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https://mathoverflow.net/questions/311028 | 13 | Fix a Lie group $G$ and a discrete subgroup $\Gamma \subset G$. Homogeneous dynamics is about studying the actions of subgroups $H \subset G$ on the quotient $G/\Gamma$.
Does anyone know of an example of a question about $\Gamma$ that was answered by considering these dynamics?
| https://mathoverflow.net/users/126543 | Has dynamics on $G/\Gamma$ ever been used to prove interesting things about $\Gamma$? | Where to begin!
The ergodicity of non-compact subgroups (singular tori) was used by Margulis to prove that higher rank lattices $\Gamma$ are arithmetic.
Once you have that $\Gamma $ is arithmetic, this has the following consequences: (1) if $Comm (\Gamma)$ is the abstract commensurator, then $Comm (\Gamma)/\Gamma... | 15 | https://mathoverflow.net/users/23291 | 311029 | 135,361 |
https://mathoverflow.net/questions/310884 | 6 | Let $\Phi$ be an irreducible crystallographic root system in a Euclidean vector space $V$. Let $S\subseteq \Phi$ be some subset of roots for which $\mathrm{Span}\_{\mathbb{R}}(S)=V$.
**Question**: How big can $[\mathrm{Span}\_{\mathbb{Z}}(\Phi):\mathrm{Span}\_{\mathbb{Z}}(S)]$ be?
If $\Phi$ is of Type A, then I bel... | https://mathoverflow.net/users/25028 | How big can the index inside the root lattice of the lattice generated by a subset of roots be? | As suggested in [my second comment](https://mathoverflow.net/questions/310884/how-big-can-the-index-inside-the-root-lattice-of-the-lattice-generated-by-a-subs#comment774980_310884), one approach to the answer is *via* Borel–de Siebenthal theory, which classifies the subgroups $H$ that can arise as in [my first comment]... | 4 | https://mathoverflow.net/users/2383 | 311039 | 135,363 |
https://mathoverflow.net/questions/311035 | 0 | Let $\mathcal{E}(\mathbb{R})$ be the space of all $C^\infty$ functions on $\mathbb{R}$ with its usual topology, and $\mathcal{E}'(\mathbb{R})$ be the dual space with the weak\* topology.
Let $(T\_i)\_{i\in I}$ be a net in $\mathcal{E}'(\mathbb{R})$ that converges to $T$ in $\mathcal{E}'(\mathbb{R})$.
Is $(T\_i)\_... | https://mathoverflow.net/users/80845 | Covergent net in $\mathcal{E}'(\mathbb{R})$ implies bounded? | Not in general. For example let $I\_0$ be an arbitrary set and $(T\_i)\_{i\in I\_0}\ \ $ be an arbitrary family of elements of $\mathcal E'$. Let $I=I\_0\cup\{\alpha\}\ \ $ for some new element $\alpha$ and define a partial order on $I$ by setting $\alpha>i$ for any $i\in I\_0$. Then $I$ is a directed set. Define a net... | 3 | https://mathoverflow.net/users/nan | 311040 | 135,364 |
https://mathoverflow.net/questions/311034 | 4 | Let $M^3$ denote the Poincare homology sphere. I am wondering what the possible orders of (smooth) automorphisms of $M$ are (I'm not sure if allowing arbitrary homeomorphisms changes things?). By presenting the space as $-1$-surgery on the trefoil, there is a automorphism of order 3 given by the 3-fold symmetry of the ... | https://mathoverflow.net/users/99414 | Possible orders of automorphisms for the Poincare homology sphere | Let $G$ act smoothly and orientably. The quotient $M/G$ is a spherical 3-orbifold. Therefore, by the elliptization theorem, it may be given a metric of constant curvature 1; pulling back, then $M$ is given a constant curvature metric for which $G$ acts by isometries. (This was known to Thurston when the action of $G$ c... | 6 | https://mathoverflow.net/users/40804 | 311042 | 135,365 |
https://mathoverflow.net/questions/311018 | 4 | Is there a limit ordinal $\kappa\_0$ with $\kappa\_0 \lt 2^{\aleph\_0}$ and such that for every limit ordinal $\lambda$ with $\kappa\_0\leq \lambda\lt 2^{\aleph\_0}$ there is a connected $T\_2$-space $X\_\lambda$ with the following property?
>
> $\lambda$ is the smallest ordinal such that $X\_\lambda$ contains no s... | https://mathoverflow.net/users/8628 | Embedding ordinals with the order topology into connected $T_2$-spaces | The answer is no: if $\lambda$ is larger than $\omega^2$ and if $X$ contains $\lambda+\omega$ then it also contains $\lambda+\omega+\omega$.
To see this observe that $\lambda+1$ is homeomorphic with $\lambda+(\omega+1)+(\omega+1)$: simply take the first two copies of $\omega+1$ and move then to the end.
So by contrapos... | 7 | https://mathoverflow.net/users/5903 | 311043 | 135,366 |
https://mathoverflow.net/questions/311032 | 2 | Every Hausdorff space is $T\_1$ and sober. Does the converse hold? I expect not. What's a counterexample?
I expected I should be able to look this up in *Counterexamples in Topology*, but unfortunately that book doesn't appear to discuss sober spaces.
| https://mathoverflow.net/users/2362 | Space which is $T_1$ and sober but not Hausdorff? | As Nate Eldredge points out in the comments, there's a counterexample on [Wikipedia](https://en.wikipedia.org/wiki/Sober_space). See there or Nate's comment for a description.
| 1 | https://mathoverflow.net/users/2362 | 311046 | 135,367 |
https://mathoverflow.net/questions/311052 | 7 | Determinant and permanent of sum of two $n\times n$ permutation matrices can be arbitrarily different.
1. What is the distribution of determinant of sum and difference of two $n\times n$ permutation matrices?
2. What is the distribution of permanent of sum and difference of two $n\times n$ permutation matrices?
How... | https://mathoverflow.net/users/10035 | Distribution of sum of two permutation matrices | I will abuse notation by identifying a permutation and the matrix it represents. We can denote by $E(\sigma), O(\sigma)$ the number of even and odd cycles that $\sigma$ decomposes into. Given two permutations $\sigma\_1,\sigma\_2$ we can compute the following:
$$\det(\sigma\_1+\sigma\_2)=\left\{
\begin{array}{ll}
(-1... | 5 | https://mathoverflow.net/users/2384 | 311055 | 135,370 |
https://mathoverflow.net/questions/311063 | 7 | I am interested in a topological classification of connected closed 3-manifold $M$ that have finite homology group $H\_1(M)$.
Since $H\_1(M)$ is the abelization of the fundamental group $\pi\_1(M)$, each closed 3-manifold with finite homotopy group has finite homology group.
It is known that each closed 3-manifold ... | https://mathoverflow.net/users/61536 | Classification of closed 3-manifolds with finite first homology group? | The answer is no by Yves' comments. Let me add that there are plenty of explicit constructions of closed hyperbolic 3--manifolds with finite homology, and this is a generic phenomenon (for example random Heegard gluings have zero first Betti number and are hyperbolic and numerical experiments on the census manifolds ex... | 17 | https://mathoverflow.net/users/32210 | 311065 | 135,374 |
https://mathoverflow.net/questions/311077 | 0 | This is a question very similar to I recently asked on [mathexchange](https://math.stackexchange.com/questions/2923959/group-action-and-invariant-measures), but different enough to get its own entry in MO.
The setting is still the same. I consider the metric space $\mathbb{R}$ and the Lebesgue-measure $\lambda$ on $\... | https://mathoverflow.net/users/128769 | Measure on group invariant under group action on metric space | You are asking about the relationship between the Haar measures on the group $G$ and its homogeneous space $X=G/H$ (which, by the way, has nothing to do with metrics on either space). It is in great detail discussed in Nachbin's book [The Haar integral](https://mathscinet.ams.org/mathscinet-getitem?mr=175995).
| 2 | https://mathoverflow.net/users/8588 | 311087 | 135,381 |
https://mathoverflow.net/questions/310712 | 3 | Let $G$, $\mathbb C[G]$ and $\text{vN}(G)$ be a torsion free group, it's group ring and group von Neumann algebra, resp.. Let $0\neq\alpha\in\mathbb C[G]$ and $0\neq p\neq1$ is a projection in the group von Neumann algebra $\text{vN}(G)$, is it true that $\alpha p\neq0$?
| https://mathoverflow.net/users/84700 | An analytical zero divisor | I guess the answer is open in general, since a positive answer to your question implies [Kaplansky's zero-divisor conjecture](https://en.wikipedia.org/wiki/Kaplansky%27s_conjecture):
Indeed, let $V \subseteq \ell^2(G)$ be the kernel of the left-multiplication with $\alpha$ on $\ell^2(G)$. Note that $V$ is a $\mathrm{... | 3 | https://mathoverflow.net/users/54441 | 311091 | 135,384 |
https://mathoverflow.net/questions/311080 | 3 | I am dealing with a sequence $\{(x\_i,y\_i)\}$ of **zero-mean** random variables. For simplicity we can assume that the sequence is i.i.d. Define $Y\_i := i^{-1} \sum\_{k=1}^i y\_k$. I would like show that
\begin{equation} \tag{1} \label{eq:1}
\sup\_{l \leq \rho \leq u} \bigg| \frac{1}{\sqrt{n}} \sum\_{i = [\rho n]}^n ... | https://mathoverflow.net/users/122927 | Convergence of $\sup_{l \leq \rho \leq u} | \frac{1}{\sqrt{n}} \sum_{i=[\rho n]}^n \sum_{k=1}^i i^{-1} x_i y_k |$ in probability to zero | $\newcommand{\E}{\operatorname{\mathsf E}}
\newcommand{\Var}{\operatorname{\mathsf Var}}
\renewcommand{\P}{\operatorname{\mathsf P}}$
Looking at the calculation of the second moment in the statement of the question, it appears that it was tacitly assumed there that $\E x\_1^2+\E y\_1^2+\E x\_1^2y\_1^2 <\infty$, whi... | 2 | https://mathoverflow.net/users/36721 | 311092 | 135,385 |
https://mathoverflow.net/questions/311085 | 11 | I had asked this question in [MSE](https://math.stackexchange.com/questions/2921560/is-this-riemann-integral-for-prime-numbers-true). It got lot of upvotes but no answer (except one which was too long to be posted as a comment) hence I am posting it in MO.
While answering another [question in MSE](https://math.stacke... | https://mathoverflow.net/users/23388 | Riemann sum formula for definite integral using prime numbers | The statement follows from the prime number theorem. By an approximation argument, we can assume that $f$ is continuously differentiable on $[0,1]$. Then,
$$\sum\_{r=1}^{n}f\left(\frac{p\_r}{p\_n}\right)=\int\_0^1 f(x)\,d\pi(p\_n x)=nf(1)-\int\_0^1 f'(x)\pi(p\_n x)\,dx.$$
We estimate the last integral for fixed $f$:
$$... | 20 | https://mathoverflow.net/users/11919 | 311099 | 135,389 |
https://mathoverflow.net/questions/311101 | 1 | Let $X$ be a smooth, projective surface in $\mathbb{P}^3$ and $p \subset X$ a closed point in $X$. How do I compute $H^1(\mathcal{O}\_{X\backslash p})$?
