[math-fun] a search for a mathematical expression using a large database
Hello everyboy, I made an experiment over a period of several months using my inverter (the home version with 610 million constants). I took the latest values of the CODATA 2002 NIST table of physical constants and tried a vast experiment to find any reasonable mathematical expression for those ratios. http://physics.nist.gov/cuu/Constants/Table/allascii.txt That data is from december 2003. I used many simple and naive models to try to find anything, any possible expression as long as it is simple, short and easily explained. Here are the results : http://www.lacim.uqam.ca/~plouffe/Search.htm Important note : that article (preprint) is an exercice in numerical analysis and an attempt to find a mathematical and simple expression and NOT any attempt to any physical theory. My knowledge of physics is naive and probably outdated. I have no idea if this is making any sense in the real physical world. It is only <the best possible mathematical expression> that could exist for those numbers that have been found using what I believe are appropriate tools. The tables I used are the ones on the Inverter and a set of specialized tables constructed from the OEIS and my own tables that are not yet public. There are 2 main findings : First I discovered a weakness in the PSLQ, LLL or integer relations algorithm that only exist for a specific type of numbers. Second, I propose a set of at least 12 values among the 28 known values (actually there are 14 + inverses). In other words, I have a mathematical expression for 12 of the 14 values. These expressions are all generated by 1 number only. The second finding is related to the first as explained in the article. The article (preprint) as been submitted to a known periodical. Simon Plouffe
a naive question : given the uncertainties (confidence intervals) in CODATA, how good is the fit, or, what is the (combined) probability that the Plouffe-values are 'real', and the variances due to chance errors? Particle Ratio ; Value; Plouffe; NumericalPlouffe Neutron-Proton; 1,001378419; L[9] L[6]^(1/2) / Phi^12; 1,001378267 Alpha particle-Electron; 7294,299536; L[5] Phi^17/L[7]^(1/2); 7294,299151 Tau-Muon; 16,8183; Phi^9 L[10]^(1/11) /L[4]; 16,81830004 Helion-Proton; 2,993152667; Phi^13 L[3]^(1/18) /L[3]/L[8]; 2,993155303 Neutron-Electron; 1838,68366; Phi^19 L[4]^(17/19) /L[7]; 1838,685785 Rest mass of alpha particle-electron; 7349,672665; Phi^(61/2)/L[12]; 7349,658428 Proton-Electron; 1836,152673; L[4]^(11/30) L[8]^(53/30); 1836,152847 Muon-Electron; 206,7682838; F[5]^(1/15) L[4]^(7/15) L[5]^(9/5); 206,7682818 Neutron-Muon; 8,89248402; L[5] F[5]^(11/5)/Phi^(39/5); 8,89248417 Proton-Muon; 8,88024333; Phi^(37/30) F[5]^(7/15)L[5]^(7/20); 8,880243302 Helion-Electron; 5495,885269; L[9] Phi^13/L[5]^(14/17); 5495,856824 Alpha-particle-Proton; 3,972599689; Phi^18 L[11]^(8/15) /L[10]/L[11]; 3,972595175 Deuteron-Electron; 3670,482965; L[2]^3 Phi^(8/29) L[8]^(36/29); 3670,482964 Wouter. ----- Original Message ----- From: "Simon Plouffe" <simon.plouffe@sympatico.ca> To: "math-fun" <math-fun@mailman.xmission.com>; "Eric W. Weisstein" <eww@wolfram.com>; <sfinch9@hotmail.com>; <D.Broadhurst@open.ac.uk>; "Jean-Paul Allouche" <Jean-Paul.Allouche@lri.fr>; <delahaye@lifl.fr>; <davis@math.toronto.edu> Sent: Friday, March 12, 2004 9:59 PM Subject: [math-fun] a search for a mathematical expression using a large database
Hello everyboy,
I made an experiment over a period of several months using my inverter (the home version with 610 million constants).
