nt.number theory - Does Ribet's level lowering theorem hold for prime powers?

I often use the following theorem (that one can state more generally) in my research.

Let E/Q be an elliptic curve of conductor N corresponding to a modular form f(E), l a prime of good or multiplicative reduction for E, and rho(l) the 2 dimensional mod l Galois representation given by the action on the l-torsion points. Suppose that the torsion subscheme E[l] extends to a finite flat group scheme over Z_l, and let p be a prime of multiplicative reduction for E such that rho(l) is unramified at p (e.g. the number field (Q(E[l]) generated by the coordinates of the l-torsion points is unramified at p). Then there exists a modular form f of conductor N/p such that f is congruent to f(E) mod l (when f has Fourier coefficients over Z then this means that all but finitely many of the coefficents are congruent mod l); one can `lower the level' from N to N/p.

Does such a result hold for powers of primes? E.g. if this holds for the mod l^n representation (instead of the mod l) does one get a congruence mod l^n?

rt.representation theory - What rings/groups have interesting quaternionic representations?

Let $mathbb{H}$ denote the quaternions. Let $G$ be a group, and define a representation of $G$ on $mathbb{H}^n$ in the natural way; that is, its a map $rho:Grightarrow Hom_{mathbb{H}-}(mathbb{H}^n,mathbb{H}^n)$ such that $rho(gg')=rho(g)rho(g')$ (where $Hom_{mathbb{H}-}$ denotes maps as left $mathbb{H}$-modules). Representations of algebras and Lie algebras can be defined in a similar way.

Any quaternionic representation of $G$/$R$/$mathfrak{g}$ can restrict to a complex representation by choosing a $gammain mathbb{P}^1$ such that $gamma^2=-1$, and using $gamma$ to define a map $mathbb{C}hookrightarrow mathbb{H}$. In this way, any quaternionic representation gives a $mathbb{C}mathbb{P}^1$-family of complex representations which parametrize the choice of $gamma$. Furthermore, since every element in $mathbb{H}$ is in the image of some inclusion $mathbb{C}hookrightarrow mathbb{H}$, this family of complex representations determines the quaternionic representation.

This observation almost seems to imply that there is nothing interesting to say about quaternionic representations that wouldn't come up while studying complex representations. However, this is neglecting the fact that there might be interesting information in how the $mathbb{C}mathbb{P}^1$-family of complex representations is put together.

For instance, any finite group will have a discrete set of isomorphism classes of complex representations, and so any quaternionic representation will have all complex restrictions isomorphic. However, the quaternion group $mathbf{Q}:= ( pm1,pm i, pm j,pm k)$ has an 'interesting' quaternionic representation on $mathbb{H}$ (more naturally, it is a representation of the opposite group $mathbf{Q}^{op}$ by right multiplication, but $mathbf{Q}^{op}simeq mathbf{Q}$).

My question broadly is: What other groups, rings and Lie algebras have quaternionic representations that are interesting (in some non-specific sense)?

This question came up when I was reading a paper of Kronheimer's, where he describes a non-canonical hyper-Kahler structure on a coadjoint orbit of a complex semisimple Lie algebra. At any point in such a coadjoint orbit determines a representation of $mathfrak{g}$ on the tangent space to the coadjoint orbit, which by the hyper-Kahler structure is naturally a quaternionic vector space. I wondered if this representation could be 'interesting', and then realized I had no real sense of what an interesting quaternionic representation would be.

gr.group theory - Largest possible order of a nilpotent permutation group?

I'm trying to obtain a bound for the order of some finite groups, and part of it comes down to the order of a permutation group of degree $n$ that is nilpotent. I imagine these have to be much smaller than the full symmetric group, and a bound that is sub-exponential in $n$ would seem reasonable (given that permutation $p$-groups fall a long way short of having exponential order), but I haven't seen this written down anywhere.

I found one reference that looks promising:

P. Palfy, Estimations for the order of various permutation groups, Contributions to general algebra, 12 (Vienna, 1999), 37-49, Heyn, Klagenfurt, 2000.

However, I can't actually find the article anywhere online. Any suggestions?

ac.commutative algebra - is there a good computer package for working with bicomplexes?

I'm interested in working with bicomplexes of modules over polynomial rings, specifically tensoring them together, and the operation of taking cohomology in one direction, and then the other. Is there any computer algebra package which will do this efficiently?

radiation - Why do black holes choke?

I read that a black hole can sometime "choke" on a star: "...the disrupted stellar matter was generating so much radiation that it pushed back on the infall. The black hole was choking on the rapidly infalling matter."

I read the report and thought it was reasonable, and in line with Rob's answer. But note that there's no certainty that this is what actually happened. They saw a very bright "optical transient" event, circa ten times brighter than a normal supernova. See the paper on the arXiv: A Luminous, Fast Rising UV-Transient Discovered by ROTSE: a Tidal Disruption Event? There's a question mark on the end of the title, they don't know for sure. And see this from the news article:

"To narrow it down from four possibilities, they studied Dougie with the orbiting Swift telescope and the giant Hobby-Eberly Telescope at McDonald, and they made computer models. These models showed how Dougie's light would behave if created by different physical processes. The astronomers then compared the different theoretical Dougies to their telescope observations of the real thing."

They came up with what they thought was the best fit. But we don't know for sure that this was actually a black hole "choking" on a star.

Why does a black hole choke?

We don't know for sure. Note in the paper on page 8 and 9 they talk about an off-axis GRB interpretation. It's possible that this is what happened. Or something else altogether. The tidal disruption idea looks like a good fit, but they don't know for certain. Anyway, have a look at the summary on page 12 for a nice round-up:

"The tidal disruption scenario was explored by fitting the event to an amended version of the model presented in Guillochon et al. (2014). The TDE model yielded a good fit to the photometric and spectral evolution of the flare, with the highest-likelihood models suggesting a disruption of a solar-mass star by a black hole."

How do astronomers manage to observe this rare event?

They didn't actually observe a black hole choking on a star. What they saw was an optical transient. Something that looked like a supernova, but didn't fit the supernova pattern. So they think it was the disruption of a star by a black hole.

dg.differential geometry - definition of the end of a manifold?

As algori points out in the comments, the definition of end may be found here. Also there you will find the definition of the neighborhood of an end.

When he says "end collared (topologically) by $S^3 times R$" he means that the end has a neighborhood homeomorphic to $S^3 times R$. Since he's assuming that $M$ has only one end, this simply means that there is a compact set whose complement is homeomorphic to $S^3 times R$ (as Mariano said in the comments).

For future reference, it's pretty common in the literature to blur the distinction between an end and a neighborhood of an end. Especially if when there are neighborhoods of the end that are products, as is the case here.

ag.algebraic geometry - What is "restriction of scalars" for a torus?

As said, the sought after concept is also known as Weil restriction. In a word, it is the algebraic analogue of the process of viewing an $n$-dimensional complex variety as a $(2n)$-dimensional real variety.

The setup is as follows: let $L/K$ be a finite degree field extension and let $X$ be a scheme over $L$. Then the Weil restriction $W_{L/K} X$ is the $K$-scheme representing the following functor on the category of K-algebras:

$Amapsto X(A otimes_K L)$.

In particular, one has $W_{L/K} X(K) = X(L)$.

By abstract nonsense (Yoneda...), if such a scheme exists it is uniquely determined by the above functor. For existence, some hypotheses are necessary, but I believe that it exists whenever $X$ is reduced of finite type.

Now for a more concrete description. Suppose $X = mathrm{Spec} L[y_1,...,y_n]/J$ is an affine scheme. Let $d = [L:K]$ and $a_1,...,a_d$ be a $K$-basis of $L$. Then we make the following "substitution":

$$y_i = a_1 x_{i1} + ... + a_d x_{id},$$

thus replacing each $y_i$ by a linear expression in d new variables $x_{ij}$. Moreover,
suppose $J = langle g_1,...,g_m rangle$; then we substitute each of the above equations into $g_k(y_1,...,y_n)$ getting a polynomial in the $x$-variables, however still with $L$-coefficients. But now using our fixed basis of $L/K$, we can regard a single polynomial with $L$-coefficients as a vector of $d$ polynomials with $K$ coefficients. Thus we end up with $md$ generating polynomials in the $x$-variables, say generating an ideal $I$ in $K[x_{ij}]$, and we put $mathrm Res_{L/K} X = mathrm{Spec} K[x_{ij}]/I$.

A great example to look at is the case $X = G_m$ (multiplicative group) over $L = mathbb{C}$ (complex numbers) and $K = mathbb{R}$. Then $X$ is the spectrum of

$$mathbb{C}[y_1,y_2]/(y_1 y_2 - 1);$$

put $y_i = x_{i1} + sqrt{-1} x_{i2}$ and do the algebra. You can really see that the
corresponding real affine variety is $mathbb{R}left[x,yright]left[(x^2+y^2)^{-1}right]$, as it should be: see e.g.
p. 2 of

http://www.math.uga.edu/~pete/SC5-AlgebraicGroups.pdf

for the calculations.

