##### Parallel session abstracts

**Kyla Baldwin**, Indiana University **Drew Reisinger**, Indiana University and University of Evansville*Representing Tilings on Closed Surfaces by Quadrilaterals *

We describe a compact, combinatorial encoding of quadrilateral tilings of closed, oriented surfaces. This representation motivates a correspondence between a certain class of closed-curve systems on such surfaces with these combinatorial data. We also adapt this method to encode periodic tilings in the plane by rhombi.

**Stephen Bates**, Indiana University and Harvard University *Maximum Likelihood Estimation of the Heterozygosity of DNA Sequences*

The heterozygosity of a DNA sample from a diploid (containing two chromosomes) organism is the fraction of sites at which the 2 chromosomes match. This quantity is of great interest in the field of genetics as it can provide information about the nature of mutations ongoing in a population. Estimation of this quantity is non-trivial because DNA sequencing methods give an incomplete picture of the data and suffer from high error rates. In this talk, we will examine a maximum likelihood approach to estimating the heterozygosity of a given DNA sequence. This approach is useful, both because it is accurate for small sampling coverage and because it provides an estimate of the sequencing error rate. We will then extend the maximum likelihood technique to the estimation of other properties of interest in a DNA sample.

**Adrienna Bingham**, Valparaiso University and Angelo State University **Denali Molitor**, Valparaiso University and Colorado College **Julie Pattyson **, Valparaiso University and University of Saint Joseph *Vertical transmission in epidemic models with isolation from reproduction*

We analyze the effect of full vertical transmission in several epidemic models involving infectious diseases that cause sterilization in the infected hosts. Under certain conditions on the parameters, we found that the sterilization effect may prevent a susceptible extinction situation regardless of how big the infection rate may be. This effect is studied under several functional forms for the infection transmission term in order to assess its robustness. The implication in pest control measures is also discussed.

**Philip Bontrager**, Goshen College *Fair Allocation Visualizations *

Two or three people have equal ownership rights for several goods for which each person may have different monetary valuations. What is a fair way to allocate the goods among the people? This work examines different notions of fairness including efficient, proportionate, envy-free, share proportionate, value proportionate. The goal is to develop visualizations to assist us and others to better understand notions of fairness and their interrelationships.

**Laura Booton**, Rose-Hulman Institute of Technology and Nebraska Wesleyan University**Chase Mathison**, Rose-Hulman Institute of Technology *Determining Properties of Metal by Analyzing Changes in Impedance*

In certain situations it is useful to identity an unknown piece of metal without contact or visual inspection. We wish to do this by inducing a current in a coil and placing the metal object in the resulting magnetic field. We have a model which gives the change in impedance of the coil based on the properties of the metal. In this talk we will analyze the inverse problem of finding the metal's properties from measuring the change in impedance.

**Matt Charnley**, Rose-Hulman Institute of Technology and University of Notre Dame **Andrew Rzeznik**, Rose-Hulman Institute of Technology and Cornell University*Thermal Detection of Inaccessible Plate Corrosion*

In engineering and industrial applications, it can be very valuable to know if an inaccessible portion of a structural element has been corroded, for both safety and efficiency. We wish to use a completely non-contact thermal method to approximate the corrosion profile, as this method would be easy to apply to a physical situation. In this talk, we will discuss the method of attacking this mathematical "inverse problem" and show how the corrosion can be calculated if the thermal properties of the metal and corroded material are known.

**Alex Chin**, Indiana University and North Carolina State University *Probabilistic Modeling Techniques for Microtubules *

We develop a discrete time model for the behavior of microtubules, a cellular filament vital for such tasks as motility and cell division. We consider integer values for growth and shortening velocities and construct a Markov chain model for a single polymer based on the nucleation rate, catastrophe frequency, and rescue frequency. This model gives rise to a set of recurrence relations for each pair of growth and shortening velocities that can be solved using linear algebra techniques to determine a length distribution for the polymer in steady state. We consider both finite and infinite models and consider the mathematical and biological implications of each.

