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https://mathoverflow.net/questions/261506 | 1 | On line 7 of page 61 of the book [a guide to quantum groups](https://books.google.de/books/about/A_Guide_to_Quantum_Groups.html?id=bn5GNkLfnsAC&redir_esc=y), a Poisson bracket is defined on $\mathbb{C}[GL\_n]$ for every classical $r$-matrix as follows.
Let $V$ be a vector space with a basis $v\_1, \ldots, v\_n$, and ... | https://mathoverflow.net/users/11877 | Construct super Poisson brackets on the coordinate rings of Lie super groups | I think that it is done in the paper of Andruskiewitsch "Lie superbialgebras and Poisson-Lie supergroups", Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 63 (1993), 147-163, <http://link.springer.com/article/10.1007/BF02941339> (the key result is Proposition 3 in that paper).
| 1 | https://mathoverflow.net/users/1306 | 261551 | 117,834 |
https://mathoverflow.net/questions/261510 | 1 | I have already posted this question [here](https://math.stackexchange.com/questions/2129590/show-0-1-knapsack-is-polynomially-reducible-to-this-problem) but have not received an answer so I am cross-posting with hope to reach a larger amount of mathematicians:
>
> Let $T=\{1,\cdots,n\}$ and consider the following ... | https://mathoverflow.net/users/80751 | Show $0-1$ Knapsack is polynomially reducible to this problem | Your constraints imply that
\begin{align\*}
y\_t=0 &\implies x\_t=0\text{ and }I\_t=I\_{t-1}+a\_t\\
y\_t=1 &\implies I\_t=0\text{ and }x\_t=I\_{t-1}+a\_t
\end{align\*}
As a consequence, if the problem is feasible then the objective value is $a\_0+a\_1+\cdots+a\_T$ for every feasible solution. Feasibility can be decided... | 2 | https://mathoverflow.net/users/12674 | 261552 | 117,835 |
https://mathoverflow.net/questions/261543 | 2 | Fix an integer vector $\mathbf m\in \mathbb Z^k$. Let $q$ be a positive integer.
Is there a "good" upper bound in terms of $q,\bf m$ for the exponential sum:
\[\sum\_{\mathbf n} e\left(\frac{\langle m, n\rangle}{q}\right).\]
Here $\langle \cdot, \cdot \rangle$ is usual inner product and $e(z) = e^{2\pi i z}$. The sum... | https://mathoverflow.net/users/99556 | Upper bound for a higher dimensional Ramanujan sum | Call this sum $c\_q(m\_1,\ldots,m\_k)$. The indicator function of the condition $(n\_1,\ldots,n\_k,q) = 1$ can be written as the sum $\sum\_{cd = (n\_1,\ldots,n\_k,q)} \mu(c)$, so that $d \mid q$, $c = \frac{q}{d}$, and $n\_j \equiv 0 \pmod{c}$ for all $j \in \{1,\ldots,k\}$. It follows that $c\_q(m\_1,\ldots,m\_k)$ ma... | 3 | https://mathoverflow.net/users/3803 | 261553 | 117,836 |
https://mathoverflow.net/questions/261412 | 6 | The [$2$-adic valuation](https://en.wikipedia.org/wiki/P-adic_order) of $n\in\mathbb{N}$, denoted $\nu(n)$, is the largest power $t$ such that $2^t$ divides $n$. The number of [integer partitions](https://en.wikipedia.org/wiki/Partition_(number_theory)) of $n$, denoted by $p(n)$, has generating function
$$\sum\_{n\geq... | https://mathoverflow.net/users/66131 | is this a familiar gen. fn. for partitions? | This can be proved from the famous:
>
> Distinct parts <-> Odd parts
>
>
>
which can be found in Hardy & Wright : An Introduction to the Theory of Numbers.
This states:
$$(1+x)(1+x^2)(1+x^3)\dots=\frac1{(1-x)(1-x^3)(1-x^5)\dots}$$
If you substitute $x\to x^{2^k}$ for $k=1,\dots$, and multiply together, t... | 4 | https://mathoverflow.net/users/70355 | 261566 | 117,842 |
https://mathoverflow.net/questions/261562 | 15 | Whenever we can interchange summation (perhaps due to Tonelli-Fubini), good things happen. Otherwise, one has to struggle evaluating double sums in just one way, because the alternative results in a divergent series. Having said that, I'm currently interested in the following:
>
> Have you encountered in your own r... | https://mathoverflow.net/users/66131 | No Tonelli or Fubini | Since there's a "number theory" tag, I suggest the quasimodular form
$E\_2(\tau)$, defined for $\tau$ in the upper half-plane as a multiple of
$\sum\_{m\in\bf Z} {\sum\_{n\in\bf Z}}' (m\tau+n)^{-2}$
where the $\prime$ indicates omission of the term $(m,n)=(0,0)$.
For even $k>2$, the corresponding sum
$\sum\_{m\in\bf Z... | 23 | https://mathoverflow.net/users/14830 | 261567 | 117,843 |
https://mathoverflow.net/questions/261535 | 5 | Is there a direct proof, using the Riemann mapping theorem for the Jordan domain, than every doubly connected domain in the complex plane can be mapped conformaly onto a round annulus.
| https://mathoverflow.net/users/102342 | Mapping the doubly connected domain to an annulus | Yes, here is a sketch. Let $A$ be your doubly connected domain, wlog $0$ and $\infty$ are in different components of the complement. Consider the preimage of
$A$ under $\exp(z)$. This is an unbounded simply connected domain $D$ with two boundary points at infinity. This domain will be periodic: $z\mapsto z+2\pi i$
will... | 5 | https://mathoverflow.net/users/25510 | 261568 | 117,844 |
https://mathoverflow.net/questions/203782 | 5 | One way to define the Monster group is to consider a conformal field theory (CFT) corresponding to central charge $c=24$ and look at the automorphism group of its vertex operator algebra. For one of the CFT's with $c=24$ this is indeed the Monster group.
What would be the corresponding group for CFT's with $c \neq 24... | https://mathoverflow.net/users/41312 | Analogues of the Monster for central charges different from 24 | As others have mentioned, there are many CFTs, but we can narrow down our list by looking at conditions that select for interesting automorphism groups. Perhaps the easiest is to consider holomorphic ($C\_2$-cofinite) vertex operator algebras. By Zhu's theorem, these necessarily have central charge given by a nonnegati... | 7 | https://mathoverflow.net/users/121 | 261571 | 117,846 |
https://mathoverflow.net/questions/261595 | 2 | Is there a simple proof of the following fact:
**Fact**. Let $S$ be a completely simple semigroup with cancellations, i.e. each of the equalities $xa=xb$, $ax=bx$ implies $a=b$. Prove that $S$ is a group.
Using Sushkevich-Rees Theorem, I can prove it, but my proof is not elegant.
Can you prove this fact using onl... | https://mathoverflow.net/users/81263 | A completely simple semigroup with cancelation is a group (simple proof) | A completely simple semigroup has by definition a (primitive) idempotent $e$ and $SxS=S$ for all $x\in S$. If $x\in S$, then $eex=ex$ and so $ex=x$ and similarly, we see that $e$ is a right identity. Thus $S$ is a monoid and the identity is the unique idempotent of $S$. Let us write $1$ for the identity. Let $x\in S$. ... | 3 | https://mathoverflow.net/users/15934 | 261597 | 117,851 |
https://mathoverflow.net/questions/261581 | 2 | What if we simply require $\mathscr{A}$ to be pre-additive, or additive? I have seen it stated without proof that if $\mathscr{A}$ and $\mathscr{B}$ are abelian categories, then any equivalence $F:\mathscr{A}\to\mathscr{B}$ must be additive, but I have not been able to prove that. This is my work so far:
Clearly, $F(... | https://mathoverflow.net/users/94022 | Let $F:\mathscr{A}\to\mathscr{B}$ be an equivalence of Abelian categories. Must $F$ be additive? | I am posting the comment above as an answer.
An equivalence of categories preserves identity morphisms, finite product, and finite coproducts. Thus, it also preserves diagonal morphisms and codiagonal morphisms. In an Abelian category, the finite product equals the finite coproduct. For every $f,g\in \text{Hom}\_{\ma... | 15 | https://mathoverflow.net/users/13265 | 261599 | 117,852 |
https://mathoverflow.net/questions/261493 | 3 | Restrict everything to a ball in $n$ dimensions, let $x$ represent the first $n-1$ variables, and $t$ the $n-$th variable. It is obvious by Holder's Inequality that
$$
\int\limits\_t\left(\int\limits\_x|f|^p\right)^{\frac1p}\leq\left(\int\limits\_t\int\limits\_x|f|^p\right)^{\frac1p}|\{f\neq0\}|^{1-\frac1p}
$$
I am int... | https://mathoverflow.net/users/96932 | Does $p$ integrability in n-1 dimensions give higher integrability in $n$ dimensions? | This can't possibly hold. You are on a bounded set, so just take your $f(t,x) = g(x)$ for any function $g$ that is in $L^{p-\epsilon}$ but not $L^p$. (Take a sequence $\hat{f}\_n \in L^p\cap L^{p-\epsilon}$ such that $\hat{f}\_n \to g$ in $L^{p-\epsilon}$. Let $f\_n = \delta \hat{f}\_n / \| \hat{f}\_n \|\_{L^p}$. Then ... | 2 | https://mathoverflow.net/users/3948 | 261603 | 117,853 |
https://mathoverflow.net/questions/261541 | 14 | $M$ is non-Kähler complex manifold. Assume that $\omega$ is $\partial$-exact and $\bar\partial$-exact $(p,q)$-form.
>
> **Question**. Is $\omega$ also $\partial\bar\partial$-exact?
>
>
>
Based on fabulous David Speyer's answer to [this MO question](https://mathoverflow.net/questions/59479/partial-bar-partial-l... | https://mathoverflow.net/users/62635 | Are $\partial$-exact and $\bar\partial$-exact forms also $\partial\bar\partial$-exact? | I think that the answer is in general *no* and that a counterexample can be constructed as follows. I will refer to the paper by D. Angella, G. Dloussky, A. Tomassini
[On Bott-Chern cohomology of compact complex surfaces](http://link.springer.com/article/10.1007/s10231-014-0458-7), *Annali di Matematica Pura ed Appl... | 11 | https://mathoverflow.net/users/7460 | 261607 | 117,854 |
https://mathoverflow.net/questions/261606 | 0 | On page 12 of [the paper](http://download.springer.com/static/pdf/137/art%253A10.1007%252FBF02941339.pdf?originUrl=http%3A%2F%2Flink.springer.com%2Farticle%2F10.1007%2FBF02941339&token2=exp=1486462161~acl=%2Fstatic%2Fpdf%2F137%2Fart%25253A10.1007%25252FBF02941339.pdf%3ForiginUrl%3Dhttp%253A%252F%252Flink.springer.com%2... | https://mathoverflow.net/users/11877 | How to prove a bracket is super anti-commutative? | This question was solved by Vladimir Dotsenko in the comments of the [question](https://mathoverflow.net/questions/261506/construct-super-poisson-brackets-on-the-coordinate-rings-of-lie-super-groups).
