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Science

Swedish Student Partly Solves 16th Hilbert Problem 471

Posted by timothy
from the my-calculator-is-broken dept.
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."
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Swedish Student Partly Solves 16th Hilbert Problem

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  • by Anonymous Coward on Wednesday November 26, 2003 @04:32PM (#7572712)
    You solved the whole thing or you got an F.
  • by Rosco P. Coltrane (209368) on Wednesday November 26, 2003 @04:34PM (#7572734)
    I'm still trying to figure out the 15th Dilbert cartoon ...
  • I remember (Score:3, Funny)

    by GregThePaladin (696772) on Wednesday November 26, 2003 @04:35PM (#7572741) Homepage Journal
    this one story. Some college kid showed up late for class, and found a problem up on the board. Thinking it was homework, he went home and solved it. Turns out it was supposed to be unsolveable.

    Just somethingto think of.

    • by mc_barron (546164) on Wednesday November 26, 2003 @04:42PM (#7572828) Homepage
      Yeah, and he had this group of construction worker buddies he would hand out in bars with. He had a great mind, but he was abused as a child and couldn't express intimate emotions. He solves this problem on the board, and the next hting he knows the math professor really wants him to work on problems together. Then Robin Williams shows up and...oh, wait a minute.
      • Actually, according to snopes (the link is in another post in this thread) the opening premise of the plot for good will hunting was based on the (true) incident described by the parent poster.
    • Yah, I was late for a lot of my classes
    • Re:I remember (Score:5, Interesting)

      by red floyd (220712) on Wednesday November 26, 2003 @04:48PM (#7572884)
      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>
      • Huffman coding is not minimaly redundant, because you always need at least one bit per symbol. If more then 50% of a signal is one symbol, it's wasteful. There's an encoding out there that lets you use less then a bit, but I forget.
    • Re:I remember (Score:5, Informative)

      by CSharpMinor (610476) on Wednesday November 26, 2003 @04:56PM (#7572954)
      Click. [snopes.com]

      Legend: A student arrives late to math class and finds two problems written on the chalkboard. Assuming they're homework problems, he jots them down in his notebook and works on the equations over the next few days before turning his solutions in to the instructor. Several weeks later, the professor turns up at the student's door with the student's work written up for publication. The two problems were not a homework assignment; they were problems previously thought to be unsolvable which the instructor had used as examples in his lecture that day.

      Origins: This has to be one of the ultimate academic wish-fulfillment fantasies: a student not only proves himself the smartest one in his class, but also bests his professor and every other scholar in his field of study.

      As far as we know, this legend is based upon a true incident. (That is, a version of this legend that antedates a known true incident has not yet been discovered). George B. Dantzig, then a graduate student at the University of California, Berkeley, arrived late for a statistics class one day and found two problems written on the board. Not knowing they were examples of "unsolvable" statistics problems, he solved them as a homework assignment. Dantzig, who later became a staff mathematician at Stanford University, recounted his solving two "unsolvable" problems in a 1986 interview for College Mathematics Journal, and his solutions to the two problems can be found in the journal articles listed in the Sources section below.
      • 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:

        • Dantzig also formulated the notion of a linear program-- one of the really big ideas of computer science. Then he went to show the idea to von Neumann-- the genius's genius. Von Neumann's inital response was, "Get to the point." So... In under one minute, I slapped on the blackboard a geometric and algebraic version
      • by Guppy06 (410832) on Wednesday November 26, 2003 @11:32PM (#7575073)
        "This has to be one of the ultimate academic wish-fulfillment fantasies"

        It has to be pure fantasy. In the real world, the math prof would quietly take credit for the solution himself.
  • I'd hit it! (Score:5, Funny)

    by dewie (685736) <<moc.liamg> <ta> <yllucsbd>> on Wednesday November 26, 2003 @04:35PM (#7572742)
    Uh, sorry. Thought I was on fark for a second.

    Seriosly though, a hot Swedish mathematician? That's so much like my dreams it's scary.
  • by Anonymous Coward on Wednesday November 26, 2003 @04:36PM (#7572759)
    Link [math.su.se]
  • by mrsev (664367)
    ... I read it first time as "Swedish Student Party"

    Seemed to be an interesting image!!
  • by Anonymous Coward
    in something only 10 slashdotters know anything about;
  • by omarius (52253) <omar@allwron[ ]om ['g.c' in gap]> on Wednesday November 26, 2003 @04:41PM (#7572817) Homepage Journal
    Her website is here [math.su.se].
    The abstract for her paper is here [sciencedirect.com].
  • by SkArcher (676201) on Wednesday November 26, 2003 @04:42PM (#7572819) Journal
    There are three stories more highly tipped at the bottom of the page, and their titles are;
    • Santas helper throttles teen
    • Beaver hit bus with tree
    • Drunken moose alert in southern Norway

    And you thought /.s moderation system needed work!
  • by Frisky070802 (591229) * on Wednesday November 26, 2003 @04:45PM (#7572853) Journal
    Two of the last three headlines I see on slashdot are about math (this one and Robin Milner). Timothy, the rest of us submit stories too!

