Mathematics Commons^™

Establishing Foundations For Investigating Inquiry-Oriented Teaching, Estrella Maria Salas Johnson

Dissertations and Theses

The Teaching Abstract Algebra for Understanding (TAAFU) project was centered on an innovative abstract algebra curriculum and was designed to accomplish three main objectives: to produce a set of multi-media support materials for instructors, to understand the challenges faced by mathematicians as they implemented this curriculum, and to study how this curriculum supports student learning of abstract algebra. Throughout the course of the project I took the lead investigating the teaching and learning in classrooms using the TAAFU curriculum. My dissertation is composed of three components of this research. First, I will report on a study that aimed to describe …

Near-Optimal Compressed Sensing Guarantees For Total Variation Minimization, Deanna Needell, R. Ward

CMC Faculty Publications and Research

Consider the problem of reconstructing a multidimensional signal from an underdetermined set of measurements, as in the setting of compressed sensing. Without any additional assumptions, this problem is ill-posed. However, for signals such as natural images or movies, the minimal total variation estimate consistent with the measurements often produces a good approximation to the underlying signal, even if the number of measurements is far smaller than the ambient dimensionality. This paper extends recent reconstruction guarantees for two-dimensional images x ∈ ℂN2 to signals x ∈ ℂNd of arbitrary dimension d ≥ 2 and to isotropic total variation problems. In this …

The Neural Ring: An Algebraic Tool For Analyzing The Intrinsic Structure Of Neural Codes, Carina Curto, Vladimir Itskov, Alan Veliz-Cuba, Nora Youngs

Department of Mathematics: Faculty Publications

Neurons in the brain represent external stimuli via neural codes. These codes often arise from stereotyped stimulus-response maps, associating to each neuron a convex receptive field. An important problem confronted by the brain is to infer properties of a represented stimulus space without knowledge of the receptive fields, using only the intrinsic structure of the neural code. How does the brain do this? To address this question, it is important to determine what stimulus space features can - in principle - be extracted from neural codes. This motivates us to define the neural ring and a related neural ideal, …

Gp Representation Space Reduction Using A Tiered Search Scheme, John Joseph Aleshunas

Dissertations

The size and complexity of a GP representation space is defined by the set of functions and terminals used, the arity of those functions, and the maximal depth of candidate solution trees in the space. Practice has shown that some means to reduce the size or bias the search must be provided. Adaptable Constrained Genetic Programming (ACGP) can discover beneficial substructures and probabilistically bias the search to promote the use of these substructures. ACGP has two operating modes: a more efficient low granularity mode (1st order heuristics) and a less efficient higher granularity mode (2nd order heuristics). Both of these …

Nguyen ’15 Places In Putnam Competition, Natalya Grabavoy

News and Events (Discontinued Series)

No abstract provided.

A Discrete Approach To The Poincare-Miranda Theorem, Connor Thomas Ahlbach

HMC Senior Theses

The Poincare-Miranda Theorem is a topological result about the existence of a zero of a function under particular boundary conditions. In this thesis, we explore proofs of the Poincare-Miranda Theorem that are discrete in nature - that is, they prove a continuous result using an intermediate lemma about discrete objects. We explain a proof by Tkacz and Turzanski that proves the Poincare-Miranda theorem via the Steinhaus Chessboard Theorem, involving colorings of partitions of n-dimensional cubes. Then, we develop a new proof of the Poincare-Miranda Theorem that relies on a polytopal generalization of Sperner's Lemma of Deloera - Peterson - Su. …

Using Partial Differential Equations To Model And Analyze The Treatment Of A Chronic Wound With Oxygen Therapy Techniques, Brandon C. Russell

Mahurin Honors College Capstone Experience/Thesis Projects

Chronic wounds plague approximately 1.3-3 million Americans. The treatment of these wounds requires knowledge of the complex healing process of typical wounds. With a system of partial differential equations, this project attempts to model the intricate biological process and to describe oxygen levels, neutrophil and bacteria concentrations, and other biological parameters with respect to time and space. Analytical solutions for the model will be derived for various frames of time in the wound-healing process. The system of equations will be numerically solved using Matlab. Numerical simulations are performed to determine optimal treatment strategies for a chronic wound.

