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https://mathoverflow.net/questions/447257 | 10 | I'd like to be able to say that a measure $\mu$ on a measurable space $X$ "is" a morphism $R \to X$, where $R$ is some incarnation of the real numbers in an appropriate category.
In other words, let $Mbl$ be the category of measurable spaces (an object is a set $X$ equipped with a $\sigma$-algebra; a morphism $f : X ... | https://mathoverflow.net/users/2362 | In which category is a measure on a measurable space a morphism? | How about this? (But I will be enlarging $Mbl$ by a functor that is not full.)
Let's make a category whose objects can be called abstract $\sigma$-algebras. I'll call the category $\Sigma$.
A $\Sigma$-object consists of
(1) a set $F$,
(2) a symmetric binary relation on $F$ called "disjointness", and
(3) a rul... | 6 | https://mathoverflow.net/users/6666 | 447283 | 180,141 |
https://mathoverflow.net/questions/445068 | 1 | I'm looking for some references to get me started on stability theory. More specifically, I want to find sources that talk about notions in stability theory, but for $\omega$-stable theories, which should hopefully be much easier to understand than general stable theories since things should be much nicer. (I guess eve... | https://mathoverflow.net/users/170461 | Stability theory in the context of $\omega$-stable theories | David Marker's textbook "Model Theory: An Introduction" largely focuses on $\omega$-stable theories. You may already be familiar with the material from chapters 1 to 4, and you can probably omit chapter 5, but chapters 6, 7, 8 are about $\omega$-stability.
Pillay's Geometric Stability Theory is better read after read... | 4 | https://mathoverflow.net/users/112484 | 447285 | 180,142 |
https://mathoverflow.net/questions/447284 | 1 | The $n$-dimensional hypercube $Q\_n$ is the graph whose vertex set is $\{0, 1\}^n$ and whose edge set is the set of pairs that differ in exactly one coordinate. A graph is called cubical if it is a subgraph of $Q\_n$ for some $n$.
We know that $|V(Q\_n)|=2^n$ and $|E(Q\_n)|=n 2^{n-1}$, so $|E(Q\_n)|=\frac{1}{2}|V(Q\_n)... | https://mathoverflow.net/users/165069 | How many edges can a t-vertex cubical graph have? | Let me try to prove that for every integer $t>0$ the number of edges of a $t$-vertex cubical graph has at most $\frac12 t\log\_2t$ edges.
This is true for $t=1$. So assume (by induction) that $t>1$ and it is proved for smaller number of vertices. Without loss of generality, the vertex set of our graph contains the el... | 2 | https://mathoverflow.net/users/4312 | 447287 | 180,143 |
https://mathoverflow.net/questions/447299 | 3 | This question was posted a long time ago on the [mathexchange](https://math.stackexchange.com/q/4240470/680679), but I didn't get any answers there, and despite having discussed it with some colleagues, I don't think I have a definitive answer.
I am wondering about **simple eigenvalue** definitions in the context of ... | https://mathoverflow.net/users/482407 | The definition of simple eigenvalue | If $\lambda$ has modulus $1$, then both definitions are equivalent for power-bounded operators, i.e., for operators $A$ that satisfy $\sup\_{n = 0,1,2,\dots} \|A^n\| < \infty$.
Indeed, if $\lvert \lambda \rvert = 1$ and if there exists a vector $x$ that is in the kernel of $(\lambda-A)^2$ but not in the kernel of $\l... | 7 | https://mathoverflow.net/users/102946 | 447300 | 180,147 |
https://mathoverflow.net/questions/447298 | 2 | All rings are assumed commutative and unital.