Any reference/idea will be most welcome.
| https://mathoverflow.net/users/32151 | Sheaf cohomology of a complement of finitely many points | For what follows, I recommend SGA 2 ([available on Arxiv](https://arxiv.org/abs/math/0511279)). There is an exact sequence
$$0\rightarrow H^1(X,\mathcal{O}\_X)\rightarrow H^1(X\smallsetminus p,\mathcal{O}\_X)\rightarrow H^2\_{p}(X,\mathcal{O}\_X)\rightarrow H^2(X,\mathcal{O}\_X)$$
where $ H^2\_{p}(X,\mathcal{O}\_X)$ is... | 8 | https://mathoverflow.net/users/40297 | 311105 | 135,392 |
https://mathoverflow.net/questions/311084 | 1 | Consider four sequences of numbers, $0 \le a\_i, b\_i, c\_i, d\_i \le 1$, suppose they satisfy the following constraints:
(1). $\sum\_{i=1}^K a\_i \ge 1/2 + \epsilon$;
(2). $\sum\_{i=1}^K d\_i \le 1/2 - \epsilon$;
(3). $a\_i d\_i = b\_i c\_i$ for all $i=1, \ldots, K$.
Is it true that
\begin{equation}
\sum\_{i=1... | https://mathoverflow.net/users/66495 | additive discrepancy under a multiplicative constraint | $\newcommand{\ep}{\epsilon}$
It is not hard to show (see the proof at the end of this answer) that for any real $a,b,c,d\ge0$ such that $ad=bc$ we have
\begin{equation}\tag{1}
|a-b|+|a-c|\ge a-d.
\end{equation}
Replacing here $a,b,c,d$ by $a\_i,b\_i,c\_i,d\_i$ and summing in $i$, we have
\begin{equation}
\sum\_i... | 4 | https://mathoverflow.net/users/36721 | 311107 | 135,393 |
https://mathoverflow.net/questions/310899 | 5 | Let $G$ be a finitely generated group and let $H$ be a subgroup of $G$. $H$ is a codimension-1 subgroup of $G$ if $C\_{G}/H$ has more than one end, where $C\_{G}$ is the Cayley graph of $G$.
Do all codimension-1 subgroups of a $3$-manifold group correspond to the fundamental group of an immersed surface in the $3$-ma... | https://mathoverflow.net/users/128762 | Codimension-1 subgroups of 3-manifold groups | Let me make some remarks on this. As far as I know, the terminology "codimension-1 subgroup" originated from a paper of Micah Sageev
*Sageev, Michah*, [**Ends of group pairs and non-positively curved cube complexes**](http://dx.doi.org/10.1112/plms/s3-71.3.585), Proc. Lond. Math. Soc., III. Ser. 71, No. 3, 585-617 (1... | 5 | https://mathoverflow.net/users/1345 | 311112 | 135,395 |
https://mathoverflow.net/questions/311008 | 6 | ~~Consider the discrete combinatorial model structure on the category of $\Delta$-generated spaces: all maps are cofibrations and fibrations and the weak equivalences are the homeomorphisms. Applying Proposition A.3.2.4 of (Higher Topos Theory, Lurie)~~ By mimicking the construction of the canonical model structure on ... | https://mathoverflow.net/users/24563 | About a canonical model structure on topologically enriched categories | Yes, the canonical model structure is unique. The uniqueness of the canonical model structure on $Cat$ was [nicely exposited](https://sbseminar.wordpress.com/2012/11/16/the-canonical-model-structure-on-cat/) by Chris Schommer-Pries on the Secret Blogging Seminar back in the day. Let's go through and mimic the proof the... | 6 | https://mathoverflow.net/users/2362 | 311120 | 135,399 |
https://mathoverflow.net/questions/311138 | 5 | Is there a good notion of distance between partitions of a (fixed, finite) set? The context is this: suppose I have a clustering algorithm, which clusters points using some method or other. Now, I perturb the positions of the points, the clustering changes, and I want some quantitative estimate of how much it has chang... | https://mathoverflow.net/users/11142 | Clustering distance | The [variation of information](https://en.wikipedia.org/wiki/Variation_of_information) seems to be the sort of thing you're looking for.
| 4 | https://mathoverflow.net/users/1847 | 311139 | 135,408 |
https://mathoverflow.net/questions/311133 | 9 | Could someone provide or point me to a family of number rings $\mathcal{O}\_{K\_n}$ that require $n$ generators (as $\mathbb{Z}$-algebra)? Second best would be a family requiring $f(n)$ generators for a strictly increasing and positive function $f:\mathbb{N}\to\mathbb{N}$.
I would also be interested in seeing several... | https://mathoverflow.net/users/94086 | Explicit family of number rings $\mathcal{O}_{K_n}$ requiring $n$ generators? | An earlier question on this type of topic was asked by Zev Chonoles in 2010 at [Which number fields are monogenic? and related questions](https://mathoverflow.net/questions/21267/) and I want to draw your attention to the comment there by BCnrd for a nice geometric analogy. When answering that question I did not addres... | 20 | https://mathoverflow.net/users/3272 | 311140 | 135,409 |
https://mathoverflow.net/questions/311135 | 13 | Consider a geodesic current $\mu$ on a closed surface $\Sigma$, as defined by Bonahon ("[The Geometry of Teichmüller space via geodesic currents](https://eudml.org/doc/143562)"). These are $\pi\_1(\Sigma)$-invariant measures on the space of geodesics on $\widetilde\Sigma$. It is well-known that $\mu$ can only have atom... | https://mathoverflow.net/users/5010 | Geodesic current supported on a pencil? | No - this is impossible because any ergodic geodesic current either has purely non-atomic marginals or corresponds to a closed geodesic. Indeed, the quoted result from Martelli implies that if one of the marginals contains atoms then it has to be concentrated on the orbit of an endpoint of a periodic geodesic (actually... | 11 | https://mathoverflow.net/users/8588 | 311142 | 135,410 |
https://mathoverflow.net/questions/311147 | 5 | Let $\sigma:\mathbb{R}^n\times \mathbb{R}^n\to \mathbb{R}$ be a bilinear symmetric form which is non-degenerate in the sense that for every $0\neq u\in \mathbb{R}^n$ there is $v\in \mathbb{R}^n$ with $\sigma\left(u,v\right)\ne 0$.
It is well-known and easy to see that there is a basis $e\_1,...,e\_n$ and $m \in \over... | https://mathoverflow.net/users/53155 | Hilbert representation of a bilinear form | The answer is no.
Let be construct such a $(V,\sigma)$.
Let $X$ be the set of functions $\mathbf{N}\to\mathbf{R}$. Let $Y\subset X$ be a subset such that
* $Y$ is linearly independent
* $Y$ contains all Dirac functions $n\mapsto\delta\_{m,n}$
* for every $g\in X$, there exists $f\in Y$ such that $\limsup(f/(|g|+1))... | 6 | https://mathoverflow.net/users/14094 | 311154 | 135,414 |
https://mathoverflow.net/questions/188379 | 8 | Firstly some definitions:
>
> 1. $B\_n$ is the braid group with $n$ strands.
> 2. $\widetilde{B\_n}$ is "homotopy braid group", which is a factor group of $B\_n$ by adding the relation that $A\_{j,k}$ commutes with
> $gA\_{j,k}g^{-1}$, where $A\_{j,k}$ are the usual generators of the pure
> braid group $P\_n$ de... | https://mathoverflow.net/users/15770 | Action of the homotopy braid groups on reduced free groups | The short answer is *yes*. I will try to explain the intuition behind appearance of the reduced free groups and the link homotopy and why they are related; for the rest, I refer to the great papers of Milnor, Goldsmith, Habegger&Lin and Bar-Natan.
Pure braids which are allowed to go ''back in time'' were given a name... | 8 | https://mathoverflow.net/users/104817 | 311155 | 135,415 |
https://mathoverflow.net/questions/311124 | 9 | Yesterday I was talking to somebody from the Haskell community.
Late in the night we found ourselves discussing possible topoi.
Lets order topoi (up to equivalence, ...) by number of objects/morphisms
* The smallest topos is a point ([this is the only finite topos](https://math.stackexchange.com/questions/795438/ar... | https://mathoverflow.net/users/14120 | What are the "smallest" topoi? | Here are some examples, that should show you that there is a lot of countable toposes and lot of things in between finite sets and sets, too much to actually hope to list or classifies.
* First as I said a lots of usual construction of Grothendieck toposes have a finistic version, which produces elementary toposes (i... | 13 | https://mathoverflow.net/users/22131 | 311169 | 135,420 |
https://mathoverflow.net/questions/311059 | 7 | If $\Omega$ is a real analytic domain in $\mathbb R^n$, is the signed distance function, $f$, defined by
\begin{equation}
f(x)=\begin{cases}d(x,\partial \Omega )&{\mbox{ if }}x\in \Omega \\-d(x,\partial \Omega )&{\mbox{ if }}x\in \Omega ^{c}\end{cases}
\end{equation}
is real analytic? where
\begin{equation} d(x,\partia... | https://mathoverflow.net/users/102331 | is signed distance function real analytic for real analytic domains | **The answer is yes,** that is $f(x)$ is real analytic in a neighborhood of any point on $\partial\Omega$.
First recall that if $f$ and $g$ are real analytic functions (of several variables), then $f+g$, $f\cdot g$, $f/g$ (when $g\neq 0$), $f\circ g$, and the inverse map $f^{-1}$, if $f$ is a diffeomorphism, are all... | 7 | https://mathoverflow.net/users/121665 | 311179 | 135,425 |
https://mathoverflow.net/questions/311183 | 0 | Is there a countable, simple, connected graph $G=(\omega, E)$ such that $\text{deg}(v)$ is infinite for all $v\in \omega$, and for all [matchings](https://en.wikipedia.org/wiki/Matching_(graph_theory)) $M\subseteq E$ the set $V\setminus (\bigcup M)$ is infinite?
| https://mathoverflow.net/users/8628 | Connected infinite graphs in which all matchings are "small" | If all degrees are infinite, it contains a perfect matching: just add edges one by one covering all vertices.
| 1 | https://mathoverflow.net/users/4312 | 311187 | 135,427 |
https://mathoverflow.net/questions/311180 | 8 | Let $\mathcal I$ be a (non-principal) ideal of subsets of $\mathbb N$. Suppose that every family $\mathcal{A} \subset \wp(\mathbb N)\setminus \mathcal I$ with the following property is countable:
$$A,B\in \mathcal A, A\neq B \Rightarrow A\cap B \in \mathcal I. $$
Let us observe that maximal ideals have this property ... | https://mathoverflow.net/users/15129 | Ideals on $\mathbb N$ and large sets that have small intersection | The denumerable atomless Boolean algebra $A$ can be isomorphically embedded in $\wp(\omega)/\mathsf{fin}$. The identity on $A$ can be extended to a homomorphism from $\wp(\omega)/\mathsf{fin}$ into the completion of $A$. This gives a homomorphic image of $\wp(\omega)/\mathsf{fin}$, hence also of $\wp(\omega)$ itself, i... | 4 | https://mathoverflow.net/users/90095 | 311191 | 135,428 |
https://mathoverflow.net/questions/311189 | 11 | **Part 1: a single finite place.** Let $K$ be a finite extension of $\mathbf{Q}\_p$.
