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https://mathoverflow.net/questions/372283 | 3 | Let $G$ be a 1-dimensional, commutative formal group over a ring $R$. Give $G$ a coordinate $x$ and let $A\subset R$ be the subring generated by the coefficients of the corresponding formal group law $F(x,y)= \sum\_{ij}a\_{ij}x^iy^j$. So $G$ is really defined over $A$.
Call a finite subgroup $K\subset G$ *special* if... | https://mathoverflow.net/users/163893 | Some special subgroups of formal groups | Let me specialize heavily to the case of formal groups (group laws) of dimension one over a $p$-adic ring $\mathfrak o$, i.e. the ring of integers of a finite extension $k$ of $\Bbb Q\_p$.
I still am uncertain about what category you’re thinking of. If we restrict further to formal groups of finite height (the endomo... | 2 | https://mathoverflow.net/users/11417 | 372374 | 155,604 |
https://mathoverflow.net/questions/372379 | 3 | The following question arose while thinking about a step in the proof of Huybrechts-Lehn, Theorem 1.3.1 (the Harder-Narasimhan filtration for the projective line $\mathbb{P}^{1}$):
**Setup**: Let $k$ be a field, let $X$ be a projective $k$-scheme, let $\mathcal{O}\_{X}(1)$ be a fixed very ample line bundle on $X$, le... | https://mathoverflow.net/users/112809 | The evaluation map on twists of a vector bundle and an induced filtration | There is a commutative diagram
$$\require{AMScd}
\begin{CD}
H^0(E(b)) \otimes H^0(\mathcal{O}(1)) \otimes \mathcal{O}(-b-1)
@>{\mathrm{ev}\_{\mathcal{O}(1)}}>>
H^0(E(b)) \otimes \mathcal{O}(-b)
\\
@VVV @V{\mathrm{ev}\_{E(b)}}VV
\\
H^0(E(b+1)) \otimes \mathcal{O}(-b-1)
@>{\mathrm{ev}\_{E(b+1)}}>>
E
\end{CD}
$$
wher... | 6 | https://mathoverflow.net/users/4428 | 372386 | 155,608 |
https://mathoverflow.net/questions/372380 | 2 | I am trying to solve a system of $9$ polynomial equations in $9$ unknowns over the non-negative reals.
Since the equations are quite large and I would like to use VBA, I prefer an algorithm that avoids partial derivatives. Hence, I tried to use the [Nelder-Mead](https://en.wikipedia.org/wiki/Nelder%E2%80%93Mead_metho... | https://mathoverflow.net/users/165801 | Using Nelder-Mead to solve system of polynomial equations | Minimum-finding routines (which is what Nelder-Mead/downhill-simplex is) are generally poorly suited to finding zeros of equation systems — if you add the squares of all equations, you get many spurious local minima in addition to the global minimum corresponding to the zero (assuming your equations have a unique zero)... | 2 | https://mathoverflow.net/users/45250 | 372399 | 155,611 |
https://mathoverflow.net/questions/372341 | 2 | Let $\mathcal{D}$ be a probability distribution with support $[0,1]$. Let $X, Y, Z$ three i.i.d. random variables with distribution $\mathcal{D}$, and $T$ a random variable uniformly distributed in $[0,1]$ independent from $X$, $Y$ and $Z$. We define $$\Delta=\mathbb{E}\left(1-|x-y|~\big|~x,y<t<z\right)$$ and $$\Delta'... | https://mathoverflow.net/users/115803 | Probability distribution optimization problem of distances between points in $[0,1]$ | Sorry, my computation in the comments was wrong. I think it leads to something with $\rho < \frac{16}{17}$.
Namely, let $\mathcal{D}$ be the distribution with $\mathrm{Pr}(\mathcal{D}=0)=\mathrm{Pr}(\mathcal{D}=3/4)=1/N$, and $\mathrm{Pr}(\mathcal{D}=1)=(N-2)/N$, where $N$ is large.
Then the possibilities for $(x,y... | 2 | https://mathoverflow.net/users/25028 | 372416 | 155,618 |
https://mathoverflow.net/questions/372414 | 5 | Consider the equation
$$f'(x)+ g(x)f(x)=0$$
This equation is an ODE and has a solution $$ f(x)=C e^{ \int\_1^x g(x) \ dx}.$$
Similarly, we can look at complex variables and consider the equation and Wirtinger derivatives
$$ (\partial\_{\bar z} f)(z) +g(z) f(z)=0.$$
Can one still write down an explicit solutio... | https://mathoverflow.net/users/119875 | First order PDE in complex variables? | You can start by looking at the [chain rule for wirtinger derivatives](https://en.wikipedia.org/wiki/Wirtinger_derivatives), from which you deduce that
$$
\partial\_{\bar z} \exp(h(z)) = \exp(h(z)) \cdot \partial\_{\bar z} h(z)
$$
Therefore, if you find a function $h$ such that $\partial\_{\bar z} h = - g(z)$ (I th... | 6 | https://mathoverflow.net/users/165826 | 372417 | 155,619 |
https://mathoverflow.net/questions/372354 | 9 | This question concerns the strictness of (co)completions, at various levels of generality.
In Blackwell–Kelly–Power's [Two-dimensional monad theory](https://www.sciencedirect.com/science/article/pii/0022404989901606), the authors state
>
> For instance, the 2-category $\mathbf{Lex}$ of small finitely-complete cat... | https://mathoverflow.net/users/152679 | 2-monads for categories with a class of (co)limits | Kelly and Lack's paper [On the monadicity of categories with chosen colimits](http://www.tac.mta.ca/tac/volumes/7/n7/n7.pdf) answers your questions (1),(2) and (3) affirmatively. The main theorems are Theorem 6.1, 6.2 and 7.1. Their main trick is Lemma 4.1, which allows them to modify a biadjunction (and so a pseudomon... | 8 | https://mathoverflow.net/users/8751 | 372428 | 155,621 |
https://mathoverflow.net/questions/372265 | 0 | Let $D$ be a domain of $\mathbb{R}^{m}$ and let
$K(x)= \log|x|$ if $m=2$, and $K(x)=|x|^{2-m}$ if $m>2$. According to Riesz decomposition theorem (Hayman and Kennedy, "subharmonic functions", vol. 1, pg 104) if $u$ is subharmonic on $D$, then there is a unique Borel measure $\mu$ such that for all compact $E$ in $D$ we... | https://mathoverflow.net/users/100746 | A simple clarification on Riesz decomposition theorem | (You need a minus sign in front of $\int\_E K(x-\zeta)\,d\mu(\zeta)$.)
The integral $\,-\!\!\int\_E K(x-\zeta)\,d\mu(\zeta)$ is subharmonic throughout $D$ and harmonic in $D\setminus E$. If $u$ admits a harmonic majorant in $D$, hence a least such majorant (call it $k$), then $h$ (which depends on $E$) can be express... | 0 | https://mathoverflow.net/users/42851 | 372435 | 155,623 |
https://mathoverflow.net/questions/372431 | 7 | By <https://arxiv.org/abs/1406.4419> (The fundamental groupoid as a terminal costack, Ilia Pirashvili), we know that for a topological space $X$, the $2$-functor
$$\text{Top}(X)\rightarrow \text{Gpd}, \quad (U\rightarrow X)\mapsto \Pi\_1(U)$$
is a $2$-cosheaf, in fact the terminal one. In particular, it follows that
$$... | https://mathoverflow.net/users/152554 | Representation of fundamental groupoid as $2$-sheaf | First of all, note that you haven't used the fact that $\Pi\_1(-)$ was the terminal $2$-cosheaf, just that it was a ($2$-)cosheaf.
Then, as I pointed out in the comments, there's a question of whether you're considering the usual $\hom$ functor into $Set$, or the internal $\hom$ with values in $Gpd$.
* If you're co... | 7 | https://mathoverflow.net/users/102343 | 372446 | 155,625 |
https://mathoverflow.net/questions/372395 | 3 | It is well-known that if a graph has maximum degree $d$, then it is $d+1$ colorable. Say we have $d+1$ graphs $G\_1,\ldots, G\_{d+1}$ **on the same vertex set** $V$, and say each $G\_i$ has maximum degree at most $d$.
A *coloring* of $\textbf{G}:=\{G\_1,\ldots, G\_{d+1}\}$ is just a labelling of the common vertex set... | https://mathoverflow.net/users/160715 | Chromatic number of a family of graphs | The concept you introduce is called a cooperative coloring. Check out, e.g., [this paper](https://arxiv.org/pdf/1806.06267.pdf). Theorem 1 (with a reference to another paper) claims a negative answer to your question; but there is other information you may find relevant.
| 4 | https://mathoverflow.net/users/17581 | 372449 | 155,627 |
https://mathoverflow.net/questions/372351 | 1 | $\DeclareMathOperator\gcd{gcd}$Take $q\in \mathbb N$ and $X>0$ ($q$ not necessarily smaller than $X$). A sum such as
$$\sum\_{d\leq X}(q,d)$$
is easily seen to be $\ll q^\epsilon (X+q)$ so that the gcd doesn't make the sum much larger than how it would be without it — the values for which $(q,d)$ are significant are ra... | https://mathoverflow.net/users/110603 | Gcd of linear function | **1.** We have
$$\sum\_{\substack{dd'\leq X\\q\mid d+d'\\d\leq d'}}1
=\sum\_{d\leq\sqrt{X}}\sum\_{\substack{d'\leq X/d\\q\mid d+d'\\d\leq d'}}1
\leq\sum\_{d\leq\sqrt{X}}\sum\_{\substack{c\leq 2X/d\\q\mid c}}1
\leq\sum\_{d\leq\sqrt{X}}\frac{2X}{qd}<\frac{2X(1+\log\sqrt{X})}{q}.$$
We get the same bound when the roles of ... | 1 | https://mathoverflow.net/users/11919 | 372452 | 155,629 |
https://mathoverflow.net/questions/372450 | 24 | Let $\operatorname{ord}\_p(2)$ be the order of 2 in the multiplicative group modulo $p$. Let $A$ be the subset of primes $p$ where $\operatorname{ord}\_p(2)$ is odd, and let $B$ be the subset of primes $p$ where $\operatorname{ord}\_p(2)$ is even. Then how large is $A$ compared to $B$?
| https://mathoverflow.net/users/165074 | Parity of the multiplicative order of 2 modulo p | This problem was asked by Sierpiński in 1958 and answered by Hasse in the 1960s.
For each nonzero rational number $a$ (take $a \in \mathbf Z$ if you wish) and each prime $\ell$, let $S\_{a,\ell}$ be the set of primes $p$ not dividing the numerator or denominator of $a$ such that $a \bmod p$ has multiplicative order d... | 39 | https://mathoverflow.net/users/3272 | 372457 | 155,633 |
https://mathoverflow.net/questions/372106 | 31 | Consider the restriction of the group cohomology $H^\*(BG,\mathbb{Z})$, where $G$ is a compact Lie group and $BG$ is its classifying space, to finite subgroups $F \le G$. If we consider the product of all such restrictions
$$H^\*(BG,\mathbb{Z}) \to \prod\_F H^\*(BF,\mathbb{Z}),$$
is this map injective?
Note, accordin... | https://mathoverflow.net/users/165135 | Is Lie group cohomology determined by restriction to finite subgroups? | After the heavy lifting done by people on MSE and in the comments, I think it's not too bad to finish off the proof that the answer is *yes*.
