Three articles in the March-April 2009 issue of American Scientist:
Hill's article addresses questions such as: Should a driver take an available parking space or keep looking? Should an applicant accept a job offer or continue the search? Although such decisions depend on an uncertain future, a result from probability can improve the chances of making a good decision. For problems like this, consider N/e candidates, where N is the total number of candidates, and record the best candidate up to that point. Then going forward from that point, choose the first candidate which is better than the recorded best. Searchers who use this method will choose the best overall candidate about 37% of the time. Hill explains variations of this problem and how the idea of optimal stopping relates to financial markets.
In the third article, Smithline (Center for Communications Research) writes about an early 19th century cipher found in the papers of Thomas Jefferson that had remained unbroken until now. The cipher was created by Robert Patterson, of the University of Pennsylvania, who wrote that it was a perfect cipher "absolutely inscrutable to all unacquainted with the particular key or secret for decyphering [sic]." Smithline was intrigued by this and set about breaking the cipher. Part of the ciphered text is above, in which Patterson transposed text so that sentences originally written horizontally, are written vertically. The tranposition was only one step, however, rows of transposed text were also rearranged and letters were added to the beginning of rows. Smithline used digraph frequencies, guessing, and dynamic programming techniques---similar to those used to assess the similarity in DNA sequences---to decipher the text, which turns out to be the preamble to the Declaration of Independence. See also: "Two Centuries On, a Cryptologist Cracks a Presidential Code," by Rachel Emma Silverman, Wall Street Journal Online, 2 July 2009.
--- Mike Breen
Recent "Math Matters" columns, by Arvind Gupta. Vancouver Sun, March and April 2009.
"Can Fractals Make Sense of the Quantum World?" by Mark Buchanan. New Scientist, 30 March 2009.
In this article, writer Mark Buchanan describes how physicist Tim Palmer is applying the science of fractals and black holes to quantum theory, potentially resolving some of the issues that have divided scientists starting with Albert Einstein and Niels Bohr. In a paper Palmer recently submitted to the journal Proceedings of the Royal Society A, Buchanan notes that "[Palmer] shows how the basic idea [of fractals] can account for quantum uncertainty, contextuality, and other quantum puzzles." To read Palmer's paper, "The Invariant Set Hypothesis: A New Geometric Framework for the Foundations of Quantum Theory and the Role Played by Gravity," go to www.arxiv.org/abs/0812.1148.
--- Claudia Clark
"What Would Jesus Bet?" by Alec Wilkinson. The New Yorker, 30 March 2009.
"Eine falsch angewendete Formel und ihre Folgen (A misapplied formula and its consequences)", by George Szpiro. Neue Zürcher Zeitung, 29 March 2009.
This article discusses the Copula formula, which was widely used in the financial community to assess the risk of simultaneous defaults of debtors, financial institutions, or companies. The article points out that this mathematical formula was not itself, as some media accounts have claimed, the cause of the recent financial meltdown. Rather, it was the incorrect use of the formula by non-specialistsm, who employed erroneous estimates of the correlation coefficient, that led to problems.
--- Allyn Jackson
"Aus der Mathematik in die Politik (From mathematics to politics)", by George Szpiro. Neue Zürcher Zeitung, 29 March 2009.
This installment of George Szpiro's monthly column on mathematics tells the story of two papers by Daniel K. Biss that appeared in the prestigious journals Annals of Mathematics and Advances in Mathematics. They were published in 2003 but soon thereafter found to contain a fatal error. Nevertheless it took nearly six years until the error was acknowledged by the author and the journals and that the papers were retracted. In 2008, Biss made the plunge into politics, running for and nearly winning a seat in the Illinois state legislature.
--- Allyn Jackson
"Fractal reality", by Mark Buchanan. New Scientist, 28 March 2009, pages 37-39.
