parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
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https://mathoverflow.net/questions/219909 | 8 | We have a graph G with maximum degree $\Delta$. The induced subgraph on vertices with degree equal to $\Delta$ is a bipartite graph (while the original graph is not).
Prove that G has a matching that covers all vertices with degree $\Delta$.
For example consider $K\_{3,3}$ and add a vertex on one edge. So the graph h... | https://mathoverflow.net/users/81011 | A matching that covers vertices with maximum degree | We use Tutte's theorem: if a graph $G$ with even number of vertices does not have a perfect matching, then there exists a set $S\subset V(G)$ such that the graph $G\setminus S$ has at least $|S|+2$ odd components.
Let $V=M\sqcup U$ be a vertex set, where $M$ is the set of vertices with maximal degree $\Delta$. We pro... | 8 | https://mathoverflow.net/users/4312 | 219978 | 103,200 |
https://mathoverflow.net/questions/219945 | 3 | Let $A\_n$ be the alternating group of $\{1,2,\cdots,n\}$.
(1). What is the cohomology ring
$$
H^\*(A\_4;\mathbb{Z}/3)
$$
and its Steenrod operation $P^i$'s?
(2). Are there general results about the cohomology ring
$$
H^\*(A\_{p+1};\mathbb{Z}/p)
$$
for general primes $p\geq 3$?
(3). What is the cohomology ring... | https://mathoverflow.net/users/65800 | mod p cohomology ring of alternating groups | An easier way for the first case
Consider $P$, a $3$-Sylow subgroup of $A\_4$. For example, take the cyclic group generated by the cyclic permutation (123). It is self normalizing, so the double coset formula for the compositions $BP\rightarrow BA\_4 \stackrel{tr}{\rightarrow}BP$ where $tr$ denotes the transfer, redu... | 1 | https://mathoverflow.net/users/43326 | 219981 | 103,202 |
https://mathoverflow.net/questions/219984 | 2 | Let $C$ be a (very) general genus 1 curve embedded in $\mathbb{CP}^1\times \mathbb{CP}^1$ as a (2,2)-divisor.
Each projection defines $C$ as a double cover of $\mathbb{CP}^1$ and induces an involution $\tau\_i:C\to C$. Let $G\simeq \mathbb Z/2 \* \mathbb Z/2$ be the subgroup of $Aut(C)$ generated by these. Note that... | https://mathoverflow.net/users/68722 | Finite orbits on an elliptic curve with two generic involutions | No. Letting $\sigma$ and $\tau$ denote the two involutions, $\sigma \circ \tau$ is a translation by an element of $\mathrm{Pic}^0(C)$. In general, this translation will not be torsion, so its orbit through any point is infinite. (In fact, if the translation IS torsion, then $G$ is a finite dihedral group, not $\mathbb{... | 5 | https://mathoverflow.net/users/297 | 219986 | 103,203 |
https://mathoverflow.net/questions/218687 | 4 | Consider the inhomogeneous linear heat equation
$$\partial\_tu-\Delta u=F$$
on $\mathbb R^n\times [0,1]$ (say) with zero initial data. Assume $F$ is very nice (say Schwarz), so that we have a nice solution $u$. It is quite standard that
$$\|\nabla^2 u\|\_{L\_x^2L\_t^2}\ll \|F\|\_{L\_x^2L\_t^2}. $$
I'm now askin... | https://mathoverflow.net/users/37103 | Mixed norm estimate for the heat equation | Yes, the estimate holds and is a very special case of theorem 2.2 in the paper by N.V. Krylov *'The heat equation in $L\_q((0,T),L\_p)$-spaces with weights', SIAM J. on Math. Anal., Vol. 32, No. 5 (2001)*.
| 1 | https://mathoverflow.net/users/14551 | 219989 | 103,204 |
https://mathoverflow.net/questions/219489 | 6 | The moduli space of K3 surfaces forms a 20-dimensional family with countably many 19-dimensional components $M\_d$ corresponding to the polarized K3s $(X,L)$ with $L^2=d$. The moduli space $M\_d$ has a natural locus $M\_d^W$ for each lattice $W$ with an embedding $W\subset H^{1,1}(X)\cap H^2(X,\mathbb Z)$. Is it known ... | https://mathoverflow.net/users/68722 | Loci in the moduli space of K3 surfaces associated to lattices | I won't completely answer your question, but will try to just rephrase it in a certain way. You are asking when two given moduli spaces of lattice-polarized K3 surfaces $M\_L$ and $M\_{L'}$ intersect. This is equivalent to the existence of a lattice with $L,L'\hookrightarrow N$ such that $M\_N$ is non-empty. Necessary ... | 3 | https://mathoverflow.net/users/8003 | 219992 | 103,205 |
https://mathoverflow.net/questions/219994 | 2 | Let $i : \mathbf P^1 \to \mathbf P^2$ be the second Veronese embedding. Clearly, $i\_\star \mathcal O\_{\mathbf P^1}$ has a locally free resolution of the form
$$
0 \to \mathcal O\_{\mathbf P^2} (-2) \to \mathcal O\_{\mathbf P^2} \to i\_\star \mathcal O\_{\mathbf P^1} \to 0
$$
More generally, for any integer $n$, w... | https://mathoverflow.net/users/nan | Veronese embeddings and locally free resolutions | The resolution looks like,
$$0\to\mathcal{O}\_{\mathbb{P}^2}(n-1)^{\oplus 2}\to \mathcal{O}\_{\mathbb{P}^2}(n)^{\oplus 2}\to i\_\*\mathcal{O}\_{\mathbb{P}^1}(2n+1)\to 0,$$ with the determinant of the $2\times 2$ matrix appearing in the left most map is just the equation of the quadric defining $i(\mathbb{P}^1)$.
| 4 | https://mathoverflow.net/users/9502 | 219997 | 103,207 |
https://mathoverflow.net/questions/219999 | 4 | <http://mathworld.wolfram.com/ChoquetTheory.html>
Is the claim in the link true? Here's the reference given there:
[https://www.renyi.hu/~p\_erdos/1934-01.pdf](https://www.renyi.hu/%7Ep_erdos/1934-01.pdf)
>
> Erdős proved that there exist at least one prime $\equiv 1\pmod{4}$ and at least one prime $\equiv 3\pm... | https://mathoverflow.net/users/81052 | Did Erdős prove there are two primes $4a+1, 4b+3$ between between $n$ and $2n$? | Yes, see the final page of
P. Erdos: Bizonyos számtani sorok törzsszámairól (On primes in some arithmetic progressions, in Hungarian), Bölcsészdoktori értekezés , Sárospatak, 1934, 1--20.
or its German translation:
P. Erdos: Über die Primzahlen gewisser arithmetischer Reihen (in German), Math. Z. 39 (1935), 473... | 6 | https://mathoverflow.net/users/16510 | 220001 | 103,209 |
https://mathoverflow.net/questions/220016 | 14 | Let $T$ be the generating function of the [Thue-Morse sequence](https://en.wikipedia.org/wiki/Thue%E2%80%93Morse_sequence); thus,
$T(x)=x+x^2+x^4+x^7+\dotsb$. It is known that $T$ satisfies the nice
congruence
$$ (1+x)^3 T^2(x) + (1+x)^2 T(x) + x \equiv 0 \pmod 2 $$
(the congruence is actually modulo the principal ide... | https://mathoverflow.net/users/9924 | Generating function of the Thue-Morse sequence | Let
$$F(x)=1-x-x^2+x^3-x^4+x^5+\ldots=(1-x)(1-x^2)(1-x^4)\ldots.$$
Then $F(x)=(1-x)F(x^2)$ and
$$F(x)=1+x+x^2+x^3+\ldots-2T(x)=\frac{1}{1-x}-2T(x).$$
So
$$T(x)-(1-x)T(x^2)=\frac{x}{1-x^2}.$$
| 14 | https://mathoverflow.net/users/5712 | 220021 | 103,219 |
https://mathoverflow.net/questions/220015 | 4 | Let $q$ be odd. If $G$ is a finite group such that $G$ has a normal subgroup $H$ of order $3$ such that $G/H\cong {\rm PGL}(2,q)$, what can we say about $G$. Is it true in general that $G\cong {\Bbb Z}\_3\times {\rm PGL}(2,q)$?
The motivation for this question: If we change ${\rm PGL}(2,q)$ to ${\rm PSL}(2,q)$, then ... | https://mathoverflow.net/users/81077 | Extensions of $\Bbb Z_3$ by $PGL(2,q)$ where $q$ is odd | Since the Schur Multiplier of the perfect group ${\rm PSL}(2,q)$ has order $2$, $G$ must have a normal subgroup $N$ of index $2$ isomorphic to $C\_3 \times {\rm PSL}(2,q)$. Since ${\rm PGL}(2,q) \setminus {\rm PSL}(2,q)$ contains an element of order $2$, we have $G = \langle N,t \rangle$, with $t^2=1$. The direct facto... | 5 | https://mathoverflow.net/users/35840 | 220023 | 103,221 |
https://mathoverflow.net/questions/220028 | 5 | Assume we are given for a transition between two time points $t\_0 = 0$ and $t\_1$ a matrix relationship, eventually describing the solution of a system of linear with non-constant coefficients,
$$Y(t\_1) = \exp(\Omega(t\_1,0))Y(0),$$
or, in a more general setting,
$$Y(t\_1) = \exp(\Omega(t\_1,0))Y(0)+c(t\_1,0),$$
wher... | https://mathoverflow.net/users/41452 | Getting out a system of linear ODEs by knowing the Magnus expansion | You can directly get the trace of $A$ from the identity
$${\rm det}\,\left[\exp\bigl(\Omega(t,0)\bigr)\right]=\exp\left[\int\_0^t {\rm tr}\,A(s)ds\right]$$
The full matrix $A$ is determined by $\Omega$ via the inverse Magnus expansion [see equation 4.2 from this [thesis](http://www.dms.uaf.edu/~bueler/tcarlsonMS.pd... | 4 | https://mathoverflow.net/users/11260 | 220033 | 103,222 |
https://mathoverflow.net/questions/220024 | 2 | Suppose $\mathbf{B}$ is a complete Boolean algebra with an infinite domain $B$. Suppose $\mathbf{B}$ is atomic (i.e. every element is the supremum of some set of atoms). This algebra contains the co-finite filter $\mathscr{F}\_c$, which in case the set of atoms of $\mathbf{B}$ is countable is generated by a chain. This... | https://mathoverflow.net/users/22019 | Boolean algebras and free filters generated by chains | For Q1, take the Boolean algebra of subsets of the rational numbers, and let the filter be generated by all of the nonempty final segments of $\mathbb Q$.
For Q3, repeat the preceding with the reals in place of the rationals. For other uncountable cardinals, repeat the preceding with the rationals and reals replaced ... | 4 | https://mathoverflow.net/users/6794 | 220041 | 103,226 |
https://mathoverflow.net/questions/220043 | 6 | What is an example of a (compact) manifold, which has two non-equivalent differential structures such that the K-homology groups are non-isomorphic? If no such example exists, i.e. "K-homology does not see the differential structure", then can someone give a heuristic explanation of why an object (originally at least) ... | https://mathoverflow.net/users/36946 | Differential structures and K-homology groups | K-homology is usually not defined in terms of pseudodifferential operators. In fact, I don't even know which definition you mean.
