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2010-01-11T13:46:00.001-08:00

'Bacterial Computers': Genetically Engineered Bacteria Have Potential To Solve Complicated Mathematical Problems

2009-07-25T00:40:00.000-07:00

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ScienceDaily (July 24, 2009) — US researchers have created 'bacterial computers' with the potential to solve complicated mathematics problems. The findings of the research demonstrate that computing in living cells is feasible, opening the door to a number of applications. The second-generation bacterial computers illustrate the feasibility of extending the approach to other computationally challenging math problems.

A research team made up of four faculty members and 15 undergraduate students from the biology and mathematics departments at Missouri Western State University in Missouri and Davidson College in North Carolina, USA engineered the DNA of Escherichia coli bacteria, creating bacterial computers capable of solving a classic mathematical problem known as the Hamiltonian Path Problem.
The research extends previous work published last year in the same journal to produce bacterial computers that could solve the Burnt Pancake Problem.
The Hamiltonian Path Problem asks whether there is a route in a network from a beginning node to an ending node, visiting each node exactly once. The student and faculty researchers modified the genetic circuitry of the bacteria to enable them to find a Hamiltonian path in a three-node graph. Bacteria that successfully solved the problem reported their success by fluorescing both red and green, resulting in yellow colonies.
Synthetic biology is the use of molecular biology techniques, engineering principles, and mathematical modeling to design and construct genetic circuits that enable living cells to carry out novel functions. "Our research contributed more than 60 parts to the Registry of Standard Biological Parts, which are available for use by the larger synthetic biology community, including the newly split red fluorescent protein and green fluorescent protein genes," said Jordan Baumgardner, recent graduate of Missouri Western and first author of the research paper. "The research provides yet another example of how powerful and dynamic synthetic biology can be. We used synthetic biology to solve mathematical problems; others find applications in medicine, energy and the environment. Synthetic biology has great potential in the real world."
According to Dr. Eckdahl, the corresponding author of the article, synthetic biology affords a new opportunity for multidisciplinary undergraduate research training. "We have found synthetic biology to be an excellent way to engage students in research that connects biology and mathematics. Our students learn firsthand the value of crossing traditional disciplinary lines."
Journal references:
Jordan Baumgardner, Karen Acker, Oyinade Adefuye, Samuel THOMAS Crowley, Will DeLoache, James O Dickson, Lane Heard, Andrew T Martens, Nickolaus Morton, Michelle Ritter, Amber Shoecraft, Jessica Treece, Matthew Unzicker, Amanda Valencia, Mike Waters, A. M. Campbell, Laurie J. Heyer, Jeffrey L. Poet and Todd T. Eckdahl. Solving a Hamiltonian Path Problem with a bacterial computer. Journal of Biological Engineering, (in press) [link]
Haynes et al. Engineering bacteria to solve the Burnt Pancake Problem. Journal of Biological Engineering, 2008; 2 (1): 8 DOI: 10.1186/1754-1611-2-8
Adapted from materials provided by BioMed Central, via EurekAlert!, a service of AAAS.

DNA Sudoku: Logic Of 'Sudoku' Math Puzzle Used To Vastly Enhance Genome-sequencing Capability

2009-06-27T08:48:00.001-07:00

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ScienceDaily (June 25, 2009) — A math-based game that has taken the world by storm with its ability to delight and puzzle may now be poised to revolutionize the fast-changing world of genome sequencing and the field of medical genetics, suggests a new report by a team of scientists at Cold Spring Harbor Laboratory (CSHL). The report will be published as the cover story in the July 1st issue of the journal Genome Research.

