matrices - (Stochastic) matrix for which a stochastic matrix logarithm exists?

Steve Hunstman's link above is good:

See the part leading up to Theorem 9 for something relevant to applications:

The main application of the following
theorem may be to establish that
certain Markov matrices arising in
applications are not embeddable, and
hence either that the entries are not
numerically accurate or that the
underlying process is not autonomous.
The theorem is a quantitative
strengthening of Lemma 8. It is of
limited value except when n is fairly
small, but this is often the case in
applications.

Also the part on regularization for best compromises when matrices are not embeddable.

real analysis - Is there a natural measures on the space of measurable functions?

Sorry for the necromancy. Here's an attempt at constructing a $sigma$-algebra using the tensor product of $sigma$-algebras. This should likely not result in a Borel structure (i.e., a $sigma$-algebra generated as the Borel $sigma$-algebra of a topological space), so I don't think it contradicts Aumann's work.

I made this answer community wiki, so feel free to edit it. If it's wrong, please correct it.

I figured I'd answer the question to provide a quick reference for the future.

---

Let $(X,Sigma_X)$ and $(Y, Sigma_Y)$ be two measurable spaces, and let $H = operatorname{Hom}(X,Y)$ be the set of measurable functions from $X$ to $Y$. Define the evaluation map $operatorname{eval} : H times X to Y$ by $$operatorname{eval}(h,x) = h(x).$$

Now, simply define $Sigma_{H}$ to be the minimal $sigma$-algebra on $H$ so that the evaluation map $operatorname{eval} : H times X to Y$ is measurable, where $H times X$ is equipped with the tensor product $sigma$-algebra $Sigma_H otimes Sigma_X$.

I think that $Sigma_H$ should be well-defined, even though it's unlikely to be Borel in most interesting situations. There should always be some minimal solution, even if it's the whole power set $2^H$.

---

Here are some general thoughts on why it is important that the evaluation function is measurable, and why this is good enough for most interesting applications, e.g., applied analysis, physics or computation. This means that f $B in Sigma_Y$ is any measurable event in $Y$, then $$operatorname{eval}^{-1}(B) = big{ (h,x) : h(x) in B big} in Sigma_H otimes Sigma_X.$$

For example, this always describes solution-sets to equations, since
$${ h(x) = y } = operatorname{eval}^{-1}({y}).$$

When $Y$ is a measurable hierarchy (i.e., a pre-ordered measurable space), then this also includes inequalities, e.g.,
$${ h(x) le y } = operatorname{eval}^{-1}(downarrow{y}),$$
where $downarrow{y} = { y' le y }$ denotes the down-set of $y in Y$. Basically, $$mbox{if you can write it down, it's probably measurable.}$$

This is very useful computationally, since the hom-set $operatorname{Hom}(H times X, Y)$ is adjoint to $operatorname{Hom}(H,Y^X)$ via the process of currying. The adjoint to the evaluation map is called function application, and in computer science is known as Apply. Ultimately, this means that anything you work out computationally is measurable, which means no more appendices full of nasty measurability proofs by hand.

Note that $Y^X$ is a measurable space when equipped with the tensor-product $sigma$-algebra, and in most cases of interest its $sigma$-algebra is not generated by a topology (reference Jochen Wengenroth's answer to this question).

Furthermore, this should be useful in measure theory, and may lead toward an answer to Kenny Easwaran's question. If you can see a way to answer it, go ahead and edit this answer.

riemannian geometry - Geodesics on zero-curvature regions of closed surfaces of genus > 1 of non-positive curvature

Suppose that $S$ is your Riemannian surface and $X subset S$ is a flat subsurface (that is, locally isometric to $mathbb{R}$ with the usual metric). Let's suppose that $X$ has some nontrivial topology. For example, $X$ is a unit disk minus one-half of a unit disk, and the core curve of $X$ is essential in $S$.

You are correct in thinking that the universal cover of $S$ is homeomorphic to $mathbb{D}$ the unit disk. However, it is impossible to choose this homeomorphism so that the universal cover of $X$, call it $bar{X}$, embeds isometrically in $mathbb{D}$. You cannot even arrange this up to homothety. To see this, choose isometric charts for $X$ that lift to give isometric charts for $bar{X}$. After choosing where any one chart of $bar{X}$ goes in $mathbb{R}^2$ (isometrically!) the positions of all others will be determined. (This is the so-called "developing map" of $bar{X}$ and you can think of it as being similar to the process of analytic continuation of a analytic function.) The point here is that the developing map will not be injective - in fact it will have image isometric to $X$ itself.

I can't think off hand of a reasonable reference (Thurston's book is perhaps an unreasonable reference). Think about it and ask any local geometers or post more questions on MO.

nt.number theory - Drawing (graphs) by numbers: a minimality question

I'm not sure this qualifies as an answer, but I hope these remarks are useful to you; they re-present your problem in a format which is likely to be answerable by experts in discrete optimization.

The abstract: I suspect the problem of computing the smallest maximum N_j is intractible, and suggest approaches to obtaining upper and lower bounds for N_j . I also make brief remarks about the case of low clique number.

Reformulation

Let K be the set of maximal cliques C_j , and consider the bipartite graph graph H with vertex-set V(G) ∪ K, and adjacency defined by
$$ v~C_j ;in; E(H) ;;;iff;;;; v in C_j ;.$$
Your weighting scheme then amounts to a weighting of the vertices of H by (co-)prime integers. Instead of considering products of such (co-)prime integers, we may consider the sum of their logarithms. So:

• weight the vertices of H with real numbers ω(v) = ln(p) for distinct primes p;


• define the "weight" Ω(v) of a neighborhood of a vertex v as the sum of ω(x) for x ranging over v and its neighbors. For vertices v ∈ V(G), its neighbors are the maximal cliques C_j to which it belongs in G; for vertices C ∈ K, its neighbors are all of the vertices in G which C contains.

We are interesting in minimizing $$large Omega(G) equiv max_{v in V(G)} Omega(v)$$ for v ∈ V(G) subject to the above definitions/constraints. The minimum N_j which you describe above is then e^Ω(G).

Now, the weights ω(v) for v ∈ V(H) form a vector of logarithms of primes. There's no reason to take any coefficient to be larger than ln(p_h), where h = |V(H)| and p_h is the h^th prime. So we may as well fix the column vector p = ( ln 2, ln 3, ... , ln p_h )^T, and describe the weight function ω in terms of permutations of the coefficients of this vector. So really, we would like to obtain
$$ Omega^ast(G) ;;=; large min_{Pi in mathfrak S_h} ;max_{v in V(G)} big(mathbf{e}_v^top A(H) : Pi ;mathbf{p} big)$$
where A(H) is the adjacency matrix of H, and $mathfrak S_h$ is the group of permutation matrices on ℝ^h.

Remarks on the reformulation

Evaluating Ω*(G) is likely to be difficult, as in computationally intractible. (Disclaimer: I am not an expert on such problems, and I have not given this instance a lot of thought; but some similar problems are NP complete.) A better question is whether you can get "nice" upper or lower bounds for Ω*(G).

• You can obtain a lower bound for Ω*(G) by taking a convex relaxation. For instance, instead of optimizing over $mathfrak S_h$, oprimize over the convex closure of that set, which is the set of doubly-stochatic matrices over ℝ^h. You can then exploit the fact that the maximum is the uniform norm of the [restriction to ℝ^V(G) of the] vector ω(H) = A(H) Π p ; as such, it is a convex function (as the uniform norm satisfies the triangle inequality). It should be possible to optimize this function efficiently using steepest descent techniques.




• Obviously, you're more interested in upper bounds for Ω*(G). The function f(x) = ln(p_x) grows asymptotically like ln(x) + ln(ln x); therefore, the contribution of a large log-prime weight to some sum Ω(v) is not much different than the contribution of a slightly larger log-prime weight. Optimizing the location of the larger primes among themselves is then unlikely to be useful; in practise it is more useful to optimize the location of the smaller primes.

  
  The weights of the clique-vertices C_j contribute to many different neighborhood weights Ω(v). This suggests that a reasonable approach is to allocate the smallest log-prime weights to cliques according (roughly) to the number of vertices they contain. Obviously this will fail if there is a very large clique which "interacts" with very few other cliques (i.e. shares vertices in common with few other cliques), and there exists elsewhere a large congregation of cliques which each share something like half of their elements with other cliques (i.e. for a large subset S of V(G), each vertex in S belongs to approximately half of a large collection of cliques). It may be worthwhile to investigate the graph of incidence of maximal cliques.

A final remark: in the case of a bipartite graph, the maximal cliques are all edges, in which case the graph H is just a subdivision of G. In this case, attributing weights to the vertices v ∈ V(G) does not aid in the representation of the graph. For graphs with low clique number, it may be worthwhile to investigate a similar scheme where only the edges or maximal cliques are given weights, or more generally where almost every vertex is given weight 1.

gr.group theory - Maximum Order of elements in $GL(n,Z)$

From the fact you're citing, it looks like

$$ f(n) = max { m : Phi(m) le n }. $$

For example, $Phi(30) = 7$ and $Phi(m) ge 11$ for all $m ge 31$ (note that since $phi(n) ge sqrt{n}$ we only need to check finitely many values!) -- so $f(7) = f(8) = f(9) = f(10) = 30$.

Now, consider the fact that
$$ lim inf phi(n) {log log n over n} = e^{-gamma} $$
which is equation (20) in this Mathworld article. Of course this holds if we replace $phi$ by $Phi$.

So $f(n)$ should grow like the inverse of the function
$$ n to {e^{-gamma} n over log log n} $$. It appears, then, that $f(n) sim e^gamma n log log n$ as $n to infty$.

Unfortunately this disagrees with your estimate. One of us is wrong somewhere.

EDIT: I believe my argument is basically right, but the original fact was stated incorrectly. From the paper of Levitt that Stanley pointed to, we should actually have

$$ Phi( p_1^{alpha_1} cdots p_k^{alpha_k}) = phi(p_1^{alpha_1}) + cdots + phi(p_k^{alpha_k}) - [k equiv 2 mod 4] $$

and so $Phi(x)$ is usually much smaller than $phi(x)$ -- therefore $f$ grows much faster than I said it did.

Orbital elements: Difference between longitude of perihelion and argument of perifocus

I know the answer now. Turns out that longitude of perihelion ($overlineomega$) is defined as the sum of longitude of ascending node ($Omega$) on the ecliptic plane and argument of perifocus ($omega$) on the orbital plane: ($Omega$ + $omega$). It's strange to add up two non-coplanar angles, even when the inclination between the two planes is very small, but that's what it is.

ca.analysis and odes - Sheaves and Differential Equations

I will start commenting on Mariano's answer. I believe it is a perfect answer for the question

How do sheaves arise in studying
solutions of differential equations ?

but not for the question

How do sheaves arise in studying
solutions to ordinary differential
equations ?

