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https://mathoverflow.net/questions/300535 | 3 | Let $X$ be a topological vector space of size $\mathfrak{c}$. Assume that there exists a countable union $X=\cup X\_n$ such that all *subsets* $X\_n$'s are relatively second countable.
Q. Does there exists a a countable union $X=\cup Y\_n$ such that all $Y\_n$'s are relatively second countable metrisable?
| https://mathoverflow.net/users/84390 | Are second-countable subsets of topological vector spaces metrizable? | This follows from general facts.
* [Topological vector spaces are (completely) regular](https://math.stackexchange.com/questions/2332760/every-topological-vector-space-is-regular).
* Subspaces of regular spaces are regular.
* A second-countable space is metrisable if and only if it is regular (see Theorems 32.1, 32.2... | 2 | https://mathoverflow.net/users/15129 | 300538 | 131,771 |
https://mathoverflow.net/questions/300431 | 21 | Let $G= \langle S \mid r \rangle$ be a one-relator presentation for a one-ended hyperbolic group, with $r$ cyclically reduced.
**Question:** Can there be a nontrivial word $w(S)$ which is trivial in the group $G$ but has length shorter than $r$? What if $r$ is the shortest possible word for a one-relator presentatio... | https://mathoverflow.net/users/nan | Can a hyperbolic, one ended, one relator group, have a shorter trivial word? | I think I found an example with shorter trivial words using a handy characterizations in a paper by Ivanov and Schupp called [*On hyperbolicity of small cancellation groups and one-relator groups*](http://www.ams.org/journals/tran/1998-350-05/S0002-9947-98-01818-2/).
Consider $\langle a,b,c \mid ab^2ac^{12}\rangle$. ... | 14 | https://mathoverflow.net/users/nan | 300541 | 131,773 |
https://mathoverflow.net/questions/300543 | 1 | Take $q\_0<q\_1<...<q\_k<q\_{k+1}<...$ positive integers, $z$ complex
From
$$T(z)=\sum\limits\_{k=0}^{\infty}\frac{1}{q\_k^z}$$
I would need to extract the first coefficient $q\_0$
It is irrelevant if this procedure is not achievable in practice. I would like to know if there is a principal solution. We do not ... | https://mathoverflow.net/users/nan | Is there a procedure for extracting first integer $q_0$ from $\sum\limits_{k=0}^{\infty}\frac{1}{q_k^z}$, all $0<q_0<q_1<...$ integers, $z$ complex? | For $z>1$, we have
\begin{equation}
q\_0^{-z}\le T(z)\le\sum\limits\_{q=q\_0}^{\infty}q^{-z}\le q\_0^{-z}+\int\_{q\_0}^\infty q^{-z}\,dq
=q\_0^{-z}(1+q\_0/(z-1)).
\end{equation}
Hence,
\begin{equation}
q\_0=\lim\_{z\to+\infty}T(z)^{-1/z}.
\end{equation}
| 2 | https://mathoverflow.net/users/36721 | 300547 | 131,776 |
https://mathoverflow.net/questions/300526 | 4 | Where can one find the proof of the following fact:
If there are two orientation-preserving diffeomorphisms $\phi\_0$ and $\phi\_1$ of $R^n$, then there exists a homotopy $\phi(t)$, such that $\phi(0)$ = $\phi\_0$, $\phi(1)$ = $\phi\_1$, and $\phi(t)$ is a diffeomorphism for any $t$ (and $\phi$ is smooth for all var... | https://mathoverflow.net/users/50311 | Why are two diffeomorphims of $R^n$ are always homotopic (in the same category)? | An explicit deformation retraction of $\mathrm{Diff}(\mathbb R^n)$ onto $\mathrm{Aff}(\mathbb R^n)$ is given by $$f\_t(x)=f(0)+\frac{f(tx)-f(0)}{t}$$ for $t\in (0,1]$ and $f\_0(x)=f(0)+f^\prime(0)x$.
| 7 | https://mathoverflow.net/users/1573 | 300548 | 131,777 |
https://mathoverflow.net/questions/300544 | 8 | Let $X$ be a topological space. Assume that there exists a sequence of simple functions $\phi\_n:X\to X$ (finite range and measurable) with $\lim\phi\_n(x)=x$.
Can we concluded $X$ may be written by a countable union $X=\cup X\_n$ where all $X\_n$'s are all relatively second countable?
| https://mathoverflow.net/users/84390 | Approximation of the identity by simple functions | The answer is no. The Moore plane is a counterexample.
For $z \in \mathbb{R}^2$ and $r \geqslant 0$ denote by $\bar D(z, r)$ the closed Euclidean disc of center $z$ and radius $r$. Recall that the Moore plane is $M = [0, + \infty) \times \mathbb{R}$, where:
* A basis of neighborhoods of $(x, y) \in (0, +\infty) \ti... | 7 | https://mathoverflow.net/users/120363 | 300549 | 131,778 |
https://mathoverflow.net/questions/300249 | 3 | We say that a real matrix is ***rotatable*** iff after turning it clockwise on $90^{\circ}$ it doesn't change.
I'm interesting about eigenvalues and eigenvectors (belonging to non-zero eigenvalues) of such type of matrices. For example, it is not hard to show that
for every tuple of real values $\lambda\_1,\ldots,\l... | https://mathoverflow.net/users/85489 | Rotatable matrix, its eigenvalues and eigenvectors | Consider the matrix $$P= \begin{pmatrix} 0 & \ldots & 1 \\ \vdots & 1 & \vdots \\ 1 & \ldots & 0 \end{pmatrix}$$ with $1$s along the "other" main diagonal and $0$s elsewhere. Then $(PA)^t$ is a rotation of the matrix $A$ by $90^\circ$ (you can check this on the basis $E\_{i,j}$ of matrices with a 1 in the $(i,j)$ slot ... | 4 | https://mathoverflow.net/users/101646 | 300558 | 131,782 |
https://mathoverflow.net/questions/299312 | 4 | If $A\subset \Bbb R^2$ then is the following statement true?
>
> $\{(x,y)\in {(A\times A)/ \sim}\,\,\,|\,\, (x,y)\sim(y,x)\}\simeq$ Möbius strip $\iff A$ is a Jordan curve.
>
>
>
| https://mathoverflow.net/users/90655 | A question on Möbius strip and Jordan curve | It seems that the answer to this problem is affirmative. We can argue as follows.
Assume that $A\subset\mathbb R^2$ is a subspace whose symmetric square $A^2/\_\sim$ is homeomorphic to the Mobius strip. In particular, $A^2/\_\sim$ is a connected 2-manifold with a boundary. Using this fact it can be shown that $A$ is ... | 5 | https://mathoverflow.net/users/61536 | 300562 | 131,783 |
https://mathoverflow.net/questions/300570 | 29 | First note that there exists a natural measure $\mu$ on $P(\omega \times \omega)$, inherited from the Lebesgue measure on the reals (by identifying the reals with $P(\omega)$ and $\omega$ with $\omega \times \omega$ in the natural ways)
Let us consider models of $ZFC$ of the form $(\omega, E),$ where $E \subseteq \om... | https://mathoverflow.net/users/11115 | On the probability of the truth of the continuum hypothesis | For any $E$ modelling ZFC and for each $n\in\omega$, we must have $(n,n)\not\in E$. Therefore $M\_{ZFC}$ is contained in the cylinder set defined by $(0,0)\not\in E,\dots,(n,n)\not\in E$ which has measure $2^{-n-1}$. Therefore $M\_{ZFC}$ has outer measure zero, thus is measurable and has measure zero.
Had we excluded... | 29 | https://mathoverflow.net/users/30186 | 300571 | 131,787 |
https://mathoverflow.net/questions/300575 | 3 | Let $\mathbb{Q}$ be the rationals, and let $\tau$ be the Euclidean topology on $\mathbb{Q}$. Is there a topology $\tau' \subseteq \tau$ such that $(\mathbb{Q},\tau')$ is connected and $T\_2$?
| https://mathoverflow.net/users/8628 | Is there a connected $T_2$-topology on $\mathbb{Q}$ that is coarser than the Euclidean one? | Using Sierpinski's topological characterization of $\mathbb Q$ (as a unique countable regular second countable space without isolated points), it can be shown that $\mathbb Q$ is homeomorphic to the set $\mathbb N$ endowed with the *Furstenberg topology* $\tau$ generated by the base consisting of all possible arithmetr... | 5 | https://mathoverflow.net/users/61536 | 300582 | 131,792 |
https://mathoverflow.net/questions/300117 | 6 | Let $f$ be an (elliptic) modular form of weight $k>0$, and consider the vertical strip $S\_m=\{x+iy\in\mathbb{C}:|x|\le 1/2, y>m$}. For every $m\ll 1$, the fundamental domain for $SL\_2(Z)$ is included in $S\_m$. We know that modular forms are bounded on the fundamental domain $\mathcal{F}$ i.e
$$|f(z)|\le c\_f\quad\fo... | https://mathoverflow.net/users/98823 | Behavior of a modular form in the lower strip | I worked out the case $n=2$, and I assume it can be generalized to any degree.
Fix any $m>0$ such that the Siegel fundamental domain is contained in $\{Z:\Im(Z)>mI\_2\}$. Take any $Z$ in the lower strip, i.e. $Y=\Im(z)\not>mI\_2$. Let $\gamma\in\Gamma\_2$ be a $4\times 4$ symplectic matrix for which $\Im(\gamma Z)>mI... | 1 | https://mathoverflow.net/users/98823 | 300583 | 131,793 |
https://mathoverflow.net/questions/300585 | 3 | Is there a good reference for the following problem? Consider any smooth bounded domain $\Omega$ and solve the heat equation
\begin{align}
\partial\_t u^\kappa &= \kappa \Delta u^\kappa,\\
u^\kappa|\_{\partial\Omega}&=0,\\
u^\kappa|\_{t=0}&=u\_0.
\end{align}
What is the rate of convergence as $\kappa\to 0$ of $... | https://mathoverflow.net/users/124658 | Rate convergence of the heat equation as diffusion tends to zero | **In a nutshell:** The rate of convergence depends on the smoothness of $u\_0$.
Here are the details.
**Setting:**
* Throughout I assume that $\kappa > 0$ is a real number, i.e. it does not depend on the spatial variable.
* I'm going to consider the equation in the space $L^2(\Omega)$ (but similar observations a... | 2 | https://mathoverflow.net/users/102946 | 300593 | 131,795 |
https://mathoverflow.net/questions/300581 | 3 | Fix $M \geq 2$. What is the smallest number $\tau = \tau(M)$ such that $$\sum\_{a,b,c,d =1\\ a + b = c+ d}^M (x\_a x\_b x\_c x\_d)^{\tau/4} \leq 1,$$ for all $x , \ldots , x\_M \in \mathbb{R}\_{\geq 0} $ satisfying $$\sum\_{j=1}^M x\_j = 1?$$ Clearly $\tau \leq 4$ and for $M = 2$, we have that $\tau = \log\_2 6$.
| https://mathoverflow.net/users/50426 | A sum over a hyperplane in $\mathbb{Z}^4$ | Fixed the gap. Now the argument should be complete. Feel free to ask questions if something is unclear.
