linear algebra - Help me with this proof: Drop a printed map of the land on the land and there must be some common point.

Hi, I have a minor in math and this is not a homework problem - my prof mentioned it 5 years ago and I could not even begin to tackle it until I took a good intro to linear algebra (after work). Please try to adjust the answer to my level.

A map is a smaller version of the land: rotated and scaled down. The prerequisite is that the map (say of USA) lands entirely inside the land (and not partly in Canada or in the ocean). Another prerequisite might be that the map has no holes (is that really necessary?) and perhaps that it must be a convex region (however I doubt that this is needed.) Please help me eliminate these doubts and prove that $p in M, p in L$. Correct my notation if needed by editing this post.

Here is the notation that I came up with (let's see if LaTex likes me):

• $L$ = land (region in $R^2$)


• $M$ = map (region in $R^2$)


• $T$ = transformation in $R^2$ (scalar times a rotation matrix)


• $vec{s}$ = shift vector (since the overall transformation is generally non-linear)


• $vec{p}$ = "pivot" - the point that does not change.

Now, I can solve for p uniquely by picking a non-trivial triangle on the land denoted by vectors $vec{l_1}, vec{l_2}, vec{l_3}$, locating the corresponding vectors $vec{m_1}, vec{m_2}, vec{m_3}$ on the map and writing out:

$vec{m_1} = T vec{l_1} + vec{s}$, $vec{m_2} = T vec{l_2} + vec{s}$, $vec{m_3} = T vec{l_3} + vec{s}$

I solve for T:
$T = [ m_1 - m_2 ; ; ; m_1 - m_3 ] [ l_1 - l_2 ;;; l_1 - l_3 ]^{-1}$
Because we picked a non-trivial triangle, it's area will be non-zero and the matrix on the right will be invertible. So, we can always solve for T.
We can also solve for s: $vec{s} = vec{m_1} - T vec{l_1}$, and finally for the pivot p: $vec{p} = (I - T)^{-1} vec{s}$. Since T = c * rotation matrix, where $c leq 1$, the only time when (I - T) does not have an inverse is when I = T (details omitted).

So, it seems that I can solve for a unique p.

• Now, how do I prove that pivot $vec{p}$ will be inside both shapes/regions L and M?


• Finally, what assumptions must I make about convexity of M, absence of holes, etc ?

P.S. Which undergraduate classes might have this as a homework problem?

lo.logic - Prime-ness checking for polynomial ideals over ACFs( algebraically closed fields).

In his paper Constructions in algebra (MR349648), Seidenberg fixes some errors in Grete Hermann's classic paper Die Frage der endlich vielen Schritte in der Theorie der Polynomideale (MR1512302). Most of the work concentrates on two problems:

1. Effectively compute the primary decomposition of an ideal.


2. Effectively compute the associated prime of a primary ideal.

Together, these do give a primality test: first check that the ideal is primary by computing its primary decomposition and then check that it equals its associated prime.

Seidenberg's paper is very well written, but he obviously has a very limited audience in mind. He does prove along the way the existence of primitive recursive degree bounds, but the task to put them together into a precise form would require a fair amount of time. Seidenberg's constructivist point of view gets in the way of pragmatism from time to time.

There has been a lot of water under the bridge since 1974 and much of the contents of Seidenberg's paper must have been redone for practical use in modern computer algebra systems. Since working out the bounds from Seidenberg's paper looks impractical, I would consider looking at the computer algebra literature to see which algorithms are currently used to carry out the two steps above and derive effective degree bounds from these.

I am not very knowledgeable in such practical matters, but there are probably computer algebra experts lurking around here. If they don't see this post, then I suggest you ask another question for algorithms or degree bounds for steps 1 and 2 specifically.

ap.analysis of pdes - The normal derivative of the Green's function

I was wondering if anything was known about the following:

Let $mathbb{D}^2=lbrace x^2+y^2< 1 rbrace subset mathbb{R}^2$ be the open unit disk.
Consider now the Green's functions $G(z; p)$ of this disk. I.e. here $pin mathbb{D}^2$ and $G(z;p)$ is smooth and harmonic in $bar{mathbb{D}}^2backslash lbrace p rbrace$, vanishes on the boundary and has the property that $H(z;p) =G(z; p)-log |z-p|$ is smooth.

Now consider the set of functions:
begin{equation}
S=lbrace fin C^infty(partial mathbb{D}^2): f= sum_{i=1}^n lambda_i partial_{nu} G(z; p_i), lambda_i in mathbb{R}, p_i in mathbb{D} rbrace
end{equation}
Here $partial_{nu} G(z; p_i)$ is the normal derivative on $partial mathbb{D}^2$.

My question is what can be said about the set $S$? In particular, is there any hope that it is dense in $L^2(partial mathbb{D}^2)$?

Playing around with things all I got was a big mess so any references would be appreciated.

Why do planets tend to rotate in the same direction although they have formed from tumbling asteroids?

You are right that the tilt of the asteroids are distributed in very random way, and that the rotation of the Solar nebula is a minor contributor to that tilt, and only skews it a little.

However, you are not right that randomness simply adds up. The randomness does in fact cancel out more and more when you combine a large amount of asteroids, until the rotation of the nebula becomes the dominant factor. This is related to the Law of large numbers.

For instance, throw a dice. The outcome is random. Throw 10 dices, calculate their sum, and divide by 10. Not so far from the average any more? You can do the same thing with thousands of dices, or millions of asteroids. When the number of asteroids that form an object is really high, the tilt is not going to be far from the average value, determined by the nebula's rotation.

The same argument goes for inclination, and the fact that even though the planets' orbits are elliptical, they are not as far from circular that a random orbit would be.

it.information theory - Untrustworthy people picking a random number

If the final outcome doesn't need to be perfectly unbiased, but instead may have a bias of up to say 10 percent, then there are good algorithms, even if the people have unlimited computational power (which renders cryptographic methods ineffective). For example, the following baton-passing algorithm works well. At the start, give the baton to an arbitrary person (say the first person). At each step, the current baton-holder gives the baton to a random person who has not yet held the baton. Whoever receives the baton last gets to make the collective coin flip.

If the number of saboteurs is fewer than $n / log n$, then this algorithm is known to produce a low-biased coin. Mike Saks proposed and analyzed this algorithm, and Miki Ajtai and Nati Linial refined the analysis. Here are the references:

M. Saks (1989), A robust non-cryptographic protocol for collective coin flipping, SIAM Journal on Discrete Mathematics 2, pages 240-244.

M. Ajtai and N. Linial (1993), The influence of large coalitions, Combinatorica 13, pages 129-145.

You can find a scanned copy of the Ajtai-Linial paper at Linial's homepage:
http://www.cs.huji.ac.il/~nati/.

Babu Narayanan and I showed that there exists an algorithm that can handle up to 49 percent of the players being saboteurs and yet still produces a low-biased coin. Here is the reference:

R. Boppana and B. Narayanan (2000), Perfect-Information Leader Election with Optimal Resilience, SIAM Journal on Computing 29:4, pages 1304-1320.

rt.representation theory - Invariant quadratic forms of irreducible representations

Let $G$ be a finite group, and $k$ be a field of characteristic zero (not necessarily algebraically closed!). Let $rho : G to mathrm{End}_k left(k^nright)$ be a irreducible representation of $G$ over $k$. Consider the vector space

$S=leftlbrace Hin mathrm{End}_kleft(k^nright) mid rholeft(gright)^T Hrholeft(gright)=Htext{ for any }gin Grightrbrace$

$=leftlbrace sumlimits_{gin G}rholeft(gright)^T Hrholeft(gright)mid Hin mathrm{End}_kleft(k^nright)rightrbrace$

and its subspace

$T=leftlbrace Hin Smid Htext{ is a symmetric matrix}rightrbrace$.

It is easy to show that, if we denote our representation of $G$ on $k^n$ by $V$, then the elements of $S$ uniquely correspond to homomorphisms of representations $Vto V^{ast}$ (namely, $Hin S$ corresponds to the homomorphism $vmapstoleft(wmapsto v^THwright)$), while the elements of $T$ uniquely correspond to $G$-invariant quadratic forms on $V$ (namely, $Hin T$ corresponds to the quadratic form $vmapsto v^THv$).

(1) In the case when $k=mathbb C$, Schur's lemma yields $dim Sleq 1$, with equality if and only if $Vcong V^{ast}$ (which holds if and only if $V$ is a real or quaternionic representation). Thus, $dim Tleq 1$, and it is known that this is an equality if and only if $V$ is a real representation. (Except of the equality parts, this all pertains to the more general case when $k$ is algebraically closed of zero characteristic).

(2) In the case when $k=mathbb R$, it is easily seen that $Tneq 0$ (that's the famous nondegenerate unitary form, which in the case $k=mathbb R$ is a quadratic form), and I think I can show (using the spectral theorem) that $dim T=1$. As for $S$, it can have dimension $>1$.

(3) I am wondering what can be said about other fields $k$; for instance, $k=mathbb Q$. If $ksubseteqmathbb R$, do we still have $dim T=1$ as in the $mathbb R$ case? In fact, $Tneq 0$ can be shown in the same way.

algebraic number theory - Conjugacy classes in the absolute galois group

I agree with the answer of Kevin Buzzard -- you better look only at quotients $Gal(K/Q)$ unramified outside some (finite) set $S$ to make the question make sense as it is.

But regardless, you can ask about the conjugacy classes in the absolute Galois group, and what they "mean". An answer was given many years ago by Ax (I guess in one of his Annals papers, 1968 or 1969), and there are elaborations and new results in the thesis of James Gray, now published in the J. of Symbolic Logic as "Coding complete theories in Galois groups" (his thesis is also available freely online).

The reason why these "other conjugacy classes" (which are essentially all of them) are difficult to construct is that the associated fixed fields are the algebraic (over $Q$) subfields of pseudofinite fields of characteristic zero. In Theorem 1.27 of his thesis, Gray states that there is a [natural, homeomorphic in the appropriate topology] bijection between the set of conjugacy classes in $Gal(bar Q / Q)$ and the Stone space of completions of $ACFA_0$ -- the theory of algebraically closed fields of characteristic zero with generic automorphism. Given such a completion of $ACFA_0$, realized by a model $(K, sigma)$ where $K$ is algebraically closed, and $sigma$ is a generic automorphism (see MacIntyre, "Generic automorphisms of fields" for the definition of generic used here), the fixed field $K^sigma$ is a pseudofinite field of characteristic zero.

