Create your personal website!  Joseph Skelton

This is a hands-on workshop! You will walk out of it with your own website! We will go over the basics of a professional website and present some examples. I will guide you through creating a google site and we will discuss how to make it stand out from other professional websites. This workshop is intended for graduate students, but everybody is welcome!

Abigail Zion. A study of mathematical models for reducing feral cat populations

We construct mathematical models of feral cat populations to determine the most effective method for humanely reducing feral cat populations over time. Cats are separated into categories based-on sex, age (reproductive age or not yet of reproductive age), and fertility status.  We developed 3 compartmental models which we use to analyze the change in population over time by with varying initial conditions, sterilization rates, and interactions with house cats. We seek to quantify the impact of each of these parameters, with a large focus on the most efficient sterilization plan.

Achouak Bekkai. On a fractional diffusion equation with nonlinear time-nonlocal source term

In this poster, we prove the local existence of a unique mild solution to a time-space fractional diffusion equation with time-nonlocal nonlinearities of exponential growth. Moreover, under some suitable conditions on the initial data, it is shown that local solutions experience blow-up.

Alvaro Carbonero & Beth Anne Castellano. Exploring Preference Orderings Through Discrete Geometry

Consider $n + 1$ points in the plane: a set $S$ consisting of $n$ points along with a distinguished vantage point $v$. By measuring the distance from $v$ to each of the points in $S$, we generate a preference ordering of $S$. The maximum number of orderings possible is given by a fourth-degree polynomial (related to Stirling numbers of the first kind), found by Good and Tideman (1977), while the minimum is given by a linear function. We investigate intermediate numbers of orderings achievable by special configurations $S$. This work is motivated by a voting theory application, where an ordering corresponds to a preference list. We also consider this problem for points on the sphere, where our results are similar to what we found for the plane. Other variants of the original problem inspired by voting theory are developed. These include using a weighted distance function and also using two vantage points.  

Joint work with Charles Kulick and Karie Schmitz. Mentors: Dr. Brittany Shelton and Dr. Gary Gordon.

Edith Lee & Van Hovenga. Quasi stationary Distributions on Invasion Model for Opinion Evolution

Utilizing Markov Chains as the basis for an invasion model for opinion evolution, we look to prove that the distribution of opinions approaches a quasi-stationary state. That is, over a long period of time, a system of (2) opinions hovers at around 50-50. We have successfully proved this phenomenon empirically.

Franziska Riepl & Logan Richard & Lucas Walls. Building Homotheties From Pinches

We show that starting from finitely many points  in $\R^m$, and successively applying "pinches" on them, it is possible to arrive at any of the configurations that result from applying, to the original points, a homothety of factor $0\leq s\leq 1$ and center the centroid of the points. Here, a "pinch'' consists in moving two points towards their common centroid. The points can have weights, which determine how the centroids are computed. Thus, we build homotheties through pinches. For three points, four pinches suffice. In general, the number of pinches is independent of the original configuration (and of $m$).

Garrett Van Beek. Game Theory

This poster is an introduction to Game Theory as a school of thought. First we will discuss what Game Theory is, and how it is useful for solving problems. Then we will introduce some different types of games (zero-sum, cooperative, and stand-offs). We will discuss the tools used by game theorists: like decision trees and defined utility. We will also discuss the applications of the theorems derived by game-theorists in different scientific and strategic endeavors. Topics include: Nim, prisoner's dilemma, 3-way stand-offs (and how it relates to nuclear diplomacy).

Karen Guthrie & Janelle Domantay. The Optimization of a Signed Tree

A signed graph G is a regular graph where each edge is assigned a + (positive edge) or a - (negative edge). The signed degree of a vertex v in a signed graph, denoted by sdeg(v), is the number of positive edges incident to v subtracted by the number of negative edges incident to v. Finally, we say G realizes the set D if: D = {sdeg(v)  :  v ∈ V(G)}. In this paper we prove that D is the signed degree set of a tree if and only if 1 ∈ D or -1 ∈ D. Further, for every valid set D, we find the smallest diameter that a tree must have to realize D.  Lastly, for valid sets D with nonnegative numbers, we find the smallest order that a tree must have to realize D.

