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https://mathoverflow.net/questions/290055 | 4 | Let $G\_d$ be the Lamplighter group $G\_d = \mathbb{Z}^d \wr \mathbb{Z}\_2 $ and $\Gamma =\{(\bar{\eta},\tilde{0}),(\bar{0},\tilde{e\_1}), \cdots,(\bar{0},\tilde{e\_d})\}$ be the generator set of $G\_d$ (where $\bar{\eta}(\tilde{m})=1 $ iff $\tilde{m}=\tilde{0}$ and $\tilde{e\_i}$ are the standard basis of the lattice ... | https://mathoverflow.net/users/8506 | Radon-Nikodym derivative of the group action on the Furstenberg-Poisson boundary of lamplighter groups | As R W explained, it depends on what you call "sharp estimates".
Maybe you will be interested in a description of the Martin boundary. For $d=1$, this is done by Brofferio and Woess in the paper Green kernel estimates and the full Martin boundary for random walks on lamplighter groups and Diestel–Leader graphs (Ann. ... | 3 | https://mathoverflow.net/users/111917 | 293191 | 129,044 |
https://mathoverflow.net/questions/293185 | 5 | This might be well known for the experts but I am not able to find a reference. I was wondering if there exists a Kahler-Einstein metric on the Fano threefold given by blow-up of $\mathbb{P}^3$ along a smooth plane cubic or not. I believe that the answer shoud be "no" by Matsushima's criterion saying that if the automo... | https://mathoverflow.net/users/31724 | Does exist a Kahler-Einstein metric on the blow-up of $\mathbb{P}^3$ along a smooth plane cubic? | No, there are no Kähler—Einstein metrics on the blow-up of $\mathbb{P}^3$ along a plane cubic.
Let $X$ be the blow-up of $\mathbb{P}^3$ along a plane cubic. Then the alpha-invariant $\alpha(X)=1/4$ by Cheltsov—Shramov[1]. On the other hand, the only K-semistable smooth Fano 3-fold with $\alpha=1/4$ is $\mathbb{P}^3$... | 8 | https://mathoverflow.net/users/42636 | 293195 | 129,046 |
https://mathoverflow.net/questions/293198 | 3 | If such a set $A$ of size $m$ exists, all its admissible pairwise sums must lie in its complement, thus
$$
\binom{m}{2} \leq 2^n-m,
$$
which gives $$m\leq 2^{(n+1)/2}\qquad (1)$$.
**Edit:** *The upper bound is wrong. See the wonderful comments and examples below the question and in the answers. I can only accept one ... | https://mathoverflow.net/users/17773 | Largest $A\subset \mathbb{F}_2^n$ such that no two $a\neq b$ in $A$ add to an element of $A.$ | The upper bound is wrong. The problem is that there is no need for the pairwise sums to be distinct. However it is true that there must be at least $m-1$ distinct sums, which gives $m \leq 2^{n-1}$.
This is sharp as we can take $A$ to consist of all elements with nonzero first entry, so all the sums of two elements o... | 11 | https://mathoverflow.net/users/18060 | 293200 | 129,049 |
https://mathoverflow.net/questions/293181 | 7 | Let $R$ be a henselian dvr, $s,\eta\in\text{Spec}(R)$ the closed and generic points, and $f : X\to \text{Spec}(R)$ a proper smooth scheme.
For a prime $\ell$ invertible on $R$, is there a specialization map
$$sp^i\_{\eta,s} : R^if\_\*(\mu\_{\ell^n})\_{s}\to R^if\_\*(\mu\_{\ell^n})\_{\eta}$$
(with $\eta$ and $s$, ... | https://mathoverflow.net/users/nan | Specialization map étale cohomology | If $\ell$ is invertible in $R$, then $R^i f\_\* (\mu\_{\ell^n}) $ is a locally constant sheaf on $R$ by smooth and proper base change. Hence it is a represenation of the fundamental group of $R$, which is equal to the Galois group of the residue field, because $R$ is Henselian.
This fundamental group is easily seen t... | 8 | https://mathoverflow.net/users/18060 | 293201 | 129,050 |
https://mathoverflow.net/questions/287640 | 6 | Suppose $X$ is a smooth variety defined over $\mathbb{Q}$. There are (at least) two automorphisms of cohomology groups of $X$ that are called "Frobenius", and I would like to understand how they are related.
* If $p$ is a prime of good reduction for $X$, then $H^n\_{dR}(X)\otimes\mathbb{Q}\_p$ depends functorially on... | https://mathoverflow.net/users/5263 | Frobenius automorphisms of cohomology of a variety | I think you are intending to take everything with smooth proper varieties (or else there is no good notion of a prime of good reduction).
In this case, the two Frobenius elements have the same characteristic polynomial.
In the smooth projective case, this is due to [Katz-Messing](http://www.math.mcgill.ca/goren/Se... | 5 | https://mathoverflow.net/users/18060 | 293203 | 129,051 |
https://mathoverflow.net/questions/292724 | 4 | I'm interested in writing GAP code to compute the inner automorphism group of a finite Lie algebra. (I'd like to be able to group conjugate subalgebras together.) I've had trouble finding good references on the topic, and I'm getting an unexpectedly large group in testing.
*UPDATE 2:*
As @Paul Levy says in his an... | https://mathoverflow.net/users/19729 | Computing the inner automorphism group of a finite Lie algebra | You are interested in finite Lie algebras, which is the same thing as finite-dimensional Lie algebras over finite fields. The usual way to understand such a problem is first to consider finite-dimensional Lie algebras algebraically closed fields of characteristic $p$.
So let's assume the ground field $k$ is algebraic... | 2 | https://mathoverflow.net/users/26635 | 293221 | 129,056 |
https://mathoverflow.net/questions/293224 | 9 | It is very well known that if $\mathfrak p\_1, \ldots,\mathfrak p\_n$ are prime ideal of an integral domain $A$, then we have the equality$$S^{-1}A=\bigcap\_{i=1}^n A\_{\mathfrak{p}\_i},$$ where $S:=A\setminus\cup\_i\mathfrak p\_i$.
For a proof of the above, see for example [this](https://math.stackexchange.com/ques... | https://mathoverflow.net/users/80084 | Localization and intersection | This is not true. Morally, the right hand side corresponds to the functions that are regular at all $\mathfrak p\_i$, and one can show that this is not always a localisation of the entire ring $A$.
**Example.** Let $(E,O)$ be an elliptic curve over an algebraically closed field $k$, and let $P \in E(k)$ be a non-tors... | 11 | https://mathoverflow.net/users/82179 | 293225 | 129,058 |
https://mathoverflow.net/questions/292799 | 2 | Let
* $\Omega$ be a set
* $\mathcal A$ be a $\sigma$-algebra on $\Omega$
* $\mu:\mathcal A\to\mathbb R$ be $\sigma$-additive
If $\mathcal S\subseteq2^\Omega$ with $\emptyset\in\mathcal S$, then $$\operatorname{Var}\_\mu^{\mathcal S}(A):=\sup\left\{\sum\_{i=1}^n|\mu(S\_i)|:n\in\mathbb N\text{ and }S\_1,\ldots,S\_n\i... | https://mathoverflow.net/users/91890 | If $\mu$ is a signed measure and $\mathcal S$ is a subfield, then the variations of $\mu$ and $\left.\mu\right|_{\mathcal S}$ coincide on $\mathcal S$ | We can prove the claim even in a more general setting. For the sake of clarity, I redeclare the objects: Let
* $\Omega$ be a set
* $\mathcal A$ be an algebra on $\Omega$
* $E$ be a $\mathbb R$-Banach space
* $\mu:\mathcal A\to E$ be bounded and $\sigma$-additive
We will need the following Lemma:
>
> Let $\nu:\m... | 1 | https://mathoverflow.net/users/91890 | 293241 | 129,064 |
https://mathoverflow.net/questions/293186 | 3 | Let $Z\_1,\ldots, Z\_n$ be independent standard gaussian random variables. Is it true that $X=\max\{Z\_1,\ldots,Z\_n\}$ has a log-concave distribution function?
| https://mathoverflow.net/users/24494 | Log-concavity of the maximum of gaussians | From the context, it appears that by "distribution function" the OP meant the pdf (usually, the distribution function is understood as the cdf). Let $F$ and $G$ denote the cdf's of $X$ and $Z\_1$, respectively, so that $F=G^n$. (Note that $1-(1-G)^n$ is the cdf of the minimum, not the maximum, of $Z\_1,\ldots,Z\_n$. Th... | 3 | https://mathoverflow.net/users/36721 | 293248 | 129,066 |
https://mathoverflow.net/questions/293226 | 3 | It's well known that when the elements of an $n \times n$ matrix $A$ are chosen independently from e.g. $U[0,1]$ distributions, then with probability $1$ the matrix $A$ will be injective (indeed, invertible). Given the Hilbert space $L^2(\mathbb{R}^d)$, a Hilbert-Schmidt kernel $k \in L^2(\mathbb{R}^d \times \mathbb{R}... | https://mathoverflow.net/users/120927 | Are injective Hilbert Schmidt operators (measure theoretically) generic? | It depends on the distribution. For instance, if we restrict our attention to the case where $\lambda\_{ij}$ is uniform on $[0, \bar{\lambda}\_{ij}]$, then we can choose $\bar{\lambda}\_{ij}$ so as to make $T$ a.s. injective, but we can also chose them so that this is not the case.
By taking transposes, it's equivale... | 3 | https://mathoverflow.net/users/4832 | 293251 | 129,068 |
https://mathoverflow.net/questions/293245 | 7 | Most true statements independent of PA that I know of is equivalent to some consistency statement. For example
* Con(PA), Con(PA + Con(PA)), Con(PA + Con(PA) + Con (PA + Con(PA)), $\dots$
* Goodstein's theorem is equivalent to Con(PA)
* Any conjunction or disjunction of the above.
Is every true statement independen... | https://mathoverflow.net/users/65915 | Is every true statement independent of $PA$ equivalent to some consistency statement? | The theory $PA + Con(PA)$ has the property you are asking for, this is the so called Friedman-Goldfarb-Harrington principle (see, e.g., [Fifty years of self-reference in arithmetic](https://projecteuclid.org/euclid.ndjfl/1093883515 "Fifty years of self-reference in arithmetic"), p. 366). Formally, for *every* $\Pi\_1$ ... | 9 | https://mathoverflow.net/users/103227 | 293261 | 129,071 |
https://mathoverflow.net/questions/293130 | 3 | Let $M,N$ be (compact) Riemannian manifolds. $N$ is viewed as an embedded submanifold of $\Bbb R^K$. The Sobolev space $W^{1,p}(M;N)$ is defined as
$$
W^{1,p}(M;N):=\{ u\in W^{1,p}(M;\Bbb R^K)\ |\ u(x)\in N \quad\text{almost every } x\in M\}.
$$
>
> How do we show that $\nabla u$, the weak derivatives of $u$, belon... | https://mathoverflow.net/users/80191 | The gradient $\nabla u$ of $u\in W^{1,p}(M;N)$ is tangent to $N$ almost everywhere | We will assume $N$ is closed, while $M$ may have boundary. Also let $n = \mathrm{dim}(M).$
If $n \leq p < \infty$ we have $C^{\infty}(M;N)$ is dense in $W^{1,p}(M;N),$ so let $u\_k$ be a sequence of smooth functions tending to $u \in W^{1,p}(M;N).$ Now let $x \in M$ and let $x\_1,\dots,x\_n$ be a choice of coordinate... | 2 | https://mathoverflow.net/users/105198 | 293262 | 129,072 |
https://mathoverflow.net/questions/293109 | 4 | Can we introduce complements on top of the standard set theory $\text{ZF}$ and have some comprehension axioms about them, like in defining a "small set" as an element of a stage of the Cumulative Hierarchy as:
$\text {Define:} \ \ small (s) \longleftrightarrow \exists f \ \exists d \ (Ord(d) \wedge dom(f)=d \\ \wedge... | https://mathoverflow.net/users/95347 | Can we add set complements on top of ZF? | Church’s “Set Theory with a Universal Set” and its variants have complements, with Replacement restricted to sets equinumerous to a well-founded set:
* Alonzo Church 1974a. “Set Theory with a Universal Set,” *Proceedings of the Tarski Symposium, Proceedings of Symposia in Pure Mathematics XXV,* ed. Leon Henkin, Ameri... | 4 | https://mathoverflow.net/users/13530 | 293278 | 129,075 |
https://mathoverflow.net/questions/293268 | 1 | Let $ (a\_{1},\cdots,a\_{k}) $ an admissible $ k $ -tuple and $ P\_{k} $ the product of the first $ k $ primes. Do we have a conjectural expression for the number of positive integers $ n $ not exceeding $ x $ such that $ (n+a\_{1},\cdots,n+a\_{k})\in\mathbb{P}^{k}\Longrightarrow(n+P\_{k}+a\_{1},\cdots, n+P\_{k}+a\_{k}... | https://mathoverflow.net/users/13625 | Admissible k-tuples and primorials | Isn't this just the question of the number (up to $x$) of prime tuples corresponding to the admissible tuple $A \cup A+P\_k$ (where $A=\{a\_1,\cdots,a\_k\}$ )?
