User a. lerario - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T02:43:49Z http://mathoverflow.net/feeds/user/16494 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/102453/map-with-prescribed-jacobian Map with prescribed Jacobian A. Lerario 2012-07-17T14:55:46Z 2012-07-17T16:02:31Z <p>Recently I came up with the following problem.</p> <p>Suppose $U$ is an open subset of $\mathbb{R}^n$ and we are given a continuous map $M:U\to GL(n;\mathbb{R})$. Does anybody know if there are conditions for the existence of a $C^1$ map $f:U\to \mathbb{R}^n$ such that $$Jf(x)=M(x)\quad \forall x \in U$$ (here $Jf(x)$ is the Jacobian matrix of $f$ at $x$)?</p> <p>I know this question is somehow a little bit general; I will really appreciate even a reference for something related to this problem. :)</p> http://mathoverflow.net/questions/89897/least-singular-value-gaussian-orthogonal-ensemble Least singular value gaussian orthogonal ensemble. A. Lerario 2012-02-29T19:20:25Z 2012-03-14T20:22:02Z <p>Hello everybody, here is my question:</p> <p>Assume A is a random symmetric $nxn$ matrix whose entries are independent, normally distributed with mean zero and variance 2 on the diagonal and 1 off diagonal (to my knowledge such a random matrix is said to belong to the 'Gaussian orthogonal ensemble'). The joint pdf for the eigenvalues of A is well known, but i was wondering:</p> <p>does there exist a precise formula for the probability that all the eigenvalues of A have norm greater then epsilon?</p> <p>Equivalently which is the probability that the least singular value of A is smaller than epsilon?</p> <p>I am computing the intrinsic volume of singular symmetric matrices of (Frobenius or trace-square) norm one and I need this precise formula to perform the epsilon limit using tubes. I am sorry if this is a well known result, but I was not able to find it in the literature. In case I would really appreciate a reference for this.</p> <p>Thanks everyone!</p> http://mathoverflow.net/questions/76015/how-much-can-a-diagonal-matrix-change-the-eigenvalues-of-a-symmetric-matrix/76197#76197 Answer by A. Lerario for How much can a diagonal matrix change the eigenvalues of a symmetric matrix? A. Lerario 2011-09-23T09:59:37Z 2011-09-23T09:59:37Z <p>In general there is no relation: for example consider the simplest case $S$ itself is diagonal and invertible. Letting $D_1=S^{-1}$ then $A$ can be any diagonal matrix $D_2$. The only considerations you can do are related to the presence of the zero eigenvalues using Binet formula for determinants. Notice also that in general $A$ itself can be nonsymmetric, and its eigenvalues can be complex. However small perturbations, i.e. small $D_1$ and $D_2$, result in a small perturbation of the eigenvalues of $S$ in the complex plane.</p> http://mathoverflow.net/questions/33199/small-neighborhoods-of-singularities-on-varieties/71059#71059 Answer by A. Lerario for Small neighborhoods of singularities on varieties A. Lerario 2011-07-23T11:25:53Z 2011-07-23T11:25:53Z <p>Indeed the following theorem to me seems exactly you were looking for (see J. Bochnak, M. Coste, M-F. Roy, "Real algebraic geometry", Theorem 9.3.6 [Local conic structure]):</p> <p>Let $E$ be a semialgebraic susbet of $\mathbb{R}^n$ and $x$ be a nonisolated point of $E.$ Let also $D_\epsilon$ be the closed $\epsilon$-ball around $x$ and $S_\epsilon$ its boundary. Set $K=S_\epsilon \cap E$. Then there for $\epsilon>0$ small enough the pair $(D_\epsilon,E∩D_\epsilon)$ is semialgebraically homeomorphic to the pair $(CS_\epsilon,CK)$, where $C$ denotes taking the cone. Moreover the semialgebraic homeomorphism can be chosen as to preserve the distance from $x.$</p> <p>Two words of remarks on the previous statement: </p> <ol> <li>Every real or complex algebraic set in $\mathbb{R}^n$ or in $\mathbb{C}^n\simeq \mathbb{R}^{2n}$ is a semialgebraic set. </li> <li>The point $x$ is any nonisolated point of $E$ (no matter singular - in whatever meaning this word has for a general semialgebraic set - or regular).</li> </ol> http://mathoverflow.net/questions/62819/canonical-form-for-a-pair-of-quadratic-forms/70473#70473 Answer by A. Lerario for Canonical form for a pair of quadratic forms A. Lerario 2011-07-16T00:39:17Z 2011-07-16T00:39:17Z <p>Hi, I just read your question. I hope it's not too late to answer, but I think the following paper contains exactly what you are looking for:</p> <p>R. C. Thompson: Pencils of complex and real symmetric and skew matrices, Linear Algebra and its Applications, Volume 147, March 1991, 323-371.</p> <p>The author classifies - up to congruence - any pairs of real symmetric matrices, which amounts to give the classification for any pair of real quadratic forms.</p> <p>You'll probably find of some interest also this paper of mine: <a href="http://arxiv.org/abs/1106.4678" rel="nofollow">http://arxiv.org/abs/1106.4678</a></p> http://mathoverflow.net/questions/102453/map-with-prescribed-jacobian/102458#102458 Comment by A. Lerario A. Lerario 2012-07-17T20:01:17Z 2012-07-17T20:01:17Z I was writing tautological for the case $M$ is differentiable. For $M$ continuous - I am not an expert - do I understand correctly that closedness of $\omega_i$ in the distributional sense is a sufficient condition? http://mathoverflow.net/questions/102453/map-with-prescribed-jacobian/102458#102458 Comment by A. Lerario A. Lerario 2012-07-17T15:52:39Z 2012-07-17T15:52:39Z This is a nice answer, though somehow tautological. Also notice that we are not allowed to take derivatives of the matrix $M$ since it is assumed to depend only continuously on $x.$ http://mathoverflow.net/questions/89897/least-singular-value-gaussian-orthogonal-ensemble/89899#89899 Comment by A. Lerario A. Lerario 2012-03-01T01:44:19Z 2012-03-01T01:44:19Z I don't see an immediate way to relate this to the GOE case... :)