Workshops

This conference is funded by a grant through the Institute of Advanced Study, School of Mathematics through the Women and Mathematics program and Lisa Simonyi, by the Tulane University Math Department and by the National Science Foundation DMS-2113829

Pamela E. Harris : Partitions and Juggling: A Story of Unifying Parts and Seeking Balance in Mathematics

Abstract: Kostant’s partition function is a vector partition function that counts the number of ways one can express a vector as a nonnegative integer linear combination of a fixed set of vectors. Multiplex juggling sequences are generalizations of juggling sequences (describing throws of balls at discrete heights) that specify an initial and terminal configuration of balls and allow for multiple balls at any particular discrete height. In this talk, I will give an introduction to a combinatorial equivalence between Kostant’s partition function and multiplex juggling sequences. Throughout I will talk about how partitioning my authentic self from my mathematical work made it impossible to juggle and find balance in my professional work.

Raegan Higgins: Modeling in and Broadening Mathematics

Abstract: My research focuses on oscillation theory of dynamic equations and mathematical modeling using time scales. As my knowledge of mathematics broadens, so does my view of the mathematics community. My experiences as a woman and as an African American influence how I serve. I am committed to creating and supporting programs and initiatives that increase the visibility and contributions of those underrepresented in mathematics. In this talk, I will describe my career path in mathematics through a survey of my research as well as my involvement in programs that broaden participation in mathematics.

Jacqueline Leonard : Facilitating Computational Participation, Place-Based Education, and Culturally Specific Pedagogy with Indigenous Students

Abstract: STEM research on Indigenous students is often deficit oriented and void of social justice. In this presentation, Leonard highlights the importance of using a decolonized approach, place-based education, and culturally specific pedagogy to broaden STEM participation among Arapaho children and youth on the Wind River Indian Reservation (WRIR) in Wyoming. Four instructors, which included two Native teachers, developed STEM curriculum that incorporated Indigenous ways of knowing. Activities included using drones, flight simulation, and original data as the contexts to learn STEM and engage in computational participation. The results of the study revealed students demonstrated an understanding of Bernoulli’s law, the principles of flight, and the importance of knowing about weather, cardinal directions, wind, and speed to fly drones. Qualitative data collected from students’ journals revealed they engaged in survivance and presencing as they used drones to tell unique stories about historical and contemporary culture on the WRIR.

Swapna Mukhopadhyay: People’s Mathematics

Abstract: When people use the word “mathematics”, typically they have in mind a body of knowledge, conceived as existing timelessly and independently of humans. As such, mathematics in general, and school mathematics in particular, fails to relate to the lives of “ordinary people”. Despite the consequent alienation and intimidation, “mathematics” is generally regarded with awe, and afforded elitist status as the highest form of intellectual activity.
By contrast, I propose a characterization of mathematics in terms of activities carried out by human beings. Moreover, these activities are not limited to the academic mathematics carried out in universities and schools. They include the mathematics of carpenters measuring, weavers designing, shopkeepers calculating, and all kinds of “just plain folk” and these activities are embedded in historical, cultural, social, and political – in short, human – contexts.
From this standpoint, I will share a conception of mathematics education that bridges the gap between mathematics classrooms and the lives of the students and their communities and that valorizes the mathematical knowledge and expertise of all.

Emma Lien: Higher Reciprocity Laws, Modular Forms of Weight One and their Galois Representations

Addie Duncan: Primitive prime divisors of the Catalan sequence

Eva Goedhart: Fun with Factoradic Happy Numbers

Diego Villamizar: On Central Factorial Numbers

Irfan Durmić: Parallel diffusion on complete graphs and a connection to polyominoes

Sarah Pritchard & Madison Ford: Exploring the Solvability of the String Link Concordance Group

Robyn Brooks: Formulas for Computing Finite Type Knot Invariants

Liza Jacoby: The Rest of the Tilings of the Sphere by Regular Polygons

Joseph Skelton: Symbolic Powers of Cover Ideals and Koszul Property

Riley Juenemann: Statistical Dashboard for Categorizing Particle Movement in Cells

Kelemua Tesfaye: Knotris Gameplay and Probability

Andrew Tawfeek: On discrete gradient vector fields and Laplacians of simplicial complexes

Mehsin Jabel Atteya: Skew-Homogeneralized Derivations of Rings

Kimberly Hadaway: An Introduction to Parking Functions

Surya Kotapati & Zia Saylor: The Connection Between Parking Functions and Reve’s Puzzle: Unpacking their Sequences

Alvaro Carbonero: Decoding Convergence: Tracing the sequences of H-Line Graphs

Josephine Reynes: The Determinant of (+1,-1)-Matrices and Oriented Hypergraphs

Isabelle Haines: Math is Beautiful: The Aesthetics of Data

Mehsin Jabel Atteya.

