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https://mathoverflow.net/questions/293841 | 2 | Let $G$ be a finite simple group of type ${\rm L}\_{n}^{\epsilon}(q)$ with the following conditions:
$n$ and $\dfrac{q^{n}-\epsilon}{(q-\epsilon)(n,q-\epsilon)}$ both prime, $n\geqslant3$ and $(n,q,\epsilon)\neq(3,4,+),(3,3,-),(3,5,-),(5,2,-)$
Show that:
1- Every minimal subgroup $L$ of order $s=\dfrac{q^{n}-\e... | https://mathoverflow.net/users/119365 | Maximal subgroups of some special cases of ${\rm L}_{n}^{\epsilon}(q)$ | It looks to me as though the answer is yes. You have excluded those cases where $C\_s \rtimes C\_n$ is not maximal in $L\_n^\epsilon(q)$.
So in any other situation, $C\_s \rtimes C\_n$ is maximal, and hence if $C\_s$ were contained in some other maximal subgroup $M$ then we would have $M \cap C\_s = C\_s$. But then $... | 2 | https://mathoverflow.net/users/35840 | 293865 | 129,284 |
https://mathoverflow.net/questions/293852 | 0 | Let $\mathcal{B}(F)$ the algebra of all bounded linear operators on an infinite-dimensional complex Hilbert $F$.
>
> Let $T,S\in\mathcal{B}(F)$. The pair $(T,S)$ is said to $\lambda$-commute if there exists $\lambda\in \mathbb{C}^\*$ such that $TS=\lambda ST$. I am looking for necessary and sufficient conditions on... | https://mathoverflow.net/users/116483 | When $\lambda$-commutativity implies commutativity? | I don't see which kind of condition you are looking for, as there are a lot of pairs $T,S$ such that $TS=\lambda ST$ and $\lambda\ne1$, even in finite dimension. Such pairs are said to $\lambda$-commute.
An interesting case happens when $\lambda$ is root of unity, say of order $r$. Then (Potter's Theorem) $(T+S)^r=T^... | 11 | https://mathoverflow.net/users/8799 | 293873 | 129,288 |
https://mathoverflow.net/questions/293760 | 4 | It is well-known that the operator $$\frac{\partial}{\partial \overline{z}} : C^{\infty}(\mathbb{C}) \to C^{\infty}(\mathbb{C})$$ is surjective. (And it also works if we replace functions by Schwartz distributions.) I wonder if $$\frac{\partial}{\partial \overline{z}} : \mathcal{S}'(\mathbb{C}) \to \mathcal{S}'(\mathbb... | https://mathoverflow.net/users/86286 | Neumann DBAR problem with tempered distributions | It is a famous theorem of Lojasiewicz and (independently) Hörmander that *every* linear partial differential operator with constant coefficients is surjective on the space of tempered distributions.
| 4 | https://mathoverflow.net/users/21051 | 293874 | 129,289 |
https://mathoverflow.net/questions/293856 | 2 | I need the following result (which I believe to be true though I was too lazy to write down a complete proof).
Let $f$ be a function of two complex variables analytic at the origin and $a\not\in\mathbb{N}$. Then the differential equation
$$z'=\frac{az}{x}+f(x,z)$$
has an unique solution $z(x)$ which is analytic at ... | https://mathoverflow.net/users/9833 | Reference request: a singular differential equation | These questions were studied for the first time by Briot and Bouquet in 19 century. For a modern reference see for example the book of E. Hille, Ordinary differential equations in the complex domain. The chapter is called Some equations of Briot and Bouguet.
Your case is actually simple: plug a formal power series for ... | 4 | https://mathoverflow.net/users/25510 | 293877 | 129,290 |
https://mathoverflow.net/questions/293880 | 0 | Assuming Goldbach's conjecture, let's define for a sufficiently large integer $ n $ the quantity $ r\_{0}(n) : =\inf\{r\geq 0,(n-r,n+r)\in\mathbb{P}^{2}\} $.
Under GRH, what is the best upper bound for $ R(x) : =\sum\_{n\leq x}r\_{0}(n) $?
| https://mathoverflow.net/users/13625 | Upper bound for $\sum r_{0}(n)$ | By a result of Jia (Three primes theorem in a short interval. VII., Acta Math. Sinica (N.S.) 10 (1994), 369-387), we have for any $\epsilon>0$,
$$\#\{n\leq x:\ r\_0(n)>x^{7/12+\epsilon}\}\ll\frac{x}{\log^2 x}. $$
As a consequence, we have the uniform bound (under the Goldbach conjecture)
$$R(x)\ll\frac{x^2}{\log^2 x}.$... | 6 | https://mathoverflow.net/users/11919 | 293891 | 129,294 |
https://mathoverflow.net/questions/293898 | 1 | We have been told that the Riesz potential in $\mathbb{R}^d$, $I\_{\alpha}(f)$, defined by
$$I\_{\alpha}(f)(x):= C\int\_{\mathbb{R}^d} \frac{f(y)}{\left| x-y \right|^{d-\alpha}}\,\mathrm{d}y $$
has the following regularizing property.
For a function $f\in C^{\beta}$ (the Hölder class), we have $I\_{\alpha}(f) \in C^{... | https://mathoverflow.net/users/121273 | Reference request: Riesz operator over Hölder class | First of all, the Riesz potential operator is not even well-defined on the Hölder class $C^\alpha$. For example, it diverges when applied to non-zero constant functions. You need to restrict to a narrower class of functions.
Second, any statement of this form requires a particular definition of $C^\alpha$ when $\alph... | 2 | https://mathoverflow.net/users/108637 | 293900 | 129,295 |
https://mathoverflow.net/questions/293887 | 1 | There are various examples of (non-separable) prime C$^\*$-algebras $A$ which are not primitive. Is there an example for which $A$ is $\sigma$-unital but non-unital?
| https://mathoverflow.net/users/121269 | Is there a non-unital $\sigma$-unital prime C$^*$-algebra which is not primitive? | It seems to me you could just tensor a unital example with the compacts. Any closed ideal of $A \otimes K$ has the form $I\otimes K$ where $I$ is a closed ideal of $A$, etc.
| 2 | https://mathoverflow.net/users/23141 | 293902 | 129,296 |
https://mathoverflow.net/questions/293867 | 4 | Is there a finite distributive lattice that is *not* isomorphic to the lattice of ideals of a finite ring?
| https://mathoverflow.net/users/8628 | Finite distributive lattices as lattice of ideals of a finite ring | The answer is **yes**, there is a finite distributive lattice which is not isomorphic to the lattice
* of right ideals in a non-commutative ring with identity,
* of ideals in a commutative ring with identity.
Let $L = \left\{ \{0, 1, 2\}, \{1, 2\}, \{1\}, \{2\}, \emptyset
\right\}$ be partially ordered by inclusion... | 4 | https://mathoverflow.net/users/84349 | 293917 | 129,298 |
https://mathoverflow.net/questions/293915 | 2 | These polynomials show up naturally in my work
$$
p\_n(x) = \sum\_{j=0}^n {n \choose j} \frac{(-x)^j}{j!}
$$
Does anyone know recognize if they belong to any class of well known polynomials. I am trying to compute
$$
\lim\_{n \rightarrow \infty} \int\_{R^2} f(|r|)e^{-\frac{a}{2}|x|^2} p\_n(a x) \, dx
$$
for some nice f... | https://mathoverflow.net/users/121284 | Does anyone recognize these polynomials? Need to compute a riemann lebesgue type limit | Maple says $$\sum \_{j=0}^{n}{\frac {{n\choose j} \left( -z \right) ^{j}}{j!}}={L}\_n \left(z \right)$$ Laguerre polynomials. See <https://en.wikipedia.org/wiki/Laguerre_polynomials> and go down to "closed form".
| 12 | https://mathoverflow.net/users/454 | 293919 | 129,300 |
https://mathoverflow.net/questions/293926 | 3 | Let $K^{\bullet}$ be a bounded complex of abelian étale sheaves on a quasi-compact and quasi-separated scheme $X$.
* For any étale cover $\mathcal{U} :=\{ U\_i\to X\}\_{i\in I}$, can we find a refinement $\mathcal{U}'$ of $\mathcal{U}$ with $\mathcal{U}'$ finite (the answer to this should be easily yes, but I just wa... | https://mathoverflow.net/users/nan | Étale hypercohomology of complexes | 1. Any étale morphism is flat and locally of finite presentation [Tag [02GR](https://stacks.math.columbia.edu/tag/02GR), Tag [02GS](https://stacks.math.columbia.edu/tag/02GS)], hence open [Tag [01UA](https://stacks.math.columbia.edu/tag/01UA)]. Quasi-compactness now immediately shows that étale covers have finite subco... | 4 | https://mathoverflow.net/users/82179 | 293928 | 129,302 |
https://mathoverflow.net/questions/293924 | 2 | If $S$ is a scheme, $X$ is a smooth quasi-projective $S$-scheme, is the $S$-projective closure of $X$ a **smooth** $S$-scheme with $X$ an **open** subscheme?
| https://mathoverflow.net/users/nan | Closure of quasi projective scheme | I don't know what you mean by *the* $S$-projective closure of $X$, but for most locally closed immersions¹ $X \hookrightarrow \mathbb P^n\_S$, the closure of $X$ in $\mathbb P^n\_S$ will not be smooth (even if $S = \operatorname{Spec} k$).
For example, if $k$ is a field of characteristic $p > 0$ and $X$ is a smooth q... | 2 | https://mathoverflow.net/users/82179 | 293930 | 129,303 |
https://mathoverflow.net/questions/293927 | 1 | Does anyone have a reference to where I may find a statement of the problem and perhaps (but not required) some elementary dicussion of Siegel's Center Problem?
| https://mathoverflow.net/users/121048 | Reference Request: Siegel Center Problem | Try this references:
<http://aip.scitation.org/doi/abs/10.1063/1.525964> (A simple proof of a particular case of C. Siegel’s center theorem, by R. de la Llave) and
<http://www.math.osu.edu/~costin.10/950/tutorial_KAM.pdf> (A tutorial on KAM theory, by the same author).
Also the following book "The KAM story: a friend... | 1 | https://mathoverflow.net/users/32389 | 293932 | 129,304 |
https://mathoverflow.net/questions/293922 | 2 | How can we prove the following conjecture?
Given any simple unweighted bipartite graph $G(V\_1, V\_2, E)$, there always exists a subgraph $G'(V\_1, V\_2, E')$ of $G$ such that the two following conditions are simultaneously satisfied:
1) $|E'| \ge \frac{1}{2}|E|$ .
2) The degree of each vertex in $G'$ is at most ... | https://mathoverflow.net/users/115803 | Existence of bipartite subgraphs satisfying degree and edge cardinality constraints | If $d\_G^{\max}$ is odd and the graph is regular, this clearly is not possible. But if $d\_G^{\max}=2k$ is even, this is possible: the edges may be properly colored with $2k$ colors [this is well-known bipartite variant of Vizing's theorem, which may be proved, for example, by induction in number of edges: color all ed... | 2 | https://mathoverflow.net/users/4312 | 293935 | 129,305 |
https://mathoverflow.net/questions/293933 | 2 | Let $X$ be a smooth projective variety over a field $k$.
