### An Analysis of Two Methods for Closing Open Chains and Identifying Their Knotting Behavior

**Advisor:** Eric Rawdon

**Abstract:** The students analyzed two different methods for identifying knotting in open chains. They wrote and ran computer code to generate random knots and analyze the knots using the two methods. They then used statistical tools to measure to what extent the two methods agree.

**Student Researchers**: Nicole Lopez, MAdeline Shogran, Emily Vecchia

### Coarse-Grain Model for Knotted Glueball Creation

**Advisor:** Eric Rawdon

**Abstract:** The students wrote and ran computer code to generate random polygons and connect them together to generate ensembles of open and closed chains. They then analyzed how the density of the initial set of polygons affected the ensemble, in particular focusing on when knots and links were created.

**Student Researchers:** John Wallace, Lance Frazier, Joseph Spitzer

### Multi-Soliton Solutions and their Properties

**Advisor: **Alicia Machuca

**Abstract:** The study of integrable nonlinear partial differential equations is interesting to mathematicians, engineers, and physicists because such equations have physically important solutions that can be expressed in terms of elementary functions. This research project will focus on studying a certain solution formula for multi-soliton solutions of the Kadomtsev-Petviashvili (KP) equation. The solution formula that will be studied is dependent on a matrix quadruplet, (*A, M, B, C*), and can be expressed in terms of matrix exponentials. The student researchers will make use of this solution formula to examine the physical properties of these multi-soliton solutions analytically.

**Student Researchers:**** **Rey Andrade-Flores and Angel Paucar

**Research Description:** In addition to the analytical study of certain solutions the student researchers will be expected to use Mathematica to animate certain solutions and to present their final results in a written report. Students will be expected to attend a conference to present their results.

### Baseline Models of Portfolio Risk

**Advisor:** Arkady Shemyakin

**Abstract:** Fluctuation in mortgage defaults provides vital information to financial institutions and a key indicator of the state of the economy. Most of these defaults can be attributed to subprime mortgages, which are often issued to borrowers with lower credit ratings. Using mortgage default data for the decade 2001-2010 provided by a major American bank, we develop a statistical model forecasting the probability of defaulting throughout the life of a mortgage. Keywords: subprime mortgages, time-to-default, mixture models, latent class, Bayesian estimation, random walk Metropolis algorithm.

**Student Researcher:** Matthew Galloway

### Characterizing Benign and Malignant Tumors using Applied Mathematics

**Advisor: **Cheri Shakiban

**Abstract:** We will use three different methods, one based on signature curves, one based on invariant histograms and the third on fractal dimension to study images which will be provided to us by the American Institute of Cancer Research (http://www.aicr.org/). We will then apply our methods to distinguish between benign and malignant tumors.

**Student Researchers: **Anna Grim, Victor Gonzalez, and Mitch Caywood

**Final Report: Applications of Signatures in Diagnosing Skin Cancer**

### Exploring Discrete Wavelet Transforms

**Advisor:** Pat Van Fleet

**Abstract:** In this project, we consider two problems. The first is the process of lifting as an alternative method for computing the discrete wavelet transform. Lifting allows us to map integers to integers which is a requirement of transforms used to perform data or image compression. The second problem is an application of wavelets to the task of pansharpening digital images. Pansharpening is used to refine satellite images (purposely recorded at a low resolution to save space) so that a higher resolution image can be produced. This technique is used by Google Maps and other applications that utilize satellite imagery.

**Student Researcher:** Naomi Latt

### Generating Arbitrary Force Profiles via Buoyancy

**Advisor**: Jeff Jalkio

**Abstract:** Buoyant forces arise from pressure gradients in fluids and the buoyant force on a partially submerged object is proportional to the volume which is submerged in the fluid. Because of this proportionality, vertical displacements of a partially submerged object will result in variations in buoyant force. This makes it possible to generate arbitrary force profiles by varying the shape of the object being submerged.

**Student Researchers: **Amanda Tenhoff and Adam Gerenz

**Research Description: **This project has both theoretical and experimental components. After reviewing the theory of buoyancy, the student researchers will develop the necessary relationships between object profile and force and present a general solution for an arbitrary force profile. They will then solve these equations analytically and/or numerically for a number of particular profiles. For the experimental portion of their work, the students will construct an experimental apparatus for measuring force and position, fabricate their buoyant objects using a 3D printer and experimentally verify their calculations.

**Final Report:** Generating Aribitrary Force Profiles via Buoyance

### Graphs and Rings

**Advisor:** Mike Axtell

**Abstract:** We will use a variety of graph-theoretic techniques to try to better understand the algebraic structure of zero divisors in commutative rings.

**Student Researchers:** Michael Driscoll, Mitchell Klein, Alexandra Ubel

**Final Report: Zero Divisor Graphs of Commutative Rings**

### Introduction to Bayesian Estimation and Copula Models of Dependence

**Advisor: **Arkady Shemyakin

**Abstract:** The main goal of the project is to develop exercises and computer problems as supporting materials for the manuscript “*Introduction to Bayesian Estimation and Copula Models of Dependence***”** (a textbook for advanced undergraduates and practitioners). Six first chapters of the book contain the basics of Bayesian theory of estimation and copula modeling. Chapters 7 and 8 discuss several applications to insurance (joint first life and last survivor insurance, automobile warranty claims), risk management (probability of default in mortgage portfolios, credit risk models), and finance (joint co-movements of international markets). We will modify existing exercises and develop new ones, and also will develop detailed solutions.

