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https://mathoverflow.net/questions/352826 | 5 | Can we find a complete noncompact Riemannian manifold $(M^n,g)$ with bounded geometry satisfying the following conditions?
1. the curvature operator $Rm>0$;
2. the scalar curvature $R \ge 1$.
Notice that any such manifold must be diffeomorphic to $\mathbb R^n$.
| https://mathoverflow.net/users/105900 | Positively curved metric with uniformly positive scalar curvature | Yes, this is possible. Note that a strictly convex hypersuface in $\mathbb R^{n+1}$ has positive $Rm$.
To get an example, consider the following graph $H\subset \mathbb R^{n+1}$ over the open unit $n$-disk in $\mathbb R^n$:
$$H:=\left(x\_{n+1}=\frac{1}{1-\sum\_{i=1}^n{x\_i^2}}\right).$$
Clearly, $H$ is convex. M... | 4 | https://mathoverflow.net/users/943 | 352888 | 149,133 |
https://mathoverflow.net/questions/352880 | 6 | I am continuing the "abc-adventure" and have a specific question, which needs some explanation:
In this [paper by Gangolli](http://www.numdam.org/article/AIHPB_1967__3_2_121_0.pdf), the term "Levy-Schoenberg" kernel is defined (Definition 2.3).
Consider the group of $G = (\mathbb{Q}\_{>0},\times)$ of positive ratio... | https://mathoverflow.net/users/nan | The abc-conjecture over the positive rationals and Levy-Schoenberg kernels? | I think I found an answer to the question above:
Let $k(a,b)$ be a (positive definite $\ge 0$, symmetric) kernel on $\mathbb{N}\times \mathbb{N}$ such that if $k^\*(a,b)$ is a function on $G \times G$ then we have:
$$k^\*(a,b) = k(a',b')$$
where $a'=\frac{a}{\gcd^\*(a,b)}, b'=\frac{b}{\gcd^\*(a,b)}$,
then $k^\*$... | 5 | https://mathoverflow.net/users/nan | 352890 | 149,134 |
https://mathoverflow.net/questions/352891 | 4 | Consider a polynomial $P\_n(x)\in\mathbb{R}[x]$, of degree $n\geq1$, of the form
$$P\_n(x)=c\_0+c\_1x+c\_2x^2+\cdots+c\_{n-1}x^{n-1}+x^n.$$
To illustrate the question, take $P\_1(x)=c\_0+x$ so that $P\_1(0)=c\_0$ and $P\_1(1)=c\_0+1$. If $\vert c\_0\vert<\frac12$ then $\vert c\_0+1\vert\geq1-\vert c\_0\vert>1-\frac12=\... | https://mathoverflow.net/users/66131 | Largest absolute value of a polynomial of degree $n$ on $\{0,1,\ldots,n\}$ | You can write your polynomial as
$$ P(x) = \sum\_{k = 0}^n P(k) L\_{n,k}(x) ,$$
where $L\_{n,k}$ are the Lagrange interpolation polynomials with nodes at $0, 1, \ldots, n$. Note that $(-1)^{n - k} L\_{n,k}$ has positive coefficient at $x^n$. Thus, with the constraint $$\max\{|P(k)| : k = 0, 1, \ldots, n\} \leqslant 1,$... | 12 | https://mathoverflow.net/users/108637 | 352892 | 149,135 |
https://mathoverflow.net/questions/352894 | 9 | In the spirit of [this question](https://mathoverflow.net/questions/96078/are-semi-direct-products-categorical-colimits), it would be interesting to give a characterization of HNN extensions as a 2-colimit. If $G$ is a group and $\alpha:H \xrightarrow{\cong} K$ is an isomorphism between two subgroups of $G$, then I thi... | https://mathoverflow.net/users/117693 | HNN-extension as a 2-colimit | Assuming your universal property is true, it exactly says that the HNN extension is the [coinserter](https://ncatlab.org/nlab/show/inserter) of $(i\_2 \circ \alpha,i\_1) : H \rightrightarrows G$.
| 10 | https://mathoverflow.net/users/2841 | 352896 | 149,136 |
https://mathoverflow.net/questions/352878 | 13 | I am looking for a (hopefully simple) example of a Calabi-Yau threefold (projective, simply connected, with trivial canonical bundle) admitting an automorphism of infinite order.
| https://mathoverflow.net/users/40297 | Calabi-Yau threefold with an automorphism of infinite order | A Schoen manifold $X$ is a generic complete intersection in $\mathbb{P}^2 \times \mathbb{P}^2 \times \mathbb{P}^1 $ of two hyper-surfaces of degree (3,0,1) and (0,3,1) respectively. Alternately, you can describe this Calabi-Yau threefold as the fiber product of two generic rational elliptic surfaces $X = S\times \_{\ma... | 13 | https://mathoverflow.net/users/9617 | 352897 | 149,137 |
https://mathoverflow.net/questions/352910 | 1 | If $G=(V,E)$ is a finite, simple, undirected graph, then by $\eta(G)$ we denote the maximum integer $n\in \mathbb{N}$ such that $K\_n$ is a [minor](https://en.wikipedia.org/wiki/Graph_minor) of $G$. If $e\in E$ we write $G\setminus e$ to denote the graph $(V, E \setminus \{e\})$.
Is there a finite graph $G=(V,E)$ and... | https://mathoverflow.net/users/8628 | Effect of removing an edge on Hadwiger number | **No,** there is no such graph. Suppose $\eta(G)=n$. Let $T\_1, \dots, T\_n$ be a collection of vertex disjoint trees in $G$ such that for all distinct $i,j \in [n]$, there is an edge $e(ij) \in E(G)$ between $T\_i$ and $T\_j$. Consider an arbitrary edge $e \in E(G)$. If $e=e(ij)$ for some $i,j$, then removing $T\_i$ (... | 6 | https://mathoverflow.net/users/2233 | 352912 | 149,141 |
https://mathoverflow.net/questions/352923 | 1 | Let $F$ be a local field of characteristic 0.
I am wondering whether an unramified principal series representation of $\operatorname{GL}\_n(F)$ can have 1-dimensional quotient when $n>1$.
In some paper, the author claims that it can’t.
Do you know the reason?
| https://mathoverflow.net/users/29422 | Subquotient of principal series | Think of $Ind\_B^G 1$ as smooth functions on $G/B$ and look at the subspace of constant functions.
| 3 | https://mathoverflow.net/users/152491 | 352926 | 149,146 |
https://mathoverflow.net/questions/352930 | 3 | It is known that the standard operator norm $\|\cdot\|\_2$ over ${\bf M}\_n({\mathbb R})$ is very flat, as is any operator norm (= subordinated norm) actually. The set of extremal points of the unit ball is the very small set ${\bf O}\_n({\mathbb R})$ and the unit sphere contains faces (convex subsets) of dimension $(n... | https://mathoverflow.net/users/8799 | Flatness directions of the operator norm | (You write $\|\cdot\|\_2$ but say "operator norm", and I assume that is what you mean.)
I haven't seen this exact statement, but it follows easily from known facts about the facial structure of operator unit balls. The basic reference is Akemann and Pedersen, Facial structure in operator algebra theory, *Proc. London... | 2 | https://mathoverflow.net/users/23141 | 352938 | 149,149 |
https://mathoverflow.net/questions/352944 | 1 | Let $C(t)$ be a symmetric, two-by-two real matrix whose entries are smooth functions of $t \in \mathbb{R}$. Suppose that $C(t)$ point-wise has eigenvalues $\lambda$ and $0$. Then $\lambda(t)$ is a smooth function too (since $\lambda(t)$ is the trace of $C(t)$). However, in general the unit-length eigenvector $v(t)$ cor... | https://mathoverflow.net/users/112378 | Matrix smoothly parametrized by t has eigenvalues (0, $\lambda$), eigenvector $v$. Is $\lambda v$ smooth? | $\newcommand{\la}{\lambda}
\newcommand{\R}{\mathbb{R}}$Such a smooth field $(\la(t)v(t))$ does not exist in general.
Indeed, let
$$C=\begin{bmatrix}f&fg\\fg&fg^2\end{bmatrix},$$
where $f$ and $g$ are the (nonnegative) functions defined in [this answer](https://mathoverflow.net/a/353108/36721).
The eigenvalues of $... | 2 | https://mathoverflow.net/users/36721 | 352949 | 149,154 |
https://mathoverflow.net/questions/262108 | 13 | I'm wondering if anyone has classified all the fields $K$ such that $Gal(\bar{K}/K)$ is abelian?
The only examples I'm aware of are: finite fields, the real numbers $\mathbb{R}$ and $k((T))$ where $k$ is any algebraically closed field of characteristic 0. (One might also add any algebraically closed field as an examp... | https://mathoverflow.net/users/24913 | Classify all the fields with abelian absolute Galois group | Geyer in [Unendliche algebraische Zahlkörper, über denen jede Gleichung auflösbar von beschränkter Stufe ist](https://doi.org/10.1016/0022-314X(69)90050-X), Satz 1.13 and the paragraph after that, gives a full characterization of which abelian profinite groups occur as absolute Galois groups: They are either $\mathbb{Z... | 18 | https://mathoverflow.net/users/50351 | 352952 | 149,155 |
https://mathoverflow.net/questions/352868 | 7 | Suppose $U$ is a non-principal ultrafilter on $\omega$, and let us define $\tau(U)$ to be the minimum cardinality of a family $\mathcal{X}\subseteq U$ such that $\mathcal{X}$ does not have an infinite pseudo-intersection, that is, there is no infinite $A$ such that $A\setminus B$ is finite for all $B\in \mathcal{X}$.
... | https://mathoverflow.net/users/18128 | Pseudo-intersections, splitting families, and ultrafilters | The answer is no -- it is consistent that every $U \in \omega^\*$ has $\tau(U) < \mathfrak{s}$.
I had an idea for proving this earlier today, using the Mathias model. I couldn't quite make things work, and I ended up talking about the problem with Alan Dow for a good part of the afternoon. (1) We still think the Math... | 7 | https://mathoverflow.net/users/70618 | 352953 | 149,156 |
https://mathoverflow.net/questions/352940 | 6 | In reading the literature one encounters countless examples of Voronoi formulas, i.e., formulas that take a sum over Fourier coefficients, twisted by some character, and controlled by some suitable test function, and spits out a different sum over the same Fourier coefficients, twisted by some different characters, and... | https://mathoverflow.net/users/152494 | Intuition about how Voronoi formulas change lengths of sums | First of all, the description of $\psi$ after the first display is confusing (assuming OP meant $\psi$ is supported around $N$, otherwise conclusion form the first display does not make sense). I went to the relevant part (end of p.318) of Li's paper and found that $\psi$ is not just any test function, it has a weight ... | 4 | https://mathoverflow.net/users/36735 | 352954 | 149,157 |
https://mathoverflow.net/questions/352699 | 42 | Some years ago I was asked by a friend if Fermat's Last Theorem was true for matrices. It is pretty easy to convince oneself that it is not the case, and in fact the following statement occurs naturally as a conjecture:
>
> For all integers $n,k\geq 2$ there exist three square matrices $A$, $B$ and $C$ of size $k\t... | https://mathoverflow.net/users/147861 | Fermat's Last Theorem for integer matrices | This problem is addressed in "[On Fermat's problem in matrix rings and groups](https://www.researchgate.net/profile/Attila_Szakacs/publication/266533344_On_Fermat%27s_problem_in_matrix_rings_and_groups/links/554b437b0cf29f836c968201.pdf)," by Z. Patay and A. Szakács, *Publ. Math. Debrecen* **61/3-4** (2002), 487–494, w... | 18 | https://mathoverflow.net/users/3106 | 352963 | 149,159 |
https://mathoverflow.net/questions/352965 | 4 | I'm looking for books where the theory (basic properties, adjoints etc.) of unbounded linear operators between locally convex spaces or at least Banach spaces is developed. In Brezis' *functional analysis, Sobolev spaces and partial differential equations* there are some limited results for the Banach space case and al... | https://mathoverflow.net/users/49733 | reference request: unbounded operators on normed spaces | You can try to use: Goldberg, Seymour Unbounded linear operators. Theory and applications. Reprint of the 1985 corrected edition [MR0810617]. Dover Publications, Inc., Mineola, NY, 2006. viii+199 pp. ISBN: 0-486-45331-6
| 5 | https://mathoverflow.net/users/85406 | 352977 | 149,167 |
https://mathoverflow.net/questions/352975 | 2 | Let $V$ be a bounded open set in $\mathbb{R}^n$ with $n>1$. According to a well known result due to Poincaré, if $x$ is a point in the boundary $\partial V$ and there exists a ball $B$ such that $x\in\partial B$ and $B\cap V=\emptyset$, then $x$ is a regular point for the Dirichlet problem.