Note that $A \cup A+P\_k$ could have as few as $k+1$ and as many as $2k$ elements. The first Hardy-Littlewood conjecture gives a formula for that. I am quite... | 3 | https://mathoverflow.net/users/8008 | 293286 | 129,078 |
https://mathoverflow.net/questions/292945 | 11 | We have the Pimsner-Voiculescu exact sequences and the Baum-Connes map
for possible computation of the $K$-theory of the reduced group $C^\*$-algebra $C^\*\_r(G)$ for a topological, locally compact, second-countable Hausdorff group $G$.
Up to now I have not seen much computations of $K(C^\*\_r(G))$.
Has anyone refe... | https://mathoverflow.net/users/88855 | Which $K$-groups $K(C^*_r(G))$ are computed? | Here are some known computations for infinite discrete groups. Basically, most of these proceed by computing the equivariant K-homology of the classifying space of proper actions and deduce the computation for K-theory of the group $C^\ast$-algebra via the assembly map.
For the Bianchi groups:
* A.D. Rahm. On the e... | 8 | https://mathoverflow.net/users/50846 | 293288 | 129,080 |
https://mathoverflow.net/questions/293259 | 2 | I am working with the [Piltz divisor problem](https://en.wikipedia.org/wiki/Divisor_summatory_function) where the number of ways in which a number $n$ can be written as a product of $k$ is of the form
$$D\_{k}(x)=xP\_{k}(\log x)+\Delta \_{k}(x)$$
where $P\_k$ is a polynomial of degree $k-1$ [defined](https://arxiv.... | https://mathoverflow.net/users/120939 | Residue of the following variant of Dirichlet function | Have a look at p.171 of this [paper](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=tm&paperid=2566&option_lang=eng) by A.F. Lavrik. You should find your answer there.
| 2 | https://mathoverflow.net/users/112959 | 293300 | 129,085 |
https://mathoverflow.net/questions/293082 | 3 | Let $T$ be stable and $M \models T$. Given two types $p(x) \in S\_x(M)$ and $q(y) \in S\_y(M)$ there is a canonical way to get a type in $S\_{xy}(M)$. Define
$$p(x) \otimes q(y) = tp\_{xy}(ab/M)$$
where $a$ realises $p$ and $b$ realises the unique non-forking extension of $q$ to $Ma$. This gives a canonical map
$$S\_x... | https://mathoverflow.net/users/57712 | Product of types in stable theories | Unfortunately, this map is never continuous. The set $[x=y]$ is clopen in $S\_{xy}(M)$, but its preimage in $S\_x(M)\times S\_y(M)$ is $\{(\mathrm{tp}(a/M),\mathrm{tp}(a/M))\mid a\in M\}$, which is not closed (for example, its closure contains the whole diagonal).
What is true is that the "nonforking extension" map i... | 1 | https://mathoverflow.net/users/2126 | 293303 | 129,086 |
https://mathoverflow.net/questions/293289 | 19 | **Question**: Suppose you have a **simply connected**, closed, orientable, smooth manifold $M$. What are some restrictions to the existence of a smooth (non-trivial) $S^{1}$-action on $M$?
**Note**: 1.There are some well known restrictions in the case that we require this action to have isolated fixed points (or pres... | https://mathoverflow.net/users/99732 | Obstruction to a general S^1-action | V. Puppe, in [Simply connected manifolds without $S^1$-symmetry. Algebraic topology and transformation groups, Proc. Conf., Göttingen/FRG 1987, Lect. Notes Math. 1361, 261-268 (1988)](https://zbmath.org/?q=an:0666.57026) proved the following:
>
> There exist a simply connected, closed, oriented smooth 6-dimensional... | 11 | https://mathoverflow.net/users/1573 | 293307 | 129,088 |
https://mathoverflow.net/questions/293320 | 2 | Let $E$ be an infinite-dimensional complex Hilbert space.
I look for an elementary proof of the following result:
>
> If $A,B \in\mathcal{L}(E)$ be two normal operators such that $AB=BA$. Then
> $$A^\*B=BA^\*\;\;\text{and}\;\; AB^\*=B^\*A.$$
>
>
>
If $E$ is finite dimensional, I found a rather difficult pro... | https://mathoverflow.net/users/113054 | Proof related to normal operators on an infinite-dimensional Hilbert spaces | A proof via the multiplication operator representation: the assumption on $A$ and $B$ ($A$ and $B$ bounded normal commuting operators on a Hilbert space $H$) are sufficient for the multiplication operator representation to hold. So up to unitary isomorphism, $A=M\_{a}:u\mapsto a(x)u(x)$ and $B=M\_{b}:u\mapsto b(x)u(x)$... | 3 | https://mathoverflow.net/users/6101 | 293330 | 129,097 |
https://mathoverflow.net/questions/293312 | 6 | We have a graph $G$. The vertices of $G$ are a measurable subset of $\mathbb{R}^n$ for some $n$. The degree of each vertex is bounded by some absolute finite constant $K$.
Q1. Does $G$ have a maximal independent set? (This is a set of mutually non-adjacent vertices such that each vertex not in the set is adjacent to ... | https://mathoverflow.net/users/9025 | Measurable maximal independent set in infinite graph of bounded degree | A relevant statement is Proposition 4.2. in
>
> **[KST1999]** A. S. Kechris, S. Solecki, S. Todorcevic, *Borel Chromatic Numbers*. Advances in Mathematics **141**, 1-44 (1999)
>
>
>
In short, the proposition shows that if you settle for *Borel* sets instead of arbitrary measurable sets, then Q1 and even Q2 *se... | 6 | https://mathoverflow.net/users/108556 | 293331 | 129,098 |
https://mathoverflow.net/questions/293328 | 5 | I would like to prove (or prove it is not true with a counter example) the following result:
Let $A$, $B$ be two squares matrices of size $n\times n$ with positive entries.
If $A \leq B$, then $\sum\_{i = 1}^n \sigma\_i(A) \leq \sum\_{i = 1}^n \sigma\_i(B)$.
The sign $\leq$ between matrices is meant elementwise an... | https://mathoverflow.net/users/101003 | Nonnegative matrices and singular values | The inequality is not true, in general. Here is a counterexample:
Let $B =
\begin{pmatrix}
1 & 1 \\ 1 & 1
\end{pmatrix}
$
and let $A =
\begin{pmatrix}
1 & 1 \\ 0 & 1
\end{pmatrix}
$ (my favourite counterexample matrix).
The singular values of $B$ are $2$ and $0$. A short computation shows that the singular v... | 8 | https://mathoverflow.net/users/102946 | 293333 | 129,099 |
https://mathoverflow.net/questions/293335 | 2 | In order to show that relative de Rham cohomology is a topological invariant, an equivalent of whitney's approximation theorem is needed for applications $f \colon (M,P) \to (N,Q)$ (i.e. $f(P) \subset Q$), where $M$, $N$ are smooth manifolds, $P \subset M$ a closed submanifold, and $Q \subset N$ a closed submanifold.
... | https://mathoverflow.net/users/74372 | Is there a Whitney's approximation theorem for applications of pairs? | A good source is:
* MR0674117 Reviewed Bröcker, Theodor; Jänich, Klaus Introduction to differential topology. Translated from the German by C. B. Thomas and M. J. Thomas. Cambridge University Press, Cambridge-New York, 1982. vii+160 pp.
They do this in detail. The relative situation is only dealt with for manifold... | 2 | https://mathoverflow.net/users/26935 | 293338 | 129,101 |
https://mathoverflow.net/questions/293324 | 6 | The bi-invariant Haar measure on the quotient $\mathrm{SL}\_2(\mathbb{Z}) \backslash \mathrm{SL}\_2(\mathbb{R}) / \mathrm{SO}\_2(\mathbb{R})$ represents the moduli space of rank two real lattices modulo rotation and is easy to write down. An element of the quotient can be written (using the Iwasawa decomposition) as:
\... | https://mathoverflow.net/users/120980 | Haar measure on $\mathrm{SL}_3(\mathbb{Z}) \backslash \mathrm{SL}_3(\mathbb{R}) / \mathrm{SO}_3(\mathbb{R})$ | Sure, and you can even do this for $\mathrm{SL}\_n$; I believe this goes back to Siegel. A good hands-on reference is Chapter 1 of *Automorphic Forms and $L$-Functions for the group $\mathrm{GL}(n,\mathbb{R})$* by Dorian Goldfeld.
More precisely, we have the Iwasawa decomposition $z = xy$ for $z \in \mathrm{SL}\_n(\m... | 14 | https://mathoverflow.net/users/3803 | 293339 | 129,102 |
https://mathoverflow.net/questions/293329 | 6 | Let $\chi : (\mathbb Z/f\mathbb Z)^\times \to K = \mathbb Q(\mu\_{\phi(f)})$ be a primitive Dirichlet character. Assume moreover that it is *not* quadratic, that is, $\chi^2$ is not the trivial character. Let $\pi\_1,\dots,\pi\_g$ be the primes lying over $2$ and $v\_1,\dots,v\_g$ be the corresponding valuations. Recal... | https://mathoverflow.net/users/58001 | 2-adic valuation of $L(0,\chi)$ for a Dirichlet character | Your question is entirely answered and with a lot of additional detail by Corollary 11.4.2 of my book Springer Graduate Texts in Math GTM 240. This must be in the literature outside of my book, but I do not know a reference.
| 9 | https://mathoverflow.net/users/81776 | 293342 | 129,104 |
https://mathoverflow.net/questions/293343 | 0 | Let $n\geq 3$ be an integer, set $[n] = \{1,\ldots,n\}$ and let $G=([n],E)$ be an undirected connected [bridgeless](https://en.wikipedia.org/wiki/Bridge_(graph_theory)) graph. Is there an orientation (explanation below) of $G$ such that for all $a\neq b\in [n]$ there is a directed path leading from $a$ to $b$?
