NC State Research Experience for Undergraduates in Mathematics:
Modeling and Industrial Applied Mathematics

May 26– August 1, 2008
Program Director:  Loek Helminck

NC State REU in Modeling and Industrial Mathematics

NSF and NSA are providing funding support for this REU program.

Where: NC State University, Raleigh, NC .
Dates: May 26– August 1, 2008. See Calendar below
Stipend and support: $3600 for ten weeks, all housing provided, as well as a partial meal allowance. Travel funds up to $300 per participant provided as needed.
Topics for 2008 REU program are: Thermal Nondestructive Evaluation in Composite Materials with Porosity,Cell Rearrangements in Tissues, Information Retrieval and Web Search, Design of Electron Devices using Computer Optimization, Menstrual Cycle Modeling, Computational Biology of Cardiac Function During Exposure to Particulate Matter and Other Toxicants, Modeling of Traffic Congestion and Mass Evacuations, Viscoelasticity of the Arterial Wall, Model Development for Multifunctional Materials, Modeling Cartilage Regeneration. Additional topics are forthcoming. Abstracts are detailed below.

This program will be similiar to those offered in the past. Past program homepages
Summer 2005 , Summer 2006 and Summer 2007.

Click on images for full size photo

Participant background, requirements and selection: Participants are expected to meet the following:

• must be a citizen or permanent resident of the United States or its possessions,
• must be a full-time undergraduate mathematics major as of September 2007,
• be committed to devote their full time to the program and not engage in any other course work or employment during the program,
  
  Participants will be selected on the basis of demonstrated mathematical creativity, motivation, and good work habits as well as meeting the above requirements, as determined from the application materials and recommendation letters.

Abstracts:

Project: Thermal Nondestructive Evaluation in Composite Materials with
Porosity
Advisors: H.T. Banks and Amanda K. Criner

In this NASA related project the goal is to develop a computational mathematical / statistical model-based methodology for thermal interrogation of aero and space structures. These structures are composed of modern composite materials containing significant porosity which must be accounted for in models for "control" or undamaged
materials. In the NDE procedure, the structure is subjected to flash heating and temperature and/or heat flux on the boundary is observed. One must then determine from this data any possible internal damage to the material. This project will involve basic mathematical and statistical first principles modeling for thermal processes (the heat
equation in a heterogeneous medium) and use of experimental data with these models in computational inverse problems. The data will be collected from both the CRSC/Math Instructional Lab at NCSU and the NDE Branch at NASA Langley Research Center.

Project: Cell Rearrangements in Tissues
Advisor: Sharon Lubkin
Developmental biology is the study of how we go from a spherically symmetric egg to a complex dynamic structure with dozens of differentiated cell types in spatially complex and useful arrangements, such as lungs, hands, and brains. Part of the mechanism of development is cell rearrangements, which are due to forces generated by the cell interacting with forces applied to the cell. We will model the rearrangements of cells in morphogenesis using a stochastic simulation technique. Some computer programming experience is necessary.

Project: Information Retrieval and Web Search
Advisor: Carl Meyer
The project concerns the mathematical technology involved in building various kinds of information retrieval systems and search engines. The material will include classical methods such as latent semantic indexing systems along with various document clustering schemes and will proceed through more recent techniques based on nonnegative matrix factorizations. A primary facet involves studying web search with particular emphasis on the Google technology. It is hoped that upon successful completion of the course students will have amassed enough knowledge to build their own search engines.

The material will be drawn from a variety of recent research papers in conjunction with the two primary references: "Google's PageRank and Beyond: The Science of Search Engine Rankings" by A. N. Langville and C. D. Meyer http://pagerankandbeyond.com/ and "Understanding Search Engines" (Second Edition) by M. W. Berry and Murray Browne http://www.ec-securehost.com/SIAM/SE17.html.

Project: Design of Electron Devices using Computer Optimization
Advisors: Lawrence Ives (Calabazas Creek Research, Inc.)
Hien Tran (NC State University)

The performance of vacuum electron devices producing radio frequency (RF) power depends on the operational characteristics of several subcomponents. For example, the efficiency of a traveling wave tube depends on the quality of an electron beam, the parameters of the RF circuit, the energy recovery characteristics of the spent beam collector, and the matching characteristics of the output waveguide system. Each subcomponent is related to one or more of the others in determining the overall performance of the device. Currently, technology exists to optimize each component's
performance in isolation from the others. Unfortunately, high performance in some components, such as the circuit, leads to reduced performance in others, such as the spent beam collector. Therefore, it is proposed to determine a mechanism for optimizing the performance of various components based on the total performance of the device rather than the performance of single components. A starting point would be to optimize both the circuit and collector together. Following a successful demonstration of this
capability, the design of the output waveguide system and electron gun could be added. Dr. Ives will be visiting NCSU several times during the REU program to discuss the physical aspects of the problem.

Project: Menstrual Cycle Modeling
Advisor: James F. Selgrade

Complex endocrine signaling between the ovaries and the pituitary gland is the key ingredient for regulating and maintaining the menstrual cycle in adult women. Previous work has resulted in an ordinary differential equations model describing the concentrations of five hormones important for this system. The model has been validated by finding a stable periodic solution which approximates data for normally cycling women. In addition, the model exhibits another stable periodic solution which has some similarities to polycystic ovarian syndrome (PCOS), the leading cause of female infertility in the US. The present project will refine and improve the existing model in the following three areas:

(i) Model simulations will be used to determine the type and range of hormone level variations needed to perturb abnormal cycles to the normal cycle.