As argued by Ben Wieland in the [comments](https://mathoverflow.net/questions/372106/is-lie-group-cohomology-determined-by-restriction-to-finite-subgroups#comment941496_372106)... | 21 | https://mathoverflow.net/users/2362 | 372461 | 155,634 |
https://mathoverflow.net/questions/372423 | 3 | If $H=(V,E)$ is a hypergraph and $\kappa$ is a cardinal,we say a map $c:V\to\kappa$ is a *coloring* if the restriction $c\restriction\_e$ of $c$ to $e$ is non-constant whenever $e\in E$ and $|e|>1$. The smallest cardinal such that there is a coloring from $V$ to that cardinal is denoted by $\chi(H)$.
By $[\omega]^\om... | https://mathoverflow.net/users/8628 | Can every number be realised as the chromatic number of a countable hypergraph? | Yes. Partition $\omega$ into $n$ infinite sets $V\_1,\dots,V\_n$. Let $H=(\omega,E)$ where $E=\{e\in[\omega]^\omega:|\{i:e\cap V\_i\ne\emptyset\}|\ge2\}$. Plainly $\chi(H)\le n$. Suppose the vertices of $H$ are colored with $m$ colors, $m\lt n$. For each $i$ choose an infinite monochromatic set $W\_i\subseteq V\_i$. By... | 2 | https://mathoverflow.net/users/43266 | 372463 | 155,635 |
https://mathoverflow.net/questions/372434 | 4 | Suppose $\Omega=[-1,1]^3$. Let $f:[-1,1]\to \mathbb R$ and $g:[-1,1]^2\to \mathbb R$ be smooth functions and suppose that given any harmonic function on $\Omega$ (i.e. $\Delta u =0$ on $\Omega$), with $u \in L^2(\Omega)$, there holds:
$$ \int\_{\Omega} u(x^1,x^2,x^2) f(x^1)g(x^2,x^3)\,dx=0.$$
Does it follow that $f$ ... | https://mathoverflow.net/users/50438 | Is a specific product function orthogonal to all harmonic functions | Looks like it is so (though the conclusion is rather that *either $f$, or $g$* is identically $0$ (one of the two is enough).
Let $v$ be the solution of the problem $\Delta v=fg$ in $\Omega$, $v|\_{\partial\Omega}=0$. Then, by Green's formula, the integral in question is (up to minus) $\int\_{\partial\Omega}u\frac{\p... | 6 | https://mathoverflow.net/users/1131 | 372467 | 155,637 |
https://mathoverflow.net/questions/372436 | 1 | **Question.** Is there a continuous curve in the plane that has a non-unique loop-erasure?
Here is the definition of a loop-erasure. A continuous curve $Y:[c,d]\to\mathbb R^2$ is a loop-erasure of a curve $X:[a,b]\to\mathbb R^2$ if there exists an increasing and right-continuous function $w:[c,d]\to [a,b]$ such that:... | https://mathoverflow.net/users/52796 | Non-uniqueness of loop-erasure for continuous-time curves | The paper that Iosif Pinelis mentioned in his answer has an example to this problem: Consider the compact space obtained by adding $\pm\infty$ to the strip $\{ z\in \mathbb C: 0\leq \mathrm{Im}(z)\leq 1\}$. Consider the curve that connects the integer points of this set (by segments) in the following order: $\ldots, n,... | 1 | https://mathoverflow.net/users/52796 | 372470 | 155,638 |
https://mathoverflow.net/questions/372468 | 12 | I am considering the following two cases:
1. Assume that there is an embedding: $D^b(\mathcal{A})\xrightarrow{\Phi} D^b(\mathbb{P}^2)$and the homological dimension of $\mathcal{A}$ is equal to $1$($\mathcal{A}$ is an abelian category), for simplicity, maybe first I assume that $\mathcal{A}$ is a module category over ... | https://mathoverflow.net/users/41650 | Embedding of a derived category into another derived category | Any fully faithful functor from $D^b(\mathcal{A})$ has adjoints (because $D^b(\mathcal{A})$ is a smooth and proper category), so its image is an admissible subcategory. A recent [result from Dmitrii Pirozhkov](https://arxiv.org/abs/2006.07643) shows that any admissible subcategory in $D^b(\mathbb{P}^2)$ is generated by... | 13 | https://mathoverflow.net/users/4428 | 372472 | 155,639 |
https://mathoverflow.net/questions/372377 | 4 | Let G be a simple algebraic group. Let H be a reductive subgroup of G which contains a regular unipotent element of G. Such subgroups were classified by [Saxl and Seitz](https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/S0024610797004808) in all good characteristics. I'm actually interested in the characteris... | https://mathoverflow.net/users/41301 | Subgroups of algebraic groups containing regular unipotent elements | I do not know a reference, but I have thought about the same question. Here is a sketch using arguments that are in the literature. For some basics about regular unipotent elements, see for example Chapter 4 of [1]. First a reminder:
>
> $(\*)$ Let $\Phi^+$ be a system of positive roots on $\Phi$. A unipotent eleme... | 3 | https://mathoverflow.net/users/10146 | 372473 | 155,640 |
https://mathoverflow.net/questions/372508 | 3 | As far as I understand, there are several ways of defining $\infty$-categories. One of them is to think of $\infty$-cateogries as $top$-enriched categories. Hence we can think of $\infty$-groupoids as generalizing topological groups. Functors between groupoids are the generalization of group homomorphisms. Hence my que... | https://mathoverflow.net/users/152554 | Morphisms of $\infty$-groupoids | Only for special kinds of topological groups (like Lie groups or groups with an underlying topological manifold). This fails horribly for e.g. profinite groups, which are detected as discrete groups by the topological nerve.
| 3 | https://mathoverflow.net/users/1353 | 372509 | 155,651 |
https://mathoverflow.net/questions/372502 | 6 | I have a collection of related (to me) questions, which stem from the fact that I feel like I have a bunch of pieces, but not a full clear picture. I'm curious about forms of reductive groups in general, so I'm only asking about $\mathbb{C}/\mathbb{R}$ for simplicity's sake and for explicit examples.
As a first fact,... | https://mathoverflow.net/users/119460 | Real forms of complex reductive groups | I answer Question 1. It is just a calculation.
Instead of a real torus, say ${\bf T}$, I consider a pair $(T,\sigma)$,
where $T$ is a complex torus and $\sigma\colon T\to T$ is an anti-holomorphic involution.
See [this question](https://mathoverflow.net/q/342300/4149) and YCor's answer.
For a complex torus $T$, con... | 7 | https://mathoverflow.net/users/4149 | 372519 | 155,654 |
https://mathoverflow.net/questions/372501 | 4 | Is the following known? It seems related to codes and/or Ramsey theory.
Given $r$, for what values of $n,k$, does there exist a collection of $n$ sets whose union contains $k$ elements such that none of these sets is contained in the union of at most $r$ of the sets.
For example given $r<3$ choosing $n=3$, $k=4$ sa... | https://mathoverflow.net/users/39187 | Collections of sets without $r$-unions covering another set | Erdos Frankl and Furedi, "Families of finite sets in which no set is covered by the union of r others} *Israel Journal of Mathematics* 51 (1–2): 79–89, 1985.
>
> Let $f\_r(n,k)$ be the maximum number of $k-$subsets of an $n$-set satisfying the condition above. Then
> $$
> f\_r(n,r(t-1)+1+d)\leq \frac{\binom{n-d}t}{... | 6 | https://mathoverflow.net/users/17773 | 372527 | 155,656 |
https://mathoverflow.net/questions/372430 | 6 | Can one build a hierarchy of stratified constructible stages? That is a hierarchy that is built in a manner similar to Godel's [constructible universe](https://en.wikipedia.org/wiki/Constructible_universe) L, but with additionally requiring that the defining formulas must be also [stratified](https://en.wikipedia.org/w... | https://mathoverflow.net/users/95347 | Can we have stratified L? | Glad to see someone taking an interest in this stuff!
All but one of the Goedel operations are stratified, and the one that isn't is the existence of $\in$ "locally." Replace this operation by one that gives you the local version of $\in$ composed with singleton, so that you get
$A \cap \{\langle \{x\},y \rangle: x \in... | 7 | https://mathoverflow.net/users/165901 | 372528 | 155,657 |
https://mathoverflow.net/questions/372507 | 2 | This is an adaptation of a Heinrich proof, but I'm missing a key ingredient.
**Conjecture.** Suppose $(x\_n)\_{n=1}^\infty$ is a Schauder basis for a Banach space $X$ whose canonical isometric copy in $X^{\*\*}$ is complemented. Then for any free ultrafilter $\mathcal{U}$ on $\mathbb{N}$, the canonical copy of $X$ in... | https://mathoverflow.net/users/73784 | gap in a Banach spaces ultrapower proof | Theorem. Suppose $T: X \to Y^\*$ is a bounded linear operator and $\mathcal{U}$ is a free ultrafilter on $\Bbb{N}$. Then $T$ extends to an operator $S:X^\mathcal{U} \to Y^\*$ with $\|S\| = \|T\|$.
Proof: Define $V:\ell\_\infty(X)$ to $Y^\*$ by letting $V(x\_n)\_n$ be the weak$^\*$ limit along $\mathcal{U}$ of $Tx\_n$... | 6 | https://mathoverflow.net/users/2554 | 372529 | 155,658 |
https://mathoverflow.net/questions/372429 | 3 | Let $X\_1, X\_2, \dots, X\_n$ be a martingale difference sequence such that
$$
X\_i \leq y \quad \text{and} \quad \sum\_{i=1}^{n} \operatorname{Var}(X\_i) \leq B^2.
$$
Question 1: Does the following hold?
$$
\mathbb{P}\left[ \sum\_{i=1}^{n}X\_i \geq x \right] \leq \exp{\left(\frac{-x^2}{2B^2 + \frac{2}{3}xy}\right)}.
$... | https://mathoverflow.net/users/165165 | Exponential inequality for the sum of martingale differences $X_1, \dots, X_n$ when $\sum_{i=1}^{n} \operatorname{Var}(X_i) \leq B^2$ | **Theorem 1:** In the known exponential bounds for martingales, the conditional variances cannot be replaced by the unconditional ones.
*Proof:* Otherwise, we would most likely have such bounds. $\Box$ :-)
This "proof" of "Theorem 1" is not so non-serious as it may look.
---
Perhaps more seriously, we have
... | 2 | https://mathoverflow.net/users/36721 | 372534 | 155,660 |
https://mathoverflow.net/questions/372514 | 17 | Let $\zeta(s)$ denote the Riemann zeta function. Is the set $\{
\zeta(s-j)\, \colon\, j\in\mathbb{Z}\}$, or even $\{\zeta(s-z)\,
\colon\, z\in\mathbb{C}\}$, algebraically independent over
$\mathbb{C}$? If not, then expanding a polynomial equation satisfied
by these functions into a Dirichlet series and taking the coeff... | https://mathoverflow.net/users/2807 | Algebraic independence of shifts of the Riemann zeta function | $\zeta(s - z)$ has an Euler product $\prod\_p \frac{1}{1 - p^{z-s}}$, and so a monomial $\prod\_i \zeta(s - z\_i)$ (with the $z\_i$ not necessarily distinct) has an Euler product
$$\prod\_i \zeta(s - z\_i) = \prod\_p \prod\_i \frac{1}{1 - p^{z\_i - s}}.$$
We want to show that these monomials are linearly independen... | 20 | https://mathoverflow.net/users/290 | 372539 | 155,663 |
https://mathoverflow.net/questions/372505 | 3 | I have accrossed a new topological space seems were derived from Hilbert Space and it used to solve some boundary value problem for PDE and ODE , Inspired by [this paper](https://ejde.math.txstate.edu/Volumes/2001/21/bouziani.pdf) (page 4, Definition 3.1) , The definition 3.1 talks about Weighted Bouziani space this me... | https://mathoverflow.net/users/51189 | What is Bouziani space and what are its applications in mathematics? | It looks to me that the ordinary Bouziani space is the space $B^1\_2(\Omega)$ discussed in the references [4] and [5] from the paper you cite.
