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https://mathoverflow.net/questions/386206 | 5 | I've been reading [Michael Shulman's](https://homotopytypetheory.org/2013/08/08/spectral-sequences/) blog posts defining cohomology in homotopy type theory, and I'd like to understand (using HoTT) why cohomology of BG is group cohomology.
if I understand correctly, given a parametrized spectrum (i.e. a fibration by s... | https://mathoverflow.net/users/56938 | Using HoTT, why is twisted cohomology of BG group cohomology? | There's not really much to say (which is why I'm marking this answer as CW):
As Phil Tosteson commented, forming $\Pi$-types is right adjoint to the constant family map $\mathcal{U} \to \mathcal{U}^{BG}$ (and $\Sigma$-types gives the left adjoint). So that's the universal property of invariants (and co-invariants).
... | 3 | https://mathoverflow.net/users/2004 | 386274 | 160,524 |
https://mathoverflow.net/questions/386282 | 8 | I've just stumbled on something that seems either too good to be true,
or else too good for me not to have heard of it before.
It has to do with the basepoint forgetting map
$$
u: [A, M] \to \langle A, M \rangle,
$$
where $A$ is a pointed space, $M$ is a topological monoid,
and $\langle A, M\rangle$
indicates unpoi... | https://mathoverflow.net/users/3634 | Pointed versus unpointed maps into a topological monoid | As is implicitly pointed out in the comments, you really want to assume that $X$ ($=M$) is path connected. And then your analysis is fine. Note that $M$ will then wish to be equivalent to $\Omega BM$, so those pointed homotopy classes then look like $\langle \Sigma A, BM \rangle$ and fundamental group issues will go aw... | 12 | https://mathoverflow.net/users/102519 | 386286 | 160,528 |
https://mathoverflow.net/questions/386299 | 3 | In the early 1930s, van Cittert published a deconvolution method. Although his method was not perfect but it is the forefather of many improved spectral deconvolution methods. The basic idea is that if we know how the instrument distorts an input, we can rectify the experimentally observed spectrum. It is an approach f... | https://mathoverflow.net/users/142414 | van Cittert deconvolution method | Let me answer your two questions in reverse order:
b) The German word [Zackenfunktion](https://books.google.nl/books?id=vai7wiq5HIYC&pg=PA1695) ("spiky function") refers to the [Dirac delta function,](https://en.wikipedia.org/wiki/Dirac_delta_function) more commonly denoted with a lower case $\delta$.
a) The delta ... | 2 | https://mathoverflow.net/users/11260 | 386300 | 160,532 |
https://mathoverflow.net/questions/386257 | 1 | Suppose $X \sim Pois(\lambda)$. I'm interested in an upper bound on the ratio, $$\dfrac{P(X \leq n)}{P(X \leq n-1)}\,,\,\,\text{for $n=1,2,3,...$}$$ Observe that, the ratio is $>1$ & as $n \to \infty,\,$ the ratio $\to 1$. Thus, It is interesting to see if there exists some constant $K>1$ (depending on whether $\lambda... | https://mathoverflow.net/users/165072 | Upper bound on the ratio of Poisson CDFs | Clearly $\Pr[X\le n-1]\ge \Pr[X=n-1]$ so you have the ratio
$$
1+\frac{\Pr[X=n]}{\Pr[X\le n-1]}\le
1+\frac{\Pr[X=n]}{\Pr[X=n-1]}
=1+\frac{\lambda^n/n!}{\lambda^{n-1}/(n-1)!}
=1+\lambda/n.
$$
Now
$$
1+\lambda/n \le \exp(\lambda/n).
$$
Hence you can take $K=e^\lambda$ in your example.
| 1 | https://mathoverflow.net/users/5429 | 386322 | 160,539 |
https://mathoverflow.net/questions/386323 | 1 | For $n\geq1$, the largest solution to this lovely equation is a local extremum on a function related to the Fibonacci sequence:
$$\sum\_{k=1}^{n} k{(-1)^{k}} \cdot \frac{\sin(\frac{k\pi}{x} )}{3+2\cos(\frac{k\pi}{x} )} = 0$$
For $n=1$, the largest solution is $1$. For $n=2$, the largest solution is:
$$\frac{\pi}{... | https://mathoverflow.net/users/174962 | Can the solutions to this beautiful equation always be expressed in terms of algebraic numbers? | Yes, you can write it in terms of an algebraic number as you desire. Both $\sin\frac{k\pi}{x}$ and $\cos\frac{k\pi}{x}$ can be written as polynomials in $\cos\frac{\pi}{x},\sin\frac{\pi}{x}$, which [can in turn be written](https://en.wikipedia.org/wiki/Weierstrass_substitution#The_substitution) as rational functions in... | 5 | https://mathoverflow.net/users/30186 | 386329 | 160,541 |
https://mathoverflow.net/questions/386166 | 1 | Let $X:=\mathbb{P}^2\_K$ with $K$ algebraically closed field. Take $p\in X$ a point and $\mathcal{I}\_p$ its ideal sheaf. One can prove (using Serre Duality and the exact sequence defining $\mathcal{I}\_p$) that $Ext^1((\mathcal{I}\_p)\_p,\mathcal{O}\_{X,p})\cong K$, so there exists a non trivial extension \begin{equat... | https://mathoverflow.net/users/129919 | Computing Ext sheaves over complex projective plane | As stated above, from the LES in $\mathcal{E}xt$ we have
$$ \mathcal{E}xt^i(\mathcal{E}, \mathcal{O}\_X) \simeq \mathcal{E}xt^i(I\_p, \mathcal{O}\_X) $$
for all $i \geq 2$. How can we compute the term on the left hand side? The key is to observe the following fact:
>
> A point in $\mathbf{P}^2$ is a complete inters... | 5 | https://mathoverflow.net/users/21278 | 386337 | 160,544 |
https://mathoverflow.net/questions/386334 | 3 | I have found that a lot of research has been done in rationality problem for algebraic tori. (For example, <https://arxiv.org/abs/1210.4525>). So I got to wonder what historical context or elementary motivation the problem has.
| https://mathoverflow.net/users/164547 | Historical context of rationality problem for algebraic torus | [Algebraic tori – thirty years after](https://arxiv.org/abs/0712.4061) gives some historical context:
>
> The rationality problem goes back to the study of [Pythagorean triples](https://en.wikipedia.org/wiki/Pythagorean_triple) : Describe the set of solutions of a given system of
> polynomial equations by rational ... | 2 | https://mathoverflow.net/users/11260 | 386353 | 160,547 |
https://mathoverflow.net/questions/386335 | 3 | While I'm still trying to understand the issues raised on my [previous question](https://mathoverflow.net/questions/380223/how-are-clifford-algebras-and-spinors-used-to-study-the-ising-model), I decided to first address the Clifford algebra used on formulating the famous Dirac equation. In this context, what is found i... | https://mathoverflow.net/users/150264 | What is the relationship between the Dirac algebra and the Clifford algebra? | Here's a partial answer.
Heads up: that Wikipedia article seems to be written by physicists for physicists, so IMHO it's not the best source to learn these things. Greub's "Multilinear algebra" has a chapter on Clifford algebras. I believe my favorite one used to be R. Shaw's "Linear algebra and group representations... | 4 | https://mathoverflow.net/users/103164 | 386361 | 160,548 |
https://mathoverflow.net/questions/386366 | 0 | Does anyone know a good reference to start studying Action-Angle coordinates?
Thank you in advance !
| https://mathoverflow.net/users/157604 | Reference for action-angle coordinates | V. I. Arnold, **Mathematical Methods for Classical Mechanics**, p. 280.
L. D. Landau and E. M. Lifshitz, **Mechanics**, p. 157.
| 4 | https://mathoverflow.net/users/13268 | 386368 | 160,550 |
https://mathoverflow.net/questions/386365 | 8 | By the work of Verdier, we know that cones in a triangulated category $\mathcal{T}$ are functorial if and only if $\mathcal{T}$ is semisimple abelian. However, in [these notes](https://pages.uoregon.edu/njp/garcia.pdf), it is said that
>
> In the context of triangulated categories, it is well known that cones are n... | https://mathoverflow.net/users/152554 | Functorial kernel in derived category | Let $\mathcal{C}$ be a stable $\infty$-category. Then $\mathcal{C}$ has a homotopy category $h \mathcal{C}$, which is triangulated. The collection of morphisms $f: X \rightarrow Y$ of $\mathcal{C}$ can be organized into an $\infty$-category $\mathrm{Fun}( \Delta^1, \mathcal{C} )$. The operation $f \mapsto \mathrm{Cone}... | 21 | https://mathoverflow.net/users/7721 | 386369 | 160,551 |
https://mathoverflow.net/questions/386348 | 5 | I am interested in systematically studying the theory of vector-valued distributions. The original two papers due to Laurent Schwartz entitled *Théorie des distributions à valeurs vectorielles. I & II* (1957-58) are written in French. Occasionally I have read mathematics papers in French when needed, with the help of G... | https://mathoverflow.net/users/175875 | English translation of Schwartz's papers on vector-valued distributions | 1. These papers have not been translated, as far as I know, however there exist lecture notes in english of courses by Schwarz on this topic:
• [Introduction to the Theory of Distributions](https://www.jstor.org/stable/10.3138/j.ctt1vxmd4v)
• [Lectures on Partial Differential Equations and Representations of Sem... | 3 | https://mathoverflow.net/users/11260 | 386374 | 160,554 |
https://mathoverflow.net/questions/386343 | 1 | In the article **AN EXAMPLE INVOLVING BAIRE SPACES** (<https://www.ams.org/journals/proc/1975-048-01/S0002-9939-1975-0362249-1/S0002-9939-1975-0362249-1.pdf>) of H. E. White Jr. it is shown that, assuming CH, there exists a Baire space $Y$ such that $Y\times Y$ is not Baire.
For the construction of this space, is use... | https://mathoverflow.net/users/138770 | CH and the density topology on $\mathbb{R}$ | According to your first reference your space is dense with respect to the density topology; this implies that, in $Y$, the closure of $Y\cap(0,\infty)$ is $Y\cap[0,\infty)$; the latter set is not open in $Y$ because its upper density at $0$ is less than or equal to $\frac12$.
That last sentence needs some amplificati... | 3 | https://mathoverflow.net/users/5903 | 386378 | 160,555 |
https://mathoverflow.net/questions/386380 | 4 | Are there any examples of finite-dimensional complex non-semisimple non-commutative symmetric Frobenius algebras? Or can one show that none exist?
I went through [this](https://arxiv.org/pdf/0910.0932.pdf) list of all complex associative algebras up to dimension $4$ and couldn't find any non-commutative non-semisimpl... | https://mathoverflow.net/users/115363 | Are there examples of finite-dimensional complex non-semisimple non-commutative symmetric Frobenius algebras? | Given any finite dimensional algebra $A$, consider the linear dual $\hat{A}= \hom(A, k)$ as an $A$-$A$-bimodule. Then $R = A \oplus \hat{A}$ may be equipped with an algebra structure as follows:
$$(a, x) \cdot (b,y) = (ab, x \cdot b + a \cdot y)$$
for $a,b \in A$ and $x,y \in \hat{A}$. The algebra $R$ has a natural... | 8 | https://mathoverflow.net/users/184 | 386381 | 160,556 |
https://mathoverflow.net/questions/386389 | 5 | I am not sure whether this was asked before, but I didn't find a reference in GAP system documentation on how to print the history of the command line (Ubuntu installation).
