Traffic Jam Mystery Solved By Mathematicians

Source:

http://www.sciencedaily.com/releases/2007/12/071219103102.htm

ScienceDaily (Dec. 19, 2007) — Mathematicians from the University of Exeter have solved the mystery of traffic jams by developing a model to show how major delays occur on our roads, with no apparent cause. Many traffic jams leave drivers baffled as they finally reach the end of a tail-back to find no visible cause for their delay. Now, a team of mathematicians from the Universities of Exeter, Bristol and Budapest, have found the answer and published their findings in the journal Proceedings of the Royal Society.

The team developed a mathematical model to show the impact of unexpected events such as a lorry (tractor trailer) pulling out of its lane on a dual carriageway (divided highway with median between traffic going in opposite directions). Their model revealed that slowing down below a critical speed when reacting to such an event, a driver would force the car behind to slow down further and the next car back to reduce its speed further still. The result of this is that several miles back, cars would finally grind to a halt, with drivers oblivious to the reason for their delay.
The model predicts that this is a very typical scenario on a busy highway (above 15 vehicles per km). The jam moves backwards through the traffic creating a so-called 'backward travelling wave', which drivers may encounter many miles upstream, several minutes after it was triggered.
Dr Gábor Orosz of the University of Exeter said: "As many of us prepare to travel long distances to see family and friends over Christmas, we're likely to experience the frustration of getting stuck in a traffic jam that seems to have no cause. Our model shows that overreaction of a single driver can have enormous impact on the rest of the traffic, leading to massive delays."
Drivers and policy-makers have not previously known why jams like this occur, though many have put it down to the sheer volume of traffic. While this clearly plays a part in this new theory, the main issue is around the smoothness of traffic flow. According to the model, heavy traffic will not automatically lead to congestion but can be smooth-flowing. This model takes into account the time-delay in drivers' reactions, which lead to drivers braking more heavily than would have been necessary had they identified and reacted to a problem ahead a second earlier.
Dr Orosz continued: "When you tap your brake, the traffic may come to a full stand-still several miles behind you. It really matters how hard you brake - a slight braking from a driver who has identified a problem early will allow the traffic flow to remain smooth. Heavier braking, usually caused by a driver reacting late to a problem, can affect traffic flow for many miles."
The research team now plans to develop a model for cars equipped with new electronic devices, which could cut down on over-braking as a result of slow reactions.
Adapted from materials provided by University of Exeter.

Fausto Intilla

www.oloscience.com

Mathematician Work To Make Virtual Surgery A Reality

Source:

