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
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. |
