User daniel - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T15:21:31Z http://mathoverflow.net/feeds/user/4526 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/123670/all-possible-linear-combinations-of-positive-half-integers-with-coefficients All possible linear combinations of positive half-integers with coefficients +/- 1 Daniel 2013-03-05T21:52:26Z 2013-03-06T01:54:45Z <p>This will be a simple problem on paper, but the brute force method is not really suitable for a computer, so I'm after a tricky algorithm that works in practice too: if $n$ positive half-integers $p_i$ are given, one can form $2^n$ linear combinations with coefficients $\pm 1$:</p> <p>$$P = e_1 p_1 + \ldots + e_n p_n$$</p> <p>Where $e_i = \pm 1$. With given $p_i$ one obtains $2^n$ values for $P$. Let's call $\mu(P)$ the multiplicity of $P$, the number of linear combinations that gives $P$. My goal is to know this $\mu(P)$. On a computer I could just loop over $2^n$ items given by n-length strings of +1 and -1's and record each $P$, but if $n$ is large ($n>64$) this becomes problematic if 64 bit integers are used (unsigned long long int in C). And also it becomes time consumong if $n$ is larger than 64 and some arbitrary precision library is used.</p> <p>Is there a tricky way to obtain $\mu(P)$ that does not require a loop over $2^n$ elements?</p> http://mathoverflow.net/questions/112496/asymptotic-expansion-of-integral-bessel-function-really Asymptotic expansion of integral (Bessel function really) Daniel 2012-11-15T16:30:47Z 2012-11-16T00:02:27Z <p>The integral</p> <p>$$I = \int_{-\infty}^\infty \frac{e^{-\varepsilon x^2}} { \sqrt{1+x^2} } dx$$</p> <p>is convergent for $\varepsilon > 0$ and can even be given in terms of the Bessel function $K_0$. As $\varepsilon \to 0$ it is divergent and $I \sim -\log \varepsilon$. What would be the simplest way to derive the above leading term in an asymptotic $\varepsilon \to 0$ expansion directly in terms of the above integral?</p> <p>Clearly, if one uses the exact result in terms of $K_0$ and then for instance uses the second order differential equation satisfied by $K_0$ it is quite simple to derive the $\log\varepsilon$ form of the divergence. But I'm looking for a way to derive this directly from the above integral. The reason is that I have a much more complicated integral to analyze which can not be given in a closed form but the above simple integral captures its difficulty so would like to understand this one first.</p> <p>Another related question: the integral</p> <p>$$J = \int_0^{2\pi} e^{-\sin(x)^2/\varepsilon^2}dx$$</p> <p>can also be evaluated exactly in terms of the Bessel function $I_0$ which result will imply $J \sim \varepsilon$ as $\epsilon\to 0$. But again, directly from the integral what is the simplest way to see this leading term in the $\varepsilon\to 0$ expansion?</p> http://mathoverflow.net/questions/110755/series-representation-of-ratio-of-two-meijer-g-functions Series representation of ratio of two Meijer G-functions Daniel 2012-10-26T14:14:30Z 2012-10-26T20:48:51Z <p>Let me use the notation from Maple <a href="http://www.maplesoft.com/support/help/Maple/view.aspx?path=MeijerG" rel="nofollow">http://www.maplesoft.com/support/help/Maple/view.aspx?path=MeijerG</a> for the Meijer G-function. Then let me define,</p> <p>$f_+(x) = MeijerG( [[+1/2],[]], [[0,0],[]], x )$</p> <p>$f_-(x) = MeijerG( [[-1/2],[]], [[0,0],[]], x )$</p> <p>Then by numerical evaluation I was able to show that</p> <p>$\lim_{x\to 0} \frac{f_-(x)}{f_+(x)} = 1/2$</p> <p>What would the further terms be in an expansion around $x=0$? I guess there will be all sorts of logarithms and other nasty singularities...</p> <p>(No, Maple 12, which I have installed, is not able to do a series expansion. I don't have access to Mathematica so I don't know about that.)</p> http://mathoverflow.net/questions/106586/simple-description-for-the-coset-sun-u1 Simple description for the coset SU(N)/U(1) Daniel 2012-09-07T09:21:14Z 2012-09-07T09:21:14Z <p>If I pick a fixed $U(1)\;\;$ subgroup in $SU(N)\;\;\;$, say a circle in the diagonal, I get the following action of $U(1)\;\;$ on $SU(N)\;\;$:</p> <p>$U \to g U g^{-1}\;\;\;\;\;$ where $U \in SU(N)\;\;\;\;$ and $\;\;g \in U(1)$</p> <p>The corresponding coset $SU(N)/U(1)\;\;\;\;\;$ is pretty simple for $N=2\;\;\;$ it is $SU(2)/U(1) = CP^1\;\;\;\;\;\;\;$. Is there a similarly simple description for $N>2\;\;\;$? What is the topology of the coset $SU(N)/U(1)\;\;\;\;\;\;$ to begin with for $N>2\;\;\;$?