http://www.sciencedaily.com/releases/2007/11/071126162542.htm

ScienceDaily (Nov. 26, 2007) — A surgeon accidentally kills a patient, undoes the error and starts over again. Can mathematics make such science fiction a reality?
The day is rapidly approaching when your surgeon can practice on your "digital double" -- a virtual you -- before performing an actual surgery, according to UCLA mathematician Joseph Teran, who is helping to make virtual surgery a viable technology. The advantages will save lives, he believes.
"You can fail spectacularly with no consequences when you use a simulator and then learn from your mistakes," said Teran, 30, who joined UCLA's mathematics department in July. "If you make errors, you can undo them -- just as if you're typing in a Word document and you make a mistake, you undo it. Starting over is a big benefit of the simulation.
"Surgical simulation is coming, there is no question about it," he said. "It's a cheaper alternative to cadavers and a safer alternative to patients."
How would virtual surgery work?
"The ideal situation would be when patients come in for a procedure, they get scanned and a three-dimensional digital double is generated; I mean a digital double -- you on the computer, including your internal organs," Teran said. "The surgeon first does surgery on the virtual you. With a simulator, a surgeon can practice a procedure tens or hundreds of times. You could have a patient in a small town scanned while a surgeon hundreds or thousands of miles away practices the surgery. The patient then flies out for the surgery. We have to solve mathematical algorithms so what the surgeon does on the computer mimics real life."
How far off is this virtual surgery?
"A three-dimensional double of you can be made, but it would now take 20 people six to nine months," Teran said. "In the future, one person will be able to do it in minutes. It's going to happen, and it will allow surgeons to make fewer mistakes on actual patients. The only limiting factor is the complexity of the geometry involved. We're working on that. Our job as applied mathematicians is to make these technologies increasingly viable."
The technology will be especially helpful with new kinds of surgeries, he said.
"A virtual surgery cannot be a cartoon," said Teran, who works with a surgeon. "It has to be biologically accurate. A virtual double needs to be really you."
Making virtual surgery a reality will require solving mathematical equations, as well as making progress in computational geometry and computer science. An applied mathematician, Teran works in these fields; he develops algorithms to solve equations. Advances by Teran and other scientists in computational geometry, partial differential equations and large-scale computing are accelerating virtual surgery.
How human tissue responds to a surgeon, Teran said, is based on partial differential equations. Teran solves on a computer the mathematical equations that govern physical phenomena relevant to everyday life. He has studied the biomechanical simulation of soft tissues.
"Most of the behavior of everyday life can be described with mathematical equations," he said. "It's very difficult to reproduce natural phenomena without math."
Tissue, muscle and skin are elastic and behave like a spring, Teran said. Their behavior can be accounted for by a classical mathematical theory.
Progress in his field is already rapid, Teran said, noting that "things in geometry that used to take days and days start to take hours and minutes."
Teran believes medical schools will increasingly train physicians using computer surgical simulation. Teran's applied mathematics can also be used to design more durable bridges, freeways, cars and aircraft.
"I would like people who design bridges to be able to use a virtual model -- I'm interested in making that a reality and in creating numerical algorithmic tools that let people who design bridges have more computational machinery at their fingertips," he said.
As an undergraduate, Teran realized "you can use math problems to solve real problems and can help people in ways that seem totally unrelated to math." He earned his doctorate at Stanford University, where he took graduate classes in partial differential equations and worked on new ways of solving the governing equations of elastic biological tissues. He was a postdoctoral scholar at New York University before joining UCLA's faculty.
"I started with math because I like problem-solving, and I like how elegant math is," Teran said. "I like how much careful analysis is required, and that there's a right answer. Now I'm completely fascinated by what you get from a simulation, the kinds of complex behavior you can reproduce on a computer and the kinds of questions you can answer. Math will tell you how the world is. It will give you an answer, and it's intellectually stimulating and fun. It really pays off."
Teran, who is teaching a course on scientific computing for the visual effects industry, said he came to UCLA because it is one of the country's best universities for applied mathematics, because its medical school is among the country's best and because it is near Hollywood, where he helps to make movie special effects.
Teran, who works with UCLA's Center for Advanced Surgical and Interventional Technology, spoke this fall as part of Intel Chief Technology Officer Justin Rattner's keynote address at the Intel Developer Forum on the rise of the "3-D Internet." Teran demonstrated virtual surgery applications.
The future 3-D Internet will include an "avatar" -- a virtual representation of you -- that could look "just like you, or better than you," Teran said.
The graphics will be astonishingly realistic and three-dimensional, he said, but the simulation needs to be much more accurate, a goal Teran is working to achieve.
"As virtual words get more realistic, modern applied mathematics and scientific computing are required," he said.
Adapted from materials provided by University of California - Los Angeles.

Fausto Intilla
www.oloscience.com

Physicists Tackle Knotty Puzzle

Source:

http://www.sciencedaily.com/releases/2007/10/071003130736.htm

Science Daily — Electrical cables, garden hoses and strands of holiday lights seem to get themselves hopelessly tangled with no help at all. Now research initiated by an undergraduate student at the University of California, San Diego has resulted in the first model of how knots form.

