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https://mathoverflow.net/questions/202472 | 0 | Cross-posted from MSE: <https://math.stackexchange.com/q/1226622/15624>.
Let $G$ be any finite group, and $S\_{\aleph\_0}$ the group of all bijections $\mathbb{Z}\rightarrow \mathbb{Z}$.
Is $G\times S\_{\aleph\_0}\cong S\_{\aleph\_0}$?
| https://mathoverflow.net/users/18335 | direct product of a finite group with an infinite symmetric group | No. The only proper normal subgroups of the infinite symmetric group is the group of permutations with finite support, and the group of even such permutations. Your group also has the normal subgroup $G.$ For the proof of the statement in the first sentence, see Pete Clark's answer to this: [Sign of infinite permutatio... | 2 | https://mathoverflow.net/users/11142 | 202473 | 97,606 |
https://mathoverflow.net/questions/202465 | 3 | I am reading Bates and Weinstein's book 'Lectures on the Geometry of Quantization'. In Chapter 6, they defined the $\hbar$-differential operator, and showed (Theorem 6.7) that the Lagrangian submanifolds sitting inside the characteristic variety quantized to 1-st order approximate eigenfunctions to the operator.
I a... | https://mathoverflow.net/users/18261 | Reference of $\hbar$-differential operator from symplectic geometry perspective | You can try:
* ["Spectral Asymptotics in the Semi-Classical Limit"](http://books.google.com/books/about/Spectral_Asymptotics_in_the_Semi_Classic.html?id=py3AjSvONMoC) by Dimassi and Sjostrand.
* ["An Introduction to Semiclassical and Microlocal Analysis"](https://books.google.com/books?id=44LQcwB1gusC&source=gbs_navl... | 1 | https://mathoverflow.net/users/7410 | 202483 | 97,611 |
https://mathoverflow.net/questions/202242 | 10 | Here's a question on probability theory from a layman (I'm a game theorist). It is very likely that the question will be a straightforward matter for someone who is a probability theorist. I guess I'm missing a very standard technique that could be used to address the problem. Any ideas would be very helpful! Thank you... | https://mathoverflow.net/users/70251 | a question on 0-1 valued stochastic process | Let $A$ be the event $A = \left\{ \limsup\_{T \to \infty}\frac{1}{T}\sum\_{t = 0}^{T - 1}X\_{t} \ge \frac{1}{2}\right\}$, so that $Y\_t = P(A \mid X\_1, \dots, X\_t)$. We will show that $P(A) = 1$ and hence $Y\_t = 1$ almost surely for all $t$.
First, Lévy's zero-one law asserts that $Y\_t \to 1\_A$ almost surely. No... | 6 | https://mathoverflow.net/users/4832 | 202492 | 97,615 |
https://mathoverflow.net/questions/202125 | 8 | Let $F$ be a free group, and let $A,B \leq F$ be two subgroups such that $AB$ contains a nontrivial normal subgroup of $F$. Must either $A$ or $B$ contain a nontrivial normal subgroup of $F$?
What if $AB = F$?
| https://mathoverflow.net/users/38889 | Products of subgroups of a free group | The answer is negative even if $A$ is finitely generated. Here is a simple construction.
Let $F$ be the free group on $\{x,y,z\}$ and let $A=\langle x,y \rangle$. Then there is a natural retraction $\rho:F \to A$ with kernel $N:=\ker \rho=\langle z^F\rangle$. Choose a subgroup $H \leqslant A$ freely generated by an i... | 7 | https://mathoverflow.net/users/7644 | 202494 | 97,616 |
https://mathoverflow.net/questions/202437 | 6 | Let $M$ be a manifold with smooth boundary. We can consider the Dirichlet or the Neumann problem on $M$. Let $(\phi\_k)$ be an orthonormal basis of eigenfunctions to the Dirichlet problem and let $(\psi\_k)$ and orthonormal basis of eigenfunctions to the Neumann problem.
What can be said about the functions $\partial... | https://mathoverflow.net/users/16702 | Boundary values of boundary value problems | Consider the Dirichlet eigenfunctions. Let $\Delta \phi\_k + \lambda\_k^2\phi\_k=0$, where $0<\lambda\_1\leq \lambda\_2\leq \lambda\_3\dots$, and $\displaystyle u\_k = \frac{\partial \phi\_k}{\partial n}\bigr|\_{\,\partial M}$. Here is a quotation from [Bäcker, Fürstberger, Schubert, and Steiner](http://arxiv.org/abs/n... | 4 | https://mathoverflow.net/users/69194 | 202495 | 97,617 |
https://mathoverflow.net/questions/202276 | 0 | I'm needing to find out if there exists an algebraic Hecke character for a number field F, $\phi: \mathbb{A}\_F \rightarrow \mathbb{C}$, for a fixed infinite part $\phi\_\infty$ and a fixed component $\phi\_p$ (for just one finite prime $p$). Maybe it would help if I can find a classification of algebraic Hecke charact... | https://mathoverflow.net/users/70270 | Algebraic Hecke characters with a given infinite part | A standard reference for algebraic Hecke characters is Chapter Zero of Schappacher's book "Periods of Hecke characters", <http://link.springer.com/book/10.1007%2FBFb0082094>
| 0 | https://mathoverflow.net/users/69244 | 202503 | 97,622 |
https://mathoverflow.net/questions/202474 | 9 | Let $G = GL\_n$ over a field $F$, and let $\gamma \in G(F)$ be a semisimple element. The characteristic polynomial $c\_\gamma(t)$ of $\gamma$ encodes a fair bit of information about $\gamma$. Importantly (for me at least), $c\_\gamma(t)$ decomposes into linear factors (over $F$) if and only if $\gamma$ lives in a maxim... | https://mathoverflow.net/users/30726 | Is there a generalization of the "characteristic polynomial" to other split/quasi-split algebraic groups? | There is a problem you will face in generality. I don't think such a minimal field extension exists, and there isn't a nice invariant.
Let $V$ be any vector space with a quadratic form, and let $-V$ be the same vector space with minus that quadratic form. Then $V \oplus -V$ is a vector space with a split quadratic f... | 6 | https://mathoverflow.net/users/18060 | 202504 | 97,623 |
https://mathoverflow.net/questions/202462 | 10 | There is the following well known and very useful heuristic principle: **Assume one has a natural map from the space of $k$-tuples of functions in $n$ variables into the space of $K$-tuples of functions in $N$ variables such that either (a) $n<N$ or (b) $n=N$ and $k<K$. Then this map cannot be onto. Also natural maps i... | https://mathoverflow.net/users/16183 | Rigorous justification that overdetermined systems do not have a solution | There is probably no single proof that would provide a rigorous justification of the OP's principle in all cases. Moreover, without specifying more clearly what is meant by a 'natural map', the principle itself turns out not to hold in general.
For example, every smooth (complex-valued) function $f$ on the unit circ... | 19 | https://mathoverflow.net/users/13972 | 202506 | 97,624 |
https://mathoverflow.net/questions/202508 | 1 | Let $M$ be a manifold.
Let $F(M,3)=\{(m\_1,m\_2,m\_3)\mid m\_1, m\_2, m\_3\in M, m\_i\neq m\_j, \text{ for any } i\neq j\}$.
Let $S\_3$ be the symmetric group of order $3$.
Let $S\_3$ act on $F(M,3)$ by $\sigma(m\_1,m\_2,m\_3)=(m\_{\sigma(1)},m\_{\sigma(2)},m\_{\sigma(3)})$, for $\sigma\in S\_3$.
Let $F(M,3)/... | https://mathoverflow.net/users/41075 | Chern classes of three (two) dimensional complex vector bundles | Consider the covering $\pi :F(M,3)\rightarrow F(M,3)/S\_3$. By construction $\pi ^\*\xi $ is the trivial bundle, so $\pi ^\*c\_i(\xi )=0$ for $i>0$. But $H^\*(F(M,3)/S\_3,\mathbb{Q})$ is just the $S\_3$-invariant subspace of $H^\*(F(M,3),\mathbb{Q})$, so $c\_i(\xi )=0$ for $i>0$.
| 3 | https://mathoverflow.net/users/40297 | 202511 | 97,627 |
https://mathoverflow.net/questions/202491 | 6 | I would like to know the tunnel number of $n$-pretzel knots. I have searched and found nothing for any $n>3$. When $n=2$, $t(K)=1$ or $2$ depending on the number of twists, which is proved in a paper by [Morimoto, Sakuma, and Yokota](https://projecteuclid.org/euclid.jmsj/1226498920). Does anyone know if this has been c... | https://mathoverflow.net/users/36934 | Tunnel number of Pretzel knots | Any pretzel knot (more generally [Monstesinos knot](http://en.wikipedia.org/wiki/Montesinos_link)) surjects a reflection orbifold in a polygon. The ranks of these groups have been [computed by Weidmann](http://doi.org/10.1112/plms/pdm018), so one may obtain a lower bound on the rank of the Montesinos knot, hence the tu... | 4 | https://mathoverflow.net/users/1345 | 202512 | 97,628 |
https://mathoverflow.net/questions/202507 | 5 | Let $X$ be a Banach Space and let $Y$ be a closed subspace of $X^{\*\*}$ such that $X\bigcap Y=0$. Let $P$ be the quotient map from $X^{\*\*}$ onto $X^{\*\*}/ Y$. I need to prove or refute that $P\left|\_{X}\right.$ has a closed range (or equivalently is bicontinuous), or is at least a semiembedding (meaning that close... | https://mathoverflow.net/users/53155 | Location of a Banach Space inside its bidual | I think that $P|\_X$ does not have to be a semi-embedding and this can be done as follows. Let $X$ be hereditary $c\_0$, $M$ be a total nonnorming subspace in $X$, and $Y=M^{\perp}\subset X^{\*\*}$. Let $P:X^{\*\*}\to X^{\*\*}/Y$ be the quotient map. Since $X^{\*\*}/Y$ can be identified with $M^\*$ we get that $||Px||=... | 6 | https://mathoverflow.net/users/37822 | 202515 | 97,629 |
https://mathoverflow.net/questions/202518 | 1 | Let $G=(V,E)$ be a finited connected graph, $V\neq \emptyset$. Let $[V]^2 := \big\{ \{v,w\}: v, w \in V\text{ and } v\neq w\big\}$. Given $F\subseteq [V]^2$ we say that $F$ is a *vertex-disjoint extension* of $E$ if
* $F\supseteq E$, and
* $f\_1\neq f\_2\in (F\setminus E)$ implies $f\_1\cap f\_2 = \emptyset$.
Given... | https://mathoverflow.net/users/8628 | Adding vertex-disjoint edges to reduce the diameter | Yes. Let $G=(V,E)=P\_k$, the $k$-point path, where $k=3\cdot2^{n-1}-1$, and let $F$ be any vertex-disjoint extension of $E$; I claim that the graph $H=(V,F)$ has radius $\operatorname{rad}(H)\ge n$.
Assume for a contradiction that $\operatorname{rad}(H)\le n-1$. Since $\Delta(H)\le3$, by the answer to [this question]... | 4 | https://mathoverflow.net/users/43266 | 202521 | 97,631 |
https://mathoverflow.net/questions/202376 | -1 | I posted this question on [MSE](https://math.stackexchange.com/questions/1221761) two days ago, but did not receive any responses. I have cross-posted it on MO, hoping it gets more attention here and that it *is* appropriate for this site.