Combining a 2,000-year-old Chinese math theorem with concepts from cryptology, the CSHL scientists have devised "DNA Sudoku." The strategy allows tens of thousands of DNA samples to be combined, and their sequences – the order in which the letters of the DNA alphabet (A, T, G, and C) line up in the genome – to be determined all at once.
This achievement is in stark contrast to past approaches that allowed only a single DNA sample to be sequenced at a time. It also significantly improves upon current approaches that, at best, can combine hundreds of samples for sequencing.
"In theory, it is possible to use the Sudoku method to sequence more than a hundred thousand DNA samples," says CSHL Professor Gregory Hannon, Ph.D., a genomics expert and leader of the team that invented the "Sudoku" approach. At that level of efficiency, it promises to reduce costs dramatically. A sequencing project that costs upwards of $10 million using conventional methods may be accomplished for $50,000 to $80,000 using DNA Sudoku, he estimates.
Originally devised to overcome a sequencing limitation that dogged one of the Hannon lab's research projects, the new method has tremendous potential for clinical applications. It can be used, says Hannon, to analyze specific regions of the genomes of a large population and identify individuals who carry mutations that cause genetic diseases – a process known as genotyping.
The CSHL team has already begun to explore this possibility via a collaboration with Dor Yeshorim, a New York-based organization that has collected DNA from thousands of members of orthodox Jewish communities. The organization's aim is to prevent genetic diseases such as Tay-Sachs or cystic fibrosis that occur frequently within specific ethnic populations. The team's new method will now allow the many thousands of DNA samples gathered by Dor Yeshorim to be processed and sequenced in a single time-saving and cost-effective experiment, which should identify individuals who carry disease-causing mutations.
The advantages of DNA Sudoku
The mixing together and simultaneous sequencing of a massive number of DNA samples is known as multiplexing. In previous multiplexing approaches, scientists first tagged each sample with a barcode – a short string of DNA letters known as oligonucleotides – before mixing it with other samples that also had unique tags. After the sample mix had been sequenced, scientists could use the barcode tags on the resulting sequences as identification markers and thus tell which sequence belonged to which sample.
"But this approach is very limiting," explains Yaniv Erlich, a graduate student in the Hannon laboratory and first author on the "DNA Sudoku" paper. "It's time-consuming and costly to have to design a unique barcode for each sample prior to sequencing, especially if the number of samples runs in the thousands."
In order to circumvent this limitation, Erlich and others in the Hannon lab came up with the idea of mixing the samples in specific patterns, thereby creating pools of samples. And instead of tagging the individual samples within each pool, the scientists tagged each pool as a whole with one barcode. "Since we know which pool contains which samples, we can link a sequence to an individual sample with high confidence," says Erlich.
The key to the team's innovation is the pooling strategy, which is based on the 2,000-year-old Chinese remainder theorem. "It minimizes the number of pools and the amount of sequencing," says Hannon of their method, which they dubbed "DNA Sudoku" because of its similarity to the logic and combinatorial number-placement rules used in the popular game.
The method, which the CSHL team has patented, is currently best suited for genotype analyses that require only short segments of an individual's genome to be sequenced to find out if the individual is carrying a certain variant of a gene or a rare mutation. But as sequencing technologies improve and researchers gain the ability to generate sequences for longer segments of the genome, Hannon envisions wider clinical applications for their method such as HLA typing, already an important diagnostic tool for autoimmune diseases, cancer, and for predicting the risk of organ transplantation.
Journal reference:
Erlich et al. DNA Sudoku--harnessing high-throughput sequencing for multiplexed specimen analysis. Genome Research, 2009; DOI: 10.1101/gr.092957.109
Adapted from materials provided by Cold Spring Harbor Laboratory, via EurekAlert!, a service of AAAS.

New Pattern Found in Prime Numbers

2009-05-10T23:19:00.001-07:00

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(PhysOrg.com) -- Prime numbers have intrigued curious thinkers for centuries. On one hand, prime numbers seem to be randomly distributed among the natural numbers with no other law than that of chance. But on the other hand, the global distribution of primes reveals a remarkably smooth regularity. This combination of randomness and regularity has motivated researchers to search for patterns in the distribution of primes that may eventually shed light on their ultimate nature.

