Abstract: In this session, we will share how inquiry based approaches can be employed to engage students in higher-level critical thinking strategies, open-ended exploration and multiple approaches in problem solving, including technology. Specifically, the participants will learn about best practices in teaching and learn to develop mathematically rich problem solving activities and tasks that can help students to develop their algebraic habits of mind. The participants will also have an opportunity to learn about effective performance based tasks with varying cognitive demand and 21st century skills designed to help educators support their students learning progression and their mathematical understanding.

Abstract: In the last decade, there have been dramatic advances in mathematical modeling, analysis and simulation techniques to understand fundamental mechanisms underlying biological, bio-inspired and engineering systems. This work will present examples of undergraduate research projects that evolved from multidisciplinary applications modeled via coupled differential equations. Some of these examples include using mathematics to understand why aneurysms rupture; understand how zika spreads; studying social dynamics and; employing mathematics to stop poaching of elephants in Africa. Mathematical analysis and computation for some benchmark model applications will also be presented. Finally, we will also discuss how such projects can provide scholarship opportunities for students at all levels to employ transformative mathematical research in multidisciplinary areas.

Abstract: Most people believe that pursuing math leads to only two options for jobs: teacher or professor. That belief is completely false. There are many exciting career paths available to mathematically inclined individuals. Dr. Paulette N. Willis will share with you her nontraditional career path as a mathematician and discuss the ups and downs of her journey. Come listen to her story and the lessons she's learned. We are sure you will gain some insight on how to choose a career you will love..

Irreducible polynomials are by definition not factorable in the conventional sense, so having another method to factor these polynomials is important. The factorization method we discuss in this paper uses matrices, and was first introduced by David Eisenbud in 1980. The method is called matrix factorization, and works whether the polynomial is irreducible or not. Regarding the question of the existence of matrix factorizations, we prove that it suffices to show every homogeneous polynomial has a matrix factorization. We then show that every homogeneous polynomial does in fact have a matrix factorization. These results have been obtained by starting with polynomials of one term and working up from there. When factoring polynomials in the conventional sense, there is one and only one way to factor them. However, using matrix factorizations, more than one factorization is possible. These factorizations will look different, so we need a precise notion of equivalence. We discuss two definitions of equivalence, Eisenbud's and homotopy equivalence, and show one implies the other, but not vice versa. Results include a theorem showing the conditions required for a polynomial to be factored using a two by two matrix, a theorem and a proposition that shows how to make a matrix factorization for a given polynomial and homogenize it for the homogenization of the polynomial, and a theorem that shows how to make an arbitrary factorization homogeneous for a homogeneous polynomial."

We discuss a recently discovered class of inverse semigroups called the inverse semigroups of self-similar graph actions. To describe this class, we characterized the idempotents, ideals, and other properties common to inverse semigroups in general. Furthermore, we provide conditions that ensure a self-similar graph action is fundamental or 0-disjunctive and, more generally, when it satisfies properties of inverse semigroups relevant to the study of C-algebras.

Construction of a cryptosystem generally requires the system to be relatively easy to solve in one direction but difficult to solve in the reverse. For example, modular multiplication is easy but modular division can be difficult. To assist in modular arithmetic, factoring methods such as Pollard's p-1 factoring algorithm can be used. An overview for the types of encryption methods are given to introduce a basic cryptosystems. With this foundation, we will present a cryptosystem based on the theory of elliptic curves.

Applications of elliptic curves include solving Fermat's Last Theorem, cryptography, and integer factorization. Using the Weierstrass equation for elliptic curves \(y^2=x^3+Ax+B\), with the condition \(4A^3+27B^2\neq 0\) is equivalent to there being no singular points on the curve allowing for a well-defined tangent line at each point. The Addition Law (or Group Law) for adding points on an elliptic curve demonstrates that (rational) points on elliptic curves form a group. This addition involves a "tangent and secant" method, which can be used to find additional rational points on a curve if one knows at the outset at least one rational point on the curve. These points, as well as the coefficients of the elliptic curve must be in a field, so this Addition Law is applied in the case where the curve is defined over a finite field. Elliptic curves can be implemented to factor an integer N, in order to determine if N is composite or prime, by reducing their coefficients and the rational points on the curve modulo N.