Suppose we have étale abelian sheaves $A, B$ on $X\_{\rm ét}$ such that
$$H^j(X\_{\rm ét}, A),\ H^j(X\_{\rm ét}, B)$$
are finitely generated abelian groups for all $j$. Is $\mathbb{H}^j(X\_{\rm ét}, A\otimes^L\_{\mathbf{Z}} B)$ finitely generated?
| https://mathoverflow.net/users/nan | Étale cohomology of tensor product | No. Take a sheaf over $\mathbb F\_\ell$ whose etale cohomology vanishes (for instance, $X$ an elliptic curve over an algebraically closed field, $\mathcal L$ the locally constant sheaf on $X$ associated to a nontrivial one-dimensional representation of its fundamental group into $\mathbb F\_\ell^\times$ ), let $A$ be t... | 4 | https://mathoverflow.net/users/18060 | 293947 | 129,308 |
https://mathoverflow.net/questions/293942 | 12 | These are five important constructions and I would like to know how they are related.
The $n$th [unordered configuration space](https://en.wikipedia.org/wiki/Configuration_space_(mathematics)#Definition) of a space $X$ is
$$
\operatorname{UConf}\_n(X):=\{\text{embeddings of $\{1,...,n\}$ into $X$}\}/(\text{$n$th symm... | https://mathoverflow.net/users/41291 | Configuration spaces, Ran spaces, free semilattices, Vietoris spaces and power objects | Too long for a comment but it is essentially a comment:
It is easy to see that for a Hausdorff space $X$ the topology on the Ran space coincides with the Vietoris topology and for a non-Hausdorff space $X$ the Ran topology is strictly weaker than the Vietoris topology.
The topology of the free topological semilatti... | 6 | https://mathoverflow.net/users/61536 | 293948 | 129,309 |
https://mathoverflow.net/questions/293949 | 3 | I already asked [this question](https://math.stackexchange.com/questions/2668737/what-is-an-half-cusps-in-hyperbolic-geometry) on math.stackexchange, but it was [suggested](https://math.stackexchange.com/questions/2668737/what-is-an-half-cusps-in-hyperbolic-geometry#comment5512579_2668737) that I post it here as well.
... | https://mathoverflow.net/users/48526 | What is a half cusp in hyperbolic geometry? | This means that the boundary is geodesic with cusps in the marked points.
The easiest example is a disk with 3 marked points on its boundary. In this case the hyperbolic metric is given by identification with an ideal triangle in ${\mathbb H}^2$, i.e., a triangle with its three vertices in $\partial\_\infty{\mathbb H... | 4 | https://mathoverflow.net/users/39082 | 293952 | 129,310 |
https://mathoverflow.net/questions/291295 | 3 | The question below has been posted on Stackexchange few days ago but I decided to share it on MO also. Hope this is not a misuse.
Fix $t\geqslant1$ and define $u\_t=\pmatrix{1 & 0\\0 & t}\otimes\pmatrix{1 & 0\\0 & 0}+\pmatrix{1 & 0\\0 & -t}\otimes\pmatrix{0 & 0\\0 & 1}\in\mathcal{M}\_2\otimes\mathcal{M}\_2$. Calculat... | https://mathoverflow.net/users/75127 | projective and Haagerup tensor norms | Here is a computation of the Haagerup norm. It is not particularly instructive, since it relies very heavily on a special feature of the case at hand. I for one would be very glad to learn some more robust techniques for computing in Haagerup tensor products.
The first step, as pointed out by Mateusz Wasilewski, is t... | 3 | https://mathoverflow.net/users/85913 | 293958 | 129,313 |
https://mathoverflow.net/questions/293896 | 2 | Is an ultrapower of an Arens regular (non-superreflexive) Banach algebra, Arens regular?
| https://mathoverflow.net/users/84700 | Arens regularity of an ultrapower of an Arens regular Banach Algebra? | I am not 100% sure what the question is asking.
>
> Is the ultrapower of an Arens regular Banach algebra also Arens regular?
>
>
>
As my comment said, if $A$ is a $C^\*$-algebra, then "yes".
$\newcommand{\mc}{\mathcal}\newcommand{\ip}[2]{\langle #1,#2\rangle}$To show it's not always true, here's a counter-e... | 4 | https://mathoverflow.net/users/406 | 293959 | 129,314 |
https://mathoverflow.net/questions/293164 | 7 | Let $E$ be a spin$^c$ bundle and $spin^c(E)$ the corresponding $spin^c(n)$-principial bundle. Let $g\_{U,V}: U \cap V \to spin^c(n)$ denote transition functions for this principial bundle and consider the map $\nu:spin^c(n) \to \mathbb{T}$ defined by $\nu(w)=w^!w$ where $(v\_1 \cdot ... \cdot v\_r)^!=v\_r \cdot ... \cd... | https://mathoverflow.net/users/24078 | First Chern class of a specific line bundle | One way to prove this is to first check that it is true when $E$ is the spin$^c$ bundle canonically associated to a complex vector bundle $F$ and then argue that complex and spin$^c$ bundles have the essentially same characteristic classes in degree up to 4.
To associate a spin$^c$-bundle to a complex vector bundle, ... | 6 | https://mathoverflow.net/users/13061 | 293960 | 129,315 |
https://mathoverflow.net/questions/293910 | 0 | Assume that $A$ is a unital $C^\*$ algebra. Is there a $C^\*$ embedding of $A$ in some $B(H)$ whose image is a hereditary $C^\*$ subalgebra of $B(H)$?
If **not**, is the answer affirmative when $A$ is commutative?
If the answer of the later question is **affirmative**, we consider the following definition:
We say... | https://mathoverflow.net/users/36688 | A property of compact topological space via certain $C^*$ embedding in operator algebras | If $A\subseteq B(H)$ is hereditary and has unit $p$, then we must have $A=pB(H)p\cong B(pH)$. So the answer is, such an embedding exists if and only if $A\cong B(H)$ for some $H$. In particular, the only commutative example is $A=\mathbb C$.
| 2 | https://mathoverflow.net/users/85913 | 293970 | 129,318 |
https://mathoverflow.net/questions/293965 | 5 | Let $X$ be the set of functions $e^{p(x)}$ of the real vector $x$, where $p$ is a multivariate polynomial with $p(0)=0$.
Is any finite subset of $X$ linearly independent? If yes, why? If no, is the answer true for other, restricted choices of $p$? (The answer is yes when the polynomials are restricted to have degree... | https://mathoverflow.net/users/56920 | linear independence of exponentials | Yes.
If you require that no difference $p\_j-p\_k$ is constant (which follows from
your assumption $p(0)=0$), then $c\_1e^{p\_1}+\ldots+c\_me^{p\_m}=0$ implies that all $c\_j=0$.
In fact a more general result is true: instead of polynomials one can take any
entire functions. This is called Borel's theorem.
(See, for ... | 6 | https://mathoverflow.net/users/25510 | 293978 | 129,320 |
https://mathoverflow.net/questions/293955 | 3 | Let $X \sim \mathcal{N}\left( {{\mu \_x},\sigma \_x^2} \right)$ and $Y \sim \mathcal{N}\left( {{\mu \_y},\sigma \_y^2} \right)$ be two univariate and independent Gaussian/normal random variables and let $Z = XY$ be their product.
It is well-known that the probability distribution of $Z$ can be approximated by the no... | https://mathoverflow.net/users/88057 | Normal approximation to the pointwise/Hadamard/Schur product of two multivariate Gaussian/normal random variables | $\newcommand{\si}{\sigma}
\newcommand{\Si}{\Sigma}
\renewcommand{\c}{\circ}
\newcommand{\tr}{\operatorname{tr}}$
Let $X\_1:=X$, $X\_2:=Y$, $\mu\_1:=\mu\_x$, $\mu\_2:=\mu\_y$, $\Si\_1:=\Si\_x$, $\Si\_2:=\Si\_y$, $N:=\mathcal{N}\_d$. Let $Y\_i:=X\_i-\mu\_i$. Then $X\_i=\mu\_i+Y\_i$, $Y\_i\sim N(0,\Si\_i)$, $Y\_1$ and $... | 1 | https://mathoverflow.net/users/36721 | 293985 | 129,323 |
https://mathoverflow.net/questions/293969 | 7 | Let $\mathcal{E}$ be a topos and $\Omega$ its subobject classifier.
Is it possible to have a nonidentity local operator (a.k.a Lawvere-Tierney topology) $j\colon\Omega\to\Omega$, a $j$-sheaf $X\in\mathcal{E}\_j\subseteq\mathcal{E}$, and an epimorphism $f\colon X\twoheadrightarrow\Omega$?
I'd be interested to see a... | https://mathoverflow.net/users/2811 | Can the subobject classifier be covered by an object in a subtopos? | Consider the Sierpinski topos, i.e., presheaves on the poset $P=\{0<1\}$, and consider the topology in which $1$ is covered by $\{0\}$, i.e., the double-negation topology. Then the presheaf $1+1+1$ (i.e., assign to each node of $P$ a 3-element set and let the transition map be a bijection) is a sheaf. It maps surjectiv... | 9 | https://mathoverflow.net/users/6794 | 293990 | 129,325 |
https://mathoverflow.net/questions/293629 | 14 | Let $S\_\omega$ be the group of bijections of the countable ordinal $\omega:=\{0,1,2,\dots\}$ and $Alt\_\omega$ be the subgroup of $S\_\omega$ consisting of even permutations of $\omega$ (i.e., the compositions of even number of transpositions). By [Onofri-Schreier-Ulam Theorem](https://math.stackexchange.com/questions... | https://mathoverflow.net/users/61536 | Is $Alt_\omega$ a dense subgroup of a non-discrete locally compact topological group? | Corollary 1.5 of [this Vissarion Belyaev's paper](http://mi.mathnet.ru/smj1670) says that
>
> A group containing an infinite inert residually finite subgroup
> embeds into a non-discrete locally compact group.
>
>
>
Here, a subgroup $H$ of a group $G$ is called *inert* if $H\cap g^{-1}Hg$ has a finite index i... | 6 | https://mathoverflow.net/users/24165 | 293991 | 129,326 |
https://mathoverflow.net/questions/293706 | 3 | Let $q = p^t$ where $p$ is prime. I am interested in estimating the complete exponential sum, which looks like
$$
\sum\_{0 \leq h < q} \chi( (h-a\_1)(h-a\_2)(h-a\_3))
\ \bar{\chi}( (h-b\_1)(h-b\_2)(h-b\_3)) \ e^{ 2 \pi i C h /q}
$$
in terms of integers $a\_i$'s and $b\_j$'s and $C$. Here $\chi$ is a Dirichlet characte... | https://mathoverflow.net/users/84272 | How to estimate a mixed character sum $\sum_{h \in \mathbb{Z}/q \mathbb{Z}} \chi(f(h)) e(Ch/q)$? | [Alpoge's comments](https://mathoverflow.net/a/293994) oversimplify the situation slightly.
There exists a unique constant $\alpha$ modulo $p^{\lfloor t/2\rfloor}$ such that $\chi(1+x)=q^{ 2\pi i \alpha x/q}$ for $\alpha$ a mulitple of $p^{\lceil t/2 \rceil}$.