The problem is equivalent to showing that the $L^4$ norm of a trigonometric polynomial $P(z)=\sum\_{k=1}^My\_kz^k$ on the unit circle $\mathbb T$ with the Haar measure $\mu$ can be bounded (with constant $1$) by th... | 5 | https://mathoverflow.net/users/1131 | 300612 | 131,800 |
https://mathoverflow.net/questions/300609 | 5 | In reducing the existence of Kähler-Einstein metrics to the complex Monge Ampere equation, the logarithm $$-\log \det (\omega + \partial \overline{\partial} \phi)$$ appears, where $\omega$ is a Kähler metric and $\phi$ is a smooth function. In Tian's book, he writes that while this function is not globally defined, we ... | https://mathoverflow.net/users/120806 | The logarithm of Kähler metric is not globally defined | **Question 1:** Note, $\omega + \partial\bar{\partial}\phi$ cannot be zero. Recall that $\phi$ is chosen so that $\omega + \partial\bar{\partial}\phi$ is another metric (in particular, a Kähler-Einstein one).
What does $\log\det(\omega + \partial\bar{\partial}\phi)$ mean? In holomorphic coordinates $(U, (z^1, \dots, ... | 10 | https://mathoverflow.net/users/21564 | 300613 | 131,801 |
https://mathoverflow.net/questions/296338 | 8 | Let $m$ be an odd integer greater than $1$. Is it true that there are positive $a, b$ such that $m=a+b$ and $a^2+b^2$ is a prime number?
It *seems* that for every odd $m$ there are many $(a,b)\in \mathbb{N}^2$ which sum to $m$ and their sum of squares give a prime number but I don't see how to prove this.
Or, equ... | https://mathoverflow.net/users/38851 | Every odd integer greater than $1$ is of the form $a+b$ with $a^2+b^2$ being prime | The question is not new. It is originally due to Ming-Zhi Zhang. One may consult <https://oeis.org/A036468> .
| 5 | https://mathoverflow.net/users/124654 | 300618 | 131,802 |
https://mathoverflow.net/questions/300606 | 6 | The curve
$$(X-16)^3=XY\tag{1}\label{1}$$
is essential to Heegner's approach to the class number one problem for imaginary quadratic fields. We have the following “modular” parametrization
\begin{equation}\tag{2}\label{2}(X,Y)=\left(2^{12}\Phi(\tau),j(\tau)\right),\end{equation}
where $\Phi(\tau)=\frac{\Delta(2\tau)}... | https://mathoverflow.net/users/122104 | Modular parametrization of a curve of Heegner and Weber | Firstly, you have a typo.
The left side is $$(X-16)^3=-4096 + 3145728q - 729808896q^2+O(q^3)$$ while the right side is $$XY=4096 + 3145728q + 880803840q^2+O(q^3).$$
Looking at (19) of [Stark's paper](https://www.sciencedirect.com/science/article/pii/0022314X69900237), I think the relevant root of $(X-16)^3=Xj(\tau)$ ... | 10 | https://mathoverflow.net/users/124666 | 300620 | 131,803 |
https://mathoverflow.net/questions/300599 | 2 | Let $f \in L^2(\mathbb R)$ then it is well-known that
$$ \widetilde{f}(x):=\sum\_{n \in \mathbb Z} \frac{1}{\varepsilon}\int\_{[n\varepsilon,(n+1)\varepsilon]} f(s) \ ds 1\_{[n\varepsilon,(n+1)\varepsilon)}(x)$$
converges in the $L^2$ sense to $f.$
But even more is true, as we can write
$$(\widetilde{f}-f)(x):=... | https://mathoverflow.net/users/124662 | Convergence rate for $L^2$ convergence | $\newcommand{\al}{\alpha}
\newcommand{\de}{\delta}
\newcommand{\De}{\Delta}
\newcommand{\ep}{\varepsilon}
\newcommand{\ga}{\gamma}
\newcommand{\Ga}{\Gamma}
\newcommand{\la}{\lambda}
\newcommand{\Si}{\Sigma}
\newcommand{\thh}{\theta}
\newcommand{\R}{\mathbb{R}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\F}{\mathcal{F}}
\n... | 7 | https://mathoverflow.net/users/36721 | 300625 | 131,804 |
https://mathoverflow.net/questions/300619 | 4 | Let $k$ be a field of characteristic $p> 0$.
Let $\mu\_{p^n}$ denote the group scheme $\mu\_{p^n}(R) =\{ x \in R: x^{p^n}=1 \}$. Then there are natural maps from $\mu\_{p^n}$ to $\mu\_{p^{n-1}}$. I'm interested in understanding what is the projective limit of this system of finite group schemes $\underset{n}{\varproj... | https://mathoverflow.net/users/124202 | What exactly is $\underset{n}{\varprojlim} \ \mu_{p^n}$? | Let me just do the simple thing and calculate the limit. Perhaps this is not what you are looking for, but then we can perhaps refine the question a bit in that case.
Write $\mu\_{p^n} = \mathrm{Spec}(k[x\_n]/(x\_n^{p^n} - 1))$. Then the natural map $\mu\_{p^n} \to \mu\_{p^{n-1}}$ that I believe you are thinking abou... | 6 | https://mathoverflow.net/users/37821 | 300626 | 131,805 |
https://mathoverflow.net/questions/300472 | 2 | Let $P$ and $Q$ be positive definite matrices. Consider the following matrix equation
$$\label{star}\tag{$\star$}
XPX^\top - P = -Q, \quad X\in\mathbb{R}^{n\times n}.
$$
>
> **My question.** Is it true that any solution of \eqref{star} can be written as $X=(P-Q)^{1/2}TP^{-1/2}$ with $T$ being an arbitrary orthogona... | https://mathoverflow.net/users/62673 | Solving a "reversed" Stein equation | I am considering the previous version of your question which contained two parts.
Part 1) We assume $Q<P$ which means that $P-Q$ is a strictly positive matrix. Of course this condition is a necessary condition for the equation to have a solution. So we assume $P, Q,P-Q$ are positive invertible matrices.
First note ... | 2 | https://mathoverflow.net/users/36688 | 300634 | 131,807 |
https://mathoverflow.net/questions/300611 | 10 | Let $M$ be an abelian monoid. For sake of simplicity we shall assume that in $M$ the cancellation law holds true. With this last assumption we define the group completion $G$ of $M$ as $$G:=M\times M/\sim$$ where $(a,b)\sim (a',b')$ if and only if $a+b'=a'+b$. It has been quite surprizing find out, reading the paper [C... | https://mathoverflow.net/users/80084 | Group completion of topological monoids | A suitable counterexample can be constructed as follows.
Let $(e\_n)\_{n\in\omega}$ be the standard orthonormal basis of the Hilbert space $\ell\_2$. For every $n\in\mathbb N$ consider the linear hull $L\_n$ of the vectors $e\_1,\dots,e\_{n}$ in $\ell\_2$. On the union $L^\infty:=\bigcup\_{n=1}^\infty L\_n$ consider... | 9 | https://mathoverflow.net/users/61536 | 300636 | 131,809 |
https://mathoverflow.net/questions/300642 | 5 | The fact that the variance of the sum of independent random variables is the sum of their variances allows one to have a good understanding of how well-concentrated each term $X\_i$ in a sum of $n$ independent random variables $S= X\_1+X\_2+\dots+X\_n$ has to be in order for $S$ to be concentraded around the mean. It s... | https://mathoverflow.net/users/14988 | Variance modulo 1 | On the one hand, the proof is very cheap. Let $Z\_j=e^{2\pi iX\_j}$. $X=\sum\_j X\_j$, $Z=e^{2\pi i X}$. Note that $\operatorname{Var}\_{\mathbb R/\mathbb Z}X\approx 1-|EZ|$ and similarly for $X\_j$ and $Z\_j$. Now just use the identity $EZ=\prod\_j EZ\_j$ to conclude.
On the other hand, finding the reference may be ... | 6 | https://mathoverflow.net/users/1131 | 300648 | 131,812 |
https://mathoverflow.net/questions/300639 | 0 | I am working on a problem that I need a simple method of identifying whether a positive integer $n$ can be expressed as
\begin{equation\*}
n=4p^2q^2-(p+q)^2
\end{equation\*}
or
\begin{equation\*}
n=4p^2q^2-(p-q)^2
\end{equation\*}
in which $p,q$ are positive integers. Do integers of these forms have any special charact... | https://mathoverflow.net/users/115637 | Characters of integers of $4p^2q^2-(p\pm q)^2$ form | It can be seen that both forms can be factored as differences of squares.
Correspondingly, $n$ can be expressed in these forms iff there exists a divisor $d\mid n$ such that $d$ and $\frac{n}{d}$ have the same parity, and
$$(\star)\qquad \left(\frac{n}{d}-d\right)^2 \mp 4\left(\frac{n}{d}+d\right)$$
is a square (where ... | 2 | https://mathoverflow.net/users/7076 | 300650 | 131,813 |
https://mathoverflow.net/questions/300640 | 6 | My aim is to understand all three coefficients arising in the Chudnovsky-Formula (see also Question [300385](https://mathoverflow.net/questions/300385/why-does-this-quasi-modular-function-have-integral-values)). Two of them are easily computed, but I failed with the third:
It is known that for all $\tau$ with $Im(\ta... | https://mathoverflow.net/users/124565 | How to compute Coefficients in Chudnovsky's Formula? | Let $\tau$ be any CM point. By basic theorems of complex multiplication, if you choose a suitable period
$\omega(\tau)$, $E\_4(\tau)/\omega(\tau)^4$, $E\_6(\tau)/\omega(\tau)^6$,
and $\sqrt{D}E\_2^\*(\tau)/\omega(\tau)^2$ (with $E\_2^\*(\tau)=E\_2(\tau)-3/(\pi\Im(\tau))$ and $D$ the discriminant of $\tau$) will be alge... | 8 | https://mathoverflow.net/users/81776 | 300656 | 131,814 |
https://mathoverflow.net/questions/300326 | 5 | On the nlab
<http://ncatlab.org/nlab/show/Reedy+model+structure#fibrant_and_cofibrant_objects>
it is claimed that a simplicial object in a model category, in which all monomorphisms are cofibrations, is always Reedy-cofibrant.
Does anyone know a reference for this?
In the case of simplicial sets the statement i... | https://mathoverflow.net/users/119240 | Reference request for statement on nlab: Reedy (co)fibrancy of (co)simplicial objects | The claim would be true if it were the case that for any simplicial object $X\_\bullet$ in some (cocomplete) category $C$, the maps $L\_nX\to X\_n$ from the latching objects are always monomorphisms. This is the case when $C$ is any topos for instance. It is also true when $C$ is any additive category (because of the D... | 3 | https://mathoverflow.net/users/437 | 300657 | 131,815 |
https://mathoverflow.net/questions/300661 | 4 | A function $f:\mathbb R\to\mathbb R$ is called *Świątkowski* if for any connected subset $C\subset \mathbb R$ and points $a,b\in C$ with $f(a)<f(b)$ there exists a continuity point $x\in C\setminus\{a,b\}$ of the function $f$ such that $f(a)<f(x)<f(b)$.
>
> **Problem.** Is each Świątkowski function $f:\mathbb R\to\... | https://mathoverflow.net/users/105651 | Is each Swiatkowski function with closed graph continuous? | In fact, this problem has been answered affirmatively in [this paper](https://arxiv.org/abs/1903.01937) of T.Banakh, M.Filipczak and J.Wodka, and also by MO-user Dap in his comment to [this MO question](https://mathoverflow.net/questions/299467/a-standard-name-for-a-function-satisfying-the-intermediate-value-theorem). ... | 3 | https://mathoverflow.net/users/61536 | 300662 | 131,817 |
https://mathoverflow.net/questions/300660 | 11 | Let $X$ be a reasonable topological space (I'd be happy to assume that $X$ is a smooth closed manifold) and let $f\colon M^n \rightarrow X$ be a continuous map from a smooth oriented $n$-manifold $M^n$ to $X$. Let $\phi\colon M^n \rightarrow M^n$ be an orientation-preserving diffeomorphism.
