User zarathustra - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T22:20:35Z http://mathoverflow.net/feeds/user/3375 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/85844/large-geodesically-convex-subsets-of-tori/85857#85857 Answer by Zarathustra for Large geodesically convex subsets of tori Zarathustra 2012-01-16T23:09:01Z 2012-01-16T23:22:24Z <p>Let me write down the steps.</p> <ol> <li>Consider the case d=2, the generalization is straightforward.</li> <li>There is an open ball B that doesn't intersect E.</li> <li>Consider two families of geodesic circles F_1 and F_2. F_1 has slope 1/p and F_2 has slope 1-1/p. Number p is chosen in such a way that each circle from F_1 nd F_2 intersects B.</li> <li>Claim: $Leb(E\cap F_1(x))\le 1/2 length(F_1(x))$ OR $Leb(E\cap F_2(x))\le 1/2 length(F_1(x))$ for each x.</li> <li>Proof: $E\cap F_1(x)$ is a proper union of open intervals that are separated by gaps of length at least $\sqrt{p^2+1}/2p$. So it remains to show that there are at least $p$ gaps. If there are less than $p$ gaps then there is an interval from $E\cap F_1(x)$ of length greater than $\sqrt{p^2+1}/2p$. It follows that one can find a simple closed curve C_1 in E which is C^0 close to a horizontal generator. Assume that in the same way we also can find C_2 in E which is C^0 close to the vertical generator. Then one can easily see that E is the torus and the claim follows.</li> <li>Apply Fubini.</li> </ol> http://mathoverflow.net/questions/28647/is-it-possible-to-partition-mathbb-r3-into-unit-circles Is it possible to partition $\mathbb R^3$ into unit circles? Zarathustra 2010-06-18T17:39:20Z 2010-11-27T14:56:50Z <p>Is it possible to partition $\mathbb R^3$ into unit circles?</p> http://mathoverflow.net/questions/36312/getting-rid-of-exceptional-fibers-by-passing-to-finite-covers Getting rid of exceptional fibers by passing to finite covers? Zarathustra 2010-08-21T18:53:47Z 2010-08-21T20:03:40Z <p>Consider a Seifert fiber space. Is it always possible to find a finite cover that is a circle bundle and the preimage of any fiber is a finite union of circles?</p> http://mathoverflow.net/questions/34188/distribution-of-fractional-parts-of-n3-2 Distribution of fractional parts of n^{3/2} Zarathustra 2010-08-02T01:23:10Z 2010-08-02T20:23:10Z <p>What can be said about the limiting distribution of the sequence of fractional parts of $\{n^{a},n>0\}$ for $a\in(1,2)$. I ran a computer experiment for $n\sqrt{n}$ and it looks like uniformly distributed. Is there a simple proof?</p> http://mathoverflow.net/questions/32656/status-of-hilbert-smith-conjecture-and-h-s-conjecture-for-holder-actions Status of Hilbert-Smith conjecture and H-S conjecture for Hölder actions Zarathustra 2010-07-20T17:09:31Z 2010-07-20T20:06:18Z <p>The Hilbert-Smith conjecture states that </p> <blockquote> <p>If $G$ is a locally compact group which acts effectively on a connected manifold as a topological transformation group then is $G$ a Lie group.</p> </blockquote> <p>It was established for actions by diffeomorphisms by Bochner and Montgomery. Later on it was also established for (compact?) actions by Lipschitz homeomorphisms (Repovs and Shchepin) and Hölder actions with very large exponent (>dim M/ dim M+2). </p> <p>I am interested if the conjecture holds for Hölder actions (with small exponents). Is it plausible these arguments can be pushed to get the conjecture for Hölder actions? Or there is a fundamental obstruction?</p> <p>Also, there is a 2001 preprint "A Proof of the Hilbert-Smith Conjecture" on arxiv that claims the full conjecture. I assume it's wrong as it wasn't published, but a comment from an expert would be highly appreciated.