Comments on: A bound on partitioning clusters
https://terrytao.wordpress.com/2017/02/05/a-bound-on-partitioning-clusters/
Updates on my research and expository papers, discussion of open problems, and other maths-related topics. By Terence TaoFri, 20 Apr 2018 04:09:28 +0000hourly1http://wordpress.com/By: Paata Ivanisvili
https://terrytao.wordpress.com/2017/02/05/a-bound-on-partitioning-clusters/#comment-481278
Sat, 27 May 2017 00:53:33 +0000http://terrytao.wordpress.com/?p=9743#comment-481278Given integer we want to find nonnegative function on such that and . And the goal is to minimize . This problem was solved when with . But for fixed the best possible does not have to be this one, it can be smaller. Is this what was meant in your first comment? The only thing that is not clear to me why this main inequality that I wrote on (and which closes induction easily) should be also necessary for the true , i.e., I mean yes I agree, it is sufficient to solve the problem and find some bound, but is it also necessary, i.e., will the result give me sharp bound?
]]>By: kaave
https://terrytao.wordpress.com/2017/02/05/a-bound-on-partitioning-clusters/#comment-480004
Fri, 21 Apr 2017 22:19:51 +0000http://terrytao.wordpress.com/?p=9743#comment-480004This is reminding me of a result of Manners: https://arxiv.org/pdf/1512.06272.pdf
Appendix B, Proposition B.1.
]]>By: Sergei Ofitserov
https://terrytao.wordpress.com/2017/02/05/a-bound-on-partitioning-clusters/#comment-479346
Sun, 26 Mar 2017 11:01:39 +0000http://terrytao.wordpress.com/?p=9743#comment-479346Dear Terence Tao! I proceed with big interest to read your notes and want to give reply on “trilinear” bound.
——Quarks in quantum chromodynamics.—— In inequality |X1|~|X2|~|X3| it is necessary to apply meanings u,d,s. Accordingly:
|X1|=u, |X2|=d, |X3|=s, because quarks u,d,s to have various quantum numbers. In equality |X1|=|X2|=|X3|=|X| it is necessary to apply meaninqs u,u,u. Accordingly:–
|X1|=u, |X2|=u, |X3|=u, because quarks u,u,u to have identical quantum numbers and directions. Matrixs of Gell-Mann (3×3) to form algebra ASU(3). Thus, in this way gauge theory, base on group SU(3), open approach for theory of Yang-Mills.Constrained quarks in groups SU(3)=(u,d,s) or SU(3)=(u,u,u) give possibility to approach for demonstration of Rieman hypothesis with other side. Fourth-display=>this metric siqnature(+—). Thanks!Sergei.
]]>By: Romain Viguier
https://terrytao.wordpress.com/2017/02/05/a-bound-on-partitioning-clusters/#comment-478460
Mon, 27 Feb 2017 18:13:19 +0000http://terrytao.wordpress.com/?p=9743#comment-478460you should make intervals, make in it a convergent series, make an increasing function on a subset n- (n- (n / i) of
the starting set for i ranging from 1 to n,
you make a deterministic decomposition (we know the arrival) of the function. Then you remove an interval, and we will have a cos and sin function.
The limit at 0 is n ^ 2 – n / i for i ranging from 1 to n.
]]>By: A Gauss-Bonnet connection - Quantum Calculus
https://terrytao.wordpress.com/2017/02/05/a-bound-on-partitioning-clusters/#comment-478253
Tue, 21 Feb 2017 01:20:26 +0000http://terrytao.wordpress.com/?p=9743#comment-478253[…] are cheap, one can find them in any library or now the internet. But it is also on the internet: An example. End Side […]
]]>By: Terence Tao
https://terrytao.wordpress.com/2017/02/05/a-bound-on-partitioning-clusters/#comment-478062
Wed, 15 Feb 2017 02:10:24 +0000http://terrytao.wordpress.com/?p=9743#comment-478062In the paper, the gap between the upper and lower bounds is of the form , which is not quite the same thing as , though in the key examples, is of exponential size in so the gap should be something like . To make the induction close, I think one should use a Bellman function that is independent of and which is smaller than by a power of when ; it has to be carefully chosen not only in the main region where but also in the extreme cases when say one of the is much smaller than the other two (this corresponds to the fact that in addition to the main critical point of the key inequality, we also have critical points at and permutations). Finding the precise Bellman function that works may require a certain amount of trial and error and iteration (trying one Bellman function, finding out the locations where the induction fails to close, tweaking the Bellman function to repair the induction in that case, and then seeing if this causes the induction to stop working at some other location, repeating as necessary).
]]>By: Paata Ivanisvili
https://terrytao.wordpress.com/2017/02/05/a-bound-on-partitioning-clusters/#comment-478060
Wed, 15 Feb 2017 01:47:59 +0000http://terrytao.wordpress.com/?p=9743#comment-478060But how to close the induction? If little is in in the expression then the “Bellman function” should also depend on . I don’t quite see how these quantities propagate, or maybe I misunderstood the question.
]]>By: SSQR (@SumantriSQR)
https://terrytao.wordpress.com/2017/02/05/a-bound-on-partitioning-clusters/#comment-478000
Tue, 14 Feb 2017 09:46:12 +0000http://terrytao.wordpress.com/?p=9743#comment-478000is this has any connection with cluster algebra that initiated by fomin-zelevinski??

[No. – T.]

]]>By: Terence Tao
https://terrytao.wordpress.com/2017/02/05/a-bound-on-partitioning-clusters/#comment-477990
Tue, 14 Feb 2017 00:45:01 +0000http://terrytao.wordpress.com/?p=9743#comment-477990Probably it can be improved. One needs to replace the right-hand side in the fourth-display by some more complicated “Bellman function” of the three sizes that grows a little bit more slowly than on the diagonal, and also gains something when the are different sizes from each other. One will probably have to do some Taylor expansion calculations near the critical point of the elementary inequality used here in order to make sure the induction closes properly.
]]>By: Matjaž Gomilšek
https://terrytao.wordpress.com/2017/02/05/a-bound-on-partitioning-clusters/#comment-477770
Sun, 12 Feb 2017 14:33:43 +0000http://terrytao.wordpress.com/?p=9743#comment-477770In the preprint, that is.
]]>