| 0 | https://mathoverflow.net/users/11877 | 261615 | 117,857 |
https://mathoverflow.net/questions/261605 | 3 | Applying the [Müntz–Szász theorem](https://en.wikipedia.org/wiki/M%C3%BCntz%E2%80%93Sz%C3%A1sz_theorem) on $[0,1]$ repeatedly, we can represent
$$
x= \sum\_{n\geq 2} c\_n x^n
$$
as a uniformly convergent series (**edit:** only over some subsequence, see edits below) on $[0,1]$ of higher powers $x^n$ for $n\geq 2$. Wha... | https://mathoverflow.net/users/91419 | Representing $x$ as a linear combination of higher powers $x^n$ | It can't be in $\ell^p$ or bounded, in fact you can't have $|c\_n| = O(t^{-n})$ for any $t > a$ where the subsequence converges on $[a,1]$. This is because if $|c\_n| = O(t^{-n})$, $\sum\_n c\_n z^n$ is analytic in $|z|<t$, and by uniqueness...
| 4 | https://mathoverflow.net/users/13650 | 261626 | 117,860 |
https://mathoverflow.net/questions/261613 | 7 | Recently I came across the following problem. Here's the setting:
Let $(M^n,g)$ be a Riemannian manifold, $\nabla$ the Levi-Civita connection, and $U$ a coordinate neighbourhood with coordinates $\{x^i\}$. $X = X^i \frac{\partial}{\partial x^i}$ is a vector field. We denote by $\nabla\_i$ the derivative $\nabla\_\fra... | https://mathoverflow.net/users/104576 | How to solve the system of PDEs defining Killing vectors | The system of PDEs in the original post actually defines a connection $D$ on the bundle $\mathcal{E} = TM \oplus \mathfrak{so}(TM)$, by
$$ D\_X \begin{pmatrix} Y \\ A \end{pmatrix} = \begin{pmatrix} \nabla\_X Y + A(X)\\ \nabla\_X A - R(X,Y) \end{pmatrix} $$
for all vector fields $X,Y$ and field $A$ of skewsymmetric end... | 8 | https://mathoverflow.net/users/394 | 261629 | 117,861 |
https://mathoverflow.net/questions/261600 | 11 | Let $I\_k$ denote the [enumeration of involutions](http://mathworld.wolfram.com/PermutationInvolution.html) among permutations in $\mathfrak{S}\_k$. I always enjoy these numbers. Of course, here is yet another cute experimental finding for which I ask validity.
>
> **Question.** Let $n!!=1!2!\cdots n!$. Is the fol... | https://mathoverflow.net/users/66131 | a Hankel matrix of involution numbers | As in arXiv:0902.1650 it suffices to show that $a(n,0)=I\_n$ if $a(n,j)$ satisfies $a(n,j)=a(n-1,j-1)+a(n-1,j)+(j+1)a(n-1,j+1)$ with $a(n,-1)=0$ and $a(0,j)=[j=0]$.
But it is easily verified that $a(n,j)=\binom{n}{j}I\_{n-j},$ because then the above recursion reduces to $I\_n=I\_{n-1}+(n-1)I\_{n-2}.$
**Edit**: Anothe... | 7 | https://mathoverflow.net/users/5585 | 261633 | 117,863 |
https://mathoverflow.net/questions/261588 | 4 | Let $X$ be a complete Riemannian space. Let us denote by $Iso(X)$ the group of isometries of $X$. It is a well-known fact that the group $Iso(X)$, when endowed with the compact-open topology, is a Lie group. Let $X\times X$ be the cartesian prodcut of $X$ with itself endowed with the product metric. Let
$G:=(Iso(X)\ti... | https://mathoverflow.net/users/11765 | On the isometry group of a self cartesian product of a Riemannian space | There is a simple answer in the case that $X$ is simply connected.
In this case, your claim is true if and only if $X$ is not flat and does not split as a Riemannian product.
"$\Longleftarrow$". If $X$ is as above, each isometry of $X\times X$ has to permute the de Rham factors. There are only two, so they are swappe... | 7 | https://mathoverflow.net/users/70808 | 261636 | 117,864 |
https://mathoverflow.net/questions/261540 | 1 | I'm interested in approximating higher derivatives of a function via values of the function only. I guess the following question has been studied, but I haven't been able to find a reference. I know that one can, via repeated application of the finite difference formula
$$f'(x)\approx \frac{f(x+\delta)-f(x)}{\delta}$$
... | https://mathoverflow.net/users/85349 | Lower bounds for finite difference formulas | I believe you are asking the following:
>
> What is the minimum number of evaluations of $f$ required to approximate $f^{(k)}$ to order of accuracy $p$?
>
>
>
In fact, this is a homework problem I often give during the first week of a numerical analysis course. The answer is that generically you need $k-p+1$ e... | 2 | https://mathoverflow.net/users/20507 | 261637 | 117,865 |
https://mathoverflow.net/questions/261638 | 2 | The original problem that gave rise to this question is to solve
$$
\sum\_{i=1}^n\frac1{1+e^{-(x-c\_i)}}=\frac n2
$$
(it comes from the need to determine difficulty of a polytomous item under the graded model in Item Response Theory).
For $n=2$ the (well, a) solution is $\frac{c\_1+c\_2}2$, so it should behave as cer... | https://mathoverflow.net/users/41291 | Does this kind of mean value have an explicit form? | Using your $y$ and $a\_i$ coordinates, one is seeking a solution to
$$\sum\_{i=1}^n\frac{1}{1+y/a\_i}=\frac{n}{2}.$$ Multiplying all this out this becomes some equation of the form $P(y)/Q(y)=0$ with $P(y)$ and $Q(y)$ polynomials in $y$ with coefficients involving the $a\_i$.
So one seeks solutions to $P(y)=0$. Now i... | 3 | https://mathoverflow.net/users/1384 | 261639 | 117,866 |
https://mathoverflow.net/questions/261482 | 22 | Let $\widehat{\mathbb{Z}}[[\widehat{\mathbb{Z}}]] := \varprojlim\_{n,m}(\mathbb{Z}/n)[x]/(x^m-1)$ be the complete group algebra of the profinite free group of rank 1. In Corollary 5.9.2 of Ribes-Zalesski's *Profinite Groups*, they state that $\widehat{\mathbb{Z}}[[\widehat{\mathbb{Z}}]]\cong \widehat{\mathbb{Z}}[[t]]$,... | https://mathoverflow.net/users/88840 | Is $\widehat{\mathbb{Z}}[[t]]\cong\widehat{\mathbb{Z}}[[\widehat{\mathbb{Z}}]]$? | There may be some things to check here, but I think the following is correct and should answer your last question.
$\newcommand{\ZZ}{\mathbb{Z}}$
$\newcommand{\Zhat}{\widehat{\mathbb{Z}}}$
I believe the final result is
$$\Zhat[[\Zhat]] \cong \prod\_q\ZZ\_q[[t]]$$
as $q$ ranges over all prime powers $p^r$ with $r$ c... | 5 | https://mathoverflow.net/users/15242 | 261642 | 117,868 |
https://mathoverflow.net/questions/261614 | 1 | When I was reading a paper about optimization, I encountered to multivariable Boolean polynomials which was undefined. What is the exact definition of a multivariable Boolean polynomial and can you give me an examples of non-negative multivariable Boolean polynomial and one that is not non-negative on $\mathbb R[x\_1,.... | https://mathoverflow.net/users/83050 | What is a multivariable Boolean polynomial? | A multivariable Boolean polynomial $P$ is an element of $\mathbb{F}\_2[x\_1,\ldots,x\_n]$ and in coding theory and cryptography usually taken to represent a map $P:\mathbb{F}\_2^n\rightarrow \mathbb{F}\_2.$
It can also be taken to mean an element $P$ of $\mathbb{Z}[x\_1,\ldots,x\_n]/(x\_1^2-x\_1,\ldots,x\_n^2-x\_n)$... | 2 | https://mathoverflow.net/users/17773 | 261648 | 117,872 |
https://mathoverflow.net/questions/261664 | 5 | A difference between finite geometries and (e.g.) Euclidean space is that "lines" in finite geometries are *unordered* subsets of the universe, while "lines" in Euclidean space are *ordered* subsets of the universe, at least insofar as they have an implicit betweenness relation attached. For example, in Euclidean space... | https://mathoverflow.net/users/25121 | Is there literature on finite geometries with ordered lines? | Yes, this has been studied and is indeed known as *ordered geometry* or the study of *betweenness spaces*:
<https://en.m.wikipedia.org/wiki/Ordered_geometry>
| 5 | https://mathoverflow.net/users/4600 | 261666 | 117,875 |
https://mathoverflow.net/questions/261659 | 16 | Let $G$ be a quasi-split connected reductive group over a $p$-adic field $F$. Let $B$ be a Borel subgroup which is defined over $F$, with $B = TU$, $T$ defined over $F$. The choice of $T$ and $B$ gives a set of nonrestricted roots $\tilde{\Delta}$, which together with an $F$-splitting and a nontrivial unitary character... | https://mathoverflow.net/users/38145 | What's the point of a Whittaker model? | This question is a bit like saying "what's the point of the theory of bases for vector spaces -- this just gives you an isomorphism of your space with $\mathbb{R}^n$. What is the point of defining this isomorphism if the thing we defined is just isomorphic to our original representation? Why would it be important to re... | 27 | https://mathoverflow.net/users/1384 | 261671 | 117,876 |
https://mathoverflow.net/questions/261662 | 3 | I am looking for references/answers that could provide guidance to the following question: Where are some counter-examples to DeGNM for the critical case $q=n/2$ for one of the coefficients of the elliptic PDE, if they exist? That is, if we consider the weak form of the PDE
$$
-\text{div }A\nabla u+Vu=f
$$
on a ball of... | https://mathoverflow.net/users/96932 | What happens to the De Giorgi-Nash-Moser estimate when the potential term lies in the critical $L^{\frac n2}$ space? | Consider $u\_{\epsilon} = \rho\_{\epsilon} \ast \log$ in $B\_{1/2} \subset \mathbb{R}^2$, where $\rho\_{\epsilon}$ are standard mollifiers. Then $|u\_{\epsilon}(0)|$ blows up as $\epsilon \rightarrow 0$, and $\Delta u\_{\epsilon} = \rho\_{\epsilon}$ (representation formula) has unit mass independent of $\epsilon$. Taki... | 6 | https://mathoverflow.net/users/16659 | 261676 | 117,877 |
https://mathoverflow.net/questions/261668 | 8 | I am interested in how I could express $\Omega^k( M \times N)$ in terms of $\Omega^i(M)$ and $\Omega^j(N)$ for $i,j = 0,1, \ldots k$. Is there a nice relation?
This question arose in the context of the Freund-Rubin solution to the bosonic equations of motion for 11-dimensional supergravity, where one posits a space-t... | https://mathoverflow.net/users/104605 | What do the differential k-forms on a product manifold look like? | Denote by $p\_M: M \times N \longrightarrow M$ and $p\_N: M \times N \longrightarrow N$ the canonical projections. Then you get an induced bilinear map from $\Omega^i(M) \times \Omega^j(N) \longrightarrow \Omega^{i+j}(M \times N)$ by taking the $\wedge$-product of the pullbacks with $p\_M$ and $p\_N$.