---
... | https://mathoverflow.net/users/8628 | Orientations in connected bridgeless graphs | Yes. This is known as [Robbins' theorem](https://en.wikipedia.org/wiki/Robbins%27_theorem).
| 3 | https://mathoverflow.net/users/108556 | 293344 | 129,105 |
https://mathoverflow.net/questions/293327 | 10 | Let $D= \{z\in \mathbb{C}:|z| < 1\}$ be the unit disk. And consider the sheaf of holomorphic functions $\mathcal{O}\_{D}$.
>
> **Question (?) :** Is there a sheaf endomorphisms $\phi : \mathcal{O}\_D \to \mathcal{O}\_D$ which is not a (possibly infinite order) differential operator. I.e. not of the form:
>
>
> $$... | https://mathoverflow.net/users/22810 | Is every endomorphism of the sheaf of holomorphic functions on a disk a differential operator? | Probably [this paper](https://www.jstage.jst.go.jp/article/kyushumfs/32/2/32_2_301/_article) answers your question negatively.
In particular it shows that if $X$ is an open subset of $\mathbb{C}^n$ and the sheaf $\mathcal{O}$ of holomorphic functions on $X$ is given the topology of compact convergence then every con... | 4 | https://mathoverflow.net/users/345 | 293346 | 129,107 |
https://mathoverflow.net/questions/293323 | 15 | I have been working with Hurwitz groups, and I came across the group $G := \langle a, b \ | \ a^2, b^3, (ab)^7, ([a,b]^2ab)^6 \rangle$. I'm trying to figure out exactly what this group is. I know it contains two copies of ${\rm PSL}(2,13)$, and there seems to be a very large 2-group in there as well, of order $2^{28}$ ... | https://mathoverflow.net/users/38744 | What is this quotient of the triangle 2-3-7 group? | It is infinite.
As you pointed out yourself, the kernels of the maps onto ${\rm PSL}(2,13)$ have $2$-quotients of apparently unbounded order, which can be computed using the $p$-quotient algorithm.
I also found homomorphisms onto the Janko sporadic groups J1 and J2. Then I found a subgroup of index $525$ in J2 of w... | 24 | https://mathoverflow.net/users/35840 | 293347 | 129,108 |
https://mathoverflow.net/questions/293112 | 6 | Let $\mathcal{P}(\{0,\dotsc,7\})$ denote the power set of $\{0,\dotsc,7\}$.
Is the following true?
>
> For any function $f: \mathcal{P}(\{0,\dotsc,7\})\rightarrow\{0,1\}$ there exists $0\leq k\leq 3$ and three mutually disjoint sets $A,B,C\subseteq \{0,\dotsc,7\}$ such that
> $\min A = 2k, \min B = 2k+1,$ and $$... | https://mathoverflow.net/users/74918 | Ramsey type theorem | Yes, your conjecture is true.
Suppose otherwise. Then there exists a counterexample $f : \mathcal{P}(8) \rightarrow \{0, 1\}$. For each set $X \in \mathcal{P}(8)$, let the proposition $P\_X$ denote $f(X) = 1$.
There are $5440$ different choices of the tuple $(A, B, C) \in \mathcal{P}(8)^3$ satisfying your constrain... | 8 | https://mathoverflow.net/users/39521 | 293359 | 129,113 |
https://mathoverflow.net/questions/293365 | 0 | Let $I\_1$ and $I\_2$ be two closed bounded intervals.
Suppose $W(x,y)$ is a smooth function whose support is contained inside $I\_1 \times I\_2$.
Suppose I have $\Phi= (\Phi\_1(x,y), \Phi\_2(x,y)) : \mathbb{R}^2 \rightarrow \mathbb{R}^2$
which is a diffeomorphism restricted to $B$, where $B$ is an open bounded box... | https://mathoverflow.net/users/84272 | Differentiablity of certain composite function | The function $\widetilde W$ is a smooth function on the open set $\Phi(B)$ and is supported in the compact set$\Phi(I\_1\times I\_2)$. As a result it can be extended by 0 as you wish and this is simply due to the continuous canonical injection
$$
C\_{comp}^\infty(\Omega)\subset C\_{comp}^\infty(\mathbb R^n),
$$
for $\O... | 2 | https://mathoverflow.net/users/21907 | 293368 | 129,115 |
https://mathoverflow.net/questions/293332 | 6 | **Question:** Let $X\_1, \ldots ,X\_n$ be $n$ iid uniformly distributed random variables, i.e., $X\_j \sim \mathcal{U}(0,1)$ for each $j=1,\ldots ,n$. What is the PDF of the maximal distance between to nearby elements? More formally, if we denote $X\_{(k)}$ as the location of the $k$-th smallest element, what is the PD... | https://mathoverflow.net/users/42864 | maximal distance of nearby iid unifrom random variables | Let us just adapt the reasoning at [stats-SE](https://stats.stackexchange.com/questions/162560/distribution-of-the-largest-fragment-of-a-broken-stick-spacings). Let $d\_0:=X\_{(1)}$ and $d\_n:=1-X\_{(n)}$.
Let $\Delta\_j:=d\_{j-1}$ for $j\in[n+1]:=\{1,\dots,n+1\}$, so that
\begin{equation}
\max\_{1\le k\le n-1}d\_k=... | 4 | https://mathoverflow.net/users/36721 | 293381 | 129,119 |
https://mathoverflow.net/questions/293388 | 4 | Suppose $B$ is an abelian group such that for every integer $n\ge 1$, the $n$-torsion subgroup $B[n]$ is finite.
Let $B\_{\rm tor} = \varinjlim\_{n\ge 1} B[n]$ be the torsion subgroup of $B$.
Is it true that, necessarily, there exists an integer $d\ge 0$ such that
$$B\_{\rm tor} \simeq (\mathbf{Q}/\mathbf{Z})^d\o... | https://mathoverflow.net/users/nan | Co-finite type abelian groups | The first question has been answered by Jeremy Rickard. Another counter-example is the Prüfer group $\mathbb Z[\ell^{\infty}]$ (this is a subgroup of a group of the form you ask for, but it is not of that form).
For the second: you are taking a torsion $\ell$-group (your $B[\ell^{\infty}]$) with finite socle, so that... | 4 | https://mathoverflow.net/users/24891 | 293393 | 129,123 |
https://mathoverflow.net/questions/293364 | 5 | Let $A$ be an abelian variety over a number field $K$, $\text{Sha}(A/K)$ its Tate-Shafarevich group, $\ell$ a prime.
Is it known that the $\ell$-primary torsion subgroup $\text{Sha}(A/K)\{\ell\}$ is trivial for almost all primes $\ell$?
| https://mathoverflow.net/users/nan | Tate-Shafarevich group over number fields | No.
It is always difficult to "prove" that something is "not known", but this may do: I claim it is not even known when $K=\mathbb Q$, $A$ is an *elliptic curve* $E$. In fact in this case, the result you ask for is not even known when $E$ *has CM*. In fact, even in this very special case, it is not known that the $l... | 16 | https://mathoverflow.net/users/9317 | 293402 | 129,127 |
https://mathoverflow.net/questions/293397 | 1 | If I pull back a cycle of codimension $c$ along a morphism of schemes I can easily see that the codimension can stay the same or the codimension can drop to all the way to $0$. But intuitively it seems the codimension can never increase. Is there a precise statement of this form?
To state it more carefully suppose $X... | https://mathoverflow.net/users/7 | Properties of codimension under pull back | This is false as stated: taking for $f$ the embedding of a closed subscheme $X\subset Y$, it would mean that $\operatorname{codim}(X\cap Z,X)\leq \operatorname{codim}(Z,Y) $. There are well-known counter-examples: let $Y\_0$ be a projective variety with two divisors $D\_1,D\_2$ which do not intersect, let $Y$ be the co... | 4 | https://mathoverflow.net/users/40297 | 293407 | 129,128 |
https://mathoverflow.net/questions/293382 | 38 | Let $X$ and $Y$ be reasonable spaces. Since $\mathbb{R}^{\infty}$ is contractible,
$$
X \times \mathbb{R}^{\infty} \cong Y \times \mathbb{R}^{\infty} \;\;\; \implies \;\;\; X \simeq Y.
$$
Is the converse also true?
My vague intuition: the factors of $\mathbb{R}^{\infty}$ provide so much extra room that there will ... | https://mathoverflow.net/users/9068 | If $X$ and $Y$ are homotopy equivalent, then are $X \times \mathbb{R}^{\infty}$ and $Y \times \mathbb{R}^{\infty}$ homeomorphic? | This question is answered by two classical theorems of infinite-dimensional topology, which can be found in the books of [Bessaga and Pelczynski](https://books.google.com.ua/books/about/Selected_Topics_in_Infinite_Dimensional.html?id=7n9sAAAAMAAJ&redir_esc=y), [Chigogidze](https://books.google.com.ua/books?id=uaIrAAAAY... | 46 | https://mathoverflow.net/users/61536 | 293409 | 129,129 |
https://mathoverflow.net/questions/293411 | 1 | How can I compute the partial derivatives of the dominant eigenvalue and eigenvectors of a real symmetric matrix $\mathbf{A}$?
In particular, given
$ \mathbf{v}^\* = \arg\max\_{\mathbf{v}} \mathbf{v}^{\top}\mathbf{A}\mathbf{v},\quad \text{subject to } ~ \|\mathbf{v}\|\_2 = 1,
$
how can I find $\partial v\_{i}^\*/... | https://mathoverflow.net/users/34445 | Gradients of the Dominant Eigenvalue and Eigenvector | See (68) [here](http://www2.imm.dtu.dk/pubdb/views/edoc_download.php/3274/pdf/imm3274.pdf): if $\lambda$ is a simple eigenvalue, $Av=\lambda v$ and $v$ is normalized to be orthogonal,
$$
\frac{\partial v}{\partial A\_{jk}} = (\lambda I - A)^+ E^{jk} v,
$$
where $E^{jk}$ is the matrix with $1$ in position $jk$ and $0$ e... | 2 | https://mathoverflow.net/users/1898 | 293413 | 129,130 |
https://mathoverflow.net/questions/293408 | 1 | Is the Brauer group $\text{Br}(K)$ of a global field $K$
* an $\ell$-divisible group for some prime $\ell$? If so, what $\ell$?
* Is $\text{Br}(K)[n]$ finite, for $n$ integer?
I know from class field theory that it fits into an exact sequence
$$0\to \text{Br}(K)\to\bigoplus\_v\text{Br}(K\_v)\xrightarrow{\sum\_v ... | https://mathoverflow.net/users/nan | Brauer group of global fields | It is $\ell$-divisible for every odd number $\ell$.
To see this, let $\alpha \in Br(K)$, and look at its image in $(\alpha\_\nu)\_\nu \in\oplus\_\nu \mathbb{Q}/ \mathbb{Z}$. You know that each component is divisible by $\ell$, so you can form, in several ways, the element $(\alpha\_\nu /\ell)\_\nu$. The problem is that... | 1 | https://mathoverflow.net/users/115052 | 293414 | 129,131 |
https://mathoverflow.net/questions/293418 | 1 | Let $f :(-1,1) \to \mathbb{R};\ \ f(x)=\sum\_{n=0}^\infty a\_n x^n$ be an analytical function expressible as a power series.
