[HN Gopher] Mathematicians prove Polya's conjecture for the eige...
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Mathematicians prove Polya's conjecture for the eigenvalues of a
disk
Author : rbanffy
Score : 187 points
Date : 2024-03-02 12:27 UTC (10 hours ago)
(HTM) web link (phys.org)
(TXT) w3m dump (phys.org)
| VyseofArcadia wrote:
| arXiv link: https://arxiv.org/abs/2203.07696
| snitty wrote:
| Mathematicians prove that you can deduce the shape of disk based
| on the frequencies it produces when you hit it with a stick.
| m3kw9 wrote:
| Useful as a magic trick, guessing the shape of your disc when
| you hit it
| kadoban wrote:
| I can deduce the disk's shape without even hitting it. It's a
| disk.
| gus_massa wrote:
| The first sentense is very misleading:
|
| > _Is it possible to deduce the shape of a drum from the sounds
| it makes?_
|
| That's a known problem with a (very nice) negative answer
| https://www.ams.org/publicoutreach/feature-column/fcarc-1997...
|
| IIUC this article is about the problem in the other direction,
| i.e. from the shape (a disk!) to the frecuencies of the sound
| (eigenvalues).It's not about an exact calculation, but about an
| aproximation of them.
|
| > _The conjecture bears on the estimation of the frequencies of
| a round drum or, in mathematical terms, the eigenvalues of a
| disk._
|
| From the research paper:
|
| > _The celebrated Polya's conjecture (1954) in spectral
| geometry states that the eigenvalue counting functions of the
| Dirichlet and Neumann Laplacian on a bounded Euclidean domain
| can be estimated from above and below, respectively, by the
| leading term of Weyl's asymptotics._
|
| <guessing> The Weyl's asymtotics is probably a good estimation
| of the very high frecuencies/eigenvalues, and the conjeture is
| probabbly that you can use the estimation as upper or lower
| bounds instead of just an aproximation.<guessing> [Sorry, not
| my area and I have not enough time to read the paper.]
| mikhailfranco wrote:
| The title words remind of an unrelated fact, Gershgorin Disks:
| The eigenvalues of any N x N matrix, A, are contained in
| the union of N discs in the complex plane. The center of
| the i_th disc is the i_th diagonal element of A. The
| radius of the i_th disc is the absolute values of the off-
| diagonal elements in the i_th row.
|
| https://blogs.sas.com/content/iml/2019/05/22/gershgorin-disc...
|
| It's rather remarkable, unexpected, and perhaps shocking, when
| you first hear it. There does not seem to be enough information
| to make it true. But it is.
|
| I used it in a real project to cancel noise.
| frutiger wrote:
| Sounds very cool. One thing I didn't understand though:
|
| > The radius of the i_th disc is the absolute values
|
| How can a radius of a single disc (i.e. a single value)
| correspond to multiple values?
| reactordev wrote:
| I'm assuming, but could be wrong as my matrix math ended with
| GameDev, that the radius of the disk = the absolute values of
| the other things. +\\- depending on which side of the disk
| they fall?
|
| That line got me as well.
| Sharlin wrote:
| It's missing the word "sum". Which apparently my brain
| auto-deduced for me because I didn't notice anything off in
| the sentence on the first reading.
| _zoltan_ wrote:
| My brain also automatically added sum and also haven't
| noticed anything, but I have a math degree, maybe it's
| just assuming things :-)
| frutiger wrote:
| I have a physics degree and did not assume it, that might
| explain the difference :)
| ximeng wrote:
| If you look at the link there's a formula showing it's the
| sum.
| abetusk wrote:
| Specifically: r_i = \sum_{i \ne j} |A_{i
| j}|
| tomrod wrote:
| Sum the absolute values of a row for all the off diagonal
| elements
| Infinity315 wrote:
| An intuitive explaination is imagine a nearly diagonal matrix
| where the values along the diagonal are much larger than
| values on the off diagonal. We know the eigenvalues of a
| diagonal matrix is simply the values on the diagonal, so for
| nearly diagonal matrices you can be pretty sure that the true
| eigenvalues are going to be pretty close to those diagonal
| entries, but a natural question to ask is how far we'd
| deviate from those diagonal entries.
|
| The answer to the above question is Gerschgorin disks and
| it's closely related cousin Brauer's Oval of Cassini.
|
| For matrices with real eigenvalues it's moreso along the real
| number line, only for cases where the eigenvalues are
| imaginary do we imagine disks.
| abetusk wrote:
| What was the noise cancelling project? How did you use this
| fact to cancel noise?
| sdenton4 wrote:
| Just guessing, but....
|
| A common noise cancellation technique is to throw away small
| eigenvalues, as in PCA. This result relates eigenvalues to
| the structure of the matrix, so might be helpful for reducing
| ev's without bothering with diagonalization?
|
| [Edit] This would presumably involve just zeroing out the
| rows with small diagonal elements and small-ish off-diagonal
| norm... Center the eigenvalue estimate disk at zero, and then
| zero out the rest of the row to make the estimate exact.
| sdenton4 wrote:
| (and after one more thought about it, one would zero out
| the row /and column/ to preserve the symmetry of the
| matrix, if applicable, and thus keep the eigenvalues real.
