March 31, 2006
Collaboration, computers changing the nature of modern mathematical proofs, |
March 31, 2006
Harvard Foundation honors mathematician Treisman Philip Uri Treisman
Noted mathematician Philip Uri Treisman was recently honored by the Harvard Foundation for his notable contributions to the teaching of mathematical skills to educationally disadvantaged youth at the annual "Advancing Minorities and Women in Science, Engineering and Mathematics" science conference at Harvard's Science Center. Treisman is a professor of mathematics and executive director of the Charles A. Dana Center at the University of Texas, Austin. He is widely known for creating the Emerging Scholars Program (ESP), designed to increase the number of minority and other underserved students who succeed in mathematics. |
March 31, 2006
Hansen to receive Nemmers PrizeBy William Harms News Office  Lars Hansen
Lars Peter Hansen, the Homer J. Livingston Distinguished Service Professor in Economics, is one of two scholars to receive the prestigious 2006 Nemmers Prizes in economics and mathematics. |
March 31, 2006
PNAS announces 2005 Paper of the YearWASHINGTON -- The PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES' first Paper of the Year prize will be awarded to Karl Mahlburg, a doctoral candidate in mathematics at the University of Wisconsin, Madison, for his paper "Partition congruences and the Andrews-Garvan-Dyson crank." Mahlburg's article solves a critical part of a mathematical puzzle in number theory. It was chosen from among 3,000 papers published in the journal in 2005. Mahlburg will be recognized on April 23 at the PNAS Editorial Board meeting in Washington, D.C. This new prize acknowledges outstanding research articles published in PNAS and it will be awarded annually. PNAS is published by the National Academy of Sciences and is one of the world's most-cited multidisciplinary scientific journals, covering the biological, physical, and social sciences. Mahlburg's paper, published in October 2005, is available online at http://www.pnas.org/cgi/content/abstract/102/43/15373. An accompanying commentary on the paper is available at http://www.pnas.org/cgi/content/extract/102/43/15277. PNAS announces 2005 Paper of the Year |
March 31, 2006
Marriage of Math and Genetics Forges New Scientific LandscapeBy James Todd Computation and biology are being combined at Duke in research, training and curricula. Durham, N.C. -- The number of different bacteria and viruses that can attack the body is practically infinite. Understanding how the body produces immune cells in sufficient variety to ward off this vast array of pathogens -- and how this process can go awry and produce leukemia -- would be a scientific triumph and medical prize. The problem, however, does not yield to a simple set of experiments because of the huge diversity of germs, as well as of the white blood cells that attack them. That's why Lindsay Cowell and Garnett Kelsoe work together. She is an assistant professor of biostatistics and bioinformatics with expertise in computational methods, and he is professor of immunology with a wet lab that experiments directly with immune cells. Together, they discovered an important piece of the puzzle of immune cell diversity. Using mathematical models by Cowell and experiments by Kelsoe, they uncovered previously-hidden genomic markers in lymphocytes, a type of white blood cell. The markers play a role in the process by which DNA breaks apart and then recombines in lymphocytes to adapt to new pathogens. By identifying the location and function of these markers, the scientists could more easily differentiate healthy cells from cancerous ones in lymphocytic leukemia. "I think we wouldn't have been as successful working separately," Cowell said. "If I were doing this in the biostatistics department without working with Garnett, there would be no way to experimentally verify [her mathematical models]." Kelsoe said that without Cowell's computer-generated predictions to guide his experiments, he would have "30 billion places" to test in the mouse genome. "Until the dean gives me more money, that's too many," he said. Such collaboration is not unique at Duke. Joe Nevins, a professor of molecular genetics and microbiology, and Mike West, a professor of statistics and decision sciences, teamed up to study breast cancer. They developed a guide for doctors to choose among options for treating breast cancer based on how a woman's genetic profile affects her likelihood of tumors recurring. Another team is Susan Murphy and Terry Furey. Murphy, an assistant research professor of gynecologic oncology, and Furey, an assistant research professor of biostatistics and bioinformatics, are searching for the genetic culprits of a disease that afflicts millions of women: ovarian cancer. By combining her data and his computational expertise, they're identifying genes likely to be turned off abnormally in ways that can lead to ovarian cancer. Furey's computer predictions guide Murphy as she experiments to clarify which genes are involved. Assistant professor of biomedical engineering Lingchong You put together his own expertise in both biology and engineering to create a system that genetically regulates the reproduction of bacteria. Such a "gene circuit" might help control the spread of malignant cells in the human body or the environment. The new approach to research even shows up among undergraduate students like senior Jackie Ou. A math and biology double major, Ou works in Professor Fred Dietrich's bioinformatics lab with graduate student Charles Hall, researching how genetic information may be passed among bacteria in ways besides traditional inheritance. "I thought it was fascinating how you could use both computational and experimental tools, bring them together in one place, to answer one question," Ou says. Duke's provost and senior academic officer, Peter Lange, sees such an approach playing an important role in the university's academic future. "Bioinformatics and computational biology offer particularly good examples of how universities must embrace new approaches to interdisciplinary research even as they strengthen their departments," he says. The new frontier between genetics and mathematics extends beyond individual research laboratories to Duke programs and curricula. In 2003, for example, Duke's Institute for Genome Sciences and Policy launched a doctoral program in computational biology and bioinformatics. "We're trying to train a whole new kind of person that didn't exist ten years ago in the vast majority of computer science or biology departments," says Terrence Oas, the program's director of graduate studies and an associate professor of biochemistry and chemistry. By enlisting faculty from both the biological and computational sciences, the program is "trying to create new scientists that are hybrids of current scientists," Oas says. Other bioinformatics courses and seminars continue to sprout up. The scientists creating the classes say dual training is crucial for students to participate in the genomic revolution. Scott Schmidler teaches a "statistical methods in bioinformatics" course in Duke's Howard Hughes undergraduate program on "Making Meaning of Genomic Information." "What's happening in the biological sciences in general is that they're becoming quantitative," says Schmidler, an assistant professor of statistics and decision sciences. "It's not sufficient just to be trained in laboratory techniques…. It's also not sufficient the other way around." Schmidler points out that a number of common steps in genomics research, such as matching sequences of DNA or sifting through a soup of DNA material, involve statistics. "You do one experiment, you [get] 10,000 data points," he says. "You're looking for one or two really different, real effects out of 10,000 data points. That's an inherently statistical problem." Computing expertise also is essential for researchers who seek to take advantage of large central databases with genomic information from laboratories around the world. Because new findings are constantly being added to these central databases and distributed to specialized databases, biologists need to be proficient in using "genome browsers" to know which experiments are worth doing, says Simon Gregory, an assistant research professor at the Duke Center for Human Genetics. A laptop "for a lot of people should be their first [lab] bench, before they dive in … and do experiments," says Gregory, who organizes annual workshops on bioinformatics. Similarly, Duke students who major in computer science can now pursue a minor in computational biology. "The scale of data that people manage is enormous," says Owen Astrachan, a professor of the practice in computer science who teaches a course for first-year students on computer science and programming from a genomics perspective. "How to deal with a terabyte of information is something computer science can teach." Marriage of Math and Genetics Forges New Scientific Landscape |
March 31, 2006
Virtual swimmer to speed up athletes'Firstly we need to understand how water interacts with the human body during competitive swimming,' CSIRO researcher Chris Glendenning says. 'We are building a virtual model of a swimmer and are using mathematical techniques known as Smoothed Particle Hydrodynamics or SPH to run simulations of the virtual model swimming in a pool. 'In contrast to traditional methods, SPH describes fluid flow as the motion of individual particles. Using this technique means we will be able to more accurately simulate the interactions of water with a swimmer, which is particularly complex at the water's surface.' To start with, the researchers will scan the skin surface of a swimmer with a laser body scanner and use motion capture information to discover how they move through the water. By combining the skin surface and motion capture information, they will be able to simulate the skin surface for all the poses the swimmer adopts while swimming. 'Once we've built the virtual swimmer, we'll need to validate the simulations to find out whether what's happening in our computer matches what happens in reality,' Mr Glendenning says. 'Then we can sit down in front of our animation and ask our questions. By making slight changes to the swimming stroke and by re-running our simulations, we'll be able to find out whether the swimmer goes faster or not. 'We'll also be able to compare swimming styles between different swimmers to gain scientific insight into how each swimmer is moving through the water and even look into the effects of superimposing the techniques from different swimmers onto one another.' Mr Glendenning is working on this joint project between Monash University, the Australian Institute of Sport and CSIRO Mathematical and Information Sciences. Source: CSIRO Virtual swimmer to speed up athletes |
March 31, 2006
Renowned mathematician to deliver lecture at UNC Ashevilleby Asheville Citizen-Times published March 30, 2006 7:45 am ASHEVILLE - Renowned mathematician Joseph Gallian will deliver the fifth annual UNC Asheville Parsons' Lecture at 7 p.m. today in UNCA's Lipinsky Auditorium. Gallian's talk, "Breaking Drivers' License Codes," is presented by the university's Mathematics Department. The event is free and open to the public. Many states use complicated algorithms or formulas to assign drivers license numbers but keep the method confidential. Just for fun, Gallian attempted to figure out how states code their license numbers. In his presentation, Gallian will discuss how he broke the codes for many states using an important problem-solving technique that is rarely emphasized in mathematics classes. He will also discuss the merits of learning simply for the sake of curiosity. Gallian is a distinguished professor of teaching and professor of mathematics at the University of Minnesota Duluth, where he has worked since 1972. A prolific writer, he is president-elect of the Mathematical Association of America and winner of its Haimo Award for distinguished teaching and the Allendoerfer and Evans Awards for expository writing. Considered one of the founders of the undergraduate research movement in mathematics, Gallian has directed a National Science Foundation-sponsored undergraduate research program for more than 20 years. Each year, UNCA's Parsons' Lecture showcases well-known mathematicians who are able to explain their field of study to a general audience. The lecture is made possible by an endowment from an alumnus in honor of UNC Asheville Professor Emeritus Joe Parsons, who served in the Mathematics Department from 1952 to 1980. Renowned mathematician to deliver lecture at UNC Asheville |
March 31, 2006
The man who inspired the greatest mathematics solutionDhananjay Khadilkar Wednesday, March 29, 2006 23:50 IST In 1995, British mathematician Andrew Wiles cracked open Fermat's last theorem, a problem that had vexed the best of mathematicians for more than 350 years. There was another man, though, who had made an important contribution in this matter: Canada-born mathematician Robert Langlands, whose functoriality conjecture became the starting point in the solution of Fermat's last theorem. Sixty-nine-year-old Langlands was in Mumbai recently to deliver a lecture at the Tata Institute of Fundamental Research. The Wolf Prize recipient seemed modest about his contribution. "It was one of the many elements in finding the proof for Fermat's theorem," he says, admitting it would have been difficult to start off without the functoriality conjecture. The utility of his conjecture in proving Fermat's Theorem still surprises Langlands. "Even I was unaware of the full consequences of my conjecture," he says. "The proof surprised me." Langlands believes it required the genius of Wiles to comprehend the link between the two. On the practical application of Fermat's last theorem, he concedes it may sound exotic and irrelevant. However, he feels the proof for difficult theorems like it continuously brings in new mathematics whose implicit nature may have important applications. "There are a number of implicit ideas in the proof which need to be studied. We don't know how to get hold of them," he says. Was Fermat's theorem the final frontier for mathematicians? "It certainly was the most famous and fascinating of all theorems because it is a difficult problem of elementary mathematics. The next challenge could be the proof of Riemann hypothesis. But it won't attract the same attention," he says. The man who inspired the greatest mathematics solution |
March 31, 2006
People-friendly mathematicsOne person you can always count on -- pun intended -- is Vera Pless, professor emeritus of mathematics, statistics and computer science at the University of Illinois-Chicago. Pless will deliver the 2006 Bullitt Mathematics Lecture at 7 p.m. tomorrow in Strickler Hall at the University of Louisville. This lecture is presented in such an engaging fashion that it should hold the fascination of all people regardless of whether they're interested in math. It's geared for the general public. The topic for tomorrow's lecture is "Error-Correcting Codes: Practical Origins and Mathematical Implications." Basically this topic has to do with the efficient and accurate transmission of information from one place to another. Error-correcting codes can help improve high fidelity on compact disc recordings, transmitting financial information, gathering information from weather or communications satellites, and transferring data between computers. The free talk is presented by the U of L mathematics department and the Bullitt Endowment. People-friendly mathematics |
March 31, 2006
Singer's Killian lecture puts geometry in perspectiveWhen Isadore Singer arrived at MIT in 1950 to teach in the Department of Mathematics, he found that his department had little contact with the physicists also housed in Building 2. More than 50 years later, mathematicians and physicists have much more to bring them together, thanks in large part to work done by Singer during his long career, which earned him this year's James R. Killian Jr. Faculty Achievement Award. (PressZoom) - Singer, who was named an Institute Professor in 1987, gave the annual Killian Lecture in Kirsch Auditorium in the Stata Center on Thursday, March 23. His talk, titled "Some Geometry of the Past Half Century and Its Historical Background," reflected on the mathematical ideas that have developed over the past 50 years, bringing together diverse areas of mathematics, including geometry, analysis and topology. That time period has also seen the development of a closer connection between math and physics. "Mathematicians and physicists -- sometimes they have some ( connection ) and sometimes they don't. Certainly back in '66 there was not much," Singer said. When the faculty awarded Singer the Killian Prize last May, Professor Marcus Thompson, chair of the selection committee, said, "Isadore Singer is one of the rare mathematicians who is able to communicate with theoretical physicists in their own language and engage them in genuine collaborations. Most important is the attitude he brings to these collaborations: not the usual mathematical disdain for the physicists' lack of rigor but a conviction that mathematicians must try to understand why the physicists' methods work and to abet them in their efforts." The Atiyah-Singer Index Theorem, developed in the early 1960s, did much to bring mathematicians and physicists together. The theorem has had applications in many areas of theoretical physics, including gauge theory, monopoles, string theory and the theory of anomalies, among others. In 2004, Singer and Sir Michael Francis Atiyah of the University of Edinburgh shared the Abel Prize, considered the equivalent of the Nobel Prize for mathematics, for the development of their theorem. The Abel Prize citation described the theorem as follows: "Scientists describe the world by measuring quantities and forces that vary over time and space. The rules of nature are often expressed by formulas, called differential equations, involving their rates of change. Such formulas may have an 'index,' the number of solutions of the formulas minus the number of restrictions that they impose on the values of the quantities being computed. The Atiyah-Singer index theorem calculated this number in terms of the geometry of the surrounding space." The index theorem "energized mathematicians and physicists and started a new liaison between the two subjects," Singer said. Singer began his Killian talk with some reminiscences about his early days at MIT, which he said has "always been a very exciting place, and it seems to me it gets more exciting every year." After earning his doctorate from the University of Chicago, Singer arrived at MIT in July 1950, ready to start teaching summer classes. His first day included a late-night trip with his new colleagues to a Boston coffeehouse "which at midnight was a meeting place for derelicts, drunks and, apparently, mathematicians." That was followed by a very early morning tour of Boston and Cambridge, and Singer started thinking that "maybe I had found a new home, and that turned out to be true." Singer then described an example of the index theorem, known as the Gauss-Bonnet formula. The formula allows the large-scale shape of an object to be calculated from the local curvature of the object. As an example, he showed a slide of Earth, which is a solid sphere. A person on Earth's surface would not be able to see that shape, but could calculate the shape by measuring the curvature of the Earth at all its points. In other words, "the average of local information gives global information," Singer said. The Killian Award was established by the faculty in 1971 as a tribute to James R. Killian, former MIT president and chairman of the MIT Corporation. It is meant to "recognize extraordinary professional accomplishments by full-time members of the MIT faculty; provide a means for the communication of these accomplishments to the faculty, students, other members of the MIT community, and to the general public; and by so doing, honor the contributions made by Dr. Killian to the intellectual life of the Institute." Singer's Killian lecture puts geometry in perspective |
March 31, 2006
Strings As Structural Elements? Engineers Devise Mathematics For New Age StructuresScientists at the University of California, San Diego (UCSD) have devised two mathematical tools considered to be major contributions to the optimal design of a new generation of deformable bridges, buildings, shape-controllable airplane wings, radio antennas, and other alternatives to current structural technologies. Two reports will be published in the International Journal of Solids and Structures, with the first appearing in the April issue. The deformable characteristic is made possible with strong, ultra-light truss-like arrangements of rods suspended by strings or wires. The resulting structure incorporates tensegrity, a combination of "tension" and "integrity." "Although tensegrity structures are not yet part of mainstream design engineering, we think their amazing properties explain why you find this arrangement in spider webs, the protein cytoskeleton of cells, and many other biological structures," said Robert E. Skelton, a professor of mechanical and aerospace engineering in UCSD's Jacobs School of Engineering. Skelton and his students have pioneered the development of rigorous scientific tools to analyze the balance of forces and movement in many types of tensegrity systems. Unlike the arms and legs of a puppet, which hang from strings, a robotic tensegrity limb would be held in tension by a system of cables. Built-in actuators could pull those cables to direct the robot to wave or pick up a block. Skelton and postdoctoral fellow Milenko Masic describe in the April issue of the International Journal of Solids and Structures a mathematical method for optimizing the initial tension of strings within defined extremes of motion. In a second paper in the journal, which is available online, Skelton, Masic, and UCSD mathematics professor Philip E. Gill describe an optimization algorithm that will help tensegrity designers maximize the strength and minimize the weight of the rods and cables. A new generation of tensile-element materials has mechanical properties that are superior to those of traditional compressive elements, and the optimization algorithm by Skelton, Masic, and Gill incorporates the strength constraints of those materials. That information is used to help specify how to design a structure with the least material while retaining the desired stiffness as the structure changes shape. "A tensegrity-based wing could change shape as an airplane gains speed, but if the stiffness was relaxed the wing would fall off" said Skelton. "In mathematical terms, our algorithm directs the tensegrity structure to maintain its stiffness as it moves from one equilibrium position to another. The beauty of this approach is we don't have to continually use energy to maintain the shape at each new equilibrium position." The optimization algorithm relies on mathematical parameters that define the pitch (upward tilt), yaw (left or right swings), and separation distance of each of a series of identical rods. "For a tensegrity-based wing to maintain its stiffness as it changes shape, the algorithm defines an optimal 'surface' in the space of our three parameters," Skelton said. "We would then very selectively make some strings shorter and others longer in order to change the wing shape as we move along a predetermined equilibrium surface." Artists such as Buckminster Fuller and Kenneth Snelson appreciated the concept of tensegrity. They created sculptures with stainless steel rods and tension wires, but most engineers have regarded tensegrity sculptures as museum curiosities. "Tensegrity, as a concept, has been around for more than 50 years, but until now we have lacked the mathematics needed to make it an engineering tool," said Skelton. "There are lots of ways to put sticks and strings together that give you nothing but a useless pile. However, our new computational tools enable us to design structures such as an airplane wing that can extend and retract like a bird's wing." Skelton said optimized tensegrity structures with Mylar, Kevlar, titanium, and specialty steels may help the next generation of engineers use those and other strong, lightweight materials to reduce costs and increase performance in a variety of new ways. "The mathematical tools we're developing could revolutionize the way engineers design all sorts of structures," Skelton said. Source: University of California, San Diego, By Rex Graham Strings As Structural Elements? Engineers Devise Mathematics For New Age Structures |
March 29, 2006
2006 mathematics prize announcedSwedish mathematician Lennart Carleson has been named as the winner of the 2006 Abel Prize for outstanding work in the field of mathematics. 
