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EspañolEMBREE-TREFETHEN CONSTANT
In mathematics, the 'Embree-Trefethen constant' is a threshold value in number theory labelled '''β
★ '''.
For a fixed real β, consider the recurrence where the sign in the sum is chosen at random for each ''n'' independently with equal probabilities for "+" and "−".
It can be proven that for any choice of β, the limit
:
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.
We have
:''σ'' < 1 for 0 < ''β'' < ''β
★ '' = 0.70258 approximately,
so solutions to this recurrence decay exponentially as ''n''→∞ with probability one, and
:''σ'' > 1 for ''β
★ '' < ''β'',
so they grow exponentially.
Regarding values of σ, we have:
★ σ(1) = 1.13198824... (Viswanath's constant), and
★ σ(β
★ ) = 1.
The constant is named after applied mathematicians Mark Embree and Lloyd N. Trefethen.
★ Embree, M., and L.N. Trefethen (1999): Growth and decay of random Fibonacci sequences. Proceedings of the Royal Society London A 455(July):2471-2485
★ Viswanath's constant
★
★ '''.
For a fixed real β, consider the recurrence where the sign in the sum is chosen at random for each ''n'' independently with equal probabilities for "+" and "−".
It can be proven that for any choice of β, the limit
:
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.
We have
:''σ'' < 1 for 0 < ''β'' < ''β
★ '' = 0.70258 approximately,
so solutions to this recurrence decay exponentially as ''n''→∞ with probability one, and
:''σ'' > 1 for ''β
★ '' < ''β'',
so they grow exponentially.
Regarding values of σ, we have:
★ σ(1) = 1.13198824... (Viswanath's constant), and
★ σ(β
★ ) = 1.
The constant is named after applied mathematicians Mark Embree and Lloyd N. Trefethen.
| Contents |
| Literature |
| See also |
| External Links |
Literature
★ Embree, M., and L.N. Trefethen (1999): Growth and decay of random Fibonacci sequences. Proceedings of the Royal Society London A 455(July):2471-2485
See also
★ Viswanath's constant
External Links
★
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