If we let $e\_q(x) = e^{2\pi i x /q}$ and $F(h) = \chi(... | 2 | https://mathoverflow.net/users/18060 | 293999 | 129,328 |
https://mathoverflow.net/questions/293986 | 2 | For integers $n\geq 1$ with $$\operatorname{rad}(n)=\prod\_{\substack{p\mid n\\p\text{ prime}}}p$$ we denote the squarefree kernel or radical of an integer $n$ (see if you want this [Wikipedia](https://en.wikipedia.org/wiki/Radical_of_an_integer)). And $\varphi(n)$ denotes the Euler's totient funciton. Then while I was... | https://mathoverflow.net/users/nan | The Euler's totient function and the product of distinct primes dividing $n$ versus the Heronian means | Your conclusion is valid, and for this we only need to assume that $\varphi(3n)$ is divisible by $3$. Indeed, this weaker condition is equivalent to the existence of a prime divisor $p\mid n$ that is either $3$ or congruent to $1$ modulo $3$. It is known that $p$ is the norm of some [Eulerian integer](https://en.wikipe... | 3 | https://mathoverflow.net/users/11919 | 294003 | 129,331 |
https://mathoverflow.net/questions/293975 | 5 | Suppose $X$ is a smooth projective variety defined over $\mathbb{Q}$, and the pure Hodge structure on $H^{2n}(X)$ is of a very simple form,
\begin{equation}
\mathbb{Q}(-n)^{b^{2n}}
\end{equation}
where $b^{2n}$ is just $\text{dim}\,H^{2n}(X)$. Under what conditions is the etale cohomology $H^{2n}\_{et}(X,\mathbb{Q}\_{\... | https://mathoverflow.net/users/87910 | When is the etale cohomology of a variety simple if its Hodge structure is simple? | Assuming the Hodge conjecture, it's true if and only if the relevant algebraic cycles are defined over $\mathbb{Q}$.
Conjecturally, the cohomologies are different realizations of the same motive. The condition on the Hodge cohomology tells you the motive is a Tate twist of an Artin motive, and you want the Artin motiv... | 5 | https://mathoverflow.net/users/121314 | 294013 | 129,333 |
https://mathoverflow.net/questions/294014 | 4 | If $\mathbf{H}$ is an $\infty$-topos, then we can define a Cartesian fibration $p : T \mathbf{H} \to \mathbf{H}$ such that the fiber of $p$ over $X$ is the $\infty$-category of spectrum objects in $\mathbf{H}/X$. Then the category $T \mathbf{H}$ is an $\infty$-topos (see [this page](https://ncatlab.org/nlab/show/tangen... | https://mathoverflow.net/users/62782 | Generalizations of tangent $\infty$-topos | This is rarely true. For example, the axioms for ∞-topoi imply that, if $T\_S\mathbf H$ is an ∞-topos, then the fibers of $p\_S$ must have [van Kampen pushouts](https://ncatlab.org/nlab/show/van+Kampen+colimit) (more generally van Kampen weakly contractible colimits). In particular, the fibers cannot be $n$-categories ... | 4 | https://mathoverflow.net/users/20233 | 294028 | 129,338 |
https://mathoverflow.net/questions/294023 | 6 | The only way I know to get a locally cartesian closed category which is not a topos is to start with a topos and then throw out some objects so that the category is not sufficiently cocomplete to be a topos. Is that the only way there is?
Well, I suppose there's another way: the category of topological spaces and loc... | https://mathoverflow.net/users/2362 | Example of a locally presentable locally cartesian closed category which is not a topos? | Every Grothendieck [quasitopos](https://ncatlab.org/nlab/show/quasitopos) is presentable and locally cartesian closed. These are categories of [separated presheaves](https://ncatlab.org/nlab/show/separated+presheaf) on a site. The simplest example of a site whose separated presheaves do not form a topos is the one-poin... | 10 | https://mathoverflow.net/users/20233 | 294031 | 129,339 |
https://mathoverflow.net/questions/294030 | 18 | Call a field $k$ *unrepeatable*$^1$ if for every field $L$ there are either zero or one field homomorphisms $k \to L$. Then the prime fields $\mathbb{Q}$ and $\mathbb{F}\_p$ for $p$ prime are clearly unrepeatable and it seems very likely to me that those are the only ones. Is that true?
Notice that an unrepeatable fi... | https://mathoverflow.net/users/644 | Is a field that never embeds twice in another field necessarily a prime field? | It seems that indeed only prime fields are unrepeatable.
**Proof:**
Let $k$ be unrepeatable and $F\subseteq k$ denote the prime field of $k$. Let $T\subseteq k$ be a transcendence base of $k/F$ and let $G=F(T)$. If $T\neq\emptyset$, then $G/F$ has non-trivial automorphisms (say take one element $t\in T$ to $t+1$). Si... | 15 | https://mathoverflow.net/users/10076 | 294032 | 129,340 |
https://mathoverflow.net/questions/294017 | 3 | I am looking for a reference explaining how to solve the Navier-Stokes equations numerically using machine learning algorithms .
Thank you in advance for your help .
| https://mathoverflow.net/users/73579 | Navier-Stokes equations and machine learning | One of the earliest papers is [Application of machine learning algorithms to flow modeling and optimization](https://web.stanford.edu/group/ctr/ResBriefs99/petros.pdf) (1999).
>
> A model reduction can be accomplished by projecting the Navier-Stokes
> equations on a properly selected lower dimensional phase subspa... | 4 | https://mathoverflow.net/users/11260 | 294033 | 129,341 |
https://mathoverflow.net/questions/293977 | 0 | A symbol $p \in S^m(\Omega)$ where $\Omega \subset \mathbb{R^n}$, or its corresponding operator $p(x,D) \in \Psi^m(\Omega)$ is said to be elliptic of order m if for every compact $A\subset \Omega$ there are positive constants $c\_A,C\_A$ such that
$|p(x,\xi)|\ge c\_A|\xi|^m$ for $x \in A$ and $|\xi|\ge C\_A$.
Usi... | https://mathoverflow.net/users/102092 | Definition of Elliptic pseudodifferential operators | In the first place, you can write
$
\mathbf 1\_\Omega=\sum\_{k\ge 1}\chi\_k, \ \chi\_k\in C^\infty\_c(\Omega).
$
Let $\nu\in C^\infty(\mathbb R^n)$, vanishing on $B(0,1/2)$, equal to 1 on $B(0,1)^c$: let us define
$$
\zeta(x,\xi)=\sum\_{k\ge 1}\chi\_k(x) \nu(\xi/R\_k),
$$
where the positive $R\_k$ is chosen so that $p(... | 3 | https://mathoverflow.net/users/21907 | 294043 | 129,347 |
https://mathoverflow.net/questions/294045 | 1 | In the book "a convenient setting for global analysis" they describe the order of an operational tangent vector on a convenient vector space.
<http://www.mat.univie.ac.at/~michor/apbookh-ams.pdf>
After which they prove that their exists those of order 2 and 3. After breaking my head on it for a bit I can't seem to ... | https://mathoverflow.net/users/120866 | What are examples of a second order operational tangent vector on an infinite dimensional Hilbert space | Let $\alpha\in B(H)'$ be a bounded linear functional on the space of all bounded linear operators on Hilbert space which vanishes on the subspace of compact operators $K(H)\supset H\otimes H'$. Then
$f\mapsto \alpha(d^2f(0))$ for $f\in C^\infty(H)$ is an operational tangent vector with this property, since it is a der... | 5 | https://mathoverflow.net/users/26935 | 294046 | 129,348 |
https://mathoverflow.net/questions/294048 | 4 | Kan extension are an incredibly useful concept when they exist. My question is: can we still derive information about a functor when Kan extensions don't exist, and if so, what information and in what way?
I can think of one way to look at it:
Let $p:C\to C'$ and $F:C\to D$ be functors. We can define a category of ... | https://mathoverflow.net/users/95265 | When Kan extensions don't exist | Yes, directions like this have been explored, for all kinds of objects with universal properties (which includes Kan extensions, since as MacLane famously wrote "all concepts are Kan extensions"). Rather than the category whose initial object would be the missing object, it's more common to consider the functor for whi... | 6 | https://mathoverflow.net/users/49 | 294052 | 129,350 |
https://mathoverflow.net/questions/293859 | 20 | I am interested in this paper which I can't read because it's in German:
*Frobenius, G.*, Über die Charaktere der mehrfach transitiven Gruppen., Berl. Ber. 1904, 558-571 (1904). [ZBL35.0154.02](https://zbmath.org/?q=an:35.0154.02).
A free online copy is [here](http://www.e-rara.ch/zut/content/titleinfo/5929009). I ... | https://mathoverflow.net/users/801 | What did Frobenius prove about $M_{12}$? | It seems to me that Frobenius is using lots of specific facts about the permutation group $M\_{12}$ (and $M\_{24}$, respectively) here, and that he has no doubts about the existence of these groups. In particular, he is using specific subgroups of $M\_{12}$ and $M\_{24}$ (and then using induced characters). Most statem... | 17 | https://mathoverflow.net/users/10266 | 294069 | 129,358 |
https://mathoverflow.net/questions/294070 | 5 | Thom's theorem states that for every homology class $\alpha \in H\_{\*}(X)$ there exists an integer $k = k(\alpha)$ such that the class $k\, \alpha$ comes from the fundamental class of an orientable closed smooth manifold, where $X$ is an arbitrary topological space. To be on the safe side we assume that $X$ is a count... | https://mathoverflow.net/users/37807 | Relative Steenrod's problem | Yes. It's quite general. If $h$ is any generalized homology theory (for example, oriented bordism) such that $h\_0(point)=\mathbb Z$ and such that $h\_n(point)=0$ for all $n<0$ then there is a map from $h$ to ordinary homology inducing an isomorphism $h\_0\to H\_0$; and after tensoring with $\mathbb Q$ this map $h\to H... | 9 | https://mathoverflow.net/users/6666 | 294072 | 129,360 |
https://mathoverflow.net/questions/294024 | 9 | If $\mathcal{M}$ is a cofibrantly generated model category and $\mathcal{C}$ is a small category, then we can give $\mathcal{M}^\mathcal{C}$ the projective model structure, in which weak equivalences and fibrations are transformations that are such on each object of $\mathcal{C}$. I recall seeing a generalization in wh... | https://mathoverflow.net/users/58888 | Looking for generalization of projective model structure | This model structure has appeared explicitly in the paper by Paul Balmer and Michel Matthey "[Codescent theory I: Foundations](https://www.sciencedirect.com/science/article/pii/S0166864104001828?via%3Dihub)". Theorem 3.5 establishes the existence of, what the authors called, the relative model structure.
But the idea... | 5 | https://mathoverflow.net/users/30641 | 294076 | 129,361 |
https://mathoverflow.net/questions/294078 | 3 | Suppose that $a,b,c$ are positive integers such that $gcd(a,b,c) = 1$. Let $X$ be the complex weighted projective space $\mathbb{C} \mathbb{P}(1,a,b,c)$. How to compute the group $T(H^{3}(X,\mathbb{Z}))$?
Notation: If $A$ is an abelian group then $T(A)$ denotes the torsion subgroup.
| https://mathoverflow.net/users/99732 | Cohomology of weighted projective spaces | Additively, the integral cohomology is the same as for the ordinary projective space (multiplication in cohomology is different though), so there is no torsion. This is Theorem 1 in the paper of Kawasaki [Cohomology of twisted projective spaces and lens complexes](https://link.springer.com/article/10.1007/BF01429212).
