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Math Digest

Summaries of Media Coverage of Math

Edited by Allyn Jackson, AMS
Contributors:
Mike Breen (AMS), Claudia Clark (freelance science writer), Lisa DeKeukelaere (2004 AMS Media Fellow), Annette Emerson (AMS), Brie Finegold (University of California, Santa Barbara), Adriana Salerno (University of Texas, Austin)


April 2008

"A life of unexpected twists takes her from farm to math department," by Billy Baker. The Boston Globe, 28 April 2008.

Gigliola Staffilani
Gigliola Staffilani. (Photo by Donna Coveney/MIT.)

 

This article describes the unlikely journey of Gigliola Staffilani from a six-year old girl in rural Italy who had never seen a book, to being the only female full professor in the MIT mathematics department at age 42. "Downplaying her own mathematical gifts," she humbly credits luck and the help of multiple people for her success. As a young girl, in part due to her older brother's influence, she dreamed of becoming a famous scientist. Despite her mother's reluctance, she went to high school and college (for the latter she had a fellowship that didn't cover housing, so she lived in the hallway of a palace run by nuns), and excelled in mathematics. She applied to graduate school at the University of Chicago and got in, but was rejected upon arrival because she hadn't passed her TOEFL (Test of English as Foreign Language). Refusing to go back to Italy, she hung around the math department until they accepted her (they expected her to quit in a couple of weeks, but she stuck). The only moment when she considered quitting was when she realized she couldn't pay for tuition, but legendary math professor Paul Sally rescued her by writing her a check for US$1000, stating that she was a "true talent" and just "needed some help." She went on to impress more people at the Institute for Advanced Study at Princeton while doing post-doc work, at Stanford on a tenure-track position, and ultimately at MIT, were she got tenure at age 36. Upon reflecting on her life's path, she simply says: "My life should have gone any other way than it did."

--- Adriana Salerno

 

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"Math Discovered or Invented?" by Joshua Hill. Canada Free Press, 28 April 2008.

Do numbers and other mathematical objects exist independently of our conception of them? In an upcoming issue of the European Mathematical Society Newsletter, many professionals will weigh in on the age-old question. According to the Platonist view, mathematical concepts are the underpinnings of our world and are therefore lying quietly until they are discovered. But some find that this leads to a theistic viewpoint and thus dismiss Platonism. As philosophical questions riddle mathematics, it will be interesting to see the musings of mathematicians on non-provable theories such as Platonism.

--- Brie Finegold

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"Bad Math = Mad Politics," by Carl Bialik. Numbers Guy Blog, Wall Street Journal Online, 25 April 2008.

In this article, "Numbers Guy" Carl Bialik writes about the mathematical formulas that have been used to determine the number of seats each state has in the U.S. House of Representatives, and the ongoing disagreement about each formula's fairness. As Bialik notes, the Constitution guarantees each state proportional representation in the House relative to its population. However, as early as 1791, a problem became apparent: what is done when the calculation results in a noninteger?

The first method used, the Jefferson method, is one of five so-called "divisor methods." Using this method, the apportionment for each state was determined by dividing each state's population by a number x such that the sum of all of the rounded-down quotients equaled the number of house seats. Unfortunately, it favored the most populous states.

The Hamilton method, used between 1852 and 1901, calculated each state's apportionment by dividing that state's population by the national population, multiplying that number by the total number of House seats, rounding down the results, then distributing y remaining seats to the y states with the largest fractional parts. The problem? It resulted in a paradox: Alabama lost a seat after the House size increased!

The Webster method, used in 1842 and from 1901 to 1941, is similar to the Jefferson method, but x was chosen so that the sum of all of the numbers, rounded up or down to the nearest whole number, equaled the number of House seats. With the Hill method---in use since 1941---x is chosen so that the sum of all of the numbers, rounded to the nearest whole number by geometric mean, equals the number of House seats.

In concluding the article, Bialik includes quotes 3 mathematicians, each of whom favors a different method.

