## The 2013 Indiana Undergraduate Research Conference

### Parallel session abstracts

*Collateral Circulation Following Acute Arterial Occlusion: A Theoretical Model of Blood Flow Compensation in the Lower Extremities***Julia Arciero**, jarciero@math.iupui.edu

Indiana University Purdue University Indianapolis **Ava S.K. Greenwood**, asgreenwood@gmail.com

Indiana University-Purdue University Indianapolis and University of Hawaii at Hilo ** Joseph Unthank**, junthank@iupui.edu

Indiana University School of Medicine

Peripheral arterial disease is characterized by the partial or complete blockage of arteries within the systemic vasculature, often due to atherosclerosis. This blockage leads to a significant reduction in blood flow to tissue and is often accompanied by symptoms such as claudication, rest pain, and critical limb ischemia. To compensate for the reduced flow, growth and new development of collateral arteries, arterioles, and capillaries has been observed. However, there remains some debate regarding which vascular segments contribute most to restoring blood flow to distal tissues following major arterial occlusion. This study presents a predictive model that identifies the primary sites of resistance, pressure changes, and blood flow in a vascular network following complete femoral artery occlusion. Theoretical predictions were obtained via a two-step investigation. First, several published studies that included measures of pressure, flow, and resistance for multiple vascular segments in the lower extremities were reviewed to obtain relevant data. Second, a theoretical model was developed in MatLab to determine which vascular segments best compensate for blood flow following complete occlusion of the femoral artery. Results indicate that the primary site of vascular resistance is shifted from the arterioles to the collateral vessels following a major arterial occlusion. The dilation of the collateral vessels is predicted to contribute most significantly to blood flow restoration to the tissues downstream of the occlusion.

*EEG Time Series Analysis and Functional Connectivity Network Measures of TD and ASD Youths ***Erik Bates**, bateser2@msu.edu

Indiana University and Michigan State University ** Katherine Coppess**, kcoppess@umich.edu

Indiana University and University of Michigan**Benjamin Seitzman**,beaseitz@imail.iu.edu

Indiana University

Graph theoretical network measures allow for investigation of the arrangement and dynamics of connections between objects in a complex system. A prominent application of graph theory is featured in the study of neural networks of the human brain. These measures may be applied to data acquired from neuroimaging methodologies, such as electroencephalography (EEG). EEG records electric potential changes in global brain activity across the scalp as a function of time. Previously recorded 32-channel EEG data of typically developing (TD) youths and youths with Autism Spectrum Disorder (ASD) during both wakeful rest and a visual task were analyzed. A normalized, cross-covariance analysis of the EEG time series was used to produce weighted and directed graphs representing functional connectivity. Several graph theoretical network measures were applied in search of the most robust measure across the two experimental conditions and between populations.

*On the Classification of size 24 Hadamard Matrices***Wade Bloomquist**, wade-bloomquist@uiowa.edu

Indiana University and University of Iowa

It is known that there exist 60 equivalence classes of Hadamard matrices of size 24. One can relate 59 of these equivalence classes through an operation known as switching. The 60th equivalence class, while not related through switching, does generate the same code, the Golay code, as one of the other matrices. The Golay code possesses an extreme amount of symmetry, evidenced by its automorphism group acting 5-transitively on its elements. This provides an opportunity to exploit this symmetry to relate the final equivalence class to the other 59. This is done by carrying over this transitivity to identify constraints on the structure of sets of 8 columns of the matrix generating the code.

*Drawing Planar Graphs via Dessins d'Enfants***Kevin Re'nard Bowman, Jr.**, mathmeisterreaderray@yahoo.com

Purdue University and Morehouse College**Sheena Chandrasekharan**, chandr12@purdue.edu

Purdue University ** Anji Li **, li723@purdue.edu

Purdue University**Amanda Marie Llewellyn**, allewellyn@g.hmc.edu

Purdue University and Harvey Mudd College

There are many examples of finite, connected planar graphs: for instance, there are paths, trees, cycles, webs, and pyramids to name a few. In 1984, Alexander Grothendieck, inspired by a result of Gennadii Belyi from 1979, constructed a finite, connected planar graph via certain rational functions by looking at the inverse image of the interval from 0 to 1. In this project, we investigate how restrictive Grothendieck's concept of a Dessin d'Enfant is in generating all planar graphs. We show that certain trees (such as paths and star graphs), certain webs (such as cycles, dipoles, and pyramids), and each of the Platonic solids (the tetrahedron, cube, octahedron, icosahedron, and dodecahedron) can all be generated as Dessins d'Enfants by exhibiting explicit Belyi maps.

