Projection onto the subspace of spherical harmonics with the same degree

Recently, I've been interested in expanding an angular function over the spherical harmonics, and particularly in retrieving the amplitude of the part corresponding to a given degree \(\ell\). More precisely, let \(F(\Omega) = F(\theta,\phi) =\sum_{\ell} \sum_{m} Y_{\ell m} (\Omega)\). The projection of \(F\) onto the subspace spanned by the harmonics with a given degree \(\ell\) (I believe this space is generally denoted by \(\mathcal{H}_{\ell}\)) is:
\begin{equation}
\label{eq:proj1}
\operatorname{Proj}_{\ell} \left [F \right ] (\Omega) = \sum_{m= - \ell}^{\ell} c_{\ell m} Y_{\ell m} (\Omega)
\end{equation}

Temporary positions in science

A very interesting article in Nature on the future of postdoc positions. Well worth reading, although the solutions proposed (increasing the proportion of permanent research staff, in one way or another) are completely impractical. The comments are even more interesting than the article itself.

Posting on arXiv

Since I'm on vacation these days, I decided to finally submit my published research papers to arXiv. A previous attempt was about as pleasant as a visit to the dentist, but somewhat longer (and unsuccessful). This time around I tried to submit other papers, and things went more or less smoothly, as soon as I learned to follow some guidelines (for documents produced using LaTeX):

• Make sure that all .eps figures are of reasonable size: no files above 6MB and no more that 10MB for the whole submission. I downsized some very large files by first converting them to .pdf and cropping to remove white margins (using Acrobat) then opened the .pdf files in Photoshop and saved a copy in .eps format (without preview and using jpeg compression).
• Check that the figures are correctly invoked in the .tex file (the name should be case-sensitive, something that is not required on Windows systems). File names should not contain special characters (more details).
• Run LaTeX locally until the compilation is error-free.
• Create a .zip archive with the .tex file, the .eps figures and the .bbl file (the compiling process on arXiv does not include Bibtex) and, if necessary, with supplementary material in .ps format.
• Hope that the compilation works without errors, otherwise wade through about three screens of output, fix things and reload files. The supplementary files are simply appended to the final .pdf; this is an easy way to include additional material or tricky parts that fail to compile on the arXiv server. This is how I managed to add to one paper a page-wide table in landscape orientation, which had resisted all other methods.

The final result is here.

 

For Word documents the process should be much easier, since one can directly submit the .pdf version. I only tried this for one paper, currently on hold because it has line numbers in the margin. This restriction is not mentioned anywhere on the arXiv site (neither is the size limit, by the way.)

Overall, the entire procedure was easier than I thought. Still, the system is far from user-friendly (metadata retrieval using the DOI would be nice), the interface is firmly stuck in the 90s, some limitations seem a bit arbitrary and the help could be more detailed.

How to read an equation

The mere formal expression of an equation is not very useful, unless complemented by a more or less intuitive understanding. Different people may have different intuitions of a given formula or different mental images of one physical systems (more on that later).

The interesting part is that putting together two such different intuitions of a relation can yield non-trivial results with almost no algebraic manipulation, as I'll show below. What is the meaning of the following formula ?

\[\frac{1}{\sqrt{2\pi} \sigma} \int_{-\infty}^{\infty} \text{d}x \exp (i q x) \exp \left (- \frac{x^2}{2 \sigma ^2} \right ) \tag{1}\]

Fundamental physics

The other day at the gym I realized I am a force of nature. The weak force, to be specific. It could have been worse, though: gravity was on a treadmill not far away.