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04-05 AR

# 2005-2006 Independent REU Projects

Tim Anderton
Hometown: Sandy, Utah
Majors: Mathematics & Physics
Faculty Mentor: Don Tucker

Summer 2006 Project Proposal:

The topic of perfect numbers has been the object of study in number theory since time immemorial. Perfect numbers were the object of a theorem by Euclid and were known to and studied by the Pythagoreans. A perfect number is a number whose factors sum to twice itself. The smallest and best known perfect number is 6 (1+2+3+6=12=2*6).

Euclid's proof showed that all numbers of the form 2^(n-1)*(2^n-1) are perfect numbers when 2^n-1 is prime. It wasn't until Euler that it was proved that in fact all even perfect numbers are of this form. It has been the prevailing wisdom that all perfect numbers are even and that odd perfect numbers do not exist. The majority of research on odd perfect numbers has been to put constraints on their existence either in an attempt to show that they do not exist or to push the lower bound of their smallest member ever higher. An organized internet search at oddperfect.org has pushed the lower bound of existence of an odd perfect number out to 10^500. That odd perfect numbers would be very large and that they would be very rare if they do exist is manifest. Despite the various restrictions on their existence, the possibility of an odd perfect number remains.

Pushing the lower boundary of their existence upward from the region of small numbers up into ever larger numbers runs into the possibility that the first odd perfect number might exist in a region of numbers which cannot be comprehensively searched by current computer technology if one exists at all. That the first odd perfect number might be in the range of 10^10^10 is not beyond consideration.

The first odd abundant number (a number whose factors add up to more than twice itself) is 945 whereas the first even abundant number is 12. If odd perfect numbers were to be as rare as even perfect numbers and start later proportionally to the late order start of odd abundant numbers compared to even abundant numbers then the first odd perfect number would appear unimaginably far out into the number field. It is known that among numbers in the regions of up to millions of digits there exist only 43 even perfect numbers. If as with abundant numbers the first odd perfect number does not appear until well after the hundredth such number it would most likely be billions or trillions of digits long. Because of the possibility that the first odd perfect number might be extraordinarily far out on the number line I would like to make a computer program which will search for odd perfect numbers as large or larger than 10^10000. Because of the vast size of the numbers under consideration simply manipulating them in simple ways can take a lot of computer time. The program would be constructed using a freeware C++ library Pari/gp which was designed for use in computational number theory. Eventually such a program might be expanded into a program capable of cooperation over networks in order to increase computational power.

Because this program is not in any way intended to make a comprehensive search of the number line up to 10^10000 one would be free to utilize heuristic methods to maximize the chances of success of the search. By focusing on likely candidates and showing no concern for whether or not the number under consideration is the first odd perfect number a gate is opened onto the number line past where searches have gone before and where those searches are not likely to go soon. The hope is that by giving up comprehensive scope for range a spot search might succeed where a comprehensive search might loose momentum.

Summer 2006 Final Report

David Arcilesi
Hometown: Sandy, Utah
Majors: Mathematics & Physics
Faculty Mentor: Ken Golden

Spring 2006 Project Proposal:

Effective Properties of Quasiperiodic Materials
Quasiperiodic materials have many unique and distinguishing properties. I plan to study quasiperiodic materials, trying to better understand the band gap structures and the propagation characteristics of such media. I hope to study many of their characteristics of such media. Then, I'll model these phenomena mathematically. Perhaps, I'll use Matlab to solve these equations and model the effective properties numerically or inclosed form.

Spring 2006 Final Report:

Minimizing the Free Energy of "Metallic" Spheres in ER Fluids
This past semester, I began to study with Dr. David Dobson the free energy minimization of "metallic" spheres in ER Fluids. To do this, a computer program was developed by Dr. David Dobson that calculated the free energy of a system of "metal" spheres in oil. The calculation of energy was done after the program attempted to move each "metal" sphere in such a way that it would result in a lower free energy for the system. So, in a given iteration, if a certain particle could move in such a way as to lower the energy of the system then the particle would move. If no such move was available for a certain particle then that particle would remain in its original position. This process was repeated a certain number of times (as determined by the operator). We did many simulations using different parameters such as: volume fraction (area of spheres/total area of space), number of particles, particle radius, and dielectric coefficient of the host material as well as that of the particles themselves. An interesting result was achieved, in which, we saw the "metal" spheres form chains in nearly the exact direction as the simulated electric field. This is exciting because it takes us a step closer to finding characteristics of configurations which mimimize the free energy of the system.