>
> Does there exist a number field $F/\mathbf{Q}$ and a finite place $v$ lying over $p$, such that for the completion of $F$ at $v$ we have a topological isomorphism of topological field extensions of $\mathbf{Q}\_p$:
>
>
> $$F\... | https://mathoverflow.net/users/nan | “Algebraization" of $p$-adic fields | The answer to part 1 is yes. Given $K/\mathbb{Q}\_p$, let $\alpha\in K$ be a primitive element, with minimal monic polynomial $f(x)=x^n+\sum\_{i=1}^n a\_i x^{n-i}$, $a\_i\in\mathbb{Q}\_p$. So we have $K\cong\mathbb{Q}\_p[x]/(f(x))$. Krasner's lemma implies there exists a positive integer $N$ such that if $b\_i\in\mathb... | 10 | https://mathoverflow.net/users/5263 | 311195 | 135,430 |
https://mathoverflow.net/questions/311131 | 2 | Question originally posted [here](https://math.stackexchange.com/questions/2925805/how-wide-is-the-minimum-gap-of-a-parameter-dependent-matrix), but I'm more likely to get an answer on MO. I'm looking for a reference on the following topic, should one exist. (Or a firm "stop looking and do it yourself".)
Let $H(s)=(1... | https://mathoverflow.net/users/43307 | The width of the minimum gap of an interpolated matrix | A bound with a function $f(G)$ that does not depend on $H(0)$ and $H(1)$ seems unlikely. In a typical situation the function $\gamma(s)$ can be extended to the complex plane and $\gamma(z)$ vanishes at a point $z=s\_0+i\delta$ for some $s\_0\in(0,1)$ close to the real axis. The inverse gap $1/\gamma(s)$ will show a pea... | 1 | https://mathoverflow.net/users/11260 | 311197 | 135,431 |
https://mathoverflow.net/questions/311218 | 13 | Let $M$ be a differentiable manifold. Let $\mu$ be a (probability) measure on $M$.
What are the conditions under which $\mu$ is given by a differential form on $M$? I imagine some sort of compatibility of the topology or the differentiable structure of $M$ with the $\sigma$-algebra of $\mu$ would be required.
(Apo... | https://mathoverflow.net/users/129125 | Measures and differential forms on manifolds | I assume that $\mu$ is a measure defined on the $\sigma$-algebra of Borel sets. First, on any manifold the notion of negligible set is well defined.
If $M$ is orientable and $\mu(N)=0$ for any negligible Borel set then the Radon-Nicodym theorem implies that, for any smooth volume form $\omega$ on $M$, there is a pos... | 14 | https://mathoverflow.net/users/20302 | 311225 | 135,438 |
https://mathoverflow.net/questions/311236 | 3 | Are there any known results about the following problem:
Given any integer $n\geqslant4$, how many $4$-elements subsets at most can we choose from $\{1,2,\cdots,n\}$ such that the intersection of any two $4$-elements subsets we have choosed has not more than $2$ element?
More generally, given any integer $n\geqslan... | https://mathoverflow.net/users/58096 | How many $2d$-elements subsets with specific property at most can we choose from $\{1,2,\cdots,n\}$ | This can be thought of as finding the largest binary code of constant weight $2d$, length $n$, minimal distant $2d$.
For $d=2$ the values are [A001843](https://oeis.org/A001843) in the OEIS. References/links on the OEIS page have some further information about both $d=2$ and other values of $d$.
| 4 | https://mathoverflow.net/users/51668 | 311240 | 135,444 |
https://mathoverflow.net/questions/311176 | 1 | Let $X$ be a smooth, projective variety, ${F}$ a quasi-coherent $\mathcal{O}\_X$-module on $X$ supported on a closed subscheme, say $Z \subset X$. Is it true that $H^i(X,F)=0$ for all $i>\dim Z$?
We know that $H^i(X,F)=0$ for all $i>\dim X$.
| https://mathoverflow.net/users/38832 | Reformulation of Grothendieck vanishing theorem | Every abelian sheaf $\mathcal{F}$ on $X$ whose support is contained in $Z$, has vanishing cohomology groups $H^i(X, \mathcal{F})$ for $i > \dim Z$.
Proof. The support of an abelian sheaf is the set of points where the stalk is nonzero. If the support of $\mathcal{F}$ is contained in $Z$, then $\mathcal{F}$ is equal t... | 2 | https://mathoverflow.net/users/129226 | 311267 | 135,450 |
https://mathoverflow.net/questions/311265 | 4 | Consider the polynomial ring $R=\mathbb C[x,y]$.
Consider the matrix $A=\begin{pmatrix} x^5+y^5&5x^5&10x^5&10x^5&5x^5\\5y^5&x^5+y^5 &5x^5&10x^5&10x^5 \\10y^5&5y^5&x^5+y^5&5x^5&10x^5\\10y^5&10y^5&5y^5&x^5+y^5&5x^5\\5y^5&10y^5&10y^5&5y^5&x^5+y^5 \end{pmatrix} \in M\_5(R)$.
Also consider the polynomial $p(x,y)=\det (A... | https://mathoverflow.net/users/127118 | On the linear factors of a polynomial obtained from the determinant of a matrix whose entries are related to Binomial expansion | Define the $\, n\times n\,$ matrix $\, A = \{a\_{i, j}\}\_{i, j=1}^n \,$ where
$\, a\_{i, j} = \binom{n}{j-i}x^n + \binom{n}{i-j}y^n. \,$
The matrix $A$ is a special [Toeplitz matrix](https://en.wikipedia.org/wiki/Toeplitz_matrix). Let $\, p(x, y) := \det(A-I).\,$ Since $\, x + y - 1 \,$
is a factor, then also $\, z ... | 6 | https://mathoverflow.net/users/113409 | 311274 | 135,454 |
https://mathoverflow.net/questions/311257 | 4 | I'm looking for a reference for the following fact: In the torus $\mathbb T^d$ let me denote by $u\_t=u(t,x)$ the (unique, distributional) solution of the heat equation
$$
\partial\_t u=\Delta u
$$
started from an arbitrary probability distribution $u\_0\in\mathcal P(\mathbb T^d)$.
I know that there is a universal cons... | https://mathoverflow.net/users/33741 | Universal decay rate of the Fisher information along the heat flow | The result actually holds with $C=d/2$ on any compact Riemannian manifold with a non-negative Ricci curvature. This can be seen by integrating the Li-Yau inequality: Theorem 1.1 in
[On the parabolic kernel of the Schrödinger operator](https://projecteuclid.org/euclid.acta/1485890415)
| 5 | https://mathoverflow.net/users/48356 | 311276 | 135,455 |
https://mathoverflow.net/questions/311282 | 5 | Let $K(G,n)$ be the Eilenberg Maclane space.
Consider the map from
$$
K(\mathbb{Z}\_2,1) \to K(\mathbb{Z}\_4,1) \overset{f}{\to} K(\mathbb{Z}\_2,1)\overset{g}{\to}K(\mathbb{Z}\_2,2) \to \dots,
$$
It looks that we can represent the map from $K(\mathbb{Z}\_2,1)\to K(\mathbb{Z}\_2,2)$
relating to the
generator of coh... | https://mathoverflow.net/users/106497 | Interesting properties in $...\to K(\mathbb{Z}_4,1) \overset{f}{\to} K(\mathbb{Z}_2,1)\overset{g}{\to}K(\mathbb{Z}_2,2) \to ...$ | Represent $p$ by the identity map $id: \mathbb{Z}\_2 \to \mathbb{Z}\_2$.
Then $(p\cup p)(a,b) = p(a)p(b)$ is non-zero only on the 2-chain $(1,1)$. Namely, as a polynomial mod 2, $(p\cup p)(a,b) = ab$. When lifted to $\mathbb{Z}\_4$, we get $p\cup p(a,b)=ab \mod 2$. Consider the function $\gamma: \mathbb{Z}\_4\to \math... | 9 | https://mathoverflow.net/users/115052 | 311283 | 135,456 |
https://mathoverflow.net/questions/311165 | 5 | Let us recall that a group $G$ is called *perfect* if it coincides with its commutator subgroup $G'$, or equivalently, if its abelianization $G/G'$ is trivial.
>
> **Question.** Is there any name for a group $G$ whose abelianization $G/G'$ is finite?
>
>
>
| https://mathoverflow.net/users/61536 | A name for a group with finite abelization? | Prompted by the OP, I'm writing my comment as an answer.
I suggest “almost perfect” group. A google search indicates this terminology has been used; for instance, here: core.ac.uk/download/pdf/61487184.pdf .
| 6 | https://mathoverflow.net/users/1463 | 311285 | 135,458 |
https://mathoverflow.net/questions/311290 | 0 | Schatten $p$ norm is convex when $p\geq1$ holds and if $p\in(0,1)$ it is quasinorm.
1. If $p\in(0,1)$ then is Schatten $p$ norm quasi convex? I am interested in definition of quasi convexity here <https://dl.acm.org/citation.cfm?id=1716350> and here <http://www.numdam.org/item/COCV_2008__14_4_795_0>.
2. Is there a g... | https://mathoverflow.net/users/10035 | Quasiconvexity property of quasinorms | No, quasinorms are in general not quasiconvex. (Well, this is true if quasiconve means that the levelsets of the function are convex; other defintions of quasiconvexity may exist…)
By positive homogeneity, a (quasi)norm is completely described by one of its levelsets. If this levelset is convex (and fulfills some oth... | 0 | https://mathoverflow.net/users/9652 | 311291 | 135,460 |
https://mathoverflow.net/questions/270770 | 4 | Let $X$ be a reduced compact complex analytic space of $\dim\_{\mathbb{C}}X\ge2$; by [KJ] definition 3.29, remark 3.44 and theorem 3.45, it admits a *strong resolution* $R(X)$ which is smooth, $E=\pi\_X^{-1}(X\_{sing})$ is a *simple normal crossing (snc) divisor*, $R(X)\setminus E$ is biholomorphic to $X\_{reg}$ and th... | https://mathoverflow.net/users/57030 | Kähler metric on compact complex manifolds with simple normal crossing divisor | Without loss of generality, let $X$ be irreducible, then it is pure-dimensional ([GH,RR] proposition 9.1.3).
Let $\left(\widehat{X},\nu\right)$ be the normalization of $X$ ([FG] chapter 2, section 16, Normalization theorem); for exact:
1. $\widehat{X}$ is a normal complex analytic space;
2. $\nu$ is a finite surjec... | 1 | https://mathoverflow.net/users/57030 | 311305 | 135,465 |
https://mathoverflow.net/questions/311318 | 2 | Do the classes of pointed Hurewicz cofibrations, pointed Hurewicz fibrations and pointed homotopy equivalences give a model structure on pointed (compactly generated weak Hausdorff) topological spaces that is compatible with the smash product?
