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04-05 AR

# 2004-2005 Independent REU Projects

Ira Burton
Highland High School
Hometown: Salt Lake City, UT
Major: Mathematics
Year: Senior

Fall 2004 Project Proposal:

Artificial Neural Networks must be designed for the problems they need to solve. This is accomplished both by modifying the logical structure of the network, and the rules used to train them. In most cases, ANNs start with little or no information on the problems they need to solve, except perhaps the types of input and output that will be provided. It is the training rule that allows ANNs to learn their desired behaviors.

My proposed project involves creating a generalized MATLAB toolbox to aid in the design of Artificial Neural Networks. Although there is at least one commercially available ANN toolbox for MATLAB, the knowledge gained by the creation of my own will be invaluable. MATLAB has been chosen because of the ease with which ANNs can be represented using linear algebra. The toolbox will be used to implement several small ANNs that will be designed to help illustrate different network structures and learning rules. The project will culminate in the creation of an ANN that plays the gamblers' card game Blackjack.

Blackjack is chosen because of the small number of variables involved while still being an open ended game. In the game of Blackjack, all players play against a dealer who has a strict set of rules dictating their play behavior. These rules will allow for the easy creation of a rule based player that can be used as a benchmark. I will compare the performance of the ANN against a rule based player who plays with the same rules as the dealer. I expect that after a proper training period, the ANN should perform better on average than the rule based player.

At a minimum, the following resources will be utilized:
• Hagan, Demuth, and Beale, Neural Network Design, PWS Publishing, 1996
• Anderson, An Introduction to Neural Networks, MIT, 1995
• MATLAB Version 7 *Student Edition*
• LaTeX2e for report creation
This project is a one semester project that will be completed by the end of Fall 2004 semester. Upon completion of the project a report summarizing the project will be provided. In addition, all applicable MATLAB code for the toolbox and test problems will be included as attachments.

Skyline High School
Hometown: Salt Lake City, UT
Major: Honors Mathematics
Year: Sophomore
Faculty Mentor: Ken Golden

Fall 2004 Project Proposal:

Sea ice influences global climatic conditions and is an indicator for global climate changes. When using sea ice as an indicator, one of the critical variables to consider is the temperature of the sea ice. There is a well-known formula that relates the temperature of the sea ice to the brine volume fraction of the sea ice. Therefore, if the temperatur is known, the brine volume fraction can be determined. Conversely, if the brine volume fraction is known, the temperature can be calculated.

The brine volume fraction of sea ice is a one component included in the composite microstructure of the sea ice. Perhaps the most feasible way to obtain data about the composite microstructure of the sea ice would be through the use of satellites. This would allow for a non-destructive and seemingly instantaneous acquisition of data. Electromagnetic waves, such as a microwave, could be sent into the sea ice, which has a comparably small microstructure, and the overall composite microstructure of the sea ice can be determined. However, this method does not reveal any immediate information about the individual components of the composite microstructure of sea ice, such as the brine volume fraction.

Research has already been done modeling the forward problem. The forward problem is collecting data about the individual components of the composite microstructure of sea ice and accurately predicting what the response of the sea ice is when an electromagnetic wave passes through it. However, research is still needed on the inverse problem. In this case, the inverse problem uses the response of the sea ice after an electromagnetic wave passes through it and attempts to determin the amoutn of the individual components present in the composite microstructure of the sea ice. Investigating this inverse problem is an important and interesting research project that I would like to study.

Fall 2004 Final Report

Spring 2005 Project Proposal:

Sea ice is a heterogeneous material that consists of ice, brine, and air pockets. This material is arranged in a random, complex, geometric scheme, where the microstructure changes dramatically depending on the temperature. As a result, it is very difficult to determine the relative volume fractions of the different materials when a comparatively long electromagnetic wave response is predicted. The inverse problem, where the electromagnetic wave response is given and the brine volume fraction is estimated, still needs significant work.

During the Fall semester of 2004, I successfully recreated formulas used in the forward problem and recreated formulas used in the forward problem and recreated graphical displays of the forward problem.

This semester I would like the opportunity to continue working on this project. The objective this semester will be the inverse problem. This initially will consist of recreating formulas and equations used in the inverse problem, and recreating graphical displays of the inverse problem. Eventually, I would like to experiment with different electromagnetic frequencies for the inverse problem, and work with actual data on the inverse problem.