A positive integer $N$ is said to be *perfect* if $\sigma(N) = 2N$, where $\si... | https://mathoverflow.net/users/10365 | On odd perfect numbers $N$ given in the Eulerian form $N = {q^k}{n^2}$, Part II | In general there is a factorization $\sigma(q^{k-1}) = 1+q+\ldots+q^{k-1} = \frac{q^k-1}{q-1} = \prod\_{d\mid k, d>1} \Phi\_d(q)$. Here $\Phi\_d$ is the $d$'th cyclotomic polynomial, defined as the minimal polynomial of a primitive $d$'th root of unity. Your example is a special case of this since $\Phi\_3(x)=1+x+x^2$ ... | 3 | https://mathoverflow.net/users/65801 | 202529 | 97,632 |
https://mathoverflow.net/questions/201311 | 9 | Let $f : X \to Y$ be an open (and continuous) map of locales. Suppose the relative diagonal $\Delta\_f : X \to X \times\_Y X$ is an open embedding of locales. Does it follow that $f : X \to Y$ is a local homeomorphism?
The answer is yes if I replace "locale" with "topological space". Indeed, by the definition of the ... | https://mathoverflow.net/users/11640 | Is an open map with open relative diagonal necessarily a local homeomorphism? | The answer is yes.
It appears for example as lemma C3.1.15 in Johnstone's sketches of an elephant.
Roughly, it can be proved by working in the internal logic of the target (hence assuming that the target is a point) and considering open subspaces $U$ of $X$ such that $U \times U \subset \Delta$, on can then show th... | 2 | https://mathoverflow.net/users/22131 | 202534 | 97,635 |
https://mathoverflow.net/questions/178195 | 4 | A search of the literature reveals that for a curve $C$ of genus $\geq 2$, determining the effective cone of $C \times C$ is hard. My question is this: do we know a single example of a curve $C$ of genus $\geq 2$ for which we understand the effective cone of $C \times C$ and its image in $\text{Num}(C \times C)$? In pa... | https://mathoverflow.net/users/15704 | Effective cone of $C \times C$ where $C$ is a Fermat curve | Since I asked this question several months ago, I wrote a paper proving that the Mori cone of $C \times C$ for a smooth projective complex curve $C$ of genus $g \geq 2$ is not polyhedral. I thought I'd include a link to that for completeness.
<http://arxiv.org/abs/1502.06061>
| 4 | https://mathoverflow.net/users/15704 | 202538 | 97,638 |
https://mathoverflow.net/questions/202525 | 5 | Edit: As David Eppstein points out (in his answer below) the assumption that the graph is non-planar is redundant.
Thank you to everyone who answered/commented.
---
I have a problem about geometric embeddings of graphs for which the case I cannot prove is when the (simple, connected) graph is 4-regular, non-pl... | https://mathoverflow.net/users/62562 | Request for examples of 4-regular, non-planar, girth at least 5 graphs | A random 4-regular graph will have large girth and will, I expect, not be planar. This suggests that that there are a lot of the graphs you want, and they have no particular special properties. Markus Mehringer's program genreg will produce 4-regular graphs quickly and, as $n$ increases. Brendan McKay's geng program ca... | 5 | https://mathoverflow.net/users/1266 | 202542 | 97,639 |
https://mathoverflow.net/questions/202544 | 5 | Is there some accepted, more concise notation for expressions like $\log \log \log n$?
I just noticed [an arXiv posting](http://arxiv.org/abs/1412.5029) that quotes the bound
$$
\frac{\log X \log \log X \log \log \log \log X}
{ \log \log \log X}
$$
and I wonder if there is room for some notational improvement in this d... | https://mathoverflow.net/users/6094 | Notation for $\log \log \cdots \log n$? | Analytic Number Theorists have been using $\log\_nx$ for the $n$-times iterated (natural) logarithm of $x$ for some time. See, e.g., the first page of [this paper](http://www.math.uiuc.edu/~ford/wwwpapers/primegaps.pdf) by Ford, Green, Konyagin, and Tao. Alternatively, Claudia Spiro used $L\_4x$ for $\log\log\log\log x... | 9 | https://mathoverflow.net/users/3684 | 202546 | 97,641 |
https://mathoverflow.net/questions/202516 | 7 | Quick Background: The $p$-series of $F$ (where $F$ is a formal group law over a graded ring $R$) will be of the form $[p](x) = px + v\_1x^{p^1} + ... + v\_nx^{p^n} + ...$ ; $(F, R)$ is Landweber-exact if the sequence $(p, v\_1, ..., v\_n)$ is $R$-regular for all $p$ and $n$.
Recall the the Landweber-Ravenel-Stong Con... | https://mathoverflow.net/users/56462 | What is an example of a formal group law that is Landweber-exact but not flat? | Consider the functor that sends $X$ to $MU\_\*(X) \otimes\_{MU\_\*} R$. If $R$ is a flat $MU\_\*$-module then this defines a homology theory. Note that the condition that $R$ is flat over $MU\_\*$ is equivalent to requiring that $\operatorname{Tor}\_1^{MU\_\*}(-,R) = 0$. But we can get away with something weaker; namel... | 12 | https://mathoverflow.net/users/16785 | 202551 | 97,645 |
https://mathoverflow.net/questions/202259 | 13 | As mentioned in the title, I would like to know a proof of the "well known" fact that the A6 preprojective algebra is of wild representation type.
Ideally, I would like to see an explicit two-parameter family of indecomposable modules.
If you happen to prefer type D to type A and want to give an answer there, that ... | https://mathoverflow.net/users/425 | Why is the A6 preprojective algebra of wild representation type? | Inspired by the reference Julian gave in his comment, here's an explicit example of a two-parameter family of indecomposable representations.
First, I'll describe a one-parameter family of indecomposable representations for the preprojective algebra of type $A\_5$.
Let $k$ be the coefficient field, and $\alpha\in k... | 14 | https://mathoverflow.net/users/22989 | 202557 | 97,648 |
https://mathoverflow.net/questions/202559 | 1 | I asked this question here:
<https://math.stackexchange.com/questions/1160134/generalized-wave-equation>
but did not get any response. I hope it is more suitable on mathoverflow.
I am interested in understanding as much as I can about the following partial differential equation, which is a generalization of the 1D w... | https://mathoverflow.net/users/2011 | Generalized wave equation | Using your coordinates, we can define a Lorentzian metric $g = - dt^2 + \frac{1}{\alpha(t)} dx^2$. Then your equation takes the form
\begin{equation}
\square u + v^i \partial\_i u + \gamma u = h ,
\end{equation}
where $\square = \frac{1}{\sqrt{-\det g}} \partial\_i \sqrt{-\det g} g^{ij} \partial\_j$ is the (d'Alambert... | 4 | https://mathoverflow.net/users/2622 | 202565 | 97,650 |
https://mathoverflow.net/questions/202569 | 1 | I need estimate the following sum:
$\sum\_{d=1}^{n}\frac{\mu(d)}{d}\sum\_{k=1}^{\lfloor n/d\rfloor}\frac{1}{k}\frac{q^k}{1-q^{-kd}}$, where $q>1$ and $\mu$ is the Möbius function.
To obtain the main term, I need to find the following sum:
$q+\frac{1}{2}q^2+\cdots+\frac{1}{n}q^n$. Though the second problem looks l... | https://mathoverflow.net/users/31134 | estimate an sum | Mathematica says that the second sum is equal to:
$$
q^{n+1} (-\Phi (q,1,n+1))-\log (1-q)
$$
What do you want to know about it? This function is also known as the Lerch Transcendent, a lot of info can be fund in [the Wikipedia article.](http://en.wikipedia.org/wiki/Lerch_zeta_function)
| 3 | https://mathoverflow.net/users/11142 | 202571 | 97,651 |
https://mathoverflow.net/questions/202563 | 2 | We know that for an effective divisor on a smooth projective variety there is a natural way of associating to it a scheme, in particular using the Cartier divisor. Can we do the same for higher codimension effective algebraic cycles? More precisely, let $X$ be a smooth projective variety and $Z\_c:=\sum\_i a\_iZ\_i$ be... | https://mathoverflow.net/users/58203 | Schemes associated to algebraic cycles and local complete intersection | 1) No, let $X$ be a plane and $Z\_i$ a point, then if $Z\_c = 2 Z\_i$, then the scehme would have to be a length $2$ subcheme supported at the point. But there are a $\mathbb P^1$ of these (the schemes defined by the ideal $(x^2,xy,y^2,ax+by)$) and there is no natural choice.
2) Locally, yes, by taking the $a\_i$th p... | 2 | https://mathoverflow.net/users/18060 | 202577 | 97,654 |
https://mathoverflow.net/questions/200780 | 10 | Throw $n$ balls into $n$ bins and let $X\_n$ be the maximum occupancy. That is the maximum number of balls found in any bin.
If you throw the balls uniformly and independently it is known that $\mathbb{E}(X\_n) = \Theta(\log{n}/\log{\log{n}})$. If the process is merely pairwise independent (but still uniform) then it... | https://mathoverflow.net/users/69586 | Maximum occupancy balls in bins with limited independence | I think <https://arxiv.org/abs/1502.05729> might be at least a partial answer to my question. It shows "a $k$-independent family of functions that imply [heaviest loaded bin] size is $\Omega(n^{1/k})$".
| 5 | https://mathoverflow.net/users/69586 | 202583 | 97,657 |
https://mathoverflow.net/questions/202590 | 5 | Fix integers $k\geq2$ and $N>1$, and let $S(k,N)$ denote the normalized new Hecke eigenforms in $S\_k(\Gamma\_1(N))$. [If it makes my question easier to answer, feel free to replace this with $\Gamma\_0(N)$].
For $f \in S(k,N)$, one can define an associated adjoint L-function $L(s,\mathrm{ad}\ f)$. Of particular inte... | https://mathoverflow.net/users/10547 | Bounding a Sum of Adjoint L-Function Values | Here is a naive way I think you can get some bounds. Warning: I have not thought through this too carefully, and the bounds from such an approach might not be as good as you want in any case.
The adjoint $L$-value is essentially the Petersson norm $(f,f)$. If $f$ is a new form, then it is $\frac{2^{k}}{N} (f,f)$ wit... | 2 | https://mathoverflow.net/users/6518 | 202617 | 97,668 |
https://mathoverflow.net/questions/202613 | 1 | Let $X$ be complete finite dimensional Alexandrov space with curvature bounded from below. **Is it true that any two points can be connected by a shortest path? If this is not true in general, it it true under some assumptions on $X$?** (E.g. this is certainly true for $X$ being a complete smooth Riemannian manifold.)
