Mathematics Commons^™

Laboratory Experiences In Mathematical Biology For Post-Secondary Mathematics Students, Matthew Lewis

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

In addition to the memorization, algorithmic skills and vocabulary which is the default focus in many mathematics classrooms, professional mathematicians are expected to creatively apply known techniques, construct new mathematical approaches and communicate with and about mathematics. We propose that students can learn these professional, higher level skills through Laboratory Experiences in Mathematical Biology (LEMBs) which put students in the role of mathematics researcher creating mathematics to describe and understand biological data. LEMBs are constructed so they require no specialized equipment and can easily be run in the context of a college math class. Students collect data and develop mathematical …

Newton’S Third Law In Karmic Warfare, Kazmier Maślanka

The Transdisciplinary STEAM+ Journal

A work entitled "Newton's Third Law in Karmic Warfare" is a mathematical visual poem which is a perfect example of a technique, that I call The Paradigm Poem. This piece makes a direct connection with the concept of karma and Newton’s Third Law of motion. I will introduce the concept of “The Mathematical Paradigm Poem” to illuminate an example of how metaphor is used in mathematical visual poetry. I will also discuss much of the process in making this aesthetic expression.

Effects Of Cell Cycle Noise On Excitable Gene Circuits, Alan Veliz-Cuba, Chinmaya Gupta, Matthew R. Bennett, Krešimir Josić, William Ott

Mathematics Faculty Publications

We assess the impact of cell cycle noise on gene circuit dynamics. For bistable genetic switches and excitable circuits, we find that transitions between metastable states most likely occur just after cell division and that this concentration effect intensifies in the presence of transcriptional delay. We explain this concentration effect with a three-states stochastic model. For genetic oscillators, we quantify the temporal correlations between daughter cells induced by cell division. Temporal correlations must be captured properly in order to accurately quantify noise sources within gene networks.

Simulating The Spread Of The Common Cold, R. Corban Harwood

Faculty Publications - Department of Mathematics

This modeling scenario guides students to simulate and investigate the spread of the common cold in a residence hall. An example floor plan is given, but the reader is encouraged to use a more relevant example. In groups, students run repeated simulations, collect data, derive a differential equation model, solve that equation, estimate parameter values by hand and through regression, visually evaluate the consistency of the model with their data, and present their results to the class.

Mathematical Optimization And Applications, Teodora Suciu

Mathematics Colloquium Series

This talk centers on mathematical optimization in the context of Calculus of Variations. Optimization involves choosing the best element from a set of choices, usually through mathematical approaches. Solving these kinds of problems is considered an essential tool in many areas of science and engineering. Additionally, various mathematics and business applications are discussed. Also explored is a real-life example with a detailed algorithm that is closely related to the Traveling Salesman problem.

Mathematical Relativity And The Nature Of The Universe, Priscila Reyes

Mathematics Colloquium Series

In this talk, I will be discussing certain space-times, which can be used to model celestial objects and events in the universe. These are solutions to Einstein's field equations, which roughly describe the relation between matter, energy and the geometry of the universe. The concept of time in relation to an observer will be demonstrated. I will also include some interesting phenomena that arise out of the unusual mathematical structure of space-times , such as Lorentz contraction, reverse Cauchy-Schwarz, and the twin paradox.

On Rational Subgroups Of Exceptional Groups., Neha Hooda Dr.

Doctoral Theses

The main theme of this thesis is the study of exceptional algebraic groups via their subgroups. This theme has been widely explored by various authors (Martin Leibeck, Gary Seitz, Adam Thomas, Donna Testerman to mention a few), mainly for split groups ([26], [27], [28], [60] ). When the field of definition k of the concerned algebraic groups is not algebraically closed, the classification of k-subgroups is largely an open problem. In the thesis, we mainly handle the cases of simple groups of type F4 and G2 defined over an arbitrary field. These may not be split over k. We first …

Essays On The Economics Of Education., Soham Sahoo Dr.

