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/391365 | 4 | Suppose $S$ is a closed, connected, oriented surface of genus at least two and $G$ is any group. Suppose further that $\Gamma$ is any group that fits into the following short exact sequence:
$$ 1 \to \pi\_1 S \to \Gamma \to G \to 1. $$
The claim is that groups $\Gamma$ correspond exactly to fundamental groups of $S$-... | https://mathoverflow.net/users/156387 | Surface bundles associated to a short exact sequence of groups | (1) It is not true that these groups are precisely the fundamental groups of $S$-bundles. The correct statement is that these groups are precisely the fundamental groups of *$S$-bundles over a base space with $\pi\_2 = 0$.* (For instance, $U(2)$ is a torus bundle over $S^2$, but its fundamental group is $\Bbb Z$, which... | 13 | https://mathoverflow.net/users/197714 | 391366 | 161,966 |
https://mathoverflow.net/questions/391180 | 6 | I've been trying to understand the asymptotic behavior of Ricci flow, and there are two facts which I am unable to square away. I'm interested in higher dimensional manifolds, but my question is easier to state for Riemann surfaces. I suspect that the solution for surfaces will also solve the case in higher dimensions,... | https://mathoverflow.net/users/125275 | Exponential convergence of Ricci flow | Having thought about this a little more, I think a more detailed explanation of the precise issue you are describing is that you are being too cavalier in "Fact 2" about what "linearizing" means. You are treating the scalar curvature as something you can freely vary, but that is not really true since $\phi$ is not an i... | 2 | https://mathoverflow.net/users/127803 | 391367 | 161,967 |
https://mathoverflow.net/questions/391364 | 7 | Let $\Sigma\_{g,n}$ denote an $n$-punctured surface of genus $g$, with $2g+n-2 > 0$. Let $\Pi\_{g,n}$ be its fundamental group (for some choice of base point), and let $\Gamma\_{g,n}$ denote its pure, orientation preserving mapping class group.
Then, by the Dehn-Nielsen-Baer theorem, the outer action of $\Gamma\_{g,n... | https://mathoverflow.net/users/88840 | Index of the mapping class group $\Gamma_{g,n}$ inside $\text{Out}(\Pi_{g,n})$ | This is surely not the most direct answer. But $\mathrm{Out}(\Pi\_{g,1}) \cong \mathrm{Out}(F\_{2g})$ surjects onto $\mathrm{GL}(2g,\mathbf Z)$, and the image of $\Gamma\_{g,1}$ lands in $\mathrm{Sp}(2g,\mathbf Z)$. So the index is infinite for $g \geq 2$.
---
In fact a more careful version of this argument shows... | 6 | https://mathoverflow.net/users/1310 | 391370 | 161,968 |
https://mathoverflow.net/questions/391360 | 2 | $(X\_k)\_k$ is a sequence of independent r.v uniformly bounded by $c.$ If $\sum\_{k}X\_k$ converges a.s then $\sum\_{k}E[X\_k]$ converges.
The above is proved using the following inequality ($X\_k$ should be centered):
$$P\left(\max\_{p \leq k}|Y\_p|>\epsilon\right) \geq 1-\frac{(\epsilon+c)^2}{E[Y^2\_k]} \ \ \ \ \ \... | https://mathoverflow.net/users/172528 | $P(\max_{1 \leq p \leq k}|Y_p| >\epsilon) \geq 1-4\frac{(\epsilon+\max_{1 \leq p \leq k } |X_p-E[X_p]|)^2}{\operatorname{Var}(Y_k)}$ | The statement
>
> If $(X\_k)$ is a sequence of independent r.v.'s uniformly bounded in absolute value by some real $c$ such that $\sum\_{k}X\_k$ converges a.s., then $\sum\_{k}EX\_k$ converges
>
>
>
follows almost immediately from the Hoffmann–Jørgensen inequality
$$EM^2\le2.4^2c^2+32t\_0^2$$
(see e.g. [Propos... | 1 | https://mathoverflow.net/users/36721 | 391381 | 161,971 |
https://mathoverflow.net/questions/336176 | 5 | In Beauville, Narasimhan, Ramanan's *Spectral curves and the generalized theta divisor*, Remark 3.7, the following exact sequence is presented:
$0 \rightarrow M(-\Delta) \rightarrow \pi^\* E \xrightarrow{\pi^\*\varphi - x} \pi^\*(L \otimes E) \rightarrow \pi^\*L \otimes M \rightarrow 0$
where $X$ is a smooth curve... | https://mathoverflow.net/users/91935 | Exact sequence involving spectral data for Higgs bundles | There is an argument in the proof of Proposition 5.17 of <https://arxiv.org/abs/2101.08583>
| 2 | https://mathoverflow.net/users/1583 | 391395 | 161,977 |
https://mathoverflow.net/questions/391369 | 4 | Let $R$ be a finite commutative ring with unity. Let $a \in R$ and define $C\_a = \{b \in R : \operatorname{ann}(a) = \operatorname{ann}(b)\}$.
I want to know the cardinality of the set $C\_a$.
For example,
If $a=0$ then $|C\_a| = 1$.
If $a$ is a unit then $|C\_a| = |U(R)|$ the set of units of $R$.
If $a$ is ... | https://mathoverflow.net/users/33047 | Elements with equal annihilators | Here is a partial answer. If $R$ is a quasi-Frobenius ring (which since $R$ is commutative is the same a a Frobenius ring), then two elements have the same annihilator if and only if they are associates (differ by multiplication by a unit) if and only if they generate the same principal ideal. This covers your examples... | 6 | https://mathoverflow.net/users/15934 | 391401 | 161,978 |
https://mathoverflow.net/questions/391400 | 3 | Suppose $K$ is an $n\times n$ Hermitian matrix and $0\leq K\leq I$, which means that $I-K$ is positive semidefinite. Let $E\subset \{1,2,\dotsc,n\}$. I wonder how to show that det$(M^{E})\geq 0$, where $M^{E}$ is defined as follows:
\begin{align\*}
M^{E}\left(i,j\right)=\begin{cases}
\delta\_{i,j}-K(i,j),&i\in E^{\co... | https://mathoverflow.net/users/197849 | Determinant of a matrix with entries specified by a set | We may assume that $E=\{1,2,\dots,k\}$. Then
$$
M^E=\begin{bmatrix}
A& C\\
-C^\*& B
\end{bmatrix},
$$
where $A$ and $B$ are PSD (and Hermitian), and $C$ is some rectangular matrix. We claim that, under these constraints, the determinant is always non-negative.
We may assume that $A$ is non-singular, the other case... | 6 | https://mathoverflow.net/users/17581 | 391409 | 161,981 |
https://mathoverflow.net/questions/391413 | 2 | Let $a\in S^d$ and $b\in S^{d-1}$ be points chosen uniformly at random on the $(d+1)$ and $d$-dimensional spheres.
I'm interested in showing some inequalities regarding their norms, the simplest being:
>
> **How to show that $\mathbb E\left[\frac{||a||\_1}{\sqrt {d+1}}\right] \le \mathbb E\left[\frac{||b||\_1}{\s... | https://mathoverflow.net/users/197231 | Let $a\in S^d$, $b\in S^{d-1}$ be uniform on the spheres. How to show $\mathbb E[\frac{||a||_1}{\sqrt {d+1}}] \le\mathbb E[\frac{||b||_1}{\sqrt d}]$? | $\newcommand{\Ga}{\Gamma}$Your first inequality is true, for each $n:=d\ge2$.
Note that $a$ and $b$ equal, respectively, $X\_{n+1}$ and $X\_n$ in distribution, where
\begin{equation\*}
X\_n:=G/|G|,
\end{equation\*}
$G=(G\_1,\dots,G\_n)$ is a standard Gaussian random vector in $\mathbb R^n$, and $|G|$ is the Euclidean ... | 4 | https://mathoverflow.net/users/36721 | 391416 | 161,983 |
https://mathoverflow.net/questions/391383 | 11 | Let $c\_n$ be a sequence of real numbers with $\sum c\_n$ converging conditionally but not absolutely. Suppose $\delta\_n > 0$ is another sequence with $\delta\_n \to 0$, and $\sum c\_n \delta\_n$ converging also conditionally but not absolutely.
Does there exist, for every $L^1$ function $f: [0, 1] \to \mathbb R$, a... | https://mathoverflow.net/users/173490 | Riemann rearrangement theorem for $L^1$ functions | The problem is trickier than I initially thought, but with the corrected condition it can be done. I need to assume that $\sum c\_n \delta\_n$ is conditionally but not absolutely convergent, $0 < \delta\_n < 1$ with $\delta\_n \to 0$ and $c\_n \to 0$ (which is currently only implied by the conditional convergence of $\... | 8 | https://mathoverflow.net/users/51695 | 391446 | 161,988 |
https://mathoverflow.net/questions/391344 | 2 | I am asking for a *reference* for the following lemma (for which I know a proof).
>
> **Lemma.** Let $f\colon X\to Y$ be a surjective morphism of irreducible *smooth* complex algebraic varieties (separated, reduced, irreducible schemes of finite type over $\Bbb C$) with *smooth* fibres over closed points of $Y$. Th... | https://mathoverflow.net/users/4149 | Smoothness of a morphism of smooth varieties with smooth fibres | Let me expand my comment into an answer, so that the question will not appear as unanswered anymore.
If $f$ is smooth, then $f$ is flat and its fibres are automatically smooth and equidimensional, see [Vakil, Theorem. 25.2.2]. This provides one implication of your statement. The converse implication is [Vakil, Exerci... | 1 | https://mathoverflow.net/users/7460 | 391450 | 161,989 |
https://mathoverflow.net/questions/391418 | 5 | It is very well known that if $A\in L^\infty(B\_1;\mathbb R ^{d\times d})$ is a positive definite symmetric matrix, the eigenvalue of the self adjoint operator $H^2(B\_1)\cap H^1\_0(B\_1)\to L^2(B\_1)$
$$T:u\to\text{div}(A Du)$$
are all real, and the corresponding eigenvectors can be chosen to form an orthonormal eigen... | https://mathoverflow.net/users/40120 | Does a suitable famlly of eigenvectors of non self-adjoint operators, sufficiently close to an adjoint one, form a basis? | In one space dimension, the answer is yes, and the eigenvalues are real and simple (Sturm-Liouville theory).
In higher space dimension, the answer is negative, because the operator needs not be diagonalisable. If you let the data $(A,C)$ depend upon a parameter, you can pass from a situation where all the eigenvalues... | 3 | https://mathoverflow.net/users/8799 | 391456 | 161,990 |
https://mathoverflow.net/questions/391441 | 4 | We shall consider the matrix-valued differential operator
$$(L u)(x) :=u'(x) - \begin{pmatrix} 0 & \sin(2\pi x-\frac{\pi}{6})\\ - 2\sin(2\pi x+\frac{\pi}{6}) & 0 \end{pmatrix} u(x).$$
This is a $1$-periodic operator. Thus, does there exist a $\lambda \in \mathbb C$ and a $1$-periodic solution to this ODE such that
... | https://mathoverflow.net/users/150549 | Existence of periodic solution to ODE | The solutions of $(L-\lambda)u=0$ are the functions $u(x)=e^{i\lambda x}v(x)$, where $v$ satisfies $Lv=0$. The periodicity amounts to $e^{i\lambda}v(1)=v(0)$. Thus your problem does admit infinitely many solutions. Just consider the monodromy matrix $M:v(0)\mapsto v(1)$, whose determinant equals $1$ (by the Wronskian).... | 6 | https://mathoverflow.net/users/8799 | 391459 | 161,991 |
https://mathoverflow.net/questions/391460 | 9 | We know that there is a cofiber sequence $S^3\xrightarrow{\eta}S^2\to\mathbb{C}\mathbb{P}^2$. It's easy to know that $\pi\_3^s(\mathbb{C}\mathbb{P}^2)=0$ so there is a surjection
$$\partial:\pi\_7^s(S^2\wedge\mathbb{C}\mathbb{P}^2)\to\pi\_7^s(\mathbb{C}\mathbb{P}^2\wedge\mathbb{C}\mathbb{P}^2)$$ by the long exact seque... | https://mathoverflow.net/users/149491 | Stable homotopy groups of complex projective plane | The question is equivalent to asking what the multiplication-by-$\eta$-map $\pi\_4\mathbb{CP}^2 \to \pi\_5 \mathbb{CP}^2$ is (which can be rewritten as $\pi\_2\mathbb{S}/\eta \to \pi\_3\mathbb{S}/\eta$). The source is generated by the lift of $2 \in \pi\_2S^2$ to $\pi\_2 \mathbb{S}/\eta$. Thus, the question translates ... | 17 | https://mathoverflow.net/users/2039 | 391464 | 161,992 |
https://mathoverflow.net/questions/391271 | 4 | A problem of Mathematical Physics that I am working on involves the computation of a certain integral. Part of the result reads:
$$
I\_k:= [\beta\_x( -1 - k, 0) + H\_{-2 - k}]x^k
$$
where $\beta\_x( -1 - k, 0)$ is the incomplete Beta function, with $x\in(0, \, 1)$, and $H\_{-2-k}$ is an harmonic number.
