bio  website  math.uvic.ca/faculty/aquas 

location  University of Victoria  
age  47  
visits  member for  4 years, 6 months 
seen  14 hours ago  
stats  profile views  3,946 
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

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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 $i=0,\ldots,n1$. Some very closely related work is due to Christian Wolf and Tamara Kucherenko. 
1d

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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 $b_na_n \le X\cap [1,b_n]\le (b_na_n)+b_{n1}$. Dividing by $b_n$ and taking the limit gives the result. 
2d

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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

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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 
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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 
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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 
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Are these inequalities for primes equivalent?
I should have said $ab\ge 2$, so that $ab\ge 2p_{n+1}$. 
May 21 
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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 $p_{n+1}^2+(ab)p_{n+1}ab$. For $L$ to succeed, but $Q$ to fail, you need $a>b$ hence $ab\ge 1$ 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 
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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 
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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 
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What is the distribution of the maximum nearestneighbor 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 $ij>1$. Given $N/2$ independent $\text{Exp}(2N)$ random variables, the maximum is approximately $\log N/(2N)$. 
May 12 
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What is the distribution of the maximum nearestneighbor 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 
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What is the distribution of the maximum nearestneighbor 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 
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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 
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Degree of permutation of hypercube
Huge, no doubt. But I don't know how huge. 
May 7 
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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 $2^{n1}$ elements is not contained in any subspace, so in general there is no linear transformation. 
May 4 
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Central limit theorem for biased random walk
Did you want $X_nX_{n1}=\pm 1$. Your $X_n$ looks like it will have $X_n\approx n$ and shouldn't have Gaussian behaviour. 
May 2 
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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 
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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. 