User magfrump - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T14:02:43Z http://mathoverflow.net/feeds/user/3572 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/121843/the-distribution-of-balls-in-a-bean-machine-that-omits-all-the-prime-pegs/121855#121855 Answer by magfrump for The distribution of balls in a Bean Machine that omits all the "prime pegs"? magfrump 2013-02-15T00:55:43Z 2013-02-15T00:55:43Z <p>Is it clear that the problem remains well defined?</p> <p>If the idea is that each ball has a 50% chance of bouncing in each direction when encountering a peg, then what happens to the ball when it would normally fall through the peg? In the picture above it appears that it would not be possible to access the outermost paths, because the paths that stay on the outside would be incomplete early on.</p> <p>My instinct here is that eventually, for every path leading anywhere but the center, there will be a prime numbered peg missing that prevents the ball from falling along that path, so for large $n$ the distribution would tend toward the discrete distribution with everything hitting the center.</p> http://mathoverflow.net/questions/44415/attractive-basins-and-loops-in-julia-sets Attractive Basins and Loops in Julia Sets magfrump 2010-11-01T04:10:43Z 2010-11-29T23:20:18Z <p>I recently learned about the Mandelbrot set for the first time from a presentation by some undergraduates in honor of Mandelbrot's death. The presentation was short and by non-experts so I left with a few questions.</p> <p>When I heard about the fundamental dichotomy, it seemed odd to me that the attractive basin at infinity was a distinguished point, in that filled Julia sets are the complement of the attractive basin of infinite. But when I naively expected the set of bounded points to be the attractive basin of the origin I was caught by the lack of duality--the filled Julia set is either simply connected or a Cantor set, and the complement of a simply connected set is simply connected, while I have only the slightest idea what the complement of a Cantor set looks like; it certainly isn't simply connected.</p> <p>Talking the subject over with a few graduate students I realized that there should be at least countably many bounded attractive basins or loops for any curve, and it seems unlikely that in the case that the basin of infinity is the complement of a Cantor set that the other attractive basins (including the loops of the roots of $f ^{n}(x) - x\$, where $f(x) = z^2+c$ ) are also complements of Cantor sets.</p> <p>When $c=0$ the picture is very nice and symmetric, with two simply connected attractive basins, countably many finite loops at the $\pi$-rational angles around the unit circle, and uncountably many infinite loops. I'm curious what the picture looks like for other $c$.</p> <p>So my questions are:</p> <p>What distinguishes the point at infinity from the other points here?</p> <p>What do the loops look like?</p> <p>Are there other attractive basins? For which $c$? What do they look like?</p> <p>Or what book(s) or paper(s) could I look through to get satisfying answers to these questions?</p> http://mathoverflow.net/questions/121843/the-distribution-of-balls-in-a-bean-machine-that-omits-all-the-prime-pegs/121855#121855 Comment by magfrump magfrump 2013-02-15T21:57:50Z 2013-02-15T21:57:50Z I agree, for any fixed $k$ there is (almost certainly) an $n$ such that the bins within $k$ of the edge will get no balls. I have no intuition for how $k$ would grow in $n$ and I guess I imagine that it would remain some kind of normal distribution between the limits.