User sasha - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T11:16:41Z http://mathoverflow.net/feeds/user/3840 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/120420/integral-representation-of-the-modified-bessel-functions-of-the-second-kind-and-a Integral representation of the modified Bessel functions of the second kind and asymptotic expansion Sasha 2013-01-31T14:45:13Z 2013-01-31T15:37:20Z <p>The modified Bessel function (Macdonald function) $K_\alpha(z)$ is known to have the following asymptotic expansion for large positive $z$: $$K_\alpha(z)=\sqrt{\frac{\pi}{2z}}e^{-z}\sum_{k=0}^\infty \frac{b_k(\alpha)}{z^ k}$$ where $b_1(\alpha)=1$, $b_2(\alpha)=\frac{\alpha^2-1^2}{1!8}$, $b_3(\alpha)=\frac{(\alpha^2-1^2)(\alpha^2-3^2)}{2!(8)^2}$ and so on. Is there any simple integral representation, for which it would be a perturbative expansion such that $$K_\alpha(z)=h(z) \int_C \exp\left(\frac{f(y)}{z}\right) g(y)^\alpha d\mu(y)$$ where $f(x)$, $g(x)$, $h(x)$ and $d \mu(x)$ are $\alpha$-independent? </p> http://mathoverflow.net/questions/116712/integral-transform-and-frac1n Integral transform and $\frac{1}{n!}$. Sasha 2012-12-18T14:54:07Z 2012-12-19T21:21:55Z <p>Probably this is a trivial question, but I am unable to find an answer: is there a function $v(x)$ such that $$\int_{0}^\infty x^n e^{v(x)} dx =\frac{1}{n!}$$ for all positiv integer n?</p> http://mathoverflow.net/questions/100936/composition-of-two-formal-series Composition of two formal series Sasha 2012-06-29T14:27:42Z 2012-07-05T12:26:51Z <p>There are two formal semi-infinite Laurent series</p> <p>$$f_+(z)=z+\sum_{k=2}^{\infty} a_k z^k$$</p> <p>and </p> <p>$$f_-(z)=z+\sum_{k=0}^{\infty} b_k z^{-k}$$</p> <p>Their composition $f_+(f_-(z))$ is a formal series infinite in both directions. Is there any way to construct two other semi-infinite series</p> <p>$$g_+(z)=z+\sum_{k=2}^{\infty} c_k z^k$$</p> <p>and </p> <p>$$g_-(z)=z+\sum_{k=0}^{\infty} d_k z^{-k}$$</p> <p>such that </p> <p>$$f_+(f_-(z))=g_-(g_+(z))$$</p> <p>How should I call this problem? Are there any non-trivial examples when it can be done?</p> http://mathoverflow.net/questions/99990/kac-peterson-paper-on-infinite-wedge-representation-and-mkp-hierarchy Kac Peterson paper on infinite wedge representation and MKP hierarchy Sasha 2012-06-19T11:31:29Z 2012-06-19T18:01:03Z <p>I need a paper:</p> <p>V. Kac, D. Peterson, Lectures on the infinite wedge representation and the MKP hierarchy, Seminaire de Math. Superieures, Les Presses de L'Universite de Montreal 102 (1986), 141-186. </p> <p>Does anyone know how I could find a copy of it?</p> http://mathoverflow.net/questions/89380/hurwitz-numbers-and-frobenius-manifolds Hurwitz numbers and Frobenius manifolds Sasha 2012-02-24T10:06:44Z 2012-04-20T16:22:01Z <p>Generating functions of the Gromov-Witten invariants (as well as some other important partition functions) are known to be related to the Frobenius manifold structure. Are there any Frobenius manifolds behind generating functions of simple (double, multiple) Hurwitz numbers?</p> http://mathoverflow.net/questions/84889/multiple-hodge-integrals-and-integrability Multiple Hodge integrals and integrability Sasha 2012-01-04T17:01:59Z 2012-02-28T17:16:53Z <p>It is known that a generating function of the linear Hodge integrals is a tau function of the KP hierarchy, namely a one-parameter deformation of the Kontsevich-Witten tau-function (see <a href="http://arxiv.org/abs/0809.3263" rel="nofollow">Kazarian</a>). Are there known any generating functions of the multiple Hodge integrals (integrals with multiple $\lambda$-classes) with nice integrable properties?