Math Lounge, Stratton Hall
WPI Chapter of SIAM Organizational Meeting
The agenda for the meeting includes:
Stratton Hall, 203
Speaker: Constantin Bacuta
2004 COMAP Mathematical Contest in Modeling (MCM)
MCM is a contest where teams of undergraduates use mathematical modeling to present their solutions to real world problems. The contest challenges teams of students to clarify, analyze, and propose solutions to open-ended problems, and attracts diverse students and faculty advisors from over 500 institutions around the world. For more information about participating in the 2004 MCM, please contact Prof. Suzanne Weekes and visit the COMAP website.
Stratton Hall, 203
An Introduction to Free Boundary Problems
In algebra and pre-calculus, typical problems involve the search for an unknown real number. For many questions which arise in engineering, the corresponding mathematical problems require the determination of a function on a given domain. (Finding the derivative of a function or solving a differential equation are examples.) Free boundary problems are mathematical models which arise when a function and at least part of the domain where the function will satisfy a differential equation are unknown at the outset. We'll discuss a few physical situations which lead to free boundary problems, and then we'll focus on the obstacle problem and see some of the mathematical challenges that it has inspired.
Stratton Hall, 203
Integral Equations on Closed Contours: Application to an Inverse Conductivity Problem
The first part of this talk focuses on the trapezoidal rule for numerical integration. The error estimate is well known to Calculus students. However, the error estimate is much improved in the case of periodic functions. A proof based on Fourier series and contour integration for functions of a complex variable, will be provided. We then apply the trapezoidal rule to the numerical solution to the conductivity equation. For the sake of simplicity, our domain will be two dimensional and the conductivity will be constant or piecewise constant. Finally, I will show how these numerics can be used in a problem of great practical importance: the location of conductivity imperfections inside a piece of material, via surface current measurements. The qualitative non-destructive testing of materials using eddy currents techniques has been employed in industry for decades. A quantitative treatment of measured data can provide additional information. This has been the focus of mathematical studies such as the one I will discuss.
Stratton Hall, 202
Introduction to No-Arbitrage Pricing Theory
In the simplest framework of a one-period binomial model (a stock driven by a coin toss) we introduce the idea of pricing stock options by no-arbitrage considerations. We point out that the model can be extended to more than one period and even to continuous time. Similar arguments lead to the famous Black and Scholes formula.
Stratton Hall, 308
Two Elementary Problems From an Elementary Viewpoint... Made Easy.
Dr. Bob LaBarre, Principal Mathematician, United Technologies Research Center
This talk is about problems and questions. We look at elementary problems as a means to understand both what questions to ask, and what the answers to these questions mean. The first problem is concerned with the ubiquitous linear system. After studying it from the natural algebraic viewpoint, we let geometry guide our understanding. The second problem under discussion involves the second easiest ODE known to man, after y'(t) = 0. We extend our knowledge by looking at this problem when uncertainty in a parameter creeps in. Finally, we build some understanding and compare our techniques to the tried and true FDH (Fat, Dumb, and Happy) approach!
Stratton Hall, 308
Jumping the Jumbo
Sid Rupani (WPI), Jesse Tippett (WPI), Dave LeRay (WPI)
This project was completed as part as the 2003 COMAP Mathematical Contest in Modelling.
In the filming of a movie stunt, it was required that a motorcycle jump over an elephant into a cushioning cardboard box apparatus. We were asked to design the optimum cardboard box configuration to satisfy the following conditions:
Specifically, we needed to determine the type of box to use, the number needed for effective cushioning, how to arrange them, suggest possible modifications to the boxes, and produce generalized results for different combinations of the impact mass and jump heights.
Through dynamic analysis of the motorcycle's trajectory, an energy balance equation was obtained that allowed us to calculate the energy that needed to be dissipated by the boxes to ensure the rider's safety. Subsequently, analysis of the energy-dissipating capacity of various boxes was undertaken, utilizing experimental data as well as empirical equations used in industry testing. ECT, or "Edge Crush Test", data was the primary attribute used in determining static and dynamic compressive strength. The final aspect of the model was practically fitting our assumption of a normal impact to this real life situation, using a simple inclined plane support mechanism for the box apparatus and determining its elevation angle.
Upon formulation of the model, we used spreadsheet software to calculate data for various box types and specifications. The real strength of our model is that in can be readily applied to any stunt of this sort due to our general approach. The model was analyzed by varying the parameters and creating a method for box selection. Using the specific design constraints (safety, low cost, low visibility), an optimum box type was determined. Following analysis, we verified our model by testing extreme situations and comparing the results to our physical intuition. In addition, various box modifications were suggested to negate the inherent weaknesses in our model to facilitate real world application.