Note the important general property that for a variety $X/L$, the dimension of the Weil restriction from $L$ down to $K$ is $[L:K]$ times the dimension of $X/L$. This is good to keep in mind so as not to confuse it with another possible interpretation of "restriction of scalars", namely composition of the map $X to mathrm{Spec} L$ with the map $mathrm{Spec} L to Spec K$ to
give a map $X to mathrm{Spec} K$. This is a much weirder functor, which preserves the dimension but screws up things like geometric integrality. (When I first heard about "restriction of
scalars", I guessed it was this latter thing and got very confused.)

ho.history overview - Historical question in analytic number theory

To extend on Matt's comment about Euler, here is something I wrote up some years ago about Euler's discovery of the functional equation only at integral points. I hope there are no typos.

Although Euler never found a convergent analytic expression for
$zeta(s)$ at negative numbers, in 1749 he published a method of
computing
values of the zeta function at negative integers by a precursor of
Abel's Theorem applied to a divergent series. The
computation led him to the asymmetric functional equation of
$zeta(s)$.

The technique uses the function
$$
zeta_{2}(s) = sum_{n geq 1} frac{(-1)^{n-1}}{n^s} =
1 - frac{1}{2^s} + frac{1}{3^s} - frac{1}{4^s} + dots.
$$
This looks not too different from $zeta(s)$, but has the advantage
as an alternating series of
converging for all positive $s$. For $s > 1$,
$zeta_2(s) = (1 - 2^{1-s})zeta(s)$.
Of course this is true for complex $s$, but
Euler only worked with real $s$, so we shall as well.

Disregarding convergence issues, Euler wrote
$$
zeta_{2}(-m) = sum_{n geq 1} (-1)^{n-1}n^m = 1 - 2^m + 3^m - 4^m +
dots,
$$
which he proceeded to evaluate as follows. Differentiate
the equation
$$
sum_{n geq 0} X^n = frac{1}{1-X}
$$
to get
$$
sum_{n geq 1} nX^{n-1} = frac{1}{(1-X)^2}.
$$
Setting $X = -1$,
$$
zeta_{2}(-1) = frac{1}{4}.
$$
Since $zeta_{2}(-1) = (1-2^2)zeta(-1)$, $zeta(-1) = -1/12$. Notice we can't set $X = 1$ in the second power series and compute $sum n = zeta(-1)$ directly. So $zeta_2(s)$ is nicer than $zeta(s)$ in this Eulerian way.

Multiplying the second power series by $X$ and then differentiating, we get
$$
sum_{n geq 1} n^2X^{n-1} = frac{1+X}{(1-X)^3}.
$$
Setting $X = -1$,
$$
zeta_{2}(-2) = 0.
$$
By more successive multiplications by $X$ and differentiations, we get
$$
sum_{n geq 1} n^3X^{n-1} = frac{X^2+4X+1}{(1-X)^4},
$$
and
$$
sum_{n geq 1} n^4X^{n-1} = frac{(X+1)(X^2+10X+1)}{(1-X)^5}.
$$
Setting $X = -1$, we find $zeta_{2}(-3) = -1/8$ and
$zeta_{2}(-4) = 0$. Continuing further, with the recursion
$$
frac{d}{dx} frac{P(x)}{(1-x)^n} = frac{(1-x)P'(x) + nP(x)}{(1-x)^{n+1}},
$$
we get
$$
sum_{n geq 1} n^5X^{n-1} = frac{X^4+26X^3+66X^2 + 26X +1}{(1-X)^6},
$$
$$
sum_{n geq 1} n^6X^{n-1} = frac{(X+1)(X^4 + 56X^3 + 246X^2 + 56X+1)}
{(1-X)^7},
$$
$$
sum_{n geq 1} n^7X^{n-1} = frac{X^6 + 120X^5 + 1191X^4 + 2416X^3 +
1191X^2 + 120X + 1}{(1-X)^8}.
$$
Setting $X = -1$, we get $zeta_{2}(-5) =
1/4, zeta_{2}(-6) = 0, zeta_{2}(-7) = -17/16$.

Apparently $zeta_{2}$ vanishes at the negative even integers, while
$$
frac{zeta_{2}(-1)}{zeta_{2}(2)} = frac{1}{4}cdotfrac{6cdot 2}{pi^2} =
frac{3cdot 1!}{1cdot pi^2},
frac{zeta_{2}(-3)}{zeta_{2}(4)} =
-frac{1}{8}cdotfrac{30cdot24}{7pi^4} = -frac{15cdot 3!}{7cdot pi^4},
$$
$$
frac{zeta_{2}(-5)}{zeta_{2}(6)} = frac{1}{4}cdot
frac{42cdot 6!}{31pi^6} =
frac{63 cdot 5!}{31cdot pi^6},
frac{zeta_{2}(-7)}{zeta_{2}(8)} = -frac{17}{16}cdot
frac{30cdot 8!}{127cdot pi^8}
= -frac{255cdot 7!}{127pi^8}.
$$

The numbers $1,
3, 7, 15, 31, 63, 127, 255$ are all one less than a power of 2,
so Euler was led to the observation that for $n geq 2$,
$$
frac{zeta_{2}(1-n)}{zeta_{2}(n)} =
frac{(-1)^{n/2+1}(2^n-1)(n-1)!}{(2^{n-1}-1)pi^n}
$$
if $n$ is even and
$$
frac{zeta_{2}(1-n)}{zeta_{2}(n)} =
0
$$
if $n$ is odd. Notice how
the vanishing of $zeta_{2}(s)$
at negative even integers nicely compensates for
the lack of knowledge of $zeta_2(s)$ at positive odd integers $> 1$ (which
is the same as not knowing $zeta(s)$ at positive odd integers $> 1$).

Euler interpreted the $pm$ sign at even $n$ and the vanishing
at odd $n$ as the single factor $-cos(pi n/2)$, and with
$(n-1)!$ written as $Gamma(n)$ we get
$$
frac{zeta_{2}(1-n)}{zeta_{2}(n)} = -Gamma(n)frac{2^n-1}{(2^{n-1}-1)pi^n}
cosleft(frac{pi n}{2}right).
$$
Writing $zeta_{2}(n)$ as $(1 - 2^{1-n})zeta(n)$ gives the
asymmetric functional
equation
$$
frac{zeta(1-n)}{zeta(n)} = frac{2}{(2pi)^n}
Gamma(n)cosleft(frac{pi n}{2}right).
$$
Euler applied similar ideas to $L(s,chi_4)$ and found its
functional equation. You can work this out yourself in
Exercise 2 below.

Exercises

1. Show that Euler's computation of zeta values at negative
   integers can be put in the form
   $$
   (1 - 2^{n+1})zeta(-n) =
   left.left(ufrac{d}{du}right)^{n}rightvert_{u=1}left(frac{u}{1+u}
   right) = left.left(frac{d}{dx}right)^{n}rightvert_{x=0}
   left(frac{e^x}{1+e^x}right).
   $$




2. To compute the divergent series
   $$
   L(-n,chi_4) = sum_{j geq 0} (-1)^{j}(2j+1)^n =
   1 - 3^n + 5^n - 7^n - 9^n + 11^n - dots
   $$
   for nonnegative integers $n$, begin with the formal identity
   $$
   sum_{j geq 0} X^{2j} = frac{1}{1-X^2}.
   $$
   Differentiate and set $X = i$ to show $L(0,chi_4) = 1/2$.
   Repeatedly multiply by $X$, differentiate, and set
   $X = i$ in order to compute $L(-n,chi_4)$ for $0 leq n leq 10$. This computational technique is not rigorous, but the answers are correct. Compare with the values of $L(n,chi_4)$ for positive $n$, if you know those, to get a formula for $L(1-n,chi_4)/L(n,chi_4)$. Treat alternating signs like special values of a suitable trigonometric function.

astrophysics - If the Sun were bigger but colder, Earth would be hotter or colder?

The equilibrium temperature of the Earth, $T_E$, scales roughly as $L^{1/4}$, which is proportional to $R^{1/2} T$, where $L$, $R$ and $T$ are the solar luminosity, radius and temperature.

The actual approximate relationship is derived by equating the power received by the Earth, which is proportional to the solar luminosity $L$, with the power radiated by the Earth, which is proportional to $T_E^4$ for a blackbody. Hence $T_E propto L^{1/4}$.

So the answer to your question depends on by how much you increase the radius compared with the decrease in temperature.

There will be second order effects that do depend on the spectrum of radiation from the Sun (and therefore its temperature) compared with the wavelength dependence of the albedo and emissivity of the Earth. So I will post a better question...

at.algebraic topology - $pi_4$ of simply-connected 4-manifold

First an important distinction: "Each element in $pi_3(vee S^2)$ has description in terms of linking number of point preimages (circles in $S^3$) of map $S^3 to S^2$" is not a fully correct statement. What has a description in terms of such linking numbers is $Hom(pi_3(vee S^2), Z).$ As you say, $pi_3(vee S^2)$ itself is described as generated by Whitehead products and $pi_3(M)$ is a quotient of this.