Spring 2005 Final Report

Summer 2005 Project Proposal:

Sea ice is a heterogeneous material that consists of ice, brine, and air pockets. When seawater freezes, the ice crystals separate from the salts, leaving brine pockets trapped in pure ice. The microstructure of Sea Ice can change dramatically depending on the temperature and age of the Sea Ice. These factors make Sea Ice a very complex material to analyze.

It is important to establish parameters for the physical properties of Sea Ice. One method called "remote sensing" collects data about Sea Ice from ships, planes, or satellites. By sending a comparatively long electromagnetic wave into the smaller microstructure of the Sea Ice, information about the physical properties of the Sea Ice can be determined by analyzing the electromagnetic response. This is the inverse problem. However, an analysis of the forward problem provides the mathematical mechanics used to develop the result of the inverse problem.

The forward or direct problem uses data about the brine volume fraction present, the temperature, the salinity, the electromagnetic response of the Sea Ice. Taken in cumulative steps, the forward problem is able to predict increasingly precise parameters for the electromagnetic response of the Sea Ice.

The forward problem has been analyzed for certain electromagnetic frequencies, but not others. The inverse problem also has been researched from a multitude of perspectives; however, the inverse problem still needs to be examined in a broader range of electromagnetic frequencies.

During the course of the summer semester 2005, I would like the opportunity to continue working with Dr. Ken Golden. I would like to conduct research on both the forward and the inverse problem at different electromagnetic frequencies where little or no research has been done. To appreciate this research, I am going to Alaska from May 26 to June 3 to collaborate with Dr. Lars Back strom about similar research. Additionally, I will experience going onto sheets of Sea Ice to collect samples for analyses.

Summer 2005 Final Report:

During the summer semester of 2005, I worked on two projects. The first project evaluated the effective complex permittivity of sea ice from data sets against their corresponding theoretical bounds. The second project involved retrieving the spectral measure for a network of conductors or resistors.

The second project began to unfold on August 08, when I attended a meeting with Dr. Golden, Dr. Dobson, Dr. Hyde, several graduate students and several other undergraduates. We discussed the roll various individuals would have in a new grant. I anticipate helping retrieve the spectral measure for a network of conductors or resistors. Currently, I am working to increase the processing speed of a program written by Dr. Dobson that will help compute the eigenvalues of the spectral measure in matrix form. Also, weekly group meetings have been scheduled to track and promote the progress others and I are making.

Zsuszanna Horvath
Brighton High School
Hometown: Salt Lake City, UT
Major: Mathematics
Year: Junior
Faculty Mentor: Davar Khoshnevisan

Fall 2004 Proposal

Fall 2004 Report

Spring 2005 Proposal

Spring 2005 Report

Ali Jabini
East High School
Hometown: Tehran, Iran
Major: Mathematics & Electrical Engineering
Year: Senior
Faculty Mentor: Ken Golden

Fall 2004 Project Proposal:

Electromagnetic Wave Propagation and Scattering in Composite Media

The behavior of electromagnetic waves in inhomogeneous and composite materials is determined mathematically by the Helmholtz equation. Methods for estimating and bounding the effective properties of the medium have been developed in the quasistatic limit where the wavelength is much longer than the microstructural scale. However, these methods have not been extended to the scattering regime where the wavelength is on the same scale as the inhomogeneities. I would like to work with Professor Golden on extending the bounding methods to the scattering regime. For propagation in a two component medium, Professor Golden has found a form for the Helmholtz equation which makes the analysis quite close mathematically to the quasistatic situation. I will begin by analyzing this problem in one dimension, where standard methods of calculus and differential and integral equations can be applied to investigate the behavior. In addition to bounding the effective properties of such media, we expect that the methods will apply to a very important class of materials, namely photonic crystals, which

Jeremy Morris
Westwood High School
Hometown: Austin, Texas
Major: Mathematics
Year: Senior
Faculty Mentor: Peter Trapa

Fall 2004 Project Proposal:

I would like to continue the research that I started with Peter Trapa this summer. This summer I wrote some software that constructs certain networks and calculates various statistics related to them. I planned to apply this software to data provided by the Office of Budget and Institutional Analysis. I received this data late in the summer and will now convert it into a form readable by my software. I will then calculate the characteristic path length, clustering coefficent, and the dominant eigenvector for its adjacency matrix. (The elements of the dominant eigenvector will be refered to as Google weights.)

I will then be able to answer some interesting questions, such as: Which students have the highest Google weights? What courses have the highest Google weights? Does this network qualify as a 'small world'?