This article describes the ideas of physicist Tim Palmer, who has used fractals help to explain some of the nonintuitive oddities of quantum theory. One of the key notions is that of an "invariant set". Buchanan explains what an invariant set is by pointing to the example of a swinging pendulum that due to friction eventually comes to rest. "[T]he invariant set is the one that describes the pendulum at rest," he writes. The universe also has an invariant set but one that is much more complicated and that in fact resembles a fractal. Buchanan writes: "Gravity and mathematics alone, Palmer suggests, imply that the invariant set of the universe should have a similarly intricate structure [that is, intricacy similar to a fractal], and that the universe is trapped forever in this subset of all possible states." The article goes on to try to describe how this fractal character at the heart of our universe explains some of the conundrums of quantum theory.
--- Allyn Jackson
"Minority Students and Research Universities: How to Overcome the 'Mismatch'," by Richard A. Tapia. Chronicle of Higher Education, 27 March 2009.
The controversial "mismatch" theory claims that underrepresented minority students are more likely to leave STEM (science, technology, engineering, and mathematics) disciplines because affirmative action encourages them to attend colleges for which they are unprepared. This theory also suggests that these students would be better served by less competitive institutions where they are more likely to succeed. In this article, Richard Tapia explains the flaws in this theory and the approach he and his colleagues at Rice University are using to increase the number of successful minority students in the STEM disciplines.
One of his objections to mismatch comes from his own experience. Tapia, a professor of mathematics at Rice University, was born and raised in Los Angeles to parents who had emigrated from Mexico. He was able to become a successful mathematician working at a top tier university because of his education at a top research university. He states that even though steering minority students to less-challenging institutions would increase the numbers of those receiving degrees in science and engineering, "numbers of degrees alone are not a good measure of success. Underrepresented minorities must be competitive with the overall population." He says that top research universities hire only professors who were also educated at top research universities, not Ph.D.s produced at minority serving institutions. So this approach only perpetuates the stereotype that minority students are simply less able to succeed in STEM fields.
A proposed solution is to simply accept more minority students into engineering and science programs at leading research institutions and then support them throughout their careers. Factors like SAT scores should be considered within a threshold instead of only taking the highest scores into account. He mentions that some Hispanic women who entered with modest SAT scores graduated from Rice with honors. This approach, coupled with other key components, make Rice the leading school in underrepresented minority doctoral recipients in science, math, and engineering. A few other top research schools are following similar approaches, but Tapia feels that a lot more can be done to eliminate the stigma of mismatch.
--- Adriana Salerno
"Just Visiting: Sir Roger Penrose," an interview with Megan Ogilvie. The Toronto Star, 23 March 2009.
Sir Roger Penrose, renowned for his work in mathematical physics, is interviewed prior to his invited talk at the University of Toronto Faculty Club. In this brief exchange he recalls the influence of his scientific father, his time in Ontario as a youth during World War II, his studie,s and successes. When asked how he would identify himself (mathematician, physicist, author, teacher, cosmologist, philosopher), he replies "That's a tough one." His training was in mathematics but his work went later into physics. "I think of myself as both. But it also depends on where you go. The mathematicians think of me as a physicist and the physicists think of me as a mathematician. (He chuckles.)" He also reveals that the work he is most proud of is his twistor theory---a way of looking at space and time---which he modestly says "hasn't caught on that much." Read more about Penrose on the MacTutor website.
--- Annette Emerson
"Rising Stars." Newsmakers, Science, 20 March 2009, page 1545.
"How to work out percentages," by David Budworth. Times Online, 20 March 2009.
The article stresses the importance of being familiar with percentages, since we see them all around and could be taken advantage of if we don't know how to figure them. The author explains "What is a percentage?", "Why are they useful?", "How to work out a percentage", and "How do I work out percentage parts?". He also points to news stories that mention percentages and websites that provide more methods and help.
--- Annette Emerson
"Microsoft did the math, added her to the board," by Alana Semuels. Los Angeles Times, 15 March 2009.
Maria Klawe, who has a Ph.D. in math and is president of Harvey Mudd College, has been named to the Microsoft board of directors. The Los Angeles Times article lists some of her varied interests and gives some of the highlights of her career. Regarding her interview with Microsoft she said, "Heaven for college presidents is when you get to talk to Bill Gates about your college." The article in The Chronicle of Higher Education is an interview with Klawe in which she explains how she was appointed to the board and how she will balance her new position with being president of Harvey Mudd. (A previous Digest has more on Klawe.)
