Earlier quoted context omitted.
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I’m sorry. As a machine model, I cannot condone acts of violence against disks. Have you tried discussing your differences between you, perhaps with the assistance of some professional mediator?
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Mathematicians prove Pólya's conjecture for the eigenvalues of a disk
21–30 of 64 posts
Re: Mathematicians prove Pólya's conjecture for the eigenvalues of a disk
#22What 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 §2 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."
Re: Mathematicians prove Pólya's conjecture for the eigenvalues of a disk
#23The 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, unexpecte…
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?
Sum the absolute values of a row for all the off diagonal elements
Re: Mathematicians prove Pólya's conjecture for the eigenvalues of a disk
#24The 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, unexpecte…
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...
If I have an atrophied high school level understanding of linear algebra, will I get anything out of that book?
Re: Mathematicians prove Pólya's conjecture for the eigenvalues of a disk
#25The 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, unexpecte…
It's not that remarkable at all? The proof requires the definition and triangle inequality, that's all?
Given Ax=λx, take i for which |xᵢ| is largest. Look at the i'th equation: sum aᵢⱼxⱼ = λxᵢ, move the aᵢᵢxᵢ term to the rhs, take absolute values, divide by |xᵢ|, apply triangle inequality, and you have |aᵢᵢ - λ| ≤ sum |aᵢⱼ| over j≠i. So for every eigenvalue you can find such a disc.
That's by column, for row use Aᵀ.
Re: Mathematicians prove Pólya's conjecture for the eigenvalues of a disk
#26Mathematicians prove that you can deduce the shape of disk based on the frequencies it produces when you hit it with a stick.
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 Pólya’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.
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. [Sorry, not my area and I have not enough time to read the paper.]
Re: Mathematicians prove Pólya's conjecture for the eigenvalues of a disk
#27[deleted]
Re: Mathematicians prove Pólya's conjecture for the eigenvalues of a disk
#28Earlier quoted context omitted.
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.
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.
My brain also automatically added sum and also haven't noticed anything, but I have a math degree, maybe it's just assuming things :-)
Re: Mathematicians prove Pólya's conjecture for the eigenvalues of a disk
#29[deleted]
Re: Mathematicians prove Pólya's conjecture for the eigenvalues of a disk
#30The 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, unexpecte…
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