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| In [[number theory]], the '''Embree–Trefethen constant''' is a threshold value labelled '''''β*'''''.<ref>{{cite doi|10.1098/rspa.1999.0412|noedit}}</ref>
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| For a fixed positive number ''β'', consider the [[recurrence relation]]
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| : <math>x_{n+1}=x_n \pm \beta x_{n-1} \, </math>
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| where the sign in the sum is chosen at random for each ''n'' independently with equal probabilities for "+" and "−". | |
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| It can be proven that for any choice of ''β'', the limit
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| :<math>\sigma(\beta) = \lim_{n \to \infty} (|x_n|^{1/n}) \, </math> | |
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| 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]].
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| We have
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| :''σ'' < 1 for 0 < ''β'' < ''β*'' = 0.70258 approximately,
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| so solutions to this recurrence decay exponentially as ''n''→∞ with probability 1, and
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| :''σ'' > 1 for ''β*'' < ''β'',
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| so they grow exponentially.
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| Regarding values of σ, we have:
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| * σ(1) = 1.13198824... ([[Random Fibonacci sequence|Viswanath's constant]]), and
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| * σ(''β''*) = 1.
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| The constant is named after [[Applied mathematics|applied mathematicians]] [[Mark Embree]] and [[Lloyd N. Trefethen]].
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| ==References==
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| {{reflist}}
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| ==External links==
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| * {{MathWorld|urlname=RandomFibonacciSequence|title=Random Fibonacci Sequence}}
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| {{DEFAULTSORT:Embree-Trefethen constant}}
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| [[Category:Mathematical constants]] | |
| [[Category:Recurrence relations]]
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