**Some context (feel free to skip right to the questions below).** I am trying to understand to what extent the property "being prime" for an ideal $I$ is intrinsic. For example, how robust "primeness" is under the relative point of view. To this end I consider only such ... | https://mathoverflow.net/users/1849 | Can a non-zero non-prime ideal become prime in a smaller ring? | As noted (implicitly) in the comments, it is not very common that $I \subseteq R$ is also an ideal in $S$. For instance, if $R$ and $S$ are domains and $I$ is nonzero, this implies that $\operatorname{Frac} R \to \operatorname{Frac} S$ is an isomorphism, and $R \to S$ is finite if $I$ is finitely generated (e.g. when $... | 6 | https://mathoverflow.net/users/82179 | 447304 | 180,148 |
https://mathoverflow.net/questions/447265 | 3 | I am trying to prove the following inequality:
$$\frac{y}{x}-1-\log\left(\frac{y}{x}\right)\geq \frac{1}{2}\frac{(x-y)^2}{x} \quad \forall x,y \in (0,1]$$
This inequality appears in the paper "[Scale-Free Adversarial Multi Armed Bandits](https://proceedings.mlr.press/v167/putta22a/putta22a.pdf)" as corollary 7, p. ... | https://mathoverflow.net/users/116451 | Proof of the inequality $\frac{y}{x}-1-\log\left(\frac{y}{x}\right)\geq \frac{1}{2}\frac{(x-y)^2}{x}$ when $x,y \in (0,1]$ | $\newcommand\p\partial$Let $d(x,y)$ denote the difference between the left- and right-hand sides of the inequality in question. We want to show that $d(x,y)\ge0$ for $x$ and $y$ in $(0,1]$.
For such $x$ and $y$, we have $xy\,\p\_y d(x,y)=(x-y)(y-1)$ and $d(x,0+)=\infty$. So, without loss of generality (wlog), either ... | 4 | https://mathoverflow.net/users/36721 | 447305 | 180,149 |
https://mathoverflow.net/questions/447292 | 15 | Let $Y \to X$ be a finite branched cover of smooth projective curves over $\mathbb{C}$, so we get a finite extension $K(Y)/K(X)$ where $K(\ )$ is the field of meromorphic functions. Say that $Y \to X$ is solvable if the normal closure of $K(Y)$ over $K(X)$ is a solvable extension of $K(X)$.
I was just wondering this ... | https://mathoverflow.net/users/297 | Is a generic genus $g \geq 7$ curve a solvable cover of $\mathbb{P}^1$? | The answer is negative by results of Zariski (weak version) and the stronger result by Guralnick and Neubauer in Theorem B of [Monodromy groups of branched coverings: the generic case](https://doi.org/10.1090/conm/186/02190). They show that for $g\ge7$, the space $\cal M\_g(\text{Sol})$ of all curves of genus $g$ admit... | 15 | https://mathoverflow.net/users/18739 | 447307 | 180,150 |
https://mathoverflow.net/questions/447279 | 8 | Suppose that $A$ is finitely-generated torsion-free abelian and $H$ is torsion-free, finitely-generated and residually nilpotent.
>
> Is the restricted wreath product $A \wr H$ necessarily residually nilpotent?
>
>
>
Residual nilpotency of a group $G$ means $\bigcap\_n G\_n = 1$ where $G\_n$ is the lower centr... | https://mathoverflow.net/users/123634 | Is a finitely-generated torsion-free wreath product of an abelian group and a residually nilpotent group itself residually nilpotent? | If $A$ is f.g. torsion-free abelian and $H$ is residually-$p$, then $A\wr H$ is residually-$p$ (easy) and hence residually nilpotent. Thus covers many cases (including the case when $H$ is residually torsion-free nilpotent, but many more).
But the answer is no in general.
**Fact.** *There are torsion-free residuall... | 8 | https://mathoverflow.net/users/14094 | 447310 | 180,151 |
https://mathoverflow.net/questions/447288 | 2 | A pre-Dynkin system is a set system $\mathcal D \subset \wp(\Omega)$ which contains $\Omega$ and is closed under complements and *finite* disjoint unions. Is it true that the monotone class generated by $\mathcal D$ equals the Dynkin system generated by $\mathcal D$,
$$
m(\mathcal D) = d(\mathcal D)?