--- Claudia Clark

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"Study Suggests Math Teachers Scrap Balls and Slices", by Kenneth Chang. New York Times, 25 April 2008.

equations

 

This article discusses new research, reported in the 25 April issue of Science magazine, ("The Advantage of Abstract Examples in Learning Math," Kaminski, Sloutsky, and Heckler) that seems to suggest that teaching mathematics through abstract principles works better than teaching with concrete problems. So-called "word problems", involving concrete situations like trains heading in opposite directions or the mixing of liquids in a cup, contain details that might simply distract students from the core mathematical principles. In a randomized, controlled experiment, Ohio State University researchers taught some students a mathematical concept using abstract mathematical ideas, and taught a control group of other students the same concept using word problems. Afterward, students were tested by seeing how well they figured out the rules of a mathematically based game. "The students who learned the math abstractly did well with figuring out the rules of the game," Chang writes. "Those who had learned through examples using measuring cups or tennis balls performed little better than might be expected if they were simply guessing." The article quotes David Bressoud of Macalester College, currently the president of the Mathematical Association of America, who called the study "fascinating".

--- Allyn Jackson

 

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"Mapmaker for the World of Influenza," by Martin Enserink. Science, 18 April 2008, pages 310-311.

Derek Smith is using mathematics to help epidemiologists keep us sniffle-free in the coming years. Smith applies his math expertise to create maps of influenza strains, which epidemiologists use when trying to determine which strain will be dominant next winter, so that the pharmaceutical companies can produce enough vaccine to combat it. In order to track flu strains and the differences between them, epidemiologists measure how well the antibodies an immune system generates to fight a known strain of the virus fare in the fight against a new strain. Smith saw these measurements—gathered into large, complex tables detailing the relative "distance" between the strains—as the perfect opportunity to create a map that would help epidemiologists by providing them with a way to visualize and understand the difference between the strains. Smith’s work led to an invitation by the World Health Organization to attend its small vaccination selection meeting each year.

--- Lisa DeKeukelaere

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"Frustration in Complexity," by P.-M. Binder. Science, 18 April 2008, pages 322-323.

There are many examples of complex systems: genetic algorithms, computers, the immune system, the brain, protein folding, and the stock market. What the author of the article points out is that even though "we know it when we see it," it is very difficult, even frustrating, to define exactly what scientists mean by "complexity." One common criterion has been cooperative behavior, or how global patterns arise from the way smaller parts interact with each other. Recently, "frustration" has emerged as a more general unifying theme, but it's a concept that's still not well-defined. A good example of dynamical frustration is a model in which three spins (just imagine arrows pointing up or down) are placed at the vertices of a triangle, and one wants to have all of them be anti-aligned with each other. Since this is impossible, we achieve "frustration." There are three well-studied manifestations of frustration. It can be of a geometric nature (like the famous Lorenz attractor), it can arise from having opposing tendencies at different scales, or it can be of a computational nature. These three groups are known not to be independent from each other, so indeed dynamical frustration might be the unifying thread that scientists who study complexity are searching for. There are other pieces to fit into the puzzle, like the fact that nonlinearity and dimensionality play an important role. Eventually, the author concludes, this might turn into the "queen of all sciences," since complex systems seem to explain so much of our world.

--- Adriana Salerno

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"Geometrical Music Theory," by Rachel Wells Hall. Science, 18 April 2008, pages 328-329.
"The shape of Beethoven's Ninth," by Davide Castelvecchi. ScienceNews, 24 May 2008, page 13.

Pitch
            classes

 

Recent research by Clifton Callendar, Ian Quinn, and Dmitri Tymoczko---all three of whom work in academic music departments---shows a novel way of using geometry to map out a musical score. Using mathematics, and even geometry, to study music is not new, but the three authors present a way to look at each moment in time in a musical score, like a vertical snapshot of the written music, as a point in n-space, where n is the number of "voices," or instrumental parts, in the piece. Callendar’s work also demonstrates how the mathematical concept of equivalence classes---a group of objects sharing a single property---can be applied to music. According to the reviewer, the new research may lead to new methods for teaching and visualizing music. (The research article, "Generalized Voice-Leading Spaces," starts on page 346 of the same issue.) At left: Musical orbifold, ordered pairs of pitch classes, courtesy of Rachel Wells Hall.