... | 10 | https://mathoverflow.net/users/1306 | 294079 | 129,362 |
https://mathoverflow.net/questions/294088 | 2 | Consider the moduli space of hyperbolic metrics on the disk with $n>3$ marked points on its boundary, $\mathcal{M}\_{D,n}$.
$\mathcal{M}\_{D,n}$ can be parametrised in terms of cross ratios of the punctures.
Equivalently, I can consider the double $D^d$ of $D$ wich is an hyperbolic surface with punctures. The leng... | https://mathoverflow.net/users/48526 | Build a Fuchsian group starting from punctures on a disk | This is a very special case of the Fock-Goncharov construction.
Divide your ideal n-gon into n-2 ideal triangles. Given one cross ratio associated to each edge (i.e., to the 4 ideal vertices of the two ideal triangles adjacent to that edge) you want to reconstruct a representation $\Gamma\to PSL(2,{\mathbb R})$ from... | 1 | https://mathoverflow.net/users/39082 | 294093 | 129,364 |
https://mathoverflow.net/questions/294011 | 13 | A short version of this question could be
>
> What are the duals of $PGL(2,\mathbf{Q}\_p)$, $PGL(2,\mathbf{R})$ and $PGL(2,\mathbf{C})$?
>
>
>
I should obviously add some precisions.
* there are different notions of duals: admissible, unitary, tempered and generic; all are of interest;
* having a parametriz... | https://mathoverflow.net/users/43737 | Cartography of the duals of GL, PGL, SL, etc | I'm assuming you're interested in complex representations of these groups. In each case the unitary, tempered and generic duals are contained in the smooth dual, so I'll start by describing that. Let $G$ be any of the groups you're interested in. Actually, most of what I'll say goes through pretty nicely for $G$ the gr... | 17 | https://mathoverflow.net/users/121342 | 294099 | 129,367 |
https://mathoverflow.net/questions/294100 | 8 | Suppose that the family $\mathrm{RO}(X)$ of regular open subsets of $(X,\mathscr{O})$ is a basis of $X$. Let the density of $\mathrm{RO}(X)$ (considered as Boolean algebra) be $\aleph\_0$.
Does $X$ have to be second-countable? If not, what if we add regularity of $X$ (both $T\_1$ and $T\_3$ separation axioms)? If ans... | https://mathoverflow.net/users/22019 | Does $\aleph_0$-density of regular open algebra entail existence of countable basis? | The answer is no, not necessarily.
For a counterexample, consider the [Sorgenfrey line](https://en.wikipedia.org/wiki/Lower_limit_topology), which is the topology on $\mathbb{R}$ with basis consisting of the half-open intervals $[a,b)$. These are each clopen and hence regular open in that topology. Furthermore, ever... | 9 | https://mathoverflow.net/users/1946 | 294104 | 129,369 |
https://mathoverflow.net/questions/294083 | 3 | Let $(W,S)$ be a Coxeter group. Then the Kazhdan-Lusztig polynomials $P\_{x,w}$ are known to satisfy the following recursive identity:
For $x < w$ and $s \in S$ satisfying $\ell(sw) < \ell(w)$, define
$$c = c\_s(x) :=
\begin{cases}
0, \,\,\,\text{ if }x < sx\\
1, \,\,\,\text{ if }x > sx.
\end{cases}$$
Then, $P\... | https://mathoverflow.net/users/47310 | Recursive formula for inverse Kazhdan-Lusztig polynomials | The case of an affine Weyl group is apparently the only one which has been looked at closely. But it may be hard to answer your specific question. As far as I know, there are two relevant papers, the first in 1986 by Andersen [*here*](https://mathscinet.ams.org/mathscinet-getitem?mr=840301) (now accessible online) and ... | 5 | https://mathoverflow.net/users/4231 | 294114 | 129,374 |
https://mathoverflow.net/questions/294051 | 2 | Let $I\_1, \dots , I\_n$ be ideals of a ring $R$ with identity having zero intersection. Assume that for some $x\in R$, $x+I\_ i$ is an element of the right socle of $R/I\_ i$, for each $ i=1,\dots , n$. My question: "Is it necessarily true that $x$ belongs to the right socle of $R$?"
I appreciate any cooperation in ... | https://mathoverflow.net/users/48889 | From socle of quotients to socle of ring itself | There’s a natural injective module homomorphism
$$R\to\bigoplus\_iR/I\_i$$
that takes $x$ into the semisimple submodule $\bigoplus\_i\text{soc}(R/I\_i)$, so the right ideal generated by $x$ is semisimple, and so $x\in\text{soc}(R)$.
| 2 | https://mathoverflow.net/users/22989 | 294116 | 129,376 |
https://mathoverflow.net/questions/294005 | 3 | I had asked [this](https://math.stackexchange.com/questions/2654532/irreducible-operators) question on MSE but did not get any response.
---
I would like some reference to books that talk about irreducible operators on Banach lattices and its properties. A quick google search did not reap fruitful results. I'm es... | https://mathoverflow.net/users/119514 | Reference request: Irreducible operators | *Preliminary remarks:*
* In the comments the OP noted that he is primarly interested in the question whether the dual operator of an irreducible operator is irreducible, so I will focus on this aspect here.
* The OP asked for a reference where this topic is treated. But to the best of my knowledge, all the standard ... | 5 | https://mathoverflow.net/users/102946 | 294120 | 129,377 |
https://mathoverflow.net/questions/273333 | 11 | **Update**: I'm happy to say that this question has been made essentially obsolete by the breakthrough result of Serge Vlăduţ, who showed that the kissing number is exponentially large: <https://arxiv.org/abs/1802.00886> !
Recall that the kissing number in $n$ dimensions is the maximal number $M$ such that there exis... | https://mathoverflow.net/users/64613 | The lattice handshake number ("nearly kissing" number)? | I'm happy to say that this question has been answered by the breakthrough result of Serge Vlăduţ, who showed that the kissing number is exponentially large: <https://arxiv.org/abs/1802.00886> !
I.e., there exists a constant $C > 0$ and a sequence of lattices $L\_n \subset \mathbb{R}^n$ such that $L\_n$ has at least ... | 5 | https://mathoverflow.net/users/64613 | 294128 | 129,381 |
https://mathoverflow.net/questions/294122 | 4 | (Intuition: in the category of non-empty sets, every function that coequalizes all points in the domain factors through the terminal object. I would like to know if something analogous happens in certain category of `connected algebraic spaces'.
I formulate the precise question in terms of commutative algebra.)
Let $... | https://mathoverflow.net/users/121350 | Do certain maps between f.g. $\mathbb{C}$-algebras factor through a local (and f.g.) algebra? | Yes it's true.
1) First assume that $B$ is reduced (non nonzero nilpotent element). Let $I$ be the set of maximal ideals of $B$. Since for a f.g. algebra over a field, the Jacobson radical equals the radical, we have $\bigcap\_{M\in I}M=0$. By the Nullstellensatz, for every $M\in I$ there is a unique $\mathbf{C}$-alg... | 5 | https://mathoverflow.net/users/14094 | 294139 | 129,384 |
https://mathoverflow.net/questions/294138 | 1 | Recently I have a conjecture on decomposing a linear program into smaller ones. I have tested it in Mathmatica by a lot of examples. However, I cannot prove it. I will appreciate if someone can give some ideas.
>
> Let $f\_1,\dotsc,f\_r \in \mathbb{R}[x\_1,...,x\_n]$ ($r>n+1$) be linear functions. Then $\{f\_i \ge ... | https://mathoverflow.net/users/121163 | a linear programming problem | The statement in the gray box is an immediate consequence of [Helly's theorem](https://en.wikipedia.org/wiki/Helly%27s_theorem): for each $i$, the set $\{ v\in\mathbb{R}^n\colon\ f\_i(v)\geq 0\}$ is convex, and because of $r>n+1$, the condition given on the right-hand side of the equivalence implies that the intersecti... | 5 | https://mathoverflow.net/users/108556 | 294142 | 129,386 |
https://mathoverflow.net/questions/294119 | 37 | When I first read Set Theory by Jech, I came under the impression that the Universe of Sets, $V$ was a fixed, well defined object like $\pi$ or the Klein four group. However as I have read on, I am beginning to have my doubts. We define
\begin{align}
V\_0 & :=\emptyset. \\[10pt]
V\_{\beta +1} & :={\mathcal {P}}(V\_\b... | https://mathoverflow.net/users/120841 | Is V, the Universe of Sets, a fixed object? | As you noticed, the iterative conception of sets requires a pre-existing universe of sets, **and** ordinals with which we can label the stages. So if you work within ZFC itself, in other words within an **existing model** of ZFC, you can perform that iterative construction to obtain $V$. Like [Asaf Karagila says here](... | 24 | https://mathoverflow.net/users/50073 | 294147 | 129,388 |
https://mathoverflow.net/questions/294063 | 2 | Consider the family of monic biquadratic polynomials given by $f\_{a,b}(x) = x^4 + 2ax^2 + b$ with $a,b$ integers. Let $K\_{a,b}$ denote the isomorphism class of quartic fields obtained by adjoining any one of the roots of $f\_{a,b}$. Is there a relatively nice list of criteria to decide whether $K\_{a,b}$ is isomorphi... | https://mathoverflow.net/users/10898 | Determining when two biquadratic polynomials generate the same field | The necessary and sufficient conditions are as follows:
1) Condition on the common quadratic subfield (as Watson Ladd's answer):
there exists $c\in\Bbb Q$ such that
$${a'}^2-b'=c^2(a^2-b)\;.$$
2) Condition on the discriminant: the polynomial discriminant being $256 b(a^2-b)^2$ we need for some $d\in\Bbb Q$
$$bb'=d^... | 5 | https://mathoverflow.net/users/81776 | 294148 | 129,389 |
https://mathoverflow.net/questions/287870 | 6 | In Bardakov, Algebra and Logic, Vol. 39, No. 4, 2000 I have found the following (page 225, see <https://link.springer.com/article/10.1007/BF02681648>)
>
> We pronounce tile validity of the following:
>
>
> **Conjecture.** For every element *z* in the derived subgroup of a free non-Abelian group *F* and for any na... | https://mathoverflow.net/users/50339 | Relation between commutator length and stable commutator length in free groups | The misconception is due to bad translation. The original text was in Russian and can be found here: <http://www.mathnet.ru/links/f40b0cf29e7b19a9b2ab1a95ef70baba/al284.pdf>.
It reads: "Можно высказать предположение о том" which means "we can phrase a guess".
I e-mailed Valeiry Bardakov about that conjecture. And... | 2 | https://mathoverflow.net/users/50339 | 294166 | 129,394 |
https://mathoverflow.net/questions/294183 | 40 | Prof. D. C. McCarty recently gave an interesting
interview (published in January 2015, and easily
found on a large video hosting site), entitled
>
> What are the limits of mathematical explanation?
>
>
>
I think this is the best interview with a
philosopher that I have heard or read in a long while,
and hope ... | https://mathoverflow.net/users/108556 | Did Hilbert laugh? | Constance Reid describes the recording on page 196 of her book [Hilbert](https://rads.stackoverflow.com/amzn/click/0387946748) (1969).