Is there a converse for t... | https://mathoverflow.net/users/100746 | The converse of a Poincaré's result on regular boundary points | The answer is no. Take the unit ball in $R^3$, and remove from it the halfplane
$x\_3=0, x\_1\geq 0$. This is regular.
| 0 | https://mathoverflow.net/users/25510 | 352989 | 149,172 |
https://mathoverflow.net/questions/352958 | 3 | Which information is currently known about $H^1\_{et}(X,\mathbb{Z}\_l)$ and $H^2\_{et}(X,\mathbb{Z}\_l)$, where $X$ is a smooth unirational variety over an algebraically closed field of finite characterstic $p\neq l$? Actually, I am interested in the case where $X$ is a quotient of a rational variety $Y$ by a free acti... | https://mathoverflow.net/users/2191 | What is known about lower etale cohomology of unirational varieties? | Concerning the $H^2$, for $X$ a smooth projective rationally chain connected variety over an algebraically closed field $k$ and $\ell \in k^\ast$, it follows from
Theorem 1.2 in <https://arxiv.org/abs/1703.05735> that
>
>
> >
> > $$\mathrm{NS}(X)\otimes \mathbb{Z}\_{\ell} = \mathrm{H}^2\_{et}(X,\mathbb{Z}\_{\el... | 4 | https://mathoverflow.net/users/4333 | 352992 | 149,174 |
https://mathoverflow.net/questions/353002 | 7 | Suppose that $f$ and $g$ are functions from $\mathbb R$ to $\mathbb R$ such that the functions $f,fg,fg^2$ are smooth, that is, are in $C^\infty(\mathbb R)$. Does it then necessarily follow that the
functions $f\sqrt{1+g^2}$ and $fg\sqrt{1+g^2}$ are smooth?
Of course, the problem here is that the function $g$ does ... | https://mathoverflow.net/users/36721 | Are $f\sqrt{1+g^2}$ and $fg\sqrt{1+g^2}$ smooth if $f,fg,fg^2$ are smooth? | **No.**
Set
$$ f(x) = \exp(-2/|x|^2) \operatorname{sign} x, \qquad g(x) = \exp(1/|x|^2) \sqrt{|x|} \operatorname{sign} x $$
for $x \ne 0$, and, of course, $f(0) = g(0) = 0$. Then clearly
$$ \begin{aligned}
f(x) & = \exp(-2/|x|^2) \operatorname{sign} x , \\
f(x) g(x) & = \exp(-1/|x|^2) \sqrt{|x|} , \\
f(x) (g(x))^2 & ... | 15 | https://mathoverflow.net/users/108637 | 353010 | 149,181 |
https://mathoverflow.net/questions/352994 | -3 | Let us consider $$f(z):=\sum\limits\_{j=1}^{j=n}a\_j\sin(\lambda\_jz) $$
where all $a\_j$ and $\lambda\_j$ (of course, $\lambda\_j$ are distinct) are real numbers and $n \in \mathbb{Z},\, n \ge 3$. The question is: are all the zeroes of $f(z)$ real?
| https://mathoverflow.net/users/35959 | Zeroes of linear combination of sines | Note that $\sinh$ is a strictly increasing positive unbounded function on $(0,\infty)$ so for say $n \ge 3$, the equation $\sinh 1+ \sinh 2+...\sinh (n-1) =\sinh x$ has a unique positive solution $x\_n> n-1$.
Using that $\sin {iy}=i\sinh y$ for real $y$, the above means that $i$ is a root of the equation $\sin z+\si... | 3 | https://mathoverflow.net/users/133811 | 353012 | 149,183 |
https://mathoverflow.net/questions/352957 | 12 | There are many interesting sequences of polynomials which contain
polynomials of arbitrarily high degree, for example classical
orthogonal polynomials. Most of them arise as characteristic polynomials
of some sequences of operators, or as polynomial solutions
of some differential equations.
>
> What are some natura... | https://mathoverflow.net/users/25510 | Examples of plane algebraic curves | How about the affine plane curves $\Phi\_n(c,t)=0$ that classify $(c,t)$ such that $t$ is a point of exact period $n$ under iteration of the quadratic map $f\_c(X)=X^2+c$? These are often called *dynatomic curves* and have been much studied in recent years, especially since describing their rational points is related t... | 13 | https://mathoverflow.net/users/11926 | 353019 | 149,185 |
https://mathoverflow.net/questions/353020 | 1 | The famous Polignac conjecture posits that the $n$-th prime gap $g\_{n}:=p\_{n+1}-p\_{n}$ attains all even positive integral values infinitely many times, which implies $\displaystyle{\zeta\_{Pol}:=s\mapsto\sum\_{n>0}(ng\_{n}/2)^{-s}}$ has an abscissa of convergence $\sigma\_{Pol}$ less or equal to $1$.
Is this absci... | https://mathoverflow.net/users/13625 | Is the abscissa of convergence of $s\mapsto\sum_{n>0}(ng_{n}/2)^{-s}$ known? | Well, the abscissa of convergence is $1$, though I see no way to deduce anything about it from the Polignac conjecture. $\sigma\_P \le 1$ is obvious from $|\zeta\_P(s)| \le \zeta(\Re s)$, while for $\sigma\_P \ge 1$ it is enough to prove that $\zeta\_P(1) = \infty$:
$$\zeta\_P(1) \ge \sum\_{k=1}^\infty \frac{1}{2^{k+... | 4 | https://mathoverflow.net/users/104330 | 353025 | 149,189 |
https://mathoverflow.net/questions/353015 | 4 | This is a follow-up on the [previous question](https://mathoverflow.net/questions/353002/smoothness-of-f-sqrt1g2-and-fg-sqrt1g2-for-functions-f-and-g-suc).
Suppose that $f$ and $g$ are functions from $\mathbb R$ to $\mathbb R$ such that the functions $f,fg,fg^2$ are smooth, that is, are in $C^\infty(\mathbb R)$. Doe... | https://mathoverflow.net/users/36721 | Are $\pm f\sqrt{1+g^2}$ and $\pm fg\sqrt{1+g^2}$ smooth if $f,fg,fg^2$ are smooth? | Still *no*. Consider the function $f(x)=(x^2+r^2)^2$, $g(x)=\frac x{x^2+r^2}$ with small $r>0$. Then $f$, $fg$, $fg^2$ are polynomials of degree $\le 4$ with bounded coefficients but $f\sqrt{1+g^2}$ is very close to $x^2|x|\sqrt{1+x^2}$ as close to $0$ as you wish when $r$ is small enough, so the maximum of the fourth ... | 11 | https://mathoverflow.net/users/1131 | 353035 | 149,192 |
https://mathoverflow.net/questions/353032 | 4 | This was a comment to the answer [here](https://mathoverflow.net/questions/351715/automorphism-groups-of-odd-order) . It is one of the series of questions about finite groups with automorphism groups of odd order and would reduce the question to nilpotent groups.
**Question.** Is it true that a finite group $G$ has ... | https://mathoverflow.net/users/nan | Automorphism groups of a group and of its Fitting subgroup | It can happen that the whole group has no automorphism of order 2 although the Fitting subgroup does:
>
> Let
>
>
> * $N$ be a non-abelian 2-generated $p$-group for some prime $p$, such that $N/Z(N)$ has no automorphism of order $2$;
> * $L$ a nontrivial group of order coprime to $2p$ with no automorphism of ord... | 5 | https://mathoverflow.net/users/14094 | 353038 | 149,195 |
https://mathoverflow.net/questions/352436 | 4 | Let $G$ be a real, connected, semisimple Lie group and $\Gamma < G$ a torsion-free lattice. Then does there exist a finite $CW$-model for $B\Gamma$?
I know this to be true in many instances (e.g. when $\Gamma$ is uniform, when $G=\mathrm{SO}\_0(n,1)$ or when $\mathrm{rank}\_{\mathbb R}(G) \geq 2$ and $\Gamma$ is irre... | https://mathoverflow.net/users/78554 | Finite models for torsion-free lattices | In fact, more is true and you do not need separate arguments for rank 1 and higher rank.
The following is Theorem 13.1(i) in the book of Ballmann, Gromov and Schroeder "Manifolds of nonpositive curvature":
Suppose that $(M,g)$ is a complete real-analytic Riemannian manifold of nonpositive curvature and finite volu... | 4 | https://mathoverflow.net/users/39654 | 353041 | 149,196 |
https://mathoverflow.net/questions/352750 | 2 | Suppose $\mathfrak{M}$ is the category of $S^1$-spectra of simplicial sheaves. I know its sequential homotopy colimits (colimit in $\mathbb{N}$ as usual) coincide with categorial colimits since stable equivalences are preserved under filtered colimits.
But I don't see why they coincide with the homotopy colimits in $... | https://mathoverflow.net/users/149491 | Definitions of sequential homotopy colimits | I've worked out.
For any two morphisms $f,g:A\to B$ in $\mathfrak{M}$, we have
$$Cone(f-g)=Cone(((id\_A,f),(id\_A,g)):A\oplus A\to A\oplus B)$$
by elementary transformations of matrices.
We prove that latter is the homotopy pushout of the diagram
$$A\xleftarrow{id+id}A\vee A\xrightarrow{f+g}B$$
, which is just $Hoc... | 0 | https://mathoverflow.net/users/149491 | 353044 | 149,197 |
https://mathoverflow.net/questions/353052 | 1 | Let $\mathbb{N}:=\omega \setminus \{0\}$. Is there $k\in \mathbb{N}$ and a map $c:\mathbb{N}\to \{1,\ldots,k\}$ such that for all $a,b\in\mathbb{N}$ the restriction $c|\_{\{a,\,b,\,a+b\}}$ is non-constant?
| https://mathoverflow.net/users/8628 | Coloring $\mathbb{N}$ such that $\{a, b, a+b\}$ is not monochromatic | No, this is the content of [Schur's Theorem](https://en.wikipedia.org/wiki/Schur%27s_theorem).
| 8 | https://mathoverflow.net/users/385 | 353053 | 149,200 |
https://mathoverflow.net/questions/353058 | 0 | I would like to find an asymptotic expansion for the hypergeometric function
$$
\_{2}F\_{2}\left(a,b;c,d;z\right),\quad a,b,c,d\in\mathbb{R}.
$$
The parameters are fixed. $z$ is real and $z\rightarrow +\infty$.