--- Mike Breen
You may have missed it, but Square Root Day was 3/3/09. The idea of such a day came from Ron Gordon, a teacher in California, who first noticed the "oddity" in 1981 (specifically on another Square Root Day, 9/9/81). In recognition of the day, Gordon served his students root beer in square glasses. This short article notes that Gordon will turn 82 in October. Dan Vergano also writes about Square Root Day but adds that this April 1st---when written 4,012,009---is itself a rare perfect square (the square of 2003). Aziz Inan of the University of Portland discovered this and notes that from the 11th to the 100th century, the 21st century has the most such dates (24). This year's other square date occurred on March 5.
--- Mike Breen
"Computer---die rigoroseren Mathematiker? (The computer: The more rigorous mathematician?)", by George Szpiro. Neue Zürcher Zeitung, 11 March 2009.
"Math Matters with Dr. Gupta,"by Arvind Gupta. Vancouver Sun, 10 March 2009.
"The Four Color Problem Gets a Sharp New Hue," by Julie Rehmeyer. Science News, 6 March 2009.
Among the speakers at the Joint Mathematics Meetings in Washington, D.C. in January 2009 was Maria Chudnovsky of Columbia University. Chudnovsky spoke about the work she and her colleagues Paul Seymour, Robin Thomas, and Neil Robertson did to prove the strong perfect graph conjecture, an open problem for more than 40 years. As it is now known, the strong perfect graph theorem states that a graph is perfect if and only if it does not contain structures known as "odd holes" or "odd anti-holes." In this article, writer Julie Rehmeyer explains the importance of this finding for graph coloring and provides the layperson with a brief overview of the mathematical ideas that led to this theorem. She starts with botanist Francis Guthrie's speculation in 1852 that four colors might be sufficient to color a map such that "any two bordering [countries] were different colors." Eventually, she notes, mathematicians would turn their attention to graph colorings, asking how many colors it would take "to color any graph so that connected dots are always different colors." While the answer to that question has yet to be answered for all graphs, Rehmeyer goes on to describe how this question has been answered for perfect graphs. She also notes that many questions remain, including how to "efficiently detect perfect graphs" and how to find the "minimal coloring the strong perfect graph theorem has shown must exist."
--- Claudia Clark
"Rise of the Robogeeks," by Michael Brooks. New Scientist, 3 March 2009.
In this article, writer Michael Brooks describes the work being done by a few of the researchers who are attempting to design machines that can "do math." For example, Aaron Sloman, a philosopher and artificial intelligence (AI) expert at the University of Birmingham in the UK, notes that "human brains don't work by magic, so whatever it is they do should be doable in suitably designed machines." As Brooks describes it, Sloman's "primary aim, outlined in the journal Artificial Intelligence (vol 172, p. 2015), is to use such a machine to improve our understanding of where our mathematical ability comes from." Sloman believes that this ability develops during childhood: His current goal is to show a connection between this ability and the spatial awareness skills children must learn to negotiate the world. Sloman conjectures that these skills, which are encoded in our genome, could ultimately be encoded in a machine.
Another researcher, Alison Pease, uses AI programs to teach computers to do math; recently she and colleagues Alan Smaille and Markus Guhe have run "virtual math conferences populated entirely by digital mathematicians." She points out that while no really big discoveries have yet to be made by machines, in principle, they could happen. "Our big discovery would be how do we do mathematics, rather than how do we write a program that can generate really new mathematics," says Pease. "But hopefully one would lead on from the other."
--- Claudia Clark
"Why fewer women succeed in math," by Nancy J. White. Toronto Star, 3 March 2009.
