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August 2011
Brie Finegold summarizes blogs on "the Code" and on Mathematics ability:
"The Code" combines a maths documentary with a treasure hunt," by Duncan Geere. Wired Blog, 27 July 2011; Last month, Marcus du Sautoy, the Simonyi Professor for the Public Understanding of Science at the University of Oxford, took BBC viewers on a threepart journey entitled "The Code," exploring the world through his lens as a mathematician. "The Code" refers to the hidden code of mathematics that "explains why things look and behave the way they do," according to du Sautoy. For instance, in the part entitled "Numbers", the series features the mating habits of cicadas, which congregate en masse every 13 years. Because 13 is prime and not very small, the cicada's exposure to predators is much lower than it would be if they mated on a composite number schedule. The next episode, "Shapes," includes a very pretty segment on soap bubbles and geometry. An associated competition challenges viewers to combine clues from the show and online games to solve a puzzle. While the clips on the show's site are not always accessible to users from the U.S., I was able to play the game Frog Hopper which requires a player to "pack" polygonal lily pads into a stream so that the frog can cross. The finalists in the challenge were announced September 1^{st}, and the winner received a mathematical sculpture from Bathsheba. Telegraph reviewers were impressed by the visual appeal of the production, and one reviewer gave the slightly backhanded compliment that "Given that he (du Sautoy) is a serious professor of mathematics at Oxford, he is remarkably engaging." But some commenters felt that the math was too dumbed down or was more of an advertisement for mathematics than it was educational.
"Math Challenged? Maybe You Were Born That Way," by Rick Green. Hartford Courant Blog, 11 August 2011; Quick! Estimate the number of candies in a jar to win a prize. Or guess which pile of potatoes is bigger. These sorts of activities test our number sense, and you can test yours in comparison to others. But does this skill relate to formal mathematical reasoning or ability? This is a question that psychologist and postdoctoral fellow Melissa Libertus set out to explore, and while she and her team found a statistically significant correlation between the number sense of preschoolers and their mathematical abilities, this finding does not establish that our mathematical abilities are predestined. The Bioblog gives a great breakdown of the major results of the paper. However, much of the media coverage pointed at the new study as an explanation for why many people are "just bad at math". Granted, the press release issued by Johns Hopkins University was titled "You Can Count on This: Mathematical Ability is Inborn, Johns Hopkins Psychologist Finds."  Brie Finegold "Math expert finds order in disorder, including stock market," by Steve Chapple. San Diego Tribune, 28 August 2011. George Sugihara is a mathematician who has followed the study of complex systems from biology to economics and back again. His domains of expertise are ecology and economicsboth complex systems characterized by dynamic networks of interactions and prone to catastrophic phase transitions. Sugihara began his professional life studying fish hatcheries, but his knowledge of instability soon led to adventures in finance and politics. As managing director of his own platoon of researchers at Deutsche Bank, Sugihara executed billions of dollars in trades to gather data for what is now known as the Sugihara Trading System. Last summer, Sugihara addressed the heads of the European Central Bank with his ideas on the European debt crisis. In Steve Chapple's biweekly column on San Diego's brightest minds, we get the highlights of Sugihara's life in the fast lane, and then Chapple pops the question: Is the economy about to improve? I'm not sure we need Sugihara to tell us that the chances are "not good". Instead, I wish Chapple had asked him to expand on his intriguing finding that "connectivity ... holds the potential for systemic destabilization, (and) hyperconnectivity can crash a system". (Photo: George Sugihara. Scripps Institution of Oceanography, UC San Diego)  Ben Polletta "The Trouble With Trillions," by Linton Weeks. National Public Radio, 22 August 2011. "A million has lost its mystique," and "a billion ain't what it used to be, either," according to this story. The premise is that, with the U.S. national debt in the trillions, most people have trouble comprehending "trillion" and have become bored by it. Author Weeks says "blame it on positional arithmetic, our everyday notational system," and gives a definition by 19^{th}century mathematician PierreSimon Laplace: "It is India that gave us the ingenious method of expressing all numbers by means of 10 symbols, each symbol receiving a value of position as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit." The author seeks thoughts of mathematician Daniel Rockmore of Dartmouth College, who says, "In a math way, there isn't really much interestingto meregarding the differences between billion, trillion, et cetera. That's the great advance of positional arithmeticjust add a few more zeroes, and for each one, get another 'order of magnitude.' [Positional notation] only grows 'logarithmically' in the size of the numberan extra multiple of 10 adds that single digit. Hardly noticeable!" As Weeks says, "the difference between 1 billion and 1 trillion is merely three additional zeroes." All this makes it difficult for people to really comprehend the high numbereven with analogies and examples. The piece concludes with Rockmore providing this perspective: "It's worth remembering that, in some cultures, the counting system goes 1, 2, many. Maybe that's the real point. Anything bigger than 2 is already confusing." No wonder the $14.4 trillion debt is confusing. The concept of positional notation is also the subject of "Magic numbers: the beauty of decimal notation," (The Conversation) adapted from a longer piece, "What if base10 arithmetic had been discovered earlier?" by David Bailey and Jonathan Borwein for their blog, Math Drudge.  Annette Emerson
