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https://mathoverflow.net/questions/75026 | 13 | Let $E\_4(q)=1+ 240q+ 2160q^2+ 6720q^3+\ldots $ be the [Eisenstein series](http://en.wikipedia.org/wiki/Eisenstein_series#Products_of_Eisenstein_series) of weight 4,
also known as the theta-series of the $E\_8$-lattice.
I'm looking for a $\mathbb N$-graded vector space $V$ of graded dimension
$(1, 240, 2160, 6720,\ld... | https://mathoverflow.net/users/5690 | I'm looking for a Virasoro-module whose character is 1+ 240q+ 2160q^2+ 6720q^3... | What I was asking for cannot be realized... for a simple and stupid reason:
The coefficients of the Verma module of the Virasoro algebra are given by the partition function $p(n)=\{1,1,2,3,5,7,11,...\}$.
Now, the growth of the partition function is given by
$$p(n) \sim \frac {1} {4n\sqrt{3}} e^{\pi \sqrt {\frac{2n}{3... | 12 | https://mathoverflow.net/users/5690 | 75292 | 45,729 |
https://mathoverflow.net/questions/75288 | 6 | Suppose a continuous map $f : S^2 \rightarrow S^2$ verifies :
$$ f(x) - f(-x) = 2 \left(f(x),x\right)x $$
where $\left(x,y\right)$ is the scalar product in $R^3$. An equivalent way of expressing $f$ could be :
$$ f(x) = X(x) + u(x)x $$
where $X$ is a tangent vector field of $S^2$ and $u$ a real-valued function with :
$... | https://mathoverflow.net/users/17767 | Degree of maps on the sphere with a property of symmetry | First, for any $x\in S^2$ we have an endomorphism $A(x)$ of $\mathbb{R}^3$ given by $A(x)(w)=x\times w$. More generally, we have an orthogonal matrix $B(t,x)=\exp(t A(x))$, which is a rotation through angle $t$ around $x$. When $w$ is perpendicular to $x$ we have $B(\pi/2,x)(w)=A(x)(w)$.
Let $F$ be the space of maps ... | 11 | https://mathoverflow.net/users/10366 | 75297 | 45,731 |
https://mathoverflow.net/questions/75301 | 1 | Do you know any good reference on Weil bounds for character sums over algebraic curves?
I prefer references which assume little previous knowledge.
| https://mathoverflow.net/users/4246 | Weil bound for characters sums | Yes, check out Chapter 8 of this paper:
*Chen, Xi; Kayal, Neeraj; Wigderson, Avi*, [**Partial derivatives in arithmetic complexity and beyond**](https://doi.org/10.1561/0400000043), Found. Trends Theor. Comput. Sci. 6, No. 1–2, 1–138 (2010). [Zbl 1278.68010](https://zbmath.org/1278.68010). [PDF](https://www.math.ias.... | 2 | https://mathoverflow.net/users/11142 | 75304 | 45,734 |
https://mathoverflow.net/questions/75277 | 1 | Can anyone give a polynomial reduction from computing the permanent of a matrix into counting the satisfying assignments for a 2SAT problem?
| https://mathoverflow.net/users/17765 | #P Complexity Question | See [The Complexity of Enumeration and Reliability Problems](http://dx.doi.org/10.1137/0208032), by Valiant. This is Problem number 7 in Valiant's list.
| 3 | https://mathoverflow.net/users/297 | 75315 | 45,739 |
https://mathoverflow.net/questions/75321 | 2 | Bonsoir/bonjour à toutes et à tous.
The title has it all, but... We know (as a consequence of the Eberlein-Šmulian theorem) that any bounded sequence, $\{x\_n\}\_{n \;\! \in \;\! \mathbb{N}}$, in a (real or complex) reflexive normed (and hence Banach) space, $\mathbf{X} \equiv (X, \|\cdot\|)$, contains a *weakly* con... | https://mathoverflow.net/users/16537 | Sufficient conditions to the existence of a weakly convergent subsequence from a Cauchy sequence in a (merely) normed space | Your condition (2) is that $T^\*$ is a surjective isomorphism, so $T$ induces a surjective isomorphism on the completion of $X$. For a counterexample, let $T$ be the right shift on $\ell\_2(Z)$ restricted to an appropriate dense subspace. What subspace? Well, it must be dense, so throw in the unit vector basis. Throw i... | 1 | https://mathoverflow.net/users/2554 | 75347 | 45,753 |
https://mathoverflow.net/questions/75213 | 13 | The title refers, of course, to Matthew (2:12) *''And being warned in a dream not to return to Herod, they departed to their own country by another way''*. To be honest, it is not that specific particular case I'm more interested in.
I'd like to have a reference, or a hint here, for a simple proof of the following f... | https://mathoverflow.net/users/6101 | Including a Jordan arc into a Jordan loop (Can the Magi go home by another way?) | Yes they can! Every Jordan arc can be included in a Jordan loop. I'll give you the simplest proof I could think of, which only uses very basic topology. It also shows that the lemma you propose does indeed hold, although I'll skip the quantitative aspect. [I see that Jim Conant linked to a very similar question in a co... | 7 | https://mathoverflow.net/users/1004 | 75350 | 45,756 |
https://mathoverflow.net/questions/75300 | 1 | Selberg trace formula <http://en.wikipedia.org/wiki/Selberg_trace_formula> says
$$\sum\_nh(r\_n)=...$$
where $1/4+r\_n^2$ is the eigenvalue of the Laplacian.
My question is "what is the geometric exlication for the transformation $r\_n\to r\_n^2+1/4$?"
Or it just makes the formula beautiful.
| https://mathoverflow.net/users/16326 | Is there some explication for the transformation of the eigenvalues in Selberg trace formula | This transformation only plays a role in the "trivial K-type" formulation of the trace formula, which is only special case of the whole story.
One considers test-functions in the functional calculus of the Laplace-Beltrami operator.
The trace formula describes the trace of a convolution operator on the group level, so ... | 1 | https://mathoverflow.net/users/nan | 75351 | 45,757 |
https://mathoverflow.net/questions/75355 | 19 | A groupoid is a category in which all morphisms are invertible.(\*) The groupoids form a very nice subclass of categories. The inclusion of the groupoids into the 2-category of small categories admits both left and right (weak) adjoints. So you can localize (or *complete*) a category to a groupoid. If E denotes the fre... | https://mathoverflow.net/users/184 | Characterizing Groupoids via Quotients? | I believe the following is an example of a category whose leanification is discrete but which is not a groupoid. It should be possible to simplify it. There may be details to work out. Let $B$ be the bicyclic monoid. It is generated by $a,b$ subject to the relations $ab=1$. (Note $ba\neq 1$). Let $K(B)$ be the Karoubi ... | 11 | https://mathoverflow.net/users/15934 | 75359 | 45,762 |
https://mathoverflow.net/questions/75329 | 21 | Let $A \to B$ be a map of commutative rings, and $d : B \to I/I^2$ be
defined by $df = f\otimes 1 - 1\otimes f$, where $I$ is the kernel of
$B \otimes\_A B \to B$, as in [Hartshorne II.8].
>
> If $df=0$, I would like to infer that $f \in A$, i.e. "if the
> derivative is zero, the function is constant".
>
>
>
... | https://mathoverflow.net/users/391 | When does the relative differential $df=0$ imply that $f$ comes from the base? | I will rather regard $I/I^2$ as $\Omega\_{B/A}$, the module of differential forms.
First some necessary conditions. If $D$ is a sub-$A$-algebra of $B$ such that $\Omega\_{D/A}=0$ (e.g. $D$ is a localization of $A$ or étale over $A$), the canonical map $\Omega\_{D/A}\otimes\_D B\to \Omega\_{B/A}$ shows that $df=0$ fo... | 30 | https://mathoverflow.net/users/3485 | 75365 | 45,764 |
https://mathoverflow.net/questions/74989 | 6 | Consider the following family of polynomials in $K[x,y]$, where $K$ has characteristic zero:
$f\_n(x,y)=(x+y)^n+(x-1)y^n,$
for $n\geq 3$. I can prove that $f\_n(x,y)$ has an irreducible factor of degree $n-1$ in $x$. I also know that the galois group of $f\_n$ over $K(y)$ is the symmetric group of degree $n-1$, but... | https://mathoverflow.net/users/2189 | A family of polynomials with symmetric galois group | [Edited mostly to incorporate references etc. from Michael Zieve]
The polynomial $(x^n-1)/(x-1) - y$ has Galois group $S\_{n-1}$ over ${\bf C}(y)$ for each $n$, as expected. This answers one of the three problems; the question statements asserts that they are equivalent, which doesn't seem to be the case (see my comm... | 18 | https://mathoverflow.net/users/14830 | 75367 | 45,765 |
https://mathoverflow.net/questions/68247 | 5 | Let l=2m+1 be prime. In my previous MO question, "What are the polynomial relations between these characteristic 2 thetas?", I defined a subring of Z/2[[x]] as follows:
The subring, S, is generated by [1],...,[m] where [i] is the sum of the x^(n^2), n running over all integers congruent to i mod l.
QUESTION...... L... | https://mathoverflow.net/users/6214 | Existence of certain identities involving characteristic 2 "thetas" | I suppose it's bad form to answer one's own MO question, but I now have an almost complete solution to this one. I can prove:
1.----H is always in the ring S generated by the [j].
2.----The same holds for G except perhaps when l=15 mod 16. (In "More questions involving characteristic 2 theta series identities" I pr... | 3 | https://mathoverflow.net/users/6214 | 75368 | 45,766 |
https://mathoverflow.net/questions/75371 | 6 | I am looking for an elementary evaluation (if one exists) of the exponential sum
$$
G\_r(a,b) = \sum\_{x \in \mathbb{F}\_{2^r}} \psi(ax^2 + bx),
$$
where $a,b \in \mathbb{F}\_{2^r}^\*$ are both units, $\psi(x) = e(Tr(x)/2)$ and $Tr : \mathbb{F}\_{2^r} \to \mathbb{F}\_2$ is the usual field Trace map
$$
Tr(x) = \su... | https://mathoverflow.net/users/15212 | Exponential sums over finite fields with even characteristic | Trace is additive, and ${\rm Tr}(u)={\rm Tr}(u^2)$ for all $u$, so $ax^2+bx$ has the same trace as $(a+b^2)x^2$. Therefore the sum is $2^r$ if $a=b^2$ and zero otherwise.
