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https://mathoverflow.net/questions/315087 | 9 | For an array $(n\_1,...,n\_k)$ of non-negative integers and non-zero reals $a\_1,...,a\_k$, define a block matrix $M$ of size $n=n\_1+\cdots+n\_k$ as follows:
The main diagonal has blocks of sizes $n\_i$ and shapes $$M\_i=J\_{n\_i}+a\_i I\_{n\_i}=\begin{pmatrix}
a\_i+1&1&\cdots&1\\
1&a\_i+1&\ddots&\vdots\\
\vdots&\ddo... | https://mathoverflow.net/users/29783 | Determinant of a block matrix with many $-1$'s | We have $M = D - e e^T$, where $D$ is a block diagonal matrix with main diagonals equal to $D\_i = \mathrm{diag}(a\_i) + 2 J\_i$, and $e$ is all ones vector with suitable dimention. By [Matrix determinant lemma](https://en.wikipedia.org/wiki/Matrix_determinant_lemma) we have
$$\det(M) = (1- e^TD^{-1}e) \det(D)$$
and
$$... | 5 | https://mathoverflow.net/users/53059 | 315093 | 136,831 |
https://mathoverflow.net/questions/315088 | 4 | Thus far, the books and articles I have read dealing with spin groups $\mathbf{Spin}(n)$ and $\mathbf{Spin}(p,q)$ consider them only in terms of either Clifford algebras or topologically as the double covers, respectively, of the special orthogonal groups $\mathbf{SO}(n)$ and $\mathbf{SO}(p,q)$. The spin groups are Lie... | https://mathoverflow.net/users/86315 | Spin groups in terms of matrices and/or linear operators | Spin groups are algebraic and so they can be presented as subgroups of general linear group that is defined by a system of algebraic equations. You can find one possible presentation in this [answer](https://mathoverflow.net/a/138179/6818) on MO.
| 7 | https://mathoverflow.net/users/6818 | 315095 | 136,832 |
https://mathoverflow.net/questions/315092 | 2 | I am little bit lost with the following (standard?) problem in model categories.
Suppose we have a Quillen adjunction between combinatorial model categories:
$$L:M\leftrightarrow N: R $$
and let $(M,Cof,Fib, WE)$ denote the model structure on $M$. I would like to know if it is possible to define a new model struc... | https://mathoverflow.net/users/129583 | Localization of a model category with respect to a class of maps | This is a very well-studied problem. Hirschhorn's book proves that, if $M$ is left proper and cellular then localization exists with respect to a set of maps. Jeff Smith figured out how to replace cellular by combinatorial; a nice write-up is in Barwick's paper [On Left and Right Model Categories and Left and Right Loc... | 8 | https://mathoverflow.net/users/11540 | 315098 | 136,834 |
https://mathoverflow.net/questions/307850 | 12 | Let $n,k\geq 2$ be positive integers. For each $1\leq i\leq n$, let $I\_i$ be a nonempty subset of $\{1,2,\dots,k\}$. Let $P\_i=\sum\_{j\in I\_i}x\_j$, and let $P=P\_1\cdot P\_2\cdot\dots\cdot P\_n$. (For example, $P=x\_1(x\_1+x\_2)(x\_1+x\_3)$.)
We want to maximize this expression subject to the constraints $x\_i\g... | https://mathoverflow.net/users/75538 | Maximize product of sums | Yes, that is true. Here is a sketch of the argument. It needs some polishing in places but, I hope, it makes clear what is going on here.
We will show that if $P\_1,P\_2,\dots, P\_m$ are of the above form and maximize $F=\sum\_k a\_k\log P\_k$ (so they are viewed as functions of $a=(a\_1,\dots,a\_m)\in (0,+\infty)^m$... | 4 | https://mathoverflow.net/users/1131 | 315100 | 136,836 |
https://mathoverflow.net/questions/315119 | 5 | Does every index $p$ subgroup of $SL(2,\mathbb{Z}\_p)$ contain the principal congruence subgroup $\Gamma(p)$?
Equivalently, must it be the preimage of an index $p$ subgroup of $SL(2,\mathbb{Z}/p\mathbb{Z})$?
| https://mathoverflow.net/users/88840 | Does every index $p$ subgroup of $SL(2,\mathbb{Z}_p)$ contain $\Gamma(p)$? | Yes.
Let $H$ be the subgroup, and let $N$ be the normal closure. The index of $N$ in $\Gamma = \mathrm{SL}(2,\mathbb{Z}\_p)$ has index dividing $p!$ which is not divisible by $p^2$. Hence either:
1. $N$ contains the principal congruence subgroup $\Gamma(p)$ and you win,
2. $N \cap \Gamma(p)$ has index $p$ inside $\... | 9 | https://mathoverflow.net/users/131269 | 315122 | 136,840 |
https://mathoverflow.net/questions/315127 | 9 | The space $( \ell^2 ,\lVert \cdot \rVert \_2 )$ is a Hilbert space. The space
$X=(\ell^2 \oplus \ell^2 , \lVert \cdot \rVert\_\infty )$ is a Banach space. Does X have fixed point property? (For any closed convex bounded subset $C\subseteq X $ and any nonexpansive map $T:C\to C $ there is a $x\in C$ such that $T(x)=x$... | https://mathoverflow.net/users/130589 | Does the Banach space $( \ell ^2 \oplus \ell ^2 )$ have F.P.P? | I think that the answer is *yes*, and that it should follow from the following facts:
1. every Hilbert space is uniformly convex, hence it has normal structure;
2. the direct sum of two Banach spaces with normal structure, endowed with the infinity norm, has again normal structure (Belluce-Kirk-Steiner, Pacific Journ... | 8 | https://mathoverflow.net/users/7460 | 315135 | 136,842 |
https://mathoverflow.net/questions/315106 | 3 | 1. Under category
Let $C$ be a locally presentable category, and let $c$ be an object of $C$. Lets denote by $C^{/c}$ the under category, objects are maps $c\rightarrow x$ and morphisms are the evident ones. I was wondering if the category $C^{/c}$ is also locally presentable ?
2. Monad
Let $T$ be a monad on a lo... | https://mathoverflow.net/users/128371 | Locally presentable categories | 1. Over and under categories of a presentable category are presentable. This is Proposition 1.57 in Adámek, Rosický, Locally Presentable and Accessible Categories.
2. If $T : \mathcal{C} \to \mathcal{C}$ preserves $\lambda$-directed colimits and $\mathcal{C}$ is $\lambda$-presentable, then the category of $T$-algebras ... | 8 | https://mathoverflow.net/users/62782 | 315143 | 136,845 |
https://mathoverflow.net/questions/315138 | 5 | Consider an array $A$ of length $n$ with $A\_i \in \{1,\dots,s\}$ for some $s\geq 1$. For example take $s = 6$, $n = 5$ and $A = (2, 5, 6, 3, 1)$. Let us define $g(A)$ as the collection of sums of all the non-empty contiguously indexed subarrays of A. In this case
$$g(A) = [2,5,6,3,1,7,11,9,4,13,14,10,16,15,17]$$
In... | https://mathoverflow.net/users/45564 | Bounds for the size of arrays with distinct subarray sums | Let $0=:x\_0<x\_1<x\_2<\dots<x\_n$ be the sums of initial segments of $A$. Then your condition is that the mutual differences of $x$'s are pairwise distinct, in other words, $X=\{x\_0,x\_1,\dots,x\_n\}$ is a [Sidon set](https://en.wikipedia.org/wiki/Sidon_sequence). Additional requirement is that consecutive elements o... | 8 | https://mathoverflow.net/users/4312 | 315153 | 136,849 |
https://mathoverflow.net/questions/315156 | 3 | If $(X,\tau)$ is a topological space, we denote by $\text{Cont}(X,X)$ the collection of all continous functions $f:X\to X$. We say that $(X,\tau)$ has the *fixed point property* if for any $f\in\text{Cont}(X,X)$ there is $x\_0\in X$ such that $f(x\_0) = x\_0$.
If $(X,\tau)$ has the fixed point property, does $\text{C... | https://mathoverflow.net/users/8628 | If $X$ has the fixed point property, what about $\text{Cont}(X,X)$? | No. In infinite dimensional normed spaces, [there exists a retraction of the unit Ball onto the unit sphere](http://www.ams.org/journals/proc/1983-088-03/S0002-9939-1983-0699410-7/S0002-9939-1983-0699410-7.pdf). Since for $X=[-1,1]$ with the usual topology, $C(X,X)$ is simply the unit Ball of $C(X)$ and since the unit ... | 10 | https://mathoverflow.net/users/35357 | 315157 | 136,851 |
https://mathoverflow.net/questions/315155 | 1 | Let $R$ be a DVR, and $f:R^n\to R^n$ a map. Suppose $f(r\_1,\dots,r\_n)=(s\_1,\dots,s\_n)$, and write $d=\min(v(s\_1),\dots,v(s\_n))$, where $v$ is the valuation on $R$. Knowing $d$, what is the best bound I can get on $\min(v(r\_1),\dots,v(r\_n))$? I'm looking for a bound in terms of the map $f$ and the individual $v(... | https://mathoverflow.net/users/64302 | How much can a map $R^n\to R^n$, $R$ a DVR, increase the valuation? | Let $\pi$ be a uniformiser, and write $r = (r\_1,\ldots,r\_n)$ and $s = (s\_1,\ldots,s\_n)$. For a vector $x = (x\_1,\ldots,x\_n) \in R^n$, write $v(x) = \min(v(x\_1),\ldots,v(x\_n))$. Note that it equals the largest integer $m$ such that $x \in \pi^mR^n$.
>
> **Lemma.** *Let $f \colon R^n \to R^m$ be a linear map,... | 3 | https://mathoverflow.net/users/82179 | 315169 | 136,857 |
https://mathoverflow.net/questions/315159 | 6 | On the page 4 of [these notes](https://pdfs.semanticscholar.org/4674/0dee0b88cafbdcc5160f9c3c60be373eb2df.pdf) it is stated that an $A\_{\infty}$-algebra $A$ is necessarily is quasi-isomorphic to an $A\_{\infty}$-algebra $HA$ with trivial differential. Moreover, $HA$ is unique up to a non-unique $A\_{\infty}$-isomorphi... | https://mathoverflow.net/users/131295 | Is the underlying vector space of the minimal model of an $A_{\infty}$-algebra canonical? | A minimal model of an $A\_\infty$-algebra $A$ is an $A\_\infty$-algebra $B$ with trivial differential together with an $A\_\infty$-quasi-isomorphism $B\to A$. This gives an isomorphism $H^\*A=H^\*B$, and we have $H^\*B=B$ since $B$ has trivial differential. Together these show that the underlying vector space of $B$ is... | 1 | https://mathoverflow.net/users/35353 | 315172 | 136,858 |
https://mathoverflow.net/questions/315116 | 8 | Given any finite relation $R$ if we let $\circ$ denote relation composition and define $R^n=\underbrace{R\circ R\cdots \circ R}\_{n\text{ times}}$ then does there exist an explicit formula for the cardinality of the set $\langle R\rangle=\{R^n:n\in\mathbb{N}\}$? If not, then are there any decent bounds for $|\langle R\... | https://mathoverflow.net/users/38626 | Given any finite relation $R$ what is the cardinality of $\langle R\rangle=\{\underbrace{R\circ R\cdots \circ R}_{n\text{ times}}:n\in\mathbb{N}\}$? | $\newcommand{\N}{\mathbb{N}}$
Recall that [Landau function](https://en.wikipedia.org/wiki/Landau%27s_function) $g(n)$ is the biggest possible $\mbox{lcm}$ of numbers wich sum up to $n$. It's asymptotic is well-studied.
I'll prove the following
$\textbf{Theorem 1.}$ There is some $C > 0$ such that we have $|\{ R^n :... | 5 | https://mathoverflow.net/users/104330 | 315173 | 136,859 |
https://mathoverflow.net/questions/314939 | 3 | Consider the following setup. Let $K$ be a compact topological space, $X$ a Fréchet space and $T:K \times X \to X$ a continuous family of linear maps (i.e. $T$ is a continuous map and $T\_k \equiv T(k, \cdot): X \to X$ is a linear for all all $k \in K$).