**Question**: Must it be t... | https://mathoverflow.net/users/124690 | Twisting bordism classes | The correct definition of bordism should have this built in. More precisely, two singular manifolds $(M\_0^n,f\_0)$ and $(M\_1^n,f\_1)$ in a space $X$ are oriented bordant if there exists a smooth manifold $W^{n+1}$ with a *diffeomorphism* $\varphi: M\_0\sqcup M\_1\stackrel{\simeq}{\to}\partial W$ and a map $F:W\to X$ ... | 14 | https://mathoverflow.net/users/8103 | 300664 | 131,818 |
https://mathoverflow.net/questions/300614 | 10 | **Definition.** A subset $K$ of a topological group $X$ is called *measure-continuous* if there exists a $\sigma$-additive Borel probability measure $\mu$ on $X$ such that for every compact subset $C\subset X$ the map $f:K\to [0,1]$, $f:x\mapsto \mu(C+x)$ is continuous.
**Remark 1.** Each measure-continuous set $K$ i... | https://mathoverflow.net/users/61536 | Are all compact subsets of Banach spaces small in a measure-theoretic sense? | I was informed by Vladimir Bogachev that the answer to Problem is negative at least for the Hilbert space $\ell\_2$ as every measure-continuous compact subset of $\ell\_2$ is contained in the image of a Hilbert-Schmidt operator $T:\ell\_2\to\ell\_2$ (for which there exists an orthonormal basis $(e\_n)\_{n\in\omega}$ in... | 6 | https://mathoverflow.net/users/61536 | 300665 | 131,819 |
https://mathoverflow.net/questions/300641 | 1 | We denote for an integer $n>1$ its square-free kernel as $$\operatorname{rad}(n)=\prod\_{\substack{p\mid n\\p\text{ prime}}}p,$$
with the definition $\operatorname{rad}(1)=1$. You can see this definition and the properties of this arithmetic function for example from this [Wikipedia](https://en.wikipedia.org/wiki/Radic... | https://mathoverflow.net/users/nan | Compare $\operatorname{rad}(an+b)$ and $\varphi(cn+d)$ in a simple and interesting inequality, for some choice of integers $a,b,c$ and $d$ | We can prove that for any real number $M>1$, the inequalities $M\mathop{\rm rad}(an+b)<\phi(cn+d)$ and $\mathop{\rm rad}(an+b)>M\phi(cn+d)$ are each satisfied for infinitely many integers $n$.
First, note that we may assume that both $\gcd(a,b)=1$ and $\gcd(c,d)=1$: dividing $an+b$ by $\gcd(a,b)$ changes the $\mathop... | 1 | https://mathoverflow.net/users/5091 | 300670 | 131,820 |
https://mathoverflow.net/questions/300653 | 1 | Let $F$ be a infinite-dimensional complex Hilbert space, with inner product $\langle\cdot\;| \;\cdot\rangle$, the norm $\|\cdot\|$, the 1-sphere $S(0,1)=\{x\in F;\;\|x\|=1\}$ and let $\mathcal{B}(F)$ be the algebra of all bounded linear operators on $F$.
>
> Let $M\in \mathcal{B}(F)$ be a bounded operator. Suppose
... | https://mathoverflow.net/users/116483 | $S_M$ is not always homeomorphic to the 1-sphere of $F$ | Taras Banakh's answer to your original question essentially answers this one too. Take $F=l^2$ and take $M$ to be the projection on the first to coordinates. Then $S\_M(0,1)=\{(a\_1,a\_2,a\_3,...)\in l^2,|a\_1|^2+|a\_2|^2=1\}$, which is homeomorphic to $S^1\times l^2$, where $S^1$ - the usual circle.
Then the unit s... | 1 | https://mathoverflow.net/users/53155 | 300688 | 131,827 |
https://mathoverflow.net/questions/300528 | 3 | Let $\Lambda$ be a finite-dimensional self-injective algebra (over an algebraically closed field, if necessary). Let $Pic(\Lambda)$ be the group of natural isomorphism classes of self-equivalences $mod(\Lambda)\rightarrow mod(\Lambda)$ of the category $mod(\Lambda)$ of finite-dimensional right $\Lambda$-modules. Simila... | https://mathoverflow.net/users/12166 | The kernel of the morphism from the Picard group to the stable Picard group of a self-injective algebra | Let $\Lambda=k[x]/(x^2)$, which has finite representation type, and take the self-equivalence of the module category induced by the algebra automorphism $x\mapsto\lambda x$ for some $\lambda\in k\setminus\{0,1\}$.
This is non-trivial, since the automorphism is not inner. But the self-equivalence of the stable module ... | 3 | https://mathoverflow.net/users/22989 | 300703 | 131,830 |
https://mathoverflow.net/questions/300687 | 4 | Let $\mathcal{F}$ be a locally free sheaf of rank $n$ on an $n$ dimensional complex manifold $X$. If the zero locus of a generic global section of $\mathcal{F}$ is $0$ dimensional, then its cycle class is equal to $c\_n(\mathcal{F})$. What can be said if the generic section has a zero locus of positive dimension?
| https://mathoverflow.net/users/64302 | Cycle class of zeroes of a global section | Let $V = H^0(X,\mathcal{F})$ be the space of global sections of $\mathcal{F}$ and let
$$
V \otimes \mathcal{O}\_X \to \mathcal{F}
$$
be the evaluation morphism. If it is surjective (i.e., $\mathcal{F}$ is globally generated), then the zero locus of a general section is zero-dimensional, and has class $c\_n(\mathcal{F})... | 5 | https://mathoverflow.net/users/4428 | 300707 | 131,831 |
https://mathoverflow.net/questions/300519 | 2 | All our rings are commutative with unity.
For an $R$-module $M$ and a submodule $N$ of $M$ and ideal $I$ of $R$, let $(N:I):=\{m\in M : Im \subseteq N\}$. Let $\mu (M)$ denote the least cardinality among the generating sets of $M$.
Now let $\alpha$ be an infinite cardinal. Let $M$ be a faithful $R$-module such tha... | https://mathoverflow.net/users/nan | On minimal generating sets of certain submodules | Are you missing some conditions? I think the following is a counterexample with $M=R$.
Let $k$ be a field and $R=k[x\_i\mid i\in I]$, with $|I|=\alpha$, a polynomial ring in $\alpha$ many variables.
Take $M=R$, $N=\langle x\_i\mid i\in I\rangle$, $m=1$, and $r$ any element of $N$.
Then
* $\mu(M)=1$,
* $\mu(N)... | 3 | https://mathoverflow.net/users/22989 | 300711 | 131,834 |
https://mathoverflow.net/questions/300715 | 6 | If $(X,\tau)$ has more than $1$ point and is $T\_2$ and connected, do we necessarily have $|X| =|\tau|$?
| https://mathoverflow.net/users/8628 | If $(X,\tau)$ has more than $1$ point and is $T_2$ and connected, do we have $|X| =|\tau|$? | Consider the topology on $\mathbb{R}^2$ generated by subsets that are open in some line from the origin. This topological space is connected, has the cardinality of continuum, and has $2^c$ open subsets.
| 11 | https://mathoverflow.net/users/6101 | 300725 | 131,838 |
https://mathoverflow.net/questions/300120 | 6 | It is a well known fact that if $(\Omega, \mathcal{F}, P)$ is a probability triple and $\{A\_i : i < k\}$ is a finite collection subsets of $\Omega$, then there is a $P' \supset P$ and $\mathcal{F'} \supset (\mathcal{F} \cup \{A\_i : i < k\})$ such that $(\Omega, \mathcal{F'}, P')$ is a probability triple. In this case... | https://mathoverflow.net/users/29231 | On the failure of extending a probability measure on uncountable $\Omega$ | Claim: Let $\mathcal{F}$ consist of all countable and co-countable subsets of $\omega\_1$ and $m: \mathcal{F} \to \{0, 1\}$ be defined by $m(X) = 0$ iff $X$ is countable. There is a countable family $\mathcal{A}$ of subsets of $\omega\_1$ such that there is no probability measure defined on the sigma-algebra generated ... | 3 | https://mathoverflow.net/users/2689 | 300747 | 131,848 |
https://mathoverflow.net/questions/300743 | 2 | Harvey Friedman is well known for investigating concrete mathematical statements that requires strong assumptions, i.e. those that can only be interpreted in a strong extension of $\text{ZF(C)}$. My first question:
1. Is there any known example of a concrete mathematical statement that requires the strength of an ext... | https://mathoverflow.net/users/95347 | Concrete mathematical statements in relation to Choice versus Reinhardt cardinals? | You're leaving yourself too much wiggle room for "concrete mathematical statement".
Is this going to be something like TREE-related statements, or about Laver tables? Is this going to be something like AD? Is this going to be something like "There exists a topological space such that something"?
Yes, there is an is... | 1 | https://mathoverflow.net/users/7206 | 300748 | 131,849 |
https://mathoverflow.net/questions/300764 | 5 | A recent issue of *American Math. Monthly* has a paper that partitions
$\mathbb{R}$ into an arbitrary finite number of uncountable sets such
that every real number is a condensation point of all the sets in the
partition. In fact, we can find uncountably many uncountable
subsets $S\_\alpha$ of $\mathbb{R}$ that are (1)... | https://mathoverflow.net/users/2807 | Nice partition of $\mathbb{R}$ into uncountably many uncountable sets | I think the following works:
Fix an uncountable subgroup $G$ of $(\mathbb{R}, +)$ - note that any such is dense in $\mathbb{R}$, and in fact intersects each nonempty open interval uncountably often - of uncountable index, and consider the set of cosets of $G$.
(Such a group can be produced by transfinite recursion ... | 8 | https://mathoverflow.net/users/8133 | 300765 | 131,855 |
https://mathoverflow.net/questions/300746 | 2 | One answer to this [Lemma on infinitely generated projective modules](https://mathoverflow.net/questions/72788/lemma-on-infinitely-generated-projective-modules) shows that every finitely generated module of a non-countably generated projective module is contained in a countably generated direct summand. Now I would lik... | https://mathoverflow.net/users/nan | When can every countably generated submodule of a a non-countably generated projective module be contained in a countably generated direct summand ? | This is an elaboration of Ralph's answer to the question linked by the OP. I use the notation from there. Moreover, I assume the axiom of countable choice, i.e. countable unions of countable sets are countable.
Let $P$ be a non-countably generated projective $R$-module with projective base $(x\_i, f\_i)\_{i\in I}$. ... | 1 | https://mathoverflow.net/users/18571 | 300766 | 131,856 |
https://mathoverflow.net/questions/300756 | 11 | *Below, I've focused on PA when lots of other theories would do. If replacing PA with a different theory leads to a more answerable question, feel free to do so.*
---
The *standard system* of a nonstandard model $M$ of PA is the set of sets of natural numbers coded by elements of $M$: $$SS(M)=\{X\subseteq\omega: ... | https://mathoverflow.net/users/8133 | Are all generalized Scott sets realized as generalized standard systems? |
>
> The question has a positive answer, not only when $M$ is a model of $PA$, but even when $M$ is a model of the fragment $I\Sigma\_1$ of $PA$.
>
>
>
The positive answer alluded to above follows from [Tanaka's self-embedding theorem](https://www.sciencedirect.com/science/article/pii/S0168007295000585), which s... | 10 | https://mathoverflow.net/users/9269 | 300771 | 131,858 |
https://mathoverflow.net/questions/300385 | 21 | It is a well-known result that the modular function $1728J(\tau) := \frac{1728E\_4(\tau)^3}{E\_4(\tau)^3-E\_6(\tau)^2}$ has integral values if $\tau$ has class number 1 - for example at $\tau\_{163}:=\frac{1+i\cdot\sqrt{163}}{2}$ you get $1728J(\tau\_{163})=-640320^3$.
