VIGRE: Former Undergraduate Research Experiences

Alicia Allen
Geoff Goehle
Edith Hand
Matt Hayden
Kristine Henderson

Deborah Hinck
Jeremey Horgan-Kobelski

Kevin Leder
Holly Lewis
Grant Macklem
Nancy Mezo
Josh Nolting

Kenzie Parton
Jonathan Peeters
Tye Rattenbury

Jocelyn Renner

Ashley Singer
Mark Snyder
Saverio Spagnolie
Elaine Spiller
Robert Thornton
Josh Wells
Stefan Wild

* Dates of participation in VIGRE are given in ().

Alicia Allen (Summer 2000)

Advisor: Jim Curry

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Geoff Goehle (Summer 2001-Summer 2002)

Fractals that Are Produced by Running Certain Variations of the de Rahm-Chaikin Curve Smoothing Algorithm

This algorithm is an iterative procedure that acts on a n-sided polygon to produce a 2n-sided polygon. Normally the polygons produced by the de Rahm-Chaikin converge to C¹ curves. Under certain conditions, however, the algorithm creates self-affine fractal sets. He studied such issues as the convergence of an initial polygon to is limiting fractal, how the fractals change over their parameter space, the issue of fractal length and dimension, and other properties held by this class of self-affine sets.

Advisors: Jim Curry & Anne Dougherty

Geoff graduated from CU-Boulder in May 2002 with a Bachelor of Science in Applied Mathematics. He is currently a graduate student at Portland State University.

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Edith Hand (Fall 2000-Spring 2001)

Automated Signature Verification System

Currently, my research is focused on an automated signature verification system. Security is an important issue, and people often prefer to use signature verification over more obtrusive methods such as fingerprinting and retinal image scanning or less personal methods like PIN numbers and passwords. The time and effort involved in human signature verification is unacceptable in situations where hundreds of signatures need to be checked daily, such as in banks and credit card companies. I am working with code for automated signature verification developed by Professor Ben Herbst. The code is currently written partly in Matlab and partly in C. I hope to develop a program written entirely in Matlab. This code will be used as an example on automated signature verification to be included in a chapter of a modeling textbook being developed by Professors Fornberg and Herbst.

Advisor: Bengt Fornberg

Edith graduated from CU-Boulder in May 2003 with a Bachelor of Science in Applied Mathematics. She is currently working at Lockheed Martin.

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Matt Hayden (Summer 2000-Fall 2000)

Modeling and Analysis of Natural Chemical Systems

My VIGRE appointment was to assist Eric Wright on the modeling and analysis of natural chemical systems. This is a significant field in applied math because certain chemical reactions have lab kinetics that do not follow the model for chemical equilibrium. By studying the stability of a generalized kinetic system, we hope to develop a theory of conditions surrounding its behavior. Understanding the stability of kinetics systems could resolve issues in environmental chemistry and biology. One such problem is the estimation of atmospheric-aquatic reaeration or mixing in the nitrogen cycle. This could be a major contribution to biogeochemistry and analysis proves that, in this case, experimental modeling based on generalized kinetics stability is preferred over biological modeling. While the long-term behavior of our test case, the carbonate system, remained stable, some reasonably natural conditions yielded unexpected short-term behavior. Considering that systems in nature rarely occur with as few as our test case (six variables), more sophisticated models may have similarly unanticipated behavior.

Advisors: Congming Li & Eric Wright

I plan to enter graduate school in biomath, biostatistics, or epidemiology.

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Kristine Henderson (Spring 2003)

Scott the Baker

There are many different steps involved in the process of making bread at the Great Harvest Bread Company. One of the first steps is to mill the wheat, which produces flour. The grinding of the stones used in the mill adds heat to the flour. Before the flour can be used, it must be returned to room temperature. Therefore, Scott the Baker places the flour into large buckets with five copper tubes, which help speed up the cooling process. Mathematically, the cooling of flour over time can be described using partial differential equations. We will discuss how Fourier series and heat transfer ideas and methods can be used to model and analyze the temperature of the flour. We will also discuss the mathematical and physical significance of the copper tubes and what other steps could be taken to further speed up the cooling.

