User john young - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T11:09:01Z http://mathoverflow.net/feeds/user/18420 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/120044/convergence-of-dirichlet-forms Convergence of Dirichlet Forms John Young 2013-01-27T19:07:45Z 2013-01-27T19:42:37Z <p>If a sequence of Dirichlet forms convergence to 0, then what about the diffusion processes associated with these Dirichlet forms? Do the finite dimensional distributions of them converges weakly? and what are the limits?</p> http://mathoverflow.net/questions/109567/eigenvalues-of-infinite-matrices Eigenvalues of infinite matrices John Young 2012-10-13T23:36:18Z 2012-10-15T11:34:12Z <p>I am trying to find some literature on infinite matrices because I want to know how to get the eigenvalues of infinite matrices. Seriously, it seems there are very few references available. Can someone tell me how to determine the spectrum of infinite matrices?</p> http://mathoverflow.net/questions/93218/convergence-of-stochastic-process Convergence of stochastic process John Young 2012-04-05T14:19:48Z 2012-04-05T14:47:13Z <p>As we know, to prove the convergence of stochastic process, we could either show the convergence of finite dimensional distribution and tightness of the process, or use techniques of martingale problems. What about the following Markov process:</p> <p>$L=\frac{1}{2}p(1-p)\frac{d^{2}}{dp^{2}}-\frac{\theta}{2}p\frac{d}{dp}+\log(\theta) p(1-p)(2p-1)\frac{d}{dp}, p\in[0,1]$</p> <p>We can see that the generator explodes when $\theta\rightarrow0$. How can we find the limit of this process as $\theta\rightarrow0$. Apparently, the techniques of martingale problems are not applicable here! </p> http://mathoverflow.net/questions/88829/additive-functional-of-markov-process additive functional of Markov process John Young 2012-02-18T13:39:18Z 2012-02-18T17:30:51Z <p>I was wondering if there is a way to figure out an explicit formula for the conditional expectation of some Markov additive functional as the following: $$E_{p}(\exp{[-\int_{0}^{t}g(X_{s})ds]}|X_{t}=q),$$ where $X_{t}$ is a diffusion process, and suppose we have an explicit transition density function of $X_{t}$. Probably Brownian motion is a perfect example for this question, but can we find an explicit formula of the above form?</p> http://mathoverflow.net/questions/77748/eigenvalue-density-of-some-random-matrices Eigenvalue Density of Some Random Matrices? John Young 2011-10-10T21:38:18Z 2011-10-15T05:36:38Z <p>Consider a class of real symmetric random matrices,$M_{n\times n}=(X_{i,j})_{n\times n},$ whose off-diagonal elements follow an exchangeable distribution, and the diagonal elements follow another exchangeable distribution and the diagonal part and off-diagonal part are independent. My question is what's the eigenvalue density of this type of random matrices? Is the eigenvalue density Universal?</p> <p>I know that Chatterjee has a paper: <a href="http://projecteuclid.org/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;handle=euclid.aop/1171377437" rel="nofollow">http://projecteuclid.org/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;handle=euclid.aop/1171377437</a></p> <p>But my matrices are different from his consideration. I do have a specific example. If we choose the off-diagonal part follow Dirichlet distribution and the diagonal part follow another Dirichlet distribution, then the eigenvalue density is some Gamma distribution. To my opinion, this is actually trivial, for after we rescale the matrix, it's asymptotic matrix is an infinite dimensional identity matrix. What about other cases?</p> http://mathoverflow.net/questions/77730/how-many-p-regular-graphs-with-n-vertices-are-there How many $p$-regular graphs with $n$ vertices are there? John Young 2011-10-10T19:30:01Z 2011-10-11T06:59:09Z <p>Suppose that there are $n$ vertices, we want to construct a regular graph with degree $p$, which, of course, is less than $n$. My question is how many possible such graphs can we get?</p> http://mathoverflow.net/questions/120044/convergence-of-dirichlet-forms Comment by John Young John Young 2013-01-28T20:38:50Z 2013-01-28T20:38:50Z yes, That is what I mean. http://mathoverflow.net/questions/120044/convergence-of-dirichlet-forms Comment by John Young John Young 2013-01-27T22:00:17Z 2013-01-27T22:00:17Z The sequence of Dirichlet forms depend on a parameter, So the convergence of the associated diffusion processes are about this parameter not time $t\rightarrow\infty$. http://mathoverflow.net/questions/109567/eigenvalues-of-infinite-matrices Comment by John Young John Young 2012-10-15T14:48:12Z 2012-10-15T14:48:12Z em interesting! Thanks a lot! http://mathoverflow.net/questions/109567/eigenvalues-of-infinite-matrices/109663#109663 Comment by John Young John Young 2012-10-15T14:45:56Z 2012-10-15T14:45:56Z Excellent example! This is indeed what I am worried about. Thanks a lot! http://mathoverflow.net/questions/109567/eigenvalues-of-infinite-matrices/109602#109602 Comment by John Young John Young 2012-10-14T16:56:23Z 2012-10-14T16:56:23Z Thank you so much! So we truncate the infinite matrix and find the eigenvalues, then we take limits. If the limits exist, then we regard the limit as the eigenvalue of infinite matrices. Do you think it is a legitimate treatment of eigenvalues of infinite matrices? Please do not advise me to read the general theory of linear operator in Hilbert space, seriously I know those stuff. But I just don't know how should we deal with infinite matrices. Do you think infinite sparse matrices are easier to deal with? Thank you so much! http://mathoverflow.net/questions/109567/eigenvalues-of-infinite-matrices Comment by John Young John Young 2012-10-14T16:41:46Z 2012-10-14T16:41:46Z Thanks a lot! Benjamin, I am really like the references! It is very helpful! http://mathoverflow.net/questions/109567/eigenvalues-of-infinite-matrices Comment by John Young John Young 2012-10-14T00:28:58Z 2012-10-14T00:28:58Z I am talking about the infinite matrix in Hilbert space. http://mathoverflow.net/questions/93218/convergence-of-stochastic-process/93222#93222 Comment by John Young John Young 2012-04-05T15:05:19Z 2012-04-05T15:05:19Z I have the similar idea, but i just don't know how to verify it. Thank you so much for your answer! http://mathoverflow.net/questions/77748/eigenvalue-density-of-some-random-matrices/78189#78189 Comment by John Young John Young 2011-10-16T00:29:08Z 2011-10-16T00:29:08Z Thanks for your comments. Yes, you are right. Probably it is not a universal case! http://mathoverflow.net/questions/77730/how-many-p-regular-graphs-with-n-vertices-are-there Comment by John Young John Young 2011-10-11T17:24:39Z 2011-10-11T17:24:39Z Thanks for your comment! http://mathoverflow.net/questions/77704/find-an-asymptotic-approximation-of-this-sequence Comment by John Young John Young 2011-10-10T18:47:33Z 2011-10-10T18:47:33Z Yes, it is indeed a recursively defined sequence, but here I write it in its general expression. http://mathoverflow.net/questions/77704/find-an-asymptotic-approximation-of-this-sequence Comment by John Young John Young 2011-10-10T17:26:39Z 2011-10-10T17:26:39Z Yes, $p$ is fixed.