The prize is worth about £520,000, and credits a discipline overlooked by the Nobel Prizes. |
March 22, 2006
Project to help pupils gain mastery over mathematicsPublished: Wednesday, 22 March, 2006, 09:17 AM Doha Time Business Reporter QATARGAS and Japanese shipping giant NYK Line have jointly launched a pilot project in Qatari schools for imparting the 'Kumon' learning system aimed at helping students gain mastery over reading and mathematics. As per the terms of the project, launched at Al-Israa Primary Girls' School, specialists from the Kumon Institute of Education will train teachers in using the method, supply materials and monitor its implementation, a Qatargas release said. The project was launched in the presence of Japanese Ambassador Masahiko Horie, NYK Line senior managing director Yasushi Yamawaki, Qatargas chief operating officer (administration) Hamad al-Baker, chief operating officer (commercial and shipping) Ahmed al-Khulaifi, director of the Education Institute at the Supreme Education Council Sabah al-Haidoos and Kumon Institute director Minoru Ota. The results of the project will be evaluated to determine how the initiative can be effectively implemented across the local education sector. Started 50 years ago by Japanese school teacher Toru Kumon, the method uses neither a classroom nor a tutoring model, but a guided "self-motivated-learning" approach, fostering improved concentration and study habits, and increased self-discipline and self-confidence. The heart of the Kumon learning system is a curriculum of over 20 clearly defined skill levels and hundreds of short assignments spanning material from pre-school all the way up to college. Each new assignment is slightly more challenging than the last. The progression is so gradual that students are able to acquire the skills to advance independently. The release said the Kumon approach allows each learner to study at the level in which they are comfortable, regardless of age or grade. Before a learner can advance from one assignment to the next, the material should be completed with a perfect score within a prescribed period of time. All work is graded and results recorded to determine the child has total command of the material and is ready to move on. Daily practice is the key to success in Kumon, the release said, adding the method is mainly used as supplement to the school curriculum. "It does not replace it (curriculum) but is a complementary and different kind of activity taking up on an average up to 30 minutes per day," the release said. Project to help pupils gain mastery over mathematics |
March 22, 2006
Marks for wrong maths calculationsAlana Buckley-Carr March 22, 2006 MATHS students will no longer be penalised for arriving at the correct answer using incorrect calculations under Western Australia's controversial outcomes-based education system. In a fundamental change to the way mathematics is assessed, the new OBE maths curriculum will reward students regardless of the process they use. Co-founder of lobby group PLATO, Greg Williams, said the move would produce high-school graduates who would not need to have a fundamental understanding of mathematical concepts. Mr Williams said that under the present system, students were awarded marks for the calculations they made, as well as the final answer. But under the OBE system, a student who gave the correct answer but made the wrong calculations to arrive at it would be given exactly the same mark. This would not equip students for a career and life in the real world, Mr Williams said. "If you're an engineer and your calculations are sloppy, the bridge that you are building falls down," Mr Williams said. PLATO's (People Lobbying Against Teaching Outcomes) concerns follow revelations that the Curriculum Council of Western Australia has turned away from the importance of spelling and grammar. The 2007 sample exams for English, media and aviation provide teachers with their first glimpse of what will be assessed under the new education system. All three samples state students should not be penalised for "poor spelling, punctuation, grammar or handwriting". Students are also permitted to draw answers or write them in dot form. "If you're not going to learn how to write English with correct grammar, spelling and continuous prose, where the hell are you going to learn it?" Mr Williams said. Mathematical Association of Western Australia president Noemi Reynolds said she did not believe the new system would result in a major change to student assessment. "But we have quite a mixture of opinions on OBE," she said. Ms Reynolds said many maths teachers had expressed concern after witnessing the confusion surrounding the implementation of a new English syllabus. "We understand and have sympathy for our fellow English teachers but maths teachers will not stand for a lack of support in the implementation (of the changes)," she said. State Education Minister Ljiljanna Ravlich said she would not speculate on how maths calculations would be marked until she had seen a sample exam. "I'm going to wait until I see a copy of an example paper until I comment," Ms Ravlich said. She said claims by PLATO that students would not be prepared for life after school was scaremongering. "Students will need to be able to demonstrate good grammar, spelling and punctuation. If they don't, it will result in students achieving lower marks in the examination," she said. "This is a pretty tough (English) examination. I think it really is quite rigorous." But federal Education Minister Julie Bishop said that while she was not attacking the concept of outcomes-based education, she did not approve of how the system was being implemented in WA. "The current debate centres around how it is working in practice and whether the (Curriculum Council) promotes sufficient guidelines to teachers," Ms Bishop said. "What I am hearing from teachers is that they need clarity on the knowledge and skills that students are to develop (under OBE)." She said spelling, grammar and punctuation had to be one of the highest priorities in the teaching and assessment of English. Marks for wrong maths calculations |
March 22, 2006
North Clayton High aims for record in Pi Day celebrationBy Jeffery Whitfield Speaking of his spot at the head of a line of 120 students — all holding a 5,000 link chain composed of multi-colored pieces of construction paper — wasn't difficult for Thomas Moore. He liked being in the front of the line. "We've been working on this a lot," said Moore, a sophomore at North Clayton High. Holding a white posterboard nearly as big as his chest with the marker-drawn "3" at its center, Moore led the chain that wrapped around nearly the entire length of the North Clayton High football field. The school celebrated Pi Day by assembling the chain, which the school's math department hopes will be the longest ever made. In mathematics, Pi establishes the ratio of the circumference of a circle to its diameter. "We're trying to break the record of 2,201 links made by a school in New Jersey," said Tarcia Jenkins, mathematics department head at North Clayton High. School officials picked the 5,000-link length after doing online research and discovering how long the chain was that the New Jersey school made. "We're praying we're correct," Jenkins said. Other classroom efforts to celebrate Pi Day included serving desserts and singing songs about Pi. Red velvet cake, sweet potato pie, pizza pie and Hawaiian pie were served in Moore's class. Students brought pies from home. "This was a departmental effort. We were looking for a way to make learning fun," Jenkins said. The school's central effort in the celebration was making the chain. Under sunny skies, with Moore at the front, students finished unveiling the chain at 1:59 p.m. during fifth period, with each link representing one digit in the non-terminating number Pi. The afternoon time was chosen because it matched the first seven digits of Pi, which starts 3.14159. "We didn't form a circle because we wanted to show a continuous line," Jenkins said. School officials plan to store the chain in a closet in the math department and call the Guinness Book of Records to see if North Clayton High has created a record. Jenkins said teachers chose to use construction paper to make the chain because it was inexpensive and each sheet could be made into 10 links. Each color in the chain represented a number. That way, when a number is repeated in Pi, it could be represented in the chain, Jenkins said. North Clayton High aims for record in Pi Day celebration |
March 22, 2006
Fry uncovers Indian maths genius's role in digital ageBy Leonard Doyle Published: 17 March 2006 Srinavasa Ramanujan, whose ideas underpin the internet revolution, was a poor Indian college dropout who nearly starved to death before he ended up at Cambridge in the early 1900s. Now acknowledged as a mathematical genius, Ramanujan, who died aged 33, is known as the most esoteric mathematical genius of the 20th century. His contribution to the digital age is to be finally recognised in film now that the writer and TV presenter Stephen Fry and India's leading writer-director, Dev Benegal, have decided to collaborate on a feature film on his life. Born in rural India, Ramanujan's life was extraordinary. By 13 he had mastered advanced trigonometry, and by 14 his teachers would stand dumbstruck in admiration at his mathematical prowess. Nicknamed "the man who knew infinity", Ramanujan excelled in number theory and modular functions. He also made significant contributions to the development of partition functions and summation formulas involving constants such as pi. But as a schoolboy he could not concentrate on other subjects besides maths and flunked his secondary school exams. He was poor and was often pushed to the point of starvation. Working as a clerk in the port of Madras, he wrote letters to Cambridge mathematicians in 1912 and early 1913. On his third attempt he found a sympathetic G H Hardy, who was keen to help the poor and disadvantaged over the "confident, booming, imperialist bourgeois English". Initially Hardy thought Ramanujan's 10-page letter, containing more than 100 statements of mathematical theorems, was a prank. He later realised that the "results must be true because, if they were not true, no one would have the imagination to invent them". Hardy, one of the pre-eminent mathematicians of the day, said that they were so advanced that "not one [theorem] could have been set in the most advanced mathematical examination in the world". Hardy said of the theorems that "many of them defeated me completely; I had never seen anything in the least like them before". He later recalled of Ramanujan: "I remember once going to see [him] when he was lying ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavourable omen. 'No,' he replied, 'it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.'" Stephen Fry came to learn about Ramanujan and G H Hardy while at Cambridge. The director Dev Benegal's passion to tell this story dates back some 20 years, to when he travelled the entire length of the river Kaveri in a small round boat made of dried palm leaves and came past the towns of Erode and Kumbakonam, where Ramanujan was born and studied. The story has haunted him for years, but never found any interest among the Indian film community. A chance encounter in 2005 led to Fry and Benegal discovering their shared passion. Benegal told the BBC: "What is amazing is that two people from two completely different backgrounds found a common language in the world of numbers and maths. "For me, Ramanujan's work and ideas are the DNA of what powers digital technology today. When your automated teller machines divide and arrange your money before coughing it up, they are all using Ramanujan's partition theory." Fry uncovers Indian maths genius's role in digital age |
March 22, 2006
Math Day is exhilaratingBy Joseph Serna, Staff writer LONG BEACH — Yolanda was ill and had to take a test a day late. Her 96 raised the class average from exactly 71 to exactly 72. How many students, including Yolanda, took the test? You have five minutes to answer. Good luck. Welcome to the seventh annual Pacific Life Math Day at The Beach, an event that pits almost 160 students from 26 high schools against each other in individual and school competitions involving algebra, geometry, trigonometry, number theory, statistics and more. Among those competing were students from Cabrillo, Millikan, Poly, Cerritos and Gahr high schools. Saturday's event at Cal State Long Beach's Walter Pyramid began with two individual tests, one multiple choice and the other fill-in-the-blank. The question above was one of the five fill-in-the-answer problems students had 25 minutes to complete. "It's not the computations so much as the problem solving that's important," said Robert Mena, chairman of the Mathematics and Statistics Department at Cal State Long Beach. While even a simple machine can do computations, Mena said these competitions emphasize the human ability to decipher a path to the solution, which can become an art form. Indeed, it would be hard to imagine a machine holding a fist full of hair or biting its lip while scrambling through the final five minutes of a test, an image that abounded among the students competing Saturday. Though the stress was obvious during the test taking, the event is not meant to be as competitive as much as it is about enjoying math, Mena said. "I worry about it becoming too intense sometimes," he said. "My message to the kids is to not worry about the clock, have fun, and learn to think on your own." A lot of the questions "require a lot of cleverness," said Ken Merryfield, associate chairman for the Mathematics and Statistics Department. "It's a way to reach for excellence, to demonstrate knowledge above and beyond grades and classes." One unique skill demonstrated to the joy of those in attendance was Shotaro "Macky" Makisumi's ability to solve a Rubik's Cube at an insurmountable speed. Makisumi, who placed 10th in the competition, solved three Rubik's Cubes, each in a different fashion. He completed one for speed, solving it in under 30 seconds, to which he apologized because it was so "slow." He solved another with his right hand while juggling two others with his left. And the third he solved behind his back, after taking a couple minutes to memorize its pattern. He is widely considered the fastest Rubik's Cube solver in the world, but said "I really think anyone with an OK memory can do it." If solving a Rubik's Cube is something everyone can do, beating Tedrick Leung of North Hollywood High School appears to be an impossibility. Though Leung was ranked third coming into the last round of competition, he came out on top following face-offs with the other competitors, retaining his title for the third year in a row. His school also won the overall competition. "Doing mathematics is exhilarating," Leung said, adding that the competition aspect also adds some fun to it. Perhaps he and his fellow competitors could help Yolanda's class raise their average. There are only 25 students in her class. Math Day is exhilarating |
March 22, 2006
Why I Quit HIV: The Aftermathby Rebecca V. Culshaw I want to start with an apology. I regret that I have not been able to individually answer every email I've received in the wake of my essay, "Why I Quit HIV," which recently appeared on Lew Rockwell. I am grateful for this forum, and I hope that I will be able to clear up some confusion people appear to have experienced. I'd also like to express my gratitude for the many, many positive and indeed inspirational letters I've received. Now I'd like to address some common questions I received. Many people inquired what impact the article would have on my job or career. I have not quit my job, nor have I been fired (so far). I've simply abandoned one area of research – I doubt I'll ever be able to publish in mathematical biology again, but that was the risk I knew I was taking. Thank you all for your concern. A few individuals kindly suggested that I inject myself with the blood of a late-stage AIDS patient. While such an act might sensationalize my viewpoint, there are a number of problems with such an "experiment." First, I can only imagine the non-HIV contaminants that might be found in such blood. Second, the data and results contained in the literature are sufficient to cast doubt on HIV. But most importantly, such an "experiment" would hardly settle anything, given the "latency period" of 10-15 years for progression to "AIDS." Many people insisted that I don't know what I'm talking about because I offer no alternative explanations for AIDS. There are many alternative explanations for "AIDS," or severe immune deficiency. The immunosuppressive effects of malnutrition, chronic drug abuse (pharmaceutical as well as recreational), parasitic infections, psychological stress, and other risks were well-established long before "AIDS" became recognized in the early 1980s. The fact is that most (but not all) AIDS patients do belong to risk groups whose members are subject to one or more of the above assaults. This fact can be checked by reading the annual CDC surveillance reports, although drug use is hidden because the CDC gives priority to "sexual transmission." And I should point out that the correlation between positive antibody tests and immune deficiency doesn't necessarily imply that HIV is the cause. To shamelessly steal an analogy from Peter Duesberg, just because long-term smokers often tend to develop yellow fingers along with lung cancer, does not mean that yellow fingers cause lung cancer. This is what we refer to in statistics as a "lurking variable" – correlated but not the cause, and hence confounding the issue. In any case, pointing out the flaws in an existing theory in no way obliges me to produce an alternative. I did receive several emails from people like myself who work or have worked with AIDS every day, people who have growing doubts or who have abandoned the theory altogether. These include doctors, pharmacists, biologists and social workers. "I volunteer in a Community Health Center, which was started twenty years ago, mainly for HIV positive people, though our clientele has expanded to all sections of our community. Also, as a former physician and then a psychiatrist, I was never able to understand this mysterious 'disease', and your writing has clarified a lot of that mystery." And there was also the following quote, from a social worker who works with HIV-positive prisoners: "Having worked with women with HIV in a prison environment, they always seemed more scared than sick." The letters that particularly affected me were those from people diagnosed with HIV, or who have lost loved ones to AIDS. I have lost count of the number of people who have told me that they are convinced their friends and lovers died from AZT poisoning rather than HIV. I have nothing to offer but my utmost sympathy. I've received mail from people who are HIV-positive and healthy for years without any AIDS medications. I have also gotten more letters than I was expecting from people whose lives have been seriously affected by false positive diagnoses, including a man who lost his position in the military after a positive HIV test, despite being at very little risk, and despite having had malaria and numerous vaccinations. He's out of work now. "I am a low-low-low-low risk group guy who has been diagnosed with HIV as a part of yearly tests (military). As a hetero[sexual], monogamous (10 years with one NEG[ATIVE] partner), non-IV drug using male...I was skeptical. However the "system" is not skeptical and it has subsequently tubed my previously successful career...The fact that I have had malaria and about a billion weird immunization shots (incl[uding] Anthrax) has not been brought up as possible source of false positive." For everyone who has been affected by AIDS in one way or another, and for those of you who have an abiding concern about doing science correctly, please know that I read all of your letters and you are in my thoughts. What I wrote was very personal, but it was also intended to serve another purpose: the average person should be aware of all the information that exists, not just what's been fed to us through the government propaganda machine. The individual citizen should be able to make informed choices about their health and their life. Let's not allow overzealous, misinformed public health agencies to take away that right from us. The article also attracted some comments from the blogosphere. The following comments appeared at a blog called Aetiology, which is owned and maintained by Seed magazine: "That's rich. First, as I mentioned, she's a mathematician. I don't know what her background is in infectious disease epi[demiology] (I contacted her but she did not respond), and she obviously shows little understanding of molecular biology in her comments about PCR (by her logic, any microbe shouldn't cause us harm because they are so tiny)." March 9, 2006 10:43 AM Yes, I am just a mathematician. I've never treated an AIDS patient, nor have I worked with HIV in the lab. But in the course of my work, I have studied both the microbiological and epidemiological aspects of AIDS, and the current HIV theory fails to explain either of these. Ever more convoluted explanations for HIV pathogenesis and epidemiology are not the signs of a mysterious virus, but rather the signs of a theory that is being shaped to fit the facts. The following quote, as well as the quote above, indicate some confusion over what I had to say about PCR. This comes from an aspiring microbiology student: "To understand my shock at the content of this article, you have to understand how incredibly steeped in the doctrine of the AIDS generation current education in Microbiology is. In the several years I have been working on my B.Sc, I have taken probably five courses that featured HIV or AIDS as prime examples of their precepts, have taken a course from one AIDS researcher, and have read about AIDS from several more. The idea of the AIDS virus has been one of the best known and studied examples of classical virology that we've ever had...I haven't read the whole article yet, but from the part I've read, it seems that it's written by a disgruntled HIV mathematician who got out of the race when she discovered that her paradigm and that of the establishment in this medical research field were radically different. From what I read, her science seems fine, except for some pretty disdainful and poorly-educated opinions on some of the best-used and most well-understood DNA techniques, such as PCR, or Polymerase Chain Reaction (the technique used by crime-scene units to amplify very small amounts of DNA so it can be identified, matched or analyzed): If something has to be mass-produced to even be seen, and the result of that mass-production is used to estimate how much of a pathogen there is, it might lead a person to wonder how relevant the pathogen was in the first place. First of all – to say this, a person needs to have absolutely no concept of how small DNA is, the degree of virulence of the pathogen being studied, and essentially no concept of how microbiology works. In short – a mathematician." The AIDS "Theory." To be very clear, I did not mean that HIV cannot be pathogenic because it is so small, I meant it cannot be pathogenic because it is so sparse; there is so little of it to be found. I was comparing PCR to a Xerox machine, rather than a magnifying glass. We need the Xerox machine because traditional virus culture techniques fail to detect HIV. Worse yet, PCR is used to measure "viral load," but this quantitative use of PCR has never been validated. As mathematician Mark Craddock has said, "If PCR is the only way that the virus can be detected, then how do you establish the precise viral load independently of PCR, so that you can be certain that the figures PCR gives are correct?" An alarmingly simple question, when you think about it; perhaps too simple for an AIDS establishment already fully committed to "surrogate markers," protease inhibitors and "combination therapies." And finally, a random blogger at LibertyPost.org appears to be lauding the toxicities of protease inhibitors: "And worse, she claims that protease inhibitors are killing HIV patients, 'And the leading cause of death in HIV-positives in the last few years has been liver failure, not an AIDS-defining disease in any way, but rather an acknowledged side effect of protease inhibitors, which asymptomatic individuals take in massive daily doses, for years,' when that's exactly what you would hope for (mortality drastically decreasing to the point that more deaths were the result of side effects) if protease inhibitors were in fact EFFECTIVE treatment for AIDS." posted on 2006-03-03 Finally, I received a series of odd emails from a prominent government HIV researcher, which includes the following quote: "The AIDS denialists are making some noise about you being the 'latest