parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
|---|---|---|---|---|---|---|---|---|---|
https://mathoverflow.net/questions/317077 | 2 | In the paper *p-adic L-functions and p-adic periods of modular forms*, Greenberg/Stevens assert that if $\sigma\_l:=\begin{pmatrix}l&0\\0&1\end{pmatrix}$ is the usual Hecke operator at $l$ double coset representative, and $\Gamma$ is the congruence group $\Gamma\_1(N)$, then $g\sigma\_lg^{-1}$ continues to lie in the s... | https://mathoverflow.net/users/120548 | When does the double coset representative for a congruence subgroup contain a $\text{SL}_2(\mathbb{Z})$-conjugacy class? | *Why the condition $\ell = 1 \bmod N$ is necessary.* Suppose $g \sigma\_\ell g^{-1}$ lies in the double coset $\Gamma \sigma\_\ell \Gamma$. Then $g \sigma\_\ell g^{-1} = r \sigma\_\ell s$ for some $r, s \in \Gamma\_1(N)$; but $r \sigma\_\ell s$ is the product of three matrices which are upper-triangular mod $N$, so it ... | 5 | https://mathoverflow.net/users/2481 | 317251 | 137,583 |
https://mathoverflow.net/questions/317231 | 2 | Let $X$ be a smooth scheme over $k[t]/(t^2),$ where $k$ is a field of characteristic 0 (the case when $X$ is a projective curve is already interesting). Let $X\_{0} \to X$ denote $X$ with the reduced induced structure, i.e. just the fiber product of $X$ with $k$ over $k[t]/(t^2).$ Let $\beta \in H^{0}(X\_{0},\Omega^{1}... | https://mathoverflow.net/users/132265 | Maps from a scheme over the dual numbers to constant schemes | Welcome new contributor. Your post still has typos, so it is not completely clear what you mean. However, my best guess of your meaning has a **negative** answer. The simplest examples have $X\_0$ a smooth projective curve of genus $g=2$ or a non-hyperelliptic curve of genus $g>2$. This follows from the proof (rather t... | 2 | https://mathoverflow.net/users/13265 | 317255 | 137,584 |
https://mathoverflow.net/questions/317240 | 8 | Suppose that I have a first-order elliptic differential operator $A: \mathrm{dom}(A) \subset L^2(E) \to L^2(E)$, where $(E,h^E) \to M$ is a hermitian vector bundle and $M$ is a compact manifold.
I know also that $A$ is $\omega$-bisectorial with $\omega < \frac{\pi}{2}$. That is to say, the spectrum $\sigma(A)$ is in ... | https://mathoverflow.net/users/132269 | Spectrum of a first-order elliptic differential operator | Consider the 0th order operator $F:=A(1+A^\*A)^{-1/2}$. The spectrum of the operator $F$ is contained in the unit disc. The symbol mapping is a $\*$-homomorphism, and therefore the spectrum of $F$ is contained in the spectrum of its symbol, call it $a$. In fact, the spectrum of $a$ coincides with the essential spectrum... | 3 | https://mathoverflow.net/users/84608 | 317266 | 137,589 |
https://mathoverflow.net/questions/317256 | 59 | What would you do/have you done in such a situation?
1. Hand out the improvement for free in your report
2. Wait until the result is published and then submit elsewhere
3. Inform the editor about the situation and ask for advice
The paper is not posted publicly so contacting the authors directly informing them and... | https://mathoverflow.net/users/130882 | What to do if you notice a substantial improvement to a result in a paper whilst refereeing it? | Option (1) is definitely the professional course of action in this case. As pointed out in the remarks, it is likely to lead to an offer of co-authorship from the original author, but that is purely within the author's discretion. If you feel that your improvement is really substantial and you are worried about credit ... | 48 | https://mathoverflow.net/users/51164 | 317270 | 137,590 |
https://mathoverflow.net/questions/317227 | 2 | Let $(V,\Phi)$ be a root system with dual root system $(V^{\ast},\Phi^{\vee})$. Let $\Delta = \{\alpha\_1, ... , \alpha\_n\}$ be a set of simple roots for $V$, and let $\Delta^{\vee} = \{\alpha\_1^{\vee}, ... , \alpha\_n^{\vee}\}$ be the coroots corresponding to $\Delta$.
We have the fundamental weights $\hat{\Delta... | https://mathoverflow.net/users/38145 | Definition of the weight lattice for a nonreduced root system | Bourbaki has the most detailed treatment, but they tend not to deal with weight lattices (or co-weight lattices) so explicitly outside their account of some of the representation theory. Thus you can make any definition you like.
The basic question here is what your *motivation* is. Weights arise in representation th... | 1 | https://mathoverflow.net/users/4231 | 317272 | 137,591 |
https://mathoverflow.net/questions/317280 | 7 | Let $\mathbf{C}$ be a category (that does not necessary have a coproduct for every collection of objects). Suppose that we have two families of objects $(A\_i)\_{i\in I}$ and $(B\_i)\_{i\in I}$ in $\mathbf{C}$ indexed by the same index set $I$. Assume further that there exist coproducts $A$ and $B$ of $(A\_i)\_{i\in I}... | https://mathoverflow.net/users/33026 | Is a categorical coproduct of epimorphisms (monomorphisms) always an epimorphism (a monomorphism)? | Question 1: Yes. The $I$-coproduct-functor $\bigsqcup\_I\colon\prod\_{i\in I}\mathbf{C}\to\mathbf{C}$ is left-adjoint (its right adjoint is the diagonal functor $\Delta\_{\mathbf{C}}^I\colon \mathbf{C}\to\prod\_{i\in I}\mathbf{C}$), hence always preserves epimorphisms.
Question 2: No, in general (even if $\mathbf{C}$... | 12 | https://mathoverflow.net/users/35349 | 317283 | 137,595 |
https://mathoverflow.net/questions/317172 | 17 | This question was communicated to me by Evgeniy Romanov.
Consider a connected polyomino $P$ that can be completely tiled in two different ways: with disjoint $2 \times 2$ square tetraminoes, and with disjoint S-shaped tetraminoes (that we allow to reflect and rotate arbitrarily). Is it true that $P$ can not be simply... | https://mathoverflow.net/users/106512 | Holes in double-tileable polynominoes | Let's color all cells like a chessboard. Then every tetramino has exactly 2 white and 2 black cells. Let's connect cells of the same color covered by the same tetramino with an edge, these edges will form 2 perfect matchings. Let's take their symmetric difference. Obviously it's not empty. Let's take a cycle such that ... | 14 | https://mathoverflow.net/users/132293 | 317284 | 137,596 |
https://mathoverflow.net/questions/317307 | 1 | This is the question that I should have asked before asking this [older question](https://mathoverflow.net/questions/317191/spatial-dimension-of-a-finite-graph).
If $(X,d)$ is a [metric space](https://en.wikipedia.org/wiki/Metric_space), we associate with it a simple, undirected graph, called its *proximity graph* $G... | https://mathoverflow.net/users/8628 | Is every finite graph isomorphic to the proximity graph of some $S\subseteq \mathbb{R}^n$? | Yes. If $n=|V|$, then for small $\varepsilon$ any metric spaces on $n$ points with distances belonging to $\{1-\varepsilon, 1+\varepsilon\} $ is embeddable to $\mathbb{R}^{n-1} $.
| 3 | https://mathoverflow.net/users/4312 | 317310 | 137,605 |
https://mathoverflow.net/questions/317331 | 4 | In [this paper](https://arxiv.org/abs/1202.2676), notation $Td\_p$ is used without explicit definition (it is stated that it is a certain combination of Chern numbers). It is claimed that HRR theorem implies
$$
Td\_p(M)=\sum\_{q}(-1)^q h^{p, q}(M)
$$
for any closed complex manifold $M$.
Am I correct assuming that $T... | https://mathoverflow.net/users/132313 | $Td_p$ notation of Kotschick | The HRR-Theorem asserts $$\int Td(M)ch(E)=\chi(M,E)=\sum\_q(-1)^q\dim H^q(M,E)$$
for every vector bundle $E$. With $E=\Omega^p$ the sheaf of holomorphic $p$-forms you get
$$\int Td(M)ch(\Omega^p)=\sum\_q(-1)^q\dim H^q(M,\Omega^p)=\sum\_q(-1)^q h^{p,q}.$$
So your guess is right.
| 4 | https://mathoverflow.net/users/39082 | 317333 | 137,609 |
https://mathoverflow.net/questions/317337 | 3 | Let $X$ be an irreducible smooth projective variety over $\mathbb{C}$.
Let $G$ be an affine algebraic group over $\mathbb{C}$.
Let $p : E\_G \longrightarrow X$ be a holomorphic principal $G$-bundle on $X$. Let $ad(E\_G) = E\_G \times^G \mathfrak{g}$ be the adjoint vector bundle of $E\_G$ associated to the adjoint rep... | https://mathoverflow.net/users/124771 | Is there any Lie algebra structure on the sheaf of sections of adjoint bundle | A principal $G$-bundle gives a **monoidal** functor from the category of representations of $G$ to the category of vector bundles. In particular, it takes the morphism
$$
[-,-] \colon \mathfrak{g} \otimes \mathfrak{g} \to \mathfrak{g}
$$
of $G$-representations (for the adjoint action) to a morphism of vector bundles
$$... | 6 | https://mathoverflow.net/users/4428 | 317340 | 137,611 |
https://mathoverflow.net/questions/317335 | 1 | Assume that $M$ is a closed connected Ricci-flat Kaehler manifold $M$ of complex dimension $n\geq 3$ with $h^{2,0}(M)=0$. Is is possible that
* $h^{n, 0}(M)\neq 1$
* $h^{p, 0}(M)\neq 0$ for some $0< p< n$?
| https://mathoverflow.net/users/132313 | Hodge numbers of compact Ricci-flat Kaehler manifold | After replacing by a finite covering space, you must have $h^{n0}=1$, because you have a parallel volume form, so nowhere zero, and any other holomorphic volume form is a holomorphic multiple, so a constant multiple (as holomorphic functions are constant).
Kobayashi, *First Chern class and holomorphic tensor fields*,... | 1 | https://mathoverflow.net/users/13268 | 317345 | 137,614 |
https://mathoverflow.net/questions/317344 | 6 | Let $A$ (resp. $B$) be a *unital* $C^\ast$-algebra, $\mathcal{Q}(A)$ (resp. $\mathcal{Q}(B)$) the compact convex subset of $A^\ast$ equipped with the $\sigma(A^\ast, A)$ (resp. $\sigma(B^\ast, B)$) topology. Suppose $\mathcal{Q}(A)$ and $\mathcal{Q}(B)$ are isomorphic in the sense that there exists a bijective affine h... | https://mathoverflow.net/users/128540 | Examples of non-isomorphic $C^\ast$ algebras with isomorphic quasi-state spaces | For any C$^\*$-algebra $A$, we can define its opposite algebra $A^{\mathrm{op}}$, which is the algebra where $ab$ is defined to be $ba$, as calculated in $A$. Let's restrict to unital algebras for simplicity. Then the identity mapping $A \rightarrow A^{\mathrm{op}}$ is linear, positive and unital, and is its own invers... | 6 | https://mathoverflow.net/users/61785 | 317346 | 137,615 |
https://mathoverflow.net/questions/317241 | 10 | [Background: I asked a [vague question](https://mathoverflow.net/q/317152/1384) the other day, but as a result of the answers, particularly Andrej Bauer's, I now have a precise question]
**Summary of question**: the inclusions are a particularly "good" class of morphisms in the category of sets. I've written down a b... | https://mathoverflow.net/users/1384 | Are inclusions "canonical" injections? | The consensus seems to be that this is an answer to the question as stated (though I didn't originally realize that it was), so I'll go ahead and post it as one.
