December 16, 2005
Mathematics Used For Discerning Immune Response To Infectious Diseases, |
December 16, 2005
Reading the MastersVictor J. Katz The Calculus Gallery: Masterpieces from Newton to Lebesgue. William Dunham. xvi + 236 pp. Princeton University Press, 2005. $29.95. Musings of the Masters: An Anthology of Mathematical Reflections. Edited by Raymond G. Ayoub. xvi + 277 pp. The Mathematical Association of America, 2004. $47.95 ($37.95 for members). The history of calculus is fairly well known and is competently described in many places. The prehistory of calculus begins with Archimedes in the third century B.C. and continues through the works of some Islamic mathematicians in the medieval period. The first two-thirds of the 17th century saw the development of integration by such mathematicians as Johannes Kepler, Bonaventura Cavalieri, Evangelista Torricelli and Pierre de Fermat; during the same period, Fermat, René Descartes and others devised approaches to finding maxima, minima and tangents. Toward the end of the century, Isaac Newton and Gottfried Leibniz wove together many of these earlier ideas into "the calculus," although they were not able to put their creation on a rigorous footing. The 18th century saw great development of the subject and its applications to more and more areas of science. But it was only in the 19th century that Augustin-Louis Cauchy, Karl Weierstrass and others gave calculus a foundation as secure as the paradigmatic mathematical topic of geometry. Although this outline is familiar, the mathematical details are less well known—and quite fascinating. Yet they are often skipped over in general histories. William Dunham, however, has remedied this situation in his brilliant book The Calculus Gallery, at least for the period beginning with Newton and Leibniz. Dunham picks out important results in calculus from the works of 13 mathematicians, sets them in the context of their times and explores the original proofs. Although some of the ideas he discusses are rather difficult, he manages to make them accessible to anyone with a background equivalent to that of a senior in college majoring in mathematics. As Dunham points out, these ideas are some of the masterworks of analysis, and anyone studying mathematics should be as familiar with these as someone studying art is with the paintings of Michelangelo and Renoir. I predict that Dunham's book will itself come to be considered a masterpiece in its field. Each chapter of The Calculus Gallery is an outstanding piece of exposition, often based on a talk Dunham has given. I will discuss just four chapters. In the one on Leonhard Euler (1707-1783), Dunham not only analyzes Euler's differential approach to the calculation of the derivative of the sine function but also shows how Euler found integrals of "bizarre" functions as well as sums of several interesting infinite series. In particular, Dunham demonstrates why the sum of the series of the reciprocals of the integral squares is p2/6, showing that Euler found this sum, along with numerous others, as a corollary to a general result on the sums of reciprocals of the roots of polynomial functions or functions represented by power series. In the chapter on Bernhard Riemann (1826-1866), we see one of the earliest examples of a "pathological" function, a function with properties that surprised many mathematicians of the day and led, along with other such functions, to the necessity for a reevaluation of intuition in analysis. Riemann's example was of a function that had infinitely many discontinuities on a finite interval and yet was Riemann-integrable. The chapter on Karl Weierstrass (1815-1897), in addition to discussing his influence on the general development of rigor in analysis, presents one of his own creations, a pathological function that is continuous everywhere but differentiable nowhere. Although Weierstrass's proof that his function satisfies those conditions is long and tricky, Dunham succeeds in making the explanation understandable to those with the patience to go through each step. Finally, in the chapter on René Baire (1874-1932), we get a marvelous discussion of the famous Baire category theorem. This result always appears in graduate courses on analysis, but for many of us, there was little besides the strange name by which to remember it. Dunham changes all that by presenting in great detail the context of the theorem. He then carefully states and proves it, using Baire's original proof. Finally Dunham demonstrates that the theorem enables one to prove easily many of the results that he had presented in the chapters on earlier mathematicians. For example, one corollary of the theorem is Georg Cantor's result that, given any sequence of distinct real numbers and any interval, there is a point in the interval not a member of the sequence. This immediately shows, as Cantor demonstrated, that the set of real numbers has cardinality larger than that of the natural numbers. Although The Calculus Gallery requires some background in advanced calculus for a complete understanding, the other book under review, Musings of the Masters, has few explicit mathematical prerequisites. The editor of this anthology, Raymond Ayoub, has collected 17 essays, originally written between 1869 and 1978, by prominent mathematicians, each reflecting the author's views on what the editor calls the "humanistic" side of mathematics. Most of the essays are, in fact, the transcripts of lectures given by these mathematicians to general audiences at such occasions as the International Congress of Mathematicians. Given that the book's title includes the word "musings," it is not surprising that these essays form a varied lot, dealing with topics as diverse as the existence of God and Goethe's opinions about mathematics. To help us put things in context, each piece is prefaced by both a biographical sketch of the author and a short summary by Ayoub. The book contains too many essays to discuss them all here individually, but let me offer my own brief musings about some of them. The oldest essay in the collection is James Joseph Sylvester's presidential address to the British Association in 1869. This noted English