... | https://mathoverflow.net/users/16183 | Existence of shortest paths in complete Alexandrov spaces | The answer is "yes", see e.g., p. 114 in [Plaut, Spaces of Wald-Berestovskii Curvature, <http://link.springer.com.sci-hub.org/article/10.1007/BF02921569]> - which is available online. It is claimed: "... If X is locally compact, Ascoli's theorem can be used to obtain the existence of minimal curves between all pairs of... | 2 | https://mathoverflow.net/users/1988 | 202620 | 97,671 |
https://mathoverflow.net/questions/202631 | 2 | I have a principal bundle $\pi: M \to B$ with structure group $G$ and a vector bundle $p: V\to M$. I need to work with $\Gamma\_G(V,M)$, the space of invariant (under the fiber action on $M$) sections of the the vector bundle $V$. I've heard one can construct a vector bundle $\tilde{p}: \tilde{V} \to B$ such that $\Gam... | https://mathoverflow.net/users/51380 | Space of invariant sections | I suppose $V\to M$ is a $G$-equivariant vector bundle. Then $\bar V$ is defined as follows: the fiber $\bar V|\_b$ over a point $b\in B$ is (canonically) equal to the space of $G$-invariant sections of $V|\_{\pi^{-1}(b)}\to \pi^{-1}(b)$. The latter space has dimension equal to the rank of $V$.
| 1 | https://mathoverflow.net/users/16183 | 202632 | 97,677 |
https://mathoverflow.net/questions/202625 | 2 | This is a follow up question of the question [Prove that $\dfrac{g(x,u\_{n})}{\left\Vert u\_{n}\right\Vert ^{p-1}}\rightarrow g\_{0}$ weakly in $L^{\overline{p}}$](https://mathoverflow.net/questions/202404/prove-that-dfracgx-u-n-left-vert-u-n-right-vert-p-1-rightarrow-g)
Let $\Omega
\subset
\mathbb{R}^{N}$
be a sm... | https://mathoverflow.net/users/70326 | Follow up question to: Prove that $\dfrac{g(x,u_{n})}{\left\Vert u_{n}\right\Vert ^{p-1}}\rightarrow g_{0}$ weakly in $L^{\overline{p}}$ | Let $\displaystyle w\_n=\frac{g(x,u\_n)}{\|u\_n\|\_{W^{1,p}}^{p-1}}$ and suppose that $\|u\_n\|\_{W^{1,p}}\geq1$ for all $n$.
We treat case $p>N$ first. In this case, $W^{1,p}(\Omega)\subset L^\infty(\Omega)$. So, the sequence $\{w\_n\}$ is bounded in $L^1(\Omega)$. In order to extract a convergent subsequence for th... | 1 | https://mathoverflow.net/users/69194 | 202637 | 97,680 |
https://mathoverflow.net/questions/202540 | 13 | Let $J$ denote the image of $J$-homomorphism spectrum and let $j$ denote its connective cover. I am interested in knowing the cohomology of $j$ i.e.
$$ [j, HZ/p]\_\*$$
as a module over Steenrod algebra. Or dually the homology of $j$ as a comodule over the dual Steenrod algebra. Is there a nice description?
A referen... | https://mathoverflow.net/users/19186 | Cohomology of the image of J spectrum | This was answered by Don Davis, 1975 Bol. Soc. Mat. Mex. In modern notation, the answer is
(at p=2) $H^\* j = (A \oplus \Sigma^7 A)/I$, where $I$ is the ideal generated by $Sq^1 \iota\_0$, $Sq^2 \iota\_0$, $Sq^4\iota\_0$, $Sq^8\iota\_0 + Sq^1 \iota\_7$, $Sq^7\iota\_7$, and $(Sq^{(0,1,1)} + Sq^{(4,2)})\iota\_7$, in Miln... | 22 | https://mathoverflow.net/users/6872 | 202647 | 97,684 |
https://mathoverflow.net/questions/150219 | 4 | **The context:**
for my research I am currently looking at parabolic systems of the type
$$
\left\{
\begin{array}{ll}
\partial\_t b(u)-\Delta u=0 \qquad & (t,x)\in \mathbb{R}^+\times\Omega\\
u=0 & x\in\Gamma=\partial\Omega\\
u(t=0)=u\_0
\end{array}
\right.\hspace{2cm}(0)
$$
where $\Omega\subset \mathbb{R}^d$ is a smoot... | https://mathoverflow.net/users/33741 | monotone parabolic systems, convex variational structure and Legendre transform | Let $\lambda>0$ and $A$ be any non-symmetric $d$-by-$d$ matrix such that $Ax\cdot x \geq \lambda |x|^2$ for every $x\in\mathbb{R}^d$. Then $x\mapsto Ax$ is uniformly monotone but is not the gradient of a convex function.
What you are looking for (duality/variational theory for monotone maps, and what replaces the Le... | 3 | https://mathoverflow.net/users/5678 | 202650 | 97,686 |
https://mathoverflow.net/questions/202586 | 14 | The question I am going to ask is really to satisfy my curiosity, as I am not at all an expert of the subject and do not plan to really work on it. Hence, if you think the question is not suitable for MO, I'll delete it. I did some browsing but could not locate any answer, but maybe I missed something simple.
Take ZF... | https://mathoverflow.net/users/29491 | Axiom of choice for sets of finite sets | I think such a set must contain a multiple of $p$ for each prime $p$. Otherwise, there is nothing to rule out the possibility of a bunch of sets of size $p$ where you are able to choose an action of the cyclic group of order $p$ on each set by. no further structure. I bet you can prove this with symmetric models.
On ... | 11 | https://mathoverflow.net/users/18060 | 202659 | 97,688 |
https://mathoverflow.net/questions/202652 | 9 | Let $G$ be a countable amenable group and $\gamma:G\to\mathbb{C}$ a positive (semi)definite function (i.e. such that $\gamma(g^{-1})=\overline{\gamma(g)}$ and
$$\sum\_{g,h\in G}f(g)\overline{f(h)}\gamma(h^{-1}g)\geq0$$
whenever $f:G\to\mathbb{C}$ is finitely supported).
Let $(F\_N)\_{N\in\mathbb{N}}$ be a Folner seque... | https://mathoverflow.net/users/18698 | Can the Cesaro limit of a positive definite function be negative? | I hope it's not frowned upon to answer one's own question, but since I just figured out the answer it doesn't make sense to keep it unanswered and there are a few upvotes so the answer may interest other people as well.
The answer is no, i.e. $L$ must be non-negative.
The proof is quite short if one uses the right t... | 7 | https://mathoverflow.net/users/18698 | 202676 | 97,694 |
https://mathoverflow.net/questions/202173 | -1 | If we consider $\omega^\omega$ as a lattice with component-wise join and meet, is there a finite distributive lattice $L$ so that there is no injective lattice homomorphism $f:L\to\omega^\omega$?
| https://mathoverflow.net/users/8628 | Finite distributive lattices not contained in $\omega^\omega$ | I believe that by $\omega^\omega$ the original poster means the poset of order-preserving maps from the chain (=totally-ordered set) $\omega$ to itself. The answer is still "no," if my proof is correct. (It is 2:09 a.m., when all conjectures are true.)
Firstly, for any $n<\omega$, $\bf{(n+1)}^\bf n$ is a sublattice o... | 3 | https://mathoverflow.net/users/51389 | 202681 | 97,696 |
https://mathoverflow.net/questions/202683 | 6 | Looks like there is counterexample to Proposition related to
abc conjecture. Confusion is likely.
From [RATIONAL AND INTEGRAL POINTS ON QUADRATIC TWISTS OF A GIVEN HYPERELLIPTIC CURVE, Andrew Granville](http://www.dms.umontreal.ca/~andrew/PDF/hyperelliptics.pdf)
---
p. 11, Proposition 2 b
Suppose that $G(x,y)... | https://mathoverflow.net/users/12481 | Counterexample to Proposition of Granville related to abc conjecture | I checked the proof of Granville. The proof only yields the bound
$\max \{ \deg(r),\deg(s)\}(\deg(G)-2)\}+1$
which covers your counterexample.
To be more detailed: The polynomials $r,s$ yield a morphism $\mathbb{P}^1\to \mathbb{P}^1$. The set $G=0$ consists of $\deg(G)$ points. The set of $\alpha$ with $G(r(\alpha),s... | 12 | https://mathoverflow.net/users/8621 | 202687 | 97,699 |
https://mathoverflow.net/questions/202686 | 16 | It is well known that for a bilinear form over an n-dimensional vector space, $n^2$ values (on all pairs of basis-vectors) determine it uniquely.
How many values do we need to specify in order to uniquely determine a norm?
The first trivial observation is that homogeneity implies it's enough to specify all the values... | https://mathoverflow.net/users/46290 | How many values determine a norm? | Well, as you have certainly already remarked (reading your post, I assume this), bilinearity makes a big difference. For the "only norm" case, what you are looking for, if I understand correctly your question, is a **set of uniqueness** for the admissible norms on a given vector space $V$. Your demonstration establishe... | 21 | https://mathoverflow.net/users/25256 | 202691 | 97,702 |
https://mathoverflow.net/questions/202698 | 0 | I encountered a group $G =\langle(1,3,2,4),(3,5,4,6)\rangle\subseteq S\_6$ in my study, but I do not know its name.
Let $f=(1,3,2,4)$ and $g=(3,5,4,6)$. We have $g^2=fg^2f$, and thus $\langle f,g^2\rangle\simeq D\_4$. I have proved that $\langle f,g\rangle=\langle f,g^2\rangle \cup \:g\langle f,g^2\rangle \cup\:fg \l... | https://mathoverflow.net/users/60561 | The name of a group of order 24 | It is an easy exercise to show that $G$ is isomorphic to $S\_4$ in its action on $6=\binom{4}{2}$ pairs of the four points of its natural action.
| 2 | https://mathoverflow.net/users/11100 | 202699 | 97,706 |
https://mathoverflow.net/questions/202566 | 7 | We're working in L(R) under AD.
We know that
$\omega\_1$ is the least measurable in HOD, $\Theta$ is the least woodin, $\delta^2\_1$ is the least strong to the woodin, etc.
My question is about characterizing other L(R) cardinals in HOD, specifically those in the projective hierarchy. What is known? Is there a k... | https://mathoverflow.net/users/31324 | Characterizing L(R) Cardinals in HOD | IMO, characterization of $\omega\_2$ should be the projective version of "$\theta$ is Woodin in HOD": replace $\theta$ with $\omega\_\omega$ and HOD with $L(HOD|\omega\_\omega)$. It has to do with Jackson's level-2 description analysis, i.e, computing $j\_\mu(\omega\_n)$ when $\mu$ is a measure on $\omega\_\omega$ (cf.... | 3 | https://mathoverflow.net/users/64308 | 202705 | 97,708 |
https://mathoverflow.net/questions/202712 | 2 | Let $\overline{M}\_{g,A}$ the moduli stack of pointed genus $g$ stable curves with weights $A = (a\_1,...,a\_n)$ introduced in
Brendan Hassett, Moduli spaces of weighted pointed stable curves, Adv. Math. 173 (2003), no. 2, 316–352
In this paper the author proves that $\overline{M}\_{g,A}$ is indeed a smooth DM-stack... | https://mathoverflow.net/users/nan | Universal curve of stacks of stable curve | There is probably not an isomorphism.
First, note a reason to be suspicious - if you had such an isomorphism, couldn't you just iterate it to get an isomorphism $\overline{M}\_{g,A}=\overline{\mathcal M}\_{g,n}$. This would make the work of Brendan Hasset in his paper somewhat superfluous.
I claim there is a natura... | 6 | https://mathoverflow.net/users/18060 | 202713 | 97,712 |
https://mathoverflow.net/questions/202717 | 2 | Consider an oriented random walk on $\mathbb Z^2$ (i.e. only steps $\rightarrow$ and $\uparrow$ with equal probability.) Say we let the walk go $2m$ steps then start guessing sites at distance $2m$ from the origin until we pick up its trail. The best first guess is at $(m,m)$ which has probability $p^\* = \binom{2m}{m}... | https://mathoverflow.net/users/49603 | Hitting probabilities for conditioned oriented random walk monotonic? | I have a brute force calculation that establishes this. Your formula for $q\_i$ is not completely correct; it should read $q\_i=\binom{2m}{m+i}/A\_i$ with
$$
A\_i = 4^m - \binom{2m}{m} - 2\sum\_{j=1}^{i-1} \binom{2m}{m+j} .
$$
To see that $q\_1\ge p^\*$, let's just write
$$
\binom{2m}{m+1}=\binom{2m}{m}\frac{m}{m+1} ... | 2 | https://mathoverflow.net/users/48839 | 202724 | 97,715 |
https://mathoverflow.net/questions/202658 | 4 | I came across a problem like this. Suppose that $\Omega$ is an open subset of $\mathbb{C}^{n}$ and $V$ is a complex submanifold of $\Omega$ of codimension 1. Now given a plurisubharmonic function $\varphi$ on $\Omega-V$ which is bounded in the sense that given any compact set $K\subset\Omega$, $\varphi$ is bounded abov... | https://mathoverflow.net/users/70418 | About extending plurisubharmonic function | $V$ is pluripolar. Let $v$ be a plurisubharmonic function which is $-\infty$ on $V$.