Could someone shed light on it?
| https://mathoverflow.net/users/78781 | Asymptotic expansion of hypergeometric 2F2 | $$\_{2}F\_{2}\left(a,b;c,d;z\right)=\frac{\Gamma (c) \Gamma (d)}{\Gamma (a) \Gamma (b)}e^z z^{a+b-c-d}\left(1+{\cal O}(z^{-1})\right)$$
As an example, the plot shows $\_{2}F\_{2}\left(a,b;c,d;z\right)$ (blue) and the asymptotics (gold) for $a=1,b=2,c=3,d=4$.
| 0 | https://mathoverflow.net/users/11260 | 353067 | 149,204 |
https://mathoverflow.net/questions/251530 | 11 | Let $R$ denote the Rado graph, and let $c$ be a fixed vertex.
**Question 1.** Is the structure obtained by extending $R$ by the constant $c$ interpretable in $R$ *without* parameters?
By *interpretable* I mean first-order interpretable; see below for an equivalent formalism.
A related group-theoretic question i... | https://mathoverflow.net/users/87983 | Eliminating constant in Rado graph | The answer to questions 1 and 2 is negative. Here is a proof.
It is based on discussions I had in 2016 with Marcello Mamino, Antoine Mottet, Manuel Bodirsky, and others at TU Dresden.
$\newcommand{\aut}[1]{\textrm{Aut}(#1)}$
$\newcommand{\set}[1]{\{#1\}}$
$\newcommand{\Nat}{\mathbb N}$
$\renewcommand{\subset}{\subse... | 2 | https://mathoverflow.net/users/87983 | 353089 | 149,216 |
https://mathoverflow.net/questions/261196 | 5 | By the fundamental work of De Concini, Eisenbud, and Procesi, an [algebra with straightening law](https://www.msri.org/~de/papers/pdfs/1980-001.pdf) (ASL) must be Cohen-Macaulay if it is built on a [Cohen-Macaulay poset](http://www-math.mit.edu/~rstan/pubs/pubfiles/52.pdf). I would like to understand the state of the a... | https://mathoverflow.net/users/12419 | If a finite poset supports a Cohen-Macaulay ASL, how far can it be from Cohen-Macaulay? | Aldo Conca and Matteo Varbaro have posted a preprint that purports to answer this question, at least for graded ASLs:
[Squarefree Gröbner degenerations](http://www.dima.unige.it/%7Evarbaro/sqfGr%2025%20Giugno%202018%20.pdf)
It appears that if $A$ is Cohen-Macaulay, then $k[P]$ must also be Cohen-Macaulay! See Corol... | 4 | https://mathoverflow.net/users/12419 | 353090 | 149,217 |
https://mathoverflow.net/questions/353077 | 7 | Brown's representability theorem gives us a very nice set of conditions to check that a (contravariant) functor $Hot^{op}\rightarrow Set$ is representable. Choose an object $X$ in $Hot$. Then it is seems natural to ask whether or not an analogue of the Brown representability theorem is true for the slice category, i.e.... | https://mathoverflow.net/users/152554 | Brown representability in slice category | Yes, sliced homotopy categories of pointed connected spaces satisfy Brown representability. (We had better be using $Hot$ to denote the homotopy category of pointed connected spaces, as Brown representability is false for the homotopy category of unbased or non-connected spaces.)
The abstract version of Brown’s repr... | 7 | https://mathoverflow.net/users/43000 | 353092 | 149,218 |
https://mathoverflow.net/questions/353085 | 2 | Let $X:(\Omega,\Sigma)\rightarrow (\mathbb{R}^d;\mathbb{B}(\mathbb{R}^d))$ be a Borel-measurable random vector. What are some general classes of such random vector for which one can give a "lower concentration inequality" of the form:
$$
\mathbb{P}(\|X\|^2>\lambda) \geq \mbox{(insert non-trivial lower bound)}
$$
where ... | https://mathoverflow.net/users/36886 | Anti-concentration inequalities: lower bound on realized second moment | Suppose e.g. that $X=X\_1+\dots+X\_n$, where the $X\_i$' are independent zero-mean random vectors with $\|X\_i\|\le1$ for all $i$. Then the Hoeffding--Azuma inequality (see e.g. [Wikipedia](https://en.wikipedia.org/wiki/Azuma%27s_inequality#A_general_form_of_Azuma's_inequality)) yields
$$P(\|X\|>u)\ge1- e^{-(E\|X\|-u)^... | 2 | https://mathoverflow.net/users/36721 | 353093 | 149,219 |
https://mathoverflow.net/questions/352341 | 1 | This question branches from Taras Banakh's [recent question](https://mathoverflow.net/questions/351827/a-new-cardinal-characteristic-related-to-partition) on a cardinal characteristic connected to families of partitions that are directed in the ordering of [partition refinement](https://math.stackexchange.com/questions... | https://mathoverflow.net/users/8628 | Thinning directed sets ${\frak P}$ of partitions of $\omega$ with no ${\frak P}$-discrete subsets | The answer to this question is negative.
Given a linearly ordered family $\mathfrak C$ of finitary partitions of $\omega$, write $\mathfrak C=\bigcup\_{n=1}^\infty\mathfrak C\_n$ where $\mathfrak C\_n=\{\mathcal P\in\mathfrak C:\sup\_{P\in\mathcal P}|P|\le n\}$.
For a partition $\mathcal P$ of $\omega$ and a point ... | 1 | https://mathoverflow.net/users/61536 | 353099 | 149,222 |
https://mathoverflow.net/questions/352962 | 11 | Let $n>1$ and $p$ be an odd prime with $p-1 \mid n-1$ such that $p^k - 1 \mid n-1$ does not hold for any $k>1$. Notice that, since $p-1 \mid n-1$, we have $T^p - T \mid T^n-T$ in $\mathbb{F}\_p[T]$ and hence also $T^p - T = (T+u)^p - (T+u) \mid (T+u)^n - (T+u)$ for all $u \in \mathbb{F}\_p$.
**Question.** Is $T^p - T... | https://mathoverflow.net/users/2841 | Greatest common divisor in $\mathbb{F}_p[T]$ with powers of linear polynomials | This is false.
Let $p$ be an odd prime, let $\ell$ be another prime, and let $m$ be a small prime divisor of $p^{\ell}-1$, that doesn't divide $p-1$. Let $n= 1 + \frac{ p^{\ell}-1}{m}$.
Then $n-1$ is a multiple of $p-1$, is not a multiple of $p^{\ell}-1$, and is not a multiple of $p^{k}-1$ for any other $k$ becaus... | 11 | https://mathoverflow.net/users/18060 | 353101 | 149,224 |
https://mathoverflow.net/questions/352743 | 4 | I would like to know to what extent the naive algebraic de Rham cohomology is a "bad" cohomology theory. If $X$ is smooth then there is a comparison theorem with singular cohomology. If $X$ is singular, then Hartshorne embeds $X$ in a smooth variety, generalises the definition and proves a similar comparison theorem si... | https://mathoverflow.net/users/122284 | Algebraic de Rham cohomology for singular varieties | Here is an answer for the naive de Rham cohomology $\mathbb{H}(X, \Omega\_X^{\bullet})$ (not the more sophisticated one of the linked article by Hartshorne (which involves chooseing an embedding $X \subset Y$ in a smooth variety $Y$, completion along $X$ etc.)).
We can use the hypercohomology spectral sequence
$$
E... | 1 | https://mathoverflow.net/users/113296 | 353106 | 149,226 |
https://mathoverflow.net/questions/353126 | 1 | Let $K$ be a compact Hausdorff space and let $C(K)$ be the space of all scalar-valued continuous functions on $K$. Let $(f\_{n})\_{n}$ be a sequence in $C(K)$ satisfying $\sup\limits\_{n}\sup\limits\_{t\in K}|f\_{n}(t)|<\infty$. We define an equivalence relation $R$ on $K$ by $$t\_{1}Rt\_{2}\Leftrightarrow f\_{n}(t\_{1... | https://mathoverflow.net/users/41619 | Metrization of quotient spaces defined by sequences of continuous functions | Yes, it is metrizable. Assume the scalar field is real; if it is complex, replace the functions $f\_n$ with their real and imaginary parts. WLOG each $f\_n$ maps $K$ into $[0,1]$. Amalgamate the $f\_n$ into a single function $f: K \to [0,1]^{\omega}$. Then $K\_1$ is homeomorphic to $f(K)$ with topology induced from $[0... | 2 | https://mathoverflow.net/users/23141 | 353127 | 149,237 |
https://mathoverflow.net/questions/353134 | 5 | Let $\mathcal X \rightarrow S$ be a flat family of projective varieties over a discrete valuation ring $S$ such that the generic fibre $\mathcal X\_{\eta}$ (say) is smooth projective variety and the special fibre $\mathcal X\_0$ (say) is a normal crossing divisor in $\mathcal X$.
Question: Does there exist a vector ... | https://mathoverflow.net/users/nan | Relative logarithmic cotangent bundle | First of all, it's unclear what you mean by $\Omega^1\_{X\_0}(\log D)$ since $X\_0$ is singular.
Second, if you make up such a definition then most probably such a vector bundle will not exist. Note that the Euler characteristics $\chi(X\_\eta, \Omega^1\_\eta)$ and $\chi(X\_0, \Omega^1\_{X\_0}(\log D)$ have to agree.... | 6 | https://mathoverflow.net/users/3847 | 353136 | 149,240 |
https://mathoverflow.net/questions/353135 | 6 | Only few strongly regular graphs with parameters $\lambda=0$ (triangle-free) and $\mu=2$ (any two non-adjacent vertices
have exactly two common neighbors) are known, see the [wikipedia page](https://en.wikipedia.org/wiki/Strongly_regular_graph): the 4-cycle, the Clebsch graph and the Sims-Gewirtz graph.
I am looking ... | https://mathoverflow.net/users/54628 | What is known about the non-existence of strongly regular graphs srg(n,k,0,2)? | [Example 1 in A.Neumaier paper](https://doi.org/10.1016/S0304-0208(08)73275-4) says in partcular that the vertex degree in this case must be $k=t^2+1$, for $t$ not divisible by 4. As well, the number of vertices is $v=1+k+\binom{k}{2}$. The examples you list correspond to $t=2,3$. The next possible parameter set corres... | 5 | https://mathoverflow.net/users/11100 | 353143 | 149,244 |
https://mathoverflow.net/questions/353124 | 2 | Define a point cloud $X=\{x\_i\}\_{1\leq i\leq n}$, for $x\_i\in\mathbb R^d$. Define the Wasserstein kernel as $$W(X,Y)=\max\_{T}\frac{1}{n}\sum\_{kl}T\_{kl}\langle x\_k,y\_l \rangle$$
where $T$ is any doubly stochastic matrix.
Consider point clouds $X\_1,...X\_m$ each of size $n$ and their Gram matrix $(W(X\_i,X\_j))\... | https://mathoverflow.net/users/104248 | Is the Wasserstein kernel positive definite? | It is **not** positive semi-definite.
Take $m=4, n=2, d=2$.
I define $u\_i = (\lfloor i / 2 \rfloor, i \% 2)$ for $i=0\dots3$.