There are other ways to choose such a class of "good maps". For instance, you can transfer any class of good maps (such as the "standard" one consisting of ... | 7 | https://mathoverflow.net/users/49 | 317354 | 137,618 |
https://mathoverflow.net/questions/317353 | 2 | (Related to [this Math.SE question](https://math.stackexchange.com/q/3012078/8157).)
For $p>1$, let $u$ be a solution to $$\tag{1}\frac{d^2 u}{dt^2} + u = |u|^{p-1}u$$ that blows up at $T>0$, that is $$\lim\_{t\nearrow T}u(t)=+\infty.$$
*Remark*. If $u$ solves (1), then the following quantity is independent on $t$... | https://mathoverflow.net/users/13042 | The blow-up rate of a nonlinear oscillator | Your heuristics is correct. Indeed, suppose that $u(t)\to\infty$ as $t\uparrow T$. Then for some real $h>0$ and all $t\in[T-h,T)$ we have $u(t)>0$ and, by the "conservation of energy" stated in your post,
\begin{equation}
u'(t)^2=2E-u(t)^2+\frac{2u(t)^{p+1}}{p+1}\sim \frac{2u(t)^{p+1}}{p+1},
\end{equation}
because... | 4 | https://mathoverflow.net/users/36721 | 317356 | 137,619 |
https://mathoverflow.net/questions/317362 | 3 | A Riemannian manifold $(X,g)$ is *hyperbolic* if the sectional curvatures are constant and negative. A theorem of Mostow says that these manifolds are determined by their fundamental group.
>
> **Theorem** (Real Mostow Rigidity) If $X$ and $Y$ are closed, hyperbolic $n$-manifolds with $n \ge 3$ and $\pi\_1 X \simeq... | https://mathoverflow.net/users/123015 | Mostow rigidity for complex hyperbolic manifolds | The general statement of Mostow-Prasad rigidity cited from <http://repository.ias.ac.in/36364/1/36364.pdf> is as follows.
Let $G$ (resp. $G^\prime$) be a semi-simple analytic group and $\Gamma$ (resp. $\Gamma^\prime$) an irreducible lattice in $G$ (resp. $G^\prime$). Assume that $G, G^\prime$ have trivial centers and... | 4 | https://mathoverflow.net/users/39082 | 317366 | 137,622 |
https://mathoverflow.net/questions/316884 | 9 | I am looking for the fundamental solution of the following PDE
$$\partial\_i (a^{ij}\partial\_j u)=f$$
where $a^{ij}(x)$ is a non-symmetric matrix with possibly non-constant coefficients.
I could find a paper [Clements - A fundamental solution for linear second-order elliptic systems with variable coefficients](h... | https://mathoverflow.net/users/32817 | Fundamental solution of an elliptic PDE in divergence form with non-symmetric matrix | This question requires an articulated answer, since the topic dealt is complex and ramified. A fundamental solution for a not necessarily divergence form $2$nd order elliptic system with $C^{2,h}$ coefficients was first constructed by [Georges Giraud](https://en.wikipedia.org/wiki/Georges_Giraud) in 1932 ([5]) by using... | 15 | https://mathoverflow.net/users/113756 | 317373 | 137,626 |
https://mathoverflow.net/questions/316939 | 4 | I have a question about the convexity of an Wasserstein ambiguity set.
Let $W\_1(\mu, \nu)$ be the Wasserstein distance of order 1 between $\mu$ and $\nu$, defined as
$$W\_1(\mu, \nu) := \min\limits\_{\gamma \in \Gamma(\mu, \nu)} \bigg \{ \int\_{\Xi \times \Xi} d(\xi, \zeta) \gamma(d\xi, d\zeta) \bigg \}, $$ where $... | https://mathoverflow.net/users/132071 | Is an ambiguity set with Wasserstein distance of order 1 is convex? | **Edit:** (This question may be better suited for math.stackexchange.)
This is true. It suffices to show the map $\mu\rightarrow W\_{1}(\mu,\nu)$ is convex. Let $\mu\_{i}\in\mathcal{P}(\Xi)$ and $\psi^{\*}\_i\in\Gamma(\mu\_{i},\nu)$ be optimal transport plans between $\mu\_{i}$ and $\nu$ for each $i=1,2$. Then $\lamb... | 4 | https://mathoverflow.net/users/118731 | 317374 | 137,627 |
https://mathoverflow.net/questions/317376 | 14 | Let $G$ be a finite solvable group of order $n$, and let $g\_1 ... g\_n$ be an enumeration of its elements. Let $a\_1 ... a\_n$ be a sequence of integers, such that $\sum a\_i$ is relatively prime to $n$.
Consider $\mathbb{C}[G]$, the group ring of $G$ with complex coefficients. Does the element $\sum a\_i g\_i$ nec... | https://mathoverflow.net/users/55916 | Units in group rings. | This is false for the cyclic group of order $6$. Let $g$ be a generator. Then $g^2-g+1$ acts by $0$ on the representations where $g$ acts by a primitive $6$-th root of $1$, and hence is not a unit in the group ring, but $1-1+1=1$ is relatively prime to $6$.
Generalizing this example, the statement is false for the cy... | 26 | https://mathoverflow.net/users/297 | 317379 | 137,628 |
https://mathoverflow.net/questions/317391 | 0 | Let $G=(V,E)$ be a simple, undirected graph. Is there a [partial ordering](https://en.wikipedia.org/wiki/Partially_ordered_set) $\leq\subseteq (V\times V)$ with the following property? $$\{v,w\} \in E \text{ if and only if } v||y$$
(We write $v||w$ in the poset $(V,\leq)$ if $v\not \leq w$ and $w\not\leq v$?)
| https://mathoverflow.net/users/8628 | Is every graph an incomparability graph? | Any [incomparability graph](https://www.sciencedirect.com/science/article/pii/S0001870812001077) is perfect (shown by Dilworth in 1950), so any non-perfect graph will be a counterexample. For an explicit counterexample, choose the cycle on $5$ vertices.
| 10 | https://mathoverflow.net/users/71028 | 317394 | 137,631 |
https://mathoverflow.net/questions/317377 | 5 | In [Three Dimensional Gravity Revisted](https://arxiv.org/abs/0706.3359), Witten studied the Abelian Chern-Simons theory in three dimensions.
Let $W$ be a three dimensional manifold. Let $\mathcal{L}$ be a non-trivial line-bundle over $W$. On page 11, Witten claims that one can pick up a four dimensional manifold $M$... | https://mathoverflow.net/users/120604 | The existence of the extension of a non-trivial line bundle | This is a bordism problem, and as such can be answered using algebraic topology. I'll answer in the unoriented setting, then indicate how to modify things if $M$ and $W$ are required to be oriented.
Complex line bundles $\mathcal{L}$ over $W$ are classified by maps $f:W\to BU(1)\simeq \mathbb{C}P^{\infty}$. We want t... | 8 | https://mathoverflow.net/users/8103 | 317396 | 137,633 |
https://mathoverflow.net/questions/317392 | 1 | Consider the optimization problem
\begin{align}
\max\_{x\in\mathbb{R}^n}~c^Tx~, \text{ s.t. } Ax=b,~x\_i\in\{0,1\}~\forall i
\end{align}where $c,b\in\mathbb{R}\_{+}^n$ and $A\in\mathbb{R}\_{+}^{n\times n}$. Thus $x$ is a boolean vector (entries can be $0$ or $1$). It is not hard to prove that this is a NP-hard problem.... | https://mathoverflow.net/users/27249 | How good is the LP relaxation? | Yes. In chapter two, section three and four of Nemhauser and Wolsey's *Integer and Combinatorial Optimization* there are conditions detailed for various relaxations of the first problem to have solutions which are within $\epsilon$ of the original problem's solution.
For the problem you address specifically, Theorem... | 1 | https://mathoverflow.net/users/118731 | 317397 | 137,634 |
https://mathoverflow.net/questions/317400 | 3 | A classical result states that the quotient $SO(4)/SO(3)$ is homotopy equivalent to $S^3$. In fact, this can be stated in more general terms since $SO(n+1)/SO(n)$ has the homotopy type of $S^n$. What I don't know is what is the homotopy type of $SO(n+1)/SO(n-1)$ or even if it has a general formultation. The concrete ca... | https://mathoverflow.net/users/104774 | Homotopy type of $SO(4)/SO(2)$ | This is the [Stiefel manifold](https://en.wikipedia.org/wiki/Stiefel_manifold) $V\_2(\mathbb{R}^4)$. It fits into a fibration $S^2\to V\_2(\mathbb{R}^4)\to S^3$ and so has trivial $\pi\_1$. By playing around with the long exact sequence of this fibration I was able to show that $\pi\_2(V\_2(\mathbb{R}^4))\cong \mathbb{... | 12 | https://mathoverflow.net/users/8103 | 317405 | 137,637 |
https://mathoverflow.net/questions/317316 | 4 | Let $A= \bigoplus\limits\_{n=0}^{\infty}{A\_n}$ be an $\mathbb{N}$-graded algebra with semisimple $A\_0$.
>
> Question: Do we have that the global dimension of $A$ is equal to $\sup \{i \geq 0 | Ext\_A^i(A\_0,A\_0) \neq 0 \}$?
>
>
>
Maybe one should ask this question under some mild further restrictions such ... | https://mathoverflow.net/users/61949 | Global dimension of a graded algebra | Your question is answered in the affirmative in the paper
*Eilenberg, Samuel*, [**Homological dimension and syzygies**](http://dx.doi.org/10.2307/1969977), Ann. Math. (2) 64, 328-336 (1956); Errata 65, 593 (1957). [ZBL0073.26003](https://zbmath.org/?q=an:0073.26003).
According to Proposition 15, the category of gra... | 6 | https://mathoverflow.net/users/18756 | 317407 | 137,639 |
https://mathoverflow.net/questions/317398 | 6 | Garner constructed (in [[1]](https://arxiv.org/abs/0810.4450) ) for the category of strict $n$-categories a comonad $Q$ as the left part of a cofibrantly generated algebraic weak factorization system such that $$n\text{-}\operatorname{Cat}(QX,Y)=\operatorname{Pseudo}(X,Y).$$
For various reasons, it is often more con... | https://mathoverflow.net/users/1353 | Comonad for normalized pseudofunctors for strict higher categories | $\require{AMScd}$Notation: for each $n \geq 0$, let $\mathbf{2}\_n$ denote the free-living $n$-cell, and let $\partial\mathbf{2}\_n$ denote its boundary. Let **$n$-Cat** denote the category of (strict) $n$-categories and (strict) $n$-functors.
Recall (see e.g. Section 7.2 of Garner's `Understanding the small object a... | 5 | https://mathoverflow.net/users/57405 | 317411 | 137,640 |
https://mathoverflow.net/questions/317121 | 17 | Let $M$ be a real analytic variety
(if someone is concerned about distinction between
"real analytic spaces" and "real analytic varieties"
in real analytic geometry, let's assume that $M$
is both "variety" and "space"). I was sure it is
well-known that higher cohomology of any
real analytic coherent sheaf over $M$ va... | https://mathoverflow.net/users/3377 | Cohomology of real analytic coherent sheaves | In a smooth case, the reference is Proposition 2.3 in Atiyah and Hirzebruch's [Analytic cycles on complex manifolds](https://www.sciencedirect.com/science/article/pii/0040938362900940).
For a non-smooth case, I don't know the general reference, but Theoreme 3 in Henri Cartan's paper [Variétés analytiques réelles et v... | 9 | https://mathoverflow.net/users/43309 | 317420 | 137,642 |
https://mathoverflow.net/questions/317390 | 11 | There may be some technical issues with the question, but hopefully what I mean is clear...
