User djoke - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T13:59:33Z http://mathoverflow.net/feeds/user/26543 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/131268/lipschitz-map-of-the-circle-onto-a-triangle Lipschitz map of the circle onto a triangle djoke 2013-05-20T20:27:18Z 2013-05-21T15:34:02Z <p>Assume that $f$ is (Euclidean) $L-$biLipchitz mapping of the unit circle onto a triangle $\Delta(A,B,C)$. Can we find a $10000 L$ bi-lipchitz extension of $f$ onto the whole plane.</p> http://mathoverflow.net/questions/131100/lipschitz-map-of-the-ellipse Lipschitz map of the ellipse djoke 2013-05-18T21:46:12Z 2013-05-19T03:09:45Z <p>Is there a L-Lipschitz homeomorphism of the Elipse $x^2/4+y^2=1$ onto the unit circle $x^2+y^2=1$ such that $L&lt;1$?</p> http://mathoverflow.net/questions/124030/boundary-behavior-of-harmonic-functions Boundary behavior of Harmonic functions djoke 2013-03-08T22:36:36Z 2013-03-08T22:36:36Z <p>Assume that $f$ is harmonic in the unit disk $|z|&lt;1$, with boundary function of bounded variation, such that $$\lim_{r\to 1}f(re^{it})= 0$$ for $t\in[0,\pi]\setminus \mathbf{Q}$, where $\mathbf{Q}$ are rational numbers. Can we then state the following $$\lim_{r\to 1}f(re^{it})= 0,\ \ \ t\in[0,\pi].$$</p> http://mathoverflow.net/questions/121750/continuity-of-integral Continuity of integral djoke 2013-02-13T22:24:10Z 2013-02-14T00:38:26Z <p>Assume that $f:[0,2\pi]\to [0,2\pi]$ is a continuous function such that $f(0)=f(2\pi)$ and define the function $$g(s)=\int_{-\pi}^\pi \frac{\sin f(t+s)-\sin f(s)}{\sin t/2} dt.$$ Is $g$ continuous or bounded? Probably not. It is related to Marcel Riesz's famous theorem.</p> http://mathoverflow.net/questions/121327/continuity-up-to-the-boundary-of-harmonic-map Continuity up to the boundary of harmonic map djoke 2013-02-09T16:00:53Z 2013-02-09T16:00:53Z <p>I have a question concerning harmonic mappings in the plane w.r.t. to a conformal non-flat metric \rho(z)|dz|. </p> <p>Assume that $f$ is a homeomorphism of the unit circle onto itself, and let $g$ be its harmonic extension inside of the unit disk. Has $g$ a continuous extension to the boundary?</p> http://mathoverflow.net/questions/117544/solution-to-differential-equation Solution to differential equation djoke 2012-12-29T17:54:00Z 2013-01-04T00:11:29Z <p>a) How to solve, or at least to prove the existence of a solution to differential equation for given initial condition $y(s)=y_0>0$ and $y'(s)=y_1$, $s&lt;0$, $$y''+(2-n)\coth(t) y'=\frac{(n-1)\sinh(2y)}{2}, t&lt;0.$$ Here $n$ is an integer $>2$.</p> <p>b) Can the previous equation have two different solution (with different initial conditions) in $(-2,-1)$, such that one is bounded and the second is not bounded?</p> http://mathoverflow.net/questions/82613/riemann-mapping-theorem-and-smoothness-on-the-boundary/115589#115589 Answer by djoke for Riemann mapping theorem and smoothness on the boundary djoke 2012-12-06T08:33:23Z 2012-12-06T08:33:23Z <p>This is well-known result by Kellogg (O. Kellogg: On the derivatives of harmonic functions on the boundary, Trans. Amer.Math. Soc. 33 (1931), 689-692.), and Warschawski (On the higher derivatives at the boundary in conformal mapping,} Trans. Amer. Math. Soc, {\bf 38}, No. 2 (1935), 310-340.), where they prove even more, that the if the boundary is C^{n,\alpha}, then the conformal parametrization is C^{n,\alpha} up to the boundary.