Mathematics Commons^™

Forms Of Knowledge Of Advanced Mathematics For Teaching, Julianna Connelly Stockton, Nicholas H. Wasserman

The Mathematics Enthusiast

In this paper, we explore in more detail why knowing advanced mathematics might be beneficial for teachers, specifically in relation to their classroom practice. Rather than by listing courses or specific advanced topics, as though those were the agents of change, we do so by considering advanced mathematical content for teachers in terms of more general forms of knowledge. In particular, we identify five forms of knowledge of advanced mathematics for teaching: peripheral, evolutionary, axiomatic, logical, and inferential. These categories were derived from analysis of an extensive mapping process linking K-12 content to relevant advanced mathematics. We connect these forms …

On Independence, Matching, And Homomorphism Complexes, Wesley K. Hough

Theses and Dissertations--Mathematics

First introduced by Forman in 1998, discrete Morse theory has become a standard tool in topological combinatorics. The main idea of discrete Morse theory is to pair cells in a cellular complex in a manner that permits cancellation via elementary collapses, reducing the complex under consideration to a homotopy equivalent complex with fewer cells. In chapter 1, we introduce the relevant background for discrete Morse theory.

In chapter 2, we define a discrete Morse matching for a family of independence complexes that generalize the matching complexes of suitable "small" grid graphs. Using this matching, we determine the dimensions of the …

Efficacy Of Acute Normovolemic Hemodilution In The Setting Of Chronic Anemia, Andrew Vieira

Undergraduate Journal of Mathematical Modeling: One + Two

A theoretical model for calculating hematocrit as it relates to blood volume loss was used to test the theoretical impact of acute normovolemic hemodilution prior to surgery in a chronically anemic patient. The technique of hemodilution is intended to decrease red blood cell loss that occurs as the result of surgical intervention, however its impact may be decreased in an anemic patient. An equation to find change in hematocrit resulting from blood volume loss was elucidated via integration. This equation was then applied to a theoretical patient with 5 L of blood and a hematocrit of 32 percent which revealed …

Diagnosing Breast Cancer With A Neural Network, John Cullen

Undergraduate Journal of Mathematical Modeling: One + Two

Fine needle aspiration (FNA) is a minimally invasive biopsy technique that can be used to successfully diagnose types of cancer, including breast cancer. Immediately, it is difficult for a human to spot any trends in the cell level data gathered during a fine needle aspiration procedure. One way to predict the type of tumor a patient has, is to use a computer to develop a mathematical model based on known data. This project utilizes the Diagnostic Wisconsin Breast Cancer Database (DWBCDB) to create an accurate mathematical model that predicts the type of a patient’s tumor (Malignant or Benign). A neural …

Alternative Typing Learning Curves, Marcel Kreuzer

Undergraduate Journal of Mathematical Modeling: One + Two

Typing is a part of everyday life for many people. However some people have difficulty in doing it. For example, for people who are blind and those who have carpal tunnel or cerebral palsy, a standard keyboard would not be effective to type. Alternative options include voice recognition and keyless keyboards. We focus on orbiTouch keyboard which has no keys, only two knobs which allow a person to type. Which alternative way of typing would be most effective given a person’s condition is discussed.

Danger Of Snow In The Sunshine State, Dmitrii Karpenko

Undergraduate Journal of Mathematical Modeling: One + Two

The main purpose of the project is to investigate the maximum deflection of a rectangular 16 x 8 inches beam, supported on both ends under uniform loading stress under snow pressure in the event that Tampa experiences snowfall. The required information for the project is the material of the beam and its dimensions, measurement of the area of the roof that would accumulate snow, and calculations of the Moment of Inertia and Uniform Distributed Load for the beam. The maximum deflection of the beam can be calculated using the information above.

The outcome of the research shows that the roof …

Application Of Simple Harmonics Modeling A Shock, Kai Raymond

Undergraduate Journal of Mathematical Modeling: One + Two

This project entails modeling a spring-shock absorber system in order to evaluate the position of a mass when this system is acted upon by outside forces. We will restrict our attention to the linear case where it is assumed that Hooke’s law gives the force exerted by the spring, and the resistance of the system will be proportional to the velocity of the mass. We analyze the time and frequency domain of the absorber system and compute its impulse response.

Approximating Surface Area Of Fluctuating Lipid Leaflets Using Weighted Grid Tessellation, Ahnaf Siddiqui

Undergraduate Journal of Mathematical Modeling: One + Two

The surface area per lipid is an important quantity useful in characterizing lipid bilayer structure and dynamics. However, simply dividing the flat surface area covered by the number of lipids does not give a sufficient characterization of surface dynamics in cases where parts of the lipid leaflets have curvaceous trajectories in three dimensions. An algorithm is proposed for modeling surface area per lipid using a three-dimensional tessellated grid. In this way both the vertical, as well as the more commonly considered the horizontal, fluctuations of lipids are used to accurately determine the area covered. This algorithm can reveal differences in …

Rudimentary Model Of Glucose Response To Stress, Nasha Rios-Guzman

Undergraduate Journal of Mathematical Modeling: One + Two

A simplistic multifactorial model was crafted to represent change in plasma glucose concentrations after increases in stress hormones above baseline. A sample application of the predictive use of this model was demonstrated, and a reflection on possible models in diabetic populations was described. The potential for use of this model in conjunction with current models is discussed, as well as the issues pertinent to such a basic model.