Hence you have a... | 13 | https://mathoverflow.net/users/12482 | 261683 | 117,879 |
https://mathoverflow.net/questions/261673 | 16 | It appears to me that there are two main ways by which algebraic geometry is applied to number theory. The first is by studying polynomials over fields of number-theoretic interest (which does not seem to be limited to number fields). Diophantine geometry is part of this circle of ideas, as well as the use of elliptic ... | https://mathoverflow.net/users/85392 | Algebraic Geometry in Number Theory | You could read Milne, Arithmetic Duality Theorems <http://jmilne.org/math/Books/ADTnot.pdf> and Neukirch-Schmidt-Wingberg, Cohomology of Number Fields <http://vg02.met.vgwort.de/na/a877a4fbfcec4aad9721c766bc577bb3?l=http://www.mathi.uni-heidelberg.de/%7Eschmidt/NSW2e/NSW2.2.pdf>
| 2 | https://mathoverflow.net/users/nan | 261691 | 117,881 |
https://mathoverflow.net/questions/261679 | 7 | Let $M\_1$ and $M\_2$ be two oriented, *connected*, closed $n$-manifolds. [It is known](http://www.map.mpim-bonn.mpg.de/Bordism#Connected_sum_and_bordism) that the disjoint union $M\_1 \sqcup M\_2$ and the connected sum $M\_1 \# M\_2$ are cobordant, via a bordism $\Sigma\_{M\_1, M\_2}$.
I'm wondering whether this bor... | https://mathoverflow.net/users/13767 | Is the bordism from disjoint union to connected sum universal for connected manifolds? | If we assume $\Sigma$ is connected, I think it's possible to prove existence, as [Mark Grant's counterexample](https://mathoverflow.net/a/261703) requires $\Sigma$ to be disconnected. The idea is that some disjoint-union-to-connect-sum bordism must appear, and then moving handles around turns it into $\Sigma\_{M\_1M\_2... | 2 | https://mathoverflow.net/users/97265 | 261709 | 117,885 |
https://mathoverflow.net/questions/261708 | 2 | I want to solve a parabolic obstacle problem, written as a variational inequality: For almost all $t\in [0,T]$
\begin{align\*}
\langle u'(t), v - u(t)\rangle +a(u(t),v-u(t)) \geq \langle f(t),v-u(t)\rangle \quad \forall v \in K
\end{align\*}
with $K = \{v \in H^1\_0(\Omega) ~\vert ~ v \geq \chi ~ \text{ f.a.a }~ x ... | https://mathoverflow.net/users/100894 | Numerical analysis of parabolic obstacle problem | There's an overview of available schemes in chapter III of
*Roland Glowinski*, MR 737005 [**Numerical methods for nonlinear variational problems**](http://dx.doi.org/10.1007/978-3-662-12613-4), ISBN: 0-387-12434-9.
(of which there is also reprint from 2008). The schemes are presented and a few references for their ... | 2 | https://mathoverflow.net/users/11512 | 261714 | 117,888 |
https://mathoverflow.net/questions/261726 | 4 | I'm interesting to know if there is general solution for the below equation and it's geometrical interpretation then my question here is :
>
>
> >
> > **Question**
> >
> >
> > When is :$\displaystyle n!=x^n-y^n$ , with , $x,y,n$ are positive integers ?
> >
> >
> >
>
>
>
| https://mathoverflow.net/users/51189 | When is :$\displaystyle n!=x^n-y^n$ , with , $x,y,n$ are positive integers? | If $n=1$, the answer is yes. Take $x=2, y=1$ so that $1!=2-1$.
If $n>1$, the answer is no. Assume $d=\gcd(x,y)$ and $x=da, y=db, \gcd(a,b)=1$. Suppose
$$n!=x^n-y^n=d^n(a^n-b^n). \tag1$$
Let $\nu\_p(z)$ denote the $p$-adic valuation of $z\in\mathbb{N}$ and $s\_p(n)$ be the sum of $p$-ary digits of $n$.
If a prim... | 16 | https://mathoverflow.net/users/66131 | 261729 | 117,891 |
https://mathoverflow.net/questions/261727 | 5 | I'm sure that this is classical--but can anyone provide a reasonable example of an $L^\infty(\mathbb{T})$ function whose Fourier series is $\ell^2$ *but no better*? Not even $L^2\log L$? Presumably one exists but nothing has come to my mind. I'm trying to understand just how far one can stretch a (version of a) particu... | https://mathoverflow.net/users/104634 | Existence of $L^\infty$ function on $\mathbb{T}$ whose Fourier series is $\ell^2$ but no better? | A theorem due to Kahane, Katznelson and de Leeuw says that for any sequence $(a\_n)\_{n\in \mathbb{Z}}$ belonging to $\ell\_2$ there is a continuous function $f \in C(\mathbb{T})$ such that $|\widehat{f}(n)| \geqslant |a\_n|$, so in general you don't get anything better than $\ell\_2$. The proof can be found in Appendi... | 4 | https://mathoverflow.net/users/24953 | 261739 | 117,894 |
https://mathoverflow.net/questions/261741 | 0 | This is somewhat unrelated to what I normally do in mathematics, which is why it may be obvious to some of you, but I was puzzled by this:
If we look for classical solutions on $[0,1]$ to
$$-y''(x) = \lambda y(x)$$ with initial conditions $y(0)=1, y'(0)=0$ then the solution is
$y\_{\lambda}(x)=\cos(\sqrt{\lambda}x... | https://mathoverflow.net/users/104662 | Solutions to Schrödinger equation parameter dependence | For the equation $-y''+V(x)y=zy$, and the solution defined by $y(0,z)=1,\; y'(0,z)=1$, if we set $f(z)=y(1,z)$, then
$$f(z)=\cos\sqrt{z}+O(|z|^{-1/2}\exp(|\Im z|),\quad z\to\infty.$$
The only assumption is that $V$ is continuous on $[0,1]$.
See, for example, Levitan, Sargsjan, Introduction to spectral theory, AMS 1975,... | 1 | https://mathoverflow.net/users/25510 | 261742 | 117,896 |
https://mathoverflow.net/questions/261699 | 4 | Let $X$ be a compact connected Riemann surface and $E \rightarrow X$ a holomorphic vector bundle of rank 2. Then for any holomorphic sub-line bundle $L \subset E$, there exists a holomorphic section $s$ of $\mathbb{P}(E)$ associated to it.The inverse is also true, that is, for any holomorphic section $s \in \Gamma(X,\m... | https://mathoverflow.net/users/40042 | Special holomorphic triples on Riemann surfaces and branch divisors | See the formula in the proof of Corollary 11.3.1 in [The Monodromy Groups of Schwarzian Equations on Closed Riemann Surfaces](https://www.math.ucdavis.edu/~kapovich/EPR/GaKaMa.pdf):
$$
deg(L)= g-1 +(deg(E) -deg(B\_s))/2.
$$
As for your first question (*can we*?), the answer is *[yes, we can](https://www.youtube.com/wat... | 6 | https://mathoverflow.net/users/21684 | 261743 | 117,897 |
https://mathoverflow.net/questions/261717 | 1 | I learned the following fact from Bruhat and Tits's paper "Homomorphismes “abstraits” de groupes algebriques simples" Section 3.18 that
Let $k$ be a local field. Suppose that a $k$-group $H$ acts $k$-rationaly on a $k$-variety $M$ and $x$ is an element of $M(k)$ such that the map $h\mapsto hx$ for $h\in H$ of the gr... | https://mathoverflow.net/users/9401 | locally closed orbits in metric Hausdorff topology | The orbits are always locally closed for the Hausdorff topology, even in positive characteristic. This follows from the appendix in the paper of Bernstein and Zelevinskii, [Representations of the group GL(n,F), where F is a local non-Archimedean field.](http://www.ams.org/mathscinet-getitem?mr=425030)
Uspehi Mat. Nauk... | 2 | https://mathoverflow.net/users/81562 | 261745 | 117,898 |
https://mathoverflow.net/questions/176976 | 8 | **Questions:** What is the asymptotic maximal size of a $4$-uniform (every set has 4 elements) set system $\mathcal{A}$ of subsets of $[n]$ such that, no two sets have size of their intersection $2$?
In general I would be interested in k-uniform set systems of size $ \tilde{} n^3$ with intersection sizes one of {0,1... | https://mathoverflow.net/users/38267 | Set system with prescribed intersection sizes | For the first question, Doesn't Frankl-Furedi give you a bound of $O(n)$?
<http://www.sciencedirect.com/science/article/pii/0097316585900354>
| 1 | https://mathoverflow.net/users/35660 | 261766 | 117,900 |
https://mathoverflow.net/questions/229747 | 14 | I'm an undergraduate student, interested in the low dimensional topology, in particular, the 4-manifold theory.
I have a question.
In the knot theory, the Reidemeister moves play fundamental roles.
For instance, to prove the fact that the Jones polynomial is an invariant of knots, we can use the Reidemeister moves.
... | https://mathoverflow.net/users/85988 | Construction of invariants of 4-manifolds with the Kirby calculus |
>
> *Disclaimer*: Shameless self-advertising.
>
>
>
Yes, it can be done, and it's really beautiful! You can define the Crane-Yetter invariant with Kirby calculus, and possibly other TQFTs ("dichromatic models"). I've written this down in this article:
<https://doi.org/10.1007/s00220-017-3012-9>
If you want a... | 11 | https://mathoverflow.net/users/13767 | 261767 | 117,901 |
https://mathoverflow.net/questions/261776 | 4 | Consider $\hat{r} = (\hat{x}\_1,\hat{p}\_1,\ldots \hat{x}\_n,\hat{p}\_n)^\intercal$ with commutation relation
\begin{equation}
[\hat{r},\hat{r}^\intercal] = i\Omega.
\end{equation}
I want a simple statement like:
>
> Any two irreducible representations of the canonical commutation relations are unitarily equivale... | https://mathoverflow.net/users/66031 | The Stone-von Neumann theorem without exponentials? | From a [random review](http://www.ams.org/mathscinet-getitem?mr=217618):
>
> There are many theorems that answer this question, due to [Rellich](http://www.ams.org/mathscinet-getitem?mr=22310), [Dixmier](http://www.ams.org/mathscinet-getitem?mr=101478), [Tillmann](http://www.ams.org/mathscinet-getitem?mr=169595), [... | 2 | https://mathoverflow.net/users/19276 | 261783 | 117,905 |
https://mathoverflow.net/questions/261707 | 11 | We have two conjectured generalizations of the question asked at
[a Hankel matrix of involution numbers](https://mathoverflow.net/questions/261600/a-hankel-matrix-of-involution-numbers)
by Tewodros Amdeberhan. Let $n!!=1!\,2!\cdots n!$.
**Conjecture 1.** Let $I\_k$ denote the number of involutions in the
symmetric g... | https://mathoverflow.net/users/2807 | Further aspects of a Hankel matrix of involution numbers | On **Conjecture 1:** As remarked by Johann Cigler in the linked question (and shown in the references linked there) the matrix $M:=\left[I\_{i+j}\right]\_{i\ge0\atop j\ge0}$ diagonalises as $M=U^TDU$ with $D:=\operatorname{diag(k!)}$, and $U$ an upper triangular integer coefficients matrix with unit diagonal elements, ... | 3 | https://mathoverflow.net/users/6101 | 261785 | 117,907 |
https://mathoverflow.net/questions/232786 | 3 | I am looking for any efforts that have been made to characterize the character kernels (equivalently, the subgroups yielding cyclic quotients) inside the lattice of subgroups of a finite abelian group. I am looking for references, but if it happens that you know a complete description that could be given in an answer, ... | https://mathoverflow.net/users/12419 | Character kernels in the lattice of subgroups of a finite abelian group | A finite group is cyclic if and only if its subgroup lattice is distributive. (A more general result was proved by Baer.) So, if you are given the subgroup lattice of a finite group that you know to be abelian, you can find those subgroups yielding cyclic quotient using only the combinatorial structure of intervals in ... | 3 | https://mathoverflow.net/users/36466 | 261792 | 117,909 |
https://mathoverflow.net/questions/261701 | 0 | In a Noetherian local ring $R$, an ideal $I$ is called an *almost complete intersection ideal* if $\mu(I)=\text{ht}(I)+1$.
