parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
|---|---|---|---|---|---|---|---|---|---|
https://mathoverflow.net/questions/356539 | 7 | There are some known criteria for the Sturm-Liouville Problem
\begin{equation} \tag{1}
\frac {\mathrm {d} }{\mathrm {d} x}\left[p(x){\frac {\mathrm {d} y}{\mathrm {d} x}}\right]+q(x)y=-\lambda w(x)y
\end{equation}
to have a spectrum discrete and bounded below (BD) in the singular case:
If singular endpoints are Lim... | https://mathoverflow.net/users/124314 | Singular Sturm-Liouville problems: criterion for discrete spectrum for zero potential ($q=0$) and Hermite Polynomials | The case $q=0$ actually has a complete answer, and the reason this works is that we can solve the equation for $\lambda =0$ explicitly then, by $u=1$ and $v=\int\_0^x \frac{dt}{p(t)}$. The spectrum is purely discrete if and only if ($w\in L^1$ and)
$$
\lim\_{x\to\infty}\int\_0^x wv^2\, dt \int\_x^{\infty} w\, dt =0 . \... | 3 | https://mathoverflow.net/users/48839 | 356592 | 150,412 |
https://mathoverflow.net/questions/356563 | 1 | Let $\Sigma$ be an embedded smooth surface in $\mathbb{R}^3$, and let $f:\Sigma\to\mathbb{R}$ be a smooth function. Suppose $f$ is square-integrable on $\Sigma$, with
\begin{align}
0<\int\_{\Sigma}f^2d\mu<\infty
\end{align}
where $d\mu$ is the area element of $g$, the induced metric on $\Sigma$ from the flat metric of ... | https://mathoverflow.net/users/137708 | Continuity of $r\mapsto\int_{\Sigma\cap B_r(x)}f^2d\mu$ | The function $\lambda:A\mapsto\int\_{\Sigma}{\mathbf 1}\_{A}f^2\mathrm d\mu$ is a measure on the Borel sets of $\mathbb R^3$, for ${\mathbf 1}\_A$ the indicator function of $A$. The quantity you are interested in is $r\mapsto\lambda(B\_r(x))$.
We can use the monotone convergence theorem to see that
$$ \lim\_{r\uparro... | 1 | https://mathoverflow.net/users/129074 | 356600 | 150,415 |
https://mathoverflow.net/questions/356573 | 2 | Consider the initial value problem
$$ \partial\_t u = \Delta u$$
$$ u(0,x) = 0$$
for the heat equation in $\mathbb R^n$, where $u: [0,T] \times \mathbb R^n \to \mathbb R$ is a smooth solution up to the time $T$. Suppose there exists a large constant $N>0$ such that
$$
|u(t,x)| \le N\exp(N|x|^2t^{-1})
$$
for any $x\in \... | https://mathoverflow.net/users/105900 | The solutions of the heat equation from $0$ datum | Have you tried the classical Tychonoff's example, $$u(t,x) = \sum\_{k = 0}^\infty \frac{g^{(k)}(t) x^{2k}}{(2k)!},$$ with $g(t) = e^{-1/t^\alpha}$ and $\alpha > 1$?
As discussed, for example, in [these lecture notes](https://users.math.msu.edu/users/yanb/847ch3.pdf), in this case $$|g^{(k)}(t)| \leqslant \frac{k!}{(\... | 8 | https://mathoverflow.net/users/108637 | 356601 | 150,416 |
https://mathoverflow.net/questions/356231 | 7 | I have a series of vague questions, related to localization of symmetric monoidal categories.
Here is the context. Say we are working over a field of characteristic zero. Then the "one category level higher" version of (DG) commutative ring is a (DG) symmetric monoidal category. It is well-known that for $X$ a schem... | https://mathoverflow.net/users/7108 | Localization of symmetric monoidal categories and geometry | Given a symmetric monoidal functor $\mathcal{C} \rightarrow \mathcal{C}'$, the property that $\mathcal{C}' \otimes\_{\mathcal{C}} \mathcal{C}' \rightarrow \mathcal{C}'$ be an isomorphism is equivalent to $\mathcal{C} \rightarrow \mathcal{C}'$ being an epimorphism in the category of symmetric monoidal dg categories. Thi... | 4 | https://mathoverflow.net/users/145919 | 356605 | 150,417 |
https://mathoverflow.net/questions/356561 | -1 | For any topological space $(X,\tau)$, let $\text{End}(X)$ denote the set of continuous functions $f:X\to X$. We say that ${\cal C}\subseteq \text{End}(X)$ *covers* $\text{End}(X)$ if for every $f\in \text{End}(X)$ there is $h\in {\cal C}$ and $x\in X$ such that $f(x) = h(x)$.
I am interested in *minimal* covers (tha... | https://mathoverflow.net/users/8628 | Minimal covering sets of continuous endomorphisms | Here is an example:
**Example.** Let $g$ be a continuous strictly increasing function such that $\lim\_{x \to -\infty} g(x) = -1$ and $\lim\_{x \to \infty} g(x) = 1$; for example
$$g(x) = \tfrac{2}{\pi}\arctan(x).$$
For $n \in \mathbf Z$, let $f\_n(x) = n$ and $g\_n(x) = g(x) + n + \tfrac{1}{2}$. Then $\{f\_n\} \cup ... | 1 | https://mathoverflow.net/users/82179 | 356609 | 150,418 |
https://mathoverflow.net/questions/356608 | 2 | Let $X\to S$ be a morphism of smooth connected varieties over an algebraically closed field $k$; let $j:\eta\to S$ be the inclusion of the generic point into $S$ (*not* a geometric generic point) and let $\mathscr F$ be a constructible étale sheaf of $\Lambda$-modules on $X\_\eta$. If $\Lambda=\mathbf{Z}/\ell^n\mathbf{... | https://mathoverflow.net/users/155654 | Help with $\mathbf{Q}_{\ell}$ sheaves | A necessary condition is that for each stratum $X\_{\eta}^i$ of some stratification of $X\_{\eta}$ on which $\mathcal F$ is lisse, the associated representation of $\pi\_1(X\_{\eta}^i)$ is unramified away from a closed subset $Z^i$ of the closure $\overline{X\_\eta^i}$ of $X\_{\eta}^i$ in $X$, such that the projection ... | 5 | https://mathoverflow.net/users/18060 | 356611 | 150,419 |
https://mathoverflow.net/questions/356593 | 0 | Let $P$ and $Q$ are sub-Gaussian distributions on $\mathbb R$, and $(X,X')$ be a coupling of $P$ and $Q$, i.e $(X,X') \sim \pi$ for some distribution on $\mathbb R^2$ with marginals $P$ and $Q$.
>
> **Question.** Does $|X-X'|$ have any concentration properties ?
> Can one reasonably bound $\mathbb P(|X-X'| > \epsi... | https://mathoverflow.net/users/78539 | Tail bounds for the absolute difference of a coupled pair of sub-Gaussian random variables | For $X\_1:=X$, $X\_2:=X'$, some positive real $c\_1,c\_2,a\_1,a\_2$, and all positive real $t$ we have
$$P(|X\_j|>t)\le c\_j e^{-a\_j t^2} \tag{1}$$
for $j=1,2$. So, for $t:=\epsilon>0$,
$$P(|X\_1-X\_2|>t)\le P(|X\_1|>t/2)+P(|X\_2|>t/2) \\
\le c\_1 e^{-a\_1 t^2/4}+c\_2 e^{-a\_2 t^2/4}.$$
---
The OP commented th... | 1 | https://mathoverflow.net/users/36721 | 356615 | 150,422 |
https://mathoverflow.net/questions/356623 | 4 | Recall that a chain complex $(C\_\*,d)$ of abelian groups is contractible if it is homotopic to the zero map. Or equivalently: there exists a degree 1 map $F: C\_\* \to C\_\*$ such that $\operatorname{Id}= dF+ Fd$.
Question: does there exist a topological space $X$ which is not contractible (in the sense of topology... | https://mathoverflow.net/users/155668 | Contractible chain complex from non-contractible space | These are known as [*acyclic spaces*](https://en.wikipedia.org/wiki/Acyclic_space) (note that since $\tilde C\_\*(X)$ is a bounded below complex of projectives, it being contractible is equivalent to its homology being trivial).
There's an extensive literature about them, starting with this [Emanuel Farjoun's paper](... | 9 | https://mathoverflow.net/users/43054 | 356625 | 150,423 |
https://mathoverflow.net/questions/356627 | 2 | I have encounter the following problem, but after trying a little I did not arrive to a good conclusion.
Suppose that $X$ is a positive random variable for which we only know that $E[X] = 2$ and $E[1/X] = 1$. We want an upper bound for $P[X \geq a]$, for $a > 2$, which must be better than $2/a$, obtained by Markov's ... | https://mathoverflow.net/users/46573 | An elementary question on probability distributions | Let $Y = X + 1/X - 2$. Then $Y \geqslant 0$ and $\mathbb{E} Y = 2 + 1 - 2 = 1$. Thus,
$$ \mathbb{P}(Y > y) \leqslant \frac{\mathbb{E} Y}{y} = \frac{1}{y} $$
for every $y > 0$. If we choose $y = a + 1/a - 2$ for some $a > 1$, then
$$ \mathbb{P}(X > a) \leqslant \mathbb{P}(X > a) + \mathbb{P}(X < 1/a) = \mathbb{P}(Y > y)... | 1 | https://mathoverflow.net/users/108637 | 356630 | 150,425 |
https://mathoverflow.net/questions/356661 | 3 | For expectation (mean), there are many useful properties such as Linearity of Expectation:
* $\mathbb{E}[X+Y]=\mathbb{E}[X]+\mathbb{E}[Y]$
* $\mathbb{E}[\alpha X]=\alpha\mathbb{E}[X]$
(The 2 equations above can be proved either by [definition](https://math.stackexchange.com/a/1810371/333208) or [convolution](https:... | https://mathoverflow.net/users/116390 | Prove or disprove the linearity of expectiles | The $\tau$-expectile, say $E\_\tau X$, of a random variable (r.v.) $X$ is the root $t$ of the equation
$$r\_X(t)=\rho(\tau),$$
where
$$r\_X(t):=\frac{E(X-t)\_+}{E(t-X)\_+}, \quad \rho(\tau):=\frac{1-\tau}\tau.$$
For any real $a>0$, we have $r\_{aX}(at)=r\_X(t)$, whence
$$E\_\tau(aX)=a\,E\_\tau X.$$
For any real $a... | 8 | https://mathoverflow.net/users/36721 | 356667 | 150,436 |
https://mathoverflow.net/questions/356642 | 9 | Suppose $e : \mathbb R \to F$ is an elementary embedding in the language of ordered fields. Can there exist an elementary embedding $e' : \mathbb R \to F$ such that $e \not= e'$? Note that it would have to be the case that for some undefinable $r \in \mathbb R$, $e(r) \not= e'(r)$, and $e(r) - e'(r)$ is infinitesimal.
... | https://mathoverflow.net/users/11145 | Can nonstandard fields contain $\mathbb R$ in different ways? | Yes, this is possible. Let $F$ be a nonarchimedean strongly $\omega$-homogeneous real-closed field such that $\mathbb R\subseteq F$ (which exists by model-theoretic general reasons). Fix an infinitesimal element $\epsilon>0$. By quantifier elimination for real-closed fields, $\pi$ and $\pi+\epsilon$ have the same type ... | 13 | https://mathoverflow.net/users/12705 | 356669 | 150,437 |
https://mathoverflow.net/questions/356673 | 5 | Let $G$ be a finite group and let $k$ be a finite field with char$(k)=p$ such that $p\mid |G|$.
If $k$ is a splitting field for $G$, then, no matter which splitting field we take, after extending scalars we always have the same number of isomorphism classes of simple modules.
However, it is not necessarily true tha... | https://mathoverflow.net/users/12826 | How to find a finite splitting field $K$ for $G$ such that every indecomposable $KG$-module is absolutely indecomposable | No.