</p> http://mathoverflow.net/questions/25541/is-geometric-realization-of-the-total-singular-complex-of-a-space-homotopy-equiva Is geometric realization of the total singular complex of a space homotopy equivalent to the space? Zarathustra 2010-05-22T00:24:16Z 2010-05-22T01:41:53Z <p>Let $X$ be a topological space and let $|Sing(X)|$ be the geometric realization of the total singular complex of $X$. </p> <p>Then $|Sing(X)|$ is a CW complex with one cell for each non-degenerate singular simplex. There's a natural map $f:|Sing(X)|\to X$ and there's a theorem that says that $f$ is a weak homotopy equivalence. That is, $f$ induces isomorphisms of homotopy groups.</p> <p>Then it seems that Whitehead theorem applies and gives that $f$ is homotopy equivalence as long as $X$ is homotopy equivalent to a CW complex (i.e. $X$ is m-cofibrant). Is that correct?</p> <p>Is there an example when $f$ is not homotopy equivalence? Any examples that come up in "real life"?</p> http://mathoverflow.net/questions/23305/how-did-gauss-discover-the-invariant-density-for-the-gauss-map How did Gauss discover the invariant density for the Gauss map? Zarathustra 2010-05-03T01:09:49Z 2010-05-03T13:39:55Z <p>The Gauss map is defined on $(0,1)$ by the formula $$f(x)=\frac1x-\Big\lfloor\frac1x\Big\rfloor$$ Then the density $$\rho(x)=\frac{1}{\log2(1+x)}$$ is $f$-invariant.</p> <p>It appeared in Gauss' diary. Gauss didn't indicate the way he had found the density. Checking invariance is straightforward.</p> <p>Is there a simple (short) way to come up with this density function?</p> http://mathoverflow.net/questions/23048/what-are-the-tricks-for-computing-estimating-gromov-hausdorff-distance What are the tricks for computing\estimating Gromov-Hausdorff distance? Zarathustra 2010-04-29T22:25:08Z 2010-05-01T15:27:36Z <p>Gromov-Hausdorff distance between two compact manifolds measures how far away the manifolds are from being isometric. (Wikipedia: <a href="http://en.wikipedia.org/wiki/Gromov%E2%80%93Hausdorff_convergence" rel="nofollow">http://en.wikipedia.org/wiki/Gromov%E2%80%93Hausdorff_convergence</a>) In many cases it is possible to do coarse estimates and conclude that a sequence of manifolds converges or diverges.</p> <p>How does one usually go about calculating GH distance precisely?</p> <p>Example: Take two spheres of different radii $r$ and $R$ with intrinsic (i.e the distance between two points is the length of the arc of a great circle that connects them) metrics obtained from standard embeddings into $\Bbb R^n$. What is the GH distance between them? </p> http://mathoverflow.net/questions/12889/lifting-a-homeomorphism-always-possible Lifting a homeomorphism, always possible? Zarathustra 2010-01-25T02:42:32Z 2010-04-06T21:30:06Z <p>Let $h:M\to M$ be a homeomorphism of a compact manifold. Let $p:\tilde M\to M$ be a covering. 1) Is it always possible to lift $h$ to $H:\tilde M\to \tilde M$ so that everything fits into the commutative diagram? 2) Given such a diagram assume additionally that p is a self-covering. Is it true that $H$ is necessarily homotopic to $h$?</p> <p>Thanks, Z.</p> <p>1/I think I can see that the answer to the second question is "no". Any additional assumptions that would make it into a "yes"?</p> <p>2/A reference to the proof of the statement in Ben's second paragraph is needed.</p> http://mathoverflow.net/questions/12266/frobenius-theorem-for-subbundle-of-low-regularity Frobenius Theorem for subbundle of low regularity? Zarathustra 2010-01-19T02:13:19Z 2010-03-17T08:33:02Z <p>Frobenius Theorem says that a subbundle $E$ of the tangent bundle $TM$ of a manifold $M$ is tangent to a foliation if and only if for any two vector fields $X, Y \subset E$ the bracket $[X,Y]\subset E$. Bracket is a second order operator, hence subbundle $E$ needs to be $C^2$.</p> <p>Are there any generalizations for subbundle which is $C^1$, $C^{1+smth}$?</p> <p>Thank you, Z.