K-homology is either defined as the dual of K-theory (i.e., it is defined as the generalized homology theory associated to the K-theory spectrum), which means that it only depends on the w... | 9 | https://mathoverflow.net/users/13356 | 220047 | 103,228 |
https://mathoverflow.net/questions/220061 | 11 | What is an example of a group $G$ which
1- is finitely generated by $S$,
2- does not have property (T),
3- admits infinitely many finite quotients which do not factor through an homomorphism $G \to H$ for some [fixed & infinite] property (T) group $H$
4- all the Cayley graphs (w.r.t. $S$) of those finite quot... | https://mathoverflow.net/users/18974 | Groups without property (T) but all finite quotients are expanders | I think your condition (4) is called "property ($\tau$)". (See Theorem 4.3.2 of Lubotzky's very nice book "Discrete Groups, Expanding Graphs and Invariant Measures".) An example is $G = \mathrm{SL}\_2\bigl(\mathbb{Z}[1/p] \bigr)$. (See Example 4.3.3E on page 52 of Lubotzky's book.)
| 13 | https://mathoverflow.net/users/68305 | 220068 | 103,240 |
https://mathoverflow.net/questions/220065 | 9 | I am looking for a paper of Weil that is published under a pseudonym, in which he proves a statement along the lines of: a singular algebraic variety cannot be deformed into a nonsingular one.
Thanks in advance.
| https://mathoverflow.net/users/48554 | Weil's paper under a pseudonym on deforming singular varieties | this pseudonomous letter mentioned by Jim Humphreys is too amusing not to summarize here:
R. Lipschitz (Ann. of Math. 69, 1959, 247-251)
*reprinted in A. Weil, Collected Papers, volume II (Springer, 1979):*
---
and for the record, here is a translation of... | 13 | https://mathoverflow.net/users/11260 | 220074 | 103,243 |
https://mathoverflow.net/questions/220088 | 3 | Could anybody come up with a cited reference for the following concept?
>
> A subset $B$ of a topological vector space $X$ is called "bornophagic" if, for every bounded $A\subset X$, there exists $\delta>0$ such that $A\subset\delta B$.
>
>
>
| https://mathoverflow.net/users/nan | "bornophagic" in cited references | Usually, such sets are called *bornivorous*. E.g., in the book *Barrelled Locally Convex Spaces* of Bonet and Perez Carreras.
| 7 | https://mathoverflow.net/users/21051 | 220092 | 103,248 |
https://mathoverflow.net/questions/220089 | 4 | I am searching for a book/lecture notes/articles where I can find the definition and properties of the $pro-p$ Iwahori subgroup of $GL\_n(F)$,(with examples if possible) the Iwahori decomposition of it, and its applications concerning the Hecke Algebra. The subgroups play an important role in the work of M.F. Vigneras ... | https://mathoverflow.net/users/69289 | Reference request for $pro-p$ Iwahori subgroup of $GL_n(F)$ | The definition is quite simple. Let $\mathcal O$ be the ring of integers of $F$, which I shall assume is a local field of residual characteristic $p$. The *standard pro-$p$-Iwahori* is the group of matrices in $GL\_n(\mathcal O)$ that are upper unipotent modulo the maximal ideal $m$ of $\mathcal O$.
Any conjugate of th... | 6 | https://mathoverflow.net/users/9317 | 220099 | 103,250 |
https://mathoverflow.net/questions/220101 | 4 | $RCA\_0$ has $\Delta\_0$-comprehension and $\Sigma\_1$ induction. Let $X\Sigma\_{n}$ be $RCA\_0$ plus $\Sigma\_n$-induction and let $X\Sigma\_{\omega}$-induction be $RCA\_0$ plus the full induction schema.
Does any of the systems $X\Sigma\_{\delta}$ for $\delta\in\omega + 1$ prove the existence of sets that are not ... | https://mathoverflow.net/users/37385 | What can be achieved by liberalizing induction for $RCA_0$? | One answer is trivially "Yes" - fix some first-order sentence $\varphi$ which is provable from $X\Sigma\_n$ but not $RCA\_0$ alone and consider the formula $$\psi(x)\equiv (\varphi \vee \neg(x\in 0'))$$ (where "$x\in 0'$" is shorthand for the standard $\Sigma^0\_1$ formula defining the Halting Problem). Then $RCA\_0$ d... | 9 | https://mathoverflow.net/users/8133 | 220102 | 103,251 |
https://mathoverflow.net/questions/220097 | 6 | $WKL\_0$ extends $RCA\_0$ with the statement that any infinite subset of the infinite binary tree has an infinite branch. Does $WKL\_0$ Prove that there are sets which are not proven to exist by the $\Delta\_0$-comprehension schema of $RCA\_0$? If so, to what level of comprehension do such sets belong?
| https://mathoverflow.net/users/37385 | Does $WKL_0$ provide more comprehension than $RCA_0$? | If I understand your question correctly, the answer is
"no" in the following sense:
Fix a model $M=(\omega\_M, \mathbb{R}\_M)$ of $RCA\_0$, and let $\mathcal{C}\_M$ be the set of structures $N=(\omega\_N, \mathbb{R}\_N)$ such that
* $N\models WKL\_0$,
* $\omega\_N=\omega\_M$ (same first-order parts), and
* $\mathb... | 6 | https://mathoverflow.net/users/8133 | 220103 | 103,252 |
https://mathoverflow.net/questions/219615 | 8 | Is it known how the number of involutions in $GL\_n(2)$, the group of $n\times n$ matrices over $\mathbb{Z}/2\mathbb{Z}$, behaves as $n\to\infty$ ?
Equivalently, one may ask this for the number of $n\times n$ matrices $A$ over $\mathbb{Z}/2\mathbb{Z}$ satisfying $A^2=0$, as $(A+I)^2=I \mod 2$.
Needless to say, ther... | https://mathoverflow.net/users/11100 | asymptotic for the number of involutions in GL(n,2) | We may write $|{\rm GL}(n,2)| = 2^{n^{2}} \prod\_{j=1}^{n}( 1- \frac{1}{2^{j}}).$
As $n \to \infty$, the rightmost factor tends to $\left( \sum\_{r=0
}^{\infty} \frac{p(r)}{2^{r}} \right)^{-1}$, where $p(r)$ is the number of partitions of $r$. Let us write $|{\rm GL}(n,2)| = 2^{n^{2}}f(n)$.
Then we see that the numbe... | 3 | https://mathoverflow.net/users/14450 | 220104 | 103,253 |
https://mathoverflow.net/questions/220032 | 61 | Recall that a dagger category is a category equipped with an involution $\*:Hom(x,y)\to Hom(y,x)$ that satisfies $f^{\*\*}=f$ and $f^\* g^\*=(gf)^\*$. A prominent example of a dagger category is the category of Hilbert spaces and continuous linear maps.
---
Now, dagger categories are *evil!*
For example, here's ... | https://mathoverflow.net/users/5690 | Are dagger categories truly evil? | I will have another go at arguing that dagger-categories are *not* evil.
Let’s look at a simpler case first. Consider the property “$1 \in X$” on sets. As a property of abstract sets, this is evil: it’s not invariant under isomorphism, e.g. any iso $\{1,2\} \cong \{2,3\}$.
But it is manifestly non-evil as a propert... | 49 | https://mathoverflow.net/users/2273 | 220111 | 103,256 |
https://mathoverflow.net/questions/220117 | 11 | Let $\xi$ be a (real) vector bundle of dimension $n$. Then the first Stiefel-Whitney class
$$
w\_1(\xi)=0
$$
if and only if $\xi$ is orientable, i.e. the structure group of $\xi$ can be reduced to $SO(n)$.
**Question 1:** Let $\xi^\mathbb{C}$ be a complex vector bundle of dimension $n$. Then the first Chern class
$... | https://mathoverflow.net/users/65800 | first Chern class of complex vector bundles and first Pontrjagin class of quaternionic vector bundles | The natural question also relates to understanding holonomy in Riemannian geometry using the idea of $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$, and Cayley numbers as scalars. There are 8 discussions, two for each of the four choices of scalars, whether non orientable or orientable in each context.
Orientable : $SO(n)$... | 9 | https://mathoverflow.net/users/31331 | 220125 | 103,262 |
https://mathoverflow.net/questions/220115 | 0 | Let $U\subset\mathbb{C}^n$ be a domain of holomorphy, we can say that $U$ is a simply connected domain?
Any hints would be appreciated.
| https://mathoverflow.net/users/40409 | Domains of holomorphy and simply connected domains | As stated by PVAL, the answer is no. Any domain in $\mathbb{C}$ is a domain of holomorphy, so in particular, for example $\mathbb{D}^\*$, the punctured disk is one. (This particular domain is also easily seen directly to be a domain of holomorphy). Thus, one obtains a counterexample in $\mathbb{C}$, and taking the prod... | 1 | https://mathoverflow.net/users/49151 | 220127 | 103,264 |
https://mathoverflow.net/questions/220112 | 3 | I've seen proofs of the fact that the probability of two random integers being coprime is $\frac{6}{\pi^2}$ (all of them leading to a use of the Riemann Zeta function and the Basel problem). In several cases it was mentioned that one can easily deduce from that the fact that the average order of $\phi(n)/n$ is also equ... | https://mathoverflow.net/users/81095 | Deduce average order of $\phi(n)/n$ from probability that two integers are coprime | So the rigorous statement that "two random integers have a $6/\pi^2$ probability of being coprime" is
$$
\lim\_{N\to\infty}\frac{1}{N^2}\sum\_{n=1}^N\sum\_{m=1}^N
\mathbf 1\_{m\text{ and }n\text{ are coprime}}\to \frac 6{\pi^2}.
$$
This is the same as
$$
\lim\_{N\to\infty}\frac 2{N^2}\sum\_{n=1}^N
\sum\_{m=1}^{n-1}\m... | 6 | https://mathoverflow.net/users/11054 | 220130 | 103,266 |
https://mathoverflow.net/questions/220138 | 5 | Let $(X,\tau)$ be a topological space. We say that $x, y \in X$ are *close* if for every neighborhood $U$ of $x$ and $V$ of $y$ we have $U\cap V \neq \emptyset$. Let $E$ be the set of $\{x,y\}$ where $x,y\in X$ with $x\neq y$ and $x,y$ are close. We say $G(X,\tau) = (X,E)$ is the *closeness graph* of $(X,\tau)$.
Give... | https://mathoverflow.net/users/8628 | Closeness graph of a topological space | There is no topology on 4 elements whose closeness graph is the $4$-cycle.
(Proof: Let's call the elements $\{A,B,C,D\}$. Define a directed graph with these elements as vertices and a directed edge $u\to v$ whenever every open set containing $u$, also contains $v$, this has to be transitive and contain all self loops... | 7 | https://mathoverflow.net/users/2384 | 220150 | 103,271 |
https://mathoverflow.net/questions/220077 | 15 | Given $n$ points of general position in $\mathbb{R}^d$ (say, $n>d$ and no $d+1$ lie in a hyperplane.) We want to draw $k$ hyperplanes not passing through those points so that they all are in different open regions of the complement of the drawn planes. What is the minimal value $k(n,d)$ for which it may be always done?... | https://mathoverflow.net/users/4312 | separating points in $\mathbb{R}^d$ by minimal number of planes | $\lceil n/d\rceil+d-2$ hyperplanes always suffice. To achieve this, we first choose $H\_1,\dots,H\_{d-1}$ in a manner that $H\_1$ separates $\lceil n/d\rceil$ points from the rest, $H\_2$ separates $\lceil n/d\rceil$ of those rest ones, and so on. Thus we get $d$ sets $S\_1,\dots,S\_d$ separated from each other, each c... | 6 | https://mathoverflow.net/users/17581 | 220151 | 103,272 |
https://mathoverflow.net/questions/220124 | 0 | Is Andre-Oort conjecture expected to hold for complex analytic topology? If yes, do the recent results on abelian type Shimura varieties cover this case?