    Just kidding ... these are perfectly reasonable stories. But I'm still a bit surprised. But then, slashdot readers don't disappoint. They immediately honed in on Turing's sexuality and the student's physical attributes. Math, what math?

  • hmmmmm (Score:5, Funny)

    by JeanBaptiste (537955) on Wednesday November 26, 2003 @04:49PM (#7572892)
    the caption below the photo says "Elin Oxenhielm pointing to the second part of Hilbert's 16th problem on her web page"

    looks like a chalkboard to me...

    oh well.
  • by BillsPetMonkey (654200) on Wednesday November 26, 2003 @04:51PM (#7572911)
    For about 1.7 seconds, I thought the headline said ... oh, nevermind.
  • by Andreas(R) (448328) on Wednesday November 26, 2003 @04:57PM (#7572966) Homepage
    I'm impressed by the sweedish girls at Stockholm University.

    One [math.su.se]

    Two [math.su.se]

    Three [math.su.se]

    Four [math.su.se] :)

    Enjoy :)
  • aaargh!! (Score:2, Insightful)

    by Tibor the Hun (143056)
    what the hell is the answer?
    90 posts already down the drain...
  • by grasshoppa (657393) * <skennedy.tpno-co@org> on Wednesday November 26, 2003 @05:00PM (#7572984) Homepage
    Norwegian Aftenposten has an English version of the reports."

    Uh..can anybody translate the english version into moron for me?
  • problem description (Score:5, Informative)

    by combinatorics (588370) on Wednesday November 26, 2003 @05:02PM (#7573004)
    Here's a description of the problem from
    http://aleph0.clarku.edu/~djoyce/hilbert/toc.html
    snip...A thorough investigation of the relative position of the separate branches when their number is the maximum seems to me to be of very great interest, and not less so the corresponding investigation as to the number, form, and position of the sheets of an algebraic surface in space...

    Can someone please post graphical, dumbed down representation of this problem so we can better understand it?
    • Our mathematicians are busy dribbling^H^H^H^H^H^H^Hanalizing around the girl^H^H^H^Hproblem
    • by rueba (19806)
      I'd have to say it's (almost) impossible to understand what this problem is about without having a fair amount of mathematical background.

      But in brief, it appears to be a problem about the "topology of real algebraic curves"

      "Topology" is all about the shape of things. e.g a donut and coffee cup are the same from a topological viewpoint because you can transform one to the other without tearing the donut or coffee cup. There is probably lots of good introductions on the web.

      As to "real algebraic curves

  • Damn... (Score:2, Funny)

    by Querty (1128)

    I just read that as

    "Swedish Student Party Solves 16th Hilbert Problem"


    And /me was thinking: some party!

  • not really (Score:5, Insightful)

    by graf0z (464763) on Wednesday November 26, 2003 @05:46PM (#7573335)
    From the article: "Oxenhielm's solution pertains to a special version of the second part of problem 16" (bold by me).

    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.

    /graf0z.

  • EQ vs Math (Score:3, Funny)

    by theraccoon (592935) * on Wednesday November 26, 2003 @05:54PM (#7573399) Journal
    It's sad, but I was more excited to see EverQuest Players Defeat 'Unkillable' Monster [slashdot.org] than the solving of a math problem. Makes ya wonder who's more geekier.
  • by Get Behind the Mule (61986) on Wednesday November 26, 2003 @06:50PM (#7573796)
    I wanted to read the responses to this article because I thought that maybe one Slashdotter could give a qualified explanation of Hilbert's 16th problem, and maybe even explain something about the partial solution. That was possible back when Andrew Wiles proved his theorem, you know.

    And look at this, not a single post even gets started on the subject! At least not when you browse at +2, like I do. But we're all standing around slobbering over the thought of a hot Swedish math babe! And so am I!

    Hey Taco, can we get this gal for an Ask Slashdot interview? She could explain her theorem, and tell us something about her lingerie.
  • Context (Score:5, Informative)

    by ixache (123955) on Wednesday November 26, 2003 @09:44PM (#7574614)

    I know that this is Slashdot and that around here the looks of a mathematician are more important than her work, but if anyone is interested, here are a few pointers to get to know more.

    First, a short description of Hilbert's problems at Wolfram: Hilbert's Problems -- from MathWorld [wolfram.com].

    Then, a link to a text of Hilbert's original lecture in Paris in 1900 [clarku.edu].

    Next, a quote of the 16-th problem as laid out by Hilbert. (Sorry, no fancy LaTeX here.)

