Anthony Quas
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Registered User
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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...
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May 8 |
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Discrete disjoint covering of integer lattices I did something related in $\mathbb R^d$ and found a family of tilings there (see <a href="mathoverflow.net/questions/77952/… question</a>). It turned out that the tilings had been previously studied under the name <i>notched cube tilings</i>. It's quite plausible that some version of Stein's ideas can be applied in your situation. |
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May 8 |
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Is it difficult to prove that nature is chaotic? The proofs of chaotic behaviour tend to rely on hyperbolic behaviour (or at least non-uniformly hyperbolic behaviour). Proving that this holds in many real systems (or even in lots of toy models) is extremely hard / apparently beyond the reach of current technology. |
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May 7 |
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Is the sequence $\{\Omega(n)\alpha\}$ uniformly distributed in $[0,1)$? If the Erdös-Kac theorem existed in a more refined version, saying that if you look at the distribution of $\Omega(n)$ over a range of $n$'s, then $\Omega(n)−\log\log n$ is close in total variation distance to a normal random variable, then the answer would be yes. |
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May 3 |
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Longest run of heads Yes. The longest run of heads is tightly concentrated. That means that almost all of the space has between $(1-epsilon)log n$ heads and $(1+epsilon)log n$ heads. All you have to do is pair the very small part of the space where $R_n(x)<(1-epsilon)\log n$ with an arbitrary $y$ in the part where $R_n(y)>= (1-\epsilon)\log n$. When this is done, pair the remaining stuff arbitrarily. |
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May 3 |
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Variant of an Expander graph: Probability that S random points cast a shadow/projection of size at most S/2 on each face of a cube. Although this probability does decay exponentially in $k$ (as long as the large deviation methods are valid, which I'm sure they are), the exponential decay is slower than the increase in the number of choices of subsets. This means that something more subtle is needed to finish off the argument. BTW: Even the 2D version of this problem seems non-trivial... |
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May 3 |
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Variant of an Expander graph: Probability that S random points cast a shadow/projection of size at most S/2 on each face of a cube. @codingTheorist: I've played with this some more. I'm sure you're right that it decays exponentially in $k$. I have tried to prove this making some assumptions that I have not justified (the applicability of large deviations techniques) together with some very crude bounds (union bounds), and have so far been unsuccessful. My idea was to pick subsets of each face of size $\alpha k$, and then to estimate the probability that for a fixed such triple, there were more than $2\alpha k$ points in the random set that project to the chosen subsets. |
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May 1 |
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Variant of an Expander graph: Probability that S random points cast a shadow/projection of size at most S/2 on each face of a cube. Hmmm... it looks like I didn't read the question as well as I should have... I hadn't thought about the 3 different projections part of the story. To get the kind of independence you need for large deviations (at least when you're looking at projection onto one face), I think you can compare the ball dropping process with putting a Poisson number of balls (actually independently) into each bin. I see about the $2/3$ now. |
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May 1 |
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Variant of an Expander graph: Probability that S random points cast a shadow/projection of size at most S/2 on each face of a cube. added 15 characters in body |
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May 1 |
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Variant of an Expander graph: Probability that S random points cast a shadow/projection of size at most S/2 on each face of a cube. added 496 characters in body |
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May 1 |
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Variant of an Expander graph: Probability that S random points cast a shadow/projection of size at most S/2 on each face of a cube. Pairwise independence is not enough; But there is much more independence than that here. This kind of argument shows up all over the place in The Probabilistic Method. |
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May 1 |
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Variant of an Expander graph: Probability that S random points cast a shadow/projection of size at most S/2 on each face of a cube. I don't think they don't need to be $k$-fold independent for this kind of argument to work. |
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May 1 |
answered | Variant of an Expander graph: Probability that S random points cast a shadow/projection of size at most S/2 on each face of a cube. |
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Apr 29 |
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Fundamental inequality of entropy in random walks $|g|$=word length, right. So that $L_{n,P}$ is the expected distance from the origin in terms of the generators after $n$ steps. Since it grows sublinearly, $l_P$ exists. |
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Apr 29 |
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How do we recognize a Markov partition? added 4 characters in body |
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Apr 29 |
answered | How do we recognize a Markov partition? |
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Apr 28 |
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order of convergence of the conditional entropy I made the substitution... |
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Apr 28 |
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order of convergence of the conditional entropy correction H(Y|X) -> H(X|Y) |
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Apr 28 |
accepted | order of convergence of the conditional entropy |
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Apr 27 |
answered | order of convergence of the conditional entropy |
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Apr 26 |
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Infinitely many planets on a line, with Newtonian gravity I think that as long as each planet is at most $\delta$ from its nearest integer, the total force on each planet is $O(\delta)$. This can be used to prove rigorously that there's a positive $\tau>0$ before any collision can occur. |
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Apr 26 |
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Infinitely many planets on a line, with Newtonian gravity Of course if you really live in a 1D world, gravitational force presumably doesn't decay at all? |
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Apr 26 |
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Infinitely many planets on a line, with Newtonian gravity I read recently about a very similar problem that appeared in a 1949 letter from Ulam to von Neumann. (In that case the particles started at points of $\mathbb Z$ with each node being occupied with probability 1/2). He showed(?) that something analogous to the universe happens: nearby groups of particles come together; and then those "solar systems" form galaxies etc. |