Advisors: Jim Curry & Adam Norris

Kristine plans to work in Denver as a financial analyst in investment banking or GMAC Commercial Mortgage.

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Deborah Hinck (Spring 2001 & Fall 2001)

Image Enhancement for Fingerprint Identification

Personal identity recognition is a crucial part of today's society. We rely on identity verification to protect personal information, find and prosecute criminals, and identify missing persons. The FBI uses a digital database of over 300 million fingerprint images to make identifications. It would be impossible for one person or even a team of people to search through a database of 300 million fingerprints; a matching program must be used to search through the database and return possible matches for an unidentified print.

A variety of image enhancement techniques have been applied to digital fingerprint images, but an optimal method has not yet been determined. This is largely due to the fact that a metric to quantify the quality of the enhancement does not exist. In this talk, I will be discussing the development of a set of standard images that represent fingerprints, methods of noise addition that model common imperfections in fingerprint images, and the non-linear thresholding image enhancement technique.

Advisor: Jim Curry & Anne Dougherty

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Jeremy Horgan-Kobelski (Spring 2000-Spring 2001)

Bifurcations In a One Parameter Family of Cubically Convergent Iterative Root Finding Algorithms

My research is focused on bifurcations in a one parameter family of cubically convergent iterative root finding algorithms first studied by Hansen and Patrick in a 1976 Numeriche Mathematick article. The goal of the Hansen-Patrick article was to obtain local convergence results for the family. The family includes such well known members as Newton’s and Euler’s methods. The goal of the current research effort is to ask and answer questions that are more global in nature. Specifically, we consider the global bifurcations that occur for a fixed family of polynomials and the parameter dependent Hansen-Patrick varies. Because of desirable symmetry properties, we are examining the cyclotomic polynomials of several degrees. Fiedler and Curry (1988) showed that for this family there is global bifurcation producing an attracting period two cycle. They also noted, using computer graphics, that the cycle seemed to be unstable under perturbation. To date we have confirmed much of that previous work and are now examining how basin boundaries change as various members of the Hansen-Patrick family are encountered. It is unclear, for example, how straight boundaries associated with Laguerre’s method become fractal in nature.

Advisors: Gareth Roberts & Jim Curry

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Kevin Leder (Fall 2000-Spring 2001 & Summer 2002)

Forward and Exact Inverse Computation of Digital Data

Kevin Leder is working with M. Ablowitz on methods employing the forward and exact inverse computation of digital data using reversible fully discrete nonlinear evolution equations. A special case of the evolution equations investigated reduce to well known linear feedback shift registers: LFSR's. LFSR's and various nonlinear versions, NLFSR's, have been used in a variety of applications such as encryption and coding theory.

Advisor: Mark Ablowitz

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Holly Lewis (Summer 2000-Spring 2001)

Sierpinski-Related Fractal Images Using Image Processing and Linear Algebra Techniques (with Josh Wells)

Holly and Josh analyzed Sierpinski-related fractal images using image processing and linear algebra techniques. The Sierpinski fractal images are generated using an iterative tiling technique and they are represented as matrices in Matlab. Using the Singular Value Decomposition (SVD) in the application of a modified Eigenface technique to our fractal images, we have proven the existence of 456 unique images out of the possible 512 Sierpinski relatives. Also using the SVD, we discovered a scaling law for the singular values that is applicable to a subset (56 of the 512 images) of the Sierpinski-related fractals. In addition, we found that this scaling law is preserved, in a fashion, under the two-dimensional non-standard Haar wavelet transform. We also investigated the use of other scaling factors in the reconstruction of the SVD for the subset of images named above. We plan to examine in further detail the images not in the subset of 56. Finally, we will apply other linear algebra decomposition methods to the entire set of fractal images in the anticipation of determining an improved classification system.

Advisors: Jim Curry & Jim Meiss

Holly graduated from CU-Boulder in May 2003 with a Bachelor of Science in Applied Mathematics. She plans to continue her education with a Master’s degree.

