论文标题
部分可观测时空混沌系统的无模型预测
The Buffon's needle problem for random planar disk-like Cantor sets
论文作者
论文摘要
我们考虑了一个自相似的有限和正面$ 1 $ -HAUSDORFF的随机性模型。我们发现，Buffon针头将$δ$ close降低到这种特殊随机性的cantor套件的概率较高的衰减率。通过Peres和Solomyak以及Shiwen Zhang的两个截然不同的随机性模型，对于Buffon针的概率似乎具有相同的衰减顺序：$ \ frac {c} {\ log \ frac {1}δ} $。在本说明中，我们证明了第三个随机性模型的衰减速率相同，这种衰减率具有一种模糊的感觉，即任何“合理的”随机康托尔（随机）的正面和有限长度都将带有$ \ frac {c} {\ log \ frac {1}δ} $的$δ$ -NEIGHIGHBOR的订单$ \ frac {c} {\ frac {\ log \ frac {1}δ} $。下面的估计值是Mattila很久以前获得的。
We consider a model of randomness for self-similar Cantor sets of finite and positive $1$-Hausdorff measure. We find the sharp rate of decay of the probability that a Buffon needle lands $δ$-close to a Cantor set of this particular randomness. Two quite different models of randomness for Cantor sets, by Peres and Solomyak, and by Shiwen Zhang, appear to have the same order of decay for the Buffon needle probability: $\frac{c}{\log\frac{1}δ}$. In this note, we prove the same rate of decay for a third model of randomness, which asserts a vague feeling that any "reasonable" random Cantor set of positive and finite length will have Favard length of order $\frac{c}{\log\frac{1}δ}$ for its $δ$-neighbourhood. The estimate from below was obtained long ago by Mattila.