--- Lisa DeKeukelaere

 

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"Edward N. Lorenz, 90; scientist developed influential chaos theory," by Thomas H. Maugh II. Los Angeles Times, 18 April 2008.
"Eye for Detail." Newsmakers, Science, 25 April 2008, page 431.

Lorenz was known for 'the butterfly effect'

 

Edward Lorenz was "the first to realize what is now called chaotic behavior in the mathematical modeling of weather systems." His obituary issued by the Massachusetts Institute of Technology (where he served many years in the Department of Meteorology) also states that "in the early 1960s, he realized that small differences in a dynamic system such as the atmosphere---or a model of the atmosphere---could trigger vast and often unsuspected results." He wrote about these conclusions in a 1972 paper, "Predictability: Does the Flap of a Butterfly's Wings in Brazil Set Off a Tornado in Texas?," which lead to what became known as "the butterfly effect." The Los Angeles Times obituary notes that his work influenced a wide range of basic sciences, and "although the chaos theory was initially applied to weather forecasting, it subsequently found its way into a wide variety of scientific and nonscientific applications, including the geometry of snowflakes and the predictability of which movies will become blockbusters." Lorenz was a member of the National Academy of Sciences and won numerous awards, honors, and honorary degrees including the Crafoord Prize (a prize established by the Royal Swedish Academy of Sciences to recognize fields not eligible for Nobel Prizes) and the Kyoto Prize "for establishing the theoretical basis of weather and climate predictability, as well as the basis for computer-aided atmospheric physics and meteorology." The MIT obituary has more information. See also: "Edward N. Lorenz (1917-2008)," an obituary by Edward Ott, Nature, 15 May 2008 (p. 300).

--- Annette Emerson

 

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"Awards." Newsmakers, Science, 18 April 2008, page 299.

Terence Tao, a mathematician at the University of California, Los Angeles, is the winner of the 2008 Alan T. Waterman Award. The prize, a three-year, US$500,000 award, recognizes outstanding mathematicians or scientists under the age of 35. Tao, 32, was honored for his contributions in partial differential equations, combinatorics, number theory, and harmonic analysis. A Nati onal Science Foundation press release has more information.

--- Mike Breen

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"The Quirky Math of Voting: Oddities and Anomolies are always possible," by John Allen Paulos. Philadelphia Inquirer, 13 April 2008.

voting

 

The theme for Mathematics Awareness Month for 2008 is the mathematics of voting. In recognition of this theme, Paulos, a professor of mathematics at Philadelphia's Temple University, writes about the varied methods of voting and how they might change the outcome of an election. He points out that only one method is currently being used in the state primaries, the plurality vote, in which the candidate with the most votes "wins". But winning Republican nominees take all the votes from that state whereas the democratic primaries split the votes proportionally. Unfortunately, no detailed description is given for other voting systems, and a celebrated but perhaps disheartening result of Kenneth Arrow is discussed. Arrow showed that there is no method to derive group preference while satisfying a set of five rules for fairness, including that individuals' rankings of candidates be taken into account and that the winning ranking not mimic any one person's vote. Although Paulos's presentation of Arrow's Theorem might lead us to believe that "No voting system can be fair", this does not necessarily mean that different rules for fairness might not yield a different outcome or that all voting methods are equal in their reflection of the public's will. Paulos closes his article by endorsing Barack Obama.

Some of the same topics are covered in "Vote of no confidence", by Phil McKenna, New Scientist, 12 April 2008, pages 30-33.

--- Brie Finegold

 

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"Random Stock Market Quite Ordered," by Vivek Kaul. Daily News and Analysis, 12 April 2008.

John Allen Paulos, who wrote A Mathematician Plays the Stock Market, discusses the stock market in an email interview, saying that "random events can frequently seem quite ordered." Paulos, a professor of mathematics at Temple University, asserts that the best investments are in low cost broad-based index funds and attributes the successes of high profile investors like Warren Buffet to randomness and to the influence of their choices on others.

