User ohad asor - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T15:57:28Z http://mathoverflow.net/feeds/user/28037 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/118687/rfc-for-definite-integral-connection-to-second-derivative RFC for definite integral connection to second derivative Ohad Asor 2013-01-12T00:31:08Z 2013-01-12T16:32:20Z <p>Hi,</p> <p>During my research I found an interesting fact, and I'd like to know if it's interesting for others as well. Find a function $g(x,t):[0,T]\times[0,T]\rightarrow[0,T]$ such that for any twice differentiable $f(x):[0,T]\rightarrow[0,T]$ such that $f(0)=f'(0)=0$, the equality $$f(x)=\intop_0^Tf''(t)g(x,t)dt$$ holds. Note that $g$ is independent of $f$.</p> <p>I found such a $g$, and I'll post it as an answer soon. I'd like to know if this is simple/known/interesting.</p> http://mathoverflow.net/questions/118396/elaborating-mercers-theorem-rkhs-on-cameron-martin-space-kx-y-minx-y Elaborating Mercer's theorem (RKHS) on Cameron-Martin space $k(x,y)=\min(x,y)$ Ohad Asor 2013-01-08T21:04:35Z 2013-01-09T11:29:13Z <p>Hi,</p> <p>I'm trying to employ Mercer's theorem on the kernel $k(x,y)=\min(x,y)$. It is known (and easy to verify) that this is a nonnegative-definite kernel over $[0,T]$ for any $T>0$.</p> <p>Fix $T>0$. Let's calculate the eigenfunctions of the transformation $\mathscr T_kf=\intop_{0}^T k(x,y)f(y)dy$:</p> <p>$$\lambda\psi(x)=\intop_{0}^T \min(x,y)\psi(y)dy=$$ $$\intop_{0}^x \min(x,y)\psi(y)dy - \intop_{T}^x \min(x,y)\psi(y)dy=$$ $$\intop_{0}^x y\psi(y)dy - x\intop_{T}^x \psi(y)dy\implies$$ $$\lambda\psi'(x)= x\psi(x)-x\psi(x)-\intop_{T}^x \psi(y)dy\implies$$ $$-\lambda\psi''(x)= \psi(x)\implies$$ $$\psi(x)=C_1\sin\frac x {\sqrt\lambda} + C_2\cos\frac x {\sqrt\lambda}$$</p> <p>it seems like we're allowed to pick $C_1=1$ and $C_2=0$. So we pick $\psi_n(x)=\sin nx$ and $\lambda_n = n^{-2}$. Then Mercer's theorem actually says that we should get: $$\min(x,y)=\sum_{n=1}^\infty n^{-2}\sin nx\sin ny$$</p> <p>this all seem very nice, but when evaluating this numerically, it doesn't work. I tried also to normalize $\psi$ by dividing by its norm which is $\sqrt {\frac 1 {4n} (2nT-\sin2nT)}$, and it didn't help.</p> <p>I also tried to substitute the original solution with $C_1,C_2$ in the original eigenvalue problem equation, and then to calculate $C_1,C_2$, but they turned out to depend on $x$, which is of cource unacceptable.</p> <p>So what's wrong here?</p> <p>I also asked the question on <a href="http://math.stackexchange.com/questions/272857/elaborating-mercers-theorem-rkhs-on-cameron-martin-space-kx-y-minx-y" rel="nofollow">http://math.stackexchange.com/questions/272857/elaborating-mercers-theorem-rkhs-on-cameron-martin-space-kx-y-minx-y</a>, but no answers there. I'll post in either forum immediately if an answer will come.</p> http://mathoverflow.net/questions/117840/injective-with-finite-discontinuities-mapping-from-mathbb-rn-to-0-1 Injective with Finite Discontinuities Mapping from $\mathbb R^n$ to $[0,1]$ Ohad Asor 2013-01-02T06:26:14Z 2013-01-02T07:28:10Z <p>Hi,</p> <p>as a continuation to the fully answered question:</p> <p><a href="http://mathoverflow.net/questions/117609/injectiveintregrable-mapping-from-mathbb-r3-to-mathbb-r" rel="nofollow">http://mathoverflow.net/questions/117609/injectiveintregrable-mapping-from-mathbb-r3-to-mathbb-r</a></p> <p>Can one think of an injective $f:\mathbb R^n\rightarrow[0,1]$ that has only a finite number of discontinuities? Or maybe one can come with some topological claim showing it is impossible?