In a recent study, Bartolo Luque and Lucas Lacasa of the Universidad Politécnica de Madrid in Spain have discovered a new pattern in primes that has surprisingly gone unnoticed until now. They found that the distribution of the leading digit in the prime number sequence can be described by a generalization of Benford’s law. In addition, this same pattern also appears in another number sequence, that of the leading digits of nontrivial Riemann zeta zeros, which is known to be related to the distribution of primes. Besides providing insight into the nature of primes, the finding could also have applications in areas such as fraud detection and stock market analysis.
“Mathematicians have studied prime numbers for centuries,” Lacasa told PhysOrg.com. “New insights and concepts coming from nonlinear science, such as multiplicative processes, help us to look at prime numbers from a different perspective. According to this focus, it becomes significant that even today it is still possible to discover unnoticed hints of statistical regularity in such sequences, without being an expert in number theory. However, the most significant issue in this work is not to unveil this pattern in primes and Riemann zeros, but to understand the reason and implications of such unexpected structure, not just for number theoretical issues but, interestingly, for other disciplines as well. For instance, these results deepen our understanding of correlations in systems composed of many elements.”
Benford’s law (BL), named after physicist Frank Benford in 1938, describes the distribution of the leading digits of the numbers in a wide variety of data sets and mathematical sequences. Somewhat unexpectedly, the leading digits aren’t randomly or uniformly distributed, but instead their distribution is logarithmic. That is, 1 as a first digit appears about 30% of the time, and the following digits appear with lower and lower frequency, with 9 appearing the least often. Benford’s law has been shown to describe disparate data sets, from physical constants to the length of the world’s rivers.
Since the late ‘70s, researchers have known that prime numbers themselves, when taken in very large data sets, are not distributed according to Benford’s law. Instead, the first digit distribution of primes seems to be approximately uniform. However, as Luque and Lacasa point out, smaller data sets (intervals) of primes exhibit a clear bias in first digit distribution. The researchers noticed another pattern: the larger the data set of primes they analyzed, the more closely the first digit distribution approached uniformity. In light of this, the researchers wondered if there existed any pattern underlying the trend toward uniformity as the prime interval increases to infinity.

The set of all primes - like the set of all integers - is infinite. From a statistical point of view, one difficulty in this kind of analysis is deciding how to choose at “random” in an infinite data set. So a finite interval must be chosen, even if it is not possible to do so completely randomly in a way that satisfies the laws of probability. To overcome this point, the researchers decided to chose several intervals of the shape [1, 10d]; for example, 1-100,000 for d = 5, etc. In these sets, all first digits are equally probable a priori. So if a pattern emerges in the first digit of primes in a set, it would reveal something about first digit distribution of primes, if only within that set.
By looking at multiple sets as d increases, Luque and Lacasa could investigate how the first digit distribution of primes changes as the data set increases. They found that primes follow a size-dependent Generalized Benford’s law (GBL). A GBL describes the first digit distribution of numbers in series that are generated by power law distributions, such as [1, 10d]. As d increases, the first digit distribution of primes becomes more uniform, following a trend described by GBL. As Lacasa explained, both BL and GBL apply to many processes in nature.
“Imagine that you have $1,000 in your bank account, with an interest rate of 1% per month,” Lacasa said. “The first month, your money will become $1,000*1.01 = $1,010. The next month, $1,010*1.01, and so on. After n months, you will have $1,000*(1.01)^n. Notice that you will need many months to go from $1,000 to $2,000, while to go from $8,000 to $9,000 will be much easier. When you analyze your accounting data, you will realize that the first digit 1 is more represented than 8 or 9, precisely as Benford's law dictates. This is a very basic example of a multiplicative process where 0.01 is the multiplicative constant.
“Physicists have shown that many processes in nature can be modeled as stochastic multiplicative processes, where the previously constant value of 0.01 is now a random variable and the data equivalent to the money of our latter example is another random variable with an underlying distribution 1/x. Stochastic processes with such distributions are shown to follow BL. Now, many other phenomena fit better to a stochastic process with a more general underlying probability x^[-alpha], where alpha is different from one. The first digit distribution related with this general power law distribution is the so-called Generalized Benford law (which converges to BL for alpha = 1).”
Significantly, Luque and Lacasa showed in their study that GBL can be explained by the prime number theorem; specifically, the shape of the mean local density of the sequences is responsible for the pattern. The researchers also developed a mathematical framework that provides conditions for any distribution to conform to a GBL. The conditions build on previous research, which has shown that Benford behavior could occur when a distribution follows BL for particular values of its parameters, as in the case of primes. Luque and Lacasa also investigated the sequence of nontrivial Riemann zeta zeros, which are related to the distribution of primes, and whose distribution of the zeros is considered to be one of the most important unsolved mathematical problems. Although the distribution of the zeros does not follow BL, here the researchers found that it does follow a size-dependent GBL, as in the case of the primes.
The researchers suggest that this work could have several applications, such as identifying other sequences that aren’t Benford distributed, but may be GBL. In addition, many applications that have been developed for Benford’s law could eventually be generalized to the wider context of the Generalized Benford’s law. One such application is fraud detection: while naturally generated data obey Benford’s law, randomly guessed (fraudulent) data do not, in general.
“BL is a specific case of GBL,” Lacasa explained. “Many processes in nature can be fitted to a GBL with alpha = 1, i.e. a BL. The hidden structure that Benford's law quantifies is lost when numbers are artificially modified: this is a principle for fraud detection in accounting, where the combinatorial mechanisms associated to accounting sets are such that BL applies. The same principle holds for processes following GBL with a generic alpha, where BL fails. Last, for processes whose underlying density is not x^(-alpha) but 1/logN, a size-dependent GBL would be the correct hallmark.”
More information: Bartolo Luque and Lucas Lacasa. “The first digit frequencies of primes and Riemann zeta zeros.” Proceedings of the Royal Society A. doi: 10.1098/rspa.2009.0126.