Doctoral Theses

This thesis consists of three empirical essays that investigate issues related to the economics of education. The main focus of this thesis is to study schooling outcomes of children in a developing country: India. The first chapter explores the effect of mothers’ labour force participation on children’s educational outcomes in the context of a large employment guarantee program in India. This study finds that workfare schemes enhancing women’s access to economic opportunities have implications for intra-household resource allocation which in turn lead to a positive effect on children’s education. The second essay investigates the effect of better access to secondary …

Convolutions And Green’S Functions For Two Families Of Boundary Value Problems For Fractional Differential Equations, Paul W. Eloe, Jeffrey T. Neugebauer

Mathematics Faculty Publications

We consider families of two-point boundary value problems for fractional differential equations where the fractional derivative is assumed to be the Riemann-Liouville fractional derivative. The problems considered are such that appropriate differential operators commute and the problems can be constructed as nested boundary value problems for lower order fractional differential equations. Green's functions are then constructed as convolutions of lower order Green's functions. Comparison theorems are known for the Green's functions for the lower order problems and so, we obtain analogous comparison theorems for the two families of higher order equations considered here. We also pose a related open question …

Race, Space, And The Conflict Inside Us, Francis Su

All HMC Faculty Publications and Research

Talking about race is hard. Our nation is wrestling with some open wounds about race. These sores have been around a while, but they have been brought to light recently by technology, politics, and an increasingly diverse population. And regardless of the outcome of the U.S. presidential election, we will all need to work at healing these sores, not just in our personal lives, but in our classrooms and in our profession.

Efficacy Of Math Video Tutorials On Student Perception And Achievement, Carol R. Kahrmann

Doctor of Education in Teacher Leadership Dissertations

Abstract

The purpose of this mixed methods research study is to explore how teacher-made video tutorials in one middle school mathematics classroom are being used and what enables/impedes their effectiveness. A quantitative, quasi-experimental design investigates how video tutorial usage relates to students’ self-efficacy and achievement in mathematics. The sample consisted of 55 students in the experimental group and 65 students in the control group. Results found parents wanted the tutorials even if they did not view them. Students did actually use the video tutorials for remediation and learning. Videos longer than ten minutes impeded the effectiveness of the videos. The …

Secondary Analysis Of Concussion Data, Martin Zwick, Stephanie Kolakowsky-Hayner, Nancy Carney, Maya Balamane, Tracie Nettleton, D. Wright

Complex Systems Faculty Publications and Presentations

Clinical studies are expensive & time-consuming. Typically in these studies specific hypotheses are subjected to confirmatory test. Yet the data may harbor evidence of unanticipated relations between variables. It is thus desirable to subject the data to secondary analyses in the hope of discovering novel & valuable associations. Exploratory analysis, however, is tentative: findings should be replicated in new data. This presentation reports some secondary analyses on concussion data. Data mining on 2 datasets will be discussed, & some unexpected findings reported. The analyses use reconstructability analysis (RA), a probabilistic graphical modeling method implemented in the Occam software package developed …

Numerical Methods For Partial Differential Equations, Yang Yang

TechTalks

Many physical phenomena can be modeled by partial differential equations, such as Euler equations and Navier–Stokes equations. However, most cases, we cannot find the exact solutions, hence numerical methods are needed to approximate them. In my work, I use discontinuous Galerkin methods (a kind of finite element method) and finite volume methods to solve hyperbolic and parabolic equations—especially for problems with strong singularities, such as delta-functions. Moreover, some of the physical bounds will be preserved, such as the positivity of the density, maximum-principle of the velocity etc. Some of the problems I am working on include compressible gas dynamics, miscible …

Minimum Number Of Distinct Eigenvalues Of Graphs, Shahla Nasserasr

Mathematics Colloquium Series

For a simple graph G on n vertices, a real symmetric nxn matrix A is said to be compatible with G, if for different i and j, the (i; j) entry of A is nonzero whenever there is an edge between the vertices i and j, it is zero otherwise. The minimum number of distinct eigenvalues, when minimum is taken over all compatible matrices with G, is denoted by q(G). In this talk, a survey of some known and new results about q(G) is presented.

2016: "Maximizing Customer Convenience With Hill Climbing", Ankit Agarwal '17, William Tong '17, Robert Lou '17, George Moe '17

Distinguished Student Work

As technology advances and the pace of life quickens, companies are expected to continually provide better services, be more accessible, and respond to requests faster. While such expectations have led to the rapid growth of retail stores and service locations throughout the United States, delivery services have arguably been the most influenced by these rising demands.

