Sports analytics is a growing field. The larger field of data analytics is exploding as a field requiring skills in mathematics and computer science. What sports projects can be tackled as an undergraduate? This talk will discuss a variety of projects Dr. Tim Chartier of Davidson College has directed with his students, who have ranged from first years to seniors. His projects have varied from helping the German National Basketball Team to his own college teams. He's also aided the NBA, NASCAR, and ESPN. Learn how to play a sport—as a sports analyst!

For centuries there have been people trying to communicate secretly, and for almost as long there have been people trying to uncover what's being said in those same secret communications. This talk will give a glimpse into the art and science of making and breaking codes, from simple substitution ciphers all the way through the German Enigma and secure online banking.

The existence of number as an entity composed of both magnitude and dimension is formalized. An isomorphic categoric relation is established between \(\Phi_g\), the set of all formulae contained within an axiomatization of arithmetic where number is considered in the absence of strict dimensionality and \(\Phi_{g'}\); the union of these two sets constituting \(\Phi_d\), the set of all statements contained within the axiomatization of arithmetic where number is rigorously defined as a measure of dimensional space. The infinite unit \(l\) is defined within \(\Phi_g\); through such definition, the existence of \(\Phi_d\) is implied, where \(\Phi_d\) is capable of providing intuition for results arising consequent to said definition otherwise inaccessible to reason. At last, provided no algebraic expression is claimed innately without value nor intuitive reason, we may consider the algebra of old to be satisfactorily established.

We discuss an optimization problem regarding the contraction ratio and the Hausdorff dimension of the outward Sierpinksi's fractals of regular N-gons. By using some known results on inward Sierpinksi's fractals and infinite geometric series, we will derive the general formulae for the maximal contraction ratio and the maximal Hausdorff dimension. We will generalize the maximal contraction ratio and the maximal Hausdorff formulae for a certain regular N-gon that is not affected by in-growth.

Numbers are often used to define more complicated numbers. For example, two integers are used to define a rational number and two reals are used to define a complex number. It might be expected that an irrational power of an irrational number would be an irrational number. Despite that, it is actually possible for an irrational power of an irrational number to be a rational number. For many generations, this circulated as a non-constructive proof by contradiction in logic for discrete mathematics textbooks and college courses. Since the '80s, a constructive proof circulated orally, such as \(\sqrt{2}^{\log_{\sqrt{2}} 3}\) equals to \(3\). A written proof was published in 2008 by Lord. The first contribution of this paper is to prove that there is an uncountable number of such pairs of irrational numbers such that the power of one to the other is a rational number. Marshall and Tan answered the question of whether there is a single irrational number \(a\) such that \(a^a\) is rational. They proved that given \(I\) = \(\frac{1}{e})^(\frac{1}{e},\infty)\), then every rational number in \(I\) is either of the form \(a^a\) for an irrational \(a\) or is in the very thin set \(\{1,4,27,256, \dots ,n^n, \dots \}\). It seems a challenging task to analytically solve the equation \(x^x=y\) for any real \(y\). To the best of our knowledge, there is no work on finding \(x\) from a given \(y\). We proved that \(\ln((\ln(y))<x\) for \(y>e\) and \(x<\ln(y)\) , for \(y>e^e\). Hence, the second contribution of this paper is to estimate the real \(x\) such that either one of the equations \(x^x=y\) or \(x^{(x^x)}=y\) holds, for a given \(y\).

We will present a math model of how a cell phone company's consumer growth will affect its popularity among consumers as well as its financial success.

We will investigate domestic chicken farming through the use of difference equations. We will also compare egg and meat production across breeds as well as the possibility of marketing.

We use past population data for Massachusetts to predict the future population. First, we graph data points of Massachusetts's population by decade, since 1880, in Maple. A well-known model in Ordinary Differential Equations, the population logistic model, is a starting point for our equation. Through Bernoulli's technique, we enhance the model to obtain more precise predictions. In this model, we leave two coefficients as variables, allowing us to utilize least squares and Maple's optimization package to optimize these values. By varying the logistic population model's exponent around 2 and changing the starting decade, we find the most optimal equation. Last, we compare this optimal equation against the predictions of the United States Census Bureau for the next twenty years.

Procedural Content Generation (PCG) is a form of content generation used in video games in which certain aspects, such as the environments and world, are generated as the player explores them without any extra input from the developers. However, to the best of our knowledge, PCG lacks a proper "checking tool" other than the predefined heuristics implemented when first programming the game. This may lead to drastic changes, such as going from a tundra to a rain forest, in a way which breaks immersion.In Graph Theory, we take a set of vertices, \(V\), and the set of corresponding edges between those vertices, \(E\), to make a graph. We believe that graph theory can be used as a "checking tool" for transitioning from one state to another within a video game. Our goal is to create a more realistic experience when transitioning from one state to another when PCG is utilized in a video game.

An open question within the network reliability community is whether or not there are reliability polynomials of a graph with multiple inflection points. In a previous paper by Dr. David Milan and Dr. Christina Graves, it was proven that multigraphs have arbitrarily many inflection points. This talk will show methods used to further investigate inflection points of simple graphs' reliability polynomials.