The recording was *not* made directly from the live address. Instead, Hilbert was asked to repeat the conclusion of his speech in the broadcasting studio of a local radio station. Th... | 55 | https://mathoverflow.net/users/11260 | 294185 | 129,399 |
https://mathoverflow.net/questions/294155 | 5 | Question 1: Let $G$ be a connected reductive group defined over a number field $K$, and $\pi$ is an irreducible cuspidal automorphic representation of $G(\mathbb{A}\_K)$. Then by a theorem of Flath, we have $\pi=\otimes^{'}\pi\_v$. All but finitely many $\pi\_v$ are unramified and they all generic by Shalike.
I want ... | https://mathoverflow.net/users/121163 | Some question about cupidal automorphic representation and supercuspidal representation | It is not necessary that a cuspidal $\pi$ have a supercuspidal local component: for example, let $\Delta(\tau)$ be Ramanujan's delta function and let $\pi = \otimes' \pi\_p$ denote the associated cupsidal automorphic representation of $\textrm{GL}\_2(\mathbb{A}\_{\mathbb{Q}})$ (see Kudla's article [here](https://mathsc... | 6 | https://mathoverflow.net/users/120578 | 294186 | 129,400 |
https://mathoverflow.net/questions/294169 | 1 | Reading through Serre's book on Galois cohomology, I encounted the following problem:
Let $n$ be an integer. Consider families of integers $c(i,j,k)$ with $i,j,k \in [1,n]$ which are alternating in $(i,j).$ Show that for every $n \geq 3,$ there exists such a family with the following property:
(\*) - If the eleme... | https://mathoverflow.net/users/nan | Serre Galois Cohomology exercise on Lie algebras | As you do, set $[x\_1,x\_2] = x\_2$, and $[x\_3,x\_1] = x\_1$, and $[x\_2,x\_n] = x\_n$ for $n\ge 3$ (and what you like for other brackets, provided the bracket is alternating).
As you have observed, these conditions (for $n\le 3$) force $x\_1=x\_2=x\_3=0$ in any Lie ring, and the remaining $[x\_2,x\_n] = x\_n$ forc... | 1 | https://mathoverflow.net/users/14094 | 294201 | 129,407 |
https://mathoverflow.net/questions/294151 | 3 | Given a Cartesian fibration $p : \mathbf{E} \to \mathbf{B}$ over an $\infty$-topos the paper by Marc Hoyois mentioned in his answer to [this question](https://mathoverflow.net/questions/294014/generalizations-of-tangent-infty-topos) gives some sufficient conditions for $\mathbf{E}$ to be an $\infty$-topos. I'd like to ... | https://mathoverflow.net/users/62782 | When the global section functor is a Cartesian fibration? | In fact I believe something quite general can be said: if $p:C\to D$ is any functor that preserves finite limits, then it is a [Street fibration](https://ncatlab.org/nlab/show/Street+fibration) if and only if it has a fully faithful right adjoint. A Street fibration is just a functor whose isofibrant replacement is a c... | 4 | https://mathoverflow.net/users/49 | 294205 | 129,409 |
https://mathoverflow.net/questions/294220 | 3 | (This question actually arose in real life when dealing with status bits with mutual influence.)
Let $G=(V,E)$ be a connected, simple, undirected graph with $|V| \geq 2$. For $v\in V$, let $N\_G(v) = \{w\in V: \{v,w\}\in E\}$. A *parity map* for $G$ is a map $f:V\to {\mathbb Z}/2{\mathbb Z}$ such that for every $v\in... | https://mathoverflow.net/users/8628 | Non-trivial parity maps in graphs | I think $K\_{n,n+1}$ works. Clearly each side must be monochromatic. So the side of odd size must be all $0$. But then the other side must be all $0$.
| 8 | https://mathoverflow.net/users/88146 | 294223 | 129,412 |
https://mathoverflow.net/questions/294214 | 2 | Let's define an iterative function $V^F$ to indicate a iterative hierarchy building function that iterates a function $F$ starting from $\emptyset$ after a well ordering relation set $R$ whose domain is a set $S$, as follows:
$$V^F\_i= \emptyset ................................................... i \in S: \not \exist... | https://mathoverflow.net/users/95347 | Does the axiom schema of Replacement follow from the abstract notion of the iterative conception of sets? | In my blog post [Transfinite recursion as a fundamental principle in set theory](http://jdh.hamkins.org/transfinite-recursion-as-a-fundamental-principle-in-set-theory/), I prove that the principle of transfinite recursion is equivalent to the replacement axiom.
| 11 | https://mathoverflow.net/users/1946 | 294226 | 129,414 |
https://mathoverflow.net/questions/294209 | 15 | Let $g$ be an $n \times n$ matrix of functions $g\_{ij}(z)$ in $\mathbb{C}(z)$. Suppose that the $g\_{ij}(z)$ have no poles on the annulus $1-\epsilon < |z| < 1+\epsilon$ and that $\det g(z)$ is nonzero on this annulus. Then I can define a holomorphic vector bundle on $\mathbb{P}^1$ by taking trivial rank $n$ bundles o... | https://mathoverflow.net/users/297 | Holomorphic line bundles on $\mathbb{P}^1$ from gluing data | I am just posting my comment above as an answer. Let $U$ and $V$ be subsets of $\mathbb{CP}^1$, or any other projective algebraic curve $C$, whose complement sets are finite, and such that $\{U,V\}$ is an open cover. Let $g=[g\_{i,j}]$ be an $n\times n$ matrix of rational functions on $C$ that are regular on $U\cap V$ ... | 6 | https://mathoverflow.net/users/13265 | 294228 | 129,415 |
https://mathoverflow.net/questions/294208 | 3 | The notion of analytic conductor of a generic representation of $\mathrm{GL}(n)$ has been defined by Iwaniec and Sarnak, and since then is at the heart of many works in analytic number theory and used for different purposes. I am interested in understanding the consistency between the different definitions arising in t... | https://mathoverflow.net/users/43737 | Consistency of the notion of conductor of a representation | In all cases, the analytic conductor is defined as the usual arithmetic conductor $q\_{\pi}$ times a product of terms coming from the archimedean places. The issue is the definition of the terms at the archimedean places. It is important to note that the analytic conductor is, in practice, a working definition: as we e... | 3 | https://mathoverflow.net/users/3803 | 294230 | 129,416 |
https://mathoverflow.net/questions/294219 | 10 | Let $q$ be a prime power and let $n\geq2$ be an integer.
Is it known what is the largest order of a unipotent upper-triangular $n\times n$ matrix over the ring $\mathbb{Z}/q\mathbb{Z}$?
I am mostly interested in upper bounds.
| https://mathoverflow.net/users/101929 | Order of unipotent matrices over $\mathbb{Z}/q\mathbb{Z}$ | This works for any unipotent matrix whose characteristic polynomial is $(x-1)^n$ (including some upper-triangular matrices without ones on the diagonal). By Cayley-Hamilton, the matrix $U$ satisfies $(U-1)^{n}=0$, so we can write $U = 1 + N$ with $N^n=0$.
Then $$ U^{p^k} = (1+N)^{p^k} = \sum\_{i=0}^{n-1} {p^k \choose... | 10 | https://mathoverflow.net/users/18060 | 294231 | 129,417 |
https://mathoverflow.net/questions/294227 | 2 | Let $\Sigma(g,n)$ be an $n$-punctured surface of genus $g$.
If we assume that $f:\Sigma(g,n)\rightarrow\Sigma(g,n)$ is a branched self-covering map of degree $d$, then the equality follows from the [Riemann-Hurwitz formula](https://en.wikipedia.org/wiki/Riemann%E2%80%93Hurwitz_formula)
$$
\chi(\Sigma(g,n)) = d\cdot\c... | https://mathoverflow.net/users/119219 | Construction of self-covering map of any surface | First of all, the Riemann-Hurwitz formula with $\chi<0$ implies that
for every self-covering $d=1$ so it is an automorphism. The only punctured surfaces with $\chi\geq 0$ are torus, sphere, and sphere with one or two
punctures. For torus and sphere with two punctures self-coverings are easy
to describe, for the rest, t... | 4 | https://mathoverflow.net/users/25510 | 294233 | 129,419 |
https://mathoverflow.net/questions/294184 | 10 | What is known about the plethysm $\text{Sym}^d(\bigwedge^3 \mathbb{C}^6)$ as a representation of $\text{GL}(6)$? It is my understanding that this should be multiplicity-free. I tried computing it using the Schur Rings package in Macaulay2 and I cannot see a pattern among the weights that appear.
If a formula is known... | https://mathoverflow.net/users/87706 | What is known about the plethysm $\text{Sym}^d(\bigwedge^3 \mathbb{C}^6)$ | $$\bigoplus\_{k \ge 0} t^k Sym^{k} V = \frac{1}{(1−tV)(1−t^2 \mathbf{g})(1−t^3 V)(1−t^4)(1−t^4 V\_2)},$$
where $V = \wedge^3 \mathbb{C}^6 = [0,0,1,0,0], V\_2 = [0,1,0,1,0]$, and $\mathbf{g} = [1,0,0,0,1].$
See section 6 of ["Series of Lie Groups" by Landsberg and Manivel](https://projecteuclid.org/euclid.mmj/10911120... | 3 | https://mathoverflow.net/users/874 | 294244 | 129,422 |
https://mathoverflow.net/questions/294234 | 16 | Suppose I have $R$ homogeneous polynomials $F\_1, ..., F\_R$ with integer coefficients. Let $V$ be the affine variety defined by these polynomials over $\mathbb{C}$. I was wondering if some bound that looks like the following was known?
$$
\#\{ \mathbf{x} \in (\mathbb{Z}/q \mathbb{Z})^n: F\_{i}(\mathbf{x}) \equiv 0 (\t... | https://mathoverflow.net/users/84272 | Number of solutions to polynomial congruences | First note that you can't do better than the trivial bound $q^n$ in general; for example if $q$ divides all the coefficients of the $F\_i$.
However this silly problem only occurs for finitely many primes (by the Chinese remainder theorem we can reduce to the case of prime powers). So let $p$ be a prime such that $V\_... | 6 | https://mathoverflow.net/users/5101 | 294253 | 129,424 |
https://mathoverflow.net/questions/294248 | 5 | I am currently studying the dynamics associated with the function $f(x)=2x$ (mod 1). In particular, if we define the orbit of an element $y \in [0,1]$
$$ orb(y)= \{ f^m(y): m \in \mathbb{Z}\}$$
it is easy to see, for example, that $orb\big(\frac{1}{2}\big)=\mathbb{Z}\big[\frac{1}{2}\big] \cap [0,1)$. My question co... | https://mathoverflow.net/users/121419 | Orbits of the function f(x)=2x (mod 1) | To be precise, $orb\big(\frac{1}{2}\big)=\mathbb{Z}\big[\frac{1}{2}\big] \cap [0,1).$
Since you say $\mathbb{Z}$ rather than $\mathbb{N}$ you mean the orbit of $y$ to include the solutions $t$ of $f^k(t)=y$ for $k \in \mathbb{Z}.$
You say that the orbit of $\frac13$ is contained in $$ \Big\{\frac{m}{3\times 2^n}\m... | 6 | https://mathoverflow.net/users/8008 | 294258 | 129,427 |
https://mathoverflow.net/questions/294225 | 5 | First, some terminology. Let $(S,x)$ be a compact surface of genus $g>0$. A *standard collection of loops* $\gamma\_1,\ldots, \gamma\_{2g}$ based at $x$ is a collection of loops that cuts $S$ into a topological disk.