I take $X\_1 = \{ u\_0, u\_1\}, X\_2 = \{u\_0, u\_2\}, X\_3 = \{u\_0, u\_3\}, X\_4 = \{u\_1, u\_2\}$
$W(X\_i, X\_j) = 0$ means that all vectors in the two sets are orthogonal. It can on... | 1 | https://mathoverflow.net/users/140058 | 353145 | 149,245 |
https://mathoverflow.net/questions/353142 | 0 | I know that for two i.i.d distributions $P$ and $Q$ the probability that $Q$ will produce a length $n$ sequence that is $\epsilon$-typical according to $P$ is bounded by
$$Q(T\_{P,\epsilon})\leq e^{-nD(P||Q)-n|\mathcal{X}|\epsilon}$$
where $D(P||Q)$ is the KL-dvergence and $T\_{P,\epsilon}$ is the (strong) $\epsilon$-t... | https://mathoverflow.net/users/103133 | Joint typicality of sequences | Essentially, yes, although it is ambiguous the way you stated it (so it is not clear to me what $T\_{P^n}$ really means). The general statement is
that you have a LDP for the empirical process $n^{-1}\sum \delta\_{\theta^i X}$ where
$X$ is the infinite sequence and $\theta$ is the (left shift), in the product topology ... | 0 | https://mathoverflow.net/users/35520 | 353156 | 149,247 |
https://mathoverflow.net/questions/353120 | 7 | Denote $\pmb{a}=(a\_1,\dots,a\_d)\in\mathbb{R}^d$ and consider the set
$$\mathcal{E}\_d=\{\pmb{a}\in\mathbb{R}^d: \text{each root $\xi$ of $x^d+a\_dx^{d-1}+\cdots+a\_2x+a\_1=0$ lies in $\vert\xi\vert<1$}\}.$$
In the reference shown below, Fam proved that the $d$-dimensional Lebesgue measure satisfies
$$\lambda\_d(\math... | https://mathoverflow.net/users/66131 | Volume of solution sets for polynomials in $\mathbb{C}[x]$ | $\def\CC{\mathbb{C}}\def\RR{\mathbb{R}}$The answer is $\tfrac{\pi^n}{n!}$. It is certainly a surprise to have the answer come out so simple!
Let $\phi : \CC^n \to \CC^n$ be the map which takes $(z\_1, z\_2, \ldots, z\_n)$ to the elementary symmetric functions $(e\_1, e\_2, \ldots, e\_n)$ where $e\_k = \sum\_{1 \leq i... | 5 | https://mathoverflow.net/users/297 | 353182 | 149,257 |
https://mathoverflow.net/questions/330233 | 8 | A *tower* is a subset $T\subset [\omega]^\omega$ of the family $[\omega]^\omega$ of all infinite subsets of $\omega$ such that $T$ is well-ordered by the relation $\supset^\*$ of almost inclusion and has no infinite pseudointersections. A tower is *regular* if its cardinality is a regular cardinal.
Consider two small... | https://mathoverflow.net/users/61536 | Relations between two tower numbers | Assuming I understand the definitions correctly, I can give you a couple of references.
(1) Dordal (see below) gives a model in which $\mathfrak{b}=\mathfrak{c}=\aleph\_2$ and all towers have cardinality $\aleph\_1$. Thus, in his model $\mathfrak{t}=\hat{\mathfrak{t}}=\aleph\_1<\aleph\_2=\mathfrak{b}=\mathfrak{c}$, a... | 2 | https://mathoverflow.net/users/18128 | 353196 | 149,260 |
https://mathoverflow.net/questions/353193 | 3 | Let $R$ be a noncommutative ring with unit, let $P$ be a projective left $R$-module, and denote $^{\vee}\!P := \,\_R\mathrm{Hom}(P,R)$. One often sees it written that projectivity implies an isomorphism
$$
\mathrm{ev}:\,^{\vee}\!P \otimes\_R P \to R, ~~ \phi \otimes p \mapsto \phi(p).
$$
But I don't see that this is we... | https://mathoverflow.net/users/143172 | Dual of a projective module | You are right. There is no such a map as the one you are trying to describe.
Here is a map that actually exists. Let $R$ and $S$ be two noncommutative rings with units, and let $P$ be an $R$-$S$-bimodule. Consider the $S$-$R$-bimodule $Q={}\_R\mathrm{Hom}(P,R)$. Then the evaluation is an $R$-$R$-bimodule map
$$
\mat... | 7 | https://mathoverflow.net/users/2106 | 353197 | 149,261 |
https://mathoverflow.net/questions/353184 | 1 | *Fix some computable bijection $b$ between $\omega$ and $2^{<\omega}$. For $r\in 2^\omega$, let $$[r]=\{f\in 2^\omega: \forall\sigma\prec f(b^{-1}(\sigma)\in r)\}$$ be the closed subset of Cantor space coded by $r$. For $M\models PA$ nonstandard, let $\mathcal{S}(M)$ be the standard system of $M$ thought of as a topolo... | https://mathoverflow.net/users/8133 | The "higher topology" of countable Scott sets | Given any topological space $X$ and subset $F\subseteq X$, define the Cantor-Bendixson sequence of $F$ in $X$ as:
* $F^{(0)} = F$
* $F^{(\alpha +1)} = F^{(\alpha)} \setminus \{x \in F^{(\alpha)} : x \text{ is isolated in }F^{(\alpha)}\}$
* $F^{(\beta)} = \bigcap\_{\alpha < \beta} F^{(\alpha)}$, $\beta$ a limit ordina... | 1 | https://mathoverflow.net/users/83901 | 353198 | 149,262 |
https://mathoverflow.net/questions/353201 | 1 | I'm currently studying the $\ell$-adic cohomology functor, i.e. the functor $$F:X \rightarrow H^i\_{ét}(X,\mathbb{Q}\_{\ell}).$$
In some sense, this is a representable functor, i.e. there exists an $\ell$-adic Eilenberg-Maclane space (see [Representability of Weil Cohomology Theories in Stable Motivic Homotopy Theory](... | https://mathoverflow.net/users/152554 | Dimension of $\ell$-adic Eilenberg-Maclane space | I see this is one of your first question on MO -- welcome! The topic of the question is certainly interesting but I think you need to put a little more effort in future questions into being clear (first and foremost with yourself) about the nature of the objects you're asking about. You don't need to understand the det... | 5 | https://mathoverflow.net/users/7108 | 353202 | 149,264 |
https://mathoverflow.net/questions/353164 | 5 | Consider a functor $F:C\to D$ between two categories $C$ and $D$. Suppose $F$ satisfies the following property: for any $a, b\in C$, $F(a)\cong F(b)\iff a\cong b$.
Of course, $a\cong b\Rightarrow F(a)\cong F(b)$, so it is the other direction tricky.
The question is then: is there a name for such functors? Have th... | https://mathoverflow.net/users/8012 | functors reflecting "isomorphism relations"? | Some authors name this "isomorphism reflecting", for example in [Noncommutative rings and their applications](https://bookstore.ams.org/conm-634) (p. 153) and [Models, Modules and Abelian Groups: In Memory of A. L. S. Corner](https://www.degruyter.com/viewbooktoc/product/37731) (p.480). But it is kind of dangerous to u... | 8 | https://mathoverflow.net/users/2841 | 353205 | 149,265 |
https://mathoverflow.net/questions/353199 | 2 | Is there is a holomorphic function $g:\mathbb{C}\to \mathbb{C}$ so that $$\frac{|g'(z)|}{1+|g(z)|^2}=\frac{c}{1+|z|^2}$$ for some $c>1$ and all $z\in \mathbb{C}$?
| https://mathoverflow.net/users/124426 | Analytic function on $\mathbb{C}$ | There is no such function. Indeed, $h(z):=g(1/z)$ satisfies the same functional equation
$$\frac{|h'(z)|}{1+|h(z)|^2}=\frac{c}{1+|z|^2}.$$
In particular, $|h'(z)|>c/2$ for $|z|<1$, hence $h'(z)$ does not have an essential singularity at $z=0$. So $g(z)$ is a polynomial, and then letting $z\to\infty$ in the original fun... | 8 | https://mathoverflow.net/users/11919 | 353208 | 149,267 |
https://mathoverflow.net/questions/353217 | 4 | Let $K$ be a compact Hausdorff space. We let $Z$ be the closed subspace generated by $\{\chi\_{F}:F$ closed $G\_{\delta}$ sets in $K\}$ in $C(K)^{\*\*}$. My question is the following:
Question 1. Is $C(K)\subseteq Z\subseteq C(K)^{\*\*}$?
Since each $f\in C(K)$ is Baire measurable, each such $f$ is the unifrom limi... | https://mathoverflow.net/users/41619 | A subspace generated by closed $G_{\delta}$ sets in $K$ between $C(K)$ and $C(K)^{**}$ | Question 2 has a positive answer. For any $f \in C(K)$, let $a = \inf\_{t\in K} f(t)$ and let $b = \sup\_{t\in K} f(t)$. For any $n \in \mathbb{N}$ with $n > 0$, we can set $$ g\_n = a + \sum\_{k=1}^n \frac{b-a}{n} \chi\_{F\_k},$$ where $ F\_k = \{t \in K : f(t) \geq a + \frac{k}{n}(b-a)\} $, which is a closed $G\_\del... | 2 | https://mathoverflow.net/users/83901 | 353221 | 149,272 |
https://mathoverflow.net/questions/353224 | -1 | Can it be shown that,for all $m\in\mathbb{Z}\_+$ there exists at least one $k$ with respect to $m$ such that
$$1\le \frac{\sum\_{i=1}^{k-1}i^m}{k^m}<2$$
*Example*: let $m=1$ then $k=\{3,4\}$
This question has already been asked on MSE but it has not received a solution so that's why I posted here. Given claim sol... | https://mathoverflow.net/users/149083 | $\exists k$ s.t. $k^m\le 1^m+2^m+...+(k-1)^m <2\cdot k^m$? | Yes. Note that for large $k$ we have $1^m+\dots+(k-1)^m\geqslant k^m$. Choose minimal $k$ with such property. Then $k>2$ and $$2k^m>2(k-1)^m>1^m+\dots+(k-1)^m$$
by minimality.
| 10 | https://mathoverflow.net/users/4312 | 353225 | 149,274 |
https://mathoverflow.net/questions/243125 | 7 | **Question**. Let $\kappa>\mathfrak c$ be a cardinal endowed with the discrete topology and $f:\kappa^\omega\to[0,1]^\omega$ be a continuous map. Is there a point $y\in[0,1]^\omega$ whose preimage $f^{-1}(y)$ is not scattered?
Let us recall that a topological space $X$ is *scattered* if each non-empty subspace of $X$... | https://mathoverflow.net/users/61536 | Does a continuous map from $\kappa^\omega$ to $[0,1]^\omega$ have a non-scattered fiber? | The continuity of $f$ is not needed. Indeed, suppose to the contrary that $\kappa^\omega$ is a union of the family $\{F\_\alpha:\alpha<\frak c\}$ of fibers of $f$. Let $\alpha<\frak c$ be any index. Since the fiber $F\_\alpha$ is scattered, there is an injective enumeration $F\_\alpha=\{f\_{\alpha,\beta}:\beta<\beta\_\... | 2 | https://mathoverflow.net/users/43954 | 353236 | 149,279 |
https://mathoverflow.net/questions/353231 | 4 | Let $X$ be a module over some ring which splits as $$X\cong M\_1\oplus S\_1\cong M\_1\oplus M\_2 \oplus S\_2 \cong M\_1\oplus M\_2 \oplus M\_3\oplus S\_3\cong \ldots$$
where the isomorphisms come from splittings $S\_i\cong M\_{i+1}\oplus S\_{i+1}$.
