December 1999 The most influential lecture ever given in mathematics? A good candidate is Bernhard Riemann's 1854 introductory lecture in Göttingen, "On the Hypotheses Which Lie at the Foundations of Geometry." Riemann's work in featured in the November 15 issue of ScienceWeek, on on-line weekly magazine of science. ScienceWeek published excerpts from Calvin C. Clawson's book: Mathematical Sorcery: Revealing the Secrets of Numbers (Plenum, New York 1999). Riemann can fairly be described as the father of modern mathematics, and more. Much of the allegedly unreasonable effectiveness of mathematics in modern physics can be explained by their both stemming from Riemann's fundamental insights.
Why does a negative times a negative equal a positive? Why do you ``invert and multiply'' when dividing fractions? These questions and more are addressed in the Fall 1999 issue of the American Educator, a publication of the American Federation of Teachers. This issue features substantial articles by Richard Askey and H. Wu, both distinguished mathematicians who have helped focus the attention of the research community on problems in mathematics education. Askey's article refers to the study Knowing and Teaching Elementary Mathematics by Liping Ma (Lawrence Erlbaum Associates, Inc., 1999). Ma's work, which emphasizes the ``profound understanding of fundamental mathematics'' as the goal toward which the education of elementary-school teachers should be directed, was reviewed by Roger Howe in the July 1999 Notices, with a survey of its implications for American schools.
Things that go bump. The September 28, 1999 New York Times carried an article by James Glanz entitled ``Scientists Discover New Clues to Earthquakes' Deadly Vibrations.'' He is referring to the similarity between the surface manifestation of certain tremors and the phenomenon of ``oscillons.'' Oscillons were named and first studied by Paul Umbanhowar (now at Northwestern University) and colleagues in 1996. They studied the behavior of spherical copper beads in a vibrating tray, and found, at certain frequencies, stable repeating patterns, as in this figure
The experiments relevant to the behavior of terrain were carried out by Jay Fineberg and O. Lioubashevski (Racah Institute of Physics), Y. Hamiel, Z. Reches, and A. Agnon (Geology Department) at the Hebrew University in Jerusalem. This time the medium was thin mud, the vibrations were in the range 60-100 cycles per second, and the following phenomena were observed. (This picture appeared in the Times).
| An oscillon in a vibrating tray of copper beads. Image courtesy of Paul Unbanhowar, Northwestern University Physics Department. |
The mathematics of oscillons is not yet completely understood. The Times article suggested that oscillons are related to solitons, another non-linear phenomenon with a somewhat similar shape, but according to Fineberg this is not the case.
| The oscillation of oscillons in a saucer-full of a suspension of potters' clay. Scale: frame width = 4cm. Time interval between pictures = 32 msec. Image courtesy of Jay Fineberg, Hebrew University of Jerusalem. |
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