I would like to have a result similar to the following statemen... | https://mathoverflow.net/users/17047 | Openness of invertibility in Fréchet spaces for families parameterized by compact spaces | The statement is indeed overoptimistic.
As a counterexample, consider the continuous family $T: [0,1] \times C^\infty([0, 1]) \to C^\infty([0, 1])$ of linear differential operators defined by
\begin{equation}
T\_r (g) (x) = g - r x g'
\end{equation}
for $r \in [0,1]$ and $ g \in C^\infty([0, 1]) $.
Clearly, $ T\... | 1 | https://mathoverflow.net/users/17047 | 315175 | 136,860 |
https://mathoverflow.net/questions/315184 | 8 | **Definition:**
A linear map $f:\mathbb C^n\to \mathbb C^n$ is called **positive** if $\langle fa,a\rangle\ge0$ for all $a\in \mathbb C^n$. Equivalently, $f\in M\_{n}(\mathbb C)$ is positive if it can be written in the form $g^\*g$ for some $g\in M\_{n}(\mathbb C)$. The unique positive element $g$ satisfying $g^2=f$ ... | https://mathoverflow.net/users/5690 | Maps which are both completely positive and positive | No. I use facts about Schur multipliers which can be found, for example in Paulsen's monograph on completely bounded maps. Basically you are searching for a positive semidefinite matrix with positive entries such that the pointwise square root of the matrix is not positive semidefinite. Specifically, let $A=\left[ \beg... | 6 | https://mathoverflow.net/users/34640 | 315196 | 136,865 |
https://mathoverflow.net/questions/314870 | 3 | (This is a follow-up question to [Positive real root separation](https://mathoverflow.net/questions/314631/positive-real-root-separation))
Let $\beta\in(1,2)$ and $\gamma\in(1,2)$ be Galois conjugates of height 1. That is, there exists a polynomial $p$ with coefficients $-1,0,1$ such that $p(\beta)=p(\gamma)=0$ (not ... | https://mathoverflow.net/users/8131 | Positive real root separation (v2) |
>
> Short Answer: the polynomials
> $$P\_{2n+1}(x) = x^{2n+1}(x^8 - x^7 - x^6 + x^4 - x^3 + x + 1) - (x^8 + x^7 - x^5 + x^4 - x^2 - x + 1)$$
> for $n \ge 7$ should have an irreducible factor with exactly two roots $\alpha\_n$ and $\beta\_n$ of modulus greater than $1$, and $\alpha\_n - \beta\_n$ is exponentially co... | 6 | https://mathoverflow.net/users/131093 | 315198 | 136,866 |
https://mathoverflow.net/questions/314810 | 3 | Suppose that $X$ is a complex algebraic variety, i.e. it is integral, separated, and of finite-type over $\mathbb{C}$. Let $\mathfrak{g}$ be a finite-dimensional complex Lie algebra.
Suppose that $\mathfrak{g}$ acts on $X$, i.e. we have a homomorphism $\phi:\mathfrak{g}\to \mathrm{Vec}(X)$, where $\mathrm{Vec}(X)$ is... | https://mathoverflow.net/users/97652 | Lie algebra action on complex algebraic variety | It seems to me that the answer should be positive.
Suppose by contradiction, that there is a singular point $x\in X$. Let us take an affine neighbourhood $U$ of $x$. Let $f\_1,\ldots, f\_n$ be the regular functions that generate the ring of functions on $U$. This functions give us an embedding $\varphi: U\to \mathbb... | 2 | https://mathoverflow.net/users/943 | 315213 | 136,870 |
https://mathoverflow.net/questions/315221 | 5 | Let $X=(X\_1,\ldots,X\_n)$ be a random vector uniformly distributed on the $n$-dimensional sphere of radius $R > 0$. Intuitively, i think that for large $p$ every coordinate $X\_i$ is normally distributed with variance $R^2/n$, but I'm not quite sure.
Question
========
More formaly, if $\Phi$ is the CDF of the stan... | https://mathoverflow.net/users/78539 | Distribution of the individual coordinates of a uniform random vector on a high-dimensional sphere | Without loss of generality, $R=1$.
Let $Z\_1,\ldots,Z\_n$ be iid standard normal random variables (r.v.'s). Then
\begin{equation}
\sqrt n\, X\_1\overset{\text{D}}=\frac{\sqrt n\,Z\_1}{\sqrt{Z\_1^2+\cdots+Z\_n^2}}
\overset{\text{D}}=
\frac{Z\_1+\cdots+Z\_n}{\sqrt{Z\_1^2+\cdots+Z\_n^2}}=:T\_1,
\end{equation}
where... | 6 | https://mathoverflow.net/users/36721 | 315232 | 136,877 |
https://mathoverflow.net/questions/315230 | 2 | Is there a treatment in the literature of the structure sets relating simple homotopy equivalences to homeomorphisms in the three dimensional case? I am aware that due to the geometrization conjecture, homotopy equivalences and homeomorphism agree. I am also aware of Jahren and kwasik's paper Three-dimensional surgery ... | https://mathoverflow.net/users/21985 | Structure sets for three dimensional surgery | It is well known that the surgery sequence completely breaks down in dimension 3, so looking at structure sets is not really the right thing to do. For instance, in the classification of high-dimensional lens spaces, a given simple homotopy type (as in Mike Miller's comment) will contain many homeomorphism types (detec... | 3 | https://mathoverflow.net/users/3460 | 315243 | 136,880 |
https://mathoverflow.net/questions/315239 | 7 | I posted this question on math.se here:<https://math.stackexchange.com/q/2996787/482732>, but I think it may be more appropriate here, sorry if I am wrong about that.
In Waldhausen's paper Algebraic K theory of Spaces(the long one) he proves the following:
$$A(X)\simeq \mathbb{Z}\times B\widehat{Gl}(\Omega^{\infty}... | https://mathoverflow.net/users/131196 | Reference request for K-Theory linearization | I claim that for every $A\_\infty$-space $A$, there is a canonical $A\_\infty$-ring structure on $\Omega^\infty\Sigma^\infty\_+A$.
First, $\Sigma^\infty\_+$ from spaces to spectra is symmetric monoidal. So it sends an $A\_\infty$-space $A$ to an $A\_\infty$-algebra in spectra $\Sigma^\infty\_+A$, that is an $A\_\inft... | 9 | https://mathoverflow.net/users/43054 | 315245 | 136,881 |
https://mathoverflow.net/questions/315254 | 5 | Let $G$ be a discrete group which acts continuously on a Stonean space $\Omega$. Consider the map $f\colon \Omega\to \{0,1\}^G$ sending $x\in \Omega$ to $\chi\_{G\_x}$, where $\chi\_{G\_x}$ denotes the characteristic function of the stabilizer $G\_x$.
Why is $f$ continuous?
We can restrict to the subbasis-sets of ... | https://mathoverflow.net/users/75338 | continuity of certain map which is defined on a Stonean space | It's indeed true: whenever a discrete group $G$ acts continuously on a Hausdorff, extremally disconnected space $X$, then the map $x\mapsto G\_x$ is continuous, where the set of subgroups of $G$ is endowed with its compact topology given by inclusion in $2^G$.
One has to show that for any $g\in G$, the map $u\_g:x\ma... | 5 | https://mathoverflow.net/users/14094 | 315275 | 136,887 |
https://mathoverflow.net/questions/315242 | 12 | If $A$ is poset with all directed suprema, it is common to consider the Scott topology on $A$, whose open subsets are the $U \subset A$ such that $U$ is upward closed and if $\bigcup\_I a\_i \in U $ for some directed supremum then $\exists i, a\_i \in U$.
It is a classical fact that the specialization order induced b... | https://mathoverflow.net/users/22131 | "Scott completion" of dcpo | I believe the paper by Johnstone linked to in the question contains the answer, and it is negative.
In that paper, Johnstone constructs a Scott topology that is not sober as a byproduct of answering in the negative the following question (marked by (a) in the paper):
>
> if $(X,\leqslant)$ has directed joins, is ... | 10 | https://mathoverflow.net/users/41291 | 315282 | 136,888 |
https://mathoverflow.net/questions/315255 | 13 | Here is what I know about Whitehead products in homotopy groups of spheres:
* $[\mathrm{id}\_{S^{2n}},\mathrm{id}\_{S^{2n}}]$ has Hopf invariant (EDIT: $\pm$) two.
* No element that survives into the stable range can be a Whitehead product, since the suspension of a Whitehead product is trivial.
* If $\alpha \in \pi\... | https://mathoverflow.net/users/59302 | Whitehead products in homotopy groups of spheres | James proves a great number of things about the Whitehead product in his paper [On the suspension sequence](https://www.jstor.org/stable/1969666) (though a number of results in that paper are stated in terms of cases rather than the stronger results that hold 2-locally). For example, he shows that there is a 2-local pa... | 8 | https://mathoverflow.net/users/360 | 315291 | 136,892 |
https://mathoverflow.net/questions/315278 | 3 | Let $(M,g)=(N,\ddot{g})\times f(B,\bar{g})$ be an Einstein warped-product manifold Ricci flat (i.e. $Ric=\lambda g$ with $\lambda=0$) where $f:N \rightarrow (0, \infty)$ (positive scalar function) and with $g= \ddot{g}+f^2 \bar{g}$.
If $(B, \bar{g})$ is Ricci flat, being $(M, g)$ Ricci flat, this means that $(N, \ddo... | https://mathoverflow.net/users/111304 | Einstein warped product manifold Ricci flat | Let $\bar M=M\_1\times\_f M\_2$ be a warped product manifold, $X\_1, Y\_1$ be two lifts of two vector fields on $M\_1$ to slides $M\_1\times p\_2$ and $X\_2, Y\_2$ be two lifts of two vector fields on $M\_2$ to slides $p\_1 \times M\_2$. Then
$$\bar Ric(X\_1,Y\_1)= Ric^1(X\_1,Y\_1)-\frac{dimM\_2}{f}Hess^f(X\_1,Y\_1)... | 2 | https://mathoverflow.net/users/46495 | 315297 | 136,893 |
https://mathoverflow.net/questions/315264 | 6 | I have an integral:
$$I(y) = \int\_0^\infty \frac{xJ\_1(yx)^2}{\sinh(x)^2}\ dx $$
and would like to asymptotically expand it as a series in $1/y$. Does anyone know how to do this? By numerically computing the integral it appears that
$$I(y) = \frac 12 - \frac 1 {\pi y}+ \frac {3\zeta(3)}{4y^3\pi^3} + O(y^{-5}) $$
but t... | https://mathoverflow.net/users/131358 | Asymptotic Expansion of Bessel Function Integral | Inserting the Mellin-Barnes representation for the square of the Bessel function ([DLMF](https://dlmf.nist.gov/10.9.E29)),
\begin{equation}
J\_{1}^2\left(xy\right)=\frac{1}{2\pi i}\int\_{c-i\infty}^{c+i
\infty}\frac{\Gamma\left(-t\right)\Gamma\left(2t+3\right)}{\Gamma^2\left(t+2\right)\Gamma%
\left(t+3\right)}\left(\f... | 7 | https://mathoverflow.net/users/46744 | 315298 | 136,894 |
https://mathoverflow.net/questions/314976 | 20 | It is easy to classify conjugacy classes in $GL\_n(\mathbb Q\_p)$ by linear algebra. How to classify $GL\_n(\Bbb Z\_p)$ conjugacy classes in a $GL\_n(\Bbb Q\_p)$ conjugacy class? For example, for general matrix $A \in GL\_n(\mathbb Z\_p)$, how many $X \in GL\_n(\mathbb Z\_p)$ are $GL\_n(\mathbb Q\_p)$ conjugated to $A$... | https://mathoverflow.net/users/102104 | $GL_n(\Bbb Z_p)$ conjugacy classes in a $GL_n(\Bbb Q_p)$ conjugacy class | The answer is yes if $A \in \mathrm{GL}\_n(\mathbb{Z}\_p)$ is semisimple.