Then $\phi+\epsilon v$ is plurisubharmonic for $\epsilon>0$ (the definition of plurisubharmonic function is easily verified for it). Therefore when $\epsilon\to 0$ the limit must be plurisubharmonic.
| 2 | https://mathoverflow.net/users/25510 | 202726 | 97,716 |
https://mathoverflow.net/questions/202728 | 4 | Suppose we have a group $G$ which is finitely generated , and let $|\cdot |$ denote some word metric on it. Must there be an element $a\in G$ such that $|a^n|\ge c\cdot n$ for some $c>0$?
My intuition says that the answer is "yes", but I wasn't able to find a proof in the general case. There are many cases that I kno... | https://mathoverflow.net/users/56465 | Must the powers of some element always grow linearly with respect to a word metric? | (Probably the question would be more suitable for MathSE)
The answer is no. Well, the trivial group is a counterexample. Also finite groups are counterexamples. So first and for all you should have specified that you assume the group to be infinite. But then the answer is still no.
In a finitely generated group, an... | 21 | https://mathoverflow.net/users/14094 | 202731 | 97,720 |
https://mathoverflow.net/questions/202740 | 6 | Let $J$ be a family of parallel closed intervals in the plane $\mathbb{R}^2$, no two different of them contained in any common line. Moreover, for arbitrary three intervals belonging to $J$ there exists a line intersecting all three of them. I would like to prove that there exists a line intersecting all segments belon... | https://mathoverflow.net/users/70458 | Segments on a family of parallel lines | Here is a proof for a finite number of parallel lines. For each segment $s\_i$, let $L\_i$ be the set of lines that intersect $s\_i$. We may regard each line $\ell \in L\_i$ as a point $(m,b) \in \mathbb{R}^2$, where $m$ is the slope of $\ell$ and $b$ is its $y$-intercept. In this way, we may regard $L\_i$ as a subset ... | 8 | https://mathoverflow.net/users/2233 | 202752 | 97,728 |
https://mathoverflow.net/questions/202749 | 3 | In Table of Integrals, Series, and Products. Seventh Edition. I.S. Gradshteyn and I.M. Ryzhik, there is
0.154.3
$$
\sum\_{k=0}^N (-1)^k {N \choose k} k^{n-1} =0, N \geq n \geq 1; 0^0 ≡ 1
$$
0.154.4
$$
\sum\_{k=0}^n (-1)^k {n \choose k} k^{n} =(-1)^n n!, n \geq 0; 0^0 ≡ 1
$$
I would like to know, there is any genera... | https://mathoverflow.net/users/70264 | A question about summation formula involving binomial coefficient | $$
\sum\_{k=0}^n (-1)^k {n \choose k} k^{n+m} = (-1)^n\cdot n!\cdot S(n+m,n),
$$
where $S(\cdot,\cdot)$ is Stirling number of the second kind. This is essentially formula (10) at <http://mathworld.wolfram.com/StirlingNumberoftheSecondKind.html>
In particular, for $m<0$, we have $S(n+m,n)=0$; and for $m=0$, we have $S... | 8 | https://mathoverflow.net/users/7076 | 202753 | 97,729 |
https://mathoverflow.net/questions/202499 | 9 | In Lurie's [DAG II](http://arxiv.org/pdf/math/0702299v5.pdf), a notion of monoidal $\infty$-category is given that differs from the notion given in his later book [Higher Algebra](http://www.math.harvard.edu/~lurie/papers/higheralgebra.pdf). In the former, the relevant structure is a cocartesian fibration of simplicial... | https://mathoverflow.net/users/11546 | When is a quasicategory over $N(\Delta)^{op}$ a planar $\infty$-operad? | So, this ends up being simpler than I realized, and is in some sense this question's existence is purely a result of me not reading the above cited DAG II closely enough.
In the first section of DAG II it's proven that if we start with a simplicial monoidal category in which the monoidal functor $C\times C\to C$ is ... | 4 | https://mathoverflow.net/users/11546 | 202758 | 97,732 |
https://mathoverflow.net/questions/202765 | 0 | Is there a countable space that is $T\_1$ and not metacompact? (A space $(X,\tau)$ is not metacompact iff there is on open cover $\cal{U}\_0$ such that for every open refinement $\cal V$ there is $x\in X$ such that $x$ is contained in infinitely many members of $\cal V$.)
Note that $(\omega,\tau)$ with $\tau=\{\empty... | https://mathoverflow.net/users/8628 | Countable, $T_1$, and not metacompact | No, every countable T$\_1$ space is metacompact. Let $\tau$ be any T$\_1$ topology on $\mathbb N.$ Let $\mathcal U$ be any open cover. For each $n\in\mathbb N$ choose $U\_n\in\mathcal U$ with $n\in U$ and let $V\_n=U\_n\setminus\{1,\dots,n-1\}.$ Then $\mathcal V=\{V\_n:n\in\mathbb N\}$ is a point-finite open refinement... | 4 | https://mathoverflow.net/users/43266 | 202766 | 97,736 |
https://mathoverflow.net/questions/202737 | 6 | Given a smooth manifold $M$, there is a vector bundle over $M$, denoted $\tau M$, known as the second-order tangent bundle. The fiber $\tau\_mM$ at $m\in M$ is the collection of linear operators $A\_m:C^\infty(M)\rightarrow\mathbb{R}$ that satisfy
$$ A\_m(f^3)=3f(m)A\_m(f^2)-3f^2(m)A\_m(f) $$
for each $f\in C^\inft... | https://mathoverflow.net/users/27121 | Making the identification $\tau M\approx TM\oplus (TM\odot TM)$ | Given a connection on the tangent bundle, you can define the second covariant derivative $\nabla^2f$ by
$$ \nabla^2f[X, Y] := \partial\_X\partial\_Y f - \partial\_{\nabla\_X Y} f.$$
Then $\nabla^2f$ is a symmetric tensor *provided that $\nabla$ was torsion-free*, which Robert pointed out but I missed in the first momen... | 6 | https://mathoverflow.net/users/16702 | 202771 | 97,740 |
https://mathoverflow.net/questions/202776 | 0 | The [Erdös-Faber-Lovasz conjecture](http://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Faber%E2%80%93Lov%C3%A1sz_conjecture) and [Hadwiger's conjecture](http://en.wikipedia.org/wiki/Hadwiger_conjecture_%28graph_theory%29) can be stated in a very similar form:
**Erdös-Faber-Lovasz conjecture**: for all finite simple undi... | https://mathoverflow.net/users/nan | Implication between Erdös-Faber-Lovasz conjecture and Hadwiger's conjecture? | Unfortunately, neither $\ell(G) \leq \eta(G)$ holds for all graphs $G$, nor $\eta(G) \leq \ell(G)$ holds for all graphs $G$.
1. **Example for a graph $G$ with $\ell(G)>\eta(G)$.** Consider the graph $G=(V,E)$ where $V = \{0,1,2\}$ and $E = \big\{\{0,1\},\{1,2\}\big\}$. Clearly $K\_3$ is not a minor of $G$, but $K\_2$... | 2 | https://mathoverflow.net/users/8628 | 202778 | 97,743 |
https://mathoverflow.net/questions/202782 | 2 | For $n\in\mathbb{N}$ we consider the set $\{1,\ldots,n\}$ and define the *line graph $L(K\_n)$ of the complete graph $K\_n$* as follows:
* $V(L(K\_n)) = \big\{\{a,b\}: a,b\in \{1,\ldots, n\}, a\neq b \big\}$;
* $E(L(K\_n)) = \big\{\{x,y\}: x,y\in V(L(K\_n)) \text{ and } x\cap y \neq \emptyset\big\}$.
For any graph ... | https://mathoverflow.net/users/8628 | The Hadwiger number of $L(K_n)$ | **Claim.** For all even $n \geq 4$, $\eta(L(K\_n)) \geq \frac{3n-4}{2}$ and for all odd $n \geq 3, \eta(L(K\_n)) \geq \frac{3n-3}{2}$
*Proof.*
A clique-minor of size $k$ in $L(K\_n)$ corresponds to $k$ edge-disjoint connected subgraphs of $K\_n$ which pairwise intersect.
Let $n$ be even. For our $\frac{3n-4}{2}$ ... | 2 | https://mathoverflow.net/users/2233 | 202784 | 97,745 |
https://mathoverflow.net/questions/202777 | 1 | There is one calculation in Chandrasekhar's "Mathematical Theory of Black Holes" that I cannot understand. Here is the setup:
We want to show that Petrov type D (i.e. two principal null directions) corresponds to the only non-vanishing Weyl scalar $\Psi\_{2}$. To see this, we first rotate the null tetrad $\lbrace l,n... | https://mathoverflow.net/users/51137 | Petrov classification/Weyl scalars | You should re-read the immediately previous section on Petrov Type II first. There he did the computation is a tiny bit more detail.
Then you will realize that since we are dealing with a fourth order polynomial in $b$ which, by assumption, has two roots each with multiplicity 2, the notion of derivative here is **d... | 3 | https://mathoverflow.net/users/3948 | 202796 | 97,748 |
https://mathoverflow.net/questions/202675 | 21 | I am soon to become a graduate student and so I started a personal project; I want to understand Faltings's proof of the Mordell conjecture.