Way To Control Chaos? Rigid Structure Discovered In Center Of Air Turbulence

2009-05-06T09:45:00.001-07:00

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ScienceDaily (May 6, 2009) — Pioneering mathematical engineers have discovered for the first time a rigid structure which exists within the centre of turbulence, leading to hope that its chaotic movement could be controlled in the future.

Dr Sotos Generalis from Aston University in Birmingham, UK and Dr Tomoaki Itano from Kansai University in Osaka, Japan, believe their discovery of the Hairpin Vortex Solution could revolutionise our understanding of turbulence and our ability to control it.
This rigid, set structure, named after its hairpin like shape was found within Plane Couette flow. This is a prototype of turbulent shear flow, where turbulence is created in fluid flow between the space of two opposite moving planar fluid boundaries, when high- and low-speed fluids collide.
Everyone from Formula One drivers experiencing drag, through to aeroplane passengers suffering a bumpy flight, will have experienced clear-air turbulence, the mixing of high- and low-speed air in the atmosphere.
This newly found turbulent state is constituted by a number of elements found in a coherent flow structure and has been described by the research team as a "tapestry of knotted vortices."
While structures, known as wall structures have been found on the ‘edge’ of turbulence, an elusive middle or wake structure has never been discovered, until now.
Dr Generalis believes that finding a regimented structure within the very heart of Couette flow could prove invaluable to controlling turbulence and the effects of turbulence between two moving boundaries, in the future. This could include working machinery parts, medical treatment involving blood flow, and turbulence in air, sea and road travel.
“Ten years ago scientists believed turbulence was in a ‘world’ of its own, until we began to find ‘wall structures’ on its side. We believed a middle or wake structure might exist, and now we can prove there is regimented structure at the very centre of turbulence. This new discovery paves the way for the ‘marriage’ between wake and wall structures in shear flow turbulence and provides a unique picture of the Couette flow turbulent eddies only observed but never understood before.
The team’s findings of this missing central link have been published in Physical Review Letters and come after nearly five years of research, created by thousands of computer generated shear flow models. The result was obtained by replicating the exposure of two opposite plates to hot and cold conditions, moving from a static to dynamic position. The research team are now aiming to find if similar structures exist within other cases of turbulent fluid flow.
“The hairpins expose an all new ‘view’ of the transition to turbulence and it is our aim to ‘unify’ this idea discovered in Couette flow, into other areas of shear flow in general,” added Dr Generalis.
Journal reference:
Tomoaki Itano and Sotos C. Generalis. Hairpin Vortex Solution in Planar Couette Flow: A Tapestry of Knotted Vortices. Physical Review Letters, 2009; 102 (11): 114501 DOI: 10.1103/PhysRevLett.102.114501
Adapted from materials provided by Aston University.

New Mathematical System Helps To Cut Bus Journey Times

2009-03-18T04:17:00.001-07:00

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ScienceDaily (Mar. 17, 2009) — A research team from the University of Burgos (UBU) has designed a system to reduce the time spent waiting at bus stops in the city, as well as the length of bus journeys. The method, which could be applied to other locations, is based on a mathematical strategy known as “taboo search.”

