Mathematics Commons^™

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.

Case Study Of Undergraduate Research Projects In Vector Analysis, Alexander Vaninsky, Willy Baez Lara, Madieng Diao, Analilia Mendez

Publications and Research

This paper presents two examples of the undergraduate research projects in vector analysis conducted under the first author’s supervision at one of the community colleges that is an integral part of a large city university. The projects were accomplished by the students pursuing associated degrees in engineering, during their sophomore year. One project was to obtain an explicit formula for the curvature of a curve in plane defined implicitly in rectangular or polar coordinates. Another project was aimed to develop an alternative procedure for finding potential function for a vector field in space based on simultaneous integration. Participation in these …

Math And Physics Activities, Maureen Miller, Hope Bragg, Christy Keefer

Integrated Math & Social Studies Lessons

Mathematics is at the core of the Hidden Figures story. These women were united by their passion for the field of mathematics. Society often portrays that there are “bad” math students, those that struggle with calculations and applications. The structure of these activities, pairing of students, permits students to support each other in working through the problems. The video clip allows students to establish connections between mathematical calculations and scientific concepts. The physics problems that students complete are motion problems that beginning rocket engineers would have solved to determine how high their rocket flew.

On Zero-Sum Rado Numbers For The Equation Ax_1 + X_2 = X_3, Nicholas Brown

Electronic Theses and Dissertations

For every positive integer $a$, let $n = R_{ZS}(a)$ be the least integer, provided it exists, such that for every coloring \[ \Delta : \{1, 2, ..., n\} \rightarrow \{0, 1, 2\},\] there exist three integers $x_1, x_2, x_3$ (not necessarily distinct) such that \[ \Delta(x_1) + \Delta(x_2) + \Delta(x_3) \equiv 0\ (mod\ 3) \] and \[ ax_1 +x_2 = x_3.\] If such an integer does not exist, then $R_{ZS}(a) = \infty.$ The main results of this paper are \[R_{ZS}(2) = 12\] and a lower bound is found for $R_{ZS}(a)$ where $a \geq 2$.

A Symbolical Approach To Negative Numbers, Paul M.E. Shutler

The Mathematics Enthusiast

Recent Early Algebra research indicates that it is better to teach negative numbers symbolically, as uncompleted subtractions or “difference pairs”, an idea due to Hamilton, rather than abstractly as they are currently taught, since all the properties of negative numbers then follow from properties of the subtraction operation with which children are already familiar. Symbolical algebra peaked in the 19th Century, but was superseded by abstract algebra in the 20th Century, because Peacock’s permanence principle, which asserted that solutions obtained symbolically would actually be correct, remained unproven. The main aim of this paper is to provide this missing proof, in …

Problem Posing In Consumer Mathematics Classes: Not Just For Future Mathematicians, Jeff Irvine

The Mathematics Enthusiast

Problem posing is recognized as a key component of mathematics (Ellerton, 2013). However, in many curricula, problem solving often dominates over problem posing (Stoyanova, 2003). This focus on problem solving exists despite research that shows that problem posing improves students' problem-solving skills, attitudes, confidence, understanding of concepts, and mathematical thinking (Singer, Ellerton, & Cai, 2013); reinforces basic mathematical skills, increases motivation, responsibility, and thinking flexibility (Ponte & Henriques, 2013); and is useful for teachers to assess students' cognitive processes, identify misconceptions, and modify instruction (Ponte & Henriques, 2013). Further, problem posing can play a large part in student motivation (McLeod, …

Teacher Development And Seventh Graders’ Achievement On Representing And Solving Equations, Sheree T. Sharpe, Analucia D. Schliemann

The Mathematics Enthusiast

We analyze the impact of a teacher development program based on a functions approach to algebra on 7th graders understanding of equations and examine how students’ score gains during the academic year relate to their teachers’ initial level of mathematical knowledge of algebra, functions, and graphs. Students from participating teachers’ and their control peers completed a mathematics assessment at the start and at the end of the school year the teachers were taking the program. We determined teachers’ initial levels of mathematics knowledge through a written assessment given at the start of the program. Although both groups of students improved …