$$
(Here $m(\mat... | https://mathoverflow.net/users/57923 | Monotone class theorem for pre-Dynkin system ("finitely additive Dynkin system/λ system) | **Part I.** $m(\mathcal{D})\subset d(\mathcal{D})$
This is true regardless of the class $\mathcal{D}$ (whether pre-Dynkin or not). This is because a Dynkin class is necessarily a monotone class -- easier to check with the [alternative equivalent definition](https://en.wikipedia.org/wiki/Dynkin_system) of $\lambda$-sy... | 2 | https://mathoverflow.net/users/138242 | 447312 | 180,152 |
https://mathoverflow.net/questions/447316 | 8 | **I. Zagier's continued fraction**
As pointed out by Gorodetsky in [his answer](https://mathoverflow.net/a/447228/12905), Zagier evaluated the continued fractions associated with his six sporadic sequences excepting the one for $(-9,-3,-27)$. Let $\color{red}{c=-27,}$ then,
$$C\_2=\cfrac{1}{-3 + \cfrac{1^4\,c}{-21 ... | https://mathoverflow.net/users/12905 | On Zagier's missing continued fraction with multiple limits? | Set $Q=(1/2)L(\chi\_{-3},2)$ (related to your Gieseking constant) and
$P=2\pi^2/81$. The limits are almost certainly (not proved),
\begin{align}
\lim\_{m\to\infty}C\_2(6m+0) &= -Q\\
\lim\_{m\to\infty}C\_2(6m+1) &= -P-Q\\
\lim\_{m\to\infty}C\_2(6m+2) &= -3P - Q\\
\lim\_{m\to\infty}C\_2(6m+3) &= \infty\\
\lim\_{m\to\in... | 10 | https://mathoverflow.net/users/81776 | 447319 | 180,154 |
https://mathoverflow.net/questions/447273 | 3 | Let $\{X\_t, t \geq 0\}$ and $\{X\_t', t \geq 0\}$ denote two markov chains on the same state space $\{1, ..., n+1\}$ with transition probability matrices $P$ and $P'$ respectively. Suppose that both chains have one absorbing state, namely state $n+1$ (the same for both chains). Let $Q$ and $Q'$ denote the transient pa... | https://mathoverflow.net/users/505212 | Comparison of time until absorption for two absorbing Markov chains | This conjecture is false in general. E.g., suppose that $n=3$,
$$Q=\frac1{10}
\begin{bmatrix}
3 & 5 & 2 \\
2 & 2 & 3 \\
4 & 5 & 1 \\
\end{bmatrix},\qquad
Q'=\frac1{10}
\begin{bmatrix}
2 & 0 & 7 \\
1 & 1 & 4 \\
4 & 2 & 4 \\
\end{bmatrix}$$
(so that
$$P=\frac1{10}
\begin{bmatrix}
3 & 5 & 2&0 \\
2 & 2 & 3&3 \\
4... | 2 | https://mathoverflow.net/users/36721 | 447324 | 180,156 |
https://mathoverflow.net/questions/446822 | 3 | Given a manifold $X$ and short exact sequence of abelian groups
$$
1\rightarrow A\_1\overset{\iota}{\rightarrow} A\_2\overset{\pi}{\rightarrow} A\_3\rightarrow 1
$$
we get the Bockstein map in cohomology $\beta : H^p(X,A\_3)\rightarrow H^{p+1}(X,A\_1)$. On cochains this is defined as follows. Choose a section $s:A\_3\r... | https://mathoverflow.net/users/495347 | Does the cohomology Bockstein homomorphism map to the homology Bockstein homomorphism under Poincarè duality? | I just found another proof you might prefer as Lemma 2.4 of this paper:
<https://arxiv.org/pdf/1708.03754.pdf>.
It's stated only for one case of groups, but I don't see why it wouldn't extend more generally.
Note that the Poincar'e duality isomorphism is given by the cap product with a fundamental class. That's the inv... | 1 | https://mathoverflow.net/users/6646 | 447337 | 180,159 |
https://mathoverflow.net/questions/447336 | 4 | Let $f: \mathbb R^n \to \mathbb R$ be a locally integrable function. Given an $\varepsilon > 0$, we say $f$ is *$\varepsilon$-times Lebesgue differentiable* if
$$\lim\_{r \to 0} \frac{\int\_{B\_r (x)} |f(y) - f(x)| \, dy}{r^{n+\varepsilon}} = 0$$
for every $x \in \mathbb R^n$.
**Question:** Suppose $f$ is $\varep... | https://mathoverflow.net/users/173490 | If a function $f$ is $\varepsilon$-times Lebesgue differentiable, is $f$ continuous? | The answer is no. Let me present a counterexample for $n=2$.