In general, for a polynomial $P(x)$ over the field of $2^r$ elements, the sum of $\psi(P(x))$ is $2^r$ less than the number of affine points on the... | 20 | https://mathoverflow.net/users/14830 | 75375 | 45,767 |
https://mathoverflow.net/questions/75369 | 4 | Let $S\_{n,r}$ denote the Stirling number of the second kind. Define $A\_{n,r}:=\frac{\binom{n+r-1}{n}(n+r)!}{S\_{n+r,r}r!}$. I want to prove:
$A\_{n,1}\ge A\_{n,2}\ge..\ge A\_{n,r}\ge \lim\_{r\to\infty} A\_{n,r}=2^n$.
I can prove this for some small particular $r$ or for some small particular $n$.
And checked this... | https://mathoverflow.net/users/2900 | A bound involving Stirling numbers of the second kind and the asymptotics | ($A\_{n,r}$ has a combinatorial interpretation: the denominator is the number of ways of partitioning the numbers $\{1,\ldots,n+r\}$ into $r$ blocks, where the list of blocks is ordered but the numbers within each block are unordered, and the numerator is the same count, except that the numbers within each block are no... | 8 | https://mathoverflow.net/users/17657 | 75379 | 45,768 |
https://mathoverflow.net/questions/75370 | 7 | Hi, all
I am working on an algorithm which uses Lanczos method to compute K smallest eigenvalue(and their eigenvectos) of a sparse matrix, just want some information or links about the complexity of Lanczos method.
Thank you
| https://mathoverflow.net/users/16675 | the complexity of Lanczos method | Complexity analysis of Lanczos seems to be hard to find in the literature. Here are two leads, that might help a bit.
For the largest eigenvalue, you might find the complexity analysis in the following paper to be useful.
[Estimating the Largest Eigenvalue by the Power and Lanczos Algorithms with a Random Start](ht... | 8 | https://mathoverflow.net/users/8430 | 75382 | 45,769 |
https://mathoverflow.net/questions/75341 | 9 | I need a solution to this problem (which is really a calculus problem) in order to prove a rigidity result for open nonnegatively curved manifolds with odd-dimensional souls:
Suppose that $f,g:\mathbf{R}\rightarrow \mathbf{R}$ are smooth functions. Assume that $f(0)=0$ and that $g(t)$ has a global maximum at $t=0$. A... | https://mathoverflow.net/users/17779 | A differential inequality needed to prove a theorem about odd-dimensional souls | We may assume $g(0)=g'(0)=0$.
For $t\in(0,1)$, we have
$$
\int\_0^t (f(s)^2+g''(s))\,ds \geq \int\_0^t f'(s)^2\,ds
\geq \left(\int\_0^t f'(s)\,ds\right)^2 \left(\int\_0^t 1^2\,ds\right)^{-1}
=f(t)^2/t\geq f(t)^2,
$$
hence $g'(t)\geq f(t)^2-\int\_0^t f(s)^2\,ds$. Let $h(t)=\int\_0^t f(s)^2\,ds$; we have $g'(t)\ge... | 16 | https://mathoverflow.net/users/17581 | 75384 | 45,771 |
https://mathoverflow.net/questions/73864 | 6 | It is well-known that the modal logic S4 is complete with respect to the class of all finite quasi-trees (where we interpret the $\Box$ modality as topological interior, and topologize a quasi-tree with the up-set topology). It is also well-known that p-morphisms (open, continuous surjections) preserve modal validity. ... | https://mathoverflow.net/users/16939 | A necessary condition for S4-completeness? | S4 is complete with respect to a Kripke frame or general frame or topological frame $F$ if and only if $F$ is an S4-frame and for every finite rooted S4-frame $G$, there exists a p-morphism of a generated subframe of $F$ onto $G$.
The left-to-right implication follows from the existence of Fine’s frame formulas: ther... | 3 | https://mathoverflow.net/users/12705 | 75401 | 45,779 |
https://mathoverflow.net/questions/75393 | 14 | Let $X$ and $Y$ be regular integral Noetherian schemes. Assume that $X$ and $Y$ are smooth and proper over a base scheme $S=Spec R$, where $R$ is a discrete valuation ring.
If $X$ and $Y$ have isomorphic generic fibres, is it also the case that their special fibres are isomorphic?
Remarks:
1. The answer is yes wh... | https://mathoverflow.net/users/5101 | Does isomorphic generic fibre imply isomorphic special fibre for smooth morphisms? | Here is an example showing the answer is no:
Start with $Z =\mathbb{P}^2\_R$, $R$ an arbitrary dvr. Let $P$ be a section of $Z \to Spec(R)$ and let $W$ be the blowup of $Z$ along the image of the section (so both fibres are $\mathbb{P}^2$ with a point blown up). Let $Q$ be a section of $W \to Spec(R)$ whose image doe... | 17 | https://mathoverflow.net/users/519 | 75405 | 45,780 |
https://mathoverflow.net/questions/75403 | 1 | I've encountered this problem, where I know everything in site (no pun intended) to be locally of finite type over my ground field, but I really need quasi-compactness.
Say I have two morphisms $a: X \to Y$ and $b: Y \to X$ such that $ba = id$ and say I know thaht $Y$ is of finite type, can I say anything about $X$?
... | https://mathoverflow.net/users/16857 | If X -> Y -> X is the identity and Y is of finite type, can I say anything about X? | Yes, it follows easily that if $Y$ is of finite type then $X$ is of finite type and one only needs the surjectivity of $b$:
Let $\{U\_{\alpha}\}\_{\alpha \in A}$ be an affine open cover of $X$, so $\lbrace b^{-1}(U\_{\alpha})\rbrace\_{\alpha \in A}$ is an open cover of $Y$. Since $Y$ is of finite type, hence quasi-co... | 5 | https://mathoverflow.net/users/519 | 75407 | 45,781 |
https://mathoverflow.net/questions/75071 | 7 | The set of all smooth maps $S^1\to M^n$ ($M$ is a smooth manifold) is a generalized manifold(see <http://ncatlab.org/nlab/show/smooth+loop+space>).
I was wondering if the set of singular loops (maps with selfcrossings or zeros of derivative) is a (Fréchet,Frolicher,diffeological)submanifold of loop space?
EDIT: it ... | https://mathoverflow.net/users/4298 | I was wondering if the set of singular loops is a (somewhere) submanifold of loop space? | Let $L(M)$ be the space of smooth maps from $S^1$ to $M$.
Let $D(M)$ be the subspace of $L(M)$ consisting of immersions $f : S^1 \to M$ where $f$ admits a unique pair of points $p,q \in S^1$, $p \neq q$ with $f(p)=f(q)$.
My understanding of your question, is you want to know if $D(M) \subset L(M)$ is a submanifol... | 4 | https://mathoverflow.net/users/1465 | 75427 | 45,790 |
https://mathoverflow.net/questions/75332 | 5 | This question is motivated by the following naive one: suppose we have a nice subset $X$ of some Euclidean space, say a polyhedron, and a nice $\mathbb{R}$-valued function $f$ on this subset, say a polynomial. Is it possible to deduce the value of the integral of $f$ along $X$ with respect to the Lebesgue measure from ... | https://mathoverflow.net/users/2349 | Approximating high-dimensional integrals by low-dimensional ones | Here's how to average $e^{sx}$ over $A\_t$. Let $r = t/2$, so at the $N$-th step of the construction of $A\_t$ we have the disjoint union $A\_t^{(N)}$ of $2^N$ intervals of length $r^N$ whose left endpoints are $\sum\_{n=1}^N (r^{n-1}-r^n) \phantom. \epsilon\_n$ with each $\epsilon\_n \in \lbrace0,1\rbrace$. Thus to ch... | 5 | https://mathoverflow.net/users/14830 | 75448 | 45,804 |
https://mathoverflow.net/questions/75444 | 0 | I'm looking for a measure to analysis the uncertainty observed in a set of variables (with multivariate Gaussian distribution). So, I've tried conventional Shanon entropy (differential entropy) which results into the following equation for MVG distributions:
**H(s) = ln(sqrt((2πe)^k\*det(cov)))**
**H(s) = 0.5\*[k\*... | https://mathoverflow.net/users/17800 | Robust entropy-like measure for analyzing uncertainity |
>
> I was wondering if there is any other measures of uncertainty
> which may result in to sum of eigs instead of sum of log(eigs)?
>
>
>
This is also known as the "total variance" and is the sum of the diagonals of the covariance matrix.
| -1 | https://mathoverflow.net/users/17806 | 75457 | 45,810 |
https://mathoverflow.net/questions/75451 | 1 | Let G be a Lie group acting on a manifold M And say that there exists a point $m \in M$ such that the stabilizer of the point is non trivial. I am unable to understand how the quotient M/G fails to be a manifold. More precisely, how is a singularity created at the point $m \in M$ while constructing the quotient. Can an... | https://mathoverflow.net/users/17804 | Examples of Lie group actions on manifolds with singular quotients | A general nonsense answer to your question would be the orbit decomposition theorem. Given a compact lie group $G$ acting on a manifold $M$, it describes a stratification of $M$ by orbit types.
The main theorem is that if $p \in M$, and $G\_p$ is the stabilizer of $p$, then there is a $G$-equivariant diffeomorphism ... | 3 | https://mathoverflow.net/users/1465 | 75465 | 45,813 |
https://mathoverflow.net/questions/41686 | 6 | **Summary**: if we are trying to use combinator logic to solve first-order logic type problems, is the best method to feed in free variables and use the standard first-order unification algorithm?
In standard first-order logic, consider the problem of deducing that the following is a contradiction
```
p(x)
~p(5)
... | https://mathoverflow.net/users/9031 | Combinator logic and unification | I think first-order unification cannot replace higher-order unification because:
(1) if a variable occurs at the predicate position, FO unification is not supposed to handle that, though it may be fixed (seems easy).
(2) more importantly, sometimes a variable can be replaced by a combinatory term containing combina... | 2 | https://mathoverflow.net/users/11555 | 75475 | 45,820 |
https://mathoverflow.net/questions/75477 | 9 | In their classic paper "Class fields of abelian extensions of $\mathbf{Q}$", Mazur and Wiles assert that
>
> "in a cyclotomic
> $\mathbf{Z}\_p$-extension only finitely
> many primes lie above any prime of
> $\mathbf{Q}$."
>
>
>
My only other source in learning this material so far has been Washington's "In... | https://mathoverflow.net/users/10547 | Prime Decomposition in Cyclotomic Z_p-extensions | The point is that the Frobenius at $\ell$ is nontrivial in this extension, so it generates an open subgroup, and the fixed field of this subgroup is precisely the field at which $\ell$ stops splitting.