We may think of a matrix $A \in M\_n(\mathbb{Z}\_p)$ as a $\mathbb{Z}\_p$-lattice of rank $n$ endowed with a $\mathbb{Z}\_p$-linear endomorphism: take the standard lattice $L=\mathbb{Z}\_p^n$ endowed with the endomorphism $\varphi$ associated t... | 7 | https://mathoverflow.net/users/6506 | 315302 | 136,896 |
https://mathoverflow.net/questions/315288 | 3 | My setup is as follows: $X$ is a projective, reduced curve (which is not integral) with a finite morphism onto $\mathbb{P}\_k^1$.
$\DeclareMathOperator{\Ann}{Ann}$
Let $R$ be a coordinate ring of $X$ which is finite free over $k[x]$ (since $X$ has more than one irreducible component, $R$ has at least two minimal prime ... | https://mathoverflow.net/users/98129 | Is the annihilator of a minimal prime ideal principal? | This is false. To see why, consider the following lemma.
>
> **Lemma.** *Let $R$ be a Noetherian ring with exactly two minimal primes $\mathfrak p$ and $\mathfrak q$ such that $\mathfrak p \mathfrak q = 0$. Then $\operatorname{Ann}(\mathfrak p) = \mathfrak q$.*
>
>
>
The assumption is in particular satisfied i... | 5 | https://mathoverflow.net/users/82179 | 315309 | 136,897 |
https://mathoverflow.net/questions/315296 | 3 | Let $Q=(Q\_0,Q\_1)$ be a quiver and $q\in Q\_0$ a chosen vertex. Let $d$ be a dimension vector with $d\_q=1$ and let $\theta\in \mathbb R^{Q\_0}$ be a $d$-generic stability parameter. Let $M$ be a $\theta$-stable $Q$-representation of dimension $d$. Let $f$ be an automorphism of $M$ which is the identity on the compone... | https://mathoverflow.net/users/88153 | Are there non-trivial automorphisms of stable framed quiver representations? | All automorphisms of the quiver representation you mention are by multiplication by scalars. So as $f$ is the identity on $M\_q$ it must be the identity on $M$.
Under your assumptions, $g=f-\mathrm{Id}\_M$ is an endomorphism of $M$ with nonzero kernel. The morphism $g\colon M\rightarrow \mathrm{Im}(g)$ is surjective,... | 3 | https://mathoverflow.net/users/21637 | 315310 | 136,898 |
https://mathoverflow.net/questions/315272 | 13 | Can the fact that all analytic sets are Lebesgue measurable be proven in $Z\_2$, or in some weak subsystem such as $\Pi^1\_1\text{-CA}\_0$? Conversely, can certain set existence axioms be derived from the assumption that all analytic sets are Lebesgue measurable?
| https://mathoverflow.net/users/83073 | reverse mathematics of the Lebesgue measurability of analytic sets | I believe $\Delta^1\_2$-CA$\_0$ does indeed suffice; however, I don't see a way to pull this down to $\Pi^1\_1$-CA$\_0$.
*This is contrary to my previous claim; my error was with respect to the strength of the relevant choice principle.*
---
Let me briefly outline the classical proof:
* First, we show a versi... | 6 | https://mathoverflow.net/users/8133 | 315311 | 136,899 |
https://mathoverflow.net/questions/314462 | 6 | Does there exist $X$ a smooth Fano manifold, $f: Y \to X$ a nontrivial ramified finite cover, $C \subseteq X$ a smooth very free rational curve, such that $f$ is étale over a neighborhood of $C$?
Without the Fano condition, an example is $X = \operatorname{Hilb^2} (\mathbb P^2)$, $Y = \mathbb P^2 \times \mathbb P^2$... | https://mathoverflow.net/users/18060 | Can free rational curves lift to ramified covers of Fano varieties? | After some helpful conversations with Johan de Jong, I came up with an example. After finding it I realize I probably should have figured it out earlier.
In fact, $\operatorname{Hilb}^2(\mathbb P^n)$ is an example for any $n>2$.
The blow-up of $\mathbb P^n \times \mathbb P^n$ at the diagonal is a two-to-one cover ... | 4 | https://mathoverflow.net/users/18060 | 315322 | 136,903 |
https://mathoverflow.net/questions/315176 | 4 | Question: Is there an infinite sequence of primes $\{q\_i\}\_{i=1}^{\infty}$ that is not too sparse ( $q\_n =O(poly(n))$ for a fixed polynomial) for which it is true that for every $k$ there is an $N(k)$ so that if $n>N$ then there is a prime $p$ in the interval $[q^{1/k}-o(q^{1/k}),q^{1/k})$ that is a quadratic residu... | https://mathoverflow.net/users/38267 | Distance between primes that are quadratic residues modulo an other prime | Let $k>2$ and $0<\epsilon<1$ be fixed real numbers (see also the Added section below). Let $x$ be a large parameter. It suffices to show that there exists a prime pair $(p,q)$ such that $x<p<2x$ and $p^k<q\leq p^k(1+\epsilon)$ and $\left(\frac{p}{q}\right)=1$. Note that for primes $p\equiv 1\pmod{4}$, the last conditio... | 4 | https://mathoverflow.net/users/11919 | 315323 | 136,904 |
https://mathoverflow.net/questions/315186 | 11 | Let $Z\_i$ be i.i.d. random variables with $\mathbb{E}[Z\_i] = 0$ and $\mathbb{E}|Z\_i|^p< \infty$ for $p=1,2,3,\cdots$. I am looking for the following type of estimate if possible, and it is not like the concentration inequalities that one normally sees.
>
> There exists $N\_0$ sufficiently large and $t\_0$ suffi... | https://mathoverflow.net/users/49551 | Maximal inequality for the average of i.i.d. random variables | I streamlined my proof a bit so it is postable now :-)
First, a disclaimer. I have no doubt that there is some slick theorem dating back to 1980's that immediately implies what you want and all one needs is to wait for a while until someone posts a reference to it. Meanwhile, here is a crude computation that gives a ... | 7 | https://mathoverflow.net/users/1131 | 315324 | 136,905 |
https://mathoverflow.net/questions/315320 | 14 | Let $MU$ be the complex bordism spectrum and let $H\mathbb{Z}$ be the Eilenberg-Maclane spectrum.
>
> Is it know what the structure of the complex cobordism cohomology $MU^{\*}(H\mathbb{Z})$ is?
>
>
>
EDIT: What if instead $H\mathbb{Z}$, one consider $H\mathbb{Z}/(p)$ for a prime $p$?
| https://mathoverflow.net/users/108963 | The structure of complex cobordism cohomology of the Eilenberg-Maclane spectrum | One can prove that $\mathrm{Map}(H\mathbf{F}\_p,MU)$ is contractible. We know that $H\mathbf{F}\_p$ is dissonant (Theorem 4.7 of Ravenel's "Localization with Respect to Certain Periodic Homology Theories"), but $MU$ is harmonic (Theorem 4.2 of that paper). Since dissonant spectra (resp. harmonic spectra) are by definit... | 16 | https://mathoverflow.net/users/102390 | 315325 | 136,906 |
https://mathoverflow.net/questions/315312 | 1 | Consider a function $f(z, w)$ of two complex variables. The function is symmetric with respect to $z$ and $w$. When $\Re(z)>0$ and $\Re(w)>0$, the function is analytic in its two variables. When $\Re(w)>0$ and $\Re(z)\leq 0$, the function has poles in $z$ at positions $q\_i$. There are countably infinitely many poles t... | https://mathoverflow.net/users/131118 | Computing the convex hull of a region of $\mathbb{C}^2$ | The result about extending to convex hull is not as general as what you state, although the erroneous generality seems irrelevant to your actual issue: it is for (connected) "tube domains", meaning sets described by conditions on the real parts only. (Further, some mild growth condition is necessary.)
A general key-w... | 2 | https://mathoverflow.net/users/15629 | 315326 | 136,907 |
https://mathoverflow.net/questions/315269 | 8 | Let $f: \mathbb R\to \mathbb R$ be a Borel measurable function. Suppose that for each $q\in \mathbb Q$, the function $f(q+x)-f(x)$ is continuous on $\mathbb R$. Is it true that there is a continuous function $g: \mathbb R\to \mathbb R$ such that $f(x)=g(x)$ for Lebesgue almost every $x\in \mathbb R$?
If the answer to... | https://mathoverflow.net/users/11056 | Is a measurable solution continuous? | Consider the 1-periodic function $f$ with Fourier series $$\sum n^{-1}\cos (2\pi n! x).$$
Note that it satisfies your property, since all but finitely many summands are $h$-periodic for any rational $h$. On the other hand, if it were a Fourier series of a continuous function $F$, its partial sums would be Cesàro conver... | 7 | https://mathoverflow.net/users/4312 | 315331 | 136,908 |
https://mathoverflow.net/questions/315318 | 3 | Let us assume that $X=\mathbb{R}\times S^1$ is given with a $G=\mathbb{Z}\_2$ action that corresponds to the symmetry $(x,e^{i\theta})\mapsto(-x,e^{-i\theta})$. I want to compute the equivariant cohomology of $X$ relative to the fixed point set $X^G=\{(0,-1),(0,1)\}$. Since the subspace $X^G$ is $G$-invariant, there is... | https://mathoverflow.net/users/35615 | Relative equivariant cohomology | The relative Borel homology of a pair $(X,A)$ of $G$-spaces is well-defined up to equivariant homotopy equivalence (actually, up to equivariant maps which are nonequivariant homotopy equivalences of pairs). So we may reduce your example to $(S^1, \pm 1)$ with the reflection action.
Now $(S^1, \pm 1)$ has two fixed p... | 3 | https://mathoverflow.net/users/40804 | 315332 | 136,909 |
https://mathoverflow.net/questions/314951 | 3 | Let $\Omega= \{(x,y) : \frac{1}{2} \leq x^2+y^2 \leq 1\}$ and $S = \{(x,y) : x^2+y^2 = 1\}$ the unit circle, and $X=w^{1.\infty}(\Omega;\mathbb{R})$ the space of Lipschitz valued functions. We denote by $\left| \cdot \right|$ the euclidean norm in $\mathbb{R}^2$ or the modulus in $\mathbb{R}$.
Given $\phi \in C^\inft... | https://mathoverflow.net/users/40120 | Prescribing a gradient direction | Here is a comment that shows possible obstructions to the existence of such an operator. I'm assuming a bit more regularity here.
Let $(M,g)$ be a compact Riemannian manifold with boundary, and let $\text{vol}\_g$ be its Riemannian volume form.
Equip the vector space $C^{\infty}(M)/\mathbb{R}$ of smooth functions int... | 3 | https://mathoverflow.net/users/69603 | 315333 | 136,910 |
https://mathoverflow.net/questions/315072 | 10 | Given Hopf $\mathbb{C}$-algebra $H$, it's Hopf dual $H^o$ is the largest Hopf algebra contained in $H^\*$, the $\mathbb{C}$-linear dual of $H$. (This is well known to be well-defined, see for example Sweedler book.)
If $j:G \to H$ is a linear map, then we have dual linear map
$$
j^\*:H^\* \to G^\*, ~~~~~~~~~ f \maps... | https://mathoverflow.net/users/125941 | Functoriality of the Hopf dual | -as suggested after the discussion in the comments-
i am understanding that the OP is asking whether a linear map $j:G \to H$ is functorial, in the sense that the image of the dual map $j^\*:H^\* \to G^\*$ preserves the finite dual hopf algebra $H^\circ$, i.e.