</p> http://mathoverflow.net/questions/88100/krichever-novikov-dubrovin-description-for-not-algebraic-spectral-curve Krichever-Novikov-Dubrovin description for not-algebraic spectral curve Sasha 2012-02-10T13:33:02Z 2012-02-24T10:15:43Z <p>Non-algebraic curves play an increasing role in string theory, sometimes they are known to be related to the integrable systems of the KP/Toda type. Are there any investigated examples of the application of the <a href="http://iopscience.iop.org/0036-0279/32/6/R04" rel="nofollow">Krichever</a>-Novikov-<a href="http://iopscience.iop.org/0036-0279/36/2/R02" rel="nofollow">Dubrovin</a> description of integrable hierarchies to non-algebraic spectral curves? </p> http://mathoverflow.net/questions/86806/best-citations-database Best citations database Sasha 2012-01-27T09:57:05Z 2012-01-27T14:16:29Z <p>Which online service provides the most complete list of citations to given mathematical paper? I mean the citations both in published papers and in preprints. I guess scholar.google is the best, maybe there is something more powerfull? </p> http://mathoverflow.net/questions/85693/from-sato-grassmannian-to-spectral-curve From Sato grassmannian to spectral curve Sasha 2012-01-14T22:08:27Z 2012-01-14T22:25:50Z <p>Assume a tau-function of the KP integrable hierarchy is fixed by the point of the Sato grassmannian (that is by a semi-infinite set of Laurant series $\varphi_k(z)=\sum_{m>0}a_{km}z^{-k+m}$). Can one restore a spectral curve that corresponds to this solution?</p> http://mathoverflow.net/questions/59321/cut-and-join-equation-and-schur-function-identity/84937#84937 Answer by Sasha for Cut-and-join equation and Schur function identity Sasha 2012-01-05T09:06:23Z 2012-01-05T09:06:23Z <p>The cut-and-join operator can be easely represented in termes of eigenvalues (see e.g. (58) <a href="http://xxx.lanl.gov/PS_cache/arxiv/pdf/1005/1005.5715v3.pdf" rel="nofollow">here</a>) so that one can act by exponential of this operator on the second formula at your question to get the required expression. </p> http://mathoverflow.net/questions/38293/virasoro-constraints-for-the-generating-function-of-hurwitz-numbers Virasoro constraints for the generating function of Hurwitz numbers. Sasha 2010-09-10T11:27:17Z 2012-01-05T08:51:12Z <p>Generating function of the simple Hurwitz numbers is known to be connected with Gromov-Witten potential of the point (Kontsevich $\tau$-function) (see e.g. <a href="http://arxiv.org/abs/math/9910004" rel="nofollow">Ian Goulden, David Jackson and Ravi Vakil</a>). On the other side Virasoro constraints are known to play an imporatant role in the Gromov-Witten theory, in particular for the point. </p> <p>Is there known the set of Virasoro constraints for the generating function of simple (or double) Hurwitz numbers? Any ideas why it should (or should not) exist? </p> http://mathoverflow.net/questions/60512/non-algebraic-curve-visualisation/63987#63987 Answer by Sasha for Non-algebraic curve visualisation Sasha 2011-05-05T11:58:38Z 2011-05-05T11:58:38Z <p>I have found a good and simple paper <a href="http://www.apmaths.uwo.ca/~djeffrey/Offprints/riemann.pdf" rel="nofollow">Graphing Elementary Riemann Surfaces</a> by Robert M. Corless and David J. Jeffrey with an explanation how to use Maple for graphing Riemann surfaces. In particular the Lambert curve is amnong the examples they consider. </p> http://mathoverflow.net/questions/60512/non-algebraic-curve-visualisation Non-algebraic curve visualisation Sasha 2011-04-04T01:46:01Z 2011-05-05T11:58:38Z <p>Is there any software which can automatically visualise a non-algebraic complex curve, I mean the structure of it's ramification points and sheet?</p> <p>I think a good test example would be the Lambert curve $y\exp y =x$ (what I really need is a bit more complicated family of the curves). </p> http://mathoverflow.net/questions/47671/matrix-integral-identity Matrix integral identity Sasha 2010-11-29T12:43:14Z 2010-11-29T12:43:14Z <p>1) How to prove that $N\times N$ matrix integral over complex matrices $Z$ $$\int d Z d Z^\dagger e^{-Tr Z Z^\dagger} \frac{x_1\det e^Z -x_2 \det e^{AZ^\dagger}}{\det(1-x_1e^Z)\det(1-x_2e^{AZ^\dagger})}$$ does not depend on the external Hermitian matrix $A$? $x_1$ and $x_2$ are numbers. The statement is trivial for $1\times1$ case.</p> <p>2)The same for </p> <p>$$\int d Z d Z^\dagger e^{-Tr Z Z^\dagger} \frac{x_1\det e^Z -x_2 \det e^{AZ^\dagger}}{\det(1-x_1e^Zg)\det(1-x_2e^{AZ^\dagger}g)}$$</p> <p>where g - arbitrary $GL(N)$ matrix.