Now, while two conjugacy classes in $Gal(bar Q / Q)$ are quite simple to describe -- the trivial conjugacy class and the conjugacy class of order 2 elements -- other conjugacy classes contain elements of infinite order. The associated fixed fields (in this infinite-order case) in $bar Q$ are psuedofinite fields, which are difficult to get your hands on. Probably the best way is to take an ultraproduct (for a non-principal ultrafilter on the set of prime numbers) of finite fields, and take the algebraic elements within. This is certainly nonconstructive, relying heavily on Zorn's lemma.

Still, such fields have arithmetic significance. The model theory related to $ACFA_0$ has had a great deal of impact on number theory lately, and Fried-Jarden also touch on related matters in their "Field Arithmetic" book.

nt.number theory - solutions to equation mod a prime

I know that characterizing the solutions to an equation in a finite field is generally difficult, but I was wondering if anyone had anything to say about the equation

(ab)^2 + a^2 + b^2 = 0 mod p

I started by writing the equation as (a^2 + 1)(b^2 + 1) = 1 mod p, and also (ab + 1)^2 + (a-b)^2 = 1 mod p, but that hasn't seemed to help much. I know how to determine the number of solutions, but what I am more interested in is the (possible) implied relation between a and b.

Thanks!

star - Why does lithium fuse at lower temperatures than hydrogen?

This is a basic question, but it's been bugging me. In the Wikipedia article for lithium burning, it states that:

Stars, which by definition must achieve the high temperature (2.5 × 10^6 K) necessary for fusing hydrogen, rapidly deplete their lithium. This occurs by a collision of lithium-7 and a proton producing two helium-4 nuclei. The temperature necessary for this reaction is just below the temperature necessary for hydrogen fusion.

I would imagine that with lithium having more protons, it would have a stronger Coulomb repulsion and require higher temperatures to fuse with hydrogen. Now, this article is pretty sketchy since it cites no sources, and I would normally dismiss this. But according to here, lithium burning occurs in protostars, before hydrogen fusion even takes place. How could lithium fuse with hydrogen at temperatures that low?

The localisation long exact sequence in K-theory over an arbitrary base

I'm not sure that what I have to say really addresses the heart of your question, but it seems at least related.

Background

The general Localization Theorem (7.4 of Thomason-Trobaugh) states the following. Suppose $X$ a quasiseparated, quasicompact scheme, suppose $U$ a Zariski open in $X$ such that $U$ is also quasiseparated and quasicompact, and suppose $Z$ the closed complement. Then the following sequence of spectra is a fiber sequence:
$$K^B(Xtextrm{ on }Z)to K^B(X)to K^B(U).$$
Here $K^B$ refers to the Bass nonconnective delooping of algebraic $K$-theory. One thus gets a long exact sequence
$$cdotsto K_n^B(Xtextrm{ on }Z)to K_n^B(X)to K_n^B(U)to K_{n-1}^B(Xtextrm{ on }Z)tocdots$$
(If one tries to work only with the connective version, then the exact sequence ends awkwardly, since $K_0(X)to K_0(U)$ is not in general surjective; indeed, the obstruction to lifting $K_0$-classes from $U$ to $X$ is precisely $K_{-1}(Z)$ by Bass's fundamental theorem.)

The term $K^B(Xtextrm{ on }Z)$ is the Bass delooping of the $K$-theory of the ∞-category of perfect complexes of quasicoherent $mathcal{O}$-modules that are acyclic on $U$. Identifying this fiber term with $K^B(Z)$ is generally a delicate matter. Let me summarize one situation in which it can be done.

Suppose that $X$ admits an ample family of line bundles [Thomason-Trobaugh 2.1.1, SGA VI Exp. II 2.2.3], and suppose that $Z$ admits a subscheme structure such that the inclusion $Zto X$ is a regular immersion (so that the relative cotangent complex $mathbf{L}_{X|Z}$ is $I/I^2[1]$, where $I$ is the ideal of definition), and $Z$ is of codimension $k$ in $d$ in $X$. Then the spectrum $K^B(Xtextrm{ on }Z)$ coincides with a nonconnective delooping of the Quillen $K$-theory of the exact category of pseudocoherent $mathcal{O}_X$-modules of Tor-dimension $leq k$ supported on $Z$. If now $Z$ and $X$ are regular noetherian schemes, then a dévissage argument now permits us to identify $K^B(Xtextrm{ on }Z)$ with $K(Z)$.

Your case

Now I'm assuming that $K(D)$ refers just to the $K$-theory of the ring $k[[t]]$ (and not, for instance, the $K$-theory of the formal scheme $mathrm{Spf}(k[[t]])$), then the discussion above applies to give you your desired localization sequence
$$K^B(X)to K^B(X[[t]])to K^B(X((t)))$$
for any scheme $X$ admitting an ample family of line bundles. If in particular $X$ is regular, then the negative $K$-theory vanishes, and we have a localization sequence
$$K(X)to K(X[[t]])to K(X((t)))$$

observation - Why does the moon sometimes appear giant and a orange red color near the horizon?

(Source, Wikipedia Commons)

The moon is generally called a "Harvest Moon" when it appears that way (i.e. large and red) in autumn, amongst a few other names. There are other names that are associated with specific timeframes as well. The colour is due to atmospheric scattering (Also known as Rayleigh scattering):

may have noticed that they always occur when the Sun or Moon is close to the horizon. If you think about it, sunlight or moonlight must travel through the maximum amount of atmosphere to get to your eyes when the Sun or Moon is on the horizon (remember that that atmosphere is a sphere around the Earth). So, you expect more blue light to be scattered from Sunlight or Moonlight when the Sun or Moon is on the horizon than when it is, say, overhead; this makes the object look redder.

As to the size, that is commonly referred to as the "Moon Illusion", which may be a combination of many factors. The most common explanation is that the frame of reference just tricks our brains. Also, if you look straight up, the perceived distance is much smaller to our brains than the distance to the horizon. We don't perceive the sky to be a hemispherical bowl over us, but rather a much more shallow bowl. Just ask anyone to point to the halfway point between the horizon and zenith, and you will see that the angle tends to be closer to 30 degrees as opposed to the 45 it should be.

University of Wisconsin discussion on the Moon Illusion.

NASA discussion on moon illusion.

A graphical representation of this:

Dr. Phil Plait discusses the illusion in detail.

solar system - If a person were to die on the Moon or Mars, would the body decompose?

Space, as Randall notes, is really dry. Mars, (recent discoveries notwithstanding) is not much moister.

In these conditions, bodies mummify.

The microbes that live in you, wouldn't survive the freezing, dessication and radiation. There is no real upper limit on how long a mummified body could exist in space. There would be nothing to cause the body to change, and so it would remain. There would be a slow breaking down of surface proteins, due to UV light, and eventually micrometeorites would erode the body, but these processes would take many millions of years.

planet - Does earth have retrograde motion?

It does not have a retrograde motion. It has a prograde rotation, and a prograde orbit of the Sun.

If you were to look down on the solar system from far above the north pole you would see the planets orbiting the sun counter-clockwise. And the planets, including the earth also spinning counter clockwise. The Moon also orbits the Earth counter-clockwise. All this motion is in the same direction, and called prograde.

There are a couple of exceptions: Venus spins clockwise. Venus spins in a retrograde motion. There is a moon of Neptune, Triton, that orbits retrograde. Uranus spins on its side.

The reason is that most objects rotate the same way is that the planets and the sun formed out of the same cloud of gas and so have the same direction of spin.

There is another sense of "retrograde". As the Earth overtakes the other planets they seem to move backwards in the sky for a while.

From the inner planets, Earth would appear to move rather like mars, with periods of retrograde motion. From the outer planets Earth would move like Venus, moving from being a "morning star" to an "Evening star".

at.algebraic topology - Principle when limits level by level don't commute with simplicial structure

Limits and colimits can be computed point-wise in functor categories, such as the category of simplicial objects of any category. That is, if you have a functor category $mathcal{C}^I$ ($I = Delta$ in your case) and a functor $F: P longrightarrow mathcal{C}^I$, in particular, for every object $p$ in $P$ you have a functor $F_p : I longrightarrow mathcal{C}$. Then $varinjlim_p F_p$ is an object of $mathcal{C}^I$; that is, a functor $I longrightarrow mathcal{C}$, whose value on objects $i$ in $I$ is

$$
(varinjlim_p F_p)(i) = varinjlim_p (F_p(i)) .
$$

This is true as far as the colimit on the right exists for every object $i$ in $I$. An analogous statement applies for limits in functor categories.

For limits, you can find the result in MacLane, "Categories for the working mathematician", chapter V, section 3, theorem 1 (see at the end of the proof also). For colimits, see "Sheaves in geometry and logic: a first introduction to topos theory", by Saunders Mac Lane,Ieke Moerdijk, p.40.

ac.commutative algebra - Finite extension of fields with no primitive element

The example given by Alon is not only a finite extension without a primitive element, but it's the simplest example of a substitute for Galois theory when the field extensions are purely inseparable. In the example, the top field is $L = {mathbf F}_p(a,b)$ and the base field is $L^p = {mathbf F}_p(a^p,b^p)$, where $a$ and $b$ are algebraically independent. Since the extension $L/L^p$ is purely inseparable, we can't describe the intermediate fields using Galois theory. Jacobson found a way to describe them using Lie algebras of differential operators: fixed points of field automorphisms are replaced by kernels of differential operators.