$
j^\*(H^\circ)\subseteq G^\circ
$
The answer is gene... | 3 | https://mathoverflow.net/users/85967 | 315338 | 136,912 |
https://mathoverflow.net/questions/315339 | 2 | Fix an integer $n\ge 2$. Let $[a,b]$ be an interval and $f: [a,b]\to \mathbb R$ be a continuous function and for $x\_1,...,x\_n$ being the Gaussian Quadrature nodes in $[a,b]$, and Gaussian Quadrature wights $w\_1,...,w\_n$ in $[a,b]$ , (everything being calculated with weight function $\omega(x)=1$ ) (see <https://en.... | https://mathoverflow.net/users/127118 | On the continuity and injective-ness of Gauss quadrature scheme for numerical integration, with weight function identically $1$ | The key here is the simple change-of-interval/rescaling formula, found e.g. at the link in the OP, according to which
\begin{equation}
T\_n(f)(x)=T\_{n,[0,x]}(f)=x\sum\_1^n w\_i f(xx\_i), \tag{\*}
\end{equation}
where the $w\_i$'s and $x\_i$'s are such that $T\_n(f)(1)=T\_{n,[0,1]}(f)=\sum\_1^n w\_i f(x\_i)$ for all ... | 2 | https://mathoverflow.net/users/36721 | 315354 | 136,914 |
https://mathoverflow.net/questions/315316 | 6 | I have two realted questions.
Let $R$ be a rational function on $\mathbb{C}$ with degree at least 2. We denote by $\mu$ the measure of maximal entropy for $R$ and recall that the Julia set coincides with the support $\mu$.
(1) If the Julia set contains a smooth curve (real 1D analytic curve), what can we say about ... | https://mathoverflow.net/users/47862 | Julia set containing smooth curve | I am not surprised that you found nothing in the literature.
On question 1: it has been studied under a much stronger assumption that the Julia set contains a smooth INVARIANT curve. Under some additional conditions, it was proved that such a curve must be algebraic. And there are such algebraic curves, other than ci... | 4 | https://mathoverflow.net/users/25510 | 315369 | 136,918 |
https://mathoverflow.net/questions/315374 | 16 | Recall that the radical of an integer $n$ is defined to be $\operatorname{rad}(n) = \prod\_{p \mid n } p$.
For a paper, I need the result that
$$\sum\_{n \leq x} \frac{1}{\operatorname{rad}(n)} \ll\_\varepsilon x^{\varepsilon} \tag{$\*$},$$
for all $\varepsilon > 0$. I have a proof of this using complex analysis and... | https://mathoverflow.net/users/5101 | Sum of the reciprocals of radicals | You can get away with elementary analytic number theory. Consider the series $\sum\_n\frac{1}{n^{\varepsilon}\rm{rad}(n)}$. It suffices to show that it converges. However, it can be written as a product of
$$
S(p)=1+p^{-1-\varepsilon}+p^{-1-2\varepsilon}+\dots=1+p^{-1-\varepsilon}\frac 1{1-p^{-\varepsilon}}\le 1+p^{-1... | 17 | https://mathoverflow.net/users/1131 | 315380 | 136,920 |
https://mathoverflow.net/questions/315329 | 10 | In the paper *[Partition theorems for systems of finite subsets of integers](https://doi.org/10.1016/0012-365X(82)90041-3)*, Pudlák and Rödl show a Ramsey-type result. The main feature of this result is that the sizes of sets in such systems are not fixed in advance (as in Ramsey's original theorem and Erdős and Rado's... | https://mathoverflow.net/users/66044 | Looking for “Set theory for a small universe” by Ketonen | A pdf version of the 27-page manuscript by Jussi Ketonen, "Set Theory for a Small Universe, I. The Paris-Harrington Axiom", is [here on Google Drive](https://drive.google.com/file/d/1xMhBMP5Ki4RDw9pIAlsUwUh9kyMUH7xf/view?usp=sharing). The date of manuscript is perhaps 1979, or 1978.
The left margin in my hard-copy is... | 6 | https://mathoverflow.net/users/16302 | 315382 | 136,921 |
https://mathoverflow.net/questions/314979 | 3 | Let $N$ be a type ${\rm II}$ factor, with trace $\tau$. Consider its fundamental group$$ \mathcal{F}(N)= \{ \tau(p)/\tau(q) \ | \ p,q \text{ non-zero finite projections in } N \text{ and } pNp \simeq qNq \}. $$
Let $\alpha$ be a free ergodic measure preserving action of a countable ICC group $\Gamma$ on a $\sigma$-fi... | https://mathoverflow.net/users/34538 | Fundamental group and group measure space construction | I asked Stefaan Vaes by email, below is his answer (reproduced with his authorization):
>
> For instance, take $G$ to be the semidirect product of $\mathbb{Z}^2$
> and ${\rm SL}(2,\mathbb{Z})$. The group von Neumann algebra $L(G)$ has
> trivial fundamental group and this was even the very first ${\rm
> II}\_1$ ... | 0 | https://mathoverflow.net/users/34538 | 315385 | 136,922 |
https://mathoverflow.net/questions/314155 | 7 | If $X$ is a smooth projective curve over $\mathbb C\_p$, then its first etale cohomology $\mathrm H^1\_{et}(X,\mathbb Q\_\ell)$ (with $\ell\neq p$) carries a certain weight filtration $W\_\bullet$ -- also known as the monodromy filtration -- which contains information about the reduction type of $X$. For instance, if $... | https://mathoverflow.net/users/126183 | The weight filtration on etale cohomology and Berkovich analytic geometry | I try to answer your question with some ideas.
First, what you said about the weight zero part was proved by Berkovich itself for any dimension in
Berkovich, V. G., An analog of Tate’s conjecture over local and finitely
generated fields, Internat. Math. Res. Notices 2000, no. 13, 665–680.
(I understand that what... | 1 | https://mathoverflow.net/users/24442 | 315387 | 136,923 |
https://mathoverflow.net/questions/315397 | 0 | (I'm trying to solve a problem for computer programming. Don't have much of a math background, so I hope I am using the right terminology)
Is there a formula for getting the partitions of a number with these three restrictions:
1. the partition numbers are from a limited nonconsecutive set.
2. Each partition number... | https://mathoverflow.net/users/131425 | Number partitions | There is no way to get a *formula* for these partitions, as it depends upon the list of available partition numbers. Nevertheless, here's *Mathematica* code that implements what you seek:
```
constrainedPartitions[s_: Integer, k_: List, t_: Integer] :=
Select[IntegerPartitions[s, {t}],
(ContainsOnly[#, k] && Dup... | 0 | https://mathoverflow.net/users/89654 | 315405 | 136,928 |
https://mathoverflow.net/questions/312432 | 18 | The literature on the representation theory of the symmetric group contains some terminology that I find puzzling, and I am wondering if someone here knows the full story.
One of the standard ways to construct the irreducible representations of the symmetric group is to define *Specht modules*. This construction prod... | https://mathoverflow.net/users/3106 | Young's natural representation of the symmetric group | Thanks to Richard Stanley for the pointer to Garsia and McLarnan's paper, Relations between Young's natural and the Kazhdan–Lusztig representations of $S\_n$, *Advances in Math.* **69** (1988), 32–92.
Young's fourth paper ("QSA IV") is:
>
> Alfred Young, On Quantitative Substitutional Analysis (Fourth Paper), *Pr... | 8 | https://mathoverflow.net/users/3106 | 315410 | 136,929 |
https://mathoverflow.net/questions/308154 | 0 | For a positive sequence $0\le\lambda\_{1}\le\lambda\_{2}\le\cdots$, consider an infinite product
\begin{equation\*}
\prod\_{i=1}^{\infty}\lambda\_{i}:=\lim\_{n\rightarrow\infty}\prod\_{i=1}^{n}\lambda\_{i}.
\end{equation\*}
The product converges if and only if $\sum\_{i=1}^{\infty}\log(\lambda\_{i})$ converges.
Also... | https://mathoverflow.net/users/119219 | Relation between infinite product and regularized product | It doesn't make sense to compare the infinite product and the zeta-regularized determinant, because they can never be finite at the same time. The assumption that the infinite product converges is incompatible with the assumption that the series defining the zeta function converges.
If the infinite product $\Pi\_i \l... | 0 | https://mathoverflow.net/users/49048 | 315413 | 136,931 |
https://mathoverflow.net/questions/315381 | 9 | Is there a closed hyperbolic $3$-manifold whose fundamental group is isomorphic to a subgroup of some compact Lie group?
It is [known](https://arxiv.org/abs/math/0602635) that every surface group can be embedded into any semisimple connected Lie group.
I would be interested in similar results for fundamental groups... | https://mathoverflow.net/users/1573 | Hyperbolic $3$-manifold groups that embed in compact Lie groups | All closed hyperbolic 3-manifold groups embed into a compact Lie group.
To prove this, note first of all that given a hyperbolic 3-manifold $M$, it suffices to show that a finite-index subgroup $G\leq \pi\_1(M)$ of index $m$ embeds into a compact Lie group. Then the representation $\rho: G\hookrightarrow O(n)$ will ... | 19 | https://mathoverflow.net/users/1345 | 315430 | 136,934 |
https://mathoverflow.net/questions/207783 | 3 | If we consider the space of integrable functions $L^1([0,1];\mathbb{R})$, it can be ordered by the convex cone of positive integrable functions $L^1([0,1];\mathbb{R}\_+)$. It is known that the corresponding dual cone is $L^\infty ([0,1];\mathbb{R}\_+)$.
I wonder if the same kind of statement holds when replacing $\math... | https://mathoverflow.net/users/73608 | Dual cone of 'positive' Bochner integrable functions | Take some element in the dual cone $g\in L^1([0,1],Y\_+)^\*\subset L^\infty\_{w^\*}([0,1],Y^\*)$.
Then, by definition, for every $f\in L^1([0,1],Y\_+)$
$$\int\_0^1 \langle f(t),g(t)\rangle\mathbb{d}t\geq0$$
In particular, for any $y\in Y\_+$ and $[a,b]\subset [0,1]$, we can take $f \equiv y\cdot \chi\_{[a,b]}$ so ... | 2 | https://mathoverflow.net/users/39517 | 315433 | 136,935 |
https://mathoverflow.net/questions/314032 | 5 | Let $Q$ be a finite simple group which may be realized as the Galois group of some extension of $\mathbb{Q}$ (like for instance $PSL\_2(\mathbb{F}\_p)$ for $p\geq 5$, or the monster group) and let $G$ be an extension of $Q$ by a cyclic group. Then is it possible to see that $G$ is realizable as a Galois group over $\ma... | https://mathoverflow.net/users/nan | Is the following variant of Shafarevich's theorem known? | It has occurred to me not too long after posting this question that the answer is obviously no, for instance the groups $\text{PSL}\_2(\mathbb{Z}/p\mathbb{Z})$ are known to be Galois over $\mathbb{Q}$ but the groups $\text{SL}\_2(\mathbb{Z}/p\mathbb{Z})$ are not known to be Galois over $\mathbb{Q}$ for large $p$.
| 1 | https://mathoverflow.net/users/nan | 315436 | 136,938 |
https://mathoverflow.net/questions/315290 | 1 | In Evans,
>
> $\textbf{Theorem} $ (Global Approximation Theorem) Assume $U$ is bounded, and $\partial U$ is $C^1$. Suppose as well that $u \in W^{k,p}(U)$ for some $1\leq p < \infty$. Then, there exist functions $u\_m \in C^{\infty}(\bar{U})$ such that
> \begin{align\*}
> u\_m \rightarrow u \quad \textrm{ in } W^{... | https://mathoverflow.net/users/131372 | Extended Global approximation theorem | A standart assumption for the density of $C^{\infty}(\overline\Omega)$ in $W^{k,p}(\Omega)$ is that the domain $\Omega$ satisfies the segment condition, namely, for every point $x\in\partial\Omega$, there exists $c\_{x}>0$ and a nonzero vector $y\_{x}$ such that
$$\forall z\in B(x,c\_{x})\cap\overline\Omega,\quad\{z+ty... | 2 | https://mathoverflow.net/users/89429 | 315445 | 136,941 |
https://mathoverflow.net/questions/306149 | 7 | The specialization map $sp:\mathfrak{X}\_\eta\to \mathfrak{X}\_{red}$ has an important role in rigid analytic geometry. I tried looking in Huber's papers ("Continuous Valuations", "A generalization of formal schemes
and rigid analytic varieties", and "Etale cohomology of Rigid Analytic Varieties and Adic Spaces") brows... | https://mathoverflow.net/users/46157 | Reference Request: Specialization map in Huber's Context | In case you're still interested: Bhatt recently proved that for any Tate-Huber pair $(A,A^+)$, the topological space $\mathrm{Spa}(A,A^+)$ is homeomorphic to an inverse limit of admissible blowups in the expected manner (Theorem 8.1.2 [here](http://www-personal.umich.edu/~bhattb/teaching/mat679w17/lectures.pdf)). For f... | 5 | https://mathoverflow.net/users/1464 | 315452 | 136,943 |
https://mathoverflow.net/questions/315355 | 2 | Let $(X,0)$ be a normal surface singularity. Suppose that it does not admit a smoothing.