We construct a family of polynomial bijections between N and \(N^m\) for any m in N. The family of bijections is based on the figurate numbers of recursively generated gnomons shaped like simplices. A beautiful advantage of this approach over simply recursively folding tuples down to a single number by chaining Cantor's pairing function is that this approach is significantly easier to reason about due to its geometric foundation. This technique also produces polynomial bijections from \(N^m\) to N of degree m, which for odd m, is lower than using Cantor's. An example application in Computer Science, iterating through infinite spaces in "list comprehensions," is lightly discussed.

Group theory has been studied for centuries by many researchers. In the case of finite groups, Lagrange proved that the cardinality of any subgroup of a given finite group is an integer divisor of the cardinality of that group. This paper defines a new concept related to both finite and infinite group theories and their subgroups, called pseudo-partitions of a group. The main result of our paper proves that the set of elements obtained from removing a subgroup from a pseudopartition of a(n) (in)finite group cannot be a subgroup of the given group.

To date, researchers investigating lunar forming impacts have only considered the impactor and proto-Earth in their simulations. This project aims to incorporate the effects of the Sun on lunar forming impact studies. The Sun makes up 99.8 percent of the solar systems mass. Hence, any valid study must include the effect caused by such a massive body. We have incorporated data from NASA's "Horizons" system at JPL to produce the correct initial conditions for the impactor and proto-Earth—relative to the Sun. The solar gravitational effect was added to complete this heliocentric model. Currently we are investigating the effect the Sun has on lunar forming impacts and comparing this with data from existing models. We hope to use this knowledge to create a more complete theory of how our Moon was formed.

The Traveling Salesman Problem (TSP) is a classical NP-hard problem in Optimization Theory that has been around since 1930. The problem is defined as follows: "Given a list of cities, what is the shortest possible route that visits each city exactly once and returns home?" The process to find the exact solution is straightforward: start with a city, then go to the next city, until all cities have been visited once and return to the initial city. Do this for all possible tours and choose the shortest tour. The difficulty lies in the fact that there are (n+1)!/2 possible tours. To understand the situation, imagine a TSP that contains 25 cities. If the first living biped (4.9 million years ago) had a super computer that could calculate 2 billion tours per second, the computer would just now finish calculating. Since finding an exact solution is not feasible, efficient approximation methods are needed. In this work, we utilize an N-body approach to extrude an accurate approximation to the TSP. The rate of complexity of an N-body problem grows at the rate of \(n^2\) which is a vast improvement over n!

We analyze yearly traffic accident data in Austin, TX in order to reveal trends in traffic accidents. We apply multiple methods in order to determine how the number of traffic accidents varies based on the hourly time of day and day of the week. We also analyze fatal and non-fatal accidents.

Pattern avoidance was first developed by Donald Knuth in 1973, while studying stack-sortable permutations. He needed a way of distinguishing which permutations on the set \(S_n\)={1, 2, … , n} could be sorted from least-to-greatest using a stack-sorting algorithm. During our research, which was done as part of the REU at the University of Texas at Tyler, we studied pattern avoidance on quasi-Stirling permutations, which are permutations on the multi-set {1, 1, 2, 2, … , n, n}. Our focus was on avoiding combinations of patterns from \(S_3\).

Given a triangle ABC with area \(\Delta\), the average area of an inscribed triangle RST whose vertices are uniformly distributed on AB, BC, and CA, is proven to be one-fourth of the area of ABC. The second and third moments of the area are shown to be \(\frac{\Delta^2}{12}\) and \(\frac{5\Delta^3}{144}\) respectively. A general formula for the n-th moment of the area is derived. Additionally, a Monte Carlo simulation, as well as a Maxima program, confirm the theoretical results. We experimentally verify that the Monte Carlo method has a \(\frac{1}{\sqrt{N}}\) convergence where N is the number of samples.

We will present a mathematical model of the motions of juggling flower sticks. We will use motion capture technology and digitization to find equations that represent the paths of the sticks and to find the rates of change in the Cartesian coordinate and angular positions.

RNA is a single-stranded molecule whose function depends on how it folds onto itself. Biologically, the folding is given by Watson-Crick Base Pairs. We consider a combinatorial abstraction of RNA where the bonding relations are given by a connection graph, instead of Watson-Crick base pairs, and foldings are represented by non-crossing matchings. Using this model, we were able to categorize graphs that guarantee a connected move graph, which is how foldings transform into one another.