Now define the quasi-modular function $s\_2(\ta... | https://mathoverflow.net/users/124565 | Why does this quasi-modular function have integral values? | Algebraicity is proven in appendix one of [1]. The function considered there is
$$ \psi(\tau) = \frac{3E\_4(\tau)}{2E\_6(\tau)} (E\_2(\tau) - \frac{3}{\pi \rm{Im} \tau}) = \frac{3}{2} s\_2(\tau).$$
The proof is by establishing, for quadratic irrationals $\tau$, the identity
$$ \psi(\tau) = 9j(\tau)\gamma + \frac{3(... | 6 | https://mathoverflow.net/users/2604 | 300776 | 131,862 |
https://mathoverflow.net/questions/300729 | 4 | Let $X$ be a topological vector space. Assume that there exists a sequence $\phi\_n:X\to X$ of finite range measurable functions with $\lim\phi\_n(x)=x$ for every $x\in X$. Can we concluded there exists a sequence $\{X\_n\}$ of ***subsets*** of $X$ with $X=\cup X\_n$ such that $X\_n$'s are all relatively second-countab... | https://mathoverflow.net/users/84390 | Approximation of the identity by finite range functions in topological vector spaces | It seems that the space $X:=C\_p(2^\omega)$ of real-valued continuous functions on the Cantor set is a counterexample to this question. The space $C\_p(2^\omega)$ is endowed with the topology of pointwise convergence.
**Claim 1.** *The space $X$ cannot be written as the countable union $X=\bigcup\_{n\in\omega}X\_n$ o... | 2 | https://mathoverflow.net/users/61536 | 300784 | 131,867 |
https://mathoverflow.net/questions/300788 | 8 | Suppose $D$ is an $\infty$-category, then we have the equivalence
$$ \text{Fib} (D) \substack{ \text{St} \\ \longrightarrow \\ \cong \\ \longleftarrow \\ \text{Un}} [ D^\text{op}, \mathbf{Kan}]$$
between (right) fibrations over $D$ and functors from the opposite of $D$ to Kan complexes ($\infty$-groupoids) via the Grot... | https://mathoverflow.net/users/119240 | Compatibility of Grothendieck construction with pullback | Yes, though it is usually written as the commutativity of *un*straightening with pullback (on the $\infty$-categorical level it doesn't matter, since straightening and unstraightening are inverse equivalences). This compatibility even holds on the point-set level if one uses a suitable model categorical presentation of... | 6 | https://mathoverflow.net/users/51164 | 300790 | 131,868 |
https://mathoverflow.net/questions/300770 | 3 | Let $X$ be a projective (not necessarily smooth) normal variety of general type over $\mathbb{C}$. Let $A$ be an abelian variety and let $A\to X$ be a surjective morphism.
>
>
> >
> > Is $X$ zero-dimensional?
> >
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I have the feeling that one can pull-back differential forms on a resolution $... | https://mathoverflow.net/users/124723 | Can an abelian variety dominate a variety of general type? | Theorem: Let $A$ be an abelian variety and $f:A\to X$ a dominant morphism to a projective variety of general type, then $\dim X=0$.
Proof: Replacing $f$ by a birational model of the Stein factorization we may assume that $f':A'\to X'$ is a projective morphism of smooth varieties with $f'\_\*\mathcal O \_{A'}=\mathca... | 9 | https://mathoverflow.net/users/19369 | 300791 | 131,869 |
https://mathoverflow.net/questions/292135 | 3 | Given a connected Artin algebra $A$ (a quiver algebra $A=kQ/I$ if it helps) with radical square zero. Can the basic strong cotilting right $A$-module $T$ be explicitly written down?
A cotilting module T over an algebra A is said to be strong in case $\widehat{\mathrm{add}(T)}$ coincides with the subcategory of module... | https://mathoverflow.net/users/61949 | Strong cotilting module for radical square zero algebras | I discuss the dual problem of how to compute the strong tilting module $T$.
Following [Auslander-Reiten (AR)](https://www.sciencedirect.com/science/article/pii/0001870891900378) let me denote by $\mathcal{P} = \mathcal{P}^\infty(A)$ the full subcategory of $\mathrm{mod}(A)$ consisting of modules with finite projectiv... | 2 | https://mathoverflow.net/users/124740 | 300797 | 131,871 |
https://mathoverflow.net/questions/300768 | 25 | Let $X$ be a projective variety (so, with some (**edit:** fixed nondegenerate closed) embedding) with the following curious property: for **every** hyperplane section $H$, we have that $X-H \cong \mathbb{A}^n$. Then is $X$ necessarily isomorphic to $\mathbb{P}^n$?
Assuming we are over the complex numbers for simplici... | https://mathoverflow.net/users/113061 | A characterisation of $\mathbb{P}^n$ | The hypotheses above are very strong, and they impose strong hypotheses on the cohomology of the complement $U$ of the universal hyperplane section. Using Leray spectral sequences for both projections, this quickly gives the result. Denote the dimension of $X$ by $n$, and denote by $m$ the dimension of the ambient proj... | 23 | https://mathoverflow.net/users/13265 | 300798 | 131,872 |
https://mathoverflow.net/questions/300818 | 5 | If $(X,\tau)$ is a topological space, we say $Y\subseteq X$ is *homeomorphism-fixing* if the only homeomorphism $\varphi:X\to X$ such that for all $y\in Y$ we have $\varphi(y)=y$ is the identity map. Moreover we say that $Y$ is *minimally homeomorphism-fixing* if for all $y\in Y$ the set $Y\setminus \{y\}$ is not homem... | https://mathoverflow.net/users/8628 | Homeomorphism-fixing subsets | The real numbers.
Any dense subset, like the rationals is homeomorphism fixing. Conversely if that subset $Y$ is not dense, we can find a point $x\in X$ and a small interval around $x$ that does not contain any element of $Y$. Now it is easy to construct a homeomorphism that is the identity outside of that interval (... | 11 | https://mathoverflow.net/users/3969 | 300819 | 131,879 |
https://mathoverflow.net/questions/300803 | 5 | I am trying to understand some part of J. Greenlees's "Four approaches to cohomology theories with reality": <https://arxiv.org/abs/1705.09365>
I have a problem with understanding $RO(Q)$-graded homotopy fixed point spectral sequence. Namely:
1. In section 2.C, proof of Lemma 2.1 - how filtration on $EQ\_+$ helps us ... | https://mathoverflow.net/users/123432 | $RO(Q)$-graded homotopy fixed point spectral sequence | For a based $G$-space or $G$-spectrum $X$, the homotopy fixed point object $X^{hG}$ is by definition $F\_G(EG\_+,X)$. Suppose we write $EG$ as the colimit of a sequence of $G$-subspaces $A\_k$. This then gives a tower of objects $F\_G((A\_k)\_+,X)$ whose inverse limit is $X^{hG}$, and the fibres of the maps in the towe... | 6 | https://mathoverflow.net/users/10366 | 300828 | 131,880 |
https://mathoverflow.net/questions/300817 | 1 | Let $A$ and $b$ be an $M\times M$ matrix and an $M\times 1$ vector, respectively.
I need to solve
$$\int\_{\|x\|^2=1} \exp\left(-x^\ast Ax + 2\mathcal{R}\{x^\ast b\}\right)
\, dx$$
where $(\cdot )^\ast$ is conjugate transpose of a vector and $\mathcal{R}\{ \cdot\}$ takes the real part.
Is this doable. Seems relate... | https://mathoverflow.net/users/124751 | integrate exponential function of quadratic form over unit norm vectors | I doubt that there is a closed-form expression for any $M$, but for large $M$ you can approximate the integral by replacing the constraint $\|x\|=1$ by a Gaussian measure
uncorrelated $x\_n$'s with mean and variance $E(x\_n)=0$, $E(|x\_n|^2)=1/M$
I assume that $A$ is positive definite, with eigenvalues $a\_n>0$, ... | 2 | https://mathoverflow.net/users/11260 | 300830 | 131,881 |
https://mathoverflow.net/questions/300800 | 8 | Let $M$ be an $n$ dimensional smooth manifold and let $j: M \to \mathbb{R}^{m}$ be an embedding. Associated to this embedding we can form the "collapse map" which is a pointed map from a sphere to the Thom space of the normal bundle $S^{m}=(\mathbb{R}^m)^{+} \to Th(N\_j)$ (which depends on the choice of tubular neighbo... | https://mathoverflow.net/users/22810 | A map of spaces implementing the Pontryagin Thom collapse map? (collapse maps in families) | The right framework of definitions is as follows.
1. You have a space $E=\text{Emb}(M,\mathbb{R}^n)$ of smooth embeddings, topologised in a way that respects all derivatives. In more detail, we give $C^\infty(M)$ the smallest topology such that the inclusion in $C(M)$ is continuous, as is the map $C^\infty(M)\to C^\i... | 10 | https://mathoverflow.net/users/10366 | 300833 | 131,882 |
https://mathoverflow.net/questions/300705 | 7 | Consider a sequence $a\_i$ defined by
$$
\begin{align\*}
a\_1&=p,\\
a\_2&=q,\\
a\_i&=a\_{i-1} \oplus a\_{i-2}+1,
\end{align\*}$$
where $\oplus$ is the bitwise xor operation. How can we give an upper bound for $a\_n$ as a function of $p,q,n$?
There are lots of $p,q$ which satisfy $\mathop {\lim }\limits\_{n \to \inft... | https://mathoverflow.net/users/120302 | A Bitwise Xor Problem | $\def\U#1{\underline{#1}}\def\O#1{\overline{#1}}$The bound $a\_n=O(n)$ is true, in fact, we have $a\_n\le\max\{8p,8q,\frac{16}3n\}$. The argument is quite elementary, but a bit tedious to write down properly, hence I will only sketch it, and rely on the reader to fill in the details.
Rather than estimating $a\_n$ dir... | 6 | https://mathoverflow.net/users/12705 | 300836 | 131,884 |
https://mathoverflow.net/questions/300785 | 6 | I have been struggling with this equation for some time and I do not seem to find any conclusive answer (it's from my research, not a homework).
It has to do with the **real** solutions $x$ to the following equation
$$ x + x f(x) = 1 + f(1),$$
where
$$ f(x) = 2\sum\_{n=1}^\infty \mathrm{e}^{(-ax^2-b) n^2} $$
with $... | https://mathoverflow.net/users/124732 | Solution of an equation with Jacobi theta function | Let $\, g(x) := \theta\_3(0,\mathrm{e}^{-ax^2 -b}).\,$ Your question about solutions to $\, x + x f(x) = 1 + f(1) \,$ is now about $\, x g(x) = g(1).\,$ Now $\,g(x)\,$ is a bell shaped curve with $\, g(x) > 0 \,$ and $\, g(-x) = g(x).\,$ If we can prove that $\,xg(x)\,$ is monotone increasing we are done.