It also seems like Bouziani, who wrote the paper you cite, is in fact the only author who uses the term ``Bouziani space''.
| 3 | https://mathoverflow.net/users/23141 | 372540 | 155,664 |
https://mathoverflow.net/questions/372548 | 3 | $f:\mathbb R\to\mathbb R$ is a convex continuous function. We have a finite or a countable set of triples: $\{(x\_n,f(x\_n),D\_n)\}\_{n\in N}$, where $D\_n$ is the slope of a tangent line $L\_n$ at $x\_n$ (if at a point $f$ is not differentiable, then multiple lines can be tangents; $L\_n$ is just one of those lines).
... | https://mathoverflow.net/users/122649 | "Mollification" of a convex function with a finite set of points unchanged | Making quantitative the assumption "the intersection of $L\_n$ and $L\_m$ cannot be the point $(x\_k, f(x\_k))$", it is possible to construct such a function $g$ (as pointed out by Jaume, the nonquantitative assumption is not sufficient).
Let us consider the problem in $\mathbb R^n$.
Given a family of indices $I$, le... | 4 | https://mathoverflow.net/users/48019 | 372553 | 155,667 |
https://mathoverflow.net/questions/372480 | 33 | This question is essentially a reposting of [this](https://math.stackexchange.com/q/3828287/10513) question from Math.SE, which has a partial answer. YCor suggested I repost it here.
---
Our starting point is a theorem of Matumoto: every group $Q$ is the outer automorphism group of some group $G\_Q$ [1]. It seems... | https://mathoverflow.net/users/35478 | Is every finite group the outer automorphism group of a finite group? | Yes.
For each finite group $Q$ I'll construct a finite group $H$ with $\mathrm{Out}(H)\simeq Q$, moreover $H$ will be constructed as a semidirect product $D\ltimes P$, with $P$ a $p$-group of exponent $p$ and nilpotency class $<p$, (with prime $p$ arbitrary chosen $>|Q|+1$) and $D$ abelian of order coprime to $p$ (ac... | 27 | https://mathoverflow.net/users/14094 | 372563 | 155,669 |
https://mathoverflow.net/questions/372496 | 4 | Let $E,F$ be Banach spaces and let $A\subset K(E,F)$ be a subset of the space of compact operators from $E$ to $F$.
A result by Kalton states that $A$ is weakly compact if and only if $A$ is WOT\* compact (here WOT\* denotes the dual weak operator topology, i.e. the topology defined by the functionals $K(E,F)\ni T\maps... | https://mathoverflow.net/users/165855 | Weak sequential compactness on the space of compact operators | The key here is the isometric embedding of $K(E,F)$ into the space of continuous functions on the compact space $M=B\_{E^{\*\*}}\times B\_{F^\*}$.
Suppose that $A$ is WOT$^\*$ sequentially compact; $A$ is bounded by the uniform boundedness principle. Then each sequence $(T\_n)$ in $A$ has a WOT$^\*$ convergent subseq... | 3 | https://mathoverflow.net/users/127871 | 372571 | 155,671 |
https://mathoverflow.net/questions/372440 | 1 | Suppose the set $S \subset \mathbb{R}^{n}$ is a smooth submanifold of dimension $k$, that is [Lee, Proposition 5.16] for every $x \in S$ there exist an open set $W \subset \mathbb{R}^{n}$ and a smooth submersion $\phi : W\to \mathbb{R}^{n - k}$ such that $W \cap S$ is a level set of $\phi$.
The submersion $\phi$ is t... | https://mathoverflow.net/users/153602 | Collection of local defining maps for smooth Euclidean submanifolds | This is a simple consequence of the finiteness of Lebesgue dimension of any manifold, i.e. for any open cover $U\_a$ of a manifold, there is a refinement $V\_{ij}$, so that $j$ runs through a finite set, and $V\_{ij}\cap V\_{ik}$ is empty if $j\ne k$; see Greub, Halperin, Vanstone, **Connections, Curvature and Cohomolo... | 1 | https://mathoverflow.net/users/13268 | 372575 | 155,673 |
https://mathoverflow.net/questions/372558 | 3 | Let $(V,Y)$ be a vertex operator algebra, and $V'$ be the graded dual of its underlying vector space. The contragredient module structure on $V'$ is given by $Y'$ defined by the formula:
$$\langle Y'(v,x)w', w\rangle = \langle w', Y(e^{xL(1)}(-x^{-2})^{L(0)}v,x^{-1})w\rangle.$$
Now, I think of the LHS as putting $v... | https://mathoverflow.net/users/105094 | Intuition behind contragredient module of a VOA | For the contragredient module vertex operator $\langle Y'(v,x)w',w\rangle$, I believe the local coordinate at the puncture $x^{-1}$ on $\mathbb{C}\mathbb{P}^1$ should actually be $z\rightarrow z^{-1}-x$. We can rewrite this coordinate as follows: $$ z\rightarrow z^{-1}-x = -x^2\cdot\frac{z-x^{-1}}{1+x(z-x^{-1})} = e^{... | 3 | https://mathoverflow.net/users/118337 | 372586 | 155,676 |
https://mathoverflow.net/questions/372565 | 6 | $\newcommand{\Rep}{\operatorname{Rep}}$
$\newcommand{\mo}{\operatorname{-mod}}$
$\renewcommand{\hat}{\widehat}$
I apologize in advance if this is a naive question but my background in algebraic geometry is fairly superficial. I mostly care about global quotients $X/G$ where $X$ is an affine scheme over $\mathbb C$ an... | https://mathoverflow.net/users/13552 | Formal completion of a quotient stack | I will assume $X$ is smooth for simplicity, but it is probably not needed. Given the stack $X/G$, there are two completions one may consider:
1. Completing along $\mathrm{B}G\rightarrow X/G$ one obtains $\hat{X}/G$.
2. Completing along $\mathrm{pt}\rightarrow X/G$ one obtains $\hat{X}/\hat{G}$.
Your next question i... | 5 | https://mathoverflow.net/users/18512 | 372587 | 155,677 |
https://mathoverflow.net/questions/372538 | 16 | Wikipedia calls [resolvent formalism](https://en.wikipedia.org/wiki/Resolvent_formalism) a useful tool for relating complex analysis to studying the spectra of a linear operator on a Banach space. Sure, I believe you because I've seen results that use the resolvent in the proof. I've also read bits of Kato's *Perturbat... | https://mathoverflow.net/users/165906 | Why should I look at the resolvent formalism and think it is a useful tool for spectral theory? | *Preliminary remark.* As mentioned in the comments, I find the notion "resolvent formalism", as well as the description in the Wikipedia article, rather misleading - resolvents are not somekind of formalism, and they are certainly not a mere "technique for applying complex analysis to spectral theory" (as claimed in th... | 27 | https://mathoverflow.net/users/102946 | 372591 | 155,679 |
https://mathoverflow.net/questions/372601 | 7 | We will suppose, for the sake of simplicity, that everything is happening within a fixed 'metacategory' $\textbf{SET}$ of sets and functions. So, from now on, a 'category' just means a category object in $\textbf{SET}$ - i.e. a small category.
Let $\mathscr{V}$ be a monoidal category. A $\mathscr{V}$-enriched categor... | https://mathoverflow.net/users/66071 | Enrichment as extra structure on a category | When your enriched categories are bicomplete enough (specifically, tensored and
cotensored over $\mathscr{V}$), you can view the extra structure of
the enrichment as a kind of action of $\mathscr{V}$ on them: this is
called a closed $\mathscr{V}$-module in Definition 10.1.3 of [Riehl's
*Categorical homotopy
theory*](ht... | 10 | https://mathoverflow.net/users/165619 | 372610 | 155,683 |
https://mathoverflow.net/questions/372617 | 8 | Let $k$ be a field and let $\operatorname{SL}\_2(k)$ act on $k[x\_1,x\_2]$ and $k[y\_1,y\_2]$ in the usual ways. These actions induce an action on the tensor product $k[x\_1,x\_2,y\_1,y\_2]$ that preserves the subspace $k[x\_1,x\_2,y\_1,y\_2]\_{s,k}$ of polynomials that are homogeneous of degree $s+k$ with total $x\_i$... | https://mathoverflow.net/users/165960 | $\operatorname{SL}_2(k)$ invariant polynomials in $k[x_1,x_2,y_1,y_2]$ | The polynomial you gave in the [comments](https://mathoverflow.net/questions/372617/operatornamesl-2k-invariant-polynomials-in-kx-1-x-2-y-1-y-2#comment942654_372617), $x\_1y\_2 - y\_2 x\_1$, after correcting the typo to $x\_1 y\_2 - x\_2 y\_1$, is invariant under $\operatorname{SL}\_2$.
Proof: It's the determinant of... | 8 | https://mathoverflow.net/users/18060 | 372619 | 155,686 |
https://mathoverflow.net/questions/372532 | 5 | I apologise if this question is unclear as I do not know much about the Ricci flow and am only asking out of curiosity. My understanding is that a neckpinch singularity is a local singularity in the sense that it occurs on a compact subset of a manifold. The classic picture is that of a dumbbell manifold, where a local... | https://mathoverflow.net/users/119114 | Neckpinch singularity of Ricci flow | It's not entirely clear what you mean by a local singularity, or what it might mean for a local singularity to be "global." I'll give one attempt to make those ideas precise. However, I won't be able to give an answer because the singularity profiles are not fully understood when the dimension is greater than three.
... | 1 | https://mathoverflow.net/users/125275 | 372625 | 155,689 |
https://mathoverflow.net/questions/372592 | 3 | Let $f:X \to \mathbb{A}^1$ be a smooth, projective morphism of relative dimension $2$. Suppose that the fiber $X\_0:=f^{-1}(0)$ contains an irreducible rational curve, say $C$ such that the restriction of the canonical bundle $K\_{X\_0}$ of $X\_0$ to $C$ is trivial. Suppose that there exists a proper, birational morphi... | https://mathoverflow.net/users/32151 | Normal bundle and small contraction in threefolds | The exact sequence
$$
0 \to N\_{C/X\_0} \to N\_{C/X} \to N\_{X\_0/X}\vert\_C \to 0
$$
in this case reads as
$$
0 \to \mathcal{O}\_C(-2) \to N\_{C/X} \to \mathcal{O}\_C \to 0
$$
which means that either $(a,b) = (0,-2)$ (if the extension class is trivial) or $(a,b) = (-1,-1)$ (if the extension class is nontrivial).
| 3 | https://mathoverflow.net/users/4428 | 372630 | 155,690 |
https://mathoverflow.net/questions/372594 | -1 | I am self studying basic topology and have trouble proving the following question.
>
> If $A$ and $B$ are compact, and if $W$ is a neighborhood of $A \times B$ in $X \times Y$, find a neighborhood $U$ of $A$ in $X$ and a neighborhood $V$ of $B$ in $Y$ such that $U \times V \subseteq W$.
>
>
>
Intuitively, in E... | https://mathoverflow.net/users/164542 | Proving neighborhood of a compact product space contains a sub-neighborhood formed by taking product | Algernon's argument seems to need a special case of Tychonoff's theorem to get to the finiteness of the union. Here is an argument which avoids that.
Lemma: Let $A\subseteq X$ be compact and $B\subseteq Y$ arbitrary. Let $W\subseteq X\times Y$ be open such that for each $x\in A$ there are a neighborhoods $U\_x$ of $x... | 1 | https://mathoverflow.net/users/165275 | 372633 | 155,691 |
https://mathoverflow.net/questions/371921 | 5 | What is an example of a pair of Hopf algebras $(A,B)$ with a surjective Hopf algebra map $\phi:A \to B$ such that $\phi$ does not admit a $B$-bi-comodule splitting $s:B \to A$? To be clear, the right $B$-comodule structure on $A$ is given by
$$
(\textrm{id} \otimes \phi) \circ \Delta\_A: A \to A \otimes B,
$$
where $\D... | https://mathoverflow.net/users/153228 | Covariant splittings of Hopf algebra projections | I'll give an example "occuring in nature." It's not the simplest possible, but you can get a simpler one by removing the generators of degrees 3, 5, and 7, which don't feature in the argument.
