Mathematics Commons^™

Supporting English Language Learners Inside The Mathematics Classroom: One Teacher’S Unique Perspective Working With Students During Their First Years In America, Amy Marie Fendrick

Research and Evaluation in Education, Technology, Art, and Design

Reflecting upon my personal experiences teaching mathematics to English Language Learners (ELL) in a public high school in Lincoln, Nebraska, this essay largely focuses on the time I spent as the only Accelerated Math teacher in my school building. From 2012 – 2017, I taught three different subjects at this high school: Advanced Algebra, Algebra, and Accelerated Math. This essay highlights why I chose to become a math and ELL teacher, as well as the challenges, issues, struggles, and successes I experienced during my time teaching. I focus on the challenges I faced teaching students who did not share my …

A Study On The Rotational B-Family Of Equations, Emel Bolat

Mathematics Dissertations - Archive

In this thesis, we study a mathematical model of long-crested water waves propagating in one direction with the effect of Earth's rotation near the equator by following the formal asymptotic procedures. Firstly, we derive a new model equation called the rotational b-family of equations by using the Camassa-Holm approximation of the two-dimensional incompressible and irrotational Euler equations. Secondly,we establish that the local well-posedness of the Cauchy problem for the rotational b-family of equations on the Sobolev space H⁸, for s > 3=2. In addition, we study the effects of the Coriolis force and nonlocal higher nonlinearities on blow-up criteria and wave-breaking …

Asymptotic Properties Of The Deconvolution Kernel Density Estimate Based On 2-Dependent Error Structure With Applications To Remaining Useful Life Problems In Reliability Theory, Geoffrey H. Schuette

Mathematics Dissertations - Archive

This thesis is motivated from an engineering question, which led us to the deconvolution problem with a dependent error structure. We establish a deconvolution kernel density estimator by adapting the methods of kernel density estimates and Fourier Transforms. In this approach, the contaminated data with additive random errors are assumed dependent and satisfying smooth or super smooth conditions. Under both smooth and supper smooth conditions, we derived: 1. optimal rates of convergence in terms of mean integrated squared error for deconvolution kernel density estimator; 2. the limiting distribution of the estimator.

Scattering And Inverse Scattering On The Line For A First-Order System With Energy-Dependent Potentials, Ramazan Ercan

Mathematics Dissertations - Archive

A first-order system of two linear ordinary differential equations is analyzed. The linear system contains a spectral parameter, and it has two coefficients that are functions of the spatial variable ��. Those two functions act as potentials in the linear system and they also linearly contain the spectral parameter λ, and hence they are referred to as energy-dependent potentials. Such a linear system arises in the solution to a pair of integrable nonlinear partial differential equations (known as the derivative nonlinear Schrödinger equations) via the so-called inverse scattering transform method. The direct and inverse problems for the corresponding first-order linear …

Secondary Mathematics Teachers’ Planned Approaches For Teaching Standard Deviation, Maryann E. Huey, Joe Champion, Stephanie Casey, Nicholas H. Wasserman

Mathematics Faculty Publications and Presentations

Research-based guidelines for learning variation exist (e.g., Franklin et al., 2007; Garfield, delMas, & Chance, 2007), but little is known about how teachers plan to teach standard deviation, or how these plans align with recent recommendations. In this article, we survey lesson plans designed by inservice and preservice secondary mathematical teachers. We report on the accuracy, technology usage, and visual representations in the lesson plans. We consider how many elements are used, the level of conceptual development, and the mathematical nature. Findings support differences between preservice and master’s level students in education, as well as a tendency by in-service teachers …

Gravitational-Wave Memory From Black Hole And Neutron Star Mergers, Matthew Karlson

Theses, Dissertations and Culminating Projects

The detection of gravitational waves from binary black hole and binary neutron star mergers has ushered in a new age of observational astronomy. Anticipation of detection from these coalescing compact binaries has led to the development of models for comparison using analytical and numerical techniques. Typically, these methods model gravitational-wave signals as small oscillations that grow over time, reach some maximum value, and eventually decay to zero. However, these models are incomplete: compact binaries can emit gravitational waves that decay to a non-zero value. This phenomenon is known as the gravitational-wave memory. In particular, the signal from compact binaries displays …