Amazon.com is an online storefront that has seen tremendous growth throughout its lifetime, and it only continues to grow. Founded in 1994 as an online bookstore, the digital retailer eventually expanded its services to include technology, home goods, and even eBooks. Amazon’s response to …

2016: Himcm International Mathematics Modeling Contest: National Outstanding Paper, Suchet Kumar '19, Tommy Vadakumchery '19, Nathan Kim '19, Bert Cao '19

Distinguished Student Work

In the modern business world, maximizing profits is the highest priority. Businesses, especially small ones, should try to save money whenever possible. Along with cutting wages, removing competition, and increasing advertising and production, efficiency of company sites can save money. For example, a cheap 50 ft. by 100 ft. warehouse costs $35,000, but when coupled with the costs of maintenance and wages, increasing the number of warehouses significantly increases costs.

Clearly, warehouses are expensive, making it necessary to place them in optimal locations. In this problem, we attempt to reduce the number of warehouses while shipping to the entirety of …

Measure And Integration, Jose L. Menaldi

Mathematics Faculty Research Publications

Abstract measure and integration, with theory and (solved) exercises is developed. Parts of this book can be used in a graduate course on real analysis.

Distributions And Function Spaces, Jose L. Menaldi

Mathematics Faculty Research Publications

Beginning with a quick recall on measure and integration theory, basic concepts on (a) Function Spaces, (b) Schwartz Theory of Distributions, and (c) Sobolev and Besov Spaces are developed. Moreover, only a few number of (solved) exercises are given. Parts of this book can be used in a graduate course on real analysis.

Estimating Propensity Parameters Using Google Pagerank And Genetic Algorithms, David Murrugarra, Jacob Miller, Alex N. Mueller

Mathematics Faculty Publications

Stochastic Boolean networks, or more generally, stochastic discrete networks, are an important class of computational models for molecular interaction networks. The stochasticity stems from the updating schedule. Standard updating schedules include the synchronous update, where all the nodes are updated at the same time, and the asynchronous update where a random node is updated at each time step. The former produces a deterministic dynamics while the latter a stochastic dynamics. A more general stochastic setting considers propensity parameters for updating each node. Stochastic Discrete Dynamical Systems (SDDS) are a modeling framework that considers two propensity parameters for updating each node …

Generalized Least-Powers Regressions I: Bivariate Regressions, Nataniel Greene

Publications and Research

The bivariate theory of generalized least-squares is extended here to least-powers. The bivariate generalized least-powers problem of order p seeks a line which minimizes the average generalized mean of the absolute pth power deviations between the data and the line. Least-squares regressions utilize second order moments of the data to construct the regression line whereas least-powers regressions use moments of order p to construct the line. The focus is on even values of p, since this case admits analytic solution methods for the regression coefficients. A numerical example shows generalized least-powers methods performing comparably to generalized least-squares methods, …

A Sampling Kaczmarz-Motzkin Algorithm For Linear Feasibility, Jesus A. De Loera, Jamie Haddock, Deanna Needell

CMC Faculty Publications and Research

We combine two iterative algorithms for solving large-scale systems of linear inequalities, the relaxation method of Agmon, Motzkin et al. and the randomized Kaczmarz method. We obtain a family of algorithms that generalize and extend both projection-based techniques. We prove several convergence results, and our computational experiments show our algorithms often outperform the original methods.

Continuous Dependence And Differentiating Solutions Of A Second Order Boundary Value Problem With Average Value Condition, Samantha A. Major

Mathematics Colloquium Series

Using a few conditions, continuous dependence, and a result regarding smoothness of initial conditions, we show that derivatives, with respect to each of the boundary data, of solutions to a second order boundary value problem with an average value integral condition solve the associated variational equation with interesting boundary conditions.

Variable Moving Average Transform Stitching Waves, Vesselin Vatchev

School of Mathematical & Statistical Sciences Faculty Publications

A moving average transform in the plane with a variable size and shape window depending on the position and the ’time’ is studied. The main objective is to select the window parameters in such a way that the new transform converges smoothly to the identity transform at the boundary of a prescribed bounded plane region. A new approximation of solitary waves arising from Korteweg-de Vries equation is obtained based on results in the paper. Numerical implementation and examples are included.