The study investigated the likelihood of knot formation and the types of knots formed in a tumbled string. The researchers say they were interested in the problem because it has many applications, including to the biophysics research questions their group usually studies.
“Knot formation is important in many fields,” said Douglas Smith, an assistant professor of physics who was the senior author on the paper. “For example, knots often form in DNA, which is a long string-like molecule. Cells have enzymes that undo the knots by cutting the DNA strands so that they can pass through each other. Certain anti-cancer drugs stop tumor cells from dividing by blocking the unknotting of DNA.”
Dorian Raymer, a research assistant working with Smith, initiated the study because he was interested in knot theory—the branch of mathematics that uses formulae to distinguish unique knots. Raymer was an undergraduate major in physics when he did the work. Smith said his own interest was piqued when he discovered that no one really knew how knots formed.
“Very little experimental work had been done to apply knot theory to the analysis and classification of real, physical knots,” said Smith. “For mathematicians, the problem is very abstract. They imagine the types of knots that can form and then classify them. In our experiments, we produced thousands of different knots, used mathematical knot theory to analyze them, and then developed a simple physics model to explain our findings.”
The experimental set up consisted of a plastic box that was spun by a computer-controlled motor. A piece of string was dropped into the box and tumbled around like clothes in a dryer. Knots formed very quickly, within 10 seconds. The researchers repeated the experiment more than 3,000 times varying the length and stiffness of string, box size and speed of rotation. They classified the resulting knots.
“It is virtually impossible to distinguish different knots just by looking at them,” said Raymer. “So I developed a computer program to do it. The computer program counts each crossing of the string. It notes whether the crossing is under or over, and whether the string follows a path to the left or to the right. The result is a bunch of numbers that can be translated into a mathematical fingerprint for a knot.
“We used the Jones polynomial—a famous math formula developed by Vaughn Jones, a mathematics professor at U.C. Berkeley—because it automatically simplifies mirror images and other knots that are identical, but look different.”
Rather than getting just a few types of knots, Smith and Raymer got all the types that mathematicians had enumerated, at least up to a certain complexity level. The longer the string, the greater was the probability of getting complex knots.
Based on these observations, the researchers proposed a simplified model for knot formation. The string forms concentric coils, like a looped garden hose, due to its stiffness and the confinement of the box. The free end of the string weaves through the coils, with a 50 percent probability of going under or over any coil. A computer simulation based on this model produced a similar pattern of simple and complex knots as observed in their experiments.
Smith and Raymer said that the model can also explain why confining a stiff string in a smaller box decreases the probability of knot formation. Increased confinement reduces the tumbling motion that facilitates the weaving of the string end through the coils. The paper cites other researchers who have proposed a similar effect to explain why knotting of the umbilical cord of fetuses is relatively rare, occurring only about one percent of the time. Confinement to the amniotic sac may restrict the probability of knotting.
Smith said that their results do not point to any magic solution to prevent knots from forming, but the project did inspire some advice for young people interested in science.
“Even today, there are still interesting scientific problems that can be studied in your garage with inexpensive, off-the-shelf materials like the ones we used in our experiments,” he said. “The most important thing is to be curious and ask good questions.”
This research was published in the journal Proceedings of the National Academy of Sciences.
Note: This story has been adapted from material provided by University of California, San Diego.

Fausto Intilla

www.oloscience.com

Math Model For Circadian Rhythm Created

Source:

http://www.sciencedaily.com/releases/2007/08/070827174303.htm

Science Daily — The internal clock in living beings that regulates sleeping and waking patterns -- usually called the circadian clock -- has often befuddled scientists due to its mysterious time delays. Molecular interactions that regulate the circadian clock happen within milliseconds, yet the body clock resets about every 24 hours. What, then, stretches the expression of the clock over such a relatively long period?

Cornell researchers have contributed to the answer, thanks to new mathematical models recently published.
In the August online edition of Public Library of Science (PLOS) Computational Biology, Cornell biomolecular engineer Kelvin Lee, in collaboration with graduate student Robert S. Kuczenski, Kevin C. Hong '05 and Jordi Garcia-Ojalvo of Universitat Politecnica de Catalunya, Spain, hypothesize that the accepted model of circadian rhythmicity may be missing a key link, based on a mathematical model of what happens during the sleeping/waking cycle in fruit flies.
"We didn't discover any new proteins or genes," Lee said. "We took all the existing knowledge, and we tried to organize it."
Using mathematical models initially created by Hong, who has since graduated, the team set out to map the molecular interactions of proteins called period and timeless -- widely known to be related to the circadian clock.
The group hypothesized that an extra, unknown protein would need to be inserted into the cycle with period and timeless, a molecule that Kuczenski named the focus-binding mediator, in order for the cycle to stretch to 24 hours.
Lee said many scientists are interested in studying the circadian clock, and not just to understand such concepts as jet lag -- fatigue induced by traveling across time zones. Understanding the body's biological cycle might, for example, lead to better timing of delivering chemotherapy, when the body would be most receptive, Lee said.
Note: This story has been adapted from a news release issued by Cornell University.