The UBU’s Research Group on Metaheuristic Techniques, working with Mexican researchers, has come up with a system to reduce the time spent waiting at bus stops in Burgos from 20 to 17 minutes, and to cut average bus journey times from 16 to 13.5 minutes.
“This translates into a 13% performance improvement of this urban transport service,” Joaquín A. Pacheco, the research group’s coordinator and director of the UBU’s Applied Economics Department, tells SINC.
“When we face a problem of this kind, we can use an exact method, which gives us an optimal solution but takes a long time to calculate – or we can use an approximate or heuristic technique, which provides a good solution with less calculation time,” says Pacheco, who chose the second method.
Heuristic algorithms are more effective in certain situations, such as when the data for a problem to be solved are imprecise, or when fast and adaptable solutions are needed. The researchers used the “taboo search” method from the most highly perfected heuristic techniques, known as metaheuristics. This strategy uses learning methods, and classifies certain options as “forbidden” or “taboo”, to avoid developing solutions that have already been previously created.
“For example, when searching for solutions, if a bus has just left a particular stop, this stop is then marked as ‘taboo’ and it cannot be included as part of that route again for a certain number of iterations (repetitions),” explains Pacheco.
The results of the “taboo search” made it possible to design slight modifications to the current lines operating in the city, for example by making a bus route run along one street instead of another, or by eliminating some twists and turns, as well as to redistribute the bus fleet, especially in the eastern part of the city. These improvements will allow urban transport users in Burgos to save time during their trips.
In order to carry out this study, the researchers maintained a set number of bus routes, as well as their starting and finishing points. The work was commissioned by Burgos City Council, which requested that no abrupt changes be made to the service. The city has 382 bus stops served by 24 routes operating on working days.
This method can also be used in other towns, according to its developers. The metaheuristic techniques are already being applied in Castilla-La Mancha and the Balearic Islands in order to reduce the cost of school transport services, and the amount of time students have to wait at bus stops.
Journal reference:
Joaquín Pacheco, Ada Álvarez, Silvia Casado, José Luis González-Velarde. A tabu search approach to an urban transport problem in northern Spain. Computers & Operations Research, 2009; 36 (3): 967 DOI: 10.1016/j.cor.2007.12.002
Adapted from materials provided by Plataforma SINC, via AlphaGalileo.

Mathematicians Reveal Secrets Of The Ancient And Universal Art Of Symmetry

2008-05-25T06:39:00.001-07:00

Source:

http://www.sciencedaily.com/releases/2008/05/080521173851.htm

ScienceDaily (May 24, 2008) — Humans have used symmetrical patterns for thousands of years in both functional and decorative ways. Now, a new book by three mathematicians offers both math experts and enthusiasts a new way to understand symmetry and a fresh way to see the world.

In The Symmetries of Things, eminent Princeton mathematician John H. Conway teams up with Chaim Goodman-Strauss of the University of Arkansas and Heidi Burgiel of Bridgewater State College to present a comprehensive mathematical theory of symmetry in a richly illustrated volume. The book is designed to speak to those with an interest in math, artists, working mathematicians and researchers.
“Symmetry and pattern are fundamentally human preoccupations in the same way that language and rhythm are. Any culture that is making anything has ornament and is preoccupied with this visual rhythm,” Goodman-Strauss said. “There are actually Neolithic examples of many of these patterns. The fish-scale pattern, for example, is 22,000 years old and shows up all over the world in all kinds of contexts.”
Symmetrical objects and patterns are everywhere. In nature, there are flowers composed of repeating shapes that rotate around a central point. Architects trim buildings with friezes that repeat design elements over and over.
Mathematicians, according to Goodman-Strauss, are latecomers to the human fascination with pattern. While mathematicians bring their own particular concerns, “we’re also able to say things that other people might not be able to say.”
Symmetries of Things contributes a new system of notation or descriptive categories for symmetrical patterns and a host of new proofs. The first section of the book is written to be accessible to a general reader with an interest in the subject. Sections two and three are aimed at mathematicians and experts in the field. The entire book, Goodman-Strauss said, “is meant to be engaging and reveal itself visually as well.”
To explain the significance within mathematics of understanding symmetry, Goodman-Strauss began by talking about mathematics in general: “Mathematicians above anything else study structure, structure for its own sake, mental structure, not necessarily physical structure. That’s why mathematics is so good at describing the world. What more fundamental kind of structure could you consider than the way patterns can be laid out in a plane?”
While mathematics may be called “a descriptive art,” Goodman-Strauss noted that mathematicians are not simply trying to describe. Rather, he said, “We’re trying to understand what inherently can be described in a quantitative, analytical way.”
For about a hundred years, mathematicians have used a system developed by crystallographers to describe symmetries, a system that didn’t easily generalize to other situations. Conway developed a notational system that is more useful for mathematicians, a flexible, intuitive system that is “much more than a naming system,” according to Goodman-Strauss.
“Conway is one of the best notation-makers in the world,” Goodman-Strauss said. “A good notation is amazing because it’s not just a way of naming things. It’s a way of making the structure of things transparent and simultaneously providing a way of enumerating them, classifying them and proving that’s what the classification is – all at once. That’s really the big exciting thing.”
The second section of the book discusses the orbifold, which is a tool for understanding symmetries. As the researchers write in the book’s introduction, Goodman-Strauss “had been preaching the gospel of the orbifold signature on his own, and was known for his gorgeous illustrations.”
Orbifolds are formed when symmetrical patterns on a surface are folded or rolled so that every distinct feature, every point on a pattern, is brought together with its corresponding point. The result is a geometrical shape, such as a sphere, a cone or a cylinder, that shows one example of the design element that was repeated to make the symmetrical pattern.
As a tool, the orbifold pattern provides an efficient way to understand patterns. Goodman-Strauss uses the example of the ancient and universal fish-scale pattern.
“Why would the fish scale pattern be so compelling and so interesting and be the basis for all kinds of other patterns? It’s very easy – because it has a very simple orbital,” Goodman-Strauss explained. “You want to get to a simple pattern on an orbifold. When you do that, then the pattern is strong and dynamic.”
Goodman-Strauss also hosts his own Web site at http://www.mathbun.com, featuring examples of symmetrical patterns and orbifolds, along with photos of other math- and art-related projects.