Let $k$ be a number field (or maybe any finitely generated field over $\mathbb{Q}$ of characteristic 0)
Let $k(\!(t)\!)$ be the field of Laurent series in $t$ with coefficients in $k$, and let $\Omega$ denote an algebraic cl... | https://mathoverflow.net/users/88840 | Does every section of the map Gal$(\overline{k(\!(t)\!)}/k(\!(t)\!))\rightarrow$ Gal$(\overline{k}/k)$ stabilize a compatible system of roots of $t$? | $\newcommand{\Gal}{\mathrm{Gal}}\newcommand{\Z}{\mathbb{Z}}$Fix a compatible system $(t\_n)$ of roots of $t$. It provides us with a section of $\rho$ thus giving an isomorphism between $\Gal(\overline{k((t))}/k((t)))$ and the semi-direct product $\Gal(\overline{k}/k)\ltimes \hat{\Z}(1)$ where $\hat{\Z}(1)$ denotes $\li... | 6 | https://mathoverflow.net/users/39304 | 317425 | 137,643 |
https://mathoverflow.net/questions/317382 | 5 | This question should be fairly elementary. I’d just like to check I’m not missing anything.
Let $\{M\_n\}\_{n\ge 0}$ be an inverse system of smooth manifolds with transition maps $f\_{t,s} : M\_t\to M\_s$, $t\ge s$, that are local diffeomorphisms.
Let $M$ be the topological inverse limit.
>
> For every $x\in M$... | https://mathoverflow.net/users/nan | On limits of manifolds | In general, this will be false. Examples are found among [solenoidal manifolds](http://www.journalofsing.org/volume9/sullivan.pdf), defined by Sullivan. For example, 1-dimensional [solenoids](https://en.wikipedia.org/wiki/Solenoid_(mathematics)).
Many of these are obtained by taking the inverse limit of finite-sheet... | 6 | https://mathoverflow.net/users/1345 | 317426 | 137,644 |
https://mathoverflow.net/questions/317393 | 8 | Let $R$ be a one-dimensional, reduced and noetherian $k$-algebra (we may also assume that $R$ is a finite $k[x]$-algebra). Let $M$ be a finitely generated, torsion-free module over $R$, i.e. no regular element of $R$ annihilates a non-zero element of $M$. Let $a \in R$ be regular.
>
> Is there an invariant $\mu(M)$... | https://mathoverflow.net/users/98129 | Is $\dim_k M/xM$ a multiple of $\dim_k R/xR$ for $M$ finitely generated, torsion-free $R$-module? | Here is a discussion only assuming that $R$ is Cohen-Macaulay. Let $G$ denote the Grothendieck group of all finitely generated $R$-modules.
Fix a regular element $a$ and consider the function $$f(M) = \dim\_k Tor\_0(M,R/aR)- \dim\_k Tor\_1(M,R/aR)= \dim\_k M/aM - \dim\_k (0:\_Ma)$$
For torsion-free module (or just... | 8 | https://mathoverflow.net/users/2083 | 317428 | 137,645 |
https://mathoverflow.net/questions/317434 | 5 | I don't understand the smoothness condition in the following theorem,
Let $f: X\longrightarrow Y$ be a projective morphism of $\underline{smooth}$ projective varieties such that $Rf\_\*\mathcal{O}\_X=\mathcal{O}\_Y$. Then the functor
\begin{equation}
Lf^\* : D^b(Y)\longrightarrow D^b(X)
\end{equation}
is fully fait... | https://mathoverflow.net/users/121526 | $Lf^*$ is fully faithful | Smoothness is not necessary. What is important is that $f$ has finite $Tor$-dimension (otherwise $Lf^\*$ does not preserve boundedness); a sufficient (but not necessary) condition for this is smoothness of $Y$.
On the other hand, it is impossible to have $X$ smooth and $Y$ singular. Indeed, in this case one can find ... | 5 | https://mathoverflow.net/users/4428 | 317437 | 137,646 |
https://mathoverflow.net/questions/317410 | 10 | Following up on [Mean minimum distance for N random points on a one-dimensional line](https://mathoverflow.net/questions/1294/mean-minimum-distance-for-n-random-points-on-a-one-dimensional-line) and [Mean minimum distance for N random points on a unit square (plane)](https://mathoverflow.net/questions/124579/mean-minim... | https://mathoverflow.net/users/132350 | Mean maximum distance for N random points on a unit square | Here is a first approximation for large $n$.
The formulas are easier if we use the diamond whose whose corners are $(\pm 1,0)$, $(0,\pm 1)$. Then the pdf of the $x$-coordinate is just $1-|x|$.
The expected maximum distance is at least $2\, E[\max x]$. If $u>0$,
$$P(\max x < u) = P(\text{all }x < u)
= (1-(1-u)^2/2)^... | 8 | https://mathoverflow.net/users/nan | 317438 | 137,647 |
https://mathoverflow.net/questions/317383 | 8 | Let $E$ be a (Hausdorff) locally convex vector space (from now on just "lcs" for short). We say that $E$ is *convenient* (also called *locally complete*, *Mackey-complete* or *$c^\infty$-complete*) if, given any disk (i.e. an absolutely convex and bounded subset) $B\subset E$ which is *closed*, the vector subspace $E\_... | https://mathoverflow.net/users/11211 | Can smoothness of curves into a convenient locally convex vector space be tested with just a dense subspace of the dual? | Here is a counter example. Let $E=\ell^2$. Consider the curve $\gamma:\mathbb R \to E$ given by
$$\gamma(t)= \Big(\frac{\sin(2^nt)}{2^n}\Big)\_{n\in \mathbb N}$$
In the dual $\ell^2$ consider the dense linear subspace of all sequences $l=(l\_n)$ with finite support. For such $l$ with $l\_n=0$ for $n\ge N$,
$$(l\circ \g... | 9 | https://mathoverflow.net/users/26935 | 317442 | 137,648 |
https://mathoverflow.net/questions/316958 | 28 | Let $\tau$ be a CM point of discriminant $D$. Assume that $D$ is not divisible by $3$. Then $j(\tau)$ is an algebraic integer of degree equal to the class number $h(D)$. Let $ \gamma\_2(\tau)=j(\tau)^{1/3}$, the cube root being chosen in such a way that $\gamma\_2(\tau)$ is positive on the imaginary axis.
Weber had s... | https://mathoverflow.net/users/122104 | Intuitive reason why the $j$-invariant is a cube? | There is a map from the $\mathbb P^1$ with coordinate $\gamma\_2$ to the $\mathbb P^1$ with coordinate $j$ given by $j= \gamma\_2^3$. We want to check that the fiber over $j$ has a $\mathbb Q(j)$ rational point. Because this is cubic covering, it can't gain a rational point over a quadratic extension if it didn't have ... | 12 | https://mathoverflow.net/users/18060 | 317447 | 137,649 |
https://mathoverflow.net/questions/317440 | 5 | Let $\xi, \eta$ be a discrete random values and $\mathbb E| ξ |$, $\mathbb E | η | < +\infty$, and any value of these
values are accepted with a non-zero probability. How to prove that from $\mathbb E (ξ \mid η) ≥ η$, $\mathbb E (η \mid ξ) ≥ ξ$ follows
$ξ = η$?
| https://mathoverflow.net/users/132026 | Сoincidence of discrete random variables | Let us prove the desired conclusion generally, without assuming that the random variables $\xi$ and $\eta$ are discrete. Let $g\colon\mathbb R\to\mathbb R$ be any strictly increasing strictly convex differentiable function such that $|g(x)|\le1+|x|$ for all real $x$, so that $|g'|\le1$ and $Eg(\xi),Eg(\eta)$ exist in $... | 7 | https://mathoverflow.net/users/36721 | 317452 | 137,652 |
https://mathoverflow.net/questions/317287 | 6 | $\require{AMScd}$
**Preliminaries:** Let $(X,\omega,J)$ be a closed Kahler manifold. That is, $X$ is a closed $2n$-manifold, $\omega$ is a symplectic form and $J$ is a compatible (integrable) complex structure. Suppose that $[\omega] = \lambda \cdot c\_1(X,J)$ with $\lambda \in \mathbb{R}$, where $[\omega] \in H^2(X)... | https://mathoverflow.net/users/123015 | Uniqueness of a compatible Kahler-Einstein structure on a symplectic manifold? | Concerning 2) one can, of course, take the product of two curves of higher genus to get a counter-example. In general, to have a statement as you want, one should look for rigid complex surfaces of general type (with ample $K$), because as soon as such a surface has a deformation, we get a counterexample (by Yau Aubin)... | 3 | https://mathoverflow.net/users/943 | 317454 | 137,653 |
https://mathoverflow.net/questions/317463 | 5 | Lavrentieff proved a Theorem which implies that every real valued continuous function defined on a dense subset $D\subseteq \mathbb R$ admits a continuous extension to some $G\_\delta $ subset of $\mathbb R$. See Theorem (4.3.20) in "General Topology" by Engelking, or this [Mathematics Stack Exchange post](https://math... | https://mathoverflow.net/users/97532 | Extending continuous functioms defined on the irrationals | Enumerate the rationals as $\{q\_n\}$ and define $f(x) = \sum\_{n : q\_n < x} 2^{-n}$. Then $f$ is continuous on $\mathbb{R} \setminus \mathbb{Q}$ but cannot be extended continuously to any proper superset of $\mathbb{R} \setminus \mathbb{Q}$.
| 12 | https://mathoverflow.net/users/4832 | 317464 | 137,657 |
https://mathoverflow.net/questions/317450 | 1 | I have 5 polynomial equations for 5 variables and I know that the set of roots is finite. All coefficients are integers. Ultimately I'd like to find all roots but finding the Groebner basis is impossible it seems (I tried maple, mathematica, sympy) probably because it takes too long or too much memory. But I found a ra... | https://mathoverflow.net/users/41312 | System of polynomial equations with a known root | If $p = (a\_1,\dotsc,a\_n)$ is a known solution and your system of equations is given by the ideal $I$, then a system of equations for all the other solutions is given as follows. Let $m\_p = (x\_1-a\_1,\dotsc,x\_n-a\_n)$ be the maximal ideal corresponding to $p$. Then the saturation $I:m\_p^\infty$ given by
$$
I:m\_p... | 4 | https://mathoverflow.net/users/88133 | 317465 | 137,658 |
https://mathoverflow.net/questions/317171 | 9 | Let $f\in\mathbb{C}[x\_1,\dots,x\_n]$, and let $V(f)$ denote the vanishing locus. Is it true that for large enough $N$, there is a homotopy equivalence
$$\mathbb{C}^n\setminus V(f)\simeq B(0,N)\setminus V(f),$$
where $B(0,N)=\{|x|<N\}$.
| https://mathoverflow.net/users/64302 | Is $\mathbb{C}^n\setminus V(f)$ homotopy equivalent with a "large ball complement"? | This is more generally true for semialgebraic subsets of $\mathbf R^n$ and follows from the fact that they are conical at infinity (see Bochnak, Coste, Roy: Real algebraic geometry, Corollary 9.3.7, p. 225)
| 7 | https://mathoverflow.net/users/85592 | 317472 | 137,662 |
https://mathoverflow.net/questions/317453 | 2 | Theorem 6.1.23 in Engelking's Topology book says that in a compact space $X$ each quasi-component is connected. Quasi-component means the intersection of all closed-and-open subsets of $X$ containing a given point. The proof uses normality of $X$, so $X$ must be Hausdorff. But what if $X$ is only $T\_1$ compact? Is it ... | https://mathoverflow.net/users/132364 | $T_1$ version of Engelking theorem? | The answer is no. Let $X$ be any totally disconnected infinite compact Hausdorff space, e.g. various projective limits of finite discret spaces. Take a point $a$ in $X$ and consider the analogue of the line of double origins: take two copies of $X$ and glue all the pairs of identified points except $a$ and its copy $a'... | 2 | https://mathoverflow.net/users/128540 | 317475 | 137,663 |
https://mathoverflow.net/questions/317473 | 5 | Assume we have a closed symplectic manifold $M$ which is the total space of a smooth fibration by half-dimensional tori. Can we infer that $M$ is the total space of a smooth fibration by Lagrangian tori?
| https://mathoverflow.net/users/132313 | Half-dimensional torus fibration vs Lagrangian torus fibration | This doesn't need to hold. For example, if one takes a $(T^4,\omega)$ with a constant symplectic structure $\omega$, in order for it to have a fibration by Lagrangian tori one should be able to find a homologically non-trivial $T^2\subset T^4$ such that $\int\_{\omega} T^2=0$ which is impossible for general $\omega$.