</p> http://mathoverflow.net/questions/81257/yang-mills-gradient-heat-flow-on-4-torus Yang Mills gradient/heat flow on 4-torus Daniel 2011-11-18T16:01:07Z 2011-11-24T23:20:16Z <p>The classic Donaldson-Kronheimer book (Geometry of 4-manifolds) uses the Yang Mills gradient flow (sometimes called heat flow) on $M$ all over the place,</p> <p>$\frac{d A}{dt} = -\frac{\delta YM(A)}{\delta A}$</p> <p>where $YM(A)$ is the Yang Mills 'action' the integral of the curvature square,</p> <p>$YM(A) = \int d^4x Tr F_{\mu\nu} F_{\mu\nu} > 0$</p> <p>The setting is quite general, either $M$ is a general 4-manifold or Kahler manifold and so all theorems, existence, uniqueness, etc, are quite general.</p> <p>I'm wondering if there are further results somewhere for the specific case of $M = T^4$ the 4-torus. For example, is it known how the long time asymptotics look like in this case? Theorems about possible blow-ups? I'd think asymptotically the gradient flow drives the connection towards a critical point but is it known how it is approached, exponentially or polynomially in $t$?</p> <p>Actually for $M=T^4$ I suspect $t^2 YM(A(t))$ goes to a constant for $t\to\infty$ as long as the initial condition for the flow is in a sufficiently small neighbourhood of the absolute minimum of $YM(A)$ but I can't prove it. What is certainly true is that $YM(A(t))$ is a decreasing function of $t$.</p> http://mathoverflow.net/questions/63345/famous-2d-riemannian-manifolds-with-non-constant-curvature "Famous" 2d Riemannian manifolds with non-constant curvature Daniel 2011-04-28T21:23:03Z 2011-05-03T10:53:51Z <p>I'm looking for "famous" or otherwise well-known 2d Riemannian manifolds which have non-constant curvatures but have a non-trivial Killing vector field. Of course there are tons of spaces like these, for instance if we parametrize the plane (or a subset of it) by $(r,\phi)$ then any conformal rescaling of the flat metric by a conformal factor which only depends on $r$ will be generally good, i.e. have non-constant curvature but the rotations generated by $\partial/\partial\phi$ will still be a symmetry.</p> <p>Are there special spaces which are somehow famous or well-studied because of some special property? Ideally, I'm looking for deformations of the Poincare disc.</p> http://mathoverflow.net/questions/34987/homotopy-group-of-space-of-gauge-fields-modulo-gauge-equivalence-on-t4 Homotopy group of space of gauge fields modulo gauge equivalence on T^4 Daniel 2010-08-09T10:08:15Z 2010-08-09T10:08:15Z <p>Singer observed in 1978 (Comm.Math.Phys. 60, 7-12) that the homotopy group of the space of gauge fields modulo gauge equivalence with gauge group $G$ on $S^4$ is given by</p> <p>$\pi_n({\cal A}/{\cal G}) = \pi_{n-1}{\cal G} = \pi_{n+3} G$</p> <p>Does anyone know what the corresponding expression is if the base manifold $S^4$ is replaced by $T^4$?</p> http://mathoverflow.net/questions/27853/infinite-dimensional-unitary-representations-of-su2-for-non-half-integer-j Infinite dimensional unitary representations of SU(2) for non-half-integer j? Daniel 2010-06-11T19:14:16Z 2010-06-12T07:27:24Z <p>The finite dimensional irreducible unitary representations of $SU(2)$ are labelled by $j$ which needs to be half-integer, the dimension of the representation is $2j+1$. This is well-known, all is good.</p> <p>If we do not require finite dimension for the representation, is it possible to make sense of representations with an arbitrary real number $j$? They will, presumably, be infinite dimensional but hopefully still unitary.</p> <p>In the half-integer case when represented on at most $2j$ degree holomorphic polynomials, the 3 basis elements of the Lie-algebra in representation $j$ act as</p> <p>$e_1 = \frac{1-z^2}{2}\frac{d}{dz} + jz$</p> <p>$e_2 = \frac{1+z^2}{2i}\frac{d}{dz} + ijz$</p> <p>$e_3 = -z\frac{d}{dz} + j$</p> <p>Clearly, if $j$ is not half-integer and we start from $f(z) = z$ and start acting on it with $e_i$, it will generate an infinite dimensional space.