mathematician criticizes remarks of the biologist T. H. Huxley to the effect that mathematicians spend their time making "subtle" deductions from a few simple, "self-evident" propositions, and that mathematics "knows nothing" of observation, experiment, induction or causation. Sylvester then brings to bear his immense erudition to show that, in fact, Huxley's ideas are totally opposite to actual practice: Mathematicians have always observed and experimented. From this Sylvester concludes that the study of Euclid in English schools should be "honorably shelved" and replaced by more "living" topics that would better stimulate the minds of students. Interestingly enough, his wishes with regard to Euclid were achieved. But more than a century later it seems doubtful that, in England or elsewhere, the replacements do a better job of providing stimulation. G. H. Hardy's 1929 lecture on "Mathematical Proof" deals with the philosophies of mathematics that were then current. Hardy explains why he is not convinced that any of those philosophies could be acceptable to the vast majority of working mathematicians, including himself. As he points out, mathematicians believe that when they have proved a theorem, they really "know" something; the philosophies he discusses apparently are too restrictive in confirming such beliefs. The only contribution by a woman in this volume is a lecture by Mary Cartwright on "Mathematics and Thinking Mathematically," given at Goucher College in 1969. In her lecture, Cartwright attempts to distinguish abstract mathematical thinking from the applications of mathematics in the real world. But, as she points out, some of the best "abstract" mathematicians formulate ideas in terms of that real world. For example, her frequent collaborator J. E. Littlewood worked on antiaircraft gunfire in the First World War and thereafter often translated more abstract problems into the language of "trajectories." Several of the essays deal, directly or indirectly, with the history of mathematics. André Weil, in a talk given at the International Congress of Mathematicians (ICM) in 1978, presents his thoughts on the reasons for studying the history of mathematics. My own opinion is that too often he looks at ideas from a past era through a modern lens, neglecting the context in which the ideas arose. Raymond Wilder, in an ICM talk from 1950, introduces us to his views on the cultural bases of mathematics, ideas that culminated in a book published in 1974. Finally, André Lichnerowicz, in a lecture from 1955, discusses the meaning of being a scientist, both historically and in modern times. Although many of the essays in this collection will stimulate thought and discussion, it would have been no great loss if some of these "musings" had remained unpublished. Nevertheless, the collection as a whole will prove valuable in bringing its readers the thoughts of well-known research mathematicians on topics outside their areas of specialty. Reviewer Information Victor J. Katz is a professor of mathematics at the University of the District of Columbia. He is the author of a well-regarded textbook, The History of Mathematics: An Introduction (HarperCollins, 1993; second edition, Addison Wesley, 1998), which is also available in a brief version (Pearson/Addison-Wesley, 2003). Reading the Masters |
December 16, 2005
A suspenseful tale of Descartes's secretsBy Dorothy Clark, Globe Staff | December 14, 2005 Descartes' Secret Notebook: A True Tale of Mathematics, Mysticism, and the Quest to Understand the Universe, By Amir D. Aczel, Broadway, 273 pp., illustrated, $24.95 Amir D. Aczel, a professor of mathematical sciences at Bentley College in Waltham, has made a name for himself unwrapping the mysteries of mathematics in ways that enlighten the uninitiated. Mathematics is intriguing, and its history is full of intrigue, the stuff of secrets, spies, and cloak-and-dagger films. The titles of Aczel's previous nine nonfiction books attest to that -- the international bestseller ''Fermat's Last Theorem: Unlocking the Secret of an Ancient Mathematical Problem" (1996) and ''The Mystery of the Aleph: Mathematics, Kabbalah, and the Search for Infinity" (2000), among them. His latest, ''Descartes' Secret Notebook," is a first-rate suspense story. It begins with an unsuspecting Aczel setting out to research the life and work of Renee Descartes when he finds the philosopher/mathematician kept a secret notebook. Aczel's project turned into a detective adventure, revealing occult science, a secret brotherhood, political and religious controversies, a locked box, romance, obsession, a jealousy that may have had fatal consequences, and Descartes's purloined skull. Best known in nonmathematical circles for his mind-body philosophical statement ''I think, therefore I am," Descartes (1596-1650) was born in west-central France. His had a privileged upbringing, and the wealth he inherited afforded him the freedom to pursue his interests, including serving as a gentleman soldier, traveling, and, of course, studying and thinking deep thoughts. The deepest of these thoughts Descartes recorded in a special notebook. ''He began to believe that mathematics held the secret to understanding the universe. . . . He worked out ancient Greek problems in geometry, but he soon concluded that the power of geometry transcended pure mathematics: geometry held the secret to all creation," writes Aczel. Not only did he keep this journal hidden in a locked box, Descartes encrypted his entries, using symbols, number sequences, and obscure figures to ensure that the nature of his work would be disguised if his notebook was ever discovered. A quarter of a century after Descartes died,Gottfried Wilhelm Leibniz, 30, whom history would recognize as one of the greatest mathematicians of all time, came to the end of his 3 1/2-year quest to find the hidden writings. The caretaker of Descartes's trove allowed Leibniz to copy only 1 1/2 pages of them, but Leibniz nevertheless was able to decipher the entire notebook. As promised, he remained silent about his findings. Descartes's secret notebook disappeared about 20 years later. All that remains is Leibniz's copy, which bears an