According to results of Borel from 1954, the mod-2 homology Hopf algebra $$H\_9 = H\_\* (\mathrm{Spin}(9);\mathbb F\_2)$$ is ... | 2 | https://mathoverflow.net/users/5792 | 372635 | 155,693 |
https://mathoverflow.net/questions/372642 | 10 | Let $c(n,k)$ denote the number of Dyck paths of semilength $n$ which are contained in the strip $0 \leq y \leq 2k + 1.$
They satisfy the recursion $\sum\_{j=0}^{k+1}(-1)^j \binom{2k+2-j}{j}c(n-j,k)=0$ for $n>k.$
We can extend the sequence to negative $n$ such that this recursion holds for all $n \in \mathbb{Z}.$
... | https://mathoverflow.net/users/5585 | Number of bounded Dyck paths with "negative length" | If $f(n)$ satisfies a linear recurrence with constant coefficients for all $n\in \mathbb{Z}$ and we set $F(x)=\sum\_{n\geq 0} f(n)x^n$, then $\sum\_{n\geq 1}f(-n)x^n = -F(1/x)$ (as rational functions). See *Enumerative Combinatorics*, vol. 1, second ed., Prop. 4.2.3.
**Addendum.** Using Exercise 3.66(d) in *Enumerati... | 10 | https://mathoverflow.net/users/2807 | 372663 | 155,698 |
https://mathoverflow.net/questions/285965 | 1 | I have asked this in MSE 8 days ago, even offered a bounty, and got nothing, so will try here.
I would like to understand the value of the skew characters of the symmetric group, $\chi\_{\lambda/\mu}$ in the particular case when both $\lambda$ and $\mu$ are hooks, i.e. $\lambda=(a,1^{n-a})$ and $\mu=(b,1^{m-b})$ with... | https://mathoverflow.net/users/83671 | Skew character with hooks | As Darij pointed out, if both $\lambda$ and $\mu$ are hooks,
your diagram will be the disjoint union of a row of size $r$ and a column of size $c$, say.
To compute the character value at $\nu$, you sum over all ways to partition the parts, $\nu = \rho \cup \eta$, such that $\rho \vdash r$, $\eta \vdash c$ and evaluat... | 1 | https://mathoverflow.net/users/1056 | 372669 | 155,699 |
https://mathoverflow.net/questions/372666 | 4 | This might be a very easy question, and it might be better for mathstackexchange in which case I apologize. I'm stuck on something an anonymous referee wrote to me about a paper of mine and I'm hoping for some clarity.
Suppose $X$ and $Y$ are Polish spaces and $A \subseteq X \times Y$ is Borel. It's well known that i... | https://mathoverflow.net/users/114946 | Do Borel subsets of the plane with null sections have Borel projections? | Notice that $\omega^\omega$ can be embedded to a null subset of itself by sending any sequence $(a\_0,a\_1,a\_2,\dots)$ to $(a\_0,0,a\_1,0,a\_2,0,\dots)$. So any Borel phenomenon that can happen in $\omega^\omega$ can also happen in a null subset.
| 13 | https://mathoverflow.net/users/6794 | 372670 | 155,700 |
https://mathoverflow.net/questions/372616 | 0 | Suppose that $f: \mathbb{R}^n \to \mathbb{R}$ is a continuous function for which we want to minimize. We may arbitrarily impose good conditions for $f$, such as Lipschitzness, smoothness, convexity, etc.
The intuitive way, if you were to teach about minimizing such a function in a first-course on optimization, would ... | https://mathoverflow.net/users/145832 | Are there search algorithms that are competitive against (gradient based) optimization routines for continuous problems? | Yes, this can be shown. What is needed is a fair ground for comparison of different algorithms and I can recommend the book "Introductory Lectures on Convex Optimization - A Basic Course" by Yurii Nesterov. There you'll find a hands-on introduction to complexity theory for continuous problems.
(A side note: Gradient ... | 0 | https://mathoverflow.net/users/9652 | 372672 | 155,701 |
https://mathoverflow.net/questions/372665 | 2 | *Note: This question is based on [a previous question](https://mathoverflow.net/q/371972/165539)*
I was continuing my research from last time, and I realized my question was too strict! Instead of the polynomial being strictly increasing, it only has to be only positive with the *maximum* smaller than $p(0)$. So, my ... | https://mathoverflow.net/users/157462 | Is it possible to create a polynomial $p(x)$ with this relation between $p(0)$ and $p(c)$? -- Part 2 | Suppose that $p$ is a polynomial with the required properties. Let $n:=\deg(p)$. Normalizing, we can assume that $p(0)=1$. Consequently, $0\le p(x)<1/b$ for any $x\in[1,c]$. As a result, the reciprocal polynomial $P(x):=x^np(1/x)$ is monic and satisfies $0<P(x)<x^n/b<1/b$ for any $x\in[c^{-1},1]$.
We now use the foll... | 3 | https://mathoverflow.net/users/9924 | 372689 | 155,706 |
https://mathoverflow.net/questions/372581 | 0 | By James's Theorem, A. Ulger (Weak compactness in $L^{1}(\mu.X)$, Proc. Amer. Math. Soc. 113(1991),143-149.) proved that a bounded subset $A$ of a Banach space $X$ is relatively weakly compact if and only if given any sequence $(x\_{n})\_{n}$ in $A$, there exists a sequence $(z\_{n})\_{n}$ with $z\_{n}\in conv(x\_{i}:i... | https://mathoverflow.net/users/41619 | A characterization of reflexivity of Banach spaces via convex block sequences | Yes, I believe the converse does hold.
Beanland/Freeman proved that an operator $T\in\mathcal{L}(X,Y)$ is weakly compact if and only if for every normalized basic sequence $(x\_n)\in\mathcal{NB}\_X$, the image sequence $(Tx\_n)$ fails to dominate the summing basis $(s\_n)$ for $c\_0$. Consequently, by considering the... | 1 | https://mathoverflow.net/users/73784 | 372697 | 155,710 |
https://mathoverflow.net/questions/372711 | 6 | This must be known or easy for some of you, but here goes:
>
> Suppose $f\_0,f\_1:[n]\to [n]$ are invertible functions, where $[n]=\{0,\dots,n-1\}$ is a set of $n$ elements.
> For a word $w=w\_1\dots w\_m\in\{0,1\}^m$ we define $f\_w=f\_{w\_m}\circ f\_{w\_{m-1}}\circ\dots\circ f\_{w\_1}$ (or make it the opposite or... | https://mathoverflow.net/users/4600 | Group action with unique word | Fix $c\in [n]$. Let $\mathcal R\_m(c)$ be $\{f\_w(c)\colon |w|=m\}$ and $r\_m(c)=|\mathcal R\_m(c)|$. We define $\mathcal R\_0(c)$ to be $\{c\}$.
*Claim*: Let $m\ge 0$. If $r\_m(c)=r\_{m+1}(c)$, then for all $M>m$ and for all $d\in\mathcal R\_M(c)$, there are two $w$'s in $\{0,1\}^M$ with $f\_w(c)=d$.
*Proof*: Let ... | 7 | https://mathoverflow.net/users/11054 | 372714 | 155,716 |
https://mathoverflow.net/questions/371504 | 7 | $\newcommand{\cat}{\mathsf} \newcommand{\fun}{\mathrm{Fun}} \newcommand{\calg}{\mathrm{CAlg}}$
Let $\cat C^\otimes,\cat D^\otimes, \cat E^\otimes$ be symmetric monoidal $\infty$-categories, and $p^\otimes: \cat D^\otimes \to \cat E^\otimes$ a map of $\infty$-operads (aka a lax symmetric monoidal functor).
Assume $p: ... | https://mathoverflow.net/users/102343 | References about "monoidal fibrations" in $\infty$-category theory | I don't know a reference but here is a not-too-long proof. The condition that $\mathsf{D} \to \mathsf{E}$ is a cartesian fibration implies that for every $\langle n \rangle \in \mathrm{Fin}\_\*$ the map $\mathsf{D}^{\otimes}\_{\langle n\rangle} \to \mathsf{E}^{\otimes}\_{\langle n\rangle}$ is a cartesian fibration and ... | 3 | https://mathoverflow.net/users/51164 | 372724 | 155,717 |
https://mathoverflow.net/questions/372699 | 1 | In order to apply the Marsden–Weinstein reduction, the action of the group $G$ must be free and proper. On the other hand, if I correctly understand, the M-W reduction obtained from a given group $G$ can be used to decrease the number of degrees of freedom of a Hamiltonian $H$, provided that the Hamiltonian flow of $H$... | https://mathoverflow.net/users/138060 | Marsden–Weinstein: example of not proper action | (Comment $\to$ answer as requested.)
Let $G=\mathbf R$ act on the 2-torus $Z=\mathrm U(1)\times\mathrm U(1)$ by $g(z\_1,z\_2)=(e^{ig}z\_1, e^{i\pi g}z\_2)$. Lift the action to $T^\*Z$ and use any $G$-invariant $H$.
Explicitly $T^\*Z=\mathbf R^2\times Z\ni(p\_1,p\_2,z\_1,z\_2)$ where $G$ acts by the flow of $K=p\_1+... | 1 | https://mathoverflow.net/users/19276 | 372729 | 155,719 |
https://mathoverflow.net/questions/372677 | 34 | I asked this question [in MSE](https://math.stackexchange.com/questions/3728967/quaternionic-and-octonionic-analogues-of-the-basel-problem) around 3 months ago but I have received no answer yet, so following the suggestion in the comments I decided to post it here.
It is a well-known fact that
$$\sum\_{0\neq n\in\m... | https://mathoverflow.net/users/115044 | Quaternionic and octonionic analogues of the Basel problem | This isn't really a full answer, but it's too long for a comment, and perhaps it's informative all the same.
Your sum $S\_k[\mathcal{O}]$ can be written as the value at $s = k$ of the sum
$$\sum\_{0 \ne \lambda \in \mathcal{O}} \frac{\lambda^k}{Nm(\lambda)^s} = \sum\_{n \ge 1} a^{(k)}\_n n^{-s},$$
where $a^{(k)}\_n :... | 22 | https://mathoverflow.net/users/2481 | 372731 | 155,720 |
https://mathoverflow.net/questions/372733 | 6 | Let $\mathbb{N}$ denote the set of positive integers. For $k\in\mathbb{N}$ let $c\_k:\mathbb{N}\to\mathbb{N}$ be defined by $x\mapsto x/2$ for $x$ even and $x\mapsto kx+1$ otherwise. The *Collatz sequence of $x\in \mathbb{N}$ with respect to $k$*, denoted by $\text{Coll}\_{x,k}:\mathbb{N}\to\mathbb{N}$ is defined by $1... | https://mathoverflow.net/users/8628 | Generalized Collatz sequences | This long comment might be helpful:
I think the answer is negative as I pointed out in the comments because almost all Collatz orbits attain almost bounded values, the result which is shown by [Terras](https://mathscinet.ams.org/mathscinet-getitem?mr=568274) and was proven by [Allouch](https://mathscinet.ams.org/math... | 5 | https://mathoverflow.net/users/51189 | 372737 | 155,723 |
https://mathoverflow.net/questions/372712 | 1 | Let $X\_0,X\_1\in [0,1]$ and $b\_1,b\_2>0$ be integers. We are going to create a numeration system for vectors $(X\_0,X\_1)$, the base being the vector $(b\_1,b\_2)$, as follows.