Jordan Robinson. Agent-Based Model of Murine Patellar Tendon Healing

The patellar tendon transmits loads from the quadriceps to the tibia, promoting locomotion. The main suggested risk factor for clinical pathologies is excessive cyclic loading, especially the type observed during athletic activity and manual labor. Tissue matrix composition and cells called tenocytes dictate tendon’s uniaxial mechanical function. Following injury, a flood of inflammatory cells and spike in expression of specific genes work in concert to remove damaged tissue, trigger cell proliferation, and deposit a provisional matrix. Unfortunately, healed tendons demonstrate significant functional deficits. Moreover, decreased cell migration and fiber alignment with aging further hampers healing outcomes; as does the relative lack of research into the age- and sex-dependent properties. This healing process can be further understood using an agent-based model (ABM). ABMs simulate individual entities, or agents, and their interactions between each other and with their environment. This approach has the advantage of building complexity from the ground up, mimicking physiology. Therefore, the objectives of this study were

Mark Curiel. Stars and Kostant's Partition Function

The enumeration of vector partitions is difficult, and, in many cases, a succinct closed form solution is effectively out-of-reach. As an alternative to enumerating vector partitions, we can seek other objects with some additional structure that agree in number. For instance, Kostant's partition function counts the number of ways to express a vector as a nonnegative integer linear combination of the positive roots of a Lie algebra. It has been shown that the number of juggling sequences with a fixed finite number of balls gives a way to enumerate Kostant’s partition function of a scalar multiple of the highest positive root of the Lie algebra. For this poster presentation, we shall define another object called an n-star, which also enumerates Kostant’s partition function of the highest root of the Lie algebra, and we provide a correspondence between juggling sequences and n-stars. 

Nikhil Sahoo. A Whole Lot of Values for Pi

I will give an account of a problem posed and solved Stanisław Gołąb in 1932: classifying the values of pi achieved by arbitrary norms on the plane. These values comprise the interval [3,4] and Gołąb also classified the norms achieving both extremes. In a 2004 account of this problem, Duncan, Luecking & McGregor also proved that the classical value π=3.1415… is a lower bound on values of pi for norms with quarter-turn symmetry. I have extended their work by classifying the norms with quarter-turn symmetry that achieve this lower bound, providing an interesting classification of inner product spaces among all normed planes.

Paige Beidelman. Game Chromatic Number of Segmented Caterpillars

Graph theory is the study of sets of points (known as vertices) connected by lines (known as edges). The graph coloring game is a game played on a graph with two players, Alice and Bob, such that they alternate to properly color a graph, meaning no adjacent vertices are the same color. Alice wins if every vertex is properly colored with k colors, otherwise Bob wins when a vertex cannot be colored using k colors. While strategies for winning this game may seem helpful, more interesting is the least number of colors needed for Alice to have a winning strategy, which is called the game chromatic number. We classified a specific tree graph noted as segmented caterpillar graphs that have vertices of degree 2, 3, and 4, for which the game chromatic number have not yet been explored.

Robert Johnson. Quantum mechanics and Schrödinger's equation

In this poster, we will cover the known analytical results for a particular PDE model of the evolution of the wave function of a quantum-mechanical system, also known as the Schrödinger’s equation. Quantum systems describe the physical properties of nature at the scale of atoms, and depend upon probability theory due to the undetermined nature of simultaneously knowing position and momentum of the particle. A methodology for solving the equation is through Sturm-Liouville Theory which proves the existence of solutions. We will also explain extended applications of this equation through quantum computing.

Sophia Natale-Short. The effects of landscape configuration and neighbor effects on coral spatial pattern

Coral reefs are exceptionally diverse and provide important ecosystem services, and the spatial distribution of corals has important consequences for the organisms it hosts. The spatial distribution of coral is partially determined by larval settlement location as coral larvae have some ability to choose a location for permanent settlement. In particular, some larvae might choose locations near adult corals (if the presence of adults indicates suitable habitat), or far from adults (if high density leads to increases in disease risk). An earlier model demonstrated that clusters of corals often form among young corals. Therefore, we sought to understand how much clustering is due to the fact that corals can only settle into open space (and different landscape configurations have different amounts of open space), and how much is due to the effects of neighbors on coral settlement. To do this, we generated landscapes using different spatial distributions (random, clustered, and overdispersed) using Poisson spatial and Matern processes and simulated negative and positive neighbor effects of adult corals on settlement. We then quantified the spatial pattern using Clark Evans R to measure the amount of clustering, dispersion, and randomness found among both new settlers and the new landscape. In our simulations, we found that the neighbor effects did not have a significant influence on the settlement pattern of the new corals. Our current and future explorations will test the relationships between the parameters of our landscape generation functions (e.g. how much clustering or overdispersion exists) and variables such as number of adult corals and the coral larvae density.

