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Mathematicians prove Pólya's conjecture for the eigenvalues of a disk

phys.org

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Re: Mathematicians prove Pólya's conjecture for the eigenvalues of a disk

#61

Post by xelxebar »

stabbles wrote: post #39
kmm wrote: post #35

Earlier quoted context omitted.

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.

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 + τI has the same eigenvalues shifted by τ, then it's a small step to consider that the magnitude of the off-diagonal may have something…

> tri-diagonal matrices with stencil [-1, 2, -1],

Just trying to understand these terms. So is a 5x5 tridiagonal matrix with your stencil look like this?

     2 -1  0  0  0
    -1  2 -1  0  0
     0 -1  2 -1  0
     0  0 -1  2 -1
     0  0  0 -1  2

Re: Mathematicians prove Pólya's conjecture for the eigenvalues of a disk

#62

Post by stabbles »

xelxebar wrote: post #61
stabbles wrote: post #39

Earlier quoted context omitted.

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 + τI has the same eigenvalues shifted by τ, then it's a small step to consider that the magnitude of the off-diagonal may have something…

> tri-diagonal matrices with stencil [-1, 2, -1], Just trying to understand these terms. So is a 5x5 tridiagonal matrix with your stencil look like this? 2 -1 0 0 0 -1 2 -1 0 0 0 -1 2 -1 0 0 0 -1 2 -1 0 0 0 -1 2

Yeah, exactly. Your matrix shows up when you do a 3-point discretization of -u''(x) = λu(x) with u(0) = u(1) = 0, at 5 equidistant (interior) grid points. That's the discrete, 1D version of the problem they're looking at in the paper.

See https://en.wikipedia.org/wiki/Compact_stencil#Three_Point_St...

Re: Mathematicians prove Pólya's conjecture for the eigenvalues of a disk

#63

Post by Ziggy_Zaggy »

homerowilson wrote: post #49
jonathan_landy wrote: post #18

Earlier 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...

You may enjoy this: https://bwlewis.github.io/cassini/

Any more details on this one? What might one use-case be for this application?
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