To see that Frob$\_\ell$ is nontrivial, recall that in the full $\mathbb{Z}\_p^\*$ extension $\mathbb{Q}(\mu\_{p^\in... | 12 | https://mathoverflow.net/users/5513 | 75479 | 45,823 |
https://mathoverflow.net/questions/75472 | 21 | For a trivial action on the coefficient, we have the following Kuenneth formula
for group cohomology:
$$
H^n(G\_1 \times G\_2; M) \cong
[\oplus\_{i= 0}^n H^i(G\_1;M) \otimes\_M H^{n-i}(G\_2;M)]
\oplus [\oplus\_{p =0}^{n+1} \text{Tor}^M(H^p(G\_1;M),H^{n+1-p}(G\_2;M))]
$$
where $G\_1$ and $G\_2$ are finite groups and... | https://mathoverflow.net/users/17787 | Kuenneth-formula for group cohomology with nontrivial action on the coefficient | I learned this through Ken Brown's textbook "Cohomology of Groups" (and studying under him): I basically use the beginning of his Chapter 5 and solve the two exercises in that section.
Let $M$ (resp. $M'$) be an arbitrary $G$-module (resp. $G'$-module), let $F$ (resp. $F'$) be a projective resolution of $\mathbb{Z}$ ... | 17 | https://mathoverflow.net/users/12310 | 75485 | 45,824 |
https://mathoverflow.net/questions/75372 | 2 | What is the fixed point set of an order two automorhism group of an Enriques surface.
| https://mathoverflow.net/users/13559 | automorphism of Enrique surface | In general, the fixed locus of an involution $\iota$ on a smooth complex surface $S$ is the union of a smooth curve $D$ and of $k$ isolated points.
This follows by Cartan's Lemma that says that in suitable holomorphic coordinates near a fixed point the action is linear.
There are trace formulae that relate these and ... | 13 | https://mathoverflow.net/users/10610 | 75488 | 45,826 |
https://mathoverflow.net/questions/75505 | 2 | I saw some remark that any locally constant function $f:\mathbb Q\_p \to \mathbb C$ can be written as
$$
f(x)=\sum\_{i=1}^{\infty}c\_i1\_{U\_i}(x), c\_i\in\mathbb C
$$
for some open subsets $U\_i\in\mathbb Q\_p$.
Why is it possible?
And furthermore, if $f$ is also compactly supported, can it be finite linear combinatio... | https://mathoverflow.net/users/17829 | Why can a locally constant function on $\mathbb{Q}_p$ be expressed as a linear combination of characteristic functions? | Suppose that $f:\mathbb{Q}\_p\to\mathbb{C}$ is locally constant. For each $k\geq 0$ the set $p^{-k}\mathbb{Z}\_p\subset\mathbb{Q}\_p$ is compact, so it can be covered by finitely many open sets on which $f$ is constant, so $f(p^{-k}\mathbb{Z}\_p)$ is finite. By taking the union over $k$, we see that the image $C=f(\mat... | 7 | https://mathoverflow.net/users/10366 | 75506 | 45,834 |
https://mathoverflow.net/questions/75484 | 8 | The first and second derivatives of the distance function (either the full $d:M\times M\to \mathbb{R}$ function or the $d(p,\cdot):M\to \mathbb{R}$ function) as well as the derivative of the exponential map (again both of the full $\exp:TM\to M$ map and of the map $\exp\_p:T\_pM\to M$) may be computed with the aid of J... | https://mathoverflow.net/users/4890 | Higher derivatives than Jacobi fields | The higher derivatives of the exponential map satisfy the corresponding higher derivative of the Jacobi equation (because the first derivative satisfies the Jacobi equation itself), which is just an inhomogeneous Jacobi equation, where the homogeneous part is the original Jacobi equation, and the inhomogeneous term inv... | 8 | https://mathoverflow.net/users/613 | 75510 | 45,837 |
https://mathoverflow.net/questions/75389 | 5 | It is known that group cohomology class $H^d[U(1),Z]$ is Z for even d and 0 for odd d.
Do we know $H^d[G,Z]$ for $G=SO(3)$, $SU(2)$ and other compact Lie group?
Also is the Borel-group-cohomology class $H^d[G,R]$ alway trivial for compact Lie group $G$?
| https://mathoverflow.net/users/17787 | Group cohomology of compact Lie group with integer coeffient | For the group $SU(2)=S^3$ we just have $H^\*(BSU(2);\mathbb{Z})=\mathbb{Z}[c\_2]$ (where $c\_2\in H^4$). More generally, for all $n$ we have
\begin{align\*}
H^\*(BU(n);\mathbb{Z}) &= \mathbb{Z}[c\_1,\dotsc,c\_n] \\\\
H^\*(BSU(n);\mathbb{Z}) &= \mathbb{Z}[c\_2,\dotsc,c\_n] \\\\
H^\*(BSp(n);\mathbb{Z}) &= \mathbb{Z}[p... | 11 | https://mathoverflow.net/users/10366 | 75512 | 45,839 |
https://mathoverflow.net/questions/75509 | 18 | This came up in a discussion I had yesterday. Since my understanding is limited,
I thought I ask here, because I know there are quite a few experts lurking about.
Recall that a holomorphic symplectic manifold $X$ is a complex manifold which comes
equipped with a nondegenerate holomorphic $2$-form $\omega$, i.e. $\omega... | https://mathoverflow.net/users/4144 | Compact holomorphic symplectic manifolds: what's the state of the art? | Here is my understanding of the situation. As you say, there are two known infinite families of irreducible holomorphic symplectic manifolds, namely:
* Hilbert schemes of $n$ points on a $K3$ surface (and deformations of these);
* generalised Kummer varieties, i.e. the fibre over $0$ of the addition morphism $T^n \ri... | 13 | https://mathoverflow.net/users/nan | 75513 | 45,840 |
https://mathoverflow.net/questions/75471 | 12 | There are examples of elliptic fiber spaces over a two-dimensional base which have infinitely many relative minimal models (where two abstractly isomorphic models connected by flops are counted separately). The one I know is given by Reid and Kawamata and works by repeatedly flopping two rational curves in a singular f... | https://mathoverflow.net/users/4834 | Infinitely many minimal models | Dear John,
One example can be found in [this paper by Fryers](http://arxiv.org/abs/math/0102055). This example is a Horrocks--Mumford quintic, in particular a Calabi--Yau threefold. He shows there are infinitely many marked minimal models, but they fall into only 8 (unmarked) isomorphism classes.
I think there is ... | 6 | https://mathoverflow.net/users/nan | 75520 | 45,843 |
https://mathoverflow.net/questions/75521 | 2 | Hurwitz spaces (or Hurwitz schemes) parametrize covers of the projective line. One can do this in many ways.
For example, one could fix the number $r$ of branch points, the degree $n$ of the cover and look only at simple covers of $\mathbf{P}^1$. This is usually denoted by $H\_{r,n}$. Fulton defined this space as a ... | https://mathoverflow.net/users/4333 | Reference request: parametrizing covers of the projective line | There is really no universal reference, unfortunately, and what you should look at depends on what you're interested in. Are you interested in arithmetic or are you working over the complex numbers? Are you interested in compactification? Do you care mostly about simple branching or about more general branching or even... | 3 | https://mathoverflow.net/users/431 | 75523 | 45,844 |
https://mathoverflow.net/questions/61274 | 4 | If R is an infinite direct product of fields, then R is an injective R-module...But I need an example of a quotient ring R, R/S, that is not injective ?? I feel that "if we take S as an infinite direct sum of fields then R/S may not be injective ??? But I couldn't show it...**Please help me to find R/S which is not inj... | https://mathoverflow.net/users/14266 | a quotient of an injective module which is not injective | (I just stumbled upon this question from a link that David White listed at a [completely different question](https://mathoverflow.net/questions/75418/commutative-algebra-final-project).)
I can give a simple argument that shows some noninjective quotient of $R$ does indeed exist. Perhaps folks already know this, but n... | 5 | https://mathoverflow.net/users/778 | 75524 | 45,845 |
https://mathoverflow.net/questions/75418 | 7 | I'm looking for a topic for a final project in commutative/homological algebra, for first year master's students (in a decent European university). During the course, they will cover the following topics: commutative rings (as in chapter 1 of Atiyah-McDonald), general module theory and structure of finitely generated m... | https://mathoverflow.net/users/1107 | Commutative algebra final project | this is similar to some other answers.
when i took basic graduate algebra from Maurice Auslander he handed out 16 pages of very terse notes the first day that he said was our Fall semester final exam. There were four sections and each of us was assigned to read, learn and write up in more detail one section. They wer... | 7 | https://mathoverflow.net/users/9449 | 75528 | 45,847 |
https://mathoverflow.net/questions/75536 | 2 | Let $R$ be a commutative ring and let $M,N$ be two finitely generated projective $R$-modules which have equal rank (not necessarily constant). What kind of general results are there concerning the question of determining whether $M$ and $N$ are isomorphic or not? This is certainly a nontrivial question (e.g., consider ... | https://mathoverflow.net/users/15488 | When are two projective modules of equal rank isomorphic? | If $R$ is noetherian of dimension d, then we have:
The Bass Cancellation Theorem: If $M$ has rank $\ge d+1$ after localizing at any prime, and if there exists a finitely generated projective module $Q$ such that $Q\oplus M\approx Q\oplus N$, then $M\approx N$. (You probably don't even need to assume $N$ projective fo... | 7 | https://mathoverflow.net/users/10503 | 75554 | 45,863 |
https://mathoverflow.net/questions/75464 | 9 | What is a good reference for a fast approach to construct affine Kac-Moody algebras from finite-dimensional simple Lie algebras?
I know that Kac's book and many others do a very detailed and progressive construction, but I mean a understandable and direct realization as in the Hong and Kang's book about Quantum grou... | https://mathoverflow.net/users/40886 | affine Kac-Moody algebras | I am a big fan of Carter's book. It's very nicely laid out and I found it quite easy to read.