Skew-Homogeneralized Derivations of Rings

The study of derivation was initiated during the 1950s and 1960s. Derivations of rings got a tremendous development in 1957 when Posner established two very striking results in the case of prime rings. In 2000, El Sofy defined a homoderivation on R as an additive mapping h: R ⟶ R satisfying h(xy)=h(x)h(y)+h(x)y+xh(y) for all x, y ∈ R while in 2020, Mehsin Jabel Atteya introduced the definition of (σ, τ)-Homgeneralized derivations of semiprime rings with some results. The main purpose of this paper is to introduce the definitions of a skew-Homoderivation (resp. a skew-AntiHomoderivation) and skew-Homogeneralized derivations (resp. a skew-AntiHomogeneralized derivation) of rings. In fact, this article divided into three sections, in the first section, we emphasize on the definitions of a skew-Homoderivation (resp. a skew-AntiHomoderivation) and skew-Homogeneralized derivations (resp. a skew-AntiHomogeneralized derivation) of rings. In the second section, we study weakly semiprime ideal and a weak zero-divisor for prime rings and semiprime rings while the final section focuses on commutativity of prime and semiprime rings admitting a skew-Homogeneralized derivations satisfy certain identities.

Robyn Brooks.

Formulas for Computing Finite Type Knot Invariants

In knot theory, finite type knot invariants can used to differentiate non-isotopic knots. In this talk, I will explain how one may compute the value of a finite type invariant from a regular knot plane diagram. This is done by using a Gauss diagram formula - a formula which counts certain sub-diagrams of crossings within a plane diagram. I will then talk about my research, which looks to extend these formulae to plane diagrams which are not regular - in particular, to plane diagrams which may have triple or other higher order crossings.

Alvaro Carbonero.

Decoding Convergence: Tracing the sequences of H-Line Graphs

Let $H$ be a connected graph on at least 3 vertices. For any graph $G$, the $H$-line graph of $G$, denoted by $HL(G)$, is that graph whose vertices are the edges of $G$ and where two vertices of $HL(G)$ are adjacent if they are adjacent in $G$ and lie in a common copy of $H$. We can iterate this procedure, and for this we use $HL^k(G)$ to denote the graph that outcomes from calculating the $H$-line graph k-times.

This presentation will focus on convergence. A sequence $\{HL^k(G) \}$ is said to converge if there exists a positive integer $k$ such that $HL^k(G) \cong HL^{k+1}(G)$. The case where $H$ is a path of order $n$ is full of surprising graphs that have a convergent sequence. This talk will start by covering some of the graphs $H$ that have been studied in the past, and it will end by going over a new technique that tries to identify every graph with a convergent sequence when $H$ is a path.

Irfan Durmić.

Parallel diffusion on complete graphs and a connection to polyominoes

In this talk we introduce the concept of parallel diffusion on graphs, which is a special version of the well-known game of chip-firing on graphs. We present a result of Mullen, Nowakowski, and Cox (https://arxiv.org/abs/2010.07745v1) that connects parallel diffusion on the complete graphs to combinatorial objects called polyominoes. We end by providing some potential directions for future study.

Addie Duncan.

Primitive prime divisors of the Catalan sequence

For a term in number sequence, we say that a prime is a primitive prime divisor if it divides that term but does not divide any of the previous terms in the sequence. One interesting question we can ask about a sequence is to determine which terms do not have any primitive prime divisors. Sometimes the set of indices for a sequence which do not have any primitive prime divisors is called the Zsigmondy set of the sequence, named after Karl Zsigmondy. In 1892, Zsigmondy proved a result on the Zsigmondy set of a certain class of divisibility sequences that implied the Zsigmondy set for the Mersenne numbers is finite. This result is often cited as heuristic evidence towards the conjecture that there are infinitely many Mersenne primes. Carmichael, in 1913, and Bilu, Hanrot, and Voutier, in 2001, have expanded on Zsigmondy's result to show that the Zsigmondy set is finite for any Lucas or Lehmer sequence and explicitly determined the elements of these Zsigmondy sets. In this talk, we will explore properties of primitive prime divisors and determine the Zsigmondy set of the Catalan sequence and some related sequences.