Dually, $Hom(pi_3(M), Z)$ will consist of a sub-module of these linking numbers, and if you want you can make it more geometric. For any collection of closed two-dimensional cochains ${ alpha_i, beta_i }$ such that $sum alpha_i smile beta_i= d theta$ one can form the generalized linking number which to some $f : S^3 to S^2$ evaluates the "integral" $int_{S^3} [ sum f^* alpha_i smile d^{-1} f^* beta_i - f^* theta]$. Here $int_{S^3}$ is evaluation on the fundamental class of $S^3$ and $d^{-1} f^* beta_i$ indicates a choice of $1$-cochain which cobounds $f^* beta_i$. If the $alpha_i$ and $beta_i$ are Poincare dual to codimension two submanifolds of $M$ this will be a linking number (with "correction" by the $theta_i$) of the preimages of those submanifolds.

So far, this is just addressing $pi_3$. But in recent work Ben Walter and I generalize such "linking numbers" and show that the resulting collection yields a finite-index subgroup of $Hom(pi_n(X), Z)$ for all $n$ for simply connected $X$. One can take the formulae there - in this case one will need to go to "weight two" - and translate them into geometric terms as I did for $pi_3$ above. We give a number of examples, and I'd be happy to provide closer analysis for simply connected $M^4$ if you think what we do is relevant and I understood better what you're looking for. The caveats are that we are representing functionals on homotopy groups rather than homotopy groups themselves, and what we can do about torsion is very limited (but is likely to suffice in this case).

soft question - What programming languages do mathematicians use?

One language I still use is PostScript. I probably need to defend that.

1. Its syntax is elegant. In fact, no language I've seen has more uniform syntax: a complete program is syntactically identical to almost any fragment of a program. There are no keywords and very few special cases.




2. It can be a lot of fun, and you can make pretty pictures.




3. It has very few data types, but some of the ones it does have are surprisingly useful. Dictionaries come to mind immediately. Also, "arrays" (which would be called "lists" in any other language) are extremely flexible. They automatically support comprehensions, not as a separate feature, but as an obvious consequence of the syntax. Functional programmers shouldn't be surprised that procedures are useful as a type; actually, due to the simplicity of the syntax and lack of keywords, any nontrivial program has to work with procedures as data.

Unfortunately, it has many drawbacks that prevent it from really being useful. Its handling of strings is abominable. Also, it has no facilities for user interaction. Its console I/O is crippled. Things like that could in principle be fixed by appropriate third-party packages, but unfortunately, to my knowledge, there are no third-party packages at all (at least for general programming). Finally, it can be very hard to debug; actually, it is more difficult to debug than any other language I know except assembly. All of those things combine to make it one of the most programmer-unfriendly languages out there. Nevertheless, some of my best work is implemented directly in PostScript, and I have done some real work in it. (Also, let's be fair: PostScript was never intended for general-purpose programming! Using a page-description language for any serious computation at all is some sort of achievement.)

(Language: PostScript. Mathematical interest: its syntax is simple enough to be interesting as a mathematical construction. It's easy to produce some mathematical illustrations, like many types of fractals.)

For real mathematical figures (such as for inclusion in papers), I use MetaPost. PostScript can be used for this purpose, but MetaPost is much better suited for this and is very TeX-friendly.

(Language: MetaPost. Mathematical interest: it's great for making mathematical figures suitable for inclusion in a LaTeX document.)

Another language that I use mostly for fun, not serious work, is x86 assembly language. In contrast to PostScript, it's an ugly language, but strangely, I think I use assembly for some of the same reasons that draw me to PostScript.

(Language: Assembly. Mathematical interest: its execution model is very simple, so expressing algorithms in it is an interesting challenge that mathematicians may enjoy.)

The rest of the languages I use need no introduction: C, C++, Python, Ruby, Java.

(Languages: C, C++, Python, Ruby, Java. Mathematical interest: none in particular, but they're useful in general programming, including mathematical programs.)

I used to use Octave, but apparently most of the world uses Matlab, and Octave has just enough incompatibilities with Matlab to make it annoying to try to use other people's code. Also, it seems to have pretty poor support for sparse matrix computations.

(Language: Octave. Mathematical interest: free approximate-clone of Matlab. It has simple syntax for matrix-centric computation.)

I used to use PHP a lot. Actually, PHP and assembly are sort of an odd couple. A while ago, for no good reason, I tried to come up with the fastest code to print out all the permutations of a string. My best solution (for strings of ~10 or more characters, IIRC) was a combination of PHP and x86 assembly. To be fair, the PHP part could have been done in another language, but PHP was almost the right tool for the job.

(Language: PHP. Mathematical interest: none in particular, but it's great for designing websites with server-side scripting, which is no less useful to mathematicians than it is to other programmers.)

I like Haskell, but I don't use it much.

(Language: Haskell. Mathematical interest: sigfpe said it.)

There are other languages I find interesting but never learned properly, like Lisp, Fortran, and Forth.

If anyone's looking for a recommendation, I don't recommend any of those. But learn all of them, and then go off into a dark corner of the universe and come back with the One Language that will rule us all.

star - How to differentiate between images of a gravitationally lensed object

In the case of multiple images of a background, distant object, the answer is relatively simple. You take a spectrum of the multiple images or parts of an extended lensed image and you see whether the spectrum looks the same, and in particular whether the redshift of the multiple images are the same.

Gravitational lensing affects light of all wavelengths equally, so regardless of the path taken, the spectrum should be unaltered, except that the different paths take different amounts of time to travel from the original object to us. So, if the background source, or its spectrum, are time-varying, then the lensed images could also appear different. However, distant galaxies do not change their redshift on such short timescales and so this should still be the same regardless.

What is the temperature 55 km beneath the surface of Mars?

If we look at Mars' possible geothermal gradient (see Earth's) which is about 25 °C per km. Using the low estimate of Mars's gradient to be 1/4 that of Earth's Source, that's a bit over 6° C per km. so 55 km, 330° C. Added that to Mars' average surface temperature of -55 C, you're talking 275° C or 527° F at 55 km underground, and that's a low estimate.

Estimates of the thickness of Mars' crust are often less than 55 km and if you're talking about digging into mantle or even, half way to the mantle, that's both an unsustainable dig and unlivable temperatures.

Source

The average thickness of the planet's crust is about 50 km (31 mi),
with a maximum thickness of 125 km (78 mi)

and

Source

Also, small point, but the 55km estimate assumes uniform temperature. The simpler pressure calculations don't take into account temperature fluctuation. With warmer temps and warmer lighter air at the bottom of the dig, which would be inevitable, you'd probably need 60 km, maybe 65 km of depth to achieve 1 atm. There's also circulation issues when you have cold heavier atmosphere above warm lighter atmosphere and there's drainage/flooding problems for the lowest part of a valley, so to avoid flooding, you might want to keep the surface temperature below freezing, to avoid runoff and silt build-up, or dig even deeper so there's a lake of sorts that catches the melting ground-ice around the dig and the living area is above the lake.

This doesn't mean that deep valley living on mars will never be possible, but it's likely impossible with it's current atmosphere. With extensive terraforming, tapping gases from underground and/or from it's ice-caps or by crashing comets into Mars (that would take a lot of comets), over time, enough of an atmosphere could be created where deep valley living might work. A simulated atmospheric pressure can also be created with a bubble-dome of sorts, or one of my favorites, a stretch-wrap around the entire planet, but for now that's way beyond our means.

For the near term, living underground on Mars is probably the way to go. Dig 50 or 100 feet underground and you'd have the means to create a near vacuum seal, which on Mars, you'd want to do, you could create earth like air pressure and you'd be protected from solar and cosmic rays and access to plenty of underground water. underground on Mars is likely the way to go, at least, in the near future. There's no shortages of articles on this. Here's one 6.

Humans can also live at less than 1 atm. People have settled as high as 16,700 feet, which is somewhere in the neighborhood of 55% of 1 atm.

ag.algebraic geometry - Groebner basis with group action

At one point my advisor, Mark Haiman, mentioned that it would be nice if there was a way to compute Groebner bases that takes into account a group action.

Does anyone know of any work done along these lines?

For example, suppose a general linear group $G$ acts on a polynomial ring $R$ and we have an ideal $I$ invariant under the group action. Suppose we have a Groebner basis $B$ of $I$. Then we can form the set $G(B) := { G(b) : b in B }$. Perhaps we also wish to form the set
$$IG(B) := { V : V text{ is an irreducible summand of } W, text{ for some }W in G(B) }$$
(note that $G(b)$ cyclic implies it has a multiplicity-free decomposition into irreducibles).

Can we find a condition on a set of $G$-modules (resp. $G$-irreducibles), analogous to Buchberger's S-pair criterion, that guarantees that this set is of the form $G(B)$ (resp. $IG(B)$) for some Groebner basis $B$?

Can the character of $R /I$ be determined from the set $IG(B)$ in a similar way to how the Hilbert series of $R /I$ can be determined from $B$?

ho.history overview - Who thought that the Alexander polynomial was the only knot invariant of its kind?