Let $G=C\_2\times C\_2$, generated by elements $g$ and $h$, and let $K$ be any finite field of characteristic $2$.
Let $V$ be a finite dimensional $K$-vector space of dimension greater than one with an endomorphism $\varphi$ with irreducible characteristic polynomial $p(t)$.
Then $V\oplus V$ can be made into ... | 8 | https://mathoverflow.net/users/22989 | 356678 | 150,439 |
https://mathoverflow.net/questions/356693 | 8 | Let $F : \mathcal{C} \to \mathbf{Cat}$ be a lax 2-functor. Then we can form a category $\int F $ which is the Grothendieck construction on F.
There's a number of resources detailing this construction, but none mentioning if we can get a 2-category, rather than a category this way. It seems like the natural construction... | https://mathoverflow.net/users/127778 | Does the Grothendieck construction produce a 2-category or a category? | The usual Grothendieck construction has for $\mathcal C$ an ordinary category, so it doesn't have any 2-cells (or at least, doesn't have any non-identity 2-cells). Moreover, we actually get not only a category out of it, but an object of the slice (2-)category $\mathcal Cat/\mathcal C$. If $\mathcal C$ weren't an ordin... | 9 | https://mathoverflow.net/users/62780 | 356697 | 150,443 |
https://mathoverflow.net/questions/356692 | 2 | The following inequality appears in the proof of certain isoperimetric-type inequalities for analytic functions in two dimensions:
$$\sum\_{m=0}^{\infty}\frac{|c\_m|^2}{m+1} \leq \pi \left(\sum\_{m=0}^{\infty}|a\_m|^2 \right)^2,$$
where
$$c\_m=a\_0a\_m+a\_1 a\_{m-1}+ \dots +a\_ma\_0.$$
It sounds like a basic ineq... | https://mathoverflow.net/users/115905 | Proof of a discrete isoperimetric inequality | Suppose you have a power series with coefficients $a\_n$
$$ f(z):= \sum\_{k=1}^\infty a\_k z^k .$$
Then the coefficients of $f^2$ are exactly $c\_n$. Also if we denote by $\odot$ the Hadamard multiplication of powerseries (coefficient-wise or equivalently convolution of the boundary values), $c\_k^2$ are the coefficien... | 4 | https://mathoverflow.net/users/153260 | 356699 | 150,444 |
https://mathoverflow.net/questions/355509 | 5 | In complex geometry, various vanishing theorems for cohomology
groups of a hermitian line bundle E over a compact complex manifold X have been found.
My question is
Is there some vanishing theorems over a **general noncompact** complex manifold exist? (Except shose on Stein manifolds)
| https://mathoverflow.net/users/124749 | Vanishing theorems on a non-compact manifold | A complex manifold of dimension $n$ is non-compact if and only if $H^n(X,{\mathcal F})=0$ for any coherent sheaf ${\mathcal F}$ on $X$. This is the only general vanishing result that I know of on non-compact manifolds.
But other than that, there are some other vanishing theorems on non-compact manifolds. For instanc... | 5 | https://mathoverflow.net/users/48958 | 356701 | 150,446 |
https://mathoverflow.net/questions/356704 | 3 | Let $f$ be an anylytic function on the unid disk $|z|<1$. It is well known that
$$\left (\int\_0^{2\pi}f(e^{i\theta})d \theta \right)^2 \geq 4\pi \iint\_{|z|<1} |f(r e^{i\theta})|^2r dr d \theta.$$
I wonder if the constant $4\pi$ cound be improved on an annulus $a<z<1$. More precisely, does the following inequality ... | https://mathoverflow.net/users/115905 | Isoperimetric inequality for analytic functions on an annulus | You can probably prove that
$$ \Big( \int\_\mathbb{A\_r} |f(z)|^2 \frac{dxdy}{\pi(1-r^2)} \Big)^{1/2} \leq \int\_{ \mathbb{T}} |f(e^{i\theta})| \frac{d\theta}{2\pi}+\int\_{ \mathbb{T\_r}} |f(re^{i\theta})| \frac{d\theta}{2\pi r}, \,\,\, \forall f\in H^1(\mathbb{A\_r}). $$
Where $\mathbb{A\_r} $ is the annulus of inter... | 0 | https://mathoverflow.net/users/153260 | 356722 | 150,450 |
https://mathoverflow.net/questions/356721 | 1 | Suppose that $A$ is a subset of a (large) finite cyclic group such that $|A|=5$ and $|A+A|=12$. Given that $g$ is a group element with $g+A\subseteq A+A$, can one conclude that $g\in A$?
| https://mathoverflow.net/users/9924 | Does $g+A\subseteq A+A$ imply $g\in A$? | This is not the case. Here is a counter example.
Let $G =Z\_m, \ \ m>20$ (say). Let $A = \{0, 1, 3, 4, 6\}; g=2$: Note that $g+A=\{2, 3, 5, 6, 8\} \subseteq \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12\}=A+A$ and that $g \not\in A$, $|A|=5$, and $|A+A|=12$. Note that this example works for all large finite cyclic groups, a... | 14 | https://mathoverflow.net/users/36707 | 356727 | 150,451 |
https://mathoverflow.net/questions/356725 | 0 | I'm currently reading the paper *["Random matrices: The distribution of the smallest singular values" by '"Terence Tao and Van Vu"](https://link.springer.com/article/10.1007/s00039-010-0057-8)* and have run into some terminology which I don't quite (rigorously) understand.
In Theorem 1.3, the authors state that $\ma... | https://mathoverflow.net/users/36886 | Terminology: "sufficiently large absolute constant" | "Absolute constant" means that it does not depend on anything. For example, $3, 10^{12},\pi$ and Feigenbaum number are absolute constants. They are real numbers. "Sufficiently large" means that
the authors did not care or could not compute or estimate it.
| 6 | https://mathoverflow.net/users/25510 | 356728 | 150,452 |
https://mathoverflow.net/questions/356492 | 8 | It is a very well-known fact that any conservation law associated with some given PDE has an associated invariance (by Noether's Theorem). However, it is completely mysterious for me how to compute/derive these conservation laws just by knowing the invariances of the equation. For example, the one-dimensional nonlinear... | https://mathoverflow.net/users/129131 | How to find the associated conservation law from a given symmetry | To be blunt, the answer to your question **is** [Noether's theorem](https://en.wikipedia.org/wiki/Noether%27s_theorem) (often precised as Noether's *first* theorem). So, essentially you already knew the answer to your own question.
However, the other answers are missing a degree of pragmatism. The calculation of the ... | 9 | https://mathoverflow.net/users/2622 | 356734 | 150,453 |
https://mathoverflow.net/questions/356306 | 10 | There might be an obvious answer to the question, but it doesn't come to mind.
Suppose we have an infinite matrix $A=(a\_{ij})$, which defines a bounded linear operator on $\ell^p$, i.e. for all sequences $(x\_i)\in \ell^p, p>1$
$$ \sum\_{i=1}^\infty\big|\sum\_{j=1}^\infty a\_{ij}x\_j\Big|^p \leq C \Vert x \Vert\_{p}... | https://mathoverflow.net/users/153260 | Is the spectrum of a "self adjoint" operator real on $\ell^p$? | It seems that I have found a counter example myself.
For the Hilbert matrix
$$ H\_\lambda:= \big( \frac{1}{1-\lambda+k+n} \big)\_{k,n\geq 0}, \lambda < 1 $$
Rosenblum in "On the Hilbert Matrix I, Proceedings of the AMS" proves that the pointspectrum considered as an operator on $\ell^p, p>2$ contains the set
$$ \{... | 4 | https://mathoverflow.net/users/153260 | 356735 | 150,454 |
https://mathoverflow.net/questions/356737 | 3 | Suppose that $E$ is a Polish space.
Portmanteau theorem asserts that a sequence $(\mu\_n)$ of Borel probability measures weakly converges to a Borel probability measure $\mu$ (shortly, $\mu\_n\overset{w}{\to\mu}$) if and only if $\limsup\_n \mu\_n(C)\le \mu(C)$ for all closed set $C\subset E$. My question is whether... | https://mathoverflow.net/users/154110 | Countable convergence-determining class for weak convergence of probability measures | Let $E$ be any separable space. The assertion is equivalent to ($\*$)
$\liminf\_{n\to \infty} \mu\_n(U) \geq \mu(U)$ for any open $U \subset E$. Since $E$ is separable, there is a countable base $\cal{U}$, which is $\cap$- and $\cup$-stable, for the open sets in $E$. This $\cal{U}$ is "convergence-determining" for open... | 1 | https://mathoverflow.net/users/100904 | 356742 | 150,458 |
https://mathoverflow.net/questions/356732 | 12 | Let $K$ be a compact subset of $\mathbb{R}^n$ (for simplicity, I am happy to take $K=\overline{B(0,1)}$ for now if it is easier).
Let $f:K \rightarrow \mathbb{R}^m$ be a **continuous** function.
Is there a **new metric** $d$ on $\mathbb{R}^m$ (compatible with the topology) such that
$$ f: K \rightarrow (\mathbb{R}^... | https://mathoverflow.net/users/155731 | Recognizing Lipschitz functions up to change of target metric | It can't be done. Here's a counterexample with $n = 2$ and $m = 1$.
For each $k \in \mathbb{N}$, define a function $f\_k: [0,1] \to [0,1]$ by linearly interpolating between the values $f\_k(\frac{i}{k}) = \frac{i}{k}$ ($0 \leq i \leq k$) and $f\_k(\frac{i}{k} + \frac{1}{k^2}) = \frac{i+1}{k}$ ($0 \leq i \leq k -1$). ... | 9 | https://mathoverflow.net/users/23141 | 356748 | 150,460 |
https://mathoverflow.net/questions/356741 | 5 | The Cuntz algebra $O\_n$ is the (universal) C\*-algebra generated by n-isometries $s\_1,...,s\_n$ such that
$$\sum\_{i=1}^n s\_is\_i^\ast =\mathbf{1}, \ \hbox{and}\ s\_i^\ast s\_j=\delta\_{ij} \mathbf{1}\ (\hbox{for all}\ i,j).$$
For example if $s\_1, s\_2$ are the generators of $O\_2$ then $s\_1^2, s\_1s\_2, s\_2$ ... | https://mathoverflow.net/users/13643 | Embedding of Cuntz algebras $O_2\subseteq O_3$? | The answer is no. According to Lemma 2.1 in (K. Kawamura, Universal algebra of sectors, Int. J. Alg. Comput. 19(3)(2009), 347–371.) we have:
$Hom(O\_m, O\_n)$ is nonempty if and only if $m=(n-1)k+1$ for some $k \geq 1$.
Applying this to the case for $Hom(O\_2, O\_3)$ gives that we must have $2=2k+1$, which is impos... | 6 | https://mathoverflow.net/users/149852 | 356760 | 150,463 |
https://mathoverflow.net/questions/356758 | 1 | I know about the Plancherel measure, but I don't know where the term "Plancherel density" is defined.
| https://mathoverflow.net/users/64244 | What is the definition of Plancherel density? | the Plancherel density is derived from the Plancherel measure, see [arXiv:1812.00047](https://arxiv.org/abs/1812.00047) for the precise definition:
| 5 | https://mathoverflow.net/users/11260 | 356764 | 150,464 |
https://mathoverflow.net/questions/356766 | 3 | Let $L$ be perfectoid field of characteristic $p$ and $L'$ be a finite extension of $L$. Then I want to prove the trace map $\text{Tr}\_{L'/L}: m\_{L'}\rightarrow m\_L$ is surjective. I find a proof in Kedlaya' paper "On categories of $(\phi,\Gamma)$-modules", but I can't understand it. The proof is as follows:
**The... | https://mathoverflow.net/users/nan | perfectoid field of characteristic $p$ | Write $I$ for the image of $\operatorname{Tr}\_{L'/L} \colon \mathfrak m' \to \mathfrak m$, which is an ideal because $\operatorname{Tr}\_{L'/L}$ is $\mathcal O\_L$-linear.