Here's an older reference: Kass, Moody, Patera, and Slansky's "Affine Lie Algebras, Weight Multiplicities, and Branching Rules"
This text is focused only on affine algebras. It is kind of light on proofs but provides a lot... | 13 | https://mathoverflow.net/users/17263 | 75555 | 45,864 |
https://mathoverflow.net/questions/75408 | 4 | Does anyone know a concrete example of a Morse function on some manifold that is perfect with respect to some field but not with respect to $\mathbb Z\_2$?
| https://mathoverflow.net/users/12137 | Perfect Morse functions | I'll expand on my comment, since you requested a "concrete example" of a Morse function. A (3-dimensional) lens space $L(p,q)$ is obtained from the unit sphere $S^3=\{(z\_1,z\_2)\in \mathbb{C}^2 | |z\_1|^2+|z\_2|^2 =1\}$ by quotienting by the action $(z\_1, z\_2) \mapsto (e^{2\pi i/p} z\_1,e^{2\pi i q/p})$, which will ... | 8 | https://mathoverflow.net/users/1345 | 75557 | 45,865 |
https://mathoverflow.net/questions/75558 | 1 | Let $0<\mu<1$ and $\alpha:=1-\mu^2$. Consider the function
$$f(x):=x\sum\_{k=-\infty}^\infty\mu^{4k}e^{-\alpha\mu^{4k}x}-\frac{1}{x}\sum\_{k=-\infty}^\infty\mu^{4k}e^{-\alpha\mu^{4k}/x},$$
defined for all $x>0$. Three properties are easy to check: $f(\mu^{2n})=0$ for every integer $n$, $f(x)=-f(1/x)$, $f(x)$ vanishes ... | https://mathoverflow.net/users/14289 | An interesting doubly infinite series | I've seen sums like this, and they can get quite amusing, e.g. the Fourier coefficients of $f(x)$ as a periodic function of $\log(x)$ involve values of the Gamma function at complex arguments (see below); but it seems that this is overkill for the question at hand: there are several ranges of $\mu$ for which $f(\mu) > ... | 7 | https://mathoverflow.net/users/14830 | 75568 | 45,869 |
https://mathoverflow.net/questions/69541 | 8 | Let $G$ be a finitely presented group. To $G$ is associated in a functorial way a Malcev Lie algebra which can be constructed in several equivalent ways. Roughly speaking, it is the quotient of the completed free Lie algebra on the same number of generators as $G$, by the (formal) logarithm of the defining relations of... | https://mathoverflow.net/users/13552 | Almost-direct product and 1-formality | Yes, the direct product of two 1-formal groups is again 1-formal, and so is the free product of two 1-formal groups. A proof is given in [arxiv:0902.1250](http://arxiv.org/abs/0902.1250), Proposition 9.2.
And no, the almost direct product of two 1-formal groups need not be 1-formal. A proof is given in the same paper... | 16 | https://mathoverflow.net/users/17846 | 75569 | 45,870 |
https://mathoverflow.net/questions/75571 | 2 | What is the Iwahori subgroup for $PGL\_2(F)$ where $F$ is a local field? I am also looking for the Levi subgroups but it seems that there is only 1 levi subgroup been the identity but this seems odd to me.
| https://mathoverflow.net/users/14744 | Iwahori for PGL_2 | The (standard) Iwahori $\bar I$ of ${\rm PGL}(N,F)$ is the image of the Iwahori subgroup $I$ of ${\rm GL}(N,F)$ formed of the matrices with integer coefficients, upper triangular mod ${\mathfrak p}\_F$, and with invertible determinant. To extract its defition from Bruhat-Tits general theory, there is an extra difficult... | 4 | https://mathoverflow.net/users/4767 | 75578 | 45,876 |
https://mathoverflow.net/questions/75374 | 7 | I would like to know what are the group cohomology classes $H^d[U(1)\rtimes Z\_2, Z]$ and $H^d[U(1)\rtimes Z\_2, Z\_T]$, and/or how to calculate them.
It can be shown that $H^d[U(1), Z]$ is $Z$ for even $d$ and 0 for odd $d$.
Here $\rtimes$ is the semidirect product: $T U T = U^{-1}$ for $U \in U(1)$, where
$T$ is th... | https://mathoverflow.net/users/17787 | Calculate the group cohomology classes $H^d[U(1)\rtimes Z_2, Z]$ and $H^d[U(1)\rtimes Z_2, Z_T]$ | The group $U(1) \rtimes \mathbb{Z}/2$ you describe is nothing but the group $O(2)$ (as $U(1) = SO(2)$).
As such I think one can see the spectral sequence for the extension does collapse, and one obtains
$$H^\*(BO(2);\mathbb{Z}) = \mathbb{Z}[x\_2, x\_3, x\_4]/(2x\_2, 2x\_3, x\_3^2-x\_2x\_4).$$
Here we can take $x\_2 =... | 9 | https://mathoverflow.net/users/318 | 75582 | 45,877 |
https://mathoverflow.net/questions/75410 | 3 | Suppose we have a projective flat non-smooth morphism of Noetherian schemes $g: X \rightarrow S$. My question regards when the sheaf of relative Kahler differentials $\Omega\_{X/S}$ is flat over $S$. In particular, I am wondering about the case where $S$ is the spectrum of a Dedekind domain, so we can just consider tha... | https://mathoverflow.net/users/18815 | Flatness of sheaf of relative Kahler differentials | Suppose that $Y$ is the spectrum of a smooth curve over perfect field $k$ and let
$D\subseteq Y$ be a finite set of closed points. Let $f:X\to Y$
be a proper morphism and let $D\subseteq X$ be a normal crossings divisor. Suppose
that $f$ is semi-stable relatively to $E,D$ and $k$, in the sense of
Illusie in par. 1.... | 5 | https://mathoverflow.net/users/17308 | 75584 | 45,878 |
https://mathoverflow.net/questions/75559 | 11 | So I was reading the American Mathematical Monthly Feb 2011 (Volume 118, number 2), and in particular, I was interested in Ravi Vakil's article about mathematics of doodling. There is a question I cannot prove (or find the proof of anywhere).
First, here is the definition of the doodle (quoted from the article):
"... | https://mathoverflow.net/users/17842 | Mathematics of doodling and the winding number | This problem is one of the easiest applications of Frenet formulas for planar curves and can be found in differential geometry textbooks.
Some minor corrections: First, $q$ is usually called "turning number" rather than "winding number". (The winding number is how many times a curve goes around a marked point; the tu... | 14 | https://mathoverflow.net/users/4354 | 75590 | 45,879 |
https://mathoverflow.net/questions/75588 | 4 | This is perhaps an ill-proposed question. Any way, thank you guys.
We have a lot of bi-stuffs, such as bi-module, bi-bundle, etc. They are basically two commuting actions from two sides, left and right. This is somehow related to we are writing horizontally. If we are writing vertically, people may talk top module, u... | https://mathoverflow.net/users/7341 | Why bi-module, bi-bundle, etc? | If you work with a single associative algebra $A$, then there is not so much sense to try to define a notion of $A$-tri-module. Namely, $A$-bimodules appear naturally as operadic $A$-modules.
---
To give you a picture, an associative algebra $A$ is in particular an algebra over the colored operad given by interv... | 2 | https://mathoverflow.net/users/7031 | 75591 | 45,880 |
https://mathoverflow.net/questions/75607 | 3 | Let $S$ be a projective surface with pseudoeffective anticanonical divisor $-K\_S$. Is it true that if $C$ is an integral curve with $C^2<0$ and $C \cdot K\_S >0$, then $\max\_C (C \cdot K\_S)$ is finte?
| https://mathoverflow.net/users/1937 | Boundedness of $C.K$ on a surface with $-K$ pseudoeffective | Let $-K=N+E$ be the Zariski decomposition of $-K$, so that $N,E$ are $\mathbb{Q}$-divisors with $N$ nef, $E$ effective, such that $N \cdot E = 0$ and the restriction of the intersection pairing to the irreducible components of $E$ is negative definite. It follows that if $C$ is an integral curve such that $C \cdot (-K)... | 7 | https://mathoverflow.net/users/8761 | 75614 | 45,890 |
https://mathoverflow.net/questions/75612 | 3 | Let $G\_1$ and $G\_2$ be topological groups. Assume that there exists a continuous homomorphism $f : G\_1 \rightarrow G\_2$ which (ignoring the group structure) is a homotopy equivalence. If $BG\_i$ is a classifying space for $G\_i$, then we get an induced map $f\_{\ast} : BG\_1 \rightarrow BG\_2$ which is a homotopy e... | https://mathoverflow.net/users/17855 | Classifying spaces, Brown representability, and homotopy equivalences | The space $\Omega BG$ is, by categorical nonsense, a classifying space for principal $G$-bundles over $\Sigma X$ with a chosen trivialisation at the basepoint. Any such bundle can be trivialised over the upper and lower halves of $\Sigma X$, and the difference between the two trivialisations gives a map $X\to G$. One c... | 8 | https://mathoverflow.net/users/10366 | 75617 | 45,891 |
https://mathoverflow.net/questions/74101 | 3 | In a 1957 paper ([Link](https://projecteuclid.org/journals/illinois-journal-of-mathematics/volume-1/issue-1/Homology-of-Noetherian-rings-and-local-rings/10.1215/ijm/1255378502.full)), Tate shows that if $I \subset R$ is an ideal of the noetherian ring R then there is a graded commutative DGA $X$ over $R$ with $H\_i X=0... | https://mathoverflow.net/users/3901 | Extension of Tate's result regarding Tor | I am not sure what the protocol is on answering ones own question so I am making it community wiki.
What I was most curious about is the following:
When does the projective resolution of a commutative algebra as a module over a commutative ring have the structure of a graded commutative (skew in Tate's paper) DGA?
... | 0 | https://mathoverflow.net/users/3901 | 75626 | 45,897 |
https://mathoverflow.net/questions/75624 | 1 | This is a simple question, and I'm sure it was a homework assignment at some point (assuming it's true) but it's one that I'm puzzled over. Suppose I have a compact domain $D \subset \mathbb{R}^n$ with area $1$ and a continuous, bounded function $g(x):D\rightarrow\mathbb{R}$. Let $F(t)$ denote the volume of the subset ... | https://mathoverflow.net/users/17860 | Integrating with sub-level sets | See
<http://en.wikipedia.org/wiki/Smooth_coarea_formula>
| 2 | https://mathoverflow.net/users/11142 | 75629 | 45,900 |
https://mathoverflow.net/questions/75618 | 3 | Here the pattern is a finite or infinite sequence of 0s and 1s, not necessarily consecutive, for example, $\lbrace 1, \*, 1, \*, 1 \rbrace$ and $\lbrace 0, \*, 0, \*, 0, \*, \ldots \rbrace$ ($ \* $, hole position not cared), which is to be moved along a given infinite binary sequence for a match.
If the pattern is f... | https://mathoverflow.net/users/17858 | Will a given pattern ever show up in an infinite random sequence of 0s and 1s? | There are a couple of ways to interpret the non-consecutive issue in your question.