I took the latest values of the CODATA 2002 NIST table of physical constants and tried a vast experiment to find any reasonable mathematical expression for those ratios. http://physics.nist.gov/cuu/Constants/Table/allascii.txt
That data is from december 2003.
I used many simple and naive models to try to find anything, any possible expression as long as it is simple, short and easily explained.
Here are the results : http://www.lacim.uqam.ca/~plouffe/Search.htm
Important note : that article (preprint) is an exercice in numerical analysis and an attempt to find a mathematical and simple expression and NOT any attempt to any physical theory. My knowledge of physics is naive and probably outdated. I have no idea if this is making any sense in the real physical world. It is only <the best possible mathematical expression> that could exist for those numbers that have been found using what I believe are appropriate tools.
The tables I used are the ones on the Inverter and a set of specialized tables constructed from the OEIS and my own tables that are not yet public.
There are 2 main findings : First I discovered a weakness in the PSLQ, LLL or integer relations algorithm that only exist for a specific type of numbers. Second, I propose a set of at least 12 values among the 28 known values (actually there are 14 + inverses). In other words, I have a mathematical expression for 12 of the 14 values. These expressions are all generated by 1 number only.
The second finding is related to the first as explained in the article.
The article (preprint) as been submitted to a known periodical.
Simon Plouffe
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Hello, concerning the analysis or error. What I used to take is v+3*e, and v-3*e, v is the value and e = error. We can assume that the error is like a bell-shaped curve it would normally means that v +/- 3*e is a 99% confidence interval. For example with 1.00137841870 +/- 0.000 000 000 58 (Spacing for readability) would give 1.00137841 or 1.00137842. --> this gives 8 valid decimal digits and maybe 9. But on the other hand some values cannot be placed into a valid interval without loosing a lot of precision as with : Tau-Muon; 16,8183; Phi9 L[10]^(1/11) /L[4]; 16,81830004 That one is right on the nose but the valid interval is 16.8264 and 16.8102 so we have 3 decimal digits for sure and maybe 4. We can hardly make any analysis since we can approach the value as near as we want. Those values with the golden ratio were the only ones that could get near a mass ratio without being too long as an expression. After all that expression is only powers of phi (in disguise). Is there a simpler answer to that ? The expressions I choose to put in the summary table are the ones I picked among the tons of candidates from which some of them were exactly 11 digits in precision. But this exercice is maybe missing the point. All of them with only powers of phi. Maybe (if I may emit this explanation) those values are average values in fact and the mass ratios are just oscillations of values among all those Lucas-Golden ratio values. There are 3 main reasons perhaps to think that way. The first obvious reference is what is found in nature about the golden ratio, as you may know perhaps, 95.6% of all plants grow with a factor of 1.618033. If you take a whole field of daisies and count the petals on each flower you will find 21 or 34 petals. But this is the point, some flowers will have 22 and some others will have 33 but in average it is 21 or 34. So the law of growth remains as being based on phi= 1.618033 but not exactly. The second principle is what is known on the hard hexagon problem, it was found recently that the piling of ice, snow is based on a critical exponent which is phi^5 = 11.0901699... The 3rd reason : the proposed values are (in my opinion) the best possible and simplest answer based on the big tables of constants and what is found in nature. Among all the real numbers, there are ripples of values around powers of phi, lucas numbers and fibonacci numbers and that phenomena is unique because of the 3 main transformations which are near 1. These transforms are the simplest possible. Actually, (I did those search also) with values like (1+sqrt(2))^n and some other simple algebraic values but only the golden ratio Fn and Ln can simplify like that. Since it was the only model having that property, I picked that one as being perhaps significant. I based my assumption also on the fact that [1,sqrt(5), phi^n] can produce a false answer when passed through the LLL algorithm and that is precisely the point, there are so many values and expressions near the point that even LLL will hang on it because the values are near integers. Simon Plouffe
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Simon Plouffe -
wouter meeussen