For instance:
```
gap> G := Group((1,2)(3,4),(1,2,3));
> irr := Irr(G);
```
How to save or print the list of my previous commands (similar ... | https://mathoverflow.net/users/101411 | Get the commands history from GAP system | There are three commands that will do all variations of this:
1. `LogTo("filename.txt")` will save all subsequent **input and output** to a file with the specified name.
2. `InputLogTo("filename.txt")` will save all subsequent **input** to a file with the specified name.
3. `OutputLogTo("filename.txt")` will save all... | 6 | https://mathoverflow.net/users/120914 | 386391 | 160,559 |
https://mathoverflow.net/questions/386393 | -2 | Is there any way to calculate the equilibrium (stationary) distribution for a **weighted directed acyclic** graph?
Some references emphasized adjacency matrix to be symmetric.
<https://arxiv.org/abs/1012.1211#content>
[Graph Example](https://i.stack.imgur.com/2lrPd.jpg)
| https://mathoverflow.net/users/166434 | Stationary distribution of a weighted directed acyclic graph | If the graph is acyclic, then either you have sinks (nodes with no edge out), or your graph is infinite.
If you have sinks, then this would indicate that you have a loop of probability $1$ at this node (you have to do something of your probability). In such a configuration, only the sink nodes can have a non-zero sta... | 1 | https://mathoverflow.net/users/174620 | 386396 | 160,561 |
https://mathoverflow.net/questions/386395 | 3 | I think there is an (possible) error in Villani's monograph titled "Hypocoercivity". To be specific, in page 48, he wrote "For the second term in (6.9), we use the identity $$\nabla\cdot(Df\nabla u) = f\nabla\cdot(D\nabla u) + \langle D\nabla u,\nabla f\rangle = f\nabla\cdot(D\nabla u) + f\langle D\nabla u,\nabla \log ... | https://mathoverflow.net/users/163454 | A possible error in Villani's monograph "Hypocoercivity" | On [the PDF version on his website](https://cedricvillani.org/sites/dev/files/old_images/2012/07/043.Hypoco.pdf), he writes the left-hand side $$-\int f\left\langle C\left(\frac{\nabla\cdot(Df\nabla u)}{f}\right), C'u\right\rangle$$
which probably means it was a typo, the $C$ should be down the fraction. And there, you... | 4 | https://mathoverflow.net/users/174620 | 386398 | 160,562 |
https://mathoverflow.net/questions/386407 | 6 | The question is in the title: The book
Godement-Jacquet "Zeta functions of simple algebras"
is from 1971. Has there ever been a textbook introduction to this material, or at least part of it? (but beyond just Tate's thesis).
| https://mathoverflow.net/users/175904 | Contemporary introduction to Godement-Jacquet "Zeta functions of simple algebras" | As you mention in your question, for $\mathrm{GL}\_1$, this is just Tate's thesis revisited.
The only exposition on this that I know of occurs in "[Automorphic Representations and $L$-Functions for the General Linear Group](https://doi.org/10.1017/CBO9780511973628)" by Dorian Goldfeld and Joseph Hundley. In particula... | 12 | https://mathoverflow.net/users/3803 | 386411 | 160,564 |
https://mathoverflow.net/questions/386412 | 3 | Is there a finitely **presented** group with exactly linear conjugacy growth which is not virtually cyclic?
Here, conjugacy growth function $c(n)$ counts the number of conjugacy classes in the $n$-ball, and it is exactly linear if $c(n) \sim An$ for some constant $A \in (0, \infty)$.
Note that by the proof of Corol... | https://mathoverflow.net/users/175905 | Linear conjugacy growth function | The conjugacy growth function $f(n)$ is the number of conjugacy classes intersecting a ball of radius $n$. So if you take Ivanov's or Osin's examples of finitely generated groups with finite number of conjugacy classes, the groups are not virtually cyclic, but the conjugacy growth function is constant. If you take the ... | 1 | https://mathoverflow.net/users/157261 | 386413 | 160,565 |
https://mathoverflow.net/questions/386400 | 3 | For any integer $k>1$ we say a hypergraph $H=(\omega,E)$ where $E\subseteq {\cal P}(\omega)$ is $k$-*regular* if $|e|=k$ for all $e\in E$. Moreover, we say $H$ is *linear* if $|e\_1\cap e\_2|\leq 1$ for all $e\_1\neq e\_2 \in E$.
Zorn's lemma shows that whenever $(\omega,E)$ is linear and $k$-regular, there is $E'\su... | https://mathoverflow.net/users/8628 | Chromatic number of maximal linear $k$-regular hypergraphs on $\omega$ | The answer is negative. Suppose $3\le k\lt\omega$. As I showed in my answer to [this question](https://mathoverflow.net/questions/363267/chromatic-number-of-regular-linear-hypergraphs-on-omega/363283), there is a linear $k$-hypergraph $(\omega,E)$ with chromatic number $\aleph\_0$; of course it can be extended to a max... | 2 | https://mathoverflow.net/users/43266 | 386419 | 160,568 |
https://mathoverflow.net/questions/386255 | 3 | The Banach-Mazur distance between two centrally symmetric convex bodies $K,L\in\mathbb{R}^n$ can be defined as
$$ d(K,L) = \inf \{ r : \exists T\colon \mathbb{R}^n \to \mathbb{R}^n \text{ linear such that } T K \subset L \subset r T K \} .$$
If $B^1=\mathrm{conv}\{\pm e\_1, \ldots, \pm e\_n\}$ and $B^\infty=[-1,1]^n$ d... | https://mathoverflow.net/users/29928 | Banach Mazur distance between the cube and the cross-polytope in the dimensions for which a Hadamard matrix exists | As pointed in comment by Bill Johnson, this equality is (trivially) **disproved** for $n=2$, since in that case $M B^1=B^\infty$ and thus the distance is $1$.
Less trivially, it also fails for $n=8$, in which case the distance is $2.5 < 2.82... = \sqrt{8} $, see [Fei Xue (2017, arXiv)](https://arxiv.org/pdf/1705.0135... | 1 | https://mathoverflow.net/users/29928 | 386424 | 160,572 |
https://mathoverflow.net/questions/386406 | 1 | Let $u(x)=\alpha+\beta U(x)$, where $U(x)=|x|^{2-m}$ ($N\geq3$) is the fondamental harmonic function, $\alpha<0$ and $\beta>0$. We know that $u$ is superharmonic on $\mathbb{R}^m$ and harmonic on $\mathbb{R}^m\setminus\{0\}$. Let $a>0$ be a given constant. Is it possible to modify $u$ on a neighborhood of infinity to o... | https://mathoverflow.net/users/100746 | Modifying a superharmonic function on a neighborhood of infinity | One way to find out that this is impossible is to note that $v = \overline u - \beta U$ is superharmonic in $\mathbb R^n$, and hence $V(r) = \int\_{\partial B\_r} v$ decreases with $r$. Since $V(r) = \alpha < 0$ for $r$ small enough, we have $V(r) < 0$ for all $r$, and so $v$ cannot have a positive limit at infinity. I... | 1 | https://mathoverflow.net/users/108637 | 386431 | 160,575 |
https://mathoverflow.net/questions/386441 | 5 | We consider finitely generated modules over an Artin algebra. Let $X$ be an indecomposable module and let $f:X \longrightarrow X \oplus X$ a monomorphism. Must $f$ always be a split monomorphism?
| https://mathoverflow.net/users/145920 | Must the inclusion of an indecomposable module in the direct sum of two copies always split? | Yes, it must be split.
Since $M$ is an indecomposable module for an Artin algebra, its endomorphism ring $E$ is a local ring with nilpotent Jacobson radical $J(E)$. Say $J(E)^n=0$.
Let the monomorphism $\varphi:M\to M\oplus M$ be given by $\varphi(m)=(\alpha(m), \beta(m))$. If either $\alpha$ or $\beta$ is an isomo... | 11 | https://mathoverflow.net/users/22989 | 386462 | 160,581 |
https://mathoverflow.net/questions/386471 | 4 | Let $X$ be a smooth projective variety over a field $k$. Then if $\ell\neq \text{char} k$, $k$ is finite, and $X$ is an abelian variety it was shown by Weil that the $\ell$-adic cohomology of $X\_{k^{sep}}$ is a semisimple representation of $G\_k$. The conjectural reason for this, if I've understood this correctly, is ... | https://mathoverflow.net/users/152554 | Semisimplicity of the étale cohomology mod $p$ | Take an elliptic curve $E$ with a (single) rational $2$-torsion point, say $y^2 = x(x^2+1)$ over $\mathbb{Q}$ (or even $\mathbb{R}$). Then the Galois action on $E[2]$ factors through a group of order $2$ and so is not semisimple.
An easier example is the variety $x^2+1=0$ and $H^0(X,\mathbb{F}\_2)$.
If you insist t... | 7 | https://mathoverflow.net/users/175951 | 386474 | 160,583 |
https://mathoverflow.net/questions/386152 | 2 | We consider finitely generated modules over an Artin algebra. Let $X$ be an indecomposable module such that the radical $\text{rad} \,X$ is a submodule of the socle $\text{soc}\,X$. What can we say about $X$? In particular, can we bound the length of $X$?
| https://mathoverflow.net/users/145920 | Indecomposable modules such that the radical is a submodule of the socle | No. Let $(R,\mathfrak m)$ be commutative local Artin ring, then the radical of $M$ is $\mathfrak mM$ and your condition is equivalent to $\mathfrak m^2M=0$. One can not bound the length of such indecomposable modules in general. In particular, if you take $R=k[x,y]/(x,y)^2$ then it has $\mathfrak m^2=0$, so any module ... | 1 | https://mathoverflow.net/users/2083 | 386480 | 160,586 |
https://mathoverflow.net/questions/386464 | 12 | I've listened to many interviews and lectures of Alain Connes, in which he says something which goes roughly as follows
*"Every non-commutative algebra has its own time (evolution of), by which I mean a one-parameter group."*
I find this statement somewhat mysterious and intriguing at the same time.
>
> **Quest... | https://mathoverflow.net/users/78539 | Making sense of "every non-commutative algebra has its own internal time evolution (aka a one-parameter group)"? | Given any von Neumann algebra $M$, we can define its noncommutative $\def\L{{\cal L}} \L^p$-spaces $\L^p(M)$ for any $\def\C{{\bf C}} p∈\C$ such that $\Re p≥0$.
Here I use the notation $\L^p:={\rm L}^{1/p}$, where the right side is the usual notation from measure theory and real analysis.
(This notation is explained in... | 13 | https://mathoverflow.net/users/402 | 386481 | 160,587 |
https://mathoverflow.net/questions/386477 | 7 | Suppose $E/\mathbb Q$ is a non CM elliptic curve and we look at the number field $K\_d$ generated by the $d$-torsion of $E$. What is known about the (complete) splitting of small primes in $K\_d$?
More precisely, since $|E(\mathbb F\_p)| \sim p+1$, if $p$ were a completely split prime, we would need $d^2 < p+1$ (appr... | https://mathoverflow.net/users/58001 | Splitting of small primes in number fields generated by the torsion of elliptic curves | Let's discuss the $\mathbb G\_m$ question first. For $p$ to split completely in the field generated by the $d$-torsion of $\mathbb G\_m$, i.e. the field generated by the $d$th roots of unity, a necessary and sufficient condition is that $\mathbb F\_p$ contains the $d$th roots of unity, i.e. $p \equiv 1\mod d$. This req... | 8 | https://mathoverflow.net/users/18060 | 386482 | 160,588 |
https://mathoverflow.net/questions/385805 | 8 | Let $\Omega$ be the set/type of truth values. We're using constructive logic. Define
$AC\_{0, 0} = \forall P : \mathbb{N}^2 \to \Omega, (\forall n \in \mathbb{N}, \exists m \in \mathbb{N}, P(n, m)) \to \exists f : \mathbb{N} \to \mathbb{N}, \forall n \in \mathbb{N}, P(n, f(n))$.
