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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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Massachusetts Institute of Technology, Cambridge, MA, 2University of Puerto Rico, Rio Piedras Campus, San Juan, 1 PR, 3Swarthmore College, Swarthmore, PA, 4Dartmouth College, Hanover, NH, 5University of Notre Dame, Notre A signed permutation is a sequence π1, π2,..., πn such that each πi ∈ {−n,...,−1,1,...,n} and {|π1|,|π2|,...,|πn|} = [n] = Dame, IN.

2,...,n−1 if πi−1 πi πi+1. Let P(π) be the set of peaks of π, P(S,n) be the set of signed permutations π ∈ Bn such that {1,2,...,n}. Let Bn be the group of all signed permutations of [n]. A signed permutation has a peak in a position i = P(π) = S, and #P(S,n) be the cardinality of P(S,n). We show #P(Ø, n) = 22n−1 and #P(S, n) = p(n) 22n−|S|−1 where p(n) is some polynomial. We also consider the case in which we add a zero at the beginning of the permutation to also allow peaks at position i = 1.

FRI-399

## ON A CLASS OF PERMUTATION POLYNOMIALS OVER FINITE FIELDS

Christian Rodriguez, Alex Santos, Ivelisse Rubio.

University of Puerto Rico, Rio Piedras Campus, San Juan, PR.

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be permutation polynomials over the finite field Fp, a,b ∈ FXp. We show that this family of polynomials is rich in elements of A. We study the coefficients a and b that make polynomials of the form Fa,b(x) = x(p+1)/2 + ax(p+5)/6 + bx permutations, and that the amount of permutation polynomials for any p is divisible by 6. Our approach in studying Fa,b(x) is to use the division algorithm to consider x = αn where n = 6k + r, r = 0,..., 5. If Fa,b(x) is a permutation, this partitions FXp into 6 classes: Fa,b(α6k+r) for r = 0,...,5 each with (p−1)/6 elements. We also conjecture that, given a finite field Fq, the number of permutation polynomials of the form Ga,b(x) = x(q+1)/2 + ax(q+d-1)/d + x is divisible by d if d is even.

FRI-404

## MUSIC OF GROUPS

University of Puerto Rico at Arecibo, Arecibo, PR.

We use math every day without even realizing it; it is no surprise that this discipline has some relationship with music. This relationship can be observed with the theory of groups. A group refers to a nonempty set which defines a binary operation that must meet certain conditions. This research seeks to investigate the behavior of changes in tones, representing them as elements of a group. For the research, a regular dodecahedron was used, where the vertices represented the 7 musical notes, each with their correspondent sustained notes. We worked with this figure rotating it about its center axis and reflecting by joining the vertices and passing through the center. Rotations and reflections represent each element of the group, allowing the formation of 12 reflection elements (identified with S) and 12 rotation elements (identified with R). Then, the changes were taken in shades of fragments of songs and their composition as elements of a group. The result of these compositions showed a particular behavior that we plan to continue observing: if the resulting element of the group was composed of a rotation and a pairing (e.g., the elements R2, R4, R6...) all other rotations that were proved par, so too would happen if it were odd (e.g., the elements R1, R3, R5,...) or if it was a reflection. Even though mathematics is routinely applied to other branches of science, it can also apply to music. This research is a demonstration of that, establishing a close relationship between these 2 disciplines.

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SAT-401

## THE ALGEBRA OF BLOCK PERMUTATIONS

Ryan Contreras1, Isabel Corona2, Matthew Sarmiento1, Rosa Orellana3. Alexander Diaz Lopez4.

Columbia University, New York, NY, 2Metropolitan State University of Denver, Denver, CO, 3Dartmouth College, 1 Hanover, NH, 4University of Notre Dame, Notre Dame, IN.