Time Scales Integral Inequalities For Superquadratic Functions, Josipa Barić, Rabia Bibi, Martin Bohner, Josip Pečarić

Mathematics and Statistics Faculty Research & Creative Works

In this paper, two different methods of proving Jensen's inequality on time scales for superquadratic functions are demonstrated. Some refinements of classical inequalities on time scales are obtained using properties of superquadratic functions and some known results for isotonic linear functionals. © 2013 The Korean Mathematical Society.

Algebro-Geometric Solutions For The Degasperis--Procesi Hierarchy, Yu Hou, Peng Zhao, Engui Fan, Zhijun Qiao

School of Mathematical & Statistical Sciences Faculty Publications

Though the completely integrable Camassa--Holm (CH) equation and Degasperis--Procesi (DP) equation are cast in the same peakon family, they possess the second- and third-order Lax operators, respectively. From the viewpoint of algebro-geometrical study, this difference lies in hyper-elliptic and non-hyper-elliptic curves. The non-hyper-elliptic curves lead to great difficulty in the construction of algebro-geometric solutions of the DP equation. In this paper, we derive the DP hierarchy with the help of Lenard recursion operators. Based on the characteristic polynomial of a Lax matrix for the DP hierarchy, we introduce a third order algebraic curve $\mathcal{K}_{r-2}$ with genus $r-2$, from which the …

Generalized Local Test For Local Extrema In Single-Variable Functions, Eleftherios Gkioulekas

School of Mathematical & Statistical Sciences Faculty Publications

We give a detailed derivation of a generalization of the second derivative test of single-variable calculus which can classify critical points as local minima or local maxima (or neither), whenever the traditional second derivative test fails, by considering the values of higher-order derivatives evaluated at the critical points. The enhanced test is local, in the sense that it is only necessary to evaluate all relevant derivatives at the critical point itself, and it is reasonably robust. We illustrate an application of the generalized test on a trigonometric function where the second derivative test fails to classify some of the critical …

Chebyshev Polynomial Approximation To Solutions Of Ordinary Differential Equations, Amber Sumner Robertson

Undergraduate Theses

In this thesis, we develop a method for finding approximate particular solutions for second order ordinary differential equations. We use Chebyshev polynomials to approximate the source function and the particular solution of an ordinary differential equation. The derivatives of each Chebyshev polynomial will be represented by linear combinations of Chebyshev polynomials, and hence the derivatives will be reduced and differential equations will become algebraic equations. Another advantage of the method is that it does not need the expansion of Chebyshev polynomials. This method is also compared with an alternative approach for particular solutions. Examples including approximation, particular solution, a class …

Effects In The Anomalistic Period Of Celestial Bodies Due To A Logarithmic Correction To The Newtonian Gravitational Potential, Omiros Ragos, Ioannis Haranas, Ioannis Gkigkitzis

Physics and Computer Science Faculty Publications

We study the motion of a secondary celestial body under the influence of the logarithmic corrected gravitational force of a primary one. This kind of correction was introduced by Fabris et al. (2009). We derive two equations to compute the rate of change of the periastron w.r.t. the eccentric anomaly and its total variation over one revolution, In a kinematical sense, this influence produces an apsidal motion. We perform numerical estimations for Mercury and for the companion star of the pulsar PSR 1913+16. We also consider the case of the artificial Earth satellite GRACE-A, but the results present a low …

Using Optimal Control Theory To Optimize The Use Of Oxygen Therapy In Chronic Wound Healing, Donna Lynn Daulton

Masters Theses & Specialist Projects

Approximately 2 to 3 million people in the United States suffer from chronic wounds, which are defined as wounds that do not heal in 30 days time; an estimated $25 billion per year is spent on their treatment in the United States. In our work, we focused on treating chronic wounds with bacterial infections using hyperbaric and topical oxygen therapies.