One way is that you are fixing a template with infinitely many positions, and in some of those positions, a definite digit value appears, and in others, there is a hole. Now, a given binary sequence conforms with the template if ther... | 4 | https://mathoverflow.net/users/1946 | 75633 | 45,901 |
https://mathoverflow.net/questions/75649 | 4 | Is it consistent to have (at the same time) a real-valued measurable cardinal and precipitous ideals on small cardinals such as $\omega\_1$? What about saturated ideals on small cardinals?
| https://mathoverflow.net/users/11145 | Real-valued measurable cardinals and strong ideals | Suppose that there are two measurable cardinals $\kappa\lt\delta$ in $V$. Perform forcing in two steps: first, we form the forcing extension $V[G]$ by the Levy collapse making $\kappa$ become $\omega\_1$. The standard arguments (I think due to Prikry) show that $\kappa=\omega\_1^{V[G]}$ now carries a precipitous ideal ... | 6 | https://mathoverflow.net/users/1946 | 75651 | 45,908 |
https://mathoverflow.net/questions/75652 | 4 | I know from two sources
that it is (or at least was) unknown whether there are infinitely
many geometrically distinct closed geodesics
for every Riemannian metric on $S^3$, the 3-sphere
(Weinberger, *Computers, Rigidity, and Moduli*, 1995, p.101;
Berger, *A Panoramic View of Riemannian Geometry*, 2003, p.461).
And from... | https://mathoverflow.net/users/6094 | Is there more than one closed geodesic on $S^3$? | I think the recent work by Huagui Duan and Yiming Long
<http://arxiv.org/abs/1008.1458>
showed the existence of at least two closed geodesics.
| 5 | https://mathoverflow.net/users/2555 | 75656 | 45,911 |
https://mathoverflow.net/questions/75667 | 0 | Consider an integer polynomial ring, $A = \mathbb{Z}[t]$, and a ring of fractions, $B = \mathbb{Z}[t, t^{-1}]$; obviously, $A$ is a subring of $B$.
Now we consider two modules over $A$ and $B$, $M$ and $N$. We want to construct a map from $N$ to $M$. But the question is that the two modules are not over the same poly... | https://mathoverflow.net/users/6577 | Question about modules, quotient rings, and polynomial rings? | I do not think you can map a module over a smaller ring into a module over a larger ring in this case.$ \phi(rx)=r\phi(x)$ will not satisfy.
| 0 | https://mathoverflow.net/users/12379 | 75668 | 45,917 |
https://mathoverflow.net/questions/75565 | 11 | Let $p$ be an odd prime number. Can a finite simple group have a conjugacy class with $2p$ elements?
| https://mathoverflow.net/users/17845 | Finite simple groups and conjugacy classes with 2p elements | I believe the answer is no. A group with a conjugacy class of degree $2p$ has a transitive permutation action of degree $2p$ on that class, and the stabilizer is the centralizer of an element.
If this action is imprimitive, then the blocks have size $p$ or 2. Wtih blocks of size $p$ there would be a subgroup of index... | 13 | https://mathoverflow.net/users/35840 | 75672 | 45,918 |
https://mathoverflow.net/questions/75676 | 1 | Let $X$ be a transient Markov chain with countable state space $S(X)$. Let $Y$ be a positive recurrent Markov chain with countable state space $S(Y)$. (Time is discrete.)
Let $A \subseteq S(X)$ be such that, with probability 1, $X$ visits $A$ infinitely many times. In fact, if it helps, let $A$ be such that, with pro... | https://mathoverflow.net/users/17883 | Product of a transient and a positive recurrent Markov chain | The answer to 1. is negative. Consider the transient Markov chain $X\_n=X\_0+n$ on $\mathbb Z$ and choose for $A$ any subset of $\mathbb Z\_+$ with different inferior and superior densities.
As regards 2., I assume you want $X$ and $Y$ to be independent. Then the answer to the first question is negative. Same $X$ as ... | 0 | https://mathoverflow.net/users/4661 | 75677 | 45,921 |
https://mathoverflow.net/questions/75632 | 13 | This seems like something that should be well-known, but as an outsider to the field, I'm having trouble locating precise statements.
Hasse-Weil zeta functions of Shimura varieties should be alternating products of automorphic $L$-functions. This seems to be known when the underlying group is $GL\_2$ (or a quaternion... | https://mathoverflow.net/users/6753 | Which Shimura varieties are known to be automorphic? | As far as I understand, what you need to do to prove that the zeta function of a Shimura variety is automorphic is (I'm ignoring the bad primes here, but I think that we now know enough about them too - I can develop later if you want) :
(1) Do some kind of point-counting over finite fields. For PEL Shimura varieties... | 14 | https://mathoverflow.net/users/12336 | 75684 | 45,926 |
https://mathoverflow.net/questions/75678 | 3 | I'm confused about some things concerning lengths of geodesics on Riemann surfaces and positive eigenvalues of the Laplacian. Moreover, I'm also interested in the relation between these two.
Let $X$ be a compact (connected) Riemann surface of genus $g\geq 2$.
Since the complex upper half plane $\mathfrak{h}$ is the... | https://mathoverflow.net/users/4333 | The smallest positive eigenvalue and the length of the shortest geodesic | You can have a sequence surfaces $X\_i$ with arbitrarily short geodesics but $\lambda\_{X\_i}$ bounded away from zero. The geodesics are non-separating, so when they shrink to length zero, the [Cheeger constant](http://en.wikipedia.org/wiki/Cheeger_constant#Cheeger.27s_inequality) stays bounded away from zero, and ther... | 9 | https://mathoverflow.net/users/1345 | 75688 | 45,929 |
https://mathoverflow.net/questions/74700 | 3 | I am attempting to understand and use the degeneration formula for GW invariants as stated in 'Gromov-Witten theory and Donaldson-Thomas theory II' (Maulik, Nekrasov, Okounkov, Pandharipande). The formula is found in section 3.4 of the aforementioned paper, but I will recall it here.
Let $\lambda:\chi\to C$ be a nons... | https://mathoverflow.net/users/17350 | Curve Splitting in the Degeneration Formula for Relative GW Invariants (MNOP II) | I have found a rough answer to a special case of my question in YP Lee's notes (roughly page 21 here <http://www.math.utah.edu/~yplee/research/grenoble.pdf>). If anyone has any further insight though I'd be eager to hear it
I will state it here for anyone who is interested.
The special case is deformation to the n... | 2 | https://mathoverflow.net/users/17350 | 75697 | 45,933 |
https://mathoverflow.net/questions/74743 | 19 | Let us denote by **ACC** the axiom of countable choice, namely the assertion that the product of countably many non-empty sets is non-empty, and denote by **UCC** the assertion that a countable union of countable sets is countable.
UCC is a simple theorem of ZF+ACC.
*Proof* Suppose for every $i\in\omega$ we have $X... | https://mathoverflow.net/users/7206 | Countable unions and the axiom of countable choice | The implication $ACC \implies UCC$ is irreversible in $ZF$. This follows again from the transfer theorem of Pincus:
1. $UCC$ is an injectively boundable statement, see note 103 in "Consequences of the axiom of choice" by Howard & Rubin. Right after the theorem in pp. 285 and its corollary, there are examples of state... | 7 | https://mathoverflow.net/users/12976 | 75711 | 45,943 |
https://mathoverflow.net/questions/75655 | 0 | This is probably not a research level question, I'm sorry if it is inappropriate. I'm reasking here [this question](https://math.stackexchange.com/questions/64179/sheaf-of-sections-and-local-triviality) on math.se.
---
Suppose that $\xi: E \to B$ is a bundle (by which I mean simply a continuous mapping), and ther... | https://mathoverflow.net/users/11248 | Sheaf of sections and local triviality | Let $E$ be the union of the coordinate axes in $\mathbb R^2$, and let $\xi:E\to\mathbb R$ be the projection on the first coordinate. Then $\Gamma(\xi)$ is a trivial sheaf, so it is isomorphic to $C(\mathbb R,F)$ with $F$ any one-point space. Yet $\xi$ is not locally trivial.
(This can be modified so that the fiber at $... | 1 | https://mathoverflow.net/users/1409 | 75713 | 45,944 |
https://mathoverflow.net/questions/75712 | 1 | Let $X \sim B(n,c/n)$ be a binomially distributed random variable with
parameter $p = c/n$, and hence mean $c$. Here $c$ is some function of $n$ such that
i) $c \geq n^{2/3}$
ii) The function $c$ grows slower than any linear function of $n$ (i.e., in big-O notation, $c = o(n)$, or equivalently $\lim\_{n \to \infty... | https://mathoverflow.net/users/11247 | Tail Conditional Expectation of a binomial random variable | CLT (sufficiently powerful version such as Berry-Esseen inequality) says that $Pr(X\ge c)\to\frac12$, so any event that has tiny probability for $X$ also has tiny probability for $X\_{\ge c}$. So $E(X|X\ge c) = c + O(c^{1/2})$. You can get a precise value for $E(X|X\ge c)$ by approximating the point probabilities of $X... | 1 | https://mathoverflow.net/users/9025 | 75723 | 45,951 |
https://mathoverflow.net/questions/75738 | 0 | Can we view a module over the ring $\mathbb{Q}[t,t^{-1}]$ to be a module over the polynomial ring $\mathbb{Q}[t]$?
where $\mathbb{Q}$ denote any rational number coefficients.
| https://mathoverflow.net/users/6577 | How to consider a module over the ring Q[t,t^(-1)] to be a module over the polynomial ring Q[t]? | Yes, via the natural inclusion $\mathbb{Q}[t] \to \mathbb{Q}[t,t^{-1}]$: let an element of $\mathbb{Q}[t]$ act the way it does when considering it as an element of $\mathbb{Q}[t,t^{-1}]$. More generally, if you have a $B$-module $M$ and a morphisms of rings $A \to B$, then you can see $M$ as an $A$-module.
| 1 | https://mathoverflow.net/users/1107 | 75739 | 45,961 |
https://mathoverflow.net/questions/75094 | 15 | Hi,
Let's denote by "semi-simplicial set" a simplicial set without degeneracies, i.e. a contravariant functor from the category $\Delta\_{inj}$ of finite linearly ordered sets and order preserving injections to sets (this is also known as $\Delta$-set).
We have an inclusion functor $j: \Delta\_{inj} \rightarrow \De... | https://mathoverflow.net/users/733 | Semi-simplicial versus simplicial sets (and simplicial categories) | The map $j\_! j^{\ast} K \rightarrow K$ is never a Joyal equivalence unless $K$ is empty.
For example, if $K = \Delta^{0}$, then $j\_{!} j^{\ast} K$ is the nerve of the category with one object $X$ and a single nonidentity morphism $e: X \rightarrow X$ satisfying $e^2 = e$.
You can think of $j\_{!} j^{\ast} K$ as "obta... | 19 | https://mathoverflow.net/users/7721 | 75740 | 45,962 |
https://mathoverflow.net/questions/75753 | 2 | When thinking of an apparently unrelated problem I stumbled upon the following question, which is certainly elementary to many readers of this site.