PhD researcher' to refute HIV as the cause of AIDS. The document they are citing...does not contain any new research, but only repeats a lot of the standard denialist disinformation." The opening of this email begins with the use of the pejorative and entirely unnecessary term "denialist," and this was followed by an "elucidation" of various aspects of virology that I imagine were intended to persuade me to change my mind, despite the fact that the arguments given were precisely those arguments that led me to doubt HIV in the first place. The arguments I presented were not intended to be "new research," but rather a short summary of the rather substantive questions that scientists such as Peter Duesberg and others have raised, which have still not been adequately answered. If the AIDS establishment is so convinced of the validity of what they say, they should have no fear of a public, adjudicated debate between the major orthodox and dissenting scientists to settle the matter once and for all. Yet all the major AIDS researchers have averted such a public debate, either by claiming that the "overwhelming scientific consensus" makes such a debate superfluous, or by saying that they are "too busy saving lives." In place of public debate, clearly politically motivated documents such as the Durban Declaration remain the establishment's standard response to dissenting voices. Even a cursory reading of this pathetic document reveals it to be a statement of faith, designed to divert attention from dissenters at the very moment when they were threatening to expose the orthodoxy in South Africa in 2000. To clarify an issue that has caused some confusion, it was not the mathematical models themselves that caused me to doubt HIV, but rather the scientific literature on which the models are based. Billions of dollars have been spent on HIV, and this has not led to a greater understanding of the virus, but rather to a series of unproven or incorrect speculations which have been widely trumpeted in both the scientific and lay press. Such a track record is indicative of institutional problems in modern biomedicine. The famous Ho/Shaw 1995 Nature papers are a typical example of this phenomenon. These were the papers largely responsible for popularizing HAART (the so-called "Highly Active Anti-Retroviral Therapy") and the "Hit hard, hit early" regime as a treatment for "HIV disease" and "viral load" as a measure of treatment success. The mathematical models used in these papers were claimed to show that HIV replicated furiously from day one – in contrast to earlier evidence suggesting it to be quite inactive. Even now, few people are aware that these conclusions were based on very poorly constructed mathematical models. Anyone who has taken a first course in differential equations can see that, if analyzed properly, the models predict the onset of AIDS within weeks or months after infection by HIV, before antiviral immunity. (For anyone interested in a mathematical refutation of the Ho paper, I refer you to Mark Craddock's analysis. Similar criticisms have been directed at the Shaw paper.) This example illustrates a central flaw in the HIV theory. The vast majority of the literature I've seen uses what is known as circular logic – you assume that something will happen, and then you mold the definitions, models, experiments, and results to support that conclusion. Craddock describes a typical example of circular logic in the Shaw paper: "They are trying to estimate viral production rates by measuring viral loads at different times and trying to fit the numbers to their formula for free virus. But if their formula is wrong, then their estimates for viral production will be wrong too." Such tactics, by definition, are excellent at maintaining the façade of a near-perfect correlation between HIV and AIDS, and of providing seemingly convincing explanations of HIV pathogenesis. But the resultant science does little to expand our actual understanding. To fully appreciate how such tactics became common, one needs to revisit the beginning of AIDS science. In 1984, HIV was announced as the cause of AIDS at a press conference before any supporting literature was published and had a chance to be critiqued by the scientific community. By the time the supporting papers were published, the lay press had all but declared HIV to be "the AIDS virus," and debate in the scientific arena was squelched. The current commonly used orthodox tactic of arguing by intimidation and forcing the conclusions to fit the facts became entrenched. Consider the time period in the scientific literature, when HIV went from being "the probable cause of AIDS" (1984) to simply "the cause of AIDS" (1985). What changed? What happened to make scientists come to such certainty? If you look at the actual papers, you'll see quite clearly that the answer is: Nothing. Returning to the Ho/Shaw papers, these have essentially been debunked by both establishment and dissenting researchers, on biological as well as mathematical grounds; they are now acknowledged to be wrong by the scientific community, and it remains a mystery how they were ever able to pass peer review in the first place. It is often asked, "Why should we care at this point? Those papers are 11 years old; our understanding has progressed since then." The short answer is that "viral load" and combination therapies are used to this day, despite the fact that they were originally based on these incorrect papers. Although current therapeutic regimens have been scaled back from the "Hit hard, hit early" dogma that was popular ten years ago, the fact remains that a large population of people have been, and continue to be, treated on the basis of a theory that is fundamentally unsupportable. Yet there is another answer to this question which is even more fundamental. It is a curious fact that few HIV researchers seem to be bothered by the events surrounding the Ho/Shaw papers. You might imagine that people might "care at this point" because of concern over the integrity of science. You might imagine that people might feel an urge to discuss how the papers got published, and if other such mistakes have happened since that time. You might imagine that the failure of the peer review process to detect such patently inept research would send off alarm bells within the HIV research community. You would be wrong. HIV researchers know the Ho/Shaw papers are wrong, yet they continue along the clinical path charted by the papers. They know that the quantitative use of PCR has never been validated, yet they continue to use "viral load" to make clinical decisions. They know that the history of HIV/AIDS is littered with documented cases of fraud, incompetence, and poor quality research, yet they find it almost impossible to imagine that this could be happening at the present moment. They know their predictions have never panned out, yet they keep inventing mysterious mechanisms for HIV pathogenesis. They know many therapies of the past are now acknowledged to be mistakes (AZT monotherapy, Hit hard, hit early), yet they never imagine that their current therapies (the ever-growing list of combination therapies) might one day be acknowledged as mistakes themselves. As a final thought, I am often asked, "How could medicine have made such a big mistake? How could so many people be wrong?." I believe the answer lies in the disintegration of scientific standards that have resulted, in large part, from the changing expectations of academic scientists. I'm an assistant professor, and my father is also a professor in the physical sciences, so I have had plenty of opportunity to see exactly how research expectations affect the quality of work we produce. It is clear to me that the pressure to obtain big government grants and to publish as many papers as possible is not necessarily helping the advancement of science. Rather, academics (and in particular, young ones) are pressured to choose projects that can be completed quickly and easily, so as to increase their publication list as fast as possible. As a result, quality suffers. This lowering of scientific standards and critical thinking has been apparent in many aspects of research for some time, and after several generations of students, it is now beginning to infiltrate the classroom – the textbooks and the undergraduate curriculum. It is germane at this point to indicate that many of the common arguments presented in response to the queries of HIV/AIDS skeptics are essentially some form of appeal to the use of low standards. (For example, "You don't need a reference that HIV causes AIDS," "The fact that HIV and AIDS are so well correlated indicates that it must be the cause," "HIV is a new virus, and new viruses will meet new standards," "Koch's postulates are outdated and don't apply in this day and age," "We don't need to worry about actual infectious virus, viral 'markers' should suffice," or "Real scientists do experiments; they don't write review articles on the literature.") All of these observations are eloquently summed up, again by Craddock: "Science is about making observations and trying to fit them into a theoretical framework. Having the theoretical framework allows us to make predictions about phenomena that we can then test. HIV "science" long ago set off on a different path...People who ask simple, straightforward questions are labeled as loonies who are dangerous to public health." It is this decline in scientific standards that I point to, when I am asked how so many people could be so wrong. Given the current research atmosphere, it was almost inevitable that a really, really big scientific mistake was going to be made. But we can still have hope for the future – hope that institutional and political pressures will no longer continue to cost lives, and hope that we will soon see honest dialogue and debate, free of name-calling and intimidation. Why I Quit HIV: The Aftermath |
March 15, 2006
Scientist John Barrow Wins Religion PrizeBy MATT CRENSONThe Associated Press Wednesday, March 15, 2006; 11:01 AM NEW YORK -- John D. Barrow, a British scientist and writer whose work explores fundamental questions about the universe and humanity's place in it, has won a religion award billed as the world's richest annual prize. In the past, the $1.4 million Templeton Prize has honored Mother Teresa, Billy Graham, and Soviet dissident Alexander Solzhenitsyn. It recognizes advancement in knowledge of spiritual matters. Barrow, 53, is known for his popular books and essays on cosmology, the study of the structure and history of the universe. His writing touches on topics such as the spiritual implications of the big bang, the nature of infinity and the limits of science in addressing some of humanity's most enduring unknowns. "People look to science to give them complete certainty, complete assurance, in the same way they look to religion," Barrow said. But in reality, he said, neither science nor religion can offer the kind of ultimate truths that humanity craves. "Religion is all about how we react to this uncertainty," he said. The award was to be announced Wednesday at a news conference at the United Nations. Barrow is the author of more than 15 books, including "Pi in the Sky," a consideration of the nature of mathematics and its relationship to the physical world. In 2002, he collaborated with Italian director Luca Ronconi on an experimental play that dramatized the concept of infinity in five parts. In his current research, Barrow is trying to demonstrate fluctuations in the value of the fine structure constant, a fundamental number related to the strength of the electromagnetic force, over the universe's history. Changes in the so-called "constant" would suggest the existence of additional dimensions besides the four familiar ones of space and time. A native of London, Barrow is currently a professor of mathematics at Cambridge University. In giving the award to Barrow, the Templeton Foundation praised the professor for having "given theologians and philosophers inescapable questions to consider when examining the very essence of belief, the nature of the universe and humanity's place in it." The Templeton prize is named for investor and mutual fund pioneer Sir John Templeton. It is administered by the Templeton Foundation, based in West Conshohocken, Pa. The award will be presented May 3 by Prince Philip in a private ceremony at Buckingham Palace. Scientist John Barrow Wins Religion Prize |
March 15, 2006
Barrow on the great basilica of natureBy John D. Barrow (March 15, 2006) Physicist John D. Barrow, winner of the 2006 Templeton Prize, reflects on the grandeur of the universe Cambridge University theoretical physicist John D. Barrow, the winner of the 2006 Templeton Prize for Progress Toward Research or Discoveries about Spiritual Realities, is a man of many talents. The author of 15 books and over 300 journal articles, as well as a play exploring the meaning of infinity, Barrow is perhaps best-known as the co-author, with Tulane University mathematical physicist Frank Tipler, of the 1986 book The Anthropic Cosmological Principle, in which he investigated whether the Earth is indeed fine-tuned for life. Reviewing the book for The New York Times, noted science journalist Timothy Ferris wrote, "I was infuriated by it, disagreed with it and loved reading it." In the following essay, written for the occasion of winning the Templeton Prize, Barrow reflects on the majesty of nature, our ever-expanding knowledge of the universe and why religion should always have a place at the table with science. A little over a year ago I was in a great church — the Basilica of St. Mark in Venice. Its predecessor was raised in the year 832 to house the mortal remains of St. Mark the Evangelist, which had supposedly been brought to Venice from Alexandria four years earlier by two Venetian merchants. They are alleged to have hidden the remains of the martyred saint under layers of pork so as to avoid the attentions of Muslim customs officials. The present Byzantine-style basilica, with its distinctive cluster of low domes, was begun in 1063 and consecrated in 1089. Today it sits next to the Doge's Palace on the edge of St. Mark's Square, attracting tourists and pigeons rather than pilgrims with a façade to launch a thousand postcards. I arrived at the church in the early evening with a small group of other scientists for a guided tour after it had closed to visitors for the day. When we entered, it was almost in total darkness. There are few windows, and they are small and far from transparent. We were asked to sit in the center, allowing just a few faint floor lights and an occasional electric candle to guide us to our seats. Above us there was only darkness. Then, very slowly, the light levels rose above us and around us, and the interior began to be illuminated by a discreet system of hidden sodium lights. The darkness around us gave way to a spectacular golden light. The arching ceilings above us were covered in a spectacular gleaming mosaic of glass and gold. Between the 11th and the 15th centuries nearly 11,000 square feet of gold mosaic was made, square by square, mixing gold with glass through a delicate process that is still not fully understood, to produce this sparkling golden sanctuary. Appearances can be deceptive. But, on reflection, what was more striking to me was the realization that the hundreds of master craftsmen who had worked for centuries to create this fabulous sight had never seen it in its full glory. They worked in the gloomy interior, aided by candlelight and smoky oil lamps to illuminate the small area on which they worked, but not one of them had ever seen the full glory of the golden ceiling. For them, like us, 500 years afterward, appearances were deceptive. Getting closer to the stars Our universe is a bit like that too. The ancient writers who celebrated the heavens' declaration of the glory of the Lord saw only through a glass darkly. Unbeknown to them and countless others who followed them, the universe has revealed itself by the instruments that modern science has made possible to be far bigger, more spectacular and more humbling than we ever imagined it to be. The universe appears big and old, dark and cold, hostile to life as we know it, dangerous and costly to explore. Many a philosopher of the past concluded that the universe was meaningless and antithetical to life: a bleak and black realm in which our little planet is a temporary outcome of the blind forces of nature. Yet, appearances may again be deceptive. Over the past 75 years, astronomers have illuminated the vault of the heavens in a completely unexpected way. The universe is not only big, but it is also getting bigger. It is expanding. Great clusters of galaxies are moving away from each other at increasing speeds. This means that the size of the universe we can see is inextricably bound up with its age. It is big because it is old. These huge periods of time are important for our own existence. We are made of complicated atoms of carbon, nitrogen and oxygen, along with many others. Maybe one day other forms of terrestrial intelligence will be made of silicon atoms. The nuclei of all these atoms do not come ready-made with the universe. They are put together by a long slow-burning sequence of nuclear reactions in the stars. It takes almost 10 billion years for this stellar alchemy to burn hydrogen to helium, and on to beryllium, and carbon and oxygen and beyond, before the dying stars explode in supernovae and spread their life-giving debris around the universe where it finds its way into grains of dust, planets, and ultimately into people. The nucleus of every carbon atom in our bodies has been through a star. We are closer to the stars than we could ever have imagined. Driven to understand Astronomy has transformed the simple-minded, life-averse, meaningless universe of the skeptical philosophers. It breathes new life into so many religious questions of ultimate concern and never-ending fascination. Many of the deepest and most engaging questions that we grapple with still about the nature of the universe have their origins in our purely religious quest for meaning. The concept of a lawful universe with order that can be understood and relied upon emerged largely out of religious beliefs about the nature of God. The atomistic picture of matter arose long before there could have been any experimental evidence for or against it. Out of these beliefs came confidence that there was an unchanging order behind the appearances that was worth studying. Great questions about the origin and end of the universe, possibly the sources of all observed complexity, and the potential infinity of space grew out of our religious focus on the great questions of existence and the nature of God. And, like all great questions, they can turn out to have answers that take us down unexpected paths, further and further away from the familiar and the everyday: multiverses, extra dimensions, the bending of time and of space – all may reveal a universe that contains more than is needed for life, more even than is needed for speculation. We see now how it is possible for a universe that displays unending complexity and exquisite structure to be governed by a few simple laws – perhaps just one law – that are symmetrical and intelligible, laws that govern the most remarkable things in our universe: populations of elementary "particles" that are everywhere perfectly identical. Reality's hidden logic It is to this simple and beautiful world behind the appearances — where the lawfulness of nature is most elegantly and completely revealed — that physicists look to find the hallmark of the universe. Everyone else looks at the outcomes of these laws. The outcomes are often complicated, hard to understand and of great significance – they even include ourselves – but the true simplicity and symmetry of the universe is to be found in the things that are not seen. Most remarkable of all, we find that there are mathematical equations, little squiggles on pieces of paper, that tell us how whole universes behave. There is a logic larger than universes that is more surprising because we can understand a meaningful part of it and, thereby, share in its appreciation. Once we thought everything in the universe was made of the things material that we find on Earth. We have now discovered that this too was only a first guess. More than 70 percent of the universe is composed of a form of dark energy whose precise identity is unknown. It reveals its presence by its dramatic effect upon the expansion of the universe. Unlike all other known forms of matter, which exert gravitational attractive forces on other forms of matter and among themselves, this dark form of energy responds repulsively to gravity, causing all material to accelerate away from it, creating an acceleration in the expansion of the universe that began to occur when it had reached about 75 percent of its presence extent. This discovery about our universe was a surprise – like discovering something totally unexpected about an old friend. Again, appearances were deceptive. So with the universe, as it was that evening in St. Mark's, things are not always as they seem when we look upward. The whole is so much more than the sum of its parts. The architects of our religious and scientific pictures of the universe, and the many commentators on their meanings that followed them, could see only a small part of what there is and knew only a small part of what it has to teach us about our place in the universe. We begin to see afresh the extraordinary nature of our local environment and the link that attaches life to the vastness of space and time. Appearances can indeed be deceptive. Knowing what we don't know There are some who say that just because we use our minds to appreciate the order and complexity of the universe around us, there is nothing more to that order than what is imposed by the human mind. That is a serious misjudgment. Were it true, we would expect to find our greatest and most reliable understanding of the world in the everyday events for which millions of years of natural selection have sharpened our wits and prepared our senses. And when we look toward the outer space of galaxies and black holes, or into the inner space of quarks and electrons, we should expect to find few resonances between our minds and the ways of these worlds. Natural selection requires no understanding of quarks and black holes for our survival and multiplication. And yet, we find these expectations turned upon their heads. The most precise and reliable knowledge we have about anything in the universe is of events in a binary star system more than 3,000 light-years from our planet and in the subatomic world of electrons and light rays, where it is accurate to better than nine decimal places. And curiously, our greatest uncertainties all relate to the local problems of understanding ourselves – human societies, human behavior and human minds – all the things that really mattered for human survival. But that is because they need to be complex: Were our minds simple enough to be understood, they would be too simple to understand. In all the science we pursue, we are used to seeing progress. Our first attempts to grasp the laws of nature are often incomplete. They capture just a part of the truth, or they see it through a glass only darkly. Some think that our progress is like a never-ending sequence of revolutions that overthrow the old order, condemned never to converge upon anything more definitive than a more useful style of thinking. But scientific progress doesn't look like that from the inside. Our new theories extend and subsume old ones. The former theories are recovered in some limited situation – slow motions, weak gravitational fields, large sizes, or low energies – from the new. Newton's 300-year-old theory of mechanics and gravity has been superseded by Einstein's, which will be succeeded by M theory or its unknown successor in the future. But in a thousand years' time schoolchildren will still study Newton's theories and engineers will still rely upon them just as they do today. They will be the simple limiting form for slow motions and weak gravity of the ultimate theory, whatever it turns out to be. In our religious conceptions of the universe, we also use approximations and analogies to have some grasp of ultimate things. They are not the whole truth, but this does not stop them being a part of the truth — a shadow that is cast in a limiting situation of some simplicity. Our scientific picture of the universe has revealed time and again how blinkered and conservative our outlook has often been, how self-serving our interim picture of the universe, how mundane our expectations, and how parochial our attempts to find or deny the links between scientific and religious approaches to the nature of the universe. Sir John Templeton has sought to encourage this impartial dialogue in the firm belief that religion and science can supply mutual illumination and appreciation of the wonders of our universe and inspire us to seek out and comprehend the truth in new ways – a truth that is unfailingly unexpected and so often not at all like it first appears. John D. Barrow is a theoretical physicist and Research Professor of Mathematical Sciences at Cambridge University and winner of the 2006 Templeton Prize for Progress Toward Research or Discoveries about Spiritual Realities. Barrow on the great basilica of nature |
