bio | website | math.uvic.ca/faculty/aquas |
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location | University of Victoria | |
age | 46 | |
visits | member for | 4 years, 1 month |
seen | 6 hours ago | |
stats | profile views | 3,701 |
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...
Dec 4 |
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Does Brownian motion immediately visit both sides of a Jordan curve?
@JoelDavidHamkins: So here's an explanation in case where the curve is a straight line. Without loss of generality, you can assume the curve is $x=0$. Now it's known that 2-dimensional Brownian motion is the same as independent Brownian motions in each of the coordinates. Also it's known that 1-dimensional Brownian motion started from 0 almost surely takes positive and negative values in any time interval $[0,\epsilon]$. |
Dec 2 |
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Rational dynamical system with nonnegative paramaters
Three comments: 1) You should wait a bit longer before cross-posting; 2) Why do you expect that this has a positive limit? 3) What is the relationship between the second difference equation and the first? |
Dec 2 |
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Minimum distance in unit cube
I think the question is trivial. The weight (sum of the entries) of any point in any such hyperplane is at least 1. The point nearest the origin of weight 1 is $(1/n,1/n,\ldots,1/n)$. The distance is $n^{-1/2}$. Done. |
Nov 30 |
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SIRS Stability Analysis
I think this question is very hard in general to do analytically. What you can do is use a computer to estimate the Lyapunov exponents along the orbit. You're computing the (logarithmic) rates at which a perturbation will grow in the various directions. If these exponents all appear to be non-positive, this is reasonable evidence for the stability of the system. |
Nov 30 |
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When does a Catalan number equal a Fibonacci number?
@Will: I don't think that a coincidence of Catalan and Fibonacci numbers would give a better rational approximation to $\log 4/\log\phi$ than occurs `in nature'. You know you can find $m$ and $n$ such that $|m\log 4-n\log\phi|<C/m$ by a pigeonhole argument. Since the $m$th Catalan number is about $4^m/(m+1)$, a coincidence would give $|m\log 4-n\log\phi|\lesssim\log m$. |
Nov 29 |
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Perron-Frobenius theory for reducible matrices
Gantmacher's book The Theory of Matrices has a section on reducible non-negative matrices. |
Nov 26 |
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A game of stones
Just a comment: I'm not sure if this is an old chestnut, but there's a related puzzle: Take an infinite 2d grid in the 1st quadrant with a stone at the origin. The moves are that you can remove a stone and replace it with two stones: one 1 square up; the other 1 square right of the original stone; same definition of overcrowded. Question: Is there a sequence of moves leaving the board in an uncrowded configuration with the bottom left 10x10 cells empty? |
Nov 25 |
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A game of stones
So isn't it sufficient to prove that you can do this starting from a single stone? |
Nov 21 |
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The upper and lower bound of the projection of a subshift of finite type
@Nikita: Yes I am sure. Write $f(a_1,\ldots,a_{n-1})$ for the maximum value of $\sum_{k\ge n}x_kq^{-k}$ over all sequences $(x_k)_{k=n}^\infty$ that are compatible with $a_1,\ldots,a_{n-1}$. The maximum occurs somewhere (maybe with ties). Define $G(a_1,\ldots,a_{n-1})$ to be the word $a_2\ldots a_{n-1}x_n$ where $x_n$ is the first letter of one of the infinite words achieving the maximum. The point is that $f(a_1,\ldots,a_{n-1})=x_nq^{-n}+(1/q)f(G(a_1,\ldots,a_{n-1}))$. This makes use of the specific form of the `projection'. |
Nov 20 |
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Infected square
@NoamD.Elkies: Good comment for the well-informed (or those with access to google like me) |
Nov 19 |
answered | The upper and lower bound of the projection of a subshift of finite type |
Nov 14 |
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Determinant of matrix from set {-1, 1}
@Noam: It seems yesterday's post was deleted. That post asked what's the probability that an 11x11 $\pm 1$ matrix has determinant above 4000. It had an answer from Robert Israel that is very close to the calculation that Neil Strickland re-did today since the previous post was deleted. |
Nov 14 |
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Infected square
The magic words are bootstrap percolation. |
Nov 14 |
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Determinant of matrix from set {-1, 1}
This question appears to be off-topic because it is essentially the same as your question from yesterday. |
Nov 13 |
answered | $P_{x}(T_{A}<\infty)<P_{x}(T_{B}<\infty)$ imply $Cap_{N}(A)<Cap_{N}(B)$, where $Cap_{N}$ is Newtonian capacity |
Nov 9 |
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Hausdorff densities
What makes you think the result is true? |
Nov 9 |
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Hausdorff densities
What if the map is constant? |
Nov 8 |
awarded | Yearling |
Nov 6 |
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Earth mover/Wasserstein distance between a pdf and an empirical distribution
Looking at the paper you cite, it seems that that paper cites an earlier work for the $W_1$ distance, which I think is the "earth-mover distance": [11] V. Dobri and J. Yukich. Asymptotics for transportation cost in high dimensions. Journal of Theoretical Probability, 8:97-118, 1995. |
Nov 5 |
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How to solve a couple of ODEs
For english speakers EDP = PDE |