Maybe, this would work with another class of fibrations? (I am interested ... | https://mathoverflow.net/users/128857 | A monoidal model structure on pointed spaces | Yes. According to [the nLab page for the Strom model structure](https://ncatlab.org/nlab/show/Str%C3%B8m+model+structure#monoidal_structure), this is proven in Section 6.4 of May's [Concise Course in Algebraic Topology](https://www.maths.ed.ac.uk/~v1ranick/papers/maybook.pdf). The point is that a closed inclusion is a ... | 2 | https://mathoverflow.net/users/11540 | 311320 | 135,470 |
https://mathoverflow.net/questions/311314 | 4 | It is well know that the Haar probability measure for the $U(N)$ group, given by
$$
\begin{align}
dX\_{U(N)} & = \frac{1}{N!(2\pi)^N}
\begin{vmatrix}
1 & 1 & \cdots & 1 & 1 \\
e^{i\lambda\_1} & e^{i\lambda\_2} & \cdots & e^{i\lambda\_{N-1}} & e^{i\lambda\_N} \\
e^{i2\lambda\_1} & e^{i2\lambda\_2} & \cdots & e^{i2\la... | https://mathoverflow.net/users/126885 | integral kernel function for the SU(N) group | The integral kernel for ${\rm U}\,(N)$, due to Dyson, has been generalized by Katz and Sarnak to other compact groups ([Random Matrices, Frobenius Eigenvalues, and Monodromy](https://web.math.princeton.edu/~nmk/RMFEM.pdf), page 121). Their result has the general form
$$d\mu=\frac{1}{n!}\det\_{n\times n}[L\_N(\lambda\_i... | 4 | https://mathoverflow.net/users/11260 | 311328 | 135,473 |
https://mathoverflow.net/questions/311281 | 8 | I'd like to understand how ordinary schemes deform or lift to spectral and derived schemes in two basic examples as well as what the structure of the space of deformations in general is.
Let $S = (X, \mathcal{O}\_X)$ be a scheme, where $X$ is the underlying topological space, and $\mathcal{O}\_X$, its structure sheaf... | https://mathoverflow.net/users/129125 | Spectral and derived deformations of schemes | In general, these are incredibly hard questions. It seems to me that one natural question to ask (if you are interested in $\pi\_0$ of ring spectra) would be about understanding *even periodic* $\mathbf{E}\_\infty$-rings $A$ with $\pi\_0 A = R$. Alternatively, you could attempt to understand those $\mathbf{E}\_\infty$-... | 6 | https://mathoverflow.net/users/102390 | 311349 | 135,477 |
https://mathoverflow.net/questions/311239 | 12 | The arcsine law for the distribution of the logarithms of the divisors of an integer $n$ states that
$$
\frac{1}{x}\sum\_{n\leq x}\frac{1}{d(n)}\sum\_{\substack{q|n\\q\leq n^{A}}}1\sim \frac{2}{\pi}\arcsin \sqrt A
$$
for $0<A\leq 1$. This was proved by Deshouillers, Dress and Tenenbaum (Acta Arithmetica (1979)
Volume ... | https://mathoverflow.net/users/10980 | Has the arcsine law been generalised to higher order divisor functions? | There's an extension of the Deshouillers-Dress-Tenenbaum theorem by Bareikis and Manstavičius, <http://doai.io/10.4064/aa126-2-5>, which almost treats the question you ask. I'm not sure why the condition $f(p^\ell) \ll 1$ is imposed there; as far as the method goes the condition $f(p^\ell) \ll \ell^C$ ($C$ absolute) sh... | 3 | https://mathoverflow.net/users/47440 | 311367 | 135,480 |
https://mathoverflow.net/questions/311370 | 2 | Let $X$ and $Y$ be Banach spaces.Let $L(X,Y)$ denote the space of all bounded linear map from $X$ to $Y$. $T:X\longrightarrow Y$ is said to be norm attaining if there exists a $x\in S\_X$(the closed unit circle in X) such that $$\|T(x)\|=\|T\|.$$Let $NA(X,Y)$ denote the set of all norm attaining maps in $L(X,Y)$. Let $... | https://mathoverflow.net/users/127674 | Regarding norm attaining functions | Let $T\in NA(X,Y)$ so there is $x\in X, \|x\|=1, \|T(x)\| = \|T\|$. By Hahn-Banach there is $f\in Y^\*, \|f\|=1, f(T(x)) = \|T(x)\| = \|T\|$. But $f(T(x)) = T^\*(f)(x)$ so
$$ \|T^\*\| = \|T\| = |T^\*(f)(x)| \leq \|T^\*(f)\| \|x\| \leq \|T^\*\| \|f\| \|x\| =\|T^\*\|. $$
Hence we have equality throughout, in particular, ... | 2 | https://mathoverflow.net/users/406 | 311375 | 135,485 |
https://mathoverflow.net/questions/311081 | 3 | Suppose $\lambda\_1,\lambda\_2,\cdots,\lambda\_n$ are algebraic numbers. $P\_1(t),P\_2(t),\cdots,P\_n(t)$ are polynomials with algebraic coefficients.
The question is to whether the following question is decidable.
$$\sum\_{i}^n P\_i(t)\exp(\lambda\_i t)$$
has a root $t\_0>0$.
| https://mathoverflow.net/users/4987 | positive root for exponential polynomial | If all numbers are real, it is decidable. This follows, for example from the general result in the paper
Vorobʹev, N. N., Jr. Deciding the consistency of a system of inequalities...
<https://link.springer.com/chapter/10.1007%2F978-1-4612-0441-1_33>.
If complex numbers are allowed it is not clear to me what the an... | 3 | https://mathoverflow.net/users/25510 | 311379 | 135,487 |
https://mathoverflow.net/questions/311362 | 4 | Is there some database that contains "all" low-index normal subgroups of the free group on two generators?
Extension: does there exist such a GAP-database?
Thank you!
| https://mathoverflow.net/users/129328 | Database subgroups of free group | I think the answer is no. There exists a Magma command $\mathtt{LowIndexNormalSubgroups}$ that does what you want, and it does indeed find generators for each of the subgroups. I believe that a version in $\mathsf{GAP}$ is being written and will be available shortly.
Using Magma I got up to index 50 without having to... | 11 | https://mathoverflow.net/users/35840 | 311380 | 135,488 |
https://mathoverflow.net/questions/311366 | 4 | Let $\mathcal{E}'(\mathbb{R})$ be equipped with its usual strong topology (being the dual space of $\mathcal{E}(\mathbb{R})$). Is convolution jointly continuous on $\mathcal{E}'(\mathbb{R})$?
| https://mathoverflow.net/users/128952 | Is convolution jointly continuous on $\mathcal{E}'$? | Yes. This is Théorème IV in §3 of Chapitre VI (page 157) in Laurent Schwartz's *Théorie des distributions*.
| 5 | https://mathoverflow.net/users/21051 | 311381 | 135,489 |
https://mathoverflow.net/questions/311395 | 2 | Denote the set of prime numbers by $P$.
Let $u,v,m,n \in \mathbb{N}-\{0\}$ satisfy:
$m \leq n$, $\gcd(u,m)=1$ and $\gcd(v,n)=1$.
>
> Is it possible to find $L \in \mathbb{N}$, such that $u+Lm \in P$
> and $v+Ln \in P$? (different primes, probably).
>
>
>
Of course, by [Dirichlet's theorem on arithmetic progres... | https://mathoverflow.net/users/72288 | Almost simultaneous Dirichlet's theorem on arithmetic progressions | This is simply a reformulation of (a weaker version of) the prime pairs conjecture, a generalization of the twin primes conjecture. For example, when $m=n=1$ and $u=1$, $v=3$, this is exactly the twin primes conjecture, except that such conjectures are usually formulated as "are there infinitely many?" rather than "doe... | 4 | https://mathoverflow.net/users/5091 | 311398 | 135,494 |
https://mathoverflow.net/questions/311397 | 5 | I've been working with Fuchsian groups and from geometrical motivations finding a cocompact normal Fuchsian subgroups of $PSL(2,\mathbb{R})$ would have intresting properties for my research.
It is known that $SL(2,\mathbb{R})$ has no connected normal subgroups other than the its centre { Id,-Id }, however it might ha... | https://mathoverflow.net/users/97192 | Normal Fuchsian subgroups | Let $\Gamma$ be a discrete subgroup of a connected Lie group $G$.
Suppose that $\Gamma$ is normal. For given $\gamma\in\Gamma$ the conjugacy class is the image of $G$ under the continuous map $x\mapsto x\gamma x^{-1}$, therefore it is connected. Since it lies in $\Gamma$, it consists of one point only, and as $x=1$ occ... | 8 | https://mathoverflow.net/users/nan | 311399 | 135,495 |
https://mathoverflow.net/questions/311392 | 1 | Let $A\in\mathbb{R}^{n\times n}$ be a diagonalizable matrix with real and strictly positive eigenvalues (note that $A$ is not required to be symmetric).
>
> **My question.** Do there exist an orthogonal matrix $T\in\mathbb{R}^{n\times n}$ and a symmetric positive definite matrix $P\in\mathbb{R}^{n\times n}$ such t... | https://mathoverflow.net/users/62673 | A "positive diagonal plus skew-symmetric" matrix decomposition | Choose any positive definite matrix $Q$. Since $A$ has eigenvalues with positive real part, the Lyapunov equation $$AP + PA^\top = Q$$
is solvable, and its solution $P$ is symmetric and positive definite.
Now decompose $AP = H+\hat{S}$, where $H$ is symmetric and $\hat{S}$ is skew-symmetric. Plugging this decompositi... | 1 | https://mathoverflow.net/users/1898 | 311403 | 135,497 |
https://mathoverflow.net/questions/311406 | 3 | Suppose $v\in R^n$ is a constant unit vector. $P\_l$ is a random projection matrix to an $l$ dimensional subspace of $R^n$ which is uniformly sampled from $G(l,R^n)$ which is the collection of all $l$-dimensional subspace in $R^n$. What is the upper bound of the following:
$$\mathbb{P}(P\_lv\leq\delta)$$
What is the or... | https://mathoverflow.net/users/123075 | Tail probability of random projection | $\newcommand{\R}{\mathbb{R}}
\renewcommand{\P}{\operatorname{\mathsf P}}
\newcommand{\Ga}{\Gamma}
\newcommand{\de}{\delta}$
In view of the spherical symmetry of the distribution of the $l$-dimensional subspace, we can fix it to be, say, the span of the first $l$ vectors of the standard basis of $\R^n$ and, accordingl... | 6 | https://mathoverflow.net/users/36721 | 311410 | 135,500 |
https://mathoverflow.net/questions/311355 | 10 | Call two Dyck paths $D\_1$ and $D\_2$ derived equivalent in case their corresponding Nakayama algebras are derived equivalent (The Dyck path of a Nakayama algebra with a linear quiver is just the top boundary of its Auslander-Reiten quiver, so we can identify Nakayama algebras with a linear quiver with Dyck paths).
De... | https://mathoverflow.net/users/61949 | Derived equivalences of Dyck paths | Question 1 has a positive answer by the answer of Gjergji Zaimi in this thread: [What are the periodic Dyck paths?](https://mathoverflow.net/questions/303487/what-are-the-periodic-dyck-paths/311511#311511) .
Here a positive answer to question 1 and 2 using a complicated classification result that uses several other d... | 6 | https://mathoverflow.net/users/61949 | 311419 | 135,503 |
https://mathoverflow.net/questions/311382 | 0 | Let's define the n-th degree Chebyshev polynomials by
$$ T\_{n} (x)=\cos(n\arccos(x)).$$
Find a polynomial $P$ such that
$$\mid y- P (x) \mid$$
is minimal, using the first three Chebyshev polynomials for the following data:
$$ \begin{bmatrix}
x & -1 & -0.5 & 0 & 0.5 & 1 \\
y & 0.6346 & 0.6565 & 1 & 1.5230 & ... | https://mathoverflow.net/users/129345 | Chebyshev interpolation | Uniform approximation by polynomials on a finite set of points is studied in Section 1.3 of
>
> T.J. Rivlin, An introduction to the approximation of functions, Dover,
> 2003.