Recursively define $X\_k=\{b\_2 X\_{k-1} + b\_1 X\_{k-2}\}$, for $k>1$. Here $\{\cdot\}$ represents the fractional part function and $X\_k\... | https://mathoverflow.net/users/140356 | Hybrid numeration system on $[0,1]^2$ | Here is a partial (negative answer) to your first question:
**Proposition 1:** Two different vectors $(X\_0,X\_1)$ and $(X\_0',X\_1')$ cannot have
the exact same digits $d\_0,d\_1,\dots$ in base $(b\_1,b\_2)$, assuming $b\_1,b\_2>0$ and $b\_1>b\_2+1$.
*Proof:* Suppose the contrary. Then for $k=0,1,\dots$ we have $X... | 1 | https://mathoverflow.net/users/36721 | 372738 | 155,724 |
https://mathoverflow.net/questions/372607 | 0 | Let x be a random variable in $\mathbb{R}^d$, $J$ a block tridiagonal $d\times d$ matrix, and probability of $x$ is defined as follows
$$p(x)\propto \exp(-x'Jx)$$
For a fixed $d\times d$ matrix $v$ compute $u$
$$u\_{ij}=\sum\_{kl}E[x\_i x\_j x\_k x\_l] v\_{kl}$$
How can this be done efficiently and what is the ... | https://mathoverflow.net/users/7655 | Fourth moment of a random-variable with block-tridiagonal structure | Using e.g. the Gauss elimination, we can diagonalize the matrix $(v\_{kl})$, that is, write
$$v\_{kl}=\sum\_{r=1}^d a\_r s\_{rk}t\_{rl}$$
for some real $a\_r,s\_{rk},t\_{rl}$ and all $k,l$; the computational complexity (CC) of this diagonalization is $O(d^3)$; cf. e.g. [this source](https://nholmber.github.io/2018/05/m... | 1 | https://mathoverflow.net/users/36721 | 372741 | 155,726 |
https://mathoverflow.net/questions/372713 | 0 | Consider the equation $$\pmb{c}+\text{sign}(G\pmb{c}) = L$$
$\pmb{c}$ is a $n\times1$ matrix.
$G$ is a $n\times n$ matrix which is also positive definite.
matrices $G$ and $c$ are real.
$L$ is a $n\times 1$ matrix whose entries are from the set $\{-1,1\}$.
Can this equation be solved for the matrix $\pmb{c}$?... | https://mathoverflow.net/users/14414 | An otherwise linear matrix equation with the presence of a signum function : reference request | You can solve the problem via mixed integer linear programming as follows. Let $\epsilon>0$ be a small constant tolerance. For $i \in \{1,\dots,n\}$, let $[\ell\_i,u\_i]$ be constant lower and upper bounds on $(Gc)\_i$, and let binary decision variables $z^-\_i$ and $z^+\_i$ indicate whether $(Gc)\_i<0$ or $(Gc)\_i>0$,... | 1 | https://mathoverflow.net/users/141766 | 372742 | 155,727 |
https://mathoverflow.net/questions/372748 | 16 | A [Markoff triple](https://en.wikipedia.org/wiki/Markov_number) $(a,b,c)$ is a solution in positive integers to the equation
$$ a^2+b^2+c^2=3abc. $$
Frobenius famously conjectured that a given integer $c$ may appear at most once as the largest coordinate of a Markoff triple $(a,b,c)$, a conjecture often referred to as ... | https://mathoverflow.net/users/11926 | Frobenius' article and the Markoff number unicity conjecture | I found Frobenius's 1913 publication in the [Biodiversity Heritage Library.](https://www.biodiversitylibrary.org/item/126263#page/528/mode/1up) (Somehow Google Scholar does not index it.) You can view the paper online and download the pdf by submitting an email address.
| 18 | https://mathoverflow.net/users/11260 | 372749 | 155,728 |
https://mathoverflow.net/questions/372598 | 4 | The first nontrivial irreducible representation of $G\_2$ is of 7-dimensional, and the first nontrivial representation of $F\_4$ is of 26-dimensional.
My question is: how much is known about the nilpotent orbits in these representations? Any classification? or the answer is very easy, there are only nilpotent orbits,... | https://mathoverflow.net/users/5082 | Nilpotent orbits in representations of exceptional groups | As per the OP's comment, we are to assume that $\mathrm{G}\_2$ and $\mathrm{F}\_4$ mean the complex simple Lie groups.
Let's start with $\mathrm{G}\_2\subset\mathrm{SO}(7,\mathbb{C})$, in its standard representation on $\mathbb{C}^7$, which is the vector space $V= \mathrm{Im}(\mathbb{O}^\mathbb{C})\subset \mathbb{O}^... | 8 | https://mathoverflow.net/users/13972 | 372756 | 155,730 |
https://mathoverflow.net/questions/372753 | 10 | Suppose we are given a univariate polynomial with rational coefficients, $p \in \Bbb Q [x]$, and are told that $p$ can be expressed as the sum of $k$ squares of polynomials with rational coefficients. It is well-known that every univariate sum of squares (SOS) polynomial can be expressed as a sum of two squares.
Can ... | https://mathoverflow.net/users/7400 | SOS polynomials with rational coefficients | In general you can't write $p = f^2 + g^2$ in ${\bf Q}[x]$ at all,
let alone do so efficiently.
For example, $2 x^2 + 3$ is positive for all $x$
(and is the sum of three squares, $(x+1)^2 + (x-1)^2 + 1^2$);
but if $2 x^2 + 3 = f(x)^2 + g(x)^2$ then $3 = f(0)^2 + g(0)^2$,
which is impossible because $3$ is not a sum o... | 20 | https://mathoverflow.net/users/14830 | 372762 | 155,732 |
https://mathoverflow.net/questions/372765 | 5 | This is probably already well-known or too big to answer. Let $G$ be a finite group and $G^{ab}$ be the abelianization of the group G. Is there any bound on $d(G)=\min\{\#S\mid G=\langle S\rangle\}$ by using $d(G^{ab})$ without considering the order of $G$?
Thanks for any answer and comments.
| https://mathoverflow.net/users/166059 | Bound the number of the minimal generating set of group G by its abelianization | Okay, so let's fill in the details on Ville's nice argument in the comments: there is no such bound, and to prove this it suffices to exhibit a sequence of finite perfect groups whose ranks are unbounded. We'll take the sequence $A\_5^n$ to be concrete although the argument applies to powers of any finite perfect group... | 9 | https://mathoverflow.net/users/290 | 372768 | 155,733 |
https://mathoverflow.net/questions/372766 | 6 | This question is related to the [last question about van der Pol's identity for the sum of divisors](https://mathoverflow.net/questions/372476/van-der-pols-identity-for-the-sum-of-divisors-and-a-quartic-polynomial-equation).
In [Touchard (1953)](https://oeis.org/A000385/a000385.pdf) it is mentioned that the sum of divi... | https://mathoverflow.net/users/165920 | Is there a similar formula like Ramanunjan's Eisenstein series identity for $\sum_{k=1}^{n-1}k^2 \sigma(k)\sigma(n-k)$? | Numerical experiments suggest that
$$A\_2(n) := \sum\_{k=1}^{n-1} k^2\sigma(k)\sigma(n-k) = \frac{n^2}{8}\sigma\_3(n) - \frac{4n^3-n^2}{24}\sigma(n).$$
PS. In fact, it directly follows from the quoted Touchard and Ramanujan identities.
A couple of similar identities:
$$A\_1(n):=\sum\_{k=1}^{n-1} k\sigma(k)\sigma(n-k)... | 8 | https://mathoverflow.net/users/7076 | 372770 | 155,734 |
https://mathoverflow.net/questions/362766 | 18 | This question is partly motivated by a few comments [here](https://mathoverflow.net/questions/362633/any-real-algebraic-variety-is-diffeomorphic-to-a-real-algebraic-variety-defined). Let me denote by $R$ the (real-closed) field of real numbers $\mathbb{R}$; everything is probably the same over an arbitrary real-closed ... | https://mathoverflow.net/users/4721 | "Real algebraic varieties" vs finite type separated reduced $\mathbb{R}$-schemes with dense $\mathbb{R}$-points | As for your first question, concerning nonaffine R-varieties as you call them, yes, there are nonaffine R-varieties. However, they are considered pathological. Example 12.1.5 on page 301
of Bochnak-Coste-Roy, Real algebraic geometry, constructs an R-line bundle over $\mathbf R^2$ whose total space is not affine. In fac... | 4 | https://mathoverflow.net/users/85592 | 372783 | 155,740 |
https://mathoverflow.net/questions/372420 | 5 | Let $X$ be a finite ultrametric space and $P(X)$ be the space of probability measures on $X$ endowed with the Wasserstein-Kantorovich-Rubinstein metric (briefly WKR-metric) defined by the formula
$$\rho(\mu,\eta)=\max\{|\int\_X fd\mu-\int\_X fd\eta|:f\in Lip\_1(X)\}$$ where $Lip\_1(X)$ is the set of non-expanding real-... | https://mathoverflow.net/users/61536 | Fast algorithms for calculating the distance between measures on finite ultrametric spaces | This is a rather more fun problem than I thought. I must apologize though, as your question is a reference request and I have no references apart from pointing at any textbook on discrete optimization. It turns out, the key is that one can rewrite your problem into a flow problem on a tree, which then is almost trivial... | 3 | https://mathoverflow.net/users/51695 | 372785 | 155,742 |
https://mathoverflow.net/questions/372787 | -1 | Fisher -Neyman Factorization Theorem is:
A statistic $T(Y)$ is sufficient for $θ$ if and only if for all $θ\in Θ$ and all $y\in \Omega$ , there is
$$
L(\theta; y) = g(T(y);\theta)h(y)
$$
where $g(.;.)$ depends on $T(y)$ and $\theta$, and $h(.)$ does not depend on $\theta$
My question is how to prove the Fisher... | https://mathoverflow.net/users/165661 | How to prove the Fisher-Neyman factorization theorem in the continuous case? | You find a proof f.i. in G.G. Roussas, A Course in Mathematical Statistics, 2. ed., Academic Press, 1997, Ch. 11, Th. 1 (p. 263).
| 0 | https://mathoverflow.net/users/100904 | 372789 | 155,744 |
https://mathoverflow.net/questions/372792 | 3 | Let $X, Y$ be Hilbert spaces and $F:X \rightarrow Y$ smooth. Assume that $M := F^{-1}(0) \subset X$ is a smooth submanifold. Is it true that for any $x\in M$, the tangent space $T\_xM$ is a Hilbert subspace of $\mathrm{ker} D\_xF$?
Of course, if $0$ is a regular value of $F$, then by the implicit function theorem, $M... | https://mathoverflow.net/users/166091 | Tangent space of smooth Hilbert submanifolds | True: via a local chart we can assume $F^{-1}(0)$ is a closed linear subspace $N$ of $X$, and since $F\_{|N}=0$, we also have $N\subset \text{ker} DF(x) $.
(A formal explanation of the latter: if we denote $i\_N:N\to X$ the (bounded, linear) inclusion map, $F\_{|N}=F\circ i\_N:N\to Y$ is the null map and by the chain... | 5 | https://mathoverflow.net/users/6101 | 372796 | 155,746 |
https://mathoverflow.net/questions/372764 | 2 | For a scheme $X$, denote by $\mathcal{Ell}\_X[\text{isog}^{-1}]$ the category of elliptic curves on $X$ localized at isogenies. Consider the functor
$$
\mathcal{Ell}^{isog}:Sch/S^{op}\rightarrow \text{Gpd}, \quad X \rightarrow \mathcal{Ell}\_X[\text{isog}^{-1}].
$$
It was asked in [this M.SE question](https://math.stac... | https://mathoverflow.net/users/152554 | Sheaf of elliptic curves up to isogeny | This does not satisfy the sheaf condition.
