词条 | Embree–Trefethen constant |
释义 |
In number theory, the Embree–Trefethen constant is a threshold value labelled β*.[1] 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:
The constant is named after applied mathematicians Mark Embree and Lloyd N. Trefethen. References1. ^{{Cite journal | last1 = Embree | first1 = M. | authorlink1 = Mark Embree| last2 = Trefethen | first2 = L. N. | authorlink2 = Lloyd N. Trefethen| doi = 10.1098/rspa.1999.0412 | title = Growth and decay of random Fibonacci sequences | journal = Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences | volume = 455 | issue = 1987 | pages = 2471 | year = 1999 | url = http://people.maths.ox.ac.uk/~trefethen/publication/PDF/1999_86.pdf| pmid = | pmc = |bibcode = 1999RSPSA.455.2471T | citeseerx = 10.1.1.33.1658 }} External links
2 : Mathematical constants|Recurrence relations |
随便看 |
|
开放百科全书收录14589846条英语、德语、日语等多语种百科知识,基本涵盖了大多数领域的百科知识,是一部内容自由、开放的电子版国际百科全书。