March 15, 2006
Odd numbers, odd chants - a successful equationBy GAYLE PEREZ THE PUEBLO CHIEFTAIN Prairie Winds Elementary one of the schools using musical approach to better comprehend and enjoy mathematics. The cheers and chants heard coming from Prairie Winds Elementary School on Monday afternoon weren't in support of the school's athletic teams. The shouts were coming from third-, fourth- and fifth-graders who were learning their latest math definitions and procedures. "Odd numbers are a little strange, 1-3-5-7-9 is their range," the third -graders roared . "Expanded notations, stretch the value out," yelled a group of fourth-graders while stretching their arms out. The chants, raps, songs and movement are all a part of the unique STAND OUT Math program designed to help students remember their math definitions and procedures. "We took the Colorado state standards and wrote raps and songs to help students acquire the math vocabulary," said Jean Sweet, an educational consultant for STAND OUT Math based in the Denver area. "We found that this is a great way to help students learn and remember the math vocabulary." Prairie Winds is one of several District 70 elementary schools that either have incorporated or soon will be incorporating the curriculum in their classrooms. On Monday, Sweet, a former classroom teacher, held an assembly at Prairie Winds to review with the school's third-, fourth- and fifth-grade students some of those definitions and procedures in preparation for the upcoming Colorado Student Assessment program exams. Sweet held mini lessons with students in each grade level, reviewing concepts ranging from multiples and factors to symmetry, expanded notation and patterns. The students knew most of the raps, having participated in STAND OUT Math sessions since the fall. Prairie Winds Principal Stephanie Russell said she implemented the poetic math program this fall to help improve the math scores. "We felt we had a pretty good handle on math but needed something to help with the vocabulary and language of math," she said. "This about language and movement and it goes along with our philosophy here at Prairie Winds." The program incorporates short, catchy phrases sung in a rap or chant format to help students better remember vocabulary. Most of the raps include movement to help students better visualize the concept. "Multiples go up, up, up," said the students as they raised their arms in increments. Fourth-grade teacher Sarah Woods said the STAND OUT Math program has been a great addition to the school's regular math curriculum. "I love it," she said. "I think this is something that has been needed - teaching the language of math. By hearing and learning the language of math, the students are better able to understand what is being asked of them." Woods compared the program to preschoolers learning the alphabet by reciting the A-B-C song. "It takes a lot of repetition for things to imprint in a child's mind," she said. "This is a way in which we can help imprint some of the vocabulary of math through the songs and movement." Woods said she has been doing a 20-minute session twice a week with her students and so far, the results have been satisfying. "When we are working on math, I hear the students break into the chants," she said. "It's neat to see that they are using these tools to help them learn and achieve." Third -graders Christen Leedom and Mattison Dabovich, said the STAND OUT program has helped them to enjoy math a lot more. "When I get stuck on a problem, I try to remember a chant that might help me out," said Christen, 9. "If there is a chant that I can do, then I can usually figure out the problem." Mattison, 8, said the raps she's learned have helped make math easier to understand and more fun to learn. "It has helped me a lot with my math problems and I'm hoping it will help me when I take my CSAP in math," she said. "My favorite one," she added, "is odd numbers are a little strange." Odd numbers, odd chants - a successful equation |
March 15, 2006
Johns Hopkins' JAMI wins Math Society of Japan awardThe Johns Hopkins University-based Japan-U.S. Mathematics Institute will be awarded the Mathematical Society of Japan's prestigious Seki-Takakazu Prize during a ceremony to be held in Tokyo on March 27. JAMI is the third recipient in the prize's 11-year history. "This is a great honor," said Steven Zucker, director of JAMI and a professor in the Krieger School's Department of Mathematics. "It shows the level of esteem with which Japanese mathematicians hold JAMI. We are very proud of the interactions with the Japanese that have developed and the bonds that have been strengthened through JAMI." The institute was founded in 1988 to further cooperation between the two countries in mathematical research. Named for a 17th-century Japanese mathematical prodigy, the Seki-Takakazu Prize was established in 1995 to honor people and organizations that have supported and encouraged the development of mathematics in Japan over a long period. The prize consists of a gold medal and a volume of Seki Takakazu's collected works, said Sadayoshi Kojima, president of the Mathematical Institute of Japan. In addition, officials from Seki's hometown of Fujioka will present JAMI with a bronze statue of the legendary Japanese figure, Kojima said. Four professors from Johns Hopkins — Jean-Pierre Meyer, Jack Morava, W. Stephen Wilson and Zucker — will travel to Tokyo to receive the award and attend a celebration the following day. Johns Hopkins' JAMI wins Math Society of Japan award |
March 15, 2006
Zerna: On Mathematics monthBy Patria L. Zerna From the e-Mail WHAT could be a more fitting theme to mark Mathematics month than "Propelling to progress through excellence in Mathematics"? This underscores the great importance of this discipline to our lives to enable us to go on with our daily business of living. Mathematics is supposed to be learner-friendly and not to be dreaded. The incontrovertible fact is that we breathe and live mathematically, e.g. pulse beat, breathing rate, speed of nerve impulse, etc. The daily grind of live is expressed in numbers. Under the auspices of the Mathematics department of City Central Elementary School, coordinated by Mts. Pedrita Amahit and chaired by Mrs. Erlinda Piñero, the month breezed through successfully and ended with a salutatory bang. This month's celebration was replete with challenging activities, which run a gamut of window card fundamental to solving practical problems encountered everyday. With the inspirational guidance of Mrs. Patria L. Zerna, Principal II, a Mathematics Bazaar was put up in the second flood of Building II. Mrs. Zerna was assisted by Mrs. Crismel G. Sarabia, Master Teacher I, Miss Debralyne Casama, and all match teachers. All mathematics teachers helped to fix and arrange various items mostly crafted by the pupils and facilitated by their respective mathematics teachers and advisers. The whole show was a veritable set-up of visually and intellectually arresting displays, which showcased the collective efforts of the pupils and the total involvement of all the faculty members concerned. The celebration was capped with a proper-finale, a culmination program that was light and easy which clearly manifested not only the mathematical skills but also demonstrated innate abilities and vocal talents of the pupils as well. Prizes galore were given individually to the deserving pupils, the winning sections and coaches. Dr. Profetiza S. Lim, OIC of the ASDS City Division of Dumaguete City delivered an inspiring message to the delight of the young audience after which she proceeded to view the exhibits. Interest in mathematics does not cease after the celebration. It is therefore a challenge to the teachers as facilitators to present the study of numbers in interesting, meaningful, and enjoyable manner to wrestle with story problems that are a reflection of daily activities, and in application of particular skills gained as evidenced in the understanding of the meaning of mathematical concepts and their relationships. It is hoped that a revitalized love of mathematics will be strengthened and deepened as our children go through experiencing with numbers not only in their academic pursuits in school but especially in the real world out there. Due to the propelling desire and enthusiasm manifested by the school stakeholders, it was then expected that the school would win major prizes in different competitions. These include the awards for First in Bulletin Board Display, First in Math Bazaar, and Champion in Division Math Challenge Grave V team orals via Febby Marton Rule and Miguel Umbac. For these, City Central Elementary School, I salute you. (Patria L. Zerna is Principal II of City Central Elementary School, Dumaguete City) (March 16, 2006 issue) Zerna: On Mathematics month |
March 15, 2006
The life of piThe United States accords pi the ultimate accolade tomorrow, its own national day. Most recall it from their school days (hazily), but here Steve Connor charts its history and celebrates a number that is irrational, transcendental ... and unique Published: 13 March 2006 In case it has escaped your attention, tomorrow is 14 March which, in American notation, is written 3/14. If you have a certain type of mind you will immediately notice that these digits bear a close approximation to one of the most important numbers in mathematics - pi. Tomorrow has therefore been declared World Pi Day in honour of the mathematical constant that has beguiled and bewildered successive generations of numerate scholars since the days of ancient Babylon. Every schoolchild is told pi is the ratio of the circumference of a circle to its diameter. In other words, divide the distance around the edge of a circle by its diameter and you always get the same or "constant" number - pi. It's a nice bit of trigonometry that we learn by gradual osmosis and forget by rapid diffusion. Yet its simple truth has provided mathematicians - ancient and modern - with a cornucopia of conundrums. The first and most interesting is working out the precise value of pi. That has proved something of a challenge since the decimal places of pi can theoretically run on for ever. For the benefit of this short history of pi we can say that the value of the constant is 3.1416. A purist would of course argue that this is a gross estimation, preferring the more precise 3.14159265358979323846. Ultra-orthodox purists would add a few thousand more digits, but even they wouldn't be quite right. A supercomputer in Tokyo once calculated pi to more than 2 billion digits. It could not, however, reach the final decimal place because as every mathematician knows, that lies somewhere beyond infinity, a place they go only in their dreams. "The mathematics of pi is often rather pretty," explained Ian Stewart, professor of mathematics at Warwick University. "All numbers are interesting but some are more interesting than others and pi is the most interesting of the lot," Professor Stewart said. The whole point about pi is it is both irrational and transcendental. Irrational because it cannot be written as a simple ratio of whole numbers and transcendental because pi is living proof you cannot square a circle. If your concentration is beginning to wander a little, then let's start from the beginning. When the town planners of Babylon began building that ancient city, they took a keen interest in geometry. It became evident to them as early as the 20th century BC that when any circle's circumference is divided by its diameter, the result was always going to be about three. In fact they calculated a value of this ration equal to 25/8 which comes within 0.5 per cent of the true value of pi. A less exact value was given by another early reference to pi, this time in the Bible (Kings 7:23), which described a round basin with the dimensions: 10 cubits in diameter and 30 cubits in circumference. Scholars point out that, although this gives us a neat and tidy value to pi of exactly three, it is unfortunately quite inaccurate. (This is perhaps why Professor Frink in an episode of The Simpsons managed to gain the full attention of a hall full of babbling scientists when he shouted "pi is exactly three!") It was, in fact, an Egyptian scribe named Ahmes who gave one of the earliest and most accurate values of pi. He documented it in a Middle Kingdom papyrus scroll written around 1650BC, which was in fact a copy of an even earlier scroll. Ahmes described pi as the result of dividing 256 by 81, or 3.160. It was however Archimedes who is credited as being the first to elevate the calculation of pi to a more theoretical discipline. It is for that reason the number is sometimes known as Archimedes' constant. Chinese, Indian and Persian scholars all had a go at calculating the constant but it was not until 1706 that someone gave it the name we know it by today. If William Jones, a Welsh mathematician, is remembered for one thing it is his suggestion to call Archimedes's constant "pi" after the Greek letter. But the real work on pi had still not begun. In 1761, Johann Lambert demonstrated the irrational nature of pi. In its simplest terms, that meant you could not describe the number as a simple ratio of two whole numbers. Schoolchildren are told that pi is about 22/7, but that this is only an approximation because pi defies mathematical rationality. The second major discovery came in 1882 when Ferdinand von Lindemann proved that pi had another unusual feature: it was transcendental. In mathematical terms, it means pi is not the root of any algebraic equation with rational coefficients. In non-mathematical terms this means pi is proof of the old adage: you cannot square a circle. In other words it is not possible with a ruler and a compass alone to find a square with exactly equal area to a given circle. But the more elegant nature of pi has been subsumed by the all important quest to crunch its numbers. The obsession perhaps began with the German mathematician Ludolph van Ceulen who, in about 1600, computed pi to the first 35 decimal places. He was so proud of his accomplishment that he had the digits inscribed on his tombstone. A life-long obsessive called William Shanks spent 20 years calculating pi to 707 decimal places. Unfortunately, his achievement was discredited when the first digital computers found that he had made a mistake at the 528th decimal place - rendering all subsequent digits meaningless. Kate Bush should perhaps have learnt this lesson before she decided to sing the song "Pi" on her album Aerial. Ms Bush sings each number of Pi to 150 decimal places - or at least that was her claim until a rather sad obsessive type decided to check each digit. "All was well for the first 53 decimal places but then Kate sang 'threeeee oneeee' when she should have sang 'zeeeeeeroooo' instead," said blogger Chris McEvoy. "She recovered for the next 24 digits but then it went to hell in a handbasket when she missed out the next 22 digits completely before finishing with a precise rendition of her final 37 digits." The infinite nature of pi has also attracted the interest of science-fiction writers, such as the great American astronomer Carl Sagan who, in his book Contact, buried a hidden signature of alien intelligence within the seemingly random digits of Pi, which have no known pattern. "It was rather naughty, because you can't in fact do this," said Professor Stewart. "You can't arrange pi to have a pattern. It was a nice little conceit on Sagan's part. In a sense a pattern within pi is not something that even God could arrange," he said. But God or no, that hasn't stopped pi from playing a central role in other science fiction plots. In one episode of Star Trek, Spock saves the Enterprise from destruction when he orders the spacecraft's computer, which has been taken over by aliens, to calculate pi to the last digit. Terry Pratchett milked the irrational nature of pi for all it was worth in his novel Going Postal, where a wheel has a value of pi that is precisely three. This new pi triggers a chain of events that eventually leads to the destruction of the universe. So as tomorrow approaches, think long, infinite thoughts of the number. Think about the actual value of numbers, and the approximate near-misses we see when we calculate pi. Pi, you see, is always going to be represented by an approximation because, like all irrational numbers, its digits never really end. And just in case you miss out on tomorrow's festivities, you'll get another chance on 22 July. This is the day - 22/7 - when European date formats permit a celebration called Pi Approximation Day. Another Life of Pi Pi took on an entirely different meaning - and found a new audience - when it featured in the title of a Booker Prize-winning novel in 2002. Born the son of a zookeeper in Pondicherry, India, the hero of Yann Martel's fable finds himself saddled with the name of Piscine as a result of his father's admiration for a swimming pool he once visited in Paris. The young lad, tired of being teased, changes his name to Pi before finding himself adrift for 227 days in a small boat with a tiger following a shipwreck. Numbers across the world By Jerome Taylor 8 Chinese culture has long placed importance on the ability of numbers to predict the future and bring luck. Eight is auspicious for the Chinese because it sounds similar to the Cantonese word for prosperous " fa" and it is considered synonymous with the transformation of bad into good. Chinese businesses will pay top money for phone numbers, number plates and addresses containing the number 8. In 2003, an airline in China paid £160,000 for the phone number 88888888. 7 The king of auspicious numbers, seven has a long and significant history. For the Abrahamic traditions, the number is of particular importance and is often referred to as the perfect number. The Old and New Testaments are littered with references to the number, while The Book of Revelations mentions it 55 times. Similarly seven is a key symbol in the Koran where it is mentioned approximately 25 times and plays a central role in forming the Islamic belief system. At the height of the Haj, Muslims circle the Ka'ba in Mecca seven times. 13 Many cultures have associated the number with bad luck but perhaps none more so than modern day America. Cities lack 13th Avenues and many buildings in the States have no 13th floor. Conspiracy numerologists are quick to point out some of the world's most notorious killers, including Jack the Ripper and Charles Manson, have 13 letters in their name. INFINITY: The ultimate impossible number, infinity has confounded mathematicians and philosophers. In Western tradition, Aristotle was one of the first to tackle this never-ending number-crunch, making the distinction between actual infinity and potential infinity. In 1895, German mathematician Georg Cantor expanded on the various theories surrounding infinity. The earliest known reference to infinity, however, appears in the Yajurveda - one of the four sacred Hindu Vedas written between 1500BC and 500BC - and was widely discussed by Jain mathematicians at least a hundred years before Aristotle. The life of pi |
March 15, 2006
Time to put circle on the calendar for national pi dayIT has been pondered over for centuries by some of the world's sharpest mathematical brains. But today, one of the most important numbers in mathematics has been given its own national day. Today's date, written in American notation - 3/14 - bears the closest approximation to pi of any date in the calendar. It has therefore been declared World Pi Day in honour of the mathematical conundrum that has baffled leading scholars since the days of ancient Babylon. As every child is taught at school, pi is the ratio of the circumference of a circle to its diameter and is always the same number, no matter what the size of the circle. For the sake of ease, pi is often approximated to 3.14159 but theoretically the decimal places of pi can run on indefinitely. The most ambitious attempt to calculate pi came from a supercomputer in Tokyo which reached more than two billion digits. It is thought that pi was first used by the Babylonians, who figured out that when the circumference of any circle is divided by its diameter the result is always going to be about three.
This article: http://news.scotsman.com/international.cfm?id=383192006 |
March 15, 2006
Math Nerds Prepare To Celebrate Pi DayBy W. David Gardner Pi Day Tuesday honors the infinite numerical challenge whose first three numbers--314--connote March 14. Math aficionados, nerds, and proud math geeks at high tech operations all over the world are preparing to celebrate Pi Day Tuesday to honor the infinite numerical challenge whose first three numbers --314 -- connote March 14. Although professional and amateur mathematicians venerate Pi, they can take a light-hearted approach to the Pi phenomenon Tuesday. "Pie-eating contests are very big," said Howard Greenspan, spokesman for MathematiciansPictures.com, a Pi-oriented Web site. "For math nerds and aficionados, it's like Christmas and New Year's all wrapped up into one. There's a lot of activity in high tech operations across-the-board -- major companies, search engines, students, professors." Greenspan added that the pie-eating contests cover all flavors from apple to strawberry, while bar games often take place in bars and restaurants near high tech centers like Silicon Valley, Boston, and New York. "There are contests to see who can reel off the biggest number of Pi -- sometimes more than 100 numbers," he added. "It's a popular bar game in some places. Computer guys are totally into these things." While Pi Day activities have a light-hearted tone to them, Greenspan observed that there are many serious computer scientists who make a life's work out of working with Pi on supercomputers. In cyberspace, a giant Pi facsimile will drop at exactly 1:59 pm Tuesday. In a nod to the first six digits of Pi, Greenspan said the Pi Drop will be repeated on the site in instant replay for those who miss the actual happening. And, he added there will be an interview with an animated Pi on the Web site. Noting that Mathematicians Pictures is a business that focuses on "the rock stars of knowledge," Greenspan said the company markets an assortment of novelty items concerning famous scientists and mathematicians. They run the gamut from Ada, the mathematician many believe was the first software programmer, to Archimedes of Syracuse, the brilliant mathematician of ancient Greece. Math Nerds Prepare To Celebrate Pi Day |
March 15, 2006
Students get a taste of pie and piBy Melody Brumble Loyola celebrates mathematical constant with variety of games Loyola College Prep School freshman Curt Chambers plunged his face into gooey chocolate filling to help classmates celebrate Pi Day on Tuesday. This is the second year the school's math club has focused on pi, which describes the ratio of the circumference of a circle to its diameter. Pi is a mathematical constant, and its short form is 3.14, the same as Tuesday's date. The actual number has an infinite amount of digits after the decimal point. March 14 also is mathematician Albert Einstein's birthday, a coincidence Loyola calculus teacher Vona Weiss called "kind of eerie." Weiss helped students organize a Mu Alpha Theta math club last year and came across the Pi Day idea while researching math activities on the Internet. The group put on a pi walk, a pi-reciting competition and a pi-eating contest. "It was a lot tastier than the average pie," Chambers said after the pi-eating contest. "I had to represent the freshman class. I was one of the very few who wanted to participate. It takes a lot of courage, but it's also just fun and games." The group added a pi-baking contest this year that attracted 20 entries. "I love math," Weiss said. "Math can be creative. People don't understand that. Math isn't about adding numbers. Math is more like problem solving." Freshman Brian Curby-Lucier likes math because "there's only one answer to a problem, and it's challenging." He won the pi-reciting contest by memorizing pi to 320 places after the decimal point. The feat sent judges scurrying to the Internet to check whether he was correct because a guide sheet contained only 200 digits after the decimal point. Curby-Lucier exchanged exuberant high fives with classmates after being declared the winner. He spent a week studying the numbers. "I memorized it like a phone number, three numbers and then four numbers at a time. "I probably could've gone farther than 320, but I worried that I would've messed up."