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Apr 26 |
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Integer dynamics hitting infinitely many primes **Pedantic comment alert**: Presumably this is not exactly what you meant: $f(x)=2x-7$ satisfies $f^n(7)$ is prime for all $n$. |
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Apr 25 |
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Expected minimum Hamming distance with overlaps I doubt that you can give an explicit function. However, I think you can get a very good approximation. For each $\ell$, I would compute $p_\ell$, the probability that a random word has distance $\ell$ from $A$ (i.e. $p_\ell=\binom{n}{\ell}2^{-n}$). You would expect $\mathbb E(X_k)\approx \ell$ where $\ell$ satisfies $p_{\ell}\le 1/k<p_{\ell+1}$. |
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Apr 25 |
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variational characterization of the average of an $L^p$ function Have you tried any examples? |
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Apr 24 |
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Entropy of edit distance This is also known in ergodic theory as the $\bar f$-distance. There are known linear lower bounds on the expected $\bar f$-distance between random strings of length $n$, so treating the $\bar f$ distance as a random variable, it will presumably be a unimodal distribution whose standard deviation is a power of $n$. Hence the entropy of this random variable will be approximately of the form $c\log n$. |
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Apr 21 |
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Sierpinski Triangle and the Chaos Game I think @Douglas Zare's answer is half of the answer. It explains why any limit points belong to the Sierpinksi triangle. You presumably also want to say that every point on the Sierpinksi triangle is the limit point of some subsequence of your $x_n$'s? In this case, given any point of the triangle, it belongs to some level $k$ sub-triangle. Code that sub-triangle by the sequence of $k$ vertices that you have to choose to fall into it. Then with probability 1, you choose that sequence of $k$ vertices infinitely many times, so you fall in the sub-triangle inf many times. |
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Apr 21 |
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Sierpinski Triangle and the Chaos Game I don't think this answers the question. This assumes you're applying the operation $S\mapsto f_1(S)\cup f_2(S)\cup f_3(S)$. The OP was asking about the chaos game, which is where you choose your favourite $x_0$ and define $x_{k+1}=f_{N_{k+1}}(x_k)$, where $N_{k+1}$ takes values in the set $\lbrace 1,2,3\\}$ with equal probability. |
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Apr 21 |
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Sierpinski Triangle and the Chaos Game For almost any sequence of randomizations, the set of limit points is the full S. triangle. BUT: This Q doesn't belong here - try math.stackexchange.com |
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Apr 20 |
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Stationary distribution of a countable state Markov chain How about the Markov chain on $\mathbb Z$ defined by $\mathbb P(X_{n+1}=k+1|X_n=k)=1$? |
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Apr 18 |
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A question from complex analysis mathoverflow.net/howtoask |
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Apr 17 |
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Obtaining conditional probabilities as pushforwards of [0,1] This sounds a lot like something that's done in the category of "Lebesgue spaces", a Lebesgue space being defined axiomatically by the existence of a sequence of finite partitions that separates points of the space. In this category, I think the answer is positive. Details can be found in the first chapter of the book by Rudolph: Fundamentals of Measurable Dynamics |
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Apr 14 |
awarded | ● Good Answer |
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Apr 4 |
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American put option pricing by “binomial trees” @George: Thanks very much indeed for the extremely helpful comments. Is there any chance you can package the most relevant ones into an answer so that it can be recorded that the original question is resolved? |
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Apr 4 |
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American put option pricing by “binomial trees” My last(?) question on this then: is there any good reason why we should expect the ball bounces to have equal and opposite slopes? Really appreciating the insight here... |
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Apr 4 |
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American put option pricing by “binomial trees” @George - nice. Thanks for pointing that out. This probably will lead to an answer to the question that I originally posed: how fast does this converge to the true option price (I had numerically observed the error seemed to be $O(1/n)$, but I think this more or less explains it: the gap between $su^id^{n-i}$ and $su^{i+1}d^{n-i-1}$ is of $O(1/n)$. |
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Apr 4 |
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American put option pricing by “binomial trees” Thanks. This gives a good explanation of the appearance of parity. Still no idea why the curves look like bouncing balls - I would have predicted some kind of monotone convergence to the limit... Is there some number theoretic property of the places where the balls "bounce"? etc... |
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Apr 4 |
asked | American put option pricing by “binomial trees” |
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Apr 3 |
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The ratio of one digits and all digits in the binary expansions of the square numbers This is probably harder than the question of whether every sufficiently large power of 2 contains a 7 in its decimal expansion, which is a fairly well-known open question. |
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Mar 29 |
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compute the waiting time for a given pattern with Kac’s lemma This question appears in David Williams' book Probability with Martingales (under the title "The Abracadabra Problem") - the question being how long before a monkey on a typewriter spells out the word abracadabra (the answer is not $26^11$). |
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Mar 27 |
answered | Chain Recurrent Set of a Isometry |
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Mar 22 |
accepted | Examples of transformations which are weak-mixing but not strong-mixing |
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Mar 4 |
answered | Probability that a random distance function is metric |
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Mar 1 |
accepted | Natural density of a set of positive integers not in certain congruence classes |
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Mar 1 |
answered | Natural density of a set of positive integers not in certain congruence classes |
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Feb 28 |
accepted | Least common period of a finite sum of exponentials |
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Feb 4 |
answered | First hit time in a graph setting |
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Feb 3 |
awarded | ● Nice Answer |
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Jan 25 |
answered | Sum of binomial coefficient |
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Jan 16 |
awarded | ● Enlightened |
