5/30/2023 0 Comments Terence tao![]() 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). The hot spots conjecture is established unconditionally for any acute-angled triangle which has one angle less than or equal to (actually a slightly larger region than this is obtained).Among other things, these methods can be used to exclude critical points occurring anywhere in the interior or on the edges of the triangle except for those points that are close to one of the vertices and in this recent preprint of Siudeja, two further partial results on the hot spots conjecture are obtained by a variant of the method: These two papers of Miyamoto introduced a promising new method to theoretically control the behaviour of the second Neumann eigenfunction, by taking linear combinations of that eigenfunction with other, more explicit, solutions to the eigenfunction equation, restricting that combination to nodal domains, and then computing the Dirichlet energy on each domain. However, in view of the theoretical advances, the precise control on the eigenfunction that we need may be different from what we had previously been contemplating. Some recent papers of Kwasnicki-Kulczycki, Melenk-Babuska, and Driscoll employ similar methods and may be worth studying further. This paper of Liu and Oshii has some promising approaches.Īfter we get good bounds on the eigenvalue, the next step is to get good control on the eigenfunction some approaches are summarised in this note of Lior Silberman, mainly based on gluing together exact solutions to the eigenfunction equation in various sectors or disks. Good upper bounds are relatively easy to obtain, simply by computing the Rayleigh quotient of numerically obtained approximate eigenfunctions, but lower bounds are trickier. On the numerical side, we have decided to focus first on the problem of obtaining validated upper and lower bounds for the second Neumann eigenvalue of a triangle. This post is the new research thread for the Polymath7 project to solve the hot spots conjecture for acute-angled triangles, superseding the previous thread this project had experienced a period of low activity for many months, but has recently picked up again, due both to renewed discussion of the numerical approach to the problem, and also some theoretical advances due to Miyamoto and Siudeja. where the updated version is and will be maintained.īelow the fold is some more technical information regarding the above calculations.The paper where these (and many more guesses of this type) are given with some background on zeta deformation etc, and.The polymath proposal is to investigate this phenomenon further (perhaps by more extensive numerical calculations) and supply a theoretical explanation for it. (But we do not know whether this 42 is the answer to everything!).įor the second sum, calculation for degrees up to 28 shows that the difference between the two sides has -valuation at least 88. Similarly, if one expands the first sum for all primes of degree (in ) up to 37, one obtains (the calculation took about a month on one computer), implying that the -valuation of the infinite sum is at least 38 in fact a bit of theory can improve this to 42. ![]() It was numerically observed in that one appears to have the remarkable cancellationĪnd all other terms in are of order or higher, so this shows that has -valuation at least 3. ![]() ![]() Then infinite series such asĬan be expanded as formal infinite power series in the variable. Let be the ring of polynomials over the finite field of two elements, and letīe the set of irreducible polynomials in this ring. I am posting this proposal on behalf of Dinesh Thakur.
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