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Grant Macklem (Fall 2000-Fall 2001)

Denoising Invariant Linear Transformations with Translation

Until recently, scientists have been unable to get clear pictures of small cellular objects. Using an electron microscope, electrons are passed through a three-dimensional sample and their energies are collected on the other side. The energy is analyzed and converted into an array of images representing slices of the object. These images are much clearer than before, but scientists would still like to make the images more clear. Traditional wavelet denoising has been successful, but it can be improved. The improvement comes from denoising with translation invariant linear transformations. Basically this amounts to finding all transformations of the image under all shifts of the image, denoising, averaging all the shifts, and transforming the average back to an image.

Advisor: Greg Beylkin

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Nancy Mezo (Spring 2002-Fall 2002)

Modeling the Transport of Solute in Water Flowing through a Fracture in a Porous Medium

The fracture is represented by two infinite parallel plates. The water flow through the plates is assumed independent of the solute's transport within the water. The concentration of the solute in the water is modeled by a (two-dimensional) partial differential equation (PDE). This PDE describes the movement of the solute due to the water's flow (described by Poiseuille's velocity profile) and the molecular advection and diffusion of the solute. The two-dimensional model describes the solute concentration as it evolves in time. This can be simulated numerically by the Random Walk Particle Method. Results from this simulated model will be shown. These results are compared to a one-dimensional model given by Taylor in 1953. Taylor's model describes the solute's average concentration. It is a one-dimensional PDE, which can be solved analytically. The simplicity of his model makes it highly desirable as a tool to predict the average solute concentration in time. This one-dimensional PDE can be broken down into two components. The advection half resembles half of the wave equation and can be solved using the Method of Characteristics. The diffusion half resembles the heat equation and can be solved using Fourie transforms. The later uses the principle of superposition and Green's theorem. The problem can be non-dimensionalized to make experimental design easier.

Advisor: Brian Bloechle

Nancy graduated from CU-Boulder in December 2002 with a Bachelor of Science in Applied Mathematics. She is currently looking for employment.

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Josh Nolting (Fall 2002-Spring 2003)

Exploring and Implementing Finite Element Methods

The presentation will encompass the process of learning and implementing finite element methods, beginning with brief descriptions of Galerkin and First Order Systems Least Squares (FOSLS) methods. The presentation will then detail a high level decomposition of the implementation procedures and describe complexities associated with these two methods. Finally, enhancements to an already implemented partial differential equation (PDE) solver will be discussed. The PDE solver (FOSPACK)* uses FOSLS method and is heavily used by the research group members. In order to make the solver extensible to a larger number of problems, the accuracy and generality needs to be improved. This means modifications to the program. A focus of the presentation is one of these modifications. It includes a module that formulates element node basis functions, derivatives to these basis functions and Gaussian quadrature node locations for varied element spaces and integration accuracy.

Advisors: Thomas Manteuffel & Stephen McCormick

Josh graduated from CU-Boulder in May 2003 with a Bachelor of Science in Applied Mathematics. He will be continuing with graduate school in Applied Mathematics at CU-Boulder in the fall.

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Kenzie Parton (Spring 2002)

The Shortest Path Between Two Points

Finding the distance between two points is a problem that comes up often in the world of mathematics. In analyzing this problem, looking at graph theory provides some useful results. Another approach to this problem is to look at the matrix form of the system where the value of a connection is placed in each position where two connected points are represented. A third approach to solving this problem is by using Dijkstra's Algorithm. Through the development of each of these algorithms, the shortest distance between any given points in a system will be easily determined. These algorithms will be useful when applied to many real world situations, including determining the distance between any two points on a sphere.

Advisor: Panos Panayotaros

Kenzie graduated from CU-Boulder in December 2002 with a Bachelor of Science in Applied Mathematics. She is currently looking for employment in San Francisco.

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Jonathan Peeters (Summer 1999-Spring 2000)

A New Method for Solving Laplace's Equation Using FFT

Given a surface profile of a water wave and the initial velocity potential values along the surface, one can advance the wave in time using conservation laws (Laplace’s equation), pressure constraints (Bernoulli’s equation), and also the claim that particles on the surface remain on the surface. Because the surface uniquely determines the solution of Laplace’s equation on the interior one can simply follow the evolution of the surface in time. Given the advantage there has been no shortage of methods developed to numerically analyze the propagation of unsteady gravity waves. For our purposes we decided to use a conformal mapping scheme initially developed by Fornberg. This efficient technique maps the deep-water wave surface to a unit disk where Laplace’s equation is easily solved by FFT. The problem with the method as is, is that due to its low accuracy in time attempts to observe a water wave for a long period of time
(40 s) become unwieldy. To avoid this difficulty, I combine a technique that Dold used with Fornberg’s code.