According to the current terminology a function $f$ satisfying $X(f)=0$ is not a solution of the vector field $X$ but a first integral. Moreover, if $X = a(x,y) partial_x + b(x,y) partial_y$ then
$$
X(f) = a partial_x f + b partial_y f .
$$
Thus $X(f)=0$ is a PDE and not an ODE. Indeed t3suji made the same point at a comment on Mariano's answer. I understand the solutions of (the ODE determined by) $X$ as functions $gamma : V subset mathbb R to U$ satisfying $X(gamma(t))=gamma'(t)$ for every $t in V$. Notice that here indeed we have a system of ODEs.

A vector field can be thought as autonomous differential equation and I do not see clearly how to consider the sheaf of its solutions.

On the other hand when we have a non-autonomous ordinary differential equation then there is its sheaf of solutions. This sheaf is a sheaf over the time variable
only and not the whole space. ( At this point it is natural to talk about connections and/or jet bundles but I will try to keep things as elementary as possible. )

Note that in general the sheaf of solutions will not be a sheaf of vector spaces: the sum of two solutions, or the multiplication of a solution by a constant need not to be a solution. This will occur only when the differential equation is linear.

The differential equations $y'(t) = y$ and $y'(t) = y^2$, both defined over the whole real line, are examples of differential equations with non-isomorphic sheaves of solutions. The solutions of the first ODE are the multiples of $exp t $ and define a sheaf of $mathbb R$-modules. The solutions of the second ODE are zero and $frac{1}{lambda - t}$ with $ lambda in mathbb R$. They do define a sheaf of sets, but not a sheaf of $mathbb R$-modules.

To obtain examples of linear differential equations with non-isomorphic sheaves, one has to have nontrivial fundamental group on the time-variable of the differential equation. Thus it is natural to consider complex differential equations over $mathbb C^{ast}$.

The equations $y'(z) = frac{ lambda y(z)}{z}$ parametrized by $lambda in mathbb C$ have non-isomorphic sheaves of solutions. More precisely,

• if $lambda in mathbb Z$ then the solution sheaf is the free $mathbb C$-sheaf of rank one (solutions of the ODE are complex multiples of $z^{ lambda }$);


• if $lambda in mathbb Q - mathbb Z $ then the solution sheaf has no global sections but some tensor power of it does;


• if $lambda in mathbb C - mathbb Q$ then the solution sheaf has no global sections nor any of its powers does.

soft question - What's your favorite equation, formula, identity or inequality?

With the stuff I've seen in the literature of sequence transformations, I've started to love the formulae for Aitken's Δ² process:

$S_n^{prime}=S_{n+1}-frac{(Delta S_n)^2}{Delta^2 S_n}$

and its generalization the Wynn ε algorithm:

$varepsilon_{k+1}^{(n)}=varepsilon_{k-1}^{(n+1)}+frac1{varepsilon_{k}^{(n+1)}-varepsilon_{k}^{(n)}}$

for the latter one especially because it is nicely represented as a lozenge diagram:

What would happen if we put a small part of White Dwarf or Neutron Star on Earth?

If we take neutron star material at say a density of $sim 10^{17}$ kg/m$^{3}$ the neutrons have an internal kinetic energy density of $3 times 10^{32}$ J/m$^{3}$. So even in a teaspoonful (say 1 cc), there is $3times10^{26}$ J of kinetic energy (similar to what the Sun emits in a second, or 10 billion or so H-bombs) and this will be released instantaneously.

The energy is in the form of around $10^{38}$ neutrons travelling at around 0.1-0.2$c$. So roughly speaking it is like half the neutrons (about 50 million tonnes) travelling at 0.1$c$ ploughing into the Earth. If I have done my Maths right, that is roughly equivalent to a 30 km radius near-earth asteroid hitting the Earth at 30 km/s.

So this material would instantly vapourise and take a large chunk of the Earth with it, probably destroying most of life on Earth.

The situation for a white dwarf is much less extreme. The density would be more like $10^{9}$ kg/m$^3$ and the energy density more like $10^{22}$ J/m$^3$ - so 10 orders of magnitude less kinetic energy density. Nevertheless that is still $10^{16}$ J, which is like a 2.5 megatonne H-bomb.

ct.category theory - Probably easy: Why is f*:A^C'->A^C continuous and cocontinuous for any functor f:C->C'?

Let $f:Cto C'$ be a functor, and let $A$ be a locally presentable, complete, and cocomplete category. Then according to the paper I'm reading, the pullback functor, $f^*:A^{C'}to A^C$ (given by precomposition with $f$), admits left and right adjoints $f_!$ and $f_*$. It's clear that the proof of this fact follows from the adjoint functor theorem, so it suffices to show that $f^*$ is continuous and cocontinuous.

However, it's not clear to me how to show this fact.

Question:

Using the notation above, why is $f^*$ continuous and cocontinuous?

Sorry if this ends up being too easy.

ag.algebraic geometry - How to use automorphisms to produce isotrivial non trivial families

If $C=B$ with the same $Z/2$ action, and $g(C/Z_2) geq 2$, then you always get a non-trivial family:

Suppose it was trivial. Then it had another projection $q : D to C$. Since $C$ is an 'etale cover of $B'$, and both have genus at least two, $g(C) > g(B')$ by the Hurwitz-formula. That is, also by the Hurwitz formula, all the maps from $B'$ to $C$ are the constant maps. So, all sections $B' to D$ of $f$ would have to be contained in a fiber of $q$, i. e. they have to be one fiber of $q$. However there are two natural sections $E$ and $F$ of $p$: $[c] to [(c,c)]$ and $[c] to [(c,-c)]$. So both of these have to be contracted to a point by $q$. Given any $b in B$, a choice $c in C$ such that $[c]=b$ gives an isomorphism $a_c : p^{-1}(d) to C$, which sends $[(c,c')]$ to $c'$. Now one can construct automorphisms $varphi_{c'}$ of $C$ for any $c' in C$ by the following composition:

$
C to^{a_c^{-1}} p^{-1}([c]) to^q C to^s p^{-1}([c']) to^{a_{c'}} C
$

where

$
S=(q|_{p^{-1}([c'])})^{-1}
$

One specialty about $varphi_{c'}$ is that it takes $c$ and $-c$ to $c'$ and $-c'$, respectively. This follows from the fact, that $E$ and $F$ are contracted to a point by $q$. So, $C$ has infinitely many different automorphisms, which is a contradiction by the assumption that $g(C) geq 2$. That is our assumption is false, therefore $Y$ is not a product family.

homological algebra - introductory book on spectral sequences

Many of the references that people have mentioned are very nice, but the brutal truth
is that you have to work very hard through some basic examples before it really makes
sense.

Take a complex $K=K^bullet$ with a two step filtration $F^1subset F^0=K$, the spectral
sequence contains no more information than is contained in the long exact sequence associated
to
$$0 to F^1to F^0to (F^1/F^0)to 0$$
Now consider a three step filtration $F^2subset F^1subset F^0=K$, write down all the short
exact sequences you can and see what you get. The game is to somehow relate $H^*(K)$
to $H^*(F^i/F^{i+1})$. Suppose you know these are zero, is $H^*(K)=0$? Once you've mastered
that then ...

co.combinatorics - Transitivity-related property of finite permutation groups

Let $cal F$ denote the group of all finitely-supported permutations of $mathbb N$.
Say that a finite subgroup $G$ of $cal F$ is singular if $G$ acts transitively on
$lbrace 1,2,3 rbrace$ but no cyclic subgroup
of $G$ acts transitively on $lbrace 1,2,3 rbrace$ (this is equivalent
to saying that some element in $G$ sends $1$ to $2$, another sends $1$ to $3$
but no element of $G$ has all of $1,2$ and $3$ in a single orbit).

The Klein group (products of disjoint transpositions on $lbrace 1,2,3,4 rbrace$)
is in example of such a subgroup.

Question 1 : are there other simple examples of minimal singular subgroups ?
Is there a parametric description of all of them up to isomorphism ?

Question 2 : Denote by ${cal F}(i to j)$ the set of all permutations in $cal F$
sending $i$ to $j$. Say that a permutation $sin {cal F}(1 to 2)$
and a permutation $tin {cal F}(1 to 3)$ are related iff
the subgroup generated by $s$ and $t$ is a minimal singular subgroup of $cal F$.
Given $s$, let $R(s)$ denoted the set of all $t$'s such that $s$ and $t$ are related.
Does $R(s)$ admit a simple description ?

Of course, any answer to question 2 automatically provides an answer to
question 1.

If we pumped an large EMP into High Earth orbit and it detinated would it act as a shield against solar radiation?

Could this pulse potentially shield spacecraft from harmful solar radiation.

I was assuming an EMP would have little effect as a radiation shield because the pulse would not be rapid enough to counter act the suns energy

f we pumped an EMP into high earth orbit and it detinated would it create a pocket of radiation-free space

soft question - What is the definition of "canonical"?

Regarding the (widely considered to be false) statement that "There is a canonical isomorphism between a finite-dimensional vector space V and its dual" in Reid Barton's answer, I think that the situation is a bit more interesting than that. It is a good illustration of the idea that an object may be defined to be "canonical" if it is constructed without making any choices, and the interesting point here is that there are various degrees of (in-)tolerance to choices. If we work with vector spaces of fixed finite dimension, then an isomorphism $i_E:Eto E^*$ between a vector space $E$ and its dual $E^*$ may be called canonical if

1. it does not depend on the choice of a basis for $E$ but we need a basis to define it, or


2. it does not depend on the choice of a basis and may be defined without choosing a basis, or


3. it does not depend on basis choices as above and does not even depend on $E$, in the sense that whenever $u:Eto F$ is an isomorphism between dim. vector spaces of the same finite dimension, then $u^*circ i_Fcirc u=i_E$ where $u^*$ is the transpose of $u$. This third notion of canonicity is essentially functoriality.

