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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."
....are always the ones to solve the major problems, and always, by accident it seems. Steve Jobs, Bill Gates, Steve Wozniak, my favorite computer college "accidents" to cite, but there are many more, such as, for example, the kids in Japan who thought they solved ?(but unconfirmed).
That never happened. click
I think that story is an urban legend, but if you've ever used Huffman coded data, Huffman himself used to tell this story:
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>
The only reason we have the rights we have is that people just like us died to gain those rights. -- Cheerio Boy
Not to mention C. F. Gauss (1777-1855)
I'd rather say it was Hardy's hypothesis, although from what I know of his character, he was probably not a sexist or prone to any other form of bigotry;
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.
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:
Speaking of hot maths chicks, Clio Cresswell was my Maths 1 tutor at Adelaide Uni a few years back. I remember thinking there probably wouldn't be too many hotter maths chicks getting around.
Here's the text, dude. I left off her phone number, because I don't want any competion:
,
;·+q2x2+q1x.
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
While we are on the topic of Scandinavian female matematicians, there is an interview in New Scientist with Norwegian mathematics professor Ragni Piene where she discusses why there are so few women mathematicians.