Also, let
$$g : (-1,1) \to \mathbb{R}; \ \ g(x)=\frac{d}{dx} \log{f(x)} = \frac{\sum\_{n=1}^\infty n a\_n x^{n-1}}{\sum\_{n=0}^\infty a\_n x^n} =\sum\_{n=0}^\infty d\_n x^n$$
Assume that $... | https://mathoverflow.net/users/100873 | Non-recursive expression for coefficients of the derivative of the logarithm of a power series | You start with Faà de Bruno formula (iterated chain rule):
We have for $I, J$ open subsets of $\mathbb R$, $f: I\rightarrow J$, $g: J\rightarrow \mathbb R$, smooth functions, $k\in \mathbb N^{\*}$,
$$
\frac{(g\circ f)^{(k)}}{k!}=\sum\_{1\le r\le k}\frac{g^{(r)}\circ f}{r!}
\sum\_{\substack{(k\_{1},\dots, k\_{r})\in {(... | 4 | https://mathoverflow.net/users/21907 | 293420 | 129,133 |
https://mathoverflow.net/questions/293102 | 2 | I am working on a project which involves learning about the zeta function (weil zeta function) for plane curves, but I do not know much algebraic geometry. (I do not know anything about schemes).
I am mostly interested in irreducible projective plane curves $C$ over finite fields $\mathbb{F}\_q$. We define the zeta f... | https://mathoverflow.net/users/120666 | Question about Zeta Function of Singular Plane Curve | For a singular curve $C$, you have a compactification $\overline{C}$ and a normalization $\tilde{C}$. Over a perfect field, such as a finite field, the normalization is smooth (and projective).
The best way to study the zeta function of $C$ is by studying the zeta function of $\tilde{C}$ and then adding additional fa... | 2 | https://mathoverflow.net/users/18060 | 293421 | 129,134 |
https://mathoverflow.net/questions/293423 | 2 | I encountered the Hochschild-Serre spectral sequence in étale cohomology
$$H^i(\text{Gal}(\overline{k}/k), H^j\_{et}(X\_{\overline{k}}, F))\Rightarrow H^{i+j}\_{et}(X\_{{k}}, F)$$
How is the filtration on $H^\*\_{et}(X\_{{k}}, F)$ "associated to the Hochschild-Serre spectral sequence" defined? Several authors quote... | https://mathoverflow.net/users/nan | Hochschild-Serre filtration and etale cohomology | Every time there is a spectral sequence, it is usually coming from a filtration. In this case it is coming from a filtration of the chain complex $C^\*(X\_k;F):=R\Gamma(X\_k;F)$. The idea behind it, is to write the functor $R\Gamma(X\_k;-)$ as $R\Gamma(k;-)\circ Rp\_\*$ where $p:X\_k\to\mathrm{Spec}\,k$ is the structur... | 3 | https://mathoverflow.net/users/43054 | 293425 | 129,136 |
https://mathoverflow.net/questions/293384 | 6 | Let $M$ be a smooth manifold.
I have been trying to figure out from the literature I know whether (any flavor of) pseudo-differential operators form a sheaf of algebras (w.r.t. the usual topology on $M$). Sadly the best results I could find showed at best that pseudo-differential form a sheaf of left $C^{\infty}$-mo... | https://mathoverflow.net/users/22810 | Do pseudo-differential operators form a sheaf of algebras? | Pseudodifferential operators are operators whose Schwartz kernel
is smooth outside of the diagonal and is conormal with respect to the diagonal
(differentiating finitely many times with respect to smooth vector
fields tangent to the diagonal produces a distribution in a certain Besov space).
From this definition one ... | 3 | https://mathoverflow.net/users/402 | 293434 | 129,138 |
https://mathoverflow.net/questions/293351 | 6 | The purpose of my question is trying to understand whether, in some cases, we can achieve greater certainty of reasoning (say when dealing with statements about natural numbers, integers or rational numbers) by using countable subsets of reals instead of using reals as a whole. I will try to explain my question in some... | https://mathoverflow.net/users/112385 | Reasoning Using Countable Subsets of Real Numbers |
>
> Suppose that during some argument (involving ℕ) one switches to real numbers and then back to discrete domain (before completing the argument). My question is, can one give examples where a switch to ℝ can be "replaced" by a switch to a suitable "countable subset of ℝ" without having to change the argument entire... | 8 | https://mathoverflow.net/users/8991 | 293435 | 129,139 |
https://mathoverflow.net/questions/293430 | 11 | I heard about the result in the theory of abelian varieties which says the following: given an abelian variety $X$ defined over a field $k$ and a purely transcendental extension $k\subset L\subset L'$ we have that the group $X(L)$ of $L$-points is the same as the group $X(L')$. Could you give me (the reference for) the... | https://mathoverflow.net/users/88385 | Points of abelian varieties over purely transcendental extensions | This follows from the following well-known lemma.
**Lemma.** Let $A$ be an abelian variety over $k$. Then any map $f \colon \mathbb P^1 \to A$ is constant.
*Proof 1.* The map $f$ induces a map on the Albanese $f\_\* \colon \operatorname{Alb}\_{\mathbb P^1} \to \operatorname{Alb}\_A$ sitting in a commutative diagram... | 16 | https://mathoverflow.net/users/82179 | 293444 | 129,145 |
https://mathoverflow.net/questions/292713 | 1 | I am reading Kashiwara's paper [GLOBAL CRYSTAL BASES OF QUANTUM GROUPS](https://projecteuclid.org/euclid.dmj/1077293577). On page 462, the action of $\tilde{e}\_i$ on an integrable $U\_q(g)$-module $M$ is defined as follows. For $u \in \ker e\_i \cap M\_{\lambda}$ and $0 \leq n \leq \langle h\_i, \lambda \rangle$, defi... | https://mathoverflow.net/users/11877 | $\tilde{e}_i$ action on $e^{(n)} u$ | The actions are defined in the Kashiwara's paper: [On crystal bases of the Q-analogue of universal enveloping algebras](https://projecteuclid.org/euclid.dmj/1077295931) in (2.2.5).
Let $M$ be an integrable $U\_q(\mathfrak{g})$-module. Then
\begin{align\*}
M = \oplus\_{0 \leq n \leq -\langle h\_i, \mu \rangle} e\_i^{(... | 2 | https://mathoverflow.net/users/11877 | 293448 | 129,146 |
https://mathoverflow.net/questions/293376 | 6 | Consider the space $X=BSL(8,\mathbb{C})/(\mathbb{Z}/2)$. The topological Brauer group of $X$ is given by $Br\_{top}(X)=Tor(H^{3}(X;\mathbb{Z}))=\mathbb{Z}/2$. I'm studying concepts of topological period and index of a class in $Br\_{top}(X)$ and I would like to calculate $ind\_{top}(\alpha)$ where $\alpha$ is the nontr... | https://mathoverflow.net/users/121001 | Calculating topological index | The index in this case is $8$. You can see that it divides $8$ as your space $X$ supports a tautological degree $8$ topological Azumaya algebra given by the map $B(SL\_8/\mu\_2) \rightarrow BPGL\_8$. If the index were lower, it would be very strange: it would imply that if $Y$ is any space with $H^2(Y,\mathbb{Z})=0$ an... | 7 | https://mathoverflow.net/users/100 | 293452 | 129,147 |
https://mathoverflow.net/questions/293341 | 1 | I remember there is a result from the early 1980s, which states that the tail probability of a binomial distribution is always at least as large as the tail probability of the normal distribution (at least when $p \leq 1/4$ or something like this for the binomial distribution is assumed). However, I cannot rememeber th... | https://mathoverflow.net/users/46852 | Reference request: binomial tail is greater than Gaussian tail | You might be thinking of Eric V. Slud (1977), Distribution inequalities for the binomial law, <https://projecteuclid.org/download/pdf_1/euclid.aop/1176995801>, and you can find further related results proved or referenced in the thesis of Jona Schulz (2016), The optimal Berry–Esseen constant in the binomial case, <http... | 4 | https://mathoverflow.net/users/26591 | 293456 | 129,149 |
https://mathoverflow.net/questions/292920 | 6 | Let $T = \mathbb{G}\_m$ be the torus, and let $\tilde{T}$ be its étale universal cover (a pro-object in schemes of finite type). Then both $T$ and $\tilde{T}$ have a well-defined étale homotopy type. Explicitly, the homotopy types are $ét(T) = B\hat{\mathbb{Z}}$ and $ét(\tilde{T}) = \*,$ for $\*$ the point. In particul... | https://mathoverflow.net/users/7108 | Homotopy equivalence between two basepoints of the etale homotopy type of the one-torus | There is a completely explicit way to do it in this case I think. The Etale homotopy type of $T$ is represented by the pro-object $\{B \mathbb{Z}/n\}\_{n \in \mathbb{N}^\times}$, ignoring for a moment the Galois action and assumeing that the field is of characteristic $0$ for simplicity. The object
$\tilde{T}$ is very... | 2 | https://mathoverflow.net/users/115052 | 293481 | 129,155 |
https://mathoverflow.net/questions/293308 | 13 | Here's an "exercise" which I thought should be easy, but which I find myself unable to do.
Let $V$ be a Banach space.
Recall that an operator $f:V\to V$ is **trace-class** if it is in the image of the natural map $V\otimes\_\pi V'\to \mathcal L(V,V)$, where $\otimes\_\pi$ denotes the projective tensor product, and ... | https://mathoverflow.net/users/5690 | Trace-class operator satisfies $\sum |\lambda_n|<\infty$? | As I mentioned in a comment, exercise 3 has a positive answer: Nuclear operators are absolutely $2$-summing, and $2$-summing operators have $2$-summable eigenvalues (see, e.g., Tomczak's book).
Exercise 2 has a negative answer, but I am not satisfied with the example and do not know what happens in various classical ... | 5 | https://mathoverflow.net/users/2554 | 293484 | 129,157 |
https://mathoverflow.net/questions/293482 | 3 | Suppose $(M, \mathcal X) \models ACA\_0$. Recall that a subset $A \subseteq M$ is $inductive$ over $M$ if $M$ satisfies all instances of induction in the expanded language with a predicate for $A$. Suppose $A$ is inductive and let us denote by $(M, \mathcal X[A])$ the model whose second order part consists of all sets ... | https://mathoverflow.net/users/114946 | If one adds an inductive subset to a model of $ACA_0$, do we always get a new model of $ACA_0$? | The answer to your (first) question is in the negative.
More explicitly: given any countable nonstandard model $M$ of PA, there are inductive subsets $A$ and $B$ of $M$ such that the expansion $(M,A,B)$ fails to satisfy induction in the extended language. I know two ways to achieve this:
**(1)** Use (Cohen) forcing... | 4 | https://mathoverflow.net/users/9269 | 293494 | 129,159 |
https://mathoverflow.net/questions/235426 | 14 | Are infinite dimensional simplicial complexes manifolds locally modeled on $\mathbb R^\infty=\operatorname{colim}\mathbb R^n$? If they are homotopy equivalent, are they homeomorphic?
Of course not. Part of such a complex might not be infinite dimensional: glue an interval to $\mathbb R^\infty$ by one endpoint and it ... | https://mathoverflow.net/users/4639 | Are infinite simplicial complexes all manifolds? | This question concerns manifolds modeled on the direct limit $\mathbb R^\infty$ of Euclidean spaces. The theory of such manifolds is well-developed. The most important results of this theory were obtained by Katsuro Sakai and can be found in Chapter 5 of his [book](https://docs.google.com/viewer?a=v&pid=sites&srcid=ZGV... | 5 | https://mathoverflow.net/users/61536 | 293501 | 129,160 |
https://mathoverflow.net/questions/234901 | 43 | Does $\mathbb C\mathbb P^\infty$ have a (commutative) group structure? More specifically, is it homeomorphic to $FS^2$, (the connected component of) the free commutative group on $S^2$?