>
>
>
An explicit solution to your (non trivial) question is given on p.36 of that book.
| 0 | https://mathoverflow.net/users/89429 | 311421 | 135,504 |
https://mathoverflow.net/questions/311415 | 4 | Let X be a smooth projective connected curve over $\mathbb{C}$ and let $n>1$ be an integer. Let $Y= Sym^n\_X$ be the $n$-th symmetric product of $X$.
>
>
> >
> > Is there, for every $i$, a nice formula for the Hodge decomposition of $H^i(Y,\mathbb{C})$?
> >
> >
> >
>
>
>
If not, what part of the Hodge dia... | https://mathoverflow.net/users/129371 | Hodge decomposition of the symmetric product of a curve | Look at Example $1.1$ in [this paper](https://www.math.wisc.edu/~maxim/sym_lib60.pdf) for a nice formula.
You can also compute them by hands (and, hopefully, prove the desired formula) by identifying $\mathrm{H}^{p,q}(\operatorname{Sym}^n X)$ with $S\_n$-invariant part of $\mathrm{H}^{p,q}(X^n)$ (use the Kunneth's f... | 2 | https://mathoverflow.net/users/115211 | 311433 | 135,509 |
https://mathoverflow.net/questions/311431 | 5 | Let $(X,B)$ be a projective log canonical pair (here I mean $B \geq 0$). Assume that the coefficients of $B$ are rational, and that $K\_X+B \equiv 0$. Is it true that $K\_X + B \sim\_\mathbb{Q} 0$? I know it is true if $(X,B)$ is klt, and "The moduli b-divisor of an lc-trivial fibration" by Ambro has a proof of it. I a... | https://mathoverflow.net/users/89459 | $K_X+B \equiv 0$ implies $K_X + B \sim_\mathbb{Q} 0$? | For lc pair or slc pair, it is true. This is Gongyo’s result. See [J. ALGEBRAIC GEOMETRY 22 (2013) 549–564].
BTW, the relative version is also true, which is not a trivial generalization of the absolute case. It is proved by Hacon and Xu [On Finiteness of B-representation and Semi-log Canonical Abundance].
| 10 | https://mathoverflow.net/users/42636 | 311434 | 135,510 |
https://mathoverflow.net/questions/311429 | 5 | The signed symmetric group $B\_n$ is a permutation group where the underlying set is $B\_n=\{\sigma \in S\_{A\_n}| \forall x \in A\_n, \sigma(-x)= -\sigma(x)\}$ with $A\_n=\{-n,-(n-1),-(n-2),\cdots,-1,1,\cdots, n-1,n\}$.
$B\_n$ can be generated by the 3 permutations $(1,2)(-1,-2) ; (1,2,\cdots, n)(-1,-2,\cdots, -n)$ ... | https://mathoverflow.net/users/129380 | Is there a size 2 generating set of the signed symmetric group $B_n$? | Yes. Take two $a,b$ generators of $S\_n$, where $b$ has odd order and fixes point $1$. For example, if $n$ is even, let $a=(1,2)$, $b=(2,3,\ldots,n)$. If $n$ is odd let $a=(1,2,3,4)$, $b=(3,4,\ldots,n)$.
Let $\bar{a},\bar{b}$ be their natural images$^\dagger$ in $B\_n$. Then $B\_n = \langle \bar{a}, \bar{b}(1,-1) \ra... | 10 | https://mathoverflow.net/users/35840 | 311449 | 135,517 |
https://mathoverflow.net/questions/311452 | 17 | Preparing to a lecture on Krein--Milman theorem I read in W. Rudin's Functional analysis textbook (1973) that it is unknown whether any convex compact set in any topological vector space has an extreme point. Is it still unknown?
| https://mathoverflow.net/users/4312 | Extreme points of convex compact sets | A counterexample is given in the following paper:
>
> Roberts, James W. "[A compact convex set with no extreme points](https://eudml.org/doc/218141)."
> Studia Mathematica 60.3 (1977): 255-266.
>
>
>
I did not see the original paper, but an exposition can be found in Section 5.6 of the book "Metric linear spa... | 22 | https://mathoverflow.net/users/35357 | 311453 | 135,520 |
https://mathoverflow.net/questions/311425 | 1 | Suppose we have a simultaneous game, that has a strong Nash equilibrium (SNA), i.e. a weak Pareto efficient Nash equilibrium (no deviation of any subset of player brings a benefit to them).
Now suppose we play this game repeatedly. Does the repeated game has a strong Nash equilibrium, too?
I would keep the questio... | https://mathoverflow.net/users/21965 | Strong Nash Equilibria in repeated games | Let's first recall the definition of a strong nash equilibrium: A strong Nash equilibrium is a Nash equilibrium in which no coalition, taking the actions of its complements as given, can cooperatively deviate in a way that benefits all of its members. In particular, some of the deviating players may profit, but then ot... | 1 | https://mathoverflow.net/users/64609 | 311454 | 135,521 |
https://mathoverflow.net/questions/311160 | 1 | It follows from Exercise 1.13 in Humphreys' Category $\mathcal{O}$ book that $M\in\mathcal{O}^\mathfrak{p}\_{\chi\_\lambda}$ has a direct sum decomposition $M=\oplus M\_i$ such that all weights of each $M\_i$ are contained in a single coset of root lattice $\Lambda\_r$ in $\mathfrak{h}^\*$. Then, the category $\mathcal... | https://mathoverflow.net/users/110229 | About subcategory of parabolic Category $\mathcal{O}^\mathfrak{p}$ | $\mathcal{O}\_\chi^{\mathfrak{p}}$ is a full subcategory of $\mathcal{O}\_\chi$, so the answer is yes. That is:
* $\mathcal{O}\_\mu = \mathcal{O}\_{\chi\_\mu}$, for all $\mu$
* $\mathcal{O}\_\mu^{\mathfrak{p}}$ is the full subcategory of $\mathcal{O}\_\mu$ of locally $\mathfrak{l}$-finite modules.
* $\mathcal{O}\_{\c... | 1 | https://mathoverflow.net/users/16384 | 311455 | 135,522 |
https://mathoverflow.net/questions/311463 | 2 | Let $\tilde W$ be a spin closed oriented manifold, $Y$ is a codimension $1$ closed oriented submanifold of $\tilde W$, and denote the $W$ the cobordism from $Y$ to
itself obtained from cutting $\tilde W$ open along $Y$. ($<[Y]>\cong H^1(\tilde W;\mathbb Z)$)
Let $X$ be a oriented manifold with periodic-end modelling ... | https://mathoverflow.net/users/95296 | Dirac operator on manifold with periodic end | Not necessarily. The condition from Taubes's paper is stronger than just requiring the vanishing of the kernel of the Dirac operator $D\_W$. Choosing $f:W \to S^1$ that is Poincaré dual to a multiple of [Y], Taubes requires that for all $r \in \mathbb{R}$, the twisted Dirac operators $D\_{W,r}= D\_W - ir f^\*d\theta$ a... | 5 | https://mathoverflow.net/users/3460 | 311468 | 135,524 |
https://mathoverflow.net/questions/311476 | 3 | Given a (separable) Hilbert space **H** and an unbounded densely defined linear operator $T:{\cal D}(T) \to $**H** such that ${\cal D}$ is **diagonalizable** (it means $\exists$ an O.N.B. of **H** such that all basis elements are eigenvectors of $T$). Is it possible for $T$ to have non-point spectrum, I mean, can exist... | https://mathoverflow.net/users/128876 | Non-point spectrum for diagonalisable self-adjoint unbounded operator | Take $T$ to be the inverse of a bounded/continuous, self-adjoint operator with eigenvalues (an orthonormal basis) all rationals between $0$ and $1$. Then $T$ has an orthonormal basis of eigenvectors, with eigenvalues all rationals above $1$. Spectra are closed...
| 7 | https://mathoverflow.net/users/15629 | 311479 | 135,528 |
https://mathoverflow.net/questions/311478 | 5 | Suppose we have a model (of $\mathsf{ZFC}$) $M$, and that $x\in 2^\omega$ is random over $M$, and that $y\in 2^{\omega}$ is Cohen over $M$. My question is whether $y$ is also Cohen over $M[x]$. In other words, if I have a real that's Cohen over a model, is it still Cohen over the model that results from performing Rand... | https://mathoverflow.net/users/13059 | Random reals preserving Cohen reals | That depends on the particular random real $x$ and Cohen real $y$. On the one hand, I could first choose $x$ random over $M$ and then choose $y$ Cohen over $M[x]$. Then $y$ is also Cohen over the submodel $M$, so it's an example where the answer to your question is yes.
On the other hand, I could first choose $y$ Coh... | 9 | https://mathoverflow.net/users/6794 | 311482 | 135,529 |
https://mathoverflow.net/questions/311045 | 3 | I would like to know whether the reduced suspension of a Hurewicz cofibration of pointed spaces (it is a Hurewicz cofibration when considered as a map of unbased spaces) is an acyclic Hurewicz cofibration.
I think that this is wrong, but I have no counterexample so far (it will probably involve a degenerately based s... | https://mathoverflow.net/users/128857 | Reduced suspension of a Hurewicz cofibration | Surprisingly enough, this is true. This follows immediately by the definition of a monoidal model category, and the fact that the category of pointed spaces under smash product and the Hurewicz model strucutre induced form the one on unbased spaces is monoidal (see More Concise Algebraic Topology for instance).
| 1 | https://mathoverflow.net/users/128857 | 311484 | 135,530 |
https://mathoverflow.net/questions/311474 | 3 | Consider a permutation group $G$ acting on an infinite set $X$. Assume $G$ has *finitely many* orbits, and every point stabiliser $G\_x$ has *finite* orbits. Now consider a permutation $\tau\in\operatorname{Sym}(X)$ of finite order, and let $H=\langle G,\tau\rangle$. Is it necessarily true that every point stabiliser $... | https://mathoverflow.net/users/57533 | Enlarging a subdegree-finite "almost transitive" permutation group to a transitive one? | No. Consider $G=\mathbf{Z}$ acting on itself by translation. Let $\tau$ be the transposition $(0,1)$. Then $H$ is the group of permutations of $\mathbf{Z}$ coinciding to translations at infinity; in particular it contains all finitely supported permutations; thus the stabilizer $H\_0$ acts transitively on the complemen... | 4 | https://mathoverflow.net/users/14094 | 311490 | 135,533 |
https://mathoverflow.net/questions/301476 | 23 | I'm looking for a proof that the following term is an **algebraic integer** whenever $\tau\_N=\frac{N+\sqrt{-N}}{2}$ is a quadratic irrationality with class number $1$:
$$A\_N:=\sqrt{-N}\cdot\frac{E\_2(\tau\_N)-\frac{3}{\pi\cdot Im(\tau\_N)}}{\eta^4(\tau\_N)}$$
Here $\eta$ denotes the Dedekind $\eta$-Function and $... | https://mathoverflow.net/users/124565 | Why are values of Eisenstein $E_2^*$ algebraic integers? | For $|q|<1$ we consider the null Jacobi theta functions
$$
\theta\_2(q):=\sum^{\infty}\_{n=-\infty}q^{(n+1/2)^2}\textrm{, }
\theta\_3(q):=\sum^{\infty}\_{n=-\infty}q^{n^2}\textrm{, }
\theta\_4(q):=\sum^{\infty}\_{n=-\infty}(-1)^nq^{n^2}.
$$
For $q=e^{-\pi \sqrt{r}}$, $r>0$ the elliptic singular modulus $k=k\_r$ is give... | 8 | https://mathoverflow.net/users/88851 | 311504 | 135,539 |
https://mathoverflow.net/questions/311516 | 7 | Let $\pi \colon X \rightarrow Y$ be a projective morphism with connected fibers between normal quasi-projective varieties. Let $N$ be a $\mathbb{Q}$-Cartier divisor on $Y$ so that $\pi^\*(N)$ is Cartier. Does it follows that $N$ is itself Cartier?
| https://mathoverflow.net/users/37338 | Pull-back divisor being Cartier | Following the clarification in the comments, I am interpreting the question as follows.
