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May 6 |
answered | Modern Mathematical Achievements Accessible to Undergraduates |
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Apr 30 |
accepted | Fundamental inequality of entropy in random walks |
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Apr 29 |
answered | Fundamental inequality of entropy in random walks |
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Apr 25 |
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An action or two of $SL_2(\Bbb Z)$? Misha, you are not quite right. The action of $SL(2,Z)$ on $R^2∖ 0$ (rather, on the quotient of this space by the central symmetry) has a clear geometrical interpretation. It is isomorphic to the action of $SL(2,Z)$ on the space of horocycles in the hyperbolic plane (i.e., to the extension of the boundary action by the Busemann cocycle). |
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Apr 18 |
accepted | Obtaining conditional probabilities as pushforwards of [0,1] |
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Apr 18 |
answered | Obtaining conditional probabilities as pushforwards of [0,1] |
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Apr 14 |
awarded | ● Fanatic |
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Apr 12 |
accepted | Absolute continuity of probabilities on Polish spaces and open sets. |
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Apr 12 |
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Absolute continuity of probabilities on Polish spaces and open sets. There is a lot of pairwise singular measures with the same support. For instance, look at the distribution of the sum $$ \sum_{k=1}^\infty x_k/2^k \;, $$ where $x_k$ are independent and take values 1 with probability $p$ and 0 with probability $1−p$. The resulting measures $m_p$ are pairwise singular (as it follows, for instance, from the law of large numbers) and have the same support $[0,1]$ (for $p=1/2$ this is the Lebesgue measure). |
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Apr 12 |
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Mean value theorem for harmonic functions on ellipsoid OK - but if you formulate the claim in terms of harmonic measures, then there is no need for any assumptions like that - right? |
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Apr 12 |
answered | Absolute continuity of probabilities on Polish spaces and open sets. |
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Apr 11 |
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Mean value theorem for harmonic functions on ellipsoid Alexandre, why do you impose the centrality condition? |
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Apr 9 |
accepted | The relations between conservative part and conservativity |
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Apr 6 |
answered | $\Delta f \le - \lambda f$ then ${\lambda _1}\left( M \right) \ge \lambda$? |
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Apr 4 |
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Methods for solving two variable recurrence Concerning the variance it's just a simple calculation (since the variance of exponential distributions is explicitly known) - then one has to apply the standard Chebyshev inequality (in the same way as in the proof of the weak law of large numbers). |
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Apr 4 |
revised |
Methods for solving two variable recurrence added 305 characters in body |
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Apr 4 |
revised |
Methods for solving two variable recurrence deleted 287 characters in body |
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Apr 4 |
answered | Methods for solving two variable recurrence |
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Apr 3 |
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The relations between conservative part and conservativity I see. I wouldn't immediately know any type III examples for smooth $\mathbb Z$ actions. The ones I know are either $\mathbb Z$ actions on pretty abstract spaces (e.g., shift on product spaces) or smooth actions of much bigger groups (e.g., boundary actions of Fuchsian or Kleinian groups). |
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Apr 3 |
answered | The relations between conservative part and conservativity |
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Apr 2 |
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Local concentration of measure on Erdos-Rényi graph Almost sure convergence a priori does not make any sense when talking about weak convergence of a sequence of measures. For a.s. convergence one needs a measure in the space of sequences (in your case in the space of sequences of graphs whose size n goes to infinity) - what is the measure with respect to which you want to consider a.s. convergence? |
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Apr 2 |
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Local concentration of measure on Erdos-Rényi graph Don't understand why you would need any concentration here. If you reverse your condition and talk about the probability of "not being a tree", then you have to deal with convergence of non-negative functions to 0. |
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Apr 2 |
answered | Local concentration of measure on Erdos-Rényi graph |
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Mar 30 |
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Finite vertex-transitive graphs that look like infinite vertex-transitive graphs Oh - are you claiming that Schreier graphs are transitive? They are not! |
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Mar 30 |