Sorry for my ignorance, but several references seem to deal with Zariski topology. Thanks.
Edit - Let $S$ be an infinite subset of special points Zariski dense i... | https://mathoverflow.net/users/81100 | Andre-Oort for conjecture | The André-Oort conjecture is about the special points of a Shimura variety that lie on a subvariety — it claims that if there is a Zariski dense set of those points, then the subvariety is special as well. (No need to know what special means, here.)
Since the Zariski topology is coarser than the complex analytic one... | 9 | https://mathoverflow.net/users/10696 | 220156 | 103,274 |
https://mathoverflow.net/questions/220157 | 4 | Suppose we have a system of linear and quadratic equations with rational coefficients and we want to find out whether this system has a real solution or not. The actual values of a solution is not important.
To be specific, let $D\in \mathbb{Q}^{m\times n}$,
$Q\_1,\ldots,Q\_r \in \mathbb{Q}^{n\times n}$, and $c\_1,\... | https://mathoverflow.net/users/81107 | Existence of real solutions for a system of linear and quadratic equations | The answer is yes, in principle, and this is explained in the reference that you cite. But a "simple" criterion probably does not exist. The state of the art is described in this book Sottile, Frank
Real solutions to equations from geometry.
American Mathematical Society, Providence, RI, 2011.
| 4 | https://mathoverflow.net/users/25510 | 220166 | 103,278 |
https://mathoverflow.net/questions/220152 | 7 | I’m interested in the question for which $n$ the special orthogonal group is homeomorphic to the product
$$ \mathrm{SO}(n) \approx S^{n-1} \times \mathrm{SO}(n-1). $$
Allen Hatcher [1, p. 293 f.] claims (?) that this is true for $n \in \{ 2, 4, 8 \}$ and wrong for all other values (although I’m not sure what is mea... | https://mathoverflow.net/users/65899 | Topological structure of SO(n) as a product | I think this is not an answer but a precisation. Assume that the statement is true for some $n$.
~~Then S^{n-1}\times SO(n-1)has a Lie group strcture and in particular a parallelizable tangent bundle. This implies that the tangent bundle of S^{n-1} is parallelizableand this is true if and only if n=2,4,8 (the original... | -1 | https://mathoverflow.net/users/41970 | 220176 | 103,280 |
https://mathoverflow.net/questions/219481 | 3 | There are several questions about transverse complete intersection arising from L. Guth's paper:
<http://www.ams.org/journals/jams/0000-000-00/S0894-0347-2015-00827-X/home.html>
We say a polynomial $P$ on $\mathbb{R}^n$ is nonsingular if for each point $x \in Z(P):=\{z\in \mathbb{R}^n\,|\, P(z)=0\}, $ we have that ... | https://mathoverflow.net/users/80791 | about transverse complete intersection | The key idea to prove these claims is Sard's theorem from differential topology. A good reference for this area is the book Differential Topology by Guillemin and Pollack.
Here is an outline of the proof of the first claim. The other two are similar.
Z(Q) is a manifold in R^3, and moreover grad Q is non-vanishing o... | 4 | https://mathoverflow.net/users/81126 | 220185 | 103,286 |
https://mathoverflow.net/questions/220100 | 5 | Let $k$ be an algebraically closed field of positive characteristic and $X$ be a smooth projective curve over $k$ of genus $g \ge 2$. Fix a polarization $L$ on $X$. Does there exist a semi-stable vector bundle on $X$ of rank $r$ and degree $d$ with gcd$(r,d)=1$?
I know that this result is true in characteristic zero... | https://mathoverflow.net/users/58203 | Existense of semi-stable vector bundles on smooth curves in positive characteristic | It seems relatively easy to construct such a bundle by induction. Namely:
**Step 0** If $r=1$, there are clearly line bundles of given degree $d$ on $X$, since $k$ is assumed to be algebraically closed.
**Step 1** Choose $(r',d')$ such that $0<r'<r$, $d'/r'>d/r$, and there are no integral points within (or on the e... | 4 | https://mathoverflow.net/users/2653 | 220194 | 103,288 |
https://mathoverflow.net/questions/220191 | 7 | I am trying to find out a closed-form formula (or a generating function at least) for the number of permutations $\sigma$ that satisfy $$ S = \sum\_{i = 1}^{n} i\sigma(i)$$ for a given value of $S$. We can find a maximum and minimum possible sum for a particular $n$ and I have observed that the intermediate sums (betwe... | https://mathoverflow.net/users/73880 | Weighted Permutation Sum | You can search this statistic (normalized so that the smallest value is 0) in www.FindStat.org and you will find that this is the rank of the permutation inside the lattice of alternating sign matrices. You find further information at <http://www.findstat.org/St000055> and the references there:
Sack, J., Úlfarsson, H... | 6 | https://mathoverflow.net/users/21291 | 220198 | 103,289 |
https://mathoverflow.net/questions/220030 | 16 | **My problem:** From the [Berry--Esseen theorem](https://en.wikipedia.org/wiki/Berry-Esseen_theorem) I know, that $$\sup\_{x\in\mathbb R}|P(B\_n \le x)-\Phi(x)|=O\left(\frac 1{\sqrt n}\right),$$ where $B\_n$ has the standardized binomial distribution and $\Phi$ is the standard normal distribution function. I can prove ... | https://mathoverflow.net/users/56668 | Normal approximation of tail probability in binomial distribution | The needed observation here is that, whereas the relative error of the Stirling approximation is worse for large $|x|$, it gets multiplied by a fast decreasing normal density function, and then the product gets integrated to produce the desired result. Here are the details.
Let $X$ be a binomial random variable with... | 7 | https://mathoverflow.net/users/36721 | 220199 | 103,290 |
https://mathoverflow.net/questions/220203 | 6 | Let $G$ be a finite abelian $p$-group, $p$ a prime. I say that a pair $(G',\varphi)$ is a *maximal cyclic quotient* (please excuse me if this definition already exists and refers to a different concept) of $G$ if $G'$ is a cyclic group and $\varphi\colon G\to G'$ is a surjective map with the following property: if $H\l... | https://mathoverflow.net/users/36370 | Maximal cyclic quotient of a $p$-group | Let $G = C\_p \times C\_{p^3} = \langle x \rangle \times \langle y \rangle$, and let $H = \langle z\rangle$, with $z = xy^p$. Then no automorphism of $G$ can map $H$ into one of the direct factors of $G$. Since $|H|=p^2$, it would have to map it into the $C\_{p^3}$ factor, and hence to $\langle y^p \rangle$. But $z$ ha... | 8 | https://mathoverflow.net/users/35840 | 220211 | 103,295 |
https://mathoverflow.net/questions/220106 | 26 | This is an embarrassingly simple question, but I was not able to find a definitive answer from literature search.
Suppose one has some collection of functions $f\_1: X \to Y\_1, \dots, f\_n: X \to Y\_n$ on a common domain $X$. Then one can form the function $(f\_1,\dots,f\_n): X \to Y\_1 \times \dots \times Y\_n$ in ... | https://mathoverflow.net/users/766 | What is the term for combining functions $f_1,f_2,\dots,f_n$ into a tuple $(f_1,\dots,f_n)$? | I was encouraged to make my comment an answer:
In the case $n = 2$, I would call it the pairing. Similarly, one has "tripling", "quadrupling", and so in general one might call it the ($n$-)*tupling* of the list $f\_1, \ldots, f\_n$. And indeed that is what the nLab calls it: see [here](http://ncatlab.org/nlab/show/p... | 17 | https://mathoverflow.net/users/2926 | 220213 | 103,297 |
https://mathoverflow.net/questions/220196 | 6 | I'd like to simplify the following expression:
$$\text{tr}\{\mathbf{A}^HE(\mathbf{C}^H \begin{bmatrix} \mathbf{0}\_{M\times M} & \mathbf{0}\_{M\times N} \\ \mathbf{0}\_{N\times M} & \mathbf{I}\_{N\times N} \end{bmatrix} \mathbf{C})\mathbf{A}\}$$, where the matrix $\mathbf{C}$ is Toeplitz and is constructed by shiftin... | https://mathoverflow.net/users/81130 | Is there a way to simplify the following trace expression? | After a cyclic permutation of the trace, the expression you need is
$$Y=\text{tr}\left\{\mathbf{A}^HE(\mathbf{C}^H \begin{bmatrix} \mathbf{0}\_{M\times M} & \mathbf{0}\_{M\times N} \\ \mathbf{0}\_{N\times M} & \mathbf{I}\_{N\times N} \end{bmatrix} \mathbf{C})\mathbf{A}\right\}=\text{tr}\left\{E(\mathbf{C}\mathbf{A}\m... | 8 | https://mathoverflow.net/users/11260 | 220219 | 103,300 |
https://mathoverflow.net/questions/220181 | 4 | Paraphrasing from Cortes' [notes](https://www2.math.hu-berlin.de/gradkoll/Cortes_vorlesung1_handout.pdf):
>
> The quaternionic Kähler condition for a manifold $M$, means that $\operatorname{End}(T(M))$ admits a
> parallel subbundle $Q$ which is locally spanned by $3$
> anticommuting skew-symmetric almost complex ... | https://mathoverflow.net/users/75217 | Homogeneous Quaternionic-Kähler Structure of the Grassmannians? | These spaces are [Wolf spaces](https://en.wikipedia.org/wiki/Quaternion-K%C3%A4hler_symmetric_space) of the form $G/K$, where $K$ or a double cover equals $H\times SU(2)$. Here, $G=SU(p+2)$ and $H=U(p)$. The isotropy representation here is the exterior tensor product of some $H$-representation
with the standard represe... | 2 | https://mathoverflow.net/users/70808 | 220223 | 103,302 |
https://mathoverflow.net/questions/220214 | 18 | Is there a complex manifold diffeomorphic to $\mathrm{SU}(3)$?
This question arises in a StackExchange discussion by HK Lee, Ted Shifrin and Jason DeVito:
<https://math.stackexchange.com/questions/488959/way-distinguishing-whether-or-not-complex-manifold>
I believe it is also related to Etesi's work on a complex s... | https://mathoverflow.net/users/30172 | Is there an integrable complex structure on $\mathrm{SU}(3)$? | It is an old theorem of Samelson that any compact Lie group $G$ of even rank has an integrable complex structure, which, in particular applies to the case of $\mathrm{SU}(3)$. Basically, one chooses a Cartan subalgebra $\frak{t}\subset\frak{g}$, which gives a splitting of the Lie algebra into (complex) root spaces, cho... | 34 | https://mathoverflow.net/users/13972 | 220230 | 103,306 |
https://mathoverflow.net/questions/220163 | 6 | Let $X$ be a complete intersection in $\mathbb{P}^n$ of multidegree $(d\_1,\ldots,d\_r)$. If we're working over a finite field $\mathbb{F}\_q$, the Ax-Chevalley-Warning theorem says that if $X$ is in the Fano range, i.e., $$ \sum d\_i \leq n,$$ then $|X(\mathbb{F}\_q)| \equiv 1 \pmod q.$
In the case that $X$ is not i... | https://mathoverflow.net/users/36254 | Smooth complete intersections and sharpness of the Chevalley-Warning theorem | Extension of **Daniel Loughan**'s example (which is also known, but not as
well-known as it should(?) be): if a prime $p$ is of the form $dn+1$ then
the Fermat hypersurface $\sum\_{i=1}^d x\_i^d = 0$ in ${\bf P}^{d-1}({\bf F}\_q)$
is smooth and its number of rational points is not congruent to $1 \bmod p$.