Fausto Intilla

www.oloscience.com

Mobius Strip: 'Endless Ribbon' Mystery Solved

Source:

http://www.sciencedaily.com/releases/2007/07/070726152816.htm

Science Daily — Dr Eugene Starostin and Dr Gert van der Heijden (both from UCL Civil & Environmental Engineering) recently published the solution to a 75-year-old mystery. The two academics have discovered how to predict the shape of a Möbius strip, the ‘endless ribbon’ which is obtained by taking a rectangular strip of paper, twisting one end through 180 degrees, and then joining the ends.

The shape takes its name from August Möbius, the German mathematician who presented his discovery of a 3D-shape with only one ‘side’ to the Academy of Sciences in Paris in 1858. The shape was rediscovered by artists and famously depicted by Escher. The first papers that attempted to work out how to predict the 3D shape of an inextensible Möbius strip were published in 1930, but the problem has remained unresolved until now.Dr Starostin and Dr van der Heijden realised that the shape can be described by a set of 20-year-old equations that have only been published online. Their letter to ‘Nature Materials’ demonstrates that these differential equations govern the shapes of elastic strips when they are at rest, and enable researchers to calculate their geometry.Möbius strips are not merely mathematical abstractions. Conveyor belts, recording tapes and rollercoasters are all manufactured in this shape, and chemists have now grown single crystals in the form of a Möbius strip. The academics believe their methods can be used to model ‘crumpled’ shapes that are not based on rectangular strips, such as screwed-up paper, the drape of fabrics and leaves.“This is the first non-trivial application of this mathematical theory,” said Dr Starostin. “It could prove to be useful to other research communities, such as mechanics and physics.”

Note: This story has been adapted from a news release issued by University College London.

Fausto Intilla

www.oloscience.com

Manhattan-Size Calculation: Mathematicians Map One Of The Most Complicated Structures

Source:

http://www.sciencedaily.com/releases/2007/03/070319090520.htm

Science Daily — Mathematicians have mapped the inner workings of one of the most complicated structures ever studied: the object known as the exceptional Lie group E8. This achievement is significant both as an advance in basic knowledge and because of the many connections between E8 and other areas, including string theory and geometry.The magnitude of the calculation is staggering: the answer, if written out in tiny print, would cover an area the size of Manhattan.