It is well-known that $AC\_{0, 0}$ i... | https://mathoverflow.net/users/175409 | A weak form of countable choice | It turns out, I believe, that there's actually a fairly simple counterexample.
The example is sheaves on the topological space given by the product of countably many copies of $\mathbb{N}$ with downwards closed set topology. Explicitly, the underlying set of the space is $\mathbb{N}^\mathbb{N}$ and a set $U \subset \... | 6 | https://mathoverflow.net/users/30790 | 386483 | 160,589 |
https://mathoverflow.net/questions/386484 | 10 | I'm not very familiar with dg algebras (not necessarily commutative) and I'm wondering if any $E\_1$ algebra in the sense of infinity categories (i.e. monoid in the stable category of $R-Mod$) over $R$ some commutative ring (discrete) can be described by a dg algebra over R? Specifically I have some $E\_1$ algebra in m... | https://mathoverflow.net/users/136287 | Can any $E_1$ algebra over $\mathbb{F}_p$ be modeled as a dg algebra? | Yes, this is precisely the content of Theorem 7.11 in [arXiv:1410.5675](https://arxiv.org/abs/1410.5675), which should be combined with §7.4 of [arXiv:1510.04969](https://arxiv.org/abs/1510.04969).
In fact, the cited results prove this for any nonsymmetric operad in chain complexes over a commutative ring,
and are also... | 9 | https://mathoverflow.net/users/402 | 386487 | 160,591 |
https://mathoverflow.net/questions/386494 | 11 | *Edit (March 24): My first question has been answered nicely, but I am still looking for an answer to the second one.*
Due to the Kan–Thurston theorem, the homology of an arbitrary group can be anything you want.
Using the Rips complex, we can see that hyperbolic groups are $F\_{\infty}$ and, if torsion-free, even ... | https://mathoverflow.net/users/14233 | Constraints on the homology of amenable groups |
>
> Is every finitely presented torsion-free amenable group of homotopical type F?
>
>
>
It's false even for metabelian groups.
For $i=1,2,3$, let $G\_i$ be a copy of a solvable Baumslag-Solitar group, say $\langle t\_i,x\_i|t\_ix\_it\_i^{-1}=x\_i^2\rangle$. Start with $H=G\_1\times G\_2\times G\_3$. It has a ... | 11 | https://mathoverflow.net/users/14094 | 386496 | 160,592 |
https://mathoverflow.net/questions/386493 | 2 | Let $M$ be a real symmetric matrix of size $N$ with its components $M\_{ij}$ following a normal distribution centered around 0.
Let $x\in\mathbb{R}^N$ be an eigenvector of $M$ with eigenvalue $\lambda\in\mathbb{R}$:
$$\sum\_j M\_{ij}x\_j=\lambda x\_i$$
I know that in that case the eigenvectors of different eigenval... | https://mathoverflow.net/users/142153 | Distribution of eigenvectors of random matrices and link with the components of the matrix | • For large $N$ the elements of an eigenvector have independent Gaussian distributions with zero mean and variance $1/N$.
• To find $\mathbb{E}[M\_{ij}x\_j]$ I decompose the matrix $M$ into eigenvalues and eigenvectors,
$$M\_{ij}=\sum\_{k} O\_{ki}\lambda\_k O\_{kj},$$
with an orthogonal matrix $O$. Then
$$\mathbb{E}[... | 1 | https://mathoverflow.net/users/11260 | 386503 | 160,594 |
https://mathoverflow.net/questions/386511 | 3 | A fundamental result in topology is that the $n$-sphere is not a retract of the $n+1$-ball. It implies that the $n$-sphere is not an absolute retract.
Is there a generalization from the sphere to closed manifolds (compact manifolds without boundary)? It would be the statement that no closed manifold is an absolute re... | https://mathoverflow.net/users/175982 | Closed manifolds are not absolute retracts? | A metrizable space is an absolute retract (AR) if and only if it is an absolute neighbourhood retract (ANR) and it is contractible.
Closed manifolds are not contractible (if $\dim M=n$ look at $H\_n(M)$) hence they are not ARs. Note however that contractibility is the only obstruction for manifolds, since every topol... | 12 | https://mathoverflow.net/users/49381 | 386513 | 160,596 |
https://mathoverflow.net/questions/386515 | 10 | [This is a question motivated by theoretical physics, so apologies if the language is rough...]
In three dimensions the spaces of invariant (or isotropic) tensors of rank $n$ have dimensions 1, 0, 1, 1, 3, 6, 15, 36, 91, 232, ... (see [MathWorld](https://mathworld.wolfram.com/IsotropicTensor.html)). Equivalently, thi... | https://mathoverflow.net/users/25145 | Basis of invariant tensors of rank n in three dimensions | Planar partitions with no singletons works. You need to pick for each $n>1$ some map with certain properties. One way to do this is to just fix a preferred trivalent tree of each size and interpret each vertex as a cross product. For example, one arbitrary choice gives the map $V^{\otimes n} \rightarrow \mathbb{C}$ giv... | 10 | https://mathoverflow.net/users/22 | 386518 | 160,597 |
https://mathoverflow.net/questions/386502 | 3 | We are given a unit origin-centered sphere $S$ in $\mathbb{R}^3$, and three points $\mathbf{x},\mathbf{y},\mathbf{z}\in S$. Let $\mathbf{h}$ be a point selected uniformly at random from $S$ and let $H$ be a hyperplane that passes through the center $\mathbf{0}$ of $S$ and has $\mathbf{h}$ as normal vector. Let $E\_x$ b... | https://mathoverflow.net/users/115803 | Random planes separating points in $\mathbb{R}^3$ | Note that the random vector $G/\|G\|$ is uniformly distributed on the unit sphere in $\mathbb R^n$, where $G$ is a standard Gaussian random vector in $\mathbb R^n$.
So, given an $n$-tuple $(x\_1,\dots,x\_n)$ of unit vectors in $\mathbb R^n$, the probability that the random hyperplane separates $x\_1$ from $x\_2,\dots... | 4 | https://mathoverflow.net/users/36721 | 386534 | 160,602 |
https://mathoverflow.net/questions/386498 | 7 | Let $K$ be the core model (below a Woodin cardinal). Let $j \colon K \to M$ be an elementary embedding, where $M$ is well founded. Under which conditions can we conclude that $j$ is an iterated ultrapower of extenders in $K$ (possibly a branch in an iteration tree)?
Are there generalizations of this result to larger ... | https://mathoverflow.net/users/41953 | The core model and elementary embeddings | Some remarks:
By Schindler's paper "Iterates of the core model", if $j:V\to N$ is elementary ($N$ transitive) and $N$ is closed under $\omega$-sequences, and $k:K\to K^N$ is the restriction of $j$, then $K^N$ is an iterate of $K$ and $k$ is the iteration map.
Note that if $M$ is a mouse modelling ZFC + ``$\delta$ i... | 8 | https://mathoverflow.net/users/160347 | 386535 | 160,603 |
https://mathoverflow.net/questions/386530 | 6 | On page 201 of Farb and Margalit's *Primer on Mapping Class Groups*, they explain why the mapping class group $\mathrm{Mod}(S)$ is torsion-free when $\partial S \neq \varnothing$. Here is my understanding of the argument:
Let $S$ be a surface with a hyperbolic metric and let $\phi\colon S \to S$ be an isometry fixing... | https://mathoverflow.net/users/153899 | Why is the mapping class group of a surface with nonempty boundary torsion-free? | I think the reason that Dehn twists enter is that we can take the differential of a (orientation-preserving) diffeomorphism $f$ of $S\_g$ that fixes a chosen basepoint $\ast \in S\_g$ at this point, and will get a map $d \colon \text{Diff}^+(S\_g,\ast) \to \text{GL}\_2^+(\mathbb R)$. The fiber of this fibration is then... | 8 | https://mathoverflow.net/users/14233 | 386538 | 160,604 |
https://mathoverflow.net/questions/386532 | 7 | Let $G$ be a finitely generated torsion-free nilpotent group. The Malcev completion of $G$ is a nilpotent Lie group $N$ into which $G$ embeds as a lattice. One way to construct this is to take the completion $\widehat{\mathbb{R}[G]}$ of the real group ring with respect to the augmentation ideal. This is a Hopf algebra,... | https://mathoverflow.net/users/175994 | Universal enveloping algebra of Malcev Lie algebra associated to nilpotent group | Yes, equivalently, for $A = \widehat{kG}$ where $k$ is of characteristic zero, the map $U = \widehat{U(\mathrm{Prim }A)} \to A$ is injective. In fact, this holds whenever $A$ is complete with respect to its augmentation ideal.
Proof: let $f: U \to A$ be the natural map, and let $J$ be the augmentation ideal of $U$.
... | 4 | https://mathoverflow.net/users/125523 | 386543 | 160,605 |
https://mathoverflow.net/questions/386449 | 4 | In a previous question [on mathoverflow](https://mathoverflow.net/questions/385992/backward-heat-equation-and-forward-perturbed-heat-equation-well-posed), I asked about the following:
Let $\Delta$ be the Laplacian on some compact interval $I$ of the real line with let's say Dirichlet boundary conditions.
The functi... | https://mathoverflow.net/users/119875 | Mapping properties of backward and forward heat equation | This is not an answer, but just an illustration of how bad things can be.
Assume for convenience and concreteness that $I = [0,\pi]$ and so we can decompose the solution $u$ in sine series $$ u(t,x) = \sum\_{k = 1}^\infty \hat{u}(t,k) \sin(k x) $$
Consider the case $f(x) = \cos(x)$ **which is real analytic**. Then
... | 2 | https://mathoverflow.net/users/3948 | 386545 | 160,607 |
https://mathoverflow.net/questions/385779 | 0 | **Question:**
given a complete symmetric graph $G(V,E)$ with $n$ vertices and edges $e\_{ij}$ having weight $\omega\_{ij}$, does there always exists a vector of vertex potentials $(\pi\_1,\,\dots,\,\pi\_n)$ that, when added to the weight of adjacent edges, guarantee that the minimum [1-tree](https://ieor.berkeley.edu... | https://mathoverflow.net/users/31310 | Relation of 1-trees to optimal tours | Just found the answer in the 1969 paper [THE TRAVELING-SALESMAN PROBLEM AND MINIMUM SPANNING TREES](https://www.cse.wustl.edu/%7Eychen/7102/Karp-TSP.pdf) by Held and Karp; in which it is shown that it is not always possible to find the otimal solution to the TSP instance by systematic variation of the vertex potentials... | 0 | https://mathoverflow.net/users/31310 | 386550 | 160,609 |
https://mathoverflow.net/questions/386539 | 2 | Given a matrix $A$ I need to select a "representative" subset of its columns so that the each non-selected column is as close as possible to a selected one.