**Q) Is it true that $I$ is generated by a $d$-sequence?**
| https://mathoverflow.net/users/9485 | Almost complete intersection ideal and $d$-sequence | Due to the comments to the previous answer (see the next answer) I give the following counterexample for the case where $d$-sequences are defined without considering permutations:
Let $R=K[x,y]$ and consider the ideal $(x^2,y^2,xy)$. It can be seen that no permutation of $x^2,y^2,xy$ is a d-sequence, e.g. $(xy):x^2y^... | 1 | https://mathoverflow.net/users/23240 | 261800 | 117,910 |
https://mathoverflow.net/questions/261782 | 3 | I'm having trouble to find references on that. Consider for instance a very simple model of a wave equation with variable coefficients:
$$\partial\_{tt}^2 u(x,t) - \nabla \cdot( a(x) \nabla u(x,t)) = f(x,t), \quad x \in \mathbb R^3,\ t>0,$$
with some initial conditions $u(\cdot,0) = u\_0$ and $\partial\_t u(\cdot,0) = ... | https://mathoverflow.net/users/80602 | Does Huygens principle holds for heterogeneous media (variable coefficients)? | A pretty extensive reference on Huygens' principle is
[G] *Günther, Paul* MR 946226 [**Huygens’ principle and hyperbolic equations**](http://www.ams.org/mathscinet-getitem?mr=946226), *Perspectives in Mathematics* ISBN: 0-12-307330-8.
First, some definitions. Denote by $A\cdot B = \delta^{ij} A\_i B\_j$ the usual E... | 2 | https://mathoverflow.net/users/2622 | 261816 | 117,916 |
https://mathoverflow.net/questions/255915 | 4 | I am reading Sergey Fomin's and Nathan Reading's paper [Root Systems and Generalized Associahedra](https://arxiv.org/pdf/math/0505518.pdf).
I need a good reference for associahedron of classical types. Besides, whether there are some conclusions for associahedron of exceptional types.
| https://mathoverflow.net/users/89288 | Reference request: Associahedron | The original source would be Fomin-Zelevinsky, <https://arxiv.org/abs/hep-th/0111053>. Note that, for them, the "associahedron" is really just a fan (the normal fan to the simple polytope associahedron). The first realization of the associahedra as polytopes was given by Chapoton-Fomin-Zelevinsky, <https://arxiv.org/ab... | 4 | https://mathoverflow.net/users/468 | 261817 | 117,917 |
https://mathoverflow.net/questions/261793 | -1 | Let $\mathbb Z\_2= \langle\sigma\rangle$ act on $\mathbb C^6$ by $(x\_1,x\_2,x\_3,x\_4,x\_5,x\_6)=-(x\_6,x\_5,x\_3,x\_4,x\_2,x\_1)$. Then what is $\operatorname{Proj}\left(\left(\frac{\mathbb C[x\_1,x\_2,x\_3,x\_4,x\_5,x\_6]}{\langle x\_1x\_6+x\_2x\_5-x\_3x\_4\rangle}\right)^{\mathbb Z\_2}\right)$ ?
I have tried to c... | https://mathoverflow.net/users/104679 | proj of an Algebra | You can diagonalise the action on $(x\_1,\ldots,x\_6)$ by the change of variables
$$(u\_1,\ldots,u\_6):=(x\_1+x\_6,x\_2+x\_5,x\_3,x\_4,x\_2-x\_5,x\_1-x\_6).$$ Then the invariants are $u\_5,u\_6$ and $A\_{ij}=u\_iu\_j$ for $1\leq i \leq j \leq4$.
Proj of this ring gives $X\subset \mathbb{P}^{11}\_{A\_{11},\ldots,A\_... | 2 | https://mathoverflow.net/users/104695 | 261822 | 117,920 |
https://mathoverflow.net/questions/261655 | 8 | For $n>2$, let $X$ be an $n$-by-$n$ invertible matrix where every row has Euclidean norm $1$. Let $Y=X^{-1}$. Let $\Vert y\_i \Vert$ be the Euclidean norm of column $i$ of $Y$.
The following conjecture seems to be confirmed by numerical evidence: for every $i$ $$ \frac{\Vert y\_i \Vert}{\sum\_{j=1}^n \Vert y\_j \Ver... | https://mathoverflow.net/users/7967 | Take a matrix with normalized rows; no column of the inverse has too large a norm | Indeed, if you think of it, it can be restated in geometric terms. What we need to prove is that the inverse altitudes of a parallelepiped spanned by $n+1$ unit vectors in $\mathbb R^{n+1}$ satisfy the "triangle inequality" (each inverse altitude does not exceed the sum of the rest). Let $v$ be one of the given unit ve... | 4 | https://mathoverflow.net/users/1131 | 261828 | 117,922 |
https://mathoverflow.net/questions/261750 | 9 | Suppose we look at all sets in ${[n] \choose k}$, for some $k \leq n/2$, and place them in layers according to the sum of their elements. Then, we say that two sets $A$ and $B$ from consecutive layers are connected by an edge, if there is some $i \in [n-1]$ for which $(i,i+1)A = B$. That is, if $A \Delta B = \{i,i+1\}$... | https://mathoverflow.net/users/35660 | Sums of sets in ${[n] \choose k}$ | You are looking at two consecutive ranks of the poset denoted
$L(k,n-k)$. I proved the existence of a matching in
<http://math.mit.edu/~rstan/pubs/pubfiles/42.pdf>. A more elementary
proof based on linear algebra was later given by R. A. Proctor,
*Amer. Math. Monthly* **89** (1982), 721-734. Another elementary proof
ba... | 15 | https://mathoverflow.net/users/2807 | 261832 | 117,923 |
https://mathoverflow.net/questions/261401 | 2 | Let $2^\alpha=\{f\mid f\colon\alpha\to2\}$.
1, Is it provable in ZFC that for all infinite ordinals $\alpha$, $2^{\alpha+1}$ cannot be embedded into $2^\alpha$, where $2^{\alpha+1}$ and $2^\alpha$ are equipped with the lexicographical ordering?
2, We can prove in ZFC that for all infinite cardinals $\kappa$ and all... | https://mathoverflow.net/users/101817 | Two questions about the order type of $2^\alpha$ equipped with the lexicographical ordering | A friend of mine proves in ZF the proposition in Question 1 as follows.
Assume towards a contradiction that $H$ is an embedding of $2^{\alpha+1}$ into $2^\alpha$. For two different $f,g\in2^\alpha$, the branching point of $f$ and $g$, which will be denoted by $\mathrm{brp}(f,g)$, is defined to be the least ordinal $\... | 2 | https://mathoverflow.net/users/101817 | 261835 | 117,925 |
https://mathoverflow.net/questions/261610 | 5 | Let $R$ be a commutative ring, and let $\mathfrak{a}\subseteq R$ be an ideal. For an $R$-module we consider the sub-$R$-modules $$\Gamma\_{\mathfrak{a}}(M)=\{x\in M\mid\exists n\in\mathbb{N}:\mathfrak{a}^n\subseteq(0:\_Rx)\}$$ and $$\widetilde{\Gamma}\_{\mathfrak{a}}(M)=\{x\in M\mid\mathfrak{a}\subseteq\sqrt{(0:\_Rx)}\... | https://mathoverflow.net/users/11025 | On the relation between two definitions of torsion functors | The conjecture is not true.
By Quý's comment, the conjecture implies that $\Gamma\_{\mathfrak{m}}$ is not a radical if $R$ is a $0$-dimensional local ring whose maximal ideal $\mathfrak{m}$ is idempotent but not nilpotent. This contradicts the following result.
**Lemma** If $R$ is a ring and $\mathfrak{a}\subseteq ... | 2 | https://mathoverflow.net/users/11025 | 261844 | 117,926 |
https://mathoverflow.net/questions/261854 | 8 | Let $k$ be a field and $G$ a connected semisimple algebraic group over $k$.
If $k$ is algebraically closed, then it is well known that all Borel subgroups of $G$ are conjugate by the action of $G(k)$. I would like to know whether this is also true over non-closed fields.
>
>
> >
> > Are all Borel subgroups of ... | https://mathoverflow.net/users/5101 | Conjugacy of Borel subgroups over arbitrary fields | Yes. This follows directly from Theorem 20.9 (i) in "Armand Borel, Linear Algebraic Groups, Second enlarged edition, 1991" which goes like this:
**Theorem**: Let $ G $ be a connected reductive group over a field $ k $. The minimal parabolic $k$-subgroups of $G$ are conjugate under $G(k)$.
| 11 | https://mathoverflow.net/users/47722 | 261859 | 117,930 |
https://mathoverflow.net/questions/261850 | 3 | Let $H$ be a (multiplicative) monoid, and denote by $H^\times$ the *set of units* of $H$ and by $\mathcal A(H)$ the *set of atoms* of $H$ (let me recall that an element $a \in H$ is an atom if (i) $a \notin H^\times$ and (ii) $a = xy$ for some $x, y \in H$ implies $x \in H^\times$ or $y \in H^\times$). Given $x \in H \... | https://mathoverflow.net/users/16537 | A BF-monoid $H$ s.t. $H^\times$ is not divisor-closed | Your monoid cannot be atomic if $H^\times$ is not divisor-closed.
Suppose $xy$ is a unit. Assume $x$ is not a unit. The other case is similar. Then $xyz=1$ for some $z$. Let $a$ be an atom. Then $a=x(yza)$. So by definition of an atom $yza$ must be a unit as $x$ is not. Thus $yzav=1$ for some $v$. Thus $yz$ is both ... | 3 | https://mathoverflow.net/users/15934 | 261870 | 117,933 |
https://mathoverflow.net/questions/261770 | 13 | It's easy to derive a presentation of the fundamental group of a 4-manifold if you have a Kirby diagram: The 1-handles are generators and the 2-handles are the relations. The 3- and 4-handles, which are invisible in the Kirby diagrams, don't contribute.
*Is there anything like this for the homotopy 2-type?*
I'm ima... | https://mathoverflow.net/users/13767 | Given a Kirby diagram of a 4-manifold, what's its homotopy 2-type? | Yes: given a CW-complex $X$ the fundamental crossed module $\Pi\_2(X,X^1)=(\partial \colon \pi\_2(X,X^1) \to \pi\_1(X^1))$, where $X^1$ is the 1-skeleton, represents the homotopy 2-type of $X$ (this can be stated in several ways). Moreover $\Pi\_2(X,X^1)$ can be calculated combinatorially: $(\partial \colon \pi\_2(X^2,... | 8 | https://mathoverflow.net/users/99088 | 261872 | 117,935 |
https://mathoverflow.net/questions/261873 | 6 | We know, if $π : X → S$ is a generically smooth family of complex projective
varieties, such that $X\_0 := π^{−1}(0)$ is an snc divisor in $X$, then the
monodromy representation is unipotent. Now assume the monodromy representation is unipotent, then is the central fibre $X\_0 := π^{−1}(0)$ simple normal crossing?
| https://mathoverflow.net/users/104720 | If monodromy representation is unipotent then special fiber is snc? | Just to be clear, the local monodromy is unipotent if $X\_0$ is *reduced* with simple normal crossings (that's probably what you meant). As Piotr pointed out, your question, as originally formulated, has an easy negative answer. However, as he suggested, the question can be modified to a something more reasonable:
>... | 8 | https://mathoverflow.net/users/4144 | 261879 | 117,938 |
https://mathoverflow.net/questions/261877 | 26 | A few months ago I came up with a proof for an old theorem. After being excited for a moment, I then tried to find my proof in the literature. Since I did not find it, then I started to wonder if it was worth publishing it.