**Student Researchers: **Matthew Galloway, Stephanie Fritz, Natalie Vandeweghe

**Research Description:** Testing and developing problem sets and computer exercises supporting the manuscript currently in preparation.

**Final Report:** Introduction to Bayesian Estimation and Copula Models of Dependence and Solutions

### Matrix Approach: The Relationship between Lagrange's and Newton's Interpolations

**Advisor:** Yongzhi Yang

**Abstract:** Lagrange’s interpolation and Newton’s interpolation are tow well-known polynomial approximations for functions. This research is the study on the relationship between Lagrange’s interpolation and Newton’s interpolation, i.e., discover matrix closed-form formulas, which connect two interpolations.

**Student Researcher:** Yi Shao

**Final Report:** Matrix Approach

### Modeling Cell Movement Using the Level Set Method

**Advisor:** Magda Stolarska

**Abstract:** This project started as part of CSUMS in 2010. During the 2010 CSUMS cohort year Matt Fox, with the help of Ben Dellaria, developed a code that allows us to simulate a biological cell moving through a series of beams, which simulates the three-dimensional extracellular matrix that cells experience in the body. All mathematical models to this point assume that a cell is moving across a flat surface, which is the case in experiments, but not in the body. This approach is significant because it is one of the first mathematical models that captures cell moving through a two-dimensional slice of the fibrous substrate the cells encounter in reality. At the end of the 2010 CSUMS cohort year, the Matlab code was well developed but still is not able to capture a cell moving through a dense fibrous network. The aim of this project is to learn about the level set method, and add additional patches to the code that will allow us to simulate cell motion through a network of fibers with arbitrary density. This will allow is to determine how the density and stiffness of the fibers affects the speed with which a cell moves through an in vivo extracellular matrix.

**Student Researcher:** Sarah Kujala

### Modeling Integrin Conservation in Cell-Substrate Interaction

**Advisor**: Magda Stolarska

**Abstract**: Cells interact with the surface on which they move via membrane-bound proteins called integrins. Activated integrins are able to bind to compounds found on surfaces, while unactivated integrins are not. The total number of integrins in the cell is conserved. By modulating activation, a cell is able to control cell-surface attachments temporally and spatially, and due to integrin conservation, the size and number of cell-substrate connections is limited. This project involves developing a partial differential equations based model describing the conservation of integrins in a cell, and their conversion between activated and inactive states. Students working on this project will use finite difference and finite element methods within Matlab to solve the model equations and simulate integrin evolution.

**Student Researcher**: Kara Huyett

**Final Report:** A Model for Integrin Binding Cells

### Propagation Time for Circulant Matrices

**Advisor:** Brenda Kroschel

**Abstract:** Students will build on the summer research of past students and will learn about the zero forcing number of circulant matrices and will explore the propagation time, and possibly throttling time, for these graphs. A goal of the summer will be to write a paper for publication that incorporates all that is known about the zero-forcing number and propagation time of circulant graphs.

**Student Researchers:** Laura Fink and Alex Bates

**Final Report:** The Zero Forcing Number of Circulant Graphs

### A Study of Sasaki's Variational Theory of Supercell Formation and Tornadogenesis

**Advisor:** Doug Dokken

**Abstract:** The students will read several papers by Sasaki and learn his theory of supercell formation and tornadogenesis. They will make code changes to the numerical weather model, ARPS, and simulate supercell thunderstorms. The code changes will help them view variables Sasaki uses to understand the processes that lead to supercell and tornado formation. They will use ViS5d to view the storms and the variables.

**Student Researchers:** Austin Swenson and Connor Theissen

**Final Report: Tornadogenesis:The Birth of a Tornadic Supercell**

### Thermodynamic Balance in Tornado Theory

**Advisor:** Misha Shvartsman

**Abstract:** While energy balance for average thunderstorm is well known, it still remains an open question how energy is redistributed on local level inside and outside tornado vortices. The project aims to investigate the energy redistribution in time and space for tornadoes. It will use theoretical and numerical modeling to calculate evolution of kinetic and internal energy of rotating vortices associated with tornado phenomena.

**Student Reserchers:** Alex Lopez, Paddy Halloran, and Tierney Dillon

**Final Report:**

### A Wavelet-Based Approach to Breaking CAPTCHAs

**Advisor:** Pat Van Fleet

**Abstract:** CAPTCHAs are Completely Automated Public Turing Test to Tell Computers and Humans Apart that are used on web sites to keep automated programs from pushing data through a form. In this research, we seek to refine and improve a CAPTCHA developed by a group of students and faculty at Grand Valley State University to break the Holiday Inn Priority Club CAPTCHA. Our method utilizes wavelet transformations along with some statistical analyses to identify letters in the CAPTCHA.

**Student Researcher:** Kathryn Wifvat

**Final Report:** CAPTCHA