</p> http://mathoverflow.net/questions/111266/simultaneous-jordanization/111274#111274 Comment by Zarathustra Zarathustra 2012-11-02T18:53:24Z 2012-11-02T18:53:24Z Now you can cite Alexandre Eremenko (mathoverflow.net/users/25510), Simultaneous Jordanization, <a href="http://mathoverflow.net/questions/111274" rel="nofollow">mathoverflow.net/questions/111274</a> (version: 2012-11-02) http://mathoverflow.net/questions/85844/large-geodesically-convex-subsets-of-tori Comment by Zarathustra Zarathustra 2012-01-16T21:51:49Z 2012-01-16T21:51:49Z Just take a small ball that is not in E then choose a small rational slope so that after we partition the torus into the circles each circle would intersect the ball... http://mathoverflow.net/questions/85844/large-geodesically-convex-subsets-of-tori Comment by Zarathustra Zarathustra 2012-01-16T21:31:03Z 2012-01-16T21:31:03Z Your question is obvious for the circle. Partition the torus into geodesic circles and apply fubini. It seems to work... http://mathoverflow.net/questions/84749/transitivity-of-a-flow-and-its-time-1-map Comment by Zarathustra Zarathustra 2012-01-02T18:48:37Z 2012-01-02T18:48:37Z You can take $c=\sqrt 2$ and your time one map will be transitive. http://mathoverflow.net/questions/71803/billiard-dynamics-under-gravity Comment by Zarathustra Zarathustra 2011-08-01T18:44:24Z 2011-08-01T18:44:24Z Not very relevant, but I recall that Chernov and Dolgopyat have results on behaviour of a billiard in the plane with convex scatterers (finite horizon) with gravity. I think the surprising thing is that the particle is recurrent. http://mathoverflow.net/questions/71803/billiard-dynamics-under-gravity Comment by Zarathustra Zarathustra 2011-08-01T17:43:43Z 2011-08-01T17:43:43Z Joseph, can you please plot a long trajectory to see if it becomes dense/equidistributed? http://mathoverflow.net/questions/71479/1-lipschitz-length-preserving-isometry Comment by Zarathustra Zarathustra 2011-07-30T03:04:35Z 2011-07-30T03:04:35Z Does higher dim. analog holds true? http://mathoverflow.net/questions/54143/whats-the-kirby-diagram-of-a-universal-mathbbr4/71352#71352 Comment by Zarathustra Zarathustra 2011-07-26T21:07:29Z 2011-07-26T21:07:29Z Welcome to MO!! http://mathoverflow.net/questions/63851/ideas-to-face-the-current-publishing-issues Comment by Zarathustra Zarathustra 2011-05-03T21:53:39Z 2011-05-03T21:53:39Z Actually arxiv does link to a commentary system <a href="http://sciencewise.info/" rel="nofollow">sciencewise.info</a> http://mathoverflow.net/questions/36312/getting-rid-of-exceptional-fibers-by-passing-to-finite-covers/36316#36316 Comment by Zarathustra Zarathustra 2010-08-23T00:59:34Z 2010-08-23T00:59:34Z How do you know that it's always possible to substitute the fibering over a bad orbifold with a &quot;good&quot; fibering? http://mathoverflow.net/questions/36312/getting-rid-of-exceptional-fibers-by-passing-to-finite-covers/36316#36316 Comment by Zarathustra Zarathustra 2010-08-21T22:32:13Z 2010-08-21T22:32:13Z Thanks!  http://mathoverflow.net/questions/36312/getting-rid-of-exceptional-fibers-by-passing-to-finite-covers/36323#36323 Comment by Zarathustra Zarathustra 2010-08-21T22:30:18Z 2010-08-21T22:30:18Z I was thinking about compact case. Thanks! And thanks for the book, it's kinda hard to read online though... http://mathoverflow.net/questions/34188/distribution-of-fractional-parts-of-n3-2/34293#34293 Comment by Zarathustra Zarathustra 2010-08-05T06:04:08Z 2010-08-05T06:04:08Z Thank you, I wasn't able to completely carry out the proof, but it looks like something doable. http://mathoverflow.net/questions/34188/distribution-of-fractional-parts-of-n3-2/34293#34293 Comment by Zarathustra Zarathustra 2010-08-02T19:47:35Z 2010-08-02T19:47:35Z Thank you. You forgot 1/N and rho must be between 1 and 2. Also I don't understand your formula for the difference (m+h)^-m^, it's wrong. Also, you forgot k in the last formula. Anyhow, if you can elaborate that'd be great. http://mathoverflow.net/questions/34188/distribution-of-fractional-parts-of-n3-2/34202#34202 Comment by Zarathustra Zarathustra 2010-08-02T17:42:07Z 2010-08-02T17:42:07Z Thank you! Do they prove the theorem in the book? If you know the proof, how hard is it?