Formally:
>
> Given $A \in \mathbb{R}^{d \times n}$ and $0<k<n$, find a subset of
> column indices $S=\{i\_1,i\_2,\ldots,i\_k\}$, so that $$
> \sum\_{j \not... | https://mathoverflow.net/users/22051 | Column subset selection with least angle optimization | You can solve this as a $k$-median (also known as $p$-median) problem. Each column is both a customer and a candidate facility location. Let $d\_{c,f}$ denote the cost of assigning customer $c$ to facility $f$. Let binary decision variable $x\_{c,f}$ indicate whether customer $c$ is assigned to facility $f$, and let bi... | 3 | https://mathoverflow.net/users/141766 | 386555 | 160,610 |
https://mathoverflow.net/questions/386298 | 1 | Kahler spaces are just certain singular spaces equipped with a Kahler metric in appropriate sense. I first came across it Demaily-Paun's classical paper [Numercical Characterization of the Kahler cone of a compact Kahler manifold](https://annals.math.princeton.edu/wp-content/uploads/annals-v159-n3-p05.pdf). However it ... | https://mathoverflow.net/users/142966 | reference request for singular Kahler space | There is a discussion of this topic in the recent book [Cycles analytiques complexes II : l'espace des cycles](https://smf.emath.fr/publications/cycles-analytiques-complexes-ii-lespace-des-cycles) by Barlet and Magnússon, chapter XII.3. An English translation (to be published by Springer) is forthcoming.
You will als... | 1 | https://mathoverflow.net/users/13168 | 386562 | 160,612 |
https://mathoverflow.net/questions/386447 | 18 | Parameters $a,b,c$ are given such that $c\leq\max(a,b)$. In an $a\times b$ board, two players take turns putting a mark on an empty square. Whoever gets $c$ consecutive marks horizontally, vertically, or diagonally first wins. (Someone must win because we use only one mark type.) For each triple $(a,b,c)$, who has a wi... | https://mathoverflow.net/users/83212 | Tic-tac-toe with one mark type | The case $a=1$ and $c=3$ is known as [Treblecross](https://en.wikipedia.org/wiki/Treblecross). It is an octal game with code .007 and there is some computational data available on [Achim Flammenkamp's webpage](http://wwwhomes.uni-bielefeld.de/achim/octal.html), but as far as I know, the game has not been analyzed compl... | 13 | https://mathoverflow.net/users/3106 | 386579 | 160,618 |
https://mathoverflow.net/questions/386064 | 2 | I have a set $F$ of vector fields. The commutator $[v, u]$ is linear in $v$ and $u$ point by point, i.e., for each couple of vector fields $v\in F$ and $u\in F$, there are two scalars $a$ and $b$ such that:
$$
[v, u] = a v + b u
$$
I remark that, by "scalar", I mean that $a$ and $b$ are functions of the place, i.e. the... | https://mathoverflow.net/users/138060 | Set of vector fields with a particular expression of the commutator | Extended comment as asked by OP.
Let's call your manifold $X$, $\dim X = n$ .
Your assumption is that there are $v\_1,\dots, v\_n\in \mathcal{T}(X)$ (vector fields) that are pointwise linearly independent and moreover
$[v\_i,v\_j] = a\_{i,j} v\_i + b\_{i,j} v\_j$ for $a\_{i,j}, b\_{i,j} \in C^{\infty}(X)$.
*Remark 1.... | 2 | https://mathoverflow.net/users/99042 | 386581 | 160,619 |
https://mathoverflow.net/questions/386586 | 1 | Write $X\_m \in \mathbb{R}^{m \times n}$ as a Gaussian ensemble, so that $(X\_m)\_{ij} \sim \mathcal{N}(0, 1)$ are independent and identically distributed. Assume that $m \geq n$. Write $X\_m = U\_m \Sigma\_m V\_m^T$ as the singular value decomposition, where $V\_m \in \mathbb{R}^{n \times n}$ are the right singular ve... | https://mathoverflow.net/users/166439 | Asymptotics of the right singular vectors as the number of rows diverge | Thanks to the rotation-invariance of the Gaussian distribution, both $U\_m$ and $V\_m$ can be taken to be Haar-distributed orthogonal matrices in $\mathbb{R}^{m\times m}$ and $\mathbb{R}^{n\times n}$ respectively. This is true for all $m$ and $n$ and not just in the limit.
| 1 | https://mathoverflow.net/users/1044 | 386600 | 160,623 |
https://mathoverflow.net/questions/386589 | 4 | Let $K$ be a local field, $K^{sep}$ its separable closure, $G = Gal(K^{sep}/ K)$ the Galois group and $C := \overline{K^{sep}}$ the completion with respect to the induced valuation.
In his paper on $p$-divisible groups, Tate proves that if $K$ is a $p$-adic field, then the continuous cohomology groups $H^{i}(G, C)$ a... | https://mathoverflow.net/users/16981 | Galois cohomology of separable closure | Yes, they are known: They vanish in degrees $i>0$, and for $i=0$ one gets the completed perfection of $K$.
Indeed, let $K'$ be the completed perfection of $K$. Then $G\_K=G\_{K'}$ as both perfection and completion do not change the etale site. But now $K'$ is already perfectoid, so the same techniques of almost mathe... | 4 | https://mathoverflow.net/users/6074 | 386607 | 160,626 |
https://mathoverflow.net/questions/386583 | 1 | Let $\mathbf{F}$ denote an M × N matrix whose entries are independent zero-mean complex random variables, the limiting eigenvalue distribution is given by the Marchenko Pastur law $MP\_{\beta}$, where $\frac{N}{M}\rightarrow \beta$.
It can be shown that the moments are giving by
\begin{equation}
\sum\_{i=1}^{k}{1 \ov... | https://mathoverflow.net/users/175894 | How to compute the first moment of the distribution of the convolution of Marcenko-Pastur law with a not iid matrix? | If I correct the typo's so that the MP distribution follows when $\mu=0$, the definitions should be:
$$f\_s(\lambda)=(1-\beta)^{+}\delta(\lambda)+{{\sqrt{(\lambda-a)^+(b-\lambda)^+}}\over{2\pi \lambda(1+\mu \lambda)}},$$
$$a=1+\beta+2\mu \beta-2\sqrt{\beta}\sqrt{(1+\mu)(1+\mu \beta)},$$
$$b=1+\beta+2\mu \beta+2\sqrt{\b... | 1 | https://mathoverflow.net/users/11260 | 386610 | 160,627 |
https://mathoverflow.net/questions/386623 | -1 | Is there is a way to recall a command from the history on the Ubuntu GAP system prompt? (something like CTL-R on Linux systems). Basically I want to reuse a command that I typed before, but the only thing that I found is the Up arrow, which is very slow...
Many thanks.
| https://mathoverflow.net/users/101411 | Recall command in GAP system | I don't know about the version bundled in Ubuntu, but if GAP is [compiled with GNU readline support](https://www.gap-system.org/Manuals/doc/ref/chap6.html#X82234FD181899530) then the standard readline search functions (Ctrl-r and Ctrl-s) should be enabled. You can use the instructions shown on the manual page to check ... | 0 | https://mathoverflow.net/users/3948 | 386625 | 160,631 |
https://mathoverflow.net/questions/386617 | 4 | Let $X$ be an $n$-dimensional Alexandrov space with curvature satisfying both $\ge 0$ and $\le 0$. Can we prove that any tangent cone of $X$ must be isometric to $\mathbb R^{k} \times C(S^{n-k-1}/\Gamma)$, where $\Gamma$ is a finite subgroup of $O(n-k)$ acting on $S^{n-k-1}$?
In general, do we have a classification f... | https://mathoverflow.net/users/105900 | Alexandrov spaces of zero curvature | No, that's too much to ask for. If you take any convex cone in $\mathbb R^n$ (any such cone is both $CAT(0)$ and Alexandrov of $curv\ge 0$) then the tangent space at the origin is just the cone itself. So it need not be any kind of quotient of $\mathbb R^n$. The above picture is general. If an $n$-dimensional space is ... | 6 | https://mathoverflow.net/users/18050 | 386648 | 160,639 |
https://mathoverflow.net/questions/386639 | 7 | Is the intersection of any two finitely generated subgroups of $\operatorname{Aut}(F\_2)$ (resp. $\operatorname{Aut}(F\_3)$) again finitely generated? That is, does $\operatorname{Aut}(F\_2)$ (resp. $\operatorname{Aut}(F\_3)$) have the Howson property? Here $F\_n$ is the free group of rank $n$.
For $n \geq 4$, it fol... | https://mathoverflow.net/users/120914 | Howson property of automorphism group of $F_2$ and of $F_3$ | According to wikipedia (<https://en.wikipedia.org/wiki/Howson_property>), any group of the form $F\_2 \rtimes \mathbb{Z}$ fails to have the Howson property. Assuming we believe wikipedia, then since $Aut(F\_2)$ contains subgroups of this form, it also fails to have the Howson property. (For example, take the subgroup g... | 7 | https://mathoverflow.net/users/164670 | 386656 | 160,640 |
https://mathoverflow.net/questions/386504 | 1 | Let $H$ be a Hilbert space and $b$ a continuous and symmetric bilinear form on $H \times H$, such that the induced operator $T \colon H \to H^{\ast}$ (the star denoting the continuous dual) is Hilbert-Schmidt. Suppose that $H$ is continuously and densely embedded in a Banach space $B$ and that
$$
\sup\_{h\_1,h\_2 \in... | https://mathoverflow.net/users/165008 | 2-summing vs Hilbert-Schmidt norm for extended operator between Hilbert and Banach space | The question can be reformulates as follows. If $H$ is a Hilbert space, $X$ ($=B^∗$) is a Banach space, $T:H\to X$ is an operator, $i:X\to H$ is a continuous injection, can one estimate the $2$-summing norm $\pi\_2(T)$ by a multiple of $π\_2(iT)$?
The following example shows that this is in general impossible. Take $... | 1 | https://mathoverflow.net/users/127871 | 386658 | 160,642 |
https://mathoverflow.net/questions/386661 | 3 | Let $X$ be a smooth plane projective curve of degree $6$ and genus $10$ (over complex numbers). Then my question is the following :
**Question :** Is it possible that there exists a special divisor $D$ of degree $9$ on $X$ admitting exactly $4$ independent sections?
**Observations :** $(i)$ From Clifford's theorem:... | https://mathoverflow.net/users/156533 | On emptiness of certain $G^r_d(X)$ on a smooth plane curve | No, this is not possible. Let $H$ be the divisor of a line. Use the base point free pencil trick to get an exact sequence
$$0\rightarrow H^0(D-H)\rightarrow H^0(D)^2\rightarrow H^0(D+H)$$
Since $\deg(D+H)=15$, we have by Riemann-Roch $h^0(D+H)\leq 7$, hence $h^0(D-H)\geq 1$. Thus $D\equiv H+E$ with $E\geq 0$ of degree ... | 3 | https://mathoverflow.net/users/40297 | 386667 | 160,645 |
https://mathoverflow.net/questions/386558 | 1 | Let $Z$ a compact set and $X$ a locally compact set. Let $p:Z\to X$ a local homeomorphism. Show that there exists $n≥1$ and $U\_1,…,U\_n$ open and closed sets of $X$ such that :
* $X=\sqcup\_{i=1}^{n}U\_i$
* the map $x\in X \mapsto \text{Card}(\rho^{-1}(\{x\}))$ is constant on $U\_i$ for all $1≤i≤n$
I have already ... | https://mathoverflow.net/users/168035 | Disjoint union of clopen sets such that the fibers has constant cardinality | The image of your continuous function is a compact subset of the discrete space $\mathbb{Z}$, so it is finite.