During my research this summer, I noticed that when randomizing an ordered graph, the range and standard deviations of the Google weights do not vary widely. I would like to investigate the possibility of grouping vertices near most of their neighbors to restore order and aid in analyzing the relationships between vertices.

I will also study the relationship between characteristic path length and the Google weights. The Characteristic path length is the median of all mean path lengths. If the Google weights could be used to estimate which vertex's mean path length will be the median, then only the vertex's mean path length would have to be calculated. This would speed up calculations for the characteristic path length considerably.

Fall 2004 Report

Spring 2005 Project Proposal:

This semester I have two goals. The first is to introduce more reality into the model I have been using. Until now, I have been connecting all students who are in the same course. This does not represent reality, rarely do students interact with everyone in the courses they are taking, especially in very large courses. I will attempt to alter the model so that students will be connected to a small number of others in their courses. There are two ways that I have thought to do this. First, when an adjacency list is generated from the data, we could decide whether two students will be connected to each other with the same propability. Or, after an adjacency list is read into the software, we could decide how many people each student will be connected to and remove edges randomly if a student is connected to too many other students.

Mainly I am interested in how many connections will be necessary to ensure that the graph will be connected. I will also be able to re-run my previous experiments to see how this will change the affect the results from the simpler model. I will also alter the software so that it extracts the largest connected component.

My second, primary, goal is to wrap up all of my results and documentation and post them on a website. Peter and I would like to set up a website where students could enter their schedule and get their Google weight, mean path length and clustering coeffecient. This will require that I come up with some interesting algorithms too, so that the the wait time will not be excessive. I will also post this software that I have written along with some documentation on how it will be used and how others can run the same expirements on their own data.

Spring 2005 Report

Megan Morris
Skyline High School
Hometown: Salt Lake City, UT
Major: Bioengineering
Year: Sophomore
Faculty Mentor: Ken Golden

Fall 2004 Project Proposal:

Composite materials such as rocks, soils, snow, and glacial ice are pervasive on the Earth's surface. The transport properties of these composite materials play a major role in large-scale geophysical systems. The fluid transport properties of one such composite material, sea ice, play an important role in global climate as it mediates energy transfer between the ocean and atmosphere. Further, this medium houses algal and bacterial life that form the foundation of the polar oceanic food web. Because of the importance of this specific composite material, understanding the transport properties and connectivity along with other complexities of sea ice microstructure is essential and can be applied to various other composite microstructures including the human body on a small scale. My goal, in working closely with Ken Golden, is to cultivate the technology to accomplish this - to develop the basic mathematics to exploit information from composiye materials such as sea ice in order to better characterize, visualize, and model the properties of these composites.

Our research, therefore, involves the basic physical and mathematical background of geophysical and medical imaging, including wave propagation in inhomogeneous media, forward and inverse electromagnetic scattering, and characterization of composite microstructures. Specifically, we will be concerned with exactly how to use given or acquired composite microstructure data to determine the three dimensional connectivity of the material. We plan to analyze connectivity extensively, further exploring the mathematics behind critical path analysis. This basic knowledge and technique development will form the foundation for a wide variety of potential research applications.

Fall 2004 Report

Summer 2005 Project Proposal:

I would like to continue my current work with Ken. This includes further investigation of composite materials and their effective properties. It also includes studying quasiperiodic waves and modeling them with Matlab. Specifically, I am working on a Matlab code that solves the Helmholtz equation.

Summer 2005 Final Report

Andrew L. Nelson
Highland High School
Hometown: Salt Lake City, UT
Major: Mathematics & Physics
Year: Senior
Faculty Mentor: Nathan Smale

Summer 2005 Project Proposal

Summer 2005 Final Report

Jeremy Pecharich
Kearns High School
Hometown: Salt Lake City, UT
Major: Mathematics & Physics
Year: Senior
Faculty Mentor: Robert Bell

Fall 2004 Project Proposal:

Curvature plays a huge role in the connection between mathematics and physics. Physical systems change dramatically when curvature is added. The area of particular interest to me is hyperbolic space because most models of general relativity are done in hyperbolic space. Hyperbolic space is the space that has sectional curvature of -1; it is most easily visualized as being a saddle. In this space triangles can be viewed as being slimmer and their angles sum to be less than 180 degrees. Yet another way to view this space is that parallel lines diverge very quickly. However, there are different methods for comparing two spaces and deciding which system has more or less curvature.