I want to get into arithmetic geometry (since I always liked both algebraic geometry and number theory) and I thought that understanding this proof might be a good start (mostly... | https://mathoverflow.net/users/44192 | Understanding Faltings's Theorem | I just wanted to make sure that you're aware that there is another proof of the Mordell conjecture that is in many ways more natural, and that has allowed great generalizations. This is the proof due to Vojta using ideas from Diophantine approximation. Vojta's proof was simplified by Bombieri, and that version does not... | 22 | https://mathoverflow.net/users/11926 | 202800 | 97,750 |
https://mathoverflow.net/questions/202773 | 13 | I have a rather soft question. Let's assume that we consider the heat equation posed in $S^1$:
$$
\partial\_t u=\partial\_x^2u.
$$
It is well known that if we define the functionals
$$
H(t)=\int\_{-\pi}^\pi u\log(u)-u+1dx,
$$
$$
E(t)=\int\_{-\pi}^\pi u^2dx,
$$
they decay. As far as I know, in the literature, a function... | https://mathoverflow.net/users/33135 | Mathematical difference between entropy and energy | First, note that entropy is well-defined only if $u$ is positive. Using integration by parts, it is in fact true on the flat torus or $\mathbb{R}^n$ (if $u$ is nonnegative and decays in space fast enough) that
$$
\frac{\partial}{\partial t}\frac{1}{p(p-1)}\log \int u^p
= -\frac{\int u^{p-2}|\nabla u|^2}{\int u^p}\le 0... | 7 | https://mathoverflow.net/users/613 | 202802 | 97,751 |
https://mathoverflow.net/questions/200881 | 25 | I am a master student in mathematics. For me a large part of doing mathematics is thinking about, reading and verifying the proof of theorems that I find them in my field of study. I can do this action in 3 ways:
* When I see a theorem I get a paper and think to prove it: this action takes time a lot and maybe I coul... | https://mathoverflow.net/users/38805 | Which way for reading the proofs? | Here is a quote of Poincaré (one of the most accomplished mathematicians of all time) regarding the reading of mathematics:
>
> I am used, when I read a memoir, to glance over first quickly so as to have a general impression, then come back to the points which seem to me obscure. I find it more convenient to do pro... | 20 | https://mathoverflow.net/users/3651 | 202807 | 97,753 |
https://mathoverflow.net/questions/202616 | 2 | Let $R$ be a local and smooth $\mathbb{Z}\_p$-algebra and $B$ an $R$-algebra of finite type which is an integral domain with $\operatorname{dim}B\leq \operatorname{dim}R$ such that
* $B/(p)$ is non-zero and finitely generated as an $R/(p)$-module and
* $B[\frac{1}{p}]$ is non-zero and finitely generated as an $R[\fra... | https://mathoverflow.net/users/467 | Does this $\mathbb{Z}_p$-algebra morphism induce a closed immersion on the generic fiber? | Still no. Consider the map $\mathbb Z\_p[s] \to \mathbb Z\_p[t]$, $s=t+pt^2$. This is a degree $1$ map on the special fiber but degree $2$ on the generic fiber. Let $R$ be the localization of $\mathbb Z\_p[s]$ at $s=1+p$ and let $B$ be the localization of $\mathbb Z\_p[t]$ at $t=1$.
Then this satisfies all your cond... | 2 | https://mathoverflow.net/users/18060 | 202812 | 97,755 |
https://mathoverflow.net/questions/202811 | 32 | Wikipedia and a few websites (and a few mathoverflow answers) say there is a constructive proof of the Brouwer fixed point theorem, some others say no. The argument for a constructive proof is always the same. The Brouwer fixed point theorem is equivalent to some other results (Miranda, Sperner) where some algorithm pr... | https://mathoverflow.net/users/6129 | Does the Brouwer fixed point theorem admit a constructive proof? | You are correct in observing the flaw in the claims for BFPT to be constructive: There is no algorithm that takes a sequence in the unit hypercube and outputs some accumulation point of it. This task is in fact LESS(1) constructive that BFPT itself. We can be slightly less wasteful, and come up with a sequence convergi... | 21 | https://mathoverflow.net/users/15002 | 202830 | 97,767 |
https://mathoverflow.net/questions/202605 | 3 | Let $S$ be an affine scheme of characteristic $p > 0$, let $E \rightarrow S$ be an elliptic curve over $S$, and let $F$ denote the absolute Frobenius. Since $E$ is its own $\mathrm{Pic}^0$ there is an identification $\mathrm{Lie}(E/S) = H^1(E, \mathscr{O}\_E)$; suppose that both of these are free $\mathscr{O}\_S$-modul... | https://mathoverflow.net/users/63877 | Interpreting Frobenius pullback as an invariant differential in the case of an elliptic curve | This is explained in the proof of Theorem 3 of section 15 of Mumford's "Abelian varieties" (pages 138-140 in the new edition) in the case when the base is an algebraically closed field of characteristic $p$. Suitably interpreted, the argument there continues to work over an arbitrary $\mathbb{F}\_p$-scheme $S$.
| 2 | https://mathoverflow.net/users/5498 | 202837 | 97,769 |
https://mathoverflow.net/questions/202836 | 3 | Consider the nonlinear mapping $\phi: \mathbb R^{2 \times 2} \to \mathbb R^3$ given by $X \mapsto \begin{pmatrix} x\_{11} x\_{21} \\ x\_{11} x\_{22} + x\_{21} x\_{12} \\ x\_{12}x\_{22} \end{pmatrix}$.
I think that the image $\phi(\mathbb R^{2 \times 2})$ is a convex cone, i.e. in particular: for all $X', X''$ there e... | https://mathoverflow.net/users/nan | Proof that image of a polynomial map is a cone | A condition for $y = (y\_1,y\_2,y\_3)$ to be in the image is that the discriminant of $t^2 y\_3 - t y\_2 + y\_1$, namely $y\_2^2 - 4 y\_1 y\_3$, is nonnegative. This is **not** a convex constraint. For example, $(1,0,1)$ is not in the image, but
$(1,2,1) = \phi\pmatrix{1 & 1\cr 1 & 1\cr}$ amd $(1,-2,1) = \phi(\pmatrix{... | 5 | https://mathoverflow.net/users/13650 | 202838 | 97,770 |
https://mathoverflow.net/questions/202711 | 6 | I apologize in advance if this question is too technical. I haven't found a reference in the literature yet, and it seems difficult enough that perhaps it has not been answered.
Let $A$, $B$, and $C$ be based topological spaces, all compactly generated and weak Hausdorff. Let $\wedge$ and $F$ denote smash product and... | https://mathoverflow.net/users/1874 | is this map a closed inclusion? | Isn't this addressed in Lewis' Appendix A (on compactly generated spaces)? His Proposition 8.5 gives a criteria (for A compact), to get a closed inclusion. But your comment on adding the disjoint basepoint is important ("The adjunction od the disjoing basepoint is essential to the [...] result"). See his CounterExample... | 4 | https://mathoverflow.net/users/70501 | 202844 | 97,773 |
https://mathoverflow.net/questions/176271 | 5 | A local martingale is a martingale iff it is in the class DL.
The condition: for every $t\in[0,\infty)$
$$E[\sup\limits\_{0\leq s\leq t} |M\_s|]<\infty\tag1$$
guarantees a local martingale $M$ is a martingale by ensuring it satisfies the condition for being in the class DL. Moreover, by Burkholder-Davis-Gundy, th... | https://mathoverflow.net/users/32325 | Examples of a continuous martingale with $E[\sup\limits_{0\leq s\leq t} |M_s|]=\infty$? | There exist indeed a uniformly integrable martingale $X$ whose max $\sup\_{k\in \mathbb{N}} |X\_k|$ is not integrable.
A simple example in discrete time can be found here <http://www.math.fsu.edu/~nichols/martingalezoo.pdf>
(see the last 2 lines of point 5).
A related construction is found in example 4.1 of the pap... | 4 | https://mathoverflow.net/users/70478 | 202846 | 97,774 |
https://mathoverflow.net/questions/202744 | 4 | Let $M$ be a complete Riemannian manifold and $\Delta$ denote the Laplacian on it. Also assume that the spectrum of $-\Delta$ lies inside $[a, \infty)$. Let $P\_t, t > 0$ denote the diffusion semigroup generated by $\Delta$, and pick a function $f \in L^2(M)$. I am looking for a proof of the following
$$\Vert P\_t f\Ve... | https://mathoverflow.net/users/70462 | Diffusion semigroup generated by Laplacian | This is essentially equivalent to boundedness of the heat kernel. I don't think you can expect it to be true for an arbitrary manifold, but it is true if the manifold has Ricci curvature bounded below.
First, note that if your inequality holds with $t=1$ then it holds with any $t \ge 1$. (Since the spectrum of $-\Del... | 3 | https://mathoverflow.net/users/4832 | 202848 | 97,775 |
https://mathoverflow.net/questions/202847 | 10 | Let $E/k$ be an elliptic curve over some algebraically closed field $k$ of characteristic $p\ge 0$. It's known that $Aut(E)$ acts faithfully on the Tate module $T\_\ell(E)$ ($\ell\ne p$) with determinant 1. Is there a complete description of the actions of $Aut(E)$ on $T\_\ell(E)$? Ie, for any elliptic curve as above, ... | https://mathoverflow.net/users/15242 | how do automorphisms of elliptic curves act on the Tate module? | How many subgroups of order $12$ and $24$ are there in $SL\_2(\mathbb Z\_l)$? By the classification of finite subgroups of $SO(3)$ there are just two of each, one abelian and one non-abelian. It is easy to see that the abelian one cannot appear because the characteristic polynomial of each element should be integral. T... | 7 | https://mathoverflow.net/users/18060 | 202850 | 97,777 |
https://mathoverflow.net/questions/202722 | 17 | Let $f:S^{n-1} \rightarrow S^n$ be a topological embedding and let $A\_f$ and $B\_f$ be the components of $S^n \setminus f(S^{n-1})$. If $\overline{A}\_f$ and $\overline{B}\_f$ are manifolds with boundary $f(S^{n-1})$, then the locally flat Schoenflies conjecture (proved by Mazur and Brown) says that $\overline{A}\_f$ ... | https://mathoverflow.net/users/70453 | $(n-1)$-dimensional sphere in $S^n$ such that the closure of a component of complement is not contractible | I want to thank Ian Agol for his comments which helped point me in the right direction. In particular, he told me that if $f:S^{n-1} \rightarrow S^n$ is a topological embedding, then the closures of the components of $S^n \setminus f(S^{n-1})$ are known as *crumpled $n$-cubes*.
It turns out that Bing originally prove... | 12 | https://mathoverflow.net/users/70453 | 202851 | 97,778 |
https://mathoverflow.net/questions/202858 | 25 | I was surprised that the numbers $\pi$, $\ln{(2)}$, $\zeta{(2)}$, and $\zeta{(3)}$ can be shown to be irrational in what seems to be "three-lined proofs" (as identified here on Overflow: [Establishing zeta(3) as a definite integral and its computation.](https://mathoverflow.net/questions/30659/establishing-zeta3-as-a-d... | https://mathoverflow.net/users/70508 | Question on the irrationality of $e$ | One has $$\int\_0^1 x^k e^x dx = A\_ke+B\_k \to 0$$ as $k\to \infty$, $A\_k,B\_k\in \mathbb{Z}$, by integration-by-parts and induction. If $e=a/b$, then $(A\_k a+B\_kb)/b \to 0$, so $A\_ke+B\_k=0$ for $k$ large, a contradiction.
| 51 | https://mathoverflow.net/users/1345 | 202860 | 97,782 |
https://mathoverflow.net/questions/202853 | 9 | Suppose I have a simplicial model category $M$. Then I can take the [homotopy coherent nerve](http://ncatlab.org/nlab/show/homotopy+coherent+nerve) of $M$ to obtain a quasicategory. This, however, only depends on the fact that $M$ is a category enriched in simplicial sets (i.e. the homotopy coherent nerve is a right Qu... | https://mathoverflow.net/users/11546 | Difference between coherent nerve of simplical model category and simplicial category | We can detect the difference between the two constructions using the homotopy category. Given any simplicially enriched category $\mathcal{C}$, we can construct an ordinary category $\pi\_0 [\mathcal{C}]$ by applying $\pi\_0$ to the hom-spaces. (This makes sense because $\pi\_0 : \mathbf{sSet} \to \mathbf{Set}$ preserv... | 8 | https://mathoverflow.net/users/11640 | 202866 | 97,785 |
https://mathoverflow.net/questions/202803 | 5 | Using *Hardy-Littlewood-Sobolev* inequality, we can prove that:
$$\left| \int\_0^1\int\_0^1 |x-y|^{-\frac{1}{2}} f(x)f(y) \mathrm{d}x \mathrm{d}y \right| \leq C \left\| f \right\|\_{L^{4/3}(0,1)}^2 \leq C \left\| f \right\|\_{L^{2}(0,1)}^2.$$
However, the left-hand side looks very similar to the singular integrals us... | https://mathoverflow.net/users/50777 | Hardy-Littlewood-Sobolev inequality using fractional sobolev norm on the RHS | One has, for $f,\,g\in \dot{H}^{-1/4}(\mathbb R)$,
$$
\begin{aligned}
\left|\int\_{-\infty}^\infty\int\_{-\infty}^\infty |x-y|^{-1/2}f(x)
\,\overline{g(y)}\,dx dy\,\right| &= C\_0 \left|\left((-\Delta)^{-1/4}f,g\right)
\right| \\
& \leq C\_0\|(-\Delta)^{-1/4}f\|\_{\dot{H}^{1/4}}\|g\|\_{\dot{H}^{-1/4}}
= C\_0\|f\... | 4 | https://mathoverflow.net/users/69194 | 202869 | 97,788 |
https://mathoverflow.net/questions/201385 | 1 | In Demailly's Analytic Methods in Algebraic Geometry (available on [his web page](https://www-fourier.ujf-grenoble.fr/~demailly/documents.html)), the definition of a (plurisubharmonic) "function with analytic singularities" is a (plurisubharmonic) function $ u: X\to \mathbb{R}$ on a complex manifold $X$ that can be loc... | https://mathoverflow.net/users/69851 | How local is the exponent in the definition of a function with analytic/algebraic singularities? | Contacting the author J.P. Demailly himself about this question has lead to the following conclusion:
In the algebraic setting, $X$ is quasi-projective, so by the Noetherian property, we have Zariski-compactness, so we can choose $\alpha$ globally.
