Curriculum Vitae

Research

Publications and

Presentations

Teaching

Activities

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• Introducing students to the idea of performing research on a mathematical topic.
• Improving students' technical communication skills, including technical document preparation, and composing and delivering presentations.
• Alerting students to some of the many applications of mathematics in the real world.
• In some courses, getting students to work together as part of a group.

• Bridge Flutter Analysis by D. Frank, D. Kammer, and F. Ding.  
  This project is concerned with approximating flutter in cable-stayed bridges. It focuses on two numerical methods to approximate the critical wind velocity needed to create the flutter phenomenon in a bridge. It discusses the methods in detail, and then gives an example that shows the accuracy of the two methods.    
  
• Applications of Monte Carlo Methods in Portfolio Analysis by M. Albrecht, J. Combes-Knoke, and M. Seminatore.
  Monte Carlo numerical methods are used extensively in areas such as pricing American derivative securities, they are also incorporated to calculate the efficient frontier of an investment portfolio as well as providing insight into investment portfolios. Other usages include the calculation of Value at Risk (VaR) as a relevant risk measure and net present value of projects.
  
• Satellite Launch and Alignment into Orbit by R. Chatterjee, V. Devaraj, F. Pineda, and J. Shoemaker.
  This project focuses on the two major aspects of space exploration: satellite launch and satellite placement into orbit. The delicate nature of these calculations necessitates numerical methods and computer calculations. The margin of error is very small, so slight errors in cancelation, among other potential pitfalls, could derail the entire launch process. The same factors apply to placing a satellite into proper orbit. Ultimately, accuracy in both launch and alignment are crucial to satellite implementation.
  
• Centroidal Voronoi Tessellations by E. Allen, Z. Girouard, P. Goswami, and B. Leary
  A centroidal Voronoi tessellation (CVT) is a partitioning of points based on minimizing distances relative to centers of mass. This paper will study three numerical methods involved in computing CVT’s. The first goal is to discuss the different applications of CVT’s, ranging from animal territories to optimal placement of resources.  The second goal is to analyze our methods of generating CVT’s. Lloyd’s method is a deterministic method while MacQueen’s method is probabilistic. We want to look at when they converge, how quickly they converge, and pros and cons of each. We will talk about how these numerical methods are utilized in finding CVT’s. Finally, we will look at the effectiveness of using a non-Euclidean method to compute anisotropic CVT’s.
  
• Numerical Modeling of a Parasite-Symbiont-Host Model by N. Komarov and M. Nemaric
  This project presents a new population model to describe the interactions of a host, parasite, and symbiont. From the equilibrium criteria, we derive restrictions on the constants in the model, and discuss certain aspects of the resulting equilibria. We then model this interaction numerically, using the forward Euler, or tangent line method, and use these approximations to discuss the implications of some of the equilibria. Finally, we discuss some possible expansions of this model.
  
• Numerical Weather Prediction by R. Barrett and E. Tucker
  This project has several focuses with the intent of surveying the development of numerical weather prediction over the past century. First, it examines the early days of weather prediction and the work of Lewis Fry Richardson, one of the pioneers in the field. In this segment of the report, some of the first techniques are studied and the reasons for their failure are explained. After this, we move on to look at the role supercomputers played in the continuing advancement of this field. Two numerical methods techniques for solving relevant equations, the Finite Difference Method and the Spectral Method, are also explained. In the final portion of this report, a more in-depth look at one of the current and most advanced numerical weather prediction systems is provided. The HIRLAM technique employs these two methods to divide weather forecasting into physics and dynamics.
  
• Financial Simulations and Variance Reduction by A. Dabholkar, V. Hidayat, F. Ho, D. Levin, and J. Qi
  Valuation of American options is one of the more difficult problems in option-pricing theory.  American options are heavily used in many financial markets, but pricing remains challenging.  Unlike European options, which can only be exercised at expiration, American options can be exercised at any point during the lifetime of the option. There exist explicit formulas to price European options (e.g. Black-Scholes-Merton), but no such formulas exist for American options due to their more complex nature. Instead numerical methods are generally used.
  
• Spectral Methods for Burger's Equation by J. Rothenberg and G. Peim 
  Burger's Equation can be solved numerically using several different spectral methods. These methods are efficient to calculate approximate solutions to
  Burger's Equation. Burger's equation is a good simplification of Navier-Stokes Equation when the velocity is in one spacial dimension and the net force acts
  in one direction. Burger's Equation is used to analyze traffic congestion and acoustics. We discuss the background of the equation, analytical solution, and
  the Galerkin and Collocation spectral approximations.
  
• Artificial Neural Network Machine Learning Through Numerical Optimization by J. Dunn and X. Li
  An antificial neural network (ANN) consists of a group of interconnected simple processing elements, known as artificial neurons, which provide a robust approach to approximating vector-valued functions. For many types of machine learning applications, such as handwriting recognition and facial recognition, ANNs are among the most accurate learning methods in existence.
  
• B-Spline Interpolation and Its Scientific Applications by A. Bakelmun, A. Charnas, G. Domville, and R. Lazrus
  B-spline interpolation is a method commonly used to approximate three-dimensional curves in space. The method estimates the curve by substituting finite, known, data points into approximate equations. The accuracy to which the interpolation can estimate a curve varies greatly depending on the field of study, the type of curves to be approximated, and the equations and data points chosen for the calculations. B-spline interpolation is applied to various fields of study such as medicine, meteorology, and chemistry.
  

• Infinite vs. Finite Dimensional Vector Spaces
• Approximating Solutions of Linear Systems of Equations
• Eigenvalues and Eigenvectors
• Self-Adjointness and Positivity
• Bilinear and Alternating Forms
• Orthogonality
• Canonical Forms
• Classical Inequalities and Their Proofs