Math in the Media 0300 *March 2000* Pebbles to Microchips. *A History of Algorithms: From the Pebble to the Microchip*, edited by Jean-Luc Chabert, is reviewed by Jeremy Gray in the February 14 2000 *Nature*. An algorithm is a recipe for solving a certain kind of problem. Many mathematical formulas are in fact algorithms, since they specify a sequence of operations to be performed on their "inputs." One of the best known ones is ("the quadratic formula") for calculating the roots of the equation ` ax`^{2}+bx+c=0 from the coefficients `a,b,c`. The recipe would run: "Take `b` and square it, then multiply `a` by `c` and subtract four times that product from the square of `b`. Process the resulting number through the square-root machine. Separate the outputs. Subtract `b` from each of them, divide the results by two times `a`, and serve." The intelligent use of algorithms is the subject of most mathematics instruction through the Sophomore year of college. Beyond Quadratic Reciprocity. Ian Porteous picks up an item by Jonathan Rogawsky from the January 2000 *AMS Notices* in the January 15 *Science News*. Quadratic reciprocity is an amazing law of number theory discovered by Gauss. In its simplest form it relates facts about arithmetic modulo `p` and arithmetic modulo `q`, where `p` and `q` are primes greater than 2, and says that `p` is a perfect square mod `q` if and only if `q` is/is not a perfect square mod `p`. "Is" unless both `p` and `q` give remainder 3 when divided by 4, "is not" when they both do. For example 13 and 17 (in the "is" category): The perfect squares mod 13 are 1, 4, 9, 3, 12 and 10; the prefect squares mod 17 are 1, 4, 9, 16, 8, 2, 15, and 13. Notice that 17 (=4 mod 13) is in the mod 13 list and 13 is in the mod 17 list. This general phenomenon is still considered ``one of the deepest and most mysterious results of elementary number theory.'' The quote is from Rogawsky, whose article concerns the recent proof of the ``local Langlands correspondence,'' a far-reaching generalization of Gauss' discovery. Larger image Driving a liquid with a vertical sinusoidal force can cause spikes to erupt from the surface. Image from Nonlinear Dynamics Laboratory, used with permission. | Theory of Liquid Spikes. The Nonlinear Dynamics Laboratory at the University of Maryland features ``a cylindrical tank partially filled with a glycerin-water mixture'' which can be subjected to vertical oscillation. Under the right conditions, singular spikes erupt from the surface, as shown in this figure. A paper ``Singularity dynamics in curvature collapse and jet eruption on a fluid surface''in the January 27 2000 *Nature*, by Daniel P. Lathrop and his associates at the Laboratory, describes an investigation of the mathematics underlying these phenomena. The partial differential equations governing the situation are completely intractable at present, but by positing a power-law scaling in times close to the singularity the team was able to derive a theoretical expression for the surface profile in excellent agreement with the experimental data. | Larger image A composite picture of a sequence of states of the liquid surface before the eruption of the spike. Image from Nonlinear Dynamics Laboratory, used with permission. | * -Tony Phillips* SUNY at Stony Brook Math in the Media Archive |