Fix some bump function $h(x)$ supported in $[-1,1]$ and such that $|h(x)|\leq 1$ and $h(0)=1$ (though those choices don't matter much). Define $f(x,y)=h(y/x^2)$ for $x>0$ and $f(x,y)=0$ elsewhere. This function is easily seen to be smooth at all points othe... | 5 | https://mathoverflow.net/users/30186 | 447339 | 180,160 |
https://mathoverflow.net/questions/446735 | 2 | Let $X$ be a compact Hausdorff space where $X$ have infinitely many points and the topology is non-discrete, $m$ be a regular probability measure defined on the Borel $\sigma$-algebra of $X$, and $g$ an homeomorphism of $X$. Suppose that the image measure $g\_{\ast}m$ (defined by $g\_{\ast}m(O) = m\big[g^{-1}(O)\big]$ ... | https://mathoverflow.net/users/151332 | Radon-Nikodym derivative in a compact Hausdorff space | I think the answer for 2 is **no**. I will state some counter-examples below, after some partial results.
A Necessary condition involving the support
-------------------------------------------
From now on, all measures are Borel probability measures. For every measure $\mu$, define its support
$$\mathrm{Supp}(\mu)... | 5 | https://mathoverflow.net/users/505464 | 447368 | 180,166 |
https://mathoverflow.net/questions/447361 | 0 | I am looking for ideas on proving the Eulerian expansion of the cotangent using residue calculation: $$\pi\cot(\pi z)=\frac{1}{z}+\sum\_{n=1}^{\infty}\left(\frac{1}{z+n}+\frac{1}{z-n}\right), \ z\in\mathbb{C}\backslash\mathbb{Z}.$$
| https://mathoverflow.net/users/109569 | Residue calculation for Eulerian expansion of the cotangent | The residue approach to the partial fraction expansion of $\cot(z)$ is explained in Freitag & Busam's "Complex Analysis", Prop. III.7.13, and probably in other books as well.
Here is a more general result regarding partial fraction expansions:
**Proposition:** Let $f \in \mathcal{M}(\mathbb{C})$ and let $(\gamma\_n... | 3 | https://mathoverflow.net/users/1849 | 447370 | 180,168 |
https://mathoverflow.net/questions/447126 | 7 | The space $\Omega SU(n)$ is homotopy-equivalent to $SL\_n(\mathbb{C}[z,z^{-1}])/SL\_n(\mathbb{C}[z])$. Using this, Steve Mitchell introduced a filtration of $\Omega SU(n)$ by subspaces $F\_k$ which can be viewed as Schubert varieties ([MR0862881](https://mathscinet.ams.org/mathscinet-getitem?mr=862881)). The homology r... | https://mathoverflow.net/users/10366 | Stable splitting of $\Omega SU(n)$ | A paper I wrote with Allen Yuan [Multiplicative structure in the stable splitting of $\Omega SL\_n(\mathbb{C})$](https://www.sciencedirect.com/science/article/abs/pii/S0001870819301525) proves a highly structured version of this splitting.
| 4 | https://mathoverflow.net/users/37734 | 447371 | 180,169 |
https://mathoverflow.net/questions/447320 | 2 | I originally asked this question on [Math StackExchange](https://math.stackexchange.com/q/4652009) a few months ago and no answers or even comments have yet been posted, so I'm asking this question again here on Math OverFlow.
---
[This Math StackExchange question](https://math.stackexchange.com/q/4643164) and [t... | https://mathoverflow.net/users/110710 | Is this a valid method of extending convergence of the Maclaurin series for $\frac{x}{x+1}$ from $|x|<1$ to $\Re(x)>-1$? | In \eqref{5}, the inner sum is obviously $x(1-x)^{n-1}$. So, the limit in \eqref{5} (equal $\dfrac x{x+1}$ indeed) exists if and only if $|1-x|<2$.
In \eqref{6}, using the substitution $k=K-n-j$ in the inner sum and then interchanging the order of summation, we see that the iterated sum under the limit sign is $\Big(... | 2 | https://mathoverflow.net/users/36721 | 447373 | 180,170 |
https://mathoverflow.net/questions/434280 | 7 | Suppose a (separable) Banach space $X$ has the following property: If $P:X\to X$ is a bounded projection different from $I$ such that $\|P\|=1$, then $\|I-P\|=1$. Does this imply that $X$ is a Hilbert space?
| https://mathoverflow.net/users/69275 | A characterization of Hilbert spaces by norm one projections | The answer is positive if $\dim X \geq 3$. Moreover it suffices to check the property for rank $1$ projections. See characterization (18.14) in the book [1].