Yanping Ma. A linear optimal feedback control for producing 1,3-propanediol via microbial fermentation

We consider a multistage feedback control strategy for the production of 1, 3-propanediol(1, 3-PD) in microbial fermentation. The feedback control strategy is widely used in industry, and to the best of our knowledge, this is the first time it is applied to 1, 3-PD. The feedback control law is assumed to be linear of the concentrations of biomass and glycerol, and the coefficients in the controller are continuous. A multistage feedback control law is obtained by using the control parameterization method on the coefficient functions. Then, the optimal control problem is transformed into an optimal parameter selection problem. The time horizon is partitioned adaptively. The corresponding gradients are derived, and finally, our numerical results indicate that the strategy is flexible and efficient.

Yuki Takahashi. Frobenius templates in certain 2×2 matrix rings

The classical Frobenius problem is to find the largest integer that cannot be represented as a linear combination of a given set of positive, co-prime integers using non-negative integer coefficients. Prior research has been done to generalize this classical Frobenius problem from a topic in number theory to a topic in ring theory; the Frobenius problem has been generalized from the ring of integers to the ring of Gaussian integers as well as to the rings Z[√m], where m is a square-free positive integer. 

In this poster, we will introduce a new generalization of the classical Frobenius problem to the commutative ring of 2 × 2 upper triangular matrices with constant diagonal. We will present answers to two research questions: for which lists of matrices is the Frobenius set non-empty, and for each list such that the Frobenius set is non-empty, what are the matrices in the Frobenius set? For the lists of two matrices, we will show a formula to determine all matrices in the Frobenius set.

Hunter Shepard & Phyllis Okwan. Characteristics of Orthogonal Projection to Global Positioning System (GPS)

The purpose of this research was to find the orthogonal projection in relation to Global Positioning System (GPS). An orthographic projection is a process of prediction in which an object is characterized or surface mapped using parallel lines to project its shape onto a plane. In this research an orthographic projection method was used to project three-dimensional plane in two measurements. Orthographic projection is a type of equal projection, where all the projection lines are symmetrical to the projection plane, bringing about each plane of the scene showing up in relative change on the review surface. The orthographic projection method was used to arrange the framework and the information on a level surface. Numerical figures were utilized to change over to arrange the framework to utilize the GPS. These projections help to navigate the coordinates of the GPS.

Maxwell Jackson & Phyllis Okwan. Using Markov Chain to predict car buying choices

Markov Chain is named after Russian mathematician Andrey Markov. A Markov Chain is a model describing a sequence of possible events in which the probability of each event depends only on the previous event. The purpose of this research was to use Markov Chain to predict car sales. This means using car sales figures from different car companies to predict what car will most likely be bought next. A 100,000 cars were selected from 4 different car companies. The research focused on the sales of Honda Accord, Toyota Corolla, Nissan Altima, and Ford Fusion. The selected cars are all sedan and 2019 models with similar features. The car sales from a 4-month period (May, June, July, and August) were used. Markov Chain model was used to (1) create a prediction about the next car to be most likely be purchased, (2) calculate how often each car is purchased, (3) carry out forecast to predict the next car to be purchased. To predict the probability of buying a car, each number of cars sold was divided by 100,000 to give a percent. The symbol p_ij was used to determine the probability of transition from one state to the next in one generation. The analysis of the calculations using transition matrix resulted in the probability of purchasing Toyota Corolla is 22.9 percent, Honda Accord is about 21.8 percent, Nissan Altima 16.5 percent, and Ford Fusion 16.4 percent in the month of June. The purchasing results in the month of August were Toyota Corolla 21.5 percent, Honda Accord 30.5 percent, Nissan Altima 22.4 percent, and Ford Fusion 13.8 percent. It was concluded that the probability of purchasing a Honda Accord in the month of August was higher than the other cars selected for this research.