From phys... | https://mathoverflow.net/users/101308 | Beta function, harmonic numbers, and integral values | TL;DR
Peter Taylor pointed out that the expressions below reduce to
$$\lim\_{n\rightarrow k}x^{-n}I\_n(x)= (x-1) \sum\_{i=0}^k \frac{H\_{k+1} - H\_i}{x^{i+1}} -2 \,\text{arctanh}\,(1-2 x)$$
---
---
$\newcommand\HGF{\_2\!\tilde{F}\_1}$Mathematica is able to compute these limits, the result is in terms of a... | 4 | https://mathoverflow.net/users/11260 | 391470 | 161,993 |
https://mathoverflow.net/questions/391454 | 14 | For two sets $O$ and $A$, we will call a *category structure* a collection of functions ${\sf dom}:A\to O,\ {\sf cod}:A\to O,\ {\sf 1}:O\to A,\ \circ:A\times\_OA\to A$ satisfying the usual axioms for a category.
>
> Can we parametrize the number of category structures (up to iso or equivalence) on two sets $O$ and ... | https://mathoverflow.net/users/92164 | How many category structures are possible on two sets? | The problem of counting semigroups and monoids of order $n$ up to isomorphism and anti-isomorphism (i.e., contravariant equvialence) is a very classical problem whose answer is conjectured but nobody has made serious progress in proving over the last 60 years. The analogous situation in group theory is that virtually a... | 25 | https://mathoverflow.net/users/15934 | 391474 | 161,994 |
https://mathoverflow.net/questions/391457 | 3 | Let $(R,\mathfrak m)$ be a local Cohen-Macaulay ring of dimension $n$ with a canonical module $\omega$. Let $M$ be a finitely generated $R$-module with $\text{depth } M=\dim M=t$. Using Bruns&Herzog's book, Cohen-Macaulay rings, Corollary 3.5.11 (a consequence of Grothendieck local-duality), I can see that $\text{Ext}^... | https://mathoverflow.net/users/174552 | For a Cohen-Macaulay module $M$ of dimension $t$ over a local CM ring of dimension $n$, is $\text{Ext}^{n-t}_R(M,\omega)$ Cohen-Macaulay? | At the request of the OP, I write down Bourbaki's proof in *Commutative algebra* X, §9, no. 1, Corollaire of Proposition 3.
The proof is by induction on $t=\dim(M)$. If $t=0$, $ \operatorname{Ext}^{n}\_{R}(M,\omega ) $ has dimension 0, hence is Cohen-Macaulay. If $t>0$, we choose an element $x$ of $\mathfrak{m}$ such... | 5 | https://mathoverflow.net/users/40297 | 391475 | 161,995 |
https://mathoverflow.net/questions/391497 | 1 | **1.** Let $X$ be a smooth irreducible $\Bbb C$-variety,
on which the algebraic $\Bbb C$-group $G={\bf G}\_{a,{\Bbb C}}$
(the additive group) acts freely on the right:
$$ X\times \_{\Bbb C} G\to X,\quad (x,g)\mapsto x\cdot g.$$
Assume that there exists a surjective morphism
onto a smooth $\Bbb C$-variety $Y$
\begin{equ... | https://mathoverflow.net/users/4149 | Taking quotient of a variety by the additive group | For 1, yes. In fact, any smooth morphism of varieties admits a section locally in the etale topology everywhere.
Proof: A generic hypersurface section is smooth of dimension one lower over any particular point. Repeat until the relative dimension is zero.
For 2, no. Any variety $Y$ with $H^1 (Y, \mathcal O\_Y) \neq... | 5 | https://mathoverflow.net/users/18060 | 391500 | 162,001 |
https://mathoverflow.net/questions/391503 | 2 | This question is related to a [previous one](https://mathoverflow.net/questions/391283/positive-scalar-curvature-on-the-double-of-a-manifold).
Let $(M^n,g)$ be a compact Riemannian manifold with boundary. Assume it has positive scalar curvature and $\partial M$ is mean convex (positive mean curvature).
>
> **Ques... | https://mathoverflow.net/users/85934 | Positive scalar curvature on the double of a manifold (assuming mean convexity of the boundary) | This is true/well-known. A reference is Gromov--Lawson "Spin and scalar curvature in the presence of a fundamental group. I" (<https://mathscinet.ams.org/mathscinet-getitem?mr=569070>) Theorem 5.7.
| 3 | https://mathoverflow.net/users/1540 | 391504 | 162,003 |
https://mathoverflow.net/questions/391435 | 4 | Let $\mu$ and $\nu$ be radially symmetric probability measures on $\mathbb R^d$. Consider the Kantorovich optimal transport problem between $\mu$ and $\nu$, with convex, nonnegative cost. Suppose there exists at least an optimal transport plan between $\mu$ and $\nu$ with finite cost.
>
> **Question.** Does it foll... | https://mathoverflow.net/users/173490 | Is the optimal transport of radially symmetric measures also radially symmetric? | *This is my commnent (which nobody sees).*
The answer is "yes".
It follows since the quotient map $\mathbb{R}^d\to[0,\infty)$ is a submetry.
The optimal plan between pushforward measures on $[0,\infty)$ lifts uniquely to an optimal plan on $\mathbb{R}^d$
| 1 | https://mathoverflow.net/users/1441 | 391513 | 162,007 |
https://mathoverflow.net/questions/391458 | 4 | Suppose a polynomial of the form
$$\prod\_i^d \sum\_j^p x\_i^{f\_j}$$
clearly symmetric, where $f\_j\in \mathbb{N}$. There is a way to find the set of $f$ numbers such that this polynomial is Schur positive? I have been trying to build a Kashiwara crystal and impose conditions on the weight function but without success... | https://mathoverflow.net/users/166314 | Schur positivity of a polynomial | Given $f\_1,\dots,f\_p$ and $d\geq \max f\_i$, a necessary and sufficient condition is that all zeros of the polynomial $\sum x^{f\_j}$ are real. See *Enumerative Combinatorics*, vol. 2, Exercise 7.91. **Note.** Your necessary condition need not hold for small $d$. If $d=1$, then $\sum x\_1^{f\_j}= \sum s\_{f\_j}(x\_1)... | 7 | https://mathoverflow.net/users/2807 | 391519 | 162,010 |
https://mathoverflow.net/questions/391417 | 6 | For a permutation $\pi\in\frak{S}\_n$, define the number of descents of $\pi$ as $$\text{des}(\pi)=\vert\{i: \pi(i)>\pi(i+1)\}\vert.$$
The following is a well-known (and interesting) identity:
$$\binom{k\ell+n-\text{des$(\pi)$}-1}n=\sum\_{\sigma\tau=\pi}
\binom{k+n-\text{des$(\sigma)$}-1}n\binom{\ell+n-\text{des$(\tau)... | https://mathoverflow.net/users/66131 | A convolution-type identity for the "major index" | Richard's identity $(\*)$ can be found as Theorem 11 (though not stated exactly this way) in my paper *Multipartite
P-partitions and inner products of skew Schur functions*, Combinatorics and algebra (Boulder, Colo., 1983), 289–317,Contemp. Math., 34, Amer. Math. Soc., Providence, RI, 1984. (It's actually $\tau\sigma=... | 6 | https://mathoverflow.net/users/10744 | 391525 | 162,015 |
https://mathoverflow.net/questions/391501 | 2 | [EDIT: The axiom of successor cardinals was found by an answer by Greg Kirmayer, not to be capturing the intended meaning of it, which is simply reflected by its name, i.e. the existence of a successor cardinal for every cardinal. A corrective note had been inserted below that axiom.
Is Z + Rank + Successor cardinals... | https://mathoverflow.net/users/95347 | Can Z + Ranks + Successor cardinals + Ordinal inaccessibility be equal to ZF? | This theory doesn't prove Replacement (assuming the consistency of an inaccessible, at least).
Assume ZFC + $\kappa$ is inaccessible and force over $V$ to add $\kappa$-many Cohen reals (i.e. the forcing is finite support product $\Pi\_{\alpha<\kappa}\mathbb{C}\_\alpha$ where each $\mathbb{C}\_\alpha$ is just Cohen fo... | 7 | https://mathoverflow.net/users/160347 | 391526 | 162,016 |
https://mathoverflow.net/questions/391536 | 2 | The title and tags say it all: I am looking for a clean statement and proof of the equivalence of categories between finite étale covers of a connected $k$-scheme and finite continuous permutation representations of the étale fundamental group $\pi\_1(X, \overline{x})$ given by the fiber functor.
I need this result f... | https://mathoverflow.net/users/175051 | Equivalence of categories between finite étale covers of connected scheme and finite continuous permutation representations of étale fundamental group | I like the notes from the 2016-2017 edition of the Stanford number theory learning seminar. The result that you want is Theorem 3.4 [here](http://virtualmath1.stanford.edu/%7Econrad/Weil2seminar/Notes/L3.pdf).
| 4 | https://mathoverflow.net/users/21278 | 391541 | 162,021 |
https://mathoverflow.net/questions/391548 | 0 | Let $\Omega \subset \mathbb{R}^n$ be a bounded Lipschitz set, and let $W^{k,p}(\Omega)$ denote the usual Sobolev space with $k \in \mathbb{N}$ being the order of the derivatives and $p \in [1, \infty)$ the rate of integrability. I know that there exists a result of this type: fix $\varepsilon >0$, then
$$\Vert f\Vert\_... | https://mathoverflow.net/users/198356 | Interpolated Sobolev norm inequality | This can be proved by contradiction. Let $\epsilon > 0$ be given, and $(f\_j \mid j \in \mathbf{N})$ be a sequence of functions in $W^{k,p}(\Omega)$ with $\lvert f\_j \rvert\_{k-1,p} \geq \epsilon \lvert f\_j \rvert\_{k,p} + j \lvert f\_j \rvert\_{0,1}$. Rescale these functions to have $\lvert f\_j \rvert\_{k-1,p} = 1$... | 2 | https://mathoverflow.net/users/103792 | 391551 | 162,022 |
https://mathoverflow.net/questions/391535 | 1 | Young's convolution inequality states that, for $1/p+1/q=1/r+1$ ($1\leq p,\, q, r\leq \infty$), we have $$\lVert f \* g \rVert\_r \leq \lVert f\rVert\_p \lVert g\rVert\_q.$$
It is implicit here that the measure under which these norms are taken is the Lebesgue measure.