Let $S=\bigcap S\_i$ and $M=\sum M\_i$. There is a canonical injectio... | https://mathoverflow.net/users/105652 | Limit of split short exact sequences | No. Take $X $ to be the direct product of nonzero modules $M\_i $ indexed by the positive integers, and $S\_i $ to be the direct product of all but the first $i $ of them. Then $S=0$ and $M $ is the direct sum of the $M\_i $.
| 8 | https://mathoverflow.net/users/22989 | 353238 | 149,280 |
https://mathoverflow.net/questions/243064 | 11 | A Tychonoff space $X$ is defined to have *countable separation* if some (equivalently, any) compactification $bX$ of $X$ contains a countable family $\mathcal U$ of open sets such that for any points $x\in X$ and $y\in bX\setminus X$ there is a set $U\in\mathcal U$ containing exactly one point of the doubleton $\{x,y\}... | https://mathoverflow.net/users/61536 | What is the smallest density of a metrizable space without countable separation? | Since [this](https://mathoverflow.net/questions/243125/has-a-continuous-map-from-kappa-omega-to-0-1-omega-a-non-scattered-fibe) problem has an affirmative answer, the last question should have a negative answer and then the smallest density of the space in the first question can be $\frak c^+$.
| 2 | https://mathoverflow.net/users/43954 | 353239 | 149,281 |
https://mathoverflow.net/questions/353245 | 3 | Let $A$ be a noncommutative unital algebra, defined over $\mathbb{C}$ say. What is an example of a left $A$-module $M$ that does not admit a right $A$-module structure giving $M$ the structure of a bimodule?
| https://mathoverflow.net/users/143172 | Left module which cannot be made into a bimodule? | Such examples are a plenty. You are asking about non-existence of an algebra map $A\rightarrow End\_AM$. Take $A$ simple, at most countably dimensional, and a simple module ${}\_{A}M$. Then $End\_AM={\mathbb C}$. Bingo!
| 9 | https://mathoverflow.net/users/5301 | 353249 | 149,284 |
https://mathoverflow.net/questions/353240 | 5 | Consider $m$ vectors $v\_1,\dots,v\_m$ in $\mathbb R^n$, drawn uniformly and independetly from unit sphere. It is pretty straightforward from Chebyshev inequality that
$$
\mathrm P (\forall i\ne j \ |v\_i \cdot v\_j|\leq \varepsilon) \to 1\ \text{as} \ n \to \infty.
$$
But what about quantitative version of this li... | https://mathoverflow.net/users/84950 | Precise probability that $m$ random vectors in $n$ dimensional space are nearly-orthogonal | $\newcommand{\ep}{\varepsilon}
\newcommand{\Ga}{\Gamma}
\newcommand{\de}{\delta}$
For $\ep\in(0,1)$, let
\begin{equation\*}
P\_{m,n}:=P\Big(\bigcap\_{1\le i<j\le m}\{|v\_i\cdot v\_j|\le\ep\}\Big)
=1-Q\_{m,n},
\end{equation\*}
where
\begin{equation\*}
Q\_{m,n}:=P\Big(\bigcup\_{1\le i<j\le m}\{|v\_i\cdot v\_j|>\ep... | 4 | https://mathoverflow.net/users/36721 | 353254 | 149,287 |
https://mathoverflow.net/questions/352544 | 2 | I am currently reading a paper by Goldstern, Kellner and Shelah, in which they, pretty nonchalantly, state *"Amoeba forcing will add a null set covering all old null sets"*, without proving this fact or giving a reference. The only thing I could find that would prove this statement was in the Bartoszynski book *"Set Th... | https://mathoverflow.net/users/138274 | Amoeba forcing adds a null set covering all old null sets | The conditions in Amoeba forcing are open sets of measure less than $1/2$. (Say, in $2^\omega$.)
A condition $q$ is stronger than $p$ iff $q \supseteq p$. (Alternatively, use closed sets of measure greater than $1/2$. Then stronger conditions will be smaller.)
For a generic filter $G$ let $U\_G$ be the union of a... | 4 | https://mathoverflow.net/users/14915 | 353263 | 149,289 |
https://mathoverflow.net/questions/353241 | 11 | This is a spin-off of my question [here](https://mathoverflow.net/questions/352212/fundamental-groups-of-complements-to-countable-subsets-of-the-plane), separated from the older question following Jeremy's suggestion.
**Definition.** Call a group $G$ *essentially freely indecomposable* if in every free product decom... | https://mathoverflow.net/users/39654 | Free product decompositions of the fundamental group of Hawaiian Earrings | This answer is courtesy of Sam Corson who kindly pointed out the following.
**Theorem:** The Hawaiian earring group $\pi\_1(\mathbb{H})$ is essentially freely indecomposable, i.e. if $\pi\_1(\mathbb{H})\cong G\_1\ast G\_2$, then one of $G\_1$ or $G\_2$ must be a finitely generated free group.
The key is to apply a... | 9 | https://mathoverflow.net/users/5801 | 353269 | 149,290 |
https://mathoverflow.net/questions/88758 | 5 | **Definition:** Let $\kappa$ be an $\aleph$ cardinal, we say that $\langle f\_\alpha\colon\alpha\to\kappa\mid\alpha<\kappa^+\rangle$ is a *ladder* if every $f\_\alpha$ is injective.
Equivalently this is the range of a choice function from every injection of $\alpha$ into $|\alpha|$ (for $\alpha<\kappa$ we can always ... | https://mathoverflow.net/users/7206 | On successive regular cardinals with no ladders | Both questions have a position answer. Which, in some sense, indicate that the non-existence of club sequences is somehow a weak property, relatively speaking.
For the first question this is quite trivial and really just requires a straightforward checking of the definitions. The point here being that if we start wit... | 3 | https://mathoverflow.net/users/7206 | 353271 | 149,291 |
https://mathoverflow.net/questions/353277 | 2 | Between polygons in $\mathbb C\cup\{\infty\}$ (including the "single side polygons", hemispheres, disks) the [Schwartz-Christoffel mappings](https://en.wikipedia.org/wiki/Schwarz%E2%80%93Christoffel_mapping) give arguably explicit conformal maps. For polygons with few angles those are well known special functions -hype... | https://mathoverflow.net/users/6575 | Conformal maps between simply connected domains with piecewise real algebraic boundary | I disagree that the Schwarz-Christoffel formula is "reasonably explicit", except in the case of triangle and rectangle, and very few other cases. The reason is that Schwarz-Christoffel formula for $n\geq 3$ contains unknown "accessory" parameters. Determination of these parameters requires inversion
of some rather comp... | 5 | https://mathoverflow.net/users/25510 | 353291 | 149,300 |
https://mathoverflow.net/questions/353094 | 2 | I have a big problem to solve this system:
$\Delta f−hf^2=0$
$p|\nabla f|^2+hf^3=0$
where $h$ and $p$ are constants (with $h \neq 0$ and $p \neq 0$, $p \neq -1$), $f$ is a scalar function defined on a 4-manifold ($f:M \rightarrow \mathbb{R}$) where $M$ is a 4-manifold not compact and where $\Delta f$ is the Lapla... | https://mathoverflow.net/users/111304 | Metric and particular system of PDE | As in my previous solution in the 3-dimensional case (discussed [here](https://mathoverflow.net/questions/334414/pde-system-problem-to-find-the-metric)), we can set $f=-(p/h)x$ for a function $x$ that satisfies
$$
\Delta x + p\,x^2 = |\nabla x|^2 - x^3 = 0.\tag 1
$$
Conversely, if $x$ satisfies this system for a metri... | 3 | https://mathoverflow.net/users/13972 | 353304 | 149,303 |
https://mathoverflow.net/questions/353303 | 4 | Let $M,N$be two smooth manifolds. Let $f,g:M\rightarrow N$ be two smooth maps. We have the notion of a homotopy (smooth homotopy) from the maps $f$ to the map $g$.
Is there a similar concept for morphisms of Lie groupoids?
Suppose $\mathcal{G}=(\mathcal{G}\_1\rightrightarrows \mathcal{G}\_0)$ and $\mathcal{H}=(\mat... | https://mathoverflow.net/users/118688 | Lie groupoids being homotopy equivalent | Yes there is! Here is one way to go.
If $X=(X\_{1}\rightrightarrows X\_{0})$ is a topological groupoid, then $X\times [0,1]=(X\_{1}\times[0,1]\rightrightarrows X\_{0}\times[0,1])$ is also a topological groupoid.
So the notion of homotopy is: if $f,f':X\rightarrow Y$ are two maps, then a homotopy between them is a m... | 3 | https://mathoverflow.net/users/148857 | 353326 | 149,309 |
https://mathoverflow.net/questions/353317 | 6 | In a braided rigid monoidal category $(\mathcal{M},\otimes)$ left and right duals coincide. What is an example of a rigid monoidal category where left and right duals coincide but there exist no braiding for the category?
| https://mathoverflow.net/users/143172 | Nonbraided rigid monoidal category where left and right duals coincide | The simplest example is G-graded vector spaces where G is a non-abelian group.
| 7 | https://mathoverflow.net/users/22 | 353327 | 149,310 |
https://mathoverflow.net/questions/353307 | 3 | **Notation and Setting**: let $\operatorname{Aff}$ denote the category of affine schemes whose objects are covariant representable functors $\operatorname{X}:\operatorname{Ring}\rightarrow\operatorname{Set}$ and $\operatorname{Spec}:\operatorname{Ring^{op}}\rightarrow\operatorname{Func(Ring, Set)} $ be the contravarian... | https://mathoverflow.net/users/142626 | Is the formal completion of an affine group necessarily a formal group? | The universal map is not a map of formal groups without some extra condition. An easy class of counterexamples comes from completions of an affine group along a closed subscheme that does not contain the identity element. In general, when you want to complete an affine group to get a formal group, you set $I\_n = I^n$ ... | 5 | https://mathoverflow.net/users/121 | 353328 | 149,311 |
https://mathoverflow.net/questions/352415 | 1 | Let $a$ and $b$ be two relatively prime positive integers. Denote by $S$ the set of all positive primes of the form $a+bn$ (where $n$ is an integer, possibly zero or negative). Let $S'$ be any set of positive primes such that $T=S\,\triangle\, S'$ is finite. Does there necessarily exist a finite Galois extension $K/\ma... | https://mathoverflow.net/users/nan | Sets of primes with a given Frobenius conjugacy class | The answer is no for $a=1$ and $b\geq 3$ arbitrary. Indeed, let $S$ be the set of (positive) primes congruent to $1$ modulo $b$. Let $S'$ be any set of (positive) primes such that $T=S\,\triangle\, S'$ is finite. Assume that $K/\mathbb{Q}$ is a finite Galois extension, and $C\subset \mathrm{Gal}(K/\mathbb{Q})$ is a con... | 1 | https://mathoverflow.net/users/11919 | 353357 | 149,320 |
https://mathoverflow.net/questions/353080 | 7 | Can we find a complete connected noncompact Riemannian manifold $(M^n,g)$ such that the curvature operator $Rm>0$ and
$$
\inf\_{p \in M} \text{Vol}\_gB(p,1)=0?
$$
| https://mathoverflow.net/users/105900 | Positively curved manifold with collapsing unit balls | The answer is negative if $\dim M=2$ and positive otherwise, as shown in the paper:
Croke, C. B., & Karcher, H. (1988). VOLUMES OF SMALL BALLS ON OPEN MANIFOLDS: LOWER BOUNDS AND EXAMPLES. AMERICAN MATHEMATICAL SOCIETY (Vol. 309). <https://www.ams.org/journals/tran/1988-309-02/S0002-9947-1988-0961611-7/S0002-9947-198... | 6 | https://mathoverflow.net/users/890 | 353364 | 149,323 |
https://mathoverflow.net/questions/353362 | 4 | Let $S$ be a finite set, and let $2^S$ be its powerset, regarded as a lattice. Let $L$ be a quotient (in the category of lattices and maps which preserve $\top,\bot,\wedge,\vee$) of $S$. What can we say about $L$?