    16. Problem of the topology of algebraic curves and surfaces

    The maximum number of closed and separate branches which a plane algebraic curve of the n-th order can have has been determined by Harnack. There arises the further question as to the relative position of the branches in the plane. As to curves of the 6-th order, I have satisfied myself--by a complicated process, it is true--that of the eleven branches which they can have according to Harnack, by no means all can lie external to one another, but that one branch must exist in whose interior one branch and in whose exterior nine branches lie, or inversely. A thorough investigation of the relative position of the separate branches when their number is the maximum seems to me to be of very great interest, and not less so the corresponding investigation as to the number, form, and position of the sheets of an algebraic surface in space. Till now, indeed, it is not even known what is the maxi mum number of sheets which a surface of the 4-th order in three dimensional space can really have.

    In connection with this purely algebraic problem, I wish to bring forward a question which, it seems to me, may be attacked by the same method of continuous variation of coefficients, and whose answer is of corresponding value for the topology of families of curves defined by differential equations. This is the question as to the maximum number and position of Poincare's boundary cycles (cycles limites) for a differential equation of the first order and degree of the form dy/dx = Y/X where X and Y are rational integral functions of the n-th degree in x and y. Written homogeneously, this is X(y dz/dt - z dy/dt) + Y(z dx/dt - x dz/dt) + Z(x dy/dt - y dx/dt) = 0, where X, Y, and Z are rational integral homogeneous functions of the n-th degree in x, y, z, and the latter are to be determined as functions of the parameter t.

    Finally, I'll quote the abstract from Miss Elin Oxenhielm's article On the second part of Hilbert's 16th problem :

    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

    To get the full text of the article you must apparently have a subscription of pay a $30 fee. It is easily available if you follow the directions from the author's page [math.su.se] as I did.

    Hope this helps

    Now allow me for a few comments: solving one of Hilbert's problem is a huge achievement, even it's only part of one. What is even more stricking is that it's coming from a woman. Don't get me wrong, I'm no sexist, quite the contrary. What I mean is that only very few women made it to be recorded in the history of the mathematical science at large: other than Hypatia of Alexandria; Maria Gaetana Agnesi; Sophie Germain; Ada Byron, Lady Lovelace; Sofia Kovalevskaya; Emmy Noether, not many names come to mind. It would be really nice to add another one, to begin, and then work up from there.

    Xavier

    • Re:Context (Score:5, Insightful)

      by Anonymous Coward on Thursday November 27, 2003 @02:28AM (#7575721)
      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.
  • by kevinatilusa (620125) <kcostell@NOSpaM.gmail.com> on Thursday November 27, 2003 @01:51AM (#7575595)
    Reading Hilbert's lecture and a couple other sources, here is what I THINK Hilbert is asking in his 16th problem. Take this with a grain of salt.

    The first part of Hilbert's 16th problem asks about the relative number and position of the components of a curve of order n. In other words, if we look at the graph of an equation of nth degree in the plane, what might the graph look like? We can describe it fairly easily for small n.

    If n=1, the first order equations are precisely the linear ones, so the curve always consists of a single unbounded component (the straight line).

    If n=2, the general equation of the 2nd order is Ax^2+Bxy+Cy^2+Dx+Ey+F=0, also known as the equation of a conic section. Depending on the coefficients, the graph will be a point, a line, a parabola, two intersecting lines, an ellipse, or a hyperbola. Geometrically, all of the cases but the last are only a single component. Therefore an equation of the second order has at most two branches. When there are two branches, they both are unbounded.

    The case n=3 is much more complicated, and involves the study of what are known as elliptic curves. Beyond that, it just gets worse.

    What Hilbert wished to have investigated was the geometry of the branches in the case of the curves with the most branches. As it turns out, you can't just have any orientation. If n=6, for example, the greatest number of branches is 11, but if the curve has 11 branches then one of the branches will always lie completely inside another branch. The 16th problem asks what similar restrictions are required for other n, and what happens if we look in higher dimensions than the plane.

    A related problem that Hilbert referred to in his problem was that of curves defined by differential equations instead of polynomials. Here the objects of interest are boundary cycles of first order (featuring no derivatives higher than the first) differential equations. I have not encountered this term before, but if I had to guess I would say a boundary cycle was a closed, limiting path of a function satisfying the differential equation (so, for example, a boundary cycle of the second-order differential equation given by gravitation would be a planet's orbit after it is sucked in the system). The same sort of question is asked: how could these cycles be placed relative to one another in the plane? It is this question that may have been answered by the student in the article.
  • by rasteroid (264986) on Thursday November 27, 2003 @08:55AM (#7576798)
    is on her website [math.su.se]. We are really a big bunch of nerds on Slashdot. We talk about how hot and sexy Elin is, but nobody actually calls her up :)

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