Eva Goedhart.

Fun with Factoradic Happy Numbers

After a quick introduction, we will briefly explore factoradic happy numbers. Then I will outline the proof that there exist arbitrarily long sequences of e-power factoradic p-happy numbers for small values of e.

Kimberly Hadaway.

An Introduction to Parking Functions

In 1966, Alan G. Konheim and Benjamin Weiss defined “parking functions” as follows: We have a one-way, one-lane street with n parking spaces, numbered in consecutive order from 1 to n, and we have n cars in line waiting to park. Each driver has a favorite (not necessarily distinct) parking spot, which we call its preference. We order these preferences in a preference vector. As each car parks, it drives to its preferred spot. If that spot is open, the car parks there; if not, it parks in the next available spot. If a preference vector allows all cars to park, we call it a parking function. In 1974, Henry O. Pollak proved the total number of parking functions of length n, meaning there are n parking spots and n cars, to be (n+1)^(n-1). In this presentation, we describe a recursive formula, expound Pollak's succinct six-sentence proof of an explicit formula, and conclude with a discussion of other parking function generalizations.

Isabelle Haines.

Math is Beautiful: The Aesthetics of Data

Through trying to render her own data in a visually appealing way, the presenter became aware of many unique and striking representations of data in history - most notably, W. E. B. DuBois's graphs for the 1900 Paris Exposition. This presentation will explore how data can be visualized creatively, as well as the significance of mathematical beauty.

Liza Jacoby.

The Rest of the Tilings of the Sphere by Regular Polygons

In the Euclidean plane, there exist beautiful infinite families of tilings using exclusively regular polygons. Moving to the surface of the sphere, we still define regular spherical polygons to be ones which are equiangular and equilateral, though we can only use finitely many polygons to tile the surface, contrary to the Euclidean plane. Gluing the tiles edge-to-edge, tilings by regular spherical n-gons for n ≥ 3 include all circumscribable polyhedra projected onto the sphere, those being the prisms and antiprisms, the five Platonic solids, the thirteen Archimedean solids, and twenty-five pf the ninety-two Johnson solids. The classification of tilings of the sphere by regular spherical polygons has since been left incomplete, as non-edge-to-edge tilings –– that is, where the edges of two adjacent polygons do not perfectly align –– were left unconsidered. However, this research completes the classification of tilings of the sphere using regular spherical polygons with three or more edges, thus completing the work of Plato, Archimedes, and Johnson and providing new beautiful families of spherical tilings.

Riley Juenemann.

Statistical Dashboard for Categorizing Particle Movement in Cells

At the intersection of nanoscience and biology lies the question of precisely how particles move within cells. In contrast to in vitro particle tracking experiments, wherein there are great controls on particle and environmental homogeneity, live-cell tracking features tremendous diversity in particle movement. Within this research area, the use of mathematics has allowed for a better description of movement categorizations and quantitative methods to differentiate between them. In this work, we have developed a first-pass statistical dashboard to categorize disparate types of particle trajectories. The tools we developed for the categorization process include the correlation between consecutive increments and effective diffusivity from a maximum likelihood estimation. The standard deviation for the major and minor axis and the creation of a parameterized path to represent a fictional moving anchor employed principal components analysis. This anchor estimation allowed the computation of the average distance the particle deviated from the anchor. Based on these data measures, machine learning was utilized to distinguish between free diffusion, stuck diffusion, directed transport, tracker error, subdiffusion, and skating diffusion. This automated categorization process proved to be successful on data simulated using stochastic differential equations and provided interesting results on the live-cell data.

Surya Kotapati & Zia Saylor.