The Positivication Of Coalgebraic Logics, Fredrik Dahlqvist, Alexander Kurz

Engineering Faculty Articles and Research

We present positive coalgebraic logic in full generality, and show how to obtain a positive coalgebraic logic from a boolean one. On the model side this involves canonically computing a endofunctor T': Pos->Pos from an endofunctor T: Set->Set, in a procedure previously defined by the second author et alii called posetification. On the syntax side, it involves canonically computing a syntax-building functor L': DL->DL from a syntax-building functor L: BA->BA, in a dual procedure which we call positivication. These operations are interesting in their own right and we explicitly compute posetifications and positivications in the case …

Features Of Agent-Based Models, Reiko Heckel, Alexander Kurz, Edmund Chattoe-Brown

Engineering Faculty Articles and Research

The design of agent-based models (ABMs) is often ad-hoc when it comes to defining their scope. In order for the inclusion of features such as network structure, location, or dynamic change to be justified, their role in a model should be systematically analysed. We propose a mechanism to compare and assess the impact of such features. In particular we are using techniques from software engineering and semantics to support the development and assessment of ABMs, such as graph transformations as semantic representations for agent-based models, feature diagrams to identify ingredients under consideration, and extension relations between graph transformation systems to …

Foreword: Special Issue On Coalgebraic Logic, Alexander Kurz

Engineering Faculty Articles and Research

The second Dagstuhl seminar on coalgebraic logics took place from October 7-12, 2012, in the Leibniz Forschungszentrum Schloss Dagstuhl, following a successful earlier one in December 2009. From the 44 researchers who attended and the 30 talks presented, this collection highlights some of the progress that has been made in the field. We are grateful to Giuseppe Longo and his interest in a special issue in Mathematical Structures in Computer Science.

Quasivarieties And Varieties Of Ordered Algebras: Regularity And Exactness, Alexander Kurz

Engineering Faculty Articles and Research

We characterise quasivarieties and varieties of ordered algebras categorically in terms of regularity, exactness and the existence of a suitable generator. The notions of regularity and exactness need to be understood in the sense of category theory enriched over posets.

We also prove that finitary varieties of ordered algebras are cocompletions of their theories under sifted colimits (again, in the enriched sense).

Multi-Model Validity Assessment Of Groundwater Flow Simulation Models Using Area Metric Approach, Omkar Aphale, David J. Tonjes

Technology & Society Faculty Publications

We demonstrate the application of the Area Metric developed by Ferson et al. (2008) for multi-model validity assessment. The Area Metric quantified the degree of models’ replicative validity: the degree of agreement between the observed data and the corresponding simulated outputs represented as their empirical cumulative distribution functions (ECDFs). This approach was used to rank multiple representations of a case study groundwater flow model of a landfill by their Area Metric scores. A multi-model approach allows for the accounting for uncertainties that may either be epistemic (from lack of knowledge), or aleatory (from variability inherent in the system). The Area …

Elementary Construction Of Lusztig’S Canonical Basis, Peter Tingley

Mathematics and Statistics: Faculty Publications and Other Works

In this largely expository article we present an elementary construction of Lusztig’s canonical basis in type ADE. The method, which is essentially Lusztig’s original approach, is to use the braid group to reduce to rank two calculations. Some of the wonderful properties of the canonical basis are already visible: that it descends to a basis for every highest weight integrable representation, and that it is a crystal basis.

Homeomorphic Property Of The Stochastic Flow Of A Natural Equation In Multi-Dimensional Case, Fatima Benziadi, Abdeldjabbar Kandouci

Communications on Stochastic Analysis

No abstract provided.

Normal Surfaces And 3-Manifold Algorithms, Josh D. Hews

Honors Theses

This survey will develop the theory of normal surfaces as they apply to the S3 recognition algorithm. Sections 2 and 3 provide necessary background on manifold theory. Section 4 presents the theory of normal surfaces in triangulations of 3-manifolds. Section 6 discusses issues related to implementing algorithms based on normal surfaces, as well as an overview of the Regina, a program that implements many 3-manifold algorithms. Finally section 7 presents the proof of the 3-sphere recognition algorithm and discusses how Regina implements the algorithm.