(1) The first sentence is just the non-discreteness of the valuation. This forces $\mathfrak m^2 = \mathfrak m$, so if $I \subseteq \mathfrak m$ ... | 3 | https://mathoverflow.net/users/82179 | 356785 | 150,468 |
https://mathoverflow.net/questions/356778 | 5 | There are loads of concentration results for sums of scalar-valued independent sums $X\_1,X\_2,\ldots, X\_N$ with $\mathbb E[X\_n]=0$. For example Hoeffding's Inequality says if all $|X\_1|\le 1$ then $\mathbb E\left[ \left|\sum\_{i=1}^N X\_i\right|\right] = O (\sqrt N)$.
These concentration results can be [generali... | https://mathoverflow.net/users/58082 | Is there an i.i.d sequence in the unit cube $[-1,1]^d$ with $\mathbb E \left[ \Big \| \sum_{i=1}^N X_N \Big \|_\infty\right] = \sqrt {dN}$? | Let $X\_i=(X\_{i,1},\dots,X\_{i,d})$, $S:=(S\_1,\dots,S\_d)$, $S\_j:=\sum\_{i=1}^d X\_{i,j}/\sqrt n$. Then, by [Hoeffding's inequality](https://en.wikipedia.org/wiki/Hoeffding%27s_inequality#General_case_of_bounded_random_variables), for $s\ge0$
$$P(|S\_j|\ge s)\le2e^{-s^2/2},$$
whence
$$E\|S\|\_\infty=\int\_0^\infty ... | 7 | https://mathoverflow.net/users/36721 | 356786 | 150,469 |
https://mathoverflow.net/questions/356776 | 12 | Let $M$ be an oriented three-manifold with $\partial M$ a torus. Suppose that two different Dehn fillings $M(r)$ and $M(r')$ are (oriented) homeomorphic to a lens space $L(p,q)$.
Does that imply that $M$ is a solid torus?
| https://mathoverflow.net/users/23935 | Two Dehn fillings yielding the same lens space? | This is a case of the oriented knot complement problem in lens spaces, also called the *cosmetic surgery* problem. If $M(r)\cong M(r')$ preserving orientation, then these form a *purely cosmetic* (or *truly cosmetic*) pair. See Problem 1.81 (A) from [Kirby's problem list](https://math.berkeley.edu/%7Ekirby/problems.ps.... | 14 | https://mathoverflow.net/users/1345 | 356793 | 150,470 |
https://mathoverflow.net/questions/356759 | 2 | In an attempt to compute cycle counts in an of a certain number theoretic graph, the following estimate was needed.
It is true that
$$\bigg|\sum\_{a,b,c\in \mathbb{Z}/p\mathbb{Z}}\bigg(\sum\_{d=1}^{p-1}\bigg(\frac{d}{p}\bigg)w^{-d-(a^2+b^2+c^2)/d}\bigg)^3\bigg| = o(p^{9/2}).$$
$w$ here is $\exp\big(\frac{2\pi i}{p}\big... | https://mathoverflow.net/users/142352 | Cancellation in a particular sum | We can actually get an explicit formula for the whole sum (I will assume $p \ne 2,3$ throughout). We start with the Salié sum:
$$\sum\_{d=1}^{p-1}\bigg(\frac{d}{p}\bigg)w^{-d-(a^2+b^2+c^2)/d} = \bigg(\sum\_d \bigg(\frac{d}{p}\bigg)w^{-d}\bigg) \sum\_{x^2 \equiv 4(a^2+b^2+c^2)} w^x.$$
The first factor is a Gauss sum... | 6 | https://mathoverflow.net/users/2363 | 356805 | 150,474 |
https://mathoverflow.net/questions/356806 | 6 | Let $(D, \succeq)$ be a directed set, and let $B$ be the space of real-valued bounded functions on $D$. A *Banach limit* $\ell$ on $D$ is a linear functional that satisfies
$$\sup\_{d \in D} \inf\_{c \succeq d} f(c) \leq \ell(f) \leq \inf\_{d \in D} \sup\_{c \succeq d} f(c)$$
for all $f \in B$.
Banach limits exist by... | https://mathoverflow.net/users/96899 | Do multiplicative Banach limits exist? | I do not think that this is the usual definition of [Banach limit](https://en.wikipedia.org/wiki/Banach_limit). (What I know under this name is linear functional on $l\_\infty$ which is positive, shift-invariant and extends the usual limit, see the linked Wikipedia article. Of course, the terminology in various sources... | 9 | https://mathoverflow.net/users/8250 | 356809 | 150,475 |
https://mathoverflow.net/questions/356595 | 3 | A vector bundle over a complex manifold is said to be **holomorphic** if its trivialization maps are biholomorphic maps. What is a "natural" example example of a vector bundle over compact complex manifold which is not holomorphic? I guess by "natural" I mean that one would be interested in such examples for reasons be... | https://mathoverflow.net/users/125941 | Vector bundle over compact complex manifold which is not holomorphic? | I interpret the question as "Are there any complex bundles over the complex manifold which do not admit the holomorphic structure". The answer to this is "yes", but I'm not sure if it is known in full generality for what manifolds the answer is positive and for what it is negative.
**Line bundles case.** Suppose X is... | 4 | https://mathoverflow.net/users/33286 | 356816 | 150,477 |
https://mathoverflow.net/questions/356762 | 14 | I want to find explicit formulas for the eigenfunctions of the Laplacian on $\mathbb{CP}^n$ endowed with the Fubini Study metric.
For the first eigenvalue $\lambda\_1 = 4(n+1)$, the eigenfunctions are given by the real and imaginary parts of $\phi^{i, j} = \frac{z\_i\bar{z\_j}}{\sum\_{k}|z\_k|^2}-\frac{\delta\_{i, j... | https://mathoverflow.net/users/104334 | Eigenfunctions of the laplacian on $\mathbb{CP}^n$ | The $k$-th eigenfunctions are actually easy to describe: In $\mathbb{C}^{n+1}$ with unitary complex coordinates $z\_0,z\_1,\ldots,z\_n$, write $Z = |z\_0|^2+\cdots+|z\_n|^2$.
Now, for a given $k\ge0$, let $H\_k$ be the (real) vector space of real-valued polynomials $p(z,\bar z)$ that are homogeneous of degree $k$ in... | 23 | https://mathoverflow.net/users/13972 | 356822 | 150,479 |
https://mathoverflow.net/questions/356618 | 27 | [Pyknotic](https://arxiv.org/abs/1904.09966) and [condensed sets](https://www.math.uni-bonn.de/people/scholze/Condensed.pdf) have been introduced recently as a convenient framework for working with topological rings/algebras/groups/modules/etc. Recently there has been much (justified) excitement about these ideas and t... | https://mathoverflow.net/users/130058 | What is the precise relationship between pyknoticity and cohesiveness? | The work on analytic geometry is all joint with Dustin Clausen!
Your main question seems a little vague to me, but let me try to get at it by answering the subquestions. See also the discussion at the nCatCafe. Also, as David Corfield comments, much of this had been observed long before: <https://nforum.ncatlab.org/d... | 28 | https://mathoverflow.net/users/6074 | 356836 | 150,484 |
https://mathoverflow.net/questions/356832 | 4 | $R$ a ring $(1\neq 0)$, $\mathbf{Perf}(R)$ is the category of perfect complexes (of right $R$-modules).
Suppose that $A\_{\bullet}\rightarrow B\_{\bullet}\rightarrow B\_{\bullet}/A\_{\bullet}$ a short exact sequence in $\mathbf{Perf}(R)$ such that
1. $A\_{\bullet}\rightarrow B\_{\bullet}$ is a cofibration in the p... | https://mathoverflow.net/users/128371 | Split cofibrations up to quasi-isomorphism | No. You can construct counterexamples by taking projective resolutions of modules in a nonsplit short exact sequence. For example, from the short exact sequence $0\to\mathbb{Z}\stackrel{2}{\to}\mathbb{Z}\to\mathbb{Z}/2\mathbb{Z}\to0$ of abelian groups you get
$$\require{AMScd}
\begin{CD}
&&0&&0&&0\\
&&@VVV&@VVV&@VVV\\
... | 7 | https://mathoverflow.net/users/22989 | 356838 | 150,485 |
https://mathoverflow.net/questions/356781 | 4 | Let $A$ be a symmetric matrix in $\mathbb R^n$ such that $A$ is positive definite and hence satisfies $0< \lambda \le A \le \Lambda < \infty.$
Let $T$ be a densely defined and closed operator from some Hilbert space $H$ into $H^n$. It is a classical theorem by John von Neumann that $T^\*T$ is self-adjoint with domain... | https://mathoverflow.net/users/150549 | Elliptic estimates for self-adjoint operators | As I understand your question now, after edit, in dimension two you ask whether
$$
\|T\_2^\* T\_1 x\| \leqslant C(\|T\_1^\* T\_1 x + T\_2^\* T\_2 x\| + \|x\|)
$$
whenever $T\_1$, $T\_2$ are densely defined closed operators; $C$ can depend on $T\_1$ and $T\_2$.
This need not be the case. Let $T\_1$ be the identity op... | 2 | https://mathoverflow.net/users/108637 | 356876 | 150,492 |
https://mathoverflow.net/questions/356884 | 14 | Given a commutative associative unital algebra over a field of characteristic zero.
>
> Is it true that any derivation of it preseves its nil-radical?
>
>
>
More explicitly, let $D$ be a derivation of an algebra $A$. Let $N$ denote the nil-radical of $A$.
>
> Is it true that $D(N)\subset N?$
>
>
>
| https://mathoverflow.net/users/16183 | Does any derivation of commutative algebra preserve its nil-radical? | Suppose $x\in N$, so that $x^n=0$ for some $n$. Then using the product rule for derivations many times, we see that
$$
0=D^n(x^n)=n! D(x)^n+Y,
$$
where $Y$ is divisible by $x$. Therefore,
$D(x)^{n^2}=(D(x)^n)^n$ is divisible by $x^n$, and therefore vanishes. Thus, $D(x)$ is nilpotent, and therefore $D(N)\subset N$.... | 24 | https://mathoverflow.net/users/1306 | 356885 | 150,493 |
https://mathoverflow.net/questions/356895 | 5 | Suppose $f\colon\mathbb{R}^{\omega}\longrightarrow\mathbb{R}$ is a function such that
$$G(f):=\{(x,y)\in\mathbb{R}^{\omega}\times\mathbb{R}\mid f(x)=y\}$$
is a Borel set. Does it necessarily follow that $f$ is a Borel function?
| https://mathoverflow.net/users/138274 | Function whose graph is a Borel relation | Yes. This immediately follows from Lusin's separation Theorem. Note that $\mathbb{R}^\omega$ is Polish.
Let $B \subseteq \mathbb{R}$ be a Borel set. The set $f^{-1}(B)=\{x \in \mathbb{R}^\omega \colon \exists y \,\,\, y \in B \land (x,y) \in G(f) \}$ is $\Sigma\_1^1$ (analytic). However, $f^{-1}(B)=\{x \in \mathbb{R}... | 11 | https://mathoverflow.net/users/134910 | 356908 | 150,499 |
https://mathoverflow.net/questions/356855 | 0 | I am reading Harold Widom's paper "Extremal Polynomials Associated with a System of Curves in the Complex Plane". At the beginning of section 11 he states that:
>
> [There is] a simple transformation which opens up the arc. Suppose $E$ is a Jordan arc which for simplicity we assume has endpoints $\pm 1$. Then if $(... | https://mathoverflow.net/users/115263 | Transformation which "opens up" an arc | I think this an issue of choice of branch of square root. Note that the paper asks you to use the branch holomorphic in $\mathbb{C} \setminus E$ and behaves as $z$ as $z \to \infty$.