Life After Calculus: 20 Years Later, Darren B. Glass

Math Faculty Publications

In 1996 Math Horizons interviewed a group of students at the Joint Mathematics Meetings; now, 20 years later, one of those students, Darren Glass, interviews another group of students.

Synchrony In A Boolean Network Of The L-Arabinose Operon In Escherichia Coli, Andy Jenkins, Matthew Macauley

Publications

The lactose operon in Escherichia coli was the first known gene regulatory network, and it is frequently used as a prototype for new modeling paradigms. Historically, many of these modeling frameworks use differential equations. More recently, Stigler and Veliz-Cuba proposed a Boolean network model that captures the bistability of the system and all of the biological steady states. In this paper, we model the well-known arabinose operon in E. coli with a Boolean network. This has several complex features not found in the lac operon, such as a protein that is both an activator and repressor, a DNA looping mechanism …

On Characterizations And Infinite Divisibility Of Recently Introduced Distributions, Gholamhossein G. Hamedani

Mathematics, Statistics and Computer Science Faculty Research and Publications

We present here characterizations of the most recently introduced continuous univariate distributions based on: (i) a simple relationship between two truncated moments; (ii) truncated moments of certain functions of the 1^th order statistic; (iii) truncated moments of certain functions of the n^th order statistic; (iv) truncated moment of certain function of the random variable. We like to mention that the characterization (i) which is expressed in terms of the ratio of truncated moments is stable in the sense of weak convergence. We will also point out that some …

New Classes Of Univariate Continuous Exponential Power Series Distributions, M. Ahsanullah, Gholamhossein G. Hamedani, M. Shakil, B.M. Golam Kibria, F. George

Mathematics, Statistics and Computer Science Faculty Research and Publications

Recently, many researchers have developed various classes of continuous probability distributions which can be generated via the generalized Pearson differential equation and other techniques. In this paper, motivated by the importance of the power series in probability theory and its applications, we derive some new classes of univariate exponential power series distributions for a realvalued continuous random variable, which we call exponential power series distributions. Various mathematical properties of the proposed classes of distributions are discussed. Based on these distributional properties, we have established some characterizations of these distributions as well. It is hoped that the findings of the paper …

Trigonometry, Ted Sundstrom, Steven Schlicker

Open Textbooks

This trigonometry textbook is different than other trigonometry books in that it is free to download, and the reader is expected to do more than read the book and is expected to study the material in the book by working out examples rather than just reading about them. So this book is not just about mathematical content but is also about the process of learning and doing mathematics. That is, this book is designed not to be just casually read but rather to be engaged.

Since this can be a difficult task, there are several features of the book designed …

The Classification Of Zp -Modules With Partial Decomposition Bases In L∞Ω, Carol Jacoby, Peter Loth

Mathematics Faculty Publications

Ulm’s Theorem presents invariants that classify countable abelian torsion groups up to isomorphism. Barwise and Eklof extended this result to the classification of arbitrary abelian torsion groups up to L∞ω-equivalence. In this paper, we extend this classification to a class of mixed Zp-modules which includes all Warfield modules and is closed under L∞ω-equivalence. The defining property of these modules is the existence of what we call a partial decomposition basis, a generalization of the concept of decomposition basis. We prove a complete classification theorem in L∞ω using invariants deduced from the classical Ulm and Warfield invariants.

Eventual Quasi-Linearity Of The Minkowski Length, Ivan Soprunov, Jenya Soprunova

Mathematics and Statistics Faculty Publications

The Minkowski length of a lattice polytope PP is a natural generalization of the lattice diameter of PP. It can be defined as the largest number of lattice segments whose Minkowski sum is contained in PP. The famous Ehrhart theorem states that the number of lattice points in the positive integer dilates tPtP of a lattice polytope PP behaves polynomially in t∈Nt∈N. In this paper we prove that for any lattice polytope PP, the Minkowski length of tPtP for t∈Nt∈N is eventually a quasi-polynomial with linear constituents. We also give a formula for the Minkowski length of coordinates boxes, degree …