Charles Cox
Major: Mathematics

Summer 2006 Final Report

Ginger Dobie
Hometown: Boise, ID
Major: Mathematics
Faculty Mentor: Nick Korevaar

Fall 2005 Project Proposal:

Investigations Related to the True Body Mass Index

Becasue the well-known B.M.I. uses the wrong power of height, special tables indicating the "normal" B.M.I. values are constructed for children of different ages. Also, the fact that very tall athletes tend to have B.M.I. values in the overweight range despite not actually being overweight, is often explained by their high muscle density rather than as an artifact of the wrong power law. We propose to study the extent to which the use of a more correct power law allows one to create a more universal B.M.I., valid for all heights (and ages). This portion of the project should be relatively straightforward. A more ambitious goal would be to access the same data which was used to validate B.M.I. values as predictors of health risks, and to see if a more correct power leads to a better correlation between extreme B.M.I.s and bad health outcomes. (Of course, a literature search shall first be done to see the extent to which this natural question has already been studied.)

A much deeper question which should be pondered, is whether there is a theoretical reason to explain Quetelet's empirical power law!

Fall 2005 Final Report:

The currently popular "Body Mass Index" is used as an indicator of whether people are overweight and underweight. Lower BMI limits are prescribed for children than for adults, and higher values are used for very tall people, making this index non-universal. I tried to find a different power law relating weight to height than the one implicit in BMI, in the hope of creating a universal body mass index for people of all sizes. I found, unfortunately, that no matter which power one uses with age-graded charts relating height to overweight, there is still too much age variation for a truly universal law.

A large part of my time was spent searching the literature on body mass index and health risks, to see what is currently known and if other people had tried to relate BMI and variants to health risks. We read several papers on this subject. Although there have been a few studies which compare different powers p in the weight to height formula, W = hp, as prototype universal BMI candidates, typically these studies restrict to the integer powers p= 1, 2, 3. There are a large number of studies which relate BMI to health risks, using the standard power of p=2, however.

An additional part of my project was to attempt a statistical analysis on actual morbidity data to find which power law would be most effective as a health risk indicator. There does not appear to be any such study in the literature. As I searched to find data sets, I found that there is a lack of publicly accessible databases. We ended up running into dead ends every time we tried to find data to work with. Due to this lack of data we were unable to test modified body mass indices as health risk predictors.

Patrick Harris
Hometown: Salt Lake City, UT
Majors: Mathematics, Physics, & Theatre
Faculty Mentor: Klaus Schmitt (Fall 2006) & Sasha Balk (Spring-Summer 2006)

Fall 2005 Project Proposal:

Beniot B. Mandelbrot wrote an article for the February 1999 Scientific American titled "A Multifractal Walk down Wall Street." In it, Mandelbrot addressess a possible method for modeling the financial wellbeing, or lack thereof, of companies via the stock market. Mandelbrot criticizes portfolio theory and convincingly advocates the use of multifractals as a means to model the stock market.

In the mouths following the release of this issue of Scientific American, several individuals entered comments as "Letters to the Editor." The general consenses was that Mandelbrot had claimed he had formulated the idea of using fractals to model the stock market when, in fact, R. N. Elliot had used fractal-like properties for his models of the stock market in the late 1930s. The outrage of the commentators spawned from the fact that Mandelbrot failed to give credit to Elliot's work. It seemed that Mandelbrot, who had been familiar with Elliot's work, blatantly ignored it and took the credit for himself.