Let's call this choice of branch $w(z)$ and call the compact set delimited by $[-1, 1] \cup E$ by $K$. Then the map we are looking for... | 1 | https://mathoverflow.net/users/124887 | 356919 | 150,502 |
https://mathoverflow.net/questions/356916 | 0 | I wonder whether this fact is true or not (if a counter-example exists, please just give a hint on how to construct it!):
Let $\Omega$ be a bounded domain in $\mathbb{R}^n$. Consider a bounded sequence $\{f\_n\}$ in $L^\infty\left([0,T];L^2(\Omega)\right)$. Suppose I also know that $\{f\_n\}$ is in $L^2\left([0,T];L^... | https://mathoverflow.net/users/105925 | Convergence in $L^\infty([0,T];L^2(\Omega))$ | It sounds like you are new to this subject, so there are some basic things for you to know. The first is that by night all Hilbert spaces are the same, so your $L^2(\Omega)$ can just be a generic $H$. Secondly, $L^\infty[0,T]$ is contained in $L^2[0,T]$ and the norm is smaller in $L^2$ up to a factor of $\sqrt{T}$, and... | 5 | https://mathoverflow.net/users/23141 | 356920 | 150,503 |
https://mathoverflow.net/questions/356823 | 1 | Consider $\Psi(x)$ to be the Chebyshev function given by
$$\Psi(x)=\sum\_{n\leq x} \Lambda(n)$$
where $\Lambda(n)$ is the Mangoldt function which is equal 0 unless $n $ is prime power, and let $(E)$ be the following equation:
$$\Psi(n!)=\Psi(A) + \Psi(A+2) \tag E \, ,$$
where $n , A $ are integers and $A$ is e... | https://mathoverflow.net/users/143969 | Equation of the Chebyshev $\psi$ function | This solution builds on @Wojowu's comment. The Mangoldt function $\Lambda(n)$ is defined as $\log p$ if $n=p^k$ is a prime power and as zero otherwise. The Chebyshev function
$$
\Psi(x)=\sum\_{n\leq x}\Lambda(n)
$$
is thus the logarithm of the least common multiple of $1,2,\dots,\lfloor x\rfloor$ because it could be wr... | 5 | https://mathoverflow.net/users/128556 | 356927 | 150,507 |
https://mathoverflow.net/questions/356931 | 2 | Looking at the von Neumann–Bernays–Gödel (NBG) axioms of Set Theory in [Wikipedia](https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory) I noticed that it has two axioms of permutation: one for circular permutations and another for transpositions.
On the other hand, it is well-know... | https://mathoverflow.net/users/61536 | In the von Neumann–Bernays–Gödel axiomatic system, the Axiom of Transposition can be simplified | I believe that your intended question was something along the lines of whether or not the standard axioms can be simplified or reduced. This is in the spirit of Exercise 13.4 in Jech's "[Set Theory](https://link.springer.com/book/10.1007%2F978-3-662-22400-7)", where one shows that some of the usual Gödel operations are... | 4 | https://mathoverflow.net/users/3199 | 356939 | 150,512 |
https://mathoverflow.net/questions/356897 | 1 | It is well-known that the (covering) dimension of countable union of compact spaces is the superium dimension of these spaces. I would like to understand certain uncountable union of compact spaces as follows. Let $b>a$. Let $\{X\_r\}\_{r\in [a,b]}$ be a family of compact spaces with index $r\in [a,b]$. Suppose that fo... | https://mathoverflow.net/users/45092 | Covering dimension of uncountable union of compact spaces | There is no upper bound for dimension in this case.
Just consider any surjective continuous map $f:[0,1]\to [0,1]^\omega$ onto the Hilbert cube. Such a map exists since the Hilbert cube is a Peano continuum. Then for every $r\in[0,1]$ put $X\_r=\{f(r)\}$. It is clear that $\dim(X\_r)=0$ for all $r\in[0,1]$ but $\dim(... | 1 | https://mathoverflow.net/users/61536 | 356942 | 150,514 |
https://mathoverflow.net/questions/356941 | 0 | Let $\mathbb{k}$ be an algebraically closed field. The symplectic algebraic group is given by
$$
\text{Sp}(2n,\mathbb{k})=\{M\in\text{Mat}\_{2n}(\mathbb{k})\mid J=M^TJ M\}\quad\text{where}\quad J=\begin{pmatrix}
0 & 1\_n \\
-1\_n & 0
\end{pmatrix}
$$
The Lie algebra of $Sp(2n,\mathbb{k})$ is given by
$$
\text{sp}(... | https://mathoverflow.net/users/150898 | Slicker computation of the Lie algebra of the symplectic group (and computing differentials of matrix equations of polynomials) | $\newcommand{\dual}{\mathbb{k}\langle\epsilon\rangle}$Let $\dual = \mathbb{k}[\epsilon]/\epsilon^2$ be the dual numbers. A $\dual$-point of a scheme $X$ is exactly a $\mathbb{k}$-point of $X$ and a tangent vector at that point.
1. In the case when $G$ is an algebraic group, we can thus obtain the $\mathbb{k}$-valued... | 5 | https://mathoverflow.net/users/125523 | 356945 | 150,515 |
https://mathoverflow.net/questions/356944 | 2 | Suppose $f: \mathcal{S}(\mathbb{R}^{d})^{n+1} \to \mathbb{C}$ is a continuous function. To each $\varphi \in \mathcal{S}(\mathbb{R}^{d})$, we can define the map $f[\varphi]: \mathcal{S}(\mathbb{R}^{d})^{n} \to \mathbb{C}$ given by:
\begin{eqnarray}
f[\varphi](\psi\_{1},...,\psi\_{n}) := f(\varphi, \psi\_{1},...,\psi\_{... | https://mathoverflow.net/users/150264 | Representation of a Schwartz map in terms of a kernel | A stronger assertion than what you ask for does hold. Namely, a continuous multilinear map of $S(\mathbb R^n) \times ... S(\mathbb R^n)$ ($d$ factors) to $\mathbb C$ is given by a tempered distribution on $\mathbb R^{nd}$.
First, the nuclearity of $S(\mathbb R^n)$ and $S(\mathbb R^{nd})$ (which is not a trivial thing... | 4 | https://mathoverflow.net/users/15629 | 356948 | 150,517 |
https://mathoverflow.net/questions/356635 | 16 | Let $\mathcal{A}$ be an Abelian category with enough injectives. Is it always possible to make the injective embedding functorial? By this I mean that there should exist a functor $I \colon \mathcal{A} \to \mathcal{A}$ and a natural transformation $\operatorname{id} \to I$ such that for all objects $A$, the mapping $A ... | https://mathoverflow.net/users/828 | Abelian category with enough injectives but not functorially | Since the dual of an abelian category is also an abelian category, the question is equivalent to the same question for projective resolutions.
I will show that the category $\mathbf{Ab}^{\operatorname{f.t.}}$ of finitely generated abelian groups has enough projectives, but no functorial projective cover. The idea is ... | 18 | https://mathoverflow.net/users/82179 | 356950 | 150,518 |
https://mathoverflow.net/questions/356965 | 1 | I know that any compact orientable 3-manifold can be obtained from the three sphere $S^3$ by an integer surgery. I am not sure why the surgery operation is completely determined by Where we map meridians of the tori. More specifically,
an integer surgery is determined by specifying a link in $S^3$ and a framing for e... | https://mathoverflow.net/users/103418 | Integer surgery on $S^3$ | You can think about attaching a solid torus as a two-step process:
First, attach $D^2 \times I$ where $D^2$ is the meridional disc of the attaching torus.
Secondly, attach the remaining of the solid torus, which is homeomorphic to a ball $B^3$.
Now if we know where the meridian of the solid torus goes, then we ... | 3 | https://mathoverflow.net/users/56571 | 356986 | 150,530 |
https://mathoverflow.net/questions/356963 | 2 | After my [previous post](https://math.stackexchange.com/questions/3616506/derivative-as-an-integral-kernel?noredirect=1#comment7433473_3616506) I got curious about the following very simple question (which I don't seem to find the answer). Given a tempered distribution $K \in \mathcal{S}'(\mathbb{R}^{n\_{1}+\cdots+n\_{... | https://mathoverflow.net/users/150264 | Integral representation of tempered distributions | Let $\mathcal L$ be a continuous linear mapping from $\mathscr S(\mathbb R^n)$ into
$\mathscr S'(\mathbb R^n)$. The Laurent Schwartz kernel theorem asserts that there exists $K\in \mathscr S'( \mathbb R^n\times \mathbb R^n)$ such that for all $\phi, \psi\in
\mathscr S(\mathbb R^n)$
$$
\langle\mathcal L\phi,\psi\rangle\... | 7 | https://mathoverflow.net/users/21907 | 356993 | 150,533 |
https://mathoverflow.net/questions/356933 | -1 | I am trying to solve the following question . Let $r\in (0,1)$ and $m\in \mathrm{N}$. Then there exists a sequence $(a\_n)\_{n\in \mathrm{N}}$ in $\mathrm{N}$ such that $$ \prod\_{n=1}^{\infty} \left ( 1- \dfrac{m}{a\_n +m} \right ) =r.$$
Is it possible that $ (a\_n)\_{n\in \mathrm{N}}$ be chosen strictly increasing?
... | https://mathoverflow.net/users/70192 | Sequence $(a_n)$ for which $ \prod_{n=1}^{\infty} \left ( 1- \dfrac{m}{a_n +m} \right ) =r.$ | By simple transformations your claim is equivalent to
the following:
For each $M\in \mathbb{N}$ and $s>0$ there is a strictly increasing sequence $(a\_n)\in\mathbb{N}^\infty$, such that
$$ \sum \limits\_{n>0} \log \left( 1+ \frac{m}{a\_n}\right) ~=~ s.$$
This can be proved by inductive construction of such a sequen... | 1 | https://mathoverflow.net/users/20804 | 356999 | 150,536 |
https://mathoverflow.net/questions/357004 | 5 | The actual question is slightly more general than that in the title:
>
> Let $p: U\to Y$ be a surjective étale morphism and $Y\to X$ be a finite morphism of schemes. Is there an étale cover $V\to X$ (surjective) such that the base change $p': V\times\_X U\to V\times\_X Y$ admits a section?
>
>
>
If necessary, ... | https://mathoverflow.net/users/42571 | Surjective étale morphisms étale locally split | We can work locally on $X$ and even (by standard limit arguments) assume that $X=\mathrm{Spec}(R)$ where $R$ is local and strictly henselian. Then $Y=\coprod\_{i=1}^{n}Y\_i$ where each $Y\_i$ is local and finite over $X$, in particular strictly henselian too. So $U\times\_Y Y\_i\to Y\_i$ has a section since it is étale... | 6 | https://mathoverflow.net/users/7666 | 357009 | 150,540 |
https://mathoverflow.net/questions/328368 | 8 | Let $A$ be a commutative ring with unit and let $P$ be a projective $A$-module finitely generated. By definition, there exists an $A$-module $P'$ such that $P\oplus P'$ is free of finite rank $r$. If $f$ is an endomorphism of $P$, its trace $\operatorname{Tr}(f|P)$ is defined to be $\operatorname{Tr}(f\oplus 0|P\oplus ... | https://mathoverflow.net/users/66686 | Property of the trace on finitely generated projective modules | This is true. It's more natural to prove a generalisation:
**Lemma.** *Let $A$ be a commutative ring, let $F\_\bullet$ be a bounded complex of finite projective $A$-modules (say in degrees $[a,b]$), and let $f\_\bullet \colon F\_\bullet \to F\_\bullet$ be an endomorphism such that $f\_\bullet^n$ is homotopic to $0$ f... | 6 | https://mathoverflow.net/users/82179 | 357022 | 150,544 |
https://mathoverflow.net/questions/357007 | 8 | Let $m,n$ be positive integers, such that $m^2\neq n^3$. I conjecture the inequality
$$|m^2-n^3|>\dfrac{1}{5}\sqrt[6]{m^2+n^3}.$$
I've tried a lot of numbers, and they all seem to work, but how do I prove it?
| https://mathoverflow.net/users/38620 | On the conjecture : $|m^2-n^3|>\frac{1}{5}\sqrt[6]{m^2+n^3}$ | As suggested by **Joe Silverman**, there are counterexamples in my paper
>
> Rational points near curves and small nonzero $|x^3-y^2|$ via lattice reduction, *Lecture Notes in Computer Science* **1838** (proceedings of ANTS-4, 2000; W.Bosma, ed.), 33-63. arXiv: math.NT/0005139 (<https://arxiv.org/abs/math/0005139>)... | 18 | https://mathoverflow.net/users/14830 | 357031 | 150,548 |
https://mathoverflow.net/questions/356729 | 1 | **Motivation.** I have a bike lock with 4 dials, and I was wondering whether I can reach any combination by always turning a fixed number $k$, say $k=2$, of the dials, by $1$ position, instead of just turning $1$ dial at the time, which makes for a boring challenge.