</p> http://mathoverflow.net/questions/42015/q-deformation-of-the-unitary-group-integral q-deformation of the unitary group integral Sasha 2010-10-13T13:44:52Z 2010-10-13T18:50:42Z <p>There is a well-known orthogonality property of $U(N)$ group characters</p> <p>$$\int d U \chi_{\mu}(U)\chi_\lambda(U^\dagger V)=\delta_{\mu\lambda}\frac{\chi_\mu(V)}{\dim_\mu}$$</p> <p>where the integral is over unitary group, $\chi_\lambda$ is a character, labeled by the partition $\lambda$ and $\dim_\mu$ is the dimension of the correspondent representation, namely $\dim_\lambda=\chi_\lambda(\bf{1})$, the value of the character on the trivial group element. </p> <p>In mathematical physics, in particular in topological strings (for example topological vertex) there appears the q-deformation of the dimension, namely $\dim^q_\lambda=\chi_\lambda(\rho)$ where $\rho$ is the diagonal matrix with the entries $1,q,q^2,\ldots$. The question:</p> <p>is there any natural deformation of the unitary integral, which gives q-deformed dimensions in the r.g.s.?</p> <p>$$\left[\int d U \right]^q \chi_{\mu}(U)\chi_\lambda(U^\dagger V)=\delta_{\mu\lambda}\frac{\chi_\mu(V)}{\dim^q_\mu}$$</p> http://mathoverflow.net/questions/39475/gromov-witten-and-integrability-2 Gromov-Witten and integrability 2. Sasha 2010-09-21T09:02:38Z 2010-09-23T10:39:20Z <p>This is a followup of my previous question <a href="http://mathoverflow.net/questions/38294/gromov-witten-and-integrability" rel="nofollow">Gromov-Witten and integrability</a>. As I have learned from the answer (but guessed before), GW potentials of the point and $P^1$ (with different modifications) are, more or less, the only examples of the GW generation functions with established integrable properties. So what about higher genera curves? Are they really so complicated to establish integrability, at least for stable sector? What is the main problem with them? </p> http://mathoverflow.net/questions/39077/online-latex-editor Online latex editor Sasha 2010-09-17T10:04:09Z 2010-09-17T18:41:34Z <p>I am not sure if this is a proper place for my question, but: can anybody recommend any good online latex editor?</p> <hr> <p>Anyone interested in this would probably benefit from the answers to this question on the tex.SE site: <a href="http://tex.stackexchange.com/questions/3/compiling-documents-online" rel="nofollow">http://tex.stackexchange.com/questions/3/compiling-documents-online</a></p> http://mathoverflow.net/questions/38294/gromov-witten-and-integrability Gromov-Witten and integrability. Sasha 2010-09-10T11:45:23Z 2010-09-16T21:25:10Z <p>The generation function of the Gromow-Witten invariants (with descendants) of the point is known to be Kontsevich-Witten tau-function of KdV, partition functions of $P^1$ and equivariant $P^1$ are known to be tau-functions of extended Toda and 2-Toda respectively. Are there any other manifolds (except of orbifolds made of mentioned previously manifolds ) for which generation functions of GW invariants are identified with tau-functions of some integrable hierarchy? </p> http://mathoverflow.net/questions/14388/jack-polynomials-as-determinants Jack polynomials as determinants Sasha 2010-02-06T13:32:52Z 2010-02-08T22:36:00Z <p>Jack symmetric polynomials are known to be generalizations of Schur functions $\chi_\lambda$, for which powerful Weyl determinant formulas are known. Are there any generalizations of two determinant formulas for general Jack symmetric $P^\alpha_\lambda(x)$ functions? </p> <p>The first determinant (Jacobi-Trudi) formula represents the character of the irrep GL(N) given by the partition $\lambda$ $$\chi_\lambda(x)=\det_{i,j} s_{\lambda_i-i+j}$$ where $s_k$ are elementary Schur function and the second one gives the same function as determinant $$\chi_\lambda(x)=\frac{\det_{i,j} x_i^{\lambda_j+N-j}}{\det_{i,j} x_i^{N-j}}$$ Jack symmetric polynomials are natural generalizations of Schur polynomials, and probably, to operate with them it would be useful to have as simple as possible analogs of Weyl formulas.