I asked a few people about journals that could publish something like this, and they gave me t... | https://mathoverflow.net/users/104725 | Where to publish a new proof of an old theorem? | If the old theorem is something commonly seen in an undergraduate math class (with the old demonstration), then this might be appropriate as a "Note" in the *American Mathematical Monthly*.
What could happen if you submit it? They may publish it. The referee may give you a reference for it. They may respond in the s... | 28 | https://mathoverflow.net/users/454 | 261880 | 117,939 |
https://mathoverflow.net/questions/261826 | 1 | **\* Question Solved \***
This question ultimately was about the conditions for violation of the Wigner-von Neumann non-crossing rule, which is still an open field of research.
Thank you very much to all those who contributed.
***Edit in response to the comments received*** (see below for initial, disproved, co... | https://mathoverflow.net/users/104693 | SOLVED: Multiplicity of the eigenvalues of the sum of two matrices | While your conjecture is wrong, it still is quite unlikely that $A+B$ has a double eigenvalue. If $A$ and $B$ are symmetric, the set of matrices $B$ such that $A+B$ has a double eigenvalue has codimension 2, not 1. So a crossing is "doubly more unlikely" than what one would expect.
A striking manifestation of this ph... | 0 | https://mathoverflow.net/users/1898 | 261881 | 117,940 |
https://mathoverflow.net/questions/261842 | 5 | Please I need a reference where I can find a version of the "The reverse Lebesgue dominated convergence theorem" for the Orlicz spaces analogous to [Theorem 1.2.7](https://books.google.com/books?id=FihFvMFIoA8C&pg=PA10)
in *Semilinear Elliptic Equations for Beginners* by Marino Badiale, Enrico Serra
>
> **Theorem 1... | https://mathoverflow.net/users/49045 | Looking for a reference for a version of the "The reverse Lebesgue dominated convergence theorem" for the Orlicz spaces | This is Theorem 1.4 in Bennet and Sharpley's "Interpolation of Operators" ([page 3](https://books.google.com/books?id=HpqF9zjZWMMC&pg=PA3)). It actually holds for Banach spaces of functions equipped with what they call "function norms" and not only Orlicz spaces.
The full result is:
A map $\rho$ on the set of posit... | 6 | https://mathoverflow.net/users/9652 | 261890 | 117,943 |
https://mathoverflow.net/questions/261886 | 9 |
>
> Let $X$ be a smooth variety of a field $k$. Then is
> $$H\_{et}^i(X, \mathbb{Q}) = 0$$
> for all $i > 0$?
>
>
>
* The result is true for $i=1$. This follows from the same argument given for $\mathbb{Z}$-coefficients given here: [Etale cohomology with coefficients in the integers](https://mathoverflow.net/... | https://mathoverflow.net/users/5101 | Etale cohomology with coefficients in $\mathbb{Q}$ | The following is surely expressing whatever is in the core non-formal aspect of Joe Berner's answer (which is above my pay grade); it is offered as an alternative version of the same ideas.
Let $X$ be a normal noetherian scheme. We'll show the higher etale cohomology with coefficients in any flat $\mathbf{Z}$-module ... | 14 | https://mathoverflow.net/users/81332 | 261911 | 117,950 |
https://mathoverflow.net/questions/260032 | 14 | Let $M$ be a Riemannian with nonempty boundary $\partial M$.
Define *multiplicity* of $x\in M$ as the number of minimizing geodesics from $x$ to $\partial M$.
The following fact seems to be standard:
>
> The set of points with multiplicity $\ge 2$ is dense in the cut locus of $M$ with respect to $\partial M$.
> ... | https://mathoverflow.net/users/1441 | Conjugate points on cut locus | *I've got a letter from Stephanie Alexander with a complete answer. Let me summarize it here.*
The first proof is given in 4.8 of "Schnittort und konvexe Mengen..." by Hermann Karcher (1968). (The formulation is slightly weaker, but from the proof proves our statement follows; the idea is the same as in the answer of... | 5 | https://mathoverflow.net/users/1441 | 261914 | 117,952 |
https://mathoverflow.net/questions/261903 | 6 | The Lagrangian Grassmannian is an important example in symplectic geometry, see [here](https://ncatlab.org/nlab/show/Lagrangian+Grassmannian) or [here](https://en.wikipedia.org/wiki/Lagrangian_Grassmannian#Maslov_index) for details. It shares many similarities with the ordinary Grassmannians (as one would expect from t... | https://mathoverflow.net/users/89074 | Combinatorics of the Cohomology Ring of the Lagrangian Grassmannians | I'm assuming you mean in $\mathbb{C}^{2n}$ (the answer for real Lagrangian Grassmannians is trickier). In this case, the answer is easy:
>
> The cohomology is isomorphic to the symmetric polynomials in $n$ variables modulo the relation killing all positive degree symmetric polynomials in the squares of the variable... | 4 | https://mathoverflow.net/users/66 | 261917 | 117,954 |
https://mathoverflow.net/questions/261866 | 2 | I posted this on MSE, but no answer is received, so I post this here.
The problem of finding the $n(n-1)/2$-dimensional volume of the set $SO(n)\subset\mathbb R^{n^2}$ is asked before in this MO [post](https://mathoverflow.net/questions/84848/volume-of-compact-simple-lie-groups-under-the-natural-euclidean-embedding).... | https://mathoverflow.net/users/74664 | Volume of $SO(n)\subset\mathbb R^{n^2}$, again | Maybe this will help: Regard $\mathrm{SO}(n)\subset M\_{n,n}(\mathbb{R})$ as the set of $n$-by-$n$ matrices $a$ that satisfy ${}^ta\,a=\mathrm{I}\_n$ and $\det(a)=1$. Then $\mathrm{SO}(n)$ is a smooth, connected submanifold of $M\_{n,n}(\mathbb{R})$ of dimension $\frac12n(n{-}1)$.
Give $M\_{n,n}(\mathbb{R})$ the pos... | 7 | https://mathoverflow.net/users/13972 | 261939 | 117,963 |
https://mathoverflow.net/questions/261941 | 6 | Consider the Hilbertspace $W^{1,2}([0,1])$ (i.e. Sobolev space) with the standard inner product which is defined by: $(f,g) = (f,g)\_{L^{2}([0,1])} + (f',g')\_{L^{2}([0,1])}$. Here $[0,1]$ is not parametrizing the circle; it is just an interval and the functions on it are not assumed to be periodic. My question is: Wha... | https://mathoverflow.net/users/104758 | Orthonormal basis in $W^{1,2}([0,1])$ | If you change the interval to $[-1,1]$ instead of $[0,1]$, the equivalent inner product would be $$\tag{$\*$} (f,g) = (f,g)\_{L^2([-1,1])} + \lambda (f',g')\_{L^2([-1,1])},$$ where $\lambda = 2$.
One way to get an orthogonal basis is to start with polynomials and apply Gram-Schmidt orthogonalization to them. You get ... | 10 | https://mathoverflow.net/users/2622 | 261946 | 117,965 |
https://mathoverflow.net/questions/261902 | 4 | I am trying to figure out the conditions under which you can glue together a collection of (differentiable) stacks by equivalences, and get a differentiable stack.
More precisely, I have a collection of stacks $U\_i$ and open substacks $U\_i|\_j\hookrightarrow U\_i$ with equivalences $\varphi\_{ij}:U\_i|\_j\to U\_j|\... | https://mathoverflow.net/users/104740 | Gluing together together differentiable stacks | Try p. 17 of notes by Breen (<http://math.uchicago.edu/~may/IMA/Breen.pdf>) *Notes on 1- and 2-gerbes*
| 4 | https://mathoverflow.net/users/25355 | 261949 | 117,967 |
https://mathoverflow.net/questions/261943 | 6 | Let $ G $ be a finite group and $ N $ be a normal subgroup of $ G $. Let $ G/N $ have two irreducible characters of degrees $ p\_1$ and $ p\_2$, where $ p\_1$ and $ p\_2$ are different primes. Let $ G/N $ have no irreducible character such that $ p\_1p\_2\mid \chi (1) $. If $ (p\_1p\_2, |N|)=1$, can we say that $G $ ha... | https://mathoverflow.net/users/31045 | A question about relation of the character degrees of $ G/N $ and $ G $ | It is possible for $G$ to have such a character. The smallest example has order $120$. Let $N = \langle k \rangle \cong C\_5$. Let $G = N \rtimes S\_4$ where the action of the symmetric group $S\_4$ on $N$ is non-trivial but factors through the sign representation. Thus $k^{(12)} = k^{-1}$, $k^{(234)} = k$ and $\langle... | 6 | https://mathoverflow.net/users/7709 | 261951 | 117,968 |
https://mathoverflow.net/questions/173095 | 3 | Schanuel's conjecture states:
* If $\alpha\_1,\alpha\_2,...,\alpha\_n$ are complex numbers linearly independent over $\mathbb{Q}$, then the transcendence degree of the field $\mathbb{Q}(\alpha\_1,e^{\alpha\_1},\alpha\_2,e^{\alpha\_2},...,\alpha\_n,e^{\alpha\_n})$ over $\mathbb{Q}$ is at least $n$.
In [What is a clo... | https://mathoverflow.net/users/21258 | Schanuel's conjecture and real root of $x+e^x=0$ | My answer is not the whole answer.
$R=-W(1)$, where $W$ is the Lambert W function.
$H(x)=x+e^{x}=0$, $H$ is an elementary function and the inverse of $H$ is $H^{-1}$ with $H^{-1}(x)=-W(e^{x})+x$.
Let us assume Schanuel's conjecture is not true and $-W(1)$ is an elementary number. That does not mean the equation $... | 1 | https://mathoverflow.net/users/94085 | 261953 | 117,969 |
https://mathoverflow.net/questions/261896 | 3 | Given positive integers $n$ and $c$, with $n>c$, is there a good way to estimate the number of ways to partition the set $\{1,\dots,n\}$ into ordered subsets, with each subset having at most $c$ elements? By "ordered subsets", I mean that the ordering *within* each subset matters, but we don't care about the order in w... | https://mathoverflow.net/users/104738 | (Approximate) Number of ways to partition $n$ elements into ordered subsets of bounded size | Let $S = \{(a\_1, \ldots , a\_c) \ : \ \sum\_i i a\_i = n\}$, and for $a \in S$, let $f(a) = 1/\prod\_i a\_i !$. Then the exact value you want is $n! \sum\_{a \in S} f(a)$ [the term $n! f(a)$ counts the number of such decompositions with $a\_i$ sets of size $i$].
We can bound this by finding some $f(a) \leq \Delta$, ... | 1 | https://mathoverflow.net/users/22512 | 261960 | 117,973 |
https://mathoverflow.net/questions/261921 | 3 | If $π : X → S$ is a family of complex projective
varieties, such that $X\_0 := π^{−1}(0)$ has simple normal singularities in $X$, then all the general fibers $X\_t$ have snc singularities at worst?
| https://mathoverflow.net/users/104720 | snc singularity is an open condition? | I am just posting my comment as an answer.