... | 9 | https://mathoverflow.net/users/943 | 317477 | 137,664 |
https://mathoverflow.net/questions/317466 | 8 | For $n\in\mathbb{N}$ and $m=\lfloor\frac{n}2\rfloor$, consider the $n\times n$ skew-symmetric matrix $A\_n$ where each entry in the first $m$
sub-diagonals below the main diagonal is $1$ and each of the remaining entries below the main diagonal is $-1$. Let $I\_n$ be the $n\times n$ identity matrix.
Next, construct ... | https://mathoverflow.net/users/66131 | Determinant of "skew-symmetric" matrices | For $n$ odd, $M\_n$ is an $n\times n$ circulant matrix, and so Theorem 17 in Krattenthaler's marvellous [text](https://arxiv.org/pdf/math/9902004.pdf) applies. Denoting by $w$ a primitive $n$th root of unity, it gives
$$\det M\_n=\prod\_{i=0}^{n-1} (x-w^i-w^{2i}-\dots -w^{mi}+w^{(m+1)i}+\dots +w^{(n-1)i}),$$
something... | 2 | https://mathoverflow.net/users/11100 | 317479 | 137,666 |
https://mathoverflow.net/questions/317462 | 1 | Let $K$ be a complete local division ring (note $v$ its valuation). For $x,y\in K$ ($y\ne0$), one puts $x^y=yxy^{-1}$. Let $r\in\mathbb N$. Consider $x,y\in K$ and $a,b\in K^\*$ such that $v(x-y)\ge r$ and $v(a-b)\ge r$. Do we have $v(x^a-y^b)\ge r$? In the commutative case, it is obvious but in the non-commutative cas... | https://mathoverflow.net/users/33128 | Valuation of congruent elements in a local division ring | $\newcommand{\Q}{\mathbb{Q}} \newcommand{\Z}{\mathbb{Z}}$
Not necessarily.
Take $K = \Q\_3 + \Q\_3 i + \Q\_3 j + \Q\_3 ij$ with $i^2=-1$ and $j^2=3$, $r=2$, $x=y=j$, $a=3i$ and $b=3(1+i)$. Then $v(j)=1$, and the maximal order of $K$ is $\Z\_3+\Z\_3 i+\Z\_3 j + \Z\_3 ij$. We have $a\equiv b \bmod 3$ so that $v(a-b) = ... | 3 | https://mathoverflow.net/users/40821 | 317486 | 137,667 |
https://mathoverflow.net/questions/317485 | 4 | Let $S\_d, S\_n$ be the permutation groups of $d,n$ elements.
An intuitive representation of the wreath product $S\_d\wr S\_n$ is $V\_1\otimes...\otimes V\_n$, where each $V\_i$ is of dimension $d$. Writing $e\_{i\_1}\otimes...\otimes e\_{i\_n}$ the canonical basis (where $i\_j=1..d$), $S\_n$ permutes the $j$ and eac... | https://mathoverflow.net/users/113692 | Decomposition into irreducible of a representation of the wreath product $S_d\wr S_n$ | Given a representation $U$ of $S\_d$ and $m \in \mathbb{N}$, we can extend the action of $S\_d \times \cdots \times S\_d$ on $U \otimes \cdots \otimes U = U^{\otimes m}$ to the wreath product $S\_d \wr S\_m$ by making $S\_m$ act on the $m$ factors by place permutation. Let $U^{\widetilde{\otimes m}}$ denote this repres... | 8 | https://mathoverflow.net/users/7709 | 317488 | 137,669 |
https://mathoverflow.net/questions/317492 | 2 | If $(X,\tau)$ is a topological space, we call $A\subseteq X$ a *retract* if there is a continous map $r:X\to A$ such that $r(a) = a$ for all $a\in A$ (we assume $A$ to be endowed with the subspace topology inherited from $X$). By $\text{Retr}(X)$ we denote the collection of retracts of $X$.
Is there a non-discrete, i... | https://mathoverflow.net/users/8628 | Non-discrete $T_2$-space $(X,\tau)$ with $2^{|X|}$ retracts | Yes.
The space of rational numbers $X=\mathbb{Q}$ is an instance.
We can view $X$ as a countable union of countably many disjoint copies of $\mathbb{Q}$.
Any nonempty subset $A$ of those copies (that is, taking all or none of each
copy) is a retract of $X$, since we can map the unused copies to a
fixed copy, and... | 4 | https://mathoverflow.net/users/1946 | 317501 | 137,670 |
https://mathoverflow.net/questions/317497 | 4 | **Why is the automorphism group of a sympelctic symmetric space a Lie group?**
$\\$
A symplectic symmetric space is a triple $(M, \omega, s)$, where $(M, \omega)$ is a symplectic manifold and $ s \; \colon M \times M \to M $, $(x, y) \mapsto s\_x(y)$, is such that $s\_x$ is an involutive symplectic diffeomorphism w... | https://mathoverflow.net/users/131790 | The automorphism group of a symplectic symmetric space | The affine group of $(M,\nabla)$ is a Lie group $G$ by Kobayashi's theorem that shows that the automorphism group of any affine connection is a Lie group (see Kobayashi and Nomizu's *Foundations of Differential Geometry*). The dimension of $G$ is at most $n+n^2$ (where $n=\dim M$).
The subgroup $H$ of $G$ consisting... | 7 | https://mathoverflow.net/users/13972 | 317502 | 137,671 |
https://mathoverflow.net/questions/317455 | 1 | Assume that the set $A$ does not have simple structures (such as the case that when all elements are odd numbers in $[1,M/2]$ then all sums are even thus there are no solutions, as pointed out by @fedja).
What is the maximum cardinality $n$ of a subset $A$ of $\{1,2,\ldots,M\}$ such that $(A+A) \cap A$ is empty and t... | https://mathoverflow.net/users/17773 | Largest cardinality $n$ of a subset $A$ of $\{1,2,\ldots,M\}$ such that $(A+A) \cap A$ is empty | As I take it, you want to describe the structure of large sum-free subsets of the interval $[1,M]$, for large values of $M$.
This is in fact a known problem, which has first appeared (in a somewhat implicit form) in a paper by Abbott and Wang some 40 years ago. In 1992 Freiman has shown that a sum-free set $A\subset... | 2 | https://mathoverflow.net/users/9924 | 317511 | 137,675 |
https://mathoverflow.net/questions/317499 | 5 | We know that equation $$s\_1+s\_2+s\_3=n-1 \quad \mbox{$s\_1,s\_2,s\_3$}\geq 1$$
has $\binom{n-2}{2}$ solution.
I want to find any good formulae for the following form :
$$\sum\_{(s\_1,s\_2,s\_3)}\prod\_{i=1}^3\binom{s\_i+s\_{i-1}-1}{s\_i}=?$$
where, $s\_0=1$ and each $(s\_1,s\_2,s\_3)$ is the solution of above equa... | https://mathoverflow.net/users/132399 | Formula for a sum of product of binomials | The generating function is $$ \sum\_{s\_1,s\_2, s\_3} {s\_1 + s\_2-1 \choose s\_2} {s\_2+ s\_3-1 \choose s\_3} x^{s\_1+s\_2+s\_3}.$$
$$ \sum\_{s\_3} {s\_2+ s\_3-1 \choose s\_3} x^{s\_3} = \left( \frac{1}{1-x}\right)^{s\_2}$$.
Then the sum over the $s\_2$ variable is
$$ \sum\_{s\_2} {s\_1 + s\_2-1 \choose s\_2}\le... | 5 | https://mathoverflow.net/users/18060 | 317524 | 137,678 |
https://mathoverflow.net/questions/317519 | 5 | Let $S$ be the set of positive integers of the form $2^a3^b 5^c 7^d$. I need information about the cardinality of the intersection of $S$ and its translates. In particular, is $S \cap (S+t)$ infinite for every integer $t$? For some values of $t$?
[Photo](https://i.stack.imgur.com/JxFaF.png) of the some of the solutio... | https://mathoverflow.net/users/132407 | Intersection of $\{2^a 3^b 5^c 7^d\}$ and its translates | This is an example of an $S$-unit equation. For ones of a shape similar to this, the solutions can be found rather easily using bounds for linear forms in logarithms and lattice basis reduction. By way of example, Theorem 5.5 of de Weger's thesis (from 1989) explicitly determines the $605$ relatively prime solutions to... | 13 | https://mathoverflow.net/users/7302 | 317527 | 137,679 |
https://mathoverflow.net/questions/317531 | 11 | I have heard of this result from Deuring 1941 paper: Given $\mathbb F\_p$ ($p$ prime number) and any number $n$ in the Hasse interval $[p+1-2\sqrt p, p+1+2\sqrt p]$ there is an elliptic curve over $\mathbb F\_p$ having $n$ points. I have minimal knowledge on more advanced topics on elliptic curves (only know a thing or... | https://mathoverflow.net/users/3949 | Deuring's result on elliptic curves. Any proof reference | You might find the following paper useful, although it proves something more general than what you are asking:
[MR0890272](https://mathscinet.ams.org/mathscinet-getitem?mr=890272),
Rück, Hans-Georg,
A note on elliptic curves over finite fields.
*Math. Comp*. **49** (1987), no. 179, 301–304,
doi:[10.1090/S0025-5718-19... | 6 | https://mathoverflow.net/users/11926 | 317538 | 137,683 |
https://mathoverflow.net/questions/317537 | 2 | Let $G$ be a reductive group over an algebraically closed field $k$. Let $T$ be a maximal torus, $B$ be a Borel subgroup and $I\_G$ is the set of simple roots. Let $P$ be a standard parabolic subgroup, $M$ be its Levi containing $T$ and let $I\_M$ be the set of simple roots of $M$ (with the natural choice of Borel of $... | https://mathoverflow.net/users/nan | Relative weight lattice | I'll use $\mathrm X^\*$ instead of $X$ for character lattices, since I can never remember which is which in the $X$/$Y$ notation. I have also updated this answer from its original wrong formulation to a hopefully correct one.
$\DeclareMathOperator\srank{srank}$Note that $\Lambda\_{G, P}$ is a lattice of rank $\srank(... | 2 | https://mathoverflow.net/users/2383 | 317542 | 137,684 |
https://mathoverflow.net/questions/317523 | 14 | Let $(M, g)$ be a Riemannian manifold, not necessarily complete. Let $x$ be a point in $M$, and let $r>0$ be such that the exponential map $\operatorname{exp}\_x$ is defined on an open ball $B=B(0,r)\subseteq T\_xM$. That is, all geodesics from $x$ exist to distance $r$.
If $y$ is a point in $M$ whose Riemannian dist... | https://mathoverflow.net/users/2819 | Minimizing geodesics in incomplete Riemannian manifolds | This is indeed the case. The basic reason is that within the domain of the exponential map, there are no issues of completeness (the minimizing geodesic needs to stay within this set.) As such, minimizing curves are indeed geodesics. I don't know of an elegant proof for this, but it's possible to just brute force the i... | 3 | https://mathoverflow.net/users/125275 | 317547 | 137,686 |
https://mathoverflow.net/questions/317539 | 2 | Let $(M,g)$ be a Riemannian manifold, geodesically complete, and assume logarithms are well defined and smooth.