enigmatic notation that he made in the margin. Leibniz died in 1716, and it would be nearly 200 years before his transcription from Descartes's notebook was discovered. The transcription continued to perplex mathematicians until 1987, when French priest and mathematician Pierre Costabel broke the code. From its dramatic first-person opening, ''Descartes' Secret Notebook" unfolds with the narrative structure of a film, employing back story and plot points that make its telling dynamic. Aczel's closing exposition on cosmology in the 21st century is necessarily a bit academic, but in this brief section he links modern scientists' efforts to discover the secrets of the universe to Descartes' work and gives readers a glimpse of the proposed new models of ''the geometry of the universe," which could suggest that Descartes was on the correct plane, so to speak, with his search for ''a divine truth about mathematics, nature, and the human condition." A suspenseful tale of Descartes's secrets |
Deember 16, 2005
Canadian wins world Scrabble titleBy DAVID LAZARUS , Staff Reporter It's not as if Adam Logan didn't know he had a real shot. Before winning it all at the recent 2005 World Scrabble Championship in London, England, the Montreal resident had competed five previous times, never finishing better than fourth in six tries. "What can I say? It was a wonderfully exciting experience, and I was utterly delighted and not a little astonished," Logan, a 30-year-old mathematician, said in an e-mail interview shortly after his decisive victory over Pakorn Nemitrmansuk of Thailand. "And now, well… realistically I'll never be world champion of anything else." With the victory, Logan, who was born in Kingston, Ont., and grew up in Ottawa, earned $15,000 (US) and became one of the disproportionately high number of Jewish Scrabble players who excel at this game of forming everyday and obscure words that use letters in unusual combinations and, of course, the holy grail – two blank tiles. Logan's mother is Jewish and was born in Israel. In 1999, another Jewish player from Montreal, Joel Wapnick, also won the world Scrabble title. Here are some more statistics: four of the eight world champions so far have been Jewish – Logan, Wapnick, the United Kingdom's Mark Nyman and United States' Joel Sherman. Three of the seven players on the 2005 Canadian team were Jewish: Logan, Wapnick and Toronto's Zev Kaufman. Out of 105 players from 41 countries (including two from Israel) who competed in London, about 10 per cent were Jewish. So what gives? Logan believes that "whatever concentrates Jews in mathematics, computer programming, law and other intellectual professions presumably has the same effect for Scrabble." He said in his case, there must be a correlation between his being a mathematician and his success in Scrabble, but he's hard-pressed to say what it is. In addition, he said, friends from common ethnic backgrounds tend to learn about the competitive Scrabble scene together. But he also noted that while some of the best Jewish Scrabble players hail from Canada, the United States and Australia, this has not been the case in the United Kingdom or New Zealand. Certainly, for Logan himself, the win capped a dramatic tournament that, at certain points, could have gone another way. Logan won 20 out of 24 games in the first round, and then the next three in a row (he had to win three out of five) in the final round. Logan said he actually clinched a ticket to the finals in his 22nd game – against Australia's Naween Fernando. "I was behind for most of it, but caught up at the end when my opponent blocked the wrong spot and I played a seven letter word on my last turn." That word was "fifteen." Logan had those two magical blank tiles and won 432-386. Logan admits he was was nervous before the first game of the finals, and not without reason, as things turned out. Logan had to play catch-up for most of the game. He was down by more than 80 points at one point, and Nemitrmansuk was ahead 432 to 386 when Logan went out with the word "twistier" in the last move. Final score in the first finals game: 524 for Logan, 409 for Nemitrmansuk. "After the first game I was never seriously threatened in the finals," Logan said. Logan prepared for the championship, in part, by playing as often as possible. For two years, he taught at a university in Liverpool and "played with a world dictionary all the time." He spent his word-study time boning up on British words. "One of the problems with playing in the world championship is that a lot of words are acceptable there that aren't good in North America. "My main method of preparation is to use a computer to practise finding the words available in a set of letters, but playing and analyzing my games is also important." You also have to be lucky, Logan said, but that's less important in a tournament because of the number of games played. Luckily for Logan, he got both blank tiles – which makes you a favourite to win – in 10 of his games, and no blank tiles in only one or two of them. "It really helps," he said. Kaufman, who was Logan's roommate during the championship and finished 58th, was delighted by his friend's victory. "I'm very happy," he said, "Adam has been in every top 10, and in the past, just missed. He was overdue in winning it." Canadian wins world Scrabble title |
December 07, 2005
