Google Search: "site:quantamagazine.org terence tao", shows a loud pro Tao bias, with at least two comments from Tao in November alone. The other article [1] follows after [2,3] on arxiv, and Tao is a coauthor on [3]. Presumably he contributed to the proof. Unfortunately, the mathematics of Ref. [3] has recently come under attack as "not so original", thus calling into question the purpose and rigour of the Quanta article, as well as the business of it. As was noticed by Predrag Cvitanović, square eigenvector components in [3] are actually just the central diagonal of the projection operators, discussed in chapter 1 of "Principles of Symmetry, Dynamics, and Spectroscopy": https://modphys.hosted.uark.edu/pdfs/PSDS_Pdfs/PSDS_Ch.1_(4.23.10).pdf The fact that the central diagonal of a particular projector works out in terms of submatrix eigenvalues follows quickly from the fact that determinant and trace are simple functions of eigenvalues. That is all. John P. Raslton, an expert on Neutrino Oscillations and matrix algebra, is mostly done with a short proof that is a lot easier to understand. His write up will also cite PSDS as having the correct theory for calculating "Eigenvectors from Eigenvalues". Here is a quick example in Mathematica: M = {{254, -38, -57}, {-38 , 17, 6}, {-57, 6, 22}} Eigenvalues[M] With[{ev = Eigenvalues@M[[ Complement[Range[3], {#}], Complement[Range[3], {#}]]]}, Expand[Times @@ ev - 7*Total[ev] + 7^2] ] & /@ Range[3] Dot[M - 273*IdentityMatrix[3], M - 13*IdentityMatrix[3]] // MatrixForm Out[] = {273, 13, 7} Out[] = {114, 456, 1026} Out[] = { {114, 228, 342}, {228, 456, 684}, {342, 684, 1026} } I personally feel like we are bikeshedding with Quanta and their cadre of well-liked, world-famous, prize winning, prime-time-tv, whoevers. There is another unanswered question--whether the authors have supported their physics thesis that "eigenvalues are the Rosetta stone for neutrino oscillations in matter". It sounds very rich, but do the authors mean to suggest that their eigenvalues were first stolen by the French, and then stolen again from the French? Any thoughts? Cheers --Brad [1] https://www.quantamagazine.org/neutrinos-lead-to-unexpected-discovery-in-bas... [2] https://arxiv.org/abs/1907.02534 [3] https://arxiv.org/abs/1908.03795 On Fri, Nov 29, 2019 at 2:45 AM Ray Tayek <rtayek@ca.rr.com> wrote:
https://science.slashdot.org/story/19/11/28/180213/mathematicians-catch-a-pa...
We finally know how big a set of numbers can get before it has to contain a pattern known as a "polynomial progression." <https://www.quantamagazine.org/mathematicians-catch-a-pattern-by-figuring-out-how-to-avoid-it-20191125/> From a report:/A new proof <https://arxiv.org/abs/1909.00309> by Sarah Peluse of the University of Oxford establishes that one particularly important type of numerical sequence is, ultimately, unavoidable: It's guaranteed to show up in every single sufficiently large collection of numbers, regardless of how the numbers are chosen. "There's a sort of indestructibility to these patterns," said Terence Tao of the University of California, Los Angeles. Peluse's proof concerns sequences of numbers called "polynomial progressions." They are easy to generate -- you could create one yourself in short order -- and they touch on the interplay between addition and multiplication among the numbers. For several decades, mathematicians have known that when a collection, or set, of numbers is small (meaning it contains relatively few numbers), the set might not contain any polynomial progressions. They also knew that as a set grows it eventually crosses a threshold, after which it has so many numbers that one of these patterns has to be there, somewhere. It's like a bowl of alphabet soup -- the more letters you have, the more likely it is that the bowl will contain words. But prior to Peluse's work, mathematicians didn't know what that critical threshold was. Her proof provides an answer -- a precise formula for determining how big a set needs to be in order to guarantee that it contains certain polynomial progressions. Previously, mathematicians had only a vague understanding that polynomial progressions are embedded among the whole numbers (1, 2, 3 and so on). Now they know exactly how to find them./
-- Honesty is a very expensive gift. So, don't expect it from cheap people - Warren Buffett http://tayek.com/
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