Let $c: I\to M $ be a smooth path in $M$, and $x\in M$. **Can we say something about** $$\Vert\nabla\_{\dot{c}(t)}\log\_{c(t)}(x)\Vert\_{c(t)} ?$$
I can easily prove that $$\Vert\nabla\_{\dot{c}(t)}\log\... | https://mathoverflow.net/users/104248 | Differentiating Riemannian logarithmic map | I believe the answer to your question is "no". I am not familiar with the term "Riemannian logarithmic map," but I imagine you mean the inverse of the Riemannian exponential map. Your condition is thus that the exponential map from any point of $M$ is a diffeomorphism (in particular this implies that $M$ is diffeomorph... | 3 | https://mathoverflow.net/users/2819 | 317548 | 137,687 |
https://mathoverflow.net/questions/317443 | 4 | Assume $\mathcal{C}$ is a monoidal category, with unit $I$. Given a monoid object $M$, I'd like to talk about modules over $M$, but couldn't find any reference. This might seem quite a stretch, but it is not so bad:
* there are notions of left, right and bimodule that "play well" with [residuals](https://ncatlab.org... | https://mathoverflow.net/users/111265 | Tensor product of modules over a monoid in a monoidal category | As from my own comment, requiring residuals to exist actually endows $X \otimes\_B Y$ with the structure of an $A$-$C$-bimodule; here's a sketch of the proof:
Assume both residuals exists, being the right adjoint to the (left and right) tensor product: this means both tensor products *are* left adjoints and [hence](h... | 3 | https://mathoverflow.net/users/111265 | 317549 | 137,688 |
https://mathoverflow.net/questions/317550 | 12 | **The *p*-adic Lindemann-Weierstrass Conjecture**: Let $\alpha\_{1},\ldots,\alpha\_{N}\in\overline{\mathbb{Q}\_{p}}$
be distinct $p$-adic algebraic numbers satisfying $\left|\alpha\_{n}\right|\_{p}<p^{-\frac{1}{p-1}}$
(so that $\exp\_{p}\left(\alpha\_{n}\right)\in\mathbb{C}\_{p}$) for all $n$. Then, $\exp\_{p}\left(\... | https://mathoverflow.net/users/120369 | Is the p-adic Lindemann-Weierstrass Conjecture still open? | Here is a 2018 paper, [A Note on One-dimensional Varieties Over the Complex p-adic Field](https://www.ejpam.com/index.php/ejpam/article/view/3281), that still lists the "full" statement as a conjecture; "half" of the statement, meaning that at least $\lfloor N/2\rfloor$ of the exponents are independent, has been proven... | 14 | https://mathoverflow.net/users/11260 | 317555 | 137,690 |
https://mathoverflow.net/questions/317551 | 7 | I can not understand Remark 12.8.8 in the preprint "SINGULAR SUPPORT OF COHERENT SHEAVES AND THE GEOMETRIC LANGLANDS CONJECTURE". I am somewhat embarrased by the degree of my confusion, hopefully someone knowledgeable could help me.
Authors claim that for any (connected) algebraic stack $Y$ and any compact object $M... | https://mathoverflow.net/users/132313 | Remark 12.8.8 in Arinkin--Gaitsgory | The answers to your questions can be found in this article: <https://arxiv.org/abs/1108.5351>. I highly recommend reading it before trying to understand Arinkin-Gaitsgory.
Let me try to resolve your difficulties. I will take for granted the existence of a dg (or equivalent, stable $k$-linear infinity-) category of $\... | 3 | https://mathoverflow.net/users/51424 | 317556 | 137,691 |
https://mathoverflow.net/questions/317554 | 15 | All matrices and vectors in this post have entries in the field $\mathbb{F}\_2$.
Fix some $n \geq 1$. For an $n \times n$ matrix $X$, write $X\_0$ for the column vector whose entries are the diagonal entries in $X$. The following curious fact arose in a paper I am writing:
**Fact**: Let $X$ be a symmetric $n \times... | https://mathoverflow.net/users/132417 | Conceptual explanation for curious linear-algebra fact in characteristic $2$ | I think one way of explaining this is via quadratic forms. The usual correspondence sends $X$ to the quadratic form $X(v)=vXv^t,$ $v$ a row vector. But in characteristic $2$, this formula simplifies to
$$X(v)=(vX\_0)^2.$$
Now examine what happens to $AXA^t$. For any characteristic, we get $AXA^t(v)=vAXA^tv^t=(vA)X(... | 6 | https://mathoverflow.net/users/51424 | 317560 | 137,693 |
https://mathoverflow.net/questions/317573 | 2 | I am searching for examples of connected locally compact group $G = N \rtimes H$, where $N$ is a simply connected nilpotent non-abelian Lie group, $H$ is linear reductive and $H$ operates on $N$ without non-trivial fixed points. Please enlighten me.
P.S. I added the ergodic theory tag because I believe such groups a... | https://mathoverflow.net/users/90755 | Examples of group $G=N \rtimes H$ where $N$ and $H$ are as below | Consider the nilpotent group $N={\mathbb R} \rtimes {\mathbb R}^2$ (the group of $3\times 3$ upper triangular unipotent matrices with real coefficients. If $v,w \in {\mathbb R}^2$, then their commutator $[v,w]$ in $N$ is simply the wedge $v\wedge w \in \wedge ^2 {\mathbb R}^2\simeq {\mathbb R}$.
The group $H=SL(2,{\... | 5 | https://mathoverflow.net/users/23291 | 317574 | 137,696 |
https://mathoverflow.net/questions/317507 | 2 | Let $a,b$ two smooth functions from the open square $I^{2}$ in $\mathbb{R}^{2}$ to $\mathbb{R}^{4}$. In particular, assume $a(t,u)$ and $b(t,u)$ be linearly independent for all $(t,u) \in I^{2}$.
I need to study the PDE problem in $ x \in C^{\infty}(I^{2},\mathbb{R}^{4})$ given by the underdetermined system
$$
\begin... | https://mathoverflow.net/users/74033 | Underdetermined system of linear PDEs | Is there anything else that you are not telling us about $a$ and $b$? The particulars of these two vector-valued functions have a great influence on what the general solution of the system
$$
a\cdot x\_t = b\cdot x\_u = a\cdot x\_u - b\cdot x\_t = 0\tag 1
$$
looks like.
For example, take the very special case in whi... | 6 | https://mathoverflow.net/users/13972 | 317580 | 137,698 |
https://mathoverflow.net/questions/317439 | 9 | Let us recall that a topological space $X$ has the *[Rothberger property](https://en.wikipedia.org/wiki/Rothberger_space)* if for any sequence $(\mathcal U\_n)\_{n\in\omega}$ of open covers of $X$ there exists a sequence $(U\_n)\_{n\in\omega}\in\prod\_{n\in\omega}\mathcal U\_n$ such that $X=\bigcup\_{n\in\omega}U\_n$.
... | https://mathoverflow.net/users/61536 | Rothberger property for finite covers | I hope I'm not messing things up in the following attempted answer.
Just for history, I seem to remember that the property you mention is called C' by Rothberger (and his main property is called C''). C alone stands for strong measure zero. So, if I'm not misquoting, your property is well known. It appears in some la... | 4 | https://mathoverflow.net/users/2415 | 317589 | 137,701 |
https://mathoverflow.net/questions/317578 | 4 | Let $\mathfrak{g}$ be a semisimple Lie algebra, $\mathfrak{g}^L$ be its Langlands dual. Feigin--Frenkel duality says
$$
W^k(\mathfrak{g})=W^{k\_L}(\mathfrak{g}^L)
$$
if $r'(k+h^{'})(k\_L+h'\_L)=1$, where $r'$ is the maximum number of edges between two vertices in the Dynkin diagram of $\mathfrak{g}$, $h'$ (resp. $h'\_L... | https://mathoverflow.net/users/132313 | Globalizing Feigin--Frenkel duality | The Feigin-Frenkel isomorphism is globalized by the global quantum geometric Langlands conjecture, proposed by Stoyanovsky, and refined by Gaitsgory and his collaborators. See [Gaitsgory's 2016 collection of conjectures](https://arxiv.org/abs/1601.05279), in particular the discussion on page 5.
At irrational level, t... | 3 | https://mathoverflow.net/users/121 | 317601 | 137,705 |
https://mathoverflow.net/questions/317565 | 10 |
>
> Do there exist infinitely many real quadratic fields $F$ such that there is an abelian surface $A$ over $\mathbb Q$ whose ring of endomorphisms, tensored with $\mathbb Q$, is $F$?
>
>
> Do there exist infinitely many real quadratic fields $F$ that are the coefficient field of a weight $2$ classical holomorphic ... | https://mathoverflow.net/users/18060 | Are there infinitely many real multiplication fields of abelian surfaces over $\mathbb Q$? | A conjecture of Coleman asserts that only finitely many rings arise as the endomorphism ring of an abelian variety of given dimension defined over a number field of given degree. See [1] for an account of this conjecture. In your case, the relevant conjecture is denoted there by $C(1,2)$. To my knowledege, the only res... | 12 | https://mathoverflow.net/users/6506 | 317606 | 137,706 |
https://mathoverflow.net/questions/317599 | 5 | Let $\xi$ be a fiber bundle $F\hookrightarrow E\to B$ (where every space is smooth, T2 and second countable), let $\Gamma(\xi)$ be the space of smooth sections. We can complete $\Gamma(\xi)$ with respect to a Sobolev $(l,2)$-norm and obtain the space of Sobolev sections $H\_l(\xi)$.
I have read that $H\_l(\xi)$ can b... | https://mathoverflow.net/users/99042 | Smooth structure on the space of sections of a fiber bundle and gauge group | Your intuition is right. To endow the space of sections of a fiber bundle $F$ with a manifold structure at $\phi \in \Gamma^\infty(F)$ you consider a tubular neighborhood (respecting the fiber structure) about the image of $\phi$ in $F$. The tube diffeomorphism serves as a linearization of every section sufficiently cl... | 8 | https://mathoverflow.net/users/17047 | 317610 | 137,708 |
https://mathoverflow.net/questions/317423 | 10 | I have some questions about the functoriality of (co)limits in $\infty$-categories, say in the framework of Lurie's Higher Topos Theory.
From the general stuff about Kan-extensions (HTT 4.3.2.6) follows that taking the colimit gives a functor $\operatorname{Map}(\mathcal{C}, \mathcal{D}) \to \operatorname{Map}(\mathcal... | https://mathoverflow.net/users/132357 | Functoriality of (co)limits in $\infty$-categories | Here is a proof of 1, which applies in any 2-category. We'll be thinking of the 2-category of quasicategories, which has homs from $Q$ to $R$ the homotopy category of the mapping quasicategory $R^Q$. With apologies for changing your notation, it would have gotten messy otherwise; I've tried to explain the connection be... | 3 | https://mathoverflow.net/users/43000 | 317617 | 137,711 |
https://mathoverflow.net/questions/317591 | 5 | I am dealing with a function $f$ of the form
\begin{equation}
f(t):=\sum\_{k=1}^Na\_ke^{\mathrm{i}\phi\_k t}
\end{equation}
and I have a promise that
\begin{equation}
0\leq f(t)\leq C\;\;\;\text{for all}\;\;\;t\in\mathbb{R},
\end{equation}
where $C>0$ is some constant. My question is the following: What bound can I fi... | https://mathoverflow.net/users/111720 | Bounds on the L^1 norm of a discrete Fourier spectrum | To avoid trivialities, I will assume the $\phi\_k$ are all distinct. Then
$$ \sum\_k |a\_k|^2 = \lim\_{R \to \infty} \frac{1}{2R} \int\_{-R}^R |f(t)|^2\; dt
\le C^2$$
so by Cauchy-Schwarz, $$\sum\_k |a\_k| \le C \sqrt{N}$$
| 5 | https://mathoverflow.net/users/13650 | 317627 | 137,716 |
https://mathoverflow.net/questions/317628 | 6 | $\def\SYT{\mathrm{SYT}}\def\RSK{\mathrm{RSK}}\DeclareMathOperator\evac{evac}$Let $\mathfrak{S}\_n$ be the symmetric group, $\SYT\_n$ be the set of standard young tableaux of size $n$.