Mathematicians are known for their solitary work style, but the assault on E8 is part of a large project bringing together 18 mathematicians from the U.S. and Europe for an intensive four-year collaboration."This is exciting," said Peter Sarnak, Eugene Higgins Professor of Mathematics at Princeton University (not affiliated with the project). "Understanding and classifying the representations of Lie Groups has been critical to understanding phenomena in many different areas of mathematics and science including algebra, geometry, number theory, Physics and Chemistry. This project will be valuable for future mathematicians and scientists."Bigger than the Human GenomeThe magnitude of the E8 calculation invites comparison with the Human Genome Project. The human genome, which contains all the genetic information of a cell, is less than a gigabyte in size. The result of the E8 calculation, which contains all the information about E8 and its representations, is 60 gigabytes in size. That is enough space to store 45 days of continuous music in MP3 format. While many scientific projects involve processing large amounts of data, the E8 calculation is very different: the size of the input is comparatively small, but the answer itself is enormous, and very dense.Like the Human Genome Project, these results are just the beginning. According to project leader Jeffrey Adams, "This is basic research which will have many implications, most of which we don't understand yet. Just as the human genome does not instantly give you a new miracle drug, our results are a basic tool which people will use to advance research in other areas." This could have unforeseen implications in mathematics and physics which do not appear for years. According to Hermann Nicolai, Director of the Albert Einstein Institute in Bonn, Germany (not affiliated with the project), "This is an impressive achievement. While mathematicians have known for a long time about the beauty and the uniqueness of E8, we physicists have come to appreciate its exceptional role only more recently --- yet, in our attempts to unify gravity with the other fundamental forces into a consistent theory of quantum gravity, we now encounter it at almost every corner! Thus, understanding the inner workings of E8 is not only a great advance for pure mathematics, but may also help physicists in their quest for a unified theory." The E8 CalculationThe team that produced the E8 calculation began work four years ago. They meet together at the American Institute of Mathematics every summer, and in smaller groups throughout the year. Their work requires a mix of theoretical mathematics and intricate computer programming.According to team member David Vogan from MIT, "The literature on this subject is very dense and very difficult to understand. Even after we understood the underlying mathematics it still took more than two years to implement it on a computer."And then there came the problem of finding a computer large enough to do the calculation. For another year, the team worked to make the calculation more efficient, so that it might fit on existing supercomputers, but it remained just beyond the capacity of the hardware available to them. The team was contemplating the prospect of waiting for a larger computer when Noam Elkies of Harvard pointed out an ingenious way to perform several small versions of the calculation, each producing an incomplete version of the answer. These incomplete answers could be assembled to give the final solution. The cost was having to run the calculation four times, plus the time to combine the answers. In the end the calculation took about 77 hours on the supercomputer Sage. Beautiful SymmetryAt the most basic level, the E8 calculation is an investigation of symmetry. Mathematicians invented the Lie groups to capture the essence of symmetry: underlying any symmetrical object, such as a sphere, is a Lie group.Lie groups come in families. The classical groups A1, A2, A3, ... B1, B2, B3, ... C1, C2, C3, ... and D1, D2, D3, ... rise like gentle rolling hills towards the horizon. Jutting out of this mathematical landscape are the jagged peaks of the exceptional groups G2, F4, E6, E7 and, towering above them all, E8. E8 is an extraordinarily complicated group: it is the symmetries of a particular 57-dimensional object, and E8 itself is 248-dimensional! To describe the new result requires one more level of abstraction. The ways that E8 manifests itself as a symmetry group are called representations. The goal is to describe all the possible representations of E8. These representations are extremely complicated, but mathematicians describe them in terms of basic building blocks. The new result is a complete list of these building blocks for the representations of E8, and a precise description of the relations between them, all encoded in a matrix with 205,263,363,600 entries. The Atlas of Lie Groups ProjectThe E8 calculation is part of an ambitious project known as the Atlas of Lie Groups and Representations. The goal of the Atlas project is to determine the unitary representations of all the Lie groups. This is one of the great unsolved problems of mathematics, dating from the early 20th century. The success of the E8 calculation leaves little doubt that the Atlas team will complete their task. The Atlas team consists of about 20 researchers from the United States and Europe. The core group consists of Jeffrey Adams (University of Maryland), Dan Barbasch (Cornell), John Stembridge (University of Michigan), Peter Trapa (University of Utah) , Marc van Leeuwen (Poitiers), David Vogan (MIT), and (until his death in 2006) Fokko du Cloux (Lyon). The Atlas project is funded by the National Science Foundation through the American Institute of Mathematics.

Note: This story has been adapted from a news release issued by American Institute of Mathematics.

Fausto Intilla

www.oloscience.com

Mathematicians Discover A Simple Way To Formulate Complex Scientific Results

Source:

http://www.sciencedaily.com/releases/2007/06/070627134355.htm

Science Daily — A new analysis of behaviour in a structured population illuminates Darwin’s theories of co-operation and competition between kin, and provides an abstract model that could simplify scientists’ quest to map behaviour among disease-causing organisms within a cell.