©The Times |
March 15, 2006
Improving math ed -- Bush right about that But where are the teachers coming from?Jonathan David Farley In his State of the Union address in January, President Bush stressed the importance of improving math education. He proposed to "train 70,000 high school teachers to lead advanced placement courses in math and science, bring 30,000 math and science professionals to teach in classrooms, and give early help to students who struggle with math." But where will these teachers come from? And will the training of teachers be sufficient to increase the number of students choosing math and science careers? And why does all this matter? Because mathematics is the foundation of the natural sciences. It is no coincidence that Isaac Newton, the man who formulated the law of gravitational attraction that revolutionized our understanding of the universe, was also the man who popularized the calculus. And the natural sciences, however pure, are what give us airplanes, cable TV and the Internet. In the 2003 Program for International Student Assessment, a test that measures math literacy, American 15-year-olds performed worse than their peers in 23 countries, as well as those in Hong Kong. It's not hard to see why. According to the National Council of Teachers of Mathematics, 40 percent of the nation's middle school math teachers do not have the equivalent of an undergraduate minor in math. The average starting salary of a teacher is only $30,000, whereas the average starting salary for a recent college graduate in computer science or engineering is $50,000. Short of following the British, who have proposed paying experienced math teachers more than $100,000, with a guaranteed minimum of $70,000, where will we find a way to attract the thousands of teachers George Bush wants? New York State initiated an innovative program to bring teachers from Jamaica for two or four years to teach in New York schools. Jamaica, a developing nation where one U.S. dollar equals 65 Jamaican dollars, is nonetheless a stable, English-speaking nation with an unbroken democratic tradition; it stands poised to beat the United States in establishing the world's first Institute for Mathematical Methods in Counterterrorism. When teachers for the New York program were recruited on the campus of the University of the West Indies, recruiters found more experienced math and science teachers than they ever dreamed they would. But you can have all the teachers in the world and still not inspire kids to learn math. My friend Autumn e-mailed me about her nephew, Joshua: "He's upset because he's asked several of the math teachers why math is important or what are certain formulas used for -- there has to be a use, correct?" Autumn told her nephew about my work in counterterrorism and for the television crime drama "Numb3rs." Autumn reported, "He's told his math teachers about you as well, and about the show 'Numb3rs.' He's informing them that through something called lattice theory you are managing to fight terrorists -- all with math." Mathematics is art, and should be appreciated for its beauty, not simply for its utility. But we cannot expect 11 year-olds to cherish totally order-disconnected topological spaces as much as professional mathematicians do. As I first proposed in January 2005, television shows like "Numb3rs" (or "Medium") -- where the main characters are mathematicians -- could work with the National Council of Teachers of Mathematics to show kids how math is really used; the council and Texas Instruments are now working together to use "Numb3rs" to promote math literacy in schools. Another way to inspire kids is to relate mathematics to something they see every day. In order to excite students and draw funding to his school, school superintendent Ronald Ross of Roosevelt, N.Y., has begun looking into the idea of creating a curriculum involving math and counterterrorism. What kinds of topics would students learn? The opening line of the Oscar-winning movie "A Beautiful Mind" is "Mathematicians won the war." During World War II, the mathematics underlying cryptography played an important role in military planning. Winston Churchill admired Alan Turing, the mathematician who had mastered the German codes, recognizing him as the man who had perhaps made the single greatest individual contribution to defeating Hitler. At Los Alamos, the lab that built the atomic bomb, Cliff Joslyn uses lattice theory to mine data drawn from thousands of reports of terrorist-related activity to discover patterns and relationships that were previously in shadow. Lattice theoretical methods developed at MIT tell us the probability that we have disabled a terrorist cell, based on how many men we have taken out and what rank they hold in the organization. Lauren McGough, a Massachusetts high school student, tested the accuracy of this model by getting her classmates to pretend they were terrorists, passing orders down a fictitious chain of command, essentially confirming what the theory predicts. High school students could learn algebra, trigonometry, calculus and logic while also learning concrete applications involving homeland security. No longer would students yawn and ask, "What is math good for?" Beauty could defeat both terror and boredom. Whatever you may think of the State of the Union address, when it comes to supporting math education, we should all see pi to pi. President Bush is correct when he says that mathematics education in America must improve if the United States is to stay economically competitive, but the stakes are much higher than that. During the Cold War, the United States would not have tolerated a military gap between itself and its adversaries. Yet today, with 61 percent of all U.S. doctorates in math going to foreigners (15 percent to Chinese), we readily accept a "math gap." Dollar for dollar, the best defense against our adversaries' weapons of mass destruction may be our allies in the Americas, armed with weapons of math instruction. Improving math education is not merely a smart idea. It is a matter of national security. Algebra is one revolutionary Islamic concept we cannot afford to neglect or ignore. Jonathan David Farley is a mathematician and science fellow at Stanford University's Center for International Security and Cooperation. Contact us at insight@sfchronicle.com. Improving math ed -- Bush right about that But where are the teachers coming from? |
March 15, 2006
Maths morale plummetsBrendan O'Keefe UNIVERSITIES are second-guessing the effect of the proposed research quality framework to run down departments they consider will not be money-spinners, especially mathematics, the discipline's national society has told the HES. Yet already employers said they could not get the graduates with the quantitative and analytical skills they needed, Australian Mathematical Sciences Institute executive officer Jan Thomas said. "The CSIRO can't get the statisticians and mathematicians it needs, BHP Billiton can't get the graduates it needs, the Australian Bureau of Statistics can't get the people it needs," Ms Thomas said. Universities were being short-sighted about maths and its sister discipline, statistics. "[They] are starting to second-guess what the RQF will be like and saying, 'If maths is not a strength, well, we won't bother too much about filling positions. We'll build up somewhere where we think we might get a few more bikkies out of the RQF,"' Ms Thomas said. "Nobody knows how the RQF is going to operate. Some universities are being very short-sighted about that because they're all dependent on having good statistical advice to do most of their research successfully." AMSI said maths and statistics were being mishandled at many universities, particularly at Central Queensland, New England, Canberra and Charles Darwin, where Ian Roberts is the last research mathematician (see box). AMSI director Philip Broadbridge said "cash-driven" universities had a "counter basic sciences and arts mentality" (see separate story). "They are looking only at their own cash supply and they seem to think that if you get rid of basic sciences and humanities, and bring in extra professional degrees, then there's a surer supply of cash from full-fee-paying students," Professor Broadbridge said. "They fail to recognise a lot of their ARC grants go to mathematicians." UCQ mathematician Russel Stonier said senior mathematicians had been replaced with more junior staff in recent years. After a restructure, mathematicians had been spread around a number of faculties. "I was never asked which faculty I would like to go to," he said. The number of mathematicians at the University of Canberra had fallen from about 12 three years ago to 5.6 now, lecturer Mary Hewett said. "We've had redundancies kind of enforced. We're very much depleted," Ms Hewett said. "It's been really difficult to be positive. The morale ... you can imagine." An internal review had recommended that the remaining mathematicians stay together as a service teaching unit with a discipline head. "One outcome of the review was to increase the profile. It did wake people up to mathematicians and statisticians here and what we can do," Ms Hewett said. The university's bachelor of science degree contained no maths and only introductory statistics, Ms Hewett said. The university's pro vice-chancellor of business, law and information science Deborah Ralston said the review had "supported the work of the maths and stats group". "They've started to talk to the divisions they are servicing and student numbers are picking up." A University of New England proposal to cut its 6.5-position maths department down to four full-timers, reported in the HES on March 1, will go to the NSW Industrial Relations Commission on Friday. Ms Thomas said the problem with maths nationwide went back to the 1980s, when a federal government "relative funding model" considered maths cheap to teach. "The model predated maths departments having to run extensive computer facilities and to offer much more differentiated courses to cater for the variety of students coming in," Ms Thomas said. "It has been stuck with a funding model that says it is cheap to teach, whereas in fact it's anything but cheap to teach and should be funded at, at least, the level of computer science." Maths morale plummets |
March 15, 2006
Foster a good attitude toward mathBy Jack Chang and Linda Chou After seven years, second-grade students will begin to learn the multiplication table again. A news report in the China Times headlined "Don't overdo it when teaching the multiplication table," was fair. Ever since the implementation of the mathematics curriculum under the Nine-Year Educational Program the public have noticed a decline in students' calculation abilities due to the gap between the old and new programs. What is calculation ability anyway? How can students improve their calculation skills? And how can teachers help their students to learn math well? What we should pay attention to is a developmental mode of math education in primary and secondary schools: Learning should be developing, teachers should teach with feeling and student learning should be meaningful. Take the instruction of the 9 x 9 multiplication table for example: One cow has four legs, so how many legs do nine cows have? During almost 30 years of teaching math, we often encountered two problems. First, should students write their answer as 4 x 9 = 36 or 9 x 4 = 36, ie, what number should be the multiplicand, the former or latter? Second, should they memorize the table? We agree that students need to memorize the multiplication table. The problem is, when and how should they learn it? When a second-grade math teacher asks that question, those who learned the table in cram schools or from their parents may answer "36" without thinking. "I have learned it already," they may say. Unfortunately, once a student knows the answer, they just close the window on the learning opportunity, believing that their job is done and that they can stop learning. As a result, they fail to develop their learning motivation, while the teacher does not know how to motivate them again. According to our professional experience, perhaps the teacher should continue to ask questions, like, "How many legs do 10 cows have? How do you know?" By doing so, he or she can diagnose the students' concept learning skill, and make them feel there is a need to continue learning. Of course, it is not so simple, as some parents may challenge teachers: "Why do you teach double-digit multiplication when the students are still learning single-digit multiplication?" The initial instruction on multiplication should not be taught hastily, and distributed learning is recommended. Teachers should consider the range, content and order of the teaching materials. What multiples of what numbers should they teach first? When should they teach terminology and symbols? How can they teach in line with student's levels and experience? Take the instruction on the multiples of four, for example. Should teachers first ask students what objects always show up as units of four? Situational questions dealing with the single-digit number four could be posed. And should they then propose appropriate questions for single-digit numbers, for example, how many legs do three cows have? Six cows? Students' problem-solving strategies are thus communicated and discussed, and they can connect back to previous problems, repeatedly thinking over various solutions. This makes teaching challenging and joyful. After this, teachers can introduce terminology and symbols, and pay attention to the need for the symbol "x". Since one cow has four legs, the total number of legs of three cows is written as 4 x 3 = 12 to show that there are three units of four. The quantity per unit is four; the number of units is three. Not teaching students this terminology simplifies teacher communication. Then, at an appropriate time, they should introduce their students to the concept that "multiplicand x multiplier = product." Putting the multiplicand 4 in the front and the multiplier 3 in the back is a convention. But the Taiwanese convention is the opposite to that of the US. In Taiwan, 3 x 4 means that there are four units of 3, and that is certainly different from 4 x 3, which means that there are three units of four. After students understand the concept of multiplication, teachers must demand that they familiarize themselves with the multiplication table through games and activities before entering third grade. As for the instruction of multiplication, students must be made aware of what they are actually doing before they familiarize themselves with the calculations. For example, the straightforward calculation 36 x 27 requires some explanation. The problem for many students when doing long multiplication is that they misalign the resulting second-line sum, 72 -- the result of 36 x 2 -- by placing the "7" in the tens column rather than the hundreds column, and the "2" in the units column rather than the tens column. It is, in fact, not 36 x 2, but rather 36 x 20. Students must be made to understand that the 72 in fact is the sum 720 with the trailing 0 left out. According to Danish academic Mogens Niss' analysis, structurally, mathematical competencies include two kinds of capabilities: "The first is to ask and answer questions about, within, and by means of mathematics. The second consists of understanding and using mathematical language and tools." The former includes: one, thinking mathematically; two, posing and solving mathematical problems; three, modelling mathematically and four, reasoning mathematically. The latter includes: one, representing mathematical entities; two, handling mathematical symbols and formalisms; three, communicating in, with and about mathematics; and four, making use of aids and tools. Students entering the classroom have their own culture and thinking. Teachers must view problems from students' perspective, and communicate with them rationally. They must not be too quick to guide them or press for answers. They must value the accumulation of experience, instead of teaching tricks. They must also emphasize the exploration of methods over results. The important thing is to build a tacit understanding between teachers and students, and to cultivate a positive and active attitude towards learning math. Jack Chang is an associate professor in the department of mathematics and information education at National Taipei University of Education. Linda Chou, his wife, is a retired math teacher. Translated by Eddy Chang Foster a good attitude toward math |
March 15, 2006
Researchers simulate complete structure of virus–on computerWhen Boeing and Airbus developed their latest aircraft, the companies' engineers designed and tested them on a computer long before the planes were built. Biologists are catching on. They've just completed the first computer simulation of an entire life form – a virus. In their quest to study life, biologists apply engineering knowledge somewhat differently: They "reverse engineer" life forms, test fly them in the computer, and see if they work in silico the way they do in vivo. This technique previously had been employed for small pieces of living cells, such as proteins, but not for an entire life form until now. The accomplishment, performed by computational biologists at the University of Illinois at Urbana-Champaign and crystallographers at the University of California at Irvine, is detailed in the March issue of the journal Structure. Deeper understanding of the mechanistic properties of viruses, the researchers say, could not only contribute to improvements in public health, but also in the creation of artificial nanomachines made of capsids – a small protein shell that contains a viral building plan, a genome, in the form of DNA or RNA. Viruses are incredibly tiny and extremely primitive life forms that cause myriad diseases. Biologists often refer to them as particles rather than organisms. Viruses hijack a biological cell and make it produce many new viruses from a single original. They've evolved elaborate mechanisms of cell infection, proliferation and departure from the host when it bursts from viral overcrowding. For their first attempt to reverse engineer a life form in a computer program, computational biologists selected the satellite tobacco mosaic virus because of its simplicity and small size. The satellite virus they chose is a spherical RNA sub-viral agent that is so small and simple that it can only proliferate in a cell already hijacked by a helper virus – in this case the tobacco mosaic virus that is a serious threat to tomato plants. A computer program was used to reverse engineer the dynamics of all atoms making up the virus and a small drop of salt water surrounding it. The virus and water contain more than a million atoms altogether. The necessary calculation was done at Illinois on one of the world's largest and fastest computers operated by the National Center for Supercomputing Applications. The computer simulations provided an unprecedented view into the dynamics of the virus. "The simulations followed the life of the satellite tobacco mosaic virus, but only for a very brief time," said co-author Peter Freddolino, a doctoral student in biophysics and computational biology at Illinois. "Nevertheless, they elucidated the key physical properties of the viral particle as well as providing crucial information on its assembly." It may take still a long time to simulate a dog wagging its tail in the computer, said co-author Klaus Schulten, Swanlund Professor of Physics at Illinois. "But a big first step has been taken to 'test fly' living organisms," he said. "Naturally, this step will assist modern medicine as we continue to learn more about how viruses live." The computer simulations were carried out in Schulten's Theoretical and Biophysics Group's lab at the Beckman Institute for Avanced Science and Technology. Other co-authors were Anton Arkhipov, a doctoral student in physics at Illinois, and Alexander McPherson, a professor of molecular biology and biochemistry, and research specialist Steven Larson, both at UC-Irvine. Source: University of Illinois at Urbana-Champaign Researchers simulate complete structure of virus–on computer |
March 15, 2006
New Mathematical Model Shows Promise for Cancer GenomicsBy: Broad Institue Cancer Some researchers work "by the numbers" to identify the genetic hallmarks of cancer. Now, a powerful mathematical tool introduced by Broad Institute scientists will facilitate this task, providing a clearer picture of how DNA runs amok in tumors. The genetic distortions that lurk within the cells of a tumor form the driving force behind malignancy. These changes, which involve either gains or losses of DNA, perturb the usual number of gene copies in a cell and can involve either of the paired chromosomes. Scientists are trying to trace the chromosomal origins of such modifications to pinpoint informative genes, forming the basis for new therapeutic targets and possible genetic predictors for cancer diagnosis. Researchers led by Matthew Meyerson, an associate member of the Broad Institute and associate professor at Harvard Medical School/Dana-Farber Cancer Institute, developed an algorithm that interprets the data from single nucleotide polymorphism (SNP) arrays, a collection of short oligonucleotides ("oligos") used to tally SNP patterns in human DNA. Probe-level allele-specific quantitation (PLASQ), described in the November issue of PLoS Computational Biology, allows scientists to approximate DNA copy number at sites throughout the genome and to assign the proportion furnished by each parental chromosome. "In cancer, genome modifications often affect only one of the two paired chromosomes, the one inherited from the father or the one contributed by the mother," said Meyerson. "PLASQ allows us to localize these changes to the culprit chromosome, which will help guide us to the most significant genes and gene mutations in the disease." In SNP arrays, oligos are parsed into probe sets and each set, comprised of 40 probes, is tailored to detect a single SNP in the human genome. These probe sets consist of one probe that perfectly matches both the target SNP and its surrounding DNA, as well as several related probes, each harboring single letter mismatches. To compute DNA copy number, PLASQ exploits the mathematical link between a probe's intensity - the readout from SNP arrays - and the known position of a mismatch relative to the target SNP. PLASQ also reveals the number of copies contributed by each parental chromosome. The researchers validated the procedure by applying it to DNA samples that had been independently analyzed by several institutions within the International HapMap Project consortium and found the results given by PLASQ to be in agreement in more than 99% of cases. Meyerson and his colleagues then used the algorithm to look for changes in DNA copy number in more than 100 lung cancer samples and discovered a multitude of gene deletions and amplifications. They noted that most of the amplifications appear to be monoallelic, which means they derive from only one parental chromosome. While this finding extends from the use of PLASQ, it also agrees with the chromosomal acrobatics that are believed to underlie gene amplification. The researchers turned their attention to an amplification ("amplicon") that covers the EGFR gene, which in addition to being amplified, is also frequently mutated and rendered abnormally active in some lung cancer samples. Combing through the surplus copies, they noted that those in excess came exclusively from the mutated allele, while the normal EGFR allele was present in typical number. Therefore, with the help of PLASQ, Broad scientists unearthed the preferential amplification of a mutant allele over its wildtype sibling. "Chromosomal segments may be targeted for amplification in tumors because they contain a heritable or germline change that confers a distinct growth advantage," said Tom LaFramboise, a computational biologist in the Broad's Cancer program and the study's lead author. "Since PLASQ provides a snapshot of the relevant SNP patterns contained in tumor amplicons, we may now be able to find these genetic variants using linkage analysis."