</p> http://mathoverflow.net/questions/120420/integral-representation-of-the-modified-bessel-functions-of-the-second-kind-and-a/120425#120425 Comment by Sasha Sasha 2013-02-01T13:06:40Z 2013-02-01T13:06:40Z I just want to have simple $1/z$ expansion for large $z$ (and not a usual steepest descent expansion). http://mathoverflow.net/questions/120420/integral-representation-of-the-modified-bessel-functions-of-the-second-kind-and-a/120425#120425 Comment by Sasha Sasha 2013-01-31T15:54:21Z 2013-01-31T15:54:21Z Thank you, I know this nice integral representation, but I want something (maybe) more complicated - with $\frac{1}{z}$ as I wrote. http://mathoverflow.net/questions/116712/integral-transform-and-frac1n/116722#116722 Comment by Sasha Sasha 2012-12-19T12:25:15Z 2012-12-19T12:25:15Z Thank you! This is precisely what I need. Don't you know if there is a constructive way to restore such measure? http://mathoverflow.net/questions/116712/integral-transform-and-frac1n Comment by Sasha Sasha 2012-12-18T17:00:38Z 2012-12-18T17:00:38Z @Pietro: Any f. http://mathoverflow.net/questions/116712/integral-transform-and-frac1n Comment by Sasha Sasha 2012-12-18T15:38:12Z 2012-12-18T15:38:12Z Alex, thank you! Is it also obvious for integrals $\int_0^\infty x^n f(x) dx$? http://mathoverflow.net/questions/100936/composition-of-two-formal-series Comment by Sasha Sasha 2012-06-29T17:23:59Z 2012-06-29T17:23:59Z That's true. Actually, in the &quot;physical&quot; example I have in my mind, coefficients are divergent. But if we assume that coefficients of the composition are well defined, what can be done? http://mathoverflow.net/questions/85693/from-sato-grassmannian-to-spectral-curve/85695#85695 Comment by Sasha Sasha 2012-01-15T09:06:05Z 2012-01-15T09:06:05Z @David: Thank you for your comment, but I do not understand this point. Many matrix models tau-functions (like multy-cut solutions) satisfy the string equations, nevertheless they are described by the finite-genus spectral curves. http://mathoverflow.net/questions/85693/from-sato-grassmannian-to-spectral-curve/85695#85695 Comment by Sasha Sasha 2012-01-15T08:59:04Z 2012-01-15T08:59:04Z As far as I know KW is described by the genus-zero spectral curve. In particular, the tau-fuction can be restored by Eynard-Orantin method starting from this genus-zero curve. http://mathoverflow.net/questions/85693/from-sato-grassmannian-to-spectral-curve/85695#85695 Comment by Sasha Sasha 2012-01-14T22:53:31Z 2012-01-14T22:53:31Z Yes, it is. Asuume I will take a special solution with a non-trivial spectral curve, for example a Kontsevich-Witten tau-function, or any other solution, for which the space W is known. What should I do to restore the spectral curve? http://mathoverflow.net/questions/85693/from-sato-grassmannian-to-spectral-curve/85695#85695 Comment by Sasha Sasha 2012-01-14T22:38:05Z 2012-01-14T22:38:05Z Maarten, thank you for your ansver. I think I have to re-read Segal-Wilson. Just to be sure: assume I know the coefficients $a_{km}$ explicitely. Is there a step-by-step procedure to construct a spectral curve? http://mathoverflow.net/questions/59321/cut-and-join-equation-and-schur-function-identity/84937#84937 Comment by Sasha Sasha 2012-01-09T08:30:58Z 2012-01-09T08:30:58Z I am not sure if there is any good infinite product expression for H(t). But I think one can obtain at least some integral transform of such infinite product. Namely, the operator $\exp (t \Delta)$ can be represented as integral of the shift operator (as in the reference I gave you), which acts on the infinite product H(0) preserving its structure. http://mathoverflow.net/questions/78658/a-question-about-the-lambert-function Comment by Sasha Sasha 2011-10-20T09:54:52Z 2011-10-20T09:54:52Z The only relevant reference I have is a paper of Kazarian, <a href="http://arxiv.org/PS_cache/arxiv/pdf/0809/0809.3263v1.pdf" rel="nofollow">arxiv.org/PS_cache/arxiv/pdf/0809/0809.3263v1.pdf</a>, but I think it is to indirect. http://mathoverflow.net/questions/78658/a-question-about-the-lambert-function Comment by Sasha Sasha 2011-10-20T09:27:36Z 2011-10-20T09:27:36Z @Igor. Thank you. One can substitute a series $z(w)$ into $f(z)$, then the constraint will give you precisely one linear equation for every $\alpha_k$. http://mathoverflow.net/questions/47671/matrix-integral-identity Comment by Sasha Sasha 2011-10-20T09:14:02Z 2011-10-20T09:14:02Z This is just a notation sometimes used in the mathphys literature to show that you integrate over $2N^2$ real variables contrary to $N^2$ for the Hermitian model. http://mathoverflow.net/questions/39077/online-latex-editor Comment by Sasha Sasha 2010-09-17T11:49:49Z 2010-09-17T11:49:49Z Thanks a lot. Now I have docs.latexlab.org and www.scribtex.com, I think second one is more convenient.