Articles on math and the lottery: New Scientist reports that statistician Mohan Srivastava of Toronto, Canada, "learned to predict which Ontario Lottery scratch cards were winners... He suspected these winning cards weren't randomly assigned but the result of a piece of software called a pseudorandom number generator. Indeed, he found that cards containing a row of three numbers that each appear only once on the card's visible section were almost always winners." Wired magazine had reported back in January that although he could have purchased many cards, kept the winners and sold the rest, he instead sent sets of unscratched cards to the Ontario Lottery and Gaming Corporation, correctly noting which were winners and which were losersafter which that game was discontinued. Meanwhile, the Daily Telegraph article notes that Harpers magazine recently reported that another statistician from Texas, who had received her Ph.D. in statistics from Stanford University, had, over several years, amassed US$20 million through "scratchies" (scratchoff lottery tickets). The theory is that she "might have exploited the use of pseudorandomness" as well. Srivastava is quoted as saying that lotteries may have another motive: "The revelation of flaws actually stokes people's appetite for the game. People are coming out of the woodwork saying, 'I can do that too! I can find the pattern!'" Thus provides some motivation for those seeking more than luck.  Annette Emerson "Big Boost for Modern Einsteins," by Melanie Grayce West. Wall Street Journal, 18 August 2011. Charles Simonyi and James H. Simons, through their personal foundations, are donating a combined US$100 million to the Institute for Advanced Study (IAS), a private, independent academic institution founded in 1930 in Princeton, NJ. "The unrestricted gift is a challenge grant to launch a $200 million campaign to bolster the institution's endowment. The challenge concludes in four years and $9 million has already been matched." Simonyi, chairman and chief technology officer of Intentional Software Corp, Bellevue, WA, also serves as chairman of the IAS board of trustees and has made many other significant contributions to the institute. Simons, nonexecutive chairman of the board of Renaissance Technologies, president of Euclidean Capital, both in New York, and founder of Math for America, also serves as a vice chairman of the IAS board of trustees. The School Of Mathematics at IAS offers seminars and special programs.  Annette Emerson Media coverage of Pierre de Fermat's 410^{th} birthday, 17 August 2011. The day, August 17, started with buzz about Google's doddle honoring the French mathematician Pierre de Fermat's 410^{th} birthday. Fermat's birthdayand Google's attention to italso inspired the following media coverage: "Pierre de Fermat: 10 things you need to know about mathematician celebrated in Google doodle," by Gaile Lubianskaite (Mirror); "Why Pierre de Fermat is the patron saint of unfinished business" by Eoin O'Carroll (The Christian Science Monitor); "Google Doodle Celebrates 410th Birthday of Mathematician Pierre de Fermat," by Chloe Albanesius (PC Magazine); "Pierre de Fermat and His 'Most Difficult Math Problem' Celebrated by Google Doodle," (International Business Times); and "Pierre de Fermat Google Doodle celebrates French mathematician's impossible math problem," by Elizabeth Flock (Washington Post Blog), among many others. Christian Science Monitor writer O'Carroll summarizes Fermat's achievement as "Fermat is best remembered not for what he did, but for what he left undone. One day in 1637, while perusing his copy of an ancient Greek text by the 3rd century mathematician Diophantus, Fermat jotted a note in the margins that would drive mathematicians crazy for the next four centuries. Fermat's marginalia, which was written in Latin and later discovered by his son after he died, read: "It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second, into two like powers. I have discovered a truly marvelous proof of this, which this margin is too narrow to contain." In other words, an + bn can never equal cn, as long as a, b, and c are positive integers and as long as n is greater than two." Fermat's Last Theorem was solved in 1994 by British mathematician Andrew Wiles, whose proof took seven years to complete.  Annette Emerson Four articles on math and sports:
* Football. Two business professors used math to create an app called DraftOpt for fantasy football, to help fans decide whom to pick in a draft. Their app incorporates the rankings of several experts and uses probability to simulate a draft's future so fans can make analytical decisions during their drafts. The professors have also created an app for baseball fantasy drafts and hope to create apps for other fantasy sports. * Baseball. When this was written, the Philadelphia Phillies' first baseman Ryan Howard led his league, the National League, in RBIs (runs batted in), which would make him a good candidate for the Most Valuable Player award. Forman tries to determine how much of Howard's success is due to his prowess and how much is due to how often Howard's teammates get on base, giving him the opportunity to drive in runs. The Phillies have many good hitters, which gives Howard more chances to drive in runschances that good hitters on other teams don't have. In the last five years, Howard has come to bat with a total of 2835 runners on base. The next highest total goes to Mark Teixeira of the New York Yankees, who is more than 100 runners behind in that same time period. When Howard is measured with some new statistics, such as wins above replacement (a measure of how many more wins a player gives his team in comparison to a replacement player), he doesn't fare as well as he does with RBIs: He is seventh on his team in wins above replacement. * Basketball. When is the best time to take a shot? That is, should a shooter pass up a shot in hopes of the team getting a better shot later on? A model based on one developed to describe traffic flow tries to answer the question by incorporating the probability that a shot will go