The Preisach model is a well-known input-output model that was originally proposed for magnetic hysteresis, characterizing transitions between magnetic dipole states as a function of the system's previous states. This dependence of a system on its history of possible states has been extended beyond magnets to obtain suitable mathematical models for a variety of hysteresis phenomena in engineering, physics, and economics applications. Transitions between \(2^N\) states of the discrete Preisach model can be described by a directed graph Γ. We consider an adjacency matrix AN associated with a random walk on this graph. Alternatively, this matrix is associated with the random evolution of an inventory under the "Last In, First Out" (LIFO) inventory management rule. By relating the self-similar nature of LIFO directed graphs to the recursive nature of Chebyshev polynomials, we are able to obtain explicit formulas for the spectrum and eigenvectors of the non-symmetric adjacency matrix \(A_N\) . This work was done in collaboration with A. Amann, D. Dmitrii, T. Kalmar-Nagy.

A Scramble Squares puzzle consists of nine square pieces with half of an image on each edge. To solve the puzzle the nine pieces are arranged in a 3-by-3 grid so that edges on adjacent pieces form a complete image. A repetition is a halfimage that appears more than once on a puzzle piece. Previous research uses a graph-theoretical approach to establish necessary and sufficient conditions for solutions without repetitions to 2-by-2 Scramble Squares puzzles. We use a similar approach to establish necessary and sufficient conditions for solutions with repetitions to 2-by-2 Scramble Squares puzzles.

Scheduling problems are commonplace in the industrial world of today. As such, being able to solve these scheduling problems serves an important role in increasing efficiency of society. Finding the solutions to these scheduling problems involves knowledge of linear optimization and graph theory. This presentation will focus on the setup of these scheduling problems in both a linear optimization and graph theory environment.

During the span of our research we studied the singular integral equation method (SIEM) that has been used in the paper "The Crack Problem for a Nonhomogeneous Plane", by F. Delale and F. Erdogan. Compared to other numerical methods (finite element, finite difference, or finite volume), SIEM is a semi-analytical method; it provides very accurate results in solving crack problems. Despite the fact that SIEM is an accurate method used in the solid mechanics community, due its heavily mathematical derivation, only a limited school of researchers are using it. Our goal in this research is to break down the mathematical details of SIEM so that more researchers can understand and use the method. We also implemented the numerical part to obtain the crack displacement profiles and the stress intensity factor (SIF) at crack tips.

In Singapore, the Central Provident Fund (CPF) serves as the national savings system. Singaporean and permanent workers are required to contribute part of their wages to a compulsory savings account to fund their retirement, healthcare and housing needs. We model the CPF based on data including workforce participation and annual income by age group.

Basketball is a multi-faceted game, the outcome of which is determined by many different components. One of these components is the concept of "shot selection," which is a major offensive focus of basketball teams and coaches. Shot selection refers to the types of offensive attempts made by a team towards the basketball goal. Good shot selection can be defined as shot attempts that have a high probability of resulting in points and give a team some type of advantage over its opponent. This mathematical model determines a function for optimal shot selection and then uses statistical data from games participated in by the Southwestern University Women's Basketball team to compare the team's points per possession to the points per possession that would be expected if the team abided by the optimal hot selection function.

In the past fifteen years, the healthcare system and cost of healthcare in the United States has become increasingly more complicated and more expensive. We examine data on the National Healthcare Expenditure (NHE) and analyze the relationship between other variables and NHE. The analysis includes linear regressions that model healthcare costs. Specifically data on the cost of different sectors in the healthcare system is used. These include prescription drugs, hospital visits, personal healthcare, GDP, and population.

In this talk, I will develop a classic formula for computing the probability of absorption in simple finite random walks with absorbing boundaries. I will also discuss the potential application of such computations to applied settings, including modeling of response times in behavioral experiments.

We will present a mathematical model of the game Mancala, a sequential two-player board game consisting of "pits" and "stones." Considering that the goal of the game is to capture the most stones, we will investigate various strategies and their effects on the outcome of the game.

American Roulette is a casino game in which players place various types of bets on subsets of 38 numbers. A wheel is then spun to determine the winning number; players who bet on this number win their bet amount plus a payout multiplier specific to each type of bet while players who did not bet on this number lose their bet amount. Previous research uses Markov chains to compute probabilities of players accumulating a desired amount of money ("success") or losing all their money ("ruin") for each type of bet. We use Markov chains to compute the probabilities of success and ruin for new betting strategies based on bold gambling and find that these strategies yield higher probabilities of success. We also use Markov chains to compute the expected game lengths until success or ruin for all betting strategies.