Let $\lVert\cdot\rVert\_{L^p\_w}$ denote the wei... | https://mathoverflow.net/users/174195 | Young's convolution inequality for weighted norms | I explained the correct generalization [in a blog post](https://qnlw.info/post/a-weighted-youngs-inequality-201910/). You version cannot hold for two reasons:
1. Your version doesn't have the right number of weights: on the left the weight only appears once and on the right it appears twice. To fix this you need to u... | 5 | https://mathoverflow.net/users/3948 | 391565 | 162,028 |
https://mathoverflow.net/questions/391563 | 2 | I came across a journal which is reputable in terms of not being listed in Beall's list, but which according to [scimagojr](https://www.scimagojr.com/) and [clarivate](https://mjl.clarivate.com/) does not have an impact factor. The journal is [Journal of Numerical
Mathematics and Stochastics](http://www.jnmas.org/index... | https://mathoverflow.net/users/51189 | Examples of reputable journals in mathematics without impact factor? And is it good to publish in them? | Newer journals often don't have impact factors because they don't have enough articles. And this can sometimes happen in odd ways. For example, the *Transactions of the American Mathematical Society* bifurcated into two pieces, where Part A is subscription journal that publishes as both print and electronic and is free... | 17 | https://mathoverflow.net/users/11926 | 391568 | 162,029 |
https://mathoverflow.net/questions/391567 | 5 | Let $U\subset\mathbb R$ be an open set. Let $n\in\mathbb N$ and suppose that $f\in\mathcal C^n(U)$, i.e. that $f$ is $n$-times continuously differentiable on $U$. The $n$-th derivative of $f$, denoted by $f^{(n)}$, then satisfies, for all $x\in U$,
$$f^{(n)}(x)=\lim\_{h\to 0}\frac{\sum\_{k=0}^n f(x+ k h) \binom nk (-1)... | https://mathoverflow.net/users/129831 | Reference for a Grünwald–Letnikov-type definition of the $n$-th derivative of a function | First, let's write out what your expression requires when $n = 2$:
$$ \lim\_{h \to 0} \frac{f(x) + f(x + 2h) - 2 f(x+h)}{h^2} $$
is required to exist for all $x \in U$.
Let $U = (-1,1)$. Take $f(x) = |x|$.
When $x \neq 0$, there exists a sufficiently small interval $I\_x$ around $x$ such that $f|\_{I\_x}$ is $C... | 6 | https://mathoverflow.net/users/3948 | 391576 | 162,032 |
https://mathoverflow.net/questions/391478 | 3 | I think there must be a standard answer to this, for people in the know.
Let $X\subseteq\mathbb{A}^{n}$ be an affine (closed) variety of dimension $\geq d$, and fix some set $\{f\_{i}\}\_{i}$ of polynomials defining $X$. Let $\mathcal{G}=\mathrm{Gr}(n-d,\mathbb{A}^{n})$ be the Grassmannian of $(n-d)$-dimensional line... | https://mathoverflow.net/users/155467 | Variety of subspaces not intersecting $X$ | $L \cap X = \emptyset$ is the same as saying that $\{f\_i\} \cup \{\ell\_j\}$ has no solution, where the $\ell\_j$ are the linear functions defining $L$. By the Nullstellensatz, that's the same as saying there exist polynomials $h\_i,g\_j$ such that $\sum\_i h\_i f\_i + \sum\_j g\_j \ell\_j = 1$. By an Effective Nullst... | 1 | https://mathoverflow.net/users/38434 | 391580 | 162,034 |
https://mathoverflow.net/questions/391586 | 3 | Consider the Dirac operator
$$ H = \begin{pmatrix} m & -i\partial\_z \\ -i\partial\_{\bar z} & -m \end{pmatrix},$$
where $\partial\_{\bar z}$ is the Cauchy-Riemann operator and $m \ge 0.$
It is not hard to see that the spectrum of this operator is symmetric with respect to zero.
However, does there exist a simple... | https://mathoverflow.net/users/108483 | Massive dirac operator symmetric spectrum | With $z=x+iy$, we use the Fourier transformation in $(x,y)$ to see that $H$ is unitarily equivalent to
$$
\frac12\begin{pmatrix}2m&\xi-i\eta\\
\xi+i\eta&-2m\end{pmatrix},
\text{whose eigenvalues are } \lambda\_\pm=\pm\sqrt{m^2+\frac{\vert\zeta\vert^2}{4}}, \ \zeta=\xi+i\eta.
$$
With $\mu=\sqrt{m^2+\frac{\vert\zeta\vert... | 2 | https://mathoverflow.net/users/21907 | 391597 | 162,040 |
https://mathoverflow.net/questions/391593 | 1 | $\DeclareMathOperator\SO{SO}\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\R{\mathbb{R}}$The holonomy of a hyperbolic surface $S$ in terms of differential geometry is either $\SO(2)$ or $\mathrm{O}(2)$ depending on orientability or a hyperbolic structure as a special $(X,G)$-structure $\pi\_1(S)\subset \PSL(2,\R)$. ... | https://mathoverflow.net/users/198622 | What is the relationship between $\mathrm{SO}(2)$ and $\mathrm{PSL}(2,\mathbb{R})$? | Consider $Q(x\_0,x\_1,x\_2)=x\_0^2-x\_1^2-x\_2^2$, and $H=\{(x\_0,x\_1,x\_2):x\_0>0,Q(x\_1,x\_2,x\_3)=1$.
The restriction of $B(x,y)={1\over 2}(Q(x+v)-Q(v))$ to the tangent space of elements of $H$ defines on $H$ a Riemannian metric whose curvature is $-1$. Its group of isometries is the restriction of $O(1,n)$ to $H... | 2 | https://mathoverflow.net/users/80891 | 391605 | 162,043 |
https://mathoverflow.net/questions/391608 | 0 | It's well known that to find a hamilton cycle is NPC, while TSP is NPH.
But it seems that for majority of graphs (**density of edge > 0.1, order > 100**) there is a fast algorithm to find different hamilton cycles if the graph is hamilton graph.
For example, g = graphs.RandomGNP(1300,0.1), it takes **31453 seconds to... | https://mathoverflow.net/users/109312 | Could you provide some TSP examples from real world to test a new algorithm? | Bill Cook's page provides lots of real-world TSP instances of various sizes:
<http://www.math.uwaterloo.ca/tsp/data/index.html>
Cook is one of the developers of the state-of-the-art TSP solver Concorde.
See also TSPLIB: <http://comopt.ifi.uni-heidelberg.de/software/TSPLIB95/>
| 2 | https://mathoverflow.net/users/141766 | 391610 | 162,046 |
https://mathoverflow.net/questions/391614 | 2 | Consider a cubic threefold $Y$ and its associated degree $14$ prime Fano threefold $X$, we have the equivalences of non-trivial components of $D^b(Y)$ and $D^b(X)$, i.e, $\mathcal{A}\_X\cong\mathcal{B}\_Y$ and this equivalence is given by a Fourier-Mukai functor $\Phi:=\Phi\_{I\_Z(H\_Y)}:D^b(X)\rightarrow D^b(Y)$, wher... | https://mathoverflow.net/users/41650 | A Fourier-Mukai equivalence between non trivial component of cubic threefold and degree 14 prime Fano threefold | This is a rational quartic curve.
Indeed, the FM kernel is induced by the HPD kernel, which is, essentially, the locus $\mathbf{Z}$ of pairs $(U,y)$, where $U$ is a 2-dimensional subspace in the fixed 6-dimensional space $V\_6$ and $y$ is a degenerate skew form on $V\_6$ such that
$$
U \cap \operatorname{Ker}(y) \ne ... | 7 | https://mathoverflow.net/users/4428 | 391624 | 162,050 |
https://mathoverflow.net/questions/391598 | 14 | Let $C$ be an abelian category, assume for simplicity that $C$ is enriched over $Vect\_k$ (vector spaces over $k$) for some fixed field $k$.
Suppose also that $C$ is both Artinian and Noetherian, so that for any object $X$ there is a sequence of objects $0=X\_0 \hookrightarrow \ldots \hookrightarrow X\_n = X$ with $X... | https://mathoverflow.net/users/49822 | Recovering an abelian category from the Ext of its simple objects | Here's a counterexample that appears in nature.
Fix a prime $p$ and a field $k$ of characteristic $p$, and let
$G=C\_{p^{n}}$ be a cyclic group of order $p^{n}$ (where $n\geq1$ if
$p$ is odd, and $n\geq2$ if $p=2$).
Then the category $\operatorname{mod}kG$ of finitely generated
$kG$-modules has only one simple modu... | 18 | https://mathoverflow.net/users/22989 | 391625 | 162,051 |
https://mathoverflow.net/questions/391622 | 4 | I know that this is maybe not a research level question, but since the topic is quite special, I thought that the chance to get some reference is higher in this community.
I am looking for a reference (book, review or research paper) of the Palatini or tetradic formulation of general relativity from a more mathematic... | https://mathoverflow.net/users/199422 | Reference for mathematical Palatini formalism of general relativity | The Palatini formalism, a variation of a Lagrangian with respect to the connection, is examined quite rigorously in
* [On the
Palatini method of variation](https://aip.scitation.org/doi/10.1063/1.523699) (1978)
* [The Palatini formulation of general relativity](https://www.cosmo-ufes.org/uploads/1/3/7/0/13701821/lect... | 6 | https://mathoverflow.net/users/11260 | 391626 | 162,052 |
https://mathoverflow.net/questions/391621 | 3 | Let $K$ be a convex set in a normed space $X$. Assume that $int(K)=\emptyset$ (norm topology). Must $K$ be contained in some (affine) hyperplane?
It's fairly easy to see that this is true in $ℝ^n$, but i couldn't generalize.
| https://mathoverflow.net/users/155342 | Convex set with no interior contained in hyperplane? | Based on Jack's comment.
The "Hilbert cube" in Hilbert space $l^2$. $$C :=\{(x\_1,x\_2,\dots) : |x\_k| \le 2^{-k}\;\forall k\}$$
$C$ is convex, compact (so it has empty interior) but has dense span (so it not contained in a closed hyperplane).
---
However, $C$ is contained in a (non-closed) hyperplane. (Axiom o... | 10 | https://mathoverflow.net/users/454 | 391632 | 162,055 |
https://mathoverflow.net/questions/372840 | 7 | For a logic $\mathcal{L}$, say that a cardinal $\kappa$ is $\mathcal{L}$-correct iff every satisfiable $\mathcal{L}$-theory of size $<\kappa$ has a model of size $<\kappa$. First-order correctness is of course boring, but quite quickly we enter the realm of strong large cardinal properties (see e.g. [here](https://math... | https://mathoverflow.net/users/8133 | Lowenheim-Skolem numbers for SOL + correctness quantifiers | Assume ZFC. Let $n$ be a meta-integer. Then $\kappa$ is $\mathcal{L}^2\_n$-correct iff $\kappa$ is $\Sigma\_{n+2}$-reflecting, i.e. $V\_\kappa\preccurlyeq\_{n+2}V$.
Proof: For $n=0$, i.e. 2nd order logic, we have: If $V\_\kappa\preccurlyeq\_2 V$ then easily $\kappa$ is $\mathcal{L}^2\_0$-correct. Suppose now that $\k... | 4 | https://mathoverflow.net/users/160347 | 391634 | 162,056 |
https://mathoverflow.net/questions/391627 | 33 | Most texts on category theory define a (small) **diagram** in a category $\mathcal{A}$ as a functor $D : \mathcal{I} \to \mathcal{A}$ on a (small) category $\mathcal{I}$, called the **shape** of the diagram. A cone from $A \in \mathcal{A}$ to $D$ is a morphism of functors $\Delta(A) \to D$, a limit is a universal cone.... | https://mathoverflow.net/users/2841 | Shapes for category theory | I think focusing on graphs is not a good idea. We focus on functors for very good reasons. Here are a few:
* Many diagrams which are used in practice are functors between categories, and forgetting that they are compatible with composition could seem artificial in many cases.