PLASQ may also alleviate a frequent problem that plagues the analysis of tumor cells. When tumors are isolated, normal cells frequently accompany their cancerous counterparts, which complicate readings from a tumor's genome. Researchers may be able to adapt the mathematic terms of PLASQ to solve this predicament.
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March 07, 2006
The Limits of MathematicsIvars Peterson At the beginning of the 20th century, the German mathematician David Hilbert (1862–1943) advocated an ambitious program to formulate a system of axioms and rules of inference that would encompass all mathematics, from basic arithmetic to advanced calculus. His dream was to codify the methods of mathematical reasoning and put them within a single framework. Hilbert insisted that such a formal system of axioms and rules should be consistent, meaning that you can't prove an assertion and its opposite at the same time. He also wanted a scheme that is complete, meaning that you can always prove a given assertion either true or false. He argued that there had to be a clear-cut mechanical procedure for deciding whether a certain proposition follows from a given set of axioms. Hence, it would be possible, though not actually practical, to run through all possible propositions, starting with the shortest sequences of symbols, and check which ones are valid. In principle, such a decision procedure would automatically generate all possible theorems in mathematics. What Hilbert was saying is that "we can solve a problem if we are clever enough and work at it long enough," mathematician Gregory J. Chaitin of the IBM Thomas J. Watson Research Center wrote in his 1998 book The Limits of Mathematics. "He didn't believe that in principle there was any limit to what mathematics could achieve." In the 1930s, Kurt Gödel (1906–1978), followed by Alan Turing (1912–1954) and others, proved that no such decision procedure is possible for any system of logic made up of axioms and propositions sufficiently sophisticated to encompass the kinds of problems that mathematicians work on every day. "More precisely, what Gödel discovered was that the plan fails even if you just try to deal with elementary arithmetic, with the numbers 0, 1, 2, 3, . . . and with multiplication and addition," Chaitin wrote in his 2005 book Meta Math! The Quest for Omega. "Any formal system that tries to contain the whole truth and nothing but the truth about addition, multiplication, and the numbers 0, 1, 2, 3, . . . will have to be incomplete." In Gödel's realm, no matter what the system of axioms or rules is, there will always be some assertion that can be neither proved nor invalidated within the system. Indeed, mathematics is full of conjectures–assertions awaiting proof–with no assurance that definitive answers even exist. Turing's argument involved mathematical entities known as real numbers. Suppose you happen upon the number 1.6180339887. It looks vaguely familiar, but you can't quite place it. You would like to find out whether this particular sequence of digits is special in some way, perhaps as the output of a specific formula or the value of a familiar mathematical constant. It turns out that the given number is the value, rounded off, of the so-called golden ratio, which can also be written as (1 + SQRT 5)/2, an example of a real number. Given that expression, which represents an infinite number of decimal digits, you can compute its value to any number of decimal places. Going in the opposite direction from the given rounded-off number to the expression, however, is much more difficult and problematic. For example, it's possible that if the mystery number were available to an extra decimal place, the final digit would no longer match the decimal digits of the golden ratio. You would have to conclude that the given number is not an approximation of the golden ratio. Indeed, the extended string of digits could represent the output of a completely different expression or formula, or even part of a random sequence. It's impossible to tell for sure. There isn't enough information available. To sort through the relationship between random sequences and the types of numbers that mathematicians and scientists use in their work, Chaitin defined the "complexity" of a number as the length of the shortest computer program (or set of instructions) that would spew out the number. "The minimum number of bits—what size string of zeros and ones—needed to store the program is called the algorithmic information content of the data," Chaitin writes in the March Scientific American. "Thus, the infinite sequence of numbers 1, 2, 3, . . . has very little algorithmic information; a very short computer program can generate all those numbers." "It does not matter how long the program must take to do the computation or how much memory it must use—just the length of the program in bits counts," he adds. Similarly, suppose a given number consists of 100 1s. The instruction to the computer would be simply "print 1, 100 times." Because the program is substantially shorter than the sequence of 100 1s that it generates, the sequence is not considered random. If a sequence is disorderly enough that any program for printing it out cannot be shorter than the sequence itself, the sequence counts as algorithmically random. Hence, an algorithmically random sequence is one for which there is no compact description. Interestingly, the number pi (the ratio of a circle's circumference to its diameter), which is expressed by an infinite number of digits, has little algorithmic information content because a computer can use a relatively small program to generate the number, digit by digit: 3.14159 . . . . On the other hand, a random number with merely 1 million digits has a much larger amount of algorithmic information. Chaitin proved that no program can generate a number more complex than itself. In other words, "a 1-pound theory can no more produce a 10-pound theorem than a 100-pound pregnant woman can birth a 200-pound child," he likes to say. Conversely, Chaitin also showed that it is impossible for a program to prove that a number more complex than the program is random. Hence, to the extent that the human mind is a kind of computer, there may be a type of complexity so deep and subtle that the intellect could never grasp it. Whatever order may lie in the depths would be inaccessible, and it would always appear to us as random. At the same time, proving that a sequence is random presents insurmountable difficulties. There's no way to be sure that we haven't overlooked a hint of order that would allow even a small compression in the computer program that produces the sequence. From a mathematical point of view, Chaitin's result suggests that we are far more likely to find randomness than order within certain domains of mathematics. Indeed, his complexity version of Gödel's theorem states: Although almost all numbers are random, there is no formal axiomatic system that will allow us to prove this fact. Chaitin's work indicates that there is an infinite number of mathematical statements that one can make about, say, arithmetic that can't be reduced to the axioms of arithmetic. So there's no way to prove whether the statements are true or false by using arithmetic. In Chaitin's view, that's practically the same as saying that the structure of arithmetic is random. "What I've constructed and exhibited are mathematical facts that are true . . . by accident," he says. "They're mathematical facts which are analogous to the outcome of a coin toss. . . . You can never actually prove logically whether they're true." This doesn't mean that anarchy reigns in mathematics, only that mathematical laws of a different kind might apply in certain situations. In such cases, statistical laws hold sway and probabilities describe the answers that come out of equations. Such problems arise when one asks whether an equation involving only whole numbers has an infinite number of whole-number solutions, a finite number, or none at all. "In the same way that it is impossible to predict the exact moment at which an individual atom undergoes radioactive decay, mathematics is powerless to answer particular questions," Chaitin states. "Nevertheless, physicists can still make reliable predictions about averages over large ensembles of atoms. Mathematicians may in some cases be limited to a similar approach." That makes mathematics much more of an experimental science than many mathematicians would like to admit. Chaitin goes further. Human creativity is absolutely necessary for mathematical work, he argues, and "intuition cannot be eliminated from mathematics." Originally posted: Feb. 21, 1998 Updated: March 4, 2006 Check out Ivars Peterson's MathTrek blog at http://blog.sciencenews.org/. References: Chaitin, G.J. 2006. The limits of reason. Scientific American 294(March):74-81. See http://www.umcs.maine.edu/~chaitin/sciamer3.html. ______. 2005. Omega and why maths has no TOEs. Plus (December). Available at http://plus.maths.org/issue37/features/omega/index.html. ______. 2005. Meta Math! The Quest for Omega. New York: Pantheon. ______. 1998. The Limits of Mathematics: A Course on Information Theory and the Limits of Formal Reasoning. Singapore: Springer-Verlag. Kleiner, I., and N. Movshovitz-Hadar. 1997. Proof: A many-splendored thing. Mathematical Intelligencer 19(No. 3):16-26. Peterson, I. 1998. The Jungles of Randomness: A Mathematical Safari. New York: Wiley. ______. 1990. Islands of Truth: A Mathematical Mystery Cruise. New York: W.H. Freeman. Velleman, D.J. 1997. Fermat's last theorem and Hilbert's program. Mathematical Intelligencer 19(No. 1):64-67. Additional information about Gregory Chaitin and his writings is available at http://www.umcs.maine.edu/~chaitin/. ********** A collection of Ivars Peterson's early MathTrek articles, updated and illustrated, is now available as the Mathematical Association of America (MAA) book Mathematical Treks: From Surreal Numbers to Magic Circles. See http://www.maa.org/pubs/books/mtr.html. The Limits of Mathematics |
March 07, 2006
Untying a math mysteryBy Margaret Wertheim, MARGARET WERTHEIM is the science columnist for the LA Weekly and director of the Institute for Figuring, which has recently been hosting a series of lectures on knot theory. YOU HAVE to hand it to mathematicians, they can turn anything into a formal problem. Balls packed into boxes, folded paper, even bits of string become, in the hands of mathematical theorists, gateways to worlds of Byzantine complexity and beauty. Take a piece of string — I mean literally, go get a piece of string and tie it into a knot. Now tape the two ends together so it makes a closed loop — necessary to fulfill the mathematical definition of a "knot." How many different knot types do you think there are? The number is infinite, and the question of how to categorize these manifestations of loopiness has engaged some of the finest mathematical minds for a century. We are nowhere near to having a complete taxonomy of knots, and some mathematicians view the problem as so inherently difficult that they think it is an impossible goal. Indeed, "knot theory" is an area of mathematics in which almost any generalized question you can think of is unlikely to be answered. Although knots in math are essentially one-dimensional objects, understanding them has turned out to be a significant challenge. Moreover, knots provide mysterious links between the mathematical continents of topology, geometry and algebra, hinting that these enigmatic twists contain secrets to powerful, deep and general truths. And yet this most esoteric branch of mathematics has also turned out to have immense application in the physical world. That's because we now know that DNA and many other long molecules arrange themselves into knotted structures. Knot theorists are suddenly in demand from biologists, who want help understanding how clumps of DNA move through different mediums, how proteins fold up and how polymers behave. The specific knottiness of a piece of DNA, for example, determines whether certain enzymes can act on it, which has important implications for understanding diseases such as cancer. Ken Millett, a knot theorist at UC Santa Barbara, is a leader in the application of this mathematics to DNA and other molecules. In the 1980s, inspired by UC Berkeley mathematician Vaughan Jones, Millett helped to revitalize knot theory when he was part of a team that discovered a strange new way of classifying knots. With this method, each knot can be associated with a particular equation that uniquely characterizes it. Still, mathematicians have no idea what the equations actually signify; they don't seem to relate to any of the usual features of knots, such as shape and form. "Do they refer to some hidden structure within the knot?" Millett asks. "We really don't know." Some physicists, however, think the equations are telling us something fundamental about the basic particles and forces of nature. They believe these arcane formulas may enable us to find the much-longed-for "theory of everything" under the umbrella of string theory. The equations also turn out to have application to the emerging field of quantum computing, which many scientists hope will usher in an era of new, more powerful computational devices. The story of knots suggests that we never know from what areas of mathematics useful applications may spring. Although mathematics has no physical substance, it can be as precious as gold or oil, and ultimately as integral to our economy. As President Bush noted in January's State of the Union speech, America's place at the top of the global technological pyramid depends on a workforce that is well educated in math and science. Yet, nationally, our schools are understaffed in these critical areas. Which brings me to the importance of Millett's other professional hat — math education. In addition to his knot research, Millett directs a program at Santa Barbara that recruits math and science undergraduates to become classroom teachers. Given that a recent report by the National Academy of Sciences revealed that nearly 60% of American eighth-graders are taught math by teachers who did not major in math or pass any kind of certification exam, efforts such as Millett's are critical. On Feb. 25, his work was honored in Washington with an award from the organization Quality Education for Minorities. In the State of the Union address, the president pledged to train 70,000 math and science teachers to handle AP courses. But the plan does not call for hiring any new teachers, which is woefully shortsighted. Math education does not require expensive equipment, specialized buildings or fancy facilities, it just needs good teachers and a supportive learning environment. The lessons of knot theory suggest that investing in this "arcane" subject will, in the long run, pay dividends. Untying a math mystery |
March 07, 2006
Professor Intermont '89 to Give Talk on Bell Ringing at Annual Sulski Memorial Lecture Intermont, who earned high honors in mathematics from Holy Cross, received her Ph.D. in mathematics from the University of Notre Dame in 1994. She taught at John Carroll University and Mesa State College before going to Kalamazoo College, where she is now an associate professor of mathematics. Her research area is algebraic topology. Intermont's lecture will introduce bell ringing (English change ringing to be more precise) as an application of algebra. The lecture will present samples of bell ringing and then describe a problem faced by those who compose for bell ringing for which algebra has the solution. The presentation is aimed at a general mathematical audience including undergraduate students. The lecture will be preceded by a dinner co-sponsored by the department of mathematics and computer science and the Mathematical Association of America. For more information about the dinner, please contact Professor Tom Cecil of the department of mathematics and computer science before March 13 at 508-793-2719 or by e-mail at cecil@mathcs.holycross.edu. The annual mathematical lecture series is a tribute to Leonard C. Sulski who taught in the mathematics department at Holy Cross from 1965 until his untimely death from leukemia in 1991. Professor Intermont '89 to Give Talk on Bell Ringing at Annual Sulski Memorial Lecture |
March 07, 2006
Mathematician and Musician Will Visit Ithaca College to Examine the Connection Between Language and MusicIthaca, NY--Manjul Bhargava, an expert on number theory and a master Indian drummer, will present "Drumming and Poetry: The Nature of Human Thought and the Origins of Mathematics" at this year's C. P. Snow Lecture Series at Ithaca College. His talk is scheduled for Thursday, March 16, at 7:00 p.m. in Textor 103. The event is free and open to the public. Held at the College annually since 1965, the C. P. Snow Lecture Series brings speakers to campus who actively combine scientific and humanistic perspectives. When Bhargava was tenured as a full professor of mathematics at Princeton in 2003, he was 28--one of the youngest of that university's faculty members to attain that rank. An acknowledged expert on number theory (the study of the property and relationship of numbers), he has garnered several honors, including the American Mathematical Society's 2005 Blumenthal Award for the Advancement of Pure Mathematics and being named one of "Popular Science" magazine's "Brilliant 10" for 2002. In addition to his work in mathematics, Bhargava is a master of the tabla, a small Indian hand drum used to create music with precise, rhythmic patterns. The relationships between those rhythms are similar, he says, to the relationships between numbers. Though mathematics pervades all the sciences, he adds, it also lies at the heart of a number of fields in the humanities, including music and linguistics. Many of the modern mathematical tools used in probability and applied to technologies such as those employed on NASA space missions originated in problems encountered by musicians and linguists thousands of years ago. Looking at some of these ancient, poetic problems and the way they have been solved through the ages reveals much about the nature of human thought and the origins of mathematics. The late physicist and novelist C. P. Snow was awarded an honorary doctor of letters degree from Ithaca College in 1967. In 1959 he wrote, "Literary intellectuals at one pole--at the other scientists, and as the most representative, physical scientists. Between the two a gulf of mutual incomprehension--sometimes (particularly among the young) hostility and dislike, but most of all, a lack of understanding." The C. P. Snow Lecture Series is an attempt to bridge that gulf. For more information, contact Michael Haaf, assistant professor of chemistry, at (607) 274-7978 or mhaaf@ithaca.edu. Mathematician and Musician Will Visit Ithaca College to Examine the Connection Between Language and Music |
March 07, 2006
Why I Quit HIVby Rebecca V. Culshaw As I write this, in the late winter of 2006, we are more than twenty years into the AIDS era. Like many, a large part of my life has been irreversibly affected by AIDS. My entire adolescence and adult life – as well as the lives of many of my peers – has been overshadowed by the belief in a deadly, sexually transmittable pathogen and the attendant fear of intimacy and lack of trust that belief engenders. To add to this impact, my chosen career has developed around the HIV model of AIDS. I received my Ph.D. in 2002 for my work constructing mathematical models of HIV infection, a field of study I entered in 1996. Just ten years later, it might seem early for me to be looking back on and seriously reconsidering my chosen field, yet here I am. My work as a mathematical biologist has been built in large part on the paradigm that HIV causes AIDS, and I have since come to realize that there is good evidence that the entire basis for this theory is wrong. AIDS, it seems, is not a disease so much as a sociopolitical construct that few people understand and even fewer question. The issue of causation, in particular, has become beyond question – even to bring it up is deemed irresponsible. Why have we as a society been so quick to accept a theory for which so little solid evidence exists? Why do we take proclamations by government institutions like the NIH and the CDC, via newscasters and talk show hosts, entirely on faith? The average citizen has no idea how weak the connection really is between HIV and AIDS, and this is the manner in which scientifically insupportable phrases like "the AIDS virus" or "an AIDS test" have become part of the common vernacular despite no evidence for their accuracy. When it was announced in 1984 that the cause of AIDS had been found in a retrovirus that came to be known as HIV, there was a palpable panic. My own family was immediately affected by this panic, since my mother had had several blood transfusions in the early 1980s as a result of three late miscarriages she had experienced. In the early days, we feared mosquito bites, kissing, and public toilet seats. I can still recall the panic I felt after looking up in a public restroom and seeing some graffiti that read "Do you have AIDS yet? If not, sit on this toilet seat." But I was only ten years old then, and over time the panic subsided to more of a dull roar as it became clear that AIDS was not as easy to "catch" as we had initially believed. Fear of going to the bathroom or the dentist was replaced with a more realistic wariness of having sex with anyone we didn't know really, really well. As a teenager who was in no way promiscuous, I didn't have much to worry about. That all changed – or so I thought – when I was twenty-one. Due to circumstances in my personal life and a bit of paranoia that (as it turned out, falsely and completely groundlessly) led me to believe I had somehow contracted "AIDS," I got an HIV test. I spent two weeks waiting for the results, convinced that I would soon die, and that it would be "all my fault." This was despite the fact that I was perfectly healthy, didn't use drugs, and wasn't promiscuous – low-risk by any definition. As it happened, the test was negative, and, having felt I had been granted a reprieve, I vowed not to take more risks, and to quit worrying so much. Over the past ten years, my attitude toward HIV and AIDS has undergone a dramatic shift. This shift was catalyzed by the work I did as a graduate student, analyzing mathematical models of HIV and the immune system. As a mathematician, I found virtually every model I studied to be unrealistic. The biological assumptions on which the models were based varied from author to author, and this made no sense to me. It was around this time, too, that I became increasingly perplexed by the stories I heard about long-term survivors. From my admittedly inexpert viewpoint, the major thing they all had in common – other than HIV – was that they lived extremely healthy lifestyles. Part of me was becoming suspicious that being HIV-positive didn't necessarily mean you would ever get AIDS. By a rather curious twist of fate, it was on my way to a conference to present the results of a model of HIV that I had proposed together with my advisor, that I came across an article by Dr. David Rasnick about AIDS and the corruption of modern science. As I sat on the airplane reading this story, in which he said "the more I examined HIV, the less it made sense that this largely inactive, barely detectable virus could cause such devastation," everything he wrote started making sense to me in a way that the currently accepted model did not. I didn't have anywhere near all the information, but my instincts told me that what he said seemed to fit. Over the past ten years, I nevertheless continued my research into mathematical models of HIV infection, all the while keeping an ear open for dissenting voices. By now, I have read hundreds of articles on HIV and AIDS, many from the dissident point of view but far, far more from that of the establishment, which unequivocally promotes the idea that HIV causes AIDS and that the case is closed. In that time, I even published four papers on HIV (from a modeling perspective). I justified my contributions to a theory I wasn't convinced of by telling myself these were purely theoretical, mathematical constructs, never to be applied in the real world. I suppose, in some sense also, I wanted to keep an open mind. So why is it that only now have I decided that enough is enough, and I can no longer in any capacity continue to support the paradigm on which my entire career has been built? As a mathematician, I was taught early on about the importance of clear definitions. AIDS, if you consider its definition, is far from clear, and is in fact not even a consistent entity. The classification "AIDS" was introduced in the early 1980s not as a disease but as a surveillance tool to help doctors and public health officials understand and control a strange "new" syndrome affecting mostly young gay men. In the two decades intervening, it has evolved into something quite different. AIDS today