What makes the Jones polynomial dramatic, I think, is not that it is a polynomial invariant per se, but that it came from an unexpected source where nobody had thought of looking. Indeed, there's still no conceptual mathematical explanation for why we should expect knot invariants to come from such a source.
Jones was working on the Potts model in statistical mechanics (how could this possibly be related to topology?). In this context, it was relevant to study representations of the braid group with n strands B_n into the Temperley-Leib algebra TL_n. The miracle now is that the Markov trace of the representation of a braid, suitably normalized, is invariant under Markov moves, and is therefore an invariant of the knot obtained by closing the braid. Why should it be a knot invariant, conceptually? Nobody knows.
What makes it even more amazing is that the Jones polynomial turns out to fit into a family with the Alexander polynomial, which is the archetypical algebraic knot invariant, which heuristically suggests that the Jones polynomial is something important which we should be looking at, and which should probably have a sensible topological interpretation.
The Jones polynomial, I think, is "mathematics we can calculate" as opposed to "mathematics we understand", even now, 25 years after its discovery. Yet it turns out to be tremendously powerful, and to have deep connections with other parts of mathematics.

nt.number theory - Expressing field inclusions by polynomial equalities on coefficients

This may ramble a bit much, but I hope it provides some help in how to think about the problem.

Let's see what your extension of fields looks like. We have 4 possible extensions (perhaps the same) So that any of them is

$mathbb Q(z_i)$

$|$

$mathbb Qleft(sqrt2right)$

$|$

$mathbb Q$

Where $z_i$ ranges of the 4 possible roots $z_1,...,z_4.$ Then $mathbb Q(z_1)$ is degree 4 (since the polynomial is irreducible), but this polynomial factors into a product of quadratics over $mathbb Qleft(sqrt2right).$ So indeed we've reduced to having only two possible extensions, in that the two roots of the same quadratic generate the same extension over $mathbb Q(sqrt2).$

However, except for this restriction, I don't see anything else to lead to a relation on the coefficients. Hopefully this will help you or someone else get a start on the problem.

One further thought:

Since the roots appear in pairs (say $z_1$ and $z_2$ are conjugate over $mathbb Qleft(sqrt 2right)$) then one can generate $sqrt 2$ with either pair, and subtract them. However, I don't immediately see a way to gather that information from the symmetric polynomials of the roots (a.k.a. the coefficients $a_1, ldots, a_4.$)

ca.analysis and odes - How many ways can we characterize gamma function?

maybe I can give you some help.
Gamma function is also called the second Euler integral.

Here comes some characterizations.

a f(s)= $$t(x)=int_{0}^{+infty}{t^(s-1)}{exp(-t)}dt$$ s>0

b f(s)=$$lim n!n^s/[s(s+1)...(s+n)] $$ $$nrightarrow +infty$$

c $$B（p，q）=Gamma(p)Gamma(q)/Gamma(pq）$$ p>0 q>0

d $$Gamma(2s)=2^(2s-1）Gamma(s)Gamma(s+1/2)/sqrt(2pi) $$ s>0

e $$Gamma(s)Gamma(1-s)=pi/sin(spi)$$ 0

May it help!

ac.commutative algebra - Completion of modules of differentials (A strange exercise in Liu's AG textbook)

A is a Noetherian ring, B is an f.g. algebra over A, I is an ideal of A. let $hat B$ be B's I-adic completion. Prove that $Omega^1_{hat B/A}$'s I-adic completion is isomorphic to $Omega^1_{B/A}$'s I-adic completion.

This is an exercise from Liu's "Algebraic geometry and arithmetic curves", Exercise VI.1.3. It seems strange because according to Part(a) of that problem, there is an exact sequence involving these two objects. And if this is true, we must prove the first term in that exact sequence is actually zero under only Noetherian condition! I feel a bit puzzled, can anyone help me? Thanks!

declination - Low precision GMST formula clarification

According to the website http://aa.usno.navy.mil/faq/docs/GAST.php, a low-precision way of calculating the GMST is via the following formula:

GMST = 18.697374558 + 24.06570982441908 D

The website doesn't actually state where the numbers come from. Another website tries to do this:

"To compute of sidereal time with low precision use of the low precision formula: GMST = 18.697374558 + 24.06570982441908 D, where 18.697… is sidereal time of the reference time 2000-01-01 at 0 UT, 24.0657… is a ratio of synodic and sidereal periods of Earth and D is days (and its fractions) since the reference time."

but I don't understand what this 'ratio of synodic and sidereal periods' means. I know what the sidereal and synodic days are and their values, but how do we use those to end up with the number 24.0657...? If anyone could help me out that'd be much appreciated!

reference request - range projection of an unbounded idempotent affiliated to a finite von Neumann algebra

To be slightly more precise: let $Msubset B(H)$ be a finite von Neumann algebra equipped with a faithful normal trace $tau$, and let $L^0(M,tau)$ be the completion of $M$ in the measure topology; this is an algebra, whose elements can be identified with those densely-defined and closed operators on $H$ that are affiliated with $M$. (See e.g. E. Nelson, Notes on noncommutative integration, JFA 1974). Let $e$ be an idempotent in $L^0(M,tau)$, not necessarily self-adjoint; then it is not hard to show that $R={ xiin H : exi=xi}$ is a closed subspace of $H$.

Question: is the orthogonal projection onto $R$ affiliated with $M$?

I suspect the answer is yes (and would like it to be, for some calculations I'm doing at the moment) but am having difficulties nailing the argument down. Given that this should, if true, be a pretty basic bit of operator algebra theory, and standard knowledge, I'd be grateful if someone could point me to a reference. (I currently have somewhat limited library access, but this might well be covered in Kadison & Ringrose for instance.)

---

Edit/update: both Martin Argerami and Jonas Meyer have given straightforward proofs of the desired result, and a quick check in Kadison & Ringrose vol. 1 has not turned up any explicit statement (probably because the result turns out to be so basic). Since I can't accept both their answers, I'm accepting Martin's on grounds of personal preference.

exoplanet - What proportion of planetary systems have been found with 'Hot Jupiters'

I completely agree with the answer from MBR. The number is actually $1.20pm 0.38$ per cent, is published by Wright et al. (2012) and is the fraction of F, G, K stars that have a hot Jupiter defined as being larger than 0.1 Jupiter masses and having an orbital period less than 10 days. Table 2 of that paper summarises results from other workers, who obtain between 0.5 and 1.5 per cent. The paper also discusses observational biases, including metallicity.

It has long been known that close-in planet incidence is higher around more metal-rich stars. There is also a bias whereby it is easier to find planets around metal-rich stars, whereas the average star in the solar neighborhood is slightly metal-poor compared with the Sun.

A study by Gonzalez (2014) accumulates our current knowledge of exoplanetary systems and their metallicities, deriving a planetary incidence rate
$$ P_{planet} = alpha 10^{beta[Fe/H]},$$
with $alpha= 0.022 pm 0.007$ (i.e. 2.2 per cent), $beta=3.0pm 0.5$ and where [Fe/H] is the usual logarithmic ratio of the metallicity of the star to the metallicity of the Sun. (i.e. the Sun has [Fe/H]=0).

This calculation is done for giant planets with orbital periods less than 4 years, so not all of them would be classed as hot Jupiters. Bottom line, the number given by Wright et al. is about right on average, but it is higher for higher metallicity stars.

lo.logic - Model theory stressing order type of universe.

I don't think many model theorists have worked on this. Granted, I'm a little unclear what Chang and Keisler were asking here, but here's one possible precisification:

Question: Suppose we are given a (complete?) theory T in a language with a binary relation < such that T proves "< is a strict linear ordering." Try to develop a theory of the "order-type spectrum" $I(alpha, T)$, which is defined as the number of nonisomorphic models $M$ of $T$ such that $(M, <^M)$ has order type $alpha$.

You could start by trying to think about what it means for $T$ to be "$alpha$-categorical" for some order type $alpha$, meaning, what are necessary and sufficient conditions for $I(alpha, T)$ to equal $1$? (I have no idea whether anybody has investigated this before.) For example, if $T$ proves that the ordering is dense without endpoints and $eta$ is the order type of the rationals, then $I(eta, T) = 1$ if and only if $T$ is $omega$-categorical, since the complete theory of $(mathbb{Q}, <)$ itself is $omega$-categorical.

An immediate complication I see to this project is that I don't know if there is any good analogue of the upward Lowenheim-Skolem theorem. It seems like it would be difficult to answer the question: "Given a theory $T$, for which infinite order types $alpha$ is $I(alpha, T) neq 0$?" (The corresponding question for cardinalities of the universe for a $T$ with infinite models is trivial, by Lowenheim-Skolem.) For example: your theory $T$ could force the order type of any model to not be Dedekind complete (e.g. take the complete theory of an densely-ordered ring with a unary predicate for a proper convex subring).

Are there Morley-like categoricity theorems? If $T$ is countable, say, and $I(alpha, T) = 1$ for some uncountable order type $alpha$, can we conclude that $I(beta, T) leq 1$ for every uncountable $beta$? Possibly this could be an interesting question; offhand I have no idea what the answer is.