Concretely, given a basis $B={e_j}$ of $E$ with dual basis $B^*={e_j^*}$, we can construct an isomorphism $i=i_{E,B}:Eto E^*$ that maps $e_j$ to $e_j^*$. This map does not depend on the choice of basis if and only if $u^*circ icirc u=i$ for all $uin text{GL}(E)$. It is easily seen that this is equivalent to the fact that $text{GL}_n(k)=text{O}_n(k)$, where $k$ is the base field, $n$ is the dimension, and $text{O}_n(k)$ is the orthogonal group of the standard (sum of squares) quadratic form. Exercise: this equality holds if and only if $n=1$ and $k$ has at most $3$ elements. Thus for $n=1$ and $text{card}(k)leq 3$ the map $i_{E,B}:Eto E^*$ does not depend on $B$, so we may write it simply $i_E$. When $text{card}(k)=2$ this is not so surprising because any two one-dimensional vector spaces over the field with two elements are uniquely (hence canonically in whichever sense you like) isomorphic, but for $text{card}(k)=3$ this is a bit more exotic. Having reached this point, we might think that we are in the funny notion 1 of canonicity (and this is what I thought some minutes ago). But in fact, still assuming that $n=1$ and $text{card}(k)leq 3$, we can exhibit an isomorphism $i:Eto E^*$ without any reference to a basis. Namely, define $i(0)=0$ and if $xin E$ is nonzero, then it is a basis of $E$, and we can define $i(x)=x^*$, the only element of the dual basis. The point is that since $a^2=1$ for all nonzero scalars in $k$, this map $i$ is linear.

Conclusion: if $n=1$ and $text{card}(k)leq 3$ there is an isomorphism $i:Eto E^*$ that is constructed without a choice of basis, and it is functorial for isomorphisms of one-dimensional vector spaces. If $nge 2$ or $text{card}(k)ge 4$, the map $i_{E,B}:Eto E^*$ is not independent of the basis $B$.

I would guess that it is possible to find examples of phenomena like 1 above.

space time - Light Cone Explanation

You are on the surface of the Earth now (I assume), as are almost all other humans. The surface of the Earth can be thought of as a two dimensional plane.

OK, let's say you do something amazing and you want to tell everyone on Earth about it, so you set up a means of broadcasting your news, at the speed of light, across the surface of the Earth. The speed of light is quick, but it still takes nearly two tenths of a second for your signal to make it round the world.

Before that signal gets to them, nobody knows your news. You can imagine a circle, defined by the speed of light times the travel time, spreading outward from your location, containing those regions that have heard the news.

Now what you do is imagine those circles stacked on top of each other, each drawn after a small increment in time, with a larger radius, and separated vertically by an amount that represents that increment in time. The circles begin at your position, when you start to broadcast. This stack of circles of increasing radius forms a light cone. Inside the cone are regions of space and time that it is possible for you to communicate your news to. Outside the cone are regions (of space and time) that can never hear the news because no signal can get there faster than the speed of light.

A concrete example. It is about 5000 km between London and New York. If the stock market crashes at 17:00 (UT) in New York, it is impossible for traders in London to hear about it for at least 0.016 seconds. For that period, they lie outside the light cone of a signal produced at 17:00 in New York. More distant cities take even longer to intercept the light cone.

The light cone idea also works in reverse. You can invert the cone to mark regions of space and time from which you could have heard some news. In the example above, the inverted light cone of the London traders does not contain New York until 0.016 seconds after 17:00.

A point of confusion is the idea of a cone, which is really only appropriate if space is defined as two dimensional - e.g. points on the surface of the Earth. Conceptually it is harder to work with in 3D (though mathematically equivalent). In 1D (points along a line), the light cone approximates to a light triangle.

What determines the configuration of orbits in a binary system?

Following the convention $mathbf{r} = mathbf{r}_2-mathbf{r}_1$, $M = m_1+m_2$, in the center-of-mass frame we have, by definition,
$$begin{eqnarray*}mathbf{r}_1 = -frac{m_2}{M}mathbf{r}text{,}quad&mathbf{r}_2 = frac{m_1}{M}mathbf{r}text{.}end{eqnarray*}$$
Hence, $ddot{mathbf{r}} = -GMhat{mathbf{r}}/r^2$ implies that the individual orbits are similar conic sections in this frame, and moreover in the bound case they are ellipses that share a common focus at the center of mass, with all three of the distinct foci being collinear and with the center between them. Although there is a degenerate case of a circular orbit.

Therefore, there's a geometrically simple condition for the intersecting configuration: there is intersection if, and only if, the distance to apoapsis of the larger mass (smaller orbit) is greater than or equal to the distance to periapsis of the smaller mass (larger orbit). Since a general ellipse in polar coordinates about a focus can be described by
$$r = frac{p}{1+ecos(phi-phi_0)}text{,}$$
where $e$ is the eccentricity, and the individual orbits are proportional, we have intersection if, and only if,
$$frac{1-e}{1+e}leq frac{m_1}{m_2}leqfrac{1+e}{1-e}text{,}$$
as the apsides occur when the cosine term is $pm 1$.

homological algebra - Why is the Hochschild homology of k[t] just k[t] in degrees 0 and 1?

Another way to write the Hochschild homology is as follows:

take A as a bimodule over itself, take a free resolution as a bimodule, and then apply the functor of coinvariants ($M mapsto M/langle rm-mr|rin Arangle$).

Your definition used the "bar-complex" resolution of the form $to A otimes A otimes A to A otimes A$
but k[t] has a much nicer resolution as a bimodule over itself, the Koszul resolution.

This is of the form $k[t] otimes k[t] to k[t] otimes k[t]$ with the map given by $1 otimes t - t otimes 1$, so when you apply coinvariants, you get two copies of $k[t]$ with trivial differential.

Actually all Koszul algebras have a nice resolution of the diagonal bimodule, and thus its easier to compute their Hochschild homology, though in general, they don't always have trivial differential after applying coinvariants.

EDIT: For the later question, probably the best answers you'll get are from HKR, though just noting that the global dimension of $k[t] otimes k[t]$ is 2 gets you halfway there.

EDIT AGAIN: Actually, any Koszul algebra has its Hochschild homology bounded above by its global dimension. This is clear from the existence of the diagonal Koszul resolution.

gr.group theory - When and how is a group of order n isomorphic to a regular subgroup of equal order?

In "Group Theory and Its Application to Physical Problems" by Morton Hamermesh, Morton states Cayley's theorem: Every group G of order n is isomorphic with a subgroup of the symmetric group S_n, which makes sense to me.

Later the book discusses regular permutations and regular subgroups, and makes this statement: "...suppose that n is a prime number. Then the group of order n is isomorphic to a regular subgroup of S_n." (page 19 in the Dover edition)

Why is the last sentence true? Is every group of any order n isomorphic to a regular subgroup of S_n?

at.algebraic topology - Are there pairs of highly connected finite CW-complexes with the same homotopy groups?

Fix an integer n. Can you find two finite CW-complexes X and Y which

* are both n connected,

* are not homotopy equivalent, yet

* $pi_q X approx pi_q Y$ for all $q$.

In Are there two non-homotopy equivalent spaces with equal homotopy groups? some solutions are given with n=0 or 1. Along the same lines, you can get an example with n=3, as follows. If $Fto Eto B$ is a fiber bundle of connected spaces such that the inclusion $Fto E$ is null homotopic, then there is a weak equivalence $Omega Bapprox Ftimes Omega E$. Thus two such fibrations with the same $F$ and $E$ have base spaces with isomorphic homotopy groups.

Let $E=S^{4m-1}times S^{4n-1}$. Think of the spheres as unit spheres in the quaternionic vector spaces $mathbb{H}^m, mathbb{H}^n$, so that the group of unit quarternions $S^3subset mathbb{H}$ acts freely on both. Quotienting out by the action on one factor or another, we get fibrations
$$ S^3 to E to mathbb{HP}^{m-1} times S^{4n-1},qquad S^3to Eto S^{4m-1}times mathbb{HP}^{n-1}.$$
The inclusions of the fibers are null homotopic if $m,n>1$, so the base spaces have the same homotopy groups and are 3-connected, but aren't homotopy equivalent if $mneq n$.

There aren't any n-connected lie groups (or even finite loop spaces) for $ngeq 3$, so you can't push this trick any further.

Is there any way to approach this problem? Or reduce it to some well-known hard problem?

(Note: the finiteness condition is crucial; without it, you can easily build examples using fibrations of Eilenberg-MacLane spaces, for instance.)

ra.rings and algebras - Name for semiring with weakened annihilation law?

If you were looking for a field with convenient terminology for its structures, I should warn you that the field of semirings is pretty bad :)

In

"Graphs, dioids and semirings: new models and algorithms" by Michel Gondran, Michel Minoux

they use "presemiring" to mean a set with two associative binary opearations, + and X where + is commutative and X distributes over + on both sides.

To make a presemiring a semiring, they require identity elements for both operations, and require that the additive identity 0 is absorbing, as you describe.

I wouldn't put too much stock in the one book though. Really there are so many authors throwing around so many terms about these things it's probably useless to find out which is the most common.

ag.algebraic geometry - Permanence of regularity in "generalised" semistable models

Given a regular ring $A$ with an element $t$, consider the "generalised" semistable model
$S := A[X_1,...,X_n]/(P_1cdots P_n - t)$ over $A$, where $P_i := {X_i}^{e_i}$ and $e_i$ are positive integers.

Question: For which $A$, $t$, and $(e_i)$ is $S$ regular?

E.g., when $n=2$ and $A$ is a discrete valuation ring and $t$ is a uniformiser, then it is easy to prove that $S$ is regular when $e_1$ and $e_2$ are both equal to 1. When $nge 3$ and $(A,t)$ is a DVR with uniformiser $t$, is there a reference?

mg.metric geometry - Soul theorem for non-negativly curved open Alexandrov manifolds?

Alexandrov manifold means Alexandrov space which happens to be a manifold, i.e. the space of directions is homeomorphic to shpere. Sorry for introducing this new term.

For such a open manifold does the soul theorem holds?
i.e. the manifold is homeomorphic to the disk bundle over it's soul?

pr.probability - Are all probabilities conditional probabilities?

In the hope of soliciting an expert opinion rather than offering one (I am neither a probabilist nor a logician), let me bring up here the topic of logical probability. Jan Lukasiewicz came up with infinitely valued propositional logic (with values in the interval $[0,1]$), which he also viewed as a way of formalizing probability, addressing the problem of interpreting conditional probabilities as well. According to Wikipedia, Richard Threlkeld Cox later showed that any extension of Aristotelian logic to incorporate truth values between 0 and 1, in order to be consistent, must be equivalent to Bayesian probability (http://en.wikipedia.org/wiki/Cox%27s_theorem). Cox's axioms deal with the notion of plausibility of a proposition A given a proposition B.

Added: this is the relevant paper by Lukasiewicz: Die logischen Grundlagen der Wahrscheinlichkeitsrechnung, Kraków, Polska Akademia Umiejętności, 1913 English translation in Selected Works, ed. by L. Borkowski, Amsterdam-London, North-
Holland Publishing Company/Warsaw, PWN, 1970, pp. 16-63

soft question - Has mathoverflow yet led to mathematical breakthroughs?