In the analytic setting, (the maximal choice of) $\alpha$ can get a... | 1 | https://mathoverflow.net/users/69851 | 202873 | 97,791 |
https://mathoverflow.net/questions/202865 | 2 | Let $H \to G$ be a homomorphism of affine algebraic groups (over characteristic $0$, if it matters). The case I care most about is when $H \to G$ is an inclusion. There is a corresponding map $f: \mathrm{B}H \to \mathrm{B}G$ of (probably I should say "derived") stacks.
I would like to describe $\mathrm{B}G$ in terms ... | https://mathoverflow.net/users/78 | Given a map of classifying spaces, can the target be described as a groupoid quotient of the source mod some action of some (co)kernel? | Yes, there is something to this effect.
In fact there is a very general context for this. Since I know you are amenable to $\infty$-categories, I will use that language.
The homotopy theory of spaces is the initial example of an $\infty$-topos and one of the basic axioms of $\infty$-topoi is that "*groupoids are ... | 3 | https://mathoverflow.net/users/184 | 202879 | 97,793 |
https://mathoverflow.net/questions/202878 | 4 | I'm reading the paper by ANGELIKA ROHDE AND ALEXANDRE B. TSYBAKOV, ESTIMATION OF HIGH-DIMENSIONAL LOW-RANK MATRICES.
And in the paper, they provide an inequation of the Schatten-p (quasi-)norm, namely, for any tow matrices $A,B \in \mathbb{R}^{m\times T}$, $\forall \ 0<p\le1$, we have
$$
\lVert A+B \rVert\_{S\_p}... | https://mathoverflow.net/users/61485 | Does Schatten-p (quasi-)norm satisfy the norm inequality for 0<p<1? | One can deduce the non-rectangular case from the rectangular one as follows. Let
$$P\colon \mathbb{R}^{m+T}\to \mathbb{R}^T,\, \iota\colon \mathbb{R}^m\to \mathbb{R}^{m+T}$$
be the obvious orthogonal projection and the isometric imbedding respectively. Then for any linear operator
$F\colon \mathbb{R}^T\to \mathbb{R}^m$... | 3 | https://mathoverflow.net/users/16183 | 202882 | 97,794 |
https://mathoverflow.net/questions/202445 | 6 | I'm reading the Hirschhorn's book *model categories and their localization* and I have a question about frames and resolutions.
Following the book (definition 16.6.1) a *cosimplicial frame* on an object $X$ in a model category $\mathcal{M}$ is a cosimplicial object $A^{\*}$ in $\mathcal{M}^{\Delta}$ such that
1)$A^... | https://mathoverflow.net/users/41970 | Technical lemma about frame and cosimplicial resolution | Consider the Hovey's definition of cosimplicial frame. A cosimplicial frame $\tilde{X}^{\*}$ on an object $X\in \mathcal{M}$ is a factorization $p^{\*}(X)\to \tilde{X}^{\*}\to ccX$ of the fold map $p^{\*}(X)\to ccX$, where the first map is a cofibration and the second is a weak equivalence that is an isomorphism in deg... | 1 | https://mathoverflow.net/users/41970 | 202892 | 97,799 |
https://mathoverflow.net/questions/202887 | 2 | As far as vanishing is concerned, the usual motivic cohomology has the following two properties (for a smooth scheme $X$ over a field):
1. $H^{p,q}(X, \mathbb Z) = 0$, if $p > q + dim(X)$; and
2. $H^{p,q}(X, \mathbb Z) = 0$, if $q<0$.
Are the analogous properties true for etale (or Lichtenbaum) motivic cohomology? ... | https://mathoverflow.net/users/70521 | Vanishing in etale motivic cohomology | Check out Chapter 10 of Mazza-Voevodsky-Weibel "Lecture notes on motivic cohomology", which discusses étale motivic cohomology. The answers to your questions can be found there:
2) yes: Immediately after Definition 10.1, you find the vanishing $H^{p,q}\_L(X,\mathbb{Z})=0$ if $q<0$. This follows directly from the def... | 4 | https://mathoverflow.net/users/50846 | 202897 | 97,801 |
https://mathoverflow.net/questions/202889 | 5 | As I am relatively new to these matters, I would like to know if you could provide me a reference for Besov spaces on unbounded domains, because when I checked the first tome of Triebel's Theory of Function Spaces, I only found the case of a smooth bounded set aside of whole or half space (in Bergh's Interpoation space... | https://mathoverflow.net/users/56191 | Reference request : Besov spaces on ubounded domains | Hormander: The Analysis of Linear Partial Differential Operators II, 1983, page 13 ff.
These spaces are $B\_{k,p}(\mathbb R^n)\cap \mathcal E'(X)$, where $X$ is open in $\mathbb R^n$.
| 3 | https://mathoverflow.net/users/26935 | 202901 | 97,802 |
https://mathoverflow.net/questions/202886 | 1 | Recall that for $t\geq2$, a partition is a $t$-core if none of its hooklengths is divisible by $t$. It is known that the $t$-cores are parametrized by ${\mathbb Z}^{t-1}$. More precisely, let $(n\_0,\dots,n\_{t-2})$ be a $(t-1)$-tuple of integers and set $n\_{t-1}:=-\sum\limits\_{i=0}^{t-2}n\_i$. Define $r\_i=\sum\limi... | https://mathoverflow.net/users/20764 | Generating function for $t$-residues of partitions using Heisenberg + $\hat{sl_t}$ representation theory | Denote by $\tilde{\mathcal F}$ the Fock space representation of $\widehat{sl}\_t$. In fact it is even a $\widehat{gl}\_t$-module, and as such irreducible. Now \begin{equation}\widehat{gl}\_t= \widehat{sl}\_t\oplus Heis \Big/ (Z\_{\widehat{sl}\_t}-Z\_{Heis}),\end{equation} where $Heis$ is the infinite dimensional Heisen... | 1 | https://mathoverflow.net/users/6107 | 202902 | 97,803 |
https://mathoverflow.net/questions/202900 | 3 | One of the applications of the holomorphic functional calculus is with regard to idempotents. For instance, if an element $a$ in a unital Banach algebra $A$ has spectrum contained in two balls, each of radius $\frac{1}{4}$ and centered at 0 and at 1 respectively, then by taking a holomorphic function that takes value 0... | https://mathoverflow.net/users/70528 | Holomorphic functional calculus and idempotents | Yes, you get your idempotent back.
You can prove it from the very definition of the functional calculus. Since $a^2=a$, we get $ker(1-a) \oplus ker(a) = A$ from elementary algebra. Now you can compute the inverse of $1-a+r e^{i \theta}$ explicitely in restriction to these two invariant subspaces.
$(1-a+re^{i\theta}... | 4 | https://mathoverflow.net/users/6129 | 202908 | 97,807 |
https://mathoverflow.net/questions/202907 | 2 | I've asked this question [here](https://math.stackexchange.com/questions/1214449/does-y-in-mathbbrn-operatornamerankx-y-ay-2-have-zero-lebesgue) on math.stackexchange, but I have been unable to solve this yet, so I'm hoping I can get some advice here.
Consider a vector $x\in \mathbb{R}^n$ and a real $n\times n$ matr... | https://mathoverflow.net/users/38168 | Lebesgue measure of set of $y\in\mathbb{R}^n$ such that $x,y,Ay$ are linearly dependent | Lemma. If $y\_1l\_2(y\_1,\dots,y\_n)-y\_2l\_1(y\_1,\dots,y\_n)\equiv 0\,$ for linear functions $l\_1,l\_2$, then $l\_i=cy\_i$ for some scalar $c$ and $i=1,2$.
Now change variables so that $x=(0,\dots,0,1)$, denote $A(y)=(l\_1(y),\dots,l\_n(y))$, $y=(y\_1,\dots,y\_n)$. If there exist $i,j$ less than $n$ such that $F(y... | 4 | https://mathoverflow.net/users/4312 | 202909 | 97,808 |
https://mathoverflow.net/questions/202911 | 0 | I'm looking for a closed-form expression for tan (q\*pi) for q rational, or an algorithm that generates one, or some other means of compactly describing the closed-form without referencing an infinite series.
It seems to me that tan(q\*pi) should at least be algebraic for q rational. So, it seems like a closed form c... | https://mathoverflow.net/users/70529 | Is there a closed form for tan(q*pi) with q rational? | $$\tan x = \frac{\exp(i x) - \exp(-i x)}{\exp(i x) + \exp(- i x)}.$$
In your case,
$$\tan (p \pi/q) = \tan (p 2\pi / (2 q).$$
Which equals
$$\frac{\omega^{2p} - 1}{\omega^{2p} + 1},$$
where $\omega$ is the primitive $2q$-th root of unity.
| 3 | https://mathoverflow.net/users/11142 | 202912 | 97,809 |
https://mathoverflow.net/questions/202923 | 11 | While reading up on quadratic reciprocity, I learned that if $p = 4k+1$ then $-1$ has a square root in $\mathbb{Z} / p \mathbb{Z}$.
Let $r\_p$ be an integer with $0\leq r\_p < p$ and $r\_p^2 \equiv -1 \mod p$. How then is $\frac{r\_p}{p} \in \mathbb{Q}$ distributed in $[0,1]$? Naively I would guess this is uniform di... | https://mathoverflow.net/users/1358 | distribution of $\sqrt{-1} \mod p$ | The equidistribution of the roots of quadratic congruences $\pmod p$ (such as $x^2+1$ in the question) was established in a famous paper of [Duke, Friedlander and Iwaniec](http://www.jstor.org/stable/2118527?seq=1#page_scan_tab_contents). The proof uses sieve ideas as well as ideas from the theory of modular forms.
| 28 | https://mathoverflow.net/users/38624 | 202926 | 97,812 |
https://mathoverflow.net/questions/202913 | 2 | The set of all binary vectors with 12 components forms a field with 2^12 elements containing 000000000000 and another 65\*63 elements. Is it possible to partition these elements into 65 subgroups of 63 vectors so that each of them is closed under binary addition (XOR)?
