Comments on: Polymath proposal: The Hot Spots Conjecture for Acute Triangles

By: Nick Rogers

Nick Rogers — Thu, 09 May 2013 20:48:06 +0000

Thanks!

By: Nilima Nigam

Nilima Nigam — Thu, 09 May 2013 20:10:46 +0000

Nicely done!

By: Nick Rogers

Nick Rogers — Thu, 09 May 2013 19:25:48 +0000

(very preliminary)

By: Nick Rogers

Nick Rogers — Thu, 09 May 2013 18:50:22 +0000

I`ve been working on computational approaches to this problem. Here is a short video illustrating my simulations: https://vimeo.com/65842093

By: Ileen

Ileen — Thu, 16 Aug 2012 14:32:08 +0000

…Take a look for more Information on that topic…

[...]Excellent blog here! Also your site so much up fast![...]…

By: Colleen

Colleen — Sun, 05 Aug 2012 01:44:45 +0000

Actually, this could be used to good realistic effcet. Sediment will tend to deposit in concavities and wind will tend to clean and abrade convexities giving them different colors. This is much the way that vegetation and sand distribution can be affected by slope. Also, if this could be applied to heightfields or masks, you could use this to flatten out the bottoms of concavities and convexities and possibly to give more of a streaky and abraded or bare rock texture to convexities. I may have to get one of those evil Windows machines just for your app.

By: Updates on the two polymath projects « What’s new

Updates on the two polymath projects « What’s new — Fri, 15 Jun 2012 22:22:54 +0000

[...] It’s been quite an active discussion in the last week or so, with almost 200 comments across two threads (and a third thread freshly opened up just now). While the problem is still not [...]

By: Three number theory bits: One elementary, the 3-Goldbach, and the ABC conjecture « mixedmath

Fri, 15 Jun 2012 13:58:30 +0000

[...] as announced at Terry Tao’s Blog, two new polymath items are on the horizon. There is a new polymath proposal at the polymath blog on the “Hot Spots Conjecture”, proposed by Chris Evans, and that [...]

By: meditationatae

meditationatae — Thu, 14 Jun 2012 15:58:09 +0000

I’ll explain why lines of steepest descent interest me. They are evevrywhere tangent to the gradient of , the lowest non-trivial eigenfunction of the Neumann Laplacian. For your height 2 , base 1 isosceles triangle, it seems the “cold spot” has about -0.09 , at the pointy vertex. Maybe we already know that the two other vertices at the base are the only two “hot spots” with being about +0.03 there, and maybe +0.025 on the base, half-way between the two “hot spot” vertices. Say we do a mirror image of the 3d graph of about the base, thus extending ; could it be that the extended has a saddle point at x=0, y=0 ? It seems that to the left of the axis of symmetry of the height 2, base 1 triangle, the steepest ascent curves should slope downwards and to the left, and to the right of the axis of symmetry, the steepest ascent curves starting at the pointy vertex should slope downwards and to the right.
For nodal lines, I meant for the eigenfunctions. They show where is average-valued. For the isosceles case, we already know where they lie, by the of Laugesen and Siudeja.

By: Nilima Nigam

Nilima Nigam — Thu, 14 Jun 2012 14:47:22 +0000

It’s easy enough for me to generate some figures showing the nodal line for a few triangles.
As for your steepest descent comment:

do you mean we should plot the nodal line in the same fixed triangle as time increases (in the heat equation), and then look at the direction of steepest descent? This is relatively easy to do.

or do you mean we should plot the nodal line for triangles with angles which are close, and somehow study the nodal lines in there? This I don’t readily see how to do, since the domain (and hence the location of the nodal lines) will change from triangle to triangle.

Let me know if this is still of interest, and I can throw up some graphics.