**Formal version.** For any integer $n>2$, let $[n]... | https://mathoverflow.net/users/8628 | Bike lock graph | **EDIT**: I've just realized that I answered a different question than was asked. The answer assumes that each of the $k$ dials needs to turn by 1 *in the same direction*. I'm leaving the answer as is in case someone will find the modified problem interesting.
TL;DR: the graph is connected iff $k$ and $n$ are co-prim... | 1 | https://mathoverflow.net/users/144261 | 357038 | 150,550 |
https://mathoverflow.net/questions/356995 | 4 | Let us have a (possibly singular) irreducible projective variety $X$ over $\mathbb{C}$, with an algebraic $\mathbb{C}^\*$-action that has finitely many fixed points $\{x\_1,\dotsc,x\_n\}$. One can define the attracting sets
$$U\_k = \{x \in X \mathrel| \lim\_{t\rightarrow \infty}t\cdot x =x\_k\}$$ that decompose $X$ i... | https://mathoverflow.net/users/114985 | Białynicki-Birula decomposition for singular projective variety | No, not at all. Take any Schubert variety, $X\_w \subset G/B$. Then for one choice of $\mathbb{C}^\* \subset T$ the BB decomposition is a cell decomposition, but for others (e.g., when the Schubert variety is singular and the torus is chosen to be attractive at the "base point" $B/B$) it will not be.
The simplest exa... | 7 | https://mathoverflow.net/users/919 | 357045 | 150,551 |
https://mathoverflow.net/questions/357046 | 3 | Let $q=2^k$. I need to explicitly construct $U\_3(q)$ as a subgroup of $G=GO\_6^-(q)$. It is well-known that
$G\cong U\_4(q)$, and as a subgroup of the latter one has $U\_3(q)$ fixing a non-isotropic point in the $\mathbb{F}\_{q^2}^4$ endowed with a sesquilinear Hermitean form preserved by $G$.
If I represent $\math... | https://mathoverflow.net/users/11100 | describing embedding $U_3(q)<O_6^-(q)$, $q$ even | This is an example of a much more general embedding. Let $q$ be a prime power and $m$ a positive integer. Let $V$ be an $m$-dimensional vector space over $F=GF(q^2)$ and let $B:V\times V\rightarrow F$ be a nondegenerate hermitian form. Then $V$ is also a $2m$-dimensional vector space over $K=GF(q)$. Also $Q(v)=B(v,v)\i... | 5 | https://mathoverflow.net/users/3214 | 357050 | 150,554 |
https://mathoverflow.net/questions/356960 | 0 | Suppose I have an analytic function $f : \mathbb{R} \to \mathbb{R}$ and I have the asymptotic expansion of some $x\_0$ up to a few terms in terms of $\epsilon$ for some $\epsilon \to 0$ which I believe is such that $f(x\_0) = 0$. How can I prove that $$f(x\_0) = 0$$
Some context - In my case, I am trying to compute t... | https://mathoverflow.net/users/155895 | How to check if you have the asymptotic solution of some equation? | Let $\bar x\_0$ be your approximation of $x\_0$. Find a small $\delta$ such that $f(\bar x\_0)$ and $f(\bar x\_0+\delta)$ have opposite sign. Then you know $x\_0$ to an accuracy of $|\delta|$ (assuming $f$ is continuous and monotonic as it usually is in the saddle-point method).
More generally, show that Newton's met... | 0 | https://mathoverflow.net/users/9025 | 357054 | 150,556 |
https://mathoverflow.net/questions/348359 | 5 | Let $A$ be a finitely generated $\mathbb C$-algebra and an integral domain. Assume also $A$ is Gorenstein. Let $M$ be a finitely generated torsion-free $A$ module.
Is it true that $Hom\_A(End\_A(M), A)\cong End\_A(M)$?
| https://mathoverflow.net/users/nan | Dual of $End_A(M)$ | As Mohan mentioned in the comments, this is false if one does not assume $M$ is reflexive, but $M$ being reflexive is still not good enough. I'll comment on the local case, and remark that this can easily be extended to more generality, including, for instance, the standard graded case.
>
> **Proposition**: Let $(A... | 2 | https://mathoverflow.net/users/155965 | 357055 | 150,557 |
https://mathoverflow.net/questions/357063 | 4 | Given co-prime $a,b$, Dirichlet's theorem states that there are infinitely many primes in the arithmetic progression $M = \{ a + bn : n \in \mathbb N\}$. Linnik's theorem asserts that the first such prime is bounded by $cb^L$ for positive constants $c,L$.
Let $M\_t = M \cap \{ n > t : n\in \mathbb N\}$ for some posit... | https://mathoverflow.net/users/79998 | Primes in arithmetic progressions above a given threshold | If $1 ≤ t ≤ c\_1 b^{L\_1}$ for some positive, absolute, and effectively computable constants $c\_1,L$, then Linnik's theorem implies that the least prime in $M\_t$ is less than $c\_2 b^{L\_2}$ for some constants $c\_2,L\_2$.
If $t$ is larger than a polynomial in $b$, then we use a stronger quantitative form of Linnik... | 4 | https://mathoverflow.net/users/111215 | 357065 | 150,558 |
https://mathoverflow.net/questions/356327 | 2 | Given an arbitrary discrete probability distribution $a = (a\_1, ..., a\_n)$ and another arbitrary discrete probability distribution $b = (b\_1, ..., b\_n)$, what is the easiest known way to find a diagonalizable $n \times n$ stochastic matrix $M$ such that $M \cdot a = b$?
| https://mathoverflow.net/users/155496 | Diagonalizable stochastic matrix that satisfies an equation | Take $M$ to be the stochastic matrix which has $b$ in every column. Then $M$ maps every probability vector to $b$, and in particular $Ma = b$.
Concerning diagonalizability, choose any basis which contains $b$ as first basis vector and such that all other basis vectors $c$ satisfy $\sum\_i c\_i = 0$. In such a basis, ... | 3 | https://mathoverflow.net/users/27013 | 357069 | 150,559 |
https://mathoverflow.net/questions/357061 | 6 | I am reading now Tyler Lawson's *$E\_n$ ring spectra and Dyer-Lashof operations* form the *Handbook of Homotopy Theory* and I've got a question on the Remark 1.4.19.
We have an operad $\mathcal{O}$ and $\Sigma\_k$ acts freely and properly discontinuosly on $\mathcal{O}(k)$. Let $V\subset\mathbb{R}^k$ consisting of el... | https://mathoverflow.net/users/123432 | Thom spectrum in the definition of power operations | If a group $G$ acts on a vector space $V$, and $X$ is a space where $G$ acts properly discontinuously, then the map
$$
(V \times X) / G \to X/G
$$
can be given the structure of a vector bundle: it inherits this structure from the vector space $V$. The pullback of this bundle to $X$ is the trivial bundle $X \times V$, b... | 14 | https://mathoverflow.net/users/360 | 357075 | 150,561 |
https://mathoverflow.net/questions/356956 | 5 | According to Theorem 1.1' in this [*paper*](https://camath.fudan.edu.cn/camb/ch/reader/create_pdf.aspx?file_no=27B603&flag=1) we have the following estimate on classical solutions $u \in C^2(\overline{B\_1^+})$ of $-\Delta u = f \text{ in } B\_1^+ = B\_1 \cap \{x \_n \ge 0 \}$ and $u = 0 \text{ on } \partial B\_1^+ \ca... | https://mathoverflow.net/users/155885 | A boundary Schauder estimate | One approach is to observe that
$$\|u\_0\|\_{L^{\infty}(B\_1^+)} \geq \frac{1}{16n}|f(0)|.$$
It suffices by linearity to prove this when $f(0) = 1$. Consider the barrier
$$b(x) = \frac{1}{2n}\left(\left|x - \frac{1}{2}e\_n\right|^2 - \frac{1}{8}\right).$$
Since $b \geq \frac{1}{16n}$ on $\partial B\_1^+$, either $u\_0 ... | 6 | https://mathoverflow.net/users/16659 | 357078 | 150,562 |
https://mathoverflow.net/questions/357056 | 4 | In the theory of stability of solitary waves I have seen many times that people mention some of the symmetries of the equation in order to introduce the "right" notion of stability. For instance, if we consider the KdV equation $$
u\_t+u\_{xxx}+uu\_x=0, \qquad (t,x)\in\mathbb{R}^2,
$$
this equation has solitary wave so... | https://mathoverflow.net/users/129131 | Stability and symmetries | **First**, you seem to be misunderstanding what orbital stability means.
The notion of **orbital stability** should be considered as in contrast against **asymptotic stability**. The latter notion is the stability that one usually expects around, say, hyperbolic fixed points of a finite dimensional dynamical system,... | 3 | https://mathoverflow.net/users/3948 | 357102 | 150,572 |
https://mathoverflow.net/questions/357097 | 2 | Let $\Gamma$ be a smooth Jordan arc, and let $\Phi \colon \hat{\mathbb C} \backslash \overline{\mathbb D} \to \hat{\mathbb C} \backslash \Gamma$ be a conformal isomorphism that fixes the point at $\infty$.
Then $\Phi$ extends continuously onto $\partial \mathbb D$ because $\Gamma$ is smooth (Caratheodory), and for eve... | https://mathoverflow.net/users/155316 | Conformal isomorphism uniquely determined by boundary identification? | For smooth curves, the answer is yes. Proof: $\Phi\_0\circ\Phi\_1^{-1}$ is a conformal map
of $C\backslash\Gamma\_0$ onto $C\backslash\Gamma$, and your condition implies that
this conformal map extends continuously to the boundary. So we have a continuous
map (in fact a homeomorphism) of the Riemann sphere, which is co... | 2 | https://mathoverflow.net/users/25510 | 357105 | 150,573 |
https://mathoverflow.net/questions/357112 | 2 | The Wikipedia page for [unusual number](https://en.wikipedia.org/wiki/Unusual_number) states that the density of that set is $\ln 2$, and that this was proven by Schroeppel in 1972. The only source that I found for that is the [HAKMEM document](https://en.wikipedia.org/wiki/HAKMEM), and there is no proof given there, j... | https://mathoverflow.net/users/35734 | How does one prove that the density of unusual numbers is $\ln 2$? | A reference is Greene and Knuth, Mathematics for the Analysis of Algorithms, 3rd ed., pages 95-98. The section of the book containing those pages may be found at <https://link.springer.com/content/pdf/bbm%3A978-0-8176-4729-2%2F1.pdf>
| 3 | https://mathoverflow.net/users/3684 | 357123 | 150,577 |
https://mathoverflow.net/questions/357101 | 4 | There exists a minimal subshift $X$ with a point $x \in X$ such that $x\_{(-\infty,0)}.x\_0x\_0x\_{(0,\infty)} \in X$?
| https://mathoverflow.net/users/134135 | Minimal subshift with some $x \in X$ such that $x_{(-\infty,0)}.x_0x_0x_{(0,\infty)} \in X$? | We can produce such a subshift by a standard hierarchical construction.