For example, if we had 4 components instead of 1... | https://mathoverflow.net/users/70533 | Binary algebra, is it possible to partition the elements in GF(2^12) into 65 subgroups closed under addition? | $\mathrm{GF}(2^{12})$ is a two-dimensional vector space over $\mathrm{GF}(2^6)$. The one-dimensional subspaces are all disjoint (barring the zero vector), and contain 63 nonzero elements each.
| 9 | https://mathoverflow.net/users/nan | 202942 | 97,820 |
https://mathoverflow.net/questions/202915 | 1 | A $C^{\*}$ algebra $A$ is graded by $\mathbb{Z}\_{n}$ iff it can be acted by $\mathbb{Z}\_{n}$. So we associate the $C^{\*}$ algebra $A\rtimes \mathbb{Z}\_{n}$ to a $\mathbb{Z}\_{n}$-graded $C^{\*}$ algebra.
Now what about if the grading group is an arbitrary finite group $G$? Is it true to say that: existence of a $... | https://mathoverflow.net/users/36688 | A $C^{*}$ algebra associated to a graded $C^{*}$ algebra | What's really happing is this:
Fact 1: The grading group and the operating group are not actually the same, it just looks that way! If you have a finite group $G$ acting on an $k$-algebra $A$ by diagonalisable automorphism, then there is a $\widehat{G}$-grading (where $\widehat{G}:=Hom(G,k^\times)$ with pointwise mul... | 5 | https://mathoverflow.net/users/3041 | 202945 | 97,821 |
https://mathoverflow.net/questions/202786 | 7 | This question is motivated by integrability of the [sequence](https://mathoverflow.net/questions/143609/does-this-sequence-always-give-an-integer/202243#202243) mistakenly arisen in the question [Does this sequence always give an integer?](https://mathoverflow.net/questions/143609/does-this-sequence-always-give-an-inte... | https://mathoverflow.net/users/5712 | On one class of Somos-like sequences | I believe this is a special case of Case (9) in Theorem 3.9 of [Allman, Cuenca and Huang](http://arxiv.org/abs/1309.0751). By the way, this paper was an REU project!
| 4 | https://mathoverflow.net/users/297 | 202946 | 97,822 |
https://mathoverflow.net/questions/202956 | 9 | Let $S^{d}$ denote the standard $d$-dimensional sphere. I heard from a physicist that from physical arguments they have been able to show that the vector bundle:
$E\_{d} = TS^{d}\oplus \Lambda ^{d-2}T^{\ast}S^{d}$
is topologically trivial, meaning that it is parallelizable. The convention is that $\Lambda ^{0}T^{\a... | https://mathoverflow.net/users/66688 | A conjecture about parallelizable generalized spheres | It is true in general that $E\_d$ is trivial. As remarked by Neil Strickland above, this boils down to showing that $TS^d\oplus\Lambda^2 TS^d$ is always a trivial bundle. To see this, represent $S^d$ as $SO(d+1)/SO(d)$. Then all bundles in question are homogeneous vector bundles (and the isomorphisms used in the commen... | 14 | https://mathoverflow.net/users/64141 | 202970 | 97,830 |
https://mathoverflow.net/questions/202954 | 8 | Let $(X,T)$ be a uniquely ergodic system (here X is compact, T is a continuous map form $X$ to itself), so for any continuous function $f:X\rightarrow\mathbb{R}$ we have for any $x\in X$, the ergodic average $$\frac{1}{n}\sum\_{i=0}^{n-1}f(T^ix)$$ is convergent pointwise (in fact it is uniformly convergent). Now I am i... | https://mathoverflow.net/users/36604 | Uniquely ergodicity and polynomial ergodic average | This is indeed true for some "nice systems", for example one can show this theorem (for say $L^{2}$-functions) for Kronecker systems simply by van-der-Corput trick.
In general, those averages converge everywhere for Nilmanifolds (Leibman, Green-Tao) with full assorted measure-classification and orbit classification r... | 3 | https://mathoverflow.net/users/8857 | 202971 | 97,831 |
https://mathoverflow.net/questions/202979 | 17 | I am asking if this variant of the weak Goldbach Conjecture is already known.
Let $N$ be an odd number. Does there exist prime numbers $p\_1$, $p\_2$ and $p\_3$ such that $p\_1+p\_2-p\_3=N$? Ideally, can we find $p\_1$, $p\_2$ and $p\_3$ so that they are small enough? For example, can we prove that for large enough $... | https://mathoverflow.net/users/18785 | A variant of the Goldbach Conjecture | Yes - the standard proof of Vinogradov's result by means of the circle method gives this result. You just need to examine an integral
$$\int\_{\mathbb{R}/\mathbb{Z}} (\widehat{f}(\alpha))^2 \widehat{f}(-\alpha) e(-\alpha N) d\alpha$$
instead of
$$\int\_{\mathbb{R}/\mathbb{Z}} (\widehat{f}(\alpha))^3 e(-\alpha N) d\alp... | 38 | https://mathoverflow.net/users/398 | 202982 | 97,835 |
https://mathoverflow.net/questions/202974 | 3 | Is there an example of a category, and a monomorphism $m:X\to Y$ between two objects such that $m$ is extremal, but not regular? (A monomorphism $m:X\to Y$ is said to be *extremal* if whenever $m=g\circ e$ with $e$ an epimorphism, then $e$ is an isomorphism.)
| https://mathoverflow.net/users/8628 | Extremal, but not regular monomorphism | [The Joy of Cats](http://katmat.math.uni-bremen.de/acc/acc.pdf) by Adámek, Herrlich, and Strecker should be your go-to book for this type of question. They note in Proposition 7.62 that if $f: X \to Y$ is extremal and $g: Y \to Z$ is regular, then $g \circ f$ is extremal. Since regular monomorphisms are extremal, it wi... | 9 | https://mathoverflow.net/users/2926 | 202984 | 97,836 |
https://mathoverflow.net/questions/202978 | 4 | I came across the following ring $A$, which appears as a Chow ring. I am wondering if it has been studied before; in particular, I am looking for a reference where this object might have been described.
The graded ring $A^n \subset \mathbb{Z}[x\_1,\dots,x\_n]$ is the subring consisting of polynomials $p$ such that
$$... | https://mathoverflow.net/users/54541 | Identify ring of polynomials symmetric under forgetting variables | The ring $A^n$ is the set of polynomials $f\in\mathbb{Z}[x\_1,\ldots,x\_n]$ such that the coefficient of $x\_{i\_1}^{a\_1}\ldots x\_{i\_k}^{a\_k}$ is equal to the coefficient of $x\_{j\_1}^{a\_1}\ldots x\_{j\_k}^{a\_k}$ whenever $i\_1<\ldots<i\_k$ and $j\_1<\ldots<j\_k$. Polynomials satisfying this condition are called... | 11 | https://mathoverflow.net/users/5263 | 202995 | 97,840 |
https://mathoverflow.net/questions/202993 | 3 | Is every monomorphism in $\mathbf{Frm}$, the category of frames, regular?
| https://mathoverflow.net/users/8628 | Is every frame monomorphism regular? | The answer is no. A category in which all monomorphisms are regular must be *balanced*, i.e., every map that is monic and epic is an isomorphism. (For, any equalizer map that is epic must be an isomorphism.) Intuitively, thinking of locales as "spaces", one should expect this fails badly, since there are many monic epi... | 8 | https://mathoverflow.net/users/2926 | 203017 | 97,846 |
https://mathoverflow.net/questions/203012 | 20 | Let $K \subseteq \mathbb{C}$ be a number field (I'm fixing an embedding), and assume $K/\mathbb{Q}$ is Galois with Galois group $G$. Let $\tau \in G$ denote complex conjugation. This question concerns the condition, let's call it (\*), that $\tau$ is in the center of $G$. In other words, the condition says that $\sigma... | https://mathoverflow.net/users/37644 | When complex conjugation lies in the center of a Galois group | Your condition is that $K$ be a kroneckerian field, namely either a totally real (as you mention) field or a totally imaginary quadratic extension of a totally real field: in this second case $K$ is said to be *a CM field*.
Since you have already treated in your question the case when $K$ is totally real, let me assu... | 20 | https://mathoverflow.net/users/18238 | 203018 | 97,847 |
https://mathoverflow.net/questions/203020 | 1 | Let $A$ and $B$ be two unbounded self-adjoint operators. From [this](https://mathoverflow.net/questions/139249/perturbation-of-unbounded-self-adjoint-operators) mathoverflow post, for instance, we know that $A + B$ is self-adjoint on $\mathcal{D}(A) \cap \mathcal{D}(B)$ if $A$ and $B$ are commuting and positive operato... | https://mathoverflow.net/users/70567 | Sum of two unbounded self-adjoint operators | This will work if you take the assumption that $A,B$ commute in a sufficiently strong sense (commuting resolvents would be enough). Then no extra assumption is needed.
There is a version of the spectral theorem that says that there is a projection valued measure that represents both $A$ and $B$:
$$
A = \int s\, dE(s,... | 1 | https://mathoverflow.net/users/48839 | 203024 | 97,848 |
https://mathoverflow.net/questions/202893 | 3 | Suppose we write $s(n,k)$ for the Stirling numbers of the *first* kind. When $p$ is a prime number, I'm interested in knowing when $s(p^a, k)$ is divisible by $p^a$. So:
>
> What is known about $s(p^a,k)$ mod $p^a$ ?
>
>
>
I have tried to ask google about this, but the literature is confusing (there are *some*... | https://mathoverflow.net/users/37021 | congruence for Stirling numbers of the first kind | As said in my comment, this article gives some partial results: Tamás Lengyel, [On p-adic properties of the Stirling numbers of the first kind](http://www.sciencedirect.com/science/article/pii/S0022314X14003102), Journal of Number Theory Volume 148, March 2015, Pages 73–94.
| 3 | https://mathoverflow.net/users/29783 | 203032 | 97,849 |
https://mathoverflow.net/questions/203028 | 3 | T. Tao in [his notes on eigenvalue inequalities uses Courant-Fischer min-max theorem to prove the eigenvalue stability inequality](https://terrytao.wordpress.com/2010/01/12/254a-notes-3a-eigenvalues-and-sums-of-hermitian-matrices/). Specifically, I am looking for proof of Eq. (13) where he states as an immediate result... | https://mathoverflow.net/users/34445 | Proof of eigenvalue stability inequality via Courant-Fischer min-max theorem | It is a simple and repeated application of $\min$ and $\max$ operators.
$$v^\*(A+B)v=v^\*Av+v^\*Bv\le v^\*Av+\|B\|\_{op},\,\forall v\in R^n\wedge |v|=1.$$
Given $V$ where $\dim(V)=i$,
$$\min\_{u\in V,|u|=1}u^\*(A+B)u\le v^\*(A+B)v\le v^\*Av+\|B\|\_{op},\,\forall v\in V\wedge |v|=1,$$
then
$$\min\_{u\in V,|u|=1}u^\*(A... | 6 | https://mathoverflow.net/users/32660 | 203034 | 97,850 |
https://mathoverflow.net/questions/201912 | 14 | Throw $n$ balls into $n$ bins, and let $X\_n$ be the max load. That is the number of balls in the fullest bin. It is known that if the balls are thrown uniformly and independently at random then $\mathbb{E}(X\_n) = \Theta(\lg{n}/\lg{\lg{n}})$.
