Nice, what are the potential use cases for this? Can I use this for my business?
This is a valid question????
Mathematicians prove Pólya's conjecture for the eigenvalues of a disk
51–60 of 64 posts
Re: Mathematicians prove Pólya's conjecture for the eigenvalues of a disk
#52From the paper "Pólya’s conjecture for Euclidean balls"
https://dms.umontreal.ca/~iossif/polya.pdf
"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. Pólya’s conjecture is known to be true for domains which tile Euclidean space, and, in addition, for some special domains in higher dimensions. In this paper, we prove Pólya’s conjecture for the disk, making it the first non-tiling planar domain for which the conjecture is verified. We also confirm Pólya’s conjecture for arbitrary planar sectors, and, in the Dirichlet case, for balls of any dimension. Along the way, we develop the known links between the spectral problems in the disk and certain lattice counting problems. A key novel ingredient is the observation, made in recent work of the last named author, that the corresponding eigenvalue and lattice counting functions are related not only asymptotically, but in fact satisfy certain uniform bounds."
Re: Mathematicians prove Pólya's conjecture for the eigenvalues of a disk
#53From the paper "Pólya’s conjecture for Euclidean balls" https://dms.umontreal.ca/~iossif/polya.pdf "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. Pólya’s conjecture is known to be true for doma…
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Re: Mathematicians prove Pólya's conjecture for the eigenvalues of a disk
#54The 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…
bee_rider already touched on this in another comment, but the theorem makes sense if you consider a matrix with large diagonal values and small off-diagonal values (in magnitude). If I have a matrix with 1,000,000 on the diagonal and 1 everywhere else, I'd expect the eigenvalues to be 1,000,000 plus or minus some small error. The Gershgorin disk theorem proves this and puts an upper bound on the error.
The diagonal elements of matrices have a lot of rather "magical" properties if you think about it. Their sum is also the sum of the eigenvalues of the matrix. And if you have a matrix A that is singular, you can choose any value x that is not an eigenvalue, and then A - xI is invertible but still mostly behaves like A.
Re: Mathematicians prove Pólya's conjecture for the eigenvalues of a disk
#55Re: Mathematicians prove Pólya's conjecture for the eigenvalues of a disk
#56Nice, what are the potential use cases for this? Can I use this for my business?
This is the sort of thing that comes up in simulations for various engineering subfields. So if your business is computer aided engineering tools or perhaps CAD, then probably yes.
Re: Mathematicians prove Pólya's conjecture for the eigenvalues of a disk
#57Nice, what are the potential use cases for this? Can I use this for my business?
Probably not - in applications, heuristics/approximations and "probably true" are king. And in this case the relevant approximations have been known for decades at least.
That's a very different beast to the game mathematicians play, which demands rigorous proof (or at least a fairly close social version of it, it's turtles all the way down).
Re: Mathematicians prove Pólya's conjecture for the eigenvalues of a disk
#58Nice, what are the potential use cases for this? Can I use this for my business?
These days, you'd probably want to know the eigenvalues of an SSD instead. /s
Re: Mathematicians prove Pólya's conjecture for the eigenvalues of a disk
#59I want a program that reconstructs the landscape from the sound of rolling thunder.
Re: Mathematicians prove Pólya's conjecture for the eigenvalues of a disk
#60Earlier quoted context omitted.
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?
For the basics, start here:
https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010...