In fact, what I'd really like to know is: which finite semilattices are retracts (via $\bot,\vee$-prese... | https://mathoverflow.net/users/2362 | Which lattices are quotients of finite powerset lattices? | The class of finite powerset lattices is closed under quotients, up to isomorphism. That is, the quotients are exactly the lattice reducts of finite Boolean algebras.
In particular, any quotient $L$ of $2^S$ has to be a bounded distributive lattice, as the class of distributive lattices is a variety. Moreover, if $x\... | 11 | https://mathoverflow.net/users/12705 | 353370 | 149,324 |
https://mathoverflow.net/questions/353375 | 4 | I am reading [this paper](https://www.worldscientific.com/doi/abs/10.1142/S0219498816501759) where the object $C\_4\times C\_2 : C\_2$ is used as a group structure. I know that $C\_n$ is a cyclic group but don't know what kind of operation between groups is identified by the symbol "$:$". Does anyone know about that? T... | https://mathoverflow.net/users/152342 | $C_4\times C_2 : C_2$: what does this mean? | The colon means "semidirect product", but it does not specify which semidirect product. This notation is a concise shorthand that gives important structural information without necessarily uniquely specifying the group. You can read more about similar notation conventions in the introduction to the ATLAS of finite grou... | 11 | https://mathoverflow.net/users/121 | 353376 | 149,325 |
https://mathoverflow.net/questions/353351 | 2 | I want to show the following:
Let $H$ be a Hilbert space and let $S:H\to H$ be a bounded operator such that
$$\|S\|<\sin\frac{\pi}{2n}.$$
Let $\mathcal{L}$ be a closed subspace of $H$ and $$u\_k:=(I-S)^ku,\;\;\;\;\text{for}\;\;\;k=0,\ldots,n\;\;\;\text{and}\;\;\;u\in\mathcal{L}\setminus\{0\}.$$
Prove that
$$\|Pu\_... | https://mathoverflow.net/users/152735 | If $\|S\|<\sin\frac{\pi}{2n}$ then $\|P(I-S)^ku\|\neq 0$ for all $k=0,\ldots,n$ | Your condition that $\| S\|<\sin\frac{\pi}{2n}$ implies that the angle between
$u\_{{k+1}}$ and $u\_{k}$ is less than $\pi/2n$, for $k=0,...,n-1$. Therefore the angle
between $u=u\_0$ and $u\_{k}$ is $<\pi/2$ for $k=1,...,n$. Since $u\in L$, $u\_1,...,u\_n$ cannot be orthogonal to $L$ that is $Pu\_k\neq 0$.
| 6 | https://mathoverflow.net/users/25510 | 353380 | 149,328 |
https://mathoverflow.net/questions/353377 | 14 | Let $F$ be a compact oriented surface and $\rho:\pi\_1(F)\rightarrow SL\_2\mathbb{C}$ be a representation. Does there exist a compact oriented three-manifold $M$ with $\partial M=F$ and a homomorphism $\tilde{\rho}:\pi\_1(M)\rightarrow SL\_2\mathbb{C}$ so that the restriction of $\tilde{\rho}$ to $\pi\_1(F)$ is equal t... | https://mathoverflow.net/users/4304 | Does every $SL_2\mathbb{C}$ representation of a closed oriented surface extend over a compact oriented three-manifold? | Here is an argument that "most" points in the $SL(2, {\mathbb C})$-character variety $X(F)$ of the surface $F$ do not correspond to representations extendible to 3-manifold groups (as in the question).
Let $M$ be a compact oriented 3-manifold with $\partial M=F$. We then have the "restriction morphism" of $SL(2, {\m... | 16 | https://mathoverflow.net/users/39654 | 353382 | 149,329 |
https://mathoverflow.net/questions/353363 | 0 | Let $K$ be a compact Hausdorff space. For a bounded Borel measurable function $f$ on $K$, we define $\phi\_{f}\in C(K)^{\*\*}$ by $\phi\_{f}(\mu)=\int\_{K}fd\mu$ for all $\mu\in C(K)^{\*}$. It is easy to see that $\|\phi\_{f}\|=\|f\|\_{\infty}$. Thus the space of all bounded Borel measurable functions on $K$ can be con... | https://mathoverflow.net/users/41619 | A closed subspace generated by open $F_{\sigma}$ sets of $K$ in $C(K)^{**}$ | Let $\epsilon > 0$. Since $a\_n$ is a bounded sequence, by compactness we can find a finite set of scalars $b\_1, \dots, b\_m$ such that for every $a\_n$ there exists a $b\_{k\_n}$ with $|a\_n - b\_{k\_n}| \le \epsilon$. Now consider the function $h = \sum\_{n=1}^\infty b\_{k\_n} \chi\_{U\_n}$, that is, $h(t) = b\_{k\_... | 1 | https://mathoverflow.net/users/4832 | 353385 | 149,331 |
https://mathoverflow.net/questions/353398 | 0 | Let $H$ be a subgroup of a finite group $G$, and let $N = N\_G(H)$ be the normalizer of $H$ in $G$.
For $x \in G$ is there a lower bound for $[ H : H \cap xHx^{-1} ]$? If $x \in N$ this index is 1, of course. If $x \notin N$ do we have $[N : H] \leq [H : H \cap xHx^{-1} ]$? What if $N/H$ is cyclic?
| https://mathoverflow.net/users/15428 | Lower bound for $[ H : H \cap xHx^{-1} ]$ | You can make $[N:H]$ as big as you want: start with an arbitrary group $E$ and non-normal subgroup $H$ (e.g. $E=C\_p\rtimes C\_2$) dihedral of order twice a prime $p$, and $H=C\_2$; for every $x\not\in H$ one has $[H:H\cap xHx^{-1}]=2$), and take $G$ to be the direct product of $E$ and any other group $U$, cyclic if yo... | 1 | https://mathoverflow.net/users/35416 | 353402 | 149,338 |
https://mathoverflow.net/questions/353404 | 1 | Assume that $\Sigma^2$ is a closed surface in $\mathbb{R}^3$ defined by the equation $\rho(x)=1$, where $\rho$ is some smooth function so that $\nabla \rho\neq 0$. Let $A=H(\rho)$ be the Hessian matrix of $\rho$. My question arises, whether $\left<Ax,x\right>>0$ for $x\neq 0$ implies that the domain $\Omega$ bounded by... | https://mathoverflow.net/users/124426 | Hessian matrix and its positiveness | Yes. your condition implies that the surface is locally convex, and the fact that locally convex implies globally convex is [a theorem of Tietze.](https://mathoverflow.net/questions/22062/to-what-extent-is-convexity-a-local-property/22091#22091)
| 2 | https://mathoverflow.net/users/11142 | 353405 | 149,339 |
https://mathoverflow.net/questions/353408 | 8 | It is known that any group $G$ can be embedded into a simple group $S$, see, e.g., the discussion at [Can any group be embedded in a simple group?](https://mathoverflow.net/questions/247402/can-any-group-be-embedded-in-a-simple-group)
My question is whether one can get an embedding such that the ambient group $S$ is... | https://mathoverflow.net/users/84626 | Embedding of a group into a simple group in which every element is a commutator | Here is a construction. Every finite group with $n$ elements embeds into $A\_{2n}$ (and even $A\_{n+2}$) which is simple if $n>2$ and of commutator width 1 (as any other finite simple group by the Ore conjecture proved by Martin W. Liebeck, E. A. O'Brien, Aner Shalev, Pham Huu Tiep, although for $A\_n$ it was probably ... | 12 | https://mathoverflow.net/users/nan | 353410 | 149,342 |
https://mathoverflow.net/questions/353414 | 8 | Let $\pi:X\to S$ be a smooth family of complex manifolds (in the sense of deformation theory) such that $\pi^{-1}(0)\cong\mathbb{C}^n$ and $S\subset \mathbb{C}$ is a neighborhood of $0$. Is $\pi$ trivial? That is, is $X\cong \mathbb{C}^n\times S$ possibly after shrinking $S$?
I know that a smooth family of *compact* ... | https://mathoverflow.net/users/123207 | Is $\mathbb{C}^n$ rigid? | Example. $\pi: \{(z,w)\in {\mathbb C}^2: |zw|<1\}\to {\mathbb C}$, $\pi(z,w)=z$.
Edit. Similarly, to get a nontrivial deformation of ${\mathbb C}^n$, consider
$$
X=\{(z\_0, z\_1,...,z\_n)\in {\mathbb C}^{n+1}: |z\_0 z\_1|<1\}
$$
and let $\pi$ be the projection of $X$ to ${\mathbb C}$ which the 1-st coordinate line ... | 17 | https://mathoverflow.net/users/39654 | 353417 | 149,343 |
https://mathoverflow.net/questions/353424 | 0 | Let $G$ be a finite non-abelian group, and consider a choice of $N$ distinct elements $g\_{0},g\_{1},\ldots,g\_{N-1}\in G$ that generate $G$. Now, let $t$ be an arbitrary positive integer, and let $d\_{1},\ldots,d\_{K}\in\left\{ 0,\ldots,N-1\right\}$ be the $N$-ary digits of $t$, so that:$$t=\sum\_{k=1}^{K}d\_{k}N^{k-1... | https://mathoverflow.net/users/120369 | Probability distribution of random products of elements of a generating set of a finite non-abelian group | To begin with, endowing the set of integers with the upper density is quite far from making it a probability space. Nonetheless, the question you ask still makes sense. Namely, you consider the $G$-valued sequence defined by your function $\chi$ and just ask whether the empirical frequencies of its values converge to t... | 3 | https://mathoverflow.net/users/8588 | 353432 | 149,346 |
https://mathoverflow.net/questions/353247 | 6 | For complex semisimple Lie algebras, the maximal dimension of an abelian subalgebra was determined by Mal'cev in 1945. For $E\_7$, for example, it is $27$, and is the radical of the $E\_6$ parabolic.
What about in characteristic $p$ for $p>0$? I suspect the answer is nearly, but not quite, the same. Maybe you have th... | https://mathoverflow.net/users/152674 | Maximal dimension of abelian subalgebra of exceptional simple Lie algebra in positive characteristic | Let $\ell\ge0$ be the characteristic of the (algebraically closed) ground field. Let $G$ be semisimple with Lie algebra $\mathfrak g$.
First, the maximal dimension of an abelian subalgebra of $\mathfrak g$ can increase for small $\ell$. Let, e.g., $\mathfrak g=sl(2)$ and $\ell=2$. Then the Borel subalgebra is abelian... | 4 | https://mathoverflow.net/users/89948 | 353436 | 149,347 |
https://mathoverflow.net/questions/353435 | 6 | Is it possible to prove that every complex number has a square root using analytic LLPO, but avoiding Weak Countable Choice or Excluded Middle? Unique Choice is allowed.
(Analytic LLPO is the statement that given any pair of real numbers $x$ and $y$, either $x \leq y$ or $x \geq y$. This statement is non-constructive... | https://mathoverflow.net/users/75761 | Every complex number has a square root via LLPO without weak countable choice | Yes it is but there is no *extensional* square root function unless we also have LPO.