**Question.** Let $(\Sigma,x)$ is a compact *hyperbolic* surface (without boundary) of genus $g$ and ... | https://mathoverflow.net/users/13441 | Short basis in $\pi_1$ on a hyperbolic surface of bounded diameter | I think that $A=2d$ will work, basically by applying Morse theory to the distance function from $x$. Morse theory for distance functions was originally considered by Gromov (and then Cheeger). For the case of surfaces, see a paper of [Gershkovich](https://books.google.com/books?id=7APzQMYf0ZsC&lpg=PA117&ots=inThz0IEpp&... | 3 | https://mathoverflow.net/users/1345 | 294261 | 129,430 |
https://mathoverflow.net/questions/294271 | 13 | Given a (for simplicity connective) spectrum $E$ and a pointed CW-space $X$ there is "the" (homological) Atiyah-Hirzebruch spectral sequence
$$E\_{pq}^2 = \tilde{H}\_p( X, \pi\_q(E)) \Rightarrow \pi\_{p+q}(X \wedge E)$$
I know of two approaches at arriving at this spectral sequence:
* One is using a CW-structure ... | https://mathoverflow.net/users/76299 | Relating two different approaches to the Atiyah-Hirzebruch Spectral Sequence | For cohomology, this is theorem 3.3 in
*Maunder, C.R.F.*, [The spectral sequence of an extraordinary cohomology theory](https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/article/spectral-sequence-of-an-extraordinary-cohomology-theory/A275BD09CBDEF45CC755FC15AA9CA... | 19 | https://mathoverflow.net/users/43054 | 294273 | 129,434 |
https://mathoverflow.net/questions/294236 | 7 | Let $G$ be a finitely generated group splitting non-trivially over
a infinite virtually cyclic subgroup $V$, namely an amalgamated free product
$$
A\*\_V B,\; V \neq A,B.
$$
Question: can the group $G$ be simple?
| https://mathoverflow.net/users/121404 | Finitely generated group splitting non-trivially over an infinite virtually cyclic subgroup | The answer is no: it's not simple.
It's an easy adaptation of the remarks done in Button's paper.
The background: (a) Minasyan-Osin: let $G$ be an amalgam $G=A\ast\_C B$ of two groups over a proper subgroup $C$, such that $C$ is weakly malnormal in $B$. Then $A\ast\_C B$ is acylindrically hyperbolic. [In case of i... | 6 | https://mathoverflow.net/users/14094 | 294282 | 129,438 |
https://mathoverflow.net/questions/294268 | 3 | From [W. Cary Huffman (2005), On the classification and enumeration of self-dual codes, Finite Fields and Their Applications
11(3) pp 451-490](https://www.sciencedirect.com/science/article/pii/S1071579705000444), I learn that there are at least 140 Type III codes of length 24, of which two have the property that all c... | https://mathoverflow.net/users/78 | How many length-24 Type III codes have no words of Hamming weight 3? | I think your question is answered in [A Complete Classification of Ternary Self-Dual Codes of Length 24](https://arxiv.org/abs/0804.0637) by Harada and Munemasa. Theorem 1 of the paper claims there are 166 inequivalent ternary self dual codes of weight 6 and 170 inequivalent ternary self dual codes of weight 3.
| 3 | https://mathoverflow.net/users/51668 | 294285 | 129,440 |
https://mathoverflow.net/questions/294289 | 3 | An element $f ∈ Homeo^+(\mathbb{R})$ is said to be of :
(1) type A, if it has a trivial germ at ∞ and it does not fix any point in an
interval of the form (−∞, r).
(2) type B, if it has a trivial germ at −∞ and it does not fix any point in an
interval of the form (s, ∞).
(3) type C, if it does not fix any point i... | https://mathoverflow.net/users/121404 | Action of homeomorphism on real line | This is not possible. I will write a proof for type A; the proofs for types B and C are similar. Without loss of generality, $n \in \mathbb Z\_{>0}$.
By the intermediate value theorem, we must either have $g(x) > x$ for all $x$, or $g(x) < x$ for all $x$. Replacing $g$ by $g^{-1}$ if necessary (and changing $f$ accor... | 5 | https://mathoverflow.net/users/82179 | 294301 | 129,445 |
https://mathoverflow.net/questions/294296 | 5 | I am trying to show that there can not be any nonvanishing Killing vector fields on a compact $G\_2$ manifold.
For the definition of a $G\_2$ manifold just see [the Wikipedia page](https://en.wikipedia.org/wiki/G2_manifold).
I know that since the manifold is Ricci-flat, any Killing vector field must be a parallel ... | https://mathoverflow.net/users/nan | Killing vector fields on a compact $G_2$ manifold | A parallel vector field implies a reduction of holonomy. Any form of $G\_2$ does not preserve any nonzero vector when acting in its nontrivial 7-dimensional representation. So the holonomy must be the subgroup of $G\_2$ preserving a nonzero vector, i.e. $SU(3)$. The reduction of holonomy group will, by deRham's splitti... | 6 | https://mathoverflow.net/users/13268 | 294307 | 129,447 |
https://mathoverflow.net/questions/294281 | 2 | Let $V$ be a finite dimensional vector space over a finite field $\mathbb{F}\_q$ (the most interesting case to me is $q=O(1)$). Let $A$ be an arbitrary subset of $V$ that spans $V$ over $\mathbb{F}\_q$. Let $\mu=|A|/|V|$ be the density of $A$. What can we say about the smallest $k$ such that every $v\in V$ can be writt... | https://mathoverflow.net/users/4162 | Smallest $k$ such that every vector is a linear combination of at most $k$ generators | A related problem was studied in the papers by Ben Klopsch and myself [How long does it take to generate a group?](https://arxiv.org/abs/0911.2908) and [Generating abelian groups by addition only](https://arxiv.org/abs/0911.2966). Describing precisely the connections with your question within the framework of an MO pos... | 3 | https://mathoverflow.net/users/9924 | 294309 | 129,448 |
https://mathoverflow.net/questions/294308 | 4 | Does anyone know any reference or proof for the following problem?
>
> Let $m$ and $n$ be positive integers, $m,n \geq 2$. Each positive integer is coloured in one of $m$ different colours. Is it possible to find $n$ positive integers $x\_1, x\_2, ..., x\_n$ such that all positive numbers of the form $$\sum\_{i=1}^... | https://mathoverflow.net/users/70464 | Colouring Positive Integers | Yes. This is a weak form of Hindman's theorem. See, for example, <https://en.wikipedia.org/wiki/IP_set> .
| 7 | https://mathoverflow.net/users/6794 | 294315 | 129,451 |
https://mathoverflow.net/questions/294303 | 7 | I am looking for a characterization of topological spaces $X,Y$ for which the function space $C\_k(X,Y)$ is Baire. Here $C\_k(X,Y)$ is the space of continuous functions from $X$ to $Y$, endowed with the compact-open topology.
It is well-known that for any complete metric space $Y$ and and compact space $X$ the space ... | https://mathoverflow.net/users/61536 | Characterizing topological spaces $X,Y$ whose function space $C_k(X,Y)$ is Baire | The Baireness of $C\_k(X)=C\_k(X,\mathbb{R})$ has been characterized for some subclasses of $k$-spaces. For example, [Gruenhage and Ma](https://www.sciencedirect.com/science/article/pii/S0166864196001630) proved that for a locally compact or first-countable space $X$, the following are equivalent:
1. $C\_k(X)$ is Bai... | 7 | https://mathoverflow.net/users/11647 | 294323 | 129,455 |
https://mathoverflow.net/questions/294322 | 4 | I am trying to improve my moderate knowledge of moduli spaces/stacks by examining the moduli stack of stable double covers of $\mathbb{P}^1$ with one marked point.
My idea is to ignore the stack structure at first, construct a moduli space and take the stack structure finally into account by displaying the stack as an ... | https://mathoverflow.net/users/121397 | moduli stack of double covers of $\mathbb{P}^1$ with one marked point | It's very important to define things carefully the problem of interest before constructing the moduli space / stack. In the most general setting you want to carefully define the moduli functor, but you should always at least have a precise idea of what the field-valued points are.
For instance, you don't specify how ... | 7 | https://mathoverflow.net/users/18060 | 294325 | 129,456 |
https://mathoverflow.net/questions/294194 | 1 | It is obvious that for a Banach space $E$, $E$ is reflexive iff $\ell^2(E)$ is reflexive. Let $\mathcal U$ be an ultrafilter. Is the reflexivity of $(E)\_\mathcal U$ equivalent to refelxivity of $(\ell^2(E))\_\mathcal U$?
| https://mathoverflow.net/users/84700 | About reflexivity of ultrapower | $\newcommand{\mc}{\mathcal}$The confusion seems to be over the following claim:
>
> **Claim:** Let $\mc U$ be a countably incomplete ultrafilter, and let $E$ be a Banach space. If $(E)\_{\mc U}$ is reflexive, then $E$ is super-reflexive.
>
>
>
To start with, we use the Eberlein-Smulian Theorem to observe that ... | 7 | https://mathoverflow.net/users/406 | 294328 | 129,457 |
https://mathoverflow.net/questions/294326 | 4 | Let $E$ be a complex Hilbert space. Let $E\overline{\otimes}E$ denotes the completion,
endowed with a reasonable uniform cross-norm of the algebraic tensor product $E\otimes E$.
>
> **Definition:** Let $A,B\in \mathcal{L}(E)$, the tensor sum of $A$ and $B$ is defined by
> $$A\oplus B:=(A\otimes I)+(I\otimes B)\in... | https://mathoverflow.net/users/113054 | Tensor sum of two operators | Let's start with a general lemma. Let $E,F$ be Banach spaces, $E\overline\otimes F$ be the completion of a reasonable crossnorm. Let $\newcommand{\mc}{\mathcal}T\in\mc L(E), S\in\mc L(F)$ be such that $T\otimes I = I\otimes S$. Then $T=S$ is a scalar multiple of the identity.
This follows, as pick $f\in F, f^\*\in F^... | 6 | https://mathoverflow.net/users/406 | 294330 | 129,458 |
https://mathoverflow.net/questions/294352 | 8 | There are infinitely many primes of the form $x^2+y^2+1$, as proved by Bredihin. Motohashi improved the result by showing that there were $\gg x/\log^2 x$ such primes up to $x$. But we expect $\Theta(x/\log^{3/2}x)$ primes up to $x$; has this result been proved? (Failing that, is anything else known about the density s... | https://mathoverflow.net/users/6043 | Primes of the form $x^2 + y^2 + 1$ | [Iwaniec](http://matwbn.icm.edu.pl/ksiazki/aa/aa21/aa21118.pdf) (Acta Arith. 1972) showed that the number of such primes has the order $x/(\log x)^{3/2}$. His result applies more generally to translates of binary quadratic forms. One can also show that short intervals contain the right density of such primes: for examp... | 16 | https://mathoverflow.net/users/38624 | 294355 | 129,466 |
https://mathoverflow.net/questions/294353 | 4 | Let $D$ a division quaternion algebra over a number field $F$, and consider $(V,q)$ be a $D$-hermitian space of $D$-dimension $2$, and introduce its group of isometries
\begin{align\*}
\mathrm{GU}(V, q) & = \left\{ g \in GL(V) \ : \forall x, y \in V, q(gx,gy) = q(x,y)
\right\}
\end{align\*}
These groups are known to... | https://mathoverflow.net/users/43737 | Are all these representations supercuspidal | First of all, you will only get a non-trivial inner form at finitely many places (where $D$ ramifies). Even then, the group is not compact mod center, so not all representations will be supercuspidal.
