User daniel - MathOverflow

All possible linear combinations of positive half-integers with coefficients +/- 1

2013-03-05T21:52:26Z

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 $\pm 1$:

$$P = e_1 p_1 + \ldots + e_n p_n$$

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.

Is there a tricky way to obtain $\mu(P)$ that does not require a loop over $2^n$ elements?

Asymptotic expansion of integral (Bessel function really)

2012-11-15T16:30:47Z

The integral

$$I = \int_{-\infty}^\infty \frac{e^{-\varepsilon x^2}} { \sqrt{1+x^2} } dx$$

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?

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.

Another related question: the integral

$$J = \int_0^{2\pi} e^{-\sin(x)^2/\varepsilon^2}dx $$

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?

Series representation of ratio of two Meijer G-functions

2012-10-26T14:14:30Z

Let me use the notation from Maple http://www.maplesoft.com/support/help/Maple/view.aspx?path=MeijerG for the Meijer G-function. Then let me define,

$f_+(x) = MeijerG( [[+1/2],[]], [[0,0],[]], x )$

$f_-(x) = MeijerG( [[-1/2],[]], [[0,0],[]], x )$

Then by numerical evaluation I was able to show that

$\lim_{x\to 0} \frac{f_-(x)}{f_+(x)} = 1/2$

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...

(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.)

Simple description for the coset SU(N)/U(1)

2012-09-07T09:21:14Z

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 $SU(N)\;\;$:

$U \to g U g^{-1}\;\;\;\;\;$ where $U \in SU(N)\;\;\;\;$ and $\;\;g \in U(1)$

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 $N>2\;\;\;$?

Yang Mills gradient/heat flow on 4-torus

2011-11-18T16:01:07Z

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,

$\frac{d A}{dt} = -\frac{\delta YM(A)}{\delta A}$

where $YM(A)$ is the Yang Mills 'action' the integral of the curvature square,

$YM(A) = \int d^4x Tr F_{\mu\nu} F_{\mu\nu} > 0$

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.

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 $t$?

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 $t$.

"Famous" 2d Riemannian manifolds with non-constant curvature

2011-04-28T21:23:03Z

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.

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.

Homotopy group of space of gauge fields modulo gauge equivalence on T^4

2010-08-09T10:08:15Z

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

$\pi_n({\cal A}/{\cal G}) = \pi_{n-1}{\cal G} = \pi_{n+3} G$

Does anyone know what the corresponding expression is if the base manifold $S^4$ is replaced by $T^4$?

Infinite dimensional unitary representations of SU(2) for non-half-integer j?

2010-06-11T19:14:16Z

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.

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.

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

$e_1 = \frac{1-z^2}{2}\frac{d}{dz} + jz$

$e_2 = \frac{1+z^2}{2i}\frac{d}{dz} + ijz$

$e_3 = -z\frac{d}{dz} + j$

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.

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?

Basically what I'm asking is whether analytic continuation in $j$ makes any sense.

L^2 space of holomorphic functions with given weight

2010-06-11T15:30:59Z

Hi folks, what is known about the $L^2$ space of holomorphic functions of 1 complex variable with the scalar product

$\langle f, g \rangle = \int dzd{\bar z} \frac{ {\bar f(z)} g(z) }{(1 + z{\bar z})^x}$

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, $f(z) = \sum_{n = -\infty}^\infty f_n z^n$.

Is this a well-known space? Is an orthogonal basis readily available?

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.

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

$(gf)(z) = (bz + d)^{2j} f\left( \frac{az+c}{bz+d} \right)$

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 $x=2j+2$.

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?

bounded homogeneous quartics

2010-03-10T12:12:36Z

If Q is a real homogeneous quartic on $R^N$,

$Q(x) = \sum_{1 <= i,j,k,l <= N} Q_{ijkl} x_i x_j x_k x_l$

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.

Comment by Daniel

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.

Comment by Daniel

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!

Comment by Daniel

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: mapleprimes.com/questions/…

Comment by Daniel

2012-10-26T20:43:10Z

Great, thanks a lot, I always forget wolframalpha!

Comment by Daniel

2012-09-07T14:01:39Z

Thanks a lot, I indeed overlooked that!

Comment by Daniel

2011-11-28T11:48:00Z

Thanks a lot Willie, this link indeed works!

Comment by Daniel

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?

Comment by Daniel

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.

Comment by Daniel

2011-11-18T22:22:10Z

That paper is only dealing with the flow on Riemann surfaces.

Comment by Daniel

2011-04-30T10:28:10Z

And there is a new version out: arxiv.org/abs/math/0505634 claiming to completely overhaul the proof. Did anyone take a look with expertise in this area?

Comment by Daniel

2011-04-30T05:23:36Z

Deane, something tells me that Dan Fox's family can perhaps be "converted" 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.

Comment by Daniel

2011-04-29T20:33:04Z

Deane, what I mean by "deformation of the Poincare disk" 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.

Comment by Daniel

2011-04-29T16:34:36Z

Are any of these deformations of the Poincare disc? (I should have added that ideally I'm looking for "famous" one or two parameter deformations of the Poincare disc such that the deformation introduces the non-constant curvature.)

Comment by Daniel

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?

Comment by Daniel

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.