Hey Tim. I'm not so sure whether my model of how "proving a theorem" works is the same as yours. But let me give some kind of an example of something and you can take it or leave it. I'm writing a paper with Toby Gee at the minute, and we're both number theorists, and the arguments in the paper are "robust" but the details need checking. We're now at the point where we're writing up technical calculations and these technical calculations are mostly in the area of representation theory of reductive algebraic groups, an area which I think it's fair to say that neither Toby nor I would call ourselves experts in. So we have this overall "robust" argument, and a write-up that exists but occasionally says "lemma: (statement in representation theory); proof: TO BE ADDED". We 'know' these lemmas are true because they fit into our overall picture, but occasionally when I write one of these things up I can't go from my intuition to a rigorous proof. Here's an example of an occasion when I got stuck:

This question of mine.

Ben Webster made a crucial remark that enabled me to finish the argument, so that lemma went from "must be true but proof not yet written or even discovered by authors" to "lemma proved".

So if I were interpreting your question in a particularly anal way, one might argue that had this lemma been the last of the lemmas we need to write up the proof of, then Ben's contribution might be "just what I needed to complete my research project". Unfortuately there are several more to go :-).

Having said all that, it's not clear to me that MO was "crucial" to solving the problem. I could have worked more on the problem until I'd done it myself. I could have asked one of the representation theorists in my own department. I could have left it and hoped that my co-author sorted it out. All of these would have been viable approaches. Why did I ask at MO? Simply because I am sick of writing this paper and asking at MO was by no means the only way of solving the problem, but I had high hopes that it would be the quickest.

Kevin

big list - Math puzzles for dinner

Here's one I saw a while ago:

A prisoner is presented with the following challenge by one of the guards of the jail. The prisoner is to be blindfolded and then the guard will place $n$ coins on a circular turntable with any combination of heads and tails facing up (with at least one tails showing initially). The prisoners goal is to flip over coins until all heads are showing.

This would be easy enough if the guard did not interfere. The prisoner could just try all $2^n$ combinations, and one of them would be guaranteed to result in all heads. However, to complicate matters, the guard may turn the table during this process. More specifically, the following process is repeated. First, the prisoner chooses a set of positions of coins to flip over. Then, before the coins are flipped, the guard turns the turntable so as to try to prevent the prisoner from flipping all of the coins to heads. Finally, the prisoner flips over the coins that are at the positions chosen in the first step. If all heads are showing, the game stops and the prisoner is set free.

The question is, for what values of $n$ does the prisoner have a winning strategy and how many moves does it take?

What if the guard uses 6-sided dice instead of coins with the goal of showing all ones (assuming the orientations of the dice are preserved relative to their positions on the turntable between rounds)?

In general, what values of $n$ allow a solution with $k$-sided dice?

ct.category theory - Cogroup Objects

This is less of an answer and more of a bit of bickering about terminology. I think that the word "group object" rightly means the following:

A group object G in a Cartesian category is a map G x G → G, satisfying an associativity axiom, such that there exists a map 1 → G satisfying the unit axiom (it is then necessarily unique) and a map G → G satisfying the inversion axiom (it is then necessarily unique).

If you want, you can consider the unit and inversion maps as part of the data. What's important in the word "group object" is that it is in a Cartesian category, i.e. a category with products. The object 1 is any choice of terminal object for the category.

From this perspective, the most obvious notion of "cogroup object" is:

A cogroup object G in a Cartesian category is a map G → G x G, satisfying a coassociativity axiom, such that there exists a map G → 1 satisfying the counit axiom (it is then necessarily unique) and a map G → G satisfying the inversion axiom (it is then necessarily unique).

But this is a rather problemmatic definition. First of all, by definition there is a unique map G → 1 for any object G. Second, there is a unique coassiciative map G → G x G; it is given by the diagonal map, which is the element of Hom(G,GxG) corresponding to (id,id) ∈ Hom(G,G) x Hom(G,G). (Remember that the definition of Cartesian product is that for any H there is a natural isomorphism Hom(H,GxG) = Hom(H,G) x Hom(H,G).) So every object of a Cartesian category is a "counital cosemigroup" in a unique way. Finally, it's rather hard to write down exactly what the inversion axiom should say.

Well, so the problem is that we're trying to use the Cartesian structure. So Wikipedia and elsewhere adopt the only nontrivial definition, which is to say that for cogroups you should use the coproduct, not the product, so you should ask for a map G → G + G. I think this is the question you're asking about. I guess I would word it "are the any natural categories C with interesting group objects in C^op?", since this definition of "cogroup object" really is "group object in the opposite category".

Finally, I'll mention that certain related constructions do show up all the time. There are good words "algebra" and "coalgebra", which refer to objects in categories with designated monoidal structures (e.g. tensor product in VECT). What's cool is that to make sense of the inverse map's axiom when you aren't using either the product (for groups) or the coproduct (for cogroups), you actually need all the data of a "bialgebra". So the correct notions of "group" and "cogroup" coincide when you give up on categorical (co)products, and become the word "Hopf algebra".

exoplanet - Likelihood of a stable system with a dwarf planet's orbit inside that of a gas giant

I have an answer for the first part of your question, because I looked it up in answering http://physics.stackexchange.com/questions/8827/question-on-the-stability-of-the-solar-system/161973#161973 on Physics SE.

If you want to see what the current state of the art on solar system simulations is you could do worse than watch Sean Raymond's presentation at "Protostars and Planets VI" from 2013. You can find the actual write up here. Or from the same conference there is Melvyn Davies' review of the long-term dynamics of planetary systems. The talk can be seen here. This review does contain the sort of information you are looking for. It discusses the past and future evolution of our solar system, as well as planetary systems in general. It presents and reviews simulations and discusses the relevant issues. Both of these guys are excellent speakers.

A brief summary would be that the solar system is probably stable for the remaining lifetime of the Sun. However, there is the intriguing possibility that Mercury could fall into the Sun or collide with Venus in the next billion years or that Mars could be ejected from the solar system on a similar timescale (e.g. from the N-body simulations of Battygin & Laughlin 2008).

ca.analysis and odes - Projection of a gradient and the gradient of a projection

I think you need to reformulate your problem.

The standard definition of a vector field being `projectible' [eg Warner.
See $pi$-projectible ] requires, in the case of the projection $(x,y,z) to (x,y)$
it to have the form $F_1 (x,y){{partial} over {partial x}} + F_2 (x,y){{partial} over {partial y}} + F_3(x,y,z) {{partial} over {partial z }}$.

For your ``projection'' to be finite, you need your $f(x,y,z)$ to depend on $z$
in such a way that the integral is finite. As a consequence, its gradient will
have $F_1$ and $F_2$ either zero, or depending on $z$.

Combining your def. of `projection of a function' with the standard def. of
projection of a vector field you get the result that the only projected vector field
you can get is the zero vector fields.

gt.geometric topology - Computing the structure of the group completion of an abelian monoid, how hard can it be?

Do people have good examples where it's "easy" to compute the group-completion of a commutative monoid, but for which the monoid itself is still rather mysterious?

This happens all the time in K-theory $K^0(X)$, both algebraic and topological. Perhaps it is even the reason that K-theory is a useful tool.

For a striking algebraic example, take $X = mathbb{A}^n_k$ where $k$ is a field. Then $K^0(X)$ is the group completion of the commutative monoid $M$ of isomorphism classes of finitely generated projective modules over $R = k[x_1, ldots, x_n]$. In 1955 Serre asked whether every such module was free, i.e., whether $M = mathbb{N}$. This question became known as Serre's conjecture. Serre proved in 1957 that every finitely generated projective $R$-module is stably free, i.e., $K^0(X) = mathbb{Z}$. However, it was not until 1976 that Quillen and Suslin independently proved Serre's original conjecture. So between 1957 and 1976, $M$ was an example of a commutative monoid whose group completion was known but which itself was not known. This is only a historical example, because $M = mathbb{N}$ turns out to be very simple; however, it illustrates the difficulty of the question in general.

A topological example where the commutative monoid is not so simple is given by $KO^0(S^n)$. Let us take $n$ congruent to 3, 5, 6, or 7 modulo 8, so that $KO^0(S^n) = mathbb{Z}$ by Bott periodicity (the generator being given by the trivial one-dimensional real vector bundle). Let $T$ be the tangent bundle to $S^n$. In $KO^0(S^n)$, of course, the class of $T$ is equal to its dimension $n$. But if we let $M$ be the commutative monoid of isomorphism classes of finite-dimensional real vector bundles on $S^n$ (so that $KO^0(S^n)$ is the group completion of $M$) then the class of $T$ is not equal to the class of the trivial $n$-dimensional vector bundle unless $S^n$ is parallelizable, which only happens when $n$ is equal to (0 or 1 or) 3 or 7. So for all other values of $n$, $M$ is not simply $mathbb{N}$; there are extra vector bundles which get killed by the group completion process. Understanding these monoids $M$ for all $n$ amounts to understanding the homotopy groups of all the groups $O(m)$, which I expect is not much easier than understanding unstable homotopy groups of spheres.

Finally, Pete's example of the monoid of cardinalities of at most countable sets and its absorbing element also makes an appearance in K-theory; here it is called the Eilenberg swindle and it explains why we restrict ourselves to finitely-generated projective modules.

software - Where can I find a set of data of the initial conditions of our solar system?

I was able to get the Cartesian orbital vectors for all the major bodies from HORIZON at the J2000 epoch only. I could extend the coverage forward thru time. It’s easy to get data overload doing this. My simulation is modeled using the Laws of Gravitation and Motion alone. This gives results that are surprisingly close to those published. Running the solar system backwards (by reversing the velocity vectors) has given me the initial vectors back to 1900. This is all I needed and the results were close enough for my purposes. I still have the CSV files.

I have also have had all sorts of problems with the horizons interface. For instance changing the date had no effect on the value of the vectors. i.e.: all specified start dates have the same values. Lately, I have not been able to duplicate this feat. There are obviously some serious problems with this interface, especially lately.

I know the data I got was correct because it correlates, perfectly, with published events, e.g.: the recent transit of Mercury.

I too am still looking for this type of data.

ho.history overview - Context for "Coronidis Loco" from Weil's Basic Number Theory

It is not hard to see that if $L/K$ is an extension of number fields, then the
discriminant of $L/K$, which is an ideal of $K$, is a square in the ideal class group of $K$.
Hecke's theorem lifts this fact to the different. (Recall that the discriminant is the norm of the different.)

