rotation - How would Earth's climate differ if it's axis were tilted around 90 degrees like Uranus?

This is a good question, and has been somewhat a topic of study by a few. So below is some consequences of if Earth had a tilt like that of Uranus (for this explanation, I am disregarding the wobbles in the Earth's current axis, just focussing on the 90 degree axis):

According to the article Not All Habitable Zones Are Created Equal (Moomaw, 1999), each day and night would be 6 months each (including an epically long dusk and dawn), with daytime temperatures reaching up to 80C, but the night time side may not reach freezing - owing to the time to take to cool down from the 6 month day. Parts of the equatorial regions would be, however, permanently encased in ice.

Pretty much like this diagram of Uranus' rotation (from Source: University of Hawaii:

A major consequence of this according to the author is:

In such an environment, life could almost certainly still appear, but it would have much more difficulty evolving into forms that could survive such grotesque temperature extremes -- which would greatly slow down its evolution into more complex forms, maybe by billions of years.

A great consequence of this, particularly the 6 month night is that photosynthesis would have been stilted, if it could have started at all. This has a great consequence on oxygen levels of the atmosphere.

Interestingly, the authors claim that if the Earth had an axial tilt of 90 degrees, but at 210 million km from the sun, then:

its climate would be positively balmy -- the equator would be 11 deg C (52 deg F), and the poles would never rise above 46 deg C (115 deg F) or fall below 3 deg C (37 deg F). Earth would have no ice anywhere on its surface, except on some of its highest mountains.

According to the article High Planetary Tilt Lowers Odds for Life? (Hadhazy, 2012), has a great way of putting it:

"Your northern pole will be boiled during part of the year while the equator gets little sunlight," said Heller. Meanwhile, "the southern pole freezes in total darkness." Essentially, the conventional notion of a scorching hell dominates one side of the planet, while an ultra-cold hell like that of Dante's Ninth Circle prevails on the other.

Then, to make matters worse, the hells reverse half a year later. "The hemispheres are cyclically sterilized, either by too strong irradiation or by freezing," Heller said.

They also describe that if life were to evolve, extremophiles (specifically, thermophiles) would be dominant - seasonally.

a infinity algebras - Is there a refinement of the Hochschild-Kostant-Rosenberg theorem for cohomology?

The HKR theorem for cohomology in characteristic zero says that if $R$ is a regular, commutative $k$ algebra ($char(k) = 0$) then a certain map $bigwedge^* Der(R) to CH^*(R,R)$ (where $wedge^* Der(R)$ has zero differential) is a quasi-isomorphism of dg vector spaces, that is, it induces an isomorphism of graded vector spaces on cohomology.

Can the HKR morphism be extended to an $A_infty$ morphism? Is there a refinement in this spirit to make up for the fact that it is not, on the nose, a morphism of dg-algebras?

ag.algebraic geometry - Eichler-Shimura isomorphism and mixed Hodge theory

Let $Y(N),N>2$ be the quotient of the upper half-plane by $Gamma(N)$ (which is formed by the elements of $SL(2,mathbf{Z})$ congruent to $I$ mod $N$). Let $V_k$ be the $k$-th symmetric power of the Hodge local system on $X(N)$ tensored by $mathbf{Q}$ (the Hodge local system corresponds to the standard action of $Gamma(N)$ on $mathbf{Z}^2$).

$V_k$ is a part of a variation of polarized Hodge structure of weight $k$. So the cohomology $H^1(Y(N),V_k)$ is equipped with a mixed Hodge structure (the structure will be mixed despite the fact that $V_k$ is pure because $Y(N)$ is not complete). The complexification $H^1(Y(N),V_kotimesmathbf{C})$ splits

$$H^1(Y(N),V_kotimesmathbf{C})=H^{k+1,0}oplus H^{0,k+1}oplus H^{k+1,k+1}.$$

There is a natural way to get cohomology classes $in H^1(Y(N),V_kotimesmathbf{C})$ from modular forms for $Gamma(N)$. Namely, to a modular form $f$ of weight $k+2$ one associates the secion

$$zmapsto f(z)(ze_1+e_2)^k dz$$

of $$Sym^k(mathbf{C}^2)otimes Omega^1_{mathbf{H}}.$$

Here $mathbf{H}$ is the upper half plane and $(e_1,e_2)$ is a basis of $mathbf{C}^2$ coming from a basis of $mathbf{Z}^2$. This pushes down to a holomorphic section of
$V_kotimes mathbf{C}$.