However the characterization fails in dimension $2$. A rank $1$ projection is of norm $1$ iff every $x$ in the range and $y$ in the kernel are Birkhoff-orthogon... | 6 | https://mathoverflow.net/users/908 | 447383 | 180,173 |
https://mathoverflow.net/questions/447352 | 2 | I'm reading the paper *"Higher Hida and Coleman theories on the modular curve"* by G.Boxer and V.Pilloni. But I'm confused with the different views towards Hecke operators.
$N$ is an integer and $p$ is a prime such that $(p,N)=1$. Let $X\rightarrow \mathrm{Spec} \mathbb{Z}\_p$ be the compactified modular of level $\G... | https://mathoverflow.net/users/470737 | Why can Hecke operators be regarded as finite flat cohomological correspondence? | The first half of the question has been answered in the comments, so let me address the second half of the question.
We want to define Hecke operators on the complex $R\Gamma(X, \omega^k)$, because this is the complex whose $H^0$ is modular forms for $k \ge 1$ (and whose $H^1$ is dual to cusp forms for $k \le 1$).
... | 3 | https://mathoverflow.net/users/2481 | 447393 | 180,176 |
https://mathoverflow.net/questions/447294 | 1 | Let $E$ be a Banach space. Let $\mathcal K(E)$ be the space of all compact (bounded linear) operators from $E$ to $E$. For a linear map $T$, we denote by $R(T)$ its range and by $N(T)$ its kernel. Let $I:E \to E$ be the identity map.
Below is the Fredholm alternative in Brezis' *Functional Analysis*.
>
> **Theore... | https://mathoverflow.net/users/477203 | Is $I-S$ in my attempt of Fredholm alternative injective? | No, in general $I-S$ will not be injective.
Indeed, suppose e.g. that $E=\mathbb R^2$ and
$$T=\begin{bmatrix}2&1\\ -1&0 \end{bmatrix};$$
here we will identify the linear operators with their matrices in the standard basis of $\mathbb R^2$.
Then (assuming $G$ and $L$ are the orthogonal complements of $N(I-T)$ and $R... | 2 | https://mathoverflow.net/users/36721 | 447402 | 180,180 |
https://mathoverflow.net/questions/447399 | 2 | Consider a convex function $f : \mathbb{R}^d \to \mathbb{R}$. Define now the set-input function $g\_f : 2^{[d]} \to \mathbb{R}$ as follows,
\begin{align}
g\_f(S) = \min \left\{ f(x) : x \in \mathbb{R}^d \text{ and } \forall i \not\in S, x\_i = 0 \right\}.
\end{align}
In other words, $g\_f(S)$ is the minimum of $f$ when... | https://mathoverflow.net/users/148528 | Submodularity of a particular function derived from a convex function? | The answer is no. E.g., if $d=2$ and
$$f(x,y)=f\_c(x,y):=(x - 1)^2 + (y - 1)^2 - 2 c (x - 1) (y - 1) \tag{1}\label{1}$$
for some real $c\in(-1,1)$ and all real $x,y$,
then the function $f$ is convex. However, for $S=\{1\}$ and $T=\{2\}$ we have
$$\Delta\_f(S,T):=g\_f(S)+g\_f(T)-g\_f(S\cup T)-g\_f(S\cap T)=2(1-c)c,$$
wh... | 4 | https://mathoverflow.net/users/36721 | 447407 | 180,182 |
https://mathoverflow.net/questions/447419 | 2 | Let $X$ be a compact Hausdorff space and $\mathcal{A}$ be a subalgebra of $C(X;\mathbb{R})$.
The Stone-Weierstrass theorem asserts that if $\mathcal{A}$ contains the constants and separates the points of $X$, then $\mathcal{A}$ is dense in $(C(X;\mathbb{R}), \|\cdot\|\_\infty)$.