Theorem (Jacobson). Let $K$ be a field of characteristic $p$ and $mathrm{Der}(K)$ be the space of derivations $K rightarrow K$, which is a Lie algebra under the bracket operation on derivations. To each field $F$ lying between $K$ and $K^p$, let $mathrm{Der}_F(K)$ be the subspace of derivations on $K$ which vanish on $F$ (the $F$-linear derivations on $K$). Then $mathrm{Der}_F(K)$ is a restricted Lie subalgebra of $mathrm{Der}(K)$ (the label "restricted" means the $p$th power of each element in the subalgebra is also in there).
If $[K:F]$ is finite then $mathrm{Der}_F(K)$ is finite-dimensional as a (left) $F$-vector space. Sending $F$ to $mathrm{Der}_F(K)$ is an inclusion-reversing bijection from the fields between $K$ and $K^p$ over which $K$ is finite-dimensional and the restricted Lie subalgebras of $mathrm{Der}_F(K)$ which are finite-dimensional over $K$, with $[K:F] = p^{dim_K(mathrm{Der}_F(K))}$. The inverse map associates to any restricted Lie subalgebra of $mathrm{Der}(K)$ which is finite-dimensional over $K$ the elements of $K$ on which the entire subalgebra vanishes.

Let's see how this theorem manifests itself in the example $K = mathbf{F}_p(u,v)$, where $u$ and $v$ are algebraically independent. Here $K^p = mathbf{F}_p(u^p,v^p)$ and $[K:K^p] = p^2$. (In other examples, $[K:K^p]$ could be infinite.) The partial derivative operators $partial_u = partial/partial u$ and $partial_v = partial/partial v$ are a basis for $mathrm{Der}(K)$ as a left $K$-vector space:
$$
mathrm{Der}(K) = Kpartial_u + Kpartial_v.
$$
An example of a field between $K$ and $K^p$ is $K^p(u^m + v^n)$ for positive integers $m$ and $n$. If $m$ and $n$ are both multiples of $p$ then $K^p(u^m + v^n) = K^p$, which is dull and all derivations are 0 on it: it corresponds to $mathrm{Der}(K) = mathrm{Der}_{K^p}(K)$. If at least one of $m$ or $n$ is not a multiple of $p$ then the set of derivations that vanish on $K^p(u^m + v^n)$ is $K(nv^{n-1}partial_u - mu^{m-1}partial_v)$. If we insist now that $m$ and $n$ are both not multiples of $p$, then changing $m$ or $n$ changes that line of derivations, so the corresponding fields $K^p(u^m+v^n)$ are not the same (you don't need Jacobson's bijection to see that: if one field is killed by a certain differential operator and another isn't, they are not the same field).

We haven't seen the restricted Lie algebra aspect playing an essential role here. In fact all one-dimensional $K$-subspaces of this two-dimensional space $mathrm{Der}(K)$ are restricted. To see this, pick a one-dimensional $K$-subspace $Kdelta$ in $mathrm{Der}(K)$. Set $F$ to be the common kernel of the operators in $Kdelta$, which is just the same thing as $ker(delta)$, so $F$ is a field between $K$ and $K^p$.
Clearly
$$
Kdelta subset mathrm{Der}_F(K) subset mathrm{Der}(K)
$$
and the top space is 2-dimensional over $K$. Thus $mathrm{Der}_F(K)$ is either $Kdelta$ or $mathrm{Der}(K)$. If $Kdelta = mathrm{Der}_F(K)$ then $Kdelta$ is restricted since the space of derivations on $K$ vanishing on a subfield is restricted and then we're done. So assume instead that $Kdelta not= mathrm{Der}_F(K)$, and then for dimensional reasons we must have $mathrm{Der}_F(K) = mathrm{Der}(K)$, which by Jacobson's correspondence implies $F = K^p$, so $ker(delta) = K^p$. But this is impossible: any nonzero $delta$ in $mathrm{Der}(K)$ is nonvanishing at either $u$ or $v$ (if $delta(u) = 0$ and $delta(v) = 0$ then $delta$ vanishes on $K^p(u,v) = K$), so $ker(delta)$ is not $K^p$ and thus we have a contradiction, which wraps up this little argument.

That all 1-dimensional $K$-subspaces of $mathrm{Der}(K)$ in this particular example are restricted Lie subalgebras sounds analogous to the fact that all one-dimensional subspaces of a Lie algebra are Lie subalgebras, although note $mathrm{Der}(K)$ is not a Lie algebra over $K$, only over $K^p$. Still, if anyone who knows about restricted Lie algebras can indicate in a comment if there is some general theorem about certain subspaces automatically being restricted subspaces, please speak up.

It is a theorem of Gerstenhaber (1964) that the Lie algebra aspect in Jacobson's theorem is always automatic from the restricted aspect: for any field $K$ of characteristic $p$, any $K$-subspace of $mathrm{Der}(K)$ which is closed under $p$th powers is closed under the Lie bracket on derivations.

This approach to a Galois theory for inseparable extensions has been extended to the case of inseparable $K/F$ where $K^{p^r} subset F$ some some $r > 1$, using higher derivations.
See "Higher derivation Galois theory of inseparable extensions" pp. 187--220 of Handbook [sic] of Algebra, Volume 1. (An account just of Jacobson's case $r = 1$ is in his Basic Algebra II, and in somewhat more detail -- including Gerstenhaber's result -- in Chapter 4 of Karpilovsky's Topics in Field Theory.)

New topic! RK asked, as a comment to Alon's answer, if there is a finite extension of $mathbf F_p(x)$ which is not primitive. No.

Theorem: If $K$ is a field of characteristic $p$ such that $[K:K^p] = p$, then every finite extension of $K$ has a primitive element.

We can apply this theorem to any rational function field over a perfect field of characteristic $p$. The conclusion of the theorem is stable under passage to finite extensions, so it applies to any function field over a perfect field of characteristic $p$.

Proof: Let $L/K$ be a finite extension. From the hypothesis that $[K:K^p] = p$, $[K:K^{p^m}] = p^m$ for all $m geq 0$. Then $[K^{1/p^m}:K] = p^m$, so if $K^{1/p^m} subset L$ we get $p^m|[L:K]$, which means $m$ is bounded. Taking $mgeq 0$ maximal such that $K^{1/p^m} subset L$, it can be shown that $L/K^{1/p^m}$ is separable.

Let $F$ be any field such that $K subset F subset L$. Letting $n geq 0$ be maximal such that $K^{1/p^n} subset F$, we have $n leq m$ and $F/K^{1/p^n}$ is separable.
Thus every field between $K$ and $L$ is a separable extension of one of the finitely many fields $K^{1/p^n}$ for $n = 0,1,dots,m$. Note the tower $F supset K^{1/p^n} supset K$, which is a separable extension on top of a purely inseparable extension, is backwards compared to what general field theory tells us: any finite extension can be expressed as a purely inseparable extension on top of a separable extension. We will now bring in that general field theory point of view, in a more precise form.

Any finite extension of fields admits a maximal separable subextension (containing all other separable extensions of the base field in the top field). Moreover, a separable extension of fields has only finitely many intermediate fields (say, by Galois theory). Therefore each $K^{1/p^n}$, for $n = 0,1,dots,m$ has only finitely many separable extensions inside $L$. Since every field between $K$ and $L$ is a separable extension of $K^{1/p^n}$ for some $n$ from 0 to $m$, there are only finitely many fields between $K$ and $L$. Therefore by Steinitz's theorem on primitive elements, $L$ is a primitive extension of $K$.

QED

fa.functional analysis - An element of $(L^{infty})^*$ which does not seem to be a finitely additive abs. cont. measure.

Hi everyone,

I have a question which I am quite stumped on. Consider the linear functional $l(f) = f(0)$ defined on $C([-1,1])$. By Hahn-Banach this linear functional can be extended to one on all of $L^{infty}([-1,1])$. Now the space $(L^{infty})^*$ is the set of all finitely additive measures which are absolutely continuous with respect to Lebesgue. Therefore $l$ must be a finitely additive measure $<< dx$ on $[0,1]$.

I apparently do not understand what this means for finitely additive measures since this element of $(L^{infty})^*$ does not appear to be absolutely continuous; it is just dirac measure. Can someone help clarify this apparent inconsistency? Are the finitely additive functionals only defined on intervals $[a,b)$ or something of this nature?

Best,
Dorian

app to locate star by coordinates

There are apps which will help guide you to more significant stars (Google skymap comes to mind if you're looking for a simple mobile system - Stellarium may also be a good option), but it's highly unlikely that any app will include the location and names of so-called 'personal' stars, as the bodies who sell such things don't have any the authority to do so and are not generally recognized in the actual world of Astronomy.

Note that the IAU (International Astronomy Union) is the body generally recognized to have the responsibility of naming and classification of stars and other bodies. They also come out strongly against such practices and services:

As an international scientific organization, the IAU dissociates itself entirely from the commercial practice of "selling" fictitious star names or "real estate" on other planets or moons in the Solar System. Accordingly, the IAU maintains no list of the (several competing) enterprises in this business in individual countries of the world. Readers wanting to contact such enterprises despite the explanations given below should search commercial directories in their country of origin.

Link to IAU page on buying of star names

amateur observing - What was the flash I saw in the night sky?

On Saturday night (April 2nd) in Sydney, Australia, somewhere between 8:30 PM and 9:00 PM my partner and I saw three flashes in the sky. We were looking at Jupiter through a pair of 7x50 binoculars and I saw the first one just slightly above the planet. My partner, without binoculars, saw nothing and I thought it had been a trick of the light.

But, about 10 minutes later, she was looking through the binoculars and saw a flash as well, that time a little to the left of Jupiter. Another few minutes later she saw it again and then said she saw something moving extremely fast towards the horizon (faster than the satellites I often witness that look like slow moving stars). I was looking in the right area, but could see nothing without the binoculars.

The flash was only in a small area, a little larger than a pinhead, not across the whole sky.

What could these flashes have been?

ag.algebraic geometry - Functoriality of Hironaka's resolution of singularities

A useful (at least for me) example is given in Kollar's article/book on resolutions of singularities about how you can't expect to get a "resolution functor": take a quadric cone $$C = {(x,y,z) in mathbb A^3: xy-z^2=0}$$ in $mathbb A^3$. Then you have the obvious map $phicolon mathbb A^2 to C$. But now suppose that $C'$ is a resolution of $C$ provided by a putative "resolution functor". Then if we let $tilde{C}$ be the minimal resolution, $C'$ factors through $C$. If we assume that $mathbb A^2$ is resolved by itself (as seems reasonable!) then we'd have to have $phi$ lifting to a map $mathbb A^2 to tilde{C}$ compatibly with the original morphism, which of course one cannot do.