Let $w\_{0,0} = 01$ and $w\_{0,1} = 011$.
For each $k \geq 0$, define $w\_{k+1,0} = w\_{k,0} w\_{k,0} w\_{k,1}$ and $w\_{k+1,1} = w\_{k,0} w\_{k,1} w\_{k,1}$.
Define $X$ by forbidding each word that doesn't occur in any $w\_{k, b}$.
Since each $w\_... | 4 | https://mathoverflow.net/users/66104 | 357125 | 150,578 |
https://mathoverflow.net/questions/357111 | 0 | Let $(\Omega,\mathcal{A},\mathbb{P})$ be a probability space.
Suppose $\{X\_n\}$ is a sequence of random variables satisfying :
$$
\sup\_{n}{\mathbb{E}(|X\_n|)} <\infty
$$
Suppose that
$$
\dfrac{M\_j}{2}<\int\_{j-1<|X\_{n}|\leq j}{|X\_{n}(t)|d\mathbb{P}(t)}\leq M\_j+\dfrac{1}{j^2} \qquad\forall n\geq 1 \text{ and }1\... | https://mathoverflow.net/users/152807 | $ \{X_n\mathbb{1}_{X_n\in[-n,n]}\}$ is uniformly integrable | Note that for any $K, n\in\mathbb{N}$ with $K<n$ we have that
$$\begin{align\*}
\mathbb{E}[|X\_n|1\_{X\_n\in[-n, n]}1\_{|X\_n|>K}] &= \mathbb{E}[|X\_n|1\_{n \geq |X\_n|>K}]\\
&=\int\_{n\geq |X\_n|>K} |X\_n|d\mathbb{P}\\
&= \sum\_{j=K+1}^n \int\_{j\geq |X\_n|>j-1} |X\_n|d\mathbb{P}\\
&\leq \sum\_{j=K+1}^n M\_j+1/j^2\\
&... | 1 | https://mathoverflow.net/users/154137 | 357136 | 150,582 |
https://mathoverflow.net/questions/357139 | 0 | *Def.[1]:*
A non-negative $n \times n$ matrix $A$ is called a *non-primitive* if there is no an integer $k$ such that all entries of $A^k$ are positive.[[1]](https://en.wikipedia.org/wiki/Perron%E2%80%93Frobenius_theorem)
*Def.[2]:*
Let ${\bf A}=(a\_{i,j})$ and ${\bf B}=(b\_{i,j})$ be two $n \times n$ non-negativ... | https://mathoverflow.net/users/64181 | The product of non-primitive matrices with zero positions in common | To decide whether a matrix with nonnegative entries is primitive or not, all you have to know is where the positive entries are – their exact values are irrelevant.
If $C$ and $D$ have positive entries in the same places, then $CD$ and $C^2$ will have positive entries in the same places. More generally, if $A\_1,A\_... | 2 | https://mathoverflow.net/users/3684 | 357144 | 150,585 |
https://mathoverflow.net/questions/355761 | 5 | In his colloquium paper "[The Structure of Selmer Groups](https://www.pnas.org/content/pnas/94/21/11125.full.pdf)" Greenberg writes the following:
>
> If $K$ is an imaginary quadratic field ... it is conjectured that for any [non-anticyclotomic] $\mathbb{Z}\_p$-extension of K at most one prime of $K$ can split comp... | https://mathoverflow.net/users/154123 | Completely split primes in non-anticyclotomic $\mathbb{Z}_p$-extensions | The article *Sur les ideaux dont l'image par l'application d'Artin dans une $\mathbb{Z}\_p$-extension est triviale* of Michel Emsalem provides a satisfactory answer to the general question of how many places can split in a $\mathbb{Z}\_p$-extension.
As such, I will mark this question as answered.
| 4 | https://mathoverflow.net/users/154123 | 357145 | 150,586 |
https://mathoverflow.net/questions/357143 | 3 | For each natural number $a$ consider the sequence $l(a):=\left(\frac{\gcd(a,b)}{a+b}\right)\_{b \in \mathbb{N}}$.
Then I have computed that for $k\ge 2, k \in \mathbb{R}$ and $p$ prime, we have:
$$|l(1)|\_k^k = \zeta(k)-1$$
$$|l(p)|\_k^k = \frac{2 p^k-1}{p^k}\zeta(k)-\left(1+\sum\_{j=1}^{p-1} \frac{1}{j^k}\right)... | https://mathoverflow.net/users/nan | Can this quantity be expressed as $x\cdot \zeta(k)+y, x,y \in \mathbb{Q}$? | Yes, and yes.
1) We have
$$|l(a)|\_k^k=\sum\_{d\mid a}\left(\prod\_{p\mid d}\left(1-\dfrac{1}{p^k}\right)\right)\left(\zeta(k)-\sum\_{j=1}^{a/d}\dfrac{1}{j^k}\right)\;,$$
which checks with your special cases.
2) Expand into partial fractions.
P.S. The coefficient of $\zeta(k)$ is equal to
$$\prod\_{p^v\Vert a}... | 3 | https://mathoverflow.net/users/81776 | 357149 | 150,588 |
https://mathoverflow.net/questions/357152 | 3 | Here is a question that popped into my head right as I fell asleep last night.
I was thinking about constructions of irrational numbers, like pi. I was wondering if there are two constructions (any constructions, the pair doesn't have to include pi itself) such that:
1. We have good reason to believe they are diff... | https://mathoverflow.net/users/156028 | Provably undecidable number inequality? | I think standard examples are like this ... Enumerate the cases of Goldbach's conjecture. Let $a\_n = 0$ if the $n$th case is true, and $a\_n = 1$ if false. Consider the number $A = \sum\_{k=1}^\infty a\_k 2^{-k}$ with the binary expansion $(a\_n)$. We may compute $A$ as accurately as we like. But $A=0$ is equivalent t... | 10 | https://mathoverflow.net/users/454 | 357154 | 150,591 |
https://mathoverflow.net/questions/354080 | 9 | The following is a known result in algebraic geometry:
Let $k$ be an algebraically closed field of characteristic zero (for example, $k=\mathbb{C}$).
Let $L$ be a field such that $k \subset L \subset k(x,y)$
and $L$ is of transcendence degree two over $k$.
Then there exist $h\_1,h\_2 \in k(x,y)$ such that $L=k(h\_1... | https://mathoverflow.net/users/72288 | Concerning $k \subset L \subset k(x,y)$ | No. $M\mathrel{:=}k(x,y)$ has a $k$-automorphism $\sigma:x\mapsto 1/x,\,y\mapsto 1/y$, of order 2. Let $G\mathrel{:=}\langle\sigma\rangle$, and put $L\mathrel{:=}M^{G}$, the fixed field. The elements $x+1/x$ and $y+1/y$ of $L$ are algebraically independent over $k$, hence $L$ has transcendency degree 2. However, no non... | 5 | https://mathoverflow.net/users/31923 | 357169 | 150,594 |
https://mathoverflow.net/questions/357168 | 3 | I am following the book *Introduction to Quadratic Forms over Fields* by T. Y. Lam. In section VI.2, the author proves that, over an arbitrary local field $F$, there is a unique quaternion division algebra, namely $D=\left(\frac{\pi,u}{F}\right)$ where $\pi$ is a uniformizer and $u$ is such that $F(\sqrt{u})$ is the un... | https://mathoverflow.net/users/50139 | Subring of quaternion algebra | That's because $w$ is a valuation (the main point being the nonarchimedean triangle inequality), which can be proved by reducing to the commutative case; see for instance Lemma 13.3.2 in <https://math.dartmouth.edu/~jvoight/quat.html>.
For the sake of completeness I will give the argument here:
* $w$ is a valuation... | 3 | https://mathoverflow.net/users/40821 | 357170 | 150,595 |
https://mathoverflow.net/questions/357174 | 2 | Let consider the birational Cremona transformation $\sigma: \mathbb{P}^2 \dashrightarrow \mathbb{P}^2 $ defined by $$(X:Y:Z) \mapsto (X^{-1}:X^{-1}: Z^{-1})=(YZ:XZ:XY)$$.
In language of birational geometry the sections $YZ,XZ,XY \in H^0(\mathbb{P}^2, O\_{\mathbb{P}^2}(2))$ induce it as
$$\sigma: \mathbb{P}^2 \bac... | https://mathoverflow.net/users/108274 | Cremona transformation $\sigma: \mathbb{P}^2 \dashrightarrow \mathbb{P}^2 $ and pushforward of divisor | I think that you need to assume that $F$ is not divisible by $X$, $Y$ or $Z$. This being done, the equation of $\sigma\_\*(F)$ on the open subset where $XYZ\not=0$ is given by $F(YZ,XZ,XY)$. Hence, $\sigma\_\*(F)$ is equal to
$$F(YZ,XZ,XY)X^aY^bZ^c$$
for some integers $a,b,c\in \mathbb{Z}$. As you want to remove $X=0$... | 2 | https://mathoverflow.net/users/23758 | 357179 | 150,599 |
https://mathoverflow.net/questions/348571 | 1 | Has anyone seen a problem like this in the literature? There are likely more generalized versions in sieve theory, which I am willing to tackle, but I would prefer a more elementary approach if available. The problem below will ask only about multiples of integers from $[n,2n)$, but I will have to face $[n,kn)$ at some... | https://mathoverflow.net/users/3402 | Counting multiples in short intervals | This is a placeholder for an answer that expands on Lucia's comment. GRP 2020.04.11.
| 0 | https://mathoverflow.net/users/3402 | 357189 | 150,601 |
https://mathoverflow.net/questions/357184 | 10 | Let $PX$ be a $\sigma$-algebra on the set $X$, and let $j : PX \to {\sf Set}\_{/X}$ be the tautological functor that sends an event $E\subseteq X$ to itself, regarded as a function with codomain $X$. Now, the category ${\sf Set}\_{/X}$ is cocomplete, thus $j$ has a unique cocontinuous extension to a pair of functors
$$... | https://mathoverflow.net/users/7952 | Are "étalé spaces" a thing for probability spaces? | Etale Spaces can be used to analyze Giry-algebras ($\mathcal{G}$-algebras), and hence (for a fixed object $X$) probability spaces on $X$ as follows. First note that your functor $j$ above should read $j: \Sigma\_X \rightarrow \mathbf{Meas}/X$, which is analogous to the topological case (requiring continuous functions r... | 9 | https://mathoverflow.net/users/143558 | 357217 | 150,615 |
https://mathoverflow.net/questions/357229 | 1 | Let $X$ be a smooth projective variety over $\mathbb{C}$. Let $E,F\to X$ be 2 holomorphic vector bundles and $D\hookrightarrow X$ be a smooth divisor. Denote by $\mathcal{O}\_X(D)$ the line bundle associated to the divisor and let $s: \mathcal{O}\_X\to \mathcal{O}\_X(D)$ be a section, such that $s^{-1}(0) = D$. Assume ... | https://mathoverflow.net/users/109370 | Explicit locally free resolution of a perfect complex $E\oplus F\to (E\oplus F)\otimes \mathcal{O}_X(D)\to (E\otimes \mathcal{O}_X(D))|_D$ | Projections to the second summands define a morphism from that complex to the complex
$$
F \stackrel{s}\to F(D)\tag{\*}
$$
of locally free sheaves. The cone of this morphism is the complex
$$
0 \to E \stackrel{s}\to E(D) \stackrel{\rho\_E}\to E(D)\vert\_D \to 0
$$
which is acyclic. Therefore, $(\*)$ is a locally free r... | 5 | https://mathoverflow.net/users/4428 | 357230 | 150,617 |
https://mathoverflow.net/questions/357093 | 2 | In the context of Goldman's paper *The symplectic nature of fundamental groups of surfaces*:
Consider a closed oriented surface $S$ with fundamental group $\pi$, and let $G$ be a connected Lie group. The space $\operatorname{Hom}(\pi,G)$ consisting of representations $\pi\to G$ is ~~a real analytic variety~~ a real a... | https://mathoverflow.net/users/155646 | Space of representations of surface group into Lie groups | The question of the structure of $Hom(\pi, G)$ and $Hom(\pi, G)/G$ is very complicated for a general connected Lie group $G$, though less so for a surface group than for a general finitely generated group. To see just how wild things can get, you should look at this paper of Kapovich and Millson <http://www.math.umd.ed... | 3 | https://mathoverflow.net/users/49247 | 357233 | 150,618 |
https://mathoverflow.net/questions/357242 | 4 | Let $X,Y$ be Banach spaces. We denote by $\mathcal{K}(X,Y)$ the space of all compact operators from $X$ into $Y$. For an operator $T:X\rightarrow Y$, we let $$\|T\|\_{e}:=\inf\{\|T-K\|:K\in \mathcal{K}(X,Y)\},$$ and $$\|T\|\_{m}:=\inf\{\|T|\_{M}\|:codim M<\infty\},$$ where $M$ represents the finite co-dimensional subsp... | https://mathoverflow.net/users/41619 | Two measures of noncompactness of operators | The answer is yes. Indeed, let us denote $Y\_h=\ell\_\infty(B\_{Y^\*})$. For every $L\in\mathcal{K}(X,Y\_h)$,
$\|T\|\_m=\|JT\|\_m=\|JT-L\|\_m\leq \|JT-L\|$. Hence $\|T\|\_m\leq \|JT\|\_e$.