(ii) The current model includes a state variable for total inhibin, Ih. However, a recent assay has been developed which distinguishes two inhibins, IhA and IhB. The different biological effects of IhA and IhB will be studied and IhA and IhB will replace Ih in the model. Available data will be used to validate the new model.

(iii) As a woman approaches menopause, production of ovarian hormones decreases. The effects of decreasing IhB on FSH levels, on cycle length and on other cycle variations will be investigated. The model will be modified to include aging and perimenopausal women.

Participants will need some background in undergraduate ordinary differential equations and linear algebra. Students will be expected to learn some endocrine biology and to do mathematical modeling and analysis, parameter identification and computer simulation. Endocrine physiologists in the RTP area will be consulted on the biology. The softwares MATLAB and XPPAUT will be used to solve delay equations and DDE-BIFTOOL, to study bifurcation and continuation.

Project: Computational Biology of Cardiac Function During Exposure to
Particulate Matter and Other Toxicants
Advisor: Dr. Marina Evans (EPA)
Predicting computational behavior of human cardiac function is a long-standing goal of systems physiology and applied biology. The biological paradigm using dissection and cataloging as major tools seems to be shifting towards integration of all available data.
Applied mathematics and computational tools are helping to provide an integrated framework where data from different systems, organs, levels, and methods is analyzed under one common denominator. Predictions can then be used to design experiments, which in turn provide feedback to the evolving models being constructed within
computers.

A correlation seems to exist between particulate matter and cardiac dysfunction, particularly in sensitive populations, particularly children and the elderly. Environmental pollutants such as byproducts of diesel in air (also known as particulate matter) seem to disrupt heart cells at the molecular level, interfering with biochemical/ electrical signals which affect major heart function, and sometimes lead to death. The goal of this project is to start analyzing differences in heart function, using spectral analysis tools, to quantify how normal heart function is affected by exposure to toxicants such as particulate matter. The overall project will utilize biologically-based computational models to describe how endothelial cells lining the heart capillaries allow transport of environmental
pollutants, and mathematically describe how pollutants may affect normal signaling, thus eventually leading to disease. A final goal of this project is to integrate other cardiac models with this computational molecular level, thus describing/predicting cardiac and
lung physiology with the experimental/and exposure measurements gathered.

Project: Modeling of Traffic Congestion and Mass Evacuations
Advisor: Pierre A. Gremaud

The mathematical study of traffic flows has recently seen some impressive breakthroughs. For instance, significant progress has been made in the last two years in our understanding of the formation of traffic jams. During the first part of the project, some of this recent work will be studied. This involves partial differential equations but also bifurcation theory and numerics. During the second part of the project, the students will put together a mathematical model of the evacuation of the North Carolina Outer Banks in case for instance of a hurricane. The model be studied, implemented and the results and predictions critically analyzed.

Project: Viscoelasticity of the Arterial Wall
Advisor: Mette S. Olufsen

Blood flow in the body is transported through a network of arteries and veins and the structure of these vessels help transforming the flow from a highly pulsatile flow to a slow almost steady flow. The arterial wall is composed of tissue that is viscoelastic allowing it to dampen out some of the wave-reflections observed during the transport of the pulse wave. During this project we will formulate viscoelastic models relating blood pressure and vessel area and we will validate this model against experimental data obtained from in-vitro studies in sheep arteries. Furthermore, we will learn about
blood pressure profiles and their relation to blood flow velocity through experimental study that we will conduct in our own laboratory. Students working with this project should have some experience with computer modeling, a solid understanding of calculus,
and some experience with differential equations. The most important component is a curiosity and interest in learning about physiological applications.

Project: Model Development for Multifunctional Materials
Advisor: Ralph Smith

Advanced aerospace, aeronautic, industrial, biomedical and nanotechnology applications increasingly rely on multifunctional materials to achieve design specifications.  These materials exhibit the capability of coupling electrical, magnetic, thermal, optical, and mechanical behavior but do so at the cost of complex and nonlinear material dynamics.  This project will focus on the development of modeling, simulation and statistical techniques for these advanced materials.  This will also involve the validation of models with experimental data.  Participants will need some background in differential equations and elementary numerical analysis along with an interest in learning certain fundamental aspects of materials science.

Modeling Cartilage Regeneration
Advisor: Haider

In this project the REU group will develop mathematical models for tissue regeneration in articular cartilage.  In such applications, natural or bioengineered scaffold materials are seeded with cartilage cells (chondrocytes) and injected into a defect site arising from injury or osteoarthritis.  Via nutrient supply and/or drug delivery, the chondrocytes proliferate and synthesize extracellular matrix constituents to turn over the gel into “regenerated” articular cartilage. The group will develop and analyze mathematical and computational models of an evolving gel-tissue material with the goal of optimizing its functional properties.    

Prerequisites: This project will be well suited to undergraduate students with a solid foundation in calculus, some experience with partial differential equations or numerical analysis, and with interests in mathematical modeling and/or mathematical biology. 

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All applicants will be notified by email about the completeness of their application a couple of days after the deadline date. Unless previously notified, a final notification that the search is closed will be emailed after all positions have been filled and confirmed in writing (this could take a month).  If you have any questions about the status of your application, especially if you are trying to make a decision on accepting another summer position, please email the program director who will happy to give you a prompt response.