* We want to compute colimits. A very fun... | 25 | https://mathoverflow.net/users/1017 | 391639 | 162,058 |
https://mathoverflow.net/questions/391603 | 3 | I am reading the cycle class map for singular projective varieties as mentioned by Laterveer in [this](https://arxiv.org/pdf/1507.04483.pdf) article (see Definition $1$). The article does not define the map but refers to an article of Totaro, which also does not define the map but refers to the article "Non-archimedean... | https://mathoverflow.net/users/45397 | Cycle class map for singular varieties | Suppose for simplicity $X$ is proper. Choose a smooth simplicial resolution $X\_\bullet \to X$. We can assume that $X\_0\to X$ is a resolution of singularities. By (a careful reading of) Deligne, *Théorie de Hodge III*, we have an exact sequence
$$0\to Gr^W\_{i} H^i(X)\to H^i(X\_0)\to H^i(X\_1)$$
where the second map i... | 2 | https://mathoverflow.net/users/4144 | 391642 | 162,060 |
https://mathoverflow.net/questions/391616 | 2 | Assume $f(x), x \in \mathbb{R}$ is a function with a compact support such that its Fourier transform $\hat{f}(\xi)$ has a decay rate
$$\hat{f}(\xi) \lesssim \frac{1}{|\xi|^\gamma + 1}$$
for some $\gamma \ge 1$.
Now set $$h(x) = xf(x).$$ Since $f$ has a compact support, $h$ should have similar or better regularity than ... | https://mathoverflow.net/users/114951 | Decay estimate of Fourier transform of a compactly supported function | The answer is positive. Since $f$ has compact support, $g:=\hat{f}$ extends to an
entire function of exponential type $\sigma$ with some $\sigma>0$.
Then your estimate on
the real line and the Phragmen - Lindelof theorem imply that
$$\log |g(z)|\leq \sigma |y|-\gamma\log|z|,\quad z=x+iy,$$
which gives that $|g(z)|=O(|z... | 6 | https://mathoverflow.net/users/25510 | 391648 | 162,062 |
https://mathoverflow.net/questions/391036 | 5 | Let $B$ be a standard Brownian motion in $\mathbb R$. Define the variables
$$\begin{align\*} X &= B\_1, & Y &= \int\_0^1B\_s\mathrm ds, & Z&= \int\_0^1B\_s^2\mathrm ds. \end{align\*}$$
It is known, see below, that $(X,Y,Z)$ admits a smooth density.
>
> Is is true that the density of $(X,Y,Z)$ is a [Schwartz functio... | https://mathoverflow.net/users/129074 | Schwartz regularity for the density of a stochastic process | Using the representation in terms of i. i. d. Gaussians $\xi\_1,\xi\_2,\dots,$
$$
B\_t=\sqrt{2}\sum\_{n=1}^\infty (-1)^{n+1}\xi\_n\frac{\sin \pi \left(n-\frac12\right)t}{\pi \left(n-\frac12\right)},
$$
we get
$$
X=\sqrt{2}\sum\_{n=1}^\infty\frac{\xi\_n}{\pi \left(n-\frac12\right)},\quad Y=\sqrt{2}\sum\_{n=1}^\infty \f... | 2 | https://mathoverflow.net/users/56624 | 391658 | 162,065 |
https://mathoverflow.net/questions/391633 | 7 | Suppose $X\subset \mathbb R^n$ is an irreducible real analytic *sub-variety* (i.e. the set of solutions of a system $f\_1=\ldots=f\_k=0$ with $f\_i$ analytic)
Let $x\in X$ be a point and let $F: X\to \mathbb R^1$ be a continuous function defined on $X$ in a neighbourhood of $x$. I want to understand whether $F$ is re... | https://mathoverflow.net/users/13441 | Real analyticity of continuous function via restriction to analytic curves | No. Take $X = \mathbb R^2$ and $F(x,y) = \frac{x^3}{x^2+y^2}$. Then $F$ is real-analytic everywhere but $(0,0)$. Moreover, at $(0,0)$, any curve passing through $(0,0)$ must have coordinates two analytic functions $x,y$ vanishing to orders $a,b$, in which case $x^2+y^2$ vanishes to order $2\min(a,b)$ and $x^3$ vanishes... | 8 | https://mathoverflow.net/users/18060 | 391659 | 162,066 |
https://mathoverflow.net/questions/391566 | 13 | Which C$^\*$-algebras admit factor states for which the von Neumann algebra it generates in the corresponding GNS representation is a type III$\_1$ factor? For example, do all purely infinite algebras admit such a state?
Even more generally: what about a type III$\_\lambda$ factor for $\lambda\in(0,1]$?
| https://mathoverflow.net/users/123905 | Factor states on C*-algebras | By the main result of [this paper of Odile Maréchal](https://mathscinet.ams.org/mathscinet-getitem?mr=430797), which was generalizing [James Glimm's famous paper](https://mathscinet.ams.org/mathscinet-getitem?mr=124756), the following extreme dichotomy holds. If $A$ is any separable C\*-algebra, precisely one of the fo... | 8 | https://mathoverflow.net/users/159170 | 391663 | 162,068 |
https://mathoverflow.net/questions/391654 | 4 | Suppose I have a finite (non-)commutative ring $R/k$ (over a field $k$ of char $0$) with a linear "trace" function $t: R \to k$. Can I always find an embedding $f: R \to M\_r(k)$ compatible with the trace functions on both sides?
One restriction I can see for the trace function on $R$ is that it should be invariant u... | https://mathoverflow.net/users/58001 | Can all finite-dimensional noncommutative algebras with trace be trace-preserving embedded into matrix rings? | You cannot always find such an embedding. Consider the ring $R=\mathbb{Q}\langle x,y\rangle$ subject only to the condition that any monomial in the letters $x$ and $y$ of degree $3$ is zero. This is a noncommutative ring, finite dimensional over $\mathbb{Q}$, and the natural factor map $t\colon R\to R/(x,y)\cong \mathb... | 6 | https://mathoverflow.net/users/3199 | 391664 | 162,069 |
https://mathoverflow.net/questions/391244 | 4 | Consider a finite, simple and undirected graph $G=(V,E)$ with $V=\{v\_1,\dots, v\_n\}$. Let us define the quantity:
$$\mathcal{I}\_k(G) := \sum\_{1\le i,j \le n} \mathbb{1}{\Big\{|\mathrm{deg}(v\_i)-\mathrm{deg}(v\_j)|\ge k}\Big\},$$
i.e. the number of all those pairs $(v\_i,v\_j)$ with degree difference greater or e... | https://mathoverflow.net/users/nan | High degree differences in bipartite graphs | Please check carefully.
Denote $d\_i=\deg(v\_i)$, $\delta\_i=\deg(w\_i)$. We assume (without loss of generality) that $d\_1\geqslant d\_2\geqslant\ldots \geqslant d\_n$ and $\delta\_1\geqslant \delta\_2\geqslant\ldots \geqslant \delta\_n$. We start with the observation similar to that of Erdős, Chen, Rousseau and Sch... | 0 | https://mathoverflow.net/users/4312 | 391672 | 162,071 |
https://mathoverflow.net/questions/391669 | 10 | Let $X\_1,\dots,X\_n$ be non commutative variables such that $\operatorname{tr} f(X\_1,\dots,X\_n) = 0$ whenever the $X\_i$ are specialized to square matrices in $M\_r(k)$ for any $r \geq 1$. Does this imply that $f$ is in the ideal generated by cyclic permutations: $g\_1\dots g\_k - g\_2\dots g\_k g\_1$ for any polyno... | https://mathoverflow.net/users/58001 | Are the trace relations among matrices generated by cyclic permutations? | The reformulation suggested by Christian Remling and Benjamin Steinberg is true (at least over a field $k$ of characteristic zero):
If $\operatorname{tr} f(X\_1,\dots, X\_n)=0$ for all $X\_1,\dots, X\_n$ in $M\_r(k)$ then $f$ is a linear combination of differences of cyclically permuted words.
An equivalent, linear... | 12 | https://mathoverflow.net/users/18060 | 391674 | 162,072 |
https://mathoverflow.net/questions/391583 | 1 | Let $X$ be a smooth quasi-projective toric variety of dimension $n$ over $\mathbb C$.
Take it to be non-compact, so its fan is not complete.
(A good example to keep in mind is a toric Calabi-Yau.)
If we denote by $D\_i$ for $i=1,\ldots,d$ the irreducible subvarieties of codimension one that are stable under the torus... | https://mathoverflow.net/users/40154 | divisors in non-compact toric varieties | As Piotr mentioned in the comments, you don't need completeness of the fan to be sure that the sequence
$$0 \to M \to \oplus \mathbb Z D\_i \to Cl(X\_{\Sigma}) \to 0$$
is exact. The only condition that you actually need is that fan $\Sigma \subset N\_{\mathbb R}$ is not contained inside (real) linear subspace of codime... | 4 | https://mathoverflow.net/users/54337 | 391679 | 162,074 |
https://mathoverflow.net/questions/391677 | 1 | Let $S$ be subset of $\mathbb{R}^n$ with perimeter 1.
Isoperimetric inequality states that then the volume of $S$ is not greater than $V\_n$,
where $V\_n$ is the volume of a ball in $\mathbb{R}^n$ with perimeter 1.
Assume that $C \cdot \text{[Volume of }S] \ge V\_n$, where $C$ is some constant.
>
> Is it true... | https://mathoverflow.net/users/31356 | Stability of isoperimetric inequality | We can assume that the set is a union of disjoint balls on a large distance from each other.
Indeed, cut the space into cubes of small fixed size $a$.
Shifting $S$ we can assume that the total area of intersection of $S$ with cutting hyperplanes is less then some constant that depends on $a$.
Now take intersection of $... | 1 | https://mathoverflow.net/users/1441 | 391684 | 162,077 |
https://mathoverflow.net/questions/382956 | 5 | The Lagarias inequality, which is equivalent to the Riemann hypothesis, is:
$$\sigma(n) \le H\_n + \exp(H\_n) \log(H\_n) =:L(n)$$
for all natural numbers $n$, where $\sigma=$ sum of divisors, $H\_n=n$-th harmonic number.
Lagarias inequality is equivalent to:
$$\sigma\left(\frac{ab}{\gcd(a,b)^2}\right) \le L\left(... | https://mathoverflow.net/users/165920 | Can the Lagarias inequality be written as a "kernel inequality"? | There are several questions being asked here. I'll just deal with the easiest one, showing that $${\sigma(a)\sigma(b)\over\sigma(\gcd(a,b))}=\sigma\left({ab\over\gcd(a,b)}\right)$$
Let $p$ be a prime, let $p^c\|a$ (meaning $a$ is a multiple of $p^c$ but not of $p^{c+1}$), let $p^d\|b$, let $r$ be the larger, and $s$ ... | 2 | https://mathoverflow.net/users/3684 | 391685 | 162,078 |
https://mathoverflow.net/questions/391670 | 14 | Let $M$ be a (loopless) matroid of rank $r$.
The *characteristic polynomial* $\chi\_M(x)$ is defined by $\chi\_M(x)=\sum\_{F \in \mathcal{L}(M)}\mu(\hat{0},F) \cdot x^{\mathrm{rk}(F)}$, where $ \mathcal{L}(M)$ is the lattice of flats of $M$ and $\mu$ its Möbius function. It is known that the signs of the characterist... | https://mathoverflow.net/users/25028 | Log-concavity of matroids: characterization of equality? | I think the following shows it's never possible for there to be equality.