Note that the squaring function is a bijection from $Q\_{+} = \{x + iy \mid x \geq 0, y \geq 0\}$ onto $H\_{+} = \{x + iy \mid y \geq 0\}$. Similarly it is a bijection from $Q\_{-} = \{x + iy \mid x \geq 0, y \leq 0\}$ onto $H\_{-} ... | 7 | https://mathoverflow.net/users/2000 | 353443 | 149,349 |
https://mathoverflow.net/questions/353439 | 4 | Stationary phase method (in the usual setup) gives asymptotic for
$$
I(\lambda)=\int\_{a}^{b} f(t) e^{i \lambda \varphi(t)} d t,
$$
when at any stationary point $x\_0$ ($\varphi'(x\_0)=0$) second derivative
does not vanishe ($\varphi''(x\_0)\ne 0$). Is it possible to find in the literature asymptotic formula for $I(\... | https://mathoverflow.net/users/5712 | Stationary phase method for $\varphi''(x_0)= 0$ | Let me assume that $a=-\infty, b=+\infty, x\_0=0$ and $f$ smooth and compactly supported near 0. Then after a suitable change of variable, you get that
$
I(\lambda)=\int g(t) e^{i\lambda t^3/3} dt,
$
with $g$ smooth and compactly supported
and applying Plancherel formula you get
$$
I(\lambda)=\int \hat g(\tau) A(\la... | 4 | https://mathoverflow.net/users/21907 | 353444 | 149,350 |
https://mathoverflow.net/questions/353440 | 10 | Let $g(t)$ be a strictly increasing differentiable function. Can it map positively measurable set to zero measurable set?
It's obviously that $\{g'>0\}$ is dense. If I can prove that the Lebesgue measure $m(\{g'=0\}) = 0$, then for every set with positive measure, there will be a positively measurable subset with $g'... | https://mathoverflow.net/users/133871 | Is there a strictly increasing differentiable function maps positively measurable set to zero measure set? | There are strictly increasing $C^1$ functions that map sets of positive measure to sets of measure zero. Here is a construction:
Let $C\subset [0,1]$ be a Cantor set of positive measure. For a construction, see <https://en.wikipedia.org/wiki/Smith-Volterra-Cantor_set>. Let $g(x)=\operatorname{dist}(x,C)$. The functio... | 17 | https://mathoverflow.net/users/121665 | 353445 | 149,351 |
https://mathoverflow.net/questions/353441 | 1 | A graph $G$ is $d$-degenerate if every subgraph of $G$ contains a vertex of degree at most $d$. It is known that an $n$-vertex $d$-degenerate graph has at most $d(n-1)$ edges. However, if we are given a bipartite $d$-degenerate graph $(H:m,n)$, what is its maximum number of edges?
| https://mathoverflow.net/users/148974 | Density of bipartite $d$-degenerate graph |
>
> **Theorem.** A $d$-degenerate $n$-vertex bipartite graph has at most $\lceil \frac{n}{2} \rceil \lfloor \frac{n}{2} \rfloor$ edges if $n < 2d$ and at most $d(n-d)$ edges if $n \geq 2d$. Moreover, both these bounds are tight.
>
>
>
*Proof.* Suppose $G$ is a $d$-degenerate $n$-vertex bipartite graph. Every $n... | 5 | https://mathoverflow.net/users/2233 | 353448 | 149,352 |
https://mathoverflow.net/questions/352295 | 8 | We are searching for rank 8 elliptic curves with the torsion subgroup $\mathbb{Z}/6$ using newly discovered families similar to Kihara's as described in
>
> A. Dujella, J.C. Peral, P. Tadić, *Elliptic curves with torsion group $\mathbb{Z}/6\mathbb{Z}$*, Glas. Mat. Ser. III 51 (2016), 321-333 doi:[10.3336/gm.51.2.0... | https://mathoverflow.net/users/95511 | Hard: One more generator needed for a Z/6 elliptic curve | A set of points that generate $E(\mathbb{Q})$ modulo torsion is given by
```
(1955516573881233507049678279 : -86467145649172260650105545143411861089140 : 1),
(49225691888888099223656060329/10201 : 67749663895993353685065159554645568700902610/1030301 : 1),
(61339810590192565389735634 : -4402893317936225229088404239... | 17 | https://mathoverflow.net/users/151977 | 353462 | 149,356 |
https://mathoverflow.net/questions/353464 | 6 | Let $R$ be a ring. Let $Mod(R)$ be the category of left $R$-modules, and let $Proj(R) \subseteq Mod(R)$ be the full subcategory of projective $R$-modules. Let's say that *$R$-projectives are closed under highly-filtered colimits* if there exists a cardinal $\kappa$ such that $Proj(R)$ is closed under $\kappa$-filtered ... | https://mathoverflow.net/users/2362 | When are projective modules closed under highly-filtered colimits? | 1. Let $R$ be a ring and $\kappa$ be a strongly compact cardinal such that $|R|<\kappa$. Then the class of all projective $R$-modules is closed under $\kappa$-filtered colimits.
This is Theorem 3.3 in the recent preprint of J. Šaroch and J. Trlifaj "Test sets for factorization properties of modules", <https://arxiv.org... | 8 | https://mathoverflow.net/users/2106 | 353468 | 149,359 |
https://mathoverflow.net/questions/350351 | 15 | The following theorem is relatively classical:
**Theorem:** Given an accessible endofunctor, (co)pointed endofunctor or (co)monad $T$ on a locally presentable category $C$, then the category of $T$-(co)algebra is also locally presentable.
The proof goes as follows: in each case the category of (co)algebra can be wr... | https://mathoverflow.net/users/22131 | presentability rank of categories of coalgebras | The case of algebras for a monad is discussed explicitly in [Gregory Bird's thesis](http://maths.mq.edu.au/~street/BirdPhD.pdf) (see theorem 6.9). The case of the categories of algebras for an endofunctor or pointed endofunctor can be deduced from the fact that if $F$ is a (pointed) endofunctor on $C$, then $F$-Alg $\r... | 8 | https://mathoverflow.net/users/22131 | 353475 | 149,362 |
https://mathoverflow.net/questions/353459 | 4 | Let $X$ be a metrizable compact space and $T\colon X\to X$ a minimal homeomorphism, i.e.
$$ \mathrm{orb}(x) := \{T^kx:k\in\mathbb{Z}\}$$
is dense in $X$ for every $x \in X$. Assume that the following condition is met:
* There exist $\varepsilon\_n \to 0$ and $s\_n \in \mathbb{N}$ such that $d(T^{s\_n}x,x) < \varepsil... | https://mathoverflow.net/users/134135 | This almost periodic condition implies equicontinuity? | In general, such a homeomorphism is not necessary equicontinuous.
The existence of such examples on $X=\mathbb{T}^2$, i.e. the $2$-torus, can be shown as follows: let $\mathcal{O}$ be the $C^\infty$ closure of the set $\{h\circ R\_\alpha\circ h^{-1} : h\in\mathrm{Diff}^\infty(\mathbb{T^2}),\ \alpha\in\mathbb{T}^2\}$,... | 1 | https://mathoverflow.net/users/889 | 353476 | 149,363 |
https://mathoverflow.net/questions/353481 | -2 | I stumbled on the following problem, if you can see a way through it.
Let $x$ be a real variable and fix a real value $\frac14\leq\nu\leq\frac34$.
>
> **QUESTION.** For $x\rightarrow0$, does there exist a constant $C>0$ (independent of $x$ that is) with the below property?
> $$\int\_{\mathbb{R}}\left\{\int\_0^{\... | https://mathoverflow.net/users/66131 | Asymptotics for certain integrals | Replacing $y$ with $y-\frac 12$, we get the integral
$$
2\int\_0^\infty\left[\int\_0^\infty z^{-\nu}e^{-\frac z2}e^{-x^2\frac{y^2+0.25}{2z^2}}
\left(e^{\frac{x^2y}{2z^2}}-e^{-\frac{x^2y}{2z^2}}\right)dz\right]^2\,dy
$$
Now, for small $x>0$, reduce the integration to $x\le z\le 2x$, $1\le y\le 2$. Then the inner integra... | 2 | https://mathoverflow.net/users/1131 | 353485 | 149,365 |
https://mathoverflow.net/questions/353478 | 2 | Let $X$ be a projective variety with only normal crossing singularity. Is there a description of the derived category or the category of perfect complexes? What about the existence of semiorthogonal decomposition?
Please provide a reference if this kind of study exists in the literature..
| https://mathoverflow.net/users/nan | Derived category of singular varieties | Let $\tilde{X}\_k$ be the normalization of the closed $k$-codimension stratum, so $\tilde{X}\_0$ is the normalization of $X$. Then there is a diagram of pullback functors between the categories $\text{Perf}(\tilde{X}\_k),$ such that $\text{Perf}(X)$ is the pullback of this diagram in the $\infty$-category of derived ca... | 4 | https://mathoverflow.net/users/7108 | 353489 | 149,368 |
https://mathoverflow.net/questions/309158 | 16 | After this question : [Does every real function have this weak continuity property?](https://mathoverflow.net/questions/309019/does-every-real-function-have-this-weak-continuity-property)
Natrualy there are an other (more difficult) :
>
> Is it true that for every real function $f:\mathbb{R}\to\mathbb{R}$, there... | https://mathoverflow.net/users/110301 | Does every real function have this weak derivation property? | the answer is yes : <http://webhome.auburn.edu/~brownj4/tatras.pdf>
PS : the answer was in the comments, but no one gave an answer
| 1 | https://mathoverflow.net/users/110301 | 353507 | 149,373 |
https://mathoverflow.net/questions/353517 | 0 | I am looking for an expression that convert calculation of given quadratic form into determinant of some block matrix. I see this in that form (that may be incorrect):
$x^T A x = \begin{vmatrix} x^T & 0 & 0 \\ 0 & A & 0 \\ 0 & 0 & x \end{vmatrix}$
To clarify. The above matrix is a block matrix with “additional” blo... | https://mathoverflow.net/users/148731 | Determinant of a block matrix with dissimilar elements | It's hard for me to understand what you are asking, but maybe this answers your question. One has
$$
-x^TAx = M / A^{-1},
$$
where
$$
M = \begin{bmatrix}A^{-1} & x\\ x^T & 0 \end{bmatrix}
$$
and the symbol $/$ denotes the [Schur complement](http://en.wikipedia.org/wiki/Special:Search?search=Schur%20complement).
One h... | 2 | https://mathoverflow.net/users/1898 | 353521 | 149,379 |
https://mathoverflow.net/questions/353520 | 3 | Let $f:X \to Y$ be a finite map from a normal projective variety to a smooth projective variety, $D$ be a Cartier divisor on $X$. Do we have any relation between $\kappa(X,D)$ and $\kappa(Y,f\_\*D)$?
| https://mathoverflow.net/users/24445 | Kodaira dimensions of push-forward via finite map | There is an obvious relation: the pushforward map $f\_\*:|mD|\rightarrow |f\_\*(mD)|=|mf\_\*D|$ is injective, hence $\kappa (X,D)\leq \kappa (X,f\_\*D)$. It is easy to see that you cannot get more: for instance, take for $f$ a general projection from a cubic surface $X\subset \mathbb{P}^3$ to $\mathbb{P}^2$, and for $D... | 7 | https://mathoverflow.net/users/40297 | 353525 | 149,381 |
https://mathoverflow.net/questions/353508 | 25 | I would like to know as curiosity how the editorial board or editors\* of a mathematical journal evaluate the quality, let's say in colloquial words the importance, of papers or articles.
>
> **Question.** I would like to know how is evaluate the quality of an article submitted in a journal. Are there criteria to e... | https://mathoverflow.net/users/142929 | Evaluation of the quality of research articles submitted in mathematical journals: how do they do that? | Assume we are talking about a good journal with a large editorial board representing a wide scope of mathematical interests. I will describe both the role of the editors and the role of the referees. This is my personal viewpoint and others might have different opinion/experience.
