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## Mathematics (General)

THU-800 THE BANQUET SEATING PROBLEM

- Alexis Jane Torre ;
- Ashley Scruse ;
- Michelle Rosado ;
- Francis Su ;

THU-800

THE BANQUET SEATING PROBLEM

__Alexis Jane Torre__^{1}, Ashley Scruse^{2}, Michelle Rosado^{3}, Francis Su^{4}.

^{1}*TheUniversity of Arizona, Tucson, AZ, ^{2}Clark Atlanta University, Atlanta, GA, ^{3}University of Puerto Rico Mayaguez Campus, Mayaguez, PR, ^{4}Harvey Mudd College, Claremont, CA.*

Suppose you are planning a banquet and want to seat n = mk people around k tables with m people at each table. Each person gives you a list of *j* people next to whom they would enjoy sitting. What is the smallest j for which you can always make a seating arrangement that would seat each person next to one of the people on their list? We show that *j *must be strictly more than half of n, the total number of people. Our key tool is a particular blue-green-red lemma that helps us construct worst-case scenario seating arrangements. We consider cases with 2 tables and more than 2 tables.

THU-809 SUBSTITUTIONS AND SIMILAR RAUZY FRACTALS

- Austin Marstaller ;
- Meghan Malachi ;

THU-809

SUBSTITUTIONS AND SIMILAR RAUZY FRACTALS

__Austin Marstaller__^{1}, Meghan Malachi^{2}.

^{1}*The University of Texas at Dallas, Richardson, TX, ^{2}Providence College, Providence, RI.*

Let *A* be an alphabet and let *S* and *T* be substitutions under *A*. There exist certain classes of substitutions that can be delineated by a fractal, specifically a Rauzy fractal. We attempted to characterize substitutions over a 3-letter alphabet that produce Rauzy fractals that are translations of one another. The methods we employed were analyzing the "staircase" constructed from the sequences created by the substitutions as well as thoroughly examining the structure of those sequences. Through these methods, we have found that should a substitution be right-conjugate to another, they will generate Rauzy fractals which are translations of one another. In future research, we wish to find more interesting characteristics of substitutions which generate similar Rauzy fractals.

THU-811 MATROID GENERALIZATION OF SPERNER'S LEMMA

- Alberto Ruiz Sandoval ;
- Gabriel Andrade ;
- Andres Rodriguez ;
- Francis Su ;

THU-811

MATROID GENERALIZATION OF SPERNER'S LEMMA

__Alberto Ruiz Sandoval__^{1}, Gabriel Andrade^{2}, Andres Rodriguez^{3}, Francis Su^{4}.

^{1}*University of Puerto Rico-Rio Piedras, San Juan, PR, ^{2}University of Massachusetts, Amherst, Amherst, MA, ^{3}Universidad de los Andes, Bogota, CO, ^{4}Harvey Mudd College, Claremont, CA.*

In a 1980 paper, Lovász generalized Sperner's lemma for matroids. He claimed that a triangulation of a d-simplex labeled with elements of a matroid M must contain at least one basis simplex. We present a counterexample to Lovász's claim when the matroid contains loops and provide a necessary condition such that Lovász's generalization holds. Furthermore, we show that under some conditions on the matroids, there is an improved lower bound on the number of basis simplices. We present further work to sharpen this lower bound by looking at M's lattice of flats and by proving that there exists a group action on the simplex labeled by M with Sn.

THU-810 USING A VOLUME ARGUMENT TO PROVE TUCKER'S LEMMA IN 2-DIMENSIONS

- Christopher Loa ;
- Beauttie Kuture ;
- Oscar Leong ;
- Mutiara Sondjaja ;
- Francis Su ;

THU-810

USING A VOLUME ARGUMENT TO PROVE TUCKER'S LEMMA IN 2-DIMENSIONS

__Christopher Loa ^{1}__, Beauttie Kuture

^{2}, Oscar Leong

^{3}, Mutiara Sondjaja

^{4}, Francis Su

^{5}.