*Robust Methods for Determining Order Parameters in Microtubule Cytoskeletal Arrays***Allison Brumfield**, brumfiel@stolaf.edu

Indiana University and St. Olaf College

The microtubule polymers inside plant cells form a variety of 2-dimensional patterns with functional implications for cell growth. We describe a robust set of analytical methods for quantifying the order parameters of these polymer arrays from live-cell microscopy images. We define a method for determining signal-to-noise ratios to account for the camera noise and the low spatial frequency noise associated with background fluorescence in the image. We develop a simulator in MATLAB to generate images with known polymer array order and signal-to-noise ratio. We then use the simulated images and signal-to-noise measurements to explore the robustness of techniques using Radon and Fourier transforms for describing the rotational and translational order parameters.

*Implementing Methods in Algebraic Graph Theory in Macaulay2***Jack Burkart**, jburkar1@nd.edu

University of Notre Dame ** Caroline Jansen**, cjansen@nd.edu

University of Notre Dame

There are significant interactions between commutative algebra and combinatorics, particularly graph theory. It is of growing importance for mathematicians to be able to do computations on graphs, among other objects. We wrote a package for Macaulay2, a program used to do algebraic computations, that implements methods for graphs and digraphs. We will discuss some of these connections and how we implemented them.

*Prime Factors of K@aacute;szonyi Numbers***Ariana Cappon**, arcappon@umail.iu.edu

Indiana University ** Emily Walther**, waltem22@wclive.westminster.edu

Indiana University and Westminster College

Snarks are a class of simple, cubic, and non-planar graphs that cannot be edge-3-colored. In the 1970's, a series of papers were published by László Kászonyi examining the number of edge-3-colorings of a snark after the removal of an edge. In the 2006 Indiana Mathematics REU, Dr. Richard Bradley and Scott McKinney were able to show that given nonnegative integers a, b, c, and d, there exists a cyclically 4-edge connected snark G with and edge e such that the Kászonyi number of G and e is equal to 2^a3^b5^c7^d. In this talk, we will discuss our extension of this research and show further prime factors of Kászonyi numbers.

*Strategy Evolution for Cooperation Games***Andrew Clemens**, andrewpc@goshen.edu

Goshen College

The world is full of many different kinds of conflict, which vary in many ways. A common type of conflict occurs when two or more participants have their own best interests in mind, but must work together, at a cost, to achieve those interests. In game theory, this is known as the prisoner's dilemma. The most basic form of this is when two players can either cooperate, which would help the opponent but hurt the cooperator by a smaller amount, or defect, which has no effect on either player. No matter what the other player does, each player always does better by defecting; however, each player does better when both cooperate than when both defect. This dilemma can be resolved with both players choosing to cooperate when the game is played in multiple rounds. This project examines what happens when there are more than two players, communication errors may occur, and opportunities for punishment of defectors are made available. Stable strategies are found using a process of evolution: different strategies play each other, the weaker ones die off, and the stronger ones remain and possibly ``mutate'' into new strategies, until the strongest strategies dominate over all the others.

*4-equitable Tree Labeling***Zena Coles**, zena_coles@yahoo.com

Valparaiso University and Bard College ** Alana Huszar**, huszara2@tcnj.edu

Valparaiso University and The College of New Jersey**Jared Miller**, jasm50@gmail.com

Valparaiso University and Bob Jones University

A tree is a vertex-edge graph that is connected and contains no cycles. A 4-equitable labeling of a graph is an assignment of labels {0,1,2,3} to the vertices. The edge labels are the absolute difference of the labels of the vertices that they are incident to. The labels must be distributed as evenly as possible amongst the vertices and they must also be distributed as evenly as possible amongst the edges. We study 4-equitable labelings of different trees: complete n-ary trees for all natural numbers n, caterpillars, T-trees, and generalized stars. We believe that proving all trees are 4-equitable will bring us one step closer to proving the famous graceful tree conjecture that has been open for half a century.