bears little or no resemblance to the syndrome for which it was named. For one thing, the definition has actually been changed by the CDC several times, continually expanding to include ever more diseases (all of which existed for decades prior to AIDS), and sometimes, no disease whatsoever. More than half of all AIDS diagnoses in the past several years in the United States have been made on the basis of a T-cell count and a "confirmed" positive antibody test – in other words, a deadly disease has been diagnosed over and over again on the basis of no clinical disease at all. And the leading cause of death in HIV-positives in the last few years has been liver failure, not an AIDS-defining disease in any way, but rather an acknowledged side effect of protease inhibitors, which asymptomatic individuals take in massive daily doses, for years. The epidemiology of HIV and AIDS is puzzling and unclear as well. In spite of the fact that AIDS cases increased rapidly from their initial observation in the early 1980s and reached a peak in 1993 before declining rapidly, the number of HIV-positive individuals in the U.S. has remained constant at one million since the advent of widespread HIV antibody testing. This cannot be due to anti-HIV therapy, since the annual mortality rate of North American HIV-positives who are treated with anti-HIV drugs is much higher – between 6.7 and 8.8% – than would be the approximately 1–2% global mortality rate of HIV-positives if all AIDS cases were fatal in a given year. Even more strangely, HIV has been present everywhere in the U.S., in every population tested including repeat blood donors and military recruits, at a virtually constant rate since testing began in 1985. It is deeply confusing that a virus thought to have been brought to the AIDS epicenters of New York, San Francisco and Los Angeles in the early 1970s could possibly have spread so rapidly at first, yet have stopped spreading completely as soon as testing began. Returning for a moment to the mathematical modeling, one aspect that had always puzzled me was the lack of agreement on how to accurately represent the actual biological mechanism of immune impairment. AIDS is said to be caused by a dramatic loss of the immune system's T-cells, said loss being presumably caused by HIV. Why then could no one agree on how to mathematically model the dynamics of the fundamental disease process – that is, how are T-cells actually killed by HIV? Early models assumed that HIV killed T-cells directly, by what is referred to as lysis. An infected cell lyses, or bursts, when the internal viral burden is so high that it can no longer be contained, just like your grocery bag breaks when it's too full. This is in fact the accepted mechanism of pathogenesis for virtually all other viruses. But it became clear that HIV did not in fact kill T-cells in this manner, and this concept was abandoned, to be replaced by various other ones, each of which resulted in very different models and, therefore, different predictions. Which model was "correct" never was clear. As it turns out, the reason there was no consensus mathematically as to how HIV killed T-cells was because there was no biological consensus. There still isn't. HIV is possibly the most studied microbe in history – certainly it is the best-funded – yet there is still no agreed-upon mechanism of pathogenesis. Worse than that, there are no data to support the hypothesis that HIV kills T-cells at all. It doesn't in the test tube. It mostly just sits there, as it does in people – if it can be found at all. In Robert Gallo's seminal 1984 paper in which he claims "proof" that HIV causes AIDS, actual HIV could be found in only 26 out of 72 AIDS patients. To date, actual HIV remains an elusive target in those with AIDS or simply HIV-positive. This is starkly illustrated by the continued use of antibody tests to diagnose HIV infection. Antibody tests are fairly standard to test for certain microbes, but for anything other than HIV, the main reason they are used in place of direct tests (that is, actually looking for the bacteria or virus itself) is because they are generally much easier and cheaper than direct testing. Most importantly, such antibody tests have been rigorously verified against the gold standard of microbial isolation. This stands in vivid contrast to HIV, for which antibody tests are used because there exists no test for the actual virus. As to so-called "viral load," most people are not aware that tests for viral load are neither licensed nor recommended by the FDA to diagnose HIV infection. This is why an "AIDS test" is still an antibody test. Viral load, however, is used to estimate the health status of those already diagnosed HIV-positive. But there are very good reasons to believe it does not work at all. Viral load uses either PCR or a technique called branched-chained DNA amplification (bDNA). PCR is the same technique used for "DNA fingerprinting" at crime scenes where only trace amounts of materials can be found. PCR essentially mass-produces DNA or RNA so that it can be seen. If something has to be mass-produced to even be seen, and the result of that mass-production is used to estimate how much of a pathogen there is, it might lead a person to wonder how relevant the pathogen was in the first place. Specifically, how could something so hard to find, even using the most sensitive and sophisticated technology, completely decimate the immune system? bDNA, while not magnifying anything directly, nevertheless looks only for fragments of DNA believed, but not proven, to be components of the genome of HIV – but there is no evidence to say that these fragments don't exist in other genetic sequences unrelated to HIV or to any virus. It is worth noting at this point that viral load, like antibody tests, has never been verified against the gold standard of HIV isolation. bDNA uses PCR as a gold standard, PCR uses antibody tests as a gold standard, and antibody tests use each other. None use HIV itself. There is good reason to believe the antibody tests are flawed as well. The two types of tests routinely used are the ELISA and the Western Blot (WB). The current testing protocol is to "verify" a positive ELISA with the "more specific" WB (which has actually been banned from diagnostic use in the UK because it is so unreliable). But few people know that the criteria for a positive WB vary from country to country and even from lab to lab. Put bluntly, a person's HIV status could well change depending on the testing venue. It is also possible to test "WB indeterminate," which translates to any one of "uninfected," "possibly infected," or even, absurdly, "partly infected" under the current interpretation. This conundrum is confounded by the fact that the proteins comprising the different reactive "bands" on the WB test are all claimed to be specific to HIV, raising the question of how a truly uninfected individual could possess antibodies to even one "HIV-specific" protein. I have come to sincerely believe that these HIV tests do immeasurably more harm than good, due to their astounding lack of specificity and standardization. I can buy the idea that anonymous screening of the blood supply for some nonspecific marker of ill health (which, due to cross reactivity with many known pathogens, a positive HIV antibody test often seems to be) is useful. I cannot buy the idea that any individual needs to have a diagnostic HIV test. A negative test may not be accurate (whatever that means), but a positive one can create utter havoc and destruction in a person's life – all for a virus that most likely does absolutely nothing. I do not feel it is going too far to say that these tests ought to be banned for diagnostic purposes. The real victims in this mess are those whose lives are turned upside-down by the stigma of an HIV diagnosis. These people, most of whom are perfectly healthy, are encouraged to avoid intimacy and are further branded with the implication that they were somehow dreadfully foolish and careless. Worse, they are encouraged to take massive daily doses of some of the most toxic drugs ever manufactured. HIV, for many years, has fulfilled the role of a microscopic terrorist. People have lost their jobs, been denied entry into the Armed Forces, been refused residency in and even entry into some countries, even been charged with assault or murder for having consensual sex; babies have been taken from their mothers and had toxic medications forced down their throats. There is no precedent for this type of behavior, as it is all in the name of a completely unproven, fundamentally flawed hypothesis, on the basis of highly suspect, indirect tests for supposed infection with an allegedly deadly virus – a virus that has never been observed to do much of anything. As to the question of what does cause AIDS, if it is not HIV, there are many plausible explanations given by people known to be experts. Before the discovery of HIV, AIDS was assumed to be a lifestyle syndrome caused mostly by indiscriminate use of recreational drugs. Immunosuppression has multiple causes, from an overload of microbes to malnutrition. Probably all of these are true causes of AIDS. Immune deficiency has many manifestations, and a syndrome with many manifestations is likely multicausal as well. Suffice it to say that the HIV hypothesis of AIDS has offered nothing but predictions – of its spread, of the availability of a vaccine, of a forthcoming animal model, and so on – that have not materialized, and it has not saved a single life. After ten years involved in the academic side of HIV research, as well as in the academic world at large, I truly believe that the blame for the universal, unconditional, faith-based acceptance of such a flawed theory falls squarely on the shoulders of those among us who have actively endorsed a completely unproven hypothesis in the interests of furthering our careers. Of course, hypotheses in science deserve to be studied, but no hypothesis should be accepted as fact before it is proven, particularly one whose blind acceptance has such dire consequences. For over twenty years, the general public has been greatly misled and ill-informed. As someone who has been raised by parents who taught me from a young age never to believe anything just because "everyone else accepts it to be true," I can no longer just sit by and do nothing, thereby contributing to this craziness. And the craziness has gone on long enough. As humans – as honest academics and scientists – the only thing we can do is allow the truth to come to light. Rebecca V. Culshaw, Ph.D. [send her mail], is a mathematical biologist who has been working on mathematical models of HIV infection for the past ten years. She received her Ph.D. (mathematics with a specialization in mathematical biology) from Dalhousie University in Canada in 2002 and is currently employed as an Assistant Professor of Mathematics at a university in Texas. Why I Quit HIV |
March 07, 2006
Physicist's Algorithm Simplifies Biological Imaging -- And Also Solves Sudoku PuzzlesBy Lauren Gold Cornell physicist Veit Elser has been engrossed recently in resolving a pivotal question in biological imaging. So he hasn't had much time for brainteasers and number games. But in discovering an algorithm critical for X-ray diffraction microscopy, Elser and colleagues solved two problems. First, they gave researchers a new tool for imaging the tiniest and most delicate of biological specimens. And second, they discovered that the same algorithm also solves the internationally popular numbers puzzle Sudoku. Not just one puzzle. All of them. The Sudoku discovery appeals to Elser's whimsical side. But the algorithm, he notes, has potential for all kinds of other endeavors. "There are a lot of problems that you can represent in terms of this language," he says. "We're providing the technique. Whatever people use it for, it's great for us." The so-called difference-map algorithm, which Elser says could have applications from productivity optimization to nanofabrication, tackles problems for which the solution must meet two independent constraints. In the case of Sudoku, the constraints are simple: Each of nine numbers, considered alone, appears nine times in the grid so that there is only one per row and column. And all nine numbers appear within each of the nine blocks. In X-ray diffraction microscopy, the constraints are more complex. But the beauty of the algorithm, as Elser demonstrates, is that complexity doesn't matter. By applying the algorithm to the jumble of raw data from such an experiment, researchers can now reconstruct from it a clear, detailed image. The result, shown in images published with a recent article in the Proceedings of the National Academy of Sciences (PNAS) with lead author David Shapiro of the State University of New York at Stony Brook and other colleagues, is a richly detailed image of a specimen as small as a single yeast cell -- taken without staining, sectioning or otherwise damaging the specimen. Unlike optical microscopes, which use a lens to focus light on a target (and are therefore only useful for specimens larger than the wavelength of visible light), imaging methods based on X-rays and electrons take advantage of the finer wavelengths provided by these forms of illumination. But some of these methods can damage the specimen with harmful radiation, require that a specimen be stained or otherwise altered or lack the penetrating power necessary for three-dimensional reconstructions. X-ray diffraction microscopy, which uses "soft" X-rays and measures the resulting diffraction pattern, is often the method of choice because it gives a detail-rich image and leaves the specimen relatively unscathed. The tricky part comes in using the diffraction pattern to reconstruct an image of the specimen. Instead of directly measuring the pixel-by-pixel contrast within the specimen, researchers are left with data that represents the object broken down into its constituent waves: a vast number of them, of different frequencies and amplitudes, waiting to be added up in a process called a Fourier synthesis to reconstruct the image. "But if it were just combining waves with a definite oscillation and adding them up, that would be a piece of cake for a computer to do," says Elser. The challenge is in the waves' phases, which are critical in reconstructing an image from the diffraction data. Without attention to that piece of the puzzle, the resulting image is reduced to noise. "People used to say that's an impossible problem," Elser says. "Then people got to thinking about the fact that there are going to be constraints coming from some rather mundane facts -- that in principle will make the problem of deducing the phase like the solution of a solvable puzzle." The mundane fact, in this case, was a basic premise of the X-ray diffraction experiment: that the object in view have a clearly defined boundary (i.e., that all pixel values outside that boundary be set to zero). "If I set up waves with known amplitudes and synthesize them, I'll find that for essentially all random combinations of phases, the thing I get is an object not confined," Elser says. "It takes a very clever combination of the phases of all those waves to add up to something in only one region; and cancel out everywhere else." For the second constraint, the researchers required that the wave amplitudes used in the Fourier synthesis matched those measured by the experiment. With the two constraints in place, the difference-map algorithm completes the job. Physicist's Algorithm Simplifies Biological Imaging -- And Also Solves Sudoku Puzzles |
March 07, 2006
How do we learn math?Devlin's Angle As a general rule, I try to stay away from the front lines of the math wars, having always felt that the real wisdom about math ed is to be found in the no man's land between the two opposing camps. But President Bush's call for more and better math teachers in his recent State of the Union address led my local newspaper, the San Jose Mercury News, to ask me to pen an Op Ed on the subject. This I duly did, and my piece appeared on Sunday February 19. Attacks (fairly mild, I have to say) predictably followed from both sides, leading me to believe that my remarks probably came out more or less as I intended them to, as a call for reason. In any event, I survived my brief skirmish into dangerous waters sufficiently to be tempted to stick my foot into the pond once again. So here goes. Much of my rationale for believing that the way forward in math ed is to be found in the DMZ of the war comes from my recognition that on both sides you find significant numbers of smart, well-educated, well-meaning people, who genuinely care about mathematics education. Unless you are of the simplistic George W. mindset, that the world divides cleanly into righteous individuals on the one side and "evil doers" on the other, it follows, surely, that both sides have something valuable to say. ("Evil doers" always stuck me as a strange phrase to hear uttered in public by a grown-up, by the way. It sounds more like the box-copy description of a band of orcs in a fantasy video game.) The challenge, then, is to reconcile the two views. In brief, the gist of my Mercury News opinion piece was this. While it sounds reasonable to suggest that understanding mathematical concepts should precede (or go along hand-in-hand with) the learning of procedural skills (such as adding fractions or solving equations), this may be (in practical terms, given the time available) impossible. The human brain evolved into its present state long before mathematics came onto the scene, and did so primarily to negotiate and survive in the physical world. As a consequence, our brains do not find it easy to understand mathematical concepts, which are completely abstract. (This is part of the theme I pursued in my book The Math Gene, published in 2000.) However, although we are not "natural born mathematicians," we do have three remarkable abilities that, taken together, provide the key to learning math. One is our language ability - our capacity to use symbols to represent things and to manipulate those symbols independently of what they represent. The second is our ability to ascribe meaning to our experiences - to make sense of the world, if you like. And the third is our capacity to learn new skills. When we learn a new skill, initially we simply follow the rules in a mechanical fashion. Then, with practice, we gradually become better, and as our performance improves, our understanding grows. Think, for example, of the progression involved in learning to play chess, to play tennis, to ski, to drive a car, to play a musical instrument, to play a video game, etc. We start by following rules in a fairly mechanical fashion. Then, after a while, we are able to follow the rules proficiently. Then, some time later, we are able to apply the rules automatically and fluently. And eventually we achieve mastery and understanding. The same progression works for mathematics, only in this case, as mathematics is constructed and carried out using our language capacity, the initial rule-following stuff is primarily cognitive-linguistic. Of course, there is plenty of evidence to show that mastery of skills without understanding is shallow, brittle, and subject to rapid decay. Understanding mathematical concepts is crucially important to mastering math. The question is: What does it take to achieve the necessary conceptual understanding, and when can it be acquired? Certainly my own experience is that conceptual understanding in mathematics comes only after a considerable amount of procedural practice (much of which therefore is of necessity carried out without understanding). How many of us professional mathematicians aced our high school or college calculus exams but only understood what a derivative is after we had our Ph.D.s and found ourselves teaching the stuff? In fact, I can't imagine how one could possibly understand what calculus is and how and why it works without first using its rules and methods to solve a lot of problems. Likewise for most other areas of mathematics. In fact, the only parts of mathematics that I find sufficiently close to the physical and social world our brains developed to handle that there are innate meanings we could tap into, are positive integer addition and subtraction for fairly small numbers, and perhaps also some fairly simple cases of division for small positive integers. Interestingly, those were the only examples cited by the readers of my Mercury News article who argued against my suggestion that understanding comes only after a lot of procedural practice. Now it may be that in those particular areas, understanding can precede, or accompany the acquisition of, procedural mastery. Personally I doubt it, but I have yet to see convincing evidence either way. But, leaving those special (albeit important) cases aside, what about the rest of mathematics? Here I see no uncertainty. Understanding can come only after procedural mastery. For example, physics and engineering faculty at universities continually stress that what they want their incoming students to have above all is procedural mastery of mathematics as a language - it is, after all, the language of science, as Galileo observed - and the ability to use various mathematical tools and methods to solve problems that arise in physics and engineering. Since even first-year physics and engineering involve use of tools such as partial differential equations, there is no hope that incoming students can have conceptual understanding of those tools and methods. But by a remarkable feature of the human brain, we can achieve procedural mastery without understanding. All it takes is practice. One of the great achievements of mathematics over the past few centuries has been the reduction of conceptually difficult issues to collections of rule-based symbolic procedures (such as calculus). Thus, one of the things that high school mathematics education should definitely produce is the ability to learn and be able to apply rule-based symbolic processes without understanding them. Without that ability, progress into the sciences and engineering is at the very least severely hampered, and for many people may be cut off. (This, by the way, is the only rationale I can think of for teaching calculus in high school. Calculus is a supreme example of a set of rule-based procedures that can be mastered and applied without any hope of anything but the most superficial understanding until relatively late in the game. Basic probability theory and statistics are clearly far more relevant to everyday life in terms of content.) Is mastery of rule-based symbolic procedures the only goal of school mathematics education? Of course not. The reason I am not focusing on conceptual issues is that much has been written on that issue - of particular note the National Research Council's excellent volume Adding it Up: Helping Children Learn Mathematics, published by the National Academy Press in 2001 (a book I have read from cover to cover on three separate occasions). My intention here is to shine as bright a light as possible on a mathematical skill that I think has, in recent years, been overlooked - and on occasion actively derided - by some in the math ed community. Life in today's society requires that we acquire many skills without associated understanding - driving a car, operating a computer, using a VCR, etc. Becoming a better driver, computer user, etc. often requires understanding the technology (and perhaps also the science behind it). But from society's perspective (and in many cases the perspective of the individual), the most important thing is the initial mastery of use. If something has been so well designed or developed that proficient use can be acquired without conceptual understanding, then the rapid acquisition of that skillful use is often the most efficient - and sometimes the only - way for an individual to move ahead. I think this is definitely the case with mathematics. I believe we owe it to our students to prepare them well for life in the highly technological world they will live in. In the case of mathematics, that means that one ability we should equip them with (not the only one by any means - Adding It Up lists several others, for instance) is being able to learn and apply rule-based symbolic processes without understanding them. That does not mean we should not provide explanations. Indeed, as a matter of intellectual courtesy, I think we should. But we need to acknowledge, both to ourselves and our students, that understanding can come only later, as an emergent consequence of use. (No shame in that. It took 300 years from Newton's invention of calculus to a properly worked out conceptual basis for its rules and methods.) How do we learn math? |
March 03, 2006