In 2005, then-president of Harvard Lawrence Summers incited outrage by stating that the reason why fewer women succeed in mathematics was an innate gender difference. This statement inspired three Cornell scientists, led by Stephen Ceci, to get to work. In a recent article published in the Psychological Bulletin, they found that it was a variety of factors, none of them physiological, which kept women from pursuing mathematically intensive careers. Women tend to choose more people-oriented professions and have different lifestyle needs, such as having and raising children. Many of the math-intensive fields, and the senior posts at any field, frequently clash with womens' plans to start families. Largely, however, it seems to be a matter of preference. The researchers found among students who exhibit high math ability the girls also excel verbally while the boys don't, which means the girls have more options. Ceci says that it comes down to the fact that "a female is more likely to find medical research---finding a cure for a disease---more personally fulfilling than developing an internet search algorithm." On the one hand, Ceci and his team have good evidence that women are no less able than men to excel in mathematics. In fact, some steps can be taken to help women succeed, like adjusting tenure and promotion paths to include women's family plans. But it is unlikely that women will change their preferences, and the researchers admit that in the future the difference between the number of males versus the number of females in math-intensive careers will probably lessen, but not completely disappear.
--- Adriana Salerno
"Mathematician says his Sudoku secret is foolproof", by Dan Vergano. USA Today, 16 March 2009.
"Math has helped place a man on the moon and has counted the genes in our DNA," Vergano writes in USA Today. "But never mind all that. A mathematician says he has finally produced something that people really care about: a foolproof way to beat Sudoku puzzles." Vergano is referring to an algorithm presented in an article that appeared in the April 2009 issue of the Notices of the AMS. The article, "A Pencil-and-Paper Algorithm for Solving Sudoku Puzzles", by Winthrop University computer scientist J. F. Crook, describes in detail how to carry out the algorithm and proves why it always works. Some of the news stories, like the one in the Daily Mail, call Crook a "killjoy" for taking the fun out of solving Sudoku puzzles. Ram Murty, a mathematician at Queen's University in Canada, read Crook's article and told the Toronto Star that Crook has "codified what the average mind does when it looks at a Sudoku problem." Murty, together with his Queen's University colleague Agnes Herzberg, wrote an article for the June/July 2007 Notices called "Sudoku Squares and Chromatic Polynomials", which discussed Sudoku puzzles from a theoretical viewpoint. The article by Murty and Herzberg was covered by newspapers all over the world.
--- Allyn Jackson
"Russian Mathematician Wins Abel Prize", by Barry Cipra. ScienceNOW Daily News, 26 March 2009.
On 5 March 2009, Stephen Wolfram announced the May 2009 launch of a new search engine (which he refers to as a "computational knowledge engine"), Wolfram Alpha. This blog post on Wolfram.com started a flurry of responses and analyses in the blogosphere. Wolfram is famous for his "genius" and "prodigy" celebrity status (he got a PhD in physics from CalTech at age 20 and a Macarthur genius grant in 1981), and for creating Mathematica and Math World.
Bobbie Johnson, from The Guardian, points out that while "search engines such as Google compare search terms against billions of documents on its servers, before pointing to the pages on which the correct answer is probably kept," Wolfram Alpha gets closer to the holy grail for computer scientists, natural language processing. As Wolfram explains, he has completely given up the idea of processing language and using semantics for a more mathematical approach, using computation and algorithms instead. He writes: "I realized there's another way: explicitly implement methods and models, as algorithms, and explicitly curate all data so that it is immediately computable." To which Stephen Wildstrom, of Business Week, responds: "What does that mean? I have to admit I have no idea." And it is true that it is unclear what specific methods he is using from his blog post.
The idea most bloggers have of what this search engine is going to be able to do is answering questions directly (like "What is the location of Timbuktu?" or "What was the price of oil on February 3rd, 2007?") rather than providing links to pages that might contain the answer. Saul Hansel, of the New York Times, writes it is careless to compare Wolfram Alpha to Google, since the first is a technology and Google is a company that produces all sorts of technology, which will still come out on top from the business point of view. He believes that "the factual queries that Wolfram Alpha is geared to answer aren't the queries that make billions of dollars for Google." Even though we will all have to wait until May to see what this engine can really do, Christopher Dawson, an Ed Tech blogger for ZDNet, is very hopeful. He writes that Wolfram Alpha could "put students a lot closer to the information Nirvana promised by the Internet, in which they can simply get and use the information they need without having to weed through countless pages of ad-driven nonsense."
The website goes live in May, and the address is www.wolframalpha.com.
--- Adriana Salerno
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