*Smooth Strictly Convex Billiards are Generically Insecure***Tom Dauer**, tjdauer@umail.iu.edu

Indiana University

A mathematical billiard is defined as a plane domain ("table") and a point mass that moves with constant speed inside the table in such a way that when the point mass hits the boundary, its angle of incidence equals its angle of reflection. Let x and y be points in a given table, either in the interior of the table or on the boundary. A blocking set for the pair x,y is a set of points in the table such that every billiard path from x to y passes through a point in the set. If a finite blocking set exists, the pair x,y is called secure; if not, it is called insecure. We show that given x and y, there exists a countable collection of dense open sets in the space of strictly convex C^2 billiard tables with x and y in the interior for which x and y are insecure. The intersection of these sets is a dense G_delta set of tables for which x and y are insecure. In this sense, the pair x,y is insecure for a "generic" smooth convex billiard table with x and y in the interior. In 2009, Tabachnikov showed that if x and y are on the boundary of such a table, they are insecure; our result sheds light on the case in which x and y are in the interior.

*Associating Finite Groups with Dessins d'Enfants***Katrina Elizabeth Eidolon**, katrina.eidolon@gmail.com

Purdue University and University of Colorado at Colorado Springs**David Heras**, dheras@email.wm.edu

Purdue University and William and Mary College ** Ahmed Tadde**, ahmedtadde@gmail.com

Purdue University and Howard University**Yuan Feng**, yuanfengjes@gmail.com

Purdue University and University of Illinois at Urbana-Champaign

Each finite, connected planar graph has an automorphism group; such permutations can be extended to automorphisms of the Riemann sphere. In 1984, Alexander Grothendieck, inspired by a result of Gennadii Belyi from 1979, constructed a finite, connected planar graph via certain rational functions by looking at the inverse image of the interval from 0 to 1. The automorphisms of such a graph can be identified with the Galois group of the associated rational function. In this project, we investigate how restrictive Grothendieck's concept of a Dessin d'Enfant is in generating all automorphisms of planar graphs. We discuss the rigid rotations of the Platonic solids (the tetrahedron, cube, octahedron, icosahedron, and dodecahedron), the Archimedean solids, and the Catalan solids via explicit Belyi maps. Conversely, we enumerate groups of small order and discuss which groups can -- and cannot -- be realized as Galois groups of Belyi maps.

*Mathematical Modeling in Ecology: What Killed the Mammoth?***Michael Frank**, michael.frank@my.simpson.edu

Valparaiso University and Simpson College ** Anneliese Slaton**, slatonam4847@mbc.edu

Valparaiso University and Mary Baldwin College**Teresa Tinta**, ttinta@umes.edu

Valparaiso University and University of Maryland Eastern Shore

One extinction theory of the Columbian mammoth (Mammuthus columbi), called "overkill" hypothesizes that early humans overhunted the animal. We will employ two different approaches to test this theory mathematically: develop a differential equations model, as well as a stochastic simulation. The system of ODEs is a modified predator-prey model that also includes immigration and emigration. The simulation is a stochastic temporospatial model based on a hexagonal grid system designed to represent North America at the end of the last ice age. Using this simulation, we model the migration of humans into North America and the response in the mammoth population. These approaches show evidence that human-mammoth interaction would have affected the extinction of the Columbian mammoth during the late Pleistocene.

*Topological bound states in one-dimensional tight binding Hamiltonians***Marvin Q. Jones**, marvinqjones@gmail.com

Indiana University and North Carolina Agricultural and Technical State University

Phases of matter are of fundamental importance to our understanding of many physical phenomena. Matter takes on numerous states, all of which differ in physical properties substantially. We know the common phases of matter: solid, liquid gas and plasma, which are all created through different conditions on a material. Variations in the conditions or parameters such as temperature, pressure and density often serve as catalysts for observing different phases of matter. Understanding the phases of matter and transitions between them is the subject of condensed matter physics. We are also interested in materials such as solids and how electrons behave in such materials. Electrons in metal have their own distinct phases such as the ability to move more or less freely depending on the properties of the metal, however, electrons in an insulator may be more tightly bound to ions. Much of the interaction stems from understanding the lattice structure of the system, which is a repeating pattern between points and a prescription (or function) about how it repeats. Many crystals and solids have unique lattice structures, which affect how electrons may behave in that material. The behavior of the phases and transitions of these materials are classified by a topological invariant that does not change if parameters of the system are changed smoothly. Understanding these new materials and their physical properties lead to many unique applications, which range from materials for computers to materials for gaming systems. We seek to investigate some of these topological properties in simple systems transitioning to more complex systems amenable to exact numerical calculations through a computer algebra system. The method involves diagonalizing the Hamiltonian matrix, which provides the eigenvalues and eigenvectors, to describe the energy spectrum of the system and the wave functions behavior at different points. Understanding the behavior of eigenvalues and eigenfunctions will help identify the topological properties of the phases of electrons.