^{1}*The University of Tennessee, Knoxville, Knoxville, TN, ^{2}Pomona College, Claremont, CA, ^{3}Swarthmore College, Swarthmore, PA, ^{4}New York University, New York, NY, ^{5}Harvey Mudd College, Claremont, CA.*

Tucker’s lemma is a statement about labeled triangulations of spheres. It is the combinatorial analog of the Borsuk–Ulam theorem. There exist many approaches to proving Tucker’s Lemma. Sperner’s lemma, the combinatorial analog of Brouwer's fixed-point theorem, is a statement about labeled triangulations of simplices. McLennan and Tourky provided a novel proof of Sperner’s lemma using a volume argument and a piecewise linear deformation of a triangulation. Using their argument, simplices maintain constant volume through the deformation. We adapted a similar volume argument to provide a new proof of Tucker’s lemma in 2-dimensions on a triangulated 2-cross-polytope P where the vertices of P have unique labels. P represents a hemisphere of a 2-sphere. The McLennan-Tourky technique does not directly apply because the natural deformation does not maintain a constant volume of P; we remedied this by inscribing P in its dual polytope D and triangulating it. The volume of D remains constant under the deformation, and we were able to prove the lemma using a volume argument with the aid of winding numbers. We then generalized the argument to triangulated 2-cross-polytopes whose vertices do not have unique labels.

FRI-815 TRIANGULATING ALMOST-COMPLETE GRAPHS

- Kim Pham ;
- Kayla Wright ;
- Landon Settle ;
- Padraic Bartlett ;

FRI-815

TRIANGULATING ALMOST-COMPLETE GRAPHS

__Kim Pham__^{1}, Kayla Wright^{2}, Landon Settle^{2}, Padraic Bartlett^{2}.

^{1}*University of California, Irvine, Irvine, CA, ^{2}University of California, Santa Barbara, Santa Barbara, CA.*

Decomposing graphs into edge-disjoint triangles is a common problem in graph theory and can be applicable to other areas such as computer graphics. It is known that a complete graph Kn can be written as a union of edge-disjoint triangles, provided a few trivially necessary conditions; namely, the number of edges is a multiple of 3, and the degree of each vertex is even. Therefore, it is natural to wonder if a similar result would hold for almost-complete graphs; that is, graphs on n vertices such that every vertex has degree of at least (1 - ɛ)n. Nash-Williams conjectured that there exists some ɛ > 0 such that these graphs admit an edge-disjoint triangle decomposition. In this presentation, we will discuss our proof of this result. Our proof uses techniques from Latin squares, design theory, and ''trades'' on graphs. No prior experience with these concepts will be necessary to follow this talk.

FRI-811 SURVIVAL ANALYSIS DIMENSION REDUCTION TECHNIQUES: A COMPARISON OF SELECT METHODS

- Ivan Rodriguez ;
- Claressa Ullmayer ;
- Javier Rojo ;

FRI-811

SURVIVAL ANALYSIS DIMENSION REDUCTION TECHNIQUES: A COMPARISON OF SELECT METHODS

__Ivan Rodriguez__^{1}, Claressa Ullmayer^{2}, Javier Rojo^{3}.

^{1}*The University of Arizona, Tucson, AZ, ^{2}University of Alaska Fairbanks, Fairbanks, AK, ^{3}University of Nevada, Reno, Reno, NV.*