$\mathbb C\mathbb P^\infty$ is well known to have the homotopy type of the classifying space of a commutative group, $\mathbb C^\tim... | https://mathoverflow.net/users/4639 | Does $\mathbb C\mathbb P^\infty$ have a group structure? | I noticed that this question still has no accepted answer and all existing answers are rather long. It seems that the answer can be easily obtained using some results of infinite-dimensional topology, namely, the theory of manifolds modeled on the direct limit $\mathbb R^\infty$ of Euclidean spaces (see Chapter 5 of [S... | 19 | https://mathoverflow.net/users/61536 | 293504 | 129,161 |
https://mathoverflow.net/questions/293509 | 4 | I'm trying to reconcile two results on the classification of principal bundles. First, we have $\mathrm{Prin}\_G(X)$ (the equivalence classes of $G$-bundles on $X$) is isomorphic to $H^1(X;G)$ (the first Cech cohomology group of X -- I'm taking $G$ to be abelian). Second, we have $\mathrm{Prin}\_G(X)$ is isomorphic to ... | https://mathoverflow.net/users/121081 | Classification of principal bundles | The cohomology group $H^1(X,\mathbb{R})$ is the Cech cohomology group of the sheaf of differentiable functions over $X$, since there exist partition of unity, $H^1(X,\mathbb{R})=0.$
[What is the right version of "partitions of unity implies vanishing sheaf cohomology"](https://mathoverflow.net/questions/11567/what-i... | 4 | https://mathoverflow.net/users/80891 | 293511 | 129,164 |
https://mathoverflow.net/questions/293521 | 1 | I am reading the paper: [ELEMENTARY CONSTRUCTION OF LUSZTIG’S CANONICAL BASIS](https://arxiv.org/pdf/1602.04895.pdf) and want to compute $F\_{\beta\_3}$ in Example 3 on page 3. Maybe there is some mistake in my computations but I could not find it. The following are my computations.
Let $U\_q(\mathfrak{g})$ be the qu... | https://mathoverflow.net/users/11877 | Question about a computation of $F_{\beta_3}$ related to canonical basis of $U_q(sl_3)$. | **The mistake happens in the last step.**
Note that
\begin{align\*}qF\_2F\_1K\_1^{-1}E\_1
&=q^{1-2+1}K\_1^{-1}F\_2F\_1E\_1\\
&=K\_1^{-1}F\_2\left(E\_1F\_1-\frac{K\_1-K\_1^{-1}}{q-q^{-1}}\right)\\
&=K\_1^{-1}E\_1F\_2F\_1-\left(\frac{q^{-1}-qK\_1^{-2}}{q-q^{-1}}\right)F\_2,
\end{align\*}
and similarly
$$-q^{2}F... | 5 | https://mathoverflow.net/users/296 | 293523 | 129,170 |
https://mathoverflow.net/questions/293495 | 8 | The notion of forcing was invented by Paul Cohen, who used
it to prove the independence of the Continuum Hypothesis. He constructed a model of set theory in which the CH fails, thus showing that CH is not
provable from ZF. Forcing was adapted from set theory to
model theory by Abraham Robinson. Robinson developed two t... | https://mathoverflow.net/users/38966 | Applications of "model-theoretic" forcing | Model theoretic forcing is extended to wider languages with some applications, for example:
1- [Model theoretic forcing in analysis](https://www.sciencedirect.com/science/article/pii/S0168007208001516?via%3Dihub) which gives an exposition of forcing for metric structures.
2- Shelah extended it to the context of abs... | 6 | https://mathoverflow.net/users/11115 | 293527 | 129,172 |
https://mathoverflow.net/questions/293524 | 6 | Let $C$ be a site, $\mathbf{S}$ some ($\infty$-? homotopy?) category of spaces.
>
> **Question.** What do you call a (covariant!) functor $F:C\to \mathbf{S}$ enjoying the following property: for every hypercovering $a\colon V\_\bullet \to U$ in $C$, the induced map $$ {\rm hocolim}\, F(V\_\bullet)\longrightarrow F... | https://mathoverflow.net/users/3847 | Homotopy cosheaf? | This is just a sheaf valued in the ∞-category $\mathrm{Space}^{op}$. It is usually called a cosheaf. A place where this kind of thing shows up is in factorization algebras, that can be described as particular cosheaves over the Ran space.
I don't think they behave significantly differently from sheaves in any other ... | 6 | https://mathoverflow.net/users/43054 | 293528 | 129,173 |
https://mathoverflow.net/questions/293502 | 2 | Please see the definition of Hardy spaces on the unit disc [here.](https://math.stackexchange.com/questions/2586370/if-a-function-belongs-to-the-hardy-space) This is regarding outer functions on a Hardy space. I know that outer functions can have no zeroes in the open unit disc since it is the exponential of something.... | https://mathoverflow.net/users/119639 | Regarding characterisation of outer functions in a Hardy space |
>
> Can we say that any function in Hp that has no zeroes in the open unit is outer?
>
>
>
No. $\exp\left(\frac{z-1}{z+1}\right)$ is inner and belongs to $H^\infty$ and thus to all $H^p$.
| 2 | https://mathoverflow.net/users/25510 | 293531 | 129,174 |
https://mathoverflow.net/questions/293540 | 1 | Let $K\subseteq G$ be compact Lie groups, with $G$ connected. Suppose that $H\_1,H\_2\subseteq K$ are two closed subgroups of $K$ that are conjugate in $G$. Are they conjugate in $K$?
| https://mathoverflow.net/users/121091 | Conjugacy in compact Lie groups | In general the answer is no. For example, in $U(n)$ you have the subgroup $U(1)^n$ which has many copies of $U(1)$ in (by choosing various "coordinate lines" and act trivially on the orthogonal complement). These specific copies of $U(1)$ are all conjugate in $U(n)$ by permutation matrices, but they are not conjugate i... | 9 | https://mathoverflow.net/users/115052 | 293543 | 129,178 |
https://mathoverflow.net/questions/293536 | 7 | Let $X$ be a normal affine variety of dimension at least two over $\mathbb{C}$ and let $U\subset X$ be a dense open. Assume that $\mathrm{codim}(X\setminus U) \geq 2$.
I think Hartog's lemma implies that every automorphism of $U$ extends to an automorphism of $X$. Is that true?
>
>
> >
> > Do we have a surjecti... | https://mathoverflow.net/users/121069 | When do automorphisms on open subsets extend | I think you have an injective homomorphism the "other way around".
As you suspect, Hartog's Lemma implies that $Hom(U,X) = Hom(X,X)$. Now, $Aut(U)$ naturally injects into $Hom(U,X)$. Also, $Aut(X)$ naturally injects into $Hom(U,X)$. The image of $Aut(U)$ in $Hom(U,X)$ lands inside the image of $Aut(X)\to Hom(U,X)$. T... | 4 | https://mathoverflow.net/users/4333 | 293548 | 129,181 |
https://mathoverflow.net/questions/293552 | 1 | Let $$u\colon B^n(0,1)\to \mathbb{R}$$ be a subharmonic function in the open unit ball in $\mathbb{R}^n$. The crucial assumption is that $u$ never equals $-\infty$.
>
> Is it true that $|u|$ is bounded on any compact subset of the unit ball?
>
>
>
Remark. (1) $u$ is bounded above on any compact subset of the ... | https://mathoverflow.net/users/16183 | Boundedness of a finite subharmonic function | No. Take a zequence $x\_k\to 0$ and a sequence of positive numbers $\epsilon\_k$
such that $\sum\_k\epsilon\_k|K\_n(x\_k)|<\infty$.
Then the function
$$\sum\_k\max\{\epsilon\_k K\_n(x-x\_k),-k\},$$
where $K\_n$ is the fundamental solution of the Laplace equation ($K\_n(x)=-|x|^{-n+2}$ when $n\geq 3$ and $K\_2=\log|x|$... | 2 | https://mathoverflow.net/users/25510 | 293555 | 129,182 |
https://mathoverflow.net/questions/293535 | 5 | Let $X$ be a Polish space, $\mathcal B$ the sigma-algebra
of Borel sets. Let $E$ be an
aperiodic countable Borel equivalence relation on
$X \times X$ (this means that every class of equivalence
is countably infinite). A set $C\in \mathcal B$ is called a complete
section for $E$, if $\forall x \in X$ $\exists y \in C... | https://mathoverflow.net/users/121090 | Equivalent of Lusin's Theorem in Borel setting | The answer is No.
A suitable counterexample can be constructed as follows.
On the real line $\mathbb R$ consider the equivalence relation $E=\{(x,y)\in\mathbb R\times \mathbb R:x-y\in\mathbb Q\}$.
Fix a countable base $\{U\_n\}\_{n\in\omega}$ of the topology on the real line. Take a countable set $X=\{x\_n\}\_{... | 4 | https://mathoverflow.net/users/61536 | 293559 | 129,184 |
https://mathoverflow.net/questions/293219 | 8 | This looks elementary, but somehow I am stuck, please bear with me:
Let $H$ be a differential 3-form, nowhere vanishing, but not necessarily closed. What is a sufficient condition that the sequence of forming wedge products with $H$ is exact, i.e. that
$$
\mathrm{ker}(H \wedge(-))\,/\,\mathrm{im}(H \wedge(-)) \;=\... | https://mathoverflow.net/users/381 | Vanishing of H-cohomology | I have not thought about the variant problem, but in the ordinary differential form case, I do not believe it is possible to have vanishing $H$-cohomology on a (finite-type) smooth manifold $M$, regardless of any conditions on $H$ such as nowhere vanishing, closed, etc.
We proceed by contradiction, supposing that the... | 4 | https://mathoverflow.net/users/66405 | 293565 | 129,188 |
https://mathoverflow.net/questions/293554 | 1 | Let $F$ be a complex Hilbert space and $\mathcal{B}(F)$ the algebra of all bounded linear operators defined on $F$.
>
> Assume that
>
>
> * $M\in \mathcal{B}(F)^+$ (i.e. $\langle Mx\;, \;x\rangle\geq 0$ for all $x\in F$).
> * $S\in\mathcal{B}(F)$ such that $\text{Im}(S^\*M)\subseteq \text{Im}(M)$.
>
>
> It is t... | https://mathoverflow.net/users/116483 | If $\text{Im}(S^*M)\subseteq \text{Im}(M)$, is $\text{Im}(SM)\subseteq \text{Im}(M)$? | No: $F = \mathbb C^2$,
$M= \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$, $S= \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}$.
| 3 | https://mathoverflow.net/users/26935 | 293571 | 129,189 |
https://mathoverflow.net/questions/293549 | 5 | The problem of giving an explicit formula for $A\_q(n,d)$ is sometimes referred to as "the main problem in coding theory." The value of $A\_q(n,d)$ is given by the maximum number of codewords in a q-ary code of length $n$ and distance $d$. More specifically, let the hamming weight of an element of $\mathbb{F}\_q^n$ be ... | https://mathoverflow.net/users/111128 | Explicit Formula of Delsarte's Linear Programming Upper Bound for $A_q(n,3)$ | The paper "Some upper bounds for codes derived from Delsarte's inequalities for Hamming schemes" by C. Roos and C. de Vroedt does give a formula for Delsarte's bound. Or at least according to Mathematical Reviews it provides such a formula. I too was unable to locate the paper, but I did find the MR which states the fo... | 2 | https://mathoverflow.net/users/51668 | 293587 | 129,193 |
https://mathoverflow.net/questions/293574 | 5 | Suppose $K$ is a bounded above complex of free abelian groups, and take its derived $\ell$-adic completion $K^{\wedge,\ell} = R\lim (K/\ell^n)$ in the derived category, for $\ell$ a prime.
