9,645 reputation
12155
bio website math.uvic.ca/faculty/aquas
location University of Victoria
age 47
visits member for 4 years, 6 months
seen 14 hours ago
The pic is the phase portrait of a simple "piecewise isometry". Define a map by sliding the two halves of the plane: the top half to the right; the lower half to the left and then rotate. Around each periodic point of the map there's a "periodic island". These are what are in the image...

20h
comment Maximizing entropy under constraints
I think the first is obvious using the relationship $H(\mathcal P^n)\le nH(\mathcal P)$ with equality iff $T^{-i}\mathcal P$ are independent for $i=0,\ldots,n-1$. Some very closely related work is due to Christian Wolf and Tamara Kucherenko.
1d
comment Reference for a lemma on the asymptotic upper density of special sets with large gaps and intervals
So I don't see why this would take 1/2 page: If $a_n<N\le a_{n+1}$, then $|X\cap [1,N]|/N \le |X\cap [1,b_n]|/b_n$ so the $\limsup$ is attained along the sequence of $b_n$'s. And $b_n-a_n \le |X\cap [1,b_n]|\le (b_n-a_n)+b_{n-1}$. Dividing by $b_n$ and taking the limit gives the result.
2d
comment The most number of points that realize only $k$ distinct distances
Doesn't the pentagon give you $f_3(2)\ge 7$ if you add a point below also?
2d
comment Link Between Birkhoff Ergodic Theorem and Strong LLN for Harris Recurrent Markov chain
So sure. There's a direct connection. If you have an invariant probability distribution for the Markov chain, then you can build from it an invariant measure (in the ergodic theory sense) on the space of paths in the space $S^\mathbb Z$. The Birkhoff ergodic theorem can then be applied to this measure (with the transformation being the shift transformation).
May
23
comment Link Between Birkhoff Ergodic Theorem and Strong LLN for Harris Recurrent Markov chain
When you stationary, do you mean that there's an invariant distribution (not necessarily a finite measure)? If so, you can build an invariant measure for the Markov chain process and apply the ergodic theorem. This is mainly useful if there is a finite invariant distribution. You can see this done for the finite state case in Walters' book for example.
May
22
comment How did Cole factor $2^{67}-1$ in 1903
Highly ignorant question: I see that if $2^{67}-1$ is written as $pq$, then $pq\equiv 1\pmod {67}$. Why are $p$ and $q$ individually congruent to 1?
May
21
comment Are these inequalities for primes equivalent?
I should have said $a-b\ge 2$, so that $ab\ge 2p_{n+1}$.
May
21
comment Are these inequalities for primes equivalent?
If you write $p_{n+2}=p_{n+1}+a$ and $p_{n+2}=p_{n+1}-b$, the product is $p_{n+1}^2+(a-b)p_{n+1}-ab$. For $L$ to succeed, but $Q$ to fail, you need $a>b$ hence $a-b\ge 1$ and so $ab\ge p_{n+1}$. This is a question about gaps between primes. Terry Tao's blog terrytao.wordpress.com/2014/08/21/… indicates that $a,b\ll p_{n+1}^{.525}$, but that this is considered a very weak upper bound. For a violation of your inequality, you would need 2 consecutive gaps almost as large as the maximum. This can probably be ruled out by some analytic # theory.
May
20
revised Prescribed values for the uniform density
added 3 characters in body
May
18
comment Counting ways to Arrange Variable Sized Objects into Fixed Number of Spaces
I think @PerAlexandersson is simply trying to understand your question (which is opaque to me also). Are you saying that the spaces are arranged in a line; and that each object takes up a different amount of space? (in that case the answer is trivially $i!$)
May
16
comment Extreme points of convex hull of Minkowski sum
It's true that $P+Q=\text{conv}(\{a_i+b_j\})$, but maybe some of those points are not extreme points?
May
12
comment What is the distribution of the maximum nearest-neighbor distance of a point cloud sampled from a solid body like?
Call the $i$th gap $X_i$. Set $Y_i=\min(X_i,X_{i+1})$. Then $Y_i$ is independent of $Y_j$ whenever $|i-j|>1$. Given $N/2$ independent $\text{Exp}(2N)$ random variables, the maximum is approximately $\log N/(2N)$.
May
12
comment What is the distribution of the maximum nearest-neighbor distance of a point cloud sampled from a solid body like?
I think that if $N$ points are uniformly chosen in $[0,1]$, the maximum separation between points is something like $\log N/(2N)$. A closely related problem is if you look at a Poisson process with rate $N$. Now instead of having exactly $N$ points, you have a Poisson random variable with expectation $N$. Conditioned on the number of points, they are uniformly distributed. The gaps are exponential random variables with parameter $N$. The distance to the nearest neighbour is the minimum of two Exp(N) r.v.s, i.e. an Exp(2$N$) r.v.
May
12
comment What is the distribution of the maximum nearest-neighbor distance of a point cloud sampled from a solid body like?
If the body is very spiky, then $D$ will scale differently (I think) than if it's a ball.
May
9
comment What are the applications of Voronoi diagrams in pure mathematics?
Here's a paper of mine that used them: math.uvic.ca/faculty/aquas/papers/paper15.pdf
May
7
comment Degree of permutation of hypercube
Huge, no doubt. But I don't know how huge.
May
7
comment Degree of permutation of hypercube
If $S_0$ is the intersection of $\{0,1\}^n$ with a subspace (e.g. $x_1=0$), then a linear transformation would have to map $S_0$ into a subspace of $\mathbb R^n$. A typical subset of $\{0,1\}^n$ with $2^{n-1}$ elements is not contained in any subspace, so in general there is no linear transformation.
May
4
comment Central limit theorem for biased random walk
Did you want $X_n-X_{n-1}=\pm 1$. Your $X_n$ looks like it will have $X_n\approx n$ and shouldn't have Gaussian behaviour.
May
2
comment existence of multiplicity of roots
So this question isn't at the right level for this site. You should probably try math.stackexchange.com instead.
May
1
comment Horizontal vs Vertical sides Exit from a Rectangle for simple symmetric Random Walk on $\textbf{Z}^{2}$
Arguments of this type (showing that this probability is bounded away from 0 if the ratio of $M$ and $N$ is bounded) are used by Bollobas and Riordan in their book on percolation theory.