Quantum leapMichelle Simmons is among the physicists racing to keep us computing as the book closes on Moore's Law. PROFESSOR Michelle Simmons' work with the tiny building blocks of the universe may one day have a big impact on how we compute. As the limits of traditional silicon and Moore's Law - the guiding principle behind 40 years of semiconductor research that suggests chips' complexity would double every 18 months - look set to hit a ceiling in the next few years, the work of Professor Simmons and other quantum computing researchers offers the best hope of ever-continuing processing improvements. The biggest barrier to the future of semiconductor manufacture is the transistors' size, which need to shrink to fit in the same amount of space. The pathways etched into the original 4004 processor in 1971 were 10 microns wide, about 1/10th the width of a human hair. But the pathways on the latest Intel processors are 65 nanometres (nm) wide, or about 1/1500th the width of a human hair. Dr Steve Duvall, a microprocessor engineer and 23-year veteran and a fellow at Intel, expects things to go even further in the future, "down to below 5 nm, possibly even 1 nm, although there's probably a practical limit at 10 nm". Given the width of your average atom is only about 1/10th of a nanometre, we're getting to the point where the atoms themselves are simply too big and clunky for our engineering purposes. A remarkable situation, to say the least. Even with continual developments in semiconductor materials, manufacturing and an emphasis on parallelism rather than producing faster and faster single chips, the days of the microprocessor as we know it appear to be numbered. The US-based Semiconductor Industry Association sees a point in about 15 years where the barriers to continued scaling of transistors get the better of us. At this point we depart from conventional semiconductor engineering and go quantum in order to see computing's future. The task for Professor Simmons and other researchers at the Australian Centre for Quantum Computer Technology is to build a quantum computer - for now, just a few qubits at a time. Professor Simmons' prototype is made of silicon, the same materials used in a traditional microprocessor. Her work is "concerned with developing the technology to build qubit test structures in silicon atom by atom with atomic precision," says Professor Simmons, federation fellow at the school of physics at the University of NSW." Our qubit is a single phosphorus atom in silicon. We need to be able to engineer devices where we can control the position of single phosphorus atoms, their environment and how to electronically manipulate them. Each of these is a challenge." Despite these not insignificant challenges, tremendous progress has been made in the past several years. Many barriers that were considered insurmountable only 10 years ago have been overcome and researchers are well on their way to developing a functional quantum computer." It is a very exciting time here at the moment," she says. What makes quantum computing so potent is the ability to leave behind conventional computing methods and use the idiosyncratic vagaries of quantum mechanics. A conventional computer performs its calculations on binary numbers, called "bits", with each transistor representing a "0" or a "1" depending on whether it holds a charge or not. A quantum computer takes a different approach. As Professor Simmons describes, quantum computing "involves a subtle and counterintuitive law of quantum physics, which tells us that atoms or electrons can exist in two different states at the same time"." In a conventional computer. a bit can only be either a 0 or a 1, while in a quantum computer a qubit can be both a 0 and a 1 at the same time. This means that the information stored in the quantum bits can exist in a huge number of different states in parallel." This "subtle but counterintuitive law" is called "superposition". The difficulty is that superposition means an atom is in two states at once until it interacts with the environment. This could be something as trivial as us observing the quantum computer in action. At this point it goes through "decoherence" and loses its quantum information, along with its computational potency. This means a quantum computer needs to be effectively isolated from the outside world, or from interacting with any observed objects that may interfere with the computational process. But here's the rub: to read the information from the quantum computer, the output eventually needs to be observed. The trick is to collapse the wave function of the atom into a conventional 0 or 1 without it going through decoherence and losing all its quantum information. The final piece in the quantum-computing puzzle is another counterintuitive phenomenon called "entanglement". In defiance of Einstein's assertion that nothing, not even information, can travel faster than the speed of light, there is plenty of evidence that indicates certain subatomic particles recklessly break this rule. Entanglement is where two particles are inexorably linked, and certain effects that apply to one will instantly apply to another. It's this phenomenon of entanglement that allows the qubits in a quantum computer to talk to each other and carry out their computations. THIS seemingly magical ability to compute over any distance instantly does not mean we'll ditch our Pentiums soon. Professor Simmons says practical large-scale quantum computers are still far off." Several small-scale quantum computers (up to 10 qubits) exist already and have performed quantum algorithms, hence the principles of quantum computation have been demonstrated. However, a largescale processor that can solve problems that cannot easily be solved using a conventional computer remains a number of years away. While it is always difficult to estimate the future, I suspect it may be still in the order of tens of years or more," Professor simmons says. Furthermore, assuming we overcome the challenges of building a large-scale quantum computer, they won't entirely replace our conventional computers altogether. Quantum computers have no peers when it comes to certain computations, such as the "travelling salesman" problem. A travelling salesman sits in his car with a bootload of goods to hawk. On his lap sits a map, showing two dozen cities, and the web of highways interconnecting them. Obviously, he'd prefer to visit each city only once and return home in the shortest possible time so he can restock and head out again. The question is, which route to take? The apparent simplicity of the problem is deceptive but is wellknown to combinatorial complexity researchers - the complexity grows exponentially as more cities are added. This means solving it for 10 cities might be a moderately tough challenge but solving it for 20 cities is nearly impossible. Such a problem involving 22 cities would take a conventional desktop computer 1600 years to solve. By that time, the solution will not be of much help to the long-dead salesman. On the other hand, "with a quantum computer, finding a solution would be almost instantaneous", Professor Simmons says. It's this prospect of unprecedented computing power, and the possibility of solving problems of staggering complexity in the blink of an eye. The travelling salesman is also part of a common class of problems called "NP-complete". As such, if we found a single efficient algorithm to solve the travelling salesman problem, it could also solve all other NP-complete problems. This has great significance because there are many practical problems that also turn out to be NP-complete, not just in the field of logistics. Efficiently designing microprocessors is also an NP-complete problem. A SIMILAR problem is code-breaking. Most encryption today uses the prime factors of very large numbers. These factors can be easily multiplied together to get the original very large number, which forms the basis of the public key encryption system. However, with only the original large number (i.e. the public key), it is practically impossible to go backwards and figure out the correct prime factors. C onventional computers have a hard time with these kinds of problems because they're forced to run through just about every possible permutation one at a time to find the right one. A quantum computer, on the other hand, can compute all the possible routes simultaneously. It can do this by using the unique properties of quantum superposition. Because a qubit can represent a 0 and a 1, and every possibility in between, it can perform calculations on all these values in one go. Professor Simmons sees a range of applications for quantum computers in this field. "Important potential applications of a quantum computer would include encryption and decryption of information (coding); weather forecasting; economic modelling; scheduling/timetabling for complex tasks such as airline schedules; fast database searches." Many of these clearly have important economic impacts." Sorting through billions of possible combinations and permutations during code-breaking might take a conventional computer years but would take a quantum computer only seconds. Needless to say, this scares many security experts around the world. But quantum computers are not as efficient as conventional computers when it comes to basic computing tasks, such as running Windows or writing an email. As such, Professor Simmons expects the future will see a fusion of the two complementary technologies." Quantum computers won't completely replace classical computers. It's more likely we'll end up using them both on one chip," she says. Moore's Law may cease to apply within 20 to 30 years for conventional silicon chips as quantum technologies continue to shift processing capability up a gear. MOORE'S LAW THE BIG TRANSISTOR SQUEEZE Intel's first microprocessor, the 4004, contained 2300 transistors and was lauded as a marvel of semiconductor engineering in its time in 1971. Intel's latest Itanium 2 processor, launched in 2004, contains nearly 600 million transistors. That's an astounding 250,000-fold increase in transistors on a single chip in only 33 years. The next generation of Itanium is expected to have no less than 1.7 billion transistors. Despite the mind-bending numbers involved, this exponential increase in transistor count was not at all unexpected. In fact, it was predicted with prescient clarity as far back as 1965 when Gordon Moore famously coined what's now known as "Moore's Law". The latest formulation of the law is now commonly accepted to mean the number of transistors in a microprocessor will double every 24 months. If the law applied to toasters, today we'd have appliances that could handle half a million slices at once and would cost only a few fractions of a cent. Whitegoods manufacturers must be lividly envious. While no other industry has seen such consistent and prolonged growth, this begs the question: when will we hit Moore's Wall? Talk of the end of Moore's Law is as old as the law itself. Yet according to Dr Steve Duvall, microprocessor engineer and 23-year veteran and a fellow at Intel, the history of semiconductor manufacturing is one of repeated technological breakthroughs that have managed to keep the industry on track with Moore's Law." Only a few years ago, the 90 nanometre mark was considered the point at which things would just fall apart." But through a combination of new technologies and materials, the 90 nm process was perfected." It all comes down to good engineering," Dr Duvall says. Yet there is reason to believe that within 15 to 20 years, even good engineering won't be enough. The barriers to continued transistor cramming do not stem from a lack of faith in human ingenuity. The problem is we can't change the laws of physics - and that is where quantum physics offers the best hope for continued computing advances. NEXT SPEAK MOORE'S LAW Intel co-founder Gordon Moore observed that the complexity of an integrated circuit at a given cost would double every 18 months. Computer chips would get faster, while their price stayed the same or fell. Moore's Law is challenged by the speed limits of the universe, which make further miniaturisation beyond a certain point impossible, with implications for industry and commerce. BIT A bit is a "binary digit", and represents a "0" or a "1". Bits make up the foundation of conventional computing. Bits can represent numbers, or values such as "true" or "false". Each transistor in a microprocessor is either switched off or on, which corresponds to the 0 or 1 of a bit. A string of eight bits is a byte. DECOHERENCE When a particle in a state of quantum superposition is observed, it "collapses" into a single fixed state. When it does so, it loses all its quantum information - essentially changing from a qubit into a bit. If decoherence occurs in a quantum computer, all the valuable superposition information is lost and the computer ceases to function. ENTANGLEMENT Normally, nothing can travel faster than the speed of light, including information. However, some quantum particles are fundamentally linked together, so if a change takes place in one particle, the other particle also changes instantaneously. Entanglement is used in a quantum computer to enable two or more qubits to interact and exchange information. QUANTUM MECHANICS Quantum mechanics is the set of laws of physics that applies to the world at a very small scale - around the level of individual atoms and subatomic particles. Quantum mechanics features many strange and counterintuitive observations, such as superposition and entanglement, but these can be used to develop quantum computers. QUBIT A qubit is a "quantum digit" and is the quantum computing equivalent of a bit. It not only represents a 0 or a 1 but, because of the vagaries of superposition, it can also represent other values in between 0 and 1. This means computations done with qubits can process much more information than conventional computers. SUPERPOSITION Superposition is counterintuitive notion that a particle, such as an atom or a photon of light, can be in two different states or two different locations at the same time. Superposition only occurs when the particle is not directly observed. As soon as it's observed, it "collapses" into just one of its states. Quantum leap |
March 01, 2006
Quantum computer solves problem, without runningBy combining quantum computation and quantum interrogation, scientists at the University of Illinois at Urbana-Champaign have found an exotic way of determining an answer to an algorithm – without ever running the algorithm. Using an optical-based quantum computer, a research team led by physicist Paul Kwiat has presented the first demonstration of "counterfactual computation," inferring information about an answer, even though the computer did not run. The researchers report their work in the Feb. 23 issue of Nature. Quantum computers have the potential for solving certain types of problems much faster than classical computers. Speed and efficiency are gained because quantum bits can be placed in superpositions of one and zero, as opposed to classical bits, which are either one or zero. Moreover, the logic behind the coherent nature of quantum information processing often deviates from intuitive reasoning, leading to some surprising effects. "It seems absolutely bizarre that counterfactual computation – using information that is counter to what must have actually happened – could find an answer without running the entire quantum computer," said Kwiat, a John Bardeen Professor of Electrical and Computer Engineering and Physics at Illinois. "But the nature of quantum interrogation makes this amazing feat possible." Sometimes called interaction-free measurement, quantum interrogation is a technique that makes use of wave-particle duality (in this case, of photons) to search a region of space without actually entering that region of space. Utilizing two coupled optical interferometers, nested within a third, Kwiat's team succeeded in counterfactually searching a four-element database using Grover's quantum search algorithm. "By placing our photon in a quantum superposition of running and not running the search algorithm, we obtained information about the answer even when the photon did not run the search algorithm," said graduate student Onur Hosten, lead author of the Nature paper. "We also showed theoretically how to obtain the answer without ever running the algorithm, by using a 'chained Zeno' effect." Through clever use of beam splitters and both constructive and destructive interference, the researchers can put each photon in a superposition of taking two paths. Although a photon can occupy multiple places simultaneously, it can only make an actual appearance at one location. Its presence defines its path, and that can, in a very strange way, negate the need for the search algorithm to run. "In a sense, it is the possibility that the algorithm could run which prevents the algorithm from running," Kwiat said. "That is at the heart of quantum interrogation schemes, and to my mind, quantum mechanics doesn't get any more mysterious than this." While the researchers' optical quantum computer cannot be scaled up, using these kinds of interrogation techniques may make it possible to reduce errors in quantum computing, Kwiat said. "Anything you can do to reduce the errors will make it more likely that eventually you'll get a large-scale quantum computer." Source: University of Illinois at Urbana-Champaign Quantum computer solves problem, without running |
March 01, 2006
Preparing for a Quantum Leap in ComputingImagine a place where anything possible always happens, like a TV screen that displays all the channels at once. If that seems beyond imagination, you are not alone. The world of quantum physics is so weird that even the scientists who study it say it challenges everyday concepts of common sense. The field has grown from a realization that at the smallest scale — the realm where atoms and molecules roam — the classical equations that Isaac Newton used to describe the physical world no longer apply. In this realm, matter behaves differently, and many realities can co-exist. Particles like electrons, for instance, occupy several locations at the same time, behaving more like fuzzy waves than solid pebbles. Fortunately, such weirdness mostly confines itself to the inner life of atoms. But a new quantum world is coming, where scientists hope to preserve the quirky diversity of the subatomic realm. This would allow them to devise superfast computers, design new drugs and guarantee security for sending secret messages. Harnessing the power of the quantum realm requires coordinated planning from experts in fields ranging from physics and chemistry to electrical engineering. And that puts USC College's Daniel Lidar in a perfect position to help prepare for the quantum future. A physicist with joint appointments in the departments of chemistry and electrical engineering, Lidar is a leader in current efforts to transform quantum physics from theoretical curiosity to cutting-edge information technology. As the son of two scientists (a biochemist and pharmacologist), Lidar was constantly exposed to scientific thinking while growing up in Israel and Holland. He earned his Ph.D. in physics from Hebrew University in Jerusalem in 1997, and soon thereafter began exploring the emerging field of quantum information theory. After a postdoctoral position at Berkeley and several years on the faculty at the University of Toronto, he migrated to USC last fall. He was drawn by Southern California's growing status as the world's leading region for the new quantum research enterprise. "This is a real hub," he said, noting that USC, Caltech and UC Santa Barbara all boast strong programs. "Southern California is probably the world capital of activity in my field." In the mid-1990s, Bell Labs mathematician Peter Shor initiated the quantum information revolution by proving that a computer using quantum programming could crack the toughest of today's secret codes, used for governmental, military and financial communication. About the same time, other research showed that only another quantum system could provide absolute protection against any illicit eavesdropping. Work by Lidar and his collaborators has focused on how to protect the delicate process of quantum computing from attack — by nature itself or malicious hackers. So far, quantum computations have been performed only in rudimentary laboratory experiments. If feasible on a larger scale, quantum computers could solve some difficult problems at a fraction of the speed of today's fastest supercomputers. The trick relies on those multiple quantum realities. Like the TV screen showing every channel at once, a quantum computer could process all the numbers in its memory simultaneously, rather than one computation at a time. It's a bit like finding which of a thousand keys opens a lock; instead of trying one at a time, you could just spin one key in the lock until it opened. Certain problems that would tax a supercomputer for a trillion years could yield to a quantum computer in minutes. But such speed is available only as long as the multiple quantum calculations can be protected from outside interference. And the same process nature uses to make rocks and people solid, instead of fuzzy like electrons, conspires to keep that time very, very short. That process, known as quantum decoherence, is usually an immediate and inevitable result of interaction with the environment — collisions with atoms or mere particles of light can cause a frail ensemble of multiple quantum realities to crash. Lidar and colleagues have shown, though, that some quantum computing set-ups are at least partially immune to the ravages of decoherence. By designing an apparatus with "decoherence free subspaces," quantum information can be preserved in the face of environmental insults. The solution is to make sure that external effects exert a symmetric effect on the quantum storage sites. (If one bit of information is altered, so is its partner, so the two together retain a record of the stored information.) A more difficult challenge may arise on a future "quantum internet" where quantum computers share data. Nobody had considered the potential for quantum viruses afflicting such a network until last year, when Lidar and post-doc Lian-Ao Wu proposed a scheme for fighting such "quantum malware" in a paper to be published in the journal Quantum Information Processing. "Essentially the proposal is to do the analog of backup," said Lidar, an associate professor hired as part of the College's Senior Faculty Initiative. Only legitimate users of a system would be told when "real" data is being transmitted. During the remaining down time, the quantum data could be stored on a secure device, off the network, while bogus transmissions serve as a decoy for intruders. A hacker would never know when the system was vulnerable, and constant intrusion attempts would be easy to detect. "It's the first look at this problem," Lidar said, and much further work will be needed to devise foolproof protection and a quantum virus cleanser if infection is successful. For now, of course, quantum viruses are of no serious concern, as there is no quantum network to attack. But Lidar foresees a growing likelihood that quantum technology will soon play a significant role in sending secure messages and eventually in computing. "It's a field that is likely to have a widespread impact in the context of secure information transmission," he said. "It is the most secure method of information transmission that we know of." As for quantum computing, its advantages are limited to certain types of problems; quantum computers are likely never to be good for word processing. But they could prove valuable in economically important realms such as designing drugs from scratch, by computing the quantum rules governing how biological molecules interact. Any such uses depend, of course, on effective hardware for building quantum computing devices, which might require advances in nanotechnology approaches for fabricating the necessary materials. Thus while Lidar focuses on theory, he emphasizes the need to develop the experimental side of the field as well. "My dream for USC would be to develop not only as a leading theoretical place, which I believe it is . . . but also to strongly develop the experimental capabilities here," he said. "That would really put us on the map." Source: University of Southern California, By Tom Siegfried Preparing for a Quantum Leap in Computing |
March 01, 2006
The Galois StoryIvars Peterson  Évariste Galois (1811–1832)
The tragic tale of Évariste Galois (1811–1832), a mathematical prodigy who died in a duel at the tender age of 20, is one of the more dramatic stories in the history of mathematics.
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March 01, 2006
A great polymath, timeless & extraordinaryV SUNDARAM  John von Neumann (1903-1957)
In his blazingly brilliant career as a mathematician, John von Neumann (1903-1957), one of the greatest polymaths of all times, had a profound impact on mathematics, quantum theory, economics, computer science, neurology and other fields. He was the brightest star in Princeton's mathematical firmament and the apostle of the new mathematical era which began in the 1940s. |