In my research I will come up with groups, G, acting on a space X which is CAT (-1). A space, X, is said to be CAT (-1) if for every triangle in X, points in the triangle are at least as close as points in a corresponding triangle on hyperbolic space. I will study the properties of an action which is properly discontinuous and acts by isometries on X moreover X/G is compact. An action is by isometries if geodesics go to geodesics. If an action satisfies these three properties then the action is called geometric.

The most common example of a CAT (-1) group is the free product on n-letters. If we let X=tree with valence 2n and G=free product on n-letters then by covering space theory G acts on X geometrically because the fundamental group of a graph is free and every graph has a tree as a universal cover. The main tool for studying these relations will be covering space theory.

A group, G, is hyperbolic if the Cayley graph is δ-hyperbolic. A metric space (X, d) is δ-hyperbolic if every triangle in X satisfies the δ-slim property. In other words every point on one edge of a triangle belongs to the union of a δ-neighborhood of the other two sides. For example, trees are 0-hyperbolic because for every triangle in the tree each side is contained in the union of the other two sides. Another example is a surface group which is the fundamental group of a compact surface. I will explore new CAT (-1) spaces and come up with examples that are my own. I will also study examples of hyperbolic groups and show they can be made to act geometrically on a CAT (-1) space.

I plan on going to graduate school in physics and knowing about hyperbolic spaces will help me when I study general relativity and more advanced topics in physics. An REU will help me when I study general relativity and more advanced topics in physics. An REU will help me because it will give me research experience. This will be invaluable when I go to graduate school. I am currently a senior seeking a double major in mathematics and physics and intend on graduating in spring 2006.

Spring 2005 Project Proposal:

I would like to continue my research with Dr. Bell into the spring semester on CAT (0) spaces. There still remain a few questions that I did not have enough time to answer. For example, proving what homotopy equivalences of the wedge of two circles give a mapping torous that is CAT (0). I will then study low-dimensional topology, namely 3-manifolds, and the connection that they have to CAT (0) spaces. I am also going to answer the question of when the mapping torus of the wedge of three circles under a homotopy equivalence is a CAT (0) space. This question is similar but much more difficult than the wedge of two circles because it will involve a 3-simplex instead of a 2-simplex. This will also force me to learn new methods to determine whether or not this is a CAT (0) space.

This semester's REU will be very helpful to me because I am going to use it as a springboard to write my senior thesis for my honor's degree this summer.

Summer 2005 Project Proposal

Summer 2005 Final Report

Kellen Petersen
Skyline High School
Hometown: Salt Lake City, UT
Major: Mathematics
Year: Junior
Faculty Mentor: Graeme Milton

Fall 2004 Project Proposal:

Composite Cylinder Assemblages with Spiral Geometry

The aim of the proposal is to study conduction, possibly with a magnetic field present, in a new microgeometry, which has some interesting features. Composite sphere assemblages and composite cylinder assemblages consisting of a core of material and surrounded by a shell of another material were first studied by Hashin and Shtrikman in the early 1960's. They attained bounds on the conductivity of two-phase mixtures. Schulgasser, among others, generalized the model to allow anisotropic material in the coating, but with the axis of symmetry radially oriented. To our knowledge, no one has studied the case where the coating material does not have radial symmetry. In particular, we aim to study the two-dimensional case where the axis of the coating material spirals into the core, using seperation of variables to solve the partial differential equation. This could have application to proving that one can combine materials with negative Hall coefficients to obtain a composite with a positive Hall coefficient.

Spring 2005 Project Proposal:

Composite Cylinder Assemblages with Spiral Geometry Continued

The aim of the proposal is to study conduction, possibly with a magnetic field present, in a new microgeometry. In the 1960's, composite sphere and cylinder assemblages of two materials were studied by Hashin and Shtrikman and they attained bounds on the conductivity of two-phase mixtures with a core of material surrounded by the coating of another. Our goal is to study composite cylinder assemblages with a core material surrounded by a shell of another, but with a coating of anisotropic material that does not have radial symmetry. Specifically, our objective is to continue the study of the two-dimensional case where the axis of the coating material spirals into the core, using seperation of variables to solve the partial differential equation. We will then extend this to the three-dimensional case, and apply a magnetic field to the problem. We also plan on finding bounds for these materials. This could have application to proving that one can combine materials with negative Hall coefficients to obtain a composite with a positive Hall coefficient.