For each point, $i$, in the range the set $\{i\}$ is clopen in $\mathbb{Z}$, hence its preimage is clopen in $X$.
| 1 | https://mathoverflow.net/users/5903 | 386671 | 160,646 |
https://mathoverflow.net/questions/386668 | 4 | Suppose that $\mathbb{A},\mathbb{B}$ are finitely complete categories and $F:\mathbb{A}\to\mathbb{B}$ is a functor which reflects finite limits. Does $F$ reflect finitely generated limits?
(Here "finite limits" means limits over finite index categories, and "finitely generated limits" means limits over finitely gener... | https://mathoverflow.net/users/94321 | Functors reflecting finitely generated limits | I think you can express a finitely generated limit as a finite limit. Given a finitely generated category $C$ with generating subgraph $G = (E \rightrightarrows V)$, let $G^\S$ be the category with objects $V+E$ and two nonidentity morphisms to each edge, from its source and target respectively. There are no nontrivial... | 4 | https://mathoverflow.net/users/49 | 386677 | 160,650 |
https://mathoverflow.net/questions/386655 | 14 | Recently Ernest Davis asked me about the following computational problem: we're given as input a composite integer $n$, a divisor $k$ of $n$, and a subset $S \subset \mathbb{Z}\_n$ of size k. The problem is to decide whether $\mathbb{Z}\_n$ can be covered with $n/k$ cyclic translations of $S$, i.e. sets of the form $S+... | https://mathoverflow.net/users/2575 | Exact coverability of $\mathbb{Z}_n$ by cyclic shifts of a given set -- easy? NP-complete? | Here are a few references; the right keywords seem to be "tiling by translation" or a "factorization of a group":
Mihail N. Kolountzakis and Máté Matolcsi, Algorithms for translational tiling, Journal of Mathematics and Music, 3:2 (2009), 85-97, <https://doi.org/10.1080/17459730903040899>
Mihail N. Kolountzakis and... | 9 | https://mathoverflow.net/users/24076 | 386685 | 160,653 |
https://mathoverflow.net/questions/386514 | 7 | Let $(X,d)$ be a metric space. For any $0<\epsilon<1$, we call the metric space $(X,d^{\epsilon})$; where $d^{\epsilon}(x,y)\triangleq (d(x,y))^{\epsilon}$ the $\epsilon$-snowflake of $(X,d)$.
My question is, given $(X,d)$ and some $\epsilon\in(0,1)$, under what conditions can we deduce that there exists some other m... | https://mathoverflow.net/users/172598 | When is a metric space a snowflake? | You may be interested in the following paper:
Jeremy T. Tyson and Jang-Mei Wu, Characterizations of Snowflake Metric Spaces. Annales Academiae Scientiarum Fennicae Mathematica. Volume 30, 2005, 313-336.
<https://www.emis.de/journals/AASF/Vol30/tyson.pdf>
The paper contains a number of equivalent characterizations o... | 7 | https://mathoverflow.net/users/176091 | 386686 | 160,654 |
https://mathoverflow.net/questions/386688 | 4 | Let $X$ be a nice space: a manifold, a variety over a field, or so on. We know that line bundles on $X$ up to isomorphisms are given by $H^1(X,\mathcal{O}\_X^\*)$.
Can it be generalized to higher rankal vector bundles on $X$ by introducing a sheaf cohomology for sheaves $GL(n, \mathcal{O}\_X)$?
Is it done somewhere... | https://mathoverflow.net/users/127260 | $H^1(X, GL(n, \mathcal{O}_X))$ and Vector Bundles | Yes, this is called non-abelian sheaf cohomology. If $X$ is a topological space and $\mathcal{G}$ is a sheaf of groups, then $H^0(X, \mathcal{G})$ is the global sections of $\mathcal{G}$, and there is also an object called $H^1(X, \mathcal{G})$. In particular, $H^1(X, GL\_n)$ classifies isomorphism classes of rank $n$ ... | 7 | https://mathoverflow.net/users/297 | 386690 | 160,656 |
https://mathoverflow.net/questions/386692 | 8 | For any topological space $A$, the cone $C(A)$ is defined to be $A \times [0,\infty)$ with $A \times 0$ identified to a point (cone point).
Let $X$ and $Y$ be two compact Hausdorff spaces such that there is a homeomorphism between $C(X)$ and $C(Y)$ which preserves the cone points. Can we prove that $X$ and $Y$ are ho... | https://mathoverflow.net/users/105900 | Does homeomorphism between cones imply homeomorphism between sections | The answer is no. Let $X$, $Y$ be any two smoothly h-cobordant closed manifolds of dimension $\ge 4$ that are non-homeomorphic. For example, we can take $X=S^2\times L(7,1)$ and $Y=S^2\times L(7,2)$ where as usual $L(p,q)$ is a 3-dimensional lens space. By the weak h-cobordism theorem the interior of the h-cobordism is... | 14 | https://mathoverflow.net/users/1573 | 386693 | 160,657 |
https://mathoverflow.net/questions/194291 | 11 | What is a good reference for cluster algebras from surfaces, with a view to their connection to Teichmuller theory?
In my view, that means it should start off with unpunctured surfaces (and in fact, it would be fine with me if it never went further).
So far as I understand, this means that the results involved mi... | https://mathoverflow.net/users/468 | What is a good introduction to cluster algebras from surfaces? | Belatedly, per comments: Lauren Williams gave a [survey course](https://math.berkeley.edu/%7Ewilliams/CA.html) with recorded lectures, referencing her paper "[Cluster algebras: an introduction](https://arxiv.org/abs/1212.6263)."
More recently: [Introduction to Cluster Algebras: Chapter 6](https://arxiv.org/abs/2008.0... | 3 | https://mathoverflow.net/users/12178 | 386694 | 160,658 |
https://mathoverflow.net/questions/386637 | 1 | Suppose $(X(t),Y(t))$ $t\in[0,1]$ is a bivariate Gaussian process. We can assume that each component is continuously differentiable, but not necessarily stationary, and that the covariance kernels of $X$, $Y$ and the correlation function between $X$ and $Y$ are non-degenerate in the sense that $C\_X(t,t)>0$, $C\_Y(t,t)... | https://mathoverflow.net/users/160467 | Zeros of a non-degenerate bivariate Gaussian Process | This is definitely true under some assumptions about the regularity of $C\_X, C\_Y$ and $\rho\_{XY}$. The easiest way to do this is to find a version of the Kac-Rice formula that applies in your case (you may need slightly more assumptions on regularity).
You could also do it more naively. Assuming $C\_X(t,t)$, $C\_Y... | 0 | https://mathoverflow.net/users/69870 | 386696 | 160,660 |
https://mathoverflow.net/questions/386629 | 4 | I'm reading Mumford's book *Geometric Invariant Theory* and confused about the proof of a lemma on Page 91&92:
>
> Lemma:
>
>
> Let $V\_0$ be a smooth surface over an algebraically closed field $k$
> with char$k=0$, and let $P$ be a closed point on $V\_0$. Let $x,y$ be uniformizing parameters at $P$ on $V\_0$ and... | https://mathoverflow.net/users/153842 | Question about valuation and blow up (a lemma in GIT book) | do you know how we cunstruct a blow up? for your first question blow up of affine $A$ at the ideal $I$ is covered by $$Spec(A[\frac{x^{r\_0}\cdot y^{s\_0}}{x^{r\_i}\cdot y^{s\_i}},...,\frac{x^{r\_n}\cdot y^{s\_n}}{x^{r\_i}\cdot y^{s\_i}}]),0\le i\le n$$, now let $Q$ be a prime ideal defining a point with positive dimen... | 1 | https://mathoverflow.net/users/65846 | 386716 | 160,662 |
https://mathoverflow.net/questions/386705 | 5 |
>
> Let $a$ and $b$ be positive numbers. Prove that:
> $$\ln\frac{(a+1)^2}{4a}\ln\frac{(b+1)^2}{4b}\geq\ln^2\frac{(a+1)(b+1)}{2(a+b)}.$$
>
>
>
Since the inequality is not changed after replacing $a$ on $\frac{1}{a}$ and $b$ on $\frac{1}{b}$ and $\ln^2\frac{(a+1)(b+1)}{2(a+b)}\geq\ln^2\frac{(a+1)(b+1)}{2(ab+1)}$ ... | https://mathoverflow.net/users/135040 | Inequality of two variables | $$
\ln\frac{(a+1)^2}{4a}\ln\frac{(b+1)^2}{4b}
=\ln\left(1-\left(\frac{a-1}{a+1}\right)^2\right)\ln\left(1-\left(\frac{b-1}{b+1}\right)^2\right)\\=
\left(\sum\_{n=1}^\infty\frac1n \left(\frac{a-1}{a+1}\right)^{2n}\right)\times
\left(\sum\_{n=1}^\infty\frac1n \left(\frac{b-1}{b+1}\right)^{2n}\right)\\
\geqslant
\left(\... | 11 | https://mathoverflow.net/users/4312 | 386717 | 160,663 |
https://mathoverflow.net/questions/375442 | 1 | Also in [SE](https://math.stackexchange.com/questions/3890935/fpqc-locally-constant-if-and-only-if-%C3%A9tale-locally-constant).
---
Let $\mathcal{F}$ be sheave over $S\_\mathrm{fpqc}$. We say $\mathcal{F}$ is a fpqc-locally constant sheaf (of finitely generated abelian groups) if there exists a fpqc covering $(S... | https://mathoverflow.net/users/110471 | Fpqc-locally constant if and only if étale-locally constant? | The answer is Yes, but it fails for some slight variants, so let me first discuss an analogue for sheaves of sets. In that case, this is closely related to the discussion of pro-etale fundamental groups in my [paper](https://www.math.uni-bonn.de/people/scholze/proetale.pdf) with Bhatt; that fundamental group classifies... | 5 | https://mathoverflow.net/users/6074 | 386723 | 160,664 |
https://mathoverflow.net/questions/368172 | 6 | $\DeclareMathOperator\Et{Et}$Let $X$ be a scheme and denote by $\Et(X)$ the associated étale homotopy type. Then by the work of Artin–Mazur, we know that for an abelian group $A$, we have
$$H^n(\Et(X),A)=H^n\_{\text{ét}}(X,A)$$
and
$$\pi^1(\Et(X))\cong \pi^1\_{\text{alg}}(X).$$
Therefore I wonder what $H\_n(\Et(X),A... | https://mathoverflow.net/users/152554 | Homology of the étale homotopy type | I'm sure there are easier and better ways to think about this, but here's how I like to think about it.
Work on the big pro-etale site on all schemes, which maps to the pro-etale site of a point, $\pi: \mathrm{Sch}\_{\mathrm{proet}}\to \ast\_{\mathrm{proet}}$. Also, let's consider (hypercomplete) sheaves of anima. Th... | 12 | https://mathoverflow.net/users/6074 | 386726 | 160,665 |
https://mathoverflow.net/questions/386728 | 4 | I am looking for literature on the question whether a randomly chosen sequence of $k$-regular graphs converges in the Benjamini-Schramm sense to the universal covering with probability one.
There are several ways to make this a precise question. I would be interested in any result. This is not my area, so it might we... | https://mathoverflow.net/users/nan | Random graphs and Benjamini-Schramm convergence | For fixed $k\ge 3$, $\frac{v(k,n)\_{\ge R}}{v(k,n)}\to 1$. First note that most such graphs have trivial automorphism groups, so it doesn't make a difference whether you ask about isomorphism classes or all labelled graphs.