Fausto Intilla - www.oloscience.com

Mathematics Simplifies Sleep Monitoring

2008-05-07T22:32:00.000-07:00

Source:

http://www.sciencedaily.com/releases/2008/05/080507105644.htm

ScienceDaily (May 7, 2008) — A UQ researcher has created a new way to measure breathing patterns in sleeping infants which may also work for adults.
The researcher, PhD student Philip Terrill, has created a mathematical formula that measures varying breathing patterns which indicate different sleep states such as active or quiet sleep.
Mr Terrill said a band, placed around the child's chest, recorded breathing rates which were then analysed using the new formula based on the maths of chaos theory.
It has been successfully tested on 30 children so far.
Current sleep monitoring involves an overnight stay in a hospital sleep lab with specialised equipment needing regular attention of a nurse, doctor or sleep technician.
Mr Terrill said he hoped his formula would form the basis of an automated sleep monitoring system that was cheaper and easier to use than current methods. "In the future, diagnosing a sleep problem may be as simple as putting on a breathing monitor during a night's sleep at home, in your own bed," Mr Terrill said.
"This would mean that those children with sleep problems could be quickly diagnosed and treated appropriately."
Minor infant sleeping problems can result in daytime sleepiness and inattention with prolonged problems causing behavioural and learning difficulties.
Mr Terrill said clinical research showed that up to 20 percent of Australian children have symptoms of sleep problems and there were very few facilities available to investigate sleep problems in Queensland children.
He said previous work analysed sleep breathing patterns using conventional statistical methods but his work used techniques from a branch of mathematics called chaos theory. The next step is to test his formula on teenagers and adults.
Adapted from materials provided by University Of Queensland.

Fausto Intilla - www.oloscience.com

Music Has Its Own Geometry, Researchers Find

2008-04-18T22:02:00.001-07:00

Source:

http://www.sciencedaily.com/releases/2008/04/080417142454.htm

ScienceDaily (Apr. 18, 2008) — The connection between music and mathematics has fascinated scholars for centuries. More than 200 years ago Pythagoras reportedly discovered that pleasing musical intervals could be described using simple ratios.