Suppose now that $\mathcal{A}$ is co... | https://mathoverflow.net/users/160714 | Stone-Weierstrass theorem: coefficients of approximating sequence bounded? | The answer is negative. Suppose we approximate a continuous function on $[-1,1]$ with ordinary
polynomialss $P\_n$. If the coefficients are bounded, say
$|a\_{n,k}|\leq C$, then
$$|P\_n(z)|\leq C(1-|z|)^{-1},\quad |z|<1$$
Therefore $\{ P\_n\}$ is a normal family in the unit disk,
and thus there is a subsequence which c... | 11 | https://mathoverflow.net/users/25510 | 447422 | 180,187 |
https://mathoverflow.net/questions/447293 | 0 | Let $E$ be a Banach space. For a linear map $T$, we denote by $R(T)$ its range and by $N(T)$ its kernel. Let $I:E \to E$ be the identity map. Let $T:E \to E$ be a compact (bounded linear) operator. Then Fredholm alternative tells us that
$$
\dim N(I-T)=\dim N(I-T^\*) < \infty.
$$
Then there are closed subspaces $G$ a... | https://mathoverflow.net/users/477203 | Is $\Lambda:= \pi_2 \circ \pi_1:E \to L$ surjective? | A counter-example from [this](https://mathoverflow.net/a/447402/477203) answer by Iosif Pinelis also works.
---
Indeed, suppose e.g. that $E=\mathbb R^2$ and
$$T=\begin{bmatrix}2&1\\ -1&0 \end{bmatrix};$$
here we will identify the linear operators with their matrices in the standard basis of $\mathbb R^2$.
Then... | 0 | https://mathoverflow.net/users/477203 | 447435 | 180,193 |
https://mathoverflow.net/questions/446923 | 3 | Let $X$ be a compact manifold of dimension $\geq k$. Denote by
\begin{equation}
h: \pi \_k(X) \rightarrow H\_k(X,\mathbb{Z})
\end{equation}
be Hurewicz homomorphism and by $\Gamma \_k(X)\subset H\_k(X,\mathbb{Z})$ its image. I want to look at the pairing
\begin{equation}
\langle \ . \ ,\ . \ \rangle : \ \Gamma \_k(X)\_... | https://mathoverflow.net/users/495347 | Pairing between cohomology and the image of the Hurewicz homomorphism | The answer is no, by the existence of nontrivial Massey products.
Call a cohomology class decomposable if it is in the subgroup generated by cup products of pairs of classes of positive dimension.
As you say, any decomposable class pairs to zero with all spherical homology classes (i.e. those in the image of the Hu... | 4 | https://mathoverflow.net/users/6666 | 447453 | 180,197 |
https://mathoverflow.net/questions/447443 | 4 | Given a smooth complex projective variety with an ample line bundle $L$, it seems to be folklore that one can get a one-point compactification of the total space $\mathbb{V}(L)$ of $L$ such that removing the zero section yields an affine variety.
I can see how to do it when $L$ is very ample, by basically exploiting ... | https://mathoverflow.net/users/120296 | One-point compactification of ample line bundle | This is EGA 2, Prop. 8.8.2. It basically says that if $L$ is ample then one can contract the zero section of the geometric realization $\mathbb V(L)$ of $L$ to a point. The result is called the affine cone of $L$. Observe that in EGA the sections of $L$ become functions on $\mathbb V(L)$. So your zero section is the "s... | 11 | https://mathoverflow.net/users/89948 | 447460 | 180,199 |
https://mathoverflow.net/questions/447458 | 7 | I'm looking for the roots of the sextic equation in $x$
$$
x^6 - (3 m) x^5 + 5 m^2 x^4 - (5 m^3) x^3 + 3 m^4 x^2 - m^5 x + L = 0.
$$
I know that at most two of the roots of this are real when $m$ and $L$ are positive integers. Also mathematica finds a closed form for all the roots (surprisingly). They are
```
x = 1/... | https://mathoverflow.net/users/265714 | Roots of this sextic | If $m=0$, then your sextic is $x^6+L$. For this to have a real root, you need $L\leq 0$, in which case those real roots are the real sixth roots of $-L$.
We may now assume $m\neq 0$. After dividing your sextic by $m^6$, and taking $y:=x/m$ and $K:=L/m^6$, you have a new (simpler) sextic $f(y):=y^6-3y^5+5y^4-5y^3+3y^2... | 20 | https://mathoverflow.net/users/3199 | 447461 | 180,200 |
https://mathoverflow.net/questions/447465 | -2 | Let $M$ be a smooth compact sub-manifold of $\mathbb R^d$. Let $p\in M$ and $x\_n,y\_n \in M$ be sequences such that $x\_n,y\_n\rightarrow p$. Does the following hold when passing to a convergent sub-sequence?
$$\lim\_{n\rightarrow \infty} \frac{x\_n-y\_n}{\|x\_n-y\_n\|} \in T\_pM$$
Indeed, I am thinking of $T\_pM$ as ... | https://mathoverflow.net/users/170501 | Is this limit a tangent vector? | I am not sure this question is appropriate for this site but here is a proof.