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Finite vertex-transitive graphs that look like infinite vertex-transitive graphs There is something I don't understand in this argument. Any, say 4-regular graph, can be edge-labelled to become a Schreier graph of the free group $F_2$. Do you claim it's embeddable into a homogeneous graph? |
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Mar 30 |
answered | Finite vertex-transitive graphs that look like infinite vertex-transitive graphs |
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Mar 27 |
answered | Ergodicity for a Probabilistic Cellular Automaton on a finite space |
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Mar 19 |
accepted | Weak convergence in measure for negligible sets. |
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Mar 19 |
answered | Weak convergence in measure for negligible sets. |
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Mar 17 |
accepted | Compactness of sigma-algebra for the $L^1$ metrics |
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Mar 17 |
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Convergence of probability measure and the *-weak convergence ? Sigma-additive probability measures are dense in finitely additive probability measures with respect to the weak topology, whereas tightness is precisely the condition which guarantees that the limit of sigma-additive measures is also a sigma-additive measure. |
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Mar 17 |
revised |
Compactness of sigma-algebra for the $L^1$ metrics deleted 1 characters in body |
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Mar 17 |
answered | Compactness of sigma-algebra for the $L^1$ metrics |
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Mar 17 |
answered | Convergence of probability measure and the *-weak convergence ? |
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Mar 16 |
answered | Equivalence of two definitions of Lyapunov exponents |
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Mar 15 |
accepted | The property of a Markov measure |
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Mar 15 |
answered | The property of a Markov measure |
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Mar 15 |
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The property of a Markov measure Another question: are you assuming that $m$ is shift invariant? |
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Mar 15 |
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The property of a Markov measure What are $a$ and $b$ in your cylinder sets? Are you talking about one-dimensional cylinders determined by first letters? |
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Mar 14 |
answered | Is the first eigenvalue of a parabolic ends of a Riemanian manifold 0? |
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Mar 14 |
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Integral of a harmonic function on a manifold with two non-parabolic ends If only one end is non-parabolic, then you may have no harmonic functions to talk about (I presume you are interested in bounded or at least positive ones). |
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Mar 14 |
accepted | Integral of a harmonic function on a manifold with two non-parabolic ends |
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Mar 14 |
answered | Integral of a harmonic function on a manifold with two non-parabolic ends |
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Mar 1 |
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How does a quasi-isometry affect Poisson or Martin boundaries? @Igor Khavkine - Your comment about comparison of Martin and Busemann compactifications is only valid for symmetric spaces. |
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Mar 1 |
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Spanning subgaph with trivial Poisson boundaries This is a theorem of Kaimanovich-Vershik-Rosenblatt: a group is amenable iff it carries a Liouville random walk. However, the step distribution of this random walk need not be finitely supported (for instance, for higher dimensional lamplighter groups). She proves in that paper that actually for these groups all measures with finite entropy have non-trivial Poisson boundary. |
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Mar 1 |
answered | Spanning subgaph with trivial Poisson boundaries |
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Feb 10 |
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Invariant measures for Cellular automata Thank you, I see now - looks like an interesting question. Is there any known a priory reason why all known $F$-invariant measures happen to be shift invariant? |
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Feb 8 |
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Invariant measures for Cellular automata Could you explain in more detail how the local rules are defined in terms of the matrix $A$? |
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Feb 4 |
accepted | Metrization of weak convergence of signed measures |
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Jan 31 |
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Is a typical path of a Brownian motion on a torus equidistributed? The Riemannian volume is stationary for the Brownian motion on any Riemannian manifold. For a compact manifold (more generally, for a finite volume one) this is the only stationary measure (which follows, for instance, from the fact that the transition probabilities are absolutely continuous with respect to the Riemannian volume), in particular, it is ergodic. Equidistribution of a.e. sample path is then essentially just the usual ergodic theorem for the Brownian motion. |