Although formal studies across many fields may obtain copious amounts of data, most of it can be collinear or redundant in terms of explaining particular and pertinent outcomes. Thus, dataset dimensionality reduction becomes imperative for facilitating the explanation of phenomena given abundant covariates. Principal component analysis (PCA) and partial least squares (PLS) are established methods used to obtain components, eigenvalues of the given data's variance-covariance matrix, such that the covariance and correlation is maximized between linear combinations of predictor and response variables. PCA employs orthogonal transformations on covariates to reduce dataset dimensionality by producing new uncorrelated variables. PLS, rather, projects both predictor and response variables into a new space to model their covariance structure. Aside from these standard procedures, 3 Johnson-Lindenstrauss-inspired random matrices were also investigated. The performance of these techniques was explored by simulating 5,000 datasets using R statistical software. The semi-parametric accelerated failure time (AFT) model was utilized to obtain predicted survivor curves. Then, bias and mean-squared error (MSE) between true and estimated survivor curves was ascertained to find the error distributions of all methods. The results herein indicate that PCA outperforms PLS, the 3 random matrices are comparable, and the 3 random matrices outdo both PCA and PLS.

THU-814 DOUBLE INTERVAL CIRCULAR SOCIETIES

- Sarah Yoseph ;
- Edwin Baeza ;
- Nikaya Smith ;
- Daniel Eckhardt ;

THU-814

DOUBLE INTERVAL CIRCULAR SOCIETIES

__Sarah Yoseph__^{1}, Edwin Baeza^{2}, Nikaya Smith^{3}, Daniel Eckhardt^{4}.

^{1}*Loyola Marymount University, Los Angeles, CA, ^{2}Purdue University, West Lafayette, IN, ^{3}University of North Carolina at Chapel Hill, Chapel Hill, NC, ^{4}Rensselaer Polytechnic Institute, Troy, NY.*

How agreeable is a society of *n* voters as *n* gets larger if their preference sets are measured on a circular spectrum? This is an extension of the work of Klawe et al., who calculated the agreement proportions of such societies on a linear spectrum, providing bounds on the fraction of voters who will agree with each other. We examine D arc-shaped double intervals on circular societies that are pairwise-intersecting and of equal length. We call such societies double-*n* circular societies and convert these circular models into double-*n* strings which helps us more easily find the maximum number of intersections of the voters' approval sets. Our question: what is the minimal agreement proportion for double-*n* circular societies? We found that the asymptotic agreement proportion is bounded between 1/4 and 6/17 and conjecture that the proportion approaches 1/3.

FRI-809 DOUBLE-N CIRCULAR SOCIETIES

- Edwin Baeza ;
- Nikaya Smith ;
- Sarah Yoseph ;
- Francis Su ;

FRI-809

DOUBLE-N CIRCULAR SOCIETIES

__Edwin Baeza__^{1}, Nikaya Smith^{2}, Sarah Yoseph^{3}, Francis Su^{4}.

^{1}*Purdue University, West Lafayette, IN, ^{2}University of North Carolina at Chapel Hill, Chapel Hill, NC, ^{3}Loyola Marymount University, Los Angeles, CA, ^{4}Harvey Mudd College, Claremont, CA.*

A society is a geometric space with a collection of subsets that represents voter preferences. We call this space the spectrum and these preference sets approval sets. The agreement proportion is the largest fraction of approval sets that intersect in a common point. Klawe et al., considered linear societies where approval sets are the disjoint union of 2 intervals, or double intervals. We examined arc-shaped double intervals on circular societies. We considered the case of pairwise-intersecting intervals of equal length and called these double-n circular societies. What is the minimal agreement proportion for double-n societies? We will show that the asymptotic agreement proportion is bounded between 0.2500 and 0.3529, and conjecture that the proportion approaches 1/3.

FRI-820 GEOMETRIC REALIZATION OF SPARSE NEURAL CODES

- Aleina Wachtel ;
- Mohamed Omar ;
- Nora Youngs ;
- R. Amzi Jeffs ;
- Natchanon Suaysom ;