By translation, we can suppose p=0. By rotation, we can suppose $\mathbb{R}^d=\mathbb{R}^{n+m}$ and $\mathbb{R}^n\times\{0\}=T\_0M$. So the inverse mapping theorem (use projection map from $M$ onto $\mathbb{R}^n$) implies that in a neighbou... | 2 | https://mathoverflow.net/users/105656 | 447469 | 180,202 |
https://mathoverflow.net/questions/447451 | 10 | A Hausdorff space is called extremally disconnected or extreme, if for every open set $U$ the closure $\overline U$ is open, too. The question, whether there are extremally disconnected topological groups which are not discrete seems difficult, see
Reznichenko, Evgenii; Sipacheva, Ol'ga Discrete subsets in topologica... | https://mathoverflow.net/users/473423 | Are there extremally disconnected groups? | Yes: every locally compact ED group is discrete.
Indeed, for topological groups, ED passes to quotients and open subgroups.
Let by contradiction $G$ be a non-discrete ED locally compact group. Passing to an open subgroup, one can suppose that $G$ is connected-by-compact. Hence it is compact-by-Lie. The Lie quotient... | 14 | https://mathoverflow.net/users/14094 | 447471 | 180,203 |
https://mathoverflow.net/questions/258115 | 14 |
>
> Here is the short version of the combinatorial problem:
>
>
> Given a positive integer $n \geq 2$. Draw a circle with $2n$ points indexed by the numbers from $\mathbb{Z}/ 2n \mathbb{Z}$. We colour the points black and white and assume that there is at least one black point. Since the problem is invariant under ... | https://mathoverflow.net/users/61949 | Special configurations on a circle from a homological algebra problem | There is a simple characterization of interesting configurations:
**Lemma.** A configuration $x\_0=0< x\_1 < x\_2 < ... <x\_r$ of Gorenstein dimension $g$ is *interesting* if and only if there exist indices $i,j$ such that $x\_i$ is even, $x\_j$ is odd, and
$$x\_{i+1} - x\_i \equiv x\_{j+1}-x\_j \equiv g\pmod{2n},$$
... | 3 | https://mathoverflow.net/users/7076 | 447475 | 180,204 |
https://mathoverflow.net/questions/447476 | 2 | There is a famous classification of the path algebras of finite acyclic quivers into finite, tame, and wild representation types. For quivers with cycles, it is standard that the 2-loop quiver (with only one vertex) is wild (perhaps the original wild problem?), and then it is not too hard to show that any quiver with t... | https://mathoverflow.net/users/38434 | Tame/wild classification of *cyclic* quivers? | Dynkin quivers are of finite type, Euclidean quivers are of tame type, and all other quivers are wild. In particular, if there is a Euclidean proper subquiver then the representation type is wild. A cycle is an $\tilde A\_n$ subquiver. So if a quiver contains a cycle but is not a cycle then it is wild.
| 5 | https://mathoverflow.net/users/460592 | 447477 | 180,205 |
https://mathoverflow.net/questions/447403 | 58 | I am currently working on a proof that would need to use the following theorem that I cannot prove:
"Let $A$ be a finite set of positive real numbers. Then, the set $A + A - A$ contains at least as many positive elements as negative elements. ($0$ is counted as a positive element)"
To clarify: The set $A + A - A$ i... | https://mathoverflow.net/users/505485 | For a finite set A of positive reals, prove that the set A + A - A contains at least as many positive as negative elements | Here is a counterexample. We first need a "[more sums than differences](https://arxiv.org/abs/math/0608131)" construction:
**Lemma**. For any $\varepsilon>0$ there exists a cyclic group ${\bf Z}/N{\bf Z}$ and a non-empty subset $A \subset {\bf Z}/N{\bf Z}$ such that $A+A = {\bf Z}/N{\bf Z}$ but $|A-A| \leq \varepsilo... | 75 | https://mathoverflow.net/users/766 | 447486 | 180,207 |
https://mathoverflow.net/questions/447473 | 0 | Let $A \to S$ be an abelian scheme over an irreducible curve $S$ over complex numbers of relative dimension $g \geq 1$. Let $s: S \to A$ be a non-constant section and denote by $X:=s(S)$. Is it true that $\mathbb{Z}\cdot X:= \bigcup\_{n \in \mathbb{Z}} nX$ is Zariski dense in A?