</p> <p>This kinda gives me the feeling that perhaps non-half-integer $j$ representations are still meaningful and are infinite dimensional. But I'm not sure, is this really the case or something will go wrong?</p> <p>Basically what I'm asking is whether analytic continuation in $j$ makes any sense.</p> http://mathoverflow.net/questions/27825/l2-space-of-holomorphic-functions-with-given-weight L^2 space of holomorphic functions with given weight Daniel 2010-06-11T15:30:59Z 2010-06-11T20:42:15Z <p>Hi folks, what is known about the $L^2$ space of holomorphic functions of 1 complex variable with the scalar product</p> <p>$\langle f, g \rangle = \int dzd{\bar z} \frac{ {\bar f(z)} g(z) }{(1 + z{\bar z})^x}$</p> <p>where $x > 2$ is a real number? The domain of integration is the entire complex plane. Poles are allowed in the functions so all possible powers in the Laurent expansion are allowed, $f(z) = \sum_{n = -\infty}^\infty f_n z^n$.</p> <p>Is this a well-known space? Is an orthogonal basis readily available?</p> <p>If $f(z)$ is a polynomial with sufficiently low degree then certainly it is in the above defined $L^2$ space. But there are much more functions that are okay, it seems, for instance $f(z) = \exp( -z )$. Or anything that falls off sufficiently fast.</p> <p>The background is this: if $x=2j+2$ where $j$ is a half-integer and the holomorphic functions can only be at most $2j$ order polynomials, then the above defined space is the $2j+1$ dimensional irreducible unitary representation of $SU(2)$. The action of $g = [ [ a, b ], [ c, d ] ] \in SU(2)$ is</p> <p>$(gf)(z) = (bz + d)^{2j} f\left( \frac{az+c}{bz+d} \right)$</p> <p>Clearly, if $f(z)$ is a polynomial at most of order $2j$ then $(gf)(z)$ is also one. And the scalar product is the one I gave above, with $x=2j+2$.</p> <p>Okay, this was the case for half-integer $j$. What is the deal with arbitrary $j$? Then I can still define the above scalar product with arbitrary $x$. The action above still preserves the scalar product. It is still a group action by $SU(2)$. Do I get an infinite dimensional representation of $SU(2)$? Is it reducible/irreducible?</p> http://mathoverflow.net/questions/17712/bounded-homogeneous-quartics bounded homogeneous quartics Daniel 2010-03-10T12:12:36Z 2010-03-10T12:18:28Z <p>If Q is a real homogeneous quartic on $R^N$,</p> <p>$Q(x) = \sum_{1 &lt;= i,j,k,l &lt;= N} Q_{ijkl} x_i x_j x_k x_l$</p> <p>what is the condition on the (totally symmetric) coefficients $Q_{ijkl}$ for Q being bounded from below? I'm looking for the simplest expression in terms of $Q_{ijkl}$. Clearly, if $Q_{ijkl}$, as considered a map from the space of real symmetric matrices to the space of real symmetric matrices is positive semi-definite, is enough. But this is a too strong condition because $x_i x_j$ is a rank-1 real symmetric matrix, so in Q(x) Q is only evaluated on rank-1 matrices, not on every real symmetric matrix.</p> http://mathoverflow.net/questions/123670/all-possible-linear-combinations-of-positive-half-integers-with-coefficients/123673#123673 Comment by Daniel Daniel 2013-03-06T10:00:41Z 2013-03-06T10:00:41Z This is very helpful, thanks. The $p_i$ numbers are not large always smaller than 10. And $n$ is always less than 100. http://mathoverflow.net/questions/112496/asymptotic-expansion-of-integral-bessel-function-really/112519#112519 Comment by Daniel Daniel 2012-11-15T21:48:16Z 2012-11-15T21:48:16Z Thanks, I didn't see your answer before adding the second part of the question. For the first part it's definitely what I was looking for, thanks a lot! http://mathoverflow.net/questions/110755/series-representation-of-ratio-of-two-meijer-g-functions/110778#110778 Comment by Daniel Daniel 2012-10-26T21:03:27Z 2012-10-26T21:03:27Z It turns out the only reason maple 12 couldn't do the expansion is the I use Fedora 16 linux distribution and there is a known bug in maple that only happens on this platform: <a href="http://www.mapleprimes.com/questions/130220-Maple-15-X86-64-LINUX-Fedora-16-Log2" rel="nofollow">mapleprimes.com/questions/&hellip;</a> http://mathoverflow.net/questions/110755/series-representation-of-ratio-of-two-meijer-g-functions/110756#110756 