I found the introduction to Kollar's article really useful in understanding what one can and cannot expect from resolution of singularities.

homological algebra - Homomorphism between exterior powers of a free module of finite rank

Here is a basis free expression.

Let the rank of $M$ be $r$. Pick an isomorphism $phi: M to M^*=hom_R(M,R)$. Now, if $min M$, define a map $f_m:Lambda^rMtoLambda^{r-1}M$ by contracting with $phi(m)$, so that if $m_1$, $dots$, $m_rin M$, then $$f_m(m_1wedgecdotswedge m_r)=sum_{i=1}^r;(-1)^i;langlephi(m),m_irangle; m_1wedgecdotswedgehat m_iwedgecdotswedge m_r.$$ Here $langlemathord-,mathord-rangle:M^*times Mto R$ is the evaluation map. It is not hard to show that $min Mmapsto f_minhom(Lambda^rM, Lambda^{r-1}M)$ is an isomorphism

(This isomorphism is not natural, because it depends on the choice of $phi$. Of course, there is a natural isomorphism $Psi_M:M^*tohom(Lambda^rM, Lambda^{r-1}M)$ given by essentially the formula, and there is no natural isomorphism $Mtohom(Lambda^rM, Lambda^{r-1}M)$, because such a thing would, when composed with the inverse of the natural isomorphism $Psi_M$, give a natural isomorphism $Mto M^*$, which does not exist.)

ct.category theory - Does linearization of categories reflect isomorphism?

In this answere I (try to) present the problem as a Algebraic Geometry one:

consider the category $mathscr{C} $ with two objects $X, Y$ and

$mathscr{C}(X, Y)$={$r_1, s'_1, r_2, s'_2$} ;
$mathscr{C}(Y, X)$={$s_1, r'_1, s_2, r'_2$} ; $mathscr{C}(X, X)$={$1_X, e_X$} ; $mathscr{C}(Y, Y)$={$1_Y, e_Y$} where $e_X, e_Y$ are idempotent, and any composition of a morphism by a a idempotent not alter the morphism, and $ 1_Y= r_1circ s_1= r_2circ s_2$, $ 1_X= r'_1circ s'_1= r'_2circ s'_2$, all other compositions give the (no identity) idempotent.
Suppose that $R$ is a commutative ring and in $Rmathscr{C} $ consider the morphims
$A:= a_1cdot r_1 + b'_1cdot s'_1 + a_2cdot r_2 + b'_2cdot s'_2: Xto Y$ and

$B:= b_1cdot s_1 + a'_1cdot r'_1 + b_2cdot s_2 + a'_2cdot r'_2: Yto X$.

Let $alpha :=a_1+b'_1+ a_2+ b'_2$, $beta :=b_1+a'_1+ b_2+ a'_2$,

Then we have $Bcirc A=1_X$ iff:

1) $ a'_1cdot b'_1+ a'_2cdot b'_2=1$ and

2) $beta cdot a_1+ (beta - a'_1)cdot b'_1+ beta cdot a_2+(beta -a'_2)b'_2=0$ i.e. $beta cdot alpha = a'_1cdot b'_1+ a'_2cdot b'_2 $

similarly we have $Acirc B=1_Y$ iff:

1') $ a_1cdot b_1+ a_2cdot b_2=1$ and

2') $alpha cdot beta = b_1cdot a_1+ b_2cdot a_2 $

all these equations are equivalent to the system of three equations:

$ a'_1cdot b'_1+ a'_2cdot b'_2=1, a_1cdot b_1+ a_2cdot b_2=1, alpha cdot beta = 1$

thinking these in $mathbb{C}[ a_1, b'_1, a_2, b'_2 , b_1, a'_1, b_2,a'_2] $ these represent three varieties on $mathbb{C}^8 $

If these varieties have an a intersections then $X, Y$ are isomorphic in $mathbb{C}mathscr{C} $ (but aren't isomorphic in $mathscr{C}) $.

at.algebraic topology - Noncommutative rational homotopy type

I can say that if A and B are commutative DGAs over Q which are connective (their homology groups are zero in negative degrees, using homological grading), then A and B are equivalent. The only proof I can come up with is a Postnikov argument based on the Hochschild-Kostant-Rosenberg theorem.

Namely, let P_nA and P_nB be Postnikov stages, which have the same homology as A and B respectively in degrees less than or equal to n and zero above; you can construct these by adding new polynomial generators to A and B with differentials that erase higher homology classes. Assume you've already constructed an equivalence from P_nA to P_nB.

Then P_n+1A and P_n+1B are classified by their next k-invariants. In the associative case this is an element in the Hochschild cohomology of P_nA with coefficients in a shift of the module M = H_n+1A = H_n+1B, and similarly for B. In the commutative case this is an element in the Andre-Quillen cohomology instead. These are classified by maps from the Hochschild homology (strictly speaking, its augmentation ideal) and Andre-Quillen homology (a shift of the derived Kahler differentials) to this shift of M.

Since we're in characteristic zero, the HKR theorem tells us that the Hochschild homology is the (derived) free graded-commutative algebra over P_nA on the Andre-Quillen homology. In particular, there is a retraction down. This implies that the collection of possible commutative k-invariants constructing an extension of P_nA by M are a subset of the possible associative k-invariants.

As a result, if P_n+1A and P_n+1B have a fixed equivalence as associative algebras, that implies that their associative k-invariants are equal. Hence that their commutative k-invariants are equal, so this allows us to lift to an equivalence of commutative algebras.

This is a sneaky problem. The fact that I needed to assume connectivity for this argument is pretty annoying, and I am not sure whether it is material or simply a failing on my part to get something general. Perhaps someone else can do better.

ADDED LATER: I believe that the same basic argument works in the coconnective case under added hypotheses (e.g. the algebra being simply connected, having only Q in degree 0 and 0 in degree 1). Unfortunately you cannot construct Postnikov stages quite as easily for general cochain algebras.

For example, let A be the commutative DGA Q[x,y,s,t] ⊗ Λ[w], a free algebra on generators x,y,s,t in degree zero and z in (cohomological) degree 1, satisfying dx = dy = dw = 0, ds = xw, dt = wy. Then there is a Massey product <x,w,y> = sy - xt in cohomology degree 0 which is nonzero, and any map from A to a DGA which has zero cohomology in positive degrees must necessarily send w to zero, and hence also this Massey product.

I have this vague feeling that in this simply-connected case (and also in the connective case) there might be a much shorter argument involving minimal models.

rt.representation theory - On the materials about cohomological induction

I am now learning induction problems in representation theory. I know David Vogan's book cohomological induction and unitary representation theory might be good references,but it is too thick.

I wonder whether there is some good and detailed notes on using derived functor to construct irreducible representations. It seems that Zuckermann ever gave a series of talks at Yale to introduce this method, but unfortunately, it was not published.

It seems that this method was motivated by Schmid's phd thesis. But I did not have schmid's phd thesis either.

All the related comments are welcomed

ag.algebraic geometry - Union of closed subschemes with the structure sheaf over it

As with all definitions, there is no "proof" that the adopted definition is the right one but only a feeling that it better corresponds to our intuition.

In the case at hand, taking $R/IJ$ as structure sheaf would make union schemes nonreduced for no good reason. For example take $R=k[x,y,z], I=(y,z), J=(x,z)$. Geometrically you are describing the union $U$ of the $x$-axis and the $y$-axis in affine three space $mathbb A^3_k$ .
It should have a reduced structure, correctly described by $Icap J$, whereas $I.J=(xy,zx, zy, z^2)$ would make the function $z$ nilpotent but not zero on $U$, which feels wrong since $U$ should be a closed subscheme of the plane $z=0$.

A more brutal objection to the idea of defining the union of two subschemes by the product of their ideals is that a subscheme $U$ would practically never be equal to its union with itself, since in general$ Ineq I^2$ : we would have (almost always)
$$Uneq U cup U$$
That looks bad!

co.combinatorics - Where do stable Kronecker coefficients live "in nature"?

Background:

For a partition $lambda$, let $lambda[N] = (N - |lambda|, lambda_1, lambda_2, lambda_3, dots)$, also let $chi_lambda$ be the corresponding irreducible character of the symmetric group $S_{|lambda|}$. Even if $lambda[N]$ is not a partition, we can make sense of $chi_{lambda[N]}$ as a class function of $S_N$ using a determinant, and Murnaghan proved that there exist coefficients $G^nu_{lambda, mu}$ (the stable, or reduced, Kronecker coefficients) such that

$chi_{lambda[N]} chi_{mu[N]} = sum_nu G^nu_{lambda, mu} chi_{nu[N]}$

for all $N ge 0$.

Question:

In a sense, the construction of $lambda[N]$ is a bit ad hoc. Is there a more representation-theoretic way to define these coefficients? In particular, if $|lambda| + |mu| = |nu|$, then $G^nu_{lambda, mu}$ coincides with the corresponding Littlewood-Richardson coefficient, so I am hopeful there is some connection. I am looking for an answer which addresses the following point: as I have defined these coefficients, it seems that the most accessible way to work with these coefficients is to use combinatorics. If I wanted to use tools from say, invariant theory or algebraic geometry, what is a more natural context for these coefficients to appear?

ag.algebraic geometry - Topological obstructions to extending algebraic vector bundles

Ariyan and Kevin Lin have asked about the problem of extending vector bundles defined on an open subvariety across the rest of the variety. There can be subtle commutative algebra obstructions, as in Sasha's answer to Kevin's question.

In another answer to Kevin's question, Donu Arapura pointed out that a coarser cohomological obstruction is almost irrelevant. If X is a complex variety and U is an open subvariety, then in order for a vector bundle E on U to extend its Chern classes must be in the image of the map $H^ast(X) to H^ast(U)$. In case X is smooth Donu sketches an argument, using Deligne's theory of weights, that this is always true if we take rational coefficients.

There is no formalism of weights in cohomology with integer coefficients. In a comment to Donu's answer I proposed an example of a line bundle on U with torsion Chern class that was not in the image of $H^2(X;Z) to H^2(U;Z)$. But this example is completely wrong. In an e-mail, Donu gave me a short argument that avoids weights and applies to integer Chern classes. In a smooth U a Chern class $c_i(E)$ can always be represented by a codimension i cycle, and to extend it to X we can take its closure.