| and one would probably want to think a bit about whether
| this kind of deletion actually makes sense for the
| problem... doing real PCA isn't /that/ hard in most cases -
| I think I would do something this janky only for extremely
| large matrices or for realtime operation on a
| microcontroller or something, and then only after thinking
| hard about it.)
| jonathan_landy wrote:
| Matrix theory by Franklin is a great, affordable book
| containing many interesting results such as this -- can highly
| recommend for those interested in linear algebra.
|
| https://www.amazon.com/Matrix-Theory-Dover-Books-Mathematics...
| packetlost wrote:
| If I have an atrophied high school level understanding of
| linear algebra, will I get anything out of that book?
| jonathan_landy wrote:
| Certainly a useful reference, if you might deal with the
| topic for some project ... but not the best refresher out
| there for the basics.
| doubloon wrote:
| or
|
| https://store.doverpublications.com/products/9780486411798
| homerowilson wrote:
| You may enjoy this:
|
| https://bwlewis.github.io/cassini/
| kwkelly wrote:
| Matrix Analysis by Horn and Johnson is another great book,
| though it is a bit pricier than a Dover book.
| stabbles wrote:
| It's not that remarkable at all? The proof requires the
| definition and triangle inequality, that's all?
|
| Given Ax=lx, take i for which |x| is largest. Look at the i'th
| equation: sum ax = lx, move the ax term to the rhs, take
| absolute values, divide by |x|, apply triangle inequality, and
| you have |a - l| <= sum |a| over j[?]i. So for every eigenvalue
| you can find such a disc.
|
| That's by column, for row use A.
| kmm wrote:
| Being easy to prove doesn't make it unremarkable. Lots of
| theorems, including this one, have straightforward proofs
| once you are given the exact formulation. The tricky part is
| coming up with the idea for the theorem itself. I remember
| being (mildly) shocked when I was taught this in undergrad,
| it just seemed too good to be true.
| stabbles wrote:
| I think that's just how textbooks present it. A fact is
| stated but you're lacking intuition.
|
| If you had played around a bit with Laplacian matrices,
| like tri-diagonal matrices with stencil [-1, 2, -1], and
| found that its eigenvalues are within 2 +- 2, and if you
| also realized that A + tI has the same eigenvalues shifted
| by t, then it's a small step to consider that the magnitude
| of the off-diagonal may have something to do with the
| spread of eigenvalues.
|
| It's likely that Gerschgorin stumbled upon it like this.
| llmzero wrote:
| Just thinking about some intuition for the result of the
| theorem: If the off diagonal elements are zero then the
| diagonal element is an eigenvalue, by continuity of the
| determinant, if the off diagonal element are small then
| $det(A-a_{ii}\lambda)$ is almost zero, that is the new
| eigenvalue is near aii. So it suggests that the off
| diagonal elements measure how far is aii from being an
| eigenvalue.
| nine_k wrote:
| The Euler's identity is also trivial to deduce, but this
| doesn't diminish its beauty.
| xanderlewis wrote:
| The Yoneda lemma is another great example of _(once you 've
| got the right setup) trivial but beautiful_
| moralestapia wrote:
| Huh?
|
| Math 101, simple is better.
| jonahx wrote:
| removed
| tsimionescu wrote:
| They are saying that the Greshgorin theorem that the OP is
| talking about is simple to prove, not the Polya conjecture
| that took 70 years from the article.
| bee_rider wrote:
| It has an intuitive element to it; we're looking for the
| eigenvalue, the vector/scalar pair where the scalar has the
| same effect on the vector as multiplying by the matrix. And
| we're comparing against something that looks vaguely like the
| magnitude of the matrix (shifted by the diagonal, and of course
| if you had a diagonal matrix, the eigenvalues would just be the
| diagonal).
|
| Not a proof or anything, of course the proof is on Wikipedia
| and nice and elegant. Just a thought on the gut feeling.
|
| I agree that it is a very nice result.
| Infinity315 wrote:
| See also Brauer's oval of Cassini which give an equivalent or
| even better approximation of the eigenvalues of a matrix.
| esafak wrote:
| I haven't read the paper, but does it apply to arbitrarily shaped
| drums? And if the basis can be arbitrarily large, is this an
| exciting result? It just says that the spectral domain is
| isomorphic?
| eigenket wrote:
| This result only applies to circular disks, spherical balls and
| their generalisations in higher dimensions. Previously it was
| known for shapes which tile the plane or tessellate the space
| in higher dimensions.
| perihelions wrote:
| What do the eigenfunctions look like? I didn't know the circular
| disk was such an exotic problem for Laplace's equation.
|
| edit: Ah, I misunderstood the problem. The eigenfunctions _are_
| exactly solved; the problem of _sorting and ordering_ their
| eigenvalues is apparently not! From their page 4,
|
| - _" Although all the eigenvalues of the Dirichlet and Neumann
| Laplacians on the unit disk are explicitly known in terms of
| zeros of the Bessel functions or their derivatives, see SS2
| below, in each case the spectrum is given by a two-parametric
| family, and rearranging it into a single monotone sequence
| appears to be an unfeasible task."_
|
| https://arxiv.org/abs/2203.07696
| colesantiago wrote:
| Nice, what are the potential use cases for this? Can I use this
| for my business?
| colesantiago wrote:
| This is a valid question????
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