Indeed the ... | 7 | https://mathoverflow.net/users/14830 | 220244 | 103,309 |
https://mathoverflow.net/questions/220247 | 1 | In some results on Hölder continuity with regards to standard Brownian motion, the following is asserted without proof.
>
> It is not hard to see that for every $k < \infty$, and every $\epsilon > 0$,$$\mathbb{P}\left\{ \sup\_{0 < s < t < 1} {{|B\_t - B\_s|}\over{\sqrt{t - s}}} \ge k\right\} > 1 - \epsilon,$$where ... | https://mathoverflow.net/users/nan | Standard Brownian motion, Hölder continuous with exponent $\gamma$ for any $\gamma < 1/2$, not for any $\gamma \ge 1/2$ | Perhaps this is what you're looking for?
Fix a sequence $t\_n \downarrow 0$ and some $k > 0$ and let $A\_n = \{ \sup\_{0 < t < t\_n} B\_t/\sqrt{t} \ge k\}$. Since $B\_{t\_n}/\sqrt{t\_n}$ has a standard normal distribution, we have $$\mathbb{P}(A\_k) \ge \mathbb{P}(B\_{t\_n}/\sqrt{t\_n} \ge k) = 1-\Phi(k) > 0$$ where ... | 6 | https://mathoverflow.net/users/4832 | 220248 | 103,311 |
https://mathoverflow.net/questions/220220 | 21 | I have recently been wondering if the following is consistent with ZFC:
>
> For every infinite ordinal $\alpha$: $|V\_\alpha\cap L|=|\alpha|$.
>
>
>
Intuitively, this states that for $L$ is very "thin", in that it doesn't branch off too much from the set of ordinals as we climb cumulative hierarchy.
I have h... | https://mathoverflow.net/users/30186 | Can $L$ be thin? | **Claim:** $|V\_\alpha \cap L| = |\alpha|$ for every $\alpha \geq \omega$ implies that $0^\#$ exists.
**Proof:** Let's assume, toward contradiction, that $0^\#$ doesn't exist that $|V\_\alpha \cap L| = |\alpha|$ for every $\alpha \geq \omega$.
Let $\mu$ be a singular cardinal in $V$. Let's apply the assumption $|V... | 29 | https://mathoverflow.net/users/41953 | 220252 | 103,313 |
https://mathoverflow.net/questions/220253 | 3 | I want to know if there exists a characterization of k-foliations of $\mathbb{R}^n$ which have all the leaves closed.
Do exists a $k$-foliation of $\mathbb{R}^n$ with a non-closed leaf?
In general, is there any characterization of manifolds in which all the foliations have only closed leaves?
Thanks in advance.
... | https://mathoverflow.net/users/37338 | Closed leaves on foliations of $\mathbb{R}^n$ | For all $n\ge 3$ and $k\in\{1,\dots, n-1\}$, there is a dimension $k$ foliation of $\mathbb{R}^n$ with non-closed leaves.
I first describe a relatively obvious construction which only works in codimension $2$ and higher. In $\mathbb{R}^3$ (thus $k=1$), consider the Hopf fibration on the sphere minus one point. It is ... | 3 | https://mathoverflow.net/users/4961 | 220258 | 103,314 |
https://mathoverflow.net/questions/220257 | 0 | Let ${\frak P}$ denote the collection of prime ideals containing the finite members of ${\cal P}(\omega)$, and order ${\frak P}$ by set inclusion.
What is the cardinality of ${\frak P}$, and what's the largest cardinality that a chain in ${\frak P}$ can have?
| https://mathoverflow.net/users/8628 | Prime ideals containing the finite members of ${\cal P}(\omega)$ | The prime ideals you ask about are the duals of the non-principal ultrafilters on $\omega$. They are all maximal, so the largest cardinality of a chain is $1$. The total number of such ultrafilters is $2^{2^{\aleph\_0}}$, the same as the total number of subsets of the real line.
| 6 | https://mathoverflow.net/users/6794 | 220260 | 103,315 |
https://mathoverflow.net/questions/220261 | 1 | Let $(X,\tau)$ be a topological space. We define the "moving" relation by setting $$ x \simeq\_m y \text{ iff there is a homemomorphism }\varphi: X\to X \text{ such that } \varphi(x) = y.$$
Clearly $\simeq\_m$ is an equivalence relation. We call a space "immovable" if $\simeq\_m$ is the diagonal $\Delta\_X=\{(x,x):x\... | https://mathoverflow.net/users/8628 | "Immovable" topological spaces | No. Let $X$ be the disjoint union of the real line and one isolated point. The quotient by "movability" collapses the real line to one point, so the quotient is a discrete space of two points, which is not immovable.
| 8 | https://mathoverflow.net/users/6794 | 220262 | 103,316 |
https://mathoverflow.net/questions/220263 | 5 | Let $T\_{0,n}$ be the Teichmuller space of $n$-punctured genus $0$ Riemann surface, and $M\_{0,n}$ the Moduli space (assume $n\geq 3$ and the punctures are numbered). What is the correct notion of the mapping class group $\Gamma\_{0,n}$, so that $M\_{0,n} \cong T\_{0,n} / \Gamma\_{0,n}$?
Now $\pi\_1(M\_{0,n}) \cong ... | https://mathoverflow.net/users/11392 | Mapping class group of a punctured genus 0 surface | Let $S\_g$ be a compact Riemann surface of genus $g$ with $n$ marked points $x\_1, \ldots, x\_n$, and set $S\_{g, \, n}:=S\_g - \{x\_0, \ldots, x\_n\}$. Also, denote by $\pi\_{g, \, n}$ the fundamental group of $S\_{g, \, n}$ (we omit the decoration $n$ when $n=0$).
Then the mapping class group $\Gamma\_{g,\, n}$ is ... | 3 | https://mathoverflow.net/users/7460 | 220264 | 103,317 |
https://mathoverflow.net/questions/220267 | -2 | Let $G = (V,E)$ be a finite, simple, undirected, connected graph, such that contracting an edge reduces the chromatic number. Does this imply that $G$ is complete?
| https://mathoverflow.net/users/8628 | Graphs such that contracting an edge decreases the chromatic number | Any odd cycle of length at least 5 is a counterexample.
| 2 | https://mathoverflow.net/users/12705 | 220269 | 103,318 |
https://mathoverflow.net/questions/220265 | 3 | I already posted a question about a sum involving the degree of a Kummer extension.
Now I'm interested in a more specific fact about Kummer extensions.
From Hooley's paper "On Artin's conjecture", we know that if $k\_n=[\mathbb{Q}(\zeta\_n,a^{1/n}):\mathbb{Q}]$
is the degree of a Kummer extension for a fixed integer $a... | https://mathoverflow.net/users/50610 | Bibliography suggestion for Kummer theory | $$\mathbb Q \left(\left( a \over b\right)^{1/n} \right)=\mathbb Q\left( \left( ab^{n-1}\right)^{1/n}\right)$$
| 3 | https://mathoverflow.net/users/76105 | 220273 | 103,320 |
https://mathoverflow.net/questions/220274 | 11 | I am looking for an example where $f:Y\to X$ and $f':Y'\to X$, are both smooth maps of smooth manifolds, but the pullback does not exist.
**Remarks:**
1) A pullback in a certain category is defined as a space satisfying a universal property, not as a fiber product. (Which is just the usual form of many pullbacks..... | https://mathoverflow.net/users/46290 | Does pullback in the category of smooth manifolds always exists? | If a pullback exists in the category of smooth manifold then, its underlying set of points has to be what you described simply by looking at morphism from the point. Moreover a map into the pullback is smooth if and only if the map to the product is smooth (because it is smooth if and only if each component is smooth b... | 12 | https://mathoverflow.net/users/22131 | 220278 | 103,322 |
https://mathoverflow.net/questions/220134 | 4 | Given an infinite distributive lattice $L$, does $L$ contain a non-principal prime ideal $I$, or a non-principal prime filter $F$? ($I$ is said to be principal if there is $x\in L$ such that $I=\{y\in L: y\leq x\}$. Dual definition for filters.)
If the answer is "yes" to the question above, what if we replace "prime"... | https://mathoverflow.net/users/8628 | Non-principal prime ideals in infinite distributive lattices | It’s not clear to me what exactly is the intended definition of prime ideals and filters in distributive lattices. Based on an analogy with other classes of structures, it seems to me that conceptually the best choice should be to make the definition correspond to subdirectly irreducible factors in subdirect products, ... | 5 | https://mathoverflow.net/users/12705 | 220281 | 103,324 |
https://mathoverflow.net/questions/220277 | 4 | It's all in the title: **Are [Wolf spaces](https://en.wikipedia.org/wiki/Quaternion-K%C3%A4hler_symmetric_space) [flag manifolds](https://en.wikipedia.org/wiki/Generalized_flag_variety)**? Both are group quotients of semi-simple Lie groups. In the Grassmannian case this is so, and I always tacitly assumed it extended t... | https://mathoverflow.net/users/12653 | Are Wolf spaces flag manifolds? | Flag manifolds have the form $G/C(S)$ where $C(S)$ is the centralizer (in $G$) of its center $S$ (a torus).
Most $G/H$ in the [list](https://en.wikipedia.org/wiki/Quaternion-K%C3%A4hler_symmetric_space) don't have this form, for $H$ has discrete center in each case except the complex Grassmannians $\mathrm{SU}(p+2)/\... | 5 | https://mathoverflow.net/users/19276 | 220284 | 103,325 |
https://mathoverflow.net/questions/220283 | 2 | <http://mathworld.wolfram.com/HamiltonDecomposition.html>
>
> In the 1890s, Walecki showed that complete graphs K\_n admit a Hamilton decomposition for odd n, and decompositions into Hamiltonian cycles plus a perfect matching for even n (Lucas 1892, Bryant 2007, Alspach 2008).
>
>
>
I was wondering if there is... | https://mathoverflow.net/users/81011 | Simple decomposition of $K_{2n}-I$ into hamiltonian cycles | In a 1-factor decomposition of $K\_{2n}$ which you describe (an edge from the center of a regular $(2n-1)$-gon to a vertex and all chords perpendicular to it), two matchings which correspond to `almost opposite' edges from the center form a Hamiltonian cycle. After you collect $n-1$ such pairs (by rotating one of them)... | 2 | https://mathoverflow.net/users/17581 | 220288 | 103,327 |
https://mathoverflow.net/questions/220128 | 8 | So, in learning about category $\mathcal{O}$ representations of a semisimple Lie algebra $\mathfrak{g}$, I've come across two natural kinds of subcategories, and I think I'm confused about their structure and the relationship between them. (In particular, if they are equivalent or not).