Illustrating the difficulties of this, here's a paper just on the possible order types of a particular theory, PA:

"Order-types of models of Peano arithmetic," by Andrey Bovykin and Richard Kaye, pp. 275-285 of Logic and Algebra, edited by Yi Zhang with a preface by Oleg Belegradek, Contemporary Mathematics 302, AMS.

A different interpretation of the original question would be: given a particular order type $alpha$, investigate structures with order type $alpha$. Along these lines, many model theorists (such as Pillay, van den Dries, Wilkie, and others) have been studying expansions of the ordered field of the real numbers under the rubric of "o-minimal theories," though generally the interest has been in definable sets rather than models per se. Chris Miller is an example of an o-minimalist who has done a significant amount of research just on structures expanding the field of reals; check out his webpage for some state-of-the-art papers in this area.

The Existence of Natural Satellites in Geostationary Orbits

While browsing through Physics SE, I noticed a question about satellites in geostationary orbit (unrelated to the one I'm asking here), and for a moment I interpreted it as referring to natural satellites (e.g. a moon). So I wondered: Could a natural satellite exist in geostationary orbit?

Then I stopped and thought. For large gas giants, such as Jupiter, having moons too close to the planet can be fatal (for the moon). If it ventures inside the planet's Roche limit, it's toast. But there is good news: the Roche limit depends on both the masses and densities of the primary body and the satellite. So perhaps this reason is non-applicable, as a high-mass natural satellite might be able to survive. So the question changes:

Could a sufficiently high-mass, high-density natural satellite occupy geostationary orbit over its primary body?

Why do we believe that the super massive black holes at the centers of two merging galaxies would themselves merge?

The SMBHs reside in the bottom of the galactic potentials, which are dominated by the galaxies' dark matter halos. But although dark matter dominates gravity, collisions between gas and dust particles in the interstellar medium causes enough friction that the baryonic component of the galaxies is decelerated. This will cause the other components of the galaxies to decelerate as well, through gravitational attraction.

Moreover, despite dark matter (and, in practice, stars and black holes since they're so small) being collisionless, there are several ways of "relaxing", i.e. to evolve towards an equilibrium. In the context of galaxy merging, the most important mechanism (I think) is "violent relaxation", where the rapid change of the gravitational potential causes particles to relax, e.g. more massive particles tend to transfer more energy to their lighter neighbors and so become more tightly bound, sinking towards the center of the gravitational potential.

Although SMBHs are…, well, supermassive, the potential will (usually) be dominated by dark matter, gas, and stars, so the new gravitational potential will also cause the SMBHs to seek towards the bottom in the same fashion, and eventually merge.

physics - Perpetuum Mobile

I think the finite size is a red herring; with ideal sources/sinks and particles it still doesn't "work".

Incorrect approach: 100% of rays from from blue focus go to the red one; 80% of rays from red focus go to the blue one; 100% > 80%; therefore imbalance. Incorrect because we must look at absolute numbers, not percentages, when saying the flow from blue to red must equal the flow from red to blue.

Incorrect approach: Initally both foci emit 100 particles, and during a time period epsilon, 100 particles go from blue to red, but only 80 go from red to blue. True so far, but Incorrect approach because the paths are not all the same length so the travel times are not all the same. When all paths are equally full of particles, the numbers arriving at each focus will be the same.

Correct approach: Short path takes T1 time units to travel; long path takes T2. At equilibrium, during a time period epsilon, 100 particles will leave blue, of which the fraction that hit the tighter-curved mirror will arrive at red after T1, and the fraction that hit the longer-curved mirror will arrive at red after T2. During the same time, 100 particles will leave red; a fraction of those will go to blue in time T1, and a fraction will go back to red in time T2. I'm too lazy to work it all out right now, but that's the correct approach.

pr.probability - Disintegrations are measurable measures - when are they continuous?

_{This is a sequel to another question I have asked.}

The notion of disintegration is a refinement of conditional probability to spaces which have more structure than abstract probability spaces; sometimes this is called regular conditional probability. Let $Y$ and $X$ be two nice metric spaces, let $mathbb P$ be a probability measure on $Y$, and let $pi : Y to X$ be a measurable function. Let $mathbb P_X(B) = mathbb P(pi^{-1} B)$ denote the push-forward measure of $mathbb P$ on $X$. The disintegration theorem says that for $mathbb P_X$-almost every $x in X$, there exists a nice measure $mathbb P^x$ on $Y$ such that $mathbb P$ "disintegrates":$$int_Y f(y) ~dmathbb P(y) = int_X int_{pi^{-1}(x)} f(y) ~dmathbb P^x(y) dmathbb P_X(x)$$
for every measurable $f$ on $Y$.

This is a beautiful theorem, but it's not strong enough for my needs. Fix a Borel set $B subseteq X$, and let $p(x) = mathbb P^x(B)$. Part of the theorem is that $p$ is a measurable function of $x$. Suppose that the map $pi : Y to X$ is continuous instead of simply measurable. My question: What is a general sufficient condition for $p(x)$ to be continuous?

To me, this is an obvious question to ask, since if $x$ and $x'$ are two close realizations of a random $x in X$, then the measures $mathbb P^x$ and $mathbb P^{x'}$ should be close too, at least in many natural situations. However, in my combing through the literature, I haven't been able to find an answer to this question. My guess is that most people are content to integrate over $x$ when they use the theorem. For my purposes, I need some estimates which I get by continuity.

At this point, I've managed to prove and write down a pretty good sufficient condition for the case I care about (Banach spaces), using an abstract Wiener space-type construction. However, I am hoping that an expert can point me toward a good reference that does this in wider generality.

night sky - How come the northern and southern lights are the same?

As most people are aware when they play with bar magnets, magnetic poles repel magnets of the same polarity and attract opposites. Magnets have two polarities: north and south.

Electric charges are classified as positive and negative, and also attract opposites and repel charges of the same sign.

However, magnetic poles and electric charges do not match up in the way your question suggests - it's not like North is positive and South is negative and they attract electric charges. Electric and magnetic fields are connected and interact (in fact, both are produced by moving electric charges) but the signs/polarity do not interact in the way you are imagining..

Aurorae (northern and southern lights) are produced because charged particles are streaming toward the Earth all the time in the solar wind. Charged particles (positive or negative) are blocked by magnetic fields - the easiest way to imagine this is to draw the Earth's magnetic field like a bar magnet and the charged particles cannot cross the field lines. At the poles the field is not parallel to the Earth's surface (blocking charged particles) but perpendicular to it, so the charged particles can move into the atmosphere without crossing magnetic field lines. In the atmosphere they interact with particles in the air and create the beautiful lights that we see.

So, in short, the lights are the same at the north and south poles because it is the same kind of particles interacting with out atmosphere.

soft question - Best online mathematics videos?

My good friend Professor Elvis Zap has the "Calculus Rap," the "Quantum Gravity Topological Quantum Field Theory Blues," a vid on constructing "Boy's Surface," "Drawing the hypercube (yes he knows there is a line missing in part 1)," A few things on quandles, and a bunch of precalculus and calculus videos. In order to embarrass all involved, he posted the series "Dehn's Dilemma" that was recorded in Italy last summer.

Does Pi change when gravity changes

$pi$ is just a number. It doesn't vary with anything.

The ratio of the circumference to the diameter of a circle varies with geometry, so in the general case, this definition doesn't hold. But there are other definitions, such as
$$
pi = int_{-1}^{1}frac{dx}{sqrt{1-x^2}},
$$
or
$$
mathrm{The,smallest,positive,number} , x , mathrm{for,which}, arccos(2x) = 0.
$$
The latter definition can also be expressed a power series, which is independent of geometry.

fa.functional analysis - Characterisation of positive elements in l¹(Z)

Consider the Banach $^* $-algebra $ell^1(mathbb Z)$ with multiplication given by convolution and involution given by $a^*(n)=overline{a(-n)}$.

I would like to find nice necessary and sufficient conditions for an element $binell^1(mathbb Z)$ to be positive, that is, to be of the form $a^* * a$ for some $ainell^1(mathbb Z)$.

By now, I have found two necessary conditions. Namely, if $binell^1(mathbb Z)$ is positive, then $$b(-n)=overline{b(n)}$$ and $$lvert b(n)rvertleq b(0)$$ for every $ninmathbb Z$.

Edit: As t3suji states in his comment below both conditions follow from the more general fact that $a$ is a positive-definite function.

Question: Is this condition also sufficient for positivity? If not, what to I have to add?

Good references would also be great.

Motivation: In the end I want to investigate the (failure of) the Gelfand–Naimark theorem for the above non-C*-algebra.

ag.algebraic geometry - Subgroup Groups and Coordinate Algebra Subalgebras

It may be possible in rare circumstances, but the natural thing is that $mathcal{O}(H)$ is a quotient of $mathcal{O}(G)$, not a subalgebra. The quotient map is dual to the inclusion $Hto G$.

I guess in the case that $G$ is a direct product $H times K$, you get
$$ mathcal{O}(G) simeq mathcal{O}(H) otimes mathcal{O}(K), $$
and in this case you have what you want. But in general, the arrow should go the other direction.

orbit - Is the duration of a sidereal year stable?