Mathematician's Insight Helps Unravel Knotty ProblemThe latest insight from Rice University assistant professor Shelly Harvey is the kind of idea that comes along rarely for a theorist in any discipline: It's an idea that is both simple and capable of explaining much. The elegance of the idea and the breadth of its descriptive power are most readily apparent to mathematicians within Harvey's chosen discipline of topology. Harvey discovered an underlying structure – which went unnoticed for more than 100 years – within the mathematical descriptions that topologists most often use to characterize complex knots. The work was described in a paper that recently appeared in the journal Geometry and Topology. "If someone comes up with a new mathematical theory that's 300 pages long with a lot of complex calculations, then you might suppose that the reason it hadn't been done previously was that it was too difficult," said fellow knot theorist and mathematics professor Tim Cochran. "However, real truth should be simpler and more beautiful than that, and this idea of Shelly's has the ring of truth to it. The moment I heard it, I knew she had hit upon something quite special." Harvey's discovery applies to a longstanding problem within knot theory, but it can best be understood within the larger context of topology. Topology is a branch of math that's sometimes called "rubber sheet geometry" because topologists study objects that retain their spatial properties even when they are twisted into odd shapes. A classic example is the topological equivalence of a donut and a coffee cup. The donut can be stretched into the shape of the cup, where the hole in the center of the donut becomes the handle on the side of the cup. Thus the property of "having one hole" is preserved. One of the underlying insights of topology is that some geometric problems depend not on the precise shape of objects but only on the way they are connected. In the classic example, 18th century mathematician Leonhard Euler proved that it was impossible to find a route through the Russian city of Königsberg that crossed each of the cities seven bridges just once. Topologically, the problem derives from the way the bridges connect the major islands of the city, so the result would be the same even if the primary shape of the town were – in the rubber-sheet analogy – twisted into a complex three-dimensional shape. In knot theory, topologists are concerned with the spatial arrangements of unbroken lines that are folded in knots – not unlike a tangled kite string or fishing line. While the study of knots may sound esoteric, it does apply to real-world problems. DNA, for example, are long, unbroken strings of amino acids that fold naturally into complex, knotted clumps. The knotting and linking of strands of DNA is a biproduct of natural cellular processes and their unknotting is necessary for the cell to survive. It is known that enzymes dubbed topoisomerases have the job of unknotting those clumps, and topologists have been collaborating with cancer researchers in recent years to attempt to find novel cancer treatments that capitalize on that. Topologists are keen to find ways to prove that two shapes, which may look very different, are truly inequivalent. One of the overarching goals in knot theory is to find a method that can determine equivalency in every case. Great attention has been paid to finding mathematical measures of a knot's complexity that can then be used to describe similarities and differences between knotted shapes. Sometimes these measures are actual numbers, like the so-called "unknotting number of a knot", and sometimes they are more sophisticated algebraic objects such as matrices or polynomials. One such measure developed 100 years ago by the Frenchman Henri Poincaré, which is reminiscent of Euler's Königsberg bridges problem, uses algebra to measure all possible paths that can be navigated in the space surrounding the knot, without ever touching the string itself. This collection of data is called the "fundamental group of the knot". "I realized that there's an algebraic structure within the fundamental group of a knot. Some of these paths are more robust than others," Harvey said. "What Tim and I subsequently determined is that this structure remains unchanged as you try to unravel the knots. It even survives in four dimensions, which turns out to be a particularly handy tool for knot theorists because four-dimensional problems – like the jiggling of a DNA strand within a cell – happen to be some of the most difficult topological problems to understand." Since Harvey's observation is so fundamental, it pertains as well to the study of many other topological objects, and these applications form part of her ongoing research program at Rice.
The research was funded by the National Science Foundation.
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December 06, 2005
Nobel laureate seeks 'cure' for warSusan Goodman, Israel 21c
 Prof Robert J. Aumann - Photo: AP
Bringing peace to the world is usually the preserve of political or religious ideologues - certainly well outside the domain of mathematicians. But this is the theme of the lecture that Prof. Robert J. Aumann of the Hebrew University of Jerusalem will deliver next week when he receives his Nobel Prize for Economic Sciences in Stockholm. |
December 06, 2005
16-Year-Old mathematics win $100,000 in Siemens Westinghouse prizesA 16-year-old, homeschooled California boy won a premier high school science competition Monday for his innovative approach to an old math problem that could help in the design of airplane wings. Michael Viscardi, a senior from San Diego, won a $100,000 college scholarship, the top individual prize in the Siemens Westinghouse Competition in Math, Science and Technology. Viscardi tackled a 19th century math problem and his new method of solving it has potential applications in the fields of engineering and physics. "He is a super-duper mathematics student," said lead judge Constance Atwell, a consultant and former research director at the National Institutes of Health. "It was almost impossible for our judges to figure out the limits of his understanding during our questioning. And he's only 16 years old," she said. Anne Lee, 17, a senior at Phoenix Country Day School in Paradise Valley, Ariz. and Albert Shieh, 16, a junior at Chaparral High School in Scottsdale, Ariz., shared the $100,000 top prize in the team category. They improved computer technology that could help locate the genetic roots of some inherited diseases like Alzheimer's, autism and bipolar disorder, reports ABC News. According to Here and Now, the Siemens Westinghouse Competition in Math, Science and Technology hands out $2 million in scholarship money and awards each year. The $100,000 scholarship winner of the individual prize is Michael Viscardi of San Diego, California. He studied a complex mathematical formula and his research could have practical applications in heat flow, magnetism and electrodynamics. The winners of the team prize are Anne Lee of Paradise Valley, Arizona and Albert Shieh of Scottsdale, Arizona. They'll split a $100,000 scholarship for developing new software that more accurately analyzes genetic data. O.Ch. 16-Year-Old mathematics win $100,000 in Siemens Westinghouse prizes |