If $K\to L$ is a map in the derived category, such that $K^{\wedge,\ell}\to L^{\wedge,\ell}$ is an isomorphism for every $\ell$, ... | https://mathoverflow.net/users/nan | Derived completion of complexes | As I mentioned in a comment, $K \to L$ must also be an isomorphism after rationalizing. For example, if $0 \to R \to F \to \Bbb Q \to 0$ is a free resolution of $\Bbb Q$, then let
$$
K = \dots \to 0 \to 0 \to R \to F \to 0 \to 0 \to \dots
$$
and $L$ be the zero complex. The map $K \to L$ is an isomorphism in the derive... | 11 | https://mathoverflow.net/users/360 | 293589 | 129,194 |
https://mathoverflow.net/questions/293516 | 4 | I have three questions.
1. I consider a sequence of metrics $h\_n$ on a two-dimensional torus which all induce the same conformal structure. Suppose that the volume of $h\_n$ is always $1$. Is it possible that the diameter of $h\_n$ tends to infinity?
2. Consider such a sequence of metrics along which the diameter is... | https://mathoverflow.net/users/25511 | Metrics with fixed conformal structure and diameter | Question 2: Yes, there are conformal metrics on a divergent sequence of tori
with Area=1 and bounded diameter:
Cutting the torus open along an embedded essential curve, you obtain a cylinder, conformally equivalent to $[0,R] \times S^1$.
The fact that the sequence of conformal structures diverges, means that
we can ... | 1 | https://mathoverflow.net/users/66777 | 293602 | 129,198 |
https://mathoverflow.net/questions/293607 | 5 | Let $G$ be the group of all permutations of $\mathbb{N}$. If I am not mistaken, this group is denoted by $S^{\infty}$.
Is there a precise locally compact topology on $G$ such that $G$ would be an amenable group? Or is $G$ isomorphic to a dense subgroup of an amenable group?
| https://mathoverflow.net/users/36688 | Amenability of $S^{\infty}$ | The answer to [this question](https://mathoverflow.net/questions/293614/is-each-locally-compact-group-topology-on-the-permutation-group-discrete/293619#293619) implies that each locally compact group topology $\tau$ on the permutation group $S^\infty$ is discrete. Since the discrete group $S^\infty$ is known to be non-... | 5 | https://mathoverflow.net/users/61536 | 293625 | 129,202 |
https://mathoverflow.net/questions/293623 | 3 | This is a variation of an [older question](https://mathoverflow.net/questions/293529/continuous-functions-sharing-a-point) in a more general setting.
Let $V$ denote the set of all functions $f:\omega\to \omega$. We say $f,g\in V$ *share* a point if there is $x\in X$ such that $f(x) = g(x)$ and set $$E = \big\{\{f,g\}... | https://mathoverflow.net/users/8628 | Graph of functions sharing a point | Let H be an uncountable independent set. Let G be the graph formed by joining every vertex of some small enough graph F to every vertex of H. (So F could be a singleton, and G could be bipartite if F were independent.) Suppose we could embed G into your shared point graph. Then no two vertices in H share a point, and s... | 5 | https://mathoverflow.net/users/3402 | 293628 | 129,203 |
https://mathoverflow.net/questions/293627 | 0 | Let $M\_{1},M\_{2}$ be (possibly non-compact) 2-dimensional, connected, smooth, orientable manifolds of finite topological type. Suppose you have smooth, surjective map $F:M\_{1} \rightarrow M\_{2}$, and the pre-image of each point in $M\_{2}$ is finite. Furthermore suppose that there exists $K>0$ such that $|F^{-1}(p)... | https://mathoverflow.net/users/99732 | Finite pre-images implies (local) branch cover? | No. Under your assumptions the map can have fold singularities (and also other kinds of singularities), namely those of the form $f(x,y) = (x, y^2)$ in local coordinates.
In order to have a local branched covering, the map needs to be open. In addition, for having a branched covering, you need also a completeness co... | 1 | https://mathoverflow.net/users/23193 | 293634 | 129,206 |
https://mathoverflow.net/questions/293614 | 5 | **Question.** Is each locally compact group topology on the permutation group $S\_\omega$ discrete?
Here $S\_\omega$ is the group of all bijections of the countable ordinal $\omega$. A *group topology* on a group $G$ is a topology turning $G$ into a Hausdorff topological group.
This question was motivated by [this... | https://mathoverflow.net/users/61536 | Is each locally compact group topology on the permutation group discrete? | Better: any homomorphism $f$ from the infinite symmetric group $S(X)$ ($X$ arbitrary set) to a locally compact group $H$ has a discrete image.
Proof: we can suppose that $f$ has a dense image.
(a) Case when $H$ is totally disconnected. Let $L$ be a compact open subgroup of $H$. Notation: $L^h=h^{-1}Lh$, $h\in H$. ... | 9 | https://mathoverflow.net/users/14094 | 293638 | 129,207 |
https://mathoverflow.net/questions/293641 | 3 | Let us suppose we have a centered Gaussian random field defined on the reals by the covariance structure. Is there a way of showing that this field has infinitely differentiable sample paths? Any theorem or result in this direction would be helpful.
| https://mathoverflow.net/users/38424 | How to show a stochastic process has infinitely differentiable sample paths | In general, you need the covariance to be smooth, especially near the diagonal. For some particular covariances there are very quick ways to prove what you want: for example if you can rewrite the field as a convolution of a random distribution with a mollifier. As for a general theorem, see Thm 2.2.3 in "Random Fields... | 3 | https://mathoverflow.net/users/7410 | 293646 | 129,210 |
https://mathoverflow.net/questions/293637 | 10 | This [question was originally posted on MSE](https://math.stackexchange.com/questions/2659397/matrix-exponential-containing-a-thermal-state), and I'm cross posting it here.
Define an infinite matrix $$ M =
\begin{bmatrix}
0 & -1 & 0 & 0 & \cdots \\
1 & 0 & -2 & 0 & \cdots \\
0 & 2 & 0 & -3 & \cdots \\
0 & 0 & 3 & 0... | https://mathoverflow.net/users/89079 | Matrix exponential, containing a thermal state | It is actually easier to compute the first column of $e^{tM}$ for each $t \in \mathbb{R}$ rather than computing only the first column of $e^M$.
Indeed, let $u(t) = e^{tM}e\_1$ for each $t \in \mathbb{R}$, where $e\_1$ denotes the first canonical unit vector. Then $u(t)$ is the first column of $e^{tM}$.
**Claim:** F... | 14 | https://mathoverflow.net/users/102946 | 293657 | 129,215 |
https://mathoverflow.net/questions/293661 | 0 | If $k$ is a perfect field (not necessarily separably closed), $f : X\to Y$ a proper birational map of smooth projective $k$-varieties, $\ell$ a prime invertible in $k$, is the induced map, for **some** $j$
$$f^\* : H^j\_{ét}(Y, \mu\_{\ell^n}^{\otimes i})\to H^j\_{ét}(X, \mu\_{\ell^n}^{\otimes i})$$
an isomorphism, fo... | https://mathoverflow.net/users/nan | Birational invariants in étale cohomology | The answer is no.
Let $Y$ be a smooth projective surface over $k$, $f : X\to Y$ the blowup of a point. Then $f^\* : \text{CH}^1(Y)\to\text{CH}^1(X)$ is *not* an isomorphism, and for some $n\ge 1$ neither is its reduction mod $\ell^n$.
Your exact sequence shows $i=1, j=2$ and the above choice of $X,Y,f$ is a counter... | 6 | https://mathoverflow.net/users/nan | 293663 | 129,218 |
https://mathoverflow.net/questions/293665 | 2 | Let $H$ be an open subgroup of a locally compact Hausdorff abelian group $G$. Assume that $G/H$ is a finitely generated abelian group. Let $\chi: H \rightarrow \mathbb{C}^{\ast}$ be a continuous homomorphism. Does $\chi$ extend to a continuous homomorphism into $\mathbb{C}^{\ast}$ defined on all of $G$?
If $\chi$ map... | https://mathoverflow.net/users/38145 | Every quasicharacter of an open subgroup extends to a quasicharacter on the whole group | The answer is affirmative and follows from the classical
**Theorem (Baer).** Any homomorphism $h:H\to Y$ from a subgroup $H$ of an Abelian group $G$ to a divisible Abelian group $Y$ extends to a homomorphism $\bar h:G\to Y$.
Observe that the multiplicative group $\mathbb C^\*$ is Abelian and divisible. So, by Baer... | 3 | https://mathoverflow.net/users/61536 | 293673 | 129,219 |
https://mathoverflow.net/questions/293668 | 8 |
>
> Let $A$ be a finitely generated $\mathbb{Z}$-algebra. Is $\operatorname{Pic}(A)$ finitely generated (as an abelian group)?
>
>
>
Thoughts:
1. We may assume that $A$ is reduced since $\operatorname{Pic}(A) = \operatorname{Pic}(A\_{\mathrm{red}})$.
2. If $A$ is reduced, then the group of units $A^{\times}$ i... | https://mathoverflow.net/users/15505 | Picard group of a finite type $\mathbb{Z}$-algebra | This is false. A counterexample is given in [Kahn06, **Rmq. 1 (6)**]. The example uses the cuspidal cubic $B = A[x^2,x^3]$ over a finite type $\mathbb Z$-algebra $A$ that is not a finitely generated $\mathbb Z$-module. For example, take $A$ to be $\mathbb Z[x]$ or $\mathbb F\_p[x]$.
As usual, one shows that (under su... | 11 | https://mathoverflow.net/users/82179 | 293675 | 129,220 |
https://mathoverflow.net/questions/293664 | 4 | Let $G$ be a finite group, $k$ be a field whose characteristic divides $|G|$, and $\rho:G\hookrightarrow\operatorname{GL}\_n(k)$ be a faithful representation of $G$. Let $V$ be a $k$-space of dimension $n$ with ring of invariants $k[V]^G$. Suppose further that $k[V]^G$ is Cohen-Macaulay, so that there are primary invar... | https://mathoverflow.net/users/32261 | Sufficient conditions for secondary invariants | Since you don't know the Hilbert series of $k[V]^G$, I don't see any way of avoiding computing all homogeneous invariants of some degrees and checking if they all lie in the $k[f\_1,\ldots,f\_n]$-module $M$ generated by the $h\_i$. I'd proceed as follows: for rising degree $d$, compute a $k$-basis of $k[V]\_d^G$. Test ... | 3 | https://mathoverflow.net/users/82616 | 293676 | 129,221 |
https://mathoverflow.net/questions/293573 | 7 | Let $\mathbb{C}\_{an}$ be the expansion of the structure $(\mathbb{C}; +,-,×,0,1)$ by adding the restricted complex analytic functions. This is the complex analog of the familiar $\mathbb{R}\_{an}$ in O-minimality.
What do we know about the model theory of $\mathbb{C}\_{an}$? Model-completeness? Quantifier-elimination?... | https://mathoverflow.net/users/27034 | Model theory of the restricted complex analytic functions | The structure $\mathbb C\_{an}$ is bi-interpretable with $\mathbb R\_{an}$. To see this, take for example the restriction of the complex exponential function to the unit square S with corners 0,1,i,1+i. Then from this restriction you can define S as the set of points where the restricted function takes a non-zero value... | 9 | https://mathoverflow.net/users/112484 | 293688 | 129,224 |
https://mathoverflow.net/questions/293651 | 4 | Let $k$ be a $p$-adic field, $T$ a torus over $k$, and $S$ an $k$-subtorus of $T$. If $\chi: S(k) \rightarrow \mathbb{C}^{\ast}$ is a smooth (resp. continuous) homomorphism, then does $\chi$ necessarily extend to a smooth (resp. continuous) homomorphism on $T(k)$?
