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## Math Digest |

- "Strings with a twist,"
*New Scientist,*31 July 2004 - "2, 3, 5, 7, 11, 13, 17, 23: What comes next?"
*International Herald Tribune*, 30 July 2004 - "Multicultural Math." NetWatch,
*Science*, 30 July 2004 - "Generous players,"
*Science News*, 24 July 2004 - "Diagrams you can count on": Review of
*Cogwheels of the Mind: the Story of Venn Diagrams*by A.W.F. Edwards,*Nature*, 22 July 2004 - "Waring Experiments,"
*Science News Online*, 17 July 2004 - "A three-player solution,"
*Nature*, 15 July 2004 - "The EU constitution is 'unfair', according to game theorists,"
*The Telegraph*, 14 July 2004 - "Building Biostatistics,"
*Nature*, 8 July 2004 - "Is Team Really as Good as Record? Do the Math,"
*The New York Times*, 4 July 2004 - "Infinite Beauty:
*Fractals and Chaos: The Mandelbrot Set and Beyond*,"*Nature*, 1 July 2004

**"Strings with a twist," by Roger Penrose. New Scientist, 31 July 2004, pages 26-29.**

This article discusses the role that twistor theory, created by Penrose in the 1960s, is now playing in the development of string theory. String theory offers the potential to resolve difficulties and paradoxes inherent in the classical theories of relativity and quantum mechanics. The basic premise of string theory says that matter is made up not of point particles but of tiny strings. One of the complications of string theory is that it says that space-time, which intuition tells us should have four dimensions, ought instead to have ten dimensions. "I have always had difficulties with these extra dimensions," Penrose writes, noting that the constructions proposed for explaining them would likely lead to instabilities and singularities. But now it seems that twistor theory could provide a way out. He describes this theory and its basis in the algebra and calculus of the complex numbers---that is, numbers of the form a + b*i*, where *i* is the square root of - 1. "The emerging link between twistors and string theory arises from the complex-space nature of twistor space," he writes. A note at the end of the article says that a new book by Penrose, *The Road to Reality: A Complete Guide to the Laws of the Universe* is being published by the British publisher Jonathan Cape.

*--- Allyn Jackson*

**"2, 3, 5, 7, 11, 13, 17, 23: What comes next?" by Michael Johnson. International Herald Tribune, 30 July 2004, page 7.**

Score one for mathematics. The Great Internet Mersenne Prime Search (GIMPS) has the author of this editorial hooked on prime numbers. "I have a lot of trouble with long division," he confesses. "High school algebra was agony for me and the concept of prime numbers never arose in the schoolhouses of small-town Indiana." But now his computer is part of the GIMPS network of about 240,000 computers around the world, which together are chugging away at calculating bigger and bigger prime numbers. When a 10 million digit prime number is found, he will be eligible for a US$100,000 prize. His odds of winning are one in 191,196, which he points out "are 70 times more favorable than my odds of winning the British lottery." So he's not so innumerate after all.

*--- Allyn Jackson*

**"Multicultural Math." NetWatch, Science, 30 July 2004, page 585.**

This short article is about the Ethnomathematics Digital Library, which provides access to over 700 items worldwide. The article begins by describing centuries-old Japanese geometric puzzles called *sangaku* and finishes by noting that "You can delve into cultures ranging from New Zealand's Maori to the Incas of South America."

*--- Mike Breen*

**"Generous players," by Erica Klarreich. Science News, 24 July 2004, pages 58-60.**

If competition is such an important part of natural selection, then why does cooperation exist in nature at all? Klarreich writes of how researchers in game theory are attempting to answer that question. The prisoner's dilemma game is often cited in the article as is the lesser-known game, snowdrift. Klarreich explains both games and how different organisms' behaviors relate to them. She also describes experiments in game theory, including a 1987 experiment with stickleback fish.

*--- Mike Breen*

**"Diagrams you can count on": Review of Cogwheels of the Mind: the Story of Venn Diagrams by A.W.F. Edwards. Reviewed by Jeremy Gray. Nature, 22 July 2004, pages 405-406.**

In this very positive review, Gray writes that Edwards has worked out a way to draw Venn diagrams for any number of sets, showing all possible intersections among the sets. The book also connects Venn diagrams with coding theory. The review concludes with, "All this and a great number of mathematically beautiful figures mean that the book deserves to become a minor classic and may well go on to make many friends for mathematics."