FRI-820

GEOMETRIC REALIZATION OF SPARSE NEURAL CODES

__Aleina Wachtel__, Mohamed Omar, Nora Youngs, R. Amzi Jeffs, Natchanon Suaysom.

*Harvey Mudd College, Claremont, CA.*

Understanding how a neural code stores information is one of the most pressing problems in neuroscience. It is vital to study these neural codes from a mathematical perspective to gain a better understanding of the brain. The combinatorial information in a neural code can allow us to determine if the code reflects the firing behavior of neurons with convex receptive fields. Thus, we seek to determine which neural codes can be realized as convex open sets in R^{2}. We restrict to codes showing sparse behavior, specifically codes in which no more than 2 neurons fire simultaneously, to reduce the intractability of determining which neural codes can be so realized. Using geometric properties of neural realizations, we find necessary and sufficient conditions for realizability of 2-sparse codes based on the associated co-firing graph. This yields a complete characterization of which 2-sparse codes can be realized in R^{3}. We also exhibit several classes of codes that are always realizable in R^{2}, and give a class of graphs never realizable in R^{2}. With a characterization of 2-sparse neural codes, we can better comprehend the manner in which biologically viable codes are spatially represented in the brain.

THU-801 A MATROID GENERALIZATION OF SPERNER'S LEMMA

- Gabriel Andrade ;
- Andres Rodriguez ;
- Alberto Ruiz Sandoval ;

THU-801

A MATROID GENERALIZATION OF SPERNER'S LEMMA

__Gabriel Andrade__^{1}, Andres Rodriguez^{2}, Alberto Ruiz Sandoval^{3}, Francis Su^{4}.

^{1}*University of Massachusetts, Amherst, Amherst, MA, ^{2}Universidad de los Andes, Bogotá·, CO, ^{3}University of Puerto Rico, Rio Piedras Campus, San Juan, PR, ^{4}Harvey Mudd College, Claremont, CA.*

In a 1980 paper, Lovász generalized Sperner's lemma for matroids. He claimed that a triangulation of a d-simplex labeled with elements of a matroid M must contain at least one "basis simplex". We present a counterexample to Lovász's claim when the matroid contains singleton dependent sets and provide an additional sufficient condition that corrects Lovász's result. Furthermore, we show that under some conditions on the matroids, there is an improved lower bound on the number of basis simplices. We present further work to sharpen this lower bound by looking at M's lattice of flats and by proving that there exists a group action on the simplex labeled by M with S_{n}.

THU-820 COMMITTEE SELECTION WITH APPROVAL VOTING IN A HYPERCUBE

- Caleb Bugg ;
- Gabriel Elvin ;
- Francis Su ;

THU-820

COMMITTEE SELECTION WITH APPROVAL VOTING IN A HYPERCUBE

__Caleb Bugg__^{1}, Gabriel Elvin^{2}, Francis Su^{3}.

^{1}*Morehouse College, Atlanta, GA, ^{2}University of California, Los Angeles, Los Angeles, CA, ^{3}Harvey Mudd College, Claremont, CA.*

In this paper we examine elections in which a committee of size *k* is to be elected, with 2 candidates running for each position. Each voter submits a ballot with his or her ideal preference for a committee, which generates their *approval set*. The approval sets of voters consist of committees that are close to their ideal preference. We define this notion of closeness with Hamming distance in a hypercube: the number of candidates by which a particular committee differs from a voter's ideal preference. We establish a tight lower bound for the popularity of the most approved committee, and consider restrictions on voter preferences that may increase that popularity. Our approach considers both the combinatorial and geometric aspects of these elections.

FRI-822 ADVANCING LASER INDUCED BREAKDOWN SPECTROSCOPY FOR ELEMENTAL ANALYSIS OF MARINE SEDIMENT SAMPLES

- Berlinda Batista ;
- Anna Michel ;

FRI-822

ADVANCING LASER INDUCED BREAKDOWN SPECTROSCOPY FOR ELEMENTAL ANALYSIS OF MARINE SEDIMENT SAMPLES

__Berlinda Batista__^{1}, Anna Michel^{2}.

^{1}*Bridgewater State University, Bridgewater, MA, ^{2}Woods Hole Oceanographic Institution, Woods Hole, MA.*