Let $\omega\_l=e^{i2\pi/l}$, and let $z\in Z[\omega\_l]$ have $|z|=1$. Can we conclude that $z=\omega\_l^k$ for some integer $k$?
Thanks in advance for any answer!
| https://mathoverflow.net/users/9890 | Elements of unit modulus in ring generated by root of unity | A.Quas already noted in his comment that ${\bf Z}[\omega\_l]$ might contain a $2l$-th root of unity (he gave $l=3$ but even $l=1$ works...). But that's the only possibility: we show that the only algebraic integers $z$ in ${\bf Q}(\omega\_l)$ satisfying $|z| = 1$ are the roots of unity in ${\bf Q}(\omega\_l)$. More gen... | 9 | https://mathoverflow.net/users/14830 | 75755 | 45,968 |
https://mathoverflow.net/questions/75744 | 13 | Hello,
Is there a solution to the following equation:
$c^n=a^{2n}+a^n b^n + b^{2n}$
where $a,b,c \in \mathbb{N}^\*$, and $n$ is an integer $\geq 2$.
The problem is due to Antoine Balan.
Thanks in advance.
| https://mathoverflow.net/users/12806 | Fermat-like equation $c^n=a^{2n}+a^n b^n + b^{2n}$ | Writing the equation as
$$
c^n+(ab)^n = (a^n+b^n)^2,
$$
one has, via results of Darmon and Merel, and Poonen, that there
are no solutions for $n \geq 4$, provided the three terms are coprime.
If $n=2$, I think Pythagorean triply stuff shown that there are again no
solutions. In the case $n=3$, I guess one could again l... | 19 | https://mathoverflow.net/users/7302 | 75756 | 45,969 |
https://mathoverflow.net/questions/75759 | 0 | Given $X$, and a locally free sheaf $\mathcal{F}$ of rank $n$ on it. We have the induced map
$f:\mathbb{P}(\mathcal{F})\rightarrow X$. Let $\phi$ be the divisor associated to the line bundle $\mathcal{O}\_{\mathbb{P}(\mathcal{F})}(1) $.
Is it true $f\_{\ast}(\phi^{n-1})=[X]$ in the Chow ring $A^\ast(X)$? Also, is tr... | https://mathoverflow.net/users/14854 | Chow ring of projective bundle | The class $\phi^{n-1}$ has degree $1$ when restricted to each fibre of $f$ since $H^{n-1}$ has degree $1$ on $\mathbb{P}^{n-1}$, where $H$ is the class of a hyperplane, i.e. a divisor associated to $\mathcal{O}(1)$. It follows that $f\_\*(\phi^{n-1})$ is indeed $[X]$ in $A^\*(X)$.
I suppose what you call the trace ma... | 1 | https://mathoverflow.net/users/519 | 75764 | 45,973 |
https://mathoverflow.net/questions/75767 | 0 | If $a, b$ are two numbers such that $(a+b)^2 = a^2 + b^2$, then $a.b = 0$.
Is there a similar statement for square matrices.
"If $A, B$ are square matrices such that $(A+B)^2 = A^2 + B^2$, then $A.B = 0$."
Note that if $(A+B)^2 = A^2 + B^2$, then $AB = -BA$, hence $tr(AB) = 0$
| https://mathoverflow.net/users/17901 | Square matrices: $(A+B)^2=A^2+B^2$ | If
$$
A=\begin{pmatrix}
0&M\\ M&0\end{pmatrix},\qquad
B =\begin{pmatrix}-I&0\\ 0&I\end{pmatrix}
$$
then $AB+BA=0$, so $A^2+B^2=(A+B)^2$ and neither $A$ nor $B$ is zero.
| 7 | https://mathoverflow.net/users/1266 | 75771 | 45,976 |
https://mathoverflow.net/questions/47043 | 15 | Local cohomology with respect to an ideal $\mathfrak{a}$ is often studied over a Noetherian ring $R$. However, the proof of a lot of basic results does not rely on noetherianity of $R$, but rather on the following two properties:
>
> (ITI) $\mathfrak{a}$-torsion submodules of injective modules are injective.
>
>
... | https://mathoverflow.net/users/11025 | injectivity of torsion submodules of injectives | @ Fred: I think the following may be help you to give an example for (ITR) question:
1. Choose a non-Noetherian local ring $(A, \frak{m})$ such that $\frak{m}= \frak{m}^n$ for all $n$.
2. Let $E(k)$ be the injective envelope of $k = R/\frak{m}$.
3. Claim: if $E(k)$ is $\frak m$-torsion, then $E(k) = k$. Indeed, let $... | 3 | https://mathoverflow.net/users/17901 | 75780 | 45,979 |
https://mathoverflow.net/questions/75776 | 8 | Say $f$ is a newform of weight $k$ and level $\Gamma\_1(N)$. $f$ is called CM if, for example, there is an imaginary quadratic field $K$ such that for all $p\nmid N$ which are inert in $K$, the $p$th Fourier coefficient $a\_p$ of $f$ is 0. (Ribet's article [*Galois representations attached to eigenforms with Nebentypus... | https://mathoverflow.net/users/1021 | Effective detection of CM modular forms | If the form is CM then it will be isomorphic to a quadratic twist of itself. So I think what I'd do with a form which I suspect is or is not CM is to just twist by all the (finitely many ) possible quadratic characters that could be involved and then to check to see if $f$ is the same as its twist, which one can do by ... | 7 | https://mathoverflow.net/users/1384 | 75781 | 45,980 |
https://mathoverflow.net/questions/75698 | 68 | I'm looking for a list of problems such that
a) any undergraduate student who took multivariable calculus and linear algebra can understand the statements, (Edit: the definition of understanding here is that they can verify a few small cases by themselves )
b) but are still open or very hard (say took at least 5 ye... | https://mathoverflow.net/users/13693 | Examples of seemingly elementary problems that are hard to solve? | This is another one where it's hard to establish a lower bound due to not much work having been done on it -- it's been open since at least the 1980's, possibly the 1950's, but it's a pretty isolated problem. I think though that we can say that it's probably hard because proving it would establish better lower bounds o... | 34 | https://mathoverflow.net/users/5583 | 75792 | 45,986 |
https://mathoverflow.net/questions/75716 | 4 | Let $\Omega$ be the interior of a compact set $K$ in the plane. Suppose $f$ and $g$ are continuous on $K$ and holomorphic in $\Omega$, and $|f(z)-g(z)|<|f(z)|$ for all $z\in K-\Omega$. Then $f$ and $g$ have the same number of zeros in $\Omega$.
PS: This problem is from Rudin's book in Ch.10. So, I just know some basi... | https://mathoverflow.net/users/13905 | General form of Rouche's Theorem | Actually this problem is from Chapter 13 (at least in my edition, see p.266). Here is a sketch. By the conditions $f$ and $g$ do not have any zeros in $K-\Omega$. In particular, zeros do not accumulate on $\partial\Omega$, hence there are only finitely many zeros in $\Omega$ by the uniqueness principle (Corollary on p.... | 5 | https://mathoverflow.net/users/11919 | 75796 | 45,988 |
https://mathoverflow.net/questions/75761 | 1 | Suppose I have a morphism $f:X\to Y$ which is a GIT quotient of $X$ with respect to some reductive, linear group.
Does the semistable $X^{ss}$ and stable locus $X^s\subset X$ determine completely the linearization (maybe up to taking a power of the linearization itself)?
Or, in better words, can two different linear... | https://mathoverflow.net/users/4096 | Does the semi-stable set determine the linearization of a GIT quotient? | The paper you want to have a look at is Thaddeus, [GIT and Flips](https://doi.org/10.1090/S0894-0347-96-00204-4 "J. Am. Math. Soc. 9, No. 3, 691-723 (1996), arXiv:alg-geom/9405004. zbMATH review at https://zbmath.org/0874.14042") ([JSTOR](https://www.jstor.org/stable/2152810)), or Dolgachev–Hu, [Variation of GIT quotie... | 4 | https://mathoverflow.net/users/7437 | 75801 | 45,991 |
https://mathoverflow.net/questions/75789 | 2 | Hi,
Is the automorphism group of a countable locally finite connected poset finite or countable?
If not, is there a way to equipp it (the uncountable group) with a topology and a measure?
Need this information for a work in progress on causal set quantization.
Thank you
| https://mathoverflow.net/users/nan | Automorphisms of locally finite countable posets | The poset consisting of a single countable antichain, with all elements incomparable, is [locally finite](http://en.wikipedia.org/wiki/Locally_finite_poset), but the automoprhism group consists of any permutation of the elements, which is an uncountable set.
But you also said "connected". If by this you mean that the... | 4 | https://mathoverflow.net/users/1946 | 75805 | 45,994 |
https://mathoverflow.net/questions/75795 | 13 | On page 55 of Weibel's Introduction to homological algebra the following passage appears:
>
> Here are two consequences that use the fact that homology commutes with arbitrary direct sums of chain complexes
>
>
>
I understand why homology commutes with arbitrary direct sums when the direct sum of a collection ... | https://mathoverflow.net/users/4002 | In an arbitrary abelian category, does chain complex homology commute with coproduct? | I couldn't think of a natural example of an abelian category in which direct sums are not exact (I think this is called axiom AB4). For example, sheaves of abelian groups and R-modules both have this property. However there are natural examples of abelian categories where direct products are not exact (i.e. not satisfy... | 25 | https://mathoverflow.net/users/7762 | 75809 | 45,997 |
https://mathoverflow.net/questions/75785 | 4 | (1) $\pi(n)$, the number of primes at most $n$, is asymptotic
to $n / \ln n$.
(2) In the Erdős-Rényi random graph model, $p = \ln n / n$
is a sharp threshold for the connectedness of the graph $G(n,p)$ on $n$
vertices with edge-probability $p$.
Is there any connection between these two, or is the
ratio $n / \ln n$ ... | https://mathoverflow.net/users/6094 | Prime number density vs. connectedness threshold: coincidence? | I'd lean towards "coincidence", for a number of reasons:
1. $\pi(n)$ is a cardinality, whereas $p$ is a density; one is comparing apples and oranges. The density of the primes is $1/\log n$, which is quite different from $\log n/n$. (It is true that the average number of divisors $\tau(n)$ of a natural number $n$ is ... | 12 | https://mathoverflow.net/users/766 | 75818 | 46,001 |
https://mathoverflow.net/questions/75774 | 6 | I've read that $\text{Br} \mathbb{P}^n\_k$ (here $\text{Br}$ is the cohomological Brauer group, i.e. $H^2\_{ét}(-,\mathbb{G}\_m)$) is just isomorphic to $\text{Br} k$. As proof of this fact seems to be not so easy in the general case, but there should be a simple and conceptual proof when $k$ has characteristic zero. D... | https://mathoverflow.net/users/1107 | Brauer group of projective space | I don't think the assumption of characteristic zero simplifies the proof a great deal. However, it does allow us to avoid having to give a more involved proof for the $p$-power torsion (where $p = char (k)$) so I will assume that below.