For $u\in \mathfrak{S}\_n$, let $\RSK:\mathfrak{S}\_n\to \SYT\_n^2$ denote the Robinson-Schensted-Knuth correspondance.
Let $P\_{u,w... | https://mathoverflow.net/users/122504 | Schutzenberger's evacuation and $\mu$-coefficient of Kazhdan–Lusztig polynomials | Let $P^\* = evac(P)$. As noted above, if $RSK(u)=(P,Q)$, then $RSK(w\_0uw\_0) = (P^\*,Q^\*)$. Conjugation by $w\_0$ induces an automorphism of the Hecke algebra sending $T\_x \mapsto T\_{w\_0 x w\_0}$ and $c\_x \mapsto c\_{w\_0 x w\_0}$, from which the result you want follows. However you might be interested that somet... | 6 | https://mathoverflow.net/users/48296 | 317635 | 137,718 |
https://mathoverflow.net/questions/317540 | 1 | Let $(X,\tau)$ be a topological space, and let $Q$ be a quasi-component of $X$. Let $S$ be a subset of $X\setminus Q$. Then is $Q$ necessarily a quasi-component of $X$ in the topology generated by $\tau\cup\{S\}$?
| https://mathoverflow.net/users/132364 | Slightly finer topology vs a quasi-component | Let $X$ be the subspace of the plane given by $X = \{ (\frac{1}{n},y) : n = 1, 2, \cdots,\ 0 \leq y \leq 1 \} \cup \{(0,0),(0,1)\}$, and let $S = \{ \frac{1}{n} : n = 1, 2, \cdots\} \times \{\frac{1}{2}\}$. Then the quasi-component of $(0,0)$ in $X$ is $\{(0,0),(0,1)\}$ but in the topology generated by $X$ and $S$ the ... | 2 | https://mathoverflow.net/users/89233 | 317649 | 137,723 |
https://mathoverflow.net/questions/317583 | 10 | The classifying space $BG$ of a topological group $G$ classifies principal $G$ bundles. I have come to appreciate this.
I hope the following question is appropriate for MathOverflow:
What does the classifying space of a topological monoid classify?
| https://mathoverflow.net/users/12156 | What does the classifying space of a topological monoid classify? | Section 5 of Segal's [Classifying spaces related to foliations](https://core.ac.uk/download/pdf/82283884.pdf) shows that for discrete monoids $M$ the space $BM$ still classifies principal $M$-bundles (in a suitable sense). In Moerdijk's *Classifying spaces and classifying topoi* there is a kind of answer for general to... | 7 | https://mathoverflow.net/users/2039 | 317657 | 137,728 |
https://mathoverflow.net/questions/317658 | 3 | The following theory is formulated in first order predicate logic with extra-logical primitives of equality $``="$, membership $``\in"$, and a single primitive constant symbol $V$ denoting the class of all sets.
The axioms are those of first order identity theory +
1. **Extensionality:** $\forall x (x \in a \leftri... | https://mathoverflow.net/users/95347 | What is the strength of adding limitation of size and a simple version of reflection to Ackermann set theory? | Let me denote as $\mathsf{K}(V)$ your system 1.+2.+3.+Super Transitivity. And as $\mathsf{K}^{+}(V)$ your system 1.+2.+3.+Limitation of size.
Note that the well-founded part translation gives an interpretation of $\mathsf{K}(V)+\mathsf{Foundation}$ in $\mathsf{K}(V)$ and $\mathsf{K}^+(V)+\mathsf{Foundation}$ in $\mat... | 4 | https://mathoverflow.net/users/36385 | 317663 | 137,729 |
https://mathoverflow.net/questions/317624 | 5 | Given a site $C$ with a Grothendieck topology and the category of presheaves $P(C)$ (either in the sense of presheaves of sets or in the $\infty$-sense), and the category $S(C)$ of sheaves with respect to the topology.
Given also a cocomplete category $D$ and a functor $F: C \to D$. Suppose $F$ has the property that ... | https://mathoverflow.net/users/18116 | Universal property of sheaf category | Given $H$ a presentable category and $S$ a set of maps in $H$ then the fullcategory $H^S$ of objects in $H$ that are right orthogonal to every arrow in $S$ is a reflective subcategory of $H$.
Moreover the reflexion $H \rightarrow H^S$ is the "cocontinuous localization of $H$ at $S$", meaning that it is universal amon... | 7 | https://mathoverflow.net/users/22131 | 317667 | 137,731 |
https://mathoverflow.net/questions/317603 | 8 | Call a function from $[0, 1]$ to itself a box function.
Given any box function $f$, define its oscillation function $Of$ as $$Of(x) = \lim \_{d \to 0} \sup \_{y, z \in B\_d (x)} |f(y) - f(z)| \, .$$ Then $Of(x)$ is itself a box function.
---
Is it true that for every box function $f$, $OOOf = OOf$?
| https://mathoverflow.net/users/132446 | Oscillation operator of a function | Yes, it is true.
For a function $h(x)$ we denote by $LS(h)$, $LI(h)$ the functions defined as $$LS(h)(x)=\max(h(x),\limsup\_{y\to x} h(y)),\\ LI(h)(x)=\min(h(x),\liminf\_{y\to x} h(y)).$$
Then $$Og=LS(g)-LI(g).$$
Denote $g(x)=Of(x)$. Note that $g=LS(g)$, i.e. $$g(x)\geqslant \limsup\_{y\to x} g(y).$$ Indeed, fo... | 3 | https://mathoverflow.net/users/4312 | 317679 | 137,737 |
https://mathoverflow.net/questions/317629 | 10 | For a prime $p$ and some $g \geq 2$, consider the adjoint representation $\mathfrak{sp}\_{2g}(\mathbb{F}\_p)$ of the symplectic group $\text{Sp}\_{2g}(\mathbb{F}\_p)$. For $p \geq 3$, it is not hard to show that this is an irreducible representation. However, it is reducible in characteristic $2$.
To explain this, we... | https://mathoverflow.net/users/132417 | The adjoint representation of the symplectic group in characteristic 2 | Let $W$ be a vector space over a field $K$ of characteristic two, let $\beta$ be a non-degenerate alternating bilinear form on $W$.
Let $$G = \operatorname{Sp}(V) = \{ g \in \operatorname{End}(W) : \beta(gv, gv') = \beta(v,v') \text{ for all } v,v' \in W \}.$$ Now the adjoint representation of $G$ that you consider ... | 2 | https://mathoverflow.net/users/38068 | 317681 | 137,738 |
https://mathoverflow.net/questions/317643 | 10 | The problem comes from a problem I encountered when I wrote the article
Find all positive integer $m$ such
$$2^{m}+1\mid5^m-1$$
it seem there no solution. I think it might be necessary to use quadratic reciprocity knowledge to solve this problem.
If $m$ is odd then $2^m+1$ is divisible by 3 but $5^m-1$ is not.
so $m... | https://mathoverflow.net/users/38620 | Find all $m$ such $2^m+1\mid5^m-1$ | Here is a proof.
>
> **Theorem.** $2^m+1$ never divides $5^m-1$.
>
>
>
Assume that there is some $m$ such that $2^m+1$ divides
$5^m-1$. We already know that $m$ must be divisible by $4$.
Let $m = 2^n a$ with an odd integer $a$ and $n \ge 2$.
The $n$th Fermat number $$F\_n = 2^{2^n} + 1$$ is congruent
to $2$ mo... | 24 | https://mathoverflow.net/users/21146 | 317684 | 137,739 |
https://mathoverflow.net/questions/317688 | 4 | I was reading the two Repka papers where he computes the leading and subleading Shalika germs for $GL\_n$ and I was wondering, where are we since then? Have these germs (and the integrals) been computed somewhere? Especially if there is a reference using similar methods or something relatively simple.
The following ... | https://mathoverflow.net/users/119736 | Reference for Shalika germs of GL(n) | This 2015 [paper](https://link.springer.com/chapter/10.1007/978-3-319-17987-2_3) (also on [arXiv](https://arxiv.org/abs/1412.3891)) by Frechette, Gordon, and Robson can serve as a summary of the status with pointers to the literature:
>
> Shalika germs first appeared in the papers of Shalika and
> Harish-Chandra. ... | 2 | https://mathoverflow.net/users/11260 | 317689 | 137,740 |
https://mathoverflow.net/questions/317623 | 3 | A pseudovariety $\mathbf{V}$ of groups is *join prime* if for any pseudovarieties $\mathbf{V}\_1, \mathbf{V}\_2, \ldots,\mathbf{V}\_m$, the implication $$\mathbf{V} \subseteq \mathbf{V}\_1 \vee \mathbf{V}\_2 \vee \cdots \vee \mathbf{V}\_m \quad \Longrightarrow \quad \mathbf{V} \subseteq \mathbf{V}\_i$$ holds for some $... | https://mathoverflow.net/users/57297 | Join prime pseudovarieties | Yes, the pseudovariety generated by $D\_4$ is join prime (and the argument
shows that the same is true for the pseudovariety generated by
$8$-element quaternion group).
The result follows from two observations:
(1) the class ${\mathbf P}$ of finite groups whose Sylow $2$-subgroups
are abelian forms a pseudovariety (i.... | 4 | https://mathoverflow.net/users/75735 | 317693 | 137,742 |
https://mathoverflow.net/questions/317686 | 2 | $A$, $B\_{i}$ are some events.
If $A$, $B\_{i}$ are independent $\forall i \in \mathbb N$ and $A \cap B\_{1}, A \cap B\_{2}, ..., A \cap B\_{k}, ...$ are independent in aggregate, how to show, that $\forall B \in \sigma \{ B\_{1}, ..., B\_{k}, ...\}$ $A$ and $B$ are independent?
Events $A\_1,..,A\_n$ are called indepen... | https://mathoverflow.net/users/132026 | About independence spread | This conjecture is false. Indeed, consider the following example.
Let $B\_1,B\_2,A$ be subsets of the ground set $\{0,1\}^3$ defined as follows:
\begin{align\*}
B\_1:=\{(1,0,0),(1,0,1),(1,1,0),(1,1,1)\}, \\
B\_2:=\{(0, 1, 0), (0, 1, 1), (1, 1, 0), (1, 1, 1)\}, \\
A:=\{(0, 0, 1), (0, 1, 1), (1, 0, 1), (1, 1, 1)\}.... | 3 | https://mathoverflow.net/users/36721 | 317696 | 137,743 |
https://mathoverflow.net/questions/317650 | 12 | Let $K$ be a knot smooth knot in a 3-manifold $M$ and fix a metric on $M$. Let $F$ be a orientable surface of genus $g$ with one boundary component. Then we can consider the family of all maps $\mathscr{F} = \{ \phi: (F, \partial F) \to (M,K) : \phi \text{ is an embedding} \}$. By pulling back the metric we can talk ab... | https://mathoverflow.net/users/99414 | Minimal area of Seifert surfaces | In question (1), if you allow $g$ to vary, then this is answered positively by [Hardt and Simon](https://mathscinet.ams.org/mathscinet-getitem?mr=554379) (see [also](http://www.ams.org/journals/bull/1979-01-01/S0273-0979-1979-14581-6/home.html)).