Decades ago, factories and textile mills surrounding New Bedford Harbor, Massachusetts dumped industrial waste into the water which, as of today, has resulted in sediments contaminated with polychlorinated biphenyls and heavy metals. Over the years, many projects have been introduced in the hopes of cleaning the harbor. With this research, we focus on using laser-induced breakdown spectroscopy (LIBS) as a technique for detecting heavy metals found in sediments samples taken from the harbor. This study used MATLAB to design a system to statistically analyze the elemental composition of the samples. With LIBS, we used a 30 mJ pulsed Nd:YAG laser to generated plasmas on the wet, dry, and slurry samples. Custom computer code was written to smooth the data in an attempt to display a clear pattern of peaks. The output is a plotted spectrum and a table of the elements found at specific wavelengths. The code focuses on distinguishing the highest peaks of the graphs which identifies elements found in the samples. With the modeled elements, we were able to quantify trends found in specific locations. Moreover, wet, dry, and slurry data are compared. The overall objective of our study is to advance the real time measurements of sediment samples in the field.

THU-802 FINDING UNIQUE HAMILTONICITY USING GRŐBNER BASES

- Vanessa Aguirre ;
- Monica Busser ;
- Kaitlyn Phillipson ;
- Aaron Wagner ;
- Kainalu Barino ;

THU-802

FINDING UNIQUE HAMILTONICITY USING GRŐBNER BASES

__Vanessa Aguirre__^{1}, Monica Busser^{2}, Kaitlyn Phillipson^{3}, Aaron Wagner^{4}, Kainalu Barino^{5}.

^{1}*University of Hawaii at Hilo, Hilo, HI, ^{2}Youngstown State University, Youngstown, OH, ^{3}Texas A&M University, College Station, TX, ^{4}Viterbo University, La Crosse, WI, ^{5}Brigham Young University-Hawaii, Laie, HI.*

Recent advances in computational algebraic geometry allow us to recognize certain graph theoretic properties, such as graph colorability and Hamiltonicity with polynomial ideals. These results allow us to test for such properties in an automated way using computers. We employ these results to develop and implement algorithms written in the mathematical software system Sage to systematically test various combinatorial conjectures in graph theory. In particular, we have used our methods to test Sheehan's conjecture which states that every Hamiltonian 4-regular graph has at least 2 distinct Hamiltonian cycles. Our results verify Sheehan's conjecture for 4-regular graphs up to 10 vertices and give support for other combinatorial conjectures. Our software can be easily modified to test other combinatorial conjectures for graphs.

FRI-810 SUBSTITUTIONS AND RAUZY FRACTALS

- Meghan Malachi ;
- Austin Marstaller ;
- Jason Saied ;
- Sara Stover ;
- Benjamin Itza-Ortiz ;

FRI-810

SUBSTITUTIONS AND RAUZY FRACTALS

__Meghan Malachi__^{1}, Austin Marstaller^{2}, Jason Saied^{3}, Sara Stover^{4}, Benjamin Itza-Ortiz^{5}.

^{1}*Providence College, Providence, RI, ^{2}The University of Texas at Dallas, Richardson, TX, ^{3}Lafayette College, Easton, PA, ^{4}Mercer University, Martinez, GA, ^{5}Universidad Autónoma del Estado de Hidalgo, Pachuca, Hidalgo, MX.*

There exist certain classes of substitutions that can be associated with a geometric representation known as the central tile or the Rauzy fractal. We define 2 distinct, primitive substitutions under a 3-letter alphabet, σ and τ, as having similar Rauzy fractals if their corresponding Rauzy fractals, T_{σ} and T_{τ}, respectively, differ by only finitely many points; that is, T_{σ} is the image of T_{τ }under a translation. We study specific cases of substitutions under a 3-letter alphabet, *A*= {a,b,c}, and conjecture that if σ^{n}(a) can be described as a permutation of τ^{n}(a) for all values of n in the set of natural numbers, then σ and τ have similar Rauzy fractals. We ultimately conjecture that if σ is right conjugate to τ, then σ and τ have similar Rauzy fractals.