We search for strongly regular graphs using the first and second subconstituents of the graph, making an assumption about automorphisms acting on those constituents.Assuming an abelian group acting on the first constituent, we find the linear representations of the group, a sum of roots of unity lying in a cyclotomic extension of the rationals.We examine parameters \((v, k, \lambda, \mu)\) where primes 5 or 3 divide the valency \(k\).Three examples of this case are (121,40,15,12), (196, 65, 24, 20), and (64,21,8,6).In the first two cases we use a 5th root of unity to determine the possibilities for the trace of the representation.Then using different methods, such as bounds for certain matrix entries and knowledge about Galois automorphisms, we eliminate some of the possibilities that survive the first test.The same process is used for the parameter set (64,21,8,6) but instead uses a cube root of unity.This project is part of an NSF REU at Sam Houston State University and is partially supported by the Avila Undergraduate Research Award and the College of Science at SHSU and by NSF grant DMS-1262897.

For centuries, number theory has been a transformative branch of pure mathematics that has enabled mathematicians to gain insight into difficult concepts-tools that have become mainstream and peaked in the 1500s-1900s, fascinating mathematicians to this day. We will be observing specific developments in the past that have inspired the mathematical community-in particular, on a congruence modulo relationship explored by Leonhard Euler and Robert Carmichael. We will explore this congruence modulo relationship, with additional properties, by deriving tools to investigate its behavior, including the types of behaviors it would entail, and understanding the connection between a familiar number theoretic function and a particular behavior---the "cycle length"---of the congruence modulo relationship.

The symmetric inverse monoid, \(SIM(n)\), is the set of all partial one-to-one mappings from the set \(\{1,2,\dots,n\}\) onto itself under the operation of composition. Earlier research on the symmetric inverse monoid delineated the process for determining whether an element of \(SIM(n)\) has a \(k\)th root. The problem of developing formulas for the number of \(k\)th roots of a given element of \(SIM(n)\) has since been posed, which this research aims to solve. The number of \(k\)th roots of a given element of an important submonoid of \(SIM(n)\), the Symmetric Group, will be delineated in this presentation and be invaluable to enumerating \(k\)th roots in \(SIM(n)\). Overall, this research will demostrate a facinating application of combinatorics to abstract algebra and will expand upon inverse semigroup theory.

Callinectes sapidus, or the blue crab, is a crustacean that is vital to the survival of the whooping cranes (Grus americana) and the ecosystem supported at Aransas National Wildlife Refuge (ANWR). The crabs serve as a keystone organism because of their direct and indirect relationships with neighboring species, including their role as a favorite food source for the cranes. We created a linear population model to predict the abundance of juvenile and adult blue crabs in ANWR. The model's key parameters are the fecundity (\(F\)), growth proportion (\(G\)), probability of staying in the juvenile stage (\(P_1\)), and probability of surviving in the adult stage (\(P_2\)). These entries are also dependent upon the values of average carapace width, larval survival probability, and growth fraction. We found that if the product of the growth fraction and the fecundity exceeds a threshold dependent upon the natural mortality rate, then the population will have long-term growth.

Texas Lakes have been invaded by a toxin releasing species known as Prymnesium parvum or golden algae. The interaction between golden algae and a competing species can be expressed by systems of nonlinear equations. Modeling was performed by utilizing the program MATLAB and incorporating chemostat model theory. Long term behavior was observed in the competition for nutrients between two species including allelopathy to simulate toxins released by the golden algae. We then consider the introduction of variable nutrient inflow using a sinusoidal function as well as a piecewise function related to rainfall data in the affected areas. Numerical results showed that the poison susceptible species survived longer in a variable nutrient environment compared to the constant nutrient environment. Coexistence between both species was never achieved but should not be presumed impossible.

The study of chaotic dynamics is defined to be the mathematics where a fixed rule or model describes how a point in a geometrical space depends on time. Another interpretation for this branch of mathematics would be to say the study of iterations. I will explore the nature of chaotic iterations and expand this method into the complex field in order to provide an algorithm for creating the Julia Set.

In order to better understand the behavior of soliton waves we analyze solutions to the Korteweg-de Vries equation for the single and multiple soliton cases. Through deliberate manipulation of entries, we notice relationships with the matrices and the representations of solutions. We analyze the matrices in the n-soliton case and its behaviors including multiplicities and minor changes to the entries of the matrices.

An example of a simple dynamical system resulting in chaotic solutions, the double pendulum is a marvelous subject for the application of Lagrange's equations of motion. In this presentation, the mathematics behind the system is explained, with a demonstration of how the chaos is a linear combination of two modes of oscillation, and a sensitivity analysis of the parameters is performed.

We discuss the dynamics of dual(outer) billiard system with finitely many singularities. First, we focus our attention to regular polygons, and study the singularity structure created by the dual billiards's path around the polygon. And then, we will expand our scope further toward irregular polygons and curvilinear shapes. We use Java programming and C programming to map our figures and analyze our visualization.

