QRS Constant
المؤلف:
Bailey, D. H.; Borwein, J. M.; and Crandall, R. E.
المصدر:
"Resolution of the Quinn-Rand-Strogatz Constant of Nonlinear Physics." Preprint. June 4, 2007. https://users.cs.dal.ca/~jborwein/QRS.pdf.
الجزء والصفحة:
...
22-4-2020
1023
QRS Constant
Quinn et al. (2007) investigated a class of 
coupled oscillators whose bifurcation phase offset had a conjectured asymptotic behavior of 
, with an experimental estimate for the constant 
as 
(OEIS A131329). Rather amazingly, Bailey et al. (2007) were able to find a closed form for 
as the unique root of 
in the interval ![[0,2]](https://mathworld.wolfram.com/images/equations/QRSConstant/Inline7.gif)
, where 
is a Hurwitz zeta function.
A related constant conjectured by Quinn et al. (2007) to exist was defined in terms of
![S(N,a)=sum_(i=1)^N[1-a^2(1-(2i-2)/(N-1))^2]^(-3/2)](https://mathworld.wolfram.com/images/equations/QRSConstant/NumberedEquation1.gif) |
(1)
|
and given by
 |
(2)
|
(OEIS A131330). Even more amazingly, the exact value of this constant was also found by Bailey et al. (2007) without full proof, but with enough to indicate that such a proof could in principle be constructed, to have the exact value
![C=1/4zeta(3/2,1/2c_1)].](https://mathworld.wolfram.com/images/equations/QRSConstant/NumberedEquation3.gif) |
(3)
|
REFERENCES:
Bailey, D. H.; Borwein, J. M.; and Crandall, R. E. "Resolution of the Quinn-Rand-Strogatz Constant of Nonlinear Physics." Preprint. June 4, 2007. https://users.cs.dal.ca/~jborwein/QRS.pdf.
Quinn, D. ; Rand, R.; and Strogatz, S. "Singular Unlocking Transition in the Winfree Model of Coupled Oscillators." Phys. Rev. E 75, 036218-1-10, 2007.
Sloane, N. J. A. Sequences A131329 and A131330 in "The On-Line Encyclopedia of Integer Sequences."
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