THU-813 PROVING TUCKER'S LEMMA USING A VOLUME ARGUMENT

- Oscar Leong ;
- Beauttie Kuture ;
- Christopher Loa ;

THU-813

PROVING TUCKER'S LEMMA USING A VOLUME ARGUMENT

__Oscar Leong__^{1}, Beauttie Kuture^{2}, Christopher Loa^{3}.

^{1}*Swarthmore College, Swarthmore, PA, ^{2}Pomona College, Claremont, CA, ^{3}The University of Tennessee, Knoxville, Knoxville, TN.*

Sperner's lemma is a statement about labeled triangulations of a simplex. Previous research provided a novel proof of Sperner's lemma using a volume argument and a piecewise linear deformation of a triangulation. We adapt a similar argument to prove Tucker's lemma on a triangulated cross-polytope P in the 2-dimensional case where vertices of P have different labels. The McLennan-Tourky technique would not directly apply because the natural deformation distorts the volume of P; we remedy this by inscribing P in its dual polytope, triangulating it, and considering how the volumes of deformed simplices behave. We then generalize the argument to apply to triangulated cross-polytopes whose vertices do not have different labels.

THU-822 AN ANALOG TO THE MEDIAN VOTER THEOREM FOR APPROVAL VOTING

- Miles Stevens ;
- Francis Su ;

THU-822

AN ANALOG TO THE MEDIAN VOTER THEOREM FOR APPROVAL VOTING

__Miles Stevens__^{1}, Francis Su.

^{1}*Morehouse College, Atlanta, GA, ^{2}Harvey Mudd College, Claremont, CA.*

The median voter theorem is a well-known result in social choice theory for majority-rule elections. We develop an analog in the context of approval voting. We consider voters to have preference sets that are intervals on a line, called approval sets, and the approval winner is a point on the line that is contained in the most approval sets. We define median voter by considering the left and right end points of each voter’s approval sets. We consider the case where approval sets are equal length. We show that if the pairwise agreement proportion is at least 3/4, then the median voter interval will contain the approval winner. We also prove that under an alternate geometric condition, the median voter interval will contain the approval winner, and we investigate variants of this result.

FRI-814 MAXIMUM POSITIVE SEMIDEFINITE NULLITY AND THE TREE COVER NUMBER

- Oscar Gonzalez ;
- Brendan Cook ;
- Carolyn Reinhart ;
- Chassidy Bozeman ;
- Minerva Catral ;
- Leslie Hogben ;

FRI-814

MAXIMUM POSITIVE SEMIDEFINITE NULLITY AND THE TREE COVER NUMBER

__Oscar Gonzalez__^{1}, Brendan Cook^{2}, Carolyn Reinhart^{3}, Chassidy Bozeman^{4}, Minerva Catral^{5}, Leslie Hogben^{4}.

^{1}*University of Puerto Rico, Rio Piedras Campus, San Juan, PR, ^{2}Carleton College, Northfield, MN, ^{3}University of Minnesota, Minneapolis, MN, ^{4}Iowa State University, Ames, IA, ^{5}Xavier University, Cincinnati, OH.*

The maximum positive semidefinite nullity of a graph G is defined to be the maximum nullity over all real positive semidefinite matrices whose graph is described by G. A simple graph consists of vertices and edges, where the set of edges are 2-element subsets of the set of vertices; no multiple edges and no loops are allowed. A tree is a connected graph with no cycles. The tree cover number of a graph is the minimum number of vertex disjoint trees occurring as induced subgraphs that cover all the vertices of the graph. This graph parameter was introduced in 2011 as a tool for the study of maximum positive semidefinite nullity, and not much is known about this parameter. It is conjectured that the tree cover number of a graph is at most the maximum positive semidefinite nullity of the graph. While studying this conjecture, we obtained several results about the tree cover number. In particular, we were able to show that, for any integer d ≥ 2, the d-dimensional hypercube has tree cover number 2. We also present results on the tree cover number of subdivided graphs, graphs resulting from edge removal, and some other families of graphs. Finally, we characterize when an edge is required in every minimum tree cover of a graph.

