[math-fun] Mystery tables
There have been no posts in response to either of my mystery tables for over a week. Is anyone still working on them, or shall I reveal what they are tables of? Here they are again: 1 2 3 4 5 6 7 8 1: all 2: -inf +inf 3: -inf 1.000000 +inf 4: -inf 0.000000 2.261406 +inf 5: -inf -0.349081 1.000000 3.074601 +inf 6: -inf -0.523305 0.523305 1.564328 3.795684 +inf 7: -inf -0.626499 0.268479 1.000000 2.000000 4.495526 +inf 8: -inf -0.694242 0.109189 0.694242 1.350067 2.385122 5.190739 +inf 9: -inf -0.741892 0.000000 0.500000 1.000000 1.643796 2.748616 5.884929 10: -inf -0.777116 -0.079600 0.364842 0.777116 1.249756 1.908997 3.102748 11: -inf -0.804148 -0.140255 0.265003 0.621297 1.000000 1.469217 2.158924 12: -inf -0.825509 -0.188046 0.188046 0.505587 0.825509 1.192579 1.671024 13: -inf -0.842792 -0.226700 0.126799 0.415933 0.695796 1.000000 1.366735 14: -inf -0.857046 -0.258628 0.076822 0.344232 0.595108 0.857046 1.155985 15: -inf -0.868994 -0.285461 0.035215 0.285461 0.514414 0.746112 1.000000 16: -inf -0.879146 -0.308343 0.000000 0.236333 0.448129 0.657173 0.879146 1 2 3 4 5 6 7 8 1: 1.000000 1.545362 2.293678 3.240906 4.248364 5.259193 6.267074 7.272747 2: 1.545362 2.000000 2.576905 3.343752 4.270644 5.262645 6.267506 7.272793 3: 2.293678 2.576905 3.000000 3.589622 4.367036 5.288115 6.272582 7.273628 4: 3.240906 3.343752 3.589622 4.000000 4.596536 5.380654 6.299598 7.279833 5: 4.248364 4.270644 4.367036 4.596536 5.000000 5.600888 6.389624 7.307724 6: 5.259193 5.262645 5.288115 5.380654 5.600888 6.000000 6.603882 7.395990 7: 6.267074 6.267506 6.272582 6.299598 6.389624 6.603882 7.000000 7.606068 8: 7.272747 7.272793 7.273628 7.279833 7.307724 7.395990 7.606068 8.000000
On 02/02/2020 18:56, Keith F. Lynch wrote:
There have been no posts in response to either of my mystery tables for over a week. Is anyone still working on them, or shall I reveal what they are tables of? Here they are again:
1 2 3 4 5 6 7 8 1: all 2: -inf +inf 3: -inf 1.000000 +inf 4: -inf 0.000000 2.261406 +inf 5: -inf -0.349081 1.000000 3.074601 +inf 6: -inf -0.523305 0.523305 1.564328 3.795684 +inf 7: -inf -0.626499 0.268479 1.000000 2.000000 4.495526 +inf 8: -inf -0.694242 0.109189 0.694242 1.350067 2.385122 5.190739 +inf 9: -inf -0.741892 0.000000 0.500000 1.000000 1.643796 2.748616 5.884929 10: -inf -0.777116 -0.079600 0.364842 0.777116 1.249756 1.908997 3.102748 11: -inf -0.804148 -0.140255 0.265003 0.621297 1.000000 1.469217 2.158924 12: -inf -0.825509 -0.188046 0.188046 0.505587 0.825509 1.192579 1.671024 13: -inf -0.842792 -0.226700 0.126799 0.415933 0.695796 1.000000 1.366735 14: -inf -0.857046 -0.258628 0.076822 0.344232 0.595108 0.857046 1.155985 15: -inf -0.868994 -0.285461 0.035215 0.285461 0.514414 0.746112 1.000000 16: -inf -0.879146 -0.308343 0.000000 0.236333 0.448129 0.657173 0.879146 I have been thinking half-heartedly (and apparently half-brainedly) about this one from time to time ... and in the process of writing down my incoherent thoughts I realise that I have solved it.
... spoiler space ... ... spoiler space ... ... spoiler space ... ... spoiler space ... ... spoiler space ... ... spoiler space ... ... spoiler space ... ... spoiler space ... The value in row m, column n is the k such that m^k = 2n^k-1. For the (1,1) entry any k will do, hence the slightly curious "all" in that spot. (I haven't looked at the second table.) -- g
On 02/02/2020 18:56, Keith F. Lynch wrote:
There have been no posts in response to either of my mystery tables for over a week. Is anyone still working on them, or shall I reveal what they are tables of? Here they are again:
[SNIP: first table, now solved]
1 2 3 4 5 6 7 8 1: 1.000000 1.545362 2.293678 3.240906 4.248364 5.259193 6.267074 7.272747 2: 1.545362 2.000000 2.576905 3.343752 4.270644 5.262645 6.267506 7.272793 3: 2.293678 2.576905 3.000000 3.589622 4.367036 5.288115 6.272582 7.273628 4: 3.240906 3.343752 3.589622 4.000000 4.596536 5.380654 6.299598 7.279833 5: 4.248364 4.270644 4.367036 4.596536 5.000000 5.600888 6.389624 7.307724 6: 5.259193 5.262645 5.288115 5.380654 5.600888 6.000000 6.603882 7.395990 7: 6.267074 6.267506 6.272582 6.299598 6.389624 6.603882 7.000000 7.606068 8: 7.272747 7.272793 7.273628 7.279833 7.307724 7.395990 7.606068 8.000000
The same thing happened with this one as with the other: I was writing up my failure to solve it when I solved it :-). At first glance it _looks_ like some sort of generalized mean thing: f(x,y) = F^-1((F(x)+F(y))/2) for some increasing function F, preferably increasing quite fast so that e.g. f(1,8) ~= 8. And it looks as if the very simple choice F(x) = exp(x) produces numbers quite similar to those in the table; small modifications, such as F(x) = sinh(0.9x) or better still sinh(x) / x^(2/3), do even better; but my admittedly crude attempts at playing with the shape of F have been unable to get the mean squared error below about 0.015, which suggests that maybe the answer _isn't_ of this form at all. (Only weakly; I haven't tried _very hard_ to optimize F.) But it turns out that the entries in the table satisfy the following (defining) equation: [... spoiler space ...] [... spoiler space ...] [... spoiler space ...] [... spoiler space ...] [... spoiler space ...] [... spoiler space ...] [... spoiler space ...] [... spoiler space ...] [... spoiler space ...] if the entry in position (x,y) is z, then x^z + y^z = 2z^z. It is not obvious to me whether this is actually equivalent to some sort of generalized mean as defined above. I'd guess not. What of course _is_ true is that z = ((x^z+y^z)/2)^1/z, so each entry is _a_ "power mean" of its row and column positions ... but with an exponent that equals the entry itself :-). -- g
participants (2)
-
Gareth McCaughan -
Keith F. Lynch