If you recall that the inverse different $mathcal D_{L/K}^{-1}$ is equal to $Hom_{mathcal O_K}(mathcal O_L,mathcal O_K),$ you see that the inverse different is the relative dualizing sheaf of $mathcal O_L$ over $mathcal O_K$; it is analogous to the canonical bundle of a curve (which is the dualizing sheaf of the curve over the ground field). Saying that
$mathcal D_{L/K}$, or equivalently $mathcal D_{L/K}^{-1}$, is a square is the same as saying that there is a rank 1 projective $mathcal O_L$-module $mathcal E$ such that
$mathcal E^{otimes 2} cong mathcal D_{L/K}^{-1}$, i.e. it says that one can take a square
root of the dualizing sheaf. In the case of curves, this is the existence of theta characteristics.

Thus, apart from anything else (and as indicated in the quotation given in the question),
Hecke's theorem significantly strengthens the analogy between rings of integers in number fields and algebraic curves.

If you want to think more arithmetically, it is a kind of reciprocity law. It expresses in some way a condition on the ramification of an arbitrary extension of number fields: however the ramification occurs, overall it must be such that the different ramified primes balance out in some way in order to have $prod_{wp} wp^{e_{wp}}$ be trivial in the class group
mod $2$ (where $wp^{e_{wp}}$ is the local different at a prime $wp$). (And to go back to the analogy: this is supposed to be in analogy with the fact that if $omega$ is any meromorphic differential on a curve, then the sum of the orders of all the divisors and poles of $omega$ is even.) Note that Hecke proved his theorem as an application of quadratic reciprocity in an arbitrary number field.

co.combinatorics - What interesting class of Matroids are there that contains the class of Gammoids?

I am looking for a (already-studied or interesting) class of Matroids such that
- Class of Gammoids are contained in it

One example would be Strongly-base-orderable Matroids. I would also be grateful if someone knows a class of Matroids such that

• Class of Gammoids are contained in it AND


• It is contained in the class of "Strongly-base-orderable" Matroids.

By the way, strongly-base-orderable is a property such that :
GIven any two bases I,J of the Matroid, there exists a bijection pi between I-J and J-I such that given any subset H of I-J, I- H +pi(H) and J - pi(H) + H is a base in the Matroid. (In Oxley's "Matroid theory" pp410 )

Motivation :
Well, I have something I can show for Gammoids but cannot for Strongly-base-orderable Matroids, although computational evidence suggests that it holds for general Matroids.

ac.commutative algebra - Torsion submodule

$A$ a commutative Noetherian domain, $M$ a finitely generated $A$-module. How can I show that the kernel of the natural map $Mrightarrow M^{**}$, where $ M^{ * *}$ is the double dual (with respect to $A$), is the torsion submodule of $M$?

I do know that in this situation torsionlessness coincides with torsion-freeness. According to Auslander this result is ``well-know'' but I can't seem to prove it or find any reference on this.

triangulated categories - If a colimit of distinguished triangles exists, is it also a distinguished triangle?

this is true in the topological setting. cofibrations can be thought of as colimits, they are actually colimits of a diagram (i have been told, but i cant recall the example) and colimits commute with colimits. In the setting i am thinking of cofibrations are the distinguished triangles. So i doubt you will find a counterexample in general since the result is definitely true for at least one triangulated category. Unfortunately, i do not see how this could be extended to other triangulated categories.

It does not seem like a result that would be true in general given my above reasons for believing the result. I will ask about the diagram it is the colimit/homotopy colimit of.

Is there a particular triangulated category you are interested in?

ag.algebraic geometry - When is the push-forward of the structure sheaf locally free

It is of course not true that for any finite morphism $f:Xto Y$ we have $f_*mathcal{O}_X$ locally free : think about a closed immersion.

In fact, your question is about the important topic of "base change and cohomology of sheaves" for proper morphisms, which is treated by Grothendieck in EGA3. The simplest answer one can give, I think, is that if $f$ is proper [EDIT : and flat, as t3suji points out] and for all $yin Y$ we have $H^1(X_y,mathcal{O}_{X_y})=0$ then $f_*mathcal{O}_X$ is locally free.

You may want to avoid going to find the exact reference in EGA3, since as Mumford says, that result is "unfortunately buried there in a mass of generalizations". In that case, go to chapter 0, section 5 of Geometric Invariant Theory (3rd ed.) by Mumford, Fogarty and Kirwan. This is where Mumford's comment is taken from.

at.algebraic topology - How does this geometric description of the structure of PSL(2, Z) actually work?

The key word here is "Bass-Serre Theory" -- using the action on the hyperbolic plane, you can easily cook up a nice action of $PSL_2(mathbb{Z})$ on a tree. This is all described nicely in Serre's book "Trees".

EDIT: Let me give a few more details. It turns out that a group $G$ splits as a free produce of two subgroups $G_1$ and $G_2$ if and only if $G$ acts on a tree $T$ (nicely, meaning that it doesn't flip any edges) with quotient a single edge $e$ (not a loop) such that the following holds. Let $e'$ be a lift of $e$ to $T$ and let $x$ and $y$ be the vertices of $e'$. Then the stabilizers of $x$ and $y$ are $G_1$ and $G_2$ and the stabilizer of $e'$ is trivial.

If you stare at the fundamental domain for the action of $PSL_2(mathbb{Z})$ on the upper half plane, then you will see an appropriate tree staring back at you. There is a picture of this in Serre's book.

EDIT 2: This point of view also explains why finite-index subgroups $Gamma$ of $PSL_2(mathbb{Z})$ tend to be free. If you restrict the action on the tree $T$ to $Gamma$, then unless $Gamma$ contains some conjugate of the order 2 or order 3 elements stabilizing the vertices, then $Gamma$ will act freely. This means that the quotient $T/Gamma$ will have fundamental group $Gamma$. Since $T/Gamma$ is a graph, this implies that $Gamma$ is free.

accretion discs - Stars at near break-up rotation rates

Accretion discs are ubiquitous in astrophisics. As a direct corollary, they are important for the following question.

Consider the following model, representing one of the most simple models for accretion discs. A central object is a star (pre-MS, WD or NS, but not a BH) of mass $M$, surrounded by a thin flat disc of material, which continuosly feeds the star at a rate $dot{M}$, such that $M/dot{M}$ is much larger than thermal and dynamical timescale of the star (i.e. accretion rate is slow).

Everywhere in the accretion disc its local motion is nearly circular and nearly Keplerian. Therefore, at the interface of the star and the disc the disc will always tend to make the star rotate at nearly-Keplerian velocities. From the other hand, if the stellar outer parts were to rotate at nearly-Keplerian velocities, these parts would become gravitationaly detached from the star, which would have significant consequences for the stellar shape and structure. Surely, though, the process is going to be slow and the acquired angular momentum will be redistributed within the star.

Now the question: What will be happening to the star if it approaches nearly break-up velocities due to such a spin-up? This involves a few subquestions: How close the rotation rate can actually get to the critical one? If it can get close enough, how would the whole process look like? That is, what would happen in the short term to the star when the effects of rotation will start to affect its structure? What would happen to the star in the long term?

I would like to keep this problem as a purely hydrodynamical one. That is, assume, that the only laws involved are hydrodynamical and gravitational ones, with some constant accretion rate supported. In reality magnetic fields would also play an important role for some stars, and stellar winds could also possibly be important.

Examples of the decribed systems are numerous. It might concern cataclysmic variables, millisecond pulsars, pre-main sequence star in a protoplanetary disc, and many more.

rotation - If Earth didn't rotate, would we feel heavier?

The centrifugal force at the equator is only 1/289 of normal gravity, or 0.35%. That's too small to be perceived.

http://books.google.com/?id=mE4GAQAAIAAJ&pg=PA358&lpg=PA358

It gradually diminishes as you move away from the equator, and becomes zero at the poles.

In any case, if Earth stopped rotating, the added weight at the equator would be the least of your worries. There would be major perturbations in terms of weather, etc, that would amount to global catastrophe.

Likewise, if Earth's rotation increased, would we feel lighter as
centrifugal force lifts us from the ground?

In theory, yes. But the rotation increase would have to be huge. Again, in that case, there would be other, much bigger effects, that would be much more important than your loss of weight. E.g. the whole planet would bulge at the equator and become more flat at the poles, with incalculable effects on plate tectonics, weather, etc.

nt.number theory - A decision problem concerning polynomial rings

Let $alpha in overline{mathbb{Q}}$ be such that all $f_i$ have coefficients in $mathbb{Q}(alpha)$ and $k in mathbb{N}$ such that the $f_i$ are in
$mathbb{Q}(alpha)[y_1, ldots, y_k]$. Then the ideal the generated by the $f_i$ in this ring corresponds to an ideal $J$ in $mathbb{Q}[x,y_1, ldots, y_k]$ which is generated by lifts of the $f_i$ together with the minimal polynomial $f$ of $alpha.$
Now clearly a neccessary condition for your problem is that $J cap mathbb{Q}[y_1 ldots, y_k]neq {0}.$ This can be checked algorithmically by computing a Groebner basis with respect to an elimination term ordering, c.f. e.g. Kreuzer/Robbiano, Computational Commutative Algebra.

For a complete solution to the original question, you are not interested in the ideal generated by the polynomials but in the subalgebra they generate. Again, sometimes it"s possible to compute the gadgets corresponding to Groebner bases for ideals; these then are called SAGBI bases, which you would need to compute with respect to an elimination term ordering. In contrast to Groebner bases, these do not neccessarily exist, though.

To summarise: SAGBI basis is the notion you're looking for, if I'm not mistaken.

fa.functional analysis - Question about projections on a Hilbert space

Sorry for the vague title, I can't think of a better one that isn't overly long.

Suppose that $S$ is a commuting set of projection operators on a Hilbert space. I'll introduce the following notation: if $p in S$, let $p^+ equiv p$ and $p^- equiv 1 - p$. Let $I equiv ${$+, -$}. The projections are ordered by defining $p leq q$ whenever the range of $p$ is contained in the range of $q$; this makes the set of all projections into a complete lattice. Is the following identity true?

$sup_{f in I^S} inf_{p in S} p^{f(p)} = 1$

In the case where $S$ is finite with elements $p_1, p_2, ldots p_n$, the left hand side of this equation is simply the product over $i$ of $p_i + (1 - p_i)$, so I'm interested in whether this can be generalised to the infinite case. It's easy to see that the following two statements are equivalent to the above:

If $inf_p p^{f(p)} x = 0$ for all $f$, then $x = 0$

If $sup_p p^{f(p)} x = x$ for all $f$, then $x = 0$

but I have no idea how to prove either of these.

My reason for asking is that I'm trying to show that, if $mathcal{H}_1$ and $mathcal{H}_2$ are Hilbert spaces, then if a projection on $mathcal{H}_1 otimes mathcal{H}_2$ is of the form $sup_i p_i otimes q_i$, with $p_i$ and $q_i$ drawn from some complete Boolean algebras of projections on $mathcal{H}_1$ and $mathcal{H}_2$ respectively, then the $q_i$ may be chosen to satisfy $q_i q_j = 0$ when ever $i neq j$. So if anybody knows of an alternative way to prove that, or knows that it's false, then by all means say so.