People make financial decisions not only on the advice of others but also according to their fixations on the importance of certain numbers, like estimated earnings, or highs and lows in stock price. Investors "anchor" on a particular value and react disproportionately to small changes in it. In one of many apt analogies, Paulos likens people's struggle to navigate the skittish stock market with a new driver's struggle to control a highly responsive Porsche. Since most people have a poor sense of what is random and what is not, investing in a stock is often more about guessing at the average person's perception of a company than about a company's actual performance. Investing in a broad-based fund might lead to just as much financial success as other options while canceling out some of the noise created by the press and others surrounding tidbits of otherwise inconsequential news.

--- Brie Finegold

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"Creeping Up on Riemann," Julie J. Rehmeyer. Science News Online, 5 April 2008.

visualization of Riemann zeta function

 

Ce Bian and Andrew Booker, two mathematicians from the University of Bristol, England, recently published a finding that could help solve the Riemann Hypothesis---one of the most challenging unsolved problems in mathematics, now that Fermat's Last Theorem and the Poincaré Conjecture have both been cracked. The Riemann Hypothesis involves a function, the Riemann zeta function, that holds information on how the prime numbers are distributed. In some places many primes are close by, but in other places there are large gaps. Decoding the distribution of the primes is an especially important problem given the nature of primes as building blocks for all other numbers. The Riemann zeta function falls into a class called L-functions, and the two mathematicians in Bristol expended over 10,000 computer hours to discover another L-function. Insights learned from their recently derived function could help them understand the secrets of Riemann's. (Image: Visualization of the Riemann zeta function, with colors indicating different function values, by Jean-Francois Colonna, CMAP/École Polytechnique.)

--- Lisa DeKeukelaere

 

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"Doing the iPod Shuffle," by Susan Stamberg. NPR Weekend Edition, 5 April 2008.

If your iPod is set on "random," why do the same songs keep popping up? A curious listener asked National Public Radio, which reached out to mathematical correspondent Keith Devlin for an explanation. According to Devlin, the answer lies in the words---not the math. Most people confuse randomness with the concept of being evenly distributed. A truly random sequence will naturally contain streaks and patterns, and therefore the repetition of certain songs on your iPod is a demonstration of the proper randomness of Apple's algorithm. To demonstrate the idea that random sequences contain repetition, Devlin cites a recent study in which a group of mathematicians assumed that the outcome of every pitch of every major league baseball game ever played had been random. The study found that under this assumption, all of the major records in baseball, such as Joe DiMaggio's hitting streak, still would have occurred.

--- Lisa DeKeukelaere

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"Creating Musical Variation," by Diana S. Dabby. Science, 4 April 2008, pages 62-63.

Johann Sebastian Bach

 

In this article, Diana Dabby, an assistant professor of electrical engineering and music at the Olin College of Engineering, writes about musical variation, "the technique of altering musical material to create something related, yet new." Adding additional notes around a theme, known as ornamentation, is one way to create variation. Joseph Haydn's F Minor Variations provide many examples of this. Composers may also create variation by using permutations and combinations of notes. Igor Stravinsky used this technique to compose his Variations: Aldous Huxley in Memoriam.

Composers have not limited themselves to rearranging groups of notes. For example, Pierre Henri mixed the recordings of three sounds---"a breathed sigh, the sung sigh of a musical saw, and a squeaking door"---to create his 1963 composition Variations pour une porte et un soupir. The composer John Cage takes the idea of variation even further: His "score" for Variations IV "consists of handwritten instructions providing a schematic that enables chance not only to decide the musical material [consisting of any sounds] but also to determine its order" so that each performance is vastly different. Another way to introduce variation is to use "chaotic mapping" with an existing work. This technique results in variations that can be "close to the original, diverge from it substantially, or achieve degrees of variability between these two extremes."

The Science website has some samples of the Haydn F Minor Variations, a chaotic mapping of the Bach Prelude in C (Bach is pictured at left), and a composition based on this chaotic mapping.

--- Claudia Clark

 

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"Building a mechanical calculator from 19th century plans," by John Cox. Computer World Australia, 4 April 2008.
"Difference Engine No. 2--No. 2," by Larry Greenemeier. Scientific American, July 2008, page 16.