</p> http://mathoverflow.net/questions/117609/injectiveintregrable-mapping-from-mathbb-r3-to-mathbb-r Injective&Intregrable Mapping from $\mathbb R^3$ to $\mathbb R$ Ohad Asor 2012-12-30T10:15:32Z 2012-12-31T07:37:01Z <p>Hi,</p> <p>Can one think of an injective and Riemann integrable $f:\mathbb R^3\rightarrow\mathbb R$? (of course it cannot be continuous)</p> http://mathoverflow.net/questions/117083/cameron-martin-like-rkhs Cameron-Martin like RKHS Ohad Asor 2012-12-23T11:57:02Z 2012-12-23T11:57:02Z <p>Hello,</p> <p>I know that $k(x,y)=min(x,y)$ is the reproducing kernel of the Cameron Martin space of all i.i.d. RVs of Brownian motion at different times, with the $cov$ inner product. </p> <p>What is the RKHS generated by $k(x,y)=min(f(x),f(y))$ for some $f$?</p> http://mathoverflow.net/questions/112249/convolutional-equation Convolutional Equation Ohad Asor 2012-11-13T03:36:50Z 2012-11-13T13:52:45Z <p>Hi,</p> <p>Given C>0. Let $f,g,h$ be $L^2$ functions such that $f,g,h$ have a compact and finite-measure support, and $f*f(x)=g*g(x)=h*h(x)=f*g(x)=g*h(x)=f*h(x)=0$ (where $*$ is the convolution) for all $|x| > C$.</p> <p>Does that imply $f(x)=g(x)=h(x)=0$ for all $x$? What is I won't require $f*f=g*g=h*h=0$? What if we had two functions $f,g$ instead of three $f,g,h$, and $f*f(x)=g*g(x)=f*g(x)=0$ for all $|x| > C$? Will it imply they both identically zero everywhere? </p> <p>If not, please provide (or hint to) a counterexample.</p> http://mathoverflow.net/questions/112152/space-of-bandlimited-functions Space of Bandlimited Functions Ohad Asor 2012-11-12T07:28:08Z 2012-11-12T16:00:51Z <p>Hi,</p> <p>I was asking myself about some necessary and/or sufficient conditions for a function to be bandlimited (i.e. its Fourier transform is zero t residing out of [-B,B] for some B>0). Of course, if a function is bounded (timelimited), it cannot be bandlimited. But for a non-bounded function, how can we tell if it's bandlimited or not? Or, how can we know if a function is both time- and band- unlimited? Any sufficient/necessary conditions. I'll be glad if you can share with me some known results.</p> <p>Thanks!</p> http://mathoverflow.net/questions/118687/rfc-for-definite-integral-connection-to-second-derivative/118726#118726 Comment by Ohad Asor Ohad Asor 2013-01-12T17:22:17Z 2013-01-12T17:22:17Z Thanks Gerald I appreciate your help http://mathoverflow.net/questions/118687/rfc-for-definite-integral-connection-to-second-derivative/118720#118720 Comment by Ohad Asor Ohad Asor 2013-01-12T15:30:21Z 2013-01-12T15:30:21Z I got $-min(x,t)$ http://mathoverflow.net/questions/118687/rfc-for-definite-integral-connection-to-second-derivative/118720#118720 Comment by Ohad Asor Ohad Asor 2013-01-12T15:06:15Z 2013-01-12T15:06:15Z can you express it in terms of max/min? http://mathoverflow.net/questions/118687/rfc-for-definite-integral-connection-to-second-derivative/118726#118726 Comment by Ohad Asor Ohad Asor 2013-01-12T13:12:55Z 2013-01-12T13:12:55Z I dont understand where $f$ comes into the picture http://mathoverflow.net/questions/118687/rfc-for-definite-integral-connection-to-second-derivative/118720#118720 Comment by Ohad Asor Ohad Asor 2013-01-12T13:11:22Z 2013-01-12T13:11:22Z so maybe I didn't get it.. doesn't Dirac and Heavyside discontinuous? http://mathoverflow.net/questions/118687/rfc-for-definite-integral-connection-to-second-derivative/118720#118720 Comment by Ohad Asor Ohad Asor 2013-01-12T11:31:16Z 2013-01-12T11:31:16Z Hint: $g$ is