FRI-800 AN ANALOG OF THE MEDIAN VOTER THEOREM FOR APPROVAL VOTING

- Kyle Duke ;
- Francis Su ;

FRI-800

AN ANALOG OF THE MEDIAN VOTER THEOREM FOR APPROVAL VOTING

__Kyle Duke__^{1}, Francis Su^{2}.

^{1}*James Madison University, Virginia Beach, VA, ^{2}Harvey Mudd College, Claremont, CA.*

The median voter theorem is a well-known result in social choice theory for majority-rule elections. We develop an analog in the context of approval voting. We consider voters to have preference sets that are intervals on a line, called approval sets, and the approval winner is a point on the line that is contained in the most approval sets. We defined median voter by considering the left and right end points of each voter's approval set. We consider the case where approval sets are equal length. We show that if the pairwise agreement proportion is at least 3/4 , then the median voter interval will contain the approval winner. We also prove that under an alternate geometric condition, the median voter interval will contain the approval winner, and investigate variants of this result.

FRI-801 A MATROID GENERALIZATION FOR SPERNER'S LEMMA

- Andres Rodriguez ;
- Gabriel Andrade ;
- Alberto Ruiz Sandoval ;
- Francis Su ;

FRI-801

A MATROID GENERALIZATION FOR SPERNER'S LEMMA

__Andres Rodriguez__^{1}, Gabriel Andrade^{2}, Alberto Ruiz Sandoval^{3}, Francis Su^{4}.

^{1}*Universidad de los Andes, Bogotá, CO, ^{2}University of Massachusetts, Amherst, Amherst, MA, ^{3}University of Puerto Rico, Rio Piedras Campus, San Juan, PR, ^{4}Harvey Mudd College, Claremont, CA.*

Previously, Lovász generalized Sperner's lemma for matroids. He claimed that a triangulation of a *d*-simplex labeled with the elements of a matroid *M, *while labeling the main vertices with a basis *B*, must contain at least 1 basis simplex on *T,* a simplex* *whose labels form a basis of the matroid. We present a counterexample to Lovász's claim when the matroid contains loops and provide a necessary condition such that Lovász's generalization holds. Under Lovász's construction one can induce a Sperner labeling by fixing the order of the basis *B.* We prove that any fully labeled simplex on the induced Sperner labeling has to be a basis simplex, giving a new proof of Lovász's claim. Furthermore, we show that for some matroids there is an improved lower bound on the number of basis simplices. Finally, we present further work to sharpen this lower bound by looking at *M'*s lattice of flats and by proving that there exists a group action on the induced Sperner's labelings by **S _{n}**.

THU-815 MATRIX COMPLETIONS FOR SYLVESTER EQUATION

- Rosa Moreno ;
- Dianne Pedroza ;
- Kirste Morris ;
- Elijah Cronk ;
- Jack Ryan ;
- Geoffrey Buhl ;

THU-815

MATRIX COMPLETIONS FOR SYLVESTER EQUATION

__Rosa Moreno__^{1}, Dianne Pedroza^{2}, Kirste Morris^{3}, Elijah Cronk^{4}, Jack Ryan^{5}, Geoffrey Buhl^{1}.

^{1}*California State University of Channel Islands, Camarillo, CA, ^{2}Ripon College, Ripon, WI, ^{3}Georgia College and State University, Milledgeville, GA, ^{4}Ithaca College, Ithaca, NY, ^{5}North Central College, Naperville, IL.*

A matrix completion problem attempts to determine if a partial matrix, one with some entries given and others freely chosen, can be completed to satisfy some property. This project focuses on determining which patterns of specified and unspecified entries for a partial matrix can be completed to solve specific Sylvester-type matrix equations. By looking at the nullspace of 2 Sylvester equations, patterns are classified as admissible or inadmissible based on their ability or inability to be completed for the given matrix equation for almost any matrix A. We completely characterize admissible and inadmissible patterns for one equation and have a partial characterization for the other equation.