Advisors: Mark Ablowitz & Bengt Fornberg

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Tye Rattenbury (Summer 1999)

Iterative Dynamics - Analyzing Simple Non-Linear Equations and Various Root-Finding Methods

Beginning with no background in research and very little in dynamics, I spent the summer reading articles and introductory books and building the tools to explore applied mathematics. Most of the articles centered on iterative dynamics-analyzing simple non-linear equations and various root-finding methods. On the tools building side, I began experimenting with computer graphics. Starting with just GL functions on Silicon Graphics computers, I wrote simple programs that drew the famous Mandelbrot Set and the corresponding Julia Sets. Further readings inspired programs for Newton’s Method applied to simple polynomials

P_n(z) = zⁿ — 1

Eventually, these graphics programs were successfully converted to openGL function calls, allowing an executable program to be created on most operating systems.

Advisors: Jim Meiss & Jim Curry

Currently enrolled in a PhD program in Computer Science at UC Berkeley.

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Jocelyn Renner (Fall 2001-Spring 2003)

Mathematical Model of Dispersion Using Lyapunov Exponents

Understanding the dynamics of systems involving dispersion of passive scalars is important in many fields of engineering and science. Dispersion can include the transport of particles such as contaminants or represent a property of the system such as heat. Ocean dynamics, weather patterns, engines, reactors, and rocket boosters all undergo mixing in different forms, and understanding the dynamics behind these systems is important for prediction of the behavior of the system. Mathematical models provide valuable insight into the complex transport processes that can occur in physical and engineering systems. Good models allow scientists and engineers to more accurately predict ocean currents and weather patterns as well as allowing them to build more efficient reactors and engines. With Dr. Keith Julien and Paul Mullowney, I am studying ways to numerically analyze dispersion in dynamical systems using Lyapunov exponents.

Advisor: Keith Julien

Jocelyn graduated from CU-Boulder in May 2003 with a Bachelor of Science in Applied Mathematics. She will be entering graduate school at Northwestern University in the fall.

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Ashlie Singer (Summer 2000-Summer 2001)

Applying the Short Pulse Asymptotic Approach to Maxwell’s Equations

The goal of this project is to apply to Maxwell’s equations the short pulse asymptotic approach developed by Alterman and Rauch. Maxwell’s equations are of particular interest to ultra-fast laser experimentalists, who produce the short pulses described by this theory. First Ashlie learned some background material, studying multiple scale expansions and basic electromagnetics relating to Maxwell’s equations. Then she derived a new set of profile equations that describe the propagation of a short pulse governed by Maxwell’s equations. We have begun the next stage of the project, also to be funded by VIGRE, where we will study analytically and numerically the behavior of solutions determined by the short pulse profile equations and compare these solutions to those arising from a traditional slowly varying amplitude approximation.

Advisor: Deborah Alterman

Ashlie plans to join the Air Force, and get a masters degree in Applied Mathematics.

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Mark Snyder (Fall 1999-Summer 2001)

Creating Discrete One-Dimensional Dynamical Systems that Satisfy Benford’s Law

If a data set satisfies Benford’s Law, then it’s significant digits will be distributed according to a logarithmic density. Our work involves creating discrete one-dimensional dynamical systems that satisfy Benford’s Law. We construct dynamical systems satisfying Benford’s Law that are conjugate to other dynamical systems that have been studied intensely (like the logistic map or Newton’s method). The dynamical systems we create inherit the measures of ergodicity from the systems to which they are conjugate. We also generate dynamical systems that satisfy Benford’s Law by solving the inverse Frobenius-Perron problem for a given density. We are able to calculate the Lyapunov exponents for these new dynamical systems. In addition to these exact results, we use Monte Carlo methods to analyze other dynamical systems. Because of the many physical data sets satisfying Benford’s Law (tax data, geophysical data, statistics from diffusion of gasses) we believe our work may have applications in mathematical modeling.