Are there any topological obstructions to extending algebraic vector bundles? That is, do there exist X, U, E, with X and U smooth, so that E does not even extend to a non-algebraic complex vector bundle on X?

ag.algebraic geometry - What do Weierstrass points look like?

I think another interpretation is in terms of Euclidean geometry instead of hyperbolic geometry. If $omega$ is a holomorphic 1-form on a Riemann surface $Sigma$, then $|omega|$ defines a Euclidean metric on $Sigma$ with singularities at the zeroes of $omega$. Think of the 1-form as locally defining a map to $mathbb{C}$, and pull back the Euclidean metric. The metric near the zero is a cone of angle $2pi k$, where $k$ is the order of the zero, since this is modeled locally on the map $zto z^k$. The number of zeroes, counted with multiplicity, must be $2g-2$. A point $Pin Sigma$ is a Weierstrass point if and only if there is a holomorphic 1-form with a zero at $P$ of order at least $g$. Equivalently, there is a flat cone metric (conformally equivalent to the underlying Riemann surface and with all cone angles of the form $2pi k$) with a cone angle of $2pi g$ at $P$ and trivial holonomy. I found this in some course notes of McMullen, but they seem to be no longer available online.

Addendum: I've added a link to McMullen's course notes on Riemann surfaces. Theorem 10.19 and the preceding discussion connect the Weierstrass points with holomorphic 1-forms.
See Section 3 of another of McMullen's papers for a discussion of holomorphic 1-forms and branched Euclidean metrics,

Correction: I forgot that a holomorphic 1-form is not quite equivalent to a flat
$2pi k$-cone metric. The problem is that if one develops the surface into the plane around a closed loop, there might be a rotational part - that is why I added that there needs to be trivial holonomy. So a flat $2pi k$ cone metric corresponds to a holomorphic 1-form twisted by a $U(1)$ character. It also corresponds to a special sort of holomorphic quadratic differential. It's still possible that high order zeroes of such twisted 1-forms correspond to Weierstrass points, but it's not equivalent to McMullen's characterization. As pointed out below by Dmitri, Troyanov has shown that there is a Euclidean structure on a surface with a cone point at any chosen point.

Addendum: Schmutz-Schaller stated this problem, which maybe is akin to what you are searching for (and as far as I know has not been answered). He gave a characterization of hyperelliptic Riemann surfaces in terms of closed geodesics.

Problem. Find a geometric characterization (based on closed geodesics) of Weierstrass
points.

On a closed hyperbolic surface, there are points which have an open neighborhood which misses every simple closed geodesic. So there is certain "focussing" of simple closed geodesics.

amateur observing - Why aren't planet orbits circular?

While this has been answered before, I'll touch on a few of your specific questions.

Common sense seems to indicate that planetary orbits should be very
close to circular, any perturbations having been eliminated over the
ages.

This seemed logical to a lot of people including some big brains throughout history like Aristotle, Ptolomy, even Copernicus and Kepler himself thought circles were neater than ellipses, but he couldn't deny that the carefully detailed planet charts and the mathematical calculations worked so much better with ellipses.

It wasn't until Newton that they knew why. Newton worked it out that ellipses are stable orbits, and in fact a circle is a type of ellipse with zero eccentricity. Somebody said once, and I'm repeating, setting a dial exactly at zero is very hard because you're always going to be off by a tiny fraction. Setting the dial between 0 and 1 is easy. That's why all orbits are ellipses.

Now when you say perturbations, that usually refers to a 3rd body in the system. For example, Earth-Sun, the Earth orbits the sun in an ellipse. If you have Earth and Venus orbiting the Sun, Venus and earth perturb each other's ellipse.

Could it be that the composition of the planet be influencing the pull
of gravity to produce a different gravitational attraction due to its
mass? I mean that if a planet were composed all of one substance,
would the planetary orbit be perfectly circular?

Gravity does change a bit at different points in an orbit for the reasons you say, planets aren't uniform. The Moon is especially unbalanced for example so this is more apparent with orbits around the moon, but all planets have some degree of non uniform mass and that has some tiny effects but that's not the cause for ellipses though, in fact, non-uniform mass will slightly warp an elliptical orbit.

I think about earth having constantly shifting water and gasses and
even solids that affect its own rotation. Might these shifts also
affect the orbit of the planet around the sun, or does the total mass
of the earth and moon together determine the orbit?

We think of water as sloshing around and causing drag cause that's our experience when we spin or move something with water in it, but that's not what happens in orbit. The Earth in orbit around the sun, for example, everything on the earth including the oceans are all falling around the sun together, so there's no "shifting" going on, in the sense that I think you mean. There are tidal bulges, but the effect of the tidal bulges on the shape of an orbit is pretty small. In fact, there's a curious side effect to tidal bulges, they tend to circularize orbits, over time. They actually have the opposite effect of what you suggest.

I tried to cover this at a basic science level. Corrections / clarifications welcome.

finite fields - roots of quadratic forms

This may be a very silly question, but I was wondering what is known about the roots of a quadratic form over variables $x_1,ldots,x_n,y_1,ldots,y_m$ in the finite field $mathbb{F}_p$. I'm not even so much interested in characterizing the solutions as I am in counting them (or showing some relation between them), but any help would be greatly appreciated (as well as a pointer to a good introductory article/text on quadratic and modular forms in general).

Thanks!

ag.algebraic geometry - What is a twisted D-Module intuitively?

NOTE: I ended up writing out a long version of this and correcting a mistake, so this is substantially changed from the old answer.

If you're taking twisted differential operators in a complex power of a line bundle, $L^c$, then you should think of them as vector bundles/sheaves on the total space $T$ of $L$ minus its zero section, endowed with a flat connection that behaves specially along the fibers of the bundle projection map.

Special how? The action of $mathbb C^*$ by fiber rotation has a differential, which is a vector field on the total space that looks like $tfrac{d}{dt}$ for any trivialization, where $t$ is the coordinate on the fiber. One should take a connection where differentiating along this vector field integrates to an equivariant structure for $mathbb C^*$ (that is, it has integral eigenvalues).

To see this, note that $mathbb C^*$ -invariant functions on $T$ are the same as functions on $X$. However, there are more $mathbb C^*$-invariant vector fields; there's a Lie algebra map from the sheaf $Y$ of $mathbb C^*$-invariant vector fields on $T$ to vector fields on $X$, but the kernel is given by functions times the vector field $t frac{d}{dt}$ (the action vector field for $mathbb C^*$). This element is central, since we're looking at $mathbb C^*$-invariant vector fields (which exactly means they commute with the action vector field). This is a central extension of Lie algebras, and Chern-Weil theory tells us that the first Chern class of the line bundle is the obstruction to splitting this extension. If you want to get very fancy, this gives a Lie algebroid over $X$, of a special type called a Picard Lie algebroid.

Now, we can think of sections of powers of this line bundle as functions on $T$ with a fixed weight under $mathbb C^*$: the sections of $L^c$ have weight $c$. So, on the sections of $L^c$, I have the relation $(tfrac{ d}{dt} -c) s = 0$, so the differential operators twisted by $L^c$ are given by taking $mathcal O_X$, the vector fields $Y$, letting them commute past each other in the usual way, and then imposing $tfrac{ d}{dt} -c=0$.

Now, if $c$ isn't an integer, then there aren't going to be any functions that satisfy this equation, but the description I gave of the TDO is fine. There's just nothing interesting to act on.

Well, except there might be. I could take some other D-module on $T$ instead, and I would still get an action of the TDO on the sheaf of solutions to $(tfrac{ d}{dt} -c) s = 0$ in that D-module; this gives a functor from D-modules on T to twisted D-modules on X. This functor is an equivalence when restricted to the subcategory of D-modules where $tfrac{ d}{dt} -c$ integrates to a $mathbb C^*$-action (since you can always pull back a twisted D-module on X and get a D-module of this form on Y). Note that whether this vector field integrates only depends on the class of $c$ modulo $mathbb Z$; if you track through these functors, the resulting equivalence between modules of TDOs is tensoring with the correct power of the line bundle.

Now, assume I have a simple holonomic twisted D-module; the corresponding $mathbb C^*$-equivariant D-module on $T$ is also simple, so this is an intermediate extension of a $mathbb C^*$-equivariant local system on a locally closed subvariety $T'$ which is the preimage of some $X'subset X$. The monodromy of a flat section of this local system around the fiber must be $e^{2 pi i c}$, so if $c$ is irrational, the fundamental group of the fiber must inject into that of $T'$, whereas if $c=a/b$ in lowest form, then the kernel of this map of fundamental groups can only contain loops at go around the fiber a multiple of $b$ times (even then, you could have trouble depending on the structure of the fundamental group).

fourier analysis - Can we extract information about how fast a function decay from its Laplace transform?

My question is whether we can extract information about how fast an integrable function converges to zero by looking at the asymptotics of its Laplace transform.

More concrete case, let $f:mathbb{R} to mathbb{R}_+$ be a smooth function in $L^1(mathbb{R})$. If we know that its Laplace transform exists on the positive real axis and:

$int_{mathbb{R}} f(x) e^{sx} {rm d}x geq e^{frac{s^2}{2}}, quad forall s > 0$,

can we conclude that the speed that $f$ converge to zero cannot be faster than $e^{-frac{x^2}{2}}$, say,

$liminf_{|x| to infty} frac{f(x)}{e^{frac{-x^2}{2} (1 - epsilon)}} > 0$

for some small $epsilon in (0, 1)$? In a more probabilistic setup, if we know the moment generating function is lower bounded by that of Gaussian, can we conclude that it is "super-Gaussian"? I know that the other direction seems to be true and is called sub-Gaussian.

If the information on the right-half real axis is not enough, do we need to know more? Will Fourier transform be more helpful? How about the other direction, i.e., lower bound on the Laplace transform and upper bound on the decay of $f$? Thanks.

quantum field theory - N=(2,2) supersymmetry in two-dimensional Euclidean space

Can anyone describe (or give a reference for) the 2d superspace formulation of N=(2,2) SUSY in Euclidean signature?

