Daily Archives: November 2, 2008

1/Problem

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Optics basics: Inverse problems at Skulls in the Stars.

Plenty of other techniques exist for measuring the internal structure of objects, using a variety of different types of waves. Magnetic resonance imaging (MRI) subjects a patient to an intense magnetic field, and makes an image by measuring the radio waves emitted when the field is suddenly switched. Ultrasound imaging uses ultrasonic waves to probe the soft tissues of the human body, and is used in mammography.

Each of these techniques is quite different in its range of application, but all require nontrivial mathematical techniques to reconstruct an image from the raw scattered wave data. These mathematical techniques are broadly grouped into a class of problems known as inverse problems, and I thought it would be worth an optics basics post to discuss inverse problems, their common features, and the challenges in solving them.

Knot Too Shabby

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Unknotting Knot Theory

One of the reasons knots have given mathematicians fits is that the same knot can appear in very different guises. Tug here, tug there, and soon a knot will become unrecognizable, but remain fundamentally unchanged. To allow a knotted string to wiggle around without danger of untying, mathematicians seal its two ends together, making it a knotted circle. The first question mathematicians have to answer is simply, when are two knots really, secretly the same?

The dream is to create a sort of machine: Send in one of these looped knots, and out pops some result that would be the same regardless of the particular configuration of the knot. Because the answer wouldn’t vary with the arrangement of the knot, such a machine is called a “knot invariant.” And indeed, in 1927, mathematician J.W. Alexander created just such a “machine,” a method that produces a polynomial (an expression like 3×2 + 4x + 1) from any knot. The good news is that Alexander’s method always gives the same polynomial for a particular knot, even if the knot has been wiggled around to look very different. The bad news is that it can also give the identical answer for knots that really are different. For example, the granny knot and the square knot have identical Alexander polynomials.

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Grab Your ACME Umbrella

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Space station trash plunging to Earth

NASA and the U.S. Space Surveillance Network are tracking the object — a 1,400-pound (635-kilogram) tank of toxic ammonia coolant thrown from the international space station — to make sure it does not endanger people on Earth. Exactly where the tank will inevitably fall is currently unknown, though it is expected to re-enter Earth’s atmosphere Sunday afternoon or later that evening, NASA officials said.

The umbrella won’t help, of course. Just ask Wile E. Coyote.