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PRACTICAL APPLICATIONS

Diverse Applications

One of Holmes’ occasional colleagues at Rensselaer is Margaret Cheney, professor of mathematics, whose research has addressed such diverse phenomena as land mines, oil deposits, birch forests, and pulmonary edema. What unifies these seemingly unrelated situations is Cheney’s mathematical specialty—inverse boundary-value problems. That’s what scientists are doing when they gather information about an inaccessible region by probing it from the outside with fields, waves, or particles.   

“They are ‘inverse,’” Cheney says, “because the order of cause and effect is reversed. The scientist observes an effect and tries to deduce the causes.”

“Boundary-value” refers to measurements made at the edge or surface of an object that reveal something about its interior. For example, by applying a current to the human skin, a doctor can measure the resulting electrical field, deduce the amount of fluid in the patient’s lungs, and potentially diagnose a case of pneumonia.

Inverse boundary-value problems usually involve differential equations. For the mathematically challenged among us, those are equations in which the unknown quantity is not a number (the electric potential at one point), but a function (the electric potential at every point). Such equations are usually too difficult to solve with paper, pencil, and unaided intellectual exertion, but solutions can be approximated by computers. That doesn’t render mathematicians extinct. Rather, their job is to come up with the appropriate algorithms for computers to use.   

Cheney described applied mathematics as “math that’s very closely connected to some physical or engineering problem. Most of these problems are already encapsulated in some mathematical equation. That’s how we discover new truths about the physical world. The math is just everywhere in the world.”   

Cheney has received funding from the Department of the Navy to research more effective means of detecting underwater mines (as in explosives, not ore deposits). “Present methods don’t work very well. A dog’s nose or a dolphin’s sonar is more reliable. If they can do it, we should be able to figure it out,” Cheney says.

 

 

 

Numerical simulation of the stability of a converging circular shock wave subject to a small perturbation in its speed. The black curves show the shock positions at successive intervals of time beginning with the outer circular shock. The colored contours indicate the speed, or Mach number, of the shock as it converges to form a pentagonal-shaped shock that ultimately focuses to a point. (Contributed by D. Schwendeman.)

Joyce McLaughlin, Ford Foundation Professor of Mathematical Sciences, is a member, like Cheney, of the board of trustees of the 9,000-member Society for Industrial and Applied Mathematics. In fact, she’s associate editor of the Journal of Mathematical Analysis and Applications.   

Her work, funded by the Office of Naval Research and the National Science Foundation, has a number of ongoing projects, including a joint research project with a Hong Kong scientist on the propagation of radio signals in the ocean.

“You can’t really compare our math department to most traditional math departments. We’re entirely applied math. We probably have the largest concentration of applied mathematicians in the same department in the country,” McLaughlin says.

Isom Herron, professor of mathematics, concentrates on the theory of fluid flows and its application to such fields as oceanography, meteorology, and the motion of ships and aircraft. He taught at Howard University for 18 years before joining the math department at Rensselaer in 1992.

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