I'm reading Hori's excellent introduction to QFT in the book 'Mirror symmetry', and my question is basically Ex. 12.1.1. page 273. What I imagine the answer is is a super version of the usual story of differential forms on complex manifolds, i.e. we complexify, find square roots of the $partial_z$ and $partial_{bar{z}}$ operators, then find a subalgebra of 'chiral' fields analogous to the subalgebra of holomorphic forms.

examples - Algebraically closed fields of positive characteristic

It's certainly not too hard to understand everything there is to understand about the algebraic closure of Fp. Perhaps the reason this is unsatisfying as an example for founding intuition is because it doesn't really have a nice topological structure; it lacks anything like a natural metric. So here's an attempt to explain why what is in some sense the next simplest example puts you in a better situation, intuition-wise.

If you have some intuition about the p-adic numbers look and feel (for example, topologically), then you secretly have intuition for the t-adic topology on the complete local field K=Fp((1/t)). Now, as far as characteristic p fields go, this sort of puts you in the position of (in your parlance) a "preschooler" who knows about R but hasn't yet gotten to kindergarten to learn about C. Why is K like R? First, it is locally compact. Second, it is at least analogous to completing Fp(t), which is very much like Q with Fp[t] as the analogue of Z, at an "infinite" valuation, namely the degree or (1/t)-adic valuation, rather than a "finite" place like a prime polynomial in Fp[t]. (The (1/t)-adic valuation corresponds to the point at infinity on the projective line over F_p. Likewise, number theorists love to say, perhaps partly to annoy John Conway, that the real and complex absolute values on Q correspond to "archimedean primes" or "primes dividing infinity". This is actually a pretty lame analogy, though, since K=Fp((1/t)) looks a lot more like Fp((t)), say, than R or C looks like Q2.)

Unfortunately there are two extra difficulties in the characteristic p case. First, upon passing to the algebraic closure L of K we lose completeness. Second, we make an infinite field extension, unlike the degree 2 extension C/R. Thus, while L is an algebraically closed field of characteristic p, it bears little resemblance to R. In fact, it's a lot more like an algebraically closed field of characteristic 0 that is a bit scarier (at least to me) than C, namely Cp, or what you get when you complete the algebraic closure of Qp with respect to the topology coming from the unique extension of the p-adic valuation.
While this may seem bad, I think it's actually good, because one can really get a handle on some of the properties of Cp. [Note that as another answerer pointed out, Cp = C as a field, but not as a topological or valued field, which is really a more interesting structure to consider from the viewpoint of intuition anyway.]

For example of some similarities, miraculously Cp turns out to still be algebraically closed, and I believe the same proof goes through for L above. Another property L and Cp share is that in addition to "geometric" field extensions K'/K obtained by considering function fields of plane curves over Fp, there are also "stupider" extensions coming from extending the coefficient field. This is like passing to unramified extensions of p-adic fields, where one ramps up the residue field. (In fact, it's exactly the same thing.) Both L and Cp are complete valued fields with residue field the algebraic closure of Fp. (But the valuation is NOT discrete; it takes values in Q.) There are some dangerous bends to watch out for topologically, however. Some cursory googling tells me that Cp is not locally compact, although it is topologically separable.

In addition, positive characteristic inevitably brings along the problem of inseparable field extensions sitting side L. This is, of course, an aspect where L/K is unlike Cp/Qp. Notwithstanding such annoyances, I would argue that the picture sketched above actually does give an example of an algebraically closed field of characteristic p for which it is possible to have some real intuition.

ag.algebraic geometry - Newton Puiseux expansions for singular surfaces?

I don't believe there is anything as general as that, but when your polynomial is over the complex numbers one can do the following. First shift and rotate coordinates so that you're working on a neighborhood of the origin where $F(0,0,0) = 0$ and $partial_z^n F(0,0,0) = 0$ for some $n$. Then you can use the Weierstrass preparation theorem and (ignoring a nonvanishing factor) write $F(x,y,z)$ as $z^n + a_{n-1}(x,y)z^{n-1} + ... + a_0(x,y)$. If the discriminant of this polynomial, viewed as a function of $x$ and $y$ has normal crossings (i.e. is a monomial times a nonvanishing factor), one can use the Jung-Abhyankar theorem to get a factorization of the form $(z - b_1(x,y))...(z - b_n(x,y))$ where the $b_i$ are analytic in fractional powers of $x$ and $y$ as you want.

Even if the discriminant is not normal crossings, you can first resolve the singularities of the discriminant to make it normal crossings and there are a few ways to do this. One way involves subdividing the $x-y$ space into wedges, and on each wedge there is a way of first doing a coordinate change of the form $(x,y) --> (x,y - g(x))$ then a "blowing up" coordinate change $(x,y) = (x',(x')^m y')$ and get the discriminant as you want. ($m$ may not be an integer, and also you might have to reverse the roles of the $x$ and $y$ variables). Then one proceeds as above. Of course the final result isn't as nice as you wanted, but this is in the nature of things.

One can iterate the above to deal with analogues in higher dimensions, but the formulas get a lot more elaborate as one might guess. There's a paper by Parusinski "On the preparation theorem for subanalytic functions" that discusses a lot of these issues.

homological algebra - Is tensor product exact in abelian tensor categories with duals?

Suppose we are in an abelian tensor category with duals, where all objects have finite length. Let $0 to A to B to C to 0$ be a short exact sequence and $Z$ an object of the category. Is
$$0 to Z otimes A to Z otimes B to Z otimes C to 0$$
exact?

Motivation: I am reading the proof of Proposition 5.7 in this paper of Deligne and trying to figure out why the lower sequence at the bottom of page 23 is exact. I believe $mathcal{H}om(X,Y)$ here is $X^{vee} otimes Y$, although I have not actually found the point in the paper where he defines it. What he is trying to prove is that the corresponding sequence of external Hom's is exact, so he can't be using that fact.

There are, of course, tons of abelian tensor categories where $otimes$ is not exact. For example, modules for any commutative ring $A$ which is not a field, with tensor product $otimes_A$. But these don't usually have duals for all of their objects.

set theory - Is Dependent Choice all we really need?

Let me adopt an extreme interpretation of your question, in order to prove an affirmative answer.

Yes, in the arena of the countable, DC suffices.

To see why, let me first explain what I mean. If one wants to consider only countable sets, then the natural set-theoretic context is HC, the class of hereditarily countable sets. These are the sets that are countable, and all members are countable, and members-of-members and so on. The class HC is the land of all-countable set theory, the land of the fundamentally countable. I am not proposing that you want to remain within HC. Rather, you want to consider the properties of the objects in HC that are expressable there and what you can prove about them in ZF+DC, versus ZFC.

Since hereditarily countable sets can be coded easily by real numbers, it turns out that the structure HC is mutually interpreted in the usual structure of the reals R and vice versa. That is, the structure HC is essentially equivalent in a highly concrete way to the usual structure of the real numbers. And so questions about the fundamentally countable are equivalent to questions in the usual projective hierarchy of descriptive set theory. Thus, in this interpretation, your question is really asking whether one needs ever needs AC as opposed to DC in order to prove a projective statement.

Question. Are the projective consequences of ZFC the same as those of ZF+DC?

The answer is Yes, by the following theorem.

Theorem. A projective statement is provable in ZFC if and only if it is provable in ZF+DC.

Proof. It suffices to show that for every model W of ZF+DC there is a model of ZFC with the same reals. The reason this suffices is that in this case, any projective statement failing in a model of ZF+DC will also fail in a model of ZFC.

So, suppose that W is a model of ZF+DC. Let L(R) be the inner model of W obtained by constructing relative to reals. This is also a model of ZF+DC. Consider the forcing notion P in L(R) consisting of well-ordered countable sequences of real numbers in L(R), ordered by end-extension. Let G subset P be L(R)-generic for P. Since P is countably closed in L(R), it follows that the forcing extension L(R)[G] has no additional real numbers. And since G adds a well-ordering of these reals in order type &omega₁, it follows that L(R)[G] has a well-ordering of the reals. From this, it follows that L(R)[G] is actually a model of ZFC, since it has the form L[A], where A is a subset of &omega₁ enumerating these reals in the order that G lists them. Thus, we have provided a model of ZFC, namely L(R)[G], with the same reals as W. It follows that W and L(R)[G] have the same projective truth, since this truth is obtained by quantifying only over this common set of reals. QED

This phenomenon generalizes by the same argument beyond projective statements, to statements of the form "L(R) satisfies phi". The point is that ZFC and ZF+DC prove exactly the same truths for L(R), since any model W of ZF+DC has the same L(R) as the model L(R)[G] in the proof above, and this satisfies ZFC.

The conclusion is that if you wish to study mathematical objects that are essentially countable only in the weak sense that they exist in L(R)---and this includes many highly uncountable objects---then the features about them in L(R) that you can prove in ZFC are exactly the same as the features you can prove in ZF+DC. So this interpretation is not so extreme after all...

galaxy - How do I modify redshifts to gain corrected line of sight velocities?

I'm currently trying to collect the data to run an N-body simulation for 11 of the galaxies in the Local Group where proper motions are known, however I don't understand how to get the required line of sight velocities for the simulation.

I have redshift data from NED for the required galaxies, however by judging what sources tell me, this won't be the required information for the simulation due to our motion around the Milky way's galactic centre.

For example, one source has given the NED redshift as 229 km/s for Carina Dwarf, but the corrected line of sight velocity is 14.3 km/s.

Does anybody know how to convert these redshifts into the desired corrected versions?

Thanks!!

lo.logic - computation, algebra, logic

From this description it is not hard to see the connection of classical computation to discrete dynamical systems and classical logic, the operations provide the dynamics by flipping bits around in a controlled fashion and since we restrict the operations to a certain subset we get classical logic.

Careful -- this observation only works for purely finite-state systems! If your idealized model of a digital computer can handle potentially unbounded quantities of input, then the connection to classical logic is lost. Happily, it fails in a way that reveals connections to topology, and explains why topological models of intuitionistic logic exist.

The basic idea is that we view a computer as realizing a function $f$. Then for any input value (from the input set $A$), if it returns an answer in a finite amount of time, it can have observed at most a finite amount of information about the input. This means that we can equip our input set with a topology in the following way: suppose our basic observations are a collection of predicates on $A$. Then we get the topological structure from the following fact: we can only take finite intersections because we can only make finitely many observations, and so can only conclude the conjunction of finitely many predicates. We can take infinite unions (i.e., existentials) because we can "get lucky" and guess the correct branch of the union (i.e., witness to the existential).