It changes some. The solar energy pressing against the Earth from the sun, the kinetic energy from coronal mass ejections hitting the Earth, tidal forces from the Earth-Sun relation and even gravitational tugs form other planets and meteors and space dust that hit the earth all change the Earth's orbit a teeny tiny bit, also, changes in the Sun's mass either by the sun accumulating mass from space dust or comets or losing mass, which most suns do over time. Each tiny orbital change changes the sidereal year (and all the other years) a tiny little bit. Perhaps this only amounts to millionths of a second per year but some of these forces are consistent and can lead to predictable changes over time. Others variable, leading to uncertainty. The long term changes to the Earth's orbital distance are currently unpredictable.

It's generally believed that the Earth is slowly moving away from the sun, but I don't believe there's much agreement on how fast this is currently happening. Articles here and here. The 2nd one estimates 1.5 CM per year or 15 KM per million years which even over a billion years (15,000 KM) is quite a small distance compared to 150 million KM. But estimates are difficult for a variety of reasons.

I asked a related question here, with a bounty and didn't get any definite answers. It's possible to mathematically model the movement of some planets (Jupiter to an extent and Neptune/Uranus and perhaps Saturn based on the Jupiter model). It's thought Jupiter moved both towards then away from the sun over the first few billion years of the solar system. The Moon's distance from the Earth is also mostly predictible and can be modeled back into the past based on the Oceanic tidal bulge, but modeling the Earth;s distance from the sun over the past is much harder. It's a tiny object compared to the Sun and Jupiter and N body systems have a lot of chaos and unpredictability to them. 4 billion years ago it's possible that the Earth was quite a bit closer to the sun, but I don't think anyone has a good model for how much closer. The faint young sun paradox and the theory that young suns eject much more matter and have more active coronal mass ejections suggests to me that the Earth was likely closer to the Sun billions of years ago and not further away.

The earth's orbital eccentricity also changes with time, on a cycle of roughly every 100,000 years but according to Wikipedia, that doesn't change the length of the year, only the shape of the orbit.

Finally, and just for fun, it makes logical sense to me that the Moon orbiting the Earth creates small apparent fluctuations at least from the point of view of the Earth, because the Earth in effect orbits the Earth-Moon barycenter in an ellipse roughly 1/81st the size of the distance to the moon, or about 4,480 KM in radius every 29 days. Depending on whether the Moon is ahead of the earth or behind it, the Earth year can change by as much as 5 minutes. I suspect that this is largely ignored when measuring Earth years though and the Earth-Moon barycenter is used. If we measured the Earth's center we should get much larger variations year to year. (see picture).

Source.

(if anyone can improve on this, feel free, I have the feeling it's only a partial answer, cause logically it makes sense that the Earth-Jupiter positions or Earth-Venus should change, if only slightly, the Sidereal year).*

star - What is the frequency distribution for luminosity classes in the Milky Way Galaxy?

I'm working on a game concept that does some mild simulation of realistic stellar classes and luminosities. In particular, I'd like to roughly model the general frequencies of the classes and luminosities of the stars in the Milky Way.

Several sources, including Wikipedia's entry on stellar classifications, show a chart that includes the frequency distribution for spectral classification: the OBAFGKM categorization. So that's fine.

What I'm having trouble finding is any frequency distribution chart similar to that one but for the Yerkes luminosity categories: Ia+, Ia, Iab, Ib, II, III, IV, V, sub-dwarf and dwarf. I have a copy of the Hipparcos database, which contains a "Spectral Types" field, but it's highly incoherent text. Still, I could write some code to parse the values in that field to try to get a rough count of luminosity categories in those roughly 116,000 stars... but I'm a little perplexed that no such chart appears to exist already somewhere in Internetland. (Either that or my search-fu is weaker than usual.)

If anyone can point me to a chart of the frequency distribution for the luminosity categories noted above, or suggest a reasonably simple way for me to calculate those values myself, I'd appreciate it.

EDIT: Out of curiosity, I went ahead and did my own simple parsing of the spectrum fields from the Hipparcos dataset.

Out of 116472 rows, only 56284 (fewer than half) provided luminosity class data in the Spectrum field. Those 56284 rows broke down this way:

Ia0     16     0.03%
Ia     241     0.43%
Iab    191     0.34%
Ib     694     1.23%
I       17     0.03%
II    1627     2.89%
III  22026    39.13%
IV    6418    11.40%
V    24873    44.19%
VI      92     0.16%
VII     89     0.16%

Note: Around 1000+ rows gave an either/or value for luminosity class (e.g., "M1Ib/II"). In these cases, I counted only the first value provided. This probably skewed the results slightly compared to counting both luminosity classes.

I'm still very curious to know whether anyone else has produced or located a similar table of frequencies for the luminosity classes, if only to see how my very trivial analysis compares.

gravity - How can a supernova cause a gravitational wave?

The key here is that to produce gravitational waves, the supernova explosion must not have perfect spherical symmetry. Technically, what you need is an "accelerating" gravitational quadrupole moment.

It is quite unlikely that supernovae explosions will be symmetric. Asymmetries can be produced by binary companions, magnetic fields, rotation...

The characteristics of a "burst" source of gravitational waves will be quite unlike the signature of coalescing binary black holes. The magnitude of the signal is expected to follow the collapsing core of the star and reach a peak at something called the "core bounce", where the equation of state of the gas "stiffens" and a shock wave propagates outwards. The signal is thus likely to gradually increase in amplitude for some tens of milli-seconds, followed by an abrupt spike at the bounce, followed by a "ring-down" as the core (a proto-neutron star) settles down towards a spherically symmetric configuration over the course of a further 10 milli-seconds or so.

The frequencies of the waves will have a broad range governed by the characteristic frequency implied by $(Grho)^{1/2}$, which is the inverse of the freefall timescale, where $rho$ is the density at the time when the GWs are emitted. For a typical core collapse, the densities could range from $10^{14}$ to a few $10^{17}$ kg/m$^3$ over the course of the event, producing GW frequencies of tens to thousands of Hz - exactly in the range at which aLIGO is sensitive.

Because GWs pass through the envelope of the star unimpeded then the GW signature of a supernova probes the very heart of a supernova explosion and should be seen some hours before the visible signature is apparent. On the other hand any neutrino emission from a core collapse supernova should be detected at a very similar time to the GW waves. Any delay here could fix the currently poorly known neutrino mass.

elliptic curves - components of E[p], E universal in char p.

I have just realised that a group scheme I've known and loved for years is probably a bit wackier than I'd realised.

In this question, in Charles Rezk's answer, I erroneously claim that his construction of the space representing Drinfeld $Gamma_1(p)$ structures on elliptic curves must be flawed, because the global properties of $Y_1(p)$ that I know from Katz-Mazur seemed to contradict global properties that his construction appeared to me to have. We took the conversation to email and I also started writing down my thoughts more carefully to check there were no problems with them. I found a problem with them---hence this question.

Let $p$ be prime, let $Ngeq4$ be an integer prime to $p$, and consider the fine moduli space $Y_1(N)$ over an algebraically closed field $k$ of characteristic $p$. The $N$ isn't important, it just saves me having to use the language of stacks. Let $Y^o$ denote the open affine of $Y_1(N)$ obtained by removing the supersingular points. Over $Y^o$ we have an elliptic curve $E$ (obtained from the universal family over $Y_1(N)$).

In brief: here's the question. The $p$-torsion $E[p]$ of $E$---it's a group scheme and its identity component is non-reduced. But (regarded as an abstract scheme) does it have a component which is reduced? I think it might! This goes against my intuition.

Now let me go more carefully. Let's consider the scheme $E[p]$ of $p$-torsion points. This is finite flat over $Y^o$ and hence as an an abstract scheme over $k$ it's going to be some sort of 1-dimensional gadget. It also sits in the middle of an exact sequence of group schemes over $Y^o$:

$0to Kto E[p]to Hto 0$

with $K=ker(F)$, $F$ the relative Frobenius map (an isogeny of degree $p$). Now at every point in $Y^o$, the fibre of $K$ is isomorphic to $mu_p$ and the fibre of $E[p]$ is isomorphic to $mu_ptimesmathbf{Z}/pmathbf{Z}$. In particular all components of all fibres are isomorphic and non-reduced. Now here is where my argument in the thread in the question linked to above must become incorrect. I wanted to furthermore claim that

(a) $K$ (as an abstract curve) is non-reduced, and then

(b) hence (because $K$ is the identity component of $E[p]$ and "all components of a group are isomorphic as sets") all components of $E[p]$ are non-reduced.

I now think that (b) is nonsense. In fact I know (b) is nonsense in the sense that $mu_p$ over $mathbf{Q}$ has only two components and they look rather different when $p$ is odd, but in some sense I feel here that the difference is more striking. In fact I now strongly suspect that $E[p]$ as an abstract scheme has two components, one being $K$ and the other being a regular scheme (an Igusa curve) mapping down in an inseparable way onto $Y^o$ (so the component isn't smooth over $Y^o$ but abstractly it's a smooth curve).