Indeed, Ardila-Denham-Huh <https://arxiv.org/abs/2004.13116> recently showed for any matroid $M$ that $T\_M(x,0)$ has log-concave coefficient sequence, and hence obviously $\frac{1}{x}T\_M(x,0)$ has log-concave coefficient sequence (Note that $... | 8 | https://mathoverflow.net/users/3404 | 391687 | 162,079 |
https://mathoverflow.net/questions/387285 | 4 | For a real number $x$, we denote
$$ \|x\|=\inf\_{m\in {\Bbb Z}}|x+m|.$$
Problem 1:
Roth's theorem states that given any irrational algebraic number $\alpha$ and for any $\epsilon>0$, there exists a constant $C(\alpha,\epsilon)$ such that
$$\|q\alpha\|>\frac{C(\alpha,\epsilon)}{q^{1+\epsilon}}$$
for each positive ... | https://mathoverflow.net/users/148253 | More about Roth's theorem: bound for the constant and multidimensional case | Prof. Lilu Zhao of Shandong University informed about the follwing paper
[Simultaneous approximation to algebraic numbers by rationals](https://projecteuclid.org/journals/acta-mathematica/volume-125/issue-none/Simultaneous-approximation-to-algebraic-numbers-by-rationals/10.1007/BF02392334.full)
in which Theorem 2 s... | 1 | https://mathoverflow.net/users/148253 | 391693 | 162,082 |
https://mathoverflow.net/questions/391700 | 5 | Let $P\_{d, n}$ be the space of polynomial maps $\mathbb{R}^n\to \mathbb{R}$ of degree at most $d$.
Is the subset $S\subset P\_{d, n}$ of nowhere negative polynomials semialgebraic?
| https://mathoverflow.net/users/200799 | Nowhere negative polynomials form a semialgebraic set | As I said in the comments this is very well known: $S$ is the complement of the projection of the semialgebraic set $\{(f,a)\in P\_{d,n}\times\mathbb{R}^n:f(a)<0\}$, hence semialgebraic by the Tarski-Seidenberg theorem.
| 10 | https://mathoverflow.net/users/50351 | 391705 | 162,085 |
https://mathoverflow.net/questions/391708 | 0 | A function $f:\mathbb{Z}\_{\geq 1}\to\mathbb{Z}\_{\geq 1}$ overwhelms $g:\mathbb{Z}\_{\geq 1}\to\mathbb{Z}\_{\geq 1}$ if for any $k\in \mathbb{Z}\_{\geq 1}$ the inequality $f(n)\leq g(n+k)$ holds only for finitely many $n\in\mathbb{Z}\_{\geq 1}$.
For example $n\to n^2$ overwhelms $n\to n$.
Does the number of non-is... | https://mathoverflow.net/users/200598 | Are there overwhelmingly more finite posets than finite groups? | On one hand, the number of groups of order $n$ is at most $2^{O((\log n)^3)}$ (see [here](https://math.stackexchange.com/a/422232/127263)). On the other hand, by considering posets which are disjoint unions of total orders, the number of posets of order $n$ is at least equal to the number $p(n)$ of partitions of $n$. S... | 9 | https://mathoverflow.net/users/30186 | 391709 | 162,086 |
https://mathoverflow.net/questions/391715 | 3 | Suppose $A$ is a $k\_1\times k\_2$ matrix with real entries, $k\_1<k\_2$. Let $M$ be the matrix
\begin{equation}
M:=\begin{pmatrix}
0\_{k\_1} & A\\ A^\top & 0\_{k\_2}
\end{pmatrix},
\end{equation}
where $0\_k$ denotes the $k\times k$ zero matrix. I know that if $\lambda$ is an eigenvalue of $M$ then $\lambda^2$ must be... | https://mathoverflow.net/users/99648 | Eigenvalues of a block matrix with zero diagonal blocks | If you decompose $M=\begin{pmatrix} X\_{q\times q}&Y\_{q\times k\_3}\\ (Y\_{q\times k\_3})^{\rm T}&0\_{k\_3\times k\_3}\end{pmatrix}$ into four block matrices, with $q=k\_1+k\_2$, then the determinant equals
$$\det M=(-1)^{k\_3}(\det X\_{q\times q})\det[(Y\_{q\times k\_3})^{\rm T}X\_{q\times q}^{-1}Y\_{q\times k\_3}].$... | 1 | https://mathoverflow.net/users/11260 | 391719 | 162,089 |
https://mathoverflow.net/questions/366103 | 2 | Zariski's Lemma is the following:
>
> Let $K$ be a field and $R$ be a $K$-algebra with $R=K[x\_1,\dots,x\_n]$
> for some $x\_1,\dots,x\_n\in R$. If $R$ is a field then $x\_1,\dots,x\_n$
> are algebraic over $K$.
>
>
>
Oskar Zariski used this Lemma to prove Hilbert's Nullstellensatz. Is there another non-trivia... | https://mathoverflow.net/users/150594 | Application of Zariski's Lemma other than Hilbert's Nullstellensatz | You can use Zariski's lemma to show that if $R$ is a finitely generated $\mathbf Z$-algebra, then its residue fields $R/\mathfrak m$ for maximal ideals $\mathfrak m$ are all finite. That leads to a description of all the maximal ideals in $\mathbf Z[x]$, for instance. See Section 5 [here](https://kconrad.math.uconn.edu... | 6 | https://mathoverflow.net/users/3272 | 391720 | 162,090 |
https://mathoverflow.net/questions/391585 | 4 | Reference: [Lectures on Analytic Geometry](https://www.math.uni-bonn.de/people/scholze/Analytic.pdf)
Let $f\colon(\mathcal A,\mathcal M)\to(\mathcal B,\mathcal N)$ be a map of analytic ring. There are several possible ways to pose the flatness:
1. Flatness as the base change functor $D\_{\ge0}(\mathcal A,\mathcal M... | https://mathoverflow.net/users/176381 | Flatness of maps of analytic rings | Flatness in analytic geometry is an interesting question! As Dustin says, it comes with several important caveats.
First, open immersions may not be flat even in the weakest sense of the word. Here is an instructive example. Let $K$ be your favourite analytic field ($\mathbb C$ or $\mathbb Q\_p$ will do) equipped wit... | 4 | https://mathoverflow.net/users/6074 | 391731 | 162,095 |
https://mathoverflow.net/questions/258270 | 10 | Thomason showed that symmetric monoidal categories model all connective spectra. But can it be done with groupoids? In order for this to be the case, the group completion prices process must be very drastic. But after all, the sphere spectrum is the group completion of the symmetric monoidal groupoid of finite sets, so... | https://mathoverflow.net/users/2362 | Do symmetric monoidal groupoids model all connective spectra? | Theorem 5.3 of
>
> Daniel Fuentes-Keuthan, *Modelling Connective Spectra via Multicategories*, [arXiv:1909.11148](https://arxiv.org/abs/1909.11148).
>
>
>
answers this question positively!
| 5 | https://mathoverflow.net/users/130058 | 391732 | 162,096 |
https://mathoverflow.net/questions/391591 | 4 | In [Lectures on Analytic Geometry](https://www.math.uni-bonn.de/people/scholze/Analytic.pdf), for complex-analytic geometry, seemingly one only considers maps $(\mathbb C,\mathcal M\_{<p})\to(\mathcal A,\mathcal M)$ of analytic rings for $0<p\le1$ where $A$ is a "structure ring" like $\mathcal O(\overline D)$ and $M$ i... | https://mathoverflow.net/users/176381 | Non-induced analytic structures in complex-analytic case | Great question! So far, we haven't been able to produce analytic ring structures on $\mathbb C$-algebras that are not induced. Similarly, if we equip $\mathbb Q\_p$ with a liquid analytic ring structure, we are also only able to use induced analytic ring structures: To "overconvergent rigid spaces"(=Größe-Klönne's dagg... | 5 | https://mathoverflow.net/users/6074 | 391733 | 162,097 |
https://mathoverflow.net/questions/391718 | 33 | The post below [first appeared](https://hsm.stackexchange.com/questions/13135/first-use-of-term-hilberts-nullstellensatz) on hsm.stackexchange over a week ago and has received no answers there yet, so by now I think it is okay to ask it here.
This year (2021) marks the 100th anniversary of Emmy Noether's 1921 paper i... | https://mathoverflow.net/users/3272 | First use of term "Hilbert's Nullstellensatz" | Below is the Dutch paper mentioned by Francois Ziegler; This paper did indeed appear before the 1927 paper mentioned in the OP, but Van der Waerden does refer to that forthcoming publication in a footnote; I translate:
>
> This theorem is a special case of the "Nulpuntenstelling" of
> HILBERT$^{5})$
>
>
> $^{5})$... | 15 | https://mathoverflow.net/users/11260 | 391734 | 162,098 |
https://mathoverflow.net/questions/391726 | 15 | I'm writing some notes for some students which just finished a first course in scheme theory. There I would like to talk about constructible sheaves, but I found it hard to give a compelling motivation for these objects. (And I don't like to give a definition without at least trying to explain why this is a nice thing ... | https://mathoverflow.net/users/131975 | How to motivate constructible sheaves | Even if you're only interested in say cohomology with coefficients in the constant sheaf, working with constructible sheaves gives you extra flexibility and is more amenable to inductive proofs.
Here is a basic theorem in the topology of algebraic varieties one of whose proofs could serve as a motivation. I discussed... | 22 | https://mathoverflow.net/users/3847 | 391737 | 162,100 |
https://mathoverflow.net/questions/391748 | 6 | By geometric theory of dynamical systems, I mean the kind found in [the book by Palis](https://www.springer.com/gp/book/9781461257059), or papers like [this one](https://arxiv.org/abs/1909.13149). In other words, dynamics on manifolds, but not specifically hyperbolic dynamical systems.
What are some recommended paper... | https://mathoverflow.net/users/173490 | What are the current research directions in the geometric theory of dynamical systems? | I think the book by Bonatti, Diaz and Viana: "Dynamics beyond uniform hyperbolicity' can give you a nice overview of one possible point of view. <https://link.springer.com/book/10.1007/b138174>
The book by Katok-Hasselblatt and their Handbook contains a lot of other points of view.
| 4 | https://mathoverflow.net/users/5753 | 391749 | 162,104 |
https://mathoverflow.net/questions/391738 | 1 | On page 92 of [these notes](https://web.math.princeton.edu/%7Eseri/homepage/courses/Analysis2011.pdf), there is a discussion on how to find the fundamental solution to the D'Alembertian operator. It is firstly proposed that $c\_n(t^2 - |x|^2 )^{-(n-1)/2}$ may be a good candidate. Then the author says that we can use th... | https://mathoverflow.net/users/121404 | Classification of homogeneous distributions | In sergiu's notes that you referred to, $j\_a$ is defined in Definition 3.2 on Page 65. See equation (134).
| 2 | https://mathoverflow.net/users/3948 | 391755 | 162,107 |
https://mathoverflow.net/questions/391776 | 34 | As a sentential logic, intuitionistic logic plus the law of the excluded middle gives classical logic.
Is there a logical law that is consistent with intuitionistic logic but inconsistent with classical logic?
| https://mathoverflow.net/users/136356 | Alternatives to the law of the excluded middle | No, every consistent propositional logic that extends intuitionistic logic is a sublogic of classical logic. (That’s why consistent superintuitionistic logics are also called *intermediate* logics.)
To see this, assume that a logic $L\supseteq\mathbf{IPC}$ proves a formula $\phi(p\_1,\dots,p\_n)$ that is not provable... | 35 | https://mathoverflow.net/users/12705 | 391778 | 162,115 |
https://mathoverflow.net/questions/391697 | 3 | I want to find the critical point of tensor $f=a\_0b\_0c\_0 + a\_1b\_1c\_1$ in $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$, and I followed this construction:
First, I take the following partial derivative:
With respect to $a$, $\cfrac{\partial f }{\partial a\_0}=b\_0c\_0, \cfrac{\partial f }{\partial a... | https://mathoverflow.net/users/117508 | Eigenvectors of a tensor in $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$ | **Recall**. Consider a $d$-dimensional rectangular tensor $T$ in $\mathbb{K}^{n\_1 \times \dots \times n\_d}$. It corresponds to a multilinear form:
$$T=\sum\_{i\_1=1}^{n\_1} \sum\_{i\_2=1}^{n\_2} \dots \sum\_{i\_n=1}^{n\_d} t\_{i\_1 \dots i\_d}x\_{i\_1}x\_{i\_2} \dots x\_{i\_d}.$$
The **singular vector tuples** (criti... | 2 | https://mathoverflow.net/users/117508 | 391782 | 162,117 |
https://mathoverflow.net/questions/391767 | 5 | $\DeclareMathOperator\bso{\beta^\*\!\omega}\DeclareMathOperator\Homeo{Homeo}$Let $\bso$ be the complement of the countable discrete space $\omega$ in its Stone-Čech compactification $\beta\omega$ (some authors denote it $\omega^\*$).