However, the representations have been classified using the theta correspondence:
>
> Wee Teck Gan and Welly Tant... | 9 | https://mathoverflow.net/users/6518 | 294370 | 129,470 |
https://mathoverflow.net/questions/294377 | 5 | A cardinal $\kappa$ is $\Sigma\_n$-correct iff $V\_\kappa \prec\_n V$. For n>1, how to force a $\Sigma\_{n+1}$-correct cardinal to be $\Sigma\_{n}$-correct but not $\Sigma\_{n+1}$-correct?
For $n=1$, we can force GCH below $\kappa$ and then violate GCH at $\kappa$.
If we assume some large cardinals, there are more... | https://mathoverflow.net/users/121464 | How to kill a $\Sigma_{n+1}$-correct cardinal softly ($n>1$)? | This is a very interesting question!
My student Erin Carmody (PhD 2015) had asked this question in
connection with her dissertation, [Force to change large cardinal
strength](https://arxiv.org/abs/1506.03432), which contains many
such killing-them-softly results. Her theorem 20 is the
$\Sigma\_2$-reflecting cardinal ... | 8 | https://mathoverflow.net/users/1946 | 294388 | 129,475 |
https://mathoverflow.net/questions/294364 | 15 | Consider a smooth cubic complex hypersurface $X\subset\mathbf{P}^{n+1}$ of dimension $n\geqslant 3$. The associated Fano variety of lines $F(X)$ is a smooth variety of dimension $2n-4$. Can one recover $X$ from $F(X)$? The answer is positive for $n=3$ (due to Clemens-Griffiths, Tjurin) and $n=4$ (due to Voisin, I think... | https://mathoverflow.net/users/104669 | Is a cubic hypersurface determined by its Fano variety of lines? | **Edit.** Regarding the Picard group, since $\rho^{-1}(F(X))$ is a $\mathbb{P}^1$-bundle over $F(X)$, the Picard group $\text{Pic}(\rho^{-1}(F(X)))$ equals $\text{Pic}(F(X))\oplus \mathbb{Z}\cdot [\pi^\*\mathcal{O}(1)]$. The fibers of the projection, $\pi:\rho^{-1}(F(X))\to X$, are complete intersections of dimension $... | 8 | https://mathoverflow.net/users/13265 | 294391 | 129,476 |
https://mathoverflow.net/questions/294351 | 1 | Given a finite permutation group $G$ (a subgroup of the symmetric group on a finite set) in terms of its generators, what is known about the decision problem of deciding if $G$ is $k$-transitive for a given $k \ge 1$?
What can we say about the complexity of this problem, and in what complexity class is it contained?
... | https://mathoverflow.net/users/37580 | Complexity of decision problem to decide if permutation group is $k$-transitive | There have been several articles on this forum discussing these topics, but here is a quick review.
A *base* $B$ for a subgroup $G \le {\rm Sym}(X)$ is a sequence $(\alpha\_1,\ldots,\alpha\_k)$ of points in $X$, such that the stabilizer $G\_B$ of the sequence (i.e. the intersections of the stabilizers of the $\alpha\... | 6 | https://mathoverflow.net/users/35840 | 294398 | 129,478 |
https://mathoverflow.net/questions/294417 | 6 | I'm migrating [this question from MSE](https://math.stackexchange.com/questions/2420887/what-is-this-property-exhibited-by-some-logical-systems) to MO, as in the span of five months, it received 6 upvotes but no answers. If my language needs to be fine-tuned in any way, constructive suggestions and guidance would be gr... | https://mathoverflow.net/users/85071 | What is this property exhibited by some logical systems? | This is equivalent to a combination of *structual completeness* with the *deduction theorem*. For a start, see [Wikipedia](https://en.wikipedia.org/wiki/Admissible_rule).
| 10 | https://mathoverflow.net/users/12705 | 294421 | 129,482 |
https://mathoverflow.net/questions/294412 | 1 | Consider the amalgamated sum $Q\_1 \rightarrow^{v\_1} Q\_1 \oplus\_P Q\_2 \leftarrow^{v\_2} Q\_2$ of $Q\_1 \leftarrow^{u\_1} P \rightarrow^{u\_2} Q\_2$ with $Q\_1,Q\_2,P$ being monoids.
Why does $v:= v\_i \circ u\_i$ factor through $(Q\_1 \oplus\_P Q\_2)^{\times}$ if either $Q\_1,Q\_2$ or $P$ is a group?
Thanks i... | https://mathoverflow.net/users/121397 | amalgamated sum of monoids | It looks like this just follows from the fact that a map of monoids $P\to Q$ where $P$ is a group factors through $Q^\times$ (the image of the inverse of some $p$ is an inverse of the image, so the image of every $p$ is invertible).
In your question, if $P$ is a group this immediately implies the conclusion, otherwis... | 2 | https://mathoverflow.net/users/5516 | 294424 | 129,483 |
https://mathoverflow.net/questions/294405 | 1 | I want to consider the following problem, which generalises the decision problem to decide if a given finite permutation group is a [Frobenius group](https://en.wikipedia.org/wiki/Frobenius_group):
>
> Given a finite permutation group in terms of its generators and a parameter $k$, decide if every element fixes at ... | https://mathoverflow.net/users/37580 | Complexity to decide for permutation group if every element fixed at most $k$ points | As I said in my comment, this problem is easily seen to be in P for fixed $k$, because we can compute all $(k+1)$-point stabilizers in polynomial time.
So let's assume that $k$ is part of the input.
I am not sure about NP, but the opposite question, does there exist a non-identity permutation fixing at least $k+1$ ... | 3 | https://mathoverflow.net/users/35840 | 294425 | 129,484 |
https://mathoverflow.net/questions/294385 | 15 | Let $Y$ be the Fermat quintic, i.e. $Y \subset \mathbb{C}P^4$ is defined by
$$
\sum\_{i=1}^5 z\_i^5 = 0
$$
In Section 5.3 of [this paper](https://arxiv.org/pdf/hep-th/0005103.pdf) by Volker Braun the author computes the K-groups of a quotient $X = Y/(\mathbb{Z}/5\mathbb{Z})$, where the generator acts by
$$
[z\_1: \dot... | https://mathoverflow.net/users/3995 | Ring structure on K-theory of a quotient of the Fermat quintic | The Atiyah-Hirzebruch spectral sequence does have a multiplicative structure, and I think this can be used to determine the multiplication on K-theory. From the paper of Braun, it follows that the spectral sequence actually degenerates integrally: the only potentially nontrivial differentials are the ones with target $... | 6 | https://mathoverflow.net/users/50846 | 294429 | 129,485 |
https://mathoverflow.net/questions/294431 | 8 | Let $(M,\omega)$ be a $4$-dimensional closed symplectic manifold. Assume there exists a Hamiltonian $S^1$-action on $M$, let $\mu:M \to \mathbb{R}^\*$ be its moment map and let $M\_{\text{red}}=\mu^{-1}(r)/S^1$ denote the symplectic reduction/symplectic quotient, where $r$ is a regular value of $\mu$ and where we assum... | https://mathoverflow.net/users/13326 | Symplectic reduction of 4-manifolds with circle actions | By a result of Hui Li <https://arxiv.org/abs/math/0605133>, the fundamental groups of $M\_{red}$ and $M$ are isomorphic, so one can deduce the genus of $M\_{red}$ from the fundamental group of $M$.
| 7 | https://mathoverflow.net/users/99732 | 294437 | 129,488 |
https://mathoverflow.net/questions/294430 | 5 | Let $k$ be an arbitrary field and $X$ be a proper scheme over $k$. Mastumura and Oort proved that the functor $S \longmapsto Aut\_{S-sch}(X \times S)$ is representable by a group scheme, locally of finite type, which I denote by $\underline{Aut}(X)$. I am interested in the forms of such a group scheme. More precisely, ... | https://mathoverflow.net/users/121495 | Forms of automorphism groups of algebraic varieties | That is not true. The basic issue is that for every coherent sheaf $\mathcal{F}$ on $X$ that is "intrinsic", and thus admits a linearization of the automorphism group, every cohomology group of $\mathcal{F}$ gives a linear representation of the automorphism group. So for every form of $X$, not only do you get a form of... | 5 | https://mathoverflow.net/users/13265 | 294441 | 129,490 |
https://mathoverflow.net/questions/294438 | 1 | I am searching for a commutative ring $R$ and a semisimple $R$-module $M$ such that the quotient ring $\frac {R}{soc(R)\cap ann\_R(M)}$ has a nonzero nilpotent element. Here, $soc(R)$ means the socle of $R$ and $ann\_R(M)$ stands for the annihilator of $M$ in $R$.
Thanks for any cooperation!
| https://mathoverflow.net/users/48889 | A nonreduced quotient ring | Take $R=K[x]/(x^3)$ and $M=S$ the unique simple $R$-module. We have $soc(R)=S$ and $ann(M)=J$ the jacobson radical of $R$. Then $soc(R) \cap ann(M)=soc(R)$ and $R/soc(R) \cap ann(M)=R/soc(R)=K[x]/(x^2)$ has nonzero nilpotent element x.
| 2 | https://mathoverflow.net/users/61949 | 294448 | 129,492 |
https://mathoverflow.net/questions/294440 | 3 | The following quotes comes from "Cantor's Attic"'s page on supercompacts:
>
> If κ is $|V\_{κ+η}|$-supercompact with $η<κ$ then it is preceeded by a stationary set of $η$-extendible cardinals. If $κ$ is $(η+2)$-extendible then it is $|V\_{κ+η}|$-supercompact. The least supercompact is not 1-extendible, in fact any ... | https://mathoverflow.net/users/78441 | Extendibility vs supercompactness | $2$-extendibility reflects $2^\kappa$-supercompactness. It suffices to show that any $2$-extendible $\kappa$ is $2^\kappa$-supercompact. Then if $\mathcal U$ is a normal fine $\kappa$-complete ultrafilter on $P\_\kappa(P(\kappa))$ and $j : V\_{\kappa+2}\to V\_{\kappa\_\* + 2}$ witnesses 2-extendibility, $\mathcal U\in ... | 5 | https://mathoverflow.net/users/102684 | 294449 | 129,493 |
https://mathoverflow.net/questions/294450 | 0 | Is the injectivity scheme present in the below axiomatic exposition of a first-order set theory provable in $\text{ZF}$?