Is it possible that there exists an isolated surface singularity $(Y,0)$ reduced near $0$ which is not irreducible and $(X,0)$ is one of its irreducible components and such that $(Y,0)$ does admit a smoothing? Is there some obstr... | https://mathoverflow.net/users/131261 | Smoothings of isolated non-irreducible surface singularities | I will adhere to Mumford's notation as much as possible.
Let $y$ denote the closed point of $Y$, $\mathcal O=\Gamma(Y,\mathcal O\_Y)$, $\widetilde{\mathcal O}=\Gamma(Y-\{y\},\mathcal O\_Y)$ and $Z=\operatorname{Spec}\widetilde{\mathcal O}$, which is the normalization of $Y$ in this situation. Then (Rim) $\widetilde{\ma... | 1 | https://mathoverflow.net/users/8726 | 315454 | 136,944 |
https://mathoverflow.net/questions/315449 | 5 | If $p\neq 2$, then the cyclic group $\mathbb{Z}\_p$ has no free continuous action on $\mathbb{C}P^n$. My question is how to prove the above fact using Leray-Serre spectral sequence associated to the Borel fibration $ \mathbb{C}P^n\hookrightarrow X\_{\mathbb{Z}\_p}\rightarrow B\_{\mathbb{Z}\_p}$.
From the Euler Chara... | https://mathoverflow.net/users/117065 | Show that if $p\neq 2$, then $\mathbb{Z}_p$ cannot act freely on $\mathbb{C}P^n$ | Consider the cohomology with $\mathbb{Z}$ coefficients (and reduce the 0-th term modulo $p$ to get uniform description of it). Then we have a spectral sequence starting from $\mathbb{F}\_p[x,y]/x^{n+1}$ with $deg(x)=deg(y)=2$ and converging to $H^\*(\mathbb{C}P^n/\mathbb{Z}\_p,\mathbb{Z})$. Since the $E^2$ term is conc... | 9 | https://mathoverflow.net/users/115052 | 315457 | 136,945 |
https://mathoverflow.net/questions/315384 | 10 | My advisor and I are working on Ricci curvature and an anonymous referee pointed out the following conjecture:
>
> Let $F\hookrightarrow M\stackrel{\pi}{\to}B$ be a fiber bundle from a compact manifold $M$ with fiber $F$, compact structure group $G$ and base $B$. Suppose that:
> i) $B$ has a metric of positive Ric... | https://mathoverflow.net/users/94097 | Positive Ricci curvature on fiber bundles | This conjecture is already proved in
*Gromoll, Detlef; Walschap, Gerard*, Metric foliations and curvature, Progress in Mathematics 268. Basel: Birkhäuser (ISBN 978-3-7643-8714-3/hbk). viii, 174 p. (2009). [ZBL1163.53001](https://zbmath.org/?q=an:1163.53001).
(page 100, Theorem 2.7.3).
| 4 | https://mathoverflow.net/users/94097 | 315461 | 136,946 |
https://mathoverflow.net/questions/315442 | 4 | $A$ is an infinite dimensional C$^\*$-algebra and $J\subset A$ is a closed right ideal. $A$ and $J$ are infinite dimensional(as a vector space). I want to find an **infinite** dimensional C$^\*$-algebra **subset of $J$**. How can I find it?
I know an infinite dimensional C$^\*$-algebra has an infinite dimensional com... | https://mathoverflow.net/users/130589 | Does a closed right ideal of a C$^*$-algebra have a C$^*$-algebra? | It need not exist. Take $A = B(l^2)$ (or the compacts if you want $A$ to be separable) and let $J$ be the set of operators of the form $$u \mapsto \langle u,v\rangle e\_1$$ for $v \in l^2$, where $e\_1$ is the first standard basis vector. It is a right ideal because for any $T \in B(l^2)$ we have $\langle Tu,v\rangle e... | 7 | https://mathoverflow.net/users/23141 | 315470 | 136,949 |
https://mathoverflow.net/questions/315482 | 6 | Let $K$ be a finite extension of $\mathbb{Q}\_p$, and let $C=\widehat{\overline{K}}$ be the completion of the algebraic closure of $K$. Let $\mathscr{O}\_C$ be the ring of integers in $C$, and let $G\_K$ be the absolute Galois group of $K$. I would like to know the following:
>
> What is the annihilator of $H^1\_{\... | https://mathoverflow.net/users/6950 | Integral Tate-Sen theory | I think the answer is "no", and that the minimal power of $p$ annihilating this module can be unbounded for fixed $i$ and varying $K$.
More precisely, fix a choice of a Galois-equivariant continuous $\mathcal{O}\_K$-linear surjection $\widehat{\mathcal{O}\_{K\_\infty}}\to \mathcal{O}\_K$
(e.g. by playing with Tate's... | 5 | https://mathoverflow.net/users/1464 | 315487 | 136,953 |
https://mathoverflow.net/questions/315480 | 0 | Consider the PDE
$$ \partial\_t f(t,x) = \Phi(x) f(t,x)+ \Psi(x)f(t,x-a)$$
$$f(0,x)=1$$
$$f(t,0)=1$$
where $a$ is a constant and $\Phi$ and $\Psi$ are some differentiable functions. To solve this I used separation of variables $f(t,x)=g(x)h(t)$ and I get
$$ \frac{\partial\_t h(t)} {h(t)}=\Phi(x)+\Psi(x)\frac{g(x-a)}{g(... | https://mathoverflow.net/users/131465 | Separation of variables for PDE | Assuming everything converges I think the correct candidate for a general solution for your PDE (leaving boundary conditions aside) is rather
$$
\int q(y) e^{yt} \prod\_{k=0}^{\infty} \frac{\Psi(x-ka)}{y-\Phi(x-ka)} dy
$$
where $q(y)$ is an arbitrary function of $y$.
Formally, it satisfies your PDE (again assuming e... | 0 | https://mathoverflow.net/users/100295 | 315492 | 136,956 |
https://mathoverflow.net/questions/315489 | 2 | Suppose $i\_Z \hookrightarrow X$ be a closed immersion, with $Z$ and $X$ being smooth varieties over $\mathbb{C}$, and $c, d$ are the dimensions of $Z$ and $X$ respectively.
$\textbf{Question}:$ Is it true that $L\_n i^\*\_{Z} \mathcal{F} = 0$ for any $n>d-c$ and $\mathcal{F}\in Coh(X)$ ?
The reason I ask is the f... | https://mathoverflow.net/users/121526 | Highest derived inverse image | Since $i\_\*$ is exact, we get isomorphisms
$$i\_\*L\_ji^\* \mathscr F \cong L\_j(i\_\*i^\*) \mathscr F.$$
Moreover, $i\_\*$ is faithful (in fact, it has a left inverse $i^\*$), thus to show that $L\_ji^\* \mathscr F$ vanishes for $j > \dim(X) - \dim(Z)$, it suffices to show the same statement for $L\_j(i\_\*i^\*)\math... | 2 | https://mathoverflow.net/users/82179 | 315496 | 136,957 |
https://mathoverflow.net/questions/315505 | 2 | We have all heard that if Goldbach's conjecture is independent, then it is true. This is because if GC is false then there is an even number which is not the sum of two primes, and hence a finite proof. What if this number is very large? Maybe beyond our reach?
If one proves that it is independent that eventually ev... | https://mathoverflow.net/users/nan | If Goldbach's Conjecture is eventually true, is it necessarily true? | We can't readily conclude that. Each $\Sigma\_1$ statement of arithmetic is provable in PA and hence in ZFC, but not every $\Pi\_2$ statement (which is what you would seem to need here) is.
Indeed the $\Pi\_1$ statement Con(ZFC) is already not provable (under the assumption that your Eventual Twin Prime Conjecture is... | 6 | https://mathoverflow.net/users/4600 | 315510 | 136,964 |
https://mathoverflow.net/questions/315512 | 4 | Is there a compact Riemann surface $X$ and a line bundle $L$ of negative degree on $X$, such that for any nontrivial extension
$$ 0 \rightarrow L \rightarrow E \rightarrow L^{-1} \rightarrow 0, $$
$E$ is a stable vector bundle on $X$? Any comment and reference is welcome, thank you.
| https://mathoverflow.net/users/40042 | Stable extensions by line bundles on Riemann surfaces | This never happens. Pick a point $p\in X$; the exact sequence $0\rightarrow L^{2}\rightarrow L^{2}(p)\rightarrow \mathbb{C}\_p\rightarrow 0$ gives rise to an exact sequence $0\rightarrow \mathbb{C}\xrightarrow{\ \partial \ } H^1(L^2)\longrightarrow H^1(L^2(p))\rightarrow 0$. The class
$e:=\partial (1)$ in $H^1(L^2)\co... | 10 | https://mathoverflow.net/users/40297 | 315515 | 136,965 |
https://mathoverflow.net/questions/252383 | 13 | It's been a long time since I tried to understand the deep meaning of the "Kan construction", or "[nerve-realization](https://ncatlab.org/nlab/show/nerve+and+realization)" adjunction
$$
\text{Lan}\_y F \dashv N\_F = \hom(F,1)
$$
that exists among the left Kan extension of $F\colon \mathcal{A}\to \mathbf{D}$ ($\cal A$ s... | https://mathoverflow.net/users/7952 | The Kan construction, profunctors, and Kan extensions | I will try to answer the second question.
>
> **Prop 1.** Let ${\bf C} \xleftarrow{i} {\bf A} \xrightarrow{f} {\bf B}$ be a span where $i$ is dense and fully faithful. Moreover $\text{lan}\_if$ is pointwise. Then, the following are equivalent.
>
>
> 1. $\text{lan}\_if \dashv \text{lan}\_fi$.
> 2. $f$ is the $i$-... | 6 | https://mathoverflow.net/users/104432 | 315519 | 136,967 |
https://mathoverflow.net/questions/315524 | 1 | A set $M$ of real values is said to be enumerable if there is a bijection between the elements of $M$ the elements of $\mathbb{N}$.
That definition does however not impose any restrictions on the order in which the elements of $M$ must be enumerated.
>
> **Questions:**
>
>
> defining two enumerable sets $A$ a... | https://mathoverflow.net/users/31310 | Enumeration Hierarchies | For the first two thirds of your question, you're really just talking about **countable linear orders**. Every set $A\subseteq\mathbb{R}$ can be viewed as a linear order in a natural way (via the usual ordering of $\mathbb{R}$), and two sets are isomorphic in your sense iff they are isomorphic when thought of as linear... | 10 | https://mathoverflow.net/users/8133 | 315526 | 136,968 |
https://mathoverflow.net/questions/315532 | 6 | Does anyone know if the following problem has ever been studied?
>
> Let $a$ and $b$ be two real numbers and consider the polynomial: $$p\_n(x,y)=\sum\_{i=0}^{n-1}x^{n-1-i}y^{i}$$
> where $n$ is a positive integer.
>
>
> Does there exist a value of $k$ such that if $p\_n(a,b)$ is an integer for $k$ consecutive ... | https://mathoverflow.net/users/70464 | $p_n(x,y)=\sum_{i=0}^{n-1}x^{n-1-i}y^{i}$ is always an integer | I will do the rational case and assume $a,b\neq 0$ otherwise the problem is trivial. You just need four consecutive values. Note that $p\_n(a,b)=\cfrac{a^n-b^n}{a-b}$.
