Mathematics Commons^™

Effects Of Single-Gender And Coeducational Learning Environments On Middle School Mathematics Achievement, Tasha Graves Henderson

Walden Dissertations and Doctoral Studies

As a result of the increased number of schools failing to meet adequate yearly progress (AYP), school districts are searching for innovative ways to raise student achievement and meet the rigorous performance standards set by state governments. Using the theoretical framework provided by brain research and the theory of multiple intelligences, the purpose of this quantitative study was to compare 2 middle school classroom structures for differences in mathematics achievement among students. The study examined whether a significant difference existed in mathematics achievement scores on the state-mandated mathematics test for 2 groups (single gender classes versus coeducational classes) in 6 …

Property D Cyclic Neofields, Scott Lacy

Mathematics Dissertations - Archive

In 1948 L. J. Paige introduced the notion of a neofield (N,\oplus,\cdot) as a set N with two binary operations, generally referred to as addition (\oplus) and multiplication (\cdot) such that (N,\oplus) is a loop with identity 0 and (N-\{0\},\cdot) is a group, with both left and right distribution of multiplication over addition. The neofield was considered a generalization of a field and its application was for the coordinatizing of projective planes and related geometry problems. In 1967 A.D. Keedwell introduced the notion of property D cyclic neofields in relation to his study of latin squares and their application to …

On M-Inverse Loops And Quasigroups Of Order N With A Long Inverse Cycle., Carl Edward Looney

Mathematics Dissertations - Archive

In 2002, A.D Keedwell and V.A Sherbacov introduced the concept of finite m-inverse quasigroups with long inverse cycles. Keedwell and Sherbacov observed that finite m-inverse loops and quasigroups with a long inverse cycle could be useful in the study of cryptology. Keedwell and Sherbacov studied the existence of these algebraic structures by determining if a Cayley table of the elements of such structures could be constructed. They showed that m-inverse loops of order 9 with a long inverse cycle do not exist for m = 2; 4 and 6; thus, there do not exist 2,4, or 6 inverse-quasigroups of order …

Numerical Methods For Spontaneous And Evoked Glutamate Release, Justin Shane Blackwell

Mathematics Dissertations - Archive

As synapses are responsible for the majority of neuron communications in the brain, the events of evoked and spontaneous synaptic vesicle release/fusion are key features of all synaptic current. These release events typically activate receptors within a single postsynaptic site and give rise to miniature postsynaptic currents through activations of $N$-methyl-$D$-asparate (NMDA) and $\alpha$-3-hydroxy-5-methyl-4-isoxazolepropionic acid (AMPA) receptors, and therefore, they have been extremely instrumental in neurotransmissions. In this paper we will use a mathematical model to simulate spontaneous and evoked neurotransmission resulted from glutamate release within a synapse. Among several issues that modeling can provide quantitative assessment, the issue of …

Chromatic Polynomials And Orbital Chromatic Polynomials And Their Roots, Jazmin Ortiz

HMC Senior Theses

The chromatic polynomial of a graph, is a polynomial that when evaluated at a positive integer k, is the number of proper k colorings of the graph. We can then find the orbital chromatic polynomial of a graph and a group of automorphisms of the graph, which is a polynomial whose value at a positive integer k is the number of orbits of k-colorings of a graph when acted upon by the group. By considering the roots of the orbital chromatic and chromatic polynomials, the similarities and differences of these polynomials is studied. Specifically we work toward proving a conjecture …

General Factoring Algorithms For Polynomials Over Finite Fields, Wade Combs

Online Theses and Dissertations

In this paper, we generate algorithms for factoring polynomials with coefficients in finite fields. In particular, we develop one deterministic algorithm due to Elwyn Berlekamp and one probabilistic algorithm due to David Cantor and Hans Zassenhaus. While some authors present versions of the algorithms that can only factor polynomials of a certain form, the algorithms we give are able to factor any polynomial over any finite field. Hence, the algorithms we give are the most general algorithms available for this factorization problem. After formulating the algorithms, we look at various ways they can be applied to more specialized inquiries. For …

Existence Of Positive Solutions To A Family Of Fractional Two-Point Boundary Value Problems, Christina Ashley Hollon

Online Theses and Dissertations

In this paper we will consider an nth order fractional boundary value problem with boundary conditions that include a fractional derivative at 1. We will develop properties of the Green's Function for this boundary value problem and use these properties along with the Contraction Mapping Principle, and the Schuader's, Krasnozel'skii's, and Legget-Williams fixed point theorems to prove the existence of positive solutions under different conditions.