>
> **Question.** Assume ZFC+CH. Does there exist a self-homeomorphism of $\bso$ w... | https://mathoverflow.net/users/14094 | Self-homeomorphism of Stone-Čech boundary with an isolated fixed point | The answer to your main question is *yes*. In fact, there is (under $\mathsf{CH}$) a self-homeomorphism of $\omega^\*$ with exactly one fixed point. Such a mapping is constructed in the proof of Theorem 5.7 in my paper:
>
> "$P$-sets and minimal right ideals in $\mathbb N^\*$," *Fundamenta Mathematicae* **229** (20... | 4 | https://mathoverflow.net/users/70618 | 391790 | 162,119 |
https://mathoverflow.net/questions/391785 | 5 | The two-dimensional complex unit ball $B$ has group of biholomorphic automorphisms $PU(2,1)$.
If $Γ$ is an arithmetic subgroup of $PU(2,1)$, the quotient $Γ\text{\\}B$ is an orbifold.
Taking its minimal resolution gives a complex manifold $X$.
**Question:** How to compute the Hodge numbers of $X$, given the group... | https://mathoverflow.net/users/125498 | Computing the invariants of ball quotient surfaces | I'm assuming that you know "where" in the commensurability class your lattice is. By this, I mean you perhaps have $\Gamma$ as a subgroup of some principal arithmetic lattice $\Lambda$ of known index, e.g., as a subgroup of $\mathrm{PU}(h, \mathcal{O}\_k)$ where $h$ is a hermitian form on $k^3$ for an appropriate numbe... | 7 | https://mathoverflow.net/users/142269 | 391792 | 162,120 |
https://mathoverflow.net/questions/391753 | 2 | Given a real number $x \in (0, 1)$, we denote by $0.x\_1x\_2\ldots$ its binary expansion, where we always choose the expansion that ends in an infinite number of $1$’s whenever a choice has to be made.
Given two real numbers $a = 0.a\_1a\_2\ldots$ and $b = 0.b\_1b\_2...$ in $(0, 1)$, denote by $S(a, b)$ the set of in... | https://mathoverflow.net/users/173490 | Well approximating sets | Let $D$ consist of all numbers $d$ in $[0,1]$ such that in their binary expansion for every $k$ the digit at location $2^k$ vanishes, i.e., $d\_{2^k}=0$. Then it is easy to check that $D$ has Lebesgue measure zero (Indeed it can be covered by $2^{2^k-k}$ intervals of length $2^{-2^k}$ for each $k$). For every $x \in (0... | 4 | https://mathoverflow.net/users/7691 | 391806 | 162,126 |
https://mathoverflow.net/questions/391760 | 5 | Let $v$ be a given vector with $\|v\|\_{\Sigma^{-1}} \leq 1$, where $\Sigma$ is a positive semi-definite matrix and $\|v\|\_{\Sigma^{-1}} = \sqrt{v^\top\Sigma v}$. Meanwhile, let $u$ be a random vector drawn from $N(0,\Sigma^{-1})$. We know that for any given vector $\phi$, it holds that
$$
P(\phi^\top u > \phi^\top v)... | https://mathoverflow.net/users/82358 | Anti-concentration of Gaussian when conditioning on event | $\newcommand{\si}{\sigma}\newcommand{\Si}{\Sigma}\newcommand{\R}{\mathbb R}\newcommand{\ep}{\varepsilon}$The answer is: in general, no -- even for convex $\mathcal C$.
Indeed, let $C:=\mathcal C=(-\infty,1)\times\R$, $v=(0,0)$, $f:=\phi=(1,0)$, and $\Si=\begin{pmatrix}\si^2&0\\0&1\end{pmatrix}$, with $\si\to\infty$. ... | 2 | https://mathoverflow.net/users/36721 | 391807 | 162,127 |
https://mathoverflow.net/questions/391822 | 5 | Let $\mathcal{C}$ and $\mathcal{D}$ be categories and let $T : \mathcal{C} \rightarrow \mathcal{D}$ be a functor. Suppose that $F : \mathcal{D}^\mathrm{op} \rightarrow \mathrm{Set}$ is a functor. (So in the older language I am more used to, $F$ is a contravariant functor.) Let $Y \in \mathcal{D}$ be an object. Under wh... | https://mathoverflow.net/users/7709 | Yoneda map for a composition of a representable functor and an arbitrary functor | This property of the functor $T$ is called being a [dense functor](https://ncatlab.org/nlab/show/dense+functor), or "densely generating functor". The notion was first introduced by Isbell in the case where $T$ is fully faithful under the name "adequate subcategory", but that usage has by now disappeared, and "dense" is... | 7 | https://mathoverflow.net/users/2362 | 391825 | 162,130 |
https://mathoverflow.net/questions/391812 | 8 | Let $R$ be a (noncommutative, associative) ring. Set $N\_2:=\{x\in R : x^2=0\}$, the set of nilpotent elements of degree $2$ (also called the square-zero elements).
If $x,y\in R$ satisfy $xy=0$, then $yx\in N\_2$, but not every element in $N\_2$ arises in this way. (See the example below.)
>
> Question: Has the s... | https://mathoverflow.net/users/24916 | Special nilpotent elements | The answer to your precise question is no: it is not always the case that $F = N\_2\cap [R,R]$. A nice way to see this is by fixing a field $k$, and constructing the universal example of a $k$-algebra equipped with a square-zero commutator. That is, $R = k\langle a,b\rangle/((ab - ba)^2)$.
In that $k$-algebra, the el... | 4 | https://mathoverflow.net/users/nan | 391835 | 162,135 |
https://mathoverflow.net/questions/391757 | 3 | Let $f\_n: [0, 1] \to \mathbb R$ be an equicontinuous sequence of functions. Does there exist a continuous function $f$ that dominates $f\_n $ in the following sense?
We say $f$ dominates the sequence $f\_n$ if for all $x \in [0, 1]$, if $\delta > 0, \varepsilon > 0$ are such that $\lvert f(x) - f(y)\rvert < \varepsi... | https://mathoverflow.net/users/173490 | Are equicontinuous function dominated by a continuous function? | It is well known that if $(f\_n)$ are equicontinuous on a compact space, then they are uniformly equicontinuous. Let $\omega(\delta)=\sup\{|f\_n(x)-f\_n(y)|\colon
|x-y|\le\delta; n\in\mathbb N\}$ be the modulus of continuity of the family, so that $\omega(\delta)\to 0$ as $\delta\to 0$. I will assume without loss of ge... | 6 | https://mathoverflow.net/users/11054 | 391841 | 162,136 |
https://mathoverflow.net/questions/391809 | 9 | Let $f: [0,\infty) \rightarrow \mathbb{R}$ be a continuous function such that $f(0) = 0$. Is it true
that if the integral
$$
\int\_0^{\pi/2} \sin(\theta) f(\lambda \sin(\theta)) \, d\theta
$$
is zero for every $\lambda > 0$, then $f$ is identically zero?
It's rather obviously true if $f$ is a polynomial and I'm hopin... | https://mathoverflow.net/users/21123 | Simple-looking problem with integrals | As suggested by Fedor Petrov, we write
$$ g(x) = f(\sqrt x) , $$
and we substitute $\lambda \sin\theta = \sqrt{x}$ and $t = \lambda^2$. This leads to
$$ \begin{aligned} 0 & = 2 \lambda \int\_0^{\pi/2} f(\lambda \sin \theta) \sin \theta \, d\theta \\ & = \int\_0^{\lambda^2} 2 f(\sqrt x) \sqrt{x} \, \frac{1}{2 \sqrt{x (\... | 12 | https://mathoverflow.net/users/108637 | 391852 | 162,138 |
https://mathoverflow.net/questions/390511 | 1 | Is there a known bound on the norm of the standard intertwining operator for the principal series of $G = \operatorname{GL}\_2(\mathbb Q\_p)$?
**Background:**
For a character $\chi = (\chi\_1,\chi\_2)$ of the standard torus $T$ in $G$, extended to a character of the standard Borel subgroup $B = TU$ we can define the ... | https://mathoverflow.net/users/38145 | The norm of the principal series intertwining operator for $\operatorname{GL}_2$ | Having a bound on $\|M\|\_1$, you get by duality a bound on $\|M\|\_\infty$
and then you get a bound on $\|M\|\_p$ for every $1\leq p\leq\infty$
by the [Riesz–Thorin theorem](https://en.wikipedia.org/wiki/Riesz%E2%80%93Thorin_theorem).
| 2 | https://mathoverflow.net/users/89334 | 391856 | 162,139 |
https://mathoverflow.net/questions/391854 | 6 | Suppose $G$ is a Lie group, with $\pi\_0(G)$ **not necessarily finite**, but might as well assume $G\_0$, the connected component of the identity, is compact.
In the case that $\pi\_0(G)$ *is* finite, then we know that there is an injection $H^\*(BG,\mathbb{Q})\to H^\*(BG\_0,\mathbb{Q})$, and this can apparently be s... | https://mathoverflow.net/users/4177 | Cohomology of BG, G non-connected Lie group, and spectral sequence relating to classifying space of connected component of the identity | Think about the case where $\pi\_0(G)=\mathbb{Z}$, so $B(\pi\_0(G))=S^1$, so we have a fibre bundle $BG\_0\to BG\to S^1$. In this case $G$ is always a semidirect product formed using an automorphism $\alpha$ of $G\_0$. By considering the preimages of the the complements of two points in $S^1$, we can express $BG$ as $U... | 8 | https://mathoverflow.net/users/10366 | 391862 | 162,142 |
https://mathoverflow.net/questions/391851 | 2 | I am interested in the study of the (semi-linear, I suppose) equation
$$\begin{cases}-\Delta u(x,y)+q(x)u(x,y)+h(x)=f(u(x,y)-kx),\;\;(x,y)\in\Omega,\\
u=g,\;\;\;\text{on }\partial\Omega.\end{cases}$$
on an open bounded domain $\Omega\subset\mathbb{R}^2$ with piecewise $C^1$ boundary. Here $k\in\mathbb{R}$ is a constant... | https://mathoverflow.net/users/105925 | Reference request for semilinear PDEs in dimension 2 | You are in dimension $2$, which is nice, because $H^{1}(\Omega)\hookrightarrow L^N(\Omega)$ for any $N<\infty$ so you can look for a normal, weak solution. The other good news is that behind your problem is a compact operator.
Let us assume $q\in L^{\infty}\left(\Omega\right)$, $q\geq0$, $g\in H^{1/2}\left(\partial\O... | 5 | https://mathoverflow.net/users/40120 | 391865 | 162,144 |
https://mathoverflow.net/questions/391866 | 1 | **Definition**: The *subgroup rank* of a finite group $G$ is the minimal natural number $n$ such that every subgroup of $G$ can be generated by $n$ elements (or fewer).