Comment by Daniel Daniel 2012-10-26T20:43:10Z 2012-10-26T20:43:10Z Great, thanks a lot, I always forget wolframalpha! http://mathoverflow.net/questions/106586/simple-description-for-the-coset-sun-u1 Comment by Daniel Daniel 2012-09-07T14:01:39Z 2012-09-07T14:01:39Z Thanks a lot, I indeed overlooked that! http://mathoverflow.net/questions/81257/yang-mills-gradient-heat-flow-on-4-torus/81818#81818 Comment by Daniel Daniel 2011-11-28T11:48:00Z 2011-11-28T11:48:00Z Thanks a lot Willie, this link indeed works! http://mathoverflow.net/questions/81257/yang-mills-gradient-heat-flow-on-4-torus/81818#81818 Comment by Daniel Daniel 2011-11-25T11:46:13Z 2011-11-25T11:46:13Z Sounds like a good idea, thanks for the reference, unfortunately our university doesn't appear to have digital subscription so can't access the article. Does anyone have a digital copy? http://mathoverflow.net/questions/81257/yang-mills-gradient-heat-flow-on-4-torus/81818#81818 Comment by Daniel Daniel 2011-11-24T23:22:17Z 2011-11-24T23:22:17Z Thanks, the Schlatter reference was really useful! The reason I'm asking about $T^4$ is that this is the case I can study numerically. When I numerically solve the gradient flow it seems to have the property that $t^2 YM(A(t))$ goes to a constant, but I'm not really sure. http://mathoverflow.net/questions/81257/yang-mills-gradient-heat-flow-on-4-torus/81267#81267 Comment by Daniel Daniel 2011-11-18T22:22:10Z 2011-11-18T22:22:10Z That paper is only dealing with the flow on Riemann surfaces. http://mathoverflow.net/questions/1973/is-there-a-complex-structure-on-the-6-sphere Comment by Daniel Daniel 2011-04-30T10:28:10Z 2011-04-30T10:28:10Z And there is a new version out: <a href="http://arxiv.org/abs/math/0505634" rel="nofollow">arxiv.org/abs/math/0505634</a> claiming to completely overhaul the proof. Did anyone take a look with expertise in this area? http://mathoverflow.net/questions/63345/famous-2d-riemannian-manifolds-with-non-constant-curvature Comment by Daniel Daniel 2011-04-30T05:23:36Z 2011-04-30T05:23:36Z Deane, something tells me that Dan Fox's family can perhaps be &quot;converted&quot; into a family on the unit disk by sticking in a couple of minus signs, but I'm not sure. The reason I think this is that the metric on the sphere $dzd{\bar z}/(1+|z|^2)^2$ and the Poincare metric on the disk $dzd{\bar z}/(1−|z|^2)^2$ are related by a minus sign. http://mathoverflow.net/questions/63345/famous-2d-riemannian-manifolds-with-non-constant-curvature Comment by Daniel Daniel 2011-04-29T20:33:04Z 2011-04-29T20:33:04Z Deane, what I mean by &quot;deformation of the Poincare disk&quot; is just a one-parameter family of metrics which for a fixed value of the parameter equals the Poincare metric. For example Dan Fox gave a reply to my question below which has a formula for a one parameter family of metrics, but unfortunately his family of metrics is given on the sphere. I'm looking for something like this, but for the unit disk. http://mathoverflow.net/questions/63345/famous-2d-riemannian-manifolds-with-non-constant-curvature/63366#63366 Comment by Daniel Daniel 2011-04-29T16:34:36Z 2011-04-29T16:34:36Z Are any of these deformations of the Poincare disc? (I should have added that ideally I'm looking for &quot;famous&quot; one or two parameter deformations of the Poincare disc such that the deformation introduces the non-constant curvature.) http://mathoverflow.net/questions/63345/famous-2d-riemannian-manifolds-with-non-constant-curvature/63382#63382 Comment by Daniel Daniel 2011-04-29T16:33:32Z 2011-04-29T16:33:32Z Thanks a lot, this was very helpful. One question: since I'm primarily interested in the non-compact case (unit disc or the entire plane) can I stick a minus sign in front of the cosh(2t) in the metric or switch the $\cosh$ and $\sinh$ to $\cos$ and $\sin$ in order to have something which looks like the one parameter deformation of the Poincare disc? http://mathoverflow.net/questions/63345/famous-2d-riemannian-manifolds-with-non-constant-curvature Comment by Daniel Daniel 2011-04-29T16:28:23Z 2011-04-29T16:28:23Z Sorry, I should have said Poincare disc, not $AdS_2$. Because of the constant negative curvature I mistakenly thought it's anti de Sitter.