FRI-802 MODELING THE INTERACTION DYNAMICS BETWEEN HONEYBEES AND FOOD AVAILABILITY

- Carlos Cruz ;
- Matthew Baca ;
- Armando Salinas ;
- Carlos Agrinsoni-Santiago ;
- Baojun Song ;

FRI-802

MODELING THE INTERACTION DYNAMICS BETWEEN HONEYBEES AND FOOD AVAILABILITY

__Carlos Cruz__^{1}, Matthew Baca^{2}, Armando Salinas^{3}, Carlos Agrinsoni-Santiago^{4}, Baojun Song^{5}.

^{1}*Loyola Marymount University, Los Angeles, CA, ^{2}New Mexico Institute of Mining and Technology, Socorro, NM, ^{3}School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ, ^{4}University of Puerto Rico in Cayey, Cayey, PR, ^{5}Montclair State University, Montclair, NJ.*

The success of honeybee (*Apis mellifera*) colonies is critical to U. S. agriculture with 35% of American diets dependent on honeybee pollination. There are various complex factors that can contribute to a colony's failure such as nutritional stress. Nutritional stressors primarily pertain to food scarcity, lack in diversity of food, and the availability of food with low nutritional value. Previous mathematical models have examined the impact of nutrition and the early recruitment on honeybee population dynamics. These models do not include the impact of a food supply with a limited storage space within a single hive. In this work, we use a mathematical model to investigate the impact of food scarcity and limited storage space on honeybee viability, early recruitment rates of workers into foragers, and the influence of these rates on the growth of a colony. A threshold, R_{d}, was found for conditions when a colony will persist or collapse. We found conditions for the stable coexistence of a honeybee population and food supply as well as conditions for periodic behavior. Through sensitivity analysis we find that a honeybee colony is most sensitive to changes in the rate at which a worker bee encounters food and the rate food is entering the food supply. There are no qualitative differences between using a Holling type I or Holling type II functional response in honeybee population persistence when modeling the interaction between a honeybee colony and the availability of food.

FRI-813 AN ANALOG OF THE MEDIAN VOTER THEOREM IN APPROVAL VOTING

- Ethan Bush ;
- Kyle Duke ;
- Miles Stevens ;
- Duane Cooper ;

FRI-813

AN ANALOG OF THE MEDIAN VOTER THEOREM IN APPROVAL VOTING

__Ethan Bush__^{1}, Kyle Duke^{2}, Miles Stevens^{3}, Duane Cooper^{3}.

^{1}*University of Michigan-Flint, Flint, MI, ^{2}James Madison University, Harrisonburg, VA, ^{3}Morehouse College, Atlanta, GA.*

The median voter theorem is a well-known result in social choice theory for majority-rule elections. We developed an analogue in the context of approval voting. We considered voters to have preference sets that are intervals on a line. These intervals are called approval sets and the approval winner is a point on the line that is contained in the most approval sets. We defined the median voter by considering the left or right endpoints of each voter’s approval sets. First, we considered the case where approval sets are equal length. We show that, if the pairwise agreement proportion is at least 3/4, then the median voter interval will contain the approval winner. Then we considered the case where approval sets vary in length. This case prompted us to use alternate geometric conditions where the median voter interval will contain the approval winner. Our results show we are able to define conditions where the median voter interval will contain the approval winner; thus, there exists an analog of the median voter theorem in approval voting.

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- Genetics Society of America
- Maryland Sea Grant
- Society for Industrial and Applied Mathematics
- St. Jude Children’s Research Hospital
- The University of Kansas-Department of Geology