Singular homology of a graph.

By a graph I will understand an undirected graph without multiple edges or loops. By a morphism of graphs I will understand a map $f$ between the underlying sets of vertices, such that if $x$ and $y$ are adjacent, then $f(x)$ and $f(y)$ are either adjacent or equal.

Let $G$ be a finite graph. One can realise $G$ as a CW-complex $|G|$ and look at topological invariants, such as singular homology. But this captures only very little information about $G$, because except from $H_0(|G|)$ and $H_1(|G|)$ all homology groups are zero.

Consider the following alternative construction: Let us write $Delta_n$ for the complete graph on $n$ vertices, and let us re-baptise this graph by the name "standard $n$-simplex". There are obvious codegeneracy and coface maps between standard simplices, so that we obtain a cosimplicial object $Delta_bullet$ in the category of graphs. Now, proceed as usual: Morphisms $Delta_bullet to G$ form a simplicial set, applying the free group construction yields then simplicial group, and the associated chain complex is the one whose homology $H_i^{mathrm{sing}}(G)$ I shall call "singular homology of $G$".

Obvious properties of $H_i^{mathrm{sing}}(G)$ are: It is a finitely generated commutative group ($G$ is finite), covariantly functorial in $G$. In particular, if we work with coefficients in a field, we obtain representations of the automorphism group of $G$. The homology of the point is $mathbb Z$ in degree $0$ and trivial in higher degrees. We can define singular cohomology accordingly, and get then a natural pairing between homology and cohomology.

The list of all natural questions one must ask after making such a definition is long, so I will not ask everything.

(a) Is there a comparison map $H_i(|G|) to H_i^{mathrm{sing}}(G)$, maybe even on the level of chain complexes? Is there some more elaborate CW-complex $||G||$ one can naturally associate with $G$ such that $H_i(||G||)$ gives back singular homology of $G$? In that case, one would ask for a natural map $|G| to ||G||$.

(b) Given a graph $G$, is there a largest integer $i$ such that $H_i^{mathrm{sing}}(G)$ is nonzero? Assuming yes, is this integer less or equal the size of the largest complete subgraph of $G$.

(c) Is there a Künneth morphism in singular cohomology? --is there a natural ring strucure on cohomology?

(d) What is a homotopy between morphisms of graphs? Given an answer to that, do homotopically equivalent morphisms induce the same maps in homology?

(z) Can you give an example of a graph with nontrivial $H_2^{mathrm{sing}}(G)$?

nt.number theory - Question about polynomials with coefficients in Z

The point of this answer is to point out that Kevin Costello's heuristic can be made rigorous. For any positive $epsilon$, if $y=O(n^{1/2-epsilon})$ then such a polynomial exists for large $n$.

Lemma: Let $G$ be a finite abelian group and let $g_1$, $g_2$, ..., $g_n$ be elements of $G$. If $2^n > |G|$ then there are integers $epsilon_i in { -1, 0, 1 }$, not all zero, such that $sum epsilon_i g_i =0$.

Proof: Consider the $2^n$ sums $sum a_i g_i$ with $a_i in { 0, 1 }$. By the pigeonhole principle, two of these are equal. Subtracting them, we get the claimed relation. QED

Now, consider the abelian group
$$G:=bigoplus_{k=1}^y (mathbb{Z}/k)^{oplus k}.$$
Let $g_i$ be the element of $G$ whose $k$-th component is $(0^i, 1^i, 2^i, 3^i, ldots, (k-1)^i)$, for $i=0$, $1$, ..., $n$. The order of $G$ is $exp( sum k log k) = exp( O(y^2 log y))$. So, if $y=O(n^{1/2-epsilon})$, then $2^{n+1} > |G|$ and the lemma tells us that there are $epsilon_i$ such that $sum epsilon_i g_i=0$. Then $sum epsilon_i x^i$ is the required polynomial.

There is a lot of slack in this argument, but Bjorn's argument shows that we can't improve the exponent of $n$ by tightening it.

stellar evolution - Are stars NML Cygni, UY Scuti, VY Canis Majoris and VV Cephei near the ends of their lives?

So far, I have been able to find information for only three of the stars: NML Cygni, VY Canis Majoris, and VV Cephei.

NML Cygni and VY Canis Majoris

Zhang et al. (2012) considered several values for $T_{text{eff}}$ (effective temperature) and $L$ (luminosity) and found that NML Cygni matches up with the evolutionary track of a star of mass $approx$ 25 M$_odot$, near the HR-diagram location of a similar star, VY Canis Majoris (denoted as VY CMa). Using these data points, the model estimates that NML Cygni is approximately 8 million years old and in the post-main sequence phase of its life.

However, NML Cygni is thought to be related to the Cygnus OB2 association, which has an age of 2-3 million years. This is strange because all the stars in the association should be nearly the same age (see also Knodlseder (2008)). That age may be questionable, however, as evidence for older stars nearby has been presented (see Wright et al. (2010)).

The exact location of VY Canis Majoris on the evolutionary tracks is not certain. Massey et al. (2006) created models that placed it on a lower-mass track to help it avoid the "forbidden zone" beyond the Hayashi limit.

VV Cephei

Based on standard parameters and stellar wind measurements, Bennett (2010) suggested that the red supergiant component of VV Cephei could be near the end of its life, assuming a mass of 20-25 M$_odot$

sg.symplectic geometry - Is a Poisson Group a group object in the category of Poisson Manifolds?

I realized that I am very confused about a certain sign in the definition of a Poisson group. I will give some definitions, and then point out my confusion.

Definitions

Group objects

Let $mathcal C$ be a category with Cartesian products. Recall that a group object in $mathcal C$ is an object $G in mathcal C$ along with chosen maps $e: 1to G$ and $m: Gtimes G to G$ (choose an initial object $1$ and a particular instance of the categorical product, and they imply all the others), such that (i) the two maps $G^{times 3} to G$ agree, (ii) the three natural maps $Gto G$ agree, and (iii) the map $p_1 times m: G^{times 2} to G^{times 2}$ is an isomorphism, where $p_1$ is the "project on the first factor" map $G^{times 2}to G$.

You may be used to seeing axiom (iii) presented slightly differently. Namely, if $p_1 times m: G^{times 2} to G^{times 2}$ is an isomorphism, then consider the map $i = p_2 circ (p_1times m)^{-1} circ (etimes text{id}) : G = 1times G to G$. It satisfies the usual axioms of the inverse map. Conversely, if $i: Gtimes G$ satisfies the usual axioms, then $p_1 times (m circ (itimes text{id}))$ is an inverse to $p_1 times m$. I learned this alternate presentation from Chris Schommer-Pries.

Poisson manifolds

A Poisson manifold is a smooth manifold $M$ along with a smooth bivector field, i.e. a section $pi in Gamma(wedge^2 TM)$, satisfying an axiom. Recall that if $vin Gamma(TM)$, then $v$ defines a (linear) map $C^infty(M) to C^infty(M)$ by differentiating in the direction of $v$. Well, if $pi in Gamma(wedge^2 TM)$, then it similarly defines a map $C^infty(M)^{wedge 2} to C^infty(M)$. The axiom states that this map is a Lie brackets, i.e. it satisfies the Jacobi identity.

A morphism of Poisson manifolds is a smooth map of manifolds such that the induced map on $C^infty$ is a Lie algebra homomorphism.

The category of Poisson manifolds has products (Wikipedia).

Poisson groups

A Poisson Group is a manifold $G$ with a Lie group structure $m : Gtimes G to G$ and a Poisson structure $pi in Gamma(wedge^2 TG)$, such that $m$ is a morphism of Poisson manifolds. Recall that a Lie group $G$ is a group object in the category of smooth manifolds.

Recall also that a Lie group is almost entirely controlled by its Lie algebra $mathfrak g = T_eG$. Then it is no surprise that the Poisson structure can be described infinitesimally. Indeed, by left-translating, identify $TG = mathfrak g times G$. Consider the adjoint action of $G$ on the abelian Lie group $mathfrak g$. Then we can define a Lie group structure $mathfrak g^{wedge 2} rtimes G$ on $wedge^2 TG$. Recall that a section $pi in Gamma(wedge^2 TG)$ is just a manifold map $G to wedge^2 TG$ that splits the projection $wedge^2 TG to G$. Then a Poisson manifold $(G,pi)$ is a Poisson group if and only if $pi : G to wedge^2 TG$ is a map of Lie groups.

Thus, a Poisson group structure is precisely the same as a Lie algebra $dpi : mathfrak g to wedge^2 mathfrak g rtimes mathfrak g$ splitting the obvious projection (here $wedge^2 mathfrak g$ is an abelian Lie algebra, and $mathfrak g$ acts on it via the adjoint action), and such that $(dpi)^* : wedge^2mathfrak g^* to mathfrak g^*$ satisfies the Jacobi identity. (Any failure of $G$ to be simply-connected, which might prevent such a map from lifting, also fails in $wedge^2 TG$, so this really is a one-to-one identification of Poisson group structures on $G$ and "Lie bialgebra" structures on $mathfrak g$.)

From this perspective, then, it is more or less clear that the inverse map $i:Gto G$ is not a morphism of Poisson manifolds (Wikipedia). Indeed, infinitesimally, $di = -1: mathfrak g to mathfrak g$, which takes $dpi$ to $-dpi$ (as $dpi$ has one $mathfrak g$ on the left and two on the right). Instead, $i$ is an "anti-Poisson map". The monoid $(mathbb R,times)$ acts on the category of Poisson manifolds by doing nothing to the underlying smooth manifolds and rescaling the Poisson structures; a smooth map is anti-Poisson if it becomes Poisson after twisting by the action of $-1$.

The unit map $e: 1 to G$, on the other hand, is Poisson; it follows from the axioms of a Poisson group that $pi(e) = 0$, and the terminal object in the category of Poisson manifolds is $1 = {text{pt}}$ with the trivial Poisson structure. ($C^infty({text{pt}}) = mathbb R$ can only support this Poisson structure.)

My question

Suppose that $G$ is a Poisson group. Then $p_1 times m: G^{times 2} to G^{times 2}$ is a Poisson map, and an isomorphism of smooth manifolds. Thus, I would expect that it is an isomorphism of Poisson manifolds. On the other hand, in the first section above I constructed the inverse map $i : Gto G$ out of this isomorphism and the other structure maps, all of which are Poisson when $G$ is a Poisson group. And yet $i$ is not a Poisson map. So where am I going wrong?

pluto - Why are there no more images revealed by New Horizon?