If it holds f... | 6 | https://mathoverflow.net/users/113409 | 300843 | 131,887 |
https://mathoverflow.net/questions/300786 | 3 | Let $k$ be a field of characteristic zero and $A$ be a $k$-algebra. A derivation on $A$ is a $k$-linear map $D: A \to A$ such that $D(ab)=aD(b)+bD(a), \forall a,b \in A$. A derivation is called locally nilpotent if for every $a\in A$, $\exists n\_a\in \mathbb N$ such that $D^{n\_a} (a)=0$, where $D^n$ means $D$ compose... | https://mathoverflow.net/users/nan | Locally nilpotent derivation on $A[X,Y]$ whose kernel is $A$; where $A$ is an affine $k$ domain, $char k=0$ | I will write it as an answer for your last comment. Assume $A$ is a field and $ker D=A$ as in the question. Then $D\neq 0$ and so $D(f)\neq 0$ for some $f\in R=A[X,Y]$. Since $D\in LND$, $D^k(f)\neq 0, D^{k+1}(f)=0$ for some $k\geq 1$. Let $s=D^{k-1}(f)$. Then $D(s)\neq 0, D^2(s)=0$, so if $ker D=A$, $D(s)\in A$ and no... | 2 | https://mathoverflow.net/users/9502 | 300844 | 131,888 |
https://mathoverflow.net/questions/300561 | 5 | I'm currently reading the book "Galois theory of $p$-extensions" by Helmut Koch.
There, we calculate the cohomological dimension of the galois group $G(K/k)$ where $K$ is the maximal (normal) $p$-extension of $k$.
(Here $p$ is a prime and $k$ is a local field or global field of finite type, i.e finite extension of ... | https://mathoverflow.net/users/123226 | A question on the injectivity of a canonical map between galois cohomology groups | I believe this map is always injective. Here is a quick argument: first note that $K'$ is normal over $k$ (because it is invariant under any automorphism of the algebraic closure of $k$ which preserves $k'$, and $k'$ is normal over $k$). This means that we can view the map $G(K'/k') \to G(K/k)$ as a composition of two ... | 2 | https://mathoverflow.net/users/51164 | 300850 | 131,892 |
https://mathoverflow.net/questions/299021 | 3 | Given a finite dimensional algebra $A$ with finite global dimension such that there are only finitely many basic tilting modules. Then every selforthogonal indecomposable module $M$ (that is a module with $\mathrm{Ext}\_A^i(M,M)=0$ for all $i \geq 1$) is a direct summand of a tilting module and thus there are only fini... | https://mathoverflow.net/users/61949 | Finding all selforthogonal indecomposable modules | Concerning your second question, it is claimed in the proof of Corollary 4.8 in a recent [preprint](https://arxiv.org/abs/1801.04738v2) by Iyama-Zhang that Kajita showed in his Master's
thesis that the Auslander algebra of the linearly oriented path with at
least 6 vertices has infinitely many classical tilting modules... | 2 | https://mathoverflow.net/users/124740 | 300858 | 131,895 |
https://mathoverflow.net/questions/88598 | 9 | While the Poincaré-Birkhoff-Witt theorem is usually proven (and sometimes even formulated) for free modules only, it is [known](https://mathoverflow.net/questions/61954) (see also [here](https://mathoverflow.net/questions/87402)) that it holds for arbitrary modules if the ground ring is a $\mathbb Q$-algebra. I am wond... | https://mathoverflow.net/users/2530 | $U\left(\mathfrak a\right) \otimes_{U\left(\mathfrak a\cap\mathfrak b\right)} U\left(\mathfrak b\right) \cong U\left(\mathfrak a + \mathfrak b\right)$ over a ring containing $\mathbb{Q}$ | For some reason, I had forgotten about this question even as I found the answer long ago.
Yes, the fact holds whenever the ground ring is a $\mathbb{Q}$-algebra. For the proof, see the First proof of Proposition 2.4.1 in my notes for [Pavel Etingof, 18.747 *Infinite-dimensional Lie Algebras*, Spring term 2012 at MIT]... | 7 | https://mathoverflow.net/users/2530 | 300874 | 131,897 |
https://mathoverflow.net/questions/300885 | 13 | Suppose that $X$ is a metric space. Is the family of all real-valued uniformly continuous functions on $X$ dense in the space of all continuous functions with respect to the topology of uniform convergence on compact sets?
| https://mathoverflow.net/users/124775 | Are uniformly continuous functions dense in all continuous functions? | Yes, and even more is true. The argument is as follows: let $f\colon X \to \mathbb R$ be a continuous function and let $K\subset X$ be a compact set. Then $f|\_K$ is uniformly continuous; let $\omega$ be its nondecreasing subadditive modulus of continuity. By [McShane-Whitney's extension formula](https://en.wikipedia.o... | 24 | https://mathoverflow.net/users/54609 | 300887 | 131,901 |
https://mathoverflow.net/questions/300892 | 2 | Given two square matrices $A$ and $B$. There are quite some results on the distance between the eigenvalues, e.g.,
$$
| \lambda\_A - \lambda\_B | \leq \| A - B \|\_F,
$$
where $A$ and $B$ are Hermitian (see [here for more](http://www.netlib.org/lapack/lawnspdf/lawn84.pdf)). I am looking for similar results for the a... | https://mathoverflow.net/users/51478 | Eigenvalue Argument Perturbation | Uhm, it does not hold even in $\mathbb{R}^{1\times 1}$. $A=\varepsilon, B = -\varepsilon$ gives $LHS=\pi$, $RHS = 2\varepsilon$.
| 1 | https://mathoverflow.net/users/1898 | 300896 | 131,903 |
https://mathoverflow.net/questions/300879 | 8 | I'm interested in the existence of several example of left Bousfield localization of model categories that are not left proper (nor simplicial). I'm relatively convince that I can construct all those I need by hand, but that got me curious about what is known in general about existence of Left Bousfield localization of... | https://mathoverflow.net/users/22131 | Left Bousfield localization without properness, what is known? | I have an unpublished note that proves Barwick's claim. Aspects of this story have appeared in some papers of mine with Michael Batanin, including one we published in the proceedings of the 2015 CRM conference in Barcelona on "Interactions between representation theory, algebraic topology, and commutative algebra." I'm... | 7 | https://mathoverflow.net/users/11540 | 300906 | 131,907 |
https://mathoverflow.net/questions/300873 | 11 | Let $\ F(n)\ (\mbox{where}\ n\in\mathbb N:=\{1\ 2\ \ldots\})\ $ be the least cardinality $\ |A|\ $ of a set $\ A\subseteq\mathbb N $ such that:
1. $\ \min A=n $
2. $\ \sum\_{x\in A}\frac 1x = 1 $
**QUESTION** Is set $\ \{n\in\mathbb N:\ \frac{F(n)}n\le 2\}\ $ finite?
>
> Background:
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Roughly speakin... | https://mathoverflow.net/users/110389 | Egyptian representations of $1$ | Lots of questions along these lines were raised by Erdos and Graham and many have been solved by Croot, Greg Martin and others. In particular, [Croot](https://eudml.org/doc/279607) has shown that any rational number $r$ can be represented as a
sum of unit fractions with denominators lying in the interval $[N, (e^r+o(1... | 14 | https://mathoverflow.net/users/38624 | 300907 | 131,908 |
https://mathoverflow.net/questions/300917 | 2 | Is there a variant of Rademacher‘s Theorem where the smallness of the points of non-differentiability is measured in terms of Baire category instead of measure?
More precisely, let X be a separable Banach space and Y be a space with the RNP. Moreover, let $U\subset X$ be an open set and $f\colon U\to Y$ be a Lipschi... | https://mathoverflow.net/users/83700 | Rademacher‘s Theorem and Baire category | No, not even in one dimension. Googling "not differentiable on a residual set" yields a citation of the paper F. Mignot, Contrôle dans les inéquations variationelles elliptiques, *J. Functional Analysis* **22** (1976), 130–185.
| 5 | https://mathoverflow.net/users/23141 | 300921 | 131,911 |
https://mathoverflow.net/questions/300919 | 1 | Does anyone know any source in which I could find a recurrence relation for the coefficients of the series solution of the Confluent Heun Equation
$$y''+\left( {\gamma\over z}+{\delta\over z-1}+\varepsilon\right)y'+{\alpha z-q\over z(z-1)}y=0$$
around the regular singular point $z=0$?
| https://mathoverflow.net/users/84866 | Confluent Heun Equation | A good place to start looking would be the book "Heun's Differential Equations" by A. Ronveaux.
| 3 | https://mathoverflow.net/users/12120 | 300923 | 131,913 |
https://mathoverflow.net/questions/300890 | 2 | In case $A$ is a symmetric finite dimensional algebra and $e$ an idempotent, $eAe$ is again symmetric.
Is there an easy counterexample for the following:
>
> In case $A$ is additionally a periodic algebra, $eAe$ is also periodic?
>
>
>
Is there a criterion when $eAe$ is still periodic depending on $A$ and $... | https://mathoverflow.net/users/61949 | Example to periodic symmetric algebras | The simplest counterexample I know is when A is the mesh algebra of generalized Dynkin type $C\_3$ (see the paper "Periodic algebras" by Erdmann and Skowronski). Its quiver is the same as the preprojective algebra of Dynkin type $D\_4$, but the relations are slightly different. It is symmetric and periodic of period $6... | 2 | https://mathoverflow.net/users/11791 | 300939 | 131,919 |
https://mathoverflow.net/questions/300925 | 2 | Let $a,d$ be polynomials of $\mathbb Z[X]$ with $\deg a>\deg d\ge0$ and $P$ be a polynomial of $\mathbb Z[X]$. Consider an infinite sequence of integers $(\lambda\_n)\_n$. Can one assert there exists a $\lambda\_n$ such that
$$(\lambda\_n +1)^2P-(\lambda\_n a+d)(a+\lambda\_n d)$$ has only simple roots?
EDIT: The pol... | https://mathoverflow.net/users/33128 | Polynomials with no multiple root | It can happen that $(\lambda+1)^2P-(\lambda a+d)(a+\lambda d)$ has multiple roots for any integer $\lambda$: namely, if $P=ad$, then
\begin{multline\*}
(\lambda+1)^2P-(\lambda a+d)(a+\lambda d) \\
= (\lambda^2+1)(P-ad) + \lambda(2P-a^2-d^2)
= -\lambda (a-d)^2,
\end{multline\*}
so that any root of $a-d$ is a multi... | 10 | https://mathoverflow.net/users/9924 | 300941 | 131,920 |
https://mathoverflow.net/questions/300940 | 3 | I have a symmetric $d \times d$ matrix $A$ and I have the following functional:
$$
\mathcal J(h) := \int\_{B\_1(0)} \vert \langle Au,u \rangle\vert \frac{\vert h'(\vert u \vert)\vert}{\vert u \vert} du,
$$
where $B\_1(0)$ is the unit ball in $\mathbb R^d$ and $h \in C\_c^\infty(\mathbb R)$ with $\text{supp}\, h\subset ... | https://mathoverflow.net/users/119793 | Infimum of an integral functional involving a symmetric matrix | $\newcommand{\al}{\alpha}
\newcommand{\de}{\delta}
\newcommand{\De}{\Delta}
\newcommand{\ep}{\varepsilon}
\newcommand{\ga}{\gamma}
\newcommand{\Ga}{\Gamma}
\newcommand{\la}{\lambda}
\newcommand{\Si}{\Sigma}
\newcommand{\thh}{\theta}
\newcommand{\om}{\omega}
\newcommand{\R}{\mathbb{R}}
\newcommand{\Z}{\mathbb{Z}}
\newco... | 3 | https://mathoverflow.net/users/36721 | 300944 | 131,921 |
https://mathoverflow.net/questions/300875 | 2 | Let $X$ be an irreducible smooth projective curve over $\mathbb{C}$. Let $G$ be a connected reductive linear algebraic group over $\mathbb{C}$. Let ${\rm M}\_{G,X}$ be the moduli space of semistable principal $G$-bundles on $X$. Are there any metrics on this moduli space?
| https://mathoverflow.net/users/124771 | Metric on moduli space of semistable principal G-bundles on curves | Let $G$ is a reductive linear algebraic group over $\mathbb{C}$, and $X$ be a connected compact Riemann surface of genus $\geq 2$.