Tying The Knot: Applications Of Topology To Chemistry, Tarini S. Hardikar

Honors Theses

Chirality (or handedness) is the property that a structure is “different” from its mirror image. Topology can be used to provide a rigorous framework for the notion of chirality. This project examines various types of chirality and discusses tools to detect chirality in graphs and knots. Notable theorems that are discussed in this work include ones that identify chirality using properties of link polynomials (HOMFLY polynomials), rigid vertex graphs, and knot linking numbers. Various other issues of chirality are explored, and some specially unique structures are discussed. This paper is borne out of reading Dr. Erica Flapan’s book, When Topology …

Some Examples Of The Interplay Between Algebra And Topology, Joseph D. Malionek

Honors Theses

This thesis presents several undergraduate and graduate level concepts in the fields of algebraic topology and topological group theory in a manner which requires very little mathematical background of the reader. It uses non-rigorous interpretations of concepts while introducing the reader to the rigorous ideas with which they are associated. In order to give the reader an idea of how the fields of algebra and topology are closely affiliated, the paper goes over five main concepts, the fundamental group, homology, cohomology, Eilenberg-Maclane spaces, and group dimension.

A Harmonic M-Factorial Function And Applications, Reginald Alfred Brigham Ii

Doctoral Dissertations

"We offer analogs to the falling factorial and rising factorial functions for the set of harmonic numbers, as well as a mixed factorial function called the M-factorial. From these concepts, we develop a harmonic analog of the binomial coefficient and an alternate expression of the harmonic exponential function and establish several identities. We generalize from the harmonic numbers to a general time scale and demonstrate how solutions to some second order eigenvalue problems and partial dynamic equations can be constructed using power series built from the M-factorial function"--Abstract, page iii.

Bootstrap-Based Confidence Intervals In Partially Accelerated Life Testing, Ahmed Mohamed Eshebli

Doctoral Dissertations

"Accelerated life testing (ALT) is utilized to estimate the underlying failure distribution and related parameters of interest in situations where the components under study are designed for long life and therefore will not yield failure data within a reasonable test period. In ALT, life testing is carried out under two or more higher than normal stress levels, with the resulting acceleration of the failure process yielding a sufficient amount of un-censored life-span data within a practical test duration. Usually one (or more) parameters of the life distribution is linked to the stress level through a suitably selected model based on …

Local Holomorphic Extension Of Cauchy Riemann Functions, Brijitta Antony

Doctoral Dissertations

"The purpose of this dissertation is to give an analytic disc approach to the CR extension problem. Analytic discs give a very convenient tool for holomorphic extension of CR functions. The type function is introduced and showed how these type functions have direct application to important questions about CR extension. In this dissertation the CR extension theorem is proved for a rigid hypersurface M in C² given by y = (Re ω)^m(Im ω)ⁿ where m and n are non-negative integers. If the type function is identically zero at the origin, then there is no CR extension. …

Zero-Dimensional Spaces And Their Inverse Limits, Sahika Sahan

Doctoral Dissertations

"In this dissertation we investigate zero-dimensional compact metric spaces and their inverse limits. We construct an uncountable family of zero-dimensional compact metric spaces homeomorphic to their Cartesian squares. It is known that the inverse limit on [0,1] with an upper semi-continuous function with a connected graph has either one or infinitely many points. We show that this result cannot be generalized to the inverse limits on simple triods or simple closed curves. In addition to that, we introduce a class of zero-dimensional spaces that can be obtained as the inverse limits of arcs. We complete by answering a problem by …

T-Closed Sets, Multivalued Inverse Limits, And Hereditarily Irreducible Maps, Hussam Abobaker

Doctoral Dissertations

"This dissertation consists of three subjects: T-closed sets, inverse limits with multivalued functions, and hereditarily irreducible maps.

For a subset A of a continuum X define T(A) = X \ {x ∈ X : there exists a subcontinuum K of X such that x ∈ int_xX(K) ⊂ K ⊂ X \ A}. This function was defined by F. Burton Jones and extensively investigated in the book [20] by Sergio Macias. A subset A of a continuum X is called T-closed set if T(A) = A. A characterization of T-closed set is given using generalized …