# Math Digest

## Summaries of Articles about Math in the Popular Press

Edited by Allyn Jackson, AMS
Contributors:
Mike Breen (AMS), Claudia Clark (Northeastern University), Annette Emerson (AMS)

### March 2004

This book was originally online and is now being published in book form. Edwin T. Jaynes, who died in 1998, was a theoretical physicist. According to Toffoli, Jaynes revived the view of early probability theorists that probability is an "extension of logic to cases in which lack of information make deductive reasoning by Aristotlean syllogism impossible." This early interpretation was replaced in the 19th century by probability as the limit of an event's relative frequency. Toffoli writes that the book "is a pleasure to read.... The bibliography is rich and well annotated. There are many exercises, and occasionally the editor has creatively turned a gap in the manuscript into an exercise for the reader."

--- Mike Breen

"Safely Rooted on Earth, Scientists Solve an Icy Martian Puzzle," by Henry Fountain, The New York Times, March 30, 2004, Section F, page 3.

With attention focused on the surface of Mars, another discovery has been made about the "Red Planet," but without the benefit of the rovers. As described in the April issue of Geology, scientist Dr. Jon Pelletier used one modified form of the Fitzhugh-Nagumo equations to model an unusual feature of the Martian landscape: the spiral troughs in its polar ice caps. Previous attempts to explain this phenomenon have included such complex factors as wind erosion and ice flows. Pelletier found that a model based upon "accublation" - the process of ice being heated enough to turn to gas on the sun-facing side of a trough, and then accumulating as ice on the opposite (shadowed) side - was sufficient to generate the spiral patterns. But the spirals on Mar's surface are not perfect: Dr. Pelletier's next step will be to create a more realistic model that addresses such factors as initial conditions, climate change, and landforms beneath the ice. Some nice images are on line.

--- Claudia Clark

"Mathematicians Caught on Film." NetWatch, Science, 26 March 2004, page 1951.

This short piece tells of the more than 3000 hours of video lectures available at the website of the Mathematical Sciences Research Institute. Many of the lectures were delivered at the Institute, but the site also has lectures that were delivered elsewhere. The lectures are available here.

--- Mike Breen

"Divine proportion and the Holy Grail," by Martin Kemp. Nature, 25 March 2004.

The author concedes that it is probable Leonardo DaVinci knew the work of Fibonacci and "was certainly aware of the divine proportion (or golden section)," but Kemp claims that no evidence reveals that Renaissance and Baroque artists "used constructional methods based upon the 'secret' division of their surfaces." He says that such attributions---that artists used mathematical secrets---with hindsight make for a good story but are not credible to serious art historians.

--- Annette Emerson

Can mathematics be used to create songs that will be sure-fire hits? A Barcelona-based artificial intelligence company, PolyphonicHMI, has created software that may do just that---and the record industry is using it, according to the article. The software is based on an algorithm that is supposed to identify the "optimal mathematical patterns" of hit songs. The software was used to analyze 50 years' worth of songs and to arrange them in a "music universe". Hit songs tend to cluster together in this universe, so a new song's hit potential can be assessed by how it compares to other songs in the universe. "The closer [the new song] lands to the center of a hit cluster, the more likely it is to be a successful song," the article says, noting that many find the whole notion of analyzing songs this way to be "sacrilege."

--- Allyn Jackson

"Top of the Top 40," by Ben Harder. Science News, 20 March 2004, page 182.

The Top 40 in this case are the top finishers in the recent Intel Science Talent Search. First place went to Herbert Mason Hedberg of North Attleboro High School (MA) for a test that could identify tumor-fighting compounds. Hedberg received a US$100,000 scholarship. Second and fourth place were awarded to math projects by Boris Alexeev of Cedar Shoals High School (GA) and Linda Brown Westrick of the Maggie L. Walker Governor's School (VA). Alexeev, who received a US$75,000 scholarship, did research into automata, while Westrick's project involved properties of the number derivative. You can read more about the top ten finishers and Westrick's project online.

--- Mike Breen

"Quantum Channel Capacities," by Charles H. Bennett and Peter W. Shor. Science, 19 March 2004, pages 1784-1787.