The answer to question (2) is no. [Almgren and Thurston](https://maths... | 10 | https://mathoverflow.net/users/1345 | 317703 | 137,745 |
https://mathoverflow.net/questions/317706 | 5 | How does the incompressible Navier-Stokes system read with heat conduction?
Where can I find an existence result for its weak solutions?
| https://mathoverflow.net/users/nan | Incompressible Navier-Stokes equation with heat conduction | There is an extensive literature, this could be helpful entry point: [Solving Navier-Stokes equations coupled with a heat transfer equation](https://arxiv.org/abs/1509.00820) (2015)
>
> In this paper, the dynamics of an incompressible fluid in a bounded
> connected domain, described by Navier-Stokes equations cou... | 4 | https://mathoverflow.net/users/11260 | 317710 | 137,747 |
https://mathoverflow.net/questions/317489 | 4 | Let $\phi(x,y)$ be an acceptable programming system (i.e., $\phi(x,y)$ is a partial recursive function such that, for every partial recursive function $f(x,y)$, there exists a recursive function $r(x)$ such that, for all $x$ and $y$, $\phi(r(x), y) = f(x,y)$). Is there an acceptable programming system $\psi$ such that,... | https://mathoverflow.net/users/132396 | Range vs Domain of computable functions | The answer is positive.
For each $n$ we can find effectively an index $s(n)$ such that $Range(\phi\_{s(n)})=Domain(\phi\_n)$ and $s(n)>s(m)$ for all $m<n$. Then $Range(s)$ is computable, and we can define
$$
\psi\_x(y)=\begin{cases}\phi\_{n}(y), & \mbox{if }x=s(n);\cr
0,& \mbox{if }x\notin Range(s)\ \& \ y\in Range(\... | 3 | https://mathoverflow.net/users/69843 | 317713 | 137,749 |
https://mathoverflow.net/questions/317704 | 15 | Is every $n \times n$ matrix with entries in $\mathbb{Z}\_p$ (or even $\mathbb{Z}$) conjugate to its transpose via a matrix in $GL\_n(\mathbb{Z}\_p)$?
On the one hand, I know the analogous fact is false for matrices over $\mathbb{Z}$ with counterexamples constructed via the Latimer-MacDuffee theorem (but the only cou... | https://mathoverflow.net/users/39120 | Is a matrix similar to its transpose over $\mathbb{Z}_p$? | No for $n\geq 3$.
If $A\in M\_n(\mathbf Z\_p)$ were similar to $A^T\in M\_n(\mathbf Z\_p)$, then going modulo $p^2$, its image in $M\_n(\mathbf Z/p^2\mathbf Z)$ would be similar to the image of its transpose.
However, Pooja Singla, Steven Spallone and I have shown in [Similarity of matrices over local rings of leng... | 17 | https://mathoverflow.net/users/9672 | 317722 | 137,750 |
https://mathoverflow.net/questions/317721 | 3 | I am thinking of forming a finer topology on a particular subset of the plane. Let $X\subseteq \mathbb R ^2$ be endowed with the Euclidean topology $\tau$. Let $A,B\subseteq X$. Let $\tau'$ be the topology generated by $\tau\cup \{A,B\}$. Then will $\tau'$ be metrizable?
If not (very sad), then what assumptions abou... | https://mathoverflow.net/users/132364 | Is a plane set still metrizable if two new subsets are declared open? | Even adding one set can break metrizability, if that set is not $F\_\sigma$.
Let $\tau'$ be generated by $\tau$ and $A$, where $A$ is not $F\_\sigma$ with respect to $\tau$. (For instance, by the Baire category theorem, $A = (\mathbb{Q} \times \mathbb{Q})^c$ would do.) Now if $\tau'$ is metrizable, then the open set ... | 3 | https://mathoverflow.net/users/4832 | 317723 | 137,751 |
https://mathoverflow.net/questions/317692 | 24 | I've believed that the answer is "yes" for years, as suggested in various sources with reference to Tóth's work. For example, the Wikipedia article for Kepler Conjecture says:
>
> The next step toward a solution was taken by László Fejes Tóth. Fejes Tóth (1953) showed that the problem of determining the maximum den... | https://mathoverflow.net/users/8429 | Is there a short proof of the decidability of Kepler's Conjecture? | I don’t believe any short proof is known for the decidability of the Kepler conjecture, or indeed any proof other than Hales’s proof and its descendants. The issue is exactly what Hales explains in the quotation: the strategy is to reduce the problem to an inequality involving only finitely many variables, but this ine... | 19 | https://mathoverflow.net/users/4720 | 317727 | 137,753 |
https://mathoverflow.net/questions/317640 | 7 | Let $\cal H$ be the Poincare upper half-plane and $\overline {\cal H}$ the union of $\cal H$ with the set of cusps $\bf P^1 (\bf Q)$, provided with its usual topology. Let $\Gamma$ a congruence subgroup acting freely on $\cal H$, $V$ an abelian group with $\Gamma$-action, and $\tilde V$ the associated local system on t... | https://mathoverflow.net/users/9317 | Invariants in relative cohomology and compact support cohomology of the quotient | In order to understand the isomorphism, I would translate everything to cohomology of sheaves and then use both Grothendieck's spectral sequences that converge to the same equivariant cohomology groups of sheaves of abelian groups on $\overline{\mathcal H}$ having a $\Gamma$-action.
A relative cohomology group $H^\st... | 1 | https://mathoverflow.net/users/85592 | 317738 | 137,755 |
https://mathoverflow.net/questions/318735 | 2 | Let $\zeta$ be the zeta function of Riemann. Is the bound for
$$I\_{T}=\int\_{0}^{T} \Big|\log|\zeta(1/2 + it)| \Big| \mathrm{d}t$$
known ?
It seems to me that $I\_{T} \ll T\log T$ since $\log|\zeta(1/2+it)|\ll \log t$ if $t$ is not an ordinate of a zero. But certainly this is too naive. Is there a better bound ?... | https://mathoverflow.net/users/132483 | Inquiry on the bound for $\int_{0}^{T} \Big|\log|\zeta(1/2 + it)| \Big| \mathrm{d}t$ | We have $I\_{T} \ll T\log T$ for $T\geq 2$. For this it suffices to verify that
$$ \int\_{T}^{T+1} \Big|\log|\zeta(1/2 + it)|\Big|\ dt\ll\log T,\qquad T\geq 2.\tag{$\ast$}$$
We can deduce this local bound from Theorem 9.6 (B) and surrounding material in Titchmarsh: The theory of the Riemann zeta-function. Indeed, this ... | 4 | https://mathoverflow.net/users/11919 | 318745 | 137,756 |
https://mathoverflow.net/questions/318736 | 7 | I am looking for ideals $I\subset \mathbb{F}\_2[x,y]$ with the following properties:
1. $I$ is generated by two homogeneous elements;
2. $I$ is invariant under the $SL\_2(\mathbb{F}\_2)$-action on $\mathbb{F}\_2[x,y]$ (given by extending the action on the two dimensional sub vector space spanned by $x,y$).;
3. The qu... | https://mathoverflow.net/users/3969 | Ideals invariant under ring automorphisms | There are tons of examples. Put $u = x^2+xy+y^2$ and $v = xy(x+y)$. Then $u$ and $v$ are $SL\_2(\mathbb{F}\_2)$ invariant. If $f(s,t)$ and $g(s,t)$ are homogenous polynomials with respect to the grading $\deg s = 2$, $\deg t=3$, then $\langle f(u,v), g(u,v) \rangle$ is a $SL\_2(\mathbb{F}\_2)$-invariant generated in th... | 5 | https://mathoverflow.net/users/297 | 318746 | 137,757 |
https://mathoverflow.net/questions/318754 | 31 | This is more of a philosophical or historical question, and I can be totally wrong in what I am about to write next.
It looks to me, that complex-analytic geometry has lost its relative positions since 50's, especially if we compare it to scheme theory. *Are there internal mathematical reasons for why that happened?*... | https://mathoverflow.net/users/13960 | Complex analytic vs algebraic geometry | Though I am not an expert on this I think that the shift toward algebraic geometry is not entirely sociological. Consider the following statement which is true in both the category of schemes and analytic spaces :
*The push-foward of a coherent sheaf by a proper map is coherent.*
In algebraic geometry, this stateme... | 22 | https://mathoverflow.net/users/37214 | 318764 | 137,762 |
https://mathoverflow.net/questions/318743 | 5 | A continuous map $d:X\to A$ is called domination if there exists a map $u:A\to X$ so that $d\circ u\simeq 1\_A$.
Is there a domination map $d:P\to P$ of a finite polyhedron $P$ so that $d$ is not a homotopy equivalence?
| https://mathoverflow.net/users/114476 | On the existence of a domination map of a finite polyhedron | This question apparently goes back to Karol Borsuk, at least in spirit. An interesting discussion together with a history of the problem can be found in a paper of Danuta Kołodziejczyk, *Polyhedra for which every homotopy domination over itself is a homotopy equivalence*, [arxiv:1411.1032](https://arxiv.org/abs/1411.10... | 3 | https://mathoverflow.net/users/17846 | 318769 | 137,766 |
https://mathoverflow.net/questions/309478 | 4 | Let $P$ be a prime ideal of a Cohen-Macaulay ring $R$. Then is the sequence $\operatorname{depth}(R/P^n)$ eventually constant ?
| https://mathoverflow.net/users/127118 | For every prime ideal $P$ of any Cohen-Macaulay ring $R$, is the sequence $\operatorname{depth}(R/P^n)$ eventually constant? | Yes, for any ideal in a Noetherian local ring. See: [this paper](https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/article/asymptotic-nature-of-the-analytic-spread/4B358F7CA56A22C1298A57229A7BFA69).
| 7 | https://mathoverflow.net/users/2083 | 318773 | 137,768 |
https://mathoverflow.net/questions/318776 | 7 | Let $f \in L^2([0,1])$ . Then [Carleson's Theorem](https://en.wikipedia.org/wiki/Carleson%27s_theorem) states that
$$\lim\_{N\to \infty} \sum\_{|n|<N} \langle f,e\_n\rangle e\_n(x)=f(x),\quad\text{a.e. } x\in[0,1],$$
where $\{e\_n\}$ is the Orthonormal basis of $L^2([0,1])$ defined by $e\_n(x)=e^{2\pi in x}$ and $\la... | https://mathoverflow.net/users/127118 | What's so special about the Orthonormal base $\{e_n\}$ of $L^2[0,1]$, where $e_n(x)=e^{2\pi i nx }$? | Yes, there are other systems that have the Carleson convergence property. Notably, [Billard](http://matwbn.icm.edu.pl/ksiazki/sm/sm28/sm28129.pdf) proved in 1967 the Walsh Paley case of Carleson's theorem. Often Carleson theorem results are phrased on the real line because one can dilate there. In this setting, for som... | 5 | https://mathoverflow.net/users/118731 | 318780 | 137,770 |
https://mathoverflow.net/questions/318742 | 7 | It is well-known that the operation of addition of two ultrafilters on the set $\mathbb{N}$ of natural numbers which extends the natural addition on $\mathbb{N}$ to $\beta\mathbb{N}$, the Cech-Stone compactification of $\mathbb{N}$, is not continuous (it is only right-continuous).