The transmission of the High Priority data set will be complete on July 20, and then image transmission will pause. For nearly two months, until September 14, New Horizons will switch to near-real-time downlinking of data from other, so-called "low-speed" instruments while it transmits just housekeeping information for all of the rest of the data. No new images will arrive on the ground during this time

Models of ZFC Set Theory - Getting Started

This answer is going to be a bit too informal, but I hope it helps.

Imagine we have the collection of all sets. Let us call them the real sets, and their membership relation the real set membership. The empty set is "actually" empty, and the class of all ordinals is "actually" a proper class.

Now that we have the real sets we can use them as the "ontological substratum" upon which everything else will be built from. And this, of course, includes formal theories and their models.

A model of any first-order theory is then only a real set. This applies to your favorite set theory too. So the models of your set theory are only real sets (but the models don't know it, just as they don't know if their empty sets are actually empty or if their set membership is the real one).

This view fits well, for example, with the idea of moving from a transitive model to a generic extension of it or to one with a constructible universe: we are simply moving from a class of models to another one, each one consisting of real sets.

But this view also leaves us with too many entities, and maybe here we have an opportunity to apply Occam's razor. It looks like we have two kind of theories: one for the real sets, which is made of things that are not sets (we can formalize our informal talk about them, but that does not make essentially any difference), another one for the models of set theory, which is made of sets.

The real sets and the theory of the real sets belong to a world where there are real sets, but there are also pigs and cows, and human languages and many other things. We don't need all that to do mathematics, do we? So why not diving into the wold of the real sets and ignore everything else?

If this story sounds too platonistic, I am sure it must have a formalistic counterpart.

With my question:

How to think like a set (or a model) theorist.

I expected to obtain an official view about all this stuff. I somehow succeeded on this, but as you can see, I'm still working on it.

Here is a related answer to a related question which I also find useful:

Is it necessary that model of theory is a set?

A Game on a Finite Projective Plane

See "Tic-Tac-Toe on a Finite Plane", Maureen T. Carroll and Steven T. Dougherty,
Mathematics Magazine, Vol. 77, No. 4 (Oct., 2004), pp. 260-274. (Preprint here: http://academic.scranton.edu/faculty/carrollm1/tictac.pdf) The second player can force a draw on the 3-by-3 and 4-by-4 projective planes.

nt.number theory - Polynomial representing all nonnegative integers

This is a cute problem! I toyed with it and didn't really get anywhere - I got the strong impression that it requires fields of mathematics that I am not expert in.

Indeed, given that the problem seems related to that of counting integer solutions to the equation $f(x,y) = c$, one may need to use arithmetic geometry tools (e.g. Faltings' theorem). In particular if we could reduce to the case when the genus is just 0 or 1 then presumably one could kill off the problem. (One appealing feature of this approach is that arithmetic geometry quantities such as the genus are automatically invariant (I think) with respect to invertible polynomial changes of variable such as $(x,y) mapsto (x,y+P(x))$ or $(x,y) mapsto (x+Q(y),y)$ and so seem to be well adapted to the problem at hand, whereas arguments based on the raw degree of the polynomial might not be.)

Of course, Faltings' theorem is ineffective, and so might not be directly usable, but perhaps some variant of it (particularly concerning the dependence on c) could be helpful. [Also, it is overkill - it controls rational solutions, and we only care here about integer ones.] This is far outside of my own area of expertise, though...

The other thing that occurred to me is that for fixed c and large x, y, one can invert the equation $f(x,y) = c$ to obtain a Puiseux series expansion for y in terms of x or vice versa (this seems related to resolution of singularities at infinity, though again I am not an expert on that topic; certainly Newton polytopes seem to be involved). In some cases (if the exponents in this series expansion are favourable) one could then use Archimedean counting arguments to show that f cannot cover all the natural numbers (this is a generalisation of the easy counting argument that shows that a 1D polynomial of degree 2 or more cannot cover a positive density set of integers), but this does not seem to work in all cases, and one may also have to use some p-adic machinery to handle the other cases. One argument against this approach though is that it does not seem to behave well with respect to invertible polynomial changes of variable, unless one works a lot with geometrical invariants.

Anyway, to summarise, it seems to me that one has to break out the arithmetic geometry and algebraic geometry tools. (Real algebraic geometry may also be needed, in order to fully exploit the positivity, though it is also possible that positivity is largely a red herring, needed to finish off the low genus case, but not necessary for high genus, except perhaps to ensure that certain key exponents are even.)

EDIT: It occurred to me that the polynomial $f(x,y)-c$ might not be irreducible, so there may be multiple components to the associated algebraic curve, each with a different genus, but presumably this is something one can deal with. Also, the geometry of this curve may degenerate for special c, but is presumably stable for "generic" c (or maybe even all but finitely many c).

It also occurs to me that one use of real algebraic geometry here is to try to express f as something like a sum of squares. If there are at least two nontrivial squares in such a representation, then f is only small when both of the square factors are small, which is a 0-dimensional set and so one may then be able to use counting arguments to conclude that one does not have enough space to cover all the natural numbers (provided that the factors are sufficiently "nonlinear"; if for instance $f(x,y)=x^2+y^2$ then the counting arguments barely fail to provide an obstruction, one has to use mod p arguments or something to finish it off...)

EDIT, FOUR YEARS LATER: OK, now I know a bit more arithmetic geometry and can add to some of my previous statements. Firstly, it's not Faltings' theorem that is the most relevant, but rather Siegel's theorem on integer points on curves - the enemy appears to be those points $(x,y)$ where $x,y$ are far larger than $f(x,y)$, and Siegel's theorem is one of the few tools available to exclude this case. The known proofs of this theorem are based on two families of results in Diophantine geometry: one is the Thue-Siegel-Roth theorem and its variants (particularly the subspace theorem), and the other is the Mordell-Weil theorem and its variants (particularly the Chevalley-Weil theorem). A big problem here is that all of these theorems have a lot of ineffectivity in them. Even for the very concrete case of Hall's conjecture on lower bounding $|x^2-y^3|$ for integers $x,y$ with $x^2 neq y^3$, Siegel's theorem implies that this bound goes to infinity as $x,y to infty$, but provides no rate; as I understand it, the only known lower bounds are logarithmic and come from variants of Baker's method.

As such, a polynomial such as $f(x,y) = (x^2 - y^3 - y)^4 - y + C$ for some large constant C already looks very tough to analyse. (I've shifted $y^3$ here by $y$ to avoid the degenerate solutions to $x^2=y^3$, and to avoid some cheap way to deal with this polynomial from the abc conjecture or something.) The analogue of Hall's conjecture for $|x^2-y^3-y|$ suggests that $f(x,y)$ goes to $+infty$ as $x,y to infty$ (restricting $x,y$ to be integers), but we have no known growth rate here due to all the ineffectivity. As such, we can't unconditionally rule out the possibility of an infinite number of very large pairs $(x,y)$ for which $x^2-y^3-y$ happens to be so close to $y^{1/4}$ that we manage to hit every positive integer value in $f(x,y)$ without hitting any negative ones. However, one may be able to get a conditional result assuming some sufficiently strong variant of the abc conjecture. One should also be able to exclude large classes of polynomials $f$ from working; for instance, if the curve $f(x,y)=0$ meets the line at infinity at a lot of points in a transverse manner, then it seems that the subspace theorem may be able to get polynomial bounds on solutions $(x,y)$ to $f(x,y)=c$ in terms of $c$, at which point a lot of other tools (e.g. equidistribution theory) become available.

Another minor addendum to my previous remarks: the generic irreducibility of $f(x,y)-c$ follows from Bertini's second theorem, as one may easily reduce to the case when $f$ is non-composite (not the composition of two polynomials of lower degree).

cosmology - How exactly can the hypothetical conformal invariance of the CMB spectrum be established by analyzing tensor modes?

In the introduction of this paper at the top of p11, it is mentioned that a hypothetical enhancement of the scale invariance of the CMB spectrum to conformal invariance could potentially be established by analyzing the higher n-point functions of the tensor modes.

First of all, why are the tensor modes needed for this? Does conformal invariance not appear in the scalar modes?
How does this work exactly (mathematically), how does conformal invariance (not just scale invariance...!) manifest itself in the n-point functions of the tensor modes?

Has this already been looked at or will it be done for the tensor modes in the BICEP2 data for example?

big list - Facts from algebraic geometry that are useful to non-algebraic geometers

Just have a look at the XIXth century. Say that you look for a primitive of an algebraic expression. The general question is whether this primitive can be written in terms of elementary functions (rational fraction and logarithms). The algebraic expression is usually associated with some algebraic curve. The answer is yes iff the curve admits a rational parametrization. When it is non-singular, this is equivalent to having genus $0$.

For instance, if $R$ is rational, then
$$int Rleft(x,sqrt{x^2+ax+b}right)dx$$
can be expressed in terms of elementary functions. On the contrary,
$$int sqrt{x^3+ax+b},dx$$
cannot, unless the polynomial $x^3+ax+b$ has a double root.

A more advanced situation is that of hyperbolic linear Partial Differential Equations. The differential operator defines a symbol, which is a polynomial in several variables. The properties of its zero set, an algebraic variety, are crucial in many aspects, for instance in determining whether Huyghens principle holds (theory of lacunas). In the Russian school, prominent researchers in PDE were also active in algebraic geometry (Petrovski, Oleinik).

A definitely more advanced situation is the use of algebraic geometry in the analysis of linear initial-boundary value problems. Let $L$ be a differential operator, for which the Cauchy problem is well-posed. A necessary condition for an IBVP to be well-posed in ${mathcal C}^infty$ is the so-called Lopatinskii Condition, which is algebraic and parametrized by frequencies (along boundary and time). If one replaces ${mathcal C}^infty$ by a Sobolev space $H^s$, then the Lopatinskii condition has to be satisfied uniformly. In several interesting cases, LC or ULC condition turns out to be sufficient for well-posedness, but this requires the construction of a so-called dissipative symmetrizer, which relies upon algebraic geometry. For hyperbolic operators, see the work of H.-O. Kreiss (ULC) and the books by R. Sakamoto (LC) or by S. Benzoni-Gavage and myself (ULC).

soft question - Most interesting mathematics mistake?

Cantor's been mentioned, but I think the lessons there should be different. First, the really big mistake was that of highly-reputed academics (including, I believe, Poincare, Kronecker and even Wittgenstein) who rejected his ideas. And (related) second, even in a wiki devoted to mistakes it seems somewhat carping to fault Cantor for failing to spot a subtlety without at the same time adequately crediting his genius.