My research project would be to research and compare Mandelbrot's multifractal modle to Elliot's Wave principle. If, in fact, they are similar, Mandelbrot would be to blame for his apparent arrogance. However, if Mandelbrot's model is innovatively unique, then his ideas are an original, creative creation. The scope of this project would be kept simply to become thoroughly familiar and fluent with Mandelbrot's and Elliot's ideas as well as portfolio theory. Then to extrapolate and interpret the validity of the commentators' claims that Mandelbrot had plagiarized Elliot's work.

Fall 2005 Final Report

Spring 2006 Project Proposal:

My REU project that I would like to research would be ocean currents. I would like to focus on the actual differential equations, or differential equation forms, that model currents. My research would also cover the effects of outside influences such as global warming, to determine how drastic small temperature changes can be. Temperature would only be one such environmental condition to consider. Other might include pollution, concentration of fresh/salt water, and, if it's prudent, the movement of the tectonic plates creating differently shaped oceans.

The main goal will be to show the chaotic effects slight temperature changes can have on delicate ocean currents by using differential equation models and accurate initial data. However, if time permits, I would like to examine the models to see the effects of other outside conditions.

Spring 2006 Final Project

Summer 2006 Final Project

Ryan Johnston
Hometown: Taylorsville, UT
Faculty Mentor: Lajos Horvath

Fall 2005 Project Proposal:

Individuals receive a credit score according to factors (variables) having to do with how well their finances are maintained. These factors are used to develop an algorithm that produces a credit score that has finitely many values. The following model is used to describe the credit scores of company i: The company is moving between credit scores like a Markovian Process with transition matrix Ai. We assume that we can observe n movements of this company. It is assumed that the transition matrices A1, A2,..., An are independent identically distributed matrices. It is also assumed that the companies are moving between credit score stages independently of each other. This means that conditionally on the transition matrices we have N time series of length n. We would be interested in estimating the common expected value of Ai. Typically n is much smaller than N, if insurance data is used then n = 10 and N = 300,000.

Of course the individual time series of companies and customers are rarely available due to security and confidentiality. Credit agencies and government agencies are, however, reporting average scores for sectors and specific industries (ie-transportation, retail). This means that we observe an aggregated data. The problem with the aggregated data is that it is a very short time series based on the average extremely large number of observations.

The goal of my project is to study and find a method of how to estimate the expected transition matrix from aggregated data, if it is at all possible. It is claimed in the literature that applying standard methods to aggregated data will lead to incorrect conclusions. I would like to study theoretically as well as empirically estimation theory with aggregated data.

Fall 2005 Final Report

Spring 2006 Project Proposal:

I wish to continue with my project from last semester. To study theoretically as well as empirically estimation theory with aggregated data. I will be exploring this idea using the first order autoregressive model when the innovations in the linear error term are heteroskedastic. The results of my research will determine whether or not applying these standard methods of estimation on aggregated data will lead to accurate conclusions. The literature claims that, although these methods are accurate for individual models, they lead to incorrect conclusions for aggregatted data.

Yuliya Kovalenko
Hometown: Donetsk, Ukraine
Major: Mathematics
Faculty Mentor: Jingyi Zhu

Fall 2005 Project Proposal:

I am interested in applying probability principles for calculating the risk faced by investors in government and corporate liabilities by analyzing the market information (e.g. telling what is the probability of the particular company to default).

Fall 2005 Final Report

Neil McBride
Hometown: Sandy, UT
Major: Mathematics & Economics
Faculty Mentor: Jingyi Zhu

Spring 2006 Project Proposal:

Mathematical modeling in credit risk
The goal of research in this area is to quantify the risk faced by investors in government and corporate liabilities. Intuitive but also sophisticated mathematical theories are developed to obtain quantitative information from the market, and then the development of credit derivatives helps investors to hedge against default risks.