*Estimating the Volatility in the Black-Scholes Formula***Rebecca Keenan**, rebecca.keenan@valpo.edu

Valparaiso University and Eastern Connecticut State University ** Rachel Lane**, rachel.lane@valpo.edu

Valparaiso University and Concordia College**Josh Matti**, josh.matti@valpo.edu

Valparaiso University and Indiana Wesleyan University

The Black-Scholes formula is one of the most popular option pricing models; however, one of the inputs, volatility, is not deterministic and available for immediate application in the formula. In this talk we will examine three different approaches for better estimating the volatility: smoothing, deriving the distribution of the volatility, and building time series models. We employ both single and double exponential smoothing techniques on European call option valuations for the S and P 100 Index. We derive a function for variance and standard deviation and calculate their expected values. Secondly, we derive the probability distributions of the volatility with a transformation technique. The expectations of the volatility from the probability distributions are then applied back to the Black-Scholes formula. Additionally, we extend the cumulative normal distribution functions in the Black-Scholes formula using a Taylor series expansion to arrive at functions of volatility. With time series volatility models, we apply Autoregressive Conditional Heteroscedasticity (ARCH) and Generalized Autoregressive Conditional Heteroscedasticity (GARCH) volatility for application into the formula. Similarly to these two well-discussed volatility models, we purpose three new time series volatility models: Moving Average Conditional Heteroscedasticity (MACH), Autoregressive Moving Average Conditional Heteroscedasticity (ARMACH), and Generalized Autoregressive Moving Average Conditional Heteroscedasticity (GARMACH).

*A Mathematical Study of Amphetamine-Induced Temperature Disregulation***Maire Kelley**, maire.kelley@gmail.com

Indiana University-Purdue University Indianapolis and University of St. Thomas-Houston **Yaroslav Molkov**, ymolkov@iupui.edu

Indiana University-Purdue University Indianapolis**Dmitry Zaretsky**, dzaretsk@iupui.edu

Indiana University School of Medicine

Recent experimental evidence shows that the body temperature control system has both excitatory and inhibitory activations. Based on these hypotheses, a mathematical model was constructed to examine the temperature responses to amphetamines in rats. This model utilizes two neural populations that feed into a single node that directly affects temperature modulation. These populations are hypothesized to be in the dorsal medial hypothalamus (DMH) and the rostral ventrolateral medulla (RVLM). DMH provides the excitatory drive while RVLM is inhibitory. The model focuses on the relationship between these nodes and their effect on the thermoeffector population in the spinal cord with the input of certain perturbations. Varying doses of methamphetamine (Meth) and ecstasy (MDMA) were administered in conjunction with changes in ambient temperatures, specifically 15-24 degrees Celsius environments. The model was manipulated to simulate different biological mechanisms in response to these disturbances to reproduce the above laboratory data. This included increasing or decreasing activation thresholds, introducing MDMA-induced vasodilation, and separating neural activated heat generation and dissipation mechanisms. In doing so, the model shows that Meth directly disturbs the thermoregulatory system through the hypothesized DMH and RVLM neural populations. Ecstasy, however, perturbs the system by evoking excessive heat dissipation and thermogenic response at the same time, putting them in conflict with each other. By using this mathematical model to explain the relationship between these neural populations, we can further elucidate the neural mechanisms that mediate mammalian thermoregulation.

*Numerical Simulation of Delay Differential Equations via PSM***Dustin Lehmkuhl**, lehmkudc@rose-hulman.edu

Rose-Hulman Institute of Technology

The Modified Picard method (PSM) for approximating IVPs (in a non-standard numerical fashion) involving ODEs or PDEs has been established as a viable option (see Sochacki and Parker et al.). The form of the approximating method allows itself to be used without much labor on delay differential equations (where the vector field at the current time relies on the state of the system at some earlier time as well as the current time). The properties of the solutions to the DDEs can be different depending on how the delay shows up, hence there are a myriad of subclasses (see Baker, Paul and Wille) and as a consequence, their numerical simulation can be delicate. This jump to DDEs via PSM appears to be possible without worrying about which subclass is involved. It would be nice to determine the limitations of PSM in this arena. Another possibility is to make the jump to delay PDEs. At the time of the talk, this will still be a work in progress.

