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Past Undergraduate Projects Supervised by UW-Math Faculty |
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Project: Vortex Filaments Student: Mikal Grant 2006-2007 UW EPSCoR fellowship Mentor: Hakima Bessaih
Description: The world around us is filled with random movements and
fluctuations that can be modeled discretely. Explaining how and why a
particular event happened is habitual and even routine for scientists.
The question, though, of future behavior is a bigger challenge, and
mathematicians have been answering the call to study predictive models
for nearly two centuries. A tool that
many modelers use is a theory from physics that small particles
suspended in fluid are in constant motion. Introduced Robert Brown in
1827, the model is given the name Brownian motion. Briefly, a Brownian
Motion is a sophisticated random number generator that is used in a wide
variety of mathematical models in physics, chemistry, engineering,
finance, and medical imagery. The
Biot-Savart model uses a 3-dimensional Brownian motion to create a
vector field that models vortices in three space. Some physical
applications for this model include magnetic fields in electrical
currents and tornado vortices.
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Project: Some Results on Stochastic
Differential Equations: Theoretical and Numerical Results.
Student: Mike Bostick
Mentor: Hakima Bessaih
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Project: Vortex Filaments: Applications to tornadoes Student: Tyler Miller Mentor: Hakima Bessaih
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Project: Fluid Mechanics, Point vortices and Turbulent flows Student: Jonathan Daraie Mentor: Hakima Bessaih
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Project: Computing Traveling-Wave
Front Solutions in a Diffusive Predator-Prey Mode
Student: Brian Kim Summer 2004 (University of Michigan REU Program) Mentor: G. Lyng  Description: This project focused on a diffusive predator-prey model. In this model the species are free to move in a one-dimensional habitat, say, along a riverbank, and the introduction of a few predators at one end will result in a wave of invasion of predators as the system transitions from one equilibrium state to another. The primary objective of this project was to write a computer code capable of computing these traveling-wave solutions with the ultimate goal of understanding the effect of the system parameters on these waves. These nonmonotone waves are found as heteroclinic orbits in a four dimensional phase space. A project report is available at http://www.math.lsa.umich.edu/undergrad/REU/projects.shtml Brian continued his studies at the Florida State Biostatistics graduate program.
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Project: An Ensemble of Anamoly
Classifiers for Identifying Cyber Attacks
Student: Christer Karlsson Mentor: P. Polyakov Description: This project concerned the use of anamaly detection software for identifying cyber attacks. The approach consisted of an ensemble of classiffiers that, together, produce a more informative output regarding the class of attack than any of the classifiers alone. Each classifier classifies based on a limited subset of possible features of network packets. The ensemble classifies based on the union of the subsets of features. Thus it can detect a wider range of attacks. In addition, the ensemble can determine the probability of the type of attack based on the results of the classifiers. Experimental results demonstrate an increase in the rate of detecting attacks as well as accurately determining their type. Publication:Kelly, C., Spears, D., Karlsson, C., and Polyakov, P. (2006). An ensemble of anomaly classifiers for identifying cyber attacks. In the Proceedings of the International SIAM Workshop on Feature Selection for Data Mining, Bethesda, MD. Available online at http://www.cs.uwyo.edu/~dspears/papers/ensemble.pdf Christer is currenly in pursuing a Ph.D in Computer Science at the Colorado School of Mines.
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Project: Analysis of solutions of the Possio integral equation in 2D aeroelasticity. Student: Westin Joy Mentor: Peter Polyakov  Description. This project studied the solvability of the generalized Possio integral equation which is a tool in the analysis of a boundary value problem in 2D subsonic aeroelasticity with the Kutta-Joukowski condition --- "zero pressure discontinuity': --- ψ(x,0,t)=0 on the complement of a finite interval in the entire real line R.