**Kristin Cordwell**, Indiana University and Princeton University **Selina Gilbertson**, Indiana University and Northern Arizona University *On the Realizability of a Critical Portrait*

Given a rational function f with fixed critical points, the associated branch data of f refers to a set of partitions of the degree of f, with each partition determined by the local degrees of points in the preimages of a corresponding distinct critical value of f. The critical portrait of f is determined by a partition of the multiplicities of the critical points of f. It is known that a critical portrait is realizable if it comes from a connected planar multigraph G, where each critical point corresponds to a vertex of G and each vertex has degree equal to the multiplicity of the associated critical point. It is also known that necessary conditions for the realizability of a critical portrait are that the associated branch data is realizable and the number of distinct critical points is at most the degree of the function. In this talk, we provide a graph-theoretic proof to conclude that these conditions are also sufficient for the realizability of a critical portrait. This answers a recent question posed by Rafe Jones and Michelle Manes.

**Emily Grube**, Rose-Hulman Institute of Technology and Carleton College **Miranda Sawyer**, Rose-Hulman Institute of Technology and Northeastern State University (Oklahoma)*Robust FBA: A Mathematical Journey*

Flux Balance Analysis is a popular tool for studying the metabolic pathways of cells and the role individual metabolites play in maintaining cell function. We introduce Robust FBA as a more accurate model of a cell's metabolism. FBA is predicated on an optimization problem in the form of a linear program. We compare the original linear program to a robust linear program in the new model. This has led us to the mathematical study of optimization problems. We introduce the concept of duality in a robust linear program and prove different facets of the stability of a robust linear program. We also examine limiting values of the robust problem and the tendency for our robust solution to approach the optimal solution of the linear model.

**Amy Ko**, Rose-Hulman Institute of Technology and Amherst College **Michael MacGillivray**, Rose-Hulman Institute of Technology and University of Notre Dame *Robust Flux Balance Analysis: Computational Results*

Flux Balance Analysis (FBA) is a computational method of analyzing the metabolic network of cells. The traditional FBA model has proven useful in predicting cellular behavior but suffers several key weaknesses. We develop a robust version of FBA that addresses these weaknesses and explain its computational successes.

**Lindsay Martin**, Indiana University *3-dimensional Gluings of Cubes *

A 3-dimensional gluing of cubes is constructed by gluing together n unit cubes. We glue the top of each cube to a bottom of another, the left of each cube to the right of another, and the front of each cube to the back of another. If we label the cubes 1 through n, the gluing pattern corresponds to a triple of permutations of {1, 2, ..., n}. In this talk, I will discuss the following question: Given a triple of permutations, can we determine whether or not the corresponding 3-dimensional gluing of cubes is a manifold? Using a formula for the Euler characteristic of a 3-dimensional gluing of cubes, the problem can be reduced to determining the number of equivalence classes of vertices in the gluing.

**Caleb McWhorter**, Rose-Hulman Institute of Technology and Ithaca College *A Foam Model for Bread Development *

Here a model for the geometry of two dimensional bread is investigated. The incorporated model takes into consideration the interior structure of the gases in a bread foam. Using the calculus of variations, a minimal configuration is found with respect to the gravitational potential and surface energy of the dough. We compare the resulting configurations to previous models of bread as a liquid drop by comparing the discrete curvatures of the convex hull from the found minimal configurations. Elementary statistics are also incorporated to compare the two models.