This invariant has been studied extensively for various families of groups. I am interested in the family of finite simple groups and I have been una... | https://mathoverflow.net/users/203598 | Subgroup rank of finite simple groups | By a result of R.Guralnick and A. Lucchini (obtained independently, not in joint work) the subgroup rank of a finite group $G$ is bounded above by $1+d$, where $d$ is the maximum over all $p$-subgroups $P$ of $G$ (and over all primes $p$) of the minimum number of generators of $P$. This requires the classification of f... | 3 | https://mathoverflow.net/users/14450 | 391868 | 162,145 |
https://mathoverflow.net/questions/391878 | 3 | Let $(M,G,\alpha)$ be a $W^\ast$-dynamical system with $G$ locally compact abelian (I am mostly interested in the case $G=\mathbb{R})$. A covariant representation of $(M,G,\alpha)$ is a pair $(\pi,u)$ consisting of a normal representation $\pi$ of $M$ on a Hilbert space $H$ and a strongly continuous unitary representat... | https://mathoverflow.net/users/95776 | Covariant representations and crossed products of von Neumann algebras | No, such a one-to-one correspondence does not hold. For instance, if $G$ is a countable infinite group and $G \curvearrowright (X,\mu)$ is an essentially free, ergodic, probability measure preserving action, the crossed product $M = L^\infty(X) \rtimes G$ is a II$\_1$ factor. At the same time, the representation $\pi :... | 4 | https://mathoverflow.net/users/159170 | 391881 | 162,149 |
https://mathoverflow.net/questions/391249 | 6 | Let us consider the free Schrödinger equation $(i\partial\_t+\Delta\_x)\psi=0$ in $\mathbb{R}\_t\times\mathbb{R}\_x^d$. I'm trying to understand the structure of the vacuum region
$$\Omega(\psi):=\{(t,x)\in \mathbb{R}\_t\times\mathbb{R}\_x^d \;\;s.t.\;\,\psi(t,x)=0\}$$
for solutions with finite energy. In particular, m... | https://mathoverflow.net/users/169603 | Vacuum region with positive measure for the Schrödinger equation | The purpose of this answer is to extend [Christian Remling's answer](https://mathoverflow.net/a/391661/3948) to dimension $d = 3$. There are two steps. (N.B. below the cut I show how to replace part 1 by a different argument that works in all dimensions, and so this should answer the question posed.)
We assume that w... | 7 | https://mathoverflow.net/users/3948 | 391889 | 162,152 |
https://mathoverflow.net/questions/391853 | 3 | I am looking for a reference for the following admittedly imprecise statement:
>
> Any projective invariant of n points in the projective plane may be
> expressed as a function of well-chosen cross-ratios.
>
>
>
By *projective invariant* I mean a rational function defined on the set of $n$-tuple of distinct po... | https://mathoverflow.net/users/6129 | Projective invariants of the plane and cross ratio | I believe the precise statement you are looking for is proposition 3.3 on page 26 of Danylo Radchenko's doctoral dissertation, [*Higher cross-ratios and geometric functional equations for polylogarithms*](https://core.ac.uk/download/pdf/223015944.pdf) (Rheinischen Friedrich-Wilhelms-Universität Bonn, 2016): the proof o... | 3 | https://mathoverflow.net/users/17064 | 391891 | 162,153 |
https://mathoverflow.net/questions/391149 | 1 | The reference for what I'm asking is page $107$ from Folland's harmonic analysis.
$G$ is a locally compact abelian group with dual $\hat{G}$. Let $H$ denote the Hilbert space $L^2(G)$.
I'm trying to understand the 'direct integral' interpretation (and it's utility) for the regular representation
$$L : G \to U(H);\ L\... | https://mathoverflow.net/users/105628 | Understanding the regular representation of an LCA group as a 'direct integral' | I have an old edition of the book.
Let $h\_1, h\_2 \in L^2(G)$, let $l\_i := F^{-1}h\_i \in L^2(\hat{G})$ and let $E\subset \hat{G}$ be Borel.
We know the Gelfand (Fourier) transform sends $L^1(G)$ into a dense subset of $C\_0(\hat{G})$.
Choose a sequence $f\_n$ such that we have pointwise (bounded) convergence $\xi(... | 0 | https://mathoverflow.net/users/105628 | 391892 | 162,154 |
https://mathoverflow.net/questions/391882 | 4 | For every $x,y\in\mathbb R$ let
$$ V(x,y) \,\equiv\, a\,x^{2n} + b\,y^{2m} - \omega(x,y)\,$$
where $a,b>0$, $n,m\in\mathbb N$, $n\geq m\geq1$, and $\omega$ is such that $\omega(x,y)/(x^{2n}+y^{2m})\to 0 \ \ \textrm{as}\ |x|+|y|\to\infty\,.$
Moreover assume $V(x,y)$ is a convex polynomial.
Is it possible to estimate t... | https://mathoverflow.net/users/58793 | Estimate of $\frac{\int x^{2p}\,e^{-x^{2n}\,+\,\omega(x,y)}\;dx}{\int e^{-x^{2n}\,+\,\omega(x,y)}\;dx}$ | $\newcommand{\R}{\mathbb R}\newcommand{\ep}{\varepsilon}
\newcommand{\om}{\omega}$We have to show that
\begin{equation\*}
\frac{I\_1+I\_2}J\ll 1+y^{2p}, \tag{1}
\end{equation\*}
where
\begin{equation\*}
I\_1:=\int\_{|x|\le|y|} dx\,x^{2p}e^{-ax^{2n}+\om(x,y)},
\end{equation\*}
\begin{equation\*}
I\_2:=\int\_{|x|>|y|}... | 2 | https://mathoverflow.net/users/36721 | 391895 | 162,155 |
https://mathoverflow.net/questions/391816 | 9 | Let $n$ be a positive integer and $G$ be a finite group of $n\times n$ matrices with integer coefficients, i.e. $G\subset\operatorname{GL}\_n(\mathbb{Z})$. It is known that for sufficiently large $n$, the maximum order of such a group is $2^nn!$ by Feit (although relying on an unpublished manuscript of Weisfeiler I bel... | https://mathoverflow.net/users/202500 | On the coefficients that appear in finite groups of matrices with integer entries | It is possible to get a number of distinct coefficients exponential in $n$. Here is an example.
Let
$$B = \begin{bmatrix}
1 & -1 & -1 & -1 & \cdots & x\_1 \\
0 & 1 & 0 & 0 & & x\_2 \\
0 & 0 & 1 & 0 & \cdots & x\_3 \\
0 & 0 & 0 & 1 & & \vdots \\
& \vdots & & & \ddots & x\_{n-1} \\
0 & 0 & 0 & 0 & 0 & 1
\end{bmatrix}... | 8 | https://mathoverflow.net/users/160416 | 391900 | 162,157 |
https://mathoverflow.net/questions/390991 | 3 | I am a hobby computer scientist and searching for an algorithm to construct a set of **n** numbers (integers) with certain properties.
**Property 1 / Step 1**
All pairwise differences of the elements should be **unique**. From what I read, I think this is called 'difference set'.
**Property 2 / Step 2**
Having ... | https://mathoverflow.net/users/195887 | Difference set of difference set | I’m assuming from the examples that you are only considering *non-negative* differences.
Let us first see what “as unique as possible” means. If the original set is $\{a\_i:0\le i<n\}$, the set of differences is
$$\{0\}\cup\{a\_i-a\_j:i,j<n,a\_i>a\_j\},$$
thus one checks easily that the set of second-order difference... | 2 | https://mathoverflow.net/users/12705 | 391901 | 162,158 |
https://mathoverflow.net/questions/391890 | 3 | By [this answer](https://mathoverflow.net/q/304708), we know that if $K/\mathbb{Q}\_p$ is a finite extension, the centralizer of $G\_K$ in $G\_{\mathbb{Q}\_p}$ is trivial. The argument there uses that the abelinization of $G\_K$ is the pro-finite completion of $K^\times$ and that the action of $G\_{\mathbb{Q}\_p}$ is t... | https://mathoverflow.net/users/152554 | Centralizer of the absolute Galois group of a number field | If $L/K$ are number fields, then the centralizer $C\_{G\_K}(G\_L)$ of $G\_L$ in $G\_K$ is trivial: As $C\_{G\_K}(G\_L)$ is normal in the normalizer $N\_{G\_K}(G\_L)$, the fixed field $E$ of $C\_{G\_K}(G\_L)$ is a Galois extension of the fixed field $F$ of $N\_{G\_K}(G\_L)$. Now $F$ lies between $K$ and $L$, in particul... | 4 | https://mathoverflow.net/users/50351 | 391904 | 162,160 |
https://mathoverflow.net/questions/391886 | 7 | I am working on sub-Riemannian geometry and try to understand what are the tools to find the equations of a sub-Riemannian problem. Here is an example:
Let us consider the system defined by a lagrangian:
\begin{equation}
L=\frac{1}{2}m(\overset{\cdot}{x}^2+\overset{\cdot}{y}^2)+\frac{1}{2}I\overset{\cdot}{\theta}^2... | https://mathoverflow.net/users/174936 | How to find equations of a sub-Riemannian problem | You can reformulate your problem in the language of geometric control (where your dynamical system is sometimes called Dubin's car). Let
$$
X\_1=\frac{1}{\sqrt{m}}(\cos\theta\partial\_x + \sin\theta\partial\_y),\qquad X\_2=\frac{1}{\sqrt{I}}\partial\_\theta$$
be a basis of your distribution. Horizontal curves are traje... | 7 | https://mathoverflow.net/users/13915 | 391910 | 162,162 |
https://mathoverflow.net/questions/391844 | 4 | This question implies that we have fixed: (i) a particular enumeration of Ordinal Turing machines; (ii) a particular way to encode an ordinal by an infinite binary sequence.
The class of $[1]$-machines is defined as the $1$st iteration of the strong jump operator for Ordinal Turing Machines. That is, a machine is equ... | https://mathoverflow.net/users/122796 | How to compare three supremums of ordinals eventually writable by Ordinal Turing Machines? | This is a rather incomplete answer because it doesn't address the harder part for the first half of question. (I hope it doesn't discourage any expert to answer the question in a more complete and techincal way.)
I have assumed that when you talk about infinite binary sequence $x$, you are talking about an $\omega$ l... | 3 | https://mathoverflow.net/users/112385 | 391917 | 162,165 |
https://mathoverflow.net/questions/391855 | 4 | I am working on Möbius disjointness for models of topological dynamic systems. In that purpose, I try to understand the notion of topological entropy. We know, for a t.d.s $(X,T)$ that it is defined by
***Definition 0***
$$h(f)=\lim\_{\varepsilon\to 0}\left(\limsup\_{n\to+\infty}\frac{1}{n}\log\left(N(n,\varepsilon... | https://mathoverflow.net/users/174936 | Are these topological sequence entropy definition equivalent? | The topological entropy I am assigning to a sequence $f$ (Definition 2) is not directly related to the similar-sounding concept of topological sequence entropy (Definition 1), but is instead related to Definition 0.
If $f(n) = F(T^n x)$ for some continuous $F: X \to {\bf C}$ on a compact metric space $X$, we see from... | 5 | https://mathoverflow.net/users/766 | 391918 | 162,166 |
https://mathoverflow.net/questions/391909 | 3 | I have a some naive questions about how to define the cohomology of a commutative monoid.
One way to express the cohomology of a group $G$ with coefficients in a module $A$ is as $\text{Ext}^i\_{\mathbb{Z}[G]}(\mathbb{Z},A)$. If we have a commutative monoid $M$ (you can also assume it's cancellative if you want), we ... | https://mathoverflow.net/users/120548 | Cohomology of commutative monoid acting on module | There are many different cohomology theories for monoids. Since you are using commutative monoids, you might be interested in Grillet's symmetric cohomology but I am not very familiar with it.
If we ignore Grillet (due to my ignorance mostly) then there are 3 cohomology theories that are popular for monoids: left/rig... | 2 | https://mathoverflow.net/users/15934 | 391921 | 162,169 |
https://mathoverflow.net/questions/391922 | 6 | Let $X$ be a finite type scheme over a field $k$.
Is it true that there exists a surjective morphism $f : Y \rightarrow X$, where $Y$ is smooth over $k$?