Even in the special case where $T$ is split over $k$,... | https://mathoverflow.net/users/38145 | For tori $S \subseteq T$, every character of $S(k)$ extends to a character of $T(k)$? | First, continuous characters on $T(k)$ are the same thing as smooth characters, by a no small subgroup argument. [This answer](https://mathoverflow.net/questions/293665/every-quasicharacter-of-an-open-subgroup-extends-to-a-quasicharacter-on-the-whol) shows that if I have an inclusion of abstract groups $H \subset G$, t... | 1 | https://mathoverflow.net/users/38145 | 293696 | 129,225 |
https://mathoverflow.net/questions/292322 | 1 | I need to know which permutation-invariant norms can be consistently decomposed in the sense that for any vector $v = (a,b,c)$ we have that
$$\|(a,b,c)\| = \|(\|(a,b)\|,c)\|.$$
More precisely, let $v = \sum\_{i=1}^n v\_ie\_i$ be a finite-dimensional
vector, and $\{P\_j\}\_{j=1}^k$ a partition of the index set $\{i\}\... | https://mathoverflow.net/users/9211 | Which norms on vectors can be consistently decomposed? | The theorem by Bohnenblust is not exactly what I wanted, but it's what I need. Adapting its statement, we have
Let $\|\cdot\|:\mathbb R^N \to \mathbb R$ be a permutation-invariant norm for $N \ge 3$ such that
* $\|x+y\| = \|(\|x\|,\|y\|)\|$ for all $x,y$ with disjoint support.
* $\|(1,1)\| \neq 1 $.
Then for any ... | 0 | https://mathoverflow.net/users/9211 | 293697 | 129,226 |
https://mathoverflow.net/questions/293684 | -3 | I have formulated the following conjecture:
Odd positive integer $ N=6n-1$ is a prime number iff neither of two diophantine equations
$6x^2+(6x−1)y=n$
$6x^2+(6x+1)y=n$
has solution. $x=1,2,3,..y=0,1,2,...n =1,2, 3..$
Odd positive integer $ N=6n+1$ is a prime number iff neither of two diophantine equations
$... | https://mathoverflow.net/users/83307 | Matrix sieve theorem | For the first proposition substitute $A=6x\pm 1$ and $B=6(x+y)\mp 1$, so that $A$ is an arbitrary positive integer coprime with $6$ and $B$ is any integer $\geq A$ with opposite sign modulo 6. For the second proposition do the same with $B=6(x+y)\pm 1$.
You are asking: is it true that $N$ coprime with 6 is composite ... | 0 | https://mathoverflow.net/users/58242 | 293699 | 129,227 |
https://mathoverflow.net/questions/293449 | 3 | Consider the parabolic problem in the cylinder of base $B$, the unit ball,
$$
\partial\_t u -\text{div}\left( A(x) D u +F(t,x)\right)=0 \text{ in } (0,T)\times B,
$$
with $(ADu +F)\cdot \nu=0$ on $(0,T)\times \partial B$, and $u(t=0)=0$,in
Suppose $A$ is smooth (let us say $C^{2}$), symmetric, $2 I\_d \geq A\geq I\_... | https://mathoverflow.net/users/40120 | Parabolic Regularity with Neumann B.C | Let's see how this goes. First, let me say that the continuity estimate you are looking for is contained in [3, Theorem 4.5] under the integrabilities as I had already mentioned, so $q > n$ and $p > 2(1-n/q)^{-1}$. Unfortunately, to get the uniformity w.r.t. $T$ of the constants, we have to take a closer look at the ar... | 2 | https://mathoverflow.net/users/85906 | 293700 | 129,228 |
https://mathoverflow.net/questions/293682 | 11 | **Problem.** Let $\mathcal F$ be a locally finite (or even discrete) family of (closed) $G\_\delta$-sets in a topological space $X$. Is the union $\cup\mathcal F$ a $G\_\delta$-set in $X$?
**Remark.** The answer to this problem is affirmative in [perfect spaces](https://arxiv.org/abs/1708.01755).
A topological space ... | https://mathoverflow.net/users/61536 | Is a locally finite union of $G_\delta$-sets a $G_\delta$-set? | I found a simple counterexample, which is however not regular.
>
> **Example.** There exists a functionally Hausdorff second-countable space $X$ containing a closed discrete subset $D$, which is not of type $G\_\delta$ in $X$.
>
>
> In this space the countable the family of singleton $\mathcal F=\{\{x\}:x\in D\}$... | 3 | https://mathoverflow.net/users/61536 | 293703 | 129,230 |
https://mathoverflow.net/questions/293694 | 3 | I am looking for an initial reference for a theorem which is known, namely:
>
>
> >
> > ***Theorem***: A Boolean algebra $A$ admits a Stone space topology (i.e. is the underlying algebra of a Stone topological algebra) **iff** $A$ is isomorphic to $\mathcal{P}(S)$, where $S$ is a set and a power-set $\mathcal{P}(... | https://mathoverflow.net/users/73577 | Stone topological Boolean algebras | I believe the paper you want is
Dona Papert Strauss, [*Topological lattices*](https://doi.org/10.1112/plms/s3-18.2.217), **Proceedings of the London Mathematical Society** (3) 18 #2 (April 1968), 217-230.
Strauss' paper is behind a paywall and I do not have access to it, without travelling to a nearby university li... | 6 | https://mathoverflow.net/users/15780 | 293707 | 129,231 |
https://mathoverflow.net/questions/293709 | 2 | I don't have experience with hypergeoemtric functions, but wish to compute the following limit:
$\lim\_{x→\infty}{F([1],[a,b];-\frac{x^2}{4})}$, where $a,b$ are non-integer real parameters.
I tried to use Maple to calculate the limit and the result is 0. I tried to prove it or calculate it by hand and use an integr... | https://mathoverflow.net/users/121112 | Limit of a hypergeometric function(1F2) | The large-$x$ limit is only zero if $a+b>3/2$:
$$\_1{F}\_2({1}; {a, b}; -x^2/4)=$$
$$=\sqrt{\tfrac{1}{\pi}}\Gamma (a) \Gamma (b) \sin \left(\tfrac{\pi}{2} (a+b-\tfrac{1}{2})-x\right)(2/x)^{a+b-3/2}+{\cal O}(1/x^2)$$
For example, when $a=b=1/2$ the amplitude of the oscillations increases as $\sqrt x$ (left plot), and ... | 1 | https://mathoverflow.net/users/11260 | 293712 | 129,232 |
https://mathoverflow.net/questions/293708 | 5 | Suppose $Y$ is a variety defined over $\mathbb{Q}$ and $pt$ is a rational point of $Y$. Let $\pi:X \rightarrow Y$ be the blow up of $Y$ at $pt$ and $D$ be the exceptional divisor. For simplicity let's assume both $X$ and $D$ are smooth. Let $CD$ be the cone of $D$ defined to be
\begin{equation}
D \times I/D \times \{0\... | https://mathoverflow.net/users/87910 | Is this Mayer-Vietoris sequence motivic? | I'm not sure if that's exactly what you mean by "being motivic", but this long exact sequence comes from a triangle in Voevodsky's category of (integral) motives $DM(\mathbb Q)$.
More generally if $X$ is a $\mathbb Q$-scheme of finite type, $Z\subset X$ a closed subscheme, $\pi : Y \to X$ the blowup at $Z$, and $E=\... | 6 | https://mathoverflow.net/users/20233 | 293722 | 129,234 |
https://mathoverflow.net/questions/293738 | 6 | I'm searching the best known upper bound for the Mertens function, but without assuming the Riemann hypothesis.
Landau, in 1901, have proved that $M(x)= O(x \exp(-c\sqrt{\ln x})$, but I am unable to find the current estimate in the classical literature.
| https://mathoverflow.net/users/83665 | Best estimate of the Mertens function without assuming the Riemann Hypothesis | As Greg Martin said in a comment, the Korobov-Vinogradov zero-free region for $\zeta(s)$ yields
$$M(x)\ll x\exp\bigl(-c(\log x)^{3/5}(\log\log x)^{-1/5}\bigr).$$
For a reference, see Satz 3 in Section V.5 of Walfisz: Weylsche Exponentialsummen in der neueren Zahlentheorie (VEB Deutscher Verlag der Wissenschaften, Berli... | 10 | https://mathoverflow.net/users/11919 | 293744 | 129,239 |
https://mathoverflow.net/questions/293743 | 7 | Bloch defines the motivic complexes $\mathbf{Z}(n)$ in his paper "Algebraic Cycles and Higher K-Theory" (1986).
Some references (that I currently am unable to track down) use $$\check{\mathbf{Z}}(n) := \mathbf{Z}(1)^{\otimes n} = \mathbf{G}\_m[-1]^{\otimes n}$$ instead of $\mathbf{Z}(n)$.
What is the relation betwe... | https://mathoverflow.net/users/nan | Two motivic complexes, compared | They are not quasi-isomorphic: your $\check{\mathbf Z}(n)$ is concentrated in a single degree. As Denis points out in the comments, your definition of $\check{\mathbf Z}(n)$ is wrong because you should use the tensor product of sheaves with transfers. In addition, you need to apply Suslin's $\mathbf A^1$-invariantifica... | 8 | https://mathoverflow.net/users/20233 | 293751 | 129,243 |
https://mathoverflow.net/questions/293750 | 2 | It's known that many problems (e.g. XOR) have the exact solutions represented by neural networks. The question is: *What kind of graph theory problems can be solved using neural networks*?
| https://mathoverflow.net/users/114039 | Solving graph theory problems using neural networks | Here is an example that has found a real-world application, in the context of [quantum error correction:](https://en.wikipedia.org/wiki/Quantum_error_correction) The decoding of [stabilizer codes](https://en.wikipedia.org/wiki/Stabilizer_code) is a problem of minimum weight-perfect matching on a [graph](https://en.wiki... | 2 | https://mathoverflow.net/users/11260 | 293753 | 129,244 |
https://mathoverflow.net/questions/293769 | 4 | I'm reading a monograph that considers the following problem:
>
> $$\inf\_{z(t) \in C^1} \int\_0^1 c\bigg(\frac{dz(t)}{dt}\bigg) dt\\ z(0) = x, z(1) = y$$
>
>
>
Here $c$ is a convex function, $z(t)$ are paths with initial and final points given. They claim the infimum is $c(y-x)$ and this follows from Jensen's... | https://mathoverflow.net/users/69441 | Inf of Jensen's inequality | Jensen's inequality $\int c(f)\,d\mu \ge c(\int f\,d\mu)$ is always equality when $f$ is constant, so you know that taking $dz/dt$ constant would saturate it. That means $z(t)$ has to be linear, and since you know the endpoints, $z$ is determined.
Also useful to know is that when $c$ is strictly convex, a constant (... | 7 | https://mathoverflow.net/users/4832 | 293770 | 129,248 |
https://mathoverflow.net/questions/293714 | 7 | Being motivated by [this problem](https://mathoverflow.net/questions/293682/is-a-locally-finite-union-of-g-delta-sets-a-g-delta-set/293703#293703), I am searching for an example of a first-countable regular topological space $X$ containing a closed discrete subset $D$, which is not $G\_\delta$ in $X$.