If someone wants a proper question, then there is one: am I right? The identity component of $E[p]$ is surely non-reduced---but does $E[p]$ have any regular components? I know how to prove this but it will be a deformation theory argument and I've got to go to bed :-/ If so then I think it's the first example I've seen, or at least internalised, of a group scheme where the behaviour of a non-identity component is in some sense a lot better than the behaviour of the identity component. I say "in some sense" because somehow it's the map down to $k$ that is better-behaved, rather than the map down to $Y^o$. Someone please tell me I'm not talking nonsense ;-)

A heart for stable equivariant homotopy theory

Let $G$ be a finite group. I wonder whether the following statement is true, known and written down:

There is a t-structure on the stable $G$-equivariant homotopy category such that the associated heart is isomorphic to the category of Mackey functors (on $B_G$).

I feel like someone has told me so, but I can't find a reference. Thanks for your help!

rt.representation theory - Decomposition of certain projectives for cyclotomic q-Schur algebras

In representation theory, a very popular set of finite dimensional algebras are the $q$-Schur algebras, which are given by looking at the endomorphisms of $V^{otimes d}$ where $V$ is the standard representation of the quantum group $U_q(mathfrak{gl}_n)$. One should think of this as kind of enhanced version of the Hecke algebra for the symmetric group that doesn't lose simple representations at roots of unity.

The cyclotomic $q$-Schur algebra is a generalization of this, where the symmetric group is replaced the complex reflection group $S_nwr C_ell$ (this is the group of monomial matrices, where one allows $ell$th roots of unity as coefficients). For more details, see the original paper of Dipper, James and Mathas.

The original definition is as the endomorphism ring of a collection of modules over the Ariki-Koike algebra called "permutation modules" $M^nu$. These are generalizations (and deformations) of the permutation representations of $S_n$ on subsets of $n$ elements, and are in bijection with $ell$-tuples $nu$ of partitions with $n$ total boxes.

This gives a natural collection of projectives $N_nu=oplus_{mu}mathrm{Hom}(M^nu,M^mu)$, which generate the category of representations, but are very far from being irreducible. On the other hand , the indecomposable projectives $P_mu$ of the cyclotomic $q$-Schur algebra are also bijection with these $ell$-tuples of partitions.

So, given a collection of multipartitions, one can ask which indecomposable projectives occur in $N_nu$ for these multipartitions; I'd be interested to know any good references for this problem, as I only know the fairly obvious things about it (i.e. what you can deduce from the generic case, etc.). However, there is one set I'm particularly interested in.

My question: If I consider only multipartitions where each constituent partition is $(1^a)$ for some $a$ (or whatever corresponds to the regular rep of $S_n$ in your conventions), which projectives appear in $N_nu$?

My conjecture: The projectives corresponding to multipartitions where all constituent partitions are $k$-restricted, where $k$ is the order of $q$ in the complex numbers.

My follow-up questions: What if I only consider multipartitions of the form $(1^a)$ and require that for some $Ssubset [1,ell]$ the corresponding partitions are empty?

My follow-up conjecture: The projectives where for each block of consecutive elements in $S$ followed by an element not in $S$, the multipartition for that piece is $k$-Kleshchev.

ag.algebraic geometry - Flatness of relative canonical bundle

It sounds like you may want Exercise 9.7 in Hartshorne's "Residues and Duality". I paraphrase the statement:

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Exercise 9.7 (RD):
Let $f: X to B$ be a flat morphism of finite type of locally Noetherian preschemes. Then, $f^!(mathcal{O}_B)$ has a unique non-zero cohomology sheaf, which is flat over $B$, iff all the fibers of $f$ are Cohen-Macaulay schemes of the right dimension. Moreover $f^!(mathcal{O}_B)$ is isomorphic to (a shift of) an invertible sheaf (on $X$) iff all the fibers of $f$ are Gorenstein schemes of the right dimension.

---

In particular, this addresses the case you mention in your comment ($f$ Gorenstein), since then $f^!(mathcal{O}_Y)$ is locally free on $X$ and, since $f$ is flat, certainly flat over $B$.

[Aside: I believe I learned this reference from Brian's book "Grothendieck Duality and Base Change", which I think also contains a proof of this.]

Variable and Multiple Stars in Hipparcos

Yes. If you have the Hipparcos data from ftp (http://vizier.u-strasbg.fr/viz-bin/VizieR?-source=I/239&-to=3), you will have several data files.

In hip_main.dat you have a field MultFlag at position 347 that indicates whether the star is a double or multiple star:

Note on MultFlag: indicates that further details are given in the Double
     and Multiple Systems Annex:
     C : solutions for the components
     G : acceleration or higher order terms
     O : orbital solutions
     V : variability-induced movers (apparent motion arises from variability)
     X : stochastic solution (probably astrometric binaries with short period)

To determine whether the star is a double or multiple star system, look in one of the files for double star solutions. For instance in h_dm_com.dat there is a field at bytes 24-25 that gives the number of components (2 for double and >2 for multiple stars).

24- 25  I2     ---     Ncomp    Number of components in this solution   (DCM2)

The Hipparcos number identifying the star can be found in bytes 43-48.

Of course this only works for multiple stars with orbital solutions.

For variability, three fields are provided in the main Hipparcos file:

314-320  F7.2  d       Period    ? Variability period (days)              (H51)
     322  A1    ---     HvarType *[CDMPRU]? variability type               (H52)
     324  A1    ---     moreVar  *[12] Additional data about variability   (H53)

with two notes (NB. if HvarType is C then the star is constant, i.e. not variable!):

Note on HvarType: Hipparcos-defined type of variability (a blank entry
     signifies that the entry could not be classified as variable or constant):
     C : no variability detected ("constant")
     D : duplicity-induced variability
     M : possibly micro-variable (amplitude < 0.03mag)
     P : periodic variable
     R : V-I colour index was revised due to variability analysis
     U : unsolved variable which does not fall in the other categories
Note on moreVar: more data about periodic variability are provided

There are two additional files on variability: hip_va_1.dat and hip_va_2.dat, which provide information such as variable type (GCVS) and max and min magnitude and also the variable star name.
You can find a description of the variable type in the General Catalogue of Variable Stars (GCVS) in the file vartype.txt if you download the GCVS at http://cdsarc.u-strasbg.fr/viz-bin/Cat?B/gcvs

amateur observing - What kind of telescope do I need to see most of the Jupiter's moons?

Unless you're really rich, unfortunately you won't be able to see all of them.

Jupiter's fifth largest Moon, Amalthea, has an apparent magnitude of $m$ = 14.1. Comparing this to the magnitude of Europa, the dimmest of the Galilean moons, which is 5.3, tells us that Amalthea is roughly 3000 times less bright. Your telescope thus needs to have an area 3000 (or radius ~55) times larger for Amalthea to have the same apparent brightness.

In general, without a camera on your telescope, the dimmest object you can see depends on your vision, but on average, humans are able to see objects of magnitude 6. That means that in order to detect an object of magnitude $m_mathrm{obj}$, you need a light-collecting area which is larger than you pupil by a factor of
$$f = 10^{(m_mathrm{obj}-6)/2.5}.$$
Thus, in order to be able to just barely detect Amalthea, you need a telescope which is larger than you pupil (radius = 6 mm) by a factor of ~1738, i.e. has a radius of 25 cm.

It quickly becomes really difficult to see them; for instance, Elara, which is the eighth largest moon, has an apparent magnitude of 16.3 and thus requires a 70 cm telescope. And remember, this is just on the threshold of what you can see.

Of course, if you mount a camera to your telescope and take images with long exposure times, you can get away with smaller telescopes. In fact all moons smaller than Amalthea were discovered this way, some of them by cameras on the Voyager space probes.

gt.geometric topology - Is there any symmetric unshellable triangulation of a tetrahedron?

M. Rudin gave a triangulation of a tetrahedron with the property that after any small tetrahedron removed, the remaining part is not homeomorphic to a ball (An unshellable triangulation of a tetrahedron, Bull. Am. Math. Soc. (64), 1958, pp.~90--91). This example is not compatible with the tetrahedral rotation group $T_{12}$. Is there an unshellable triangulation invariant under the rotation group's action?

Some information (maybe usefull): P. Alfeld told me that RH Bing had another unshellable triangulation example using knot theory (Some aspects of the topology of 3-manifolds related to the Poincare conjecture, in "Lectures on Modern Mathematics", vol. 2, pp. 93-128, Wiley, New York, 1964). Bing's example is to prove that such triangulation does exist, but not a plicit construction.

maximal compact subgroup as fixed points of some involution on p-adic group?

Here is how the real and p-adic situations are the same.

Let $G$ be a connected reductive algebraic group defined over a field $F$ not of characteristic two. Let $theta$ be an involution of $G$ defined over $F$. Then the group $G^theta$ of fixed points is a reductive algebraic subgroup of $G$.

Here are two ways in which they are different.