Deligne had conjectured (Formes modulaires et repr'esentations l-adiques, Bourbaki talk, 1968/69) that the above correspondence gives a bijection between the cusp forms of weight $k+2$ and $H^{k+1,0}$. (This was before he had even constructed the Hodge theory, so strictly speaking this can't be called a conjecture, but anyway.) Subsequently this was proved by Zucker (Hodge theory with degenerating coefficients, Anns of Maths 109, no 3, 1979). See also Bayer, Neukirch, On automorphic forms and Hodge theory, (Math Ann, 257, no 2, 1981).

The above results concern cusp forms and it is natural to ask what all modular forms correspond to in terms of Hodge theory. It turns out that all weight $k+2$ modular forms give precisely the $k+1$-st term of the Hodge filtration on $H^1(Y(N),V_kotimesmathbf{C})$ i.e. $H^{k+1,0}oplus H^{k+1,k+1}$.

The proof of this is not too difficult but a bit tedious. So I would like to ask: is there a reference for this?

upd: The original posting contained non-standard notation; this has been fixed.

big picture - Why is it useful to study vector bundles?

I think many of the other answers boil down to the same underlying idea: Sections of vector bundles are "generalized functions" or "twisted functions" on your manifold/variety/whatever.

For example, Charles mentions subvarieties, which are roughly "zero loci of functions". However, there are no non-constant holomorphic global functions on, say, a projective variety. So how can we talk about subvarieties of a projective variety? Well, we do have non-constant holomorphic functions locally, so we can still define subvarieties locally as being zero loci of functions. But the functions $f_i$ which define a subvariety on one open set $U$ and the functions $g_i$ which define a subvariety on another open set $V$ won't necessarily agree on $U cap V$. We need some kind of "twist" to make the $f_i$'s and the $g_i$'s match up on $U cap V$. Upon doing so, the global object that we obtain is not a global function (because, again, there are no non-constant global functions) but a "twisted" global function, in other words a section of a vector bundle whose transition functions are described by these "twists".

Similarly, sections of vector bundles and line bundles are a nice way to talk about functions with poles. Meromorphic functions then become simply sections of a line bundle, which is nice because it allows us to avoid having to talk about $infty$. This is essentially why line bundles are related to maps to projective space $X to mathbb{P}^n$; intuitively, $n+1$ sections of a line bundle over $X$ is the same as $n+1$ meromorphic functions on $X$, which is the same as a map "$X to (mathbb{C} cup infty)^{n+1}$" which becomes a map "$X to mathbb{P}^n$" after we "projectivize".

One way to think of vector bundles and their sections as being invariants of your manifold/variety/whatever is to think of them as describing what kinds of "generalized" or "twisted" functions are possible on your manifold/variety.

The view of sections of vector bundles as being "twisted functions" is also useful for physics, as in e.g. David's answer. For instance, suppose we have a manifold, which we think of as being some space in which particles are moving around. We have local coordinates on the manifold, which are used to describe the position of the particles. Since we are on a manifold, the transitions between the local coordinates are nontrivial. We may also be interested in studying, say, the velocities or momenta (or acceleration, etc.) of the particles moving around in space. On local charts we can describe these momenta easily in terms of the local coordinates, but then for a global description we need transitions between these local descriptions of momenta, just like how we need transitions between the local coordinates in order to describe the manifold globally. The transitions between local descriptions of momenta are not the same as that between the local coordinates (though the former depends on the latter); phrased differently, we obtain a non-trivial (ok, not always non-trivial, but usually non-trivial) vector bundle over our manifold.

distances - Why has Moving Cluster Method been successful only for Hyades?

Do you know this paper? Mamajek (2005). "A Moving Cluster Distance to the Exoplanet 2M1207b in the TW Hydrae Association".

I'll risk an opinion:
There is about a thousand clusters with kinematic measurements, but besides no more than half a dozen, all pm components are under 20 mas (milli-arcseconds) with errors more or less in the range 0.3-6 mas. Such errors are too high in relation to the pm values and as a result, the vectors will not converge.

sequences and series - Uniquely generate all permutations of three digits that sum to a particular value?

