NC State Research Experience for Undergraduates Mathematics:
Summer 2006 Experience

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Research and Education Activities:
The REU program in Modeling and Industrial Applied Mathematics was held in the Department of Mathematics at North Carolina State University from May 30 to August 4, 2006. A total of 27 students (15 female and 12 male including one African-American student) participated in the REU program. Of the 27 students, 16 were supported by NSF, 9 were supported by NSA, 1 was supported partially by both NSF and NSA, and one student came with her own funding. The students were divided to work on the following 7 projects:

Project 1: Mathematical Modeling of Bird Flu and Pandemic Influenza

REU Students: Ian Appel (Duke), Nancy Ho (Mills College), BreAnne Pickelsimer (Elizabethtown College), Thu Tran (Bryn Mawr)
Faculty Advisor: Alun Lloyd
Graduate Student Assistant: Alex Capaldi

The main focus of the project was mathematical modeling of influenza epidemics. Mathematical models for epidemics are important because they can be used to help public health authorities to plan their responses to the appearance of deadly variants of the flu virus. A wide variety of mathematical approaches and tools were employed during the project, including dynamical systems theory, stochastic modeling, numerical simulation, including Monte Carlo techniques, inverse problem methodology and optimization theory.

A particular issue of interest was the emergence and spread of drug resistant variants of the influenza virus. Drug resistance is of major concern: many of the responses that have been proposed if (when) the next influenza pandemic strikes rely on the use of antiviral drugs to mitigate the impact of the virus. In order to analyze mitigation strategies, we extended a mathematical model proposed by Stilianakis et al (1997). Their model concluded that chemoprophylaxis of susceptible persons, as opposed to treatment of those who are symptomatic, is the optimal strategy for minimizing the incidence of influenza in a closed population. However, the authors did not account for the supply of antiviral drugs being finite and for the death of symptomatic persons. Our model, which includes both of these factors, predicts that chemoprophylaxis is not necessarily the optimal strategy, especially when minimizing the number of deaths for a given amount of antiviral drugs.

Project 2: A Mechano-chemical Model for Chondron Deformation in Articular Cartilage Subjected to Osmotic Loading
REU Students: Kadie Elliot (Blackburn College), Ka-Ho Lo (Stony Brook Univ), Ken Hedrick (California State Poly, Pomona)
Faculty Advisor: Mansoor Haider
Graduate Student Assistant: Brandy Benedict



The chondron is a microstructural mechano-biological unit in articular cartilage consisting of the cell (chondrocyte) and its surrounding pericellular matrix. In this project, an inhomogeneous triphasic (solid-fluid-ion) model for chondron deformation within the tissue extracellular matrix was formulated under conditions of confined osmotic swelling of a cartilage sample. Model solutions were evaluated numerically and used to quantify, separately, the effects of collagen stiffness and proteoglycan density on the micromechanical environment of the cells in cartilage. Effects of alterations due to osteoarthritis were also simulated.

Project 3: Cell Based Model of Convergent Extension in the Ascidian Notochord
REU Students: Tracy M. Backes (Harvey Mudd College), Russell Latterman (Arizona State), Stephen A. Small (Norfolk State University)
Faculty Advisor: Sharon Lubkin
Graduate Student Assistant: Janine Haugh


We present a three-dimensional cellular Potts model (a discrete cell based Monte-Carlo model) of convergent extension of the ascidian notochord. Our work derives from recent research that explores the coupling of invagination and convergent extension in ascidian notochord formation. Modeling cells individually allows us to assign different physical properties and behaviors to each cell type and to study how these properties affect notochord elongation. We have tested the roles of cell-cell adhesion, cell-extracellular matrix adhesion, random motion, and extension of individual cells, as well as the presence or absence of various tissue types, and determined which factors are necessary and/or sufficient for convergent extension.

Project 4: Analysis of Models for Blood Flow in the Cardiovascular System
REU Students: Kelly Koser (Carnegie Mellon), Julia Moore (RPI), Cheryl Zapata (NC State)
Faculty Mentor: Mette Olufsen
Colloborator: Vera Novak (Beth Israel Deaconess Medical Center and Harvard Medical School)
Graduate Student Assistant: Laura Ellwein

Blood flow in the cardiovascular system can be predicted using compartmental models based on coupled nonlinear ordinary differential equations. Such models include compartments for arteries, veins, and a pulsating heart. These models can predict dynamics observed in experiments in which finger blood pressure and cerebral blood flow velocity are measured during postural change from sitting to standing.

Such models often comprise a large number of parameters. Some of these parameters can be determined based on physiological properties such as height and weight, while others are harder to determine. To validate these models against data from individual people, model parameters must be adjusted appropriately. In this project, we discuss how to predict model parameters for individual data sets using nonlinear least squares optimization methods, including the Levenberg-Marquardt method, that minimize the difference between computed and measured quantities. Furthermore, we discuss how solutions to sensitivity equations derived using automatic differentiation can be used to predict which parameters can be identified given a limited number of data sets and how sensitivity information can be used to reduce the complexity of the model.

Project 5: Applications of Monte Carlo Simulations in Financial Mathematics
REU Students: Shi Chen (Amherst College), Gene Kim (Rutgers), Aolin Li (Univ of California, Berkeley), Erin Koehler (Clemson), Martina Mincheva (Franklin and Marshall College)
Faculty Advisor: Tao Pang
Graduate Student Assistant: Yuqing Lu


In the area of financial mathematics, derivative pricing has been a big challenge for a long time. Among all methods, Monte Carlo method is widely used by both researchers and practitioners. This project considers some applications of Monte Carlo method in derivative pricing issues that arise from the equity market and the fixed-income market.

