Published electronically January 24, 2020

DOI: 10.1137/19S019073

**Authors:** Aidan Bowman (Socrates Preparatory School, Casselberry, FL) and Jonathan H. Yu (Homeschooled) **Sponsor:** Dr. Neal Gallagher and Kristina Vuong (Socrates Preparatory School Casselberry, FL)

**Abstract: **Multiple methods are brought together here. Change of base transformation for fractions, continued products, mixed radix representations, the binary Spigot Algorithm, and repeating decimals are all used to compute a million binary digits of π, and to investigate interesting properties of repeating decimals in arbitrary bases. An Excel spreadsheet utilizes the Spigot algorithm to compute binary digits of π. A Java program listing is also included that can be used to compute a million binary digits of π.

Published electronically January 24, 2020

DOI: 10.1137/19S017621

**Authors:** Jacob Householder (Whittier College)**Sponsor:** Fredrick Park (Whittier College)

**Abstract: **Image segmentation is the task of finding salient regions of importance in an image. In this work, we take a curve evolution approach to this problem where we deform an initial curve in the inward normal direction with the objective of finding the boundaries of objects in an image. To achieve this, we propose a variational image segmentation model that incorporates a clique based shape signature with a geodesic active contours energy. The model scheme consists of evolving a parametric representation of an active contour to minimize the penalty that the model induces. This penalty is minimized when the curve is on the boundaries of objects in the image, areas with sudden change in pixel intensity i.e. light to dark. We demonstrate successful capture of illusory contours, segmentation of objects in a cluttered background, and segmentation of occluded objects.

Published electronically March 11, 2020

DOI: 10.1137/19S128274X

MATLAB Codes

**Author:** Amy Ngo (Rensselaer Polytechnic Institute) **Sponsor:** Dr. Richard Moeckel (University of Minnesota)

**Abstract: **In this paper, we aim to develop models of the action potentials of healthy human myocardial and pacemaker cells using the periodically forced Fitzhugh-Nagumo system. Pacemaker cells generate impulses which cause myocardial cells to contract, producing a heartbeat. Such impulses both cause and result from changes in membrane potential. Using eigenvalue stability analysis and the Hopf Bifurcation Theorem, we determined ranges of the two constants intrinsic to the system and the forcing amplitude for which the system has a unique, stable limit cycle. From simulations in MATLAB, we discovered that at forcing amplitudes near and greater than the maximum value which induces the limit cycle, the square wave and the cosine wave forced systems describe the behaviors of myocardial and pacemaker action potentials, respectively, with high fidelity.

Published electronically April 14, 2020

DOI: 10.1137/18S017302

**Authors:** So Mang Han, Taylor Okonek, Nikesh Yadav, and Xiaojun Zheng (St. Olaf College) **Sponsor:** Matthew Wright (St. Olaf College)

**Abstract: **Topological data analysis seeks to discern topological and geometric structure of data, and to understand whether or not certain features of data are significant as opposed to random noise. While progress has been made on statistical techniques for single-parameter persistence, the case of two-parameter persistence, which is highly desirable for real-world applications, has been less studied. This paper provides an accessible introduction to two-parameter persistent homology and presents results about matching distance between 2-parameter persistence modules obtained from families of simple point clouds. Results include observations of how differences in geometric structure of point clouds affect the matching distance between persistence modules. We offer these results as a starting point for the investigation of more complex data.

Published electronically April 29, 2020

DOI: 10.1137/19S1280211

**Authors:** Luciano Vinas (University of California, Berkeley) **Sponsor:** Atchar Sudhyadhom (University of California, San Francisco)

**Abstract: **We explore the effects of interleaved shuffling on the rate of convergence for the discrete heat equation with Dirichlet boundary conditions. We derive a closed form for the expected value of the shuffled discrete heat equation and establish bounds on its rate of convergence. In particular for any connected region D _ Zd with volume jDj and a non-negative initial state h0 2 RjDj, there is an upper bound on the spectral radius associated with the shuffled discrete heat equation that grows on the order of 1 (1=jDj1=d). An analogous lower bound for the standard discrete heat equation is also derived which grows on the order of 1O(1=jDj2=d).

Published electronically June 17, 2020.

DOI: 10.1137/19S1308475

**Author:** Jack Hughes (University of Guelph) **Sponsor:** Herman Eberl (University of Guelph)

**Abstract:** We derive a mathematical model for biofilm growth on a degradable substratum. The starting point is the one-dimensional Wanner-Gujer biofilm model. The processes included are diffusion of a dissolved growth controlling substrate into the biofilm, cell lysis, detachment of biomass, and growth from substrate conversion and substratum degradation. The resulting model is a system of two non-linear ordinary differential equations, the evaluation of the right-hand side of which requires the solution of a semi-linear two-point boundary value problem. We perform standard analysis from the existence and uniqueness of solutions to equilibrium and stability analysis. Finally, we study our model through numerical simulations and investigate how biofilms evolve on degradable substratum. We find in our setting that the addition of substratum degradation has no effect on the long-term behaviour of the biofilm, but does affect the transient behaviour.