Women In Science: A Course, Krystyna Krupinski

Senior Honors Projects

Ask any student to name five women who have made advancements in scientific fields and most likely they will not be able to name more than two. With a lack of courses highlighting women scientists’ achievements, this lack of knowledge is hardly surprising I am addressing this issue in my work. Through the researching of scientific advancements made by women, I have proposed a method to teach students about these scientists so names like Johnson, Franklin, and Curie become as commonplace as Einstein, Hawking, and Bohr. Frequently in science classes in middle schools and high schools, the focus is only …

The Dispersion Process For Particles On Graphs, Adam Cartisano

Theses, Dissertations and Culminating Projects

In this thesis, we study a process called Dispersion, in which M particles are dispersed among the vertices of a graph G. All particles initially occupy a single vertex called the origin vertex. At each discrete time step, all particles which share a vertex with at least one other, move to a randomly (though not necessarily uniformly) chosen neighbor of the currently occupied vertex. The process ends when each vertex is occupied by at most one particle. We will explore various aspects of the Dispersion process. One of these is the expected time to completion, E[T_Disp] for 3 …

Candy Sharing And Chip Firing Games On Graphs, Joseph Degaetani

Theses, Dissertations and Culminating Projects

The Candy Game begins with a finite number of players sitting in a circle, each with an initial amount of candy. At each time step, each player passes half of their pile to the player on their left (with odd sized stacks receiving an extra piece of candy). The original question was whether every initial distribution of candy results in every player holding the same number of pieces after a finite number of turns. For arbitrary initial distributions, we prove asymptotically tight bounds on the final amount of candy. The diffusion chip firing game assigns integral chip amounts to each …

Magic Squares Of Squares Of Order Three Over Finite Fields, Giancarlo Labruna

Theses, Dissertations and Culminating Projects

A magic square M over an integral domain D is a 3 x 3 matrix with entries from D such that the elements from each row, column, and diagonal add to the same sum. If all the entries in M are perfect squares in D, we call M a magic square of squares over D. Martin LaBar raised an open question in 1984, which states, “Is there a magic square of squares over the ring Z of the integers which has all the nine entries distinct?” We approach to answering a similar question in case D is a finite field. …

Divergence Of Cat(0) Cube Complexes And Coxeter Groups, Ivan Levcovitz

Dissertations, Theses, and Capstone Projects

We provide geometric conditions on a pair of hyperplanes of a CAT(0) cube complex that imply divergence bounds for the cube complex. As an application, we characterize right-angled Coxeter groups with quadratic divergence and show right-angled Coxeter groups cannot exhibit a divergence function between quadratic and cubic. This generalizes a theorem of Dani-Thomas that addressed the class of 2-dimensional right-angled Coxeter groups. This characterization also has a direct application to the theory of random right-angled Coxeter groups. As another application of the divergence bounds obtained for cube complexes, we provide an inductive graph theoretic criterion on a right-angled Coxeter group's …

The Distribution Of Totally Positive Integers In Totally Real Number Fields, Tianyi Mao

Dissertations, Theses, and Capstone Projects

Hecke studies the distribution of fractional parts of quadratic irrationals with Fourier expansion of Dirichlet series. This method is generalized by Behnke and Ash-Friedberg, to study the distribution of the number of totally positive integers of given trace in a general totally real number field of any degree. When the number field is quadratic, Beck also proved a mean value result using the continued fraction expansions of quadratic irrationals. We generalize Beck’s result to higher moments. When the field is cubic, we show that the asymptotic behavior of a weighted Diophantine sum is related to the structure of the unit …

The Structure Of Models Of Second-Order Set Theories, Kameryn J. Williams

Dissertations, Theses, and Capstone Projects

This dissertation is a contribution to the project of second-order set theory, which has seen a revival in recent years. The approach is to understand second-order set theory by studying the structure of models of second-order set theories. The main results are the following, organized by chapter. First, I investigate the poset of T-realizations of a fixed countable model of ZFC, where T is a reasonable second-order set theory such as GBC or KM, showing that it has a rich structure. In particular, every countable partial order embeds into this structure. Moreover, we can arrange so that these embedding preserve …

