![]() The arguments of Miyamoto also allow one to rule out critical points occuring for most of the interior points of a given triangle it is only the points that are very close to one of the three vertices which we cannot yet rule out by Miyamoto’s methods. ![]() (For instance, the Miyamoto method relies on upper bounds on, and these can be obtained numerically.) In particular, some hybrid of the Miyamoto method and the numerical techniques we are beginning to discuss may be a promising approach to fully resolve the conjecture. to keep eigenfunctions monotone on the edges on which they change sign), we may be able to establish the hot spots conjecture for a further range of triangles. So if we can develop more techniques to rule out critical points occuring on edges (i.e. The hot spots conjecture is also established for any acute-angled triangle with the property that the second eigenfunction has no critical points on two of the three edges (excluding vertices).In particular, the case of very narrow triangles have been resolved (the dark green region in the area below). ![]() ![]()
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