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Jun
25
comment The expected outcome of a random game of chess?
Two suggestions though, the actual counts used by the simulations and a two sided p-value. Since you are much, much closer to 1/2, it be interesting to see what the statistics say. "perfectly consistent" is not quite precise enough!
Jun
25
comment The expected outcome of a random game of chess?
Nice catch, chalk this one up to the power of peer review! To be honest, this library was simply the easiest to install and get up and running, I imagine that a C library may be much faster to run long simulations. Also, mate is possible with two moves.
Jun
25
comment The expected outcome of a random game of chess?
@nbubis I'm going to let Winther take the credit for finding the flaw in the code above. I happy enough, my initial answer, flawed as it was, was enough to spur a better investigation. I'll update my answer and point to his, IMHO he answers the question correct and should get the check.
Apr
13
comment Forbidden toroidal minors
Wow, that is a much larger set than I expected! Do you know, off-hand, if there are simple ways to test if a graph is toroidal? If not, I can ask in a new question.
Apr
2
comment Fourier decomposition of the Mandelbrot set
Thanks for the article!
Feb
26
comment Graph diameter of a single vertex?
@Wayne so the question boils down to, is the trivial path considered a path in the calculation of the diameter?
Feb
11
comment Expected number of clusters on chessboard
@MarnixKlooster For diagonals, you can have at most N=61, as you need the squares (0,1),(1,0),(1,1) to be filled (or any other corner).
Feb
11
comment Expected number of clusters on chessboard
@MarnixKlooster According to the def. by OP, you can have a two independent clusters with even 62 squares filled. Consider the situation where all squares are filled on except for (0,1),(1,0). The single conner square and the remaining large 61-size cluster are not connected "sideways". Now if you count diagonals as a connection, the story is different.
Feb
6
comment Are there any infinites not from a powerset of the natural numbers?
Thank you for the informative answer, I think it clears up a few mistakes in both my nomenclature and my understanding of the ordinals. I was trying to lookup the Hebrew "B" transfinite number and couldn't simply google for the letter: to help those who follow, they are called Beth numbers.
Oct
16
comment Trying to work out the probabilities of a dice game I used to play
While the expected number of turns is a useful number, sometimes the distribution of games can be helpful, especially when your friends might give up playing the games after 50 rolls!
Oct
8
comment Determine if a set of points on a sphere come from a uniform distribution?
@JohnMoeller For some reason I thought you were referring to something different. If the $L_2$ norm just the standard Euclidean distance, and the other $L_p$ norms are just a different power in the summation and the radical, how do these relate a uniform distribution across a sphere?
Oct
7
comment Determine if a set of points on a sphere come from a uniform distribution?
@JohnMoeller I figured as much since multi-dimensional KS tests seem to be difficult (and often called into question, asaip.psu.edu/Articles/beware-the-kolmogorov-smirnov-test). I'll look into the Wasserstein metric, but I can't search for the $L_p$ distance if I don't know its name as a word, not a symbol.
Sep
29
comment Are the primes compressible?
Ah I think I see now, despite the representation (e.g. the variants purposed by Ted) a compression algorithm that relies on a block scheme (like zlib) will naturally fall into a length commensurate with the block. Nice insight - thanks!
Sep
29
comment Are the primes compressible?
@Ted I've reran the results with the more efficient representation you've suggested. In this case we still have the same ~80% ratio between the prime sequence and the permuted sequence, but now both sequences compress far better than the random sequence (presumably due to the fact that there are more 0's than 1's). So I really appreciate the effort, but neither explanation describes why the prime sequence compresses over a permutation of the same sequence.
Sep
29
comment Are the primes compressible?
Just tested this idea by removing the last bit of each prime numbers binary representation and the ratio is still ~.80, so this doesn't quite answer the question.
Sep
24
comment Distribution of primes remainders
@fretty If you want to collect these comments into an answer I'll accept and upvote it. I've learned a lot reading through - thanks!
Sep
23
comment Distribution of primes remainders
@fretty Your comment seems to provide more information than this answer. Could you expand upon it and (now that I know what I'm looking for) and describe how it fits in with Chebyshev's bias? en.wikipedia.org/wiki/Chebyshev%27s_bias
Aug
27
comment Small angular displacements with a quaternion representation
This is essentially the same idea voiced by @RahulNarain, correct? If so, I'll accept this so we can mark it closed. If not, please let me know the difference.
Aug
23
comment Small angular displacements with a quaternion representation
@RahulNarain That's a good idea. If you write it as an answer I'll accept it. Just to be clear, given your $q_R(\Delta)$ sampled from the hyperspherical cap, would the distribution of new perturb quaternions simply be $q q_R(\Delta)$?
Aug
22
comment Small angular displacements with a quaternion representation
@RahulNarain The end goal is a rotation step in a Monte-Carlo simulation, so the emphasis would be on the ease of implementation. I worry however, that my method above wouldn't sample evenly around my starting $q$. In regards to your other comment, I guess my question could be reformulated to, "how does one generate a distribution around the unit (did you mean identity?) quaternion?"