyond traditional epidemic simulations by providing a simple and powerful way to represent and analyze policy-based as well as individual-based adaptive interventions. Users can also stop the simulation at any point, assess the state of the simulated system, and add additional interventions. Indemics is available to end-users via a web-based interface.

Detailed performance analysis shows that Indemics greatly enhances the …

Using Topology To Explore Mathematics Education Reform, Carling Sugarman

HMC Senior Theses

Mathematics education is a constant topic of conversation in the United States. Many attempts have been made historically to reform teaching methods and improve student results. Particularly, past ideas have emphasized problem-solving to make math feel more applicable and enjoyable. Many have additionally tackled the widespread problem of “math anxiety” by creating lessons that are more discussion-based than drill-based to shift focus from speed and accuracy. In my project, I explored past reform goals and some added goals concerning students' perceptions of mathematics. To do so, I created and tested a pilot workshop in topology, a creative and intuitive field, …

There And Back Again: Elliptic Curves, Modular Forms, And L-Functions, Allison F. Arnold-Roksandich

HMC Senior Theses

L-functions form a connection between elliptic curves and modular forms. The goals of this thesis will be to discuss this connection, and to see similar connections for arithmetic functions.

Zp-Modules With Partial Decomposition Bases In L Δ ∞Ω, Carol Jacoby, Peter Loth

Mathematics Faculty Publications

We consider the class of mixed Zp-modules with partial decomposition bases. This class includes those modules classified by Ulm and Warfield and is closed under L∞ω-equivalence. In the context of L∞ω- equivalence, Jacoby defined invariants for this class and proved a classification theorem. Here we examine this class relative to Lδ∞ω, those formulas of quantifier rank ≤ some ordinal δ, defining invariants and proving a classification theorem. This generalizes a result of Barwise and Eklof.

Euler, Reader Of Newton: Mechanics And Algebraic Analysis, Sébastien Maronne, Marco Panza

MPP Published Research

We follow two of the many paths leading from Newton’s to Euler’s scientific productions, and give an account of Euler’s role in the reception of some of Newton’s ideas, as regards two major topics: mechanics and algebraic analysis. Euler contributed to a re-appropriation of Newtonian science, though transforming it in many relevant aspects. We study this re-appropriation with respect to the mentioned topics and show that it is grounded on the development of Newton’s conceptions within a new conceptual frame also influenced by Descartes’s views sand Leibniz’s formalism.

Applications Of Cubical Arrays In The Study Of Finite Semifields, Kelly Casimir Aman

Mathematics Dissertations - Archive

It is well known that any finite semifield, S, can be viewed as an n-dimensional vector space over a finite field or prime order, Fp, and that the multiplication in S defines and can be defined by an n x n x n cubical array of scalars, A. For any element a E S, the matrix, La, corresponding to left multiplication by a can be determined from A. In this paper we show that there exists a unique monic polynomial of minimal degree, f E Fp[x], such that f(a) = 0, and which divides the minimal polynomial of La. Furthermore, …

Brain Control Of Functional Reach In Healthy Adults And Stroke Survivors, Janis J. Daly, Ken Hrovat, John Holcomb, Svetlana Pundik

Mathematics and Statistics Faculty Publications

Purpose: Recovery of the most basic shoulder-flexion/elbow-extension components of functional reach is critical for effective arm function following stroke. In order to understand the mechanisms of motor recovery, it is important to characterize the pattern of brain activation during the reach task. Methods: We evaluated 11 controls and 23 moderately to severely impaired chronic stroke survivors (>6 months), with impaired shoulder flexion and elbow extension. Measures were acquired for Arm Motor Ability Test (AMAT) and functional Magnetic Resonance Imaging (fMRI) during the basic shoulder/elbow reach. Results: First, in controls, lateralization of fMRI signal during the reach task was less …