Spring 2006 Final Project

Summer 2006 Project Proposal:

This semester I wish to continue my work with Dr. Jingyi Zhu on credit migration modeling. The purpose of credit migration modeling is to develop explicit probabilities that a bond rating will move into a different state given that it currently has a specific rating. The most common way to approach this problem is to formulate a Markov chain that uses past data to predict future events. Early on, most of the models were discrete-time and homogenous. This created problems because many rating transitions are rare events and show up as having zero probability under this framework. This is intuitively incorrect since rare events should always be possible. To fix this problem, a continuous-time homogenous Markov chain was developed that used log-likelihood estimation techniques to estimate the generator matrix of the Markov chain. This approach was shown to give strictly positive probabilities for the transition matrix. In addition, this method has a number of desirable statistical properties including tight confidence intervals. This has caused it to become very popular with practitioners. The downfall of this approach, however, is that it does not correctly account for business cycle effects that have been shown to greatly influence the probabilities in past research. Therefore, the purpose of my research will be to develop a continuous-time model that weights past information based on business cycle information. This will produce a model that can use predictions about future states of the economy to determine which past periods should influence forecasts on future events. This model should be much more dynamic than previous continuous-time homogenous Markov chains.

Kellen Petersen
Hometown: Sandy, UT
Major: Mathematics & Physics
Faculty Mentor: Ken Golden

Spring 2006 Project Proposal:

Two-Componenet Composite Materials
The goal of this project is to study two-component composite materials and their properties. In many heterogeneous materials the microstructure is not currently known and therefore properties of the material (permittivity, conductivity, etc.) are averaged over, or homogenized. It has been shown that there exists an integral representation for the effective permittivity of a material involving measure μ. Finding the correct spectral measure is difficult, but it gives a great amount of information about the medium. We desire to study the relationship of the measure to underlying microgeometry to gain insight into the microstructure and effective properties of materials.

Matthew Reimherr
Hometown: Salt Lake City, UT
Major: Mathematics
Faculty Mentor: Lajos Horvath

Fall 2005 Project Proposal:

For my REU project I would like to examine the risk associated with corporate credit scores. There are some voices out there that believe that the scores reflect too much risk. It would not be surprising if this were true since the scoring system is designed for the loaners to analyze risk. For them it is better to error on the side of too much risk than not enough. There are also orginizations that do very well by loaning to those groups that are deemed too high risk by normal banks. This would again suggest that there may be validity to the claim corporate credit scores reflect an inaccurate amount of risk.

To analyze this problem I must first fully understand how the credit scores are designed and implemented. This will be a bit difficult since the companies that design these credit scores try to keep their methods under lock and key. They are in fact businesses and you cannot fully expect them to make all their data public. There are, however, two papers by Galiardini and Gourieoux (Stochastic Migration Models with Application to Corporate Risk and Efficient Derivative Pricing by Extended Mehtod of Moments) that will be valuable tools in evaluating the risk associated with corporate credit scores.

After I have developed a firm understanding of the methods and concepts associated with credit scores, I shall try and determine if there is indeed too much risk associated with the scores. After all, it could very well be the case that some unhappy people were rejected for a loan after a valid and accurate risk assessment and decided the scores needed to be changed.

After I have determined my course I shall try to do one of two things. Either I will finish my paper explaining that the credit scores are accurate and give reasons why, or I shall try my best to examine inaccurate risk in these scores and how much risk they should reflect.

Mid way through my project I will attempt contact with businesses and organizations that may have interesting input on this topic. Companies such as MFS Investment Management have done well by picking up investors deemed too high risk by banks. Also meeting with banks to understand their policy on loans would provide some valuable information.

My project may take up to two semesters, but I believe that there is a great deal that can be done in this area. It will take some time to fully understand how these scores work and are implemented. It may also take some time to see if these scores reflect too much risk or not enough. However, I believe that it is a promising project and I look forward to working on it.

Fall 2005 Final Report

Spring 2006 Project Proposal:

This semester I wish to continue my research on corporate credit risk. Much of the literature on this topic is written by economists and not by mathematicians. Therefore, a great deal of the literature is concerned with isolating a particular factor or group of factors and using them to define a credit score. My interest however, is not on deciding which factors should be used for the score, but with modeling the behavior of a company as it moves through different scores.