A Thorough And Accessible Proof Of The Erdo˝S-Kac Theorem Following Granville And Soundararajan, Thomas John Scheithauer

Online Theses and Dissertations

The Erd\H{o}-Kac Theorem states that, as $n$ tends to infinity, the distribution of $\omega(n)$, the number of distinct prime divisors of $n$, becomes normally distributed with mean and variance $\log\log n$. Granville and Soundararajan gave a proof of the Erd\H{o}s-Kac Theorem that avoided many of the specialized techniques present in several earlier approaches. This thesis includes considerable detail, prerequisite theorems, and instructive background in order to provide a self-contained exposition of the proof of Granville and Soundararajan.

Advancements And Applications Of Nonstandard Finite Difference Methods, Daniel Wood

Mathematics Dissertations - Archive

A class of dynamically consistent numerical methods are analyzed for general n-dimensional productive-destructive systems (PDS). Using this analysis, a methodology for constructing positive and elementary stable nonstandard numerical methods is established. The nonstandard approach results in qualitatively superior numerical methods when compared to the standard ones. PDS model a wide range of dynamical systems, including ones with biological, chemical and physical interactions. Building upon this, a nonstandard finite difference method for solving autonomous dynamical systems with positive solutions is constructed. The proposed numerical methods are computationally efficient and easy to implement. Several examples are given which show that the numerical …

On Solving Finitely Reflected Backward Stochastic Differential Equations, Wilber Alexander Ventura

Mathematics Dissertations - Archive

Classical theory gives a closed form representation of the density p(t,x), a solution to a linear parabolic PDE, via the Feynman-Kac Formula of the underlying diffusion process. In the non-linear PDE case there is no closed form representation for p(t,x) and instead one solves a SDE running back in time whose initial (deterministic value) coincides with p(t,x). This method of solving semi-linear parabolic PDEs is an effective alternative to known numerical schemes. Furthermore, the FBSDE approach allows for treatment of non-smooth coefficients in the PDE that cannot be handled by classical deterministic methods. One of the most important extensions of …

Numeracy Infusion Course For Higher Education (Niche), 1: Teaching Faculty How To Improve Students' Quantitative Reasoning Skills Through Cognitive Illusions, Frank Wang, Esther I. Wilder

Publications and Research

We describe one of the eight units of a professional development program, the Numeracy Infusion Course for Higher Education (NICHE), which introduces research on cognition, including dual-processing theories, to university faculty. Under the dual-processing framework, System 1 (intuition) quickly proposes intuitive answers to judgment problems as they arise, while System 2 (deliberation) monitors the quality of these proposals, which it may endorse, correct, or override. We present several classic questions that demonstrate the pitfalls of overreliance on intuition without analytical thinking, then describe faculty participants’ responses to these questions and their ideas on how to apply cognitive illusion research to …

Key Factors Driving Personnel Downsizing In Multinational Military Organizations, Ilksen Gorkem, Resit Unal, Pilar Pazos

Engineering Management & Systems Engineering Faculty Publications

Although downsizing has long been a topic of research in traditional organizations, there are very few studies of this phenomenon in military contexts. As a result, we have little understanding of the key factors that drive personnel downsizing in military setting. This study contributes to our understanding of key factors that drive personnel downsizing in military organizations and whether those factors may differ across NATO nations’ cultural clusters. The theoretical framework for this study was built from studies in non-military contexts and adapted to fit the military environment.

This research relies on historical data from one of the largest multinational …

Image Denoising By A Local Clustering Framework, Partha Sarathi Mukherjee, Peihua Qiu

Mathematics Faculty Publications and Presentations

Images often contain noise due to imperfections in various image acquisition techniques. Noise should be removed from images so that the details of image objects (e.g., blood vessels, inner foldings, or tumors in the human brain) can be clearly seen, and the subsequent image analyses are reliable. With broad usage of images in many disciplines—for example, medical science—image denoising has become an important research area. In the literature, there are many different types of image denoising techniques, most of which aim to preserve image features, such as edges and edge structures, by estimating them explicitly or implicitly. Techniques based on …

Extension Of Chebfun To Periodic Functions, Grady B. Wright, Mohsin Javed, Hadrien Montanelli, Lloyd N. Trefethen

Mathematics Faculty Publications and Presentations

Algorithms and underlying mathematics are presented for numerical computation with periodic functions via approximations to machine precision by trigonometric polynomials, including the solution of linear and nonlinear periodic ordinary differential equations. Differences from the nonperiodic Chebyshev case are highlighted.

