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.