Consider a curve that is the union of two $\mathbb P^1$s, glued at $0$ and $\infty$. We can form an open cover consisting of the complement of $0$ and the complement of $\infty$, each two $\mathbb A^1$s glued at a point. The intersection of the cover is two disjoint $\mathbb... | 4 | https://mathoverflow.net/users/18060 | 372798 | 155,747 |
https://mathoverflow.net/questions/372804 | 5 | If $A$ is a C\*-algebra, we say that a subset $I\subseteq A$ is hereditary if
$$
0\leq x \leq y \in I \Rightarrow x\in I.
$$
It is is well known that closed 2-sided ideals are hereditary.
Would it also be true for arbitrary 2-sided ideals? What about self-adjoint 2-sided ideals?
| https://mathoverflow.net/users/110570 | Is every 2-sided ideal in a C*-algebra hereditary? | No. Take $A = C[0,1]$ and let $I$ be the (unclosed) ideal generated by the function $f(t) = t$. This ideal is self-adjoint, but it does not contain the function $g(t) = t\sin^2(\frac{1}{t})$, so it is not hereditary. (Example II.5.2.1 (iii) in Bruce Blackadar's fantastic book [*Operator Algebras: Theory of C${}^\*$-Alg... | 10 | https://mathoverflow.net/users/23141 | 372807 | 155,748 |
https://mathoverflow.net/questions/339958 | 4 | Let $\Omega$ be a bounded smooth domain of $\mathbb{R}^n$, $0<s<1$ and $(-\Delta)^s$ denotes the restricted fractional Laplacian. Let consider the following fractional Heat equation:
$$
\begin{cases}
u\_t = (-\Delta)^s u + f(x,t) & \quad \mathrm{in} \Omega \times (0,T),\\ u(x,0)=u\_0 & \quad \mathrm{in} \ma... | https://mathoverflow.net/users/76453 | Question about the regularity of fractional Heat equation | It depends on $s$, if $1/2\leq s \leq 1$, then you get analyticity. If $0<s<1/2$, then you get Gevrey class only. See Section 8.3, arXiv:1606.00873
| 1 | https://mathoverflow.net/users/155359 | 372809 | 155,749 |
https://mathoverflow.net/questions/372613 | 11 | Given $n$ quadratic polynomials in $n-1$ variables over the complex field, what is the maximum number of common zeros? Can we have $2^{n-1}-1$ common zeros? I assume that a linear combination of the polynomials is always different from zero and the number of zeros is finite.
With $4$ polynomials, the maximum is not s... | https://mathoverflow.net/users/4274 | Maximum number of common zeros of n polynomials in n-1 variables | There is a bound for the multiplicity of a (homogenous) almost complete intersection in Theorem 1 of this [paper](https://arxiv.org/pdf/0802.0469.pdf) by Engheta. In case of $n$ quadrics in $n-1$ variables, that bound is $2^{n-1}-(n-2)$. So for $n\geq 4$, you can not get $2^{n-1}-1$.
(In Theorem 1 there was a conditi... | 14 | https://mathoverflow.net/users/2083 | 372813 | 155,750 |
https://mathoverflow.net/questions/372808 | 2 | Let $k$ be a non-archimedean field and denote by $\mathbb{A}\_k^n$ the analytic affine space of $n$ dimensions over $k$ (analytic in the sense of Berkovich). There is a natural injective map of sets $\mathbb{A}\_k^n(k) \to \mathbb{A}\_k^n$. Is $\mathbb{A}\_k^n(k)$ mapped onto a dense subset of $\mathbb{A}\_k^n$? Clearl... | https://mathoverflow.net/users/112369 | Is $\mathbb{A}_k^n(k)$ dense in the Berkovich analytification of $\mathbb{A}_k^n$? | If $k$ is not algebraically closed, then $\mathbb A^n\_k(k)$ is not doing to be dense with $\mathbb A\_k^n$. I'll show this for $n=1$ for simplicity. Take any point $P$ in $\mathbb A^1\_k$ with a residue field $K$ which is a proper finite extension $k$. Since $k$ is complete, $k$ is closed as a subspace of $K$, and hen... | 5 | https://mathoverflow.net/users/30186 | 372818 | 155,751 |
https://mathoverflow.net/questions/371355 | 6 | I have read according list of below papers a basic connection between [Jones polynomial](https://en.wikipedia.org/wiki/Jones_polynomial) and statistical mechanics is that the Kauffman bracket or Kauffman polynomial a polynomial invariant of knots is in different special cases the Jones polynomial for knots and the part... | https://mathoverflow.net/users/51189 | What are applications of Jones polynomial on von Neumann algebras? | I don't think it's quite right to think of knot polynomials as having applications to von Neumann algebras. Instead I think it's more accurate to say that the Temperley-Lieb-Jones algebras (and more generally "towers of algebras with Markov traces" or equivalently quantum groups or tensor categories) have applications ... | 4 | https://mathoverflow.net/users/22 | 372828 | 155,754 |
https://mathoverflow.net/questions/372815 | 3 | $\DeclareMathOperator{\complex}{\mathbb{C}}$
Let $\bigvee^m(\complex^n)\subseteq (\complex^n)^{\otimes m}$ denote the space of symmetric tensors, i.e. the set of $x \in (\complex^n)^{\otimes m}$ that are invariant under permutations of the $m$ factors. The (cone over) the Veronese variety of $\bigvee^m(\complex^n)$ is ... | https://mathoverflow.net/users/150898 | A different notion of a decomposable symmetric tensor (besides Veronese) | Your $\vee$ is essentially multiplication of polynomials. The variety of tensors $x\_1 \vee \dotsb \vee x\_m$ corresponds to polynomials that factor as products of linear factors. Points of the (projective) variety correspond to hyperplane arrangements. Dually, they correspond to *cycles* of $m$ points (in the dual pro... | 9 | https://mathoverflow.net/users/88133 | 372839 | 155,757 |
https://mathoverflow.net/questions/372821 | 5 | Let $G$ be a Lie group (paracompact, not necessarily compact), and $A$ an abelian Lie group. I want to write down cocycles in $\mathrm{H^n}(\mathbf{B}G,A)$, the cohomology in the cohesive $\infty$-topos of smooth $\infty$-stacks. This topos is presented by the model category ${[\mathrm{CartSp}^\mathrm{op},\mathrm{sSet}... | https://mathoverflow.net/users/130827 | Computing cohomology using bounded hypercovers |
>
> I would like to know an example of an abelian Lie group A where the statement is wrong (and why).
>
>
>
There is no such example because the statement is true for all $A$.
(This also implies trivial answers for the other two questions: always and no.)
Any abelian Lie group $A$ fits into an exact sequence
$... | 4 | https://mathoverflow.net/users/402 | 372842 | 155,758 |
https://mathoverflow.net/questions/372835 | 4 | Let $\mathfrak{A}$ be a [separable] unital C\*-algebra and let $Q$ be a dense subset of the state space of $\mathfrak{A}$. Suppose that for each $f\in Q$ the associated GNS representation is faithful. Is $\mathfrak{A}$ simple?
| https://mathoverflow.net/users/166106 | Characterization of simple C*-algebras via GNS representations | I think the answer is negative.
If $A$ and $B$ are C\*-algebras then the state space $S(A\oplus B)$ contains a natural copy of $S(A)$ and one of $S(B)$ such
that
$S(A\oplus B)$ is the convex hull of $S(A)\cup S(B)$. Moreover, the convex combinations
$$
\sigma =\alpha \varphi +\beta \psi ,
$$
with $\varphi \in S(A)$... | 5 | https://mathoverflow.net/users/97532 | 372846 | 155,759 |
https://mathoverflow.net/questions/372831 | 42 | I am a young PhD student (24) at a Germany university and I am not sure whether this is the right place to ask this kind of question. If not feel free to move it elsewhere or delete it completely.
Currently, I a have a half time position in Analysis and my doctoral advisor more and more turns out to be not very invol... | https://mathoverflow.net/users/91126 | How to deal with an advisor that offers you nearly no advising at all? | I second Nate's suggestion to look at <https://academia.stackexchange.com>, there are already many similar questions (with answers, some of them specifically from mathematicians) on that site that may help you.
But since "go somewhere else" is not exactly the kind of answer someone in your situation needs, here are a... | 49 | https://mathoverflow.net/users/30516 | 372856 | 155,760 |
https://mathoverflow.net/questions/372814 | 5 | Let $X=(V,E)$ be a graph, and to each vertex $v \in V$, associate a group $G\_v$. The graph product of the groups $G\_v$ (as defined e.g. [here](https://groupprops.subwiki.org/wiki/Graph_product_of_groups)) is $F/R$; the quotient of the free product of the $G\_v$ by the by the normal subgroup generated by commutators $... | https://mathoverflow.net/users/103150 | "Simplicial complex" product of groups? | I don't know if this will help, but in
<https://arxiv.org/pdf/math/0101220.pdf>,
we used graph products of spaces (which I think had been introduced by Danny Cohen but I'm not sure) This leads to a graph tensor product of crossed resolutions of the groups. This will give a resolution of the graph product. I'm not sure ... | 4 | https://mathoverflow.net/users/3502 | 372859 | 155,761 |
https://mathoverflow.net/questions/372874 | 3 | Let me first give a vague definition of "theory"/"physical theory", [see also](https://ncatlab.org/nlab/show/theory+%28physics%29). A (physical)theory is a collection of rules and notions that were successful in predicting a behaviour of an idealised physical system. This question is about the theories that have stood ... | https://mathoverflow.net/users/163521 | Mathematical formalization of physics | [John Baez](https://arxiv.org/abs/quant-ph/0404040v2) (2004) discusses quantum theory and general relativity from a unified perspective provided by category theory.
>
> Faced with the great challenge of reconciling general relativity and
> quantum theory, it is difficult to know just how deeply we need to
> rethink... | 4 | https://mathoverflow.net/users/11260 | 372890 | 155,767 |
https://mathoverflow.net/questions/372880 | 11 | Let $M$ be the $E\_8$ manifold. Is there a closed manifold $N$ such that $M\times N$ is smoothable? What is the smallest possible dimension of $N$?
| https://mathoverflow.net/users/nan | Smooth structure on direct product | Extending Michael Albanese's answer above, $M \times N^k$ will never be smoothable. For if it were then choose a point $p\in N$ and a chart U around $p$. Then $M \times U$ is an open subset of $M \times N$, and hence is smoothable. But as argued in Scorpan (p. 219, Lemma), $M \times \mathbb{R}^k$ is not smoothable.
S... | 11 | https://mathoverflow.net/users/3460 | 372895 | 155,769 |
https://mathoverflow.net/questions/372891 | 5 | Let $B\subset \mathbb{R}^n$ be a open ball. Let $\{f\_i\}$ be a sequence of functions bounded in the Hölder norm $C^{k,\alpha}(B)$ for a given integer $k\geq 0$ and $\alpha\in (0,1)$.