**Question.** For an effective Weil divisor $N$ on $Y$, for an effective Cartier divisor $A$ on $X$, for a positive integer $\ell$ such that the effective Weil divisor $\ell N$ is Cartier and such that the pullback effective Cart... | 5 | https://mathoverflow.net/users/13265 | 311521 | 135,542 |
https://mathoverflow.net/questions/311424 | 3 | Let $G$ be a reductive group acting on the smooth affine variety $X$ such that the stabilizers are finite. Is it true that the quotient $X/G$ is a local complete intersection (LCI)? In particular, is the quotient of a smooth affine variety to the algebraic action of a finite group LCI? If no, is there any condition on ... | https://mathoverflow.net/users/128556 | Are quotient varieties local complete intersections? | As discussed in the comments, $X/G$ need not in general even be Gorenstein, let alone a local complete intersection.
Actually, if the ground field has positive characteristic, $X/G$ may not even be Cohen-Macaulay (let alone Gorenstein, let alone LCI). For example, if $X = \mathbb{A}\_k^4$, where $k = \overline{\math... | 6 | https://mathoverflow.net/users/12419 | 311525 | 135,544 |
https://mathoverflow.net/questions/311456 | 9 | Given a principal $G$ bundle $P(M,G)$ and a manifold $F$ with an action of $G$ on it from left, we construct a fiber bundle over $M$ with fiber $F$ and call this the associated fiber bundle for $P(M,G)$.
I do not get the motivation behind the construction given in Kobayashi and Nomizu which I will write down below.
... | https://mathoverflow.net/users/118688 | Motivation for construction of associated fiber bundle from a principal bundle | This construction reverses the construction of the frame bundle from a vector bundle (e.g., the tangent bundle). The idea is that each point $f \in F\_p$ in the frame bundle of a vector bundle $E$ is, by definition a basis of $E\_p$. This therefore defines a natural map of $F \times \mathbb{R}^k \rightarrow E$, where $... | 8 | https://mathoverflow.net/users/613 | 311526 | 135,545 |
https://mathoverflow.net/questions/311524 | 4 | I am wondering if the boundedness of growth can be characterized by sequences. I am not sure if I use the term "growth" correctly, or use the correct tags for this question. Here is what I mean.
Let $X$ be an uncountable set and let $F$ be a collection of real-valued functions on $X$ with the following property:
Fo... | https://mathoverflow.net/users/53155 | Bounded growth of functions vs bounded growth of functions on countable sets | This was a fun question! The answer is no.
Let $\Omega$ be the set of all countable ordinals. For each limit ordinal $\alpha \in \Omega$ let $f\_\alpha: \Omega \to [1,\infty)$ be a function which increases to infinity on $[0,\alpha)$ and is constantly zero on $[\alpha,\Omega)$.
For any limit ordinal $\alpha \in \Om... | 6 | https://mathoverflow.net/users/23141 | 311527 | 135,546 |
https://mathoverflow.net/questions/311486 | 15 | Suppose $X$ is a smooth projective variety defined over an arbitrary algebraically closed field $k$, and consider the action of $\Sigma\_n$ on the $n$-fold product $X^n$. Is it true that $H\_{\acute{e}t}^i(\mathrm{Sym}^n(X),\mathbb{Q}\_\ell)\cong H\_{\acute{e}t}^i(X^n,\mathbb{Q}\_\ell)^{\Sigma\_n}$? In particular, what... | https://mathoverflow.net/users/122812 | When is the etale cohomology of $\mathrm{Sym}^n(X)$ isomorphic to the $\Sigma_n$-invariants in the étale cohomology of $X^n$? | One can give a spectral-sequence free argument. Let $X$ be an algebraic variety and $G$ a finite group acting on $X$, acting freely on a dense open subset. Let us say that $X$ is quasi-projective so that $X/G$ exists as a scheme (rather than an algebraic space), but this is not essential. We have $\pi \colon X \to X/G$... | 8 | https://mathoverflow.net/users/1310 | 311531 | 135,550 |
https://mathoverflow.net/questions/311535 | 10 | I will just repeat the title:
>
> Is there a closed non-smoothable 4-manifold with zero Euler
> characteristic?
>
>
>
I am guessing yes simply based on other existence theorems I have seen for 4-manifolds.
| https://mathoverflow.net/users/21848 | Is there a closed non-smoothable 4-manifold with zero Euler characteristic? | The Kirby-Siebenmann invariant in $H^4(M;\Bbb Z/2)$, an obstruction to smoothability, is additive under connected sum in dimension 4. In even dimensions, $\chi(M \# N) = \chi(M) + \chi(N) -2$.
To construct manifolds with nontrivial Kirby-Siebenmann invariant we should apply Freedman's theorem: simply connected topol... | 18 | https://mathoverflow.net/users/40804 | 311536 | 135,551 |
https://mathoverflow.net/questions/311537 | 3 | Let $S$ be the set of integers which are a product of $k$ distinct primes, $k$ a fixed positive integer (the condition that the primes are distinct is not crucial). Landau used the Prime Number Theorem to prove that
$$\frac{1}{x}\sum\_{n \in S:\, n \le x} = \frac{\log^{k-1} \log x}{(k-1)!\log x} (1+o(1)),$$
and he obta... | https://mathoverflow.net/users/31469 | Almost-Primes in Short Intervals | [Kátai](https://link.springer.com/chapter/10.1007%2FBFb0075758), building on an important paper of Ramachandra, was able to prove that
$$\sum\_{n \in S: x \le n \le x+x^c} 1 \sim x^c \frac{\sum\_{n \in S: n \le x}}{x}$$
as $x \to \infty$, for $c=\frac{1}{2}+\varepsilon$ conditionally and $c=\frac{7}{12}+\varepsilon$ un... | 3 | https://mathoverflow.net/users/31469 | 311542 | 135,553 |
https://mathoverflow.net/questions/311203 | 1 | There was a similar thread on the neighbour forum [StackExchange](https://math.stackexchange.com/questions/114053/are-topological-vector-spaces-completely-regular) on sufficient conditions for a topological space to be *completely* regular $T\_{3^1/\_2}$.
Please, let me know any known **condition(s) that a topologica... | https://mathoverflow.net/users/113768 | Sufficient conditions for a topological space to be regular $T_3$ | The list of sufficient conditions in the question neatly avoids addressing the real issue I think. From the comments I see that it is regularity of a very specific topology that you are after: Jakubowski's $S$-topology on Skorokhod space. None of the conditions that you have will help you very much in that case.
The re... | 6 | https://mathoverflow.net/users/5903 | 311547 | 135,555 |
https://mathoverflow.net/questions/311552 | 7 | Let $X$ be a set, and let $\text{Part}(X)$ denote the collection of all partitions of $X$. For $A, B\in \text{Part}(X)$ we set $A\leq B$ if $A$ refines $B$, that is for all $a\in A$ there is $b\in B$ such that $a\subseteq b$. This relation defines a partial order on $\text{Part}(X)$.
If $X$ is an infinite set, is the... | https://mathoverflow.net/users/8628 | Surjective order-preserving map $f:{\cal P}(X)\to \text{Part}(X)$ | It may be clarifying to work with equivalence relations $E$ on $X$ rather than partitions on $X$. The two are in natural bijection, with $E$ inducing a partitioning quotient map $q: X \to X/E$, and $X/E$ refines $X/E'$ iff $E \subseteq E'$ as subsets of $X \times X$.
Next, there is a surjective order-preserving map ... | 18 | https://mathoverflow.net/users/2926 | 311556 | 135,559 |
https://mathoverflow.net/questions/292707 | 1 | Let $(K,\sigma)$ be a *difference field* of characteristic $0$, *i.e.* equiped with field morphism $\sigma:K\rightarrow K$. Assume that $K$ satisfy a non-trivial univariate difference polynomial identity $\delta=0$ for some $\delta\in K\{x\}$, where $$K\{x\}=K\left[\sigma^i(x):i\in\mathbb N\right].$$ Let $F=\{x\in K:\s... | https://mathoverflow.net/users/18583 | On difference identities and $[K:F]$ | The answer is yes and appears in R. Cohn's book 'Difference algebra' Lemma II p. 201.
| 1 | https://mathoverflow.net/users/18583 | 311565 | 135,564 |
https://mathoverflow.net/questions/311564 | 7 | I am teaching a course in basic differential topology, and, following e.g. Milnor, I defined functions of class $C^k$ on subsets of the Euclidean space $\mathbb{R}^n$ as follows.
Let $f\colon X\to \mathbb{R}$ be a function, where $X\subseteq \mathbb{R}^n$. Then $f$ is of class $C^k$ if for every point $x\_0\in X$ the... | https://mathoverflow.net/users/6206 | Smooth functions on subsets of $\mathbb{R}^n$ | The answer is yes for functions defined on closed sets $X\subset\mathbb{R}^n$.
In Section 1.5.5 in [1] we have a necessary and a sufficient condition of the existence of an extension to a $C^m$ function for a finite $m$ and in Section 1.5.6 in [1] we have a necessary and sufficient condition for the existence of an ext... | 6 | https://mathoverflow.net/users/121665 | 311568 | 135,565 |
https://mathoverflow.net/questions/311567 | 2 | I am looking for a reference on the paper on compact Sobolev embeddings.
If we define the Sobolev space $$X\_{0}(A):=\{u\in H^s(\mathbb R^N): u=0\quad \text{in}\quad \mathbb R^N \setminus A\}$$ where $A$ is an annulus and $s\in(0, 1)$.
Is it true that the class of radial functions in $X\_{0}(A)$ is compact in $L^... | https://mathoverflow.net/users/111999 | compactness of fractional Sobolev spaces | You have the compact embedding of radial Sobolev functions in $X\_0(A)$ to $L^p(A)$ for all $1\leq p<\infty$ if and only if $s\geq 1/2$.
The proof goes as follows.
A radial function $F(x)$ on the annulus $A=\{ a<|x|<b\}$ is a function of the form $F(x)=f(|x|)$ for some $f$ defined on the interval $(a,b)$. Now $F\in... | 5 | https://mathoverflow.net/users/121665 | 311569 | 135,566 |
https://mathoverflow.net/questions/311575 | 3 | I have about twenty five (multilinear) polynomials $f\_1(\mathbf{x}), f\_2(\mathbf{x}), \dots, f\_{25}(\mathbf{x})$ all in fifteen variables and I would like to decide if there is a $\mathbf{y} \in [0,1]^{15} \subset \mathbb{R}^{15}$ such that $f\_i(\mathbf{y}) \ge 0$ for all $1 \le i \le 25$.