Next, for any $\ell\ge 3$, if an increasing sequence of random $k$-regular graphs is taken, th... | 3 | https://mathoverflow.net/users/9025 | 386729 | 160,666 |
https://mathoverflow.net/questions/386740 | 4 | Recall the adjunction formula $$ g(\alpha) = 1 + \frac{1}{2}\left( \alpha^2 -c\_1(X)\cdot \alpha \right)$$ where $g(\alpha)$ is the genus of a pseudoholomorphic representative of the Poincaré dual of $\alpha$, $A=PD(\alpha)\in H^2(X;\Bbb Z)$, for a symplectic $4$-manifold $(X,\omega, J)$. The expected dimension of the ... | https://mathoverflow.net/users/93538 | Compactness as a consequence of the adjunction formula for genus second homology class | Assume your 4-manifold is minimal (otherwise there are multiply covered exceptional spheres which potentially give noncompactness). Then it's a computational check to see that: If $d(\alpha)=0$ with $\alpha=k[C]$ represented by a $J$-curve that is a $k$-fold cover of the underlying $J$-curve $C$ (assume connected for s... | 2 | https://mathoverflow.net/users/12310 | 386750 | 160,669 |
https://mathoverflow.net/questions/386742 | 1 | In [Estimates of fractional heat kernel](https://mathoverflow.net/questions/382309/estimates-of-fractional-heat-kernel), it was stated that $$ \partial\_{x\_j} p\_t^{(n)}(x) = -\frac{x\_j}{2 \pi} \, p\_t^{(n+2)}(\tilde x) $$
where $x = (x\_1, \ldots, x\_n) \in \mathbb R^n$, $\tilde x = (x\_1, \ldots, x\_n, 0, 0) \in \m... | https://mathoverflow.net/users/140379 | Integrability of fractional heat kernel | Of course we can!
---
And if we actually do, then we find that $\phi(t, x) = |\nabla\_x p\_t^{(n)}(x)|$ satisfies $$\phi(t, x) = t^{-(n+1)/(2s)} \phi(1, t^{-1/(2s)} x)$$ and, with $\tilde x = (x, 0, 0) \in \mathbb R^{n+2}$), $$\phi(1,x) = |x| p\_1^{(n+2)}(\tilde x) \approx C |x| \min\{1, |x|^{-(n+2+2s)}\} = C \mi... | 1 | https://mathoverflow.net/users/108637 | 386751 | 160,670 |
https://mathoverflow.net/questions/386653 | 29 | The notion of trace of a matrix can be generalized to trace of an endomorphism of a dualizable objects in a symmetric monoidal category. (See [Ponto & Shulman](https://arxiv.org/abs/1107.6032v2) for a nice description.)
Is there a categorification of the notion of determinant as well? If it exists, where can I read a... | https://mathoverflow.net/users/176076 | Categorification of determinant | There is a notion of determinant functor, they were introduced for abelian and exact categories by P. Deligne in his paper "Le déterminant de la cohomologie" (<https://publications.ias.edu/sites/default/files/Number58.pdf>).
There is an extension to categories of bounded complexes by F. Knudsen and D. Mumford.
More r... | 15 | https://mathoverflow.net/users/27816 | 386753 | 160,671 |
https://mathoverflow.net/questions/386760 | 4 | **Context and Notation**
Let $X$ be a manifold and $\mathcal{C} = Op(X)$ be the category of open subsets of $X$ with inclusions. I then consider the projective model structure on the (simplicial model) category, $sPre$, of simplicial presheaves, $\mathcal{C}^{op} \to sSet$, and its simplicial mapping space (i.e. homo... | https://mathoverflow.net/users/19926 | Checking that (hyper) sheafification is fibrant in local projective model structure on simplicial presheaves | There are two ways to make this construction work.
The first way is to iterate the step $F↦F^†$ transfinitely many times.
The reason that a single iteration of $F↦F^†$ is not sufficient
is that while $F^†$ does add the missing data that prevents $F$
from satisfying the lifting property for elements of $S$,
the newly ... | 2 | https://mathoverflow.net/users/402 | 386767 | 160,675 |
https://mathoverflow.net/questions/386752 | 1 | Sorry if this turns out to be a silly question, but I am having difficulties in both understanding it and finding other references for it. I hope that someone can clear my concepts here on overflow.
Recently, I am reading the book "Degenerate Complex Monge-Ampère Equations" by Guedj and Zeriahi and having some confus... | https://mathoverflow.net/users/164604 | How to understand subharmonic functions, distributions, and measure? | Before reading such advanced books on "degenerate Monge Ampere equations", you need a background in subharmonic functions and potential theory (and distributions if you do not have it). Some good introductory books are Notions of Convexity by L. Hormander,
and Subhrmonic Functions, v. 1, by W. Hayman and P. Kennedy.
| 4 | https://mathoverflow.net/users/25510 | 386769 | 160,677 |
https://mathoverflow.net/questions/386775 | 4 | I came across this post [Coefficients of the Inflated Eulerian Polynomial](https://arxiv.org/pdf/1504.01089.pdf) by AULI-GRAHAM-SAVAGE. In particular, the polynomials related to **descents** interested me
$$P\_n(x)=\sum\_{\pi\in\mathfrak{S}\_n}x^{n\cdot\text{des}(\pi)+\pi\_n}.$$
Here $\pi=\pi\_1\pi\_2\cdots\pi\_n\in\ma... | https://mathoverflow.net/users/66131 | A close reative of "Inflated" Eulerian polynomials | Yes, this is true. First of all, we denote $x^a=t$, then the relation to prove reads as $$\sum\_{\pi} t^{{\rm maj}(\pi)}x^{\pi\_n}=(x^n+x^{n-1}t+\ldots+xt^{n-1})\prod\_{j=0}^{n-2}(1+t+\ldots+t^{j}).\quad\quad\quad (1)$$
Taking the coefficients of the fixed power of $x$, say $x^s$, in both sides of (1), we reduce (1) to... | 6 | https://mathoverflow.net/users/4312 | 386780 | 160,681 |
https://mathoverflow.net/questions/386785 | 2 | All the matrices in this statement are in the field $\mathbb{F}\_2$. Let $I$ be the identity matrix of size $10 \times 10$. What are all the possible $n$ ($\geq 6$) for which
there exists a matrix $X$ of size $n \times 10$, such that any $6$ rows of $X$ jointly with any $4$ rows of $I$ form a $10 \times 10$ invertible ... | https://mathoverflow.net/users/nan | Existence of a matrix in $\mathbb{F}_2$ with some invertibility properties | There is no such $n$. Assume that it exists, then $n=6$ also satisfies this property. Denote the rows of $X$ by $r\_1,\ldots,r\_6$, the columns of $X$ by $c\_1,\ldots,c\_{10}$, the rows of $I$ by $e\_1,\ldots,e\_{10}$. Assume that, say, $c\_1,\ldots,c\_{6}$ are linearly dependent. Then the minor formed by the first 6 r... | 1 | https://mathoverflow.net/users/4312 | 386786 | 160,682 |
https://mathoverflow.net/questions/385664 | 0 | Let $Y$ be a nonempty region in $\mathbb{R}^n$. I am designing an algorithm which given a point $x\_0$ outside $Y$ in a finite number of steps lead to a point $x\_n∈ Y$. The way I do it is that I have a smooth function $f(x)$ which has the property that it is negative in $Y$ and positive outside $Y$ and it has only one... | https://mathoverflow.net/users/38654 | What to call a function that is negative on a set | In the context of [Level-set method](https://en.wikipedia.org/wiki/Level-set_method), such a function is called a ***level set function***. The reason is that the boundary $\partial Y$ corresponds to the zero-level set of the function $f$.
| 1 | https://mathoverflow.net/users/176090 | 386798 | 160,685 |
https://mathoverflow.net/questions/386807 | 4 | Suppose $F$ is a field. I want to know whether the map $GL\_n(GW(F))\to GL\_n(W(F))$ is surjective, where $GW$ means Grothendieck-Witt and $W$ means Witt. In the case $F$ is algebraic closed, it reduces to the surjectivity of $GL\_n(\mathbb{Z})\to GL\_n(\mathbb{Z}/2\mathbb{Z})$. I know the case $n=1$ is true.
| https://mathoverflow.net/users/149491 | Is the map $GL_n(\mathbb{Z})\to GL_n(\mathbb{Z}/2\mathbb{Z})$ surjective? | For the question in your title, yes: $\operatorname{GL}\_n(\mathbb Z/2\mathbb Z) = \operatorname{SL}\_n(\mathbb Z/2\mathbb Z)$ is generated by transvections, and these obviously lift to $\operatorname{GL}\_n(\mathbb Z)$.
| 8 | https://mathoverflow.net/users/2383 | 386808 | 160,688 |
https://mathoverflow.net/questions/386814 | 5 | There are five real forms of the exceptional Lie group, $E\_6$. Four of them are notated as in the following:
* The split form as EI or $E\_{6(6)}$
* The quasi-split form as EII or $E\_{6(2)}$
* EIII or $E\_{6(-14)}$
* EIV or $E\_{6(-26)}$
What do the annotations to $E\_6$ actually indicate and are they also used f... | https://mathoverflow.net/users/35706 | What is meant by this notation of the real forms of $E_6$? | The notation is a bit complicated to make precise, but the number in parentheses is the *character*, which is defined on page 353 section C of Helgason, **Differential Geometry, Lie Groups and Symmetric Spaces**. The character is the difference $\dim \mathfrak{p}\_0 - \dim \mathfrak{k}\_0$ in dimensions in a Cartan dec... | 8 | https://mathoverflow.net/users/13268 | 386819 | 160,691 |
https://mathoverflow.net/questions/386800 | 1 | Consider the periodic strip $\Omega=\mathbb{T}\times[0,1]$ where $\mathbb{T}$ is the 1D torus with period 1. We consider the mixed Dirichlet/Neumann problem
$$-\Delta u=f$$
with boundary conditions
$$u(x,0)=0,\,\partial\_y u(x,1)=0$$
for all $x\in \mathbb{T}$. I'd like to say that the Laplacian associated with these bo... | https://mathoverflow.net/users/166785 | Poisson equation in a periodic strip | There are probably different (and more general) ways of doing that, but in your specific case I think you can prove everything with rather elementary arguments. (Maybe that's more or less what Giorgio Metafune had in mind).
Start by "extending" $f$ into a function $\tilde f$ in the following way:
* $\tilde f(x,y) =... | 3 | https://mathoverflow.net/users/150933 | 386824 | 160,693 |
https://mathoverflow.net/questions/386793 | 1 | All the matrices in this statement are in the field $\mathbb{F}\_2$. Let $I$ be the identity matrix of size $10 \times 10$ and let $e\_1$, $e\_2$, $\ldots$, $e\_{10}$ denote its rows. For $i\in \{1,5 \}$, define the $2 \times 10$ matrix $A\_{i} = \left( \begin{matrix} e\_{2i-1} \\ e\_{2i}\end{matrix} \right)$.
Withou... | https://mathoverflow.net/users/nan | Existence of matrices in the field $\mathbb{F}_2$ with some invertibility properties | You may set the rows of $X\_i$ to be $e\_{2i-1-k}+e\_{2i+k}$, for $k=1,2,3$. Due to symmetry, there are only two cases to check, and both work.
| 1 | https://mathoverflow.net/users/17581 | 386840 | 160,696 |
https://mathoverflow.net/questions/386802 | 5 | It is well-known that the harmonic series is not summable. In some sense this means that it takes a lot of rather large values.