On Some Geometry Of Graphs, Zachary S. Mcguirk

Dissertations, Theses, and Capstone Projects

In this thesis we study the intrinsic geometry of graphs via the constants that appear in discretized partial differential equations associated to those graphs. By studying the behavior of a discretized version of Bochner's inequality for smooth manifolds at the cone point for a cone over the set of vertices of a graph, a lower bound for the internal energy of the underlying graph is obtained. This gives a new lower bound for the size of the first non-trivial eigenvalue of the graph Laplacian in terms of the curvature constant that appears at the cone point and the size of …

The Advection-Diffusion Equation And The Enhanced Dissipation Effect For Flows Generated By Hamiltonians, Michael Kumaresan

Dissertations, Theses, and Capstone Projects

We study the Cauchy problem for the advection-diffusion equation when the diffusive parameter is vanishingly small. We consider two cases - when the underlying flow is a shear flow, and when the underlying flow is generated by a Hamiltonian. For the former, we examine the problem on a bounded domain in two spatial variables with Dirichlet boundary conditions. After quantizing the system via the Fourier transform in the first spatial variable, we establish the enhanced-dissipation effect for each mode. For the latter, we allow for non-degenerate critical points and represent the orbits by points on a Reeb graph, with vertices …

Geometry And Analysis Of Some Euler-Arnold Equations, Jae Min Lee

Dissertations, Theses, and Capstone Projects

In 1966, Arnold showed that the Euler equation for an ideal fluid can arise as the geodesic flow on the group of volume preserving diffeomorphisms with respect to the right invariant kinetic energy metric. This geometric interpretation was rigorously established by Ebin and Marsden in 1970 using infinite dimensional Riemannian geometry and Sobolev space techniques. Many other nonlinear evolution PDEs in mathematical physics turned out to fit in this universal approach, and this opened a vast research on the geometry and analysis of the Euler-Arnold equations, i.e., geodesic equations on a Lie group endowed with one-sided invariant metrics. In this …

Physical Applications Of The Geometric Minimum Action Method, George L. Poppe Jr.

Dissertations, Theses, and Capstone Projects

This thesis extends the landscape of rare events problems solved on stochastic systems by means of the \textit{geometric minimum action method} (gMAM). These include partial differential equations (PDEs) such as the real Ginzburg-Landau equation (RGLE), the linear Schroedinger equation, along with various forms of the nonlinear Schroedinger equation (NLSE) including an application towards an ultra-short pulse mode-locked laser system (MLL).

Additionally we develop analytical tools that can be used alongside numerics to validate those solutions. This includes the use of instanton methods in deriving state transitions for the linear Schroedinger equation and the cubic diffusive NLSE.

These analytical solutions are …

Constant Symplectic 2-Groupoids, Rajan Amit Mehta, Xiang Tang

Mathematics Sciences: Faculty Publications

We propose a definition of symplectic 2-groupoid which includes integrations of Courant algebroids that have been recently constructed. We study in detail the simple but illustrative case of constant symplectic 2-groupoids. We show that the constant symplectic 2-groupoids are, up to equivalence, in one-to-one correspondence with a simple class of Courant algebroids that we call constant Courant algebroids. Furthermore, we find a correspondence between certain Dirac structures and Lagrangian sub-2-groupoids.

Nonparametric Estimation Of Time Series Volatility Model Estimation, Teng Tu

Arts & Sciences Graduate Student Theses and Dissertations

In this article we consider two estimation methods of a non-parametric volatility model with autoregressive error of order two. The first estimation method based on the two- lag difference. To get a better result, we consider the second approach based on the general quadratic forms. For illustration, we provided several data sets from different simulation models to support the procedures of both two methods, and prove that the second approach can make a better estimation.

Matrix Methods Of Data Analysis, Kaiye Yu

Renée Crown University Honors Thesis Projects - All

Nowadays, data is playing a bigger part than ever before in the history. In order to get more useful information, methods involving matrix are powerful. There are different algorithms that are able to help one to learn the information they need from data. In this study, there are two main algorithms that I will focus on. One is Pagerank algorithm, a traditional algorithm that was applied to searching engine decades ago in Google. However, Pagerank algorithm has certain limits in providing information like covariance between different factors. Thus, another method is also studied, which is principal component analysis (PCA), while …