A set partition of [n] = {1,2,..., n} is a collection of nonempty disjoint subsets of [n], called blocks, whose union is [n]. A block permutation of [n] consists of two set partitions A and B of [n] having the same number of blocks, and a bijection f: A to B. We consider the set BPn= {f:A to B|f is a block permutation}. The elements in BPn can be visualized as graphs having two rows of n-labeled vertices, corresponding to A and B. The connected components of each row are determined by connecting the vertices within each block of A and B. We then connect each block of A to the block of B that it maps to under f. The product gf of 2 block permutations f: A to B and g: C to D of [n] is obtained by gluing the bottom of a graph representing f to the top of a graph representing g, and connecting each block of A to a block in D.

We show that BPn is closed under this operation, and hence is a monoid. We have found a set of generators and seek to find a presentation for BPn. We also describe a Hopf algebra structure on BPn.

FRI-402

## AN UPPER BOUND ON THE NUMBER OF ARCS IN A DIGRAPH GIVEN ZERO FORCING NUMBER

Jason Hu1, Leslie Hogben2.

University of California, Berkeley, Berkeley, CA, 2Iowa State University, Ames, IA.

1 A simple directed graph (digraph) is a mathematical diagram that contains vertices and arcs, which represent individual objects and the connections between them, respectively. Given a digraph, we can always find a zero forcing set, defined as a set of vertices that can be colored blue, such that after repeated application of the color change rule, the entire graph becomes blue. The color change rule specifies that for any blue vertex b, if w is the only white vertex that b points to, then w should be colored blue. The size of the smallest zero forcing set that can be defined on a particular graph G is then called its zero forcing number, Z(G). In our research, we have found an upper bound on the number of arcs that a digraph G can have in terms of n (the number of vertices in the graph) and Z(G) (the zero forcing number). In addition, we have created a technique to construct graphs realizing the maximum number of arcs, given n and Z(G). Related results are also presented.

SAT-398

## IMPACT OF CLIMATIC VARIABLES IN PREDICTING INCIDENCE OF VISCERAL LEISHMANIASIS

Germaine Suiza, George Alexiades, Anuj Mubayi.

Northeastern Illinois University, Chicago, IL.

Leishmaniasis is a vector-borne disease that is caused by a protozoan parasite belonging to the genus Leishmania.

Visceral leishmaniasis (VL) is the most severe form of the disease; the parasite affects the internal organs. VL is transmitted between humans through the bite of the vector, sandflies. The distribution of disease is based on the vector abundance and survival in the environment. Temperature and humidity plays an important role in survival, development, and activity of sandflies. The dynamics of VL in Bihar, India are essentially driven by climate.

The objective of the study is to develop and analyze a mathematical model that can predict the incidence of VL based on various climatic factors. Climate and incidence data from 2000 to 2007 are obtained and analyzed in 3 epidemiologically different regions of Bihar, India: Gaya, Patna, and Bhagalpur. Using seasonal autoregressive moving average (SARMA), a time series analysis, we were able to study how much each climatic factor can explain incidence in Bihar. We used Akaike information criterion (AIC) to provide the best model. Auto-correlations were carried out to determine any potential dependency in the independent variables. A lag of 3 months in average temperature is considered to best explain VL incidence. The study results will be helpful in identifying best times during the year to distribute resources for implementing control programs.

FRI-397

## INFINITELY MANY SOLITON SOLUTIONS TO THE KORTEWEG-DE VRIES EQUATION

Edward Duran, Tuncay Aktosun.

University of Texas at Arlington, Arlington, TX.

The Korteweg-de Vries equation is a nonlinear partial differential equation with important applications in propagation of water waves and of acoustic waves in plasmas. We investigate solutions containing n solitary wave components (solitons) to the Korteweg-de Vries equation. We express n-soliton solutions by providing a compact formula that uses

three constant matrices A, B, and C of sizes n × n, n × 1, and 1 × n, respectively. The compact formula is transformed into another equivalent form so that the limit n → + can be applied, yielding an expression for infinitely many soliton solutions. This is done by exploiting the relationship between the eigenvalues and the matrix trace and the relationship between the eigenvalues and the determinant, which provides the proper mathematical justification for the limit n → +.