We define the operator $F\_{\varepsilon}: \ell^{\infty}(\mathbb N) \rightarrow [0,\infty]$ by $$F\_{\varepsilon}(x) = \sum\_{i=1}^{\infty} 2^{-\varepsilon \vert x\_i \vert^{-1}} \text{ for }... | https://mathoverflow.net/users/119875 | Is the harmonic series worse than any summable series? | $\newcommand\ep\varepsilon\newcommand\de\delta$
Let us show more:
\begin{equation\*}
\frac{F\_{\ep}(x)}{F\_{\ep}((1/n))}\to0\tag{$\*$}
\end{equation\*}
(as $\ep\downarrow0$).
Indeed,
\begin{equation\*}
F\_{\ep}((1/n))=\sum\_{i=1}^\infty 2^{-\ep i}=\frac{2^{-\ep}}{1-2^{-\ep}}\asymp\frac1\ep. \tag{0}
\end{equation\*}
... | 6 | https://mathoverflow.net/users/36721 | 386842 | 160,697 |
https://mathoverflow.net/questions/386020 | 8 | It is well known that the ring of modular forms over $\mathbb{C}$ is $$
\mathbb{C}[c\_4,c\_6]
$$ where $$
c\_4 = 1+240 q + \cdots,\qquad
c\_6 = 1-504 q - \cdots
$$ are the standard Eisenstein series, and the discriminant $$
\Delta= q - 24 q^2 + \cdots
$$ satisfies $$
c\_4^3-c\_6^2=1728 \Delta.
$$
You can use ellipti... | https://mathoverflow.net/users/5420 | Modular forms over $\mathbb{Z}$ vs modular forms with integral Fourier coefficients | Apparently it is a classic result, e.g. [a paper of Igusa](https://doi.org/10.2307/2373943) starts by stating this fact saying it's classic without citing any. A big more googling turned up that there's something called Victor Miller's basis which exactly does the job, implemented in Sage, see [here](https://doc.sagema... | 6 | https://mathoverflow.net/users/5420 | 386843 | 160,698 |
https://mathoverflow.net/questions/386833 | 2 | $\newcommand{\div}{\operatorname{div}}$$\newcommand{\SO}{\operatorname{SO(2)}}$$\newcommand{\R}{\operatorname{\mathbb{R}}}$$\newcommand{\bdx}{\partial\_x}$$\newcommand{\bdy}{\partial\_y}$$\newcommand{\til}{\tilde}$
Let $\Omega \subseteq \R^2$ be an open connected domain, and let $U,V :\Omega \to \SO$ be smooth maps.
... | https://mathoverflow.net/users/46290 | The only rotation fields satisfying this PDE are constant | *NB*: Evidently I used a different convention on divergence. In terms of matrix components my convention is $(\mathrm{div} M)\_j = \sum\_{i = 1}^2 \partial\_{x^i} M\_{ij}$. If your convention is $\sum\_{i = 1}^2 \partial\_{x^i} M\_{ji}$, then everywhere I multiplied the divergence on the right by something, you should ... | 3 | https://mathoverflow.net/users/3948 | 386848 | 160,700 |
https://mathoverflow.net/questions/386851 | 1 | Schubert variety $V$ is a special type of (possibly singular) subvarieties of a Grassmannian. Since the Grassmannians are Kähler manifolds (in fact projective varieties) are we able to conclude that any smooth Schubert subvariety of a Grassmannian is in fact also a Kähler manifold?
| https://mathoverflow.net/users/176218 | Are smooth Schubert varieties Kähler? | Yes, smooth closed subvarieties of projective varieties are projective, and hence Kahler. Smooth Schubert varieties are very rare though, see the sources below for a description of them:
*Ryan, Kevin M.*, [**On Schubert varieties in the flag manifold of Sl(n,({\mathbb C}))**](http://dx.doi.org/10.1007/BF01450738), Ma... | 8 | https://mathoverflow.net/users/297 | 386853 | 160,702 |
https://mathoverflow.net/questions/386556 | 9 | Let $G = \operatorname{GL}\_2$, and let $V = L^2(Z(\mathbb A)G(\mathbb Q) \backslash G(\mathbb A),\omega)$, for $\omega$ a character of the ideles $\mathbb A^{\ast}$, identified with a central character. For a character $\mu$ of $\mathbb A^{\ast}/\mathbb Q^{\ast}$ such that $\mu^2 = \omega$, let $\chi = \mu \circ \oper... | https://mathoverflow.net/users/38145 | Why are characters orthogonal to cusp forms? | For the second measure theoretic question, I don’t know the answer, but I think, for the first question about the orthogonality is resolved as follows:
We may assume that $\omega$ is unitary by tensoring a central character.
By the Gelfand-Piatetski-Shapiro theorem, the space of L^2 cusp forms is decomposed to a di... | 3 | https://mathoverflow.net/users/163485 | 386855 | 160,703 |
https://mathoverflow.net/questions/386856 | 3 | I was interested in the Grigorieff forcing (you can read the definition here: [How "much" does (Grigorieff) forcing destroy an ultrafilter?](https://mathoverflow.net/questions/62981/how-much-does-grigorieff-forcing-destroy-an-ultrafilter))
I couldn't prove that it destroys ultrafilters, and I couldn't find the proof... | https://mathoverflow.net/users/123559 | Grigorieff forcing and destruction of ultrafilters | Let $g$ be the generic subset of $\omega$ added by the Grigorieff forcing, $G(F)$, where $F$ was a free ultrafilter.
It is easy to see that for every $A\in F$, $g\cap A$ is non-empty, since if $f$ is any condition in $G(F)$, then there is some $n\in A\setminus\operatorname{dom} f$ and we can simply take $f\cup\{(n,1)... | 1 | https://mathoverflow.net/users/7206 | 386863 | 160,704 |
https://mathoverflow.net/questions/386613 | 7 | Given $M$ a symmetric matrix in $\mathbb F\_2^{n\times n}$ having $\mathsf{det}\_\mathbb R(M)\neq0$ (*non-singular in reals*) and satisfying $PMP'=(M+J+I)$ or $P(M+J+I)P'=M$ where $P$ is a permutation matrix in $\mathbb F\_2^{n\times n}$ and where $J$ is all $1$s and $I$ is identity matrix in $\mathbb F\_2^{n\times n}$... | https://mathoverflow.net/users/10035 | On a matrix problem in the field $\mathbb F_2$ | Most of the question has already been addressed. As regards the last part, I just poit out that if ${\rm det} M = 0$ and ${\rm det}(M+I) = 1$, then we must have ${\rm rank}(M) = n-1$ ( I mean here the $\mathbb{F}\_{2}$ rank). This follows since $M + I + J$ is (by assumption) similar to $M$, and hence has the same rank.... | 1 | https://mathoverflow.net/users/14450 | 386864 | 160,705 |
https://mathoverflow.net/questions/386850 | 14 | I can calculate the derivation of the octonions and I clearly find the 14 generators that form the algebra of $G\_2$. However, when I do the same calculation for the quaternions, I end up with the three generators of $SO(3)$, which basically tells me that I can rotate the set of imaginary units anyway I like.
Intuiti... | https://mathoverflow.net/users/89713 | Why is the automorphism group of the octonions $G_2$ instead of $SO(7)$ | The quaternions are generated by any *two* imaginary elements $x$ and $y$ that are orthonormal, i.e., $\bigl(1,\, x,\, y,\, xy\bigr)$ is an orthonormal basis of the quaternions. Moreover, the multiplication table using such a pair does not depend on which pair you choose. That's why the automorphism group of the quater... | 26 | https://mathoverflow.net/users/13972 | 386866 | 160,707 |
https://mathoverflow.net/questions/386847 | 2 | Let $\mathcal{H}$ be a complex finite dimensional Hilbert space and let $\mathcal{A}\subseteq \mathcal{B}(\mathcal{H})$. I am looking to understand the different decompositions of $\mathcal{H}$ induced by $\mathcal{A}$ and the connections between them (if any). For example if $\mathcal{A}$ is a multiplicative abelian g... | https://mathoverflow.net/users/176157 | Decomposition of Hilbert spaces via groups and algebras representations | This is basic C${}^\*$-algebra theory and should be found in most introductory texts --- I feel sure it's in *C${}^\*$-algebras by Example* by Davidson, for instance.
| 5 | https://mathoverflow.net/users/23141 | 386867 | 160,708 |
https://mathoverflow.net/questions/386809 | 8 | Suppose I have a triangle
$$A \to B \to C \to A[1]$$
in $D(Ab(X))$, the derived category of abelian sheaves on some topological space $X$. For each $x \in X$, there is an exact functor $D(Ab(X)) \to D(Ab)$ that takes a complex of sheaves to the complex of stalks at $x$.
Is my triangle an exact triangle if the ima... | https://mathoverflow.net/users/94696 | Checking exactness of a triangle on stalks | The answer is no, but we can say a bit more : it can become true if you pass to the derived $\infty$-category and replace the words "distinguished triangle" with "cofiber sequence" (modulo the choice of a nullhomotopy)
I'll assume we already know that the composite $A\to C$ vanishes - this cannot be deduced from a st... | 4 | https://mathoverflow.net/users/102343 | 386877 | 160,711 |
https://mathoverflow.net/questions/386042 | 4 | For each (not necessarily planar) embedding of a graph in $\mathbb{R}^k$ one can calculate the ratio
$$\gamma = \frac{\textsf{mean Euclidean length of edges}}{\textsf{mean Euclidean distance between nodes}} \times \textsf{diameter}$$
where the diameter is the maximal Euclidean distance between nodes. An embedding c... | https://mathoverflow.net/users/2672 | How to show that random graphs cannot be embedded with short edges | Perhaps you could look into [Spectral Graph Drawing](http://www.cs.yale.edu/homes/spielman/561/2015/lect02-15.pdf).
Since a single comment with a link does not qualify as an answer in the views of many on this site, and truly I don't have enough reputation to just post a comment, let me try to expand on this post to ... | 1 | https://mathoverflow.net/users/175761 | 386900 | 160,716 |
https://mathoverflow.net/questions/386893 | 8 | What is known about the largest set of $k$-wise linearly independent vectors in $\mathbb F\_q^n$? I am especially interested when $q=2$, and in the regime where $k$ is fixed an $n\to\infty$. Here are some partial results:
* The largest pairwise independent set has $\frac{q^n-1}{q-1}$ vectors.
* The largest size of a ... | https://mathoverflow.net/users/59232 | Largest set of $k$-wise linearly independent vectors in $\mathbb F_q^n$? | $\newcommand{\F}{{\mathbb F}}$
Let's focus on the case $q=2$ which you are primarily interested in. Suppose that $v\_1,\dotsc,v\_m\in\F\_2^n$, and let $V$ be the matrix over $\F\_2$ of size $n\times m$ with $v\_1,\dotsc,v\_m$ (written in the standard basis) as its columns. It is readily seen that for $v\_1,\dotsc,v\_m$... | 9 | https://mathoverflow.net/users/9924 | 386901 | 160,717 |
https://mathoverflow.net/questions/386876 | 6 | Given a category $\mathcal{C}$, the category of elements functor sets up an equivalence of categories
$$
\mathsf{DFib}(\mathcal{C})
\cong
\mathsf{PSh}(\mathcal{C}),
$$
whereas the Grothendieck construction sets up a $2$-equivalence
$$
\mathsf{CartFib}(\mathcal{C})
\cong
\mathsf{PseudoPSh}(\mathcal{C}).
$$
**Question:... | https://mathoverflow.net/users/130058 | Is there a Grothendieck correspondence for sheaves/stacks? | The essential image of the Grothendieck construction from (weak) functors $C\to\operatorname{Cat}$ which are sheaves with respect to a Grothendieck topology $\mathcal T$ is described in [Section 8.4 of the Stacks project](https://stacks.math.columbia.edu/tag/0268), with the full subcategories of sheaves of groupoids an... | 4 | https://mathoverflow.net/users/35687 | 386902 | 160,718 |
https://mathoverflow.net/questions/386871 | 4 | Cross-post from [MSE](https://math.stackexchange.com/q/4064240/272127).