I would like to know wh... | https://mathoverflow.net/users/49446 | Solving polynomial inequalities -- efficient Positivstellensatz on a computer | You might try an optimization-based approach: give an optimizer the problem
maximize $z$ subject to $z - f\_i(y) \le 0$, $i = 1 \ldots 25$, and $0 \le y\_j \le 1$, $j = 1 \ldots 15$.
This might be too difficult for a global optimizer (but it might be worth a try).
If it's a local optimizer, you're not guaranteed to... | 1 | https://mathoverflow.net/users/13650 | 311588 | 135,572 |
https://mathoverflow.net/questions/256496 | 9 | The Freyd-Mitchell embedding theorem is a very useful tool for dealing with small abelian categories. However, it does not allow to use "elements" of objects of an abelian category $A$ in those statements that involve "infinite constructions".
So I wonder: for which Grothendieck abelian $A$ (this certainly implies t... | https://mathoverflow.net/users/2191 | Objects of which Grothendieck abelian categories have elements? | I am not an expert of the abelian world, but I think I can answer.
Since my background is not precisely abelian, I will start with an example in category theory.
>
> **Thm.** Let $\mathcal{K}$ be a locally $\kappa$-presentable category, then there is a faithful and conservative functor to Set that preserves $\kap... | 1 | https://mathoverflow.net/users/104432 | 311589 | 135,573 |
https://mathoverflow.net/questions/311347 | 4 | Let $\mathcal{C}$ be a category equipped with a Grothendieck topology $\tau$, i.e., a site.
Under which conditions on $\mathcal{C}$ can one construct a Borel $\sigma$-algebra, $\sigma\_\tau$, for $\tau$?
If one *can* construct $\sigma\_\tau$, can one then sensibly define a "finitely additive measure" $\mu$ on $\ma... | https://mathoverflow.net/users/129125 | Measures on sites | I guess that a good reference might be Olivier Leroy's theory, developing measure theory for locales (a special kind of topoi). Unfortunately Leroy passed away dramatically in 1996 and the paper has been typeset by Claire Voisin and Jean Malgoire, and can be dowloaded from [arXiv](https://arxiv.org/abs/1303.5631). The ... | 4 | https://mathoverflow.net/users/18238 | 311591 | 135,574 |
https://mathoverflow.net/questions/311598 | 4 | Fix a prime number $p$. If $n$ is a positive integer, then denote
$$\text{$\omega\_{p,k}(n):=\#$ of $k$'s in the $p$-ary expansion of $n$}$$
and the total sum of all its $p$-ary digits by
$$\Omega\_p(n):=\sum\_{k=0}^{p-1}k\cdot\omega\_{p,k}(n).$$
>
> **Question.** Given a prime $p$ and for each $n\in\Bbb{N}$, is ... | https://mathoverflow.net/users/66131 | Polynomial expansions via prime-base digits | First we notice that the equality $\Omega\_p(a)+\Omega\_p(b)=\Omega\_p(a+b)$ happens if and only if there are no carries when adding $a+b$ in base $p$. Indeed the number of carries is equal to
$$\frac{\Omega\_p(a)+\Omega\_p(b)-\Omega\_p(a+b)}{p-1}$$
which is also equal to $\nu\_p\left(\binom{a+b}{a}\right)$ by [Kummer... | 6 | https://mathoverflow.net/users/2384 | 311605 | 135,576 |
https://mathoverflow.net/questions/311600 | 8 | My question is that whether the following statement is true or not.
---
*In a complete metric space $(X, d)$, if a sequence of open balls $\{B(x\_i, r\_i)\}\_{i=1}^\infty$ satisfies
$$
\exists \epsilon > 0 ~~s.t.~ B(x\_{i+1}, (1+\epsilon)r\_{i+1}) \subset B(x\_i, r\_i), \forall i \ge 1 \tag{1}
$$
then $\bigcap\_{... | https://mathoverflow.net/users/114996 | Intersection of nested open ball in complete metric spaces is nonempty? | I think this statement is true.
Suppose we had a counterexample $\{B(x\_i,r\_i)\}\_{i=1}^\infty$ satisfying condition (1) for some $\epsilon > 0$ but whose intersection was empty. Observe that any subsequence will still be a counterexample.
For each $i$, the point $x\_i$ does not belong to some $B(x\_j,r\_j)$, as o... | 8 | https://mathoverflow.net/users/23141 | 311607 | 135,577 |
https://mathoverflow.net/questions/311603 | 1 | Let $R$ be a commutative Noetherian hereditary ring (<https://en.wikipedia.org/wiki/Hereditary_ring>) of Krull dimension $1$. Then is it true that $R$ is a finite direct product of Dedekind domains ?
| https://mathoverflow.net/users/127118 | Decomposing Noetherian hereditary rings of Krull dimension $1$ into product of hereditary domains (i.e. Dedekind domains) | Certainly finite products of Dedekind rings with fields work as well. These are the only ones, even without the assumption on the Krull dimension:
>
>
> >
> > **Lemma.** *Let $R$ be a Noetherian (commutative) ring. Then $R$ is hereditary if and only if $R$ is a finite product $\prod\_{i = 1}^r R\_i$ where each $R... | 3 | https://mathoverflow.net/users/82179 | 311608 | 135,578 |
https://mathoverflow.net/questions/310728 | 4 | Let $[a,b]$ be an interval in real line . Given any function $f:[a,b]\to \mathbb R$ and set $A \subseteq [a,b]$ of size $n+1$, there exists a unique polynomial $p\_{f,A,n}(x)$ of degree $n$ such that $f(a)=p\_{f,A,n}(a),\forall a\in A$. Such a polynomial is called the interpolating polynomial of $f$ with nodes $A$. Due... | https://mathoverflow.net/users/127118 | Find $p$ s.t. there is a sequence of nodes in $[0,1]$ s.t. sequence of interpolating polynomials of every continuous function converges in $p$-norm | The answer to the question is yes. The property even holds for any $0<p<\infty$.
The first such result is due to Erdös and Turan (1936) : let $f$ be a continuous function and $w(x)$ a weight on $[-1,1]$. Denote by $L\_{n}(f,w)$ the Lagrange interpolant to $f$ at the zeros of the orthogonal polynomials with respect t... | 2 | https://mathoverflow.net/users/89429 | 311613 | 135,581 |
https://mathoverflow.net/questions/306588 | 15 | I'll first explain what Mobius inversion says, and then state what I am fairly sure the equivariant version is. I can write out a proof, but I also can't believe this hasn't been done already; this is a request for references to where it has already been done.
**Ordinary Mobius Inversion** Let $P$ be a finite poset w... | https://mathoverflow.net/users/297 | Equivariant Mobius inversion | Sami Assaf and I prove this in section 5 of our paper [Specht modules decompose as alternating sums of restrictions of Schur modules](https://arxiv.org/abs/1809.10125). It is surprising that we couldn't find a reference!
| 4 | https://mathoverflow.net/users/297 | 311623 | 135,586 |
https://mathoverflow.net/questions/311625 | 5 |
>
> **Question**: Let $\omega\_k$ be the number of distinct prime divisors of k.
> What is the asymptotic growth of $C\_n := \sum\_{k=1}^n 2^{\omega\_k}$?
>
>
>
Thank you for considering this elementary question. Below I give some motivation for this problem and some of my progress.
**Motivation:**
There are ... | https://mathoverflow.net/users/3970 | What is the asymptotic growth of $\sum_{k=1}^n 2^{\omega_k}$? | As you observe,
$$C\_n=\sum\_{k=1}^n\sum\_{d\mid k}|\mu(d)|=\sum\_{k=1}^n\sum\_{d\mid k\text{ squarefree}}1.$$
Exchanging the order of summation,
$$C\_n=\sum\_{d\leq n\text{ squarefree}}\sum\_{d\mid k\leq n}1=\sum\_{d\leq n\text{ squarefree}}\left\lfloor\frac{n}{d}\right\rfloor=n\sum\_{d\leq n\text{ squarefree}}\frac{1... | 10 | https://mathoverflow.net/users/30186 | 311629 | 135,588 |
https://mathoverflow.net/questions/311590 | 0 | Let $1<p<\infty$, and $f\_n$ be a bounded sequence in $C(0,T;H^2(0,L))$. It looks obvious to me that $f\_n^p$ is also bounded in $C(0,T;H^2(0,L))$. When we take the derivative of $f^p(t)$ twice we get $$(f^p)\_{xx}(t)=p(p-1)f^{p-2}(t)(f\_x(t))^2+pf^{p-1}(t)f\_{xx}(t).$$
Since for every $t\in[0,T]$,$f(t)$ and $f\_x^2(t)... | https://mathoverflow.net/users/113264 | $f_n$ is bounded in $C(0,T;H^2(0,L))$ so is $f_n^p$? | Yes, your justification is correct but needs to be expanded upon to be rigorous. In particular, it is important that not only are $f$ and $f\_x$ continuous, but their $C^0$ norm is controlled by their Sobolev norm, so that a uniform bound can be derived. By Sobolev embedding theorem (since $k = 2 > n/2$ with $k$ the de... | 3 | https://mathoverflow.net/users/7378 | 311648 | 135,594 |
https://mathoverflow.net/questions/311583 | 5 | Let $\mathscr{X}$ be a smooth proper DM stack over a field $k$ (perhaps assumed to be separably closed and/or of char. $0$) and let $\pi \colon \mathscr{X} \rightarrow X$ be its coarse moduli space.
What are some general results on the relationship between $H^i\_{\mathrm{et}}(\mathscr{X}, \underline{\mathbf{Z}\_\ell... | https://mathoverflow.net/users/56878 | What is the relationship between the $\ell$-adic cohomology of a DM stack and that of its coarse moduli space? | The result you want is this.
>
>
> >
> > Let $f : \mathscr{X} \to S$ be a proper tame DM stack with $S$ a scheme, and $g : S' \to S$ any morphism of schemes. Let $\mathscr{F}$ be a torsion sheaf on $\mathscr{X}$. Then the natural base change morphism
> > $$g^\ast R^i f\_\ast \mathscr{F} \to R^if'\_\ast g'^\ast \... | 3 | https://mathoverflow.net/users/21278 | 311649 | 135,595 |
https://mathoverflow.net/questions/311534 | 2 | What are the sufficient conditions for a von Neumann algebra to have a first countable set of states with respect to the weak \* operator topology?
| https://mathoverflow.net/users/95697 | Topology of state space in von Neumann algebras | Every von Neumann algebra is a C$^\*$-algebra. So the usual theorem that a C$^\*$-algebra $A$ is (norm) separable iff its state space is first countable in the weak-\* topology (*i.e.* the topology $\sigma(A^\*,A)$) applies. As von Neumann algebras are norm separable iff they are finite-dimensional, we conclude that th... | 4 | https://mathoverflow.net/users/61785 | 311653 | 135,597 |
https://mathoverflow.net/questions/311630 | 11 | It seems that the smooth isometric embedding theorem by Nash is true also for noncompact manifolds.
>
> Is it true that any (complete, connected) Riemannian manifold $(M^n,g)$ admits a **proper** smooth isometric embedding $\iota:M^n\hookrightarrow\mathbb R^N$ into some Euclidean space?
>
>
>
This is equivalen... | https://mathoverflow.net/users/36952 | Nash isometric embedding for noncompact manifolds | Nash's theorem states that any smooth embedding $f$ with Lipschitz constant less than 1 can be approximated by smooth isometric embedding (if the dimension of Euclidean space is sufficiently large).