Advisors: Jim Curry & Anne Dougherty

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Saverio Spagnolie (Fall 1999-Summer 2001)

Numerical Studies of a Non-Analytic Hamiltonian PDE

Many conservative partial differential equations are actually Hamiltonian systems. When this energy level is conserved over time (in finite dimensions), the Poincaré recurrence theorem states that solutions to the partial differential equation must be recurrent in time. The aforementioned research project is to build a computer program capable of determining, quickly and accurately, the recurrence times for these solutions while preserving the Hamiltonians of the partial differential equations. The program has already shown itself accurate in analyzing recurrence times for the non-linear Schrödinger equation. Similar equations are currently being tested using exact solutions as comparisons to numerical solutions. When the program is shown accurate in these cases, it may then be effective in determining the recurrence times of non-exact solutions to the equations similar to the non-linear Schrödinger equation.

Advisors: Harvey Segur & John Carter

Currently a graduate student at Courant.

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Elaine Spiller (Summer 1999-Spring 2000)

Determining How the Four-Wave-Mixing Terms Propagate.

The growth of the Internet and computer communications has put a serious strain on the long distance telecommunications industry. This need for space has generated new technology known as wavelength division multiplexing (WDM) which adds a tremendous data carrying capacity to fiber optic cables. WDM allows transmission of data on several different wavelengths through the same fiber, similar to radio stations sending out different signals on different frequencies. Solitons are a kind of "large" light wave, which work well with WDM. Solitons are advantageous in the fact that when two solitons collide, they retain their original characteristics with only slight spatial shifts.

Soliton propagation is modeled by the nonlinear Schrödinger equation

iu_z + ¹/₂u_tt + |u|²u = 0.

Unfortunately, a collision by waves of two different frequencies say, u₁ and u₂, generates two new waves knownas four-wave-mixing terms. Although these terms are not as large as the original waves, they add undesired noise to the system. I have numerically solved the linear Schrödinger equation and I am in the process of solving the nonlinear equation in order to determine how the four-wave-mixing terms propagate.

Advisor: Mark Ablowitz & Rudy Horne

Elaine is currently at Northwestern in Applied Math.

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Robert Thornton (Spring 2002)

The Trig Identity of the Future

An effort to discover ways to reduce the cost of working with functions of several dimensions by using separable functions produced an unexpected discovery: a new trigonometric identity that allows us to write the sine of a sum of N variables as a sum of N separable functions. We are considering what happens if we take a sine of a sum of N distinct variables, such as sin(x+y+z) and multiply it by a sine of a sum of N different variables, such as sin(u+v+w). Simply multiplying out would result in a sum of N^2 terms, but we can now write it with 4N terms and it may be possible to reduce the sum to 2N, or even N terms.

Advisor: Martin Mohlenkamp & Lucas Monzón

Robert graduated from CU-Boulder in May 2002 with a Bachelor of Science in Applied Mathematics.

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Josh Wells (Summer 2000-Spring 2001)

Sierpinski-Related Fractal Images Using Image Processing and Linear Algebra Techniques (with Holly Lewis)

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Stefan Wild (Spring 2001-Summer 2002)

Seeding Non-Negative Matrix Factorizations with the Spherical K-Means Clustering

Over the past year I have continued my research in the areas of applied linear algebra and statistics under professors James Curry and Anne Dougherty. My research at Colorado has culminated in my Master's thesis entitled "Seeding Non-Negative Matrix Factorizations with the Spherical K-Means Clustering" which I successfully defended in April. In July I will be presenting a portion of the work at the SIAM International Conference on Applied Linear Algebra in Williamsburg, VA. In addition to these proceedings, we are currently in the late stages of preparing a paper for the Journal of Pattern Recognition. Much of the two and a half years I spent investigating this topic were under VIGRE support and I am truly grateful that I was given opportunity the do cutting edge research in applied mathematics as an undergraduate student. My experiences have thus far been encouraging and I will be venturing off to Cornell University in Ithaca, NY next fall to pursue a PhD in Operations Research.

Advisors: Jim Curry & Anne Dougherty

Stefan will begin graduate school at Cornell in Fall 2003.

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revised June 2003