December 06, 2005
NJIT mathematician receives Fulbright to study thin film science in ArgentinaEver wonder how manufacturers produce the thinnest and finest materials for cell phone displays and even smaller electronic products? If so, you are entering the burgeoning new world of "thin film" science and the life work of theoretical physicist and applied mathematician Lou Kondic, PhD, associate professor, department of mathematics at New Jersey Institute of Technology (NJIT). Kondic recently received a 2005-2006 Fulbright Scholar grant to study a dimension of thin film science focusing on the thinnest fluids. Kondic will travel next spring to Argentina for three months, where he will help physicists discover better ways to coat very delicate, almost invisible glass fibers. For almost two decades, scientists around the world have been searching for better polymers (more commonly known as plastics) to accomplish this task. These fibers are used to conduct electrical signals in microelectronics, optics and nanotechnology applications. Interest in thin film science has sky-rocketed because of recent scientific and technological breakthroughs. Aside from coating glass fibers, these new thin microscopic coatings are used to enhance the durability of products ranging from the outer covering of NASA space ships to army tanks in Iraq. In Argentina, Kondic will focus on how polymers are used to create a thin coating of a fluid film around an existing surface. Scientists consider fluids made of polymers to be complex. "My research will concentrate," said Kondic, "on the instabilities and patterns that form during the flows of these complex thin films." Kondic's work especially interests researchers in the computer industry who want to know more about how to reach uniform coverage of a rotating silicon surface with a thin film polymer. Kondic hopes his work in Argentina will shed more light. To achieve results, Kondic will use large scale numerical simulations to analyze the problems and find solutions. These simulations will be performed on a large computer located at NJIT, purchased with funds from the National Science Foundation. This kind of work is often referred to as mathematical modeling. Mathematical modelers, like Kondic, formulate mathematical equations that are believed to describe physical, biological, or sociological phenomena. The modelers take the known and accepted formulas of physics and/or chemistry and create mathematical equations that described unexplained phenomenon--such as why two fluids may adhere. While mathematical modeling may not always validate a fundamental physical or chemical principle, if the answer matches most of the presumed data, then researchers know they may be on the right path. Scientists in many fields, including biology, chemistry, physics, and engineering use mathematical modeling in their research. Economists, sociologists, and political scientists also utilize sophisticated mathematical modeling to deal with detailed problems associated with human behavior. Kondic is the author of more than 50 research articles. His most recent scholarly article, "On Nontrivial Traveling Waves in Thin Film Flows, Including Contact Lines" appeared in September of 2005 in Physica D. The National Science Foundation, NASA and the International Exchange of Scholars have supported Kondic's work. Kondic received his doctorate in physics from City College of City University of New York. New Jersey Institute of Technology, the state's public technological research university, enrolls more than 8,100 students in bachelor's, master's and doctoral degrees in 100 degree programs offered by six colleges: Newark College of Engineering, New Jersey School of Architecture, College of Science and Liberal Arts, School of Management, Albert Dorman Honors College and College of Computing Sciences. NJIT is renowned for expertise in architecture, applied mathematics, wireless communications and networking, solar physics, advanced engineered particulate materials, nanotechnology, neural engineering and eLearning. NJIT mathematician receives Fulbright to study thin film science in Argentina |
December 06, 2005
Mathematicians go where computers can'tFor all the advances in computing in recent years, many real-world problems still defy the capability of even the most advanced supercomputers and to address one such problem a team of mathematicians has been called for help. As part of the US government's $20-million initiative to have advanced mathematics pick up where sheer computing power is inadequate, the Oregon State University recently received a $647,000 grant from the Department of Energy. In this project, varsity mathematicians will try to model the flow of fluids through a porous medium, such as water through soil. It might sound simple but in practice this could be so extraordinarily complex that there are still more questions than answers, a university statement said. "The use of models that are suitable for laboratory experiments to describe processes on the scale of a watershed will bring any computer to its knees," said Ralph Showalter, professor and head of the university's maths department. "We're trying to connect information at the micro-scale to the big picture and for that we need new mathematical systems that at least give the computers a chance." The federal initiative will cover many topics ranging from the production of energy to pollution clean-up, manufacturing smaller computer chips and making new "nano-materials". The Oregon State University is one of 17 universities and eight Department of Energy laboratories participating in the initiative. The programme tackles problems of "multi-scale mathematics" - questions that span time scales from fractions of a second to years, and the atomic level to whole watersheds. The problems are so vast that they cannot easily be broken down into simpler questions that could be solved using traditional