And the so-called musica universalis or "music of the spheres" emerged in the Middle Ages as the philosophical idea that the proportions in the movements of the celestial bodies -- the sun, moon and planets -- could be viewed as a form of music, inaudible but perfectly harmonious.
Now, three music professors -- Clifton Callender at Florida State University, Ian Quinn at Yale University and Dmitri Tymoczko at Princeton University -- have devised a new way of analyzing and categorizing music that takes advantage of the deep, complex mathematics they see enmeshed in its very fabric.
Writing in the April 18 issue of Science, the trio has outlined a method called "geometrical music theory" that translates the language of musical theory into that of contemporary geometry. They take sequences of notes, like chords, rhythms and scales, and categorize them so they can be grouped into "families." They have found a way to assign mathematical structure to these families, so they can then be represented by points in complex geometrical spaces, much the way "x" and "y" coordinates, in the simpler system of high school algebra, correspond to points on a two-dimensional plane.
Different types of categorization produce different geometrical spaces, and reflect the different ways in which musicians over the centuries have understood music. This achievement, they expect, will allow researchers to analyze and understand music in much deeper and more satisfying ways.
The work represents a significant departure from other attempts to quantify music, according to Rachel Wells Hall of the Department of Mathematics and Computer Science at St. Joseph's University in Philadelphia. In an accompanying essay, she writes that their effort, "stands out both for the breadth of its musical implications and the depth of its mathematical content."
The method, according to its authors, allows them to analyze and compare many kinds of Western (and perhaps some non-Western) music. (The method focuses on Western-style music because concepts like "chord" are not universal in all styles.) It also incorporates many past schemes by music theorists to render music into mathematical form.
"The music of the spheres isn't really a metaphor -- some musical spaces really are spheres," said Tymoczko, an assistant professor of music at Princeton. "The whole point of making these geometric spaces is that, at the end of the day, it helps you understand music better. Having a powerful set of tools for conceptualizing music allows you to do all sorts of things you hadn't done before."
Like what?
"You could create new kinds of musical instruments or new kinds of toys," he said. "You could create new kinds of visualization tools -- imagine going to a classical music concert where the music was being translated visually. We could change the way we educate musicians. There are lots of practical consequences that could follow from these ideas."
"But to me," Tymoczko added, "the most satisfying aspect of this research is that we can now see that there is a logical structure linking many, many different musical concepts. To some extent, we can represent the history of music as a long process of exploring different symmetries and different geometries."
Understanding music, the authors write, is a process of discarding information. For instance, suppose a musician plays middle "C" on a piano, followed by the note "E" above that and the note "G" above that. Musicians have many different terms to describe this sequence of events, such as "an ascending C major arpeggio," "a C major chord," or "a major chord." The authors provide a unified mathematical framework for relating these different descriptions of the same musical event.
The trio describes five different ways of categorizing collections of notes that are similar, but not identical. They refer to these musical resemblances as the "OPTIC symmetries," with each letter of the word "OPTIC" representing a different way of ignoring musical information -- for instance, what octave the notes are in, their order, or how many times each note is repeated. The authors show that five symmetries can be combined with each other to produce a cornucopia of different musical concepts, some of which are familiar and some of which are novel.
In this way, the musicians are able to reduce musical works to their mathematical essence.
Once notes are translated into numbers and then translated again into the language of geometry the result is a rich menagerie of geometrical spaces, each inhabited by a different species of geometrical object. After all the mathematics is done, three-note chords end up on a triangular donut while chord types perch on the surface of a cone.
The broad effort follows upon earlier work by Tymoczko in which he developed geometric models for selected musical objects.
The method could help answer whether there are new scales and chords that exist but have yet to be discovered.
"Have Western composers already discovered the essential and most important musical objects?" Tymoczko asked. "If so, then Western music is more than just an arbitrary set of conventions. It may be that the basic objects of Western music are fantastically special, in which case it would be quite difficult to find alternatives to broadly traditional methods of musical organization."
The tools for analysis also offer the exciting possibility of investigating the differences between musical styles.
"Our methods are not so great at distinguishing Aerosmith from the Rolling Stones," Tymoczko said. "But they might allow you to visualize some of the differences between John Lennon and Paul McCartney. And they certainly help you understand more deeply how classical music relates to rock or is different from atonal music."
Adapted from materials provided by Princeton University.

Fausto Intilla - www.oloscience.com

Traffic Jam Mystery Solved By Mathematicians

2007-12-19T23:29:00.000-08:00

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

2007-11-27T11:23:00.000-08:00

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

2007-10-08T03:06:00.000-07:00

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

2007-08-31T11:42:00.000-07:00

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

2007-08-19T11:11:00.000-07:00

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

2007-08-19T11:00:00.000-07:00

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

2007-08-19T10:57:00.000-07:00

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

2007-08-19T10:46:00.000-07:00

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