in, the quality of future shots, and the time left to shoot. Brian Skinner, a graduate student in physics at the University of Minnesota, developed a function he calls the shooter's sequence, which is a "recursive, quadratic map with no known analytical solution." He says that when he plays basketball he often wonders whether a player is shooting too much, which led him to this research. Basketball coaches are using models to analyze the game and determine strategy, but are reluctant to share their methods or their conclusions. Read Skinner's article in the ArXiv. * Golf. Robert Grober, a physicist at Yale University who has had a lifelong passion for golf, has found a better way to putt on a slope. He says to line up a putt from the position of the ball and from a few nearby points that are equidistant to the hole, which will lead to a small, diamondshaped figure centered on the "fall line" directly above the hole. Using the collection of points instead of only one should increase the golfer's chance of choosing the correct target line. The online article has a nice diagram of the process. See Grober's research article in the ArXiv.  Mike Breen "How Fibonacci Introduced The World To Numbers," an interview with Keith Devlin and Ira Flatow. National Public Radio, Science Friday, 12 August 2011. When you hear the phrase "personal computing revolution" you probably think about how mainframe computers the size of an entire room have changed into slim laptops and elegant smartphones. Turns out that the 13th century had its own "personal computing revolution". NPR's math guy Keith Devlin tells Ira Flatow the story of how Fibonacci revolutionized the world of international trade by introducing the western world to ten numbers, 0 through 9. Using the new numbers allowed any ordinary person to do the essential arithmetic needed for a business transaction, such as currency conversion and interest rate charges. Before this revolution only highly trained mathematicians were able to perform such calculations, since they were done using roman numerals. Essentially Fibonacci turned each person willing to learn some mathematics into his own personal computer. (Photo courtesy of Keith Devlin.)  Baldur Hedinsson "Explainer: the point of pure mathematics," by Lashi Bandara. The Conversation, 2 August 2011. Australian website The Conversation asks Ph.D. student Lashi Bandara to answer no small question, "What is the point of pure mathematics?" Lashi does a good job answering the question in a clear and understandable manner. She describes the main difference between pure and applied mathematics and gives examples of what type of problems pure and applied mathematicians undertake. To illustrate the importance of pursuing pure mathematics Lashi chronicles how research in pure mathematics played an essential role in creating one of the most important tools today, the modern computer.  Baldur Hedinsson "Ultimate logic: To infinity and beyond," by Richard Elwes. New Scientist, 1 August 2011. The continuum hypothesis may be hypothesis no more, as Richard Elwes reports in this ambitious tour of the metaphorstraining infinite architecture of modern set theory. The continuum hypothesis was first stated by Cantor, who discovered that the set of real numbers is strictly larger than the set of natural numbers, and hypothesized that there were no sets of intermediate size. It became infamous when it was shown to be unprovable in the rigorous axiomatic framework for set theory developed by Zermelo and Fraenkel. Godel, the father of unproveable propositions, suggested that the continuum hypothesis and other logical holes in the fabric of set theory could be mended by defining larger and larger theories, themselves in correspondence with infinite sets. Unfortunately, Godel's tower of theories tops out. Worse, it isn't unique. A multitude of towers of theoriesand other infinite axiomatic constructionsexist. In some of them, the continuum hypothesis is true; in others, there is one set intermediate in size between the reals and the integers; in still others, there are infinitely many such sets. Hugh Woodin's breakthrough is a "map" which may allow some of these multiple theories to be glued together into a supertheory. In the gluing, the continuum hypothesis comes out true. Even better, this supertheory would allow further theories to be built on top of it ad infinitum, so the logical holes in the underpinnings of mathematics can be made arbitrarily small.  Ben Polletta "Why Math Works," by Mario Livio. American Scientist, August 2011, page 80. In this article, theoretical astrophysicist Mario Livio explores an ageold question about the nature of mathematics: does it "exist in some abstract realm, with humans merely discovering its truths?" he asks. Or is it, in the words of Albert Einstein, a "product of human thought that is independent of human existence"albeit one that "fits so excellently the objects of physical reality." To Livio, both discovery and invention are critical: "humans invent mathematical concepts by way of abstracting elements from the world around themshapes, lines, sets, groups, and so fortheither for some specific purpose or simply for fun. They then go on to discover the connections among those concepts." At the same time, "our mathematics is ultimately based on our perceptions and the mental pictures we can conjure... We adopt mathematical tools that apply to our worlda fact that has undoubtedly contributed to the perceived effectiveness of mathematics." Mathematicians and scientists also "tend to select problems that are amenable to mathematical treatment." However, Livio argues, "this careful selection of problems and solutions only partially accounts for mathematics' success in describing the laws of nature. Such laws must exist in the first place! Luckily for mathematicians and physicists alike, universal laws appear to govern our cosmos." Why these laws exist, Livio cannot say, "except to note that perhaps in a universe without these properties, complexity and life would never have emerged, and we would not be here to ask the question."  Claudia Clark

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