I am thus looking for examples of co... | https://mathoverflow.net/users/15860 | Continuous binary operations on $\beta\mathbb{N}$ | In this paper, [Dimension phenomena associated with $\beta\mathbb{N}$-spaces](https://doi.org/10.1016/S0166-8641(01)00281-4), Ilijas Farah proved that continuous maps from $\beta\mathbb{N}^2$ (and other powers) to $\beta\mathbb{N}$ are quite simple: there is a finite disjoint cover such that the map depends on one coor... | 7 | https://mathoverflow.net/users/5903 | 318790 | 137,772 |
https://mathoverflow.net/questions/317735 | 2 | Let $p,q$ be odd primes. Consider the polynomial ring $\mathbb C[x\_0,...,x\_{q-1}]$. For $m=0,1,...,p-1$, let
$$\sigma\_m=\sum\_{0\le j\_0\le p;...;0\le j\_{q-1}\le p; j\_1+...+j\_{q-1}=p; 1.j\_1+...+(q-1)j\_{q-1}\equiv m (\mod p)} \dfrac {p!}{j\_0!...j\_{q-1}!} x\_{0}^{j\_0}...x\_{q-1}^{j\_{q-1}}$$.
Notice that ... | https://mathoverflow.net/users/127118 | On a special type of subring of $\mathbb C[x_0,...,x_{q-1}]$ | Since $K$ is generated by $p$ elements, $\mathbb{C}(x\_0,\ldots,x\_{q-1})$ cannot be an algebraic extension of $K$ if $q>p$. I claim that $\mathbb{C}(x\_0,\ldots,x\_{q-1})$ is a finite Galois extension of $K$ whenever $p\geq q$.
Define $\zeta=e^{2\pi i/p}\in\mathbb{C}$, and for $0\leq k\leq p-1$, define
$$
\omega\_k:... | 1 | https://mathoverflow.net/users/5263 | 318796 | 137,774 |
https://mathoverflow.net/questions/318777 | 3 | Given a set $X$ and $k\in\mathbb{N}$ we call a subset of $X$ a $k$-*subset* if its cardinality is $k$. If ${\cal S}$ is a collection of subsets of $X$ and $x\in X$ we set ${\cal S}\_x=\{S\in {\cal S}: x\in S\}$.
Let $1<k<\ell$ be integers. Is it possible to find infinitely many integers $n>\ell$ such that there is a... | https://mathoverflow.net/users/8628 | A set coverage problem | At first, the second condition follows from the first by averaging over all $k$-sets containing $a$. Moreover, for any $m\leqslant k$ and any $m$-set $A\subset \{1,\dots,n\}$ we may count the number $N$ of pairs $B\subset C$ where $A\subset B$, $B$ is a $k$-set and $C$ is an $\ell$-set from $\mathcal{L}$. For any fixed... | 3 | https://mathoverflow.net/users/4312 | 318801 | 137,776 |
https://mathoverflow.net/questions/318785 | 15 | *Disclaimer: I'm far from an expert on any of the topics of this question. I apologize in advance for any horrible mistakes and/or inaccuracies I have made and I hope that the spirit of the question will still be clear despite them.*
The (integral) representation rings of the symmetric groups can be packed together i... | https://mathoverflow.net/users/22810 | Schur-Weyl duality and q-symmetric functions | As Sam Hopkins says, the category of all representations of $GL\_n(\mathbb F\_q)$ is too large to give what you want. Instead, let's consider the category of *unipotent representations*, i.e. those appearing in the irreducible decomposition of $\mathbb Q [GL\_n(\mathbb F\_q)/B\_n(\mathbb F\_q)]$.
Unipotent represent... | 15 | https://mathoverflow.net/users/52918 | 318802 | 137,777 |
https://mathoverflow.net/questions/318748 | 6 | Classical work by Casselman shows that for an irreducible admissible representation $\rho$ of $GL\_2$ over a non-archimedean field $k$, there is a minimal power $n\geq 0$ of the prime ideal $\mathfrak{p}$ such that $\rho$ has a fixed vector under $(\begin{smallmatrix} \* & \* \\ \mathfrak{p}^n & 1+\mathfrak{p}^n\end{sm... | https://mathoverflow.net/users/448 | Conductor of quaternionic representation | I consider a Casselman type of local newform theory on quaternion algebras in [my paper on the basis problem](https://arxiv.org/abs/1804.04234) (sections 2 and 3), which gives you a positive answer to your question half of the time (Cases 1 and 2 below). Here is a brief summary. For simplicity, I'll assume trivial cent... | 4 | https://mathoverflow.net/users/6518 | 318818 | 137,781 |
https://mathoverflow.net/questions/283003 | 11 | I asked this question a while ago [on MSE](https://math.stackexchange.com/questions/1596348/getting-the-most-general-form-of-mayer-vietoris-from-the-axioms-of-homology), got no answer, put a bounty on it, still got no answer, was advised to ask here instead, hesitated, forgot about the question for a while and now reme... | https://mathoverflow.net/users/3041 | Getting the most general form of Mayer-Vietoris from the Eilenberg-Steenrod axioms | If you are willing to work with mapping cones, then this follows from looking at the triple (= threefold iterated) mapping cone for the cube with vertices $A\_{12} = A\_1 \cap A\_2$, $A\_1$, $A\_2$, $A$, $X\_{12} = X\_1 \cap X\_2$, $X\_1$, $X\_2$ and $X$ in two different ways.
Let us use your notation $C\_A^X = X \cu... | 6 | https://mathoverflow.net/users/9684 | 318823 | 137,782 |
https://mathoverflow.net/questions/318826 | 9 | Let $M\_n$ be the $n\times n$ matrix with entries
$$\binom{i}{2j}+\binom{j}{2i}, \qquad \text{for $1\leq i,j\leq n$}.$$
>
> **QUESTION.** Is this true? There is some evidence. The determinant $\det(M\_{2n+1})=0$ and
> $$\det(M\_{2n})=(-1)^n\binom{2n}n^22^{n(n-3)}.$$
>
>
>
| https://mathoverflow.net/users/66131 | Certain matrices of interesting determinant | Noam Elkies in the comments reduces the problem to the identity $$\det\left(\binom{(n+1)+i}{2j+2}\right)\_{i,j=0}^{n-1}=\binom{2n}n2^{n(n-3)/2}.$$ In general the determinant $\binom{N+i}{c\_j+j}$, $i,j=0,\dots,n-1$ for integers $0\leqslant c\_0\leqslant c\_1\leqslant \dots \leqslant c\_{n-1}\leqslant N$ may be calculat... | 14 | https://mathoverflow.net/users/4312 | 318835 | 137,786 |
https://mathoverflow.net/questions/318798 | 1 | As is shown in ***Representations of Semisimple Lie Algebras in the BGG Category $\mathcal{O}$***, every nonzero module $M \in \mathcal{O}^\mathfrak{p}$ has a finite filtration with nonzero quotients, each of which is a highest weight module in $\mathcal{O}^\mathfrak{p}$.
Thus the action of $Z(g)$ on $M$ is finite.
... | https://mathoverflow.net/users/110229 | About finite direct sum of full subcategory of category $\mathcal{O}^\mathfrak{p}$ | There is infinitely many linkage classes each containing some $\Phi^+\_I$-dominant elements. But since any module from $\mathcal{O}$ is finitely generated it will decompose only into finitely many modules from $\mathcal{O}\_\chi$.
I hope I understood your question correctly. There are some problems, e.g. $\mathfrak{s... | 1 | https://mathoverflow.net/users/6818 | 318845 | 137,787 |
https://mathoverflow.net/questions/318855 | 4 | Consider a two-form $\gamma \in \Lambda^2(V)$ where $V$ is a real vector space. Now I would like to know the necessary and sufficient conditions for $\gamma$ to be expressible as an exterior product of two one-forms, $\gamma=\alpha \wedge \beta,\, \alpha,\beta \in \Lambda(V)$.
Obviously, a necessary condition for the... | https://mathoverflow.net/users/133579 | What are the necessary and sufficient conditions for a two-form to be an exterior product of two one-forms? | While Plücker relations give the general theory, a direct answer to your question is the following: a 2-form $\gamma$ is decomposable (is a product of two 1-forms) iff $(\iota\_v \gamma) \wedge \gamma = 0$ for any vector $v$, where $\iota\_v \gamma$ is the usual contraction of a vector with a 2-form, that is, $(\iota\_... | 10 | https://mathoverflow.net/users/2622 | 318858 | 137,792 |
https://mathoverflow.net/questions/318854 | 0 | An intuitionistic Kripke model is a triple $\langle W,\leq, \Vdash \rangle$, where $\langle W,\leq \rangle$ is a preordered Kripke frame, and
$\Vdash$ satisfies the following condition of hereditariness (or monotonicity):
if $P$ is a propositional variable, $w\leq u$, and $w\Vdash P$, then
$u\Vdash P$.
* Are there... | https://mathoverflow.net/users/122435 | Superintuitionistic logics which are not hereditary/monotonic: impossible or possible? | Every propositional logic $L$ weaker than classical logic (i.e., any logic whose provable propositions are a subset of classically provable propositions) has a Kripke model. Just take $W = \{\star\}$, the frame with a single element, and the trivial preorder. The model is equivalent to the boolean algebra $\{\bot, \top... | 3 | https://mathoverflow.net/users/1176 | 318878 | 137,805 |
https://mathoverflow.net/questions/318838 | 7 | I just today realized that the concept of ordinal definability is defined in a different way by vopenka-Balcar-Hajek ``The notion of effective sets and a new proof of the consistency
of the axiom of choice'' (see the end of this question for the definition).
As they stated, their definition is equivalent to the one g... | https://mathoverflow.net/users/11115 | Effective set= ordinal definable set | Regarding question 1, I interpret the phrase
>
> the class of transfinite power sets of the empty set
>
>
>
to refer to the sets $V\_\alpha$ appearing in the cumulative hierarchy. On this reading, the phrase
>
> the closure of the class of transfinite power sets of the empty set with respect to the fund... | 7 | https://mathoverflow.net/users/1946 | 318888 | 137,812 |
https://mathoverflow.net/questions/318875 | 2 | Does there exist a non-unital nuclear $C^\*$ algebra $A$ of $\prod\_nM\_n(\Bbb C)$ such that $A$ properly contains $\oplus\_n M\_n(\Bbb C)$ and each element $(x\_n)\not \in \oplus\_n M\_n(\Bbb C)$ we have $\lim\_ntr\_n(x\_n)=0$,where $tr$ is the unique tracial state on $M\_n(\Bbb C)$.
| https://mathoverflow.net/users/63864 | a nuclear $C^*$-subalgebra in $\prod_n M_n(\Bbb C)$ | Sure, let $A = (\bigoplus M\_n ) + \mathbb{C}\cdot P$ where $P$ is a projection of the form $P = (p\_n)$ with each $p\_n \in M\_n$ a rank 1 projection. This is a one-dimensional extension of $\bigoplus M\_n$, and so it is nuclear. (If $I$ and $A/I$ are nuclear then so is $A$.)
| 7 | https://mathoverflow.net/users/23141 | 318889 | 137,813 |
https://mathoverflow.net/questions/316594 | 4 | The following question arose in some discussions recently as a misunderstanding of another problem.
**Question:** Which subsets $E\subset \mathbb{F}\_{p^k}$ satisfy the property that $ \sum\limits\_{x\in E}Q(x)=0$ for all $Q(x)$ polynomials in $\mathbb{F\_{p}}[x]$ of degree less or equal to $t$?
Zero here is the ad... | https://mathoverflow.net/users/118731 | Subsets $E$ of $\mathbb{F}_{p^k}$ with vanishing polynomial subset sums | Let $E$ be a subset of any field. Then $\sum\_{x \in E} x^m=0$ for all $m \leq t$ if and only if $$\sum\_m \sum\_{x \in E} u^{-m-1} x^m = \sum\_{x \in E} \frac{1} {u-x} = \sum\_{x \in E} \frac{d}{du} \log (u-x) = \frac{d}{du} \log \left(\prod\_{x \in E} (u-x)\right) $$ has degree $\leq -t -2$ in $u$.
Now, $$\frac{d}{... | 2 | https://mathoverflow.net/users/18060 | 318902 | 137,817 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.