SIAM Undergraduate Research Online

Volume 13

SIAM Undergraduate Research Online Volume 12

Periodic Properties of Expansions for Fractions into any Base

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.

Geodesic Active Contours with Shape Priors for Segmentation, Disocclusion, and Illusory Contour Capture

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.

The Fitzhugh-Nagumo System as a Model of Human Cardiac Action Potentials

Published electronically March 11, 2020
DOI: 10.1137/19S128274X

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.

Distributions of Matching Distances in Topological Data Analysis

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.

Bounds on Rate of Convergence for the Shuffled Discrete Heat Equation in Zd

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).

A Mathematical Model of Biofilm Growth on Degradable Substratum

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.

Stochastic Automata Networks and Tensors with Application to Chemical Kinetics

Published electronically July 28, 2020
DOI: 10.1137/20S1316263

Authors: Marie Neubrander (University of Alabama)
Sponsor: Roger Sidje (University of Alabama)

Abstract: Often, given a system of biochemical reactions, it is useful to be able to predict the system's future state from the initial quantities of the involved molecules. There are methodologies for developing such predictions, ranging from simple approaches such as Monte Carlo simulations to more sophisticated higher-order tensors and stochastic automata networks. Many revolve around solving the chemical master equation that arises in the modeling of the underlying biochemical kinetics. This work considers the case of dealing with the resulting high-dimensional data and shows how tensor representations allow us to cope with the \curse of dimensionality" that significantly complicates such problems. A key outcome in this work is the demonstration of the inherent differences and similarities between two prominent modeling methods, by computational examples on one hand and a mathematical proof on the other hand. Applications where biochemical reactions occur are found in a variety of scenarios, including enzyme kinetics and genetics. Tensor-based solutions may have applications in dealing with many other high dimensional data outside of strictly chemical reaction systems.