Does there exist a subsequence which converges to a function $f$ (necessarily $f\in C^{k,\alpha}(B)$) in the norm $C^{k,\alpha/2}(\bar... | https://mathoverflow.net/users/16183 | Arzelà-Ascoli theorem and Hölder spaces | At first, if partial derivatives of order at most $k$ of $f\_{n\_i}$ converge to those of $f$, than automatically $f\in C^{k,\alpha}(B)$, since $$|(D^k f)(x)-(D^k f)(y)|\leqslant \limsup\_i |(D^k f\_{n\_i})(x)-(D^k f\_{n\_i})(y)|\leqslant c\cdot |x-y|^\alpha$$
unformly over $x,y\in B$ (here $D^k$ denotes the vector of ... | 8 | https://mathoverflow.net/users/4312 | 372900 | 155,773 |
https://mathoverflow.net/questions/372909 | 1 | Let $H$ be a Hilbert space and $B(H)$ denotes the space of all continuous linear operators on $H$. I am looking for a class/example of bounded linear functionals $B(H)\to \mathbb C$ which cannot be reduced to the type $$T\mapsto\sum\_{i=1}^\infty k\_i\left<Tx\_i,y\_i\right>$$ for some fixed $x\_i$'s and $y\_i$'s in $H$... | https://mathoverflow.net/users/145729 | Example of linear functionals on $B(H)$ | So, you are asking about non-normal functionals on $B(H)$. This is very similar to the question of what are the functionals on $\ell\_\infty$ that are not in $\ell\_1$?
Fix an ultrafilter $U$ on $\mathbb N$ and let $(e\_n)$ be an ONB for $H$. Define
$$\langle \phi, T\rangle = \lim\_{n,U} \langle Te\_n, e\_n\rangle\... | 9 | https://mathoverflow.net/users/15129 | 372920 | 155,782 |
https://mathoverflow.net/questions/372879 | 1 | Consider the $\mathbb Z^2$ lattice, we then define for $u=(u\_{ij})\_{i,j \in \mathbb Z}$ the discrete Laplacian
$$(\Delta u)\_{i,j}=u\_{i+1,j}+u\_{i-1,j}+ u\_{i,j+1}+u\_{i,j-1}$$
and the weight which pushes the mass down at every point
$$ (T u)\_{i,j}=\frac{1}{\sqrt{i^2+j^2+1}} u\_{i,j}.$$
We then define $v\_{ij}=... | https://mathoverflow.net/users/119875 | $\ell^1$-bound on graph laplacian with weight | Direct calculation shows $((T\Delta)^2 v)\_{0,0} = 2\sqrt{2}$.
This implies that
$$((T\Delta)^{2n}v)\_{0,0} \ge (2\sqrt{2})^n$$
for $n = 1,2,\ldots$.
You might be interested in studying the iterates of $P\_{\lambda} = \frac{1}{4}\lambda T\Delta = \lambda T\Delta'$ where $\Delta' = \frac{1}{4}\Delta$ and $\lambda > ... | 2 | https://mathoverflow.net/users/7631 | 372923 | 155,783 |
https://mathoverflow.net/questions/372911 | 6 | Consider the Reedy category $2\rightarrow 1 \leftarrow 0$. Consider a map of diagrams of topological spaces $D\to E$ over this Reedy category:
The maps which are fibrations are depicted with the symbol $\twoheadrightarrow$: the map of diagrams $D\to E$ is a pointwise fibration... | https://mathoverflow.net/users/24563 | About the dual of the cube lemma in homotopy theory | Yes, $D\_0 \times\_{D\_1} D\_2 \to (E\_2 \times\_{E\_1} E\_0) \times\_{E\_0} D\_0$ is a fibration.
First, observe that
$$(E\_2 \times\_{E\_1} E\_0) \times\_{E\_0} D\_0 \cong E\_2 \times\_{E\_1} D\_0 \cong (E\_2 \times\_{E\_1} D\_1) \times\_{D\_1} D\_0$$
by the pullback pasting lemma. Also,
$$D\_2 \times\_{D\_1} D\_0 ... | 8 | https://mathoverflow.net/users/11640 | 372925 | 155,785 |
https://mathoverflow.net/questions/372922 | 6 | Maybe I'm wrong, but I just noticed that the different permutations of $(1,2)(2,3)(3,4),\dots,(n-1,n)$ seem to be $2^{n-2}$ and I don't know why this is true. Can someone help if I'm right about this and explain a little bit?
e.g.: $n=4$, $(1,2)(3,4)(2,3) = (3,4)(1,2)(2,3)$ and $(2,3)(1,2)(3,4) = (2,3)(3,4)(1,2)$ but... | https://mathoverflow.net/users/165064 | Why does the number of permutations of $n-1$ adjacent transpositions where the outputs are different equal $2^{n-2}$? | These are the [Coxeter elements](https://en.wikipedia.org/wiki/Coxeter_element) of the symmetric group, and they correspond to orientations of the Type A Dynkin diagram, of which there are $2^{n-2}$.
| 14 | https://mathoverflow.net/users/25028 | 372926 | 155,786 |
https://mathoverflow.net/questions/372938 | 4 |
>
> **Note:** This question came from MSE, but since I've received some useful observations I posted it here. [Post on MSE](https://math.stackexchange.com/a/3845228/717872)
>
>
>
Consider $1 \leq k < n$ positive integers, and denote by $G(\mathbb{P}^k,\mathbb{P}^n)$ the Grassmannian of $\mathbb{P}^k$'s in $\math... | https://mathoverflow.net/users/165918 | Describe $\mathcal{N}_{G(\mathbb{P}^1,\mathbb{P}^k)\mid G(\mathbb{P}^1,\mathbb{P}^n)}$ [from MSE] | (A more general answer valid not only for the Grassmannian of lines).
Denote by $U$ the rank $k$ tautological bundle on $Gr(k,n)$, using the convention that $det(U^{\vee})= \mathcal{O}\_{Gr(k,n)}(1)$ (in the Pluecker embedding). Then the zero locus of a general global section of $U^{\vee}$ is naturally identified wit... | 5 | https://mathoverflow.net/users/52811 | 372941 | 155,790 |
https://mathoverflow.net/questions/372930 | 3 | Let $[a,b]$ be an interval and $X$ a Banach space (for starters). We know that continuous functions $f:[a,b]\to X$ are Riemann integrable. Suppose now that $X$ is a quasi-Banach space, that is, its norm satisfies $\|x+y\|\leq K (\|x\|+\|y\|)$ for all $x,y\in X$ and some $K\geq 1.$
I found that, in general, quasi-Bana... | https://mathoverflow.net/users/161393 | Integration on quasi-Banach spaces and Schatten ideals | No, there are such continuous functions, which are continuous with values in $\mathcal{L}^p(H)$ for any $p$ but such that $\int\_a^b f$ (which is well defined in the Banach space $\mathcal{L}^1(H)$) does not belong to $\mathcal{L}^p(H)$ for any $p<1$.
An almost counter-example is given as follows on $H=\ell^2$. Take ... | 4 | https://mathoverflow.net/users/10265 | 372945 | 155,793 |
https://mathoverflow.net/questions/372487 | 27 | [ZBmath](https://zbmath.org/) (formerly Zentralblatt für Mathematik) will become "Open access" in 2021 (see for instance at [EMS site](https://euro-math-soc.eu/news/19/12/17/zbmath-become-open-access) and at [FIZ Karlsruhe site](https://www.fiz-karlsruhe.de/en/nachricht/zbmath-open-informationen-fuer-die-mathematik-wer... | https://mathoverflow.net/users/14094 | Sustainability of ZBmath unrestricted access | Thanks a lot for the question! On behalf of Klaus Hulek (as zbMATH Editor-in-Chief) we can confirm that it is correct that we will go Open Access as of 1st January 2021. This will mean that the database is freely accessible by everybody worldwide. It also means that most of the data will become open via a CC-BY-SA lice... | 21 | https://mathoverflow.net/users/100979 | 372954 | 155,795 |
https://mathoverflow.net/questions/372952 | 9 | Let us say that an algebra $A$ over a field $k$ is Picard-surjective if the canonical map
$$ \mathrm{Aut}(A) \rightarrow \mathrm{Pic}(A)$$
is surjective. Here $\mathrm{Pic}(A)$ denotes the group of isomorphism classes of invertible $A$-$A$-bimodules and the map sends an automorphism $\alpha$ to the $A$-$A$-bimodule $A\... | https://mathoverflow.net/users/16702 | Picard-surjectivity and Morita-equivalence | Yes, the basic algebra of $A$ will be Picard-surjective.
The basic algebra is the endomorphism algebra $\operatorname{End}\_A(\bigoplus\_{i=1}^{n}P\_i)$ of the direct sum of indecomposable projective (right) modules, one from each isomorphism class. It is Morita equivalent to $A$.
Suppose $A$ is basic. Then as a le... | 8 | https://mathoverflow.net/users/22989 | 372959 | 155,796 |
https://mathoverflow.net/questions/372947 | 0 | Let $f\in L^\infty(\mathbb{R})\cap C(\mathbb{R})$, that is $f$ is continuous and bounded on $\mathbb{R}$. Let $S\_r$ denote the shift by $r\in \mathbb{R}$: $S\_r f=f(\cdot-r)$.
Suppose $S\_{r} f $ converges to $ f$ as $r\rightarrow 0$ in the weak dual topology $\sigma(L^\infty, L^1)$, for a that is, for each $\varphi... | https://mathoverflow.net/users/166196 | Convergence in weak dual topology $\sigma(L^\infty, L^1)$ | It isn't research level, but $f(t) = \sin(e^{t^2})$ is a counterexample. (The uniform distance between $f$ and any shift of $f$ is $1$.)
| 3 | https://mathoverflow.net/users/23141 | 372961 | 155,798 |
https://mathoverflow.net/questions/371472 | 5 | Suppose that $S$ is an infinite set and that $\alpha$ and $\beta$ are metrics over $S$ such that the topology induced by $\alpha$ is everywhere strictly finer than the metric induced by $\beta$, meaning that every open set $U$ in $\beta$ contains a set $V$ that is open in $\alpha$ but not in $\beta$. Suppose further th... | https://mathoverflow.net/users/22344 | Is the space of metric topologies over a given set dense (in the order sense)? | $\def\cl{\operatorname{cl}}$
A large family of counterexamples can be constructed using the following
proposition:
Let $(S, T\_1)$ be a topological space with two complementary dense
subspaces $A, B$.
Define $T\_3 = \{ (A \cap U) \cup (B \cap V) \mid U, V \in T\_1 \}$,
in other words $(S, T\_3)$ is the topological su... | 2 | https://mathoverflow.net/users/10075 | 372964 | 155,800 |
https://mathoverflow.net/questions/372871 | 10 | Let $M$ be a closed orientable smooth 4-manifold. Assume $\pi\_1(M)=\{0\}$ and $b\_2(M)>0$.
Let $S$ be a closed orientable surface. Denote $P=M\times S$.
Can it so happen that there is no complex projective manifold homotopy equivalent to $P$?
Is it possible to rule out the existence of a closed symplectic 6-mani... | https://mathoverflow.net/users/nan | Topological factors of complex projective manifolds | Let $M=\mathbb CP^2\#\mathbb CP^2$ and let $S=T^2$ be the $2$-dimensional torus. I think this gives an example for the original question. As for the symplectic version of the question, I am sure it is an open problem.
*Proof.* Suppose by contradiction $P=M\times S=\mathbb CP^2\#\mathbb CP^2\times T^2$ is homotopic to... | 2 | https://mathoverflow.net/users/943 | 372991 | 155,808 |
https://mathoverflow.net/questions/372751 | 1 | Let $K \in M\_+(R\_+^2), f \in M\_+(R\_+)$. Consider operator
$$
(T\_k)(x)=\int\_{R\_+}K(x,y)f(y)dy, \quad y\in R\_+.
$$
Denote by $f^\*(t)=\inf\{\lambda>0: \alpha x \in R\_+: \mu\_f(y)>\lambda\}$ the non-increasing rearrangement of $f$. Here $\mu\_f(y)=\{\alpha x\in R\_+: |f(x)|>y\}$.