If instead, for each ball considered sequentially we look at two bins cho... | https://mathoverflow.net/users/45564 | The power of two random choices with pairwise independence | Will Sawin's idea seems good. Here is a slightly simpler way to get a similar pairwise independent distribution where the maximum load with under a binary choice is still $\Theta(\sqrt{n}).$
Let $\lbrace x\_i \rbrace$ be a random sequence of bins that is symmetric under permuting bins and indices so that a random set... | 3 | https://mathoverflow.net/users/2954 | 203037 | 97,851 |
https://mathoverflow.net/questions/203038 | 7 | For any topological space $X$ we have a natural functor
$\text{Cov}\_X \rightarrow \text{Fun}(\pi\_1(X),\text{Set})$
from the category of coverings of $X$ to the category of functors $\pi\_1(X) \rightarrow \text{Set}$ from the fundamental groupoid to sets, which to a covering $p: Y \rightarrow X$ associates the fib... | https://mathoverflow.net/users/3824 | Coverings/Cech cohomology of totally disconnected spaces | Here is a reference for the sheaf cohomology of totally disconnected spaces: [R. Wiegand, 1969](http://www.ams.org/proc/1969-020-02/S0002-9939-1969-0253324-8/S0002-9939-1969-0253324-8.pdf). This should answer your questions when the sheaf is abelian.
In short, the cohomological dimension of a locally compact totally ... | 1 | https://mathoverflow.net/users/6129 | 203040 | 97,852 |
https://mathoverflow.net/questions/203044 | 24 | I'm working on a problem where I need information on the size of $E\_n=|S\_n-n\mu|$, where $S\_n=X\_1+\ldots+X\_n$ is a sum of i.i.d. random variables and $\mu=\mathbb EX\_1$. For this to make sense, the $(X\_i)$ have to be integrable. In that case, the weak law of large numbers says $E\_n/n$ converges to 0 in probabil... | https://mathoverflow.net/users/11054 | Rate of convergence in the Law of Large Numbers | An early occurence of such bounds is in the theorem of Theorem of vonBahr and Eseen
vonBahr, B., Esseen C.-G.: Inequalities for the rth absolute moment of a sum of random
variables, $1\leq r \leq 2$. Ann. Math. Statist. 36, No.1, 299-393 (1965).
Theorem: Let $X\_i$ be independent (not necessarily i.i.d.) zero mean ... | 16 | https://mathoverflow.net/users/35520 | 203064 | 97,863 |
https://mathoverflow.net/questions/202976 | 9 | Let $X$ be a nice topological space and denote by $\pi\_1(X)$ its fundamental group.
It is well-known that there is a well-defined map
$$
0 \rightarrow H^2(\pi\_1(X),A) \rightarrow H^2(X,A),$$
where $A$ is an abelian group, seen as a trivial $\pi\_1(X)$-module. I am looking for a concrete interpretation of this map... | https://mathoverflow.net/users/34256 | Interpretation of the monomorphism $H^2(\pi_1(X),\mathbb{Z}) \rightarrow H^2(X,\mathbb{Z})$ | First, let's see what is this map $H^2(\pi\_1(X),A) \rightarrow H^2(X,A)$.
$\pi\_1(X)$ is characterized by the following property: A set endowed with a left action of $\pi\_1(X)$ is the same thing as a locally constant sheaves over $X$.
This mean in particular that there is a natural functor from $\pi\_1(X)$-sets t... | 9 | https://mathoverflow.net/users/22131 | 203068 | 97,865 |
https://mathoverflow.net/questions/203072 | 2 | If a space is hyperconnected (that is, the every non-empty open sets intersect), is it also path-connected?
| https://mathoverflow.net/users/nan | Does hyperconnected imply path-connected | No - take $(\mathcal{P}(\mathbb{N}), \tau)$ where $\tau = \{\emptyset\}\cup\{A\subseteq\mathbb{N}: \mathbb{N}\setminus A \text{ is finite}\}$. Clearly, every two non-empty open sets have non-empty intersection, so the space is hyperconnected, but is not path-connected, see [this post](https://mathoverflow.net/questions... | 7 | https://mathoverflow.net/users/8628 | 203073 | 97,867 |
https://mathoverflow.net/questions/202835 | 11 | I don't expect to find an explicit counterexample to my question, because ~~any example which was known to have the Haagerup property yet not have AP would have given an exact group without AP, and the existence of such groups~~
the existence of exact groups without AP was open until the recent(ish) work of Lafforgue-... | https://mathoverflow.net/users/763 | For discrete groups, does the Haagerup property imply the AP of Haagerup-Kraus? | The answer is no. It's solved in the following paper by Osajda.
<http://arxiv.org/abs/1406.5015>
| 7 | https://mathoverflow.net/users/7591 | 203074 | 97,868 |
https://mathoverflow.net/questions/203039 | 5 | This is a naive question and I'm afraid it might be better placed on math.se. I would like to leave it to your judgement.
I would like to know what is known about sets $A$ of natural numbers such that $A$ contains $0$ and there exists a natural number $n$ such that the sum of $n$ $A$s is a submonoid of $\Bbb N$ or
... | https://mathoverflow.net/users/20803 | Sets of natural numbers such that sums of a bounded number of its elements form a semigroup | This question is a special case of a problem proposed by John Brzozowski in 1966 during the seventh SWAT (now FOCS) Conference. Let $A$ be a finite alphabet and let $A^\*$ be the free monoid on $A$. A subset $L$ of $A^\*$ has the *finite power property* if there exists a natural number $n$ for which $L^\* = L^n$. Brzoz... | 6 | https://mathoverflow.net/users/38236 | 203077 | 97,869 |
https://mathoverflow.net/questions/203063 | 1 | Let $X$ be locally compact Alexandrov space whose curvature satisfies both inequalities $\geq K$ and $\leq K$. What can be said about such a space? Is it locally isometric to the standard Riemannian manifold of constant curvature $K$ (e.g. sphere, Euclidean, or hyperbolic space)?
(Recall that if $X$ is a smooth Riema... | https://mathoverflow.net/users/16183 | Alexandrov spaces of constant curvature | See Theorem 10.10.13 in the book A Course in Metric Geometry by Burago-Burago-Ivanov. (The statement is on google books [here](https://books.google.com/books?id=dRmIAwAAQBAJ&pg=PA404&lpg=PA404&dq=nikolaev%20upper%20curvature&source=bl&ots=NJcxT8fec8&sig=je65fJfMMACKDCz8xDSqKQE7VQc&hl=en&sa=X&ei=96kvVZ-sPMm5ggT7loGoAg&v... | 5 | https://mathoverflow.net/users/70595 | 203080 | 97,870 |
https://mathoverflow.net/questions/202746 | 12 | According to Joyal, Street ("*An Introduction to Tannaka Duality and Quantum Groups*"), any $k$-linear abelian category $\mathcal{C}$ admitting a faithful, exact functor $U: \mathcal{C} \rightarrow \mathcal{V}ect\_{k}$ into finite-dimensional $k$-vector spaces arises as a category of finite-dimensional comodules over s... | https://mathoverflow.net/users/70463 | k-linear abelian categories which are not categories of modules | If $A$ is a $k$-algebra, and $M$,$N$ are finite-dimensional $A$-modules, then
$$\operatorname{Ext}^i\_A(M,N)\cong\operatorname{Tor}^A\_i(M,N^\*)^\*$$
(where $\*$ denotes $k$-dual).
So $\operatorname{Ext}^i\_A(M,N)$ must be the dual of a vector space, and so in particular its dimension can't be countably infinite.
F... | 12 | https://mathoverflow.net/users/22989 | 203081 | 97,871 |
https://mathoverflow.net/questions/203005 | 1 | Let $M$ be a smooth manifold and $\tau M$ its second-order tangent bundle. A second-order vector field $A\in \Gamma(\tau M)$ can locally be expressed as a finite sum of operators $C^\infty(M)\rightarrow C^\infty(M)$ of the form
$$ L\_X+L\_YL\_Z,$$
where $L\_V:C^\infty(M)\rightarrow C^\infty(M)$ denotes the Lie der... | https://mathoverflow.net/users/27121 | Are sections of $\tau M$ differential operators on the exterior algebra? | If I understand correctly, you ask if any second order linear differential operator $A:C^\infty(M)\to C^\infty(M)$ of the form $A=L\_X+L\_Y L\_Z$ can be made to act on differential forms by choosing any such decomposition and applying it by Lie derivatives to differential forms. This does not work, as it depends on the... | 1 | https://mathoverflow.net/users/745 | 203082 | 97,872 |
https://mathoverflow.net/questions/203083 | 6 | Let $\pi:\mathcal{C} \to B$ be a (flat) family of complex projective schemes of pure dimension $1$ with fixed Hilbert polynomial, in particular, for some $n \ge 3$, $\mathcal{C} \hookrightarrow \mathbb{P}^n\_B$, the composition of this closed immersion with the natural projection $\mathbb{P}^n\_B \to B$ is $\pi$ and th... | https://mathoverflow.net/users/58203 | Deformation of curves and closed immersions | The answer to the first question is no. In the moduli space $\mathcal{M}\_{10}$ of curves of genus 10, the complete intersections $(3,3)$ in $\Bbb{P}^3$ form a strict subvariety $\mathcal{CI}$. Pick for $B$ a subvariety of $\mathcal{M}\_{10}$ which intersect $\mathcal{CI}$ transversally at one point $[C]$. Embed the co... | 7 | https://mathoverflow.net/users/40297 | 203086 | 97,873 |
https://mathoverflow.net/questions/203079 | 2 | Let $A, B :[a,b]\subset\mathbb{R}\to\mathbb{R}^2$ be two functions of class $C^{1}([a,b])$ such that two segments (or intervals) $[A(t\_1),B(t\_1)]$ and $[A(t\_2), B(t\_2)]$ never intersect for $t\_1, t\_2\in [a,b], \ t\_1\neq t\_2$. Prove that the area of the region described by the segment $[AB]$, $R=\{[A(t),B(t)] \ ... | https://mathoverflow.net/users/61629 | Old Peano theorem (demonstration is missing details) | $\newcommand{\bR}{\mathbb{R}}$
I think that there is something wrong with the formula you wrote. Suppose that
$$ A(t)=(t,0),\;\;B(t)=(t,1),\;\;\;t\in [0,1]. $$
In this case the segment $[A(t), B(t)]$ is the vertical segment from $(t,0)$ to $(t,1)$. The region $R$ is thus the unit square $[0,1]\times [0,1] \subset\b... | 2 | https://mathoverflow.net/users/20302 | 203088 | 97,875 |
https://mathoverflow.net/questions/203113 | 10 | Let $d\ge 2$ and let
$$
\sqrt d =[a\_0; \overline{a\_1,\dots, a\_\ell, 2a\_0}]
$$
be its continued fraction expansion. Clearly, if $d=n^2+1$, then $\ell=0$, which gives the lower bound for $\ell$.
**Question.** What is the best known upper bound for $\ell=\ell(d)$ as a function of $d$?
For instance, $\ell(d)=O(d)... | https://mathoverflow.net/users/8131 | An upper bound for the length of the continued fraction expansion of $\sqrt d$ | It is known that $\ell(d)=O(\sqrt{d}\log d)$ and $\ell(d)=\Omega(\sqrt{d}/\log\log d)$.
See [Cohn's paper](http://projecteuclid.org/euclid.pjm/1102811631) (free access) for more details.
For numerical results and some further historical comments see [Williams's paper](http://www.jstor.org/stable/2007664).
| 13 | https://mathoverflow.net/users/11919 | 203116 | 97,884 |
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