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# «Strengthening the Nation through Diversity, Innovation & Leadership in STEM San Antonio,Texas · October 3-6, 2013 Get Connected! Connect with the ...»

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1 A graph G = (V,E) is a collection of vertices V and edges E where an edge is an unordered pair of vertices. An orientation G’ of a graph G is the same collection of vertices, but, if {i,j} is an edge in G, we can have directed edge (i,j) or (j,i) but not both, where (i,j) means there is an arc from vertex i to vertex j. Matrix A corresponds to oriented graph G’ if,for i and j distinct, the entry aij is nonzero exactly when (i,j) is a directed arc in G’. Note that the diagonal entries in A are free. The maximum nullity M(G’) is defined to be the largest possible nullity over all real matrices corresponding to G’. For an oriented graph G’, the zero forcing number Z(G’) is the minimum number of blue vertices needed to force all vertices in G’ blue according to the color change rule. The color change rule is, for oriented graph G’ with vertices initially colored blue or white, a blue vertex b forces a white vertex w blue if w is the only white outneighbor of b. We investigate M(G’) and Z(G’) for an oriented graph G’ and its underlying, unoriented graph G (the parameters M(G) and Z(G) have related definitions), including establishing that, for certain families of graphs, it is always possible to find an orientation G’ of G so that Z(G’) = Z(G) and M(G’) = M(G).

FRI-400

## TOWERS OF REPEATED EXPONENTIAL SEQUENCES

Caroline VanBlargan1, Justin Peters2.

St. Mary’s College Maryland, St. Mary’s City, MD, 2Iowa State University, Ames, IA.

1 Euler proved which numbers have convergent, iterated exponential towers. That is, given a real number x, he looked at the convergence of a sequence where the nth term in the sequence is x exponentiated by itself n times.

We have generalized this question from dealing with a constant number x to considering any sequence of positive real numbers. We provide results about regions of convergence, existence of certain types of sequences, and relationships between a sequence and its sequence of exponential towers.

SAT-406

## ON THE SCHUR POSITIVITY OF DIFFERENCES OF PRODUCTS OF SCHUR FUNCTIONS

Nadine Jansen1, Jeremy Meza2, Rosa Orellana3, Jeremiah Emidih4, Samuel Ivy5.

North Carolina Agricultural and Technical State University, Greensboro, NC, 2Carnegie Mellon University, Pittsburgh, 1 PA, 3Dartmouth College, Hanover, NH, 4University of California, Riverside, Riverside, CA, 5North Carolina State University, Raleigh, NC.

The Schur functions are a basis for the ring of symmetric functions indexed by partitions of nonnegative integers. A symmetric functions f is called Schur positive if, when expressed as a linear combination of Schur functions, f=Σcλsλ, each coefficient cλis nonnegative. We wish to investigate expressions of the form sλcsλ - sμcsμ (1) where λ partitions n and μ partitions n-1, and the complements sλc and sμc are taken over a sufficiently large m x m square. We give a necessary condition that if (1) is Schur positive, then μ is contained in λ. Furthermore, we show how conjugating partitions preserves Schur positivity. Lastly, we incorporate the Littlewood-Richardson rule to show that particular classes of λ and μ are never Schur positive.

FRI-396

## SPECTRAL PROPERTIES OF WEIGHTED CAYLEY DIGRAPHS

Christopher Cox1, Hannah Turner2, Gregory Michel3, Sung Song1, Katy Nowak1.

Iowa State University, Ames, IA, 2Ball State University, Muncie, IN, 3Carleton College, Northfield, MN.

1 From a subset S of a finite group G, we can define the Cayley digraph of G with connector set S to be the directed graph Cay(G,S) with vertex set G and arc set (x,xs) for some s in S. Naturally arising from this definition is the Cayley isomorphism (CI) property. A Cayley digraph Cay(G,S) is said to be a CI graph if, for every subset T in G such that Cay(G,T) is isomorphic to Cay(G,S), there exists a group automorphism of G that maps S to T. If no such group automorphism exists for any Cay(G,T) isomorphic to Cay(G,S), the graph is called a non-CI graph. Finally, if every Cayley graph of G is CI, the group is called a CI group. In this presentation, we provide a new view of Cayley

digraphs by studying the adjacency matrix of the weighted Cayley graph obtained by weighting the edges of the digraphs corresponding to group character representations of G. Given a non-CI group, we examine the spectra of the weighted adjacency matrices of two isomorphic non-CI graphs and investigate the relation between the CI property of Cayley graphs over G and the spectra of the weighted adjacency matrices of the graphs. We present our findings on the spectral properties of the weighted Cayley digraphs and show how they can be applied to better understand the CI property and the structures of finite groups in general. In particular, we show how to determine if each isomorphism class of Cayley graphs of G are CI classes or not.