*--- Mike Breen*

**"Waring Experiments," by Ivars Peterson. Science News Online, 17 July 2004.**

In this article, writer Ivars Peterson discusses the arithmetic of whole numbers, as well as some of the mathematicians who have contributed to this field. He points out that mathematicians, professional and amateur, have long been interested in the ways in which integers can be expressed as the sum of different numbers. For example, Joseph-Louis Lagrange proved in 1770 that any positive integer could be written as the sum of no more than four squares. What about expressing any whole number as a sum of cubes or fourths? Edward Waring (1736-1798) proposed that at most nine cubes or 19 fourths would suffice. This was followed by other mathematicians proving related results, such as David Hilbert's proof of the existence of some minimum number of terms to represent every whole number. But it would take until 1912 for the proof of nine cubes, and over 200 years before 19 fourths was proven.

Peterson offers additional examples of propositions and proofs following from Waring's original proposition, which has led, and undoubtedly will continue to lead, to new questions. This is part of the tradition of mathematical experiment. Less typical, as Peterson notes, is the difference between the accessible nature of these questions and conjectures, and the difficulty of proving them.

*--- Claudia Clark*

**"A three-player solution," by Lewi Stone. Nature, 15 July 2004, pages 299-300.**

Stone summarizes research by Greg Dwyer et al ("The combined effects of pathogens and predators on insect outbreaks," pages 341-345 of the same issue), which offers a mathematical model of populations of insect pests. Traditional models rely on either climactic factors or competition and predation. These models fail to account for the cyclic nature of some insect populations. After more than a decade of studying the gypsy moth, Dwyer et al present a model that incorporates the moth, its predators and a virus that infects gypsy moth larvae. Stone writes that the model's predictions conform to observed gypsy moth data.

*--- Mike Breen*

**"The EU constitution is 'unfair', according to game theorists," by Roger Highfield. The Telegraph, 14 July 2004.**

The author reports on researchers who claim that the new European Constitution voting process will give Germany undue influence ("37 percent more clout than the UK") and that it favors the biggest and smallest states over medium-size states. The research on the flaws in the voting system, published by physicist Karol Zyczkowski and mathematician Wojciech Slomczynsky (Jagiellonian University, Krakow), was analyzed and supported by 50 other European scientists. "The scientists use a branch of mathematics called game theory to calculate how much power each country will have to sway the Council of Ministers if the new Constitution is adopted..." Highfield briefly describes the point made in 1949 by Lionel Penrose, that "voting power is not the same thing as voting weight... To represent true voting power, Penrose devised the 'square root law,' where the influence of each country is proportional to the square root of its population size." Highfield reports that the scientists have adopted this law into "the Jagiellonian Compromise" and hope that the EU will consider it.

*--- Annette Emerson*

**"Building Biostatistics," by Anne Gimalac. Nature, 8 July 2004.**

In the NatureJobs section of this issue, Gimalac emphasizes that drug companies are now incorporating "pharmacogenetics" into their strategic plans and are looking for scientists who are comfortable with statistics, mathematics, and informatics as well as genetics and biology. Serono Genetics Institute in Evry, France, for example, is particularly interested in developing and expanding its workforce of scientists in biostatistics and bioinformatics. Gimalac stresses that the set of skills needed in the pharmaceutical industry requires the integration of mathematics and biology, two disciplines that over the years have become separated in academia.

*--- Annette Emerson*

**"Is Team Really as Good as Record? Do the Math," by Alan Schwarz. The New York Times, 4 July 2004, page B8.**

The math in this case is comparing a baseball team's actual winning percentage to the quantity (*s*^{2}/(*s*^{2}+*a*^{2}) where *s* is the number of runs the team has scored and *a* is the number of runs the team has allowed. Bill James devised the statistic (often referred to as "Pythagorean" because of its similarity to the Pythagorean Theorem) in the early 1980s. Teams whose winning percentage is substantially better than the Pythagorean ratio may be playing way over their heads and thus may be expected to come down to earth. The article compares team's actual records with their Pythagorean records, as of July 1, 2004.

*--- Mike Breen*

**"Infinite Beauty: Fractals and Chaos: The Mandelbrot Set and Beyond," by Kenneth Falconer. Nature, 1 July 2004.**

In this article, after defining and discussing the importance of the Mandelbrot set, mathematician Kenneth Falconer provides a brief review of Benoit Mandelbrot's most recent book, *Fractals and Chaos: The Mandelbrot Set and Beyond*. This fourth book in Mandelbrot's *Selecta* contains edited reprints of his papers, including many seminal works, mainly from the 1980s. The book also provides overviews of Mandelbrot's work, historical background, and an explanatory chapter for the non-expert. In Falconer's view, this accessible book presents "a fascinating insight into [Mandelbrot's] journey of seeing and discovering... [and] gives a feeling for his philosophy and approach of experimental mathematics."

*--- Claudia Clark*

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