Then, amazingly, we can use continuity as a stand-in for computability! This is pretty much the idea Dana Scott had when he invented domain theory (which gets a different name because the topologies that we get this way are "weird" -- e.g., they're typically not Hausdorff -- and so the theorems you want develop a bit differently).

Proving a hypergeometric function identity

While playing around with the fractional calculus, I got stuck trying to show that two different ways of differintegrating the cosine give the same result. DLMF and the Wolfram Functions site don't seem to have this "identity" or something that can obviously be transformed into what I have, so I'm asking here.

The "identity" in question is

$(alpha-1)left({}_1 F_2 left(1;frac{1-alpha}{2},frac{2-alpha}{2};-frac{x^2}{4}right)-{}_1 F_2 left(-frac{alpha}{2};frac12,frac{2-alpha}{2};-frac{x^2}{4}right)cos(x)right)stackrel{?}{=}alpha x sin(x),{{}_1 F_2 left(frac{1-alpha}{2};frac32,frac{3-alpha}{2};-frac{x^2}{4}right)}$

Expanding the LHS minus the RHS in a Taylor series shows that the coefficients up to the 50th power are 0; trying out random complex values of $alpha$ and $x$ seems to verify the identity. I would however like to see a way to confirm the identity analytically. How do I go about it?

soft question - Fundamental Examples

It is not unusual that a single example or a very few shape an entire mathematical discipline. Can you give examples for such examples? (One example, or few, per post, please)

I'd love to learn about further basic or central examples and I think such examples serve as good invitations to various areas. (Which is why a bounty was offered.)

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Related MO questions: What-are-your-favorite-instructional-counterexamples,
Cannonical examples of algebraic structures, Counterexamples-in-algebra, individual-mathematical-objects-whose-study-amounts-to-a-subdiscipline, most-intricate-and-most-beautiful-structures-in-mathematics, counterexamples-in-algebraic-topology, algebraic-geometry-examples, what-could-be-some-potentially-useful-mathematical-databases, what-is-your-favorite-strange-function; Examples of eventual counterexamples ;

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To make this question and the various examples a more useful source there is a designated answer to point out connections between the various examples we collected.

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In order to make it a more useful source, I list all the answers in categories, and added (for most) a date and (for 2/5) a link to the answer which often offers more details. (~year means approximate year, *year means a year when an older example becomes central in view of some discovery, year? means that I am not sure if this is the correct year and ? means that I do not know the date. Please edit and correct.) Of course, if you see some important example missing, add it!

Logic and foundations: $aleph_omega$ (~1890), Russell's paradox (1901),
Halting problem (1936), Goedel constructible universe L (1938), McKinsey formula in modal logic (~1941), 3SAT (*1970), The theory of Algebraically closed fields (ACF) (?),

Physics: Brachistochrone problem (1696), Ising model (1925), The harmonic oscillator,(?) Dirac's delta function (1927), Heisenberg model of 1-D chain of spin 1/2 atoms, (~1928), Feynman path integral (1948),

Real and Complex Analysis: Harmonic series (14th Cen.) {and Riemann zeta function (1859)}, the Gamma function (1720), li(x), The elliptic integral that launched Riemann surfaces (*1854?), Chebyshev polynomials (?1854) punctured open set in C^n (Hartog's theorem *1906 ?)

Partial differential equations: Laplace equation (1773), the heat equation, wave equation, Navier-Stokes equation (1822),KdV equations (1877),

Functional analysis: Unilateral shift, The spaces $ell_p$, $L_p$ and $C(k)$, Tsirelson spaces (1974), Cuntz algebra,

Algebra: Polynomials (ancient?), Z (ancient?) and Z/6Z (Middle Ages?), symmetric and alternating groups (*1832), Gaussian integers ($Z[sqrt -1]$) (1832), $Z[sqrt(-5)]$,$su_3$ ($su_2)$, full matrix ring over a ring, $operatorname{SL}_2(mathbb{Z})$ and SU(2), quaternions (1843), p-adic numbers (1897), Young tableaux (1900) and Schur polynomials, cyclotomic fields, Hopf algebras (1941) Fischer-Griess monster (1973), Heisenberg group, ADE-classification (and Dynkin diagrams), Prufer p-groups,

Number Theory: conics and pythagorean triples (ancient), Fermat equation (1637), Riemann zeta function (1859) elliptic curves, transcendental numbers, Fermat hypersurfaces,

Probability: Normal distribution (1733), Brownian motion (1827), The percolation model (1957), The Gaussian Orthogonal Ensemble, the Gaussian Unitary Ensemble, and the Gaussian Symplectic Ensemble, SLE (1999),

Dynamics: Logistic map (1845?), Smale's horseshoe map(1960). Mandelbrot set (1978/80) (Julia set), cat map, (Anosov diffeomorphism)

Geometry: Platonic solids (ancient), the Euclidean ball (ancient), The configuration of 27 lines on a cubic surface, The configurations of Desargues and Pappus, construction of regular heptadecagon (*1796), Hyperbolic geometry (1830), Reuleaux triangle (19th century), Fano plane (early 20th century ??), cyclic polytopes (1902), Delaunay triangulation (1934) Leech lattice (1965), Penrose tiling (1974), noncommutative torus, cone of positive semidefinite matrices, the associahedron (1961)

Topology: Spheres, Figure-eight knot (ancient), trefoil knot (ancient?) (Borromean rings (ancient?)), the torus (ancient?), Mobius strip (1858), Cantor set (1883), Projective spaces (complex, real, quanterionic..), Poincare dodecahedral sphere (1904), Homotopy group of spheres, Alexander polynomial (1923), Hopf fibration (1931), The standard embedding of the torus in R^3 (*1934 in Morse theory), pseudo-arcs (1948), Discrete metric spaces, Sorgenfrey line, Complex projective space, the cotangent bundle (?), The Grassmannian variety,homotopy group of spheres (*1951), Milnor exotic spheres (1965)

Graph theory: The seven bridges of Koenigsberg (1735), Petersen Graph (1886), two edge-colorings of K_6 (Ramsey's theorem 1930), K_33 and K_5 (Kuratowski's theorem 1930), Tutte graph (1946), Margulis's expanders (1973) and Ramanujan graphs (1986),

Combinatorics: tic-tac-toe (ancient Egypt(?)) (The game of nim (ancient China(?))), Pascal's triangle (China and Europe 17th), Catalan numbers (18th century), (Fibonacci sequence (12th century; probably ancient), Kirkman's schoolgirl problem (1850), surreal numbers (1969), alternating sign matrices (1982)

Algorithms and Computer Science: Newton Raphson method (17th century), Turing machine (1937), RSA (1977), universal quantum computer (1985)

Social Science: Prisoner's dilemma (1950) (and also the chicken game, chain store game, and centipede game), the model of exchange economy, second price auction (1961)

Statistics: the Lady Tasting Tea (?1920), Agricultural Field Experiments (Randomized Block Design, Analysis of Variance) (?1920), Neyman-Pearson lemma (?1930), Decision Theory (?1940), the Likelihood Function (?1920), Bootstrapping (?1975)

nt.number theory - Where stands functoriality in 2009?

Here are few remarks which might be relevant, although I understand almost nothing of the global Langlands program.

Lafforgue is currently working on problems relating to functoriality. There are a number of recent preprints and notes on his webpage, see for example "Quelques remarques sur le principe de fonctorialité". If you don't read French, maybe the lectures of Lafforgue in Cambridge a few months ago would be useful. They are available in various video formats at the Newton Institute webpage. To find them, see this list, and scroll down to May - there is a total of 5 talks by Lafforgue, the first one on May 5th.

My impression of Lafforgue's work is that he aims for a proof of functoriality in a fairly general setting, and (amazingly!) he hopes that the method would work also in the number field case and not only for function fields (although I might have misunderstood this). The method has at least some vague similarity with Tate's thesis, I think.

For more general background on functoriality and related things, see maybe Knapp's survey on the Langlands program, the Clay Summer School Proceedings from 2003 (here is the Google Books page), and this short note of Rapoport on Lafforgue's earlier work.

Edit: Thanks to "unknown" and David for pointing out the work of Ngo! I should have added that Laumon also gave a talk in May at the Newton Institute, on Ngo's proof, this is available here (both video and slides). See also the discussion at SBS. On the functoriality principle in general, there is also this 15-page expository presentation of Arthur.

nt.number theory - A question on liftings of supersingular elliptic curves over the prime fields

I don't know a result that says what you want, but I'll give you some pointers that might help. The "number" of elliptic curves over $mathbb{F}_p$ with $p-t+1$ points is the "number" of binary quadratic forms with discriminant $H(t^2-4p)$. I put number in quotes because, on both sides of the formula, one must weight the objects being counted by one over the size of their automorphism group. (This will usually be $2$, in both cases.) I got this from section 1 of Lenstra's paper Factoring Integers with Elliptic Curves, which is very readable, he says that the result is essentially due to Deuring.

In chapter 13, section 5, of Lang's Elliptic Functions, I find the following statement, again attributed to Deuring: Given an elliptic curve $E_0$ over $mathbb{F}_p$, and an endomorphism $a_0$ of $E_0$, there is a number field $K$, an elliptic curve $E$ over $K$, an endomorphism $a$ of $E$ and a place $mathfrak{p}$ of $mathcal{O}_K$, lying over $p$, such that the reduction of $(E, a)$ modulo $mathfrak{p}$ is the base change of $(E_0, a_0)$ to $mathcal{O}_K/mathfrak{p}$. (Lang doesn't mention this base change explicitly, so maybe I am missing something; it seems to me to be what he is proving.) So, in particular, if $E_0$ has $p-t+1$ points, it can be lifted to a curve with complex multiplication by $mathbb{Z}[F]/(F^2-tF+p)$.

What Lang doesn't say, though, is anything about the uniqueness of this lift. And, indeed, there are some issues here, because any quadratic twist of $E$, by a $D$ for which $left( frac{D}{p} right) =1$, would have the same properties.

Do other stars have Oort clouds?