Let $\Phi(x)=\int\_0^x \phi... | https://mathoverflow.net/users/122182 | Example when Kantorovich condition would not hold | Maybe the simplest classical example is a weakly singular kernel
$$K(x,y) = |x-y|^{-\lambda}$$
with some fixed $\lambda\in(0,1)$.
In this example $\int\_{\mathbb R^2}K(x,y)^qdx=\infty$ for every $q>0$ by Fubini-Tonelli (and also all mixed norms from my earlier comment are infinite).
However, for constant $u$ an... | 2 | https://mathoverflow.net/users/165275 | 372994 | 155,809 |
https://mathoverflow.net/questions/372962 | 12 | Let $P\subset \mathbb{R}^2$ be a positive Lebesgue measure set. Then $P$ does not necessarily contain a subset of the form $A\times B$ where $A,B\subset \mathbb{R}$ are of positive Lebesgue measure.
For example consider $P=\{(x,y)\in [0,1]\times[0,1]:x-y\notin \mathbb{Q}\}.$
This example leads me to ask:
Given an... | https://mathoverflow.net/users/166207 | Regarding a positive Lebesgue measure set in $\mathbb{R}^2$ | The question was answered by Robert Israel 1995 on Usenet, essentially by the set mentioned in fedja's comment. The proof that this set has the required property is carried out in detail in Example 4.3.1 of M. Väth, Ideal Spaces, [Springer 1997](https://www.springer.com/de/book/9783540631606).
Here is a sketch of the... | 7 | https://mathoverflow.net/users/165275 | 373002 | 155,810 |
https://mathoverflow.net/questions/373003 | 1 | Let $(e\_{j})\_{j=1}^{\infty}$ be a basis for the Banach space $X$. If there exist constants $\zeta\_{1},\zeta\_{2}>0$ such that for all $N\in\mathbb{N}$,
\begin{equation\*} \zeta\_{1}\left(\sum\_{i=1}^{N}\|x\_{i}\|^{p}\right)^{\frac{1}{p}}\leq\left\Vert\sum\_{i=1}^{N}x\_{i}\right\Vert\leq\zeta\_{2}\left(\sum\_{i=1}^{N... | https://mathoverflow.net/users/165007 | Definition question: asymptotic-$\ell_{p}$ versus coordinate-free asymptotic-$\ell_{p}$ | Because according to your definition even $\ell\_p$, $p\neq 2$ is not asymptotic-$\ell\_p$. You can pick your finite dimensional subspaces from a larger Euclidean subspace (by Dvoretsky's theorem). I also don't see how one could salvage your definition.
But you are right that one can define asymptotic structures with... | 2 | https://mathoverflow.net/users/3675 | 373011 | 155,811 |
https://mathoverflow.net/questions/373008 | 10 | Let $M$ be a connected closed orientable 3-manifold. Assume $M$ is not the direct product of a surface and the circle.
Can there be a symplectic or Kähler manifold homeomorphic to $M\times M$? I think this might work if $M$ is a non-trivial circle bundle over the torus.
| https://mathoverflow.net/users/nan | Symplectic structure on the square of a 3-manifold | Let $M$ be a 3-manifold fibering over $S^1$, so there exists a fibration $\Sigma \to M \to S^1$.
Then $M\times M$ will admit a symplectic structure.
There is a symplectic structure on $M\times S^1$, associated to the fibration $\Sigma \to M\times S^1 \to S^1\times S^1=T^2$ which is trivial in the second factor, by a ... | 12 | https://mathoverflow.net/users/1345 | 373017 | 155,814 |
https://mathoverflow.net/questions/373012 | 5 | Let $X$ be a Banach space, $H\subseteq X$ be a dense hyperplane, and $f$ be a **continuous** linear functional defined on $H$. Then $f$ is uniformly continuous and hence it admits a unique continuous extension to $X$.
However, let us instead choose a **discontinuous** linear functional $g$ extending $f$ to the whole ... | https://mathoverflow.net/users/110570 | Wildly discontinuous linear functionals | No non zero linear functional has the property you ask for. Suppose $F$ is a non zero linear functional. Choose $x$ s.t. $F(x)=1$. Let $G$ be a continuous linear functional s.t. $G(x)=1$. Let $Y$ be the intersection of the kernels of $F$ and $G$, so that $Y$ has codimension $2$. Then $F$ is continuous on the linear spa... | 11 | https://mathoverflow.net/users/2554 | 373019 | 155,815 |
https://mathoverflow.net/questions/373018 | 2 | Let $S$ be a connected open complex manifold. Let $\phi:X\to S$ be a proper holomorphic submersion with connected fibers. Can the fibers of $\phi$ have infinitely many distinct Hodge diamonds?
An example of varying Hodge diamonds is given in the paper "Complex parallelisable manifolds and their small deformations".
... | https://mathoverflow.net/users/nan | Infinitely many deformation equivalent Hodge diamonds | No, because $h^i(\Omega\_{X,s}^j)$ is upper semicontinuous in the analytic Zariski topology. So it can attain only a finite number of possible values over $S$.
| 4 | https://mathoverflow.net/users/4144 | 373023 | 155,818 |
https://mathoverflow.net/questions/372688 | 3 | Let $\mathcal{D}$ be a probability distribution with support $[0,1]$. Let $x, y, z$ the outcomes of three i.i.d. random variables $X, Y, Z$ with distribution $\mathcal{D}$, sorted in increasing order, i.e., $x\le y\le z$, . Let $a=y-x$ and $b=z-y$. We define
$$\Delta=1-\mathbb{E}\left[\frac{b}{a+b}\cdot a+\frac{a}{a+... | https://mathoverflow.net/users/115803 | Symmetric distribution optimization problem of distances between points in $[0,1]$ | If $\mathcal{D}$ has density of $16/13$ on $[0,13/32]\cup[19/32,1]$, with no support elsewhere, then $\Delta=0.840$, $\Delta'=0.887$, and the ratio is $0.947$. This is less than the $20/21 = 0.952$ from the uniform distribution.
This may not be close to minimal overall, but it's close to minimal for distributions sup... | 3 | https://mathoverflow.net/users/nan | 373026 | 155,819 |
https://mathoverflow.net/questions/372910 | 2 | At the very beginning of Chapter 11 of Larry Guth's book, we are given the following theorem which is supposed to be proved within the chapter:
Theorem 11.1. There is a constant K so that the following holds. If ℒ is a set of L lines in R^3 with |P\_3(ℒ)| >= KL^(3/2), then there is a plane that contains at least 10L^... | https://mathoverflow.net/users/101271 | Question involving an incidence geometry theorem from Larry Guth's book Polynomial Methods in Combinatorics [2016] | Yes, it can be proven using Theorem 11.7 and in fact it is almost done in the book.
In the Corollary 11.8 it is proven that
There exists constant $C$ s.t. the following holds. Suppose that $\mathfrak{L}$ is a set of $L$ lines in $\mathbb{R}^3$ that contains at most $B$ lines in any plane. If $B\ge L^{1/2}$, then
$$... | 1 | https://mathoverflow.net/users/134387 | 373027 | 155,820 |
https://mathoverflow.net/questions/373043 | 7 | My question pertains to the journal "American Mathematical monthly" published by the MAA.
I wish to ask whether a paper as a part of a PhD thesis (subject: Combinatorics ) can be submitted to the AMM. Like, how does the community take publications in AMM to be? Most articles there are like extensions to Putnam/ IMO o... | https://mathoverflow.net/users/100231 | Can American Math. Monthly be used to publish hard research? | Of course read the description on the AMM web page about what sort of thing they publish. <https://www.maa.org/press/periodicals/american-mathematical-monthly>
*The Monthly's readers expect a high standard of exposition; they look for articles that inform, stimulate, challenge, enlighten, and even entertain. Monthly ... | 18 | https://mathoverflow.net/users/454 | 373061 | 155,826 |
https://mathoverflow.net/questions/373055 | 9 | Let $M$ be a closed complex manifold that is not deformation equivalent to a complex projective manifold.
Can $M$ be orientedly diffeomorphic to a complex projective manifold? What if $M$ is moreover Kähler?
| https://mathoverflow.net/users/nan | Deformation equivalent vs diffeomorphic to projective manifold | I believe the answer is yes and follows from the combination of Theorem 4.6 here <https://arxiv.org/pdf/math/0111245.pdf>
and Theorem 1.3 here <https://arxiv.org/pdf/math/0111245.pdf>
The first result shows that deformations of standard complex tori are complex tori (i.e. $\mathbb C^n/\Gamma$ where $\Gamma\cong \math... | 10 | https://mathoverflow.net/users/943 | 373063 | 155,827 |
https://mathoverflow.net/questions/373049 | 8 | I came across this question when I was discussing the rather wonderful [Devil's Chessboard Problem](https://twitter.com/jamestanton/status/1273607099709186050) with my colleague, Francis Hunt.
We realised that there is a nice connection to a packing question in $(\mathbb{F}\_p)^n$ and I want to ask what is known abou... | https://mathoverflow.net/users/801 | Perfect sphere packings (as opposed to perfect ball packings) |
>
> Question: Is this the only way to construct a perfect sphere packing in a finite vector space?
>
>
>
No. Take the linear space $V$ generated by the following vectors in $\mathbb{F}\_2^8$:
$(0,0,0,1,1,1,1,0)$
$(0,0,1,0,1,1,0,1)$
$(0,1,0,0,1,0,1,1)$
$(1,0,0,0,0,1,1,1)$.
One can see that any two diffe... | 5 | https://mathoverflow.net/users/125498 | 373073 | 155,832 |
https://mathoverflow.net/questions/373053 | 3 | I need to find a solution (all solutions, or at least upper and lower bounds) in positive integer numbers to the system $Ax \ge f$, where $A$ is an integer matrix.
With SageMath, I solved it with the function
```
Polyhedron.integral_points()
```
But, this is very slow and can take about 2-3 hours for a matrix wh... | https://mathoverflow.net/users/108188 | How to find a solution of a large system of linear diophantine inequalities? | Essentially this is an integer linear programming problem (e.g. for finding bounds on a variable, your objective could be to maximize or minimize that variable). Although integer linear programming is NP-complete, there is well-developed software for this which should be quite fast for a problem the size you mentioned.... | 2 | https://mathoverflow.net/users/13650 | 373077 | 155,834 |
https://mathoverflow.net/questions/373084 | 10 | I'm looking for a proof of the existence of the Joyal model structure -- with its usual description -- which uses Cisinski theory directly. The closest thing I know of is Theorem 5.26 of Ara's [Higher quasi-categories vs higher Rezk spaces](https://arxiv.org/abs/1206.4354), but even there it seems he needs to *assume* ... | https://mathoverflow.net/users/2362 | Proof of existence of Joyal model structure via Cisinski theory? | Such a proof is given in Chapter 3 of Cisinski's book [Higher categories and homotopical algebra](http://www.mathematik.uni-regensburg.de/cisinski/CatLR.pdf), see Definition 3.3.7 and Theorem 3.6.1. (Note that Cisinski's proof uses as the interval object not the nerve of the free-living isomorphism, but the simplicial ... | 10 | https://mathoverflow.net/users/57405 | 373087 | 155,835 |
https://mathoverflow.net/questions/373095 | 4 | Let $M$ be a complex projective manifold with an antiholomorphic involution. Can $M$ be defined by equations with real coefficients then?
| https://mathoverflow.net/users/nan | Complex projective manifold with an antiholomorphic involution | **Corrected.** As Robert Bryant points out, it is not enough to show that the manifold can be realised as a submanifold of $\mathbb CP^n$ invariant under some anti-holomorphic involution of $\mathbb CP^n$. For this reason the answer is extended.
---
Let $\sigma: M\to M$ be the anti-holomorphic involution. Take a ... | 8 | https://mathoverflow.net/users/943 | 373099 | 155,838 |
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