Visualizing this problem, as unique ways to hand out ninja stars to ninjas. This also shows how each larger solution is made up of its neighboring, more simple solutions.

Here is how to implement it in php: (might help you understand it too)

function multichoose($k,$n)
{
 if ($k < 0 || $n < 0) return false;
 if ($k==0) return array(array_fill(0,$n,0));
 if ($n==0) return array();
     if ($n==1) return array(array($k));
     foreach(multichoose($k,$n-1) as $in){ //Gets from a smaller solution -above as (blue)
  array_unshift($in,0);  //This prepends the array with a 0 -above as (grey)
      $out[]=$in;
     }
     foreach(multichoose($k-1,$n) as $in){ //Gets the next part from a smaller solution too. -above as (red and orange)
  $in[0]++; //Increments the first row by one -above as (orange)
      $out[]=$in;
     }
     return $out;
}

print_r(multichoose(3,4)); //How many ways to give three ninja stars to four ninjas?

Not optimal code: Its more understandable that way.

Our output:

(0,0,0,3)
(0,0,1,2)
(0,0,2,1)
(0,0,3,0)
(0,1,0,2)
(0,1,1,1)
(0,1,2,0)
(0,2,0,1)
(0,2,1,0)
(0,3,0,0)
(1,0,0,2)
(1,0,1,1)
(1,0,2,0)
(1,1,0,1)
(1,1,1,0)
(1,2,0,0)
(2,0,0,1)
(2,0,1,0)
(2,1,0,0)
(3,0,0,0)

Fun use to note: Upc relies upon this exact problem in barcodes. The sum of the whitespace and blackspace for each number is always 7, but is distributed in different ways.

//Digit   L Pattern  R Pattern  LR Pattern (Number of times a bit is repeated)
    0   0001101    1110010    2100
    1   0011001    1100110    1110
    2   0010011    1101100    1011
    3   0111101    1000010    0300
    4   0100011    1011100    0021
    5   0110001    1001110    0120
    6   0101111    1010000    0003
    7   0111011    1000100    0201
    8   0110111    1001000    0102
    9   0001011    1110100    2001

Note only 10 of the 20 combinations are used, which means the code can be read upside-down just fine. All 20 can be used however, and are in EAN13, with a bit more complexity.

http://en.wikipedia.org/wiki/EAN-13

http://en.wikipedia.org/wiki/Universal_Product_Code

http://www.freeimagehosting.net/uploads/58531735d3.png

How large must an object be to be seen through a telescope?

A few clarifications.

Telescopes in general operate at very large distances - "at infinity" is the term used in optics parlance.

A bright enough object can be seen from any distance, no matter what its size is. All that matters is that:

1. It's bright enough to produce an impression on whatever sensor
   you're using (or your eye)




2. The background is dark enough to produce sufficient contrast

But then it would be just a bright but tiny spot.

I believe what you're really asking for is: what is the combination of factors that shows the object as bigger than a simple dot? In that case, it's two factors: aperture of the telescope, and angular size of the object.

Assuming a flawless telescope, its aperture is what determines its resolving power. The resolving power is the angle at which two dots can be separated by the telescope. The formula is:

resolving power = 1 / (10 * aperture)

where resolving power is in arcseconds, and aperture is in meters. Examples:

aperture         resolving power
10 cm            1 arcsec
20 cm            0.5 arcsec
1 m              0.1 arcsec

As long as the object's angular size is bigger than the resolving power, it will appear bigger than a dot.

That's all. In astronomy, we don't speak of an object's absolute size, we only speak of the angular size. But that should be enough. As soon as you have the angular size, and say the distance, then you could deduce the absolute size, it's a simple matter of trigonometry.

absolute size = distance * tangent(angular size)

E.g., this is the size of an object of 1 arcsec angular size, situated at 384,000 km (the orbit of the Moon):

http://www.wolframalpha.com/input/?i=384000+km+*+tangent%281+arcsec%29

It's 1.8 km (in case the link above is unavailable).

In other words, that's the minimum absolute distance resolved by a telescope 10 cm in aperture, for objects on the Moon. Any two dots closer together than 1.8 km, placed on the Moon, are seen as one dot in a 10 cm telescope.