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Estimates from below of the Buffon noodle probability for undercooked noodles | | Matthew Bond
; Alexander Volberg
; | | Date: |
9 Nov 2008 | | Abstract: | Let $Cant_n$ be the $n$-th generation in the construction of the middle-half
Cantor set. The Cartesian square $K_n$ of $Cant_n$ consists of $4^n$ squares
of side-length $4^{-n}$. The chance that a long needle thrown at random in the
unit square will meet $K_n$ is essentially the average length of the
projections of $K_n$, also known as the Favard length of $K_n$. A result due
to Bateman and Volberg cite{BV} shows that a lower estimate for this Favard
length is $c frac{log n}{n}$.
We may bend the needle at each stage, giving us what we will call a noodle,
and ask whether the uniform lower estimate $c frac{log n}{n}$ still holds for
these so-called Buffon noodle probabilities. If so, we call the sequence of
noodles undercooked. We will define a few classes of noodles and prove that
they are undercooked. In particular, we are interested in the case when the
noodles are circular arcs of radius $r_n$. We will show that if $r_n geq
4^{frac{n}{5}}$, then the circular arcs are undercooked noodles. | | Source: | arXiv, 0811.1302 | | Services: | Forum | Review | PDF | Favorites | |
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