mathematical techniques and models. Even in the study of something as basic as water moving through soil, what you see depends on what window you look through, Showalter said. "You look through a microscope at a liquid moving for a few moments between soil particles and you observe a certain behaviour," he said. "Study the same process at the scale of a bucket or barrel and longer time scales, and the picture is incredibly different. And for our purposes, we might need to effectively model this process on the scale of a reservoir or a polluted field of groundwater over a period of decades." Showalter said that conceptually it is similar to trying to describe the path of a butterfly on a long migration rather than the up-and-down motion of its body with each cycle of its wings. Existing mathematics is able to do this averaging or "upscaling" in many cases, he said, but not yet in the more complex problems the Department of Energy initiative plans to address. Showalter and a colleague will try to create new mathematical models that are able to tackle these topics and then do analysis and simulation to study their accuracy. With success, they said, some day the problems may be simplified enough that a supercomputer can handle them. (IANS) Mathematicians go where computers can't |
December 06, 2005
Mathematical study backs asthma theoryA mathematical study has provided independent backing for warnings that regular use of some asthma medications may worsen the condition. A decade ago, a New Zealand professor reported that regular or frequent use of salbutamol or Ventolin, is associated with asthma "instability". Writing in the journal Nature, a team of international researchers applied the latest mathematical techniques to the data gathered by Otago Professor Robin Taylor. They calculated the risk of severe asthma attacks occurring within a month, based on the condition of a patient's airways. Dr Taylor says the study reinforces the message that long-acting medication, such as salmeterol, are more effective longer term than short-acting treatments. Copyright © 2005 Radio New Zealand Mathematical study backs asthma theory |
December 06, 2005
Nature teaches engineers new tricksBy Robert C. Cowen, Christian Science Monitor To help solve design problems, we should look to nature. For example, ants could help with traffic patterns, bees could provide insights on aerodynamics, and skunk cabbage may reveal new ways to regulate temperature. Through millions of years of evolution, many species have come up with elegant solutions to problems that crop up today in various engineering fields. We should learn from this biological wisdom, says Francis Ratnieks at the University of Sheffield in England. The biologist cited the foraging strategy of pharaoh's ants to illustrate this point in the July 28 issue of Nature. How these foragers move efficiently to and from a food supply provides an answer to what Dr. Ratnieks calls one of those "simple to state but hard to solve problems" that confront designers of traffic flow, electronic messaging, electricity transmission, and other network systems. How do pharaoh's ants know which way to go? By laying out a chemical trail in a pattern in which the junction of three trails forms a Y. The stem of the Y leads to or away from the nest. It also intersects the two arms at a wide angle while the arms form a smaller interior angle of about 60 degrees. The ants have enough geometrical instinct to sense the difference in angles and follow the right trail. Insects probably have evolved a variety of solutions to the foraging problem, Ratnieks notes. We should care about them "because human life depends more and more on engineering systems that must solve similar problems to function efficiently," he explains. In the field of aerodynamics, honey bees and bumble bees have some wisdom to share, according to research published this week in the Proceedings of the National Academy of Sciences. The flapping flight of these small insects carries aerodynamics into a region where conventional theory fails. These bees have evolved a flight system different from that of most small flying insects, find Michael Dickinson at the California Institute of Technology in Pasadena and colleagues. While some insects fly by swinging their wings in large arcs, the bees' wings move in much shorter arcs while flapping at a relatively high frequency. This short-arc high-speed wing motion gives the bees a much wider power range than other insects enjoy. Then there's the mystery of skunk cabbage temperature control. The plant's internal temperature remains around 20 degrees C. even when it's freezing — 0 degrees C. or below — outside. How the plant does it is still unknown. But research suggests that the regulation operates in accord with an algorithm based on mathematical chaos theory, explain Japanese scientists Takanori Ito and Kikukatsu Ito in the November Physical Review E. New Scientist magazine reports that the two scientists have now built and are testing a temperature control that incorporates the algorithm. One of the most important lessons we can learn from highly evolved biological systems is how to make our own engineered systems stronger, Ratnieks notes. "If there's one thing natural selection is good at, it is eliminating solutions that are not robust," he says. Copyright 2005, The Christian Science Monitor Nature teaches engineers new tricks |
December 06, 2005