So you only need to find a proper embedding $f$.
Take any embedding with Lipshitz constant $<\tfrac13$, and add one mor... | 8 | https://mathoverflow.net/users/1441 | 311656 | 135,599 |
https://mathoverflow.net/questions/311666 | 1 | Here's two random $(0,1)$-matrices:
$$
A=
\begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 1 \\
\end{bmatrix}
\qquad
B=
\begin{bmatrix}
1 & 1 \\
0 & 1 \\
\end{bmatrix}.
$$
They can be interpreted as biadjacency matrices: the graph $G[A]$ has vertices $\{r\_1,r\_2\} \cup \{c\_1,c\_2,c\_3\}$ and undirected edges $r\_i c\_j$ if and o... | https://mathoverflow.net/users/48278 | The Kronecker product of two bipartite graphs' biadjacency matrices: what's it called? | For bipartite graphs $G[A]$ and $G[B]$, their tensor product $G[A] \times G[B]$ is the disjoint union of bipartite graphs $G[A \otimes B]$ and $G[A \otimes B^T]$. See, for example, p.56 of Hammack, Imrich, Klavžar - Handbook of Product Graphs.
One possible reason why this extensive book doesn't mention this "product"... | 3 | https://mathoverflow.net/users/106512 | 311668 | 135,604 |
https://mathoverflow.net/questions/201107 | 2 | In [1], Propisition 6.1.9(2), it said that if $R$ is a perfectoid ring such that $pR^\circ$ is closed in $R^\circ$ (this includes the case if $R$ is of character $p$, or if $p$ is invertible in $R$, in particular if $R$ is a perfectoid $K$-algebra, $K$ a field), then the Frobenius $\Phi:R^\circ/p\to R^\circ/p$ is surje... | https://mathoverflow.net/users/66614 | Is Frobenius on $R^\circ/p$ surjective for general perfectoid rings $R$? | [Probably this question is no longer interesting to the author. But since I faced the same problem while trying to learn basics of perfectoid spaces I decided to write down an argument here]
We start with a perfectoid ring $R$ and a pseudo-uniformizer $\varpi$ s.t. $\varphi:R^{\circ}/\varpi \to R^{\circ}/\varpi^p$ is... | 3 | https://mathoverflow.net/users/115211 | 311669 | 135,605 |
https://mathoverflow.net/questions/311674 | 7 | Background of my question is the following: I have found a solution for my question [Smoothness Conditions for Planar “Mock-parametric” Spline Interpolation](https://mathoverflow.net/questions/271043/smoothness-conditions-for-planar-mock-parametric-spline-interpolation) and while developing the solution, I encountered ... | https://mathoverflow.net/users/31310 | How Much Flesh to the Bones does an Initial Online Publication need? | I presume that with "online" publication you have arXiv in mind. (For a journal it does not really make a difference whether it is online or not.) With regards to your question, the key difference between arXiv and a journal is not so much the absence of a refereeing process, but the fact that on arXiv you can post mul... | 6 | https://mathoverflow.net/users/11260 | 311678 | 135,608 |
https://mathoverflow.net/questions/311182 | 20 | Suppose a group $\Gamma$ acts by isometries on the Hilbert space $\mathbb{H}^\infty$ and it fixes the origin. So $\Gamma$ acts on the unit sphere $\mathbb{S}^\infty$ as well.
Assume that the action $\Gamma$ on $\mathbb{S}^\infty$ has no dense orbits. Is there a universal constant $\varepsilon >0$ such that there are... | https://mathoverflow.net/users/1441 | Diameter of a quotient of the infinite dimensional sphere | There is no such universal constant $\epsilon > 0$. Work with the complex Hilbert space $L^2[0,1]$ (which of course is also a real Hilbert space). Fix $n \in \mathbb{N}$.
Let $\Gamma\_0$ be the set of continuous piecewise linear increasing bijections from $[0,1]$ to itself. [1] It is a group with composition as produ... | 16 | https://mathoverflow.net/users/23141 | 311681 | 135,609 |
https://mathoverflow.net/questions/311703 | 3 | Assuming that the diagonal map $X\rightarrow X\times X$ is a cofibration. Is it true that the diagonal map $\Sigma X\rightarrow \Sigma X\times \Sigma X$ is a cofibration? (Where $\Sigma X$ is the reduced suspension of $X$.)
| https://mathoverflow.net/users/128857 | Is the following map a cofibration? | Yes. See Gaunce Lewis' paper
<http://www.ams.org/journals/tran/1982-273-01/S0002-9947-1982-0664034-8/>
By definition $X$ is locally equiconnected (LEC) iff the diagonal map $X \to X \times X$ is a cofibration (of unbased spaces). It follows that $X \times I$ is LEC, which implies that $\Sigma X$ is LEC, by the Dyer... | 7 | https://mathoverflow.net/users/9684 | 311704 | 135,616 |
https://mathoverflow.net/questions/311636 | 6 | Assume we are working over $\mathbb{C}$, and we have a projective morphism with connected fibers $f: X \rightarrow Z$ whose geometric generic fiber $X\_\overline{\eta}$ is isomorphic to a Hirzebruch surface $\mathbb{F}\_n$.
Thus, $X\_\overline{\eta}$ admits a morphism to $\mathbb{P}^1$, and this is defined over some ... | https://mathoverflow.net/users/89459 | Breaking a morphism with generic fiber $\mathbb{F}_n$ | If $n>0$, and if you have a rational section (for instance when $Z$ is a curve), then you do not need the finite extension. The reason is that the field $K(Z)$ is perfect (as you work in characteristic zero), and that the Galois group acts on $\mathbb{F}\_n$ preserving the exceptional curve (unique curve of negative se... | 4 | https://mathoverflow.net/users/23758 | 311705 | 135,617 |
https://mathoverflow.net/questions/311709 | 2 | I am reading [this](https://dl.acm.org/citation.cfm?id=1328795) paper related to an algorithm for nonsmooth optimization problems. After many simplifications, I was able to formalize the method as follows: let $\Bbb B $ denote the unit ball in $\Bbb R^n$ and consider a set $D\subseteq \Bbb B.$ We have an operator $T: \... | https://mathoverflow.net/users/114128 | Convergence of a stochastic sequence? | $\newcommand{\F}{\mathcal{F}}$
It appears that by $\mu(D)=\mu(B)$ you meant $\mu(D)=\mu(\Bbb B)$; otherwise, this condition would not make sense.
It also appears that the sentence "Sample $u^k$ from $\Bbb B$ according to the probability space described", which you quoted, means that $u^0,u^1,\dots$ are independent... | 2 | https://mathoverflow.net/users/36721 | 311712 | 135,619 |
https://mathoverflow.net/questions/311713 | 1 | I'm researching method of biprime number factoring. I have a biprime number 1012322327 \* 1115382761 (19 decimal digits= 1129126872111204847). I'd like to know how many iterations (or trials) the best method has to perform to obtain a solution factors of this number. I'd like to have an estimation to compare to my own ... | https://mathoverflow.net/users/129560 | How many iterations the best biprime factoring method has to factor a number | Well there are about $5\times 10^7$ primes which are less than the square root of your input, since $$\pi(x)\sim x/\log(x),$$ so even a brute force search for prime divisors would be faster. Also your number is too small to try state of the art algorithms on.
The magma online calculator [here](http://magma.maths.usyd... | 0 | https://mathoverflow.net/users/17773 | 311721 | 135,621 |
https://mathoverflow.net/questions/311619 | 3 | Just recently (September 24 - 26) there was a conference at Oxford dedicated to 20th anniversary of CMI. (<https://www.claymath.org/events/cmi-20>) The program looks interesting. Does anyone know if there will be videos (or, at least, proceedings) published?
| https://mathoverflow.net/users/9833 | CMI at 20 conference | All of the lectures were filmed, so I have to figure they will be putting them online at some point.
| 0 | https://mathoverflow.net/users/658 | 311737 | 135,624 |
https://mathoverflow.net/questions/311110 | 0 | Suppose you have $100$ coins whose probabilities of obtaining the outcome "head" are $p\_1,\ldots,\,p\_{100}$. These probabilities are not necessarily equal each other. Consider the following random experiment divided into rounds.
* **Round 1:** Throw simultaneously the $100$ coins and observe the number of heads.
* ... | https://mathoverflow.net/users/113347 | Bivariate Poisson-Binomial distribution | I found a solution to my problem. This solution builds on the paper
Nelsen, R. B. (1987). Discrete bivariate distributions with given marginals and correlation. *Communications in Statistics-Simulation and Computation*, 16(1), 199-208.
Since $\mathbb{P}(Y\_1=y\_1,\,Y\_2=y\_2)=\mathbb{P}(Y\_1=y\_1,\,X\_2=y\_2-y\_1)... | 0 | https://mathoverflow.net/users/113347 | 311749 | 135,628 |
https://mathoverflow.net/questions/311750 | 3 | **Note:** Since what I am asking about below touches on a potentially controversial subject, let me emphasize that I am only asking for a specific reference, and I am not asking for a discussion of the controversy itself, or even for other references besides the specific one I'm asking about.
---
Some years ago, ... | https://mathoverflow.net/users/3106 | Looking for an erratum (reference request) | Thanks to Sofie Verbeek for the answer. The reference is
>
> Erratum to "Galois Representations and Modular Forms" by Kenneth A. Ribet, *Bulletin (New Series) of the American Mathematical Society* **33** (1996), p. 43.
>
>
>
| 3 | https://mathoverflow.net/users/3106 | 311751 | 135,629 |
https://mathoverflow.net/questions/311647 | 8 | Let $C\subset \mathbb{P}^2$ be a smooth conic without $k$-points.
Call the Chow $k$-motive in *zero-dimensional* if it is a sum of $M\mathbb{L}^n$ where $M$ is an Artin motive, i.e. a part of a motive of zero-dimensional scheme.
**Q.** How to see that a motive of $C$ is not (or is) zero-dimensional?
**P.S.** By Ch... | https://mathoverflow.net/users/103054 | Motive of a conic without points | This is true with $\mathbb Q$-coefficients; see the proposition below. The reason it's difficult is that it's hard to compute Chow groups over non-algebraically closed fields. However, we have the following:
>
>
> >
> > **Lemma 1.** *Let $k \subseteq \ell$ be a separable algebraic field extension, and let $X$ be ... | 7 | https://mathoverflow.net/users/82179 | 311766 | 135,636 |
https://mathoverflow.net/questions/311781 | 6 | Consider the *discrete* monoid $M$ of nondecreasing continuous maps from $[0,1]$ to itself preserving the extremities. Note that the monoid is right-cancellative ($x.z=y.z$ implies $x=y$, since $z$ is always onto).
>
> What do we know about the homotopy type of this monoid (viewed as a
> one-object category) ? In ... | https://mathoverflow.net/users/24563 | Homotopy type of a specific discrete monoid | This space is contractible, and so all of its homotopy groups are trivial.
Define two elements in $M$ by:
$$
\begin{align\*}
A(x) &=
\begin{cases} 2x &\text{if }x \leq 1/2\\1 &\text{if }x \geq 1/2\end{cases}\\
B(x) &=
\begin{cases} 0 &\text{if }x \leq 1/2\\2x-1 &\text{if }x \geq 1/2\end{cases}
\end{align\*}
$$
Define... | 11 | https://mathoverflow.net/users/360 | 311786 | 135,642 |
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