Fix $\lambda \in \mathfrak{h}^... | https://mathoverflow.net/users/81101 | Confusion about Subcategories of Category $\mathcal{O}$ | The source of your confusion is the difference between B-equivariance and being smooth along B-orbits. Since in step 3, you want only smoothness along B-orbits, you have to do the same thing in step 1. In order to do that, you need to take some modules not in category $\mathcal{O}$, and instead consider modules where t... | 10 | https://mathoverflow.net/users/66 | 220294 | 103,330 |
https://mathoverflow.net/questions/220296 | 5 | Let $A$ be a unital $C^\*$-algebras and $K(H)$ is $C^\*$-algebras of compact operators on separable Hilbert space $H$. Is it true that $(A \otimes K(H))^{\*\*}= A^{\*\*} \overline{\otimes}B(H)$?
| https://mathoverflow.net/users/73660 | second dual of minimal tensor products of $C^*$-algebras | Yes (but maybe there is a more direct argument ? ):
$A$ and $K(A) = A \otimes K(H)$ are Morita equivalent so they have equivalente categories of representations, moreover this equivalence is implemented as following: any representation of $K(A)$ is of the form $H \otimes R$ with $R$ a representation of $A$.
$A^{\*\... | 4 | https://mathoverflow.net/users/22131 | 220301 | 103,333 |
https://mathoverflow.net/questions/220300 | 5 | I need an algorithm for getting the order of the group in random elliptic curves mod n, being n a composite module. As far as I know, usual algorithms like Schoof's algorithm only works with prime modulus. What are the alternatives?
| https://mathoverflow.net/users/81180 | What is the fastest algorithm for counting points in elliptic curves mod n? | I don't expect that there will be a known polynomial time algorithm. If I recall correctly, we don't know a polynomial time algorithm to decide, given $n=pq \equiv 1 \pmod 4, p,q$ primes, whether $p,q \equiv 1 \pmod 4$ or not. If we had a polynomial time algorithm to compute the number of points in $y^2 =x^3+x \pmod n$... | 8 | https://mathoverflow.net/users/2290 | 220303 | 103,334 |
https://mathoverflow.net/questions/220236 | 2 | We say that a sequence $(x\_{n})\_{n}$ in a Banach space $X$ is unconditionally $p$-summable ($1\leq p<\infty$) if
$$\sup\_{x^{\*}\in B\_{X^{\*}}}(\sum\_{n=m}^{\infty}|\langle x^{\*},x\_{n}\rangle|^{p})^{\frac{1}{p}}\rightarrow 0\quad (m\rightarrow \infty).$$ We denote the set of all unconditionally $p$-summable seque... | https://mathoverflow.net/users/41619 | A question on unconditionally $p$-summable sequences | Yes. Choose $n\_1<n\_2<\dots$ s.t. for all $m$
$$
\sup\_{x^{\*}\in B\_{X^{\*}}}\sum\_{k=n\_m}^\infty |\langle x^\*, x\_k \rangle|^p < \epsilon^p/4^m.
$$
Set $\lambda\_k = 1$ for $k<n\_1$ and $\lambda\_k = 2^{-k}$ for $n\_m\le k < n\_{m+1}$.
| 0 | https://mathoverflow.net/users/2554 | 220308 | 103,337 |
https://mathoverflow.net/questions/220268 | 10 | A function $f\colon \mathcal{P}(\mathbf{N})\to [0,1]$ is said to have the *Darboux property* whenever for all $X \subseteq \mathbf{N}$ and $y \in [0,f(X)]$, there exists $Y \subseteq X$ such that $f(Y)=y$.
Moreover, $f$ is said to be *monotone* if $f(X)\le f(Y)$ whenever $X\subseteq Y$ and *subadditive* if $f(X\cup ... | https://mathoverflow.net/users/32898 | Does a monotone subadditive $f: \mathcal{P}(\bf N)\to [0,1]$ admit a finite partition with values in $(0,1)$? | $\let\eps\varepsilon$It seems that the following function fits:
$$
f(A)=\inf\left\{\alpha\colon \quad \sum\_{x\in A}x^{-\alpha}<\infty\right\}.
$$
(I assume that $\mathbb N$ starts with 1.)
Clearly, it is monotone, and $f(\mathbb N)=1$. Moreover, it is more than subadditive: we have $f(A\cup B)=\max\{f(A),f(B)\}$. Thi... | 7 | https://mathoverflow.net/users/17581 | 220314 | 103,338 |
https://mathoverflow.net/questions/220290 | 2 | I wonder if it is possible to find (and if yes, where?) an electronic copy of the following monograph:
**Author:** Schmickler-Hirzebruch, Ulrike
**Title:** *Elliptische Flächen über $\mathbb P^1(\mathbb C)$ mit drei Ausnahmefasern und die hypergeometrische Differentialgleichung.*
Schriftenreihe des Mathematisch... | https://mathoverflow.net/users/36575 | Looking for Schmickler-Hirzebruch' monograph on elliptic surfaces | This is the journal that is now called the [Münster Journal of Mathematics.](http://wwwmath.uni-muenster.de/mjm/index.html) Only recent volumes, from 2008, are online. The complete journal, including the volume 33 you are looking for, has actually been digitized by [Hathitrust](http://catalog.hathitrust.org/Record/0006... | 2 | https://mathoverflow.net/users/11260 | 220316 | 103,339 |
https://mathoverflow.net/questions/220313 | 1 | Consider a two-component tame link in 3-space, consisting of an arc from $(-1,1,0)$ to $(1,1,0)$ and an arc from $(-1,-1,0)$ to $(1,-1,0)$, confined to the slab $-1 \leq x \leq 1$. Call such a link trivial if it can be deformed so that the arcs admit parameterizations in which the $x$-coordinate is strictly increasing.... | https://mathoverflow.net/users/3621 | Cancellation of 2-component links | What you are talking about are called *string links*. String links have a stacking monoid operation. It's non-commutative (provided you have two or more strands). The invertible elements are precisely the string links that are braids. There are a few references on this topic.
I forget who first proved the result abo... | 1 | https://mathoverflow.net/users/1465 | 220317 | 103,340 |
https://mathoverflow.net/questions/220322 | 11 | Recently I was talking to an alien who does not know complex function theory. I was trying to convince her that the set of conformal equivalence classes of smooth embedded tori in $R^3$ is two parametric. Is there a nice two-parametric family of tori in $R^3$ which are pairwise conformally non-equivalent?
I do not h... | https://mathoverflow.net/users/25510 | Tori in three-space | It is a 1961 result by Adriano Garsia that *every* conformal class can be represented by an embedded surface. His proof seems to be reasonably constructive, see [this question](https://mathoverflow.net/questions/53999/conformal-embedding-of-riemann-surfaces-into-3-space) (and answers thereto).
**EDIT** Pinkall, in [t... | 8 | https://mathoverflow.net/users/11142 | 220325 | 103,342 |
https://mathoverflow.net/questions/220315 | 2 | Let $\operatorname{Ad}:\operatorname{SL}\_n(\mathbb{R}) \to \operatorname{GL}(\mathfrak{sl}\_n(\mathbb{R}))$ be the adjoint representation (i.e. $\operatorname{Ad}(g)X=gXg^{-1}$) of $SL\_n(\mathbb{R})$. Then is the image of $\operatorname{Ad}$ a Zariski closed subgroup?
In general, are the any sufficient conditions ... | https://mathoverflow.net/users/47704 | When is the image of the adjoint representation of a real algebraic group Zariski closed? | Let $G$ be a connected linear algebraic group over the field of real numbers $\mathbb{R}$.
By "connected" I mean "connected over $\mathbb{C}$".
Let $G(\mathbb{R})$ denote the group of $\mathbb{R}$-points of $G$.
It does not have to be connected.
Let $\Gamma\subset G(\mathbb{R})$ be a subgroup of finite index in $G(\mat... | 3 | https://mathoverflow.net/users/4149 | 220329 | 103,344 |
https://mathoverflow.net/questions/220327 | 3 | Let $B\_t$ be a standard Brownian motion. Does there with probability one exist a sequence of partitions $\{t\_{k, n} : k = 0, 1, \dots, k\_n\}$ $$0 = t\_{0, n} < t\_{1, n} < \dots < t\_{k\_n, n} = 1,$$with$$\lim\_{n \to \infty} \max\{t\_{j, n} - t\_{j - 1, n} : j = 1, \dots, k\_n\} = 0$$and$$\liminf\_{n \to \infty} \s... | https://mathoverflow.net/users/81192 | Brownian motion, quadratic variation, existence of partitions? | If the sequence of partitions is fixed/deterministic, the answer is no by an old result of Paul Lévy (P. Levy , Theorie de l'addition des variables aleatoires, Paris, 1937).
If one allows the partitions to depend on the element of the probability space than the quantity will get as large as $2\log \log n$ almost sure... | 4 | https://mathoverflow.net/users/630 | 220334 | 103,345 |
https://mathoverflow.net/questions/220328 | 5 | The title says it all.
I suspect that the answer in general is no, although my intuition tells me that a jump in the dimension of the fibre of the nilradical at some point of Spec(A) can occur only when one meets an associated prime cycle.
Note the easiest case: when the ring is generically reduced then it has to b... | https://mathoverflow.net/users/17988 | In a noetherian commutative ring with only one associated prime, is the nilradical locally free? | Actually, this *never* happens unless the nilradical is $0$ (i.e., unless the ring is a domain). Let $A$ be a Noetherian ring with only one associated prime whose nilradical $N$ is locally free. Then $A$ has only one minimal prime, so $N$ is prime. Now localize at $N$. We now have a $0$-dimensional Noetherian local rin... | 6 | https://mathoverflow.net/users/75 | 220335 | 103,346 |
https://mathoverflow.net/questions/220349 | 5 | Dumb question: Let $X\_n:\Omega \to \mathbf{R}$ be a sequence of $L^2(\Omega,\Sigma,\mathbf{P})$ random variables that has a weak limit $X$ in $L^2$.
Suppose also that $\mu\_n$, the distributions of $X\_n$ on $\mathbf{R}$ converge vaguely to $\mu$.
Does the distribution of $X$ have to be $\mu$? If the $X\_n$ are u... | https://mathoverflow.net/users/47510 | Weak convergence of random variables in $L^2$ and vague convergence | No, it is not true. Take $\Omega = [0,1]$ equipped with the Lebesgue measure and let $(X\_n)\_{n \in \mathbb{N}}$ be the sequence of Rademacher functions -- all of these have the same distribution (uniform on $\{-1,1\}$) but they converge weakly in $L^{2}[0,1]$ to $0$, since they form an orthonormal system. They are al... | 7 | https://mathoverflow.net/users/24953 | 220351 | 103,351 |
https://mathoverflow.net/questions/220352 | 7 | Let $M$ be a $m$-dimensional manifold whose cohomology ring and cell structure are well-understood, such that there is a free action of the symmetric group $S\_n$ on $M$. Then we have a $n!$-sheeted covering space
$$
\pi:M\to M/S\_n.
$$
Let $S\_n$ act on the Euclidean spaces $\mathbb{R}^n$ and $\mathbb{C}^n$ by permuti... | https://mathoverflow.net/users/65800 | vector bundles associated to a covering space | When you have an Euclidean vector space of type:
$$\mathbb{R}^n\rightarrow M\times\_{S\_n}\mathbb{R}^n\rightarrow M/S\_n$$
Then its classifying map $\phi:M/S\_n\rightarrow BO(n)$ factors as:
$$M/S\_n\rightarrow BS\_n\stackrel{\rho\_n}{\rightarrow} BO(n)$$
where $\rho\_n$ is induced by the regular reprsentation $S\_n\ri... | 11 | https://mathoverflow.net/users/27816 | 220362 | 103,354 |
https://mathoverflow.net/questions/220354 | 13 | Given two natural numbers $n\geq 1$ and $b\geq 2$, denote by $S\_b(n)$ the sum of the digit of $n$ in its representation in base $b$. Clearly $S\_b(n)$ varies from 1 (when $n$ is a power of $b$) to $(b-1)\lfloor\ln{n}/\ln{b}\rfloor$ (when $n=b^k-1$ for some integer $k$). However, for a fixed $n$, the value of $S\_b(n)$... | https://mathoverflow.net/users/54552 | Average digit sum in different bases | Numerics suggest that $S(n) \sim Cn^2$ for some constant $C$ between $0.175$ and $0.18$.