Driving Challenges revealed by new roundabout formulaeA new formula by mathematicians at the University of Surrey shows an ideal trajectory for a car tackling a typical UK roundabout… something esure's new analysis shows is rarely achievable! In the last three years, more than one in every dozen UK car accidents has occurred while motorists approach or drive around roundabouts – with a quarter involving collisions while drivers change lanes - according to new research by internet insurer, esure.com. The annual bill is estimated at over £75m per year. esure.com's report, Roundabout Trajectories and Intersections, researched by Dr Anne Skeldon of the Mathematics and Statistics Department at the University of Surrey reveals the complex formulae that underpin roundabout trajectories and manoeuvres. It casts light on why roundabouts can be so potentially hazardous by showing that the 'textbook' way of negotiating a roundabout becomes near impossible when a number of cars are using a multi-lane roundabout at a time. The report describes the formula for the 'ideal' path that a driving school might recommend a driver follows to enter and negotiate a roundabout smoothly (above). However, further investigation shows that this formula becomes inadequate very quickly on a typical well-used roundabout. Dr Anne Skeldon said: "A driving instructor may show a learner how to approach, enter and leave a roundabout smoothly, but plotting the paths of multiple cars on a multi-lane roundabout makes it clear just how difficult negotiating roundabouts can be. There will be numerous points where the paths of different cars intersect - each of which is a potential accident point. In real life, this adds up to danger. "Cars can only follow textbook routes around a roundabout if there is a precise relationship between their times of entry and speeds. Unfortunately, with modern traffic this is rarely the case. "The criss-crossing of paths on busy roundabouts turns the desired smooth passage into a much more complex manoeuvre – both mathematically and in terms of the driving skills required. Drivers must be prepared to use their mirrors, switch lanes, signal and brake very precisely to ensure a safe passage." She added. To test the research, esure.com also analysed drivers' reports from 15,000 insurance claims where accidents had occurred on roundabouts. The results show that more than a quarter (25.3%) involved accidents during lane changes - although interestingly 20% occurred while one driver was stationary on - or in the entry lanes of – a roundabout. Mike Pickard, Head of Risk and Underwriting at esure, said: "Changing lanes on any road requires care. When you are doing this on a circular road with cars trying to enter, exit and manoeuvre accordingly it is easy to see why a roundabout becomes a recipe for problems. "Roundabout accidents result in claims totalling over £75m each year, so we would urge people to be especially careful when using them, particularly in bad weather." Driving Challenges revealed by new roundabout formulae |
December 06, 2005
Millennium problemsBy Queena Lee-Chua Inquirer BRENT Buela [BUDZ_012000@yahoo.com] has this intriguing question: "What is the hardest math problem you ever encountered?" There are a lot of math problems that fascinate me, but I will have to agree with most mathematicians that the hardest unsolved puzzles are the so-called Millennium Problems. In 2000, the Clay Foundation in Cambridge, Massachusetts, challenged experts to try their hand at seven difficult puzzlers, with $1 million for each at stake. The problems span applied and pure math, from topology and number theory to computer science and particle physics. The problems are too complex to be published here, but let me give you an idea of what they are about. Prime numbers The Riemann Hypothesis, formulated in 1859 by German mathematician Bernhard Riemann, makes a conjecture about prime numbers and the way they are distributed. It includes a zeta function and complex numbers. Interestingly, it became the basis for a plot in the TV series "Numbers" (Mondays at 9 p.m. on AXN). The plot got it right: unraveling the Riemann hypothesis may affect code-forming and code-breaking, as prime numbers are the basis of our most secure cipher systems. A related problem is the Birch and Swinnerton-Dyer Conjecture, posed by British mathematicians Brian Birch and Peter Swinnerton-Dyer in the 1960s. Involving elliptic curves (used by Andrew Wiles to prove Fermat's Last Theorem), the conjecture also helps us better understand prime numbers. The Yang-Mills Theory and the Mass Gap Hypothesis were formed half a century ago by physicists Chen-Ning Yang and Robert Mills to describe all forces of nature, except gravity. The physics works, but the math is not yet refined. Solving these will further our understanding of how the quantum realm (and ultimately, our world) works, and explain why, for instance, electrons have mass. The P versus NP Problem (posed by US mathematician Stephen Cook and countless others) concerns computers--specifically, how efficiently computers can solve problems. Type P tasks are easily done, type E can take millions of years, and type NP, well, seems to be between them, but no one really knows for sure. Are P and NP the same? If not, how different are they? (This came out in another episode of "Numbers.") The solution has big implications for industry, commerce and electronic communications, including the Internet. Differential equations The Navier-Stokes Equations, by French Claude Henri Navier and Irishman George Gabriel Stokes, both mathematicians, describe the motion of liquids and gases, such as water around a boat and air over a plane wing. These partial differential equations may look simple, but no one has yet found a formula to get the exact solutions. Thankfully, engineers use approximations to design better watercraft and aircraft. Knowing the exact solution will lead to advances in nautical and aeronautical science. The Poincare Conjecture, first raised by the French mathematician Henri Poincare a century ago, is about topology, the study of surfaces. For instance, stretch a rubber band around an apple's surface, and you can shrink it without tearing or allowing it to leave the surface. This same cannot be done with a donut. Poincaire was interested in four-dimensional analogues of apples and donuts, and understanding them better can help us design better silicon chips, understand the brain, and improve transportation and media. Even harder to understand is the Hodge Conjecture, about how complex math objects can be built from simpler ones. There is another question in topology, posed by Scottish mathematician William Hodge in 1950, and it is the most abstract problem of all. If you want to impress your friends, then say this: for projective algebraic varieties, Hodge cycles are algebraic cycles. However, like the Poincare Conjecture, solving this problem may revolutionize our world. Intrigued? For details about these puzzles, read Keith Devlin's "The Millennium Problems" (Basic Books, NY, 2002). Millennium problems |