The study by Queen’s Mathematics and Statistics professor Peter Taylor, and co-authors Troy Day (Queen’s) and Geoff Wild (University of Western Ontario) presents a simple formula for balancing the benefit and cost in altruistic acts, allowing researchers to predict behaviour and summarize disparate results in a simple framework.“Although our main focus is on cooperation, these graph-theoretic relationships can apply to the evolution of other traits,” says Dr. Taylor. For example, at Queen’s we are particularly interested in the behaviour of pathogens competing within a host, in their capacity to cause disease, and we expect applications of our results to these models.” The study titled Evolution of cooperation in a finite homogeneous graph is published in Nature. It provides a system that can be applied to any species within its natural environment to gain an understanding of its behaviour and interactions.“One can imagine interacting individuals playing a game,” says Dr. Taylor. “With fitness determined by the game payoffs and the competition between offspring for space, our model predicts which strategies will emerge under the forces of evolution.”The research was funded by NSERC’s program of discovery grants.

Note: This story has been adapted from a news release issued by Queen's University.

Fausto Intilla

www.oloscience.com

Indians Predated Newton 'Discovery' By 250 Years, Scholars Say

Source:

http://www.sciencedaily.com/releases/2007/08/070813091457.htm

Science Daily — A little known school of scholars in southwest India discovered one of the founding principles of modern mathematics hundreds of years before Newton -- according to new research.Dr George Gheverghese Joseph from The University of Manchester says the 'Kerala School' identified the 'infinite series '- one of the basic components of calculus - in about 1350.

The discovery is currently - and wrongly - attributed in books to Sir Isaac Newton and Gottfried Leibnitz at the end of the seventeenth centuries.The team from the Universities of Manchester and Exeter reveal the Kerala School also discovered what amounted to the Pi series and used it to calculate Pi correct to 9, 10 and later 17 decimal places.And there is strong circumstantial evidence that the Indians passed on their discoveries to mathematically knowledgeable Jesuit missionaries who visited India during the fifteenth century.That knowledge, they argue, may have eventually been passed on to Newton himself.Dr Joseph made the revelations while trawling through obscure Indian papers for a yet to be published third edition of his best selling book 'The Crest of the Peacock: the Non-European Roots of Mathematics' by Princeton University Press.He said: "The beginnings of modern maths is usually seen as a European achievement but the discoveries in medieval India between the fourteenth and sixteenth centuries have been ignored or forgotten."The brilliance of Newton's work at the end of the seventeenth century stands undiminished -- especially when it came to the algorithms of calculus."But other names from the Kerala School, notably Madhava and Nilakantha, should stand shoulder to shoulder with him as they discovered the other great component of calculus- infinite series."There were many reasons why the contribution of the Kerala school has not been acknowledged - a prime reason is neglect of scientific ideas emanating from the Non-European world - a legacy of European colonialism and beyond."But there is also little knowledge of the medieval form of the local language of Kerala, Malayalam, in which some of most seminal texts, such as the Yuktibhasa, from much of the documentation of this remarkable mathematics is written.He added: "For some unfathomable reasons, the standard of evidence required to claim transmission of knowledge from East to West is greater than the standard of evidence required to knowledge from West to East."Certainly it's hard to imagine that the West would abandon a 500-year-old tradition of importing knowledge and books from India and the Islamic world."But we've found evidence which goes far beyond that: for example, there was plenty of opportunity to collect the information as European Jesuits were present in the area at that time."They were learned with a strong background in maths and were well versed in the local languages."And there was strong motivation: Pope Gregory XIII set up a committee to look into modernising the Julian calendar."On the committee was the German Jesuit astronomer/mathematician Clavius who repeatedly requested information on how people constructed calendars in other parts of the world. The Kerala School was undoubtedly a leading light in this area."Similarly there was a rising need for better navigational methods including keeping accurate time on voyages of exploration and large prizes were offered to mathematicians who specialised in astronomy."Again, there were many such requests for information across the world from leading Jesuit researchers in Europe. Kerala mathematicians were hugely skilled in this area."

Note: This story has been adapted from a news release issued by University of Manchester.

Fausto Intilla

www.oloscience.com