Spring 1993

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The Painlev Transcendent
By Anne Noonburg

The differential equation x''=x^2-t has interested mathematicians since early in the 20th century. It is a variation of the first Painlevé transcendent, one of a group of nonlinear second-order differential equations having solutions whose only movable singularities are poles.

Computing Changes Core Mathematics
By David Arney; Frank Giordano

The New Core: The curriculum at USMA provides four semesters of core mathematics given to all students during their first two years. Using the recent advances in bringing computing and experimenting into the classroom, USMA has devised a new core mathematics curriculum. This new curriculum has been in place since Fall 1990 and was designed using a systems engineering, top-down approach which tries to fit material from several desired topics courses into the following four course sequence: Discrete Dynamical Systems (difference equations), Calculus I (differential and integral), Calculus II (multivariable), and Probability and Statistics.

The Flight of a Ski Jumper
By Ernest True

Modeling the Forces: When a ski jumper leaves the ramp of a ski jump and becomes airborne, the dominant forces that determine the success of the jump are the force due to gravity and the air resistance, which can be resolved into a lift force and a drag force. The lift and drag forces can be altered by the jumper’s posture over the skis and by the position of the skis relative to the ground. This combination of lift and drag can make the difference between a “good” jump and a short or even dangerous jump.

Heating and Cooling of Buildings
By Steven Dunbar

Our goal is to formulate and analyze a mathematical model that describes the 24-hour temperature variation inside a building. The interior variation will result from the outside temperature variation and the heat generated by the people and machines inside the building. We ignore the heating and cooling of the interior with furnaces or air-conditioning, so the situation modeled would best apply in spring or fall.

The Lorenz Attractor
By Courtney Coleman

Determinism or Chaos: Who would have believed in 1963 that a newly published article in a technical journal would lead to a revolution in scientific thought? Yet that was the effect of E. N. Lorenz’s note, Determinate Non-Periodic Flow, which appeared that year in the Journal of Atmospheric Science. Lorenz used a system of three mildly nonlinear ordinary differential equations as a simplified model of the convection currents and thermal variations in air cells beneath a thunderhead.

The Savvy Solver III
By Larry Shampine

The third in a series of articles on various implementations of numerical methods for solving ODEs and their difficulties.

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