Last semester I ended up studying stochastic migration models and their application to corporate credit risk. The stochastic migration model for modeling credit risk was introduced by Christian Gourieroux and Patrick Gagliardini (Gourieroux, Christian and Gagliardini, Patrick, "Stochastic Migration Models with Application to Corporate Risk" Journal of Financial Econometrics, Vol. 3, No. 2, pp. 188-226, 2005) as a general model that limits neither the number of factors needed to compute credit scores nor their economic interpretations. However, Gourieroux and Gagliardini do not provide a great deal of analysis on their model. Even the existence of such a model is assumed and never proven.

Over the lest semester I have been examining their model and trying to give it a more mathematical foundation. I wish to continue that work this semester and also start providing simulations for the model.

Spring 2006 Final Project

Ben Richards
Major: Mathematics
Faculty Mentor: Davar Khoshnevisan

Summer 2006 Project Proposal:

Information is regularly transmitted through many different mechanisms. Whether the mechanism be words as in written and spoken language, notes as in music, or electrical impulses in a nervous system, they all serve the same purpose-transferring information. I am particularly interested in the transfer of information in the form of musical notes.

It is generally agreed that most of the great classical composers have unique styles present in their compositions that are often described with sensory rhetoric such as 'airy' or 'fluid'. I want to research a method of quantifying these characteristics through statistical and probabilistic analysis of a composer's work.

Specifically I propose to approach the problem from the standpoint of authentication, in which successful trials and methods have been developed and implemented on the works of Shakespeare. The idea behind this method is that, when a known author or composer leaves behind a large sampling of work which is unquestionably their own, an analysis of that work can provide quantitative measures and descriptions of which words or notes appear most often and how often they appear in certain patterns. With that information in hand, a work of unknown or disputed authorship can be compared to the characteristic description of the known works, and a measure of confidence of potential authorship can be made.

The techniques I will apply for analysis are those of information theory. Specifically I expect the bulk of my research to involve constructing entropy tables for a specific composer. Several crucial questions will be investigated during this process such as, what is the appropriate unit to analyze music in? Though a note may be the most basic, it is likely coherent patterns will be recognized more readily with larger units such as measures, intervals, or phrases.

My most ambitious hope for application of this research is to achieve a confidence measure of Mozart's authorship of his Requiem, a piece whose authorship is still disputed. In long-term goals, I hope this research will help me develop the knowledge to analyze styles of music and then synthesize the most probable examples of each of those styles. That is to say, to create the quintessential example of a style of music through mathematical analysis.

Summer 2006 Final Project

Sandy Schaefer
Major: Mathematics
Faculty Mentor: David Goulet

Summer 2006 Final Project

Daniel Sjoberg
Hometown: Cedar Hills, Utah
Majors: Mathematics & Economics
Faculty Mentor: Jingyi Zhu

Spring 2006 Project Proposal:

Mathematical Modeling in Credit Risk
The goal of research in this area is to quantify the risk faced by investors in government and corporate liabilities. Intuitive but also sophisticated mathematical theories are developed to obtain quantitative information from the market, and then the development of credit derivatives helps investors to hedge against default risks.

Ban Tran
Hometown: Salt Lake City, UT
Major: Mathematics
Faculty Mentor: Daniele Arcara

Fall 2005 Project Proposal:

Tautological rings belong to some deep aspects of algebraic geometry and topology. Finding equations in tautological rings used to employ very difficult mathematics. Recently, it has been proposed that there might be an algorithm which can be used to discover these equations. This algorithm has been put on test in a few cases and has so far been successful.

However, the complexity of the algorithm grows fast, and it would be very hard and time-consuming to test it in other cases by hand. My research will focus on writing a computer program to perform all of the complicated computations to test this algorithm in many cases. If successful, the algorithm should produce new equations which are still unknown.

Steven Lee Ward
Hometown: Salt Lake City, UT
Major: Mathematics
Faculty Mentor: Kim Montgomery

Fall 2005 Project Proposal
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