Difference Engine No. 2

 

It is 7 feet high and 11 feet long, weighs about 3 tons, contains a multitude of gears, is operated by turning a crank, and can calculate solutions involving large polynomials without multiplying. It was designed in the 1840s by Charles Babbage but not built until the late 1980s by London's Science Museum. A second unit has been built and has been unveiled in the United States at the Computer History Museum in Mountain View, CA. "It" is Difference Engine No. 2, one of Babbage's attempts to eliminate human error by automating the creation of tables of values. These include logarithmic and trigonometric tables, widely used in his time for navigation, banking, and engineering.

The Difference Engine performs multiplication by using the method of differences---a means of finding successive values by adding previously calculated values. With the input of some initial values and muscle power, this machine can calculate and print the solutions to equations with 7th degree polynomials to 31 digits of accuracy, notes Andrew Carol, designer of a simpler LEGO Difference Engine. Carol also contends that, among all the machines designed and built to automate the process of creating tables of values, Babbage's design was "the most advanced and the most famous." (Photo of Difference Engine No. 2, courtesy of the Science Museum, London.)

--- Claudia Clark

 

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"Aztec Math Decoded, Reveals Woes of Ancient Tax Time," by Brian Handwerk. National Geographic News, 3 April 2008.
"Aztecs were whizzes at math," by Clara Moskowitz. MSNBC.com, 3 April 2008.
"Aztec Math Used Hearts and Arrows," by David Biello. Scientific American, 3 April 2008.
"How the Aztecs could count hand on heart," by Roger Highfield. Telegraph (UK), 3 April 2008.
"Aztecs did fractions, study says," by Neil Bowdler. BBC News, 4 April 2008.
"Aztec math finally adds up," by Alan Zarembo. Los Angeles Times, 4 April 2008.

 

Aztec map

It seems like the Aztecs had as much trouble as present-day Americans do when it came to figuring out their taxes. Two ancient codices (from A.D. 1540 to 1544) from the city-state of Tepetlaoztoc, recently deciphered by scientists Maria del Carmen Jorge y Jorge (National Autonomous University in Mexico City, Mexico) and B.J. Williams (University of Wisconsin-Rock County), give records of each household and its number of members, amount of land owned and soil types. Since landowners often had to pay tribute according to the value of their holdings, Jorge y Jorge explains, these texts were very detailed. The report, "Aztec Arithmetic Revisited: Land-Area Algorithms and Acolhua Congruence Arithmetic," published in the 4 April 2008 issue of Science, shows how the size of each parcel was calculated using a series of five algorithms, one of which coincidentally was also used by ancient Sumerians. One of the most puzzling features of the arithmetic in these records involved fractional symbols like hearts, hands, and arrows. But as Jorge y Jorge explains, these were actually very natural measurements related to the human body. The hand, for example, likely symbolizes the distance from the tips of the fingers of one outstreched arm to those of the other. (Image: Map (circa 1540) depicting parcels of land, courtesy of Library of Congress, Geography and Map Division.)

--- Adriana Salerno

 

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"At the Edge of Life's Code," by Thania Benios. Scientific American, April 2008, pages 106-109.

protein-protein interactions

 

Chris Wiggins (Columbia University) is using machine learning to model genes' protein-making activities. Biologists have a great deal of data about gene activity, but making sense of the data is difficult. Wiggins is trying to understand the fundamental rules of gene activity and describe mathematically the network of interactions between genes and proteins. He thinks that machine learning algorithms can find meaning in the vast amount of data, because "machine learning lets the data decide what's worth looking at." Higgins' work began with yeast and has now progressed to networks in higher organisms, including some human cells. His ultimate goal is investigating cancerous cells and learning which interactions lead to a diseased cell. (At left: Visualization of the modular structure (represented by color) inferred via Bayesian inference in an E. Coli protein-protein interaction network, where nodes represent proteins and edges represent the interactions between them. Image courtesy of Jake Hofman.)

--- Mike Breen

 

 

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