Firstly, by Proposition 1.4 of Grothendieck's "Groupe de Brauer II", $H^2(X, \mat... | 16 | https://mathoverflow.net/users/519 | 75823 | 46,004 |
https://mathoverflow.net/questions/75793 | 3 | I would like to know whether there is a specific relationship between the deformation quantizations of the Poisson algebras of the functions on a symplectic manifold, say $M$, and of the functions on an open subset of $M$, in other words an openly embedded $N\to M$. I'm only starting to get familiar with the literature... | https://mathoverflow.net/users/2622 | Open symplectic embeddings and deformation quantization | Hi Igor,
there is a quite elementary way to see that star products restrict to open subsets: it's essentially part of the definition of a star product. Here, I will focus on the case of smooth (symplectic/Poisson) manifolds, where you do not have to rely too much on sheaf-theoretic notions. Now, $\star$ being a star ... | 4 | https://mathoverflow.net/users/12482 | 75824 | 46,005 |
https://mathoverflow.net/questions/75701 | 2 | Let $Q = \{(x,y): x,y\geq 0\} $ be the 1st quadrant of $\mathbb R^2$, and $f$ is a function defined on it such that all the partial derivative(any order) of $f$ exists and continuous. By Whitney extension theorem (1934 Proceedings of AMS) we know there exists a functions $\tilde{f}$ and open set $\tilde{U}$, such that ... | https://mathoverflow.net/users/16031 | Conformal Extension from a closed set to open | Ah, so you just mean that, when you regard $\mathbb{R}^2$ as $\mathbb{C}$ and you have a complex-valued function $f$ on $Q$, the closed first quadrant of $\mathbb{C}$, that satisfies the Cauchy-Riemann equations up to and including the boundary of $Q$, then does it extend holomorphically across the boundary. (I've neve... | 3 | https://mathoverflow.net/users/13972 | 75826 | 46,006 |
https://mathoverflow.net/questions/75829 | 13 | Let $X\_1, X\_2, \cdots$ be i.i.d. random variables with $E(X\_1) = 0, E(X\_1^2) = \sigma^2 >0, E(|X\_1|^3) = \rho < \infty$.
Let $Y\_n = \frac{1}{n} \sum\_{i=1}^n X\_i$ and let us note $F\_n$ (resp. $\Phi$) the *cumulative distribution function* of $\frac{Y\_n \sqrt{n}}{\sigma}$ (resp. of the standard normal distribut... | https://mathoverflow.net/users/1606 | Berry Esseen type result for probability density functions | The magic words are "local limit theorem".
| 9 | https://mathoverflow.net/users/11142 | 75838 | 46,011 |
https://mathoverflow.net/questions/75787 | 5 | A *tower of algebras* is a sequence of algebras
$$A\_0 \hookrightarrow A\_1 \hookrightarrow \cdots \hookrightarrow A\_n \hookrightarrow \cdots$$
with embeddings $A\_n \otimes A\_m \hookrightarrow A\_{n+m}$ satisfying an associativity condition.
I found this definition recently in papers in representation theory such... | https://mathoverflow.net/users/13251 | Where does the definition of "Tower of Algebras" come from? | I'm a bit out of my comfort zone with this, but Goodman, de la Harpe and Jones "Coxeter graphs and towers of algebras" might be a place to start. The work here was motivated by questions about von Neumann algebras.
| 2 | https://mathoverflow.net/users/1266 | 75843 | 46,013 |
https://mathoverflow.net/questions/75848 | 2 | As we know, the albanese map $Alb$ assoicates a smooth proper variety $X$ of dimension $n$ to an abelian variety $Alb(X)$ of dimension $g=H^0(X,\Omega^1\_X)$. Another well known fact is the moduli spaces $A\_g$, $g\leq 3$ have dense subsets consisting of $Alb(X)$ for those $X$ satisfying $n=1$, which is no longer true ... | https://mathoverflow.net/users/5661 | on a lower bound related to albanese map | One can always take $n=2$ and this is the best possible:
Given a principally polarised abelian variety $(A, \Theta)$ of dimension $g$, if $S$ is the intersection of $g-2$ general elements of the linear system $|3\Theta|$, then $S$ is a smooth surface. The Lefschetz hyperplane section theorem implies that the natural... | 3 | https://mathoverflow.net/users/519 | 75851 | 46,015 |
https://mathoverflow.net/questions/67235 | 10 | As we know, a finite group $G$ is a rational group if $\chi (g)\in\mathbb{Q}$, where $\chi$ is every irreducible charahter and $g\in G$. I have an interesting question that is "Is 2-Sylow subgroup of a rational group also a rational group?"
Any hints will be appreciated :)
| https://mathoverflow.net/users/13898 | Is 2-sylow subgroup of a rational group also a rational group? | This had been a long standing conjecture, but it has now been answered negatively. Isaacs and Navarro have found a counterexample.
| 6 | https://mathoverflow.net/users/17926 | 75858 | 46,019 |
https://mathoverflow.net/questions/75695 | 1 | **Upload:** the general question has been answered in the negative. Now I am proposing a more specific question, which is actually the one that I am interested in.
I think that this is a quite natural question, but I was able neither to find anything in literature, nor to find an argument by myself.
Let $G=(V,E)$ b... | https://mathoverflow.net/users/13809 | How much local is the information encoded by the isoperimetric constant of a graph? | As far as I understand, you suppose that $\partial A$ is disjoint from $A$, and in the process you define $A\_n=A\_{n-1}\cup \partial A\_{n-1}$,
Under these assumptions, the answer to both questions is negative.
Consider a binary tree with a root vertex $a$ (that is, $a$ has degree 1, and each other vertex has degr... | 2 | https://mathoverflow.net/users/17581 | 75862 | 46,021 |
https://mathoverflow.net/questions/75854 | 2 | What is the relationship between (g,K)-module and Maass forms for GL(2)?
(g,K)-modules are defined in chapter 2 of Bump, Automorphic forms and representations.
There is a classification of (g,K)-modules.
What is the relationship between (g,K)-module and Maass forms for GL(2)? and what does the classification of (... | https://mathoverflow.net/users/2666 | What is the relationship between (g,K)-module and Maass forms? | So you've seen that there are essentially three types of (g, K)-modules: finite-dimensional ones; principal series; and discrete series. The finite-dimensional ones don't interest us, since they are never unitary. So that leaves two cases. And these correspond precisely to the two kinds of cuspidal automorphic represen... | 12 | https://mathoverflow.net/users/2481 | 75868 | 46,023 |
https://mathoverflow.net/questions/75870 | 4 | I rephrase my last question. Given a locally finite countable connected (as a graph) poset which satisfies the following further condition: the intersection of any antichain with the set of elements preceding (or succeding) any element is a finite set.
1. Is the automorphism group of this poset countable or uncounta... | https://mathoverflow.net/users/nan | Automorphisms of locally finite countable posets | For question 1, the automorphism group needn't be countable. To see this, consider the poset consisting of infinitely many diamonds stacked on top of one another.
```
.
.
.
3
/ \
* *
\ /
2
/ \
* *
\ /
1
/ \
* *
\ /
0
```
This is locally f... | 1 | https://mathoverflow.net/users/1946 | 75872 | 46,024 |
https://mathoverflow.net/questions/75875 | 7 | I am asking in the sense of isometry groups of a manifold. SU(3) is the group of isometries of CP2, and SO(5) is the group of isometries of the 4-sphere. Now, it happens that both manifolds are related by Arnold-Kuiper-Massey theorem: $\mathbb{CP}^2/conj \approx S^4$; one is a branched covering of the other, the quotie... | https://mathoverflow.net/users/4037 | Why SU(3) is not equal to SO(5)? | Why would you think that they are related?
The map $\mathbb{CP}^2/\text{conjugation} \to S^4$ is only $SO(3)$-equivariant, where $SO(3) \subset SU(3)$ consists of the real matrices and $SO(3) \subset SO(5)$ is the maximal subgroup acting irreducibly on the 5-dimensional vector representation of $SO(5)$.
Not sure if... | 10 | https://mathoverflow.net/users/394 | 75877 | 46,025 |
https://mathoverflow.net/questions/75864 | 4 | I've been thinking about $SU(n)$-invariant metrics on the odd-dimensional spheres $S^{2n-1} \simeq SU(n)/SU(n-1)$. For $S^1$, all such metrics are in correspondence with the positive reals. For $S^{3} \simeq SU(2)$, the tangent space is parallelizable, and so, such metrics are in correspondence with metrics on $R^2$. D... | https://mathoverflow.net/users/1648 | Invariant Metrics on the Sphere | Assume $n>2$. The $SU(n)$-invariant metrics on the sphere $S^{2n-1}$ are in bijection with $SU(n-1)$ invariant metrics on $T\_x S^{2n-1}$ where $SU(n-1)$ is realized as the stabilizer of some $x\in S^{2n-1}$. This action is the sum of the defining $SU(n-1)$-module $V$ and the trivial real 1-dimensional module $T$. The ... | 4 | https://mathoverflow.net/users/2349 | 75880 | 46,026 |
https://mathoverflow.net/questions/75878 | 1 | Hello,
I have the following proof of Cayley's Theorem: [Proof](http://imageshack.us/photo/my-images/821/sformulaproof.png).
This proof counts orderings of directed edges of rooted trees in two ways and concludes the number of rooted trees with directed edges of order $n$.