Somewhat along the same lines, one might mention Fourier's difficulties in getting his ideas accepted.

ag.algebraic geometry - Kähler manifold which is not algebraic

It doesn't in itself give specific examples, but the theoretical answer to the question "When is a compact Kahler manifold algebraic?" is given by the Kodaira embedding theorem. A nice exposition is given in Richard Wells' Differential Analysis on Complex Manifolds. Indeed, the KET is the crescendo to which the entire book builds.

co.combinatorics - Sum of $n$ vectors in $(mathbb Z/n)^k$

There was some discussion of the case n=3 in sci.math around 1994. There is a card game called Set with an 81 card deck so that each card is naturally a point in $(mathbb Z/3)^4$. Several cards are dealt out, and your task is to identify triples of cards called Sets which form a line, or equivalently, which add up to the 0 vector. A natural question is how many points you can deal out without the existence of a line. It's not too hard to construct 9 distinct points in affine 3-space, or 20 distinct points in affine 4-space over $mathbb Z/3$ so that there is no line contained in the points, and these are the maximums. These correspond to $N=19$ for $(n,k) = (3,3)$ and $N=41$ for $(n,k) = (3,4)$, as in the reference Ricky Liu linked, by repeating each point twice.

The maximal configurations are highly symmetric. The 9 points in dimension 3 correspond to a nondegenerate conic, which is unique up to symmetry. The 20 points in dimension 4 actually correspond to a nondegenerate conic containing 10 points in projective 3-space viewed as lines passing through the origin in affine 4-space.

For example, there are 9 points in dimension 3 satisfying $z=x^2 + y^2:$
${(0,0,0),(pm1,0,1),(0,pm1,1),(pm1,pm1,-1)}$
and this set contains no lines.

quantum field theory - Mathematics of path integral: state of the art

First, there are several rigorous definitions of integration in infinite dimensional spaces, like the Bochner integral in Banach spaces (see Wikipedia), or see the book by Parthasarathy: "Probability measures on metric spaces" (this includes the Gaussian probability measures used by constructive QFT already mentioned).

These cannot be used to make the Feynman path integral into a rigorous defined mathematical entity with finite values, i.e. the problem is to get an integral that spits out finite numbers in physically interesting models.

For starters, there cannot be a translationally invariant measure (on the Borel sigma algebra) other than the one that assigns infinite volume to every open set in an infinite dimensional metric space (hint: a ball of radius r contains infinitly many pairwise disjunct copies of the ball of radius r/2). So the path integral, as it is written by physicists, certainly has no interpretation via a translationally invariant measure, contrary to what the notation usually employed may suggest.

While there currently is no mathematically rigorous definition of a Feynman path integral applicable to an interesting subset of physical models, here are some books that give some hints at the current state of the affair:

Huang and Yan: "Introduction to Infinite Dimensional Stochastic Analysis" (this contains a description of the Feynman path integral from the viewpoint of "white noise analysis"),

Sergio Albeverio, Raphael Hoegh-Krohn; Sonia Mazzucchi:"Mathematical theory of Feynman path integrals. An introduction",

Pierre Cartier, Cecile DeWitt-Morette: "Functional integration: action and symmetries".

BTW: This is in a certain sense a "one million dollar" question because one of the millenium problems of the Clay Mathematics Institute is a rigorous construction of Yang-Mills theories.

gr.group theory - Automorphisms of the totally ordered group Z^n with lexicographical order

Yes. Any group homomorphism fixes zero. Now, look at the set of things greater than zero. If the homomorphism is order preserving, then it must take the least thing there to the least thing in the image. However, as the image is surjective, it must then be fixed. Then, inductively, everything larger than zero is fixed. A similar argument works for things less than zero. In fact, this proof appears to merely require a totally ordered group, and not the lexicographic ordering, so this should work for any monomial ordering (that is, total order on Z^n.

mathematical writing - Using TikZ in papers

It is not exactly answering either of your two questions, but here is another work-around. I had problems putting papers on the arxiv which used pgf/tikz because the version of pgf/tikz they used at the arxiv was not as up to date as my version. The admin at the arxiv told me to do the following. LaTeX your file with the option -recorder. This will create a .fls file containing a list of all of the FiLeS used by LaTeX when typesetting your document. Choose all of the files in the list containing "pgf" or "tikz" and move them into the directory containing your document. You can then send that directory to your collaborator/the arxiv/the journal without worrying about how up to date their set up is.

[Unfortunately, I then had the problem that the tikz graphics I produced required an enormous amount of working memory which was greater than the allocation on the arxiv server, so I resorted to using 'grab' on my mac to take a high resolution snap-shot of the graphic, which I then incorporated into the LaTeX file :-(. However, I have used this including-all-the-files technique for subsequent uploads to the arxiv.]

Diagonalization of quadratic forms over euclidean rings

Let $A$ be a commutative euclidean ring, (probably) with 2 a unit in $A$. I'm trying to compute Witt and Grothendieck-Witt rings and since $A$ is a PID any f.g.p. module over it is free so I only need think about forms on $A^n$.

Question: Is every (non-degenerate) quadratic form over $A$ diagonalizable?

A form $q$ is diagonalizable we can perform a base change on $A^n$ such that the matrix for $q$ becomes diagonal. Edit: Non-degenerate here means that any matrix associated to the form is invertible.

From Milnor-Husemoller's book I know this is true if $A$ is local. If I can show that any non-degenerate quadratic form on $A^n$ represents some unit ($q(x) = u$ a unit for some $x in A^n$) then the statement holds by cor. I.3.3 in Mil-Hus.

big list - Examples of common false beliefs in mathematics

$pi$ is equal to 22/7.

This was touched upon in the comments to a totally unrelated answer but I think this false belief is important enough to warrant its own answer (and as far as I could tell it does not have one yet, my apologies if I overlooked one.)

Of course, it's unlikely anyone on this site believes this, or ever believed it, which is why I think it's important to insist on this: it does not really resonate with us, we are unlikely to warn students against it, yet we probably see in front of us many students who have that false belief and then will move on to spread it around.

A Piece of Evidence

Let me offer as evidence this gem taken off the comments section of an unrelated (but quite thought-provoking) article on Psychology Today, of all places! When Less is More: The Case for Teaching Less Math in Schools (The title is a misnomer, it's a case for starting math later, but I think that with such a scheme you should be able to teach more math overall; anyway, read it for yourselves.)

Some years ago, my (now ex-) wife was involved in a "trivia night" fundraiser at her elementary school, and they wanted me on their "teacher team" to round out their knowledge. They had almost everything covered except some technology-related topics and I was an IT guy. In round four, my moment to shine arrived, as the category was "Math & Science" and one of the questions was, "give the first five digits of pi." I quickly said, "3.1415." The 9 teachers at the table ignored me and wrote down "22/7" on scrap paper and began to divide it out. I observed this quietly at first, assuming that 22/7ths gave the right answer for the first 5 digits, but it doesn't. It gives something like 3.1427. I said, "Whoops, that won't work." They ignored me and consulted among themselves, concluding that they had all done the division properly on 22/7ths out to five digits. I said, "That's not right, it's 3.1415."
[...]

I'm cutting it off here because it's a long story:
hilarity ensues when the non-teacher at the table stands up for the truth (when he finds out that the decimals of 22/7 were the expected answer!) The final decision of the judges:

"We've got a correction on the 'pi' question, apparently there's been confusion, but we will now be accepting 3.1415 as a correct answer as well" [as 3.1427].

The Moral of the Story

I used to dismiss out of hand this kind of confusion: who could be dumb enough to believe that $pi$ is 22/7? (Many people apparently: in the portion of the story I cut was another gem - "I'm sorry, but I'm a civil engineer, and math is my job. Pi is 22/7ths.")

Now, I treat this very seriously, and depending on where you live, you should too. Damage wrought during the influential early years is very hard to undo, so that the contradictory facts "$pi$ is irrational" and "$pi$=22/7" can coexist in an undergraduate's mind. And when that person leaves school, guess which of the two beliefs will get discarded: the one implanted since childhood, or the one involving a notion (rational numbers) which is already getting fuzzy in the person's brain? I'm afraid it's no contest there, unless this confusion has been specifically addressed.

So if you have any future teachers in your classes (and even if you don't, cf. the civil engineer above), consider addressing this false belief at some point.

planet - Do planetary surface temperatures change in unison in a solar system?

The simple answer to your question is yes. Taking a simplified equation from Carroll & Ostlie, An Introduction to Modern Astrophysics Second Edition, the temperature of a planet can be estimated as:
$$
T_{p} = T_{odot}(1-a)^{frac{1}{4}}sqrt{frac{R_odot}{2D}}
$$
Where $T_p$ is the predicted temperature of a planet in a circular orbit of radius $D$ with an albedo of $a$ around a star with a temperature of $T_odot$ and a radius of $R_odot$. If the energy output of the star were to increase, raising $T_odot$, then there would be a corresponding increase in the temperature of all planets orbiting said star.

In practice there are factors which can make this correlation difficult to measure. The albedo of a planet during the course of a day can vary greatly and the distance of a planet from the host star changes throughout the year. This equation also assumes that the planet is a perfect black body which most are not which can also change a planets temperature and obscure any changes caused by the host star.

gr.group theory - Countable open subgroup

I believe the question the poster is trying to ask is, "Why is theorem 3.6 of the article http://link.springer.com/article/10.1023%2FA%3A1010466924961#page-1 true?" Certainly this seems to be something the OP cares about, so I'll address it. (I think this question is borderline for MO, since the answer seems to be rather trivial - feel free to downvote this answer if you think the question is definitely inappropriate, although please say that's why you're downvoting.)

Definition (1.1 in the cited paper): If $G$ is a group, a $T$-sequence $alpha=langle a_nrangle_{ninomega}$ is a sequence of elements of $G$ which converges to 0 (the authors say "vanishes;" I presume that's what this means) in some non-discrete topology on $G$. A topology $tau$ on $G$ is determined by $alpha$ if $tau$ is a maximal topology in which $alpha$ converges to 0.

The theorem the OP is asking about is:

Theorem (3.6): If $tau_1$ is a topology on an infinite group $G$ determined by some $T$-sequence,then $tau_1$ is complemented by some topology $tau_2$ also determined by a $T$-sequence.

The proof of this theorem, in its entirety, is:

Note that $(G, tau_1)$ has a countable open subgroup. Now, apply Theorems 1.6 and 3.5 and Lemma 2.3.

The part the OP seems to be asking about is the first sentence. The key is that in the theorem's hypothesis the topological group $(G, tau_1)$ is assumed to be generated by some $T$-sequence $alpha$. The reason this matters is that if $(G, tau_1)$ is determined by $alpha$, then clearly $Acuplbrace 0rbrace$, where $A$ is the underlying set of $alpha$, must be open - since $tau_1$ is maximal among the topologies in which $alpha$ converges to 0. Now the desired countable open subgroup is just the group generated by $A$.