That is not correct. Already in projective $3$-space, cones over smooth plane cubics specialize to cones over unions of three lines, i.e., a simple normal crossings divisor consisting of union of three hyperplanes.
| 3 | https://mathoverflow.net/users/13265 | 261962 | 117,974 |
https://mathoverflow.net/questions/261958 | 0 | Consider the metric space $X = \mathbb{R}$, $\mathcal{B}$ the Borel $\sigma$-algebra on $\mathbb{R}$ and $\mu$ a probability measure on $X$. Let $A \in \mathcal{B}$ and $\tau\_n \nearrow \infty$ a sequence of positive numbers.
I can't demonstrate that the following sequence of measures is or is not convergent in the... | https://mathoverflow.net/users/70760 | Is the following sequence convergent in the weak topology? | Let $\mu\_n$ be the measure defined by $$\mu\_n(A) = \frac{1}{\tau\_n} \int\_0^{\tau\_n} \mu(A-t)\,dt.$$
In general, this sequence doesn't converge weakly. Suppose for instance that $\mu$ is a point mass at $0$. Then $\mu\_n$ is the uniform probability measure on $[0, \tau\_n]$. Let $f$ be any bounded, continuous, st... | 1 | https://mathoverflow.net/users/4832 | 261963 | 117,975 |
https://mathoverflow.net/questions/261944 | 1 | Let $f:X\rightarrow \mathbb{P}\_1$ be a smooth fibration from a smooth rationally connected manifolds to the smooth rational curve $\mathbb{P}\_1$. Assume further that the generic fiber of $f$ is also rationally connected.
Question: Besides the trivial product of $\mathbb{P}\_1$ with a rationally connected manifolds... | https://mathoverflow.net/users/62735 | Non-Trivial Families of rationally connected manifolds | I am just posting my comment as an answer.
The simplest examples are obtained as follows. Begin with $Y=\mathbb{P}^2\times \mathbb{P}^1$ together with its projection $$\text{pr}\_2: \mathbb{P}^2\times \mathbb{P}^1 \to \mathbb{P}^1.$$ For each integer $N\geq 5$, let $(\sigma\_i:\mathbb{P}^1\to \mathbb{P}^2\times \math... | 1 | https://mathoverflow.net/users/13265 | 261967 | 117,976 |
https://mathoverflow.net/questions/261438 | 12 | Let $G$ be a finite group. In [1], Bredon defines an equivariant cohomology theory for $G$-CW complexes $H^\*\_G(X;M)$. The coefficients are taken in modules over the orbit category of $G$, that is, contravariant functors $M:\mathcal{O}\_G\to \mathcal{Ab}$ from the category of finite $G$-sets and $G$-maps to the catego... | https://mathoverflow.net/users/8103 | Calculations of cup products in Bredon cohomology | Frankly, there aren't many calculations out there. Most of the work I know of is on the calculation of the $RO(G)$-graded cohomology of a point, of a projective space, or of $B\_GO(n)$. Here are some references and notes on them:
[1] L. G. Lewis, Jr., *The $RO(G)$-graded equivariant ordinary cohomology of complex pro... | 5 | https://mathoverflow.net/users/58888 | 261969 | 117,977 |
https://mathoverflow.net/questions/261930 | 8 | I apologize if this question is elementary: Let $A$, $B$, and $A^{\prime}$ be groups such that $A^{\prime}$ is an elementary extension of $A$. Is it true that $A^{\prime}\times B$ is an elementary extension of $A\times B$? Clearly this is true for ultrapowers.
| https://mathoverflow.net/users/44949 | Elementary extensions of direct product | Yes. Let $T$ be the theory of two disjoint groups, in the language $(\cdot\_1, \cdot\_2, U\_1, U\_2)$. Note that if $(G\_1, G\_2) \models T$ then the group operation on $G\_1 \times G\_2$ is definable without parameters. Thus we can recover the theory of $G\_1 \times G\_2$ from the theory of $(G\_1, G\_2)$, which is cl... | 8 | https://mathoverflow.net/users/26705 | 261973 | 117,978 |
https://mathoverflow.net/questions/261965 | 2 | Let $E\_1$ and $E\_2$ be elliptic curves over $\mathbb{C}$. A (primary) Kodaira surface is a principal bundle $X \to E\_1$ with fibre $E\_2$. $X$ is a compact complex surface with trivial canonical bundle and so it has Kodaira dimension $0$. In general $X$ is not algebraic (not even Kaehler because $b\_1=3$).
The qu... | https://mathoverflow.net/users/1220 | Analogue of Kodaira surfaces | You could follow Suwa's construction of the Kodaira surfaces from his paper *Compact quotients of $C^2$ by affine transformation groups*, and define various surfaces which are quotients of $k^2$ by groups of affine transformations of the special form that Suwa arrives at in his paper. I don't know any applications, but... | 2 | https://mathoverflow.net/users/13268 | 261983 | 117,982 |
https://mathoverflow.net/questions/261574 | 6 | Let $\{S\_i\}\_{i \in I}$ be a directed projective system and $S = \varprojlim S\_i$.
1.How to prove that $S$ is non-empty in the following case: all $S\_i$ are nonempty compact Hausdorff spaces.
2.I know that when $S\_i$ are non-empty sets and $f\_{ij}$ are surjective, $S$ may still be empty. But if we furthermor... | https://mathoverflow.net/users/104306 | When is the projective limit non-empty? | This answer is to supplement Fred Rohrer's answer, copying out some details extracted from Bourbaki (in writing for instance TG.I.9.6, he was referring to a French edition of Topologie Générale, but I'll just refer to the English language edition of Bourbaki's Set Theory Treatise, starting [here](https://archive.org/st... | 9 | https://mathoverflow.net/users/2926 | 261985 | 117,983 |
https://mathoverflow.net/questions/257862 | 41 | For every convex compact set $K$ of area $1$ in $\mathbb{R}^2$, among all ellipses of area $1$ there exists an ellipse $E$ such that the area of the symmetric difference between $K$ and $E$ is smallest possible.
>
> **Questions.**
>
>
> (a) Is $E$ unique?
>
>
> (b) If the answer is "yes", does the same hold for... | https://mathoverflow.net/users/36904 | Approximating a convex disk by an ellipse | Question (a) has a positive answer in the centrally symmetric case. The proof is involved and I will only summarize the strategy here. Full details can be found in [this ArXiv paper](https://arxiv.org/abs/1702.03808). Comments, suggestions, corrections etc are welcome.
Let $K \subset \mathbb{R}^2$ be a compact convex... | 5 | https://mathoverflow.net/users/1516 | 261986 | 117,984 |
https://mathoverflow.net/questions/261982 | 5 | The title has it all. Is there any consolidated terminology for referring to a (multiplicative) monoid $H$ such that $xy \in H^\times$ (if and) only if $x, y \in H^\times$? Here is a short list of monoids with this property:
* Commutative monoids.
* [Unit-cancellative](https://mathoverflow.net/a/259790/16537) monoids... | https://mathoverflow.net/users/16537 | Terminology for a monoid $H$ s.t. $xy \in H^\times$ only if $x, y \in H^\times$ | The correct term is Dedekind finite. A monoid is Dedekind finite of $xy=1$ implies $yx=1$. This is clearly equivalent to your condition.
| 10 | https://mathoverflow.net/users/15934 | 262001 | 117,989 |
https://mathoverflow.net/questions/262003 | 4 | Is there a positive integer $N$, besides 1 and 2, such that there is a permutation $a\_1=1,a\_2,a\_3,\dots,a\_N$ of $1,2,3,\dots,N$ in which for each $k>1$, $a\_k=a\_{k-1}\div k,a\_k=a\_{k-1}-k,a\_k=a\_{k-1}+k,\textrm{or }a\_k=a\_{k-1}\times k$?
| https://mathoverflow.net/users/88300 | Can one permute the positive integers with just the four arithmetical operations? | This is implicit in Gerhard Paseman's answer but it should be made explicit: what you ask for is not possible, for a fairly simple reason. Consider $a\_N$. There are four possibilities:
$$
(1)\qquad a\_N = a\_{N-1}/N
$$
but this can't be satisfied, because $a\_{N-1}\le N$, so either $a\_{N-1}$ doesn't divide $N$ or $a\... | 6 | https://mathoverflow.net/users/46551 | 262009 | 117,992 |
https://mathoverflow.net/questions/262008 | 4 |
>
> **Question.** Numerically, the following is convincing. However, is there a proof?
> $$\left(\sum\_{k\geq1}\frac1{\sqrt{2^k+3^k}}\right)^4
> <\pi^2\left(\sum\_{k\geq1}\frac1{2^k+3^k}\right)\left(\sum\_{k\geq1}\frac{k^2}{2^k+3^k}\right).$$
>
>
>
This comes up in some recent work and the inequality seems need... | https://mathoverflow.net/users/66131 | seeking proofs: infinite series inequalities | This may serve as a different approach. By Cauchy-Schwarz inequality, $$\left(\sum\_{k\geq 1}\frac 1{\sqrt{2^k+3^k}}\right)^2\leq \left(\sum\_{k\geq 1}\frac 1{k^2}\right)\left(\sum\_{k\geq 1}\frac{k^2}{2^k+3^k}\right),$$ which shows that $$\left(\sum\_{k\geq 1}\frac 1{\sqrt{2^k+3^k}}\right)^2\leq \frac{\pi^2}6\left(\su... | 18 | https://mathoverflow.net/users/104791 | 262012 | 117,993 |
https://mathoverflow.net/questions/260747 | 8 | Let $\{V\_i\}\_{i=1}^N$ be a set of $n\times m$, $n\geq m$, real matrices of full column rank and let $X=X^\top\in\mathbb{R}^{n\times n}$ be a positive definite trace-one matrix. Moreover, let $A^{1/2}=(A^{1/2})^\top$ denote the (unique) symmetric square root of a positive semi-definite matrix $A$, $\mathrm{tr}\, A$ th... | https://mathoverflow.net/users/62673 | A generalized log inequality for positive definite trace-one matrices | Fedor's proof can be generalized :
Let $S = X^{1/2}$, $U\_i = V\_i S^{1/2}$ and $W\_i = U\_i (U\_i^T S U\_i)^{-1/2}$ .
Then $W\_i^T S W\_i = I\_m$ where $I\_m$ is the $m\times m$ identity .
Since $log(x) \ge 1 - 1/x$ it is enough to show that
$$tr \sum\_{i=1}^N \mu\_i \le 1$$
where
$$\mu\_i = (\sum\_{j=1}^N W\_i... | 3 | https://mathoverflow.net/users/17261 | 262016 | 117,994 |
https://mathoverflow.net/questions/260872 | 5 | I'm looking for a reference to a book which develops an It\^{o} lemma for semi-martingales with values in infinite dimensional Hilbert-Manifolds. I expect the techniques to be the same but still I appreciate a reference as there are hints in some of the literature that these things are known.
So far I've only found ... | https://mathoverflow.net/users/36886 | Reference: Stochastic Analysis on Hilbert Manifolds | As the question is general and observes the wide of the subject; I have chosen some reference.
[Path Integrals on a Compact Manifold with Non-negative Curvature](https://arxiv.org/pdf/math/0612711)
[Foundations of the Theory of Semilinear Stochastic Partial Differential Equations](https://www.hindawi.com/journals/i... | 2 | https://mathoverflow.net/users/19072 | 262024 | 117,995 |
https://mathoverflow.net/questions/262030 | 10 | In my [previous MO question](https://mathoverflow.net/questions/262008/seeking-proofs-infinite-series-inequalities), the inequality was about a specific series and nicely answered by Cherng-tiao Perng. After testing with a few more numerical infinite sums, I came to realize that perhaps more is true.