Nika Polevaya
Juab High School
Hometown: Obninsk, Russia
Major: Mathematics
Year: Senior
Faculty Mentor: Ken Bromberg

Fall 2004 Project Proposal:

During this year I will be working with Dr. Ken Bromberg in three-dimensional topology and geometry. The goal of this project is to explicitly construct convex cores for certain hyperbolic 3-manifolds. These manifolds come from quasifuchsian Klainian punctured torus groups. In the paper "Pleating Invariants for Punctured Torus Groups",Keen and Series have described a parameterization of these groups by the pleating locus of their convex core boundaries. In this project I will construct these convex cores when each componenet of the boundary of the convex core is pleated along a single curve and these curves intersect exactly once on the torus.

This paper requires a substantial knowledge of hyperbolic geometry and familiarity with hyperbolic 3-manifolds. During the summer of 2004 I have been striving to improve my background necessary for embarking upon the above construction by studying hyperbolic geometry. I have learned about the origins of hyperbolic plane as a natural covering space for the surface of genus 2. I have had a detailed overview of different models of the hyperbolic spcae, such as Poincare disc model, Upper Half-Plane model, the Hemisphere model, and Klein Model, as well as the usefulness of each of these models and the dependence of these models from each other (like the dependence of the Klein model on the Hemisphere model, or the dependence of Hemisphere model on the Poincare disc model).

I've studied the different operations intrinsic to the hyperbolic space such as inversions, as well as their connection to Euclidian operations such as reflections through the concept of the complex sphere. If one imagines that a line in a complex plane, through which the reflection is made, is a circle througn infinity, then that reflection is an inversion about that circle. I have also seen a generalization of the inversion in a complex 2-sphere to an inversion about a sphere of dimension greater than one. As I studied the properties of the hyperbolic lines, I have also learned about the equidistant curves and the hyperbolic metric as compared to the Euclidian metric, and the area of the ideal hyperbolic triangles.

During this year I plan to further solidify my background in hyperbolic geometry by studying in depth the "Hyperbolic Geometry" by James W. Anderson, from which I hope to gain familiarity with the Mobius group and its properties and further understanding of length and distance and area in hyperbolic plane. After that I will thoroughly study the Keen and Series' paper and try to explicitly construct the boundaries of rational laminations.

Fall 2004 Report

Spring 2005 Project Proposal:

2 D Case: I hope to complete the construction of a general right-angled hyperbolic hexagon with three arbitrary sides a, b, c (as the other three will be determined by these three) that I have started in the Fall semester. As that task is completed, I will use specific side-pairing transformations to construct a pair of pants (which will explicitly provide a hyperbolic structure on a pair of pants).

3 D Case: After that I will embark upon a construction of certain hyperbolic octahedrons. These octahedrons turn out to be fundamental domains for certain Kleinian punctured torus groups. An additional motivation for the problem is provided by the paper "Pleated Invariants for Punctured Torus Groups" by Linda Keen and Caroline Series. To gain further insight into the problem, I will thoroughly study this paper.

Spring 2005 Report

David Seal
Judge Memorial High School
Hometown: Salt Lake City, UT
Major: Mathematics
Year: Junior
Faculty Mentor: Nathan Smale

Spring 2005 Project Proposal:

In the Fall of 2004, I completed an introductory course on applied Partial Differential Equations. I am interested in exploring PDE's (Partial Differential Equations), and in particular the Laplacian from a more theoretical perspective including three major branches of mathematics: Real Analysis, Complex Analysis and Fourier Series Analysis. The Laplacian has many applications, ranging from fluid dynamics to electricity and magnetism and therefore is an important topic to study.

I propose beginning my REU with an analysis of Laplace's equation in Euclidian space. These will be equations of the form:

a) del2(u)=0 on ball of radius r in Rn

b) u=f; some given function on the boundary of ball of radius r in Rn

c) del2(u)=g on interior of ball radius r in Rn

Partial Differential Equations of the form (a)-(c) are known as Poisson's equation. By taking a superposition of functions with zero boundary conditions that satisfy (c), along with those that satisfy the Laplacian, one can solve any PDE of the form (a)-(c), for "well behaved" functions f and g. In my applied PDE's course, we explored Poisson's equation in R2 for discs and rectangles, I would like to explore Poisson's equation on other surfaces.

Complex analysis will play an important role in my study of PDE's because functions that have a complex derivative are unique in the sense that they automatically satisfy :aplace's equation on whatever region thay are analytic (or have a complex derivative). Conformal mapping theorems allow one to take a function which is analytic on desired complicated region and map it onto a simpler region such as a ball or rectangle in order to readily solve it.
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