coordinate - How to calculate distance from object to point on celestial sphere

You can use SkyCoord.separation to compute the separation between coordinates:

c1 = SkyCoord('08h55m10s',' -7d14m42s', frame='icrs')  # your coords
c2 = SkyCoord(133.782, -7.241, unit='deg', frame='icrs')  # first object in table
sep = c1.separation(c2)

print c1
<SkyCoord (ICRS): ra=133.791666667 deg, dec=-7.245 deg>
print c2
<SkyCoord (ICRS): ra=133.782208333 deg, dec=-7.24123611111 deg>
print sep
0d00m36.3947s

The table you printed is truncated in between. If you enlarge your terminal emulator, you will be able to see a column called dist, which is the distance between your input coordinates and the corresponding object in arcsecs:

print table['dist']
       dist       
------------------
36.420772999999997
44.138542000000001
51.987606999999997
         52.838003
49.937266999999999
22.372699000000001
53.621690000000001
         12.375482
          25.27176
53.707365000000003

Then the nearest object is the one whose separation is:

print table['dist'].min()
12.375482

Or equivalently, the one whose index is:

print table['dist'].argmin()
7

The separation is calculated using the equation that @Rob Jeffries pointed out.

mg.metric geometry - Compute the Centroid of a 3D Planar Polygon

In response to JBL's comment, I offer this answer merely to close out this topic. It has been effectively answered in the comments:
Simply project to xy and to xz and compute the centroid there. (One tiny wrinkle not addressed is if the polygon lies in a plane perpendicular to xy or to xz. But then simply chose the coordinate planes in which it does not lie.)

On the advice of Andrew Stacey, I am designating this answer "community wiki," and hope that someone will vote it up so it will no longer be bumped to the top of the active list by the MO background process.

space - What would happen if a rocket traveling at speed of light would collide with a planet (like earth)?

Let's assume it is possible for a spacecraft to travel at the speed of light (I've read the interstellar book by Kip Thorne, apparently this is theoretically possible if you swing around two black holes)

I have watched the video where the traveling of a light particle is simulated and while planets are certainly big enough so you can adjust your route, what would happen if a rocket (or spaceship in general) would collide with an earth-like planet (i.e. a planet having a massive surface). Would we burst through the whole planet, make it only to a core, or just a few km? I know what happens when a meteor lands, but obviously they aren't traveling at the speed of light.

And one similar another question - when a spaceship is traveling at the speed of light, is there a problem with collision with smaller objects? I think you can't avoid millions of rockets in outer space, it would surely damage the spaceship, wouldn't it?

Why do we have so few neutrinos to study?

It is because the neutrinos cross section is very small.

Roughly, the cross section is a measure of the interaction probability (for a given force) of the given particle with your target.

Since the probability for interaction depends upon the strength and range of the interaction, the cross-sections for neutrino interactions are very small since they interact only by the weak interaction.

Hence, it is not that we receive few neutrinos, rather it is our (in)ability to detect them, due to our current knowledge of their intrinsic nature, which prevent us from measuring high fluxes of neutrinos.

For this very reason, neutrinos detectors are made of huge volume targets, in order to maximize the detection probability.

gr.group theory - А generalization of Gromov's theorem on polynomial growth

My paper with Shalom does settle the question when $S = S_n$ is known to have size polynomial in n (and maybe is allowed to grow just a little bit faster than this, something like $n^{(log log n)^c}$ or so), but I doubt that the result is known yet if S is allowed to be arbitrarily large.

Note that even the bounded case is nontrivial - it's not obvious why having $|S_n^n| leq n^{O(1)} |S_n|$ implies polynomial growth. (There is no reason why growth has to be uniform for fixed cardinality of generators; for instance, I believe it is a major open problem (due to Gromov?) as to whether exponential growth is the same as uniform exponential growth for finitely generated groups.)

If we had a good non-commutative Freiman theorem, then one may possibly be able to settle your question affirmatively (note from the pigeonhole principle that if $|S_n^n| leq n^C |S_n|$ for some large n, then there exists an intermediate $m=m_n$ between 1 and n such that the set $B := S_n^m$ has small doubling, in the sense that $|B^2| = O(|B|)$). The best result in this direction for general groups currently is due to Hrushovski, which does show that sets of small doubling contain some vaguely "virtually nilpotent" structure, but it is not yet enough to give Gromov's theorem, let alone the generalisation you mention above. More is known if one already has some additional structure on the group (e.g. it has a faithful linear representation of bounded dimension, or if it is already known to be virtually solvable).

How often does Earth cross the galaxy regions with higher supernovae activity?

Actually i read some where that in the entire time of humankind on earth, the earth has just covered 1/10th of a percent around the milky way galaxy, so assuming earth would survive that long and humankind would survive to witness it, i would say it would take billions-billions of year, before earth would move to closer to giant stars.

As for the supernova activity, humankind hasn't seen a supernova (excluding some theories), but our best bet to witness on would be Betelgeuse, it is already old for its size class and is expected to explode relatively soon (it may explode tomorrow or after a million years nobody knows exactly when) compared to its age.

So, i would say, we would be lucky if we witness a supernova in our lifetime.

co.combinatorics - Constructing a smooth lattice from a discrete one.

I have the standard lattice L defined over partitions of $1ldots n$ under the split-merge relation. I also have an antimonotone function from L to R that's submodular, and so gives me a metric on L via the construction $d(x,y) = 2f(x wedge y) - f(x)-f(y)$.

In general, but on this lattice specifically, I'm wondering if there's a construction that allows me to define a continuous lattice C with the property that L behaves like the "integer points" on C, and a metric $d'$ can be induced on C that when restricted to L behaves like $d$.

For example, if I have two points $x le y$ in $L$, I'd like to construct a point $z$ "halfway" between $x$ and $y$ such that under the metric $d', d'(x,z) = d'(z,y)$, and under the ordering that defines $C, x le z le y$.