---
Suppose $(M,g)$ be a closed Riemannian manifold. Because every parallel (nontrivial) $p$-form $\omega$ is harmonic so the $p$-th Betti number should be positive i.e. $b\_p\geq 1$. How to see this using De Rham cohomology? (of course witho... | https://mathoverflow.net/users/90655 | Existence non-trivial parallel $p$-form implies non-triviality of $p$-th cohomology group using De Rham cohomology | Here is an argument in the orientable case: If $\omega$ is a $g$-parallel $p$-form on an orientable Riemannian $n$-manifold $(M^n,g)$, then it is closed since $\mathrm{d}\omega = \pi\_{p+1}(\nabla^g\omega) = 0$, where $$\nabla^g:\Omega^p(M)=C^\infty(\Lambda^p(T^\*M))\to C^\infty(T^\*M\otimes\Lambda^p(T^\*M))$$ is the L... | 7 | https://mathoverflow.net/users/13972 | 386909 | 160,719 |
https://mathoverflow.net/questions/386915 | 1 | Let $A$ be a ring and $\text{mod} A$ the category of finitely generated (right) modules over $A$. Is the class of isomorphism types of $\text{mod} A$ always a set? In particular, is it the case if $A$ is an Artin algebra?
EDIT: The answer is true for an arbitrary ring $A$. However, can we bound the cardinality if we ... | https://mathoverflow.net/users/145920 | Is the class of isomorphism types of a module category always a set? | Yes, because you're only considering finitely generated modules. No assumptions on $A$ are required.
Indeed, a finitely generated module is always a quotient of some $A^n$ by some submodule. For each $n$, there is a set of submodules of $A^n$ (of size $\leq 2^{|A^n|}$), and there are countably many integers (by defin... | 9 | https://mathoverflow.net/users/102343 | 386916 | 160,720 |
https://mathoverflow.net/questions/386921 | 14 | I am 16 years old at the time of writing (so I have no supervisors to seek advice from) and I have written a mathematics research paper, which I plan on submitting to a journal for publication. I asked an [identical question](https://academia.stackexchange.com/questions/164114/how-to-structure-a-proof-by-induction-in-a... | https://mathoverflow.net/users/167114 | How to structure a proof by induction in a maths research paper? | Writing a proof for school is very different from writing a proof for a research paper. Perhaps the most important distinction is that the audiences are completely different. In school, your audience is your instructor, whose job is to assess your ability to learn and apply a principle. The audience of a research artic... | 26 | https://mathoverflow.net/users/111215 | 386928 | 160,725 |
https://mathoverflow.net/questions/386904 | 5 | I am trying to get a lower bound (or even the exact value) of
$$
\min\_{X \in \mathbb{R}^{n\times n}} \|X - I\_n\|\_{\infty} \enspace \text{s.t.} \enspace \mbox{Rank}(X) = m
$$
where $m \leq n$, and the infinity norm is
$$ \| X \|\_{\infty} := \max\_{ij}|X\_{ij}| $$
I have a very simple lower bound, obtained wi... | https://mathoverflow.net/users/173967 | Distance of low-rank matrices to the identity for the $\infty$-norm | Theorem 1.1 from Alon's paper ["Perturbed identity matrices have high rank: proof and applications"](https://www.tau.ac.il/%7Enogaa/PDFS/identity1.pdf) says that if $\|X-I\_n\|\_\infty<c$ with $1/(2\sqrt n)<c<1/4$, then $m\gg \log n/(c^2\log(1/c))$ with an absolute implicit constant. This is exactly what you need: if $... | 4 | https://mathoverflow.net/users/9924 | 386931 | 160,726 |
https://mathoverflow.net/questions/386912 | 9 | Let $\Gamma$ be a group, say finitely generated if it helps. Does $\Gamma$ admit a largest Hopfian quotient? That is, does there exist a Hopfian quotient $H$ of $\Gamma$, such that every surjective homomorphism from $\Gamma$ onto a Hopfian group factors through $H$?
I first thought to define $H = \Gamma / K$, where $... | https://mathoverflow.net/users/145915 | Largest Hopfian quotient | The answer is no even for finitely generated groups.
Here's a construction of a finitely generated residually Hopfian, non-Hopfian group. It is even solvable (actually center-by-metabelian).
---
Denote by $M(u,v,x,y,z)$ the matrix $$\begin{pmatrix}u & x & z\\ 0 & v & y \\ 0 & 0 & 1\end{pmatrix}.$$
Fix a prime $... | 9 | https://mathoverflow.net/users/14094 | 386933 | 160,728 |
https://mathoverflow.net/questions/386896 | 3 | I've been reading the article [*Forms of $\operatorname{GL}(2)$ from the analytic point of view*](http://www.ams.org/books/pspum/033.1/546600) by Gelbart and Jacquet in [Corvallis](http://dx.doi.org/10.1090/pspum/033.1) and am confused on a particular claim (equation 5.17 on page 232).
The claim essentially boils dow... | https://mathoverflow.net/users/38145 | Calculating the residue of Eisenstein series from the residue of the intertwining operator | I seem to have solved it, modulo some convergence issues I'm not yet confident about. We have
$$\langle E(\hat{f}\_1(s),g), F\_2(g) \rangle = \int\limits\_{G(\mathbb Q)Z(\mathbb A) \backslash G(\mathbb A)} E(\hat{f}\_1(s),g)\overline{F\_2(g)}dg$$
where a well known calculation expresses this as as an integral over th... | 1 | https://mathoverflow.net/users/38145 | 386938 | 160,731 |
https://mathoverflow.net/questions/386839 | 1 | This cropped up in a research question I'm tackling.
I wish to solve the following optimization problem:
$$
\text{minimize}\ \sum\_{i=1}^\infty f\_i \sum\_{j=1}^i \sqrt{f\_j}
\quad\text{subject to}\ \sum\_{i=1}^\infty f\_i=1\ \text{and}\ f\_i \in [0,1]\ \forall i.
$$
My attempt was to apply calculus of variation, and... | https://mathoverflow.net/users/131653 | Calculus of variations for double sum with Lagrange multiplier | The basic heuristic starts with looking at the continous analog
$$
\int\_0^\infty f(x) \int\_0^x \sqrt{f(y)} \,dy \, dx ,
$$
which has Dirac sequences at 0 as minimizing sequences with value 0. Regarding the discrete system as a regularization of the continuum one, one might think that $f\_1=1$ and $f\_i=0$ for $i\geq ... | 1 | https://mathoverflow.net/users/13400 | 386941 | 160,732 |
https://mathoverflow.net/questions/385911 | 11 | Let $M$ be a smooth manifold, let $\mathcal{P}$ be a Whitney stratification of $M$ and let $S\subset M$ be a stratum with closure $\overline{S}$.
**Question:** Does there exist an open neighborhood $U\subset M$ of $\overline{S}$ such that $U$ deformation retracts onto $\overline{S}$?
In the case when $\overline{S}\... | https://mathoverflow.net/users/9581 | Local topology of Whitney stratified spaces | In fact, any Whitney stratified set admits a stratification in the sense of Thom/Mather, cf Mather's notes on topological stability, published in B.A.M.S Volume 49, Number 4, October 2012, Pages 475–506, cf section 8 in particuler p 492 :
"Hence it follows from Proposition 7.1 that any Whitney stratified set admits the... | 3 | https://mathoverflow.net/users/27816 | 386942 | 160,733 |
https://mathoverflow.net/questions/386908 | 4 | Consider the usual de Rham CDGA $(\Omega^\* Sym^\*(V),d)$ for a free $\mathbb{Z}\_{(p)}$-module $V$. What is known about its cohomology?
It is easy to compute ranks of primary summands in $H^\*(\Omega^\* Sym^\*(V),d)$ through the Bockstein spectral sequence or by applying the Kunneth's formula. Meanwhile a truly func... | https://mathoverflow.net/users/8906 | The integral cohomology of the de Rham complex | Following David Speyer's suggestion, fix exponents $e\_1,\dots, e\_n$ and consider the subcomplex consisting of sums of those monomials of the form $$\prod\_{i=1}^n \left( x\_i^{e\_i - \epsilon\_i} (d x\_i)^{\epsilon\_i} \right)$$ for $\epsilon\_i \in \{0,1\}$. The point of doing this is that this subcomplex is stable ... | 6 | https://mathoverflow.net/users/18060 | 386943 | 160,734 |
https://mathoverflow.net/questions/378877 | 4 | I am trying to read the following [paper](https://arxiv.org/pdf/1706.03755.pdf) by Granville-Harper-Soundararajan and I had a few questions regarding the paper. They prove, for $S(x):=\sum\_{n\leq x}f(n)$ and $F\_x(s):=\prod\_{p\leq x}\left(1+\prod\_{p\leq x}\frac{f(p^k)}{p^{ks}}\right)$, the following theorem:
Suppo... | https://mathoverflow.net/users/nan | On a sharp version of Halasz's theorem | Let h(x)=log(100 log x/L(x)) which is >3 for large x. Then the first term in (1) is << x h(x)/e^{h(x)}. This is evidently << x always, and significantly better than that the larger h(x) gets
| 2 | https://mathoverflow.net/users/70432 | 386953 | 160,737 |
https://mathoverflow.net/questions/386805 | -1 | In this [article](https://www.jstor.org/stable/2046117?seq=1) the author proves the following lemma:
**LEMMA:** $\forall N \in \Bbb N$, there exists $v=v\_N$ with compact support so that
$$[M\_S(M\_S v)^\delta(x)]^{1/\delta} \geq c\_\delta NM\_S v(x), \forall x \in [0,1]^2.$$
Here $M\_S$ is the strong maximal opera... | https://mathoverflow.net/users/176160 | What does $O(N)$ mean in this article and how does it imply this lemma? | The $O(1)$ is indeed an abuse of notations, but I personally find pretty clear what is meant by it, given the context. In fact the statement would have been crystal clear and rigorously meaningful if the author had simply written $\geq CN$ instead of $=O(N)$... This is precisely the definition of $\Omega(N)$. But choos... | 3 | https://mathoverflow.net/users/33741 | 386974 | 160,743 |
https://mathoverflow.net/questions/386979 | 1 | Let $L$ and $M$ be matrices over a commutative ring $R$ equipped with an involution "$\*$". Define $L \oplus M$ (the "direct sum" of $L$ and $M$) to be $\begin{bmatrix}L & 0 \\ 0 & M \end{bmatrix}$. If $K$ is a matrix over $(R, \*)$, define $K^\*$ to be the "conjugate-transpose" of $K$, i.e. $(K^\*)\_{ij} = (K\_{ji})^\... | https://mathoverflow.net/users/75761 | If the direct sum of $L$ and $M$ has a pseudoinverse, then do $L$ and $M$ have pseudoinverses? | I propose an answer to my own question:
The following observation is based on Section 3.1 of "Theory of generalized inverses over commutative rings" by K.P.S. Bhaskara Rao.
The matrix $ALA$ is a pseudoinverse of $L$. Likewise, $DMD$ is the pseudoinverse of $M$. Both claims are easily verified.
| 1 | https://mathoverflow.net/users/75761 | 386980 | 160,746 |
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