SAT-399

## EGYPTIAN FRACTIONS, THE GREEDY ODD ALGORITHM, AND GROUPOIDS

Andres Vindas Melendez1, Julia Bergner2.

University of California, Berkeley, Berkeley, CA, 2University of California, Riverside, Riverside, CA.

1 There has been a vast amount of research performed on Egyptian fractions and groupoids separately. In the recent past, Baez and Dolan defined the notion of groupoid cardinality; it is a topic that can be used to see the connection between Egyptian fractions and groupoids. A recent paper by Bergner and Walker shows that any positive rational number occurs as the groupoid cardinality of some groupoid, and this problem can be reduced to the question of whether any positive rational number has an Egyptian fraction decomposition. This result, obtained through the use of the greedy algorithm, implies the fact that any positive real number is the cardinality of a groupoid with no 2 components having the same cardinality. However, if a different algorithm is used, the decompositions are not alike.

This research seeks to investigate the differing decompositions that are obtained when applying the greedy odd algorithm. For example, under this algorithm, some rational numbers might have infinite decompositions or repeated summands. These results can be used to explore groupoids that have a particular cardinality relating to the outcomes of the greedy odd algorithm.

FRI-405

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West Point, NY, 4George Washington University, Washington, DC, 5Dartmouth College, Hanover, NH.

Given any simple graph, there is a corresponding symmetric function called the chromatic symmetric function (CSF).

Introduced by Richard Stanley in 1995, the CSF of a graph G = (V(G), E(G)) is defined as follows: ΧG = ∑κ∏v∈V(G)Xκ(v) where the sum is over all proper colorings κ of G. A proper coloring is a labeling of a graph such that no 2 adjacent vertices have the same label. In 2008, Scott presented several open problems in graph theory. In our poster, we investigate these open problems and generalize some of his results. Our goal is to find necessary conditions for any 2 graphs that will ensure that they have the same CSF. We first consider 2 special types of graphs: trees and unicycles and write a program in SAGE to compare the CSF for any simple graph and compile a library of graphs with a small number of vertices alongside their CSFs.

FRI-403

## MIDDLE SCHOOL STUDENTS’ OPINIONS REGARDING STEM BEFORE AND AFTER A SUMMER ROBOT CAMP

EXPERIENCE Leanne Cohn, Chrystal Johnson, Gregg Gold.

Humboldt State University, Arcata, CA.

Science, technology, engineering, and mathematics (STEM) are critically important fields. A number of studies have looked at various metrics, including how many students enroll in STEM courses, how many of them drop out, etc.

However, as far as we know, research has not been conducted looking at students’ potential attitude changes toward STEM in the context of attending a summer camp devoted to a highly STEM oriented subject (robots). Here, we administered a paper-based entrance and exit survey regarding STEM attitudes to middle school participants in a summer robot camp. Our data will be statistically analyzed to determine the extent to which robot camp had an effect on participants’ attitudes towards STEM subjects. Hopefully analysis will show us a positive correlation between STEM camp attendance and positive changes in participants’ attitudes towards STEM. We are hopeful this research will provide evidence that camps like this will motivate and encourage students to have more positive attitudes toward STEM.

–  –  –

SAT-407

## SUBGROUP STRUCTURES ON CAYLEY GRAPHS AND THE CAYLEY ISOMORPHISM PROPERTY

Hannah Turner1, Sung Song2, Christopher Cox2, Katy Nowak2, Gregory Michel3.

Ball State University, Muncie, IN, 2Iowa State University, Ames, IA, 3Carleton College, Northfield, MN.