**Extensionality:** $\forall A,B \ [\forall x \ (x \in A \leftrightarrow x \in B) \to A=B]$
**Empty:** $\exists O \ \forall x \ (x \not \in O)$
**Union:** $\forall A \ \exists U \ \forall x \ (... | https://mathoverflow.net/users/95347 | Is the following injectivity schema provable in ZF-foundation? | No, some instances of the injectivity scheme are refutable in ZF. For example, let $\phi(x,A)$ say "either $x$ is an ordinal $\geq2$ or $x=\{y\}$ for some $y\in A$. (To verify the second hypothesis of injectivity for this $\phi$, use that an ordinal $\geq2$ is never a singleton.)
| 5 | https://mathoverflow.net/users/6794 | 294452 | 129,494 |
https://mathoverflow.net/questions/294382 | 9 | Let $X$ be a smooth projective threefold with $h^{0,1}(X) = h^{0,2}(X)=0$ that has a smooth anticanonical section $D$.
Then $D$ is necessarily a $K3$ surface.
Consider a subgroup
$$Pic\_X(D) = i^\*(Pic(X))$$
of $Pic(D)$, where $i:D \hookrightarrow X$ is the inclusion map.
I tried several examples of $X$, including s... | https://mathoverflow.net/users/38823 | Primitivity of subgroups in the Picard groups of anticanonical $K3$ surfaces | I think the following may give an example $X$ where the image of the map in question is not primitive.
Let $S$ be the blow-up of $\mathbb{P}^2$ in 9 points that are the intersection of two cubics (so $|{-}K\_S|$ is an elliptic pencil). Let $Y \subset \mathbb{P}^1 \times S$ be a general element of $|2H -2K\_S|$, for $... | 5 | https://mathoverflow.net/users/13061 | 294459 | 129,495 |
https://mathoverflow.net/questions/294200 | 3 | What can I say about the eigenvalues and eigenvectors of the tridiagonal matrix $T$ given as
$T = \begin{pmatrix}
a\_1 & b\_1 \\
c\_1 & a\_2 & b\_2 \\
& c\_2 & \ddots & \ddots \\
& & \ddots & \ddots & b\_{n-1} \\
& & & c\_{n-1} & a\_n
\end{pmatrix}$.
If I set $a\_i = 0$, do you know any previous results?
I know som... | https://mathoverflow.net/users/121398 | Eigenvalues and eigenvectors of tridiagonal matrices | It seems that your question has already been answered here:
[Eigenvalues of Symmetric Tridiagonal Matrices](https://mathoverflow.net/questions/131527/eigenvalues-of-symmetric-tridiagonal-matrices)
No results for general tridiagonal matrices.
| 1 | https://mathoverflow.net/users/88057 | 294464 | 129,497 |
https://mathoverflow.net/questions/294462 | 1 | Does there exists a non-zero function $$f\in C\_0([0,1]):=\{f:[0,1]\to \mathbb R:\ f\text{ is continuous and } f(0)=f(1)=0\},$$ such that $(-\Delta)^{\frac\alpha 2}f\in C\_0([0,1]) $, where $(-\Delta)^{\frac\alpha 2}$ is the Dirichlet fractional Laplacian defined by
$$
(-\Delta)^{\frac\alpha 2}f(x):=\int\_0^1(f(x)-f(y)... | https://mathoverflow.net/users/120741 | Dirichlet fractional Laplacian and zero boundary conditions | Yes, there are many functions with this property. In fact, these functions are dense in $C\_0([0,1])$.
---
For every $g \in C\_0([0,1])$ there is a unique $f \in C\_0([0,1])$ such that $(-\Delta)^{\alpha/2} f(x) = g(x)$ for $x \in [0,1]$. This $f$ is given more-or-less explicitly: $$f(x) = \int\_0^1 G\_{(0,1)}(x,... | 2 | https://mathoverflow.net/users/108637 | 294467 | 129,499 |
https://mathoverflow.net/questions/294272 | 1 | Is there a Riemannian metric $g$ on $\mathbb{R}^2$ with inducing distance
$d$ which is not isometric to the standard metric but satisfy the property quoted bellow?
>
> For every two distinct points $p,q\in \mathbb{R}^2$, the locus of all points $z$ with $d(z,p)=d(z,q)$ is a geodesic.
>
>
>
As a higher dimensio... | https://mathoverflow.net/users/36688 | A possible characterization of Euclidean geometry via the curvature of the Median-submanifold | The answer to your question is in the affirmative, even in the more general setting of general (pseudo)-Riemannian manifolds.
In the Riemannian case this is a theorem due to Busemann. Consider a Riemannian manifold $(M,g)$. The set $\Sigma\_{p,q}$ consisting of all points equidistant from $p$ and $q$ on a Riemannian ... | 5 | https://mathoverflow.net/users/3948 | 294470 | 129,500 |
https://mathoverflow.net/questions/294404 | 3 | Let $X$ be a smooth curve over $\mathbb{C}$. I wish to compute the deformation of the following data $(E,F,x)$. $E$ and $F$ are locally free sheaves over $X$ and $x$ is a point on $X$. They satisfy:
1) $E\longrightarrow F$ is a injective map of sheaves over $X$.
2) $F/E$ has length 1 and is supported at $x$.
How... | https://mathoverflow.net/users/58689 | Deformation of "Hecke modification" | A morphism $F \to E$ can be thought of as a representation of the quiver
$$
A\_2 = \{ \bullet \to \bullet \}
$$
in the category $Coh(X)$. The category $Rep(A\_2,Coh(X))$ is abelian, so what you need is $Ext^1$ in this category. If $S\_1$ and $S\_2$ denote the simple modules of the first and the second vertices of the ... | 4 | https://mathoverflow.net/users/4428 | 294479 | 129,505 |
https://mathoverflow.net/questions/294484 | 4 | It is a well known result that with $\tau(n) := \sum\_{d|n} 1$ that
$$\sum\_{n\le X}\tau(n)\sim X\log X,\\ \sum\_{n\le X}\tau(n)^2\sim C\_1X\log^3X$$
for some $C\_1 > 0$. In general, it is also known that for $k\ge 0$,
$$\sum\_{n\le X} \tau(n)^k\ll X\log^{C(k)}X$$ (I think one can take $C(k) = 2^k - 1$). Are asymptoti... | https://mathoverflow.net/users/40983 | Asymptotics for sums of powers of divisor function | Yes, these are standard things in analytic number theory. We have
$$ \sum\_{n=1}^\infty\frac{\tau(n)^k}{n^s}=\zeta(s)^{2^k}F\_k(s),\qquad\Re(s)>1,$$
where $F\_k(s)$ is an explicit Dirichlet series that converges absolutely for $\Re(s)>1/2$. The exponent $2^k$ is dictated by $\tau(p)^k=2^k$ for primes $p$. It follows, b... | 9 | https://mathoverflow.net/users/11919 | 294486 | 129,508 |
https://mathoverflow.net/questions/294477 | 2 |
>
> Definition: Let $\{a\_1,\dots,a\_n\}$ be any $n$ distinct points on the
> Riemann sphere $\mathbb{C}\cup\{\infty\}$ with coordinate $s$, and let
> $R$ be the ring of rational functions with poles allowed only in
> $\{a\_1,\dots,a\_n\}$. This ring is called **$n$-point ring**.
>
>
>
It is well known that
... | https://mathoverflow.net/users/119743 | Is $\mathbb{C}[x^{\pm 1},y]/\langle y^3-(x^2+ax+b) \rangle$ a $n$-point ring? | No (assuming that $a^2 \neq 4b$, so the quadratic is not a square). In fact, the fields $\mathrm{Frac}(R)$ and $\mathbb{C}(t)$ are not isomorphic. From a sophisticated perspective, the point is that the former has genus one and the latter has genus $0$. In general, this is a hard notion to define in an elementary way; ... | 4 | https://mathoverflow.net/users/297 | 294492 | 129,510 |
https://mathoverflow.net/questions/294371 | -1 | Take $A,B,C,D$ pairwise coprime with $$n<A,B,C,D<2n$$ $$
n/4<|A−B|,|C−D|,|A−C|,|A−D|,|B−C|,|B−D|$$ and consider the space of solutions to $ACa+ADb+BCc+BDd=0$ spanned by $3\times 4$ matrix
$$N=\begin{bmatrix}
-D&C&0&0\\
-B&0&A&0\\
0&0&-D&C
\end{bmatrix}.$$
Denote the $\Bbb Q$-linear space spanned by rows of $N$ by $T\... | https://mathoverflow.net/users/10035 | On distribution of size of integer points in a subspace associated to a linear diophantine equation | We will assume that $n<A,B,C,D<2n$ and $\gcd(A,B)=\gcd(C,D)=1$.
For $x,y,z\in\mathbb{Q}$, we write $v(x,y,z)$ for the vector
$$\pmatrix{\frac{x}{C}&\frac{y}{C}&\frac{z}{C}}N\in\mathbb{Q}^4.
$$
**Lemma 1.** For $x,y,z\in\mathbb{Q}$, it holds that
$v(x,y,z)\in\mathbb{Z}^4$
iff $x,y,z\in\mathbb{Z}$ and the following con... | 1 | https://mathoverflow.net/users/120173 | 294493 | 129,511 |
https://mathoverflow.net/questions/294513 | 3 | Let $H$ be a connected, unipotent linear algebraic group defined over a local field $k$. Let $H \times\_k X \rightarrow X$ be an action of $H$ on an irreducible, affine $k$-variety $X$ which is defined over $k$. Then the orbits of the group action $H(\overline{k}) \times X(\overline{k}) \rightarrow X(\overline{k})$ are... | https://mathoverflow.net/users/38145 | Orbits of unipotent groups over local fields are closed? | Edit: This answer is valid only in characteristic 0!
Yes. For a unipotent group $H$ the orbits of $H(k)$ in $X(k)$ coinside with the intersections of the form $H(\bar{k}).x \cap X(k)$, hence even Zarizki closed. In general, the orbits of $H(k)$ on sets of the form $H(\bar{k}).x \cap X(k)$ are classified by the first ... | 5 | https://mathoverflow.net/users/115052 | 294514 | 129,520 |
https://mathoverflow.net/questions/294501 | 10 | Let $G$ be a finite 2-generated group. Let $p$ be a prime dividing the order of $G$. Must there exist a generating pair $(g,h)$ of $G$ such that $|g|$ is divisible by $p$?
If not, is this true at least for finite simple groups?
| https://mathoverflow.net/users/15242 | For every prime divisor $p$ of a finite 2-generator group $G$, is there a generating pair containing an element of order divisible by $p$? | The answer is yes for finite simple groups $G$. Every element of $G$ is an element of a generating pair. This is proved in
Guralnick, Robert, Kantor, William, *[Probalistic generation of finite simple groups](http://dx.doi.org/10.1006/jabr.2000.8357)*, J. Algebra 234 (2000), p. 743–792. ([MR1800754](http://www.ams.or... | 12 | https://mathoverflow.net/users/35840 | 294518 | 129,522 |
https://mathoverflow.net/questions/294523 | -1 | Is there an operator that transform a Multiset into a set by choosing all the distinct elements? Is there a better/more accurate way to write it than this:
We denote the set $S$ by choosing the distinct elements in the Multiset $A$. I.e., $S=distinct(A)$
For example assume $A=\{2,3,3,3,5\}$ then $S=distinct(A)=\{2,... | https://mathoverflow.net/users/121536 | Multiset to set operator | [Wikipedia](https://en.wikipedia.org/wiki/Multiset) says that this operator is the *support* operator, $S=\operatorname{supp} A$.
| 4 | https://mathoverflow.net/users/13268 | 294525 | 129,524 |
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