Fix a topological type $\tau$. Then the moduli space of semistable principal $G$-bundles over $X$ of type $\tau$, denoted $\mathcal{M}\_\tau(X,G)$, is a projective variety, and also is i... | 2 | https://mathoverflow.net/users/12218 | 300953 | 131,925 |
https://mathoverflow.net/questions/300918 | 3 | We can define the [signature of a manifold](https://en.wikipedia.org/wiki/Signature_(topology)) in $4k$ dimensions.
1) If I understand correctly, the signature $\sigma$ of the manifold of the product space of spheres would always be zero:
>
> $$\sigma(S^n \times S^m \times S^p \times S^q \times \dots )=0$$
>
>
... | https://mathoverflow.net/users/27004 | Signature of the manifold of the multiple fibrations over spheres | The signature of an orientable bundle over a sphere with closed fiber vanishes.
Here is an argument via Novikov additivity. Any bundle over $S^k$ with fiber $N^{4n-k}$ is obtained by gluing two copies of $N \times D^k$ (clutching construction) along $N \times \partial D^k$. Now the signature of $N \times D^k$ is zer... | 5 | https://mathoverflow.net/users/3460 | 300963 | 131,927 |
https://mathoverflow.net/questions/300946 | 6 | Consider an algebraic manifold whose number of points is $q^n ([n+1]\_q)$. Is there a geometric relation to $A^n (P^n)$? In particular, is there an equivalence in the Grothendieck ring of varieties or could there be a birational equivalence?
If there is no such equivalence in general, might some additional reasonable... | https://mathoverflow.net/users/10446 | If number of points on a manifold is $q^n ( [n+1]_q )$ does it imply a geometric relation to $A^n (P^n)$? | The Russell Cubic $R:=V(x + x^2 y + z^2 + t^3)\subset \mathbb{A}^4$ is not isomorphic to $\mathbb{A}^3$ although over $\mathbb{C}$ they are both diffeomorphic to $\mathbb{R}^{6}$ (see this [Wikipedia](https://en.wikipedia.org/wiki/Exotic_affine_space) page).
I ran a *Mathematica* program I quickly wrote to compute th... | 6 | https://mathoverflow.net/users/12218 | 300971 | 131,930 |
https://mathoverflow.net/questions/300973 | 7 |
>
> Prove that for all positive integers $ n$ different from
> $ 3$ and $ 5$, $ n!$ is divisible by the number of its positive
> divisors.
>
>
>
I tried some things,such as the number of divisors of $ n!$ is just $ \prod\_p a\_p$ where $ a\_p=1+\sum\_{j\ge 1}\left\lfloor\frac n{p^j}\right\rfloor\le\min\left(1... | https://mathoverflow.net/users/38620 | Prove $ n!$ is divisible by the number of its positive divisors | The sequence, number of divisors of $n$-factorial, is tabulated at the [OEIS](https://oeis.org/A027423). It says there that it divides $n!$ for all $n\ge6$, giving a reference to Florian Luca and Paul Thomas Young, On the number of divisors of $n!$ and of the Fibonacci numbers, Glasnik Matematicki, Vol. 47, No. 2 (2012... | 12 | https://mathoverflow.net/users/3684 | 300979 | 131,933 |
https://mathoverflow.net/questions/300976 | 6 | This question begins with a sort of mysterious comment at the bottom of [this Wikipedia page](https://en.wikipedia.org/wiki/Injective_cogenerator#In_general_topology) on injective cogenerators. There, it is said, without citation or proof, that as a result of the [Tietze Extension Theorem](https://en.wikipedia.org/wiki... | https://mathoverflow.net/users/11546 | Cogenerator of Categories of Topological Spaces Satisfying Some Separation Axiom | Perhaps I should have been a little better at Googling before posting this question, but it seems to be answered, to a degree, in a paper from 1980 by Giuli, cited below. In particular, any epi-reflective subcategory, i.e. one that is closed under products and subspaces (hence whose inclusion is limit preserving), has ... | 4 | https://mathoverflow.net/users/11546 | 300982 | 131,935 |
https://mathoverflow.net/questions/300992 | 1 | Let $X$ and $Y$ be two Fano varieties of the same dimension embedded into a same projective space $\mathbb P^N$, assume $Pic X= \mathbb Z\mathcal O\_X(1)$ and $Pic Y=\mathbb Z\mathcal O\_Y(1)$, where $\mathcal O\_X(1)$ means the restriction of $\mathcal O\_{\mathbb P^N}(1)$ on $X$.
Then does it hold that $X$ and $Y$... | https://mathoverflow.net/users/119184 | Minimal embeddings of certain Fano varieties with Picard number one | This is definitely not true; just consider two smooth cubic threefolds in $\mathbb{P}^4$.
Of course many other examples exist, e.g. Fano hypersurfaces in projective space which are not quadrics nor cubic surfaces (here the Picard group is generated by the hyperplane class by the Lefschetz hyperplane section theorem).... | 5 | https://mathoverflow.net/users/5101 | 300994 | 131,940 |
https://mathoverflow.net/questions/300539 | 4 | Setting
-------
Let us regard the Hilbert space $L^2(0,1)$ and the $C\_0$-semigroup $(T(t))\_{t\geq 0}$ defined by
$$
T(t):\left\{
\begin{array}{rml}
L^2(0,1) & \to & L^2(0,1), \\
[f]\_{\sim} &\mapsto &\left[x \mapsto
\begin{cases}
f(x+t), & \text{if}\; x+t<1\\
0, & \text{else}
\end{cases}
\right]\_{\sim}.
\end{array... | https://mathoverflow.net/users/114751 | $L^2$-valued integral as parameter integral | The trick is to show that both functions $\Big(\int\_0^1 T(t)f \,\mathrm{d}t\Big) (x)$ and $\int\_0^1 \big(T(t)f\big)(x)\,\mathrm{d}t$ induce the same element in the dual space. Let $h \in L^2(0,1)$ be arbitrary. Since the scalar product is continuous in both arguments, we have
$$
\Big\langle h, \int\_0^1 T(t)f \,\math... | 0 | https://mathoverflow.net/users/114751 | 300996 | 131,942 |
https://mathoverflow.net/questions/300999 | 7 | Let $i: X \hookrightarrow Y$ be a closed embedding of smooth algebraic varieties. In the book *D-modules, perverse sheaves and representation theory* the authors say that there exists a locally free resolution of the $i^{-1}\mathcal{O}\_Y$-module $\mathcal{O}\_X$ called the *Koszul resolution*. Namely, they say it is
$... | https://mathoverflow.net/users/91572 | Kozsul resolution of $\mathcal{O}_X$ | It exists locally, and more generally when $X$ is the zero locus of a global section $s$ of a rank $r$ vector bundle $E$ on $Y$, where $r$ is the codimension of $X$ in $Y$. Then the resolution is given by the celebrated *Koszul complex*
$$0\rightarrow \bigwedge^rE^\*\xrightarrow{\ i(s)\ }\bigwedge^{r-1}E^\*\rightarrow ... | 10 | https://mathoverflow.net/users/40297 | 301005 | 131,944 |
https://mathoverflow.net/questions/300750 | 2 | Let $P$ be a non-finitely generated projective module over a commutative Noetherian ring. Is every finitely generated submodule of $P$ contained in some finitely generated direct summand of $P$ ? Or at least , is every finitely generated submodule of $P$ contained in some proper direct summand of $P$ ?
This question... | https://mathoverflow.net/users/nan | Finitely generated submodule of non-finitely generated projective module is contained in some proper direct summand ? | Bass showed, in Corollary 4.5 of
*Bass, H.*, Big projective modules are free, Ill. J. Math. 7, 24-31 (1963). [ZBL0115.26003](https://zbmath.org/?q=an:0115.26003),
that every projective module for a connected Noetherian commutative ring is either finitely generated or free (and hence a direct sum of finitely generat... | 4 | https://mathoverflow.net/users/22989 | 301007 | 131,946 |
https://mathoverflow.net/questions/301013 | 4 | We are given a complete (separable) metric space $X$ and a dense subset $D\subset X$. Consider a sequence of continuous functions $f\_n\colon X\to \mathbb R$ such that $$\int\limits\_D f\_n \, {\rm d}\mu\to 0$$ for every compactly supported finite signed Borel measure $\mu$ on $D$. It seems to me that it is asking for ... | https://mathoverflow.net/users/124775 | Weak convergence of measures on dense sets | Take $X:=\mathbb{R}$ and $D:=\mathbb{R}\setminus\{0\}$. Consider any sequence of continuous functions $(f\_n)\_n$ that converges uniformly to $0$ on compact sets of $D$, but with $\langle \delta\_0,f\_n\rangle:=f\_n(0)=1$, like e.g. $f\_n(x):=(1-n|x|)\_+$ .
**rmk.** Of course the same example works for any $D\subset\... | 4 | https://mathoverflow.net/users/6101 | 301018 | 131,948 |
https://mathoverflow.net/questions/301019 | 10 | This question is inspired by [this](https://mathoverflow.net/q/300946/12218) MO question; indeed it is a special case on which to focus.
An *exotic affine space* is an affine variety $V$ whose $\mathbb{C}$-points are diffeomorphic to $\mathbb{R}^{2n}$ yet $V$ is not algebraically isomorphic to $\mathbb{A}^n$.
Say t... | https://mathoverflow.net/users/12218 | Are all exotic affine spaces count equivalent to affine space? | For all but finitely many primes, yes. Any such $V$ has $V\_{\mathbb C}$ smooth and has $H^i(V\_{\mathbb C}, \mathbb Q\_\ell)=0$ for $i\neq 0$ and $=\mathbb Q\_\ell$ for $i=0$. Both these properties are known to be constructible, so they hold for $V\_{\mathbb F\_q}$ for all but finitely many $q$.
For any such $q$, th... | 11 | https://mathoverflow.net/users/18060 | 301023 | 131,950 |
https://mathoverflow.net/questions/301017 | 3 | In a message of the 29 th March 2008 edited on the FOM list "[AC and strongly inaccessible cardinals](https://cs.nyu.edu/pipermail/fom/2008-March/012783.html)", Robert Solovay shows that the so-called Tarski-Grothendieckset set theory can be equivalently axiomatized as:
(1) ZFC + "There exists a proper class of strongl... | https://mathoverflow.net/users/30395 | Tarski's axiom A, MK set theory and the Global Choice axiom | Question (1) is answered by the observation that pairing follows easily from replacement, once a two-element set exists.
For question (2), the answer is negative. I claim that from a suitable consistency assumption, it is consistent that we have the version of KM without global choice, but with AC for sets, plus a p... | 4 | https://mathoverflow.net/users/1946 | 301025 | 131,951 |
https://mathoverflow.net/questions/301026 | 2 | Let $X$ be a $\mathrm{CAT}(0)$ space, $p\in X$ and $v\in T\_pX$. Let $N\subset T\_pX$ be the set of tagent vectors making an angle greater than or equal to $\pi/2$ with $v$.
Is it true that the set $\exp(N)\subset X$ is convex (that is, for every couple of points, the minimal geodesic between them is contained in $\... | https://mathoverflow.net/users/31015 | Convexity of set of normal directions in a CAT(0)-space | This is not true even at an ordinary conical singularity with total angle greater than $2\pi$ on a surface.
| 3 | https://mathoverflow.net/users/28128 | 301029 | 131,953 |
https://mathoverflow.net/questions/300988 | 2 | This is a simple question that I direfully need an answer for. If the response is in the negative, I can work with it. If the response is in the positive, I can also work with it. I just can't seem to find an answer, and I need to direct my proof in one direction or the other.