The other question has some very good detailed answers, worth reading, but briefly and more specific to your question:

Would there not be interactions between their Oort cloud objects and
the planets of the Solar system? For example, the outer edge of the
Sun's Oort cloud is approximately 100,000 AU away. So, if another star
comes at a distance of 1 light year from the Sun, its Oort cloud
should similarly go through the inner solar system.

While very little is actually known, it's not crazy to think that smaller stars might have smaller oort clouds and larger stars, larger ones, though there's a great deal of uncertainty on what an average oort cloud is, how far an average one extends, and how densely packed they are.

First part first, how often does a star pass within one light year of our sun?

Not very often. Ballpark, maybe every million to couple million years or so. The last time was 70,000 years ago, a small star, 15% the mass of the sun. See here.

In the fly-by section of that same wiki-article, it says a star passes that close (0.82 light years or less) about once every 9 million years.

Scholz's star is quite small, only about 15% the mass of our sun, and it's unclear if it had an Oort cloud that extended .82 light years at all, so while it passed through our Sun's Oort cloud, it's unclear if the inner solar-system passed through it's Oort cloud, assuming it has one. It's also possible that the sun stole much of Scholz's star's oort cloud during the fly-by, but that's just speculation.

Did Scholz's star send Oort cloud objects heading towards Earth? That's a good question and the answer is, probably, but we don't know, because if it did, it would take longer than 70,000 years for any re-directed Oort cloud comets to reach the inner solar-system. By the same article, an estimated 2 million years for re-directed Oort cloud objects to reach the inner solar system, so, in about 1.93 million years (give or take), we'll start to see what Scholz's sent our way. :-)

In 1.36 million years, Gliese 710 could pass within 1 light year of Earth. The estimate is 1.1 light years plus/minus .577 light years. Gliese 710 is larger than Scholz's, about 60% the mass of the sun but we know nothing about how large or how dense it's Oort cloud might be. But, very generally, it seems likely that the Sun has passed through another stars Oort cloud at some point, perhaps every 1-10 million years, perhaps every hundred million years or so, but it seems reasonable to me that has happened and will continue to happen, on rare occasion.

2nd question, what would happen if the inner planets passed through another star's Oort cloud.

If we use our Oort cloud as a model, and, warning, these estimates are enormously ballpark, but, a trillion Oort cloud objects 1 KM or larger and the Oort cloud extends between 100,000 and 200,000 AU (lets say 150,000 as a middle estimate), that works out to one 1 KM or larger object per cubic region of space about 15 AU across. That's 3/4ths the distance between the Sun and Neptune, so if you figure just one, 1 KM or larger comet or object in a 3D space over such a vast distance, most would pass by the inner planets unaffected. The occasional one would get it's orbit measurably altered with a fly-by of one of the gas giant planets and rarely, there would be an impact. Now presumably there's more smaller stuff in Oort clouds too, so there should be smaller impacts more frequently, though there's almost nothing that's actually known about how dense Oort clouds are, so anything beyond "infrequent", is impossible to say.

Passing through another Stars Oort cloud should take tens of thousands, perhaps 100,000 years years or so, so there probably would be the occasional impact on a planet here and there, but on a human time scale, significant impacts should be very rare.

at.algebraic topology - Are the strata of Nakajima quiver varieties simply-connected? Do they have odd cohomology?

Nakajima defined a while back a nice family of varieties, called "quiver varieties" (sometimes with "Nakajima" appended to the front to avoid confusion with other varieties defined in terms of quivers). These are most concisely defined as the moduli of certain representations of certain preprojective algebras.

I'll be interested in the affine (in the sense of "affine variety" not "affine Lie algebra") version of these, which is the moduli space of semi-simple representations of a certain preprojective algebra. This variety is singular, but can be divided into smooth strata which correspond to fixing the size of the automorphism group of the representation (i.e. one stratum for simple representations, one for sums of pairs of non-isomorphic simples, etc.). I would like to know a bit about the geometry of these strata. One basic (and important for me) question is

Are these strata simply connected or equivariantly simply connected for the action of a group?

Actually, I know that they are not simply connected from some very simple examples (such as the nilcone of $mathfrak{sl}_2$), but in those examples there is an action of a group such that there are no equivariant local systems (if you're willing to think of the quotient as a stack, the quotient is simply connected), so they are "equivariantly simply connected."

Similarly, I'm interested in the cohomology of these strata; I'd like its odd part to vanish. Again, there's no hope of this in the most obvious way. The only way it could happen is equivariantly.

Does the odd (equivariant) cohomology of these strata vanish for any group action?

let me just note in closing: I would be entirely satisfied if these results were only true in the finite type case; I know a lot of geometric statements for quiver varieties go a little sour once you're outside the finite type case.

If anyone knows these or any other results in the literature about the geometry of these strata, I would be very happy to hear them.

Sundial (solar clock) in a binary star system

The sun forms a useful basis for measuring time because to a reasonable approximation it stays put, and the rotation of the Earth causes it to traverse the sky once every 24 hours.

Provided the planet is spinning, a normal sundial can be used, but may not be quite as regular as on Earth.

A binary stellar system can have one of three possible configurations: Two stars in a close binary, with a planet orbiting at a much greater distance. This is the "Tattooine" configuration. Everything has a double shadow. A regular sundial can be used but it is not as regular, as the positions of the two stars changes as they orbit each other.

The second configuration is a planet orbiting a star with a second star at a much greater distance. If Jupiter had been a star this would be the second configuartion. The secondary star would be much weaker, and a regular sundial could be used. The shadow from the secondary would just be ignored.

The last configuration is like the second, except the planet is in orbit around the smaller star. The more distant star is stronger, and compariable in brightness. In this situation it is quite likely that the planet is tidally locked to the star, so a sundial isn't useful for telling the time. The slow motion of the distant star might give a long day (which could last many years) to the side of the planet that faces away from the nearby star, followed by a long cold night. This would be a challenging planet on which to live.

ra.rings and algebras - Relations in matrix semigroups

You are asking whether there is a faithful matrix representation of a finitely generated semigroup with no finite presentation. I don't know the answer to that. Note that only finitely many relations are needed to specify the representation variety $V_n$, so if the answer is yes, you would get a sequence of finitely presented semigroups with the same representation variety $V_n$, and thus a family of finitely presented semigroups with no faithful representation in dimension $n$. That's not a surprise, but it may be a way to look for an example.

Edit: Let me correct an example I had here before.

There is no bound on the length of relations possible for matrices with entries whose entries are at most 1, since you can represent the additive semigroup generated by 1 and $0le p/q le 1$, which is determined by commutativity and $p*1 = q * (p/q)$ in additive notation.

$$1to left( begin{array}{cc} 1 & 1 \ 0 & 1end{array}right), ~~~~frac pq to left( begin{array}{cc} 1 & frac pq \ 0 & 1end{array}right) $$.

Perhaps a bound would be possible not in terms of the magnitudes of the entries, but some other height function.

This type of example can be embedded in a slightly more complicated fashion when n=1, since you can have relations between 2, 3, and $2^p/3^q$ in the multiplicative semigroup of rationals. If that seems contrived because one of the generators is redundant, take 4, 9, and $2^{2p+1}/3^{2q+1}$.

matrices - Describing $SU(n,C)$

Use the fact that matrices act on vectors. $SU(n)$ acts transitively on the space of unit-length vectors; the stabilizer of a point is $SU(n-1)$ by Thorny's argument. For example, for the vector $(1,0,...,0)$ the stabilizer is the subgroup $left(begin{matrix}1&0\\ 0&Aend{matrix}right)approx SU(n-1)$. Now by the orbit-stabilizer theorem, the space of unit-length vectors is identified with $SU(n)/SU(n-1)$. Fixing one vector $(1,0,...,0)$ fixes this identification, and then each other vector corresponds to a coset $gSU(n-1)$ which is the family you describe.

This isn't really using much about matrices or geometry; I referred to this as the "orbit-stabilizer theorem" above, but it is really just the basic structural feature of group actions. It's certainly something you can understand by yourself; if it's not immediately obvious, you can think about some simpler examples. In the group of permutations $S_n$, consider the family of permutations that map $1mapsto 3$ -- how do elements of this family differ from each other? You could also try extending your argument to understand the first column of matrices in $GL(n,mathbb{R})$; you will of course find a similar answer, but the details are interestingly different. Another fun example is to consider linear functions from $mathbb{R}tomathbb{R}$, and look at the family of linear functions taking $2mapsto 7$.

at.algebraic topology - Is any interesting question about a group G decidable from a presentation of G?

I don't have a complete answer, but here are some thoughts.

The Rips Construction takes an arbitrary finitely presented group Q and produces a 2-dimensional hyperbolic group $Gamma$ and a short exact sequence

$1to Kto Gammastackrel{q}{to} Qto 1$

such that the kernel $K$ is generated by 2 elements. It turns out, by a result of Bieri, that $K$ is finitely presentable if and only if $Q$ is finite.

One can improve the finiteness properties of $K$ using a fibre product construction. Let

$P={(gamma,delta)inGammatimesGammamid q(gamma)=q(delta)}$.

By the '1-2-3 Theorem', if $Q$ is of type $F_3$ then $P$ is finitely presentable.

I would guess that $P$ has good higher finiteness properties if and only if $Q$ is finite. Perhaps one can use the fact that $Pcong K rtimesGamma$.

Even if this is true then it still doesn't quite solve your problem, as we don't have a presentation for $P$. To do this, one needs to be given a set of generators for $pi_2$ of the presentation complex of $Q$, which enable one to apply an effective version of the 1-2-3 Theorem. (In the absence of this data, presentations for $P$ are not computable. Indeed, $H_1(P)$ is not computable.)

Question: Does there exist a list of presentations for groups $Q_n$ such that:

1. each group $Q_n$ is of type $F_3$;




2. the set ${ninmathbf{N}mid Q_ncong 1}$ is recursively enumerable but not recursive;




3. but generators for $pi_2(Q_n)$ (as a $Q_n$-module) are computable?

If so, and if I'm right that the higher finiteness properties of $P$ are determined by $Q$, then higher finiteness properties are indeed undecidable. Simply apply the Rips Construction and the effective version of the 1-2-3 Theorem to the list $Q_n$.