*Using mathematics to investigate the role of integrin activation in cell migration***Lauren Lembcke**, llembcke@iupui.edu

Indiana University-Purdue University Indianapolis

Wound healing requires the migration of cells into the wounded region to restore the function of the tissue. In disease cases such as NEC, cell migration has been observed to be impaired, in part due to an overexpression of integrin proteins. In NEC, components of bacterial walls (LPS) bind with TLR4 receptors on the intestinal epithelium and trigger a signaling cascade that yields integrin activation. An increased presence of bacteria is hypothesized to cause increased integrin activation and decreased cell migration. This study aims to elucidate the steps in the signaling cascade which contribute most significantly to the overactivation of integrins. A mathematical model will be developed to simulate different scenarios that may lead to impaired cell migration. Model results will be compared with experimental data and will provide insight into the mechanism of reduced cell motility in NEC and other diseases.

*Generalized cyclotomy for finding supplementary difference sets***Yancy Liao**, yyl5107@psu.edu

Indiana University and The Pennsylvania State University

Supplementary difference sets of a finite abelian group are a collection of subsets where every non-zero element of the group is represented the same number of times as a difference of distinct elements. When the group in question is the integers modulo n, a special relationship exists between supplementary difference sets and binary sequences of length n. Cyclotomial cosets have been used to find supplementary difference sets when n is prime. A generalization of cyclotomy for composite n is not well-developed. We considered when n is twice a prime and present some findings about the structure of its generalized cosets. We also discovered some classifications of supplementary difference sets in this case and how they are related by their identification with binary sequences.

*Complex Dynamics and the Smale Mean Value Conjecture for quartic polynomials***Nicholas Miller **, ntmdm6@mail.missouri.edu

Indiana University and University of Missouri - Columbia ** Max Zhou**, maxzhou@umail.iu.edu

Indiana University

Let f(z)=z+a_{2}z^2+a_{3}z^3+a_{4}z^4+... be a complex polynomial. One version of the Smale Mean Value Conjecture (SMVC) states that there exists a critical point c of f such that |f(c)/c|≤ 1. The SMVC bound has been proved for polynomials up to degree 10 (Sendov, Marinov 2007). We are interested in a stronger form of this conjecture (SC) which states there exists a critical point that both satisfies the SMVC bound and converges to the origin under iterations of f. In 2010, Hayley Miles-Leighton and Kevin Pilgrim proved SC for quadratic and cubic polynomials. We prove SC holds for quartics with a repeated critical point. Currently, we are identifying regions in the two complex dimensional parameter space where the stronger conjecture holds for a general quartic polynomial.

*The classification of critically fixed rational functions***Nicholas Nuechterlein**, nknuecht@umich.edu

Indiana University and University of Michigan ** Samantha Pinella**, s.pinella@sms.ed.ac.uk

Indiana University and University of Edinburgh

A natural problem in complex dynamical systems is to classify self mappings of the Riemann sphere, i.e. the complex plane with the point at infinity. In 1989, Tischer used previous results of Thurston to explicitly classify, as branched mappings, the set of complex polynomials whose critical points are also fixed. We ask if the same can be done for rational functions, f, from the Riemann sphere to the Riemann sphere where f(z)=p(z)/q(z) with degree d:=max(deg(p), deg(q)) and gcd(p,q)=1. Nekrashevych showed that, up to conjugation by Mobius transformation, the dynamics of such maps f are determined by an algebraic invariant called a wreath recursion on a free group G, where G is the fundamental group of the Riemann sphere modulo the critical points of f . We develop an algorithm to find the wreath recursions of certain branched mappings which we then use as inputs for a newly developed computer program by Bartholdi that provides numerical approximations of these maps.

*The Effects of Bloc Formation on Relative Power in Weighted Voting Games***Peter Schrock**, peterrs@goshen.edu

Goshen College

In a nine person committee where every member has one vote and a majority, or five votes, is required to reach a consensus, five members would stand to increase their power by agreeing to form a bloc. By secretly voting before meeting as a whole committee, the five could come to a consensus by majority and then all agree to vote the same way in front of the full committee. Because the bloc effectively controls the committee, each member would have one fifth of the full committee's power, instead of the one ninth they would receive without the bloc. Three particularly Machiavellian members of the five could then theoretically form a bloc within the bloc of five and control the whole committee. If the full committee required unanimity instead of a majority, forming blocs that require a majority decreases the power of individual voters. Members of such a bloc would lose their ability to single handedly veto a proposal and thus would lose power. The aim of this paper is to characterize games and groups of players within those games for which voting power would be increased by forming blocs.