Consider the exponential functions $\alp... | https://mathoverflow.net/users/nan | Are the immediate basin of these exponential maps simply connected? | Yes. All periodic components of the set of normality of any transcendental entire function are simply connected. This is a theorem of Baker,
The domains of normality of an entire function.
Ann. Acad. Sci. Fenn. Ser. A I Math. 1 (1975), no. 2, 277–283.
| 1 | https://mathoverflow.net/users/25510 | 301037 | 131,956 |
https://mathoverflow.net/questions/300820 | 2 | Let $\mathbb{H} \subset \mathbb{C}$ be the upper half plane. First recall the following statement: if $f^\* \colon \mathbb{H} \rightarrow \mathbb{H}$ is quasi-conformal (qc), then there exists an extension $\overline{f^\*} \colon \overline{\mathbb{H}} \rightarrow \overline{\mathbb{H}}$ of $f^\*$. Note that this is an e... | https://mathoverflow.net/users/117619 | Equality on $\partial \mathbb{H}$ of lifts for isotopy to a conformal map | I found a proof and post it here for completeness/further reference.
Suppose $f \circ g^{-1}$ is isotopic to a conformal map $h \colon \mathbb{H} / \Gamma\_g \rightarrow \mathbb{H} / \Gamma\_f$. Abbreviate $g\_0 = f \circ g^{-1}$. Let $[g\_0]\_\*$ and $[h]\_\*$ denote the induced maps $\Gamma\_g \rightarrow \Gamma\_f... | 0 | https://mathoverflow.net/users/117619 | 301038 | 131,957 |
https://mathoverflow.net/questions/301041 | 5 | Suppose we have a product space $(X\_1\times X\_2,\mu\_1\otimes\mu\_2)$, with finite measures $\mu\_1,\mu\_2$ and $p>1$.
Is there a possibility that an inequality of this form holds on the product space?
$$\|f\|\_{L^pL^p}\leq C\_1\|f\|\_{L^1L^p} + C\_2\|f\|\_{L^pL^1},$$
where $\|f\|\_{L^pL^q}=\big(\int\_X\big(\int\_Y |... | https://mathoverflow.net/users/89806 | Mixed norm inequality | The answer is no. E.g., suppose that $X\_1=X\_2=[0,1]$, $\mu\_1=\mu\_2=$ Lebesgue measure, $f(x\_1,x\_2)=g(x\_1)g(x\_2)$, $g=1\_{[0,u]}$, $u\in(0,1)$. Then your proposed inequality becomes
$$u^{2/p}\le(C\_1+C\_2)u^{1+1/p},$$
which fails to hold for any given real $p>1$, $C\_1$, $C\_2$ if $u$ is small enough.
| 8 | https://mathoverflow.net/users/36721 | 301047 | 131,960 |
https://mathoverflow.net/questions/301050 | 8 | Hard as I tried, I couldn't find a proof of Remark 2.2.2.11 in Higher Topos Theory, or prove it myself. It seems to need an explicit formulation for the unstraightening functor, so my question is: is an explicit expression known for the unstraightening? Anyway, is it possible to obtain Remark 2.2.2.11 without having on... | https://mathoverflow.net/users/124841 | Explicit expression of the unstraightening functor | Wow, I have always thought that unstraightening has to be easier than straightening, but I've never actually looked at Lurie's treatment before, so I'm surprised to realize he defines straightening directly while defining unstraightening as its right adjoint.
Anyway, I'm pretty sure that Remark 2.2.2.11 follows from ... | 12 | https://mathoverflow.net/users/2362 | 301053 | 131,961 |
https://mathoverflow.net/questions/300938 | 7 | Consider a $d$-dimensional convex rational polytope $P\subset\mathbb{Q}^d\subset\mathbb{R}^d$. Then, it's a standard fact that in general the function counting the number of lattice points inside the multiples $t\cdot P$ of $P$ is a quasi-polynomial instead of a polynomial, as in the integral case. This is the so-calle... | https://mathoverflow.net/users/103164 | How different can the constituents of an Ehrhart quasi-polynomial be? | Let $d=\dim(P)$. First, since $L(t,P)$ is non-decreasing in $t$, for any positive integer $n$ we have
$$f\_i((n-1)D+i) \leq f\_j((n-1)D+j) \leq f\_i(nD+i) \leq f\_j(nD+j)$$
whenever $i \leq j$. Thus it is easy to see that the coefficients of $t^d$ in $f\_i,f\_j$ must be the same ([in fact](https://en.wikipedia.org/wik... | 5 | https://mathoverflow.net/users/33089 | 301056 | 131,962 |
https://mathoverflow.net/questions/301012 | 3 | $
\newcommand{dist}{\operatorname{dist}}
\newcommand{B}{\mathbb{B}}
$
Let $\mathcal {M}$ be a Riemannian manifold, $p \in S \subset \mathcal{M}$ and $r>0$. Denote $S\_{r} := S \cap\B(p,r)$.
**Question 1:** For small values of $r$, Is there a relation similar to the following
$$
\dist(\exp\_p^{-1}(u);\exp\_p^{-1}(S\... | https://mathoverflow.net/users/53059 | Relation between a distance function and normal coordinations | Lemma 3.24 at page 87 of Alexander Grigor'yan's "Heat Kernel and Analysis on Manifolds" says that (I reformulate it slightly)
>
> For any point $p \in M$ and chart $(U', h)$ around $p$ there exist a $U \subseteq U'$ and a constant $C \ge 1$ such that for all $x,y \in U$ we have
>
>
> $$\frac 1 C \| h(x) - h(y) \|... | 2 | https://mathoverflow.net/users/54780 | 301064 | 131,968 |
https://mathoverflow.net/questions/300991 | 6 | If $(X,\tau)$ is a $T\_2$-space such that [all non-empty open sets are homeomorphic](https://mathoverflow.net/questions/300253/t-2-spaces-where-all-non-empty-open-sets-are-homeomorphic) (with the subspace topology) to $X$, is $(X,\tau)$ necessarily [homogeneous](https://en.wikipedia.org/wiki/Homogeneous_space)?
| https://mathoverflow.net/users/8628 | Homeomorphic open sets and homogeneity | For an infinite Hausdorff space the *diversity* of a space is the number of homeomorphism types of non-empty open sets, so if all non-empty open sets are homeomorphic, the space is said to be of diversity one.
According to [this paper](https://ac.els-cdn.com/0166864195000739/1-s2.0-0166864195000739-main.pdf?_tid=d186... | 7 | https://mathoverflow.net/users/2060 | 301065 | 131,969 |
https://mathoverflow.net/questions/301048 | 3 | Let $X$ a binomial variable of parameter $(N,p)$, with $0<p<0.5$
I would like to lower bound $\mathbb{P}\left(X <Np \right)$ by a constant ($\frac{1}{5}$ seems true and is enough for me).
Thank you by advance
| https://mathoverflow.net/users/90197 | Lower bound for the probability of binomial variable to be less than her expectation | $\newcommand{\al}{\alpha}
\newcommand{\de}{\delta}
\newcommand{\De}{\Delta}
\newcommand{\ep}{\varepsilon}
\newcommand{\ga}{\gamma}
\newcommand{\Ga}{\Gamma}
\newcommand{\la}{\lambda}
\newcommand{\si}{\sigma}
\newcommand{\Si}{\Sigma}
\newcommand{\thh}{\theta}
\newcommand{\om}{\omega}
\newcommand{\R}{\mathbb{R}}
\newcomm... | 3 | https://mathoverflow.net/users/36721 | 301070 | 131,970 |
https://mathoverflow.net/questions/301014 | 10 | I asked [this question](https://math.stackexchange.com/q/2783558/660) on Mathematics Stackexchange, but got no answer.
Is the product
$$
\prod\_{i\in I}A\_i
$$
of a family $(A\_i)\_{i\in I}$ of [Jacobson rings](https://en.wikipedia.org/wiki/Jacobson_ring)
a Jacobson ring?
(Here "ring" means "commutative ring wit... | https://mathoverflow.net/users/461 | Is the product of Jacobson rings a Jacobson ring? | The answer is **no** in general.
Take $R = \prod\_{n \in \mathbb{N}\_{> 0}} \mathbb{Z}/2^n\mathbb{Z}$. Then the Jacobson radical of $R$ is $\prod\_{n \in \mathbb{N}\_{> 0}} 2\mathbb{Z}/2^n\mathbb{Z}$, and it contains a non-nilpotent element, namely $(2 + 2^n \mathbb{Z})\_n$. Therefore the Jacobson radical of $R$ does... | 12 | https://mathoverflow.net/users/84349 | 301073 | 131,972 |
https://mathoverflow.net/questions/301049 | 6 | **Question:** Given a quadratic irrational $x = a + b\sqrt{D}$ ($a,b \in \Bbb{Q}$, $D \in \Bbb{N}\_{> 0}$ square-free) and its Galois conjugate $x' = a - b\sqrt{D}$, is it true that the continued fraction expansions of $x$ and $x'$ have the same period?
Computations of a few random example seems to suggest that is in... | https://mathoverflow.net/users/3824 | Periods of the continued fraction expansions of Galois-conjugate quadratic-irrationals | **No.** For example:
$-\frac{2}{3} + \frac{5}{7}\sqrt{6} = [1; \overline{12, 18, 1, 32, 1, 1, 2, 171, 15, 3, 1, 1, 1, 18, 2, 2, 2, 3}]$
but
$-\frac{2}{3} - \frac{5}{7}\sqrt{6} = [-3; 1, 1, \overline{2, 2, 18, 1, 1, 1, 3, 15, 171, 2, 1, 1, 32, 1, 18, 12, 3, 2}]$
As it turned out, my "random" examples all had $D ... | 3 | https://mathoverflow.net/users/3824 | 301078 | 131,973 |
https://mathoverflow.net/questions/301076 | 3 | Let $f=N(\mu,\sigma^2)$ be a univariate normal distribution with mean $\mu$ and variance $\sigma^2$ and let $f\_1 = N(\mu+\epsilon,\sigma^2)$ and $f\_2=N(\mu,(\sigma+\epsilon)^2)$ be some small perturbations to $f$. Are there any statistical metrics $D(\cdot,\cdot)$ (e.g. Kolmogorov-Smirnov, Wasserstein, Prokhorov, etc... | https://mathoverflow.net/users/70190 | Are there any statistical metrics that satisfy this kind of condition? | According to Proposition 7 on p. 236 of [Givens and Shortt](https://projecteuclid.org/euclid.mmj/1029003026), the $L^2$ Wasserstein distance between $N(\mu\_1,\sigma\_1^2)$ and $N(\mu\_2,\sigma\_2^2)$ is the Euclidean distance between the points $(\mu\_1,\sigma\_1)$ and $(\mu\_2,\sigma\_2)$. So, the derivative of this ... | 4 | https://mathoverflow.net/users/36721 | 301081 | 131,974 |
https://mathoverflow.net/questions/301010 | 6 | Let $X\_N$ denote the Fermat curve defined over $\mathbb{Q}$ by the equation $x^N+y^N-z^N=0$ and let $X\_{N,\mathbb{Q}(\mu\_N)}$ be the base change. Let $G$ be the Galois group of $\mathbb{Q}(\mu\_N)/\mathbb{Q}$, hence $G \cong (\mathbb{Z}/N\mathbb{Z})^\times$. Consider the motivic cohomology $H^2\_\mathcal{M}(X\_N,\ma... | https://mathoverflow.net/users/124826 | Galois descent in motivic cohomology | What you need is the existence of the transfer map $N : K\_2(L) \to K\_2(K)$ for any finite field extension $L/K$, which is due to Bass and Tate, see *Introduction to algebraic $K$-theory* by Milnor. I don't know of any definition of the transfer map using Matsumoto's decription of $K\_2$, one should rather use Milnor'... | 4 | https://mathoverflow.net/users/6506 | 301087 | 131,978 |
https://mathoverflow.net/questions/300691 | 7 | I have a somewhat technical question about the concept of graph limits:
Suppose that $G\_n$ is a sequence of labelled, simple, unweighted graphs, and let $W\_n$ denote the graphon of $G\_n$ (i.e. $W\_n(x,y) = 1$ for all $\frac{i-1}{n}<x\leq \frac{i}{n}$ and $\frac{j-1}{n} < y \leq \frac{j}{n}$, whenever $(i,j)$ or $(... | https://mathoverflow.net/users/123578 | A Question on Graph Limits | It looks so. $W$ may be approximated with prescribed accuracy $\varepsilon$ in a cut-norm (and even in $L^1$) by a graphon $W\_\varepsilon$ which corresponds to a certain finite graph (that is, the function $W\_\varepsilon(x,y)$ depends only on integer parts of $Nx,Ny$ for certain large $N$.) And for large $n$ the dist... | 3 | https://mathoverflow.net/users/4312 | 301105 | 131,988 |
https://mathoverflow.net/questions/301115 | 1 | I am reading about distributions in the context of differential geometry.