In other words, is every such scheme a quotient of a smooth scheme over $k$?
| https://mathoverflow.net/users/127260 | Is every variety an image of a smooth variety? | Presumably, you want to say that $X$ is reduced, since $Y$ will be reduced and any map from a reduced scheme lands in $X^{\mathrm{red}}$. Once you've said that, this follows from De Jong's alterations theorem, appearing in
* A.J. De Jong, *Smoothness, semi-stability and alterations*, Publications Mathématiques de l'I... | 8 | https://mathoverflow.net/users/297 | 391923 | 162,170 |
https://mathoverflow.net/questions/391924 | 7 | Suppose that we have $A\_{\infty}$ algebras $A,B$ (over a field of characteristic $0$), with $A\_{\infty}$ maps $f,g: A \rightarrow B$. In the paper <https://arxiv.org/abs/math/0401007> (top of page 4, item $(6)$) Markl defines a notion of $A\_{\infty}$ homotopy, which can analogously be used to define a homotopy betwe... | https://mathoverflow.net/users/102819 | Comparing notions of $A_{\infty}$ homotopy (in char 0): Markl's definition versus "Sullivan homotopy" | $\newcommand{\dd}{\mathrm d}$Markls notion is the same as an $A\_\infty$-morphism from $A$ to $B\otimes C^\bullet\_{[0,1]}$, where the second factor are the cellular cochains on the interval with (non-commutative!) dga structure defined via the Alexander-Whitney map. The two notions are then related by producing $A\_\i... | 3 | https://mathoverflow.net/users/35687 | 391933 | 162,171 |
https://mathoverflow.net/questions/391929 | 2 | Recall that a *Motzkin path* is a piece-wise linear planar path
connecting points in the integer lattice quadrant
$\Bbb{Z}\_{\geq 0} \times \Bbb{Z}\_{\geq 0}$ beginning at the origin $(0,0)$ and
ending at $(n,0)$ for some $n \in \Bbb{Z}\_{>0}$ whose *steps*
are either
\begin{equation}
\begin{array}{ll}
\nearrow & = (... | https://mathoverflow.net/users/70119 | Fibonacci-Motzkin paths and J-type continued fractions | Grouping the terms of $F(z)$ by the height reached, we get
$$F(z) = \frac{1}{(1 - z\gamma\_0)} + \frac{z^2 \beta\_1}{(1 - z\gamma\_0) (1 - z\gamma\_1)} + \frac{z^4 \beta\_1 \beta\_2}{(1 - z\gamma\_0) (1 - z\gamma\_1) (1 - z\gamma\_2)} + \cdots \\
$$
This has the form of Euler's continued fraction $$a\_0 + a\_0a\_1 + a\... | 3 | https://mathoverflow.net/users/46140 | 391937 | 162,172 |
https://mathoverflow.net/questions/391756 | 1 | A proper vertex coloring of a graph $G$ is acyclic if there is no bicolored cycle. A graph is 2-degenerate if its every subgraph has a vertex of degree at most 2. I think every 2-degenerate graph has an acyclic proper vertex coloring using 3 colors, but I did not find any source stating this. Does anyone know this? Am ... | https://mathoverflow.net/users/148974 | Acyclic proper coloring of 2-degenerate graphs | Corollary 3 in <https://dmtcs.episciences.org/344> says that for every graph $G$, if $G'$ is the 1-subdivision of $G$, then the acyclic chromatic number of $G'$ is at least $\sqrt{\frac12 \chi(G)}$. Apply this result with $G$ the complete graph $K\_n$. Then the acyclic chromatic number of $K'\_n$ is at least $\sqrt{\fr... | 2 | https://mathoverflow.net/users/25980 | 391939 | 162,173 |
https://mathoverflow.net/questions/390790 | 3 | Let $J(C)$ be the jacobian of a hyperelliptic curve $C$ of genus 2 defined over finite field $\mathbb{F}\_q$. Let $\Theta$ be the image of the curve on the Jacobian under the embedding $P \mapsto P - \mathcal{O}$, which is also known as the theta divisor.
Do we know something about the structure of the following set:... | https://mathoverflow.net/users/131249 | Generalization of torsion points on Jacobian of genus 2 over finite fields (with respect to the theta divisor) | The set $J(C)\_{\Theta}[n]$ has the structure of a smooth irreducible algebraic curve, and the restriction of $J(C)\xrightarrow{\times n } J(C)$ to $C$ defines a morphism $J(C)\_{\Theta}[n]\rightarrow C$ which is a finite etale cover with Galois group $J(C)[n]$.
I don't think it is good to think of $J(C)\_{\Theta}[n]$ ... | 4 | https://mathoverflow.net/users/110362 | 391964 | 162,180 |
https://mathoverflow.net/questions/391540 | 7 | Let $M$ be a connected projective complex manifold with a smooth anticanonical divisor $D$ ($D \sim -K\_M$).
Let $k$ be the number of components of $D$.
Some cheap thoughts give:
If $M$ is a Fano manifold of dimension higher than one, $k=1$.
If $M=\mathbb P^1$ or $M= \mathbb P^1 \times X$, where $X$ is a projecti... | https://mathoverflow.net/users/69559 | Possible number of components of anticanonical sections of projective manifolds | (Let me turn my comment into an answer to give a correct attribution.)
Yes, in the setup you describe, the maximum number of connected components of the divisor $D$ is 2. This is related to the so-called *connectedness principle* in birational geometry.
The statement that you want follows from Proposition 5.1 of [*... | 4 | https://mathoverflow.net/users/121595 | 391967 | 162,181 |
https://mathoverflow.net/questions/391955 | 6 | In the "The Rising Sea" by Vakil one can find the base change theorem for proper morphisms over a locally Noetherian base (28.1.6). He later indicates (28.2.M) how one could exchange the locally Noetherian condition by finitely presented using a result of Grothendieck. And indeed, it does not seem too hard to show this... | https://mathoverflow.net/users/164782 | Cohomology and base change without Noetherian assumption | There is a fairly general version of base change for schemes in Lipman's "yellow book":
Lipman, Joseph; Hashimoto, Mitsuyasu: *Foundations of Grothendieck duality for diagrams of schemes*. Lecture Notes in Mathematics, **1960**. Springer-Verlag, Berlin, 2009.
Also available at:
[https://www.math.purdue.edu/~jlipm... | 4 | https://mathoverflow.net/users/6348 | 391971 | 162,183 |
https://mathoverflow.net/questions/391956 | 6 | Let $G$ be a finite group, $S \subset G$ a generating set, closed under taking inverses, and $\lvert\cdot\rvert$ the word length with respect to this set $S$.
>
> **Question.** Is the function $k(g,h) = \frac{1}{1+\lvert gh^{-1}\rvert}$ positive definite, for $g,h \in G$?
>
>
>
A positive answer would allow ev... | https://mathoverflow.net/users/165920 | Is the function $k(g,h) = \frac{1}{1+\lvert gh^{-1}\rvert}$ positive definite? | ****[NB. Addendum below gives a counterexample to OP's question.]****
Computationally, this appears to be true for $S\_n$, $2 \le n \le 6$.
The MATLAB code I used to check this is below (using the case $n = 4$ to eyeball the Cayley graph). By Cayley's theorem and appropriately tweaking the code (essentially, the ar... | 6 | https://mathoverflow.net/users/1847 | 391977 | 162,185 |
https://mathoverflow.net/questions/391980 | 5 | I would like to know if the following recurrence relation for Catalan numbers (see [mathoverflow.net/questions/191524](https://mathoverflow.net/questions/191524/a-recurrence-relation-on-catalan-numbers) and also [math.stackexchange.com/questions/2113830](https://math.stackexchange.com/questions/2113830/recurrence-relat... | https://mathoverflow.net/users/68593 | Reference request: recurrence relation for Catalan numbers | T. Koshy, *Catalan Numbers with Applications* (Oxford, 2009), [page 322](https://books.google.com/books?id=o50RDAAAQBAJ&pg=PA322), proves a very similar identity:
$$C\_n=\sum\_{k=1}^{\left\lfloor\frac{n+1}{2}\right\rfloor}(-1)^{k-1} \binom{n-k+1}{k} C\_{n-k}$$
$$\Leftrightarrow \sum\_{k=1}^{\left\lfloor\frac{n+1}{2}\ri... | 7 | https://mathoverflow.net/users/11260 | 391985 | 162,187 |
https://mathoverflow.net/questions/391983 | 1 | I want to prove or find a counterexample that there exist constants $\mu>0, \rho>0$ such that the following inequality holds:
\begin{align}
(H + \mu M)^2 \succeq \rho M^2,
\end{align}
where $\mu>0, \rho>0$ are constants to be chosen, $H\in \mathbb{R}^{n\times n}$ is a fixed symmetric matrix with bounded eigenvalues $\l... | https://mathoverflow.net/users/178204 | Prove or disprove a matrix inequality (positive semidefinite) | This is true. Change coordinates so that $M$ is the projection to first $n-1$ coordinates: $Me\_n=0$, $Me\_i=e\_i$ for $i<n$.
If $He\_n=0$, then $H,M$ both have the same invariant orthogonal decomposition $e\_n\oplus e\_n^{\perp}$, they vanish on the first component and $M$ acts as ${\rm Id}$ on the second component,... | 4 | https://mathoverflow.net/users/4312 | 391987 | 162,188 |
https://mathoverflow.net/questions/391988 | 19 | This is a question that I need to answer in order to resolve an issue for my dissertation and I am looking for a reference. Here is the precise statement of the question.
>
> Suppose we have two three-dimensional lens spaces $L(n;r)$ and $L(n;s)$ which are homotopy equivalent but not diffeomorphic. Can their produc... | https://mathoverflow.net/users/205582 | Can the product of a 3-dimensional lens space with a circle be diffeomorphic to another such product when the lens spaces aren't diffeomorphic? | The answer is no. If $L$, $L^\prime$ are 3-dimensional lens spaces and $S^1\times L$ is diffeomorphic to $S^1\times L^\prime$, then the covering space of $S^1\times L$ corresponding to the torsion subgroup defines an
h-cobordism between $L$ and $L^\prime$ (we have embeddings of L and L′ in the covering space with disjo... | 28 | https://mathoverflow.net/users/1573 | 391992 | 162,189 |
https://mathoverflow.net/questions/391451 | 3 | This might be stupid question to some experts who works in the realm of automorphic form.
Let $K$ be a number field and $\mathbb{A}$ is a adele ring of $K$. Let $G$ be a connected reductive group defined over $K$. Moeglin and Waldspurger defined the norm function on $G(\mathbb{A})$ as follows:
For $GL\_{2n}(\mathbb... | https://mathoverflow.net/users/35898 | Questions on norms on Adelic group | For the first question,(and hence for the second,) it is false.
For example, you can take elements that are upper triangular unipotent and have trivial non-archimedean components as its counterexamples.
However, it is true that the norm is lower bounded. It is easily proven if you use the fact that the norm is inve... | 1 | https://mathoverflow.net/users/163485 | 391998 | 162,191 |
https://mathoverflow.net/questions/392004 | 0 | Let $u \in C^0(-T,T; L^2(B\_R))$ be a measurable function, then is the following true?
$$
\int\_0^R \sup\_{-T<t<T} \int\_{S\_r} |u(\sigma ,t)|^2 \ d \sigma \ dr = \sup\_{-T<t<T}\int\_0^R \int\_{S\_r} |u(\sigma ,t)|^2 \ d \sigma \ dr.
$$
Here $S\_r$ denotes the sphere of radius $r$ and $d\sigma$ is the standard meas... | https://mathoverflow.net/users/100801 | Interchange of integration and supremum | I don't think your left hand side is well defined for the class of $u$ you are considering, I can change each $u(.,t)$ to a large value on the zero-set $S\_{|t|}$, which will result in the supremum picking $t=r$ and changing the value of the left hand side.
But even for smooth functions, there is a counterexample in ... | 3 | https://mathoverflow.net/users/51695 | 392007 | 162,194 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.