Say you have $p\_k$, $p\_{k+1}$, $p\_{k+2}$, $p\_{k+3}$ are all rational.
Note that $p\_{k+1}^2-p\_kp\_{k+2}=(ab)^k$ and $p\_{k+2}^2-p\_{k+1}p\_{k+3}=(a... | 11 | https://mathoverflow.net/users/41010 | 315534 | 136,970 |
https://mathoverflow.net/questions/315536 | 6 | In dimension one, it is well known that $\mathcal{F}\chi\_{(-1,1)}=\frac{\sin{x}}{x}$. This implies, in particular, that $\frac{\sin{x}}{x}$ is a definite positive function. I wonder if a similar result holds in dimension three. Moreover, it would be nice to actually get strictly positive definiteness. So my question i... | https://mathoverflow.net/users/54552 | Positive-definiteness of radial sinc function in three dimensions | It’s the Fourier transform of the rotation-invariant probability [measure](https://mathoverflow.net/questions/149692/fourier-transform-of-the-unit-sphere) on the unit sphere, and as such is positive definite.
Strict positive definiteness also holds, by the theorem in zu Castell, Filbir and Szwarc ([2005](//ams.org/ma... | 5 | https://mathoverflow.net/users/19276 | 315544 | 136,973 |
https://mathoverflow.net/questions/315398 | 6 | Let $\pi\colon E \rightarrow B$ be a fiber bundle with fiber $F$. I am not assuming that $B$ is simply-connected. We then have Serre spectral sequences in both rational homology and rational cohomology:
$$E^2\_{pq} \cong H\_p(B;H\_q(F;\mathbb{Q})) \Longrightarrow H\_{p+q}(E;\mathbb{Q})$$
$$E\_2^{pq} \cong H^p(B;H^q... | https://mathoverflow.net/users/131427 | Serre spectral sequence degeneration in homology vs cohomology | The answer is yes, and this follows from the comments, but it's worth spelling out the (non-obvious) details.
By Will Sawin's comment, the claim will follow if we can show that
$$
\dim\_\mathbb{Q} H^p(B;H^q(F;\mathbb{Q})) = \dim\_\mathbb{Q} H\_p(B;H\_q(F;\mathbb{Q}))
$$
for all $p,q\ge0$. The Universal Coefficient T... | 3 | https://mathoverflow.net/users/8103 | 315547 | 136,975 |
https://mathoverflow.net/questions/315539 | 1 | On $\mathbb{P}^2$, we have two full strong exceptional collections:
$\{\mathcal{O}\_{\mathbb{P}^2},\Omega\_{\mathbb{P}^2}(2),\mathcal{O}\_{\mathbb{P}^2}(1)\}$ and $\{\mathcal{O}\_{\mathbb{P}^2}(-2), \mathcal{O}\_{\mathbb{P}^2}(-1), \mathcal{O}\_{\mathbb{P}^2}\}$.
We use $B\_1,B\_2$ to denote the endomorphism algeb... | https://mathoverflow.net/users/48616 | Morita equivalence of quivers from related exceptional collections | The algebras $B\_1$ and $B\_2$ are NOT Morita-equivalent. Indeed, their categories of modules have exactly three simple (i.e., without non-trivial subobjects) objects, that correspond to the vertices of the quivers, and the dimensions of $Ext$-spaces between these simple objects is $(3,3,3)$ for one quiver and $(3,3,6)... | 5 | https://mathoverflow.net/users/4428 | 315548 | 136,976 |
https://mathoverflow.net/questions/315558 | 2 | Consider $H\_1,H\_2,H\_3\subset\mathbb{P}^{2m+1}$ three general linear subspaces of projective dimension $m$.
Then there exists a quadric hypersurface $Q^{2m}\subset\mathbb{P}^{2m+1}$ containing $H\_1,H\_2,H\_3$. This is just a dimension count.
*Does there exist a **smooth** quadric hypersurface $Q^{2m}\subset\math... | https://mathoverflow.net/users/14514 | Linear subspaces in quadric hypersurfaces | This is true when $m$ is odd and false otherwise.
Indeed, let $\mathbb{P}^{2m+1} = \mathbb{P}(V)$. Assuming $H\_1 \cap H\_2 = \emptyset$ (by genericity), we have $H\_1 = \mathbb{P}(V\_1)$, $H\_2 = \mathbb{P}(V\_2)$ and
$$
V = V\_1 \oplus V\_2.
$$
Furthermore, assuming $H\_3 \cap H\_1 = H\_3 \cap H\_2 = \emptyset$, w... | 6 | https://mathoverflow.net/users/4428 | 315560 | 136,981 |
https://mathoverflow.net/questions/315552 | 4 | Let $e=[1,1,\ldots,1]\in\mathbb{Z}^n$. I am looking for a way to find a vector $a\in\mathbb{Z}^n$ such that:
1. $\langle a,e\rangle=0$ and
2. for every nonnegative $v\in\mathbb{Z}^n$ such that $\langle e,v\rangle=n$ and $v\neq e$ we have $\langle a,v\rangle\neq 0$
Here the vector $v$ is nonnegative if it has nonneg... | https://mathoverflow.net/users/16678 | Separate the trivial partition by a linear hyperspace | There is no chance you can keep $a$ of polynomial size. Indeed, let $a$ be any vector with the sum of absolute values less than ${n\choose [n/2]}/2$. Then among the sums $\sum\_{k\in I}a\_k$ where $I\subset\{1,\dots,n\}$, $|I|=[n/2]$ there are two equal, meaning that there is a non-trivial vector $w$ consisting of $0,\... | 8 | https://mathoverflow.net/users/1131 | 315565 | 136,982 |
https://mathoverflow.net/questions/315441 | 11 | Let $X$ be an algebraic variety over an algebraically closed field. Consider the two subsets $X\_0\subseteq X\_1 \subseteq X$:
$$X\_0 = \{a\in X| a \mbox{ is a scheme-theoretic complete intersection in }X\},$$
$$X\_1 = \{a\in X| a \mbox{ is a set-theoretic complete intersection in }X\}.$$
**Question: what do we know ... | https://mathoverflow.net/users/5301 | The Locus of Complete Intersection Points | This should be difficult in general. However there are some easy remarks to get going:
First, if $p$ is a CI point, then $X\_p$ is regular. That is because when you localize, the number of generators can only drops, and it is still have to be at least $n=\dim X$. So they are equal.
Now let's try to answer number 3... | 5 | https://mathoverflow.net/users/2083 | 315579 | 136,986 |
https://mathoverflow.net/questions/315564 | 1 | Let $R$ be a commutative $\mathbb{Q}$-algebra which is not an integral domain,
for example: $R=\frac{\mathbb{Q}[t]}{(t^2-1)}$.
Let $k$ be an algebraically closed field of characteristic zero, and let $p,q \in k[x,y]$ such that $\operatorname{Jac}(p,q)=1$.
A [known result](https://eudml.org/doc/152524) by T. T. Moh ... | https://mathoverflow.net/users/72288 | The Jacobian Conjecture over a commutative $\mathbb{Q}$-algebra which is not an integral domain | The most general situation I can cover is the following:
>
> **Proposition.** *Let $k$ be a field of characteristic $0$, and let $R$ be a $k$-algebra of finite type. If $f,g \in R[x,y]$ are polynomials such that $\det\operatorname{Jac}(f,g) = 1$ but $(f,g) \colon R[x,y] \to R[x,y]$ is not an isomorphism, then $\deg... | 2 | https://mathoverflow.net/users/82179 | 315586 | 136,988 |
https://mathoverflow.net/questions/315531 | 3 | **Definition:**
Let $G$ be a graph. A subset $C \subseteq V(G)$ is a vertex cover of
$G$ if for each $e \in E(G)$, $e\cap C \neq \phi$. If $C$ is minimal
with respect to inclusion, then $C$ is called minimal vertex
cover of $G$.
Let $G$ be a graph with $V(G)=\{z\_1,\ldots,z\_n\}$.
Let $H$ be the new graph $H$ on new... | https://mathoverflow.net/users/68302 | Minimal vertex cover | Not necessarily. E.g., the set $\{z\_{i,1}\colon 1\leq i\leq n\}$ is also a minimal vertex cover of $H$ (whenever $G$ has no isolated vertices), and it may have more elements (e.g., if $G$ is a star, and $M\_1$ consists of its center).
In fact, all minimal covers of $H$ can be pbtained in the following way. Notice th... | 2 | https://mathoverflow.net/users/17581 | 315590 | 136,989 |
https://mathoverflow.net/questions/315576 | 6 | Let $E$ be an elliptic curve over $\mathbb{Q}$ and $p$ a prime. It is conjectured in the book of Coates and Sujatha "Galois Cohomology of Elliptic Curves" (Conjecture 2.5) that the corank of the Selmer group of $E$ over the cyclotomic $\mathbb{Z}\_p$ extension is $1$ when $E$ has potentially supersingular reduction at ... | https://mathoverflow.net/users/nan | What is the current status on the corank conjecture for Selmer groups? | Yes, the corank conjecture is a theorem for elliptic curves over $\mathbb{Q}$. The key to the proof is the following:
>
> **Theorem** (Kato, 2004): For any $E$ and any $p$, the "fine Selmer group" $Sel\_p^0(E / \mathbb{Q}\_\infty) = \operatorname{ker}\Big(Sel\_p(E / \mathbb{Q}\_\infty) \to H^1(\mathbb{Q}\_{p, \inft... | 5 | https://mathoverflow.net/users/2481 | 315599 | 136,993 |
https://mathoverflow.net/questions/315595 | 12 | In SGA 4, Grothendieck introduced a set-theoretic device called a *Grothendieck universe*. He and his collaborators worked in Bourbaki set theory, which is practically similar to $\mathsf{ZFC}$ but fundamentally different. To ease the work with universes, he introduced two addition axioms to the theory: they are mostly... | https://mathoverflow.net/users/83143 | Translating Grothendieck axiom UB into ZFC | If we work with some given universe $U$, we have to make sure that we do not leave it accidentally. The definition of a universe does this for most operations. But there is still a way to leave the universe, namely by the (global) axiom of choice, i.e., Hilbert's symbol $\tau$.
Consider a relation $R$ with a variable... | 8 | https://mathoverflow.net/users/11025 | 315608 | 136,995 |
https://mathoverflow.net/questions/303183 | 3 | I am reading [Orbifolds as Groupoids: an Introduction](https://arxiv.org/pdf/math/0203100.pdf) by Ieke Moerdijk.
In page $8$ when explaining local charts, it says the following :
>
> Let $\mathcal{G}$ be a Lie groupoid. For an open set $U\subseteq \mathcal{G}\_0$, we write $\mathcal{G}|\_U$ for the full subgroupo... | https://mathoverflow.net/users/118688 | Proper and etale groupoid is locally a translation groupoid | In the following I will use the notation you cited in the problem statement.
The point of the proof should be that you want to construct an open neighborhood $N\_x$ which is stable under the $\delta$-action of the stabiliser group $G\_x$ and (here comes the crucical property!) that
$$
s^{-1} (N\_x) = \coprod\_{\gamm... | 1 | https://mathoverflow.net/users/46510 | 315619 | 136,999 |
https://mathoverflow.net/questions/315613 | 7 | Does there exist $f \in L^1(\mathbb{R}^n)$ s.t., $\displaystyle \widehat{f}(\xi) = \frac{\sin |\xi|}{|\xi|}$ in case of dimension $n \ge 3$?