However, I know a version of Cayley's The... | https://mathoverflow.net/users/17931 | Cayley's Theorem regarding marked trees | What is the question? This proof is in "Proofs from the Book" and credited to Jim Pitman.
| 0 | https://mathoverflow.net/users/8008 | 75882 | 46,028 |
https://mathoverflow.net/questions/75742 | 32 | **Background**
Let $(M,g)$ be a finite-dimensional riemannian (or more generally pseudoriemannian) manifold. Suppose that I know that a certain Lie group $G$ acts transitively and isometrically on $M$ and after a little bit more work I exhibit $M$ as a homogeneous space $G/H$. Let
$$
\mathfrak{g} = \mathfrak{h} \oplu... | https://mathoverflow.net/users/394 | Isometry group of a homogeneous space | Here is an algorithm to compute the Lie algebra of the group of isometries of a homogeneous space $G/H$ endowed with a $G$-invariant (pseudo-)Riemannian metric $g$. It is phrased in terms of essentially algebraic computations using the left-invariant forms on $G$, but it could be reduced completely to computations with... | 38 | https://mathoverflow.net/users/13972 | 75887 | 46,029 |
https://mathoverflow.net/questions/75170 | 9 | Given a group $Q$ and an abelian group $C$, I want to determine the number $I(Q,C)$ of isomorphism classes of all groups $G$ having a central subgroup $C'$ isomorphic to $C$ such that the quotient $G/C'$ is isomorhic to $Q$. Thus $I(Q,C)$ equals the number of groups (up to isomorphism), that fit into a central extensio... | https://mathoverflow.net/users/17734 | Counting isomorphism classes via extensions | Let $Q$ be a group and $A$ a $Q$-module. Call two extensions $G, G'$ of $Q$ by $A$ weakly equivalent, if there is a commutative diagramm
$$ 0 \to A \to G \to Q \to 1 $$
$$ \hspace{2pt} \downarrow \hspace{20pt} \downarrow \hspace{20pt} \downarrow $$
$$ 0 \to A \to G' \to Q \to 1 $$
with vertical isomorphisms. Denote th... | 8 | https://mathoverflow.net/users/10194 | 75889 | 46,030 |
https://mathoverflow.net/questions/75884 | 11 | I was reading the abstract of a recent preprint ([Division Algebras and Supersymmetry III](http://arxiv.org/abs/1109.3574) by Juhn Huerta), and I wondered if something much simpler than what he was talking about had been worked on: have [Lie $2$-groups](http://ncatlab.org/nlab/show/Lie+2-group) been applied to the reso... | https://mathoverflow.net/users/3993 | Lie $2$-groups and differential equations | A well-studied special case of higher symmetries of differential equations is that of differential equations that arise as [Euler-Lagrange equations](http://ncatlab.org/nlab/show/Euler-Lagrange+equation) of local action functionals. The symmetries and symmetries-of-symmetries and symmetries-of-symmetries-of-symmetries ... | 11 | https://mathoverflow.net/users/381 | 75892 | 46,031 |
https://mathoverflow.net/questions/75879 | 8 | In July, Asaf Karagila asked [three questions](https://mathoverflow.net/questions/71524/how-elementary-can-we-go) about elementary substructures of the universe of sets. The latter two were answered, the upshot being that the hypothesis $V\_\kappa \prec V$ doesn't alone endow $\kappa$ with any large cardinal properties... | https://mathoverflow.net/users/8547 | Are there large cardinals for $n$-elementarity? | The $\Sigma\_n$ reflecting cardinals, of course, have
precisely the property that you mention, by definition. A
cardinal $\kappa$ is $\Sigma\_n$-reflecting if
$V\_\kappa\prec\_{\Sigma\_n} V$. One sometimes sees this term
defined in such a way to insist also that $\kappa$ is
inaccessible, but to my way of thinking, thes... | 11 | https://mathoverflow.net/users/1946 | 75895 | 46,034 |
https://mathoverflow.net/questions/75888 | 3 | I've learned when you have a integral smooth scheme line bundles are the same as Cartier divisors are the same as Weil divisors. My question is to what extent does this continue to hold (if at all) when you are talking about ind objects. Since things become more subtle in positive characteristic lets say we are working... | https://mathoverflow.net/users/7 | Line bundles on Ind Schemes | I don't know the general story, but I think the correspondence between Weil divisors and line bundles breaks down already for $\mathbf{A}^\infty$, which is pretty much the smoothest ind-variety out there. Using the standard ind-structure on $\mathbf{A}^\infty$, the example I have in mind is the union of all hyperplanes... | 5 | https://mathoverflow.net/users/5081 | 75898 | 46,036 |
https://mathoverflow.net/questions/75856 | 2 | If one takes a group presentation then one can ask various questions of it, such as "is this element equal to the identity", "are these elements conjugate" etc. I was wondering if the solution to such a problem in a representation of a group always yields a solution to the problem with respect to the presentation.
Fo... | https://mathoverflow.net/users/6503 | Decision problems and group representations | Decidability of the word problem for a finitely generated group has nothing to do with a presentation of the group. It is not part of the input. If G is a finitely generated group with solvable word problem and H is a finitely generated group of automorphisms of G, then H has decidable word problem. There exists a Turi... | 4 | https://mathoverflow.net/users/15934 | 75899 | 46,037 |
https://mathoverflow.net/questions/75841 | 1 | Let $X$ be a manifold, $L$ be a complex line bundle over $X$, and $L^{\*}$ be the associated principal bundle. Suppose $\alpha$ is a connection form on $L^{\*}$, with associated connection $D$ on $L$. Suppose we wish to define a Hermitian product $\langle,\rangle$ on $L$ compatible with $D$ in the sense that for every ... | https://mathoverflow.net/users/17913 | Connections with compatible Hermitian products on complex line bundles | Wanted to clarify one potentially confusing difference between Śniatycki's and Prof. Figueroa-O'Farrill's notation: $\beta$ is actually the *local representative* of the connection $2\pi i~\alpha$ with respect to the non-vanishing local section $\sigma$, i.e., its pullback under $\sigma$
$$
\beta = 2\pi i~\sigma^\*\alp... | 1 | https://mathoverflow.net/users/17945 | 75904 | 46,038 |
https://mathoverflow.net/questions/75873 | 6 | I'm interested in $n \times m$ joint probability tables with prescribed row and column marginals. Such tables form a convex set known as the *transportation polytope*. What are the extreme points of this set?
For example, for a $2 \times 2$ case of
$$\begin{bmatrix} x\_{11} & x\_{12}\\\\
x\_{21} & x\_{22} \end{bmatri... | https://mathoverflow.net/users/14974 | Extreme points of transportation polytope | A complete solution with references can be found in Section 8.1 of Brualdi, Combinatorial Matrix Classes, Cambridge University Press, 2006.
Here is how to make an extreme point, and all extreme points can be made in this way. Suppose $\{r\_i\}$ and $\{c\_j\}$ are the required row and column sums. Start with a zero ma... | 10 | https://mathoverflow.net/users/9025 | 75905 | 46,039 |
https://mathoverflow.net/questions/61592 | 3 | Suppose $L$ is a lattice (free abelian group) and $\sigma$ is a (pointed) spanning rational cone in $L\otimes\mathbb Q$. Then $M=L\cap \sigma$ is a monoid with $M^{gp}=L$. A monoid of this form is called a *toric monoid*. Toric monoids are are precisely the finitely generated, commutative, integral (cancellative), shar... | https://mathoverflow.net/users/1 | Can cones (toric monoids) be built as colimits of their faces? | Yes, assuming that $D$ includes all faces (i.e. if $D\_i$ is in $D$, the so are all of its faces). This is Corollary 2.12 of [Toric Stacks II: Intrinsic Characterization of Toric Stacks](http://arxiv.org/abs/1107.1907).
The basic solution to both problems is to note that a toric monoid $M$ is determined by its dual ... | 3 | https://mathoverflow.net/users/1 | 75912 | 46,042 |
https://mathoverflow.net/questions/75908 | 7 | Cross-posted from <https://math.stackexchange.com/questions/65195/minimum-cardinality-of-a-difference-set-in-mathbb-rn>.
Given a finite set $S$ of $m$ points in $\mathbb R^n$ that do not all lie in the same $(n-1)$-dimensional hyperplane, consider the set of difference vectors:
$\{x-y \, | \, x,y \in S\}$
What is... | https://mathoverflow.net/users/9021 | Minimum cardinality of a difference set in $R^n$ | A basic inequality proved in 1987 by Freiman, Heppes, and Uhrin ("A lower estimation for the cardinality of finite difference sets in $R^n$", Number theory, Vol. I (Budapest, 1987), 125–139, Colloq. Math. Soc. János Bolyai, 51, North-Holland, Amsterdam, 1990) is that $|S-S|\ge(n+1)|S|-n(n+1)/2$. A number of improvement... | 8 | https://mathoverflow.net/users/9924 | 75918 | 46,044 |
https://mathoverflow.net/questions/75859 | 21 | Let $S\_\infty$ denote the full symmetric group on countably many points and index the points by $\mathbb{N}$. For any weight (for lack of a better name) function $w:\mathbb{N}\rightarrow\mathbb R^+$, define a subgroup $S\_\infty(w)\subset S\_\infty$ as follows:
$$S\_\infty(w) := \{\pi\in S\_\infty|\sum\_{\pi(k)\neq ... | https://mathoverflow.net/users/12301 | Is there an easy description of the structure of this infinite group? | Not an answer, just a series of observations which will hopefully be of some use, and too long for a comment. The last observation *might* be relevant to your question about whether $G$ is the direct sum of countably many copies of $S\_\infty$, unless I've over-simplified something.
1. For each weight $w$, we may ass... | 5 | https://mathoverflow.net/users/7521 | 75920 | 46,046 |
https://mathoverflow.net/questions/75827 | 1 | Hi guys, I have a question. Prove or disproof the statement:
---
Any inner horn $\Lambda[n,k],0< k< n $ admits a filtration $\mathbf{n}<\cdots<\Lambda[n,k]$, such that each step is filling an inner horn.
---
where $\mathbf{n}$ is the simplicial set $0\to 1 \cdots \to n$ (this notation is bad), its 1-cells are $... | https://mathoverflow.net/users/7341 | Decomposition of inner horns | Joyal and Tierney also prove (extending the result of Cordier and Porter) a large class of statements of this form (including, in addition to this fact, that the inclusion of the spine into a simplex (that is, the inclusion of $n$ into $\Delta^n$ in your notation) is inner anodyne). This is in the paper by Joyal and Ti... | 2 | https://mathoverflow.net/users/1353 | 75927 | 46,048 |
https://mathoverflow.net/questions/75921 | 7 | In [Linear logic, monads and the lambda calculus (DRAFT)](http://citeseer.ist.psu.edu/viewdoc/summary?doi=10.1.1.36.3645), Proposition 3.0.2 says that the Eilenberg-Moore category for a commutative monad has the structure of a symmetric monoidal closed category. My question is: How do you construct the tensor product f... | https://mathoverflow.net/users/4085 | What is the tensor product for the Eilenberg-Moore category of a commutative monad? | Caveat: this construction only works if your category of algebras has coequalizers of reflexive pairs, i.e. coequalizers of parallel pairs of arrows with a common right inverse.
Let $T \colon C \to C$ be your monad. Being commutative, it comes with maps $\mathrm{dst} \colon T(A) \otimes T(B) \to T(A \otimes B)$.
Let ... | 13 | https://mathoverflow.net/users/10368 | 75929 | 46,049 |
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