Generative describes a mode of art in which works are produced by an artist strictly bound by a systematic process. In this presentation, we will explore and analyze different iterative processes to create algorithmically generated line drawings. In addition, we will attempt to simulate organic, naturalistic line drawings that imitate the organic, naturalistic style of an artist's hand.

A recurrence matrix is a matrix whose entries are the terms of a sequence defined by a recurrence relation. A recurrence relation is an equation that recursively defines a sequence of numbers, once one or more initial terms are given. The rank of a matrix is the maximum number of linearly independent columns or rows of the matrix. In 2014, Christopher Lee and Valerie Peterson proved that the maximum rank of a recurrence matrix is the order of the corresponding recurrence relation but that for order-two recurrence relations the rank drops whenever the ratio of the two initial terms of the recurrence relation is an eigenvalue of the relation. We prove their findings using the method of fundamental solutions in hopes of extending their results to recurrence relations of orders higher than two.

This study reports the results of an investigation conducted to assess a team's performance in the game of cricket. We consider the effectiveness of handling bowlers by a team as a measure to represent a team's performance. Allocation of bowlers to bowl a certain number of overs is seen as an assignment problem in operations research and the Hungarian method is used to perform the analysis. Instead of the popular measurement indicators, we consider a statistic, which is based on the number of runs conceded by the bowler, the number of wickets taken, and the number of overs bowled by the bowler. Using the data from international Cricket tournament, team's performance is analyzed and compared with the rest of the teams who took part in the tournament.

Marfan Syndrome is a genetic disorder, which can affect multiple systems such as the cardiovascular, skeletal, pulmonary and ocular. We will present a stochastic model of the Ghent criteria to diagnose Marfan Syndrome based on physical characteristics. We will model the probability of complications, separately and in different combinations.

We study the virtual generalization of classical knots in relation to connected sums and graph theory. The focus is on invariants of classical prime knots to verify the generalized properties for virtual prime knots. The main purpose of this work is to explore the transition, and to gain a better understanding of the connection, between classical and virtual knots.

We define the lattice sphere number of a lattice, \(L\), to be the minimum radius of a lattice sphere containing a non-trivial knot. We use a variety of arguments to establish the lattice sphere numbers for the simple cubic lattice (sc), face-centered cubic lattice (fcc), two variations of the body-centered cubic lattice (bcc-14 and bcc-8), and the simple hexagonal lattice (sh).

In this work, we do an analysis of Dowker's codes of an alternating knot. Then we construct a bijective function which generates a set that can be partitioned into two classes, that is, a class of classical knots and a class of virtual knots.

In this talk, we will describe new polynomial knot invariants that occur in the Melvin-Morton-Rozansky expansion of the colored Jones polynomial. This expansion is of particular interest as it provides a relationship between the Jones polynomial and the Alexander polynomial of a knot. In addition to explaining this relationship, we will share how our results support a conjecture by Rozansky about amphichiral knots. We will be presenting a sample of our results and discussing the implications they may have on future work.

A tiling is said to be unilateral if no two equal sides of polygons meet corner to corner and equitransitive if any tile can be mapped via a symmetry of the tiling to any other congruent tile. It has been shown that a unilateral and equitransitive (UE) tiling can be made with any arbitrary number of sizes of squares. My research group and I have shown that there are exactly two UE tilings by equilateral triangles: one with two sizes of triangles and one with three sizes of triangles.

We'd all like to know what makes mathematics (or science) students successful, but how do you go about 'studying' the answer to this question rather than just 'having an opinion'? In this talk, we'll look at how to design studies on mathematics success, and what various studying say.

Take a visual journey through multi-variable calculus with Calc Plot 3D as your tour guide. This powerful java-based program has an intuitive interface providing easy access to visualizations of all things Calc III. We'll specifically show you how to plot out 3-dimensional surfaces, space curves and vectors. We'll also provide information for enhancing your presentations and/or lectures with image capture and CalcPlot3D scripting. Best of all, CalcPlot3D is free! You'll leave with the access instructions to the website of CalcPlot3D creator, Paul Seeburger, Professor of Mathematics at Monroe Community College, Rochester, NY. His work has been supported by the National Science Foundation Grant, DUE-CCLI 0736968. In Summer 2015, faculty from El Centro received STEM grant funding for their curriculum development with CalcPlot3D.

The Mathematical Sciences Department at the University of Texas at Dallas (UTD) has received an award from the National Science Foundation (NSF) through the new Enriched Doctoral Training (EDT) program in the Division of Mathematical Sciences. The goal of NSF's EDT program in general, and the UTD project in particular, is to ensure that while they are in graduate school, mathematics and statistics Ph.D. students gain marketable skills and research experience developing mathematics and statistics to improve understanding of problems in science and engineering. Students involved in EDT will work on an interdisciplinary research problem posed by one of our external partners from industry, a government lab, or a research institution. They will perform research in a team of 2-3 students, one mathematics and one statistics faculty mentor at UTD, and the external partner. Students will bee exposed to the entire life cycle of an industrial research problem from starting with an open-ended problem description to obtaining final results.And finally, students will be well positioned to continue work on the project in a summer internship at the external partner's organization.In this talk I will describe the new EDT program at UT Dallas.