</p> http://mathoverflow.net/questions/114959/counterpart-of-weierstrass-theorem Counterpart of Weierstrass theorem djoke 2012-11-30T07:03:14Z 2012-11-30T14:21:58Z <p>Assume that $(X,\tau)$ is a topological space and assume that every continuous mapping $f$ of $X$ into real line $\mathbb{R}$ achieves its maximum. Under which conditions on $\tau$, the space $X$ is compact. It can be easily prove that, $X$ is compact, provided that $\tau$ is a metric topology in $X$. </p> <p>Is for example this true for the Hausdorff spaces?</p> http://mathoverflow.net/questions/114913/dini-condition-and-integrability-condition Dini condition and integrability condition djoke 2012-11-29T18:52:49Z 2012-11-30T07:26:49Z <p>Assume that $A$ is an arbitrary positive integrable function on $[0,1]$. Whether exists a convex function $f_A(x)=x g(x)$ of $(0,+\infty)$ into itself (depending on $A$) such that $\lim_{x\to +\infty} g(x)=+\infty$ and $$\int_0^1 A(x) g(1/x^2) dx &lt;+\infty.$$ This question is related to membership of $g$ to some Dini class.</p> http://mathoverflow.net/questions/112432/schwarz-type-inequality Schwarz type inequality djoke 2012-11-14T23:02:15Z 2012-11-19T19:48:32Z <p>a) Is true the following statement. Let $h$ be analytic in the unit disk such that $$|h(z)|\le \frac{|z|^2}{1-|z|^2},$$ then $$|h'(z)|\le \frac{2}{(1-|z|^2)^2}.$$ a') Is true the following statement. Let $h$ be analytic in the unit disk such that $$|h(z)|\le \frac{|z|^2}{1-|z|^2},$$ then the inequality $$|h'(z)|\le \frac{8}{\pi(1-|z|^2)^2}$$ is sharp. The inequality can be proved by using Schur test, and Riesz-Thorin convexity type theorem (Dunford &amp; Schwartz 1958, §VI.10.11).</p> <p>b) If $$|h(z)|\le \frac{|z|^2}{|1-z^2|}$$ then we have better conclusion $$|h'|\le \frac{2|z|}{(1-|z|^2)|1-z^2|}$$ and this follows by using Schwarz lemma. Namely in this case $$|H(z)|=|(1-z^2) h(z)/z^2|\le 1.$$ Then $$|H'(z)|\le \frac{1-|H(z)|^2}{1-|z|^2}.$$</p> <p>As $$H'(z)=(1-z^2) h'(z)/z^2-2/z^3 h(z),$$ it follows that $$|(1-z^2) h'(z)/z^2|\le \frac{2(1-|z|^2)/|z|^3 h(z)+1-|H(z)|^2}{1-|z|^2}$$ $$\le \frac{2|H(z)|/|z| +1-|H(z)|^2}{1-|z|^2}\le \frac{2|z|^{-1}}{1-|z|^2}.$$</p> <p>The question a) is related to precise estimation of norm of a Bergman projection into Bloch space and is far for being a homework.</p> http://mathoverflow.net/questions/110110/boundary-of-star-shaped-domain Boundary of star-shaped domain djoke 2012-10-19T18:46:25Z 2012-10-20T01:57:47Z <p>Assume that $M\subset R^n$, $n\ge 3$, is a boundary of an open bounded set $D$ containing $0$, which is starlike w.r.t. 0, meaning that each ray $[0,x]$ from $x\in M$ to $0$ meets $M$ only once. Is $M$ smooth almost everywhere?</p> http://mathoverflow.net/questions/109613/criteria-for-lipschitz-continuity Criteria for Lipschitz continuity djoke 2012-10-14T13:54:57Z 2012-10-17T07:54:36Z <p>Is the following statement true. Assume that $f:[0,1]\to [0,1]$ is a continuous function such that $$\sup_t\lim\sup_{s\to t}\frac{|f(s)-f(t)|}{|t-s|}&lt;\infty,$$ then $f$ is Lipchitz continuous.