**The role of the editors.**
Good jo... | 36 | https://mathoverflow.net/users/121665 | 353532 | 149,384 |
https://mathoverflow.net/questions/343699 | 2 | I am reading [Differentiable stacks, gerbes, and twisted K-Theory](http://www.personal.psu.edu/pxx2/book.pdf) by Ping Xu.
To talk about (twisted) K-theory of differentiable stacks, author introduced (page $41$) the set up of $C^\*$-algebras. All I know about $C^\*$-algebras is their definition and one or two results.... | https://mathoverflow.net/users/118688 | $C^*$-algebras appearance in study of Lie groupoids and differentiable stacks | 1. Consider first the case of a manifold $M$ seen as the space of units of the groupoid structure $M\to M$ where $s=t=id\_M$ is the projection and no pairs are composable, so that you have only identities. Then the corresponding groupoid $C^\*$-algebra is nothing but the standard $C^\*$-algebra of continuous functions ... | 2 | https://mathoverflow.net/users/6032 | 353549 | 149,388 |
https://mathoverflow.net/questions/352626 | 4 | Is there a reference discussing in an organized way (with a proof) the Weyl integration formula for a reductive group over a local field (Archimedean or not), expressing the Haar integral on the group as a sum over Levi's of integrals over the elliptic elements in the Levi of orbital-like integrals?
Thank you!
Sasha
... | https://mathoverflow.net/users/2095 | Reference request - Weyl's integration formula | Section 7 of Kottwitz's article in the 2003 Clay proceedings [here](http://www.claymath.org/library/proceedings/cmip04.pdf) has what you are looking for.
| 3 | https://mathoverflow.net/users/136176 | 353558 | 149,391 |
https://mathoverflow.net/questions/353564 | 8 | Birkhoff's completeness theorem (see [here](http://www.math.uwaterloo.ca/~snburris/htdocs/ualg.html), Theorem 14.19) states that an equation which is true in all models of an algebraic theory can be proven in [equational logic](https://en.wikipedia.org/wiki/Equational_logic).
**Question.** Does the proof of Birkhoff'... | https://mathoverflow.net/users/2841 | Birkhoff's completeness theorem put into practice | Let me try to restate the question.
I consider an identity to be a pair, written $(s,t)$ or $s\approx t$. I also consider a set of identities to be a set of pairs.
Birkhoff's Theorem compares three things, namely
(1) $\Sigma\models s\approx t$,
(2) The pair $(s,t)$ belongs to the fully invariant congruence $\The... | 10 | https://mathoverflow.net/users/75735 | 353575 | 149,393 |
https://mathoverflow.net/questions/352564 | 2 | Suppose we have a graph **G**. Say **B** a fundamental basis of the cycle space of **G**. Say *LP* a linear programming problem where there is a variable for each vertex of **G**, each variable can take value $\geq 0$, for each odd cycle of **B** we add to *LP* the constraint $x\_{a} + x\_{b} + x\_{c} + ... + x\_{i} \g... | https://mathoverflow.net/users/152281 | Odd cycle transversal | **No.** Let $G$ be the graph obtained by gluing a $3$-cycle $abc$ and a $5$-cycle $cdefg$ together at vertex $c$. Then $(x\_a, x\_b, x\_c, x\_d, x\_e, x\_f, x\_g)=(0,0,3,1,1,0,0)$ is an optimal solution of the LP. However, neither $d$ nor $e$ are contained in a minimum odd cycle transversal of $G$, since $\{c\}$ is the... | 3 | https://mathoverflow.net/users/2233 | 353576 | 149,394 |
https://mathoverflow.net/questions/339533 | 2 | Consider the following problem: given a [variety](https://en.wikipedia.org/wiki/Variety_(universal_algebra)) of algebras (a class of algebras in a given algebraic signature defined by some set of equations), describe its semisimple subvarieties. That is, describe its subvarieties $\mathsf{K}$ such that each algebra $\m... | https://mathoverflow.net/users/145176 | Jacobson semisimple varieties of commutative rings | I think this question asks: which varieties of commutative rings have the property that every member is a subdirect product of simple rings (= fields). These are the varieties satisfying some identity of the form $x=x^n$ for some $n>1$. They are exactly the congruence distributive varieties of commutative rings. These ... | 1 | https://mathoverflow.net/users/75735 | 353586 | 149,397 |
https://mathoverflow.net/questions/353562 | 2 | Suppose $(M,g, \omega)$ is a Kähler manifold with $\text{Ric}(g) = g$, i.e., $M$ is a Fano manifold. Is $M$ necessarily compact? If not, perhaps complete and Fano implies compact? I'd like to build a catalog of examples of Fano manifolds, so any input is appreciated.
| https://mathoverflow.net/users/nan | Do non-compact Fano manifolds exist? | By the [Bonnet Myers theorem](https://en.wikipedia.org/wiki/Myers%27s_theorem), bounded positive Ricci curvature and complete Riemannian metric forces compact. David Wraith once explained to me that if the Ricci decays more slowly than quadratically in distance from a given point, on a complete Riemannian manifold, the... | 4 | https://mathoverflow.net/users/13268 | 353587 | 149,398 |
https://mathoverflow.net/questions/353258 | 8 | The following correspondence between algebraic theories and monads on $\mathbf{Set}$ is well-known (see, for example, *Algebraic Theories: A Categorical Introduction to General Algebra*).
>
> The category of (finitary) $S$-sorted algebraic theories is equivalent to the category of (finitary) monads on $\mathbf{Set}... | https://mathoverflow.net/users/152679 | Characterisation of essentially algebraic theories as monads | I'm going to give a partial answer to my question, which addresses a misconception I had and illustrates why many of the existing generalisations of theory–monad correspondence are not sufficient to provide a monadic correspondence with essentially-algebraic theories. As far as I know, this indicates that a corresponde... | 3 | https://mathoverflow.net/users/152679 | 353597 | 149,402 |
https://mathoverflow.net/questions/353609 | 2 | To make use of the Lie algebra action of $\mathsf{gl}\_2(\mathbb{C})$ to establish a isomorphism in modular representation theory, I would like an answer to this question:
>
> Let $K$ be a field of prime characteristic. When is there a subring $R$ of the complex numbers and a maximal ideal $M$ of $R$ such that $R/M... | https://mathoverflow.net/users/7709 | When is there a subring of the complex numbers surjecting onto a given field of prime characteristic? | The complex numbers have transcendence degree the continuum over $\mathbb Q$ so contain a copy of the field of rational functions in continuum many variables over $\mathbb Q$. This in turn contains the ring of polynomials in continuum many variables over $\mathbb Z$, which surjects onto any ring of cardinality at most ... | 7 | https://mathoverflow.net/users/112113 | 353614 | 149,405 |
https://mathoverflow.net/questions/353312 | 2 | This is probably a known problem but I was not able to find exactly what I am looking for.
I have the following linear heat equation with zero-flux boundary conditions:
\begin{equation}
\begin{cases}
\dot{u} - \Delta u = u \quad \text{in} \quad \Omega;\\
\nabla u \cdot \boldsymbol{n} = 0 \quad \text{on} \quad \part... | https://mathoverflow.net/users/152665 | Regularity on the boundary for the heat equation with linear source | The estimate $(2)$ is false even in a half-plane. Indeed, let $w$ be any solution to the heat equation on $\mathbb{R}^2 \times [0,\,\infty)$ that is even in $y$ (so $w$ solves the Neumann problem for the heat equation in the upper half-plane), and vanishes on the $x$-axis at $t = 0$. Then $w\_R(x,\,y,\,t) := w(Rx,\,Ry,... | 1 | https://mathoverflow.net/users/16659 | 353634 | 149,408 |
https://mathoverflow.net/questions/353630 | 2 | Consider a uniform random tournament with $n$ vertices. (Between any two vertices $x,y$, with probability $0.5$ draw an edge from $x$ to $y$; otherwise draw an edge from $y$ to $x$.) Let $S$ be the set of all out-degrees. Let $s\_1$ be the largest element of $S$, and $s\_2$ the next largest. (If $S$ is a singleton, let... | https://mathoverflow.net/users/83212 | Difference between two largest degrees | You guess is correct, assuming that by $x\_1$ and $x\_2$ you meant $s\_1$ and $s\_2$.
Indeed, the probability in question is $1-p\_n$, where
\begin{equation}
p\_n:=P(\exists i\in[n]\ D\_i-\max\_{j\in[n]\setminus\{i\}}D\_j\ge cn),
\end{equation}
where $[n]:=\{1,\dots,n\}$ and $D\_i$ is the out-degree of the $i$th ... | 4 | https://mathoverflow.net/users/36721 | 353639 | 149,410 |
https://mathoverflow.net/questions/353620 | 6 | Let $Idem = Idem^{(\infty)}$ be the walking idempotent [1], and let $Idem^{(n)}$ be its n-skeleton. Note that $Idem$ has one nondegenerate simplex in each dimension. Let $\iota\_n^m: Idem^{(n)} \to Idem^{(m)}$ be the inclusion. Lurie has shown [2] the following:
* If $X$ is a quasicategory, and if $Idem^{(3)} \xright... | https://mathoverflow.net/users/2362 | Is the inclusion of its 2-skeleton into the walking idempotent homotopy cofinal? | I don't know what's wrong with the following computation, but the answer is clearly *no*: if there were a cofinal functor from a finite simplicial set to $Idem$, then any $\infty$-category with finite colimits would have split idempotents, which is not the case (witness finite spaces).
Somewhat surprisingly, this se... | 2 | https://mathoverflow.net/users/2362 | 353645 | 149,413 |
https://mathoverflow.net/questions/353638 | 0 | As well known to us, K.Y. Liang and S. Zeger proposed GEE for longitudinal data analysis in their famous paper[1]. At the appendix of the paper, authors show the proof of Theorem 2. I tried to reproduce that proof following their idea, but when I tried to derive $A\_{i}$, I failed. Here, I want to add two comments:
1... | https://mathoverflow.net/users/126313 | Deriving asymptotic variance of generalized estimating equation estimator (GEE) | The key point is Law of Large Numbers.
Answer:
$\partial \frac{1}{K}\sum A\_{i}/\partial \beta = \frac{1}{K}\sum \partial A\_{i}/\partial \beta = \frac{1}{K}\sum \{[\partial(D\_{i}^{T}V\_{i}^{-1})/\partial \beta \times S\_{i}] + [D\_{i}^{T}V\_{i}^{-1} \times \partial S\_{i}/\partial \beta]\}$.
Let $E\_{1} = \fra... | 0 | https://mathoverflow.net/users/126313 | 353652 | 149,415 |
https://mathoverflow.net/questions/353648 | 2 | I have the following integral
$$
I(\varepsilon) = \iint\_D \frac{\sqrt{1+|\nabla h(u,v)|^2}}{[(h(u,v)+\varepsilon)^2+u^2+v^2]^2} du dv,
$$
where $h$ is a smooth function with
$h(0,0)=0 = h\_u(0,0) = h\_v(0,0)$, $D$ is a disk centered at the origin.
It seems like that the asymptotic of $I(\varepsilon)$ depends only on... | https://mathoverflow.net/users/129135 | Asymptotic of an area integral | $\newcommand{\ep}{\varepsilon}$
This is indeed a matter of splitting the integral. In polar coordinates, for some real $R>0$,
\begin{equation}
I(\ep) = \int\_0^{2\pi}(J\_t(\ep)+K\_t(\ep))\,dt,
\end{equation}
where
\begin{equation}
J\_t(\ep):=\int\_0^{r\_\*}\frac{\sqrt{1+|\nabla h|^2}}{\big((h+\ep)^2+r^2\big)^2}\,r... | 6 | https://mathoverflow.net/users/36721 | 353657 | 149,419 |
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