>
> Does the f... | https://mathoverflow.net/users/66131 | An attempt to generalize the previous inequality | This is known to be Carlson's inequality from 1935 (for $t\_k\geq 0$, and not all $t\_k$ are $0$). The Swedish mathematician Fritz Carlson (1888-1952) also proved the optimality of the constant $\pi^2$. For an elegant elementary proof of the inequality see G. H. Hardy, A note on two inequalities, J. London Math. Soc. 1... | 27 | https://mathoverflow.net/users/13034 | 262038 | 117,998 |
https://mathoverflow.net/questions/261711 | 11 | Consider the $n$-element subsets $\{a\_1<a\_2<\cdots <a\_n\}$ of $\{1,\ldots ,2n\}$ satisfying $a\_i\geq 2i$ for all $i=1,\ldots ,n$. The number of such subsets is given by $${2n\choose n}-{2n\choose n-1}=\frac{1}{n+1}{2n\choose n},$$
which is the $n$th Catalan Number.
I want to know if the $q$-Catalan number $$\frac... | https://mathoverflow.net/users/23980 | Does $q$-Catalan number count subspaces? | An answer to your question was given in ``Rank Polynomials" by Brandt, Dipper, James, and Lyle, published in Proc. London Math. Soc. (3) 98 (2009), 1-18. A special case of Theorem 2.6 in that paper answers your question.
| 5 | https://mathoverflow.net/users/104811 | 262041 | 118,000 |
https://mathoverflow.net/questions/262031 | 3 | Let $f:X\to \mathbb \Delta$ be family of projective varities which fibers are smooth, we know central fiber can be singular and may not be mild. Kollar introduced [semi-log-canonical singularities](https://web.math.princeton.edu/~kollar/book/chap3.pdf) to get mild singularity on central fiber. Under which assumption on... | https://mathoverflow.net/users/104720 | semi-log canonial singularities is an open condition? | Your first question is hard to answer as posed. In general, one can say very little about what kind of singularities a particular fiber might have even if the total space and hence the nearby fibers are non-singular.
For example, let $X\_0$ be the projectivized cone over a hypersurface of degree $d$ in $\mathbb P^n$... | 6 | https://mathoverflow.net/users/10076 | 262045 | 118,003 |
https://mathoverflow.net/questions/262063 | 4 | By a classical result of Singer (1938), for a prime number $p$ the cyclic group $C\_n$ of order $n=1+p+p^2$ contains a subset $D$ of cardinality $|D|=1+p$ such that $DD^{-1}=C\_n$. Such set $D$ is called a *difference set*. The cardinality restrictions imply that each non-unit element $x\in C\_n$ can be uniquely writte... | https://mathoverflow.net/users/61536 | Large gaps in Singer planar difference sets? | No you cannot have very big gaps in a perfect difference sets -- the elements in a perfect difference set will be pretty uniform. Suppose $A$ is a perfect difference set $\mod n$ with $n=1+p+p^2$ as above. Put
$$
{\hat A}(k) = \sum\_{a \in A} e(ak/n).
$$
Since $A$ is a perfect difference set, for any $k\neq 0$ one h... | 9 | https://mathoverflow.net/users/38624 | 262064 | 118,005 |
https://mathoverflow.net/questions/262065 | -1 | I am trying to learn how to compute the projective bundle $\mathbb{P}(\mathcal{O}(a\_1)\oplus \cdots \mathcal{O}(a\_k))$ over some projective space using relative proj. How can I find a presentation for the ideal $I$ giving the closed subscheme of $\mathbb{P}^n\times\mathbb{P}^m$? For example, I want to understand how ... | https://mathoverflow.net/users/78824 | How do I find the algebra representing the projective bundle of a direct sum of line bundles over a projective space? | OK, I guess I will write it here as it might need more room.
You know that $\mathbb P(\mathscr E)$ remains the same if you twist it by a line bundle. So, choose a sufficiently ample line bundle $\mathscr L$ such that $\mathscr E\otimes \mathscr L$ is generated by global sections and switch $\mathscr E$ with $\mathscr... | 8 | https://mathoverflow.net/users/10076 | 262072 | 118,006 |
https://mathoverflow.net/questions/261920 | 11 | **Problem.** What is the smallest cardinality $d(n)$ of a set $A$ of integer numbers such that the difference set $A-A=\{a-b:a,b\in A\}$ contains $n$ consequtive integer numbers?
It can be shown that $(1+\sqrt{4n-3})/2\le d(n)\le \frac32p(\sqrt{n})=\frac32\sqrt{n}+O(n^{21/80})$ where $p(x)$ is the smallest prime numb... | https://mathoverflow.net/users/61536 | What is the smallest cardinality of a set A whose difference A-A contains $n$ consequtive integer numbers? | Since you say that only Question 2 is open, I'll only address that. The answer is no, and $d(n)$ must be at least $(1+\delta)\sqrt{n}$ for some positive $\delta$. I won't compute this, but it shouldn't be too hard to find some bound.
Suppose for contradiction that $|A|\le (1+\delta) \sqrt{n}$. Since the difference s... | 8 | https://mathoverflow.net/users/38624 | 262073 | 118,007 |
https://mathoverflow.net/questions/262076 | 1 | Let $K$ be a covariance matrix. It is positive semidefinite, its diagonal elements are all 1, and its off-diagonals are between -1 and 1. Let $K.^2$ be its element-wise power (Hadamard power). Can we show that maximum eigenvalue of $K$ are great or equal than the maximum eigenvalue of $K.^2$?
| https://mathoverflow.net/users/104833 | Maximum eigenvalue of Hadamard power of a positive semidefinite matrix | Theorem 3 in ["On majorization and Schur products"](https://www.sciencedirect.com/science/article/pii/0024379585901478/pdf?md5=f8866f6d5d885524d837052e8c42719d&pid=1-s2.0-0024379585901478-main.pdf) by Bapat and Sunder provides a stronger result. Take $A=B=K$ and specialize to $k=1$ in the majorization.
| 3 | https://mathoverflow.net/users/69850 | 262081 | 118,009 |
https://mathoverflow.net/questions/261704 | 5 | Let $M$ be a co-admissible module over a Frechet Stein Algebra $A=\varprojlim A\_{q\_n}$ as in [this paper](https://arxiv.org/pdf/math/0206056.pdf). Let $N$ be a closed submodule of $M$. I have some difficulty in understanding lemma $3.6$ of the above paper. That is I want to show that $N$ is co-admissible.
For that... | https://mathoverflow.net/users/69289 | Coherent subsheaf of co-admissible modules of Schneider and Teitelbaum | The map $A\_{q\_n} \otimes\_{A\_{q\_{n+1}}} N\_{n+1} \rightarrow N\_n$ is surjective, by definition of $N\_n$. To show that it is injective, it suffices to show that the composition $A\_{q\_n} \otimes\_{A\_{q\_{n+1}}} N\_{n+1} \rightarrow N\_n \rightarrow M\_n$ is injective. But this composition factors as
$$
A\_{q\_n}... | 5 | https://mathoverflow.net/users/21724 | 262084 | 118,010 |
https://mathoverflow.net/questions/262080 | 1 | I was wondering if there is an analogue to the classical Riesz Thorin theorem for Schatten classes. I suppose the answer is yes, since Schatten classes are so similar to $\ell^p$ spaces for which the theorem holds. However, I could not find a reference online. In particular, does one have to take the bounded or compact... | https://mathoverflow.net/users/104662 | Interpolation between Schatten classes | A Riesz-Thorin interpolation theorem (and a Marcinkiewicz one) is known to hold for Schatten classes (with the case $p=\infty$ corresponding to the *compact* operators, think of the canonical duality $c\_0' = \ell^1$): cf. [1, Thm. 13.1], [2, Thms. 2.9-10] and [2, Remark 1 on p. 23].
*References*
[1] I.C. Gohberg,... | 2 | https://mathoverflow.net/users/13034 | 262095 | 118,014 |
https://mathoverflow.net/questions/262004 | 6 | For $\alpha < \beta$, where $\alpha$ is regular and $\beta$ is measurable, let $M(\alpha, \beta)$ be the Mitchell forcing for making $2^\alpha=\beta=\alpha^{++}$ and forcing the tree property at $\beta.$
Now suppose $\alpha<\beta<\gamma$, with $\alpha$ regular and $\beta, \gamma$ measurable. The product forcing $M(\a... | https://mathoverflow.net/users/11115 | Questions about Mitchell forcing and tree property | At least in a variant of Mitchell's forcing, the iteration forces the tree property at both cardinals.
Let $M(\alpha, \beta)$ be the iteration $Add(\alpha, \beta) \ast \mathbb{C}(\alpha, \beta)$ where $p \in \mathbb{C}(\alpha, \beta)$ iff $p$ is a sequence of length $\beta$, with at most $\alpha$ non-trivial coordin... | 4 | https://mathoverflow.net/users/41953 | 262103 | 118,017 |
https://mathoverflow.net/questions/262049 | 6 | Does anyone have a good explanation of the name, and why Doob chose it? It states the following: if $T$ is a stopping time such that $\mathbb{P}(T < \infty)$, and $M\_n$ is a uniformly integrable martingale, then
$$
E[M\_T] = E[M\_0].
$$
| https://mathoverflow.net/users/47510 | History of optional sampling/stopping theorem | "Optional stopping" and "optional sampling" refer to a strategy of peeking while you are sampling and then, based on what you find, exercise the option to quit sampling.
Doob's theorem states that in a fair casino, where your return is a martingale, you cannot increase your expected return if you are given the option... | 4 | https://mathoverflow.net/users/11260 | 262104 | 118,018 |
https://mathoverflow.net/questions/262091 | 6 | Is there an established term for the following type of square matrices?
$\begin{pmatrix}
c & c & c & c & \cdots & c & c \\
c & a & b & b & \cdots & b & b \\
c & b & a & b & \cdots & b & b \\
c & b & b & a & & b & b \\
\vdots & \vdots & \vdots & & \ddots & & \vdots \\
c & b & b & b & & a & b \\
c & b & b & b & \cdots ... | https://mathoverflow.net/users/66043 | What is the term for this type of matrix? | When $b = 0$, we have an $n \times n$ symmetric [arrowhead matrix](https://en.wikipedia.org/wiki/Arrowhead_matrix). When $b \neq 0$, we have
$$\begin{bmatrix}
c & c & c & \cdots & c & c \\
c & a & b & \cdots & b & b \\
c & b & a & \cdots & b & b \\
c & b & b & \cdots & b & b \\
\vdots & \vdots & \vdots & \ddots & \vd... | 7 | https://mathoverflow.net/users/91764 | 262111 | 118,021 |
https://mathoverflow.net/questions/261857 | 11 | The question title basically has my question. What is the precise definition/construction of$$\text{Sym}(\text{Sym}^2),$$as mentioned in the following?