Identifying Similar Polygons: Comparing Prospective Teachers’ Routines With A Mathematician’S, Sasha Wang

Mathematics Faculty Publications and Presentations

This paper reports two prospective teachers' and a mathematician's ways of identifying similar triangle and hexagons through the analysis of routines, a characteristic of geometric discourse. The findings show that visual recognition was a common approach for the mathematician as wells as the two prospective teachers. However, when asked for justification, their routines of identifying similar polygons diverged. The paper also discusses the implication of classroom discourse practices to enhance prospective teachers' communication and reasoning skills while learning geometric concepts such as similarity.

The Numerical Solution Of The Exterior Impedance (Robin) Problem For The Helmholtz’S Equation Via Modified Galerkin Method: Superllipsoid, Hy Dinh

Mathematics Theses

This thesis focuses on finding the solution for the exterior Robin Problem for the Helmholtz Equation and therefore, determines how a convergent smooth surface depending on its outer shape, in this case the superellipsoid, responds to different outer waves. The primary purpose is to calculate the possibility of a certain object, acquiring sufficient conditions, to either submerge under respectively high water pressure or maintain in outer space; if applicable, this approach can be used for a new efficient design of a portion of a submarine or part of a space craft, the second of more interest to NASA, one of …

Discrete And Dynamic Population Models With Logistic Growth Rate, Sabrina Heike Streipert

Doctoral Dissertations

"The Beverton-Holt difference equation defines a discrete relation describing a population model. Considering periodic carrying capacity and periodic inherent growth rate, a population with seasonal changing life cycle and environment is reflected. The so-called periodically forced Beverton-Holt equation is investigated and its unique periodic solution is derived. This provides the first Cushing-Henson conjecture, while a counterexample proves that the classical second Cushing-Henson conjecture is not satisfied. Modifications of the conjecture are formulated. To extend the studies, the Beverton-Holt equation is investigated in the quantum calculus time setting. The existence of the globally attracting periodic solution of the Beverton-Holt q-difference …

Deletion-Induced Triangulations, Clifford T. Taylor

Theses and Dissertations--Mathematics

Let d > 0 be a fixed integer and let A ⊆ ℝ^d be a collection of n ≥ d + 2 points which we lift into ℝ^d+1. Further let k be an integer satisfying 0 ≤ k ≤ n-(d+2) and assign to each k-subset of the points of A a (regular) triangulation obtained by deleting the specified k-subset and projecting down the lower hull of the convex hull of the resulting lifting. Next, for each triangulation we form the characteristic vector defined by Gelfand, Kapranov, and Zelevinsky by assigning to each …

Polyhedral Problems In Combinatorial Convex Geometry, Liam Solus

Theses and Dissertations--Mathematics

In this dissertation, we exhibit two instances of polyhedra in combinatorial convex geometry. The first instance arises in the context of Ehrhart theory, and the polyhedra are the central objects of study. The second instance arises in algebraic statistics, and the polyhedra act as a conduit through which we study a nonpolyhedral problem.

In the first case, we examine combinatorial and algebraic properties of the Ehrhart h*-polynomial of the r-stable (n,k)-hypersimplices. These are a family of polytopes which form a nested chain of subpolytopes within the (n,k)-hypersimplex. We show that a well-studied unimodular triangulation of the (n,k)-hypersimplex restricts to a …

A Refinement Of Michener’S Example Classification, Laurie O. Cavey, Margaret T. Kinzel

Mathematics Faculty Publications and Presentations

In this paper we propose a refinement of Michener’s (1978) well-known example classification based on data from university mathematicians. The refinement takes into account the mathematician’s perspective on the role of examples in doing mathematics. More specifically, our work provides insight into the ways in which mathematicians talk about using examples in their scholarly work and their work with students. The proposed classification has the potential to inform our work as teachers as we strive to create opportunities to engage students in authentic mathematical work.