</p> http://mathoverflow.net/questions/131268/lipschitz-map-of-the-circle-onto-a-triangle/131283#131283 Comment by djoke djoke 2013-05-21T06:34:54Z 2013-05-21T06:34:54Z Then you obtain $L^9$ growth of bi-Lipschitz constant. http://mathoverflow.net/questions/131100/lipschitz-map-of-the-ellipse Comment by djoke djoke 2013-05-19T07:05:39Z 2013-05-19T07:05:39Z The metric from the Euclidean plane is assumed and Mixon has the answer. http://mathoverflow.net/questions/124030/boundary-behavior-of-harmonic-functions Comment by djoke djoke 2013-03-09T10:17:22Z 2013-03-09T10:17:22Z $f=P[g]$, where $g(t):[0,2\pi]\to \gamma\subset \mathbf{C}$ is of bounded variation. http://mathoverflow.net/questions/117544/solution-to-differential-equation/117980#117980 Comment by djoke djoke 2013-01-04T21:16:08Z 2013-01-04T21:16:08Z Thanks! One more question. If the initial speed $y'(-2)=0$, can you have a similar conclusion? http://mathoverflow.net/questions/117544/solution-to-differential-equation/117980#117980 Comment by djoke djoke 2013-01-04T16:38:43Z 2013-01-04T16:38:43Z Why you can assume that $f\in[0,1]$. You should prove that $f&gt;0$? http://mathoverflow.net/questions/117544/solution-to-differential-equation/117980#117980 Comment by djoke djoke 2013-01-03T18:17:00Z 2013-01-03T18:17:00Z I updated the formulation. The question is can we find a global solution for example in (-2,-1). http://mathoverflow.net/questions/117544/solution-to-differential-equation/117545#117545 Comment by djoke djoke 2012-12-30T13:17:10Z 2012-12-30T13:17:10Z Thanks, but I need to prove more than the existence. http://mathoverflow.net/questions/117544/solution-to-differential-equation Comment by djoke djoke 2012-12-30T06:52:13Z 2012-12-30T06:52:13Z @Shahrooz Probably you have &quot;ask matlab&quot; to solve slightly different equation with the right hand side containing sinh(2t)! http://mathoverflow.net/questions/117544/solution-to-differential-equation Comment by djoke djoke 2012-12-29T20:59:19Z 2012-12-29T20:59:19Z @Shahrooz And which is the solution? Can you spell it? Thanks. http://mathoverflow.net/questions/114913/dini-condition-and-integrability-condition/114920#114920 Comment by djoke djoke 2012-11-30T13:21:07Z 2012-11-30T13:21:07Z Very nice solution. http://mathoverflow.net/questions/114913/dini-condition-and-integrability-condition/114930#114930 Comment by djoke djoke 2012-11-30T13:21:01Z 2012-11-30T13:21:01Z Thanks for the solution. http://mathoverflow.net/questions/114959/counterpart-of-weierstrass-theorem/114961#114961 Comment by djoke djoke 2012-11-30T09:40:24Z 2012-11-30T09:40:24Z @Pietro Majer. See the email. The metric space has this property, and need not be a countably compact space. http://mathoverflow.net/questions/114959/counterpart-of-weierstrass-theorem Comment by djoke djoke 2012-11-30T07:18:23Z 2012-11-30T07:18:23Z @Budney This question was for me interesting from long time ago when I was a student. The question is not related to any research; it is only solely interesting. http://mathoverflow.net/questions/112432/schwarz-type-inequality/112647#112647 Comment by djoke djoke 2012-11-26T20:39:04Z 2012-11-26T20:39:04Z @Fedja: Just to remind you. How you can improve the constant 4, and what about your counterexample? http://mathoverflow.net/questions/112432/schwarz-type-inequality/112647#112647 Comment by djoke djoke 2012-11-19T19:49:42Z 2012-11-19T19:49:42Z @Fedja: Your &quot;blow up&quot; solution is suspicious.