December 06, 2005
God Created The IntegersBest-selling author and physicist Stephen Hawking explores the "masterpieces" of mathematics, 25 landmarks spanning 2,500 years and representing the work of 15 mathematicians, including Augustin Cauchy, Bernard Riemann, and Alan Turing. This extensive anthology allows readers to peer into the mind of genius by providing them with excerpts from the original mathematical proofs and results. It also helps them understand the progression of mathematical thought, and the very foundations of our present-day technologies. Each chapter begins with a biography of the featured mathematician, clearly explaining the significance of the result, followed by the full proof of the work, reproduced from the original publication. About the Author Stephen William Hawking was born on 8 January 1942 in Oxford, England and is one of the world's leading theoretical physicists. Hawking is the Lucasian professor of mathematics at the University of Cambridge (a post once held by Sir Isaac Newton), and a fellow of Gonville and Caius College. That he holds this post while almost completely incapacitated by severe motor neurone disease has made him a worldwide celebrity. Stephen Hawking has worked on the basic laws which govern the universe. With Roger Penrose he showed that Einstein's General Theory of Relativity implied space and time would have a beginning in the Big Bang and an end in black holes. His many publications include The Large Scale Structure of Spacetime with G F R Ellis, General Relativity: An Einstein Centenary Survey, with W Israel, and 300 Years of Gravity, with W Israel. Stephen Hawking has two popular books published; his best seller A Brief History of Time, and his later book, Black Holes and Baby Universes and Other Essays. God Created The Integers |
December 06, 2005
It all adds up: mathematical model shows which couples will divorceThere are no general laws of human relationships as there are for physics, but a leading marital researcher and group of applied mathematicians have teamed up to create a mathematical model that predicts which couples will divorce with astonishing accuracy. The model holds promise of giving therapists new tools for helping couples overcome patterns of interaction that can send them rushing down the road toward divorce. Psychologist John Gottman and applied mathematicians James D. Murray and Kristin Swanson will describe how the model was developed and how it enables Gottman to predict with 94 percent accuracy which couples will divorce after viewing just the first few moments of a conversation about an area of martial contention. They will discuss their work today at a press briefing during the annual meeting of the American Association for the Advancement of Science in Seattle "When Newton invented calculus it put science on a mathematical foundation and physics really took off," said Gottman who is a University of Washington emeritus professor of psychology and director of the Relationship Research Institute. "But psychology is a field that has lagged behind in using mathematics and there is no math in social psychology." Murray, who is an emeritus professor of applied mathematics at the UW and Oxford University, agreed, noting that a lot of people are phobic about mathematics and that psychology has not been exposed to models. "What we did is extract key elements into a model so that it is interpretive and predictive," Murray said. "The mathematics we came up with is trivial, but the model is astonishingly accurate." The model was developed using data collected from hundreds of videotaped conversations between couples in Gottman's laboratory. Physiological data, such as pulse rates also was collected and analyzed. The conversation reflected underlying problems the couple had and that is why the model is so predictive, according to Murray. "Before this model was developed divorce prediction was not accurate," Gottman added, "and we had no idea how to analyze what we call the masters and disasters of marriage -- those long-term happily married and divorced couples." The key turned out to be quantifying the ratio of positive to negative interactions during the talk. The magic ratio is 5 to 1, and a marriage can be in trouble when it falls below this. The mathematical model charts this interaction into what the researchers call a "Dow-Jones Industrial Average for marital conversation." "When the masters of marriage are talking about something important, they may be arguing, but they are also laughing and teasing and there are signs of affection because they have made emotional connections," Gottman said. "But a lot of people don't know how to connect or how to build a sense of humor, and this means a lot of fighting that couples engage in is a failure to make emotional connections. We wouldn't have known this without the mathematical model. "It gives us a way to describe a relationship and the forces that are impelling people that we never had before The math is so visual and graphical that it allows us to visualize what happens when two people talk to each other." It also is allowing researchers to simulate what a couple might do under different circumstances. For example, the model permits them to see what happens if a behavior changes, say a husband allowing himself to be influenced by his wife, and how that increases the number of positive interactions. Ultimately, this will allow therapists to do micro experiments with couples to strengthen their relationships, he believes. Gottman, Murray and Swanson, who is a UW research assistant professor of pathology and adjunct research assistant professor of applied mathematics, also will participate in an AAAAS symposium on the science of love and marriage that runs from 11 a.m. to 12:30 p.m. Saturday. ### For more information, contact Gottman at (206) 832-0300 or johng@gottmanresearch.com Murray at (206)-842-3909 or murrayjd@amath.washington.edu Swanson at (206) 221-6577 or swanson@amath.washington.ed It all adds up: mathematical model shows which couples will divorce |