Posted on 11/29/2003 8:28:51 PM PST by SteveH
Historic maths problem 'cracked'
By Dr David Whitehouse
BBC News Online science editor
A 22-year-old student at Stockholm University, Elin Oxenhielm, may have solved part of one of mathematics' greatest unsolved problems. Called Hilbert's problem 16, it has confounded workers for over a century.
But in a few hours of inspiration she may have seen the light. Her solution is to be published in a maths journal.
Her research into so-called planar polynomial vector fields may have practical applications for computer simulations in science and economics.
Passion for maths
"I solved it before I knew its significance," Elin Oxenhielm told BBC News Online.
"It took a few months of thinking about it at first, but then the solution came remarkably quickly," she says.
Her breakthrough comes a century after the problem was posed by Prussian mathematician David Hilbert. In 1900 he gave a lecture in Paris where he laid down the 23 greatest problems for maths in the 20th century.
They were a varied selection that had confounded the greatest mathematical minds of the age.
Couched in language that only mathematicians appreciate, they included such questions as: can the continuum of numbers be regarded as a well ordered set, and can space be constructed by congruent polyhedra?
Over a century later only three of Hilbert's problems remain unconquered, numbers six, eight and 16.
Number eight is the famous Riemann hypothesis to do with prime numbers, regarded by many as the most difficult problem in maths today.
Recently problems eight and 16 on Hilbert's list have been placed on a list of the 18 biggest challenges for 21st century mathematicians.
But problem 16 may now have a partial answer.
A few hours' work
Mathematicians describe it as a question of the "topology of algebraic curves and shapes." Non-technically it deals with the way solutions to equations are arrived at.
Elin Oxenhielm's solution is of a special version of the second part of the problem, called the "boundary cycles for polynomial differential equations".
"It only took me a few hours to solve the problem once I expressed it in the correct way," she told BBC News Online.
"It is difficult to describe for non-mathematicians but the way I solved the problem may have practical applications."
It may improve the way scientists use computers to simulate such diverse phenomena as global warming and economies.
Her work is to be published in the mathematical journal Nonlinear Analysis.
Oxenhielm believes her method can be used to unlock the mystery of the entire 16th problem, and plans to write a popular book about her work.
Story from BBC NEWS:
http://news.bbc.co.uk/go/pr/fr/-/2/hi/science/nature/3243736.stm
Published: 2003/11/27 15:22:54 GMT
© BBC MMIII
Are you suggesting that these things depend upon spatial orientation?
Also, the theory may not apply in the vacinity of a black hole, for instance.
That redounds to my point. In the vicinity of a black hole, lengths may well not be rotationally invariant.
But then neither would the length of your tape measure. Or would it?
It wouldn't, but you can determine that by measuring the angle it subtends from some distance away. In other words, the angles of a triangle in a curved space won't add up to 180 degrees (which presumably is why Consort picked the black hole example in the first place).
And if the whole of human logic is flawed, then isn't this an indictment of our educational institutions in general? I put it to you, Consort - isn't this an indictment of our entire American society? Well, you can do whatever you want to us, but we're not going to sit here and listen to you badmouth the United States of America. Gentlemen!
Is that a rant, a whine, or a bit of satire? Better yet, are you proving my point? Is it an inductment of our educational institution or of American society? No...it's an indictment of the whole human race.
Well, you can do whatever you want to us, but we're not going to sit here and listen to you badmouth the United States of America. Gentlemen!
Is that a tirade, a bloviation, a diatribe, or a banter?
Is that part of the language of math and have we yet decided if math represents flawed human logic?
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