**John Shrontz **, Valparaiso University and University of Alabama, Huntsville *A Partial Ordering of Knots ***Arazelle Mendoza**, Valparaiso University and Christopher Newport University**Tara Sargent**, Valparaiso University and Clarke University

Knot theory is the study of the different ways to embed a circle in three-dimensional space. In this talk we study how knots behave under crossing changes, which are local operations that one can perform to diagrams of knots. In particular, we investigate a partial ordering of alternating knots which says that K > K' if every diagram of K can be transformed into K' via some number of simultaneous crossing changes. This partial ordering was originally introduced by Kouki Taniyama in the paper "A Partial Order of Knots". We expanded upon Taniyama's partial ordering and present theorems about the structure of the partial ordering for more complicated knots. Our approach is largely graph theoretic, as we begin by translating each knot diagram into one of two planar graphs by checkerboard coloring the plane. Of particular interest are the class of knots known as pretzel knots, as well as knots that have only one "direct" minor in the partial ordering.

**Annie Murphy**, Indiana University and Clarkson University **Ashley Weber**, University of Michigan *Developing a Phylogenetic ANCOVA: Analyzing Multiple Variables in the Evolution of Continuous Traits*

Phylogenetic comparative methods are a class of statistical methods in evolutionary biology that incorporate the dependence of a phylogenetic tree structure into the framework of the traditional ANOVA or ANCOVA model. Phylogenetic comparative methods have been developed to evaluate the relationships between two continuous evolutionary variables, or traits, and a continuous trait coupled with a categorical trait. We worked to develop a new phylogenetic ANCOVA method to evaluate the relationship between two continuous traits and a categorical trait. Our method incorporates dependence on the phylogenetic tree structure and uses a maximum likelihood approach to estimate a regression line and a variance for the predicted continuous trait in each category. Our model also evaluates unique variances for each level of the categorical variable.

**Kenneth Reed**, Rose-Hulman Institute of Technology and Texas A and M University *Modeling Dough Expansion During The Rising Process*

Bread and dough can be modeled as a weighted foam. During the rising process gas is generated in the dough causing the bubbles increase in volume and the dough to expand. Taking into consideration gas production, diffusion, and gravity it is possible to model this process. This model sheds light on the shape of the exterior surface of dough as it rises as well as the distribution of bubble sizes seen in bread.

**Ted Samore**, Rose-Hulman Institute of Technology *Numerical Simulation of Estrogen Protein Dimer Exchange in a Chromatography Column*

The estrogen receptor protein has important regulatory functions and contributes to breast cancer development. The purified protein exists in equilibrium between monomers and dimers. We performed size-exclusion chromatography and dimer exchange assay experiments for the ligand-binding domain of the protein. We also mathematically modeled the chromatography using nonlinear convection diffusion equations and then compared the results of the experiments with the simulations to estimate the association and dissociation rate constants for the purified protein.

**Connor Scholten**, Valparaiso University and Grand Valley State University **Tyler Schrock**, Valparaiso University and Troy University **Alexa Serrato**, Valparaiso University and Harvey Mudd College *Generalized Pattern Containment in Trees *

In this talk, we first define what it means for a full binary ordered tree to contain another binary tree in a non-contiguous sense. We then prove that any two j-leaf trees are both contained within all i-leaf trees the same number of times, regardless of their shapes. Finally, we determine a generating function G_t(x,y) for any path tree t where the coefficient of x^iy^n represents the number of trees with i leaves that have n copies of t.

**Molly Stillman**, Rose-Hulman Institute of Technology and Western State College of Colorado *Modeling the Inner Structure of a Cookie: 2D *

A model for the interior structure of a cookie or of bread is a cluster of gas bubbles shaped by the force of gravity and the tension of the dough. Building off of the work of Jordan Finch who last year considered a hexagonal structure for the interior, we look at structures that can be regarded as decorations of an underlying hexagonal grid to improve the model. This talk focuses on the topological structure in 2D where given different configurations that decorate an underlying hexagonal structure; we will see what patterns arise due to the force of gravity and the surface tension of the dough.

**John Strieff**, Wabash College

Eutocius of Ascalon, a 6th century AD Greek Mathematician, wrote commentaries on Archimedes and Apollonius. His works show us how students were introduced to these writings in Late Antiquity. Our project's focus is the diagrams in manuscripts and their relation to the text. In this presentation we will give the historical context, overview of the manuscript tradition, and examples of our studies.