It is easy to s... | https://mathoverflow.net/users/61536 | Is there a first-countable space containing a closed discrete subset which is not $G_\delta$? | EDIT: fixed some error in the proof that the diagonal is not a $G\_\delta$, and added details. I hope that the proof is now correct.
I think that the following works. First I'll describe a first countable non-regular example, and then explain how it can be modified to obtain a regular one.
Endow $\omega\_1$ with th... | 6 | https://mathoverflow.net/users/29491 | 293779 | 129,252 |
https://mathoverflow.net/questions/293776 | 5 | If $X$ is a smooth projective variety over a field, $Z\subset X$ a smooth closed subvariety of codimension $d$, $X'\to X$ the blowup of $X$ along $Z$, there's the blowup formula
$$H^j(X'\_{et},\mathbf{Z}(n)) = H^j(X\_{et},\mathbf{Z}(n))\oplus\bigoplus\_{r=1}^{d-1}H^{j-2r}(Z\_{et},\mathbf{Z}(j-r)).$$
I must be missi... | https://mathoverflow.net/users/nan | Blowup formula for motivic cohomology | There is simply a typo in this formula, the expression
$H^{j - 2r}(Z\_{et},j-r)$ should be replaced by $H^{j - 2r}(Z\_{et},n-r)$. This is closesly related to the fact that as motives
$M(\mathbb{P}(E)) = \oplus\_{r = 0}^{rank(E)} M(X)(r)[2r]$ for a vector bundle $E$ over a smooth variety $E$. In this case the sum in t... | 2 | https://mathoverflow.net/users/115052 | 293780 | 129,253 |
https://mathoverflow.net/questions/293773 | 3 | Let $X$ be a scheme, $K^{\bullet}$ and $P^{\bullet}$ bounded complexes of abelian sheaves on $X\_{\rm ét}$.
I want to compute the hypercohomology:
$$\mathbb{H}^\*(X\_{\rm ét}, K^{\bullet}\otimes^L\_{\mathbf{Z}}P^{\bullet})$$
Is there a spectral sequence relating this to the étale hypercohomologies of $K^{\bullet}... | https://mathoverflow.net/users/nan | Spectral sequence for tensor product of complexes | I think that this situation is not quite suitable for spectral sequences. Having such a spectral sequence would be related to some compatibility of (derived) global sections and tensor product. In general there is a map
$R\Gamma(X,A) \otimes^L R\Gamma(X,B) \to R\Gamma(X, A \otimes^L B)$ but I don't think that it is ev... | 0 | https://mathoverflow.net/users/115052 | 293782 | 129,254 |
https://mathoverflow.net/questions/293767 | 11 | This is different from $C$ being dualizable ($[C,D] = C^\vee \otimes D$). (**EDIT:** It turns out to be the same -- see Mike Shulman's answer!) But for example, if $C$ is a locally free sheaf of finite rank on a scheme/locally ringed space $X$, then $C$ has this property in the category of quasicoherent sheaves, or in... | https://mathoverflow.net/users/2362 | What do you call $C$ if $[D,C] = D^\vee \otimes C$ for all $D$? | First of all, a nitpick: the condition "$[D,C] = D^\vee\otimes C$" should be stated more precisely as "the canonical map $D^\vee\otimes C \to [D,C]$ is an isomorphism". Now as you mentioned in a comment, it's well-known that the $\forall C$ version of this condition is equivalent to dualizability of $D$, and indeed the... | 12 | https://mathoverflow.net/users/49 | 293791 | 129,259 |
https://mathoverflow.net/questions/293799 | 2 | If $A\_i$ are commuting normal matrices, then
$$\displaystyle\sup\_{\|y\|=1}\bigg(\displaystyle\sum\_{i=1}^n|\langle A\_iy,y\rangle|^2\bigg)=\displaystyle\sup\_{\|y\|=1}\bigg(\displaystyle\sum\_{i=1}^n\|A\_iy\|^2\bigg)\;.$$
>
> Is the converse of this result true?
>
>
>
| https://mathoverflow.net/users/113054 | Converse of a result related to normal matrices | no, just take $A\_1$ and $A\_2$ a pair of unitary operators with one common eigenvector $x\_0$; the right-hand side of the equation equals $2$ for all $x$ and the left-hand side reaches this maximum of $2$ for $x=x\_0$, but there is no reason why $A\_1$ and $A\_2$ should commute.
---
example:
$$A\_1=\left(
\begin... | 4 | https://mathoverflow.net/users/11260 | 293800 | 129,262 |
https://mathoverflow.net/questions/293793 | 6 | I heard of a result on Riemannian manifolds but I couldn't find a good reference:
>
> For every point $p$ on a complete Riemannin manifold there exists a closed geodesic passing through $p$ with length two times the injective radius at $p$.
>
>
>
Thanks!
| https://mathoverflow.net/users/51663 | A question on closed geodesics | The statement is not true.
First, a comment on terminology. There are two related notions for a geodesic that passes twice through a point $p$: a *periodic geodesic* and a *geodesic loop at $p$*. The former is smooth at all points, while the latter may be non-smooth at $p$. Sometimes periodic geodesics are called *... | 15 | https://mathoverflow.net/users/1573 | 293801 | 129,263 |
https://mathoverflow.net/questions/293823 | 0 | As a follow-up to [What are the Applications of Hypergraphs](https://mathoverflow.net/questions/13750/what-are-the-applications-of-hypergraphs), the linked article:
[Learning with Hypergraphs: Clustering, Classification, and Embedding](https://www.microsoft.com/en-us/research/publication/learning-hypergraphs-clusteri... | https://mathoverflow.net/users/13313 | Why not have many edges instead of one edge connecting to many nodes (in hypergraphs) | The hypergraph gives strictly more information than the graph: it can distinguish between the case where Larry, Curly, and Moe wrote a joint paper from the case where Larry and Curly wrote a paper and Curly and Moe wrote a different paper.
| 3 | https://mathoverflow.net/users/6043 | 293824 | 129,269 |
https://mathoverflow.net/questions/293803 | 3 | A topological space $(X,\tau)$ is said to be *homogeneous* if for $x,y\in X$ there is a homeomorphism $\varphi: X\to X$ such that $\varphi(x) = y$. Let us call a topological space $(X,\tau)$ *homogenizable* if there is a homogeneous space $X\_h$ and a continous map $e\_X: X \to X\_h$ such that for any homogeneous space... | https://mathoverflow.net/users/8628 | Non-homogenizable topological spaces | This problem is classical and is discussed in $\S 3$ of [this paper](http://www.mathnet.ru/links/6dbb3e53dec0d3d2a368270f9aee4a2e/rm2424.pdf) of Arkhangelski. In particular, in (3.24) he mentions the following
**Theorem (Okhromeshko, 1983).** Each topological space $X$ is a retract of its free homogeneous space $H(X... | 6 | https://mathoverflow.net/users/61536 | 293825 | 129,270 |
https://mathoverflow.net/questions/293826 | 6 | Let $G$ be the rational points of a connected, reductive group over a $p$-adic field $F$. Let $S$ be a maximal split torus of $G$ with $\Delta$ a set of simple roots corresponding to a minimal parabolic. Let $(\pi,V)$ be an irreducible, admissible representation of a Levi subgroup $M$ of $G$. For $P = MN$ corresponding... | https://mathoverflow.net/users/38145 | Making sense out of intertwining operators defined by a vector valued integral | It's not really that the space of locally constant, compactly-supported (complex-valued) functions on a p-adic (or other totally disconnected) group has *no* topology. Rather, it has a canonical topology with some convenient features. That is, that space is a "strict" (filtered) colimit of finite-dimensional spaces. Th... | 6 | https://mathoverflow.net/users/15629 | 293833 | 129,272 |
https://mathoverflow.net/questions/293828 | 6 | Let f be a smooth map from a (compact,oriented) surface S to a (compact, oriented) 3-manifold M. Suppose that I have an embedded (non-contractible) loop $\gamma$ in my surface $S$, can I find an (immersed) loop $\gamma'$ freely homotopic to $f \circ \gamma$ which is disjoint from $im(S)$?
| https://mathoverflow.net/users/120035 | General position for map from surface to 3-manifold | In general, no you cannot. Consider one dimension down. Take two curves on a torus, intersecting transversely in a point. One of the curves cannot be homotoped to be disjoint from the pair of curves. Now, cross with a circle $S^1$ to get $T^3$. The pair of curves crossed with $S^1$ is an immersed surface (two immersed ... | 9 | https://mathoverflow.net/users/1345 | 293835 | 129,273 |
https://mathoverflow.net/questions/293832 | 7 | Consider the collection of symmetric groups $\{\Sigma\_n\}\_{n\geq1}$ as a semi-simplicial set (i.e. a simplicial set without degeneracies) as follows. Consider $i\in\{1,\dots,n+1\}$ and $\pi\in\Sigma\_{n+1}$ represented as a sequence $(\pi(1),\dots,\pi(n+1))$, then
$$d\_{i-1}(\pi)=(\pi(1)-\epsilon\_1,\dots, \widehat{\... | https://mathoverflow.net/users/43574 | Homotopy type of the semi-simplicial set of symmetric groups | It is contractible. To see this first observe that it is simply connected. It has 2 arcs $a = (1,2)$ and $b = (2,1)$, however the tirangle $(3,1,2)$ gives us the relation that $ab = b$ and symmetrically $ba = a$ so $\pi\_1 = \{1\}$ for this space. To see that the homology vanishes you can use the homotopy operator on t... | 11 | https://mathoverflow.net/users/115052 | 293837 | 129,274 |
https://mathoverflow.net/questions/293842 | 0 | let $\Omega:=\left( a,b\right) \subset\mathbb{R}$, and suppose
$f:\Omega\rightarrow\left[ 0,\infty\right) $ is a bounded (Lebesgue)
measurable function, with $f\not \equiv 0$ almost everywhere. It is *not* true
in general that there exists some interval $I\subset\Omega$ such that
*essinf*$\_{I}f>0$.
My question is: ... | https://mathoverflow.net/users/115976 | On the essential infimum (over subdomains) of nonnegative measurable functions | One can construct a measurable $f$ with $f > 0$ almost everywhere yet $\operatorname{essinf}\_I f = 0$ for every interval. In particular we have $\operatorname{supp} f = \Omega\_+ = (a,b)$.
Enumerate as $(q\_n, r\_n)$ the subintervals of $(a,b)$ having rational endpoints. For each $n$, let $I\_n$ be a subinterval of ... | 1 | https://mathoverflow.net/users/4832 | 293849 | 129,277 |
https://mathoverflow.net/questions/293831 | 2 | An *overring* of an integral domain is a domain lying between it and its quotient field. Is it possible to have an overring $S$ of an integral domain $R$ such that $S$ has two maximal ideals $\mathfrak{m}$ and $\mathfrak{n}$ with $R \cap \mathfrak{m} \subsetneq R \cap \mathfrak{n}$? I think this should be possible but ... | https://mathoverflow.net/users/17218 | An overring of an integral domain having maximal ideals contracting to strictly comparable primes | Let $k$ be a field and start with $S\_0:=k[x,y]$. (Ultimately, $S$ will be a localization of $S\_0$). Let $\mathfrak{m}\_0=(y)$ and $\mathfrak{n}\_0=(x,y-1)$. In the affine plane $P:=\mathrm{Spec}(S\_0)$, these are respectively the generic point of the $x$-axis and the closed point $(0,1)$.
Put $z=xy\in S\_0$ and $R... | 2 | https://mathoverflow.net/users/7666 | 293854 | 129,279 |
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