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Swedish Student Partly Solves 16th Hilbert Problem
An anonymous reader writes "Swedish media report that 22-year-old Elin Oxenhielm, a student at Stockholm University, has solved a chunk of one of the major problems posed to 20th century mathematics, Hilbert's 16th problem.
Norwegian Aftenposten has an English version of the reports."
Based on the photo alone. I would say she is engaged or even *gasp* married. Yup, when your single and on the prowl...the "ring finger" is the first thing you look at. Why bother wasting hers and your time?
Life is not for the lazy.
In other word's, problem no 16 is still unsolved besides special cases.
Special versions of fermats theorem were already proofed by fermat himself. But it took 300 years until Andrew Wiles and one of his students proved it generally. If You look at the history of famous mathematical conjectures (ie fermats, poincares) You'll see: prooving a special case will probably not really help prooving the general case. If You are very lucky, You get a hint how to solve the "real" problem.
I've taken a look at her article (downloaded it via an institutional subscription). It's eight pages long, with a lot of figures, and is short and easy to read. It's also categorically not an important theoretical contribution to Hilbert's 16th problem.
The author tries to determine the number of limit cycles for the Lienard equation. This would not solve the full 16th problem, but it would deal with an interesting special case, and it would likely take powerful new techniques to solve even this case. She tries to do so as follows:
She notes that numerical calculations show that the solution is well approximated by a simple trig function. (The figures are evidence in support of this assertion.) She then bounds the number of limit cycles, under this approximation, in a straightforward and elementary way. I have not carefully checked this bound, but I see no reason to doubt it (or to believe there's anything novel about it, for that matter). However, there is no attempt whatsoever at a rigorous justification of the approximation, or even a rigorous formulation of it. Therefore this simply does not constitute a full proof, although the article refers to it as a proof. Hilbert's 16th problem is already well understood in simple cases, and any attempt to reduce the more complex cases to simple cases must justify all approximations.
Incidentally, if this were an important theoretical paper on Hilbert's 16th problem, the journal "Nonlinear analysis" would be a strange place for it (it's more interdisciplinary, and is not a mainstream outlet for theoretical mathematics). That's no reason it couldn't be true, but it's some cause for initial suspicion as well as explanation for why the article was accepted. Probably the editors and referees were applied scientists unfamiliar with the problem, who were perfectly happy to accept an approximation justified by some numerical data.