<http://aimpl.org/repnstability/1/>
If possible, I would like this to be self-contained and not defer to other stuff about twisted commutative algebras, as much as ... | https://mathoverflow.net/users/104715 | Precise definition/construction of $\text{Sym}(\text{Sym}^2)$? | As Dan Petersen points out, it's impossible to answer this question without talking about twisted commutative algebras. Moreover, to talk about TCAs over $\mathbb{Z}$, you need a more general definition of TCAs (which becomes equivalent to the earlier Sam-Snowden definitions when you're over $\mathbb{C}$). I'm not sure... | 4 | https://mathoverflow.net/users/250 | 262113 | 118,022 |
https://mathoverflow.net/questions/261934 | 8 | There are many sources cite that simply typed lambda calculus extended with [fixed-point combinator](https://en.wikipedia.org/wiki/Fixed-point_combinator) is Turing complete. For example, [Does there exist a Turing complete typed lambda calculus?](https://cs.stackexchange.com/questions/2638/does-there-exist-a-turing-co... | https://mathoverflow.net/users/104754 | Is simply typed lambda calculus with fixed-point combinator Turing-complete? | The simply-typed $\lambda$-calculus with the fixpoint combinator but *without* a primitive integer type is *not* Turing-complete, at least not in the usual sense (Church integers and computation as $\beta$-reduction). This is a consequence of a result of Statman [1], stating that (somewhat surprisingly) termination in ... | 6 | https://mathoverflow.net/users/45027 | 262123 | 118,028 |
https://mathoverflow.net/questions/262082 | 0 | I already asked this on "Mathematics", but I guess (at least) Question 2 could be appropriate here as well.
Let's consider a selfadjoint compact operator $C\colon L^2[0,1] \rightarrow L^2[0,1],$ where $(L^2[0,1], ||\cdot||\_{L^2[0,1]})$ stands for the hilbert space of measurable function $f\colon [0,1] \rightarrow \m... | https://mathoverflow.net/users/66236 | Multiplicity of eigenvalues of a compact operator and explicit decay rate of the eigenvalues | I doubt that you'll find a very general condition that requires multiplicity $1$, other than those that are essentially restatements of that fact.
There will be conditions related to particular forms of operators, e.g.
Sturm-Liouville integral operators.
Similarly, for explicit decay rates
you'll want something mo... | 1 | https://mathoverflow.net/users/13650 | 262129 | 118,031 |
https://mathoverflow.net/questions/262134 | 6 | I know that for any Fuchsian group $\Gamma$, there is a spectral gap, which leads to
$$ \left| \int\_0^1 F(x + iy) \, dx - \int\_{\Gamma \backslash \mathbb{H}} F \, \frac{dx \, dy}{y^2} \right| < C\_F y^\delta $$
this is related to the equidistribution of the horocycle flow in the hyperblic plane.
Possibly I ne... | https://mathoverflow.net/users/1358 | rate of equidistribution of the horocycle flow for $SL(2, \mathbb{Z})$ | A standard reference for this kind of thing for congruence subgroups is Iwaniec's "Spectral Methods of Automorphic Forms". Let $\Gamma = \Gamma\_0(q)$. The natural functions $F$ to consider on $\Gamma \backslash \mathbb{H}$ are Hecke-Maass cusp forms $f$ and Eisenstein series $E\_{\mathfrak{a}}(z,1/2+it)$, where $\math... | 9 | https://mathoverflow.net/users/3803 | 262139 | 118,033 |
https://mathoverflow.net/questions/262124 | 1 | The answer is probably well-known, but I cannot find anything definite in the literature.
Suppose we have the usual ingredients of a CLT, i.e. the series
$$X\_N = \sum\_{n=1}^N x\_n $$
where $x\_n$ are i.i.d.'s. The CLT says that $X\_N/ \sqrt{N}$ approaches a normal distribution.
Some of the literature states... | https://mathoverflow.net/users/40588 | Does a Central Limit Theorem imply a series is $O(\sqrt{N})$? | The sharp general result in this direction is the classical [law of the iterated logarithm](https://en.wikipedia.org/wiki/Law_of_the_iterated_logarithm) (LIL). Suppose, after renormalizing if necessary, that the $x\_n$ are iid with zero mean and unit variance. Then the LIL states that
$$\limsup\_{n \to \infty} \frac{X\... | 8 | https://mathoverflow.net/users/4832 | 262140 | 118,034 |
https://mathoverflow.net/questions/261316 | 4 | **UPDATE - Feb. 9, 2017:** The original title of this post was
"The $\text{isometry}^+$ group of hyperbolic $n$-space as $\mathrm{PSL}\_2$ of a Clifford group."
The original question, which appears below,
did not receive answers, but I discovered in the meantime that the confusion arrises from conflicting notation in t... | https://mathoverflow.net/users/14835 | How should we define $\mathrm{PSL}_2$ of a Clifford group? | *I think made this a little too complicated, and I don't think there is a lot of interest in the topic in the first place. So let me just add something more conclusive for the sake of resolving the post.*
To think about how to extend the notation $\mathrm{PSL}$,
we should start by looking at what $\mathrm{GL}$
means.... | 0 | https://mathoverflow.net/users/14835 | 262142 | 118,035 |
https://mathoverflow.net/questions/262141 | 3 | **The question**
I consider the Laplacian $\Delta = \partial\_1^2 + \partial\_2^2 + \partial\_3^2$ in $\mathbb{R}^3$. By the "standard" fundamental solution of the Laplacian, I mean the function
$$ \displaystyle E(x) = C|x|^{-1} $$
where $C$ is some normalization constant. *I would like to know if one can constru... | https://mathoverflow.net/users/103168 | Are there fundamental solutions of the laplacian that decay rapidly? | No. You want the Fourier transform to satisfy $-|\xi|^2 \widehat{E}(\xi)=1$, so $\widehat{E}=-1/|\xi|^2 + \widehat{F}$, with $\textrm{supp}\:\widehat{F}=\{0\}$, but this says that $\widehat{F}$ is a linear combination of $\delta$ and its derivatives, so $F$ is a polynomial.
| 7 | https://mathoverflow.net/users/48839 | 262147 | 118,037 |
https://mathoverflow.net/questions/261972 | 4 | Let $\mathfrak g:=(V,d,[\cdot,\cdot])$ be a differential graded Lie algebra (DGLA) where $d$ is the zero differential.
Intrinsic formality: The DGLA $\mathfrak g$ will be said intrinsically formal if any $L\_\infty$-algebra $l$ admitting $\mathfrak g$ in cohomology (i.e. $H(l)=\mathfrak g$) is formal (i.e. there exi... | https://mathoverflow.net/users/104743 | Intrinsic formality versus rigidity of a differential graded Lie algebra | As I said in the comments, the answer to question 1 is that the two notions are equivalent (at least in characteristic zero, which I will assume throughout).
Assume $\mathfrak{g}=(V,0,[\cdot,\cdot])$ is intrinsically formal. Now let $(V,l')=(V,0,l'\_2=[\cdot,\cdot],l'\_3,\ldots)$ be an $L\_\infty$ algebra as in the ... | 3 | https://mathoverflow.net/users/3075 | 262152 | 118,038 |
https://mathoverflow.net/questions/262153 | 17 | Let $M$ be the $n\times n$ matrix, known as the *GCD matrix*, of entries $M\_{ij}=\gcd(i,j)$. In the paper
H J S Smith, *On the value of a certain arithmetical determinant*, Proc. London Math. Soc. **7**:208-212 (1875-76)
it is shown that $\det M=\prod\_{k=1}^n\varphi(k)$; where $\varphi(k)$ is the [Euler totient f... | https://mathoverflow.net/users/66131 | The GCD-matrix: generalizing a result of Smith? | For Pell numbers, the answer appears to be
$$\det\left[\gcd(P\_i,P\_j)\right]\_{i,j=1}^n = \prod\_{k=1}^n \sum\_{d|k} \mu(\frac{k}{d})\cdot P\_d,$$
i.e. the product of first $n$ terms of the Moebius transform of Pell numbers. At least, this equality holds for all $n\leq 100$.
---
**UPDATE**
A more general state... | 16 | https://mathoverflow.net/users/7076 | 262158 | 118,041 |
https://mathoverflow.net/questions/262146 | 5 | **Problem 1.** For which $n$ does the cyclic group $C\_n$ admit a *difference set* $D\subset C\_n$, i.e., a set such that each non-unit element $x\in C\_n$ can be uniquely written as the difference $x=ab^{-1}$ for some $a,b\in D$?
A necessary condition is that $n=1+d+d^2$ for some $d$.
If $p$ is prime, then for $... | https://mathoverflow.net/users/61536 | Which cyclic groups admit a difference set? | What you called difference sets in cyclic groups are usually called PLANAR cyclic difference sets, namely those with \lambda equal to 1. The question you asked here has been studied for many years. Singer's construction from 1938 shows that for any prime power n, there is a planar cyclic difference set of order n. In t... | 9 | https://mathoverflow.net/users/104811 | 262159 | 118,042 |
https://mathoverflow.net/questions/262131 | 5 | Let $K$ be a (locally compact) local field and $G$ be a linear algebraic $K$-group.
Does the topological group $G(K)$ have a cocompact solvable closed subgroup?
If $\mathrm{char}(K)=0$, it is true that $G(K)$ has a cocompact solvable closed subgroup, see Proposition 9.3 in "A. Borel and J. Tits: Groupes reductifs" ... | https://mathoverflow.net/users/9401 | Does the group G(K) have a cocompact solvable closed subgroup? | Yes. By replacing $G$ with its maximal smooth closed subgroup (whose formation commutes with any separable extension on $K$; see Lemma C.4.1 and Remark C.4.2 in the book *Pseudo-reductive Groups*), we may and do assume $G$ is $K$-smooth. Alternatively, if you prefer, replace $G$ with the Zariski-closure of $G(K)$ to ac... | 8 | https://mathoverflow.net/users/81332 | 262163 | 118,044 |
https://mathoverflow.net/questions/262167 | 22 | Let $K$ be a finite extension of $\mathbf{Q}\_p$. Let $F\_d$ be the unramified extension of $\mathbf{Q}\_p$ of degree $d$. I would like to know whether there exists some $d \geq 1$ and some $L \subset K \cdot F\_d$ such that $L/\mathbf{Q}\_p$ is totally ramified and $K \cdot F\_d / L$ is unramified.
If $K/\mathbf{Q}... | https://mathoverflow.net/users/5743 | Totally ramified subextension in a finite extension of $\mathbf{Q}_p$ | This is not a complete answer, but perhaps it's a roadmap to a counterexample.
My strategy is to consider some non-Galois $K/\mathbf{Q}\_p$ for which the result is *true*, and let's make some deductions about what $K$ must look like in this situation. Then let's find a $K$ that doesn't look like this.
Let $M/\mathb... | 14 | https://mathoverflow.net/users/1384 | 262178 | 118,047 |
https://mathoverflow.net/questions/262182 | 4 | I am looking for exact references for the comparison theorem for the étale fundamental group.
I mean the following result:
>
> **Theorem** (Grothendieck). For a pointed algebraic variety $(X,x)$ over $\mathbb{C}$ there is a canonical isomorphism
> between the étale fundamental group $\pi\_1^{\text{ét}}(X,x)$ and t... | https://mathoverflow.net/users/4149 | Reference request: the comparison theorem for the étale fundamental group | This is in SGA1, Exposé XII, Section 5.
| 7 | https://mathoverflow.net/users/1310 | 262183 | 118,050 |
https://mathoverflow.net/questions/262197 | 3 | Margulis' normal subgroup theorem states that any normal subgroup of a higher rank irreducible lattice is either finite or of finite index.
What are the known counter-examples in rank $1$ ?
I am especially interested by $PU(2,1)$.
| https://mathoverflow.net/users/76193 | Counterexamples to Margulis Normal subgroup theorem in rank 1 | Any cocompact lattice in a rank-one Lie group is *word-hyperbolic*. Olshanskii proved that such groups are SQ-universal, meaning in particular that they have uncountably many normal subgroups. Similar results are known for non-uniform lattices, which are *relatively hyperbolic*.
As Yemon Choi suggests in comments, th... | 5 | https://mathoverflow.net/users/1463 | 262199 | 118,054 |
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