It is known that for $n = 2$, the function $\displaystyle f(x) = \frac{\chi\_{\{|x| < 1\}}}{\sqrt{1-|x|^2}}$ curiously has some constant multiple of $\dfrac{\sin |\xi|}{|\xi|}$ ... | https://mathoverflow.net/users/74787 | Is $\frac{\sin |\xi|}{|\xi|}$ in range of Fourier Transform for $n \ge 3$? | Here's a formula from Duistermaat and Kolk book *Distributions: Theory and Applications*, Chapter 17, Eq, (17.13). We denote by $\newcommand{\eF}{\mathscr{F}}$ $\eF$ the Fourier transform. Then $\newcommand{\ii}{\boldsymbol{i}}$ $\newcommand{\ve}{\varepsilon}$
$$
\eF^{-1}\left(\frac{e^{\ii t\Vert \xi\Vert}}{\Vert\xi\... | 10 | https://mathoverflow.net/users/20302 | 315623 | 137,001 |
https://mathoverflow.net/questions/315618 | 6 | A set $\mathscr{U}$ is a universe if the following conditions are met:
>
> * For any $x \in \mathscr{U}$ we have $x \subseteq \mathscr{U}$
> * For any $x,y \in \mathscr{U}$ we have $\{x,y\} \in \mathscr{U}$,
> * For any $x \in \mathscr{U}$ we have $\mathcal{P}(x) \in \mathscr{U}$,
> * For any family $(x\_i)\_{i \in... | https://mathoverflow.net/users/83143 | How do we formally construct the successor universe $\mathscr{U}^+$ of a universe $\mathscr{U}$ in $\mathsf{ZFC}$? | The intersection of a nonempty set of universes is a universe. Now, let $U$ be a universe, and suppose that there exists a universe $V$ with $U\in V$. Then, the set of all universes $W$ with $U\in W\subseteq V$ exists and is nonempty. Its intersection is a universe, and this is readily checked to be the smallest univer... | 11 | https://mathoverflow.net/users/11025 | 315630 | 137,004 |
https://mathoverflow.net/questions/315626 | 2 | Let $H\_0$ and $H\_1$ be two distributions. The Neyman-Pearson lemma says that of all rejection regions $R$ with fixed probability $\alpha$ under $H\_0$, the one with maximal probability under $H\_1$ is the set of the form $R = \{x: \frac{p\_1(x)}{p\_0(x)} \ge c\}$ with $c$ chosen such that $\mathbb{P}\_{x \sim H\_0}(x... | https://mathoverflow.net/users/127773 | upper bound on power of neyman-pearson hypothesis test | $\renewcommand{\P}{\operatorname{\mathsf P}}
\newcommand{\al}{\alpha}
\newcommand{\be}{\beta}
\newcommand{\la}{\lambda}
\newcommand{\D}{\overset{\text{D}}=}
\newcommand{\eD}{\overset{\text{D}}\to}
\newcommand{\si}{\sigma}
\newcommand{\tZ}{\tilde Z}$
Let $(X\_1,\dots,X\_d)$ be the observed normal random vector. Then ... | 0 | https://mathoverflow.net/users/36721 | 315644 | 137,011 |
https://mathoverflow.net/questions/301877 | 26 | In [this great question](https://mathoverflow.net/q/301844/102946) by Nathaniel Johnston, and in its answers, we can learn the following remarkable inequality: For all $v,w \in \mathbb{R}^n$ we have
\begin{align\*}
\|v^2\| \, \|w^2\| - \langle v^2, w^2 \rangle \le \|v\|^2 \|w\|^2 - \langle v,w \rangle^2; \quad (\*)
\e... | https://mathoverflow.net/users/102946 | Symmetric strengthening of the Cauchy-Schwarz inequality | $(\*\*)$ is always equality. By homogenuity we may suppose that $\|v\|=\|w\|=1$, then $Uv,Uw$ are any two unit vectors with prescribed inner product $\langle Uv,Uw\rangle =\langle u,v\rangle$. I claim that you may find two two-dimensional vectors with prescribed given inner product such that equality in $(\*)$ holds. T... | 7 | https://mathoverflow.net/users/4312 | 315659 | 137,014 |
https://mathoverflow.net/questions/315647 | 5 | Let $\Delta$ be a flag complex on $n$ vertices. Let $r$ be the smallest size of the facets of $\Delta$. Suppose that $2r>n$. Must $\Delta$ be acyclic?
| https://mathoverflow.net/users/2083 | Criterion for acyclicity of flag complexes | It looks to me like you can prove the stronger property of contractibility by induction, as follows. Let $\Delta$ be the independence complex of graph $G$, as guaranteed by the flag property.
If $\Delta$ is a cone, then $\Delta$ is contractible. This will be the base case, along with dimension 0 (where $\Delta$ has a... | 3 | https://mathoverflow.net/users/19729 | 315665 | 137,016 |
https://mathoverflow.net/questions/315009 | 7 | I am new to StackExchange and I am currently going through Gaudry's paper on counting points on hyperelliptic curves (see <https://hal.inria.fr/inria-00512403/document>). As a part of the generalization of Schoof s algorithm for genus 2, we have to compute l-torsion points. For this we use Cantor s division polynomials... | https://mathoverflow.net/users/131249 | Schoof's Algorithm for Hyperelliptic curves over $\mathbb{F}_q$ : Question regarding computation of resultant: Gaudry | *Answer:*
The resultant $R(x\_1)$ vanishes at $x\_1=X\_1$ iff $E\_1(X\_1,x\_2)=0=E\_2(X\_1,x\_2)$ for some $x\_2=X\_2$. Take $X\_2$ to be one of the $\delta={\rm deg}\,(d\_2)=2l^2-3$ roots of the polynomial $d\_2$. Then $E\_1(x\_1,X\_2)$ and $E\_2(x\_1,X\_2)$ vanish if $d\_2(x\_1)=0$, so $d\_2(x\_1)$ is a factor of ... | 4 | https://mathoverflow.net/users/11260 | 315666 | 137,017 |
https://mathoverflow.net/questions/314875 | 14 | It's well known that for Noetherian separated regular schemes the canonical map $$K(X) \longrightarrow G(X)$$ (Quillen uses $K'$ instead of $G$, though) is a weak equivalence.
This statement is usually called Poincaré duality.
One can also define $K$-theory with compact support (for sufficiently nice schemes $X$) b... | https://mathoverflow.net/users/40883 | Why $K(X) \longrightarrow G (X)$ is a Poincaré duality for K-theory? | To my knowledge, one can only make this analogy fully consistent with Weibel's homotopy invariant $K$-theory $KH$ and $G$-theory (although the proofs of what I claim below rely heavily on our understanding of classical algebraic $K$-theory). Then, using the canonical map $K(X)\to KH(X)$, the pairing relating $KH$ and $... | 16 | https://mathoverflow.net/users/1017 | 315676 | 137,021 |
https://mathoverflow.net/questions/315673 | 5 | Is there any Bernstein type theorems for CMC hypersurfaces in $\mathbb{R}^{n+1}$ in the literature?
More precisely I would like to know if there is an answer to the following
**QUESTION**: Let $f : \mathbb{R}^n \to \mathbb{R}$ be a smooth function such that $\mathrm{graph}(f)$ is a constant mean curvature hypersur... | https://mathoverflow.net/users/86341 | Bernstein type theorems for CMC hypersurfaces in $\mathbb{R}^{n+1}$ | This was solved in a series of articles in the 1960s.
De Giorgi, Almgren, and Simons have shown that in $\mathbb{R}^{\le 8}$ every CMC graph is a hyperplane. Then Bombieri - De Giorgi - Giusti have shown that in $\mathbb{R}^{\ge 9}$ there are minimal graphs which are not hyperplanes.
Here is a link to the latter ar... | 7 | https://mathoverflow.net/users/98590 | 315677 | 137,022 |
https://mathoverflow.net/questions/315661 | 2 | Lebesgue's differentiation theorem states that if $x$ is a point in $\mathbb{R}^n$ and $f:\mathbb{R}^n\rightarrow\mathbb{R}$ is a Lebesgue integrable function, then the limit of $\frac{\int\_B f d\lambda}{\lambda(B)}$ over all balls $B$ centered at $x$ as the diameter of $B$ goes to $0$ is equal almost everywhere to $f... | https://mathoverflow.net/users/5017 | Does the Lebesgue Differentiation Theorem hold for regular polytopes? | The Lebesgue differentiation theorem holds for *regular families* of sets. A regular family is a family of sets that contain sets of arbitrarily small diameters and have the property that there is a constant $C\geq 1$ such that for each $E$ set in the family there is a ball $B(x,r)$ such that
$$
E\subset B(x,r),
\quad
... | 6 | https://mathoverflow.net/users/121665 | 315683 | 137,024 |
https://mathoverflow.net/questions/315682 | 6 | I'm reading 《Algebraic geometry and statistical learning theory》.My problem is why the Fisher information matrix is equal to the Hessian matrix of the Kullback–Leibler distance at the true parameter?In the page 8
[detail](https://i.stack.imgur.com/XeQLH.png)
[detail](https://i.stack.imgur.com/6EoGe.png)
| https://mathoverflow.net/users/128758 | Why the Fisher information matrix is equal to the Hessian matrix of the Kullback–Leibler distance at the true parameter? | $\newcommand{\R}{\mathbb{R}}
\newcommand{\p}{\partial}$
Let $(p\_w)\_{w\in W}$ be a family of pdfs, where $W$ is an open subset of $\R^k$. Take any $w\_\*\in W$. The Kullback-–Leibler "distance" from $w=(w\_1,\dots,w\_k)\in W$ to $w\_\*$ is
\begin{equation}
d(w):=D(w,w\_\*):=\int p\_{w\_\*} \ln\frac{p\_{w\_\*}}{p\_... | 8 | https://mathoverflow.net/users/36721 | 315687 | 137,026 |
https://mathoverflow.net/questions/315697 | 7 | Let $X$ be an irreducible normal projective scheme over $\mathbb{C}$. Let $U$ be the open subscheme of smooth points of $X$. Consider the closed subscheme $Z = X \setminus U$. Suppose that the codimension of $Z$ in $X$ is at least $2$. Is it true that the fundamental group of $U$ and $X$ are isomorphic?
Edit: Is it ... | https://mathoverflow.net/users/124771 | Fundamental group of an open subscheme of a normal scheme | Let me expand my comment into an answer.
Take as $X$ the cone of vertex $v$ over an elliptic curve $E$. Then $X$ is simply connected (this is a general property of projective cones). However, $U = X-\{v\}$ is *not* simply connected: in fact, the projection $\pi \colon U \to E$ onto the basis gives $X$ the structure ... | 7 | https://mathoverflow.net/users/7460 | 315710 | 137,030 |
https://mathoverflow.net/questions/315717 | 13 | You are gracious enough to host me for a few days while I attend a conference. After I leave, you're surprised to see a gift on the kitchen table. It's a box with a category inside! The objects aren't labeled so it's a little hard to tell what's going on with it, but you can see, for example, that there's a terminal ob... | https://mathoverflow.net/users/113369 | Can one recover an algebraically closed field $k$ from the dots and arrows of its category of finitely generated $k$-algebras? | I think the following works for any commutative ring: Consider the category of abelian group objects in the overcategory above the initial element. This is equivalent to the category of finitely generated $R$-Modules (see for example <https://ncatlab.org/nlab/show/module>). By the general theory of Morita equivalence, ... | 16 | https://mathoverflow.net/users/131560 | 315724 | 137,036 |
https://mathoverflow.net/questions/315723 | 1 | Let $X$ be a Banach space with a Schauder basis, and $Y$ a Banach space. Let $P\_N$ denote the coordinate projections relative to the basis of $X$, and let $X\_N$ denote their ranges. Specifically, $\{x\_n\}\_{n=1}^{\infty}$ is a basis in $X$ and
$$P\_N(x)=\sum\_{n=1}^Na\_nx\_n,\quad\hbox{with}\quad x=\sum\_{n=1}^{\in... | https://mathoverflow.net/users/82382 | If the ranges of all f.d basis projections in $X$ are isometric to subspaces of $Y$, does it follow that $X$ is isometric to a subspace of $Y$? | It is false for $L\_p$. Consider the Haar basis. The span of the first $n$ basis vectors is isometrically isomorphic $\ell\_p^n$, but $L\_p$ does not embed even isomorphically into $\ell\_p$ if $p\not=2$.
| 3 | https://mathoverflow.net/users/2554 | 315726 | 137,038 |
https://mathoverflow.net/questions/315690 | 9 | Is it true that for prime $p\neq 2 $, $k > 1$ and $n\_1,n\_2,\dots,n\_k\geq 1$, the cyclic group $\mathbb{Z}\_p$ has no continuous free action on $ \mathbb{C}P^{n\_1} \times \mathbb{C}P^{n\_2} \times \cdots \times \mathbb{C}P^{n\_k}$?
