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In number theory, the Embree–Trefethen constant is a threshold value labelled ''β *''. For a fixed positive number ''β'', consider the recurrence relation : 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 1, 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. ==References== 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Embree–Trefethen constant」の詳細全文を読む スポンサード リンク
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