Conversely, since $Y\_h$ is $1$-injective, for every finite codimensional closed subspace $M$ of $X$, then there exists an exte... | 1 | https://mathoverflow.net/users/39421 | 357246 | 150,623 |
https://mathoverflow.net/questions/357240 | 1 | Suppose $x\_{1},...,x\_{n} \in \mathbb{R}^{d}$ are fixed and $f: \mathcal{S}(\mathbb{R}^{d}) \to \mathbb{C}$ is given by:
$$ f(\phi) = e^{\sum\_{j=1}^{n}\alpha\_{j}\phi(x\_{j})}$$
with $\alpha\_{1},...,\alpha\_{n} \in \mathbb{C}$. I'd like to know if there is any kind of derivative whose derivative of $f$ would be:
$$... | https://mathoverflow.net/users/152094 | Pointwise functional derivative as partial derivative | A [functional derivative](https://mathoverflow.net/questions/349057/question-about-functional-derivatives/349584#349584) is what you need:
$$
\begin{split}
\frac{\partial f}{\partial \phi} &\triangleq \left.\frac{\mathrm{d}}{\mathrm{d}\varepsilon} f(\phi+\varepsilon\varphi)\right|\_{\varepsilon=0} = \left.\frac{\mathrm... | 2 | https://mathoverflow.net/users/113756 | 357248 | 150,624 |
https://mathoverflow.net/questions/357132 | 6 | Let $G$ be a discrete group with universal principal bundle $EG\to BG$, and let $X$ and $Y$ be left $G$-spaces. An equivariant map $\overline{f}:X\to Y$ induces a fibre-preserving map $f:EG\times\_G X\to EG\times\_G Y$ between Borel constructions, as in the following diagram.
$\require{AMScd}$
\begin{CD}
X @>\overli... | https://mathoverflow.net/users/8103 | Fibre preserving maps of Borel constructions | The answer to the question in the edit is no. Take $G=\mathbb Z$, $X$ to be a point and $Y=\mathbb R$ with the action $n\cdot x = x+n$. Then there are no equivariant maps from $X$ to $Y$ but there are many maps from $EG \times\_G X = BG = S^1$ to the cylinder $EG \times\_G Y$ over $BG$.
In this special case when $X$ ... | 4 | https://mathoverflow.net/users/33141 | 357263 | 150,630 |
https://mathoverflow.net/questions/357245 | 0 | In $1969$, Margulis proved, for suitable constant $h>0$and $r$ is a positive constant that :
$a(p)=\lim\_{r\to \infty} \frac{VolS(p,r)}{e^{h r}}$ with ($(S(p,r)$ is geodesic spheres), exists at each point $p$ in manifolds of negative curvature which it is the main result implies purely exponential growth of volume of... | https://mathoverflow.net/users/51189 | Is this $a(p)=\lim_{r\to \infty} \frac{VolS(p,r)}{e^{h r}}$ exists and applied for manifolds with positive curvature? | When $M$ has positive curvature, the limit should always be 0. Indeed if $M$ has non-negative Ricci curvature, Bishop-Gromov tells you that $\frac{Vol S(p, r)}{n\omega\_nr^{n-1}}\leq 1$, so the volumes grow at most polynomially.
| 1 | https://mathoverflow.net/users/104334 | 357271 | 150,634 |
https://mathoverflow.net/questions/357256 | 2 | Let $\Omega=[0,1]^2$ be the unit square and $a>0$.
1) I would like to know one estimate of the constant $C(a)$ such that
$$
\forall u\in W^{1,2}(\Omega)\quad \int\_\Omega u^2\le C(a)\int\_\Omega |\nabla u|^2,
$$
under the weighted average and periodicity conditions
$$
\int\_\Omega e^{-ay}u(x,y)\, dx dy=0,\qquad u... | https://mathoverflow.net/users/155950 | Poincaré inequality under weighted average condition | For $u\in W^{1, 2}(\Omega)$ set $L(u) = \int\_\Omega e^{-ay}u(x, y)dxdy$. We want to prove that if $L(u) = 0$ then $||u||\_{L^2}\le C||\nabla u||\_{L^2}$ for some universal constant $C$. We will first deal with the more interesting case $a \ge 1$ for simplicity.
Since $W^{1, 2}(\Omega)$ is a Hilbert space and $L$ is ... | 2 | https://mathoverflow.net/users/104330 | 357273 | 150,635 |
https://mathoverflow.net/questions/357275 | 2 | Let $K$ be a knot with genus $g$. Seifert's algorithm produces a surface of genus $k$ whose boundary is $K$. In general $k$ may be larger than $g$, but are there any bounds on how much larger it can be? That is, does there exist a function $\phi \colon \mathbb{N} \rightarrow \mathbb{N}$ such that $k \leq \phi(g)$?
| https://mathoverflow.net/users/156106 | Upper bounds on the genus of the surface produced by Seifert's algorithm | Just a note about terminology: the minimal genus of a Seifert surface arising from Seifert's algorithm to a diagram of $K$ is called the *canonical genus* of $K$.
[This paper](https://arxiv.org/abs/math/0608765) by Brittenham and Jensen proves that the Whitehead double of an alternating pretzel knot $K$ has canonical... | 5 | https://mathoverflow.net/users/13119 | 357289 | 150,639 |
https://mathoverflow.net/questions/357219 | 1 | Say $a^2+b^2=c^2$ is a primitive Pythagorean triple. Then consider the Linear Diophantine Equation
$$ua^2+vb^2+xab+ybc+zca=0$$ where $(u,v,x, y, z)\in\mathbb Z^4$ are variables. If $(u,v,x, y, z)\neq(0,0,0,0,0)$ then can we say anything about $\|(u,v,x, y, z)\|\_\infty$ or the probability distribution of $\|(u,v,x, y, ... | https://mathoverflow.net/users/136553 | Small linear relations between primitive Pythagorean triples $\mathsf I$ | Yes, when $m>n>0$ and
$$ a = m^2 - n^2 $$
$$ b = 2mn $$
$$ c = m^2 + n^2 $$
then
$$ -n a^2 +(m-n)b^2 - n ab +(m-n)bc - n ca = 0 $$
or quintuple
$$ -n, m-n, -n, m-n, -n $$
There is a second pattern that gives the same optimum when $n$ is small, quintuple
$$ -2n, m-n, m-n, m-n, -2n $$
| 4 | https://mathoverflow.net/users/3324 | 357292 | 150,640 |
https://mathoverflow.net/questions/357290 | 9 | I've been trying to prove (maybe even disprove) the following inequality:
$$
\sum\_{n=1}^{N} \frac{a\_n}{\sqrt{\sum\_{i=1}^{n}a\_i}} \leq C \sqrt{\sum\_{n=1}^{N}a\_n}
$$
Where $ a\_1,...,a\_N\geq 0 $ are some non-negative numbers, and $C$ is an absolute constant.
Help will be much appreciated.
| https://mathoverflow.net/users/156112 | An inequality involving square roots and sums | For every $n\in\{1,\dotsc,N\}$, we have
$$2\sqrt{\sum\_{i\leq n} a\_i}-2\sqrt{\sum\_{i\leq n-1} a\_i}=\frac{2a\_n}{\sqrt{\sum\_{i\leq n} a\_i}+\sqrt{\sum\_{i\leq n-1} a\_i}}>\frac{a\_n}{\sqrt{\sum\_{i\leq n} a\_i}}.$$
Summing these up, we obtain the inequality with $C=2$. It is also straightforward to see that for $C<2... | 18 | https://mathoverflow.net/users/11919 | 357296 | 150,642 |
https://mathoverflow.net/questions/357294 | 1 | Let $f:X\to Y$ be a flat, surjective, smooth morphism between smooth algebraic varieties (over $\mathbb C$). We assume that $f$ has relative dimension $n$ and we assume also that $\dim Y\ge 2$ (just to avoid the case of a curve that might be easier).
Let $L$ be a line bundle on $Y$, then we have a homomorphism in she... | https://mathoverflow.net/users/65980 | Injectivity of the cohomology map associated to the pullback of line bundles | The unit of adjunction map $L\rightarrow f\_\ast f^\ast L$ is an isomorphism if the fibres of $f$ are connected (I assume this in what follows). Then the Leray spectral sequence $E^{pq}\_2=H^p(R^q f\_\ast f^\ast L)\Rightarrow E^{p+q}=H^{p+q}(f^\ast L)$ has edge maps $e^p:E^{p0}\rightarrow E^p$ which are exactly the map... | 1 | https://mathoverflow.net/users/104669 | 357303 | 150,646 |
https://mathoverflow.net/questions/357287 | 7 | Suppose someone came up with an algorithm that could take any snark and perform edge contraction to result in the Peterson graph. If an inspection of the algorithm reveals that it works as claimed, would the algorithm be sufficient to prove the 4CT?
| https://mathoverflow.net/users/154202 | Could the 4-color theorem be proven by contracting snarks? | Yes, the 4-colour theorem is true if and only if every snark is non-planar (this is due to Tait).