In this article, researchers Bennett and Shor consider the progress that has been made in quantum information theory. They begin by providing an overview of classical information theory, defining entropy---which quantifies the amount of uncertainty in a source---and capacity, "the maximum rate at which a channel can carry information without spoiling it." They discuss the extension of these two concepts to quantum systems. But while the quantum extension of entropy is straightforward, the same cannot be said of capacity. For one thing, there are several ways to define the capacity of quantum channels. Then there's the role of "entanglement," which they discuss at length, and refer to here as "the strangest quantum phenomenon." They conclude that "although several important questions remain unanswered, and undoubtedly further phenomena remain to be discovered, it is fair to say that we now have a broadly accurate understanding of quantum information as the natural generalization of its classical forebear."

--- Claudia Clark

This was one of many obituaries that appeared in newspapers worldwide about John A. Pople, a mathematician who became a chemist and won a Nobel Prize in 1998 "for developing computer techniques to test the chemical structure of matter."

--- Annette Emerson

"Stretchy strings make `fuzzy' black hole give up its secrets," by Stephen Battersby. New Scientist, 13 March 2004, pages 10-11.

This article discusses recent calculations by a physicist at Ohio State, Samir Mathur, that use string theory ideas to shed new light on understanding of black holes. Stephen Hawking's work on black holes says that any energy falling into a black hole will cause emission of so-called Hawking radiation, which carries no information about what went into the hole. Using string theory, Mathur has calculated that there may be a way for that information to be emitted after all. Whether this new calculation reflects reality is difficult to say. "The best test of Mathur's idea in the short term may be to work through the maths and see whether it really does make quantum mechanics work for black holes," the article says.

--- Allyn Jackson

"Lady of longitude," by Stephanie Pain. New Scientist, 13 March 2004, pages 48-49.

This article tells the story of Mary Edwards, a woman who lived in the 18th century and did arithmetic computations used in longitude calculations. Her husband contracted for the computing work, but it was Mary actually carried it out. When he died in 1784, she continued to support her family by earning money as a "human computer". At that time women typically did not learn much mathematics, and it is not known how she learned the arithmetic skills to do these computations. The article is based on new research presented in "Mary Edwards: Computing for a living in 18th-century England," by Mary Croarken, IEEE Annals of the History of Computing, volume 25, page 9.

--- Allyn Jackson

"Hurricanes and butterflies," by Thomas C. Halsey and Mogens H. Jensen. Nature, 11 March 2004, pages 127-128.

Systems may seem chaotic, but such a property can be hard to verify. According to the authors of this article, Sam Gratrix and John N. Elgin "have developed a powerful new method to determine from experimental observation of a system whether it is chaotic, and, if it is, what the precise quantitative nature of that chaos is." Gratrix and Elgin have used recurrence times, the time it takes points to return within a certain distance of its starting point, to verify the multifractal properties of strange attractors. The new method is preferable to the "box-counting" method, which can classify non-multifractals as multifractals. The research is published in Physical Review Letters, "Pointwise Dimensions of the Lorenz Attractor."

--- Mike Breen

"Forever and Ever," by John Allen Paulos. abcnews.com, 7 March 2004.

In this month's "Who's Counting" column, Paulos explores the interplay between mathematics and literature: "Increasingly, fiction seems to come with a mathematical flavor, mathematical exposition with a narrative verve." He illustrates the phenomenon in brief analyses of David Foster Wallace's book Everything and More: A Compact History of Infinity and John Barrow's play (performed to date only in Europe) Infinities, and applauds such attempts to bridge the the cultures of mathematics and literature. Paulos (recently given the 2003 AAAS Award for Public Understanding of Science and Technology) concludes that "mathematical exposition is too important to be left only to mathematicians, and the wide variety of literary forms available should not be off-limits to mathematicians and physicists."

--- Annette Emerson

"Committing just one crime moves youngsters into a different mathematical world," is the conclusion of a team who analyzed two very different studies of youths in North London and Pittsburgh. "The researchers expected that the number of crimes committed per person would fit a statistical distribution shaped like a bell if the criminal acts were committed by random people in the selection...instead they found that crime rates fell into a mathematical pattern called a power law, in which large deviations from average behaviour are more common. In both studies, most of the boys committed no crimes at all." The conclusion was that people who commit no crimes at all live in a different world than those who do, so the best strategy is to deter people from committing their first crime.

--- Annette Emerson

The article attempts to explain the concepts of space, Calabi-Yau manifolds, how mathematicians perceive circles and spheres, string theory, brane theory, gravity---all part of the search for the "True theory of the universe."

--- Annette Emerson

The authors present the various systems used in elections. They give examples of problems and perplexing election results in the 2000 U.S. and 2002 French presidential elections, explain how "true majority rule and rank-order voting result in dramatically different outcomes," and propose changes that would more accurately represent voters' wishes.

--- Annette Emerson