**by Monte Basgall**

Four days of intensive research, analysis, programming and writing on a long February weekend netted three Duke undergraduates both an "outstanding" rating and a winning paper in the 2000 Mathematical Contest for Modeling (MCM).

MCM is an annual international tournament where about 500 university teams use math and ingenuity to solve real-world problems in a race against the clock.

The winning team senior Jeff Mermin, a mathematics and classical languages major from Chapel Hill; junior Daniel Neill, an electrical engineering and computer science major from Tampa; and sophomore Sam Malone, a mathematics and economics major from Zebulon won for their 35-page paper that formulated and tested five different mathematical models for analyzing collision avoidance in air traffic control.

The Federal Aviation Administration is now actually evaluating new computer software that would automatically alert controllers to the threat of potential collisions between planes.

"It was an extremely sophisticated project, and this paper was the best that I have seen from a Duke team," said David Kraines, the associate professor in the department of mathematics who served as the contest's adviser.

This is the third year in a row that a Duke team has received an outstanding rating in the MCM. Mermin was also on the 1998 team, and Malone was on the 1999 team.

The air traffic problem came in two parts, the first asking: "Given two airplanes flying in space, when should the air traffic controller consider the objects to be too close and to require intervention?" The second part upped the ante, asking contestants to assess how complex any airspace sector is from the perspective of an air traffic controller's workload.

After developing five different models to address the problem's first part, the Duke students rated three models as most promising. Those were a "probabilistic simulation" model that projects flight paths in an uncertain environment, a "close approach" model that computes danger as a function of aircraft distance, and a "space-time" model that considers closely approaching aircraft in four dimensions.

To address the second part, the Duke undergraduates developed an algorithm a set of mathematical rules for evaluating the decision processes that air traffic controllers follow to detect potential collisions and resolve conflicts in judgment.

The contest began just after midnight on Friday, Feb. 4, and the students' first decision was to choose between two different problems. Instead of air traffic control, they could have opted for Problem B, which asked contestants to model "the assignment of radio channels to a symmetric network of transmitter locations over a large planar area, so as to avoid interference."

"We didn't really like the way that problem was set up," Mermin said of Problem B in a recent interview. "It looked like they had basically already decided what they wanted the model to be." The Duke group reasoned that "Problem A probably would allow us to be more creative since it was a little more open-ended maybe," Malone added.

After selecting Problem A, the students immediately "slept on it," Mermin said. "We came back at 10 or so Friday morning and started working." At that point, "to some extent, we divided the problem up," said Neill. "While Sam (Malone) focused more on the second part, Jeff (Mermin) and I focused more on the first."

The group did research until about Friday evening, after which Mermin spent most of the night typing up possible solutions to the problem's first part. Meanwhile, Neill came up with another model, the probabilistic one.

On Saturday morning, the team went though government documents on air traffic controllers. Then Neill and Mermin started the programming for the problem's first part while Malone began deriving the rest of the equations for the second part.

All of Sunday and until shortly before the 5 p.m. Monday deadline was spent writing. "All you can do is get plenty of sleep in the preceding days because you know you're going to be missing out on sleep over the weekend," Mermin said.