Showing that a snark has a Petersen minor would be enough to show that it is non-planar.
| 15 | https://mathoverflow.net/users/1492 | 357308 | 150,649 |
https://mathoverflow.net/questions/357251 | 5 | Let $R$ by a commutative ring with $1$, and $I \subset R$ a non-zero integral ideal in $R$. When $R$ has finite quotients, and $I = P$ is prime in $R$, the group of units $(R/P)^{\times}$ of the finite ring $R/P$ is cyclic as $R/P$ is a finite field. Do there exist known sufficient and necessary conditions on $R$ and $... | https://mathoverflow.net/users/149325 | Do there exist general conditions for cyclicity of unit groups of quotient rings (generalizations of the primitive root theorem)? | Throughout, let $R$ be a Noetherian ring and $I \subseteq R$ an ideal such that $R/I$ is finite. Then $R/I$ is Artinian, so we may write $I = I\_1 \cdots I\_r$ with $I\_i = \mathfrak m\_i^{n\_i}$ where $\mathfrak m\_1, \ldots, \mathfrak m\_r \subseteq R$ are pairwise distinct prime ideals and $\mathfrak m\_i^{n\_i} \su... | 5 | https://mathoverflow.net/users/82179 | 357315 | 150,651 |
https://mathoverflow.net/questions/357307 | 1 | Let $R$ be a reductive, finite-dimensional Lie algebra over a field of characteristic 0, and let $V$ be a semisimple $R$-module (also finite dimensional). I have seen a reference to the fact that $H^i(R,V) \cong H^i(R,V^R)$, which seems like a stronger version of Theorem 10 of [Hochschild and Serre's paper](https://www... | https://mathoverflow.net/users/156119 | Lie algebra cohomology: $H^i(R,V)=H^i(R,V^R)$ with $R$ reductive and $V$ an $R$-module | (This is meant to be a comment, but I don't have enough reputations) If I'm not mistaken, this follows almost directly from the paper you cited. By semisimplicity of $V$, it suffices to prove for irreducible $V$. If $V$ is not the trivial module then $V^R =0$ and the statement follows from Theorem 10 of the paper. If $... | 2 | https://mathoverflow.net/users/99342 | 357318 | 150,652 |
https://mathoverflow.net/questions/357316 | 1 | Consider a one-dimensional Itô diffusion:
$$\mathrm{d} X\_{t}=b\left(X\_{t}\right) \mathrm{d} t+\sigma\left(X\_{t}\right) \mathrm{d} B\_{t}$$
where $X\_0 = 0$ and $B\_t$ is the standard Brownian Motion. The exit time $\tau$ is defined as:
$$\tau = \inf (t >0 : X\_t \notin (a, b) )$$
Now I want to calculate the ... | https://mathoverflow.net/users/153595 | Question about the exit time of a time-homogeneous Itô diffusion | You can solve this by reducing it to a problem of Brownian motion: Define the scale function
$\varsigma(x) = \int\_{X\_0}^x e^{-2\int\_{X\_0}^y \frac{b(z)}{\sigma^2(z)} dz} dy$
the process
$M\_t = \varsigma(X\_t)$
is a local martingale. Therefore, by Dambis-Dubins-Schwarz it is a time changed Brownian motion
... | 3 | https://mathoverflow.net/users/20026 | 357323 | 150,654 |
https://mathoverflow.net/questions/357187 | 1 | Are there any refences for conjugacy classes of involutions in finite simple group $E\_7(q)$?
| https://mathoverflow.net/users/152963 | What do conjugacy classes of involutions like in finite simple group $E_7(q)$? | For $q$ odd see:
D. Gorenstein, R. Lyons and R. Solomon,The classification of the finite simple groups, Number 3,Mathematical Surveys and Monographs, vol. 40, Amer. Math. Soc., 1998 MR1490581
For $q$ even:
Aschbacher, Michael M., and Gary M. Seitz. "Involutions in Chevalley Groups over Fields of Even Order." Nago... | 1 | https://mathoverflow.net/users/152333 | 357325 | 150,655 |
https://mathoverflow.net/questions/357300 | 2 | In which books we can find representations or character tables, Sylow 2-subgroups and conjugacy classes for finite simple groups and their Schur covering groups and properties for Schur multiplier of finite simple groups?
The following websites may be useful to my questions:
<https://math.stackexchange.com/question... | https://mathoverflow.net/users/152963 | In which books we can find structure information for finite simple groups and their Schur covering groups? | Beyl-Tappe [4] say on p. 119:
>
> 5.9 REMARK. The Schur multiplicators of all finite simple groups have been found, often by exhibiting a universal perfect cover (representation group). For the results see the summary by GRIESS [2] and follow the references given there.
>
>
>
[1] *Griess, Robert L. jun.*, [**S... | 2 | https://mathoverflow.net/users/19276 | 357326 | 150,656 |
https://mathoverflow.net/questions/356088 | -1 | Is this special case known?
For $\lambda(q)$ -- well-factorable function and $q|P(z)$, $\pi(x;q,a)$ $a=1$.
$\displaystyle \sum\_{q\leq x^{1-\epsilon}} \lambda(q) ( \pi (x;q,1)-\frac{\pi(x)}{\varphi (q)} )\leq \frac{Cx}{\log^A(x) }$
| https://mathoverflow.net/users/155294 | BV analogue with well-factorable function. (Primes in arithmetic progression) | This type was proved by Bombieri, Friedlander and Iwaniec in the paper "*Primes in arithmetic progressions to large moduli*.
| 0 | https://mathoverflow.net/users/156150 | 357347 | 150,660 |
https://mathoverflow.net/questions/357166 | 5 | My question is that
**Is every stabilizer of the canonical boundary action of a hyperbolic group on its Gromov boundary a finitely generated group?**
I guess every stabilizer is a (finitely generated) virtually cyclic group, but I do not have a proof nor a reference.
**More generally, let G be a countable group ... | https://mathoverflow.net/users/9401 | The stabilizers of the canonical boundary action of hyperbolic groups | This is following my comment, which was getting too long.
Using the notations of the paper you are citing, there are two kinds of points in the boundary : elements of the Gromov boundary $\partial \Gamma$ of the fine graph $\Gamma$ on which $G$ acts and elements in $V\_\infty$, which are vertices of $\Gamma$ of infinit... | 4 | https://mathoverflow.net/users/111917 | 357350 | 150,662 |
https://mathoverflow.net/questions/357343 | 1 | Do you know what is the uncertainty on the Pearson correlation coefficient **as a function of the uncertainty on the measurement in the data set**.
I know of an expression giving the uncertainty related to the limited size of the data set, but I'm looking for the uncertainty related to the measurement of the data it... | https://mathoverflow.net/users/155651 | What is the uncertainty on the (Pearson) correlation coefficient? | Suppose that the variables $X,X'$ are fixed with means $\mu, \mu'$, standard deviations $\sigma, \sigma'$, and correlation $\rho$. Suppose they are observed with errors $Z,Z'$ that are normally distributed with mean 0 and standard deviations $\epsilon, \epsilon'$. Then to a first approximation:
\begin{align}
Var[\tex... | 1 | https://mathoverflow.net/users/nan | 357356 | 150,666 |
https://mathoverflow.net/questions/357383 | 11 | Physicists Some people like to define the "Fourier transform" on [Minkowski space](https://en.wikipedia.org/wiki/Minkowski_space) as $\hat f(\xi) = \int e^{i \eta(x,\xi)} f(x) dx$, where $\eta(x,\xi)$ is the Minkowski form. I'm used to thinking of the Fourier transform as a canonical isomorphism $L^2(K) \to L^2(\hat ... | https://mathoverflow.net/users/2362 | Fourier transform on Minkowski space | Well you seem to have worked it out but I wrote most of this before your comment happened: I claim there isn't any material difference between your "Minkowski space Fourier transform" and the usual Fourier transform on ${\mathbb R}^n$: in fact write $$ \hat f(\xi)\equiv \int e^{i\eta(x,\xi)} f(x) dx $$ for *any* non-de... | 11 | https://mathoverflow.net/users/22757 | 357399 | 150,679 |
https://mathoverflow.net/questions/357286 | 0 | The invariant-subspace problem is probably an open problem for reflexive spaces which asks:
>
> Does every bounded linear operator on an infinite-dimensional separable Hilbert space have a non-trivial closed invariant subspace?
>
>
>
This problem is closely related to properties of the orbits, which are
called... | https://mathoverflow.net/users/51189 | Consequences of invariant-subspace problem to Li–Yorke chaos | Li Yorke chaos means there is an uncountable scrambled set. Linear operators can be Li-York chaotic (see the work of A. Peris and his collaborators, for example) and for the case of a linear operator $T:X \to X$, Li-York chaos is equivalent to there being an irregular vector: i.e., a vector $x\in X$ such that $\liminf\... | 0 | https://mathoverflow.net/users/20484 | 357407 | 150,682 |
https://mathoverflow.net/questions/357210 | 5 | For ODEs, the standard theorem of continuous dependence of initial parameters deals only with functions that are Lipschitz. Do there exist more general results holding for non-Lipschitz functions? If so, could someone direct me to some resources on the topic?
| https://mathoverflow.net/users/140681 | Continuous dependence on initial parameters of an ODE for non-Lipschitz functions? | The continuous dependence on initial conditions and parameters (and even on the right-hand side in the compact-open topology) is a consequence of the uniqueness. See Theorem 3.2 of Chapter II in Hartman's "Ordinary differential equations" (I call this statement the Kamke lemma). Note that without the uniqueness, the qu... | 5 | https://mathoverflow.net/users/85336 | 357414 | 150,684 |
https://mathoverflow.net/questions/357370 | 1 | My question is: is there a standard name for a structure like the following?
For positive integers $n$, $k < n$ define a "$k$-set-free test for $n$" as a set $C$ of subsets of the integers $\{0, \dots, n-1\}$ such that: for every $i \in \{0, \dots, n-1\}$ and every $S \subset \{0, \dots, n-1\} \setminus \{i\}$ with $... | https://mathoverflow.net/users/56665 | What is a reference for this sort of test set system that avoids all sets of size $\le k$? | This is an instance of the [set cover problem](https://en.wikipedia.org/wiki/Set_cover_problem). I used integer linear programming to obtain the minimum $|C|$ for $1\le k < n \le 12$. Do any of these results surprise you?
\begin{matrix}
n\backslash k &1 &2 &3 &4 &5 &6 &7 &8 &9 &10 &11 \\
\hline
2 &2 \\
3 &3 &3 \\
... | 2 | https://mathoverflow.net/users/141766 | 357417 | 150,685 |
https://mathoverflow.net/questions/357369 | 5 | Let $A$ and $B$ be two C$^{\*}$-algebras and suppose we have a non-zero projection $p\in A\otimes B$. (We can assume $A$ is nuclear, so that there is only one possible tensor product.)
Does there exist a choice of elements $a\_{1},\ldots,a\_{n}\in A$ and $b\_{1},\ldots,b\_{n}\in B$ such that:
1. $\left\|\left(\sum... | https://mathoverflow.net/users/126776 | Approximating a projection by a sum of elementary tensors with a certain property | Yes, this is possible, assuming that $A$ (or $B$) is nuclear. The same argument below (using that exact $C^\ast$-algebras are locally reflexive) also works if $A$ or $B$ is exact and the tensor product is the spatial (aka minimal) tensor product.
The role of nuclearity is that any two-sided closed ideal $J \subseteq ... | 6 | https://mathoverflow.net/users/126109 | 357420 | 150,687 |
https://mathoverflow.net/questions/357358 | 3 | Let $H$ be a Hopf algebra, and $V$ and $W$ a left, and a right,
$H$-comodule respectively. The tensor product
$$
V \otimes W
$$
has an obvious $H$-$H$-bicomodule structure.
If $V$ and $W$ are irreducible as left and right comodules, then
is $V \otimes W$ irreducible as a $H$-$H$-bicomodule?
| https://mathoverflow.net/users/153228 | Irreducibility of product bicomodules | No, off course, not. You need a slightly stronger condition. One of your comodules need to be absolutely irreducible.
To prove it, apply the fundamental theorem of coalgebra. The comodules give you simple subcoalgebras $C\_M$ and $C\_N$ in the coradical of $H$. You are asking for the algebra $(C^{\ast}\_M)^{op}\otime... | 3 | https://mathoverflow.net/users/5301 | 357421 | 150,688 |
https://mathoverflow.net/questions/357412 | -3 | Let $f$ be a function such that :$f:\mathbb{R+}\to \mathbb{R+}$ and $f^{-1}$ is a compositional inverse of $f$ , I have tried to find solution of the following functional $f(x)^{f^{-1}(x)}=x^2$, I took $f(x)=x$ but it doesn't work it coincide only for $x=1$ and for $f(x)=\exp(x)$ I come up to $x^x=x^2$ then coincide on... | https://mathoverflow.net/users/51189 | Function satisfying $f(x)^{f^{-1}(x)}=x^2$ with $ f^{-1}$ is a compositional inverse of $f$ and $f:\mathbb{R+}\to \mathbb{R+}$? | When $f(x) < 0$ (which must happen for uncountably many $x$ if $f^{-1}$ is defined on $\mathbb R$), the fact that $f(x)^{f^{-1}(x)} > 0$ requires $f^{-1}(x)$ to be a rational number (presuming we define $a^b = \exp(b \log(a))$ for some branch of the logarithm; note all branches of logarithm of a negative number have im... | 6 | https://mathoverflow.net/users/13650 | 357422 | 150,689 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.