How to prove it?
Thank you so much in advance.
| https://mathoverflow.net/users/117065 | Does there exist a free action of $\mathbb Z_p$ on this space? | Consider the action of the automorphism on $H^2(\prod\_i \mathbb C P^{n\_i} , \mathbb Z)$ by linear automorphisms and $H^\*(\prod\_i \mathbb C P^{n\_i} , \mathbb Z)$.
Inside $H^2(\prod\_i \mathbb C P^{n\_i} , \mathbb Z)$ consider the set of nonzero integral multiples of hyperplane classes of the different factors. Th... | 8 | https://mathoverflow.net/users/18060 | 315731 | 137,041 |
https://mathoverflow.net/questions/315011 | 2 | Let $M$ be a symplectic manifold equipped with a hamiltonian action of a compact Lie group $G$ with moment map $\mu\colon M\to \mathfrak g^\*$. Assume $c\in \mathfrak g^\*$. Then the symplectic reduction theorem tells us that if $G$ acts freely on $\mu^{-1}(c)$ then we can take a quotient of $\mu^{-1}(c)$ by $G$-action... | https://mathoverflow.net/users/88385 | Global symplectic reduction | In the general case, the reduced space $\mu^{-1}(c) / G\_c$ is what is called a stratified symplectic space. This means, that for every orbit type $(H)$ the orbit type subset $\mu^{-1}(c)\_{(H)} / G\_c$ is a smooth symplectic manifold and the symplectic forms on these strata fit together in the sense that there is a co... | 2 | https://mathoverflow.net/users/17047 | 315738 | 137,045 |
https://mathoverflow.net/questions/315752 | 2 | Let $X$ be a normal projective (or, quasi-projective) variety over $\mathbb{C}$. Let $U \subset X$ be an open subscheme whose complement $Z = X \setminus U$ has codimension at least $2$ in $X$. Let $L$ be a line bundle on $U$. Is it possible to extend $L$ to a line bundle $\widetilde{L}$ on $X$ such that $\widetilde{L}... | https://mathoverflow.net/users/124771 | Extension of line bundle defined over an open subscheme | As explained here [Extending vector bundles on a given open subscheme](https://mathoverflow.net/questions/22111/extending-vector-bundles-on-a-given-open-subscheme?rq=1) the only possible such extension is
$$
\tilde{L} = (i\_\*L)^{\vee\vee},
$$
where $i \colon U \to X$ is the embedding. The sheaf $\tilde{L}$ is a refle... | 9 | https://mathoverflow.net/users/4428 | 315753 | 137,048 |
https://mathoverflow.net/questions/315748 | 7 | In symplectic geometry, the Grassmannian of isotropic planes for a symplectic vector space is a well known and well studied object; for example, one can realize it as a homogeneous space with a known stabilizer subgroup of the symplectic group, and one can also realize a Schubert cell decomposition.
However, one can ... | https://mathoverflow.net/users/113061 | Grassmannians of planes isotropic with respect to general tensors | What you are asking about is very classical in the theory of exterior differential systems. The subspaces of $D$ that you are calling `isotropic' are what Élie Cartan called the *integral elements* of the differential ideal $\mathcal{I}$ generated by the sections of $D^\perp\subset T^\*M$ (the annihilator subbundle of ... | 11 | https://mathoverflow.net/users/13972 | 315765 | 137,052 |
https://mathoverflow.net/questions/270616 | 15 | A map $f\colon X\to Y$ between metric spaces is *uniformly open* whenever for each $\varepsilon >0$ there is $\delta >0$ such that for any $x\in X$ one has
$$B\_Y\big(f(x),\delta\big)\subseteq f\big(B\_X(x,\varepsilon)\big), $$
where $B\_X, B\_Y$ denote open balls in the respective spaces.
Uniformly open maps have t... | https://mathoverflow.net/users/15129 | Open bilinear maps that are not uniformly open | Recently it was proved that multiplication in the space of functions of bounded variation is open but not uniformly so it is a counterexample to the above question. This also reflects the above comments, which predicted that.
>
> Stanisław Kowalczyk, Małgorzata Turowska, [Multiplication in the space of functions of... | 2 | https://mathoverflow.net/users/15129 | 315768 | 137,055 |
https://mathoverflow.net/questions/315776 | 3 | Let $\chi(s)=\int\_{0}^{1}x(t)^{s}f(t)dt$,
where $x(t)$ and $f(t)$ are real valued continuous functions for
$t\in[0,1]$, and $f(t)\geq0$.
Is it possible to show that
$\left(\chi(0)\chi(2)-\chi(1)^{2}\right)\left(\chi(4)\chi(2)-\chi(3)^{2}\right)-\left(\chi(3)\chi(1)-\chi(2)^{2}\right)^{2}\geq0$
Note: I believe t... | https://mathoverflow.net/users/131587 | A moment inequality | This inequality is false in general, by homogeneity considerations. Indeed, it can be rewritten as $L(x)\ge R(x)$, where $L(x):=\left(\chi(0)\chi(2)-\chi(1)^{2}\right)\left(\chi(4)\chi(2)-\chi(3)^{2}\right)$ and $R(x):=\chi(3)\chi(1)-\chi(2)^{2}$. Take any non-constant positive $x$, so that, by the Cauchy--Schwarz ineq... | 6 | https://mathoverflow.net/users/36721 | 315778 | 137,057 |
https://mathoverflow.net/questions/315793 | 2 | The Komlós–Major–Tusnády Approximation (see [Wikipedia](https://en.wikipedia.org/wiki/Koml%C3%B3s%E2%80%93Major%E2%80%93Tusn%C3%A1dy_approximation)) considers the sum of uniform variables in $(0,1)$. There are also version where instead the sum of equiprobable $0/1$ variables is used ($p=1/2$).
Is there are generaliz... | https://mathoverflow.net/users/126079 | Generalization of Komlós–Major–Tusnády Approximation | The Wikipedia article that you are looking at is, not about the Komlós–Major–Tusnády (KMT) approximation for sums of iid r.v.'s, but about the KMT approximation for the empirical process. However, the second paper (KMT 1976) cited in the Wikipedia article contains the following result.
Let $S\_n:=\sum\_1^n X\_i$, wh... | 2 | https://mathoverflow.net/users/36721 | 315795 | 137,062 |
https://mathoverflow.net/questions/315779 | 1 | Let $F$ be a global function field with full constant field $\mathbb{F}\_q$. We fix a place $\infty$ and let $A$ be the ring of elements of $F$ regular away from $\infty$. We denote by $v\_\infty$ the normalized valuation associated to $\infty$. If $a\in A$ and $v\_\infty(a)\geq 0$ then $a$ has no poles. Therefore (see... | https://mathoverflow.net/users/23207 | On the maximal value of the valuation at infinity of elements in the ring of integers of a global function field | 1. Typically, no. If $\infty$ has degree $1$ then $D\_{F,\infty}=-1$ if and only if $F$ is rational, since in this case $a$ would have degree $1$ so $F=\mathbb{F}\_q(a)$. When $\infty$ has degree $1$, $-D\_{F,\infty}$ is the first non-gap in the Weierstrass semigroup of $\infty$. Lots of literature on that. It can take... | 1 | https://mathoverflow.net/users/2290 | 315807 | 137,063 |
https://mathoverflow.net/questions/315814 | 2 | Let $(X, L)$ and $(Y, M)$ be two polarized abelian varieties .
According to Birkenhake C. and Lange H. in Complex Abelian Varieties a homomorphism of polarized abelian varieties $f:(Y, M)\longrightarrow (X, L)$ is a homomorphism of complex tori $f:Y\longrightarrow X$ such that $f^{\*}c\_1(L) = c\_1(M)$.
Question:... | https://mathoverflow.net/users/29836 | Pull-back of polarization | Yes! This is sometime called naturality of Chern classes. You can find it in many books, for instance Complex Geometry - An Introduction | Daniel Huybrechts, or Differential forms in algebraic topology by Bott and Tu.
| 1 | https://mathoverflow.net/users/48866 | 315815 | 137,065 |
https://mathoverflow.net/questions/185980 | 9 | Suppose I have an $(\infty,1)$-topos $\mathcal{X}$ and a (small) set of maps $S$ in $\mathcal{X}$, which therefore generates an accessible localization $S^{-1}\mathcal{X}$. Is there any "small" condition $P$ on $S$ which implies that this localization is left exact (hence a sub-$(\infty,1)$-topos), and such that any ac... | https://mathoverflow.net/users/49 | A "small" definition of sub-(∞,1)-topoi | This follow from some recent (I heard about this a year ago) results by Anel, Biedermann, Finster and Joyal.
Unfortunately their work is not available yet, but You have some slide of Mathieu Anel on the topic on his web page presenting this (<http://mathieu.anel.free.fr/mat/doc/Anel-LexLocalizations.pdf>). I don't k... | 5 | https://mathoverflow.net/users/22131 | 315817 | 137,066 |
https://mathoverflow.net/questions/315816 | 46 | Morel and Voevoedsky developed what is now called motivic homotopy theory, which aims to apply techniques of algebraic topology to algebraic varieties and, more generally, to schemes. A simple way of stating the idea is that we wish to find a model structure on some algebraic category related to that of varieties or th... | https://mathoverflow.net/users/117073 | Why is Voevodsky's motivic homotopy theory 'the right' approach? | (Don't be afraid about the word "$\infty$-category" here: they're just a convenient framework to do homotopy theory in).
---
I'm going to try with a very naive answer, although I'm not sure I understand your question exactly.
The (un)stable motivic ($\infty$-)category has a universal property. To be precise the... | 30 | https://mathoverflow.net/users/43054 | 315818 | 137,067 |
https://mathoverflow.net/questions/315811 | 9 | Assume that $1<a<2$ and $c\ne 0$ is a real number. What is known about the distribution of the sequence $cn^a$ modulo 1? Say, is it true that for certain $\theta<1$ (depending on $a$ and $c$) we have $|\sum\_{k=1}^n \{ck^a\}-\frac{n}2|=O(n^\theta)$?
| https://mathoverflow.net/users/4312 | Distribution of $\{cn^a\}$ | Exercise 2.23 in Kuipers and Niederreiter, Uniform Distribution of Sequences, goes: Use Theorem 2.7 to show that the sequence $\{\alpha n^{\sigma}\}$, $n=1,2,\dots$, $\alpha\ne0$, $1<\sigma<2$, is uniformly distributed modulo one.
Theorem 2.7 is as follows. Let $a$ and $b$ be integers with $a<b$, and let $f$ be twic... | 6 | https://mathoverflow.net/users/3684 | 315820 | 137,069 |
https://mathoverflow.net/questions/315674 | 7 | Let $f \in \mathbb{C}[[x,y]]$ be the germ of an isolated plane curve singularity. Then the Hilbert-Samuel multiplicity $e\_f$ of $f$ is given as follows:
$$e\_f = \lim\_{s \to \infty}\frac{1}{s} \cdot \dim\_{\mathbb{C}} \mathbb{C}[[x,y]]\bigg/\left(f, \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}\r... | https://mathoverflow.net/users/76440 | Relationship between Hilbert-Samuel multiplicity and polar multiplicity | First, the quantity you defined is the Hilbert-Samuel multiplicity of the ideal $J= (f\_x,f\_y)$ in $R=\mathbb C[[x,y]]/(f)$. The multiplicity of $f$ usually refers to the multiplicity of the maximal ideal $m$ of $R$.
As you noted, it is enough to show that a generic combination of the generators of $J$ is a reducti... | 4 | https://mathoverflow.net/users/2083 | 315835 | 137,074 |
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