Analysis Of Rainfall In Tampa, Amy Polen

Undergraduate Journal of Mathematical Modeling: One + Two

Rainfall for a region is very important to define, because it affects an ecosystem health, wildlife, and even human behavior. Using data obtained from National Oceanographic and Atmospheric Administration (NOAA) at the Tampa Bay International Airport and graphing it, the total rainfall for a year was estimated by both a rough trapezoidal Riemann’s sum approximation and a definite integral generated by polynomial regression. It was determined that both approximations gave an underestimate of the total rainfall that was measured, but the polynomial integral gave a reasonable estimate with a small percent error. It was seen through the graphical representation of …

Analyzation Of The Resistor-Inductor-Capacitor Circuit, Martin D. Elton

Undergraduate Journal of Mathematical Modeling: One + Two

Starting with the basics of what a Resistor-Inductor-Capacitor circuit (RLC) is, i.e. its fundamental components, and ending with practical applications using advanced calculus to aid in predetermining the results and circuit design, this paper analyzes the RLC circuit via an advanced calculus based approach.

The Solow Model And Standard Of Living, Eric Frey

Undergraduate Journal of Mathematical Modeling: One + Two

All across the world, living standards vary significantly. The Solow growth model, developed by Nobel Prize winning economist Robert Solow in 1956, is still one of the most commonly used models in economics to explain economic growth. This paper will outline the Solow growth model, and its assertion that increases in total factor productivity (TFP) can lead to limitless increases in the standard of living in a country. Much of the mathematical notation and explanation has been derived from Stephen Williamson of Washington University. Additionally, it will provide empirical examples illustrating the model’s ability to match real-world data.

The Forces Affecting A Sailboat, Kelly Stukbauer

Undergraduate Journal of Mathematical Modeling: One + Two

The objective of this project is to determine the specific effects of the various forces acting on a sailboat. Through this project, a greater understanding of both the physics behind sailing, and the calculus behind that physics, shall be gained.

Optimization Of A Fuel Cell, Eduardo Gines

Undergraduate Journal of Mathematical Modeling: One + Two

Fuel cells are devices that generate energy from a chemical reaction that takes place inside the cell. The main parts of these devices are two electrodes and an electrolyte solution. The project consists of determining the area of the electrodes for the fuel cell at which the cell produces its maximum amount of power. This was done with the performance curve of the fuel cell which was in terms of voltage vs current density. The performance curve was turned into terms of power density vs current density, and through this curve the maximum power was determined by identifying the maximum …

Gallai-Ramsey Numbers For C7 With Multiple Colors, Dylan Bruce

Honors Undergraduate Theses

The core idea of Ramsey theory is that complete disorder is impossible. Given a large structure, no matter how complex it is, we can always find a smaller substructure that has some sort of order. One view of this problem is in edge-colorings of complete graphs. For any graphs G, H₁, ..., H_k, we write G → (H₁, ..., H_k), or G → (H)_k when H₁ = ··· = H_k = H, if every k-edge-coloring of G contains a monochromatic H_i in color i for some i ∈ …

Influences Of Probability Instruction On Undergraduates' Understanding Of Counting Processes, Kayla Blyman

Theses and Dissertations--Education Sciences

Historically, students in an introductory finite mathematics course at a major university in the mid-south have struggled the most with the counting and probability unit, leading instructors to question if there was a better way to help students master the material. The purpose of this study was to begin to understand connections that undergraduate finite mathematics students are making between counting and probability. By examining student performance in counting and probability, this study provides insights that inform future instruction in courses that include counting and probability. Consequently, this study lays the groundwork for future inquiries in the field of undergraduate …

Crossed Product Algebras Over Dihedral Field Extensions, Kaitlyn Lee

Undergraduate Honors Thesis Collection

Let F be the field of fractions of R, a ring of power series with coefficients in some field. Let K/F be a finite Galois extension, and assume the integral closure S of R in K is also a power series ring.

In this paper, we consider a construction from ring theory called a crossed product algebra and an associated function f: G × G →K called a cocycle that is used to define a multiplication operation for the crossed product algebra. Consider a crossed product algebra whose cocycle f takes values in S. We give a proof concerning which …

An Improved Imaging Method For Extended Targets, Sui Zhang

Doctoral Dissertations

The dissertation presents an improved method for the inverse scattering problem to obtain better numerical results. There are two main methods for solving the inverse problem: the direct imaging method and the iterative method. For the direct imaging method, we introduce the MUSIC (MUltiple SIgnal Classification) algorithm, the multi-tone method and the linear sampling method with different boundary conditions in different cases, which are the smooth case, the one corner case, and the multiple corners case. The dissertation introduces the relations between the far field data and the near field data.

When we use direct imaging methods for solving inverse …

Bounding And Stabilizing Realizations Of Biased Graphs With A Fixed Group, Nancy Ann Neudauer, Dan Slilaty

Mathematics and Statistics Faculty Publications

Given a group Γ and a biased graph (G, B), we define a what is meant by a Γ-realization of (G, B) and a notion of equivalence of Γ-realizations. We prove that for a finite group Γ and t ≥ 3, that there are numbers n(Γ) and n(Γ, t) such that the number of Γ-realizations of a vertically 3-connected biased graph is at most n(Γ) and that the number of Γ-realizations of a nonseparable biased graph without a (2Ct , ∅)-minor is at most n(Γ, t). Other results pertaining to contrabalanced biased graphs are presented as well as an analogue …

Chain Transitive Homeomorphisms On A Space: All Or None, Ethan Akin, Juho Rautio

Mathematics and Statistics Faculty Research & Creative Works

Extending earlier work, we consider when a compact metric space can be realized as the omega limit set of a discrete time dynamical system. This is equivalent to asking when the space admits a chain transitive homeomorphism. We approach this problem in terms of various conditions on the connected components of the space. We also construct spaces where all homeomorphisms are chain transitive.