In the real case, one can always choose $theta$ so that the group of rational points of $G^theta$ is compact. In the p-adic case, compact reductive groups are quite rare, and so in most cases there is no analogous way to choose $theta$.

Second, compact subgroups do not play the same roles in the real and p-adic cases. Think of the fields themselves. In the p-adic case, the maximal compact subring is the ring of integers. In the real case, there are no nontrivial compact subrings. There is a ring of integers, but it is not compact. Moreover, since $G^theta$ has smaller dimension than $G$, it cannot be an open subgroup, and maximal compact subgroups are always open in the p-adic case. Thus, even in the rare cases where $G^theta$ is compact, it is not maximal.

st.statistics - Population Spearman Rank Correlation Coefficient

Let p(x,y) be the joint probability density function of the random variables X and Y. Let P_x(x) and P_y(y) the marginial cumulative distribution functions respectively. The key observation is that the normalized rank of a sample of x (i.e., its rank divided by the number of observations R(x_i)/n) is just a sample of the random variable P_x(X). Thus, it is not hard to convince oneself that the statistic:

Rho = 1-6(P_x(X)-P_y(Y))^2 is an estimator of the Spearman rank correlation, and its population mean is the population's Spearman rank coefficient is given by:

rho = 1 - 6 int ((P_x(x)-P_y(y))^2 p(x,y) dxdy)

The following article performs the same calculation for a weighted version of the Spearman's correlation coefficient:

http://www.ine.pt/revstat/pdf/rs060301.pdf

I think that the sample Spearman is unbiased because of the averaging by n*(n-1)*(n+1), but I still don't know how to prove that.

Please, notice that the population mean of the statistic (the population Spearman correlation coefficient) becomes zero when the random variables are independent, i.e., p(x,y) = p(x)*p(y).

venus - Occurence of Venusian transits

Quoting http://eclipse.gsfc.nasa.gov/transit/catalog/VenusCatalog.html

"When a transit of Venus occurs, a second one often follows eight years later. This is because the orbital periods of Venus (224.701 days) and Earth (365.256 days) are in an 8 year (2922 days) resonance with each other. In other words, in the time it takes Earth to orbit the Sun eight times, Venus completes almost exactly thirteen revolutions about the Sun. As a result, Venus and Earth line up in the same positions with respect to the Sun. Actually, the two orbital periods are not quite commensurate with each other since Venus arrives at the eight year rendezvous about 2.45 days earlier that Earth. After the third eight-year cycle, Venus arrives too early for a transit to occur."

generating functions - Solving recurrence equation with indexes from negative infinity to positive infinity

In many cases, the recurrence equations that people are solving involves index of only non-negative values. Here I have a recurrence equation which arises from transport of light in an infinite 1D chain:

$a_m=sum _{j=1}^{infty } left(T_ja_{m+j}+T_ja_{m-j}right) + delta _{m,0}$

where $delta_{m,0}$ is the Kronecker delta function. i.e.:

$delta_{i,j} = begin{cases} & 1 text{ if } i=j \ & 0 text{ if } i neq j end{cases}$

Here I would like to solve $a_m$, where the index of m is from negative infinity to positive infinity, while $T_j$ is a given sequence, and p is just a given constant.

Defining the generating function $G(z)=sum _{k=-infty }^{infty } a_kz^k$, I found that:

$G(z)=frac{1}{1-sum _{k=1}^{infty } t_kleft(z^{-k}+z^kright)}$

The problem is, how am I going to do series expansion on G? Doing a simple expansion of $frac{1}{1-sum _{k=1}^{infty } t_kleft(z^{-k}+z^kright)}=sum _{j=0}^{infty } left(sum _{k=1}^{infty } t_kleft(z^{-k}+z^kright)right){}^j$ won't help. Since the power is too difficult to expand out.

And contour integration isn't helping as well, since it is too difficult to compute analytically or numerically too.

Here I would like to ask about direction in obtaining analytical solution, or approximated one.

And in my case, my function G is given by:

$G(z)=left(1+frac{3i}{2r^3}left(r^2left(ln left(1-frac{e^{i r}}{z}right)+ln left(1-e^{i r}zright)right)right)-i rleft(text{Li}_2left(frac{e^{i r}}{z}right)+text{Li}_2left(e^{i r}zright)right)+text{Li}_3left(frac{e^{i r}}{z}right)+text{Li}_3left(e^{i r}zright)right){}^{-1}$

p.s.:I have posted the same problem in Voofie.

units - Is the Earth-Sun distance 1.012 AU?

As of this moment (2016 May 18, 13:15 UTC), the Earth is 1.0116 astronomical units from the Sun. WA is smart enough to know that "Sun Earth distance in AU" is a time-dependent question.

Update: The stricken text that follows from my original answer is incorrect. I am leaving it present (but stricken) for the sake of humility.

It interpreted your query to mean Sun Earth distance today in AU, and because today is 2016 May 18 and because you didn't specify a time of day, it picked noon (UTC), it in turn interpreted your query to mean Sun Earth distance on 2016 May 18 at noon UTC in AU.

The correct answer: I happened to ask WA the very question posed in the OP a couple of minutes apart and got two different answers (1.014 and 1.015 AU), and this did not occur across a day boundary. WA apparently interpreted your query to mean the current distance between the Sun and the Earth.

You have to be very careful of what you ask WA.

A better query is to ask WA What is the mean distance between the earth and the sun in AU? That query will give you an answer of 1.0000010178 au. An even better question is What is the semimajor axis of Earth's orbit? That will give you an answer of 1.00000011 au. Note the extra zero, but also note that for some reason, the precision is reduced.

the sun - Does Dark Matter affect the motion of the Solar System?

The dark matter model that is used to explain the "missing mass" problem relating to our Galactic rotation curve, consists of a pseudo-spherical distribution that is much more extended than the visible stars and gas. Even though this "halo" contains more than ten times the mass of the visible matter, when you work out what it's density should be in the solar neighbourhood, it turns out to be around 100 times less than the "normal" interplanetary medium.

This means, that inside the Earth's orbit, there could be as much as $10^{10}$ kg of dark matter. This sounds a lot and in principle, yes, this would speed up the orbits of planets in a way that depended on their distance from the Sun. However, the predicted effects of this are still about 6 orders of magnitude below the current precision of measurement.

co.combinatorics - Let G be a graph such that for all u, v ∈ V (G), u no equal to v , |N (u) ∩ N (v )| is odd. Then show that the number of vertices in G is odd

I proved over here that statement ii) holds when we make the stronger assumption that $|N(u) cap N(v)|$ is exactly $1$ for every $u, v$.

Most of the argument probably does not generalize, but at least one piece of it does. That part is that if $G$ is a minimal counterexample, then the complement graph is connected. I'll prove this by contradiction:

Let $X$ and $Y$ be a partition of the vertices into two nonempty parts such that every vertex in X is connected to every vertex in Y. If $X$ has even size, then we see that $Y$ has the same property $G$ has and is smaller than $G$, so $Y$ has odd size and $G$ has an odd number of vertices. If $X$ has odd size, then if we collapse $X$ to a point $x$ we still have the property that $|N(u) cap N(v)|$ is odd for $u, v$ in $Y$, and also for any $u$ in $Y$, $|N(x)cap N(u)|$ is odd by your step i), so $Ycup x$ satisfies the same properties $G$ does and contains a vertex that is connected to everything, so $Ycup x$ has odd size, so $G$ has odd size.

Unfortunately, I can't see any obvious nice relationship that must be satisfied by two nonadjacent vertices in our graph $G$...

Why are Lie Algebras/ Lie Groups so much like crossed modules, and not?

A crossed module consists of a pair of groups $G$ and $H$ with a group homomorphism, $t:H rightarrow G$, and $alpha: G times H rightarrow H$ that defines an action of $G$ on $H$, $tilde{alpha}$: via
$alpha(g,h)=tilde{alpha}(g) (h)$. These maps satisfy,
$t(alpha(g,h))= g t(h) g^{-1}$, and $alpha(t(h), h')= h h' h^{-1}$.

According to Baez and Lauda HDA 5 example 48, page 64, a Lie group and a Lie algebra can be used to construct a crossed module. The construction is to let $alpha$ be defined via the adjoint map, and to let $t$ be defined as the trivial map $t(v)=1$ for all $v in g$.

To simplify my question, suppose that $g = su(2)$, and $G=SU(2)$. Now consider $alpha(g,v) = g v g^{-1}$ for $gin G$ and $vin g$. Define $t(v) = exp{v}$. We can compute that $t(alpha(g,h))= g t(h) g^{-1}$. However,

1. The Baker-Campbell-Hausdorff formula precludes that $t$ is a homomorphism, and


2. $tilde{alpha}(exp{h}) (h')$ is rotation of $h'$ about the vector $h$ through an angle $ 2 |h|$.
   In other words, $tilde{alpha}(exp{h}) (h') = (exp{h}) h' (exp{(-h)}).$

So the lack of homomorphism and the wrong Peiffer identity puts a damper on things. Is there a ``crossed module" like name for such a structure? For example, is it related to a $2$-group in any sense?

This question may be related to Theo's about Bernoulli numbers.