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Project: Coin weighing problems Student: Kim Creaser Description: This project concerned the all-equals coin problem: given a collection of n coins, determine, using only a pan-balance, in the smallest number of weighings whether or not the coins all have the same weight. Fast algorithms for solving this problem are needed in many applications including spectroscopy, tomography, and GIS. In essence, the all-equals coin problem is the fundamental problem than one faces whenever one needs to detect anomalies. Recently, Kozlov and Vu shocked the mathematical world by giving an algorithm that can weigh significantly more the 2^n coins in just n weighings–provided that weights of the coins are “generic”. In practice the assumption of generic is not a constraint. Unfortunately, the Kozlov and Vu algorithm, while mathematically nice, does not automatically produce weighing schemes that are practical. The project investigated ways to create inputs to the Kozlov and Vu method that output practical, and realistic weighing schemes. Kim is currently a Ph.D. student in UW's Electrical and Computer Engineering department working on designing an electronic eye modeled after a fly's eye.
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Project: Electronic Voting Schemes Student: Brenda Christensen Mentor: B. Shader Description: This project concerned the timely, and increasingly important issue of electronic voting. In order to move from punch cards and dangling chads to ATM-type or even internet voting, it will be necessary to develop a voting scheme that voters can trust, and one that is verifiably secure against cyber-attacks. Several schemes have already been proposed, and even marketed. Most notably, Chaum has marketed a “Visual Electronic Voting” scheme that provides both the voter and the election commission a receipt. The project took Chaum’s scheme a step further. Using the mathematics of cryptography and shared secret schemes its developed an electronic voting scheme that provides receipts to a variety of entities (e.g. the election commission, the individual voter, the League of Women Voters, and perhaps each political party). The receipts were designed so that only certain collections of the parties can access (or verify) the information. Brenda is currently an MS student in UW's Statistics Program.
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Project: Robustness of Small-world networks Student: Andy Curtis Mentor: B. Shader Description. Small-world phenomena were initially studied in the 1960s through a series of social network experiments, and are, as evidenced by the game "The six degrees of Kevin Bacon", even part of our pop-culture. Recently, mathematicians and physicists have shown that most small-world phenomena are expected consequences of the mathematical properties of certain networks--known as small-world networks. In this paper, we survey some recent mathematical developments dealing with small-world networks, as well as present a new small-world network model and discuss some new ideas for decentralized searching. The goal is to give the reader a sense of the importance of small-world networks, and some of the useful applications dealing with these networks. Resulting publication: Andy Curtis, Small-worlds: Beyond Social Networking, Rose-Hulman Undergraduate Math Journal, Article 7, 2004.
Article available at
http://www.rose-hulman.edu/mathjournal/download.html
Andy received his MS in Computer Science from Colorado State University in 2007. He is currently a PhD student in Computer Science at the University of Waterloo, and his research concerns algorithms, networks, combinatorial optimization and data structures.
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Project: External memory algorithms for
graphs and digraphsStudent: Andy Curtis Mentor: B. Shader Description. This project concerned the development of external-memory algorithms for fundamental graph algorithms. In external-memory algorithm design, memory is managed as part of the algorithm to minimize the number of input/outputs (I/Os) performed between internal memory and secondary storage. The number of operations performed by the CPU is generally ignored in external algorithm analysis, since a typical transfer of data from disk is about one million times slower than from internal memory. In this project, we developed the first I/O efficient algorithm to determine if a general directed graph is acyclic. In addition to checking acyclicity our algorithm returns a topological ordering of the vertices. I/O efficient topological sorting has been a long-standing open problem, and our results help to further progress work on this problem. Our acyclicity testing algorithm has an I/Ocomplexity of O((V + E B)log2(VB)+ sort(E)). Resulting publication: Andy Curtis and B. Shader, Testing acyclicity and topological sort in external memory. Andy received his MS in Computer Science from Colorado State University in 2007. He is currently a PhD student in Computer Science at the University of Waterloo, and his research concerns algorithms, networks, combinatorial optimization and data structures. |