>
> A distribution $S$ of dimension $r$ on a manifold $M$ is an assignment to each point $p \in M$ of an $r$-dimensional subspace $S\_p$ of $T\_pM$.
>
>
>
(1) Is this $r$ called *rank* of the distribution?
Further, I introduce a clos... | https://mathoverflow.net/users/117515 | Rank of a distribution | (1) The boxed sentence would be better written as "A distribution $S$ of *rank* $r$ on a manifold $M$ is an assignment of an $r$-dimensional subspace $S\_p$ of $T\_pM$ to each point $p\in M$." Confusing 'dimension' and 'rank' in this context is careless writing since, assuming that $S\subset TM$ is a smooth subbundle, ... | 8 | https://mathoverflow.net/users/13972 | 301116 | 131,989 |
https://mathoverflow.net/questions/301072 | 0 | It is well-known that there are finitely many indecomposable module over the preprojective algebra associated to a quiver $Q$ if and only if $Q=A\_2,A\_3,A\_4$ and tame type for $A\_5$ and wild for others.
Now let say $Q$ is a ADE Dynkin diagram, and $V$ be an indecomposable preprojective algebra. What can we say about... | https://mathoverflow.net/users/41979 | dimension vector of indecomposable module over preprojective algebra | For every finite dimensional algebra that does not have finite representation type, there is no bound on the dimension of indecomposable modules. This is known as the first Brauer-Thrall conjecture (now a theorem).
| 3 | https://mathoverflow.net/users/22989 | 301121 | 131,991 |
https://mathoverflow.net/questions/301123 | 1 | In this old question of mine
<https://math.stackexchange.com/questions/1651906/spheres-as-symplectic-homogeneous-spaces>
the presentation of spheres as symplectic group homogeneous spaces was discussed very well, giving
$$
S^{4n-1} \simeq Sp(n)/Sp(n-1).
$$
Going us back now to the presentation of the spheres most... | https://mathoverflow.net/users/42100 | Torus actions on $Sp(n)$-spheres | The presentation $\mathbb{S}^{2n-1} = \mathit{U}\_n/\mathit{U}\_{n-1}$ is tantamount to considering $\mathbb{S}^{2n-1}$ as the unit sphere in $\mathbb{C}^n$ (as $\mathit{U}\_n$ is the group of $\mathbb{C}$-linear maps of $\mathbb{C}^n$, acting, say, from the right, preserving the Hermitian inner product and $\mathit{U}... | 7 | https://mathoverflow.net/users/17064 | 301128 | 131,992 |
https://mathoverflow.net/questions/18454 | 61 | I got fantastic answers to my previous question (about modern references for the fact that surfaces can be triangulated), so I thought I'd ask a related question. A basic fact about surface topology is that if $S$ is a noncompact connected surface, then $\pi\_1(S)$ is a free group (possibly trivial or $\mathbb{Z}$). I'... | https://mathoverflow.net/users/317 | Fundamental groups of noncompact surfaces | In case anyone is interested, I wrote up a detailed account synthesizing the various answers here and correcting some issues I ran into. It is entitled "Spines of manifolds and the freeness of fundamental groups of noncompact surfaces" and can be downloaded from my page of notes [here](http://www.nd.edu/~andyp/notes/).... | 11 | https://mathoverflow.net/users/317 | 301131 | 131,993 |
https://mathoverflow.net/questions/301117 | 2 | Let $T$ be a self-adjoint operator (possibly unbounded) and $S$ a bounded self-adjoint operator.
Then one can study the unitary groups $R\_T(t):=e^{itT}$ and $R\_S(t):=e^{itS}.$
Now if you think about the function $f\_t(x):=e^{itx}$ then this one is Lipschitz continuous with
$$\left\lvert f\_t(x)-f\_t(y) \right\... | https://mathoverflow.net/users/124879 | Lipschitz bound on semigroups | No. Consider for instance periodic $L^2$ functions and let $T=i\frac{d}{dx}$, and let $S$ be multiplication by $\sin x$. The function $\exp(i\cos x)$ is in the nullspace of $T-S$, but not in the nullspace of $R\_T-R\_S$.
| 3 | https://mathoverflow.net/users/12120 | 301133 | 131,994 |
https://mathoverflow.net/questions/301138 | 3 | I'm not sure if this is a research level question, but:
Let $F:Rep\_A \to Rep\_B$ be an exact cocomplete functor between representation categories of finite dimensional $k$ algebras, where $k$ has charecteristic zero and is algebraically closed. By Eilenberg-Watts $F$ has both left and right adjoint. Are they natural... | https://mathoverflow.net/users/58211 | Question on Eilenberg-Watts theorem | No. Suppose $F=-\otimes\_AM$, where $M$ is a finite dimensional $B$-$A$-bimodule that is projective as a left $A$-module (so that $F$ is exact).
The right adjoint is $\text{Hom}\_B(M,-)$. But unless $M$ is also projective as a right $B$-module this is not right exact, and so can’t be a left adjoint.
So, for example... | 5 | https://mathoverflow.net/users/22989 | 301140 | 131,996 |
https://mathoverflow.net/questions/301125 | 12 | Apologies for the title.
Miller's theorem (formerly [Sullivan's conjecture](https://en.wikipedia.org/wiki/Sullivan_conjecture)) gives that for a finite group $G$ and a finite dimensional connected CW complex $X$, the based mapping space $\operatorname{Map}\_\*(BG,X)$ is weakly contractible. In particular, by consider... | https://mathoverflow.net/users/8103 | Extending a weak version of Sullivan's generalized conjecture | The answer to the first question is negative by a result of Dror Farjoun and Zabrodsky. In [Fixed points and homotopy fixed points](https://eudml.org/doc/140123 "Fixed points and homotopy fixed points"), they prove that if a finite group $G$ is not a $p$-group, then there exists a finite $G$-CW complex $X$ with empty f... | 13 | https://mathoverflow.net/users/80403 | 301141 | 131,997 |
https://mathoverflow.net/questions/301127 | 2 | Let $(X,\tau)$ be a topological space, and let ${\cal U}$ be an open cover. We say that ${\cal U}$ is *thick* if for all $x\in X$ we have $$|\{V\in {\cal U}: x\in V\}| = |X|.$$
Is there a Hausdorff space $(X,\tau)$ with $|X|>1$ and an open cover ${\cal U}$ of $X$ such that every [refinement](https://en.wikipedia.org/wi... | https://mathoverflow.net/users/8628 | Thick refinements of covers | This is impossible for any $T\_1$ space $X$. For suppose $X$ is $T\_1$ and $\mathcal U$ is an open cover of $X$, and fix $x \in X$. If $U$ is some member of $\mathcal U$ containing $x$, then $\{U\} \cup \{V \setminus \{x\} : V \in \mathcal U,\, V \neq U\}$ is a refinement of $\mathcal U$, but it is not thick because on... | 4 | https://mathoverflow.net/users/70618 | 301144 | 131,999 |
https://mathoverflow.net/questions/263429 | 8 | Topological Hochschild homology is a generalization of Hochschild homology from rings to $E\_\infty$-ring spectra. On the other hand, there is a natural way to extend the notion of Hochschild homology to dg rings (either by explicitly writing down the bar complex, which is now a double complex, or by defining it as a d... | https://mathoverflow.net/users/6059 | Topological Hochschild homology and Hochschild homology of dg algebras | The notion of Hochschild homology can be defined abstractly in any suitable homotopical context in which a tensor product exist (say, in any presentably symmetric monoidal $\infty$-category). Given an associative algebra object $A$ in such a context, its Hochschild homology is the (suitably defined) tensor product of $... | 4 | https://mathoverflow.net/users/51164 | 301156 | 132,004 |
https://mathoverflow.net/questions/301148 | 15 | I am wondering about the following version of the Banach-Steinhaus theorem.
Let $A$ be a closed convex subset contained in the unit ball of a Banach space $X$ and consider bounded operators $T\_n \in \mathcal L(X).$
Assume we know that for every $x \in A$ the sequence $\left\lVert T\_n x \right\rVert$ is bounded u... | https://mathoverflow.net/users/119875 | Version of Banach-Steinhaus theorem | The answer is **yes**, as a close inspection of the standard proof of the uniform boundedness principle/Banach-Steinhaus theorem shows. The standard proof (or at least the proof which I would consider to be the standard one) can e.g. be found on [Wikipedia](https://en.wikipedia.org/wiki/Uniform_boundedness_principle).
... | 17 | https://mathoverflow.net/users/102946 | 301158 | 132,005 |
https://mathoverflow.net/questions/301165 | 1 | Suppose that $T$ is a consistent first order theory. Now let the language of $T$ be $L\_T$.
**Question:** is it always consistent to add a new primitive constant $D$, and a new primitive binary relation $\in^\*$ called 'class membership', and add a new symbol $\epsilon$ and axiomatize that for each formula $\phi$ th... | https://mathoverflow.net/users/95347 | Can we have a nearily unrestricted class comprehension over predicates that do not mention the class membership symbol | If I understand you correctly, the answer is yes, providing that you don't just add those axioms directly to $T$, but instead add the assertion $\phi^D$ for each axiom of $T$.
Your theory is simply the syntactic analogue of taking a model $M$ in the language $L\_T$, and adding a second-order part with an object repr... | 3 | https://mathoverflow.net/users/1946 | 301174 | 132,009 |
https://mathoverflow.net/questions/301191 | 1 | How can one see that $I$ is an infinite projection in $B(\mathcal{H})$, where $\mathcal{H}$ is an inseparable Hilbert space?
| https://mathoverflow.net/users/125816 | On projection theory for inseparable Hilbert spaces | Let $\beta$ be a Hilbert basis for $H$. If $\tilde\beta$ is a finite subset of $\beta$ and $f:\beta\rightarrow \beta\setminus\tilde\beta$ is a bijection, then
$$U:H\rightarrow H,\quad v\mapsto f(v)$$
introduce a partial isometry in $B(H)$ whose initial projection is $I$ and its final projection is $I-P$, where $P$ is ... | 1 | https://mathoverflow.net/users/84700 | 301192 | 132,013 |
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