Note that the bases $\frac n2<b\le n$ are easy to calculate: we get a sequence of two-digit numbers of the form $1x$, where $x$ runs from $\frac n2$ or so to $0$ (covering all integers in between). The sum of all these digits is ... | 14 | https://mathoverflow.net/users/5091 | 220365 | 103,355 |
https://mathoverflow.net/questions/220363 | 4 | "The symmetric power $L$-functions are a powerful tool for studying algebraic or geometric objects through analytic methods." I read this sentence in the introduction of a Master thesis. I want to find examples when one can use the analytic properties of the symmetric power $L$-functions to establish results in analyti... | https://mathoverflow.net/users/76102 | Examples when one can use the the symmetric power $L$-functions to study topics related to the number theory | [Applications of Symmetric Power $L$-functions](http://www.mast.queensu.ca/~murty/Murty.pdf), a set of lecture notes by Ram Murty on number theory applications including the Sato-Tate conjecture, the Ramanujan conjecture, the Selberg eigenvalue conjecture, and Artin's conjecture on the holomorphy of non-Abelian $L$-ser... | 8 | https://mathoverflow.net/users/11260 | 220377 | 103,358 |
https://mathoverflow.net/questions/220366 | 2 | Recently i'm reading a paper,there is a inequality that confuse me. L is a symmetric,irreducible and semi-positive definite matrix with eigenvalues of $0=\lambda\_{1}(L)<\lambda\_{2}(L)\leq...\leq\lambda\_{m}(L)$(counting the multiplicities),P is a positive definite matrix with maximum eigenvalues $\lambda\_{m}(P)$,bot... | https://mathoverflow.net/users/61978 | an inequality about kronecker product with eigenvalues question | We use $L^t \otimes P^t = (L\otimes P)^t$ twice below. The proof follows by observing that
\begin{eqnarray\*}
\sup\_{x\neq 0}\frac{x^T(L\otimes P)^2 x}{x^T(L\otimes P)x} = \sup\_{z=(L\otimes P)^{1/2}x} \frac{z^T(L\otimes P)z}{z^Tz} \le \sup\_{z \neq 0}\frac{z^T(L\otimes P)z}{z^Tz} = \|L\otimes P\|.
\end{eqnarray\*}
| 3 | https://mathoverflow.net/users/8430 | 220386 | 103,362 |
https://mathoverflow.net/questions/220388 | 2 | Let $X$ be a smooth projective surface and $C$ a Cartier divisor on $X$. Denote by $\mathcal{H}^1\_C(\mathcal{O}\_X)$ the sheaf associated to the presheaf $U \mapsto H^1\_{C \cap U}(\mathcal{O}\_X|\_U)$. Let $j:X\backslash C \to X$ be the natural immersion. Using the local cohomology sequence (see Hartshorne Ex. III.$2... | https://mathoverflow.net/users/46578 | On conflicting descriptions for tor of a local cohomology group | I believe I understand now your first construction. You are comparing the following two short exact sequences. $$\begin{array}{ccccccccc} 0 & \rightarrow & \mathcal{O}\_X & \rightarrow & \mathcal{O}\_X(C) & \rightarrow & \mathcal{O}\_X(C)/\mathcal{O}\_X & \rightarrow& 0 \\ & & =\downarrow & & \downarrow & &\downarrow \... | 1 | https://mathoverflow.net/users/13265 | 220397 | 103,366 |
https://mathoverflow.net/questions/220323 | 7 | Let $G$ be a reductive group over a finite field $k$, let $F$ be a Frobenius morphism on $G$.
I'll start with a somewhat vague question and make my question more specific further down:
*How do cuspidal representations fit into Deligne-Lusztig characters $R\_{T,\theta}$?*
A little about what is known classically: ... | https://mathoverflow.net/users/30726 | Structure of Deligne-Lusztig representations $R_{T,\theta}$ for ministropic $T$ and cuspidal representations | EDIT: It's easy to answer Question 2 affirmatively by pointing to the groups $G=\mathrm{Sp}(4,q)$ of Lie type $B\_2= C\_2$ in odd characteristic. Bhama Srinivasan first worked out the ordinary irreducible characters of $G$ using *ad hoc* methods in her thesis work at Manchester: the resulting paper is [here](http://www... | 5 | https://mathoverflow.net/users/4231 | 220401 | 103,367 |
https://mathoverflow.net/questions/220400 | 7 | Let $X$ be some smooth scheme over $\mathbf C$ equipped with an action of $\mu\_n$ (the group of $n$th roots of unity).
The étale cohomology groups of X are therefore equipped with an action of $\mu\_n$.
Now, let's suppose that the action of $\mu\_n$ on $X$ extends to an action of $\mathbf G\_m$.
Then, the analytic ... | https://mathoverflow.net/users/5239 | Algebraic proof without using comparison theorem for étale cohomology | Smoothness of $X$ is not needed (neither for the comparison isomorphism nor for the result in question). Let $X$ be any quasi-separated scheme over a separably closed field $k$, equipped with an action by a connected $k$-group scheme $G$ of finite type. Let $n > 0$ be an integer not divisible by the characteristic of $... | 7 | https://mathoverflow.net/users/70739 | 220405 | 103,370 |
https://mathoverflow.net/questions/220396 | 21 | Let $f:[0,1]\to[0,1]$ be given. The level sets of $f$ (ie the collection of all sets of the form $\{x\in[0,1]:f(x)=y\}$, for each fixed $y\in[0,1]$) partition the domain of $f$. I am curious for set theoretic or point set topology criteria for which partitions of $[0,1]$ could be the level sets for a continuous functio... | https://mathoverflow.net/users/35158 | Which partitions of $[0,1]$ are collection of level sets of a real continuous function? | Any map $[0,1]\to[0,1]$ is a quotient map onto its image, and the image must be either a point or a closed interval. So a partition comes from such a map iff the quotient of $[0,1]$ by the equivalence relation associated to the partition is homeomorphic to either a point or an interval. In particular, given any charact... | 25 | https://mathoverflow.net/users/75 | 220407 | 103,371 |
https://mathoverflow.net/questions/220387 | 4 | Can the number of points at integral distance to all three points of a non-degenerate triangle of area $A$ be bounded by $1+cA$ for some suitable constant $c$?
Remark: Since it is easy to bound this number by $4(D+1)^2$ where $D$ is the diameter of the triangle, a sequence giving rise to counterexamples must consist... | https://mathoverflow.net/users/4556 | Bounding the number of points at integral distance from vertices of a triangle | Even a baby version turns out to be wrong: there exist triangles with arbitrarily small area such that there are two points at integral distances to all three vertices; this already shows that there is no such $c$.
Let $R$ be a large integer, and take a triangle $PAQ$ with $PA=R$, $PQ=R+1$, $AQ=2$. Choose a point $B... | 2 | https://mathoverflow.net/users/17581 | 220412 | 103,374 |
https://mathoverflow.net/questions/220192 | 3 | Is there a parallel algorithm for doing modular multiplication of polynomials over Z/nZ? n is a very large number (for hundreds and thousands of bits).
Normally, the method used is binary exponentiation, but it's not a good idea for parallelization.
| https://mathoverflow.net/users/69999 | Parallel algorithm for modular multiplication of polynomials over Z/nZ | This problem is covered in great detail in Knuth's "The art of computer programming, volume II: Seminumerical algorithms". If the degree of the polynomials is $k$, then generalized Karatsuba schemes give the product of these polynomials in $O(k^{1+\varepsilon})$ multiplications modulo $n$, and these schemes parallelize... | 3 | https://mathoverflow.net/users/37555 | 220418 | 103,377 |
https://mathoverflow.net/questions/220195 | 4 | Imagine you have a shift invariant ($\sigma$-invariant) probability measure $\eta$
in the Bernoulli space $\{0,1\}^{\mathbb{N}}$. Define
$\mathcal{P} = \{[0],[1]\}$;
$\mathcal{P}^{n} = \mathcal{P}\vee...\vee \sigma^{-n+1}(\mathcal{P})$ (cylinders of length $n$);
$k\_{n} = \#\{P \in \mathcal{P}^{n}:\eta(P)\leq\fra... | https://mathoverflow.net/users/66009 | Entropy equals zero? | EDIT - The answer below deals with an ergodic m.p.s
As this question got up-voted, I've decided to fuly write a solution, based on the sketch I've made in the comments.
Fix some $\varepsilon>0$ small, and $n \gg \_\varepsilon 0$, and denote by $C\_{n}$ to be the cylinders of length $n$.
Let $h=h\_{\mu}(\sigma)$ be ... | 1 | https://mathoverflow.net/users/8857 | 220422 | 103,379 |
https://mathoverflow.net/questions/220378 | 4 | There is known theorem of Laguerre, that every linear ordinary differential equation of second order
$$y''+A(t)y'+B(t)y=0$$
by point transformation could be mapped into
$$y'' = 0,$$ that in few words means that we may get rid of two last terms.
Also there is generalization for higher order ODE. But is there any gener... | https://mathoverflow.net/users/nan | Generalized Theorem of Laguerre | Yes, there is a generalization that covers this case, and much more general second order systems. For example, you can consult L. P. Eisenhart's 1927 book *Non-Riemannian Geometry*, where he develops the geometry of paths (which is what you are asking about) along the line of his research on the subject with Veblen. He... | 8 | https://mathoverflow.net/users/13972 | 220425 | 103,380 |
https://mathoverflow.net/questions/220426 | 5 | An [interesting question](https://puzzling.stackexchange.com/questions/22846/88/) came up in the Puzzling Stack Exchange a few days ago about "queen-connected sets". When trying to solve this problem, I came across an arrangement of five colours of queens that would not attack each other on a toroidal 5×5 chessboard, a... | https://mathoverflow.net/users/27459 | $n$ groups of $n$ queens on a toroidal chessboard | The "queen numbers" are precisely those numbers whose smallest prime factor is at least 5.
It is a theorem of Polya that
>
> You can place n queens on an nxn toroidal board such that no two queens can attack each other if and only if the smallest prime factor of n is at least 5.
>
>
>
I couldn't find Polya's... | 8 | https://mathoverflow.net/users/70618 | 220433 | 103,385 |
https://mathoverflow.net/questions/54219 | 0 | Please help to find books about orders and algebras on trees.
If there is no modern books, please advice good old ones!
I'm more interested in finite trees (my current problem), but infinite ones are very appreciated too (as probably there is no difference between fin/inf in some contexts).
I'm especially interested ... | https://mathoverflow.net/users/3315 | Modern books about orders and algebras on trees | 1) Trees (Springer Monographs in Mathematics) 1st ed. 1980. Corr. 2nd printing 2002 Edition by Jean-Pierre Serre
| 1 | https://mathoverflow.net/users/3315 | 220437 | 103,388 |
https://mathoverflow.net/questions/220312 | 1 | Let $X$ be a noetherian scheme over $\mathbb{C}$, and let $E$ be a locally free sheaf of finite rank over $X$. Then we have the projective bundle $f: \mathbb{P}(E)\rightarrow X$.