SAT-405

## THE LATTICE OF SET PARTITIONS AND TRANSITION MATRICES OF SYMMETRIC FUNCTIONS

Alexandria Burnley1, Rosa Orellana2, Candice Price3, Auia Richburg4, Simone Thiry5.

University of Illinois, Urbana-Champaign, Champaign, IL, 2Dartmouth College, Hanover, NH, 3United States Military 1 Academy, West Point, NY, 4Morehouse College, Atlanta, GA, 5University of Maine at Farmington, Farmington, ME.

Brick tabloids are combinatorial objects, introduced by O. Egecioglu and J. Remmel, through which the transition matrices between the bases m, e, p, and h of the commuting symmetric functions may be defined. There is an alternative way to describe the transition matrices using symmetric functions in noncommuting variables and the lattice of set partitions. Our goal is to study functions on the lattice of set partitions that arise as entries in the transition matrices. Our research explores the relationship between brick tabloids and functions on the lattice of set partitions.

For example, we study Nλ(μ), the number of set partitions of type μ that are larger than or equal to a set partition of type λ, and nλ(μ), the number of set partitions of type μ that are less than or equal to a set partition of type λ.

SAT-403

## CALCULATION OF THE MAPPING CLASS GROUP OF A GENUS 2 SURFACE

Alyssa Loving1, Michael Andrews2.

University of Hawaii at Hilo, Hilo, HI, 2Massachusetts Institute of Technology, Cambridge, MA.

1 A useful tool for studying an object with symmetry is the automorphism group of the given object. For example, one might consider the space of homeomorphisms from a genus 2 surface to itself. This group is too big to study, but a

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SAT-396

## DO TRANSITIVE TOURNAMENTS MINIMIZE GRAPH COSTS?

Tynan Lazarus1, Daniel Pritikin2.

University of Hawaii at Hilo, Hilo, HI, 2Miami University, Oxford, OH.

1 Let G be an undirected graph and let T be a tournament on the same vertex set as G. Define the cost of G relative to T to be the sum of two-step paths in T from u to v plus two-step paths from v to u for any pair of vertices u and v.

In our research, we determine, for several classes of graphs, which tournaments minimize the cost. Pelsmajer, et al, conjecture that for each graph there is a transitive tournament that minimizes the graph’s cost. We prove that a transitive tournament minimizes the cost for complete graphs, nearly complete graphs, paths, star graphs, and cycles.

SAT-402

## ALGORITHMIC CONSTRUCTION OF PERMUTATIONS OF HIGH DISPERSION

Juan Carrillo, Carlos Cruz, Edward Mosteig.

Loyola Marymount University, Los Angeles, CA.

Given a collection of objects, a permutation is a function that reorders the objects into a potentially new arrangement.

An example of a permutation of the first 5 positive integers is (3, 5, 1, 2, 4). Here the number of objects being permuted is called the block length of the permutation. We investigate the notion of dispersion, which measures the extent to which objects are “scattered” or “dispersed” under the action of a permutation. Dispersion is a numerical value between 0 and 1, where larger values indicate a low level of redundancy in linear patterns found in a graph of the permutation. The purpose of our two-fold investigation is to shed light on the behavior of dispersion as block length increases and to construct permutations with dispersions as large as possible. For small block lengths of at most 12, we compute the average value of dispersion via computer for all permutations of a given block length. The calculation is intractable for large block lengths, so we produce estimates using sampling techniques. Empirically, dispersion appears to follow a distribution with mean 0.81 and extraordinarily small standard deviation. This small standard deviation poses an obstacle to finding permutations with large dispersion via pseudo-random sampling, so

we developed and analyzed heuristic algorithms to overcome this challenge. Permutations play a role in applications such as coding theory, experimental design, and radar equipment, so our research has the potential to impact our understanding of some fundamental questions in these areas.

SAT-400

## ZERO FORCING NUMBER IN A GRAPH AND ITS ORIENTATION

Kathryn Manternach1, Leslie Hogben2.

Central College, Pella, IA, 2Iowa State University, Ames, IA.

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