Embree-Trefethen constant
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In mathematics, the Embree-Trefethen constant is a threshold value in number theory labelled β*.
For a fixed real β, consider the recurrence
- xn+1=xn±βxn-1
where the sign in the sum is chosen at random for each n independently with equal probabilities for "+" and "-".
In can be proven that for any choice of β, the limit
- <math>\beta(\sigma) = \lim_{n \to \infty} (|x_n|^{1/n})<math>
exists almost surely. In informal words, the sequence behaves exponentially with probability one—and σ(β) can be interpreted as its almost sure rate of exponential growth.
For
- 0 < β < β* = 0.70258 approximately,
solutions to this recurrence decay exponentially as n→∞ with probability one, whereas for
- β > β*
they grow exponentially.
Regarding values of σ, we have:
- σ(1)=1.13198824... (Viswanath's constant), and
- σ(β*)=1 .
External link
- Proc. Roy. Soc. London Series A 455 (1999) 2471-2485 (http://taddeo.ingentaselect.com/vl=896274/cl=134/nw=1/rpsv/cgi-bin/linker?ini=rsl&reqidx=/catchword/rsl/13645021/v455n1987/s5/p2471)