Now $f$ is a flat morphism and we have $f\_{\*}\mathcal{O}\_{\mathbb{P}(E)}=\mathcal{O}\_X$ and $R^if\_{\*}\mathcal{O}\_{\mathbb{P}(E)}=0$ ... | https://mathoverflow.net/users/70593 | Can one drop the locally free assumption in projection formula on a projective bundle? | Pick an affine covering of $X$ over which $E$ trivialises. For an affine $U = Spec(A)$ from the covering $\Gamma(U, R^if\_\*f^\*H) = H^i(f^{-1}(U), f^\*H)$. Now, use the standard covering of $f^{-1}(U)=Proj(A[x\_0,x\_1,\ldots,x\_n])$ to compute Cech cohomology.
| 2 | https://mathoverflow.net/users/10941 | 220439 | 103,390 |
https://mathoverflow.net/questions/220447 | 19 | $$
x \cdot y = \frac{1}{2 \cdot 2 !} \left( (x + y)^2 - (x - y)^2 \right)
$$
$$
\begin{eqnarray}
x \cdot y \cdot z &=& \frac{1}{2^2 \cdot 3 !} ((x + y + z)^3 - (x + y - z)^3 \nonumber \\
&-& (x - y + z)^3 + (x - y - z)^3 ),
\end{eqnarray}
$$
$$
\begin{eqnarray}
x \cdot y \cdot z \cdot w &=& \frac{1}{2^3 \cdot 4 !} ( (... | https://mathoverflow.net/users/81243 | Are the following identities well known? | Although not the exactly the same due to $2^{n-1}$ instead of $2^n$ terms, the OP's formula seems to be essentially the well-known *polarization* formula for homogeneous polynomials, which is stated as following:
>
> Any polynomial $f$, homogeneous of degree $n$ can be written as $f(x)=H(x,\ldots,x)$ for a specific... | 33 | https://mathoverflow.net/users/8430 | 220450 | 103,393 |
https://mathoverflow.net/questions/220449 | 7 | Let $p$ be prime and $q = p^n$. Let $E$ be an elliptic curve over $\mathbb{F}\_q$, and let $E^{(p)}$ be the pullback of $E$ by the $p$-power Frobenius of $\mathbb{F}\_q$. If $E$ is isomorphic (over $\mathbb{F}\_q$) to its Galois conjugate $E^{(p)}$, then does it follow that $E$ is the base change of an elliptic curve o... | https://mathoverflow.net/users/63877 | Is an elliptic curve that is isomorphic to its Frobenius conjugate defined over $\mathbb{F}_p$? | Almost, but not quite. If $j(E)$ is the $j$-invariant of $E$ then, under your hypothesis, $j(E)=j(E^{(p)})=j(E)^p$, so $j(E) \in \mathbb{F}\_p$. Hence $E$ is either defined over $\mathbb{F}\_p$ or is a twist of such a curve. If you take the quadratic twist in $\mathbb{F}\_{p^2}$ of an elliptic curve defined over $\math... | 9 | https://mathoverflow.net/users/2290 | 220452 | 103,394 |
https://mathoverflow.net/questions/220241 | 5 | Let $L=\Delta + c$ in 3 dimensions, where $c$ is a positive constant.
I met this modified mean value property of a solution $u$ of $Lu=0$ as
$$u(\xi)=\frac{\sqrt{c}\rho}{sin(\sqrt{c}\rho)}\frac{1}{4\pi \rho^2}\int\_{\partial B(\xi,\rho)} u(x)d\sigma(x)$$
where $sin(\sqrt{c}\rho) \ne 0$, and $\sigma$ denotes the surf... | https://mathoverflow.net/users/51546 | Modified mean value property | Ah, it is one of these cases when you better know your formula in all dimensions.
It is
$$
\int\_{|x-y|=\rho} u(x) d\sigma(x) = (2\pi)^{n/2} \cdot \frac{J\_{n/2-1}(\sqrt{c}\rho)}{(\sqrt{c}\rho)^{n/2-1}} \cdot u(y)
$$
One of the ways to prove it:
$$
\Delta u + c u = 0,
$$
$$
\int\_{|x-y|\leq R} \Delta u(x... | 2 | https://mathoverflow.net/users/16623 | 220458 | 103,397 |
https://mathoverflow.net/questions/220459 | 1 | Consider two closed convex cones $A$ and $B$ in $\mathbb{R}^3$. Assume that they are convex even without zero vector, i.e. $A \setminus \{0\}$ and $B \setminus \{0\}$ are also convex (it helps to avoid weird cases like a plane being convex cone).
Suppose that they do not have common directions, i.e. $A \cap B = \{0\}$.... | https://mathoverflow.net/users/81250 | Convex cones: strict separation | I think you may be misreading the Hahn-Banach theorem in this case, as it should give you a strict separating plane here. Anyway, you can avoid this problem by enlarging the cones a bit before applying Hahn-Banach. This is no problem in the finite dimensional case.
Here's what I believe is a direct proof: Find a codi... | 7 | https://mathoverflow.net/users/2622 | 220467 | 103,398 |
https://mathoverflow.net/questions/140554 | 10 | Let $\mathfrak{g}$ be a complex simple Lie algebra with bracket $[x,y]$. For **which** $z\in \mathfrak{g}$ does the formula
$$
\mu(x,y)=ad (z)([x,y])=[z,[x,y]]
$$
define another Lie bracket on the same vector space ? For $\mathfrak{g}=\mathfrak{sl}(2,\mathbb{C})$ this holds for every $z$. Explicit computation seems to ... | https://mathoverflow.net/users/32332 | Homotopes of simple Lie algebras | In the paper [Derivation Double Lie Algebras](https://homepage.univie.ac.at/Dietrich.Burde/papers/burde_49_double_lie.pdf) the following result is proved:
*Theorem $3.2$*: Let $\mathfrak{g}$ be a simple Lie algebra of rank $r\ge 2$ over an algebraically closed field $K$ of characteristic
zero, and $z\in \mathfrak{g}$... | 7 | https://mathoverflow.net/users/32332 | 220475 | 103,403 |
https://mathoverflow.net/questions/220461 | 6 | Let $S\_n$ be the symmetric group of $n$ points. I want to find references (or proofs) for the following statement (1).
(1). There does not exist any faithful orthogonal representation
$$
S\_n\longrightarrow O(n-2).
$$
In order to prove (1), I want to use the result that any unitary representation of $S\_n$ is ove... | https://mathoverflow.net/users/80110 | references for faithful orthogonal (or unitary) representation of symmetric groups | The smallest degree faithful representation of $S\_n$ is $n-1$ in characteristic does not divide $n$. It is Theorem 22 of Chapter 19, Section 8 of Y. G. Berkovich, E. M. Zhmud; Characters of finite groups. Part 2.
Translated from the Russian manuscript by P. Shumyatsky, V. Zobina and Berkovich. Translations of Mathemat... | 5 | https://mathoverflow.net/users/15934 | 220478 | 103,405 |
https://mathoverflow.net/questions/220476 | 9 | Let $G$ be a finite subgroup of $\textrm{Gl}\_{n+1}(k)$ (where $k$ is an algebraically closed field). My question is: do there exist examples of $G$ such that the corresponding quotient $P$ of $\mathbb{P}^n$ by $G$ is not isomorphic to $\mathbb{P}^n$ but yet is locally a set-theoretic complete intersection (if we embed... | https://mathoverflow.net/users/2191 | Which weighted projective spaces (and their finite quotients) are local complete intersections? | Regarding your question about weighted projective spaces, a lot is known about them, see for instance **[1]** and **[2]**.
In particular, any weighted projective space $\mathbb{P}(\mathcal Q)$ is irreducible, normal, Cohen-Macaulay and has at most cyclic quotient singularities (hence rational singularities), see **[2... | 15 | https://mathoverflow.net/users/7460 | 220483 | 103,408 |
https://mathoverflow.net/questions/220484 | 2 | Let $S$ be a non-trivial simple group and suppose $S \trianglelefteq G$ if $C\_G(S)=1$ then $S$ is characteristic in $G$. To prove this let $\phi$ be an automorphism of $G$ and note that the intersection $S\cap \phi(S)$ can't be trivial since otherwise $S$ commutes with $\phi(S)$ in $G$. Therefore since both $S$ and $\... | https://mathoverflow.net/users/79888 | Perfect centerless normal subgroups | Let $S$ be a finite simple group and $V$ a faithful absolutely irreducible module for $S$. Then $W = V \otimes V$ is a faithful irreducible module for $S \times S$.
Let $G = W \rtimes (S \times S)$ be the corresponding semidirect product of $W$ with $S \times S$. Then $G$ has two normal subgroups of the form $W \rtim... | 2 | https://mathoverflow.net/users/35840 | 220487 | 103,410 |
https://mathoverflow.net/questions/220440 | 17 | Let $P\_n$ be the set of degree $n$ polynomials that pass through $(0,1)$ and $(1,1)$ and are non-negative on the interval $[0,1]$ (but may be negative elsewhere).
Let $a\_n = \min\_{p\in P\_n} \int\_0^1 p(x)\,\mathrm{d}x$ and let $p\_n$ be the polynomial that attains this minimum.
Are $a\_n$ or $p\_n$ known sequen... | https://mathoverflow.net/users/50796 | Non-negative polynomials on $[0,1]$ with small integral | Following Robert Israel's answer, we also scale everything to $[-1,1]$ (thus multiplying the result by 2). As he mentions, the optimal polynomial is always a square of some other polynomial, $p\_{2n}=p\_{2n+1}=q\_n^2$, and $q\_n$ is either even or odd (see Lemma below). So we are left to find the minimal $L\_2[-1,1]$-n... | 20 | https://mathoverflow.net/users/17581 | 220489 | 103,411 |
https://mathoverflow.net/questions/220486 | 7 | It is well-known by Grothendieck (or earlier by Dedekind-Weber) that every vector bundle on $\mathbb{P}^1\_k$ for $k$ a field decomposes into a sum of the line bundles $\mathcal{O}(k)$.
As investigated by [Hübl and Sun](http://www.tandfonline.com/doi/abs/10.1080/00927879908826642), this fails if we replace $k$ by a d... | https://mathoverflow.net/users/2039 | Vector bundles on open (affine) curves | By a theorem of Serre, see Theorem 1 in [this paper](http://www.numdam.org/item?id=SD_1957-1958__11_2_A9_0), vector bundles over smooth affine curves over fields are direct sum of a line bundle and a trivial bundle. Then the classification problem is given by the line bundle classification. In particular, both question... | 7 | https://mathoverflow.net/users/50846 | 220492 | 103,412 |
https://mathoverflow.net/questions/220496 | 5 | I think I can prove the following using the compactness of first order logic and I am wondering what a purely algebraic proof would look like.
>
>
> >
> > Let $R$ be a unital ring (not necessarily commutative but definitely associative) whose additive group is finitely generated. Suppose that $K\otimes R$ is sem... | https://mathoverflow.net/users/15934 | Lefschetz Principle for semisimplicity | Over a perfect field, or one with characteristic larger than $\dim\_F A$, semisimple is the same as [separable](https://en.wikipedia.org/wiki/Separable_algebra).
