Swedish Student Partly Solves 16th Hilbert Problem 471
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."
It's funny that college kids.... (Score:0, Interesting)
Re:It's funny that college kids.... (Score:1, Interesting)
Re:I remember (Score:0, Interesting)
Re:I remember (Score:5, Interesting)
He was flunking information theory at MIT, and his prof told him he'd pass if he solved mimimal redundancy coding. So he did, and invented Huffman codes.
<HUMOR>
Of course, as his students at UCSC, we used to believe that his roommate solved it, and Huffman killed him for the solution (and hid the body)...
</HUMOR>
Re:It's funny that college kids.... (Score:5, Interesting)
1. It was her job. (she is a grad student and a teaching asst, therefore has a JOB even if it way underpaid).
2. Just the other day
3. She is not a "college kid" as you put it, but a PhD student (she does not fit into the same drug-imbibing, all-night partying picture)
Re:Maybe math, then.. (Score:2, Interesting)
Not to mention C. F. Gauss [st-and.ac.uk] (1777-1855)
Re:Maybe math, then.. (Score:3, Interesting)
He was an atheist and [most likely] a homosexual, and was therefore very much an 'outsider' himself in his times)
There simply weren't very many women in math 100 years ago.
And while I'm on the topic, it is interesting to note that Stockholm University was one of the first universites to give a chair in mathematics to a woman;
The great Sonya Kovalevskaya [st-and.ac.uk].
This one is true, AND... (Score:3, Interesting)
This is apparently a true story. At least, I have Dantzig's account here in "History of Mathematical Programming -- A Collection of Personal Reminiscences." Two interesting side nodes:
Re:It's funny that college kids.... (Score:4, Interesting)
That's very typical. As people get older, they get less creative. As people get married, they become unimaginative dolts. [abc.net.au]
Of course, I'm happily married, and I'd like to think that I still have *some* creative spark, but then, I *am* here, at 6:33 PM on Turkey-Day eve, reading slashdot...
Maybe they're right, after all?
Re:I'd hit it! (Score:2, Interesting)
Re:SwedishHot at SlashDot (Score:2, Interesting)
On the second part of Hilbert's 16th problem
Elin Oxenhielm,
Department of Mathematics, Stockholm University, Stockholm 10691, Sweden
Received 3 July 2003; accepted 3 October 2003. ; Available online 18 November 2003.
Abstract
Let k be an integer such that k is larger than or equal to zero, and let H be the Hilbert number. In this paper, we use the method of describing functions to prove that in the Lienard equation, the upper bound for H(2k+1) is k. By applying this method to any planar polynomial vector field, it is possible to completely solve the second part of Hilbert's 16th problem.
Author Keywords: Second part of Hilbert's 16th problem; Hilbert number; Lienard equation; Describing function; Limit cycle; Polynomial vector field
Article Outline
1. Introduction
2. Preliminaries
3. Result
Acknowledgements
References
1. Introduction
In 1900, Hilbert presented a list consisting of 23 mathematical problems (see [1]). The second part of the 16th problem appears to be one of the most persistent in that list, second only to the 8th problem, the Riemann conjecture. The second part of the 16th problem is traditionally split into three parts (see [5]).
Problem 1. A limit cycle is an isolated closed orbit. Is it true that a planar polynomial vector field has but a finite number of limit cycles?
Problem 2. Is it true that the number of limit cycles of a planar polynomial vector field is bounded by a constant depending on the degree of the polynomials only?
Denote the degree of the planar polynomial vector field by n. The bound on the number of limit cycles in Problem 2 is denoted by H(n), and is known as the Hilbert number. Linear vector fields have no limit cycles, hence H(1)=0.
Problem 3. Give an upper bound for H(n).
Let k be an integer such that k0. In 1977, Lins et al. [2] found examples with k different limit cycles in the Lienard equation
(1)
where
F(x)=q2k+1x2k+1+q2kx2k+··
The degree of this polynomial vector field is denoted by 2k+1. The coefficients qi (for integers i such that 1i2k+1) are real constants. Lins et al. [2] conjectured the number k as the upper bound for the number of limit cycles of the Lienard equation (1). Their conjecture thus states that in the Lienard equation (1), the upper bound for H(2k+1) is k.
In his list of mathematical problems for the next century, published in 1998, Smale [4] mentioned the Lienard equation (1) as a simplified version of the second part of Hilbert's 16th problem (see [3]).
In the present paper, we will prove the conjecture stated by Lins et al. [2] in 1977, thereby solving the simplified version of the second part of Hilbert's 16th problem stated by Smale [3] in 1998.
2. Preliminaries
In this section, we will introduce the method of describing functions, which may be used to calculate limit cycles in nonlinear dynamic systems (see [4]).
Consider a dynamic system
where x is the m-dimensional vector of state variables, M is an mxm constant matrix and h(x) is an m-dimensional vector of nonlinear functions.
Assume that the state variables are dominated by a harmonic term of a specific order
xa0+a1 sin(t),
where a0 is the m-dimensional vector of center values, a1 is the m-dimensional vector of amplitudes and is the frequency. a0, a1 and are assumed to be real. a1 and are nonzero.
Then, approximate the vector of nonlinear functions by discarding higher harmonic terms (terms of the form cos(rt) and sin(rt) for integers r such that r2)
h(x)+Na1 sin(t),
where is an m-dimensional constant vector and N is an mxm constant matrix. The components of N are called describing functions.
The system becomes
and solutions for a0, a1 and satisfy
Re:Context (Score:2, Interesting)
While we are on the topic of Scandinavian female matematicians, there is an interview [newscientist.com] in New Scientist [newscientist.com] with Norwegian mathematics professor Ragni Piene [math.uio.no] where she discusses why there are so few women mathematicians.
Re:It's funny that college kids.... (Score:3, Interesting)
Well, you're actually technically wrong on both counts. First according to the dept's webpage she's not a PhD student, she's a teaching assistant (amanuens). And thus her job is actually to teach, not to do research.
No doubt she was given the amanuensis position in anticipation of becoming a PhD student, but since Sweden changed their PhD acceptance criteria, departments have become wary of accepting students (there aren't as many positions available these days). (My own department for example had 120 applicants for four positions this year, you basically had to have published papers to even get in as a PhD student). Hence departments like to pull stunts such as these, i.e. hiring someone beforehand as e.g. a TA (or similar) to see if they can do the work before comitting to taking them on. I'd say she passed... :-)
As to why students (as in undergrads) have come up with breakthroughs as of late my own theory is that they are the ones that can actually work on these problems, having nothing to lose. As a PhD student that's not a smart thing to do, see my other post [slashdot.org] on this topic.