SET is a card game which requires cognitive, logical, and spatial reasoning skills to complete a set. A set is a group of three cards which are either all the same or all different in four distinct features: shape, shading, number, and color. The game of SET has connections to a variety of topics in mathematics, including counting, probability, modular arithmetic, and geometry. We will discuss some of these connections while improving our own game strategy.

Origami is the art of paper folding to construct a sculpture from a square. This art usually results in the paper having many shapes and patterns which we intend to use to explain mathematical concepts. We will go over TEKS from the primary level and use origami as a manipulative to teach them.

In this presentation, we will discuss performance of middle school students in mathematics courses. In particular, we study data on how gender, availability of home support systems and other additional school support systems effect failure rates.

Neural codes, represented by binary strings, encode neural activity and show relationships between stimuli, like distance between locations in an environment. Certain neurons, called place cells, have shown experimentally to fire in convex regions in space. An open convex code is a code that can be realized as open convex sets. It was initially believed that all open convex codes can also be realized with closed sets; however, counterexamples with six neurons show that this is not the case. Codes that are open convex but not closed convex are called unstable neural codes. In this talk, we will show that five neurons is the minimal number of neurons necessary to construct unstable neural codes, as well as discuss common elements of such codes.

Personality psychologists often apply clustering techniques on questionnaire data to model personality structure. Inspired by this work, we apply techniques from topological data analysis (TDA) to understand the structure of this data. The data comes from Cattell's Sixteen Personality Factor Questionnaire (collected by Bell, Rose, & Damon in 1972). Subjects were 969 adult male volunteers divided into three age groups: 25 to 34, 35 to 54, and 55 to 82. We use persistent homology (a TDA tool) to cluster the data and identify that personality structure is slightly different between the age groups. It is also curious to note that data from the youngest age group appears to have a topological "hole", which raises questions of the psychological significance. This work suggests that additional research, including applying TDA tools to other questionnaire data sets can provide insights to the study of personality. Preliminary work on current research will also be included.

The ordinary van der Waerden number W(k, l) is the least positive integer W such that any k-coloring of the initial segment of positive integers [W] contains a monochromatic arithmetic progression of length l. In this paper we consider tilings of rectangles by T-tetrominos, and ask when arithmetic progressions (AP) of tiles must occur, that is, tiles that are equally spaced and all in the same orientation. Define T(4, k) to be the least T such that in any tiling of a \(4\times T\) rectangle by T-tetrominos, a k-term AP must appear. And let the notation \((M,N) \to k\) mean that any tiling of an \(M\times N\) rectangle by T-tetrominos must contain a k-term AP. When evaluating T(4, 2) and T(4, 3), the combinations of tilings can be sorted out by hand. As the sizes increase, the tilings get longer and the computer becomes useful to sort through the millions of tilings in order to find or avoid APs. We will present exact values of T(4, k) for some small k, and characterize those pairs (M, N) for which \((M, N)\to 3\). Our initial results show that there is a suprisingly tight correlation between our tiling numbers and the ordinary van der Wearden numbers for the coloring of the integers.

A new class of non-standard Lagrangians that explicitly depend on the special functions of mathematical physics is discovered and this requires adding some auxiliary conditions to the Euler-Lagrange equations. Several examples with applications of the new non-standard Lagrangians are presented and discussed. Moreover, the relationships between the obtained results and Lie algebras and Lie groups are also briefly addressed.

We analyze a mathematical model for traffic based on fluid dynamics and, specifically, hyperbolic conservation laws. These conservation laws are a set of partial differential equations which define the dependency between changes in traffic density and traffic flux. We investigate the theoretical relationship between these fundamental traffic variables.

Complex Matter Space (CMS), Antiparticles, Wave functions, and Tachyons. The Special Theory of Relativity cannot recognize velocities faster than the speed of light. However, there are phenomena that do appear to transmit data and maybe energy much faster than the speed of light. New assumptions will be postulated that two of matter's intrinsic components, 1) mass, and 2) charge, are a single complex quantity: M=m+iq. The mass will be measured by real number system and the charge on the imaginary axis. We will review the history of Complex Matter Space, introduce complex numbers, vectors, and vector spaces, then explain how momentum and energy are reformulated in Complex Matter Space, and finally, use the Complex Matter Space concept to help explain antiparticles and to conjecture the existence of tachyons which are implied by CMS and may help explain how the wave functions of entangled particles collapse so quickly.