Over a field $F$ of characteristic larger than $\dim\_F A$, separable is the same as strongly separable. So, discarding finitely many characteristics, we can... | 3 | https://mathoverflow.net/users/297 | 220505 | 103,416 |
https://mathoverflow.net/questions/220494 | 5 | I asked this on mathstackexchange but didn't get any response (or many views) so I'm asking it here, although clearly it belongs over there.
In the answer to [this](https://mathoverflow.net/questions/117684/are-spectra-really-the-same-as-cohomology-theories) question on mathoverflow, it says:
"The integral homology g... | https://mathoverflow.net/users/80739 | unwinding the definition of $H_i(KU)$ as a map of spectra $\mathbb{S}^i \to HZ \wedge KU$ | I'll define a *naive prespectrum* to be a system of spaces $X\_i$ with maps $\Sigma X\_i\to X\_{i+1}$. That seems to be what you want to work with, but it is not technically very satisfactory. However, if $\mathcal{C}$ is one of the fancier categories of spectra, then one can define $FX$ to be the homotopy colimit in $... | 7 | https://mathoverflow.net/users/10366 | 220506 | 103,417 |
https://mathoverflow.net/questions/220502 | 7 | As is well-known (see Friedrich's book for example) every Kähler manifold is spin (or at least spin$^c$) and the Dirac is given (up to a twist) by $\partial + \partial^\*$. What happens in the quaternionic-Kaehler and hyper-Kähler cases? Are they spin$^c$, does the Dirac admit a nice description? I'm particularly inter... | https://mathoverflow.net/users/12653 | Spin Structures for Quaternionic-Kaehler and Hyper-Kaehler Manifolds | A very natural Dirac operator for Wolf spaces is discussed in Köhler, K, Weingart, G.,
Quaternionic analytic torsion,
Adv. Math. 178 (2003), 375–395. It acts on subcomplexes of the de Rham complex, in analogy with the Kähler situation.
EDIT: There is a hierarchy of groups
$$Spin(n)\hookrightarrow Spin^c(n)=Spin(n)\cd... | 6 | https://mathoverflow.net/users/70808 | 220523 | 103,423 |
https://mathoverflow.net/questions/220501 | 12 | Assume GCH and that $\kappa$ is a regular uncountable cardinal. Let $\mathbb{P}$ be a separative, $<\kappa$-directed closed, nowhere trivial, $\kappa^+$-cc poset of size $\kappa^+$. Must $\mathbb{P}$ be forcing equivalent to $\text{Col}(\kappa, < \kappa^+)$? (which I believe is equivalent to adding $\kappa^+$ many Cohe... | https://mathoverflow.net/users/26319 | Forcings that are not equivalent to Levy collapse | First note that in the special case $\kappa = \omega\_1$, countably closed implies countably directed-closed.
**Answer 1:** No. In any model of CH, there is a countably closed, $\omega\_2$-c.c. forcing that is inequivalent to $\mathbb P = \text{Add}(\omega\_1,\omega\_2)$.
Proof: Let $\mathbb Q$ be Jensen's partial ... | 8 | https://mathoverflow.net/users/11145 | 220524 | 103,424 |
https://mathoverflow.net/questions/220511 | 7 | Consider the space $C\_c(\mathbb{R})$ of continuous real-valued functions on $\mathbb{R}$ equipped with the inductive limit topology by $C\_c(\mathbb{R}) = \bigcup\_{n \in \mathbb{N}} C\_c(\mathbb{R}, K\_n)$ where $K\_n$ is some compact exhaustion of $\mathbb{R}$ by compact sets and $C\_c(\mathbb{R}, K\_n)$ is the Bana... | https://mathoverflow.net/users/58682 | Questions on topologies on space of Radon measures |
>
> It's not sequential because its closed subspace $M[0,1] = (C[0,1])^\ast$ is not sequential.
>
>
>
Here is an example of a set $A \subset M[0,1]$ that is sequentially $\tau\_v$-closed but not $\tau\_v$-closed:
Consider sequence of functions $f\_n \in C[0,1]$, $\Vert f \Vert\_{C[0,1]} = 1$ and $\operatorname... | 2 | https://mathoverflow.net/users/22758 | 220528 | 103,425 |
https://mathoverflow.net/questions/220532 | 3 | Let $\Omega^{\*}\_{\text{poly}}\: : \: sSet\to dg\_{\geq 0}Comm\_{+}$ be the polynomial De Rahm functor on simplicial sets, where the codomain is the category of commutative differential graded algebras over a field of charachteristic zero. Let $G$ be a group, what is $\Omega^{\*}\_{\text{poly}}(BG)$? Its cohomology? D... | https://mathoverflow.net/users/41970 | Polynomial differential forms on $BG$ | Let $\mathbb{F}$ be a field of characteristic zero.
For any simplicial set $Y$ denote by $RY$ its topological realization then you have an isomorphism of graded algebras:
$$H^\*(\Omega\_{poly}^\*(Y,\mathbb{F}))\cong H^\*\_{Sing}(RY,\mathbb{F}).$$
Any book about rational homotopy theory will be a good reference:
* A... | 5 | https://mathoverflow.net/users/27816 | 220534 | 103,427 |
https://mathoverflow.net/questions/220533 | 7 | I'm trying to sort out the history of spectral methods in the study of real analytic $GL\_2$-Eisenstein series. From what I read so far, I would say that the subject was really kicked off by the seminal work of Selberg who developped a a very broad theory which applies to any weakly symmetric Riemannian spaces and not ... | https://mathoverflow.net/users/11765 | History of spectral methods to the study of real analytic $GL_2$-Eisenstein series | R. Rankin's 1939 paper giving a non-trivial estimate on Ramanujan's $\tau$ function used the "real-analytic Eisenstein series" for $SL\_2(\mathbb Z)$, at least. Selberg's related paper just-slightly later seemed to express the same awareness for such cases, as opposed to the more general situations treated in the 1950'... | 11 | https://mathoverflow.net/users/15629 | 220535 | 103,428 |
https://mathoverflow.net/questions/220541 | 5 | With a graduate student, I'm going through the paper (Proc. London Math. Soc. (3) 47 (1983), no. 2, 193–224.)
Here's the background and notation.
We have a quadratic character $\chi$ modulo $q$, with Siegel zero $\beta\_0$. $\eta=((1-\beta\_0)\log q)^{-1}$, so $3\le \eta\ll q$ is known. Let $L=\log q$. We take
$$... | https://mathoverflow.net/users/6756 | Another question on Heath-Brown's "Prime twins and Siegel zeros" | "Those oft are stratagems which errors seem,
Nor is it Homer nods, but we that Dream."
Heath-Brown's proof is fine. It needs just one more line of explanation.
Note that
$$
\sum\_{\substack{\rho \neq \beta \\ |\gamma| \le 1}} \frac{1}{|\rho -1|^2} \ll
\int\_{r\_0}^{2} \# \{ \rho \neq \beta: |\rho -1| \le x \} \... | 4 | https://mathoverflow.net/users/38624 | 220542 | 103,430 |
https://mathoverflow.net/questions/220543 | 2 | Given two PDE(s): $F(x,y,z,p,q)=0$
and $G(x,y,z,p,q)=0$
In I.A.N Sneddon's "Elements of Partial Differential Equations",If every solution of $F=0$ is a solution of $G=0$,then $F=0$ and $G=0$ are said to be compatible.
But according to another textbook if $F=0$ and $G=0$ have atleast one common solution then they... | https://mathoverflow.net/users/81290 | Proving compatibility of two Partial differential equations | Actually, you have described three entirely different notions of 'compatibility' of a pair of first order PDE for a single function of two variables. The first two that you have listed are not the standard ones (and, in fact, are not very useful), and I am surprised that you were able to find them stated that way in te... | 3 | https://mathoverflow.net/users/13972 | 220548 | 103,433 |
https://mathoverflow.net/questions/220556 | 3 | This question was asked at [MSE](https://math.stackexchange.com/questions/1465274/is-2n-1-finitely-many-times-the-product-of-consecutive-primes) but recieved no attention at all.
Here it is:
Are there finitely many $(n,k) \in \mathbb{N}^2$ with $2^n-1=p\_1p\_2\cdots p\_k$ ?
$p\_1=3,p\_2=5 , ...,p\_k$ are consecu... | https://mathoverflow.net/users/38851 | Is $2^n -1$ finitely many times the product of consecutive primes? | $\let\dvds\mid$Yes. By Zsigmondy's theorem, $2^{12}-1$ has some prime divisor $p\_s$ not dividing $2^i-1$ for $i<12$ (in fact, $p\_s=13$). Now, if $2^n-1=p\_1p\_2\dots p\_k$ with $k\geq s$, then $p\_s\dvds 2^n-1$, so $12\dvds n$ and hence $3^2\dvds 2^n-1$, which is impossible. Thus only the cases with $k<s$ are left.
... | 7 | https://mathoverflow.net/users/17581 | 220558 | 103,437 |
https://mathoverflow.net/questions/220471 | 5 | Let $k$ be an algebraically closed field of characteristic zero. and let $$\sigma: SL\_n(k)\rightarrow SL\_n(k)$$
be an involution.
My questions are:
* How could one calculate the fundamental group of $SL\_n(k)^\sigma$ ? (the invariant subgroup)
* In particular, what is $\pi\_1(SO\_n(k))$ and $\pi\_1(Sp\_{2n}(k))... | https://mathoverflow.net/users/75343 | What is the algebraic fundamental groups of $SO(n)$ and $Sp(2n)$? | Let $k$ be an algebraically closed field of characteristic 0.
Let $G$ be a connected reductive group over $k$.
The notion of the algebraic fundamental group of $\pi\_1(G)$
was introduced in
[my memoir here](http://www.math.tau.ac.il/~borovoi/papers/galofile.pdf)
and generalized to arbitrary characteristic
[here](http... | 7 | https://mathoverflow.net/users/4149 | 220568 | 103,440 |
https://mathoverflow.net/questions/220554 | 8 | From my understanding the proof of Thurston's hyperbolization theorem for Haken $3$--manifolds consists of cutting the manifold along a hierarchy (collection of incompressible, $\partial$-incompressible surfaces) to obtain a collection of $3$--balls. A hyperbolic structure is put on the $3$--balls, and then a bootstrap... | https://mathoverflow.net/users/37434 | Conditions on the hierarchy for Thurston's hyperbolization theorem | Yes, there are conditions; basically you want to maintain the hypothesis of not having incompressible annuli at each stage. A really good reference for this is Morgan's essay, "On Thurston's uniformization theorem for three-dimensional manifolds", in Morgan, John W.; Bass, Hyman, The Smith conjecture (New York, 1979), ... | 5 | https://mathoverflow.net/users/3460 | 220572 | 103,441 |
https://mathoverflow.net/questions/220562 | 4 | It's a well-known open problem (Sophie-Germain primes) whether there are infinitely many primes $p$, $2p+1$. What about $p$, $2p-1$?
Seemingly it's also an open problem (see [here](https://math.stackexchange.com/questions/1472622/pair-of-primes-of-the-form-4k3-and-8k5) and the linked question).
I am aware that it is... | https://mathoverflow.net/users/81052 | Primes $p$ for which $2p-1$ is prime | I agree with Felipe Voloch that this question is not quite suitable for MO. At any rate, it is well-known (folklore) among number theorists that solving any single $k=2$ case of [Dickson's conjecture](https://en.wikipedia.org/wiki/Dickson's_conjecture) would be a major breakthrough. The best approximations we know are ... | 5 | https://mathoverflow.net/users/11919 | 220573 | 103,442 |
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