1 For a finite group G and a symmetric subset S of G, the Cayley (undirected) graph Cay(G,S) is the graph whose vertex set is G and such that two vertices x and y are adjacent if y is xs for some s in S. A group G is said to have the Cayley-isomorphism (CI) property if for any 2 isomorphic Cayley Graphs Cay(G,S) and Cay(G,T), there exists an automorphism of the group that sends S to T. We know that if a group G has a subgroup H that does not show the CI property, then G also does not have the CI property. In this poster, we analyze CI and non-CI subgroups of a non-CI group G and we characterize certain Cayley graphs of G where the connector sets generate these subgroups. We study the relation between the particular non-CI isomorphism classes of non-CI groups and their non-CI subgroups, and we also characterize irreducibly non-CI groups, those non-CI groups for which every subgroup of the group is CI.

FRI-407

## CLASSIFICATION OF THE CAYLEY GRAPHS OF SYMMETRIC GROUPS

Gregory Michel1, Hannah Turner2, Christopher Cox3, Katy Nowak3, Sung Song3.

Carleton College, Northfield, MN, 2Ball State University, Muncie, IN, 3Iowa State University, Ames, IA.

1 In this poster, we completely classify the Cayley graphs of the symmetric group on 4 letters. We characterize all possible Cayley graphs that come from symmetric subsets and we identify the individual classes of isomorphic graphs for which the Cayley isomorphism (CI) property holds. In studying the CI property, we analyze whether or not certain graphs are connected, planar, bipartite, edge-transitive, and/or strongly regular. We further investigate whether some of the strongly regular graphs can be decomposed by a pair of directed regular graphs, especially into a pair of directed strongly regular graphs, doubly regular tournaments, or normally regular digraphs that are realized as directed Cayley graphs of the same group. By analyzing patterns among the isomorphism classes, we can generalize certain properties for arbitrarily large symmetric groups.

–  –  –

FRI-409

## STUDY OF THE HILL ESTIMATOR

Maria Correa1, Rebekah Starks2, McKenna Mettling3, Javier Rojo4.

St. Mary’s University, Midland, TX, 2University of Arizona, Tucson, AZ, 3Regis University, Denver, CO, 4Rice 1 University, Houston, TX.

Heavy-tailed distributions are used for modeling in many of the popular fields, for example telecommunications and finance. Thus, the development and study of methods to estimate the tail index for these distributions is highly significant. For our research, we studied the well-known Hill estimator developed in 1975 by Bruce M. Hill. We studied its accuracy and flaws, which include Hill horror plots, by using various heavy-tailed distributions and real datasets to create simulations in R studio.

SAT-409

## SURVIVAL ANALYSIS

Adrian Carballeira1, Noel Martinez2, Kourtney Howell3, Javier Rojo4.

University of Arizona, Tucson, AZ, 2University of Texas at El Paso, El Paso, TX, 3Xavier University of Louisiana, New 1 Orleans, LA, 4Rice University, Houston, TX.

In recent years, scientists have exponentially increased the amount of data that can be gathered from a given experiment. The “curse of dimensionality” along with computational restrictions can hinder attempts to extract meaningful information from the data. Principal components analysis is one common way to reduce dimensionality.

Alternately, the Johnson-Lindenstrauss theorem guarantees that, given the number of data points and a specified error tolerance, there exists a mapping, realized as a linear transformation, into a lower dimension with the property

that pairwise distances between points in the original data set are preserved up to a small error bound. We compare several suggested lower bounds and types of random projection matrices found in the literature and see that these bounds are quite conservative. We use the software R to generate different random projection matrices to determine which type of random projection matrix most frequently upholds the conclusions of the Johnson-Lindenstrauss theorem. Also using R, we use both random projections and PCA to reduce data in order to estimate survival curves for randomly-generated data and compute bias and mean squared error in order to compare these 2 methods.

Knowing which dimension reduction method yields more accurate results allows scientists to analyze summarized data without losing precision, a much more computationally efficient task.

SAT-408

## ESTIMATING TAIL INDICES: THE ACCURACY OF THE ROJO1 ESTIMATOR

McKenna Mettling1, Maria Correa2, Rebekah Starks3, Javier Rojo4.

Regis University, Denver, CO, 2St. Mary’s University, Texas, San Antonio, TX, 3University of Arizona, Tucson, AZ, 1 Rice University, Houston, TX.

4 The focus of our project is to investigate the accuracy of the current estimators used for finding the tail indices for heavy-tail distributions. After studying both the Hill and Pickands estimators and discovering that both had certain flaws, we noticed there is no true consensus on which of the classical estimators we should use for research.

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