REU students worked on this project for ten weeks. First they used Monte Carlo simulation to price European and Asian options in the equity market. Then they considered the applications of Monte Carlo simulation in the fixed-income market and the numerical results of some derivatives of bond, such as bond options, interest rate caps and floors, were obtained. They also tried various variance-reducing techniques to make their algorithms more efficient and the results were very promising. More importantly, they tested different models for the interest rates by calibrating the models to the real market data.

Project 6: A Hybrid Optimization Approach for the Optimal Design of Traveling Wave Tubes
REU Students: Adam Attarian (NC State), Laura Tarko (Mount Holyoke College), Jeremy Zuckero (Wilkes Univ)
Faculty Advisor: Hien Tran
Industrial Mentor: Lawrence Ives (Calabazas Creek Research, Inc.)
Graduate Student Assistant: John David


Traveling Wave Tubes (TWTs) are vacuum devices invented in the early 1940Æs for amplification of radio frequency (RF) power. They are critical for radar, communications and electronic warfare missions of all Armed Services, as well as in certain commercial applications. Because of their high power, broad-bandwidth, compact size, and high efficiency, TWTs are used for satellite communications, airborne, ship borne, and ground-based radars, jamming, and decoy applications. Commercial applications include satellite communications, radar, and materials processing. In this project, we studied a hybrid optimization approach for the optimal design of TWTs using two independent physics-based design and simulation codes. In particular, we used the CHRISTINE suite of large signal codes to model the slow wave circuit, in conjunction with Beam Optics Analysis to model a multi-stage depressed collector. These two simulation codes were successfully integrated with a sampling optimization procedure, the Nelder-Mead algorithm, to automate the process of optimizing several physical parameters while realizing several design constraints.

Project 7: Posets of Involutions in Weyl Groups

	REU Students: Juan Li (Smith College), Erin Oliver (Smith College), Spyro Roubos (University of Rochester), Nathan Williams (Carleton College), Jeremy Schwartz (Brandeis Univ), Patricia Cahn (Smith College), Chelsey Cooley (NCSU RISE Program) Faculty Mentors: Aloysius Helminck, Ruth Haas (Smith College) Graduate Student Assistant: Stacy Beun

Weyl groups generalize permutation groups and are useful in many branches of mathematics. Understanding which elements are in the set of involutions is important for many applications to symmetric spaces, representation theory, network problems, etc. The elements of the set of involutions are characterized by sequences in the set of reflections which induce a poset relationship between them called the Bruhat poset. There exists an algorithm to compute this poset which is inefficient due to the fact that in Weyl group packages elements are expressed as reduced words in the generators and there can be many words representing the same element. The students worked on a new data structure which gives a unique representation of the elements of Weyl groups and which improves the efficiency of the algorithm exponentially. In addition they proved a number of combinatorial results about this poset and developed and implemented algorithms to compute this combinatorial structure.

Presentations:
The feedback from faculty mentors and collaborators are that the projects were very successful and that the REU students have obtained promising and in some cases significant results. Indeed, results from the 7 projects were presented as poster presentation at the 5th Annual NC State Undergraduate Research Summer Symposium on Thursday, August 3, 2006. Project 7 was presented both orally and in poster format at the Young Mathematicians Conference, Department of Mathematics, Ohio State University, right after the REU program, August 4-6, 2006. In addition, the four projects in biomathematics (Projects 1-4) were presented in a special mini-symposium as well as poster presentation at the Joint SIAM-SMB Conference on the Life Sciences, July 31-August 4, 2006. Results of Projects 4, 6, and 7 were presented as poster presentation at the Joint Mathematics Meetings (photos below) in New Orleans, January 5-8, 2007. Finally, Project 3 was submitted for presentation at the Berkeley Undergraduate Mathematics Conference, Department of Mathematics, University of California, Berkeley, April 7, 2007.

Recognitions:
The scientific results obtained from the REU projects have already been well received by the mathematical community. In particular, the poster presentation from Projects 3 and 4 won poster presentation awards at the Joint SIAM-SMB Conference on the Life Sciences, July 31-August 4, 2006. The poster presentation from Projects 6 and 4 were prize winners at the Joint Mathematics Meetings in New Orleans, January 5-8, 2007; they were ranked 4th and 5th, respectively (http://www.maa.org/students/undergrad/07winners.html)

Journal Publications
Laura M. Ellwein, Hien T. Tran, Cheryl Zapata, Johnny Ottesen, and Mette S. Olufsen, "Sensitivity Analysis and Model Assessment: Mathematical Models for Arterial Blood Flow and Blood Pressure", Special Issue of Journal of Cardiovascular Engineering, p. , vol. , ( ).
Submitted

Adam Attarian, John David, Hien T. Tran, Lawrence Ives, "A Hybrid Optimization Approach for the Optimal Design of Traveling Wave Tubes", IEEE Transactions on Plasma Science, p. , vol. , ( ).
In preparation

P. Cahn, R. Haas, A.G. Helminck, J. Li, J. Schwartz, "On a EL-labeling for the poset of twisted involutions", Congressus Numerantium, p. ,
vol. , ( ). In preparation

P. Cahn, R. Haas, A.G. Helminck, J. Li, J. Schwartz, "A Data Structure for the Exceptional Weyl Groups of type $F_4$ and $G_2$", Communications in Algebra, p. , vol. , ( ).
In preparation

C. Cooley, R. Haas, A.G. Helminck, N. Williams, "Combinatorial Properties of the Richardson-Springer Involution Poset", Adv. in Appl. Math., p. , vol. , ( ).
In preparation