We used a mathematical model describing the interaction between bacteria, neutrophils and oxygen. Optimal control theory was then employed to study oxygen treatment strategies with the mathematical model. Existence of a solution was shown for both therapies. Uniqueness was also …

Analyzing And Solving Non-Linear Stochastic Dynamic Models On Non-Periodic Discrete Time Domains, Gang Cheng

Masters Theses & Specialist Projects

Stochastic dynamic programming is a recursive method for solving sequential or multistage decision problems. It helps economists and mathematicians construct and solve a huge variety of sequential decision making problems in stochastic cases. Research on stochastic dynamic programming is important and meaningful because stochastic dynamic programming reflects the behavior of the decision maker without risk aversion; i.e., decision making under uncertainty. In the solution process, it is extremely difficult to represent the existing or future state precisely since uncertainty is a state of having limited knowledge. Indeed, compared to the deterministic case, which is decision making under certainty, the stochastic …

Floquet Theory On Banach Space, Fatimah Hassan Albasrawi

Masters Theses & Specialist Projects

In this thesis we study Floquet theory on a Banach space. We are concerned about the linear differential equation of the form: y'(t) = A(t)y(t), where t ∈ R, y(t) is a function with values in a Banach space X, and A(t) are linear, bounded operators on X. If the system is periodic, meaning A(t+ω) = A(t) for some period ω, then it is called a Floquet system. We will investigate the existence …

Hypergraph Capacity With Applications To Matrix Multiplication, John Lee Thompson Peebles Jr.

HMC Senior Theses

The capacity of a directed hypergraph is a particular numerical quantity associated with a hypergraph. It is of interest because of certain important connections to longstanding conjectures in theoretical computer science related to fast matrix multiplication and perfect hashing as well as various longstanding conjectures in extremal combinatorics.

We give an overview of the concept of the capacity of a hypergraph and survey a few basic results regarding this quantity. Furthermore, we discuss the Lovász number of an undirected graph, which is known to upper bound the capacity of the graph (and in practice appears to be the best such …

The Torsion Angle Of Random Walks, Mu He

Masters Theses & Specialist Projects

In this thesis, we study the expected mean of the torsion angle of an n-step
equilateral random walk in 3D. We consider the random walk is generated within a confining sphere or without a confining sphere: given three consecutive vectors ^→e₁ , ^→e₂ , and ^→e₃ of the random walk then the vectors ^→e₁ and ^→e₂ define a plane and the vectors ^→e₂ and ^→e₃ define a second plane. The angle between the two planes is called the torsion angle of the three vectors. Algorithms are …

Minimizing Travel Time Through Multiple Media With Various Borders, Tonja Miick

Masters Theses & Specialist Projects

This thesis consists of two main chapters along with an introduction and
conclusion. In the introduction, we address the inspiration for the thesis, which
originates in a common calculus problem wherein travel time is minimized across two media separated by a single, straight boundary line. We then discuss the correlation of this problem with physics via Snells Law. The first core chapter takes this idea and develops it to include the concept of two media with a circular border. To make the problem easier to discuss, we talk about it in terms of running and swimming speeds. We first address …

Chip Firing Games And Riemann-Roch Properties For Directed Graphs, Joshua Z. Gaslowitz

HMC Senior Theses

The following presents a brief introduction to tropical geometry, especially tropical curves, and explains a connection to graph theory. We also give a brief summary of the Riemann-Roch property for graphs, established by Baker and Norine (2007), as well as the tools used in their proof. Various generalizations are described, including a more thorough description of the extension to strongly connected directed graphs by Asadi and Backman (2011). Building from their constructions, an algorithm to determine if a directed graph has Row Riemann-Roch Property is given and thoroughly explained.

On Well-Quasi-Orderings, Forrest Thurman

HIM 1990-2015

A quasi-order is a relation on a set which is both reflexive and transitive, while a well-quasi-order has the additional property that there exist no infinite strictly descending chains nor infinite antichains. Well-quasi-orderings have many interesting applications to a variety of areas which includes the strength of certain logical systems, the termination of algorithms, and the classification of sets of graphs in terms of excluded minors. My thesis explores how well-quasi-orderings are related to these topics through examples of four known well-quasi-orderings which are given by Dickson's Lemma, Higmans's Lemma, Kruskal's Tree Theorem, and the Robertson-Seymour Theorem. The well-quasi-ordering conjecture …