continuous. http://mathoverflow.net/questions/118396/elaborating-mercers-theorem-rkhs-on-cameron-martin-space-kx-y-minx-y/118436#118436 Comment by Ohad Asor Ohad Asor 2013-01-09T10:22:43Z 2013-01-09T10:22:43Z now it really works :) http://mathoverflow.net/questions/118396/elaborating-mercers-theorem-rkhs-on-cameron-martin-space-kx-y-minx-y/118414#118414 Comment by Ohad Asor Ohad Asor 2013-01-09T08:56:47Z 2013-01-09T08:56:47Z Thanks for your help! Unfortunatly, this doesn't work as well. Here's my code: #include &lt;iostream&gt; #include &lt;cmath&gt; using namespace std; long double phi(long double x, long double n, long double t = 2) { static long double pi = acos((long double)-1); return sin((n+.5)*pi*x/t)*sqrt(t*2.)/(pi*(n+.5)); } int main(int, char**) { long double sum = 0, x=.8,y=.3; for (long double n = 1;n&lt; 100000;n++) { sum += phi(x,n)*phi(y,n); if (((int)n)%1000) cout&lt;&lt;n&lt;&lt;' '&lt;&lt;sum&lt;&lt;endl; } return 0; } http://mathoverflow.net/questions/118396/elaborating-mercers-theorem-rkhs-on-cameron-martin-space-kx-y-minx-y Comment by Ohad Asor Ohad Asor 2013-01-08T22:01:05Z 2013-01-08T22:01:05Z here's my code: ## #include &lt;iostream&gt; #include &lt;cmath&gt; using namespace std; long double phi(double x, double n) { return sin(n*x)/n; } int main(int, char**) { long double sum = 0, x=8,y=1; for (long double n = 1;n&lt; 100000;n++) { sum += phi(x,n)*phi(y,n); if (((int)n)%1000) cout&lt;&lt;n&lt;&lt;' '&lt;&lt;sum&lt;&lt;endl; } return 0; } ## http://mathoverflow.net/questions/118396/elaborating-mercers-theorem-rkhs-on-cameron-martin-space-kx-y-minx-y Comment by Ohad Asor Ohad Asor 2013-01-08T21:50:05Z 2013-01-08T21:50:05Z wow that's amazing. you actually got $\min(x,y)=\sum_{n=1}^\infty n^{-2}\sin nx\sin ny$? http://mathoverflow.net/questions/118051/conjecture-on-primal-representaion-of-vectors Comment by Ohad Asor Ohad Asor 2013-01-04T20:09:26Z 2013-01-04T20:09:26Z Of course, but once it's answered in either locations, I'll immediately post the answer on the second one. I have no intention to abuse (and will not, of course). http://mathoverflow.net/questions/118051/conjecture-on-primal-representaion-of-vectors Comment by Ohad Asor Ohad Asor 2013-01-04T13:48:19Z 2013-01-04T13:48:19Z Most Significant Bit (the leftest one) http://mathoverflow.net/questions/117840/injective-with-finite-discontinuities-mapping-from-mathbb-rn-to-0-1/117844#117844 Comment by Ohad Asor Ohad Asor 2013-01-02T14:59:29Z 2013-01-02T14:59:29Z yes, I do understand now, many thanks!! btw when I practically want to use what we learnt here (especially in the previous question), I encounter a problem that $f(x)$ for some $x\in\mathbb R^n$ requires $nd$ digits to be represented numerically, when $d$ is the maximum number of digits in the elements of $x$. so this can be numerically unstable. I'm now trying to find such injective&amp;integrable function that would be more numerically stable for computers. http://mathoverflow.net/questions/117840/injective-with-finite-discontinuities-mapping-from-mathbb-rn-to-0-1/117844#117844 Comment by Ohad Asor Ohad Asor 2013-01-02T07:25:59Z 2013-01-02T07:25:59Z i agree i dont have a full understanding, but this will come with time... i'm doing my best http://mathoverflow.net/questions/117840/injective-with-finite-discontinuities-mapping-from-mathbb-rn-to-0-1/117844#117844 Comment by Ohad Asor Ohad Asor 2013-01-02T07:12:34Z 2013-01-02T07:12:34Z Thanks! Please note that in the mentioned question (in that link), there is an example with only countably many discontinuities.