The Connection Between Parking Functions and Reve’s Puzzle: Unpacking their Sequences

Consider this. There is a one-way street with a dead end and n parking spots, and n cars that each have a preferred parking spot (which is not necessarily distinct) on that street. These cars cannot reverse and can only move forward in search of an empty spot if their preferred space is taken. If all their preferred parking spots are collected into a preference vector, and the preference vector allows all cars to successfully park in a spot, you have a parking function. An important extension of this problem involves counting “bumps,” or the m spaces between a car’s desired parking spot and the spot in which it actually parks. We first used Python code to study the relationship between the size n of parking functions and the total number of bumps m, or the sum of each car’s bumps. We found that the sequence of results of size n at m=1 bumps is analogous to that described by Hinz et al. in Chapter 5 of “The Tower of Hanoi—Myths and Maths.” In this text, the authors explore the puzzling Reve’s Puzzle: a wooden stack of three discs, each balanced on top of another and ordered by size, on the leftmost of four pegs. The goal of this puzzle is to move the whole stack from the leftmost peg to the rightmost peg without violating the size ordering of the original stack. The similar sequence appeared in the deconstruction of the moves required to solve Reve’s Puzzle. Our research attempts to uncover the connection between these two seemingly disparate problems by linking their corresponding objects and placeholders: cars to bumps and disks to pegs.

Emma Lien.

Higher Reciprocity Laws, Modular Forms of Weight One and their Galois Representations

The law of quadratic reciprocity allows us to determine how a quadratic polynomial decomposes into irreducible components in any finite field. While answering this same question for higher degree polynomials is largely unsolved in general, we can examine a few specific cases, specifically polynomials related to weight one eta products. We use a theorem by Serre and Deligne to determine that there is some polynomial whose splitting behavior in finite fields is determined by the Fourier coefficients of the eigenform. In this paper we aim to determine which specific polynomial is related to each eta product.

Sarah Pritchard & Madison Ford.

Exploring the Solvability of the String Link Concordance Group

Suppose you have n knotted-up pieces of string tangled together. Is there a mathematical way to describe just how tangled they are? The Milnor’s invariants of a string link provides us with information about how "linked” components are. In this talk, we’ll discuss two ways to calculate the Milnor’s invariants of a string link: a group-theoretic method and a method that involves generating surfaces bounded the link. Then, we’ll discuss some results concerning the solvability of the string link group; by calculating the Milnor’s invariants of string link commutators, we can learn more about how things commute in the string link group.

Josephine Reynes.

The Determinant of (+1,-1)-Matrices and Oriented Hypergraphs

Hadamard's maximum determinant problem aims to find the maximum determinant of a matrix H of size n with entries +1 and -1. An oriented hypergraphic method is introduced to calculate the determinant of any {±1}-matrix by using the incidence-based notion of cycle-covers of its associated Laplacian. This provides a locally signed-graphic interpretation of the maximum determinant problem, where signed circle conditions for three different fundamental sets of circles are obtained relating orthogonality, entries in the equivalent {0,+1}-matrix, and a characterization of n! different classes of fundamental circles that are equivalent to the maximum determinant problem.

Joseph Skelton.

Symbolic Powers of Cover Ideals and Koszul Property

We show that adding a whisker at the vertices of a cycle cover of a graph results in a new graph with the following property: all symbolic powers of its cover ideal are Koszul or, equivalently, componentwise linear. This extends previous work where the whiskers were added to all the vertices or to the vertices of a vertex cover of the graph. The presented results are from the joint work with Yan Gu, and Huy Tài Hà.

Andrew Tawfeek.

On discrete gradient vector fields and Laplacians of simplicial complexes

Discrete Morse theory, a cell complex-analog to smooth Morse theory, has been developed over the past few decades since its original formulation by Robin Forman. Today, it plays a large role in topological data analysis, particularly persistent homology. We prove that the characteristic polynomial of the Laplacian of a simplicial complex serves as a generating function for the discrete gradient vector fields on the complex. We show this holds for all 1-dimensional complexes, but only for particular classes of simplicial complexes in higher dimensions, through the aid of higher dimensional rooted forests.

Kelemua Tesfaye.

Knotris Gameplay and Probability

Knotris is a new game we have developed based on mosaic knot theory and inspired by Tetris. In this talk, I discuss probabilities related to gameplay and how they give us insight into game strategies. As well as describe how code has indicated which game states are more difficult to play within and informed our game design.

Diego Villamizar.

On Central Factorial Numbers

In this talk, we will make use of set partitions, permutations, and their generating functions to give combinatorial relations for the generalized central factorial numbers. Moreover, we will present relationships between these interpretations with Bernoulli polynomials. This is a joint work with T. Komatsu and J. L. Ramirez.

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.