I expect it is false in general but ca... | https://mathoverflow.net/users/146212 | Zariski dense in abelian scheme | How about if $A = E\_1\times E\_2$ with $E\_i$ non isotrivial elliptic curves and $s: S \to P\times 0$. Then your $\mathbb Z\cdot X$ is contained in $E\_1\times 0$ and so is not Zariski dense. In fact, this is the only thing that can happen.
Let $Z$ be the Zariski closure of $\mathbb Z\cdot X$ in $A$. Then $Z$ is clo... | 4 | https://mathoverflow.net/users/58001 | 447496 | 180,210 |
https://mathoverflow.net/questions/447447 | 2 | There is a prevalent method called the "Nonlinear adjoint method" in the study of viscosity solution and Hamilton--Jacobi equation, especially equations of the form
$$
u^\varepsilon + H(x,Du^\varepsilon) - \varepsilon \Delta u^\varepsilon = 0
$$
It usually requires taking derivative
$$
\frac{d u^\varepsilon}{d\var... | https://mathoverflow.net/users/124759 | Smooth dependence of parameter of PDE - viscosity solutions | You need to justify this with finite differences. I will give only a sketch of proof. What you do is formally differentiate the equation to get an equation for $v^\epsilon = (\partial/\partial \epsilon) u^\epsilon$. You prove uniqueness of this equation, which is presumably implied by your assumptions and some regulari... | 4 | https://mathoverflow.net/users/5678 | 447498 | 180,211 |
https://mathoverflow.net/questions/447499 | 1 | Let $\Sigma^n \subset S^{n+1}$ be a codimension one, embedded minimal hypersurface in the sphere. As the sphere has positive Ricci curvature, this must be unstable. In particular, perturbing $\Sigma$ in the direction of the first Jacobi eigenfunction produces an embedded surface $\Sigma'$ that is mean-convex, with mean... | https://mathoverflow.net/users/103792 | Singularities of mean-convex MCF in the sphere? | For topological reasons you can see that any minimal surface $\Sigma\subset \mathbb{S}^3$ that is not a sphere or a torus has to give rise to a mean convex flow that becomes singular before it disappears. This is essentially because the only possible singularities are spheres and cylinders which prevents the flow from ... | 3 | https://mathoverflow.net/users/127803 | 447502 | 180,213 |
https://mathoverflow.net/questions/447509 | 0 | I am looking for references studying orthonormal systems of functions $\{h\_n\}\_{n\geq0}$ on a sphere $S^d$ ($d=2$ or $d\geq2$) with respect to weights that are not uniform (unlike spherical harmonics). That is, such that
$$
\int\_{S^d} h\_n(x)h\_m(x)w(x)\,\mathrm{d}x=\delta\_{nm}.
$$
I am particularly interested in w... | https://mathoverflow.net/users/54638 | Relatively explicit orthogonal systems on the sphere that are not spherical harmonics | This problem immediately reduces to one on orthogonal systems on the interval $[-1,1]$.
Indeed, suppose that $h\_n(x)=H\_n(x\_1)$ for all integers $n\ge0$ and all $x\in S^d$, where $H\_n$ is a polynomial of degree $n$. Then
$$\int\_{S^d}h\_n(x) h\_m(x)\,dx=\int\_{-1}^1 H\_n(x\_1) H\_m(x\_1) W(x\_1)\,dx\_1,$$
where $W... | 2 | https://mathoverflow.net/users/36721 | 447523 | 180,219 |
https://mathoverflow.net/questions/447474 | 1 | Consider an orthonormal matrix $W\in\mathbb{R}^{2n\times 2n}$ that satisfies the "abs property" $$|w\_i|^T |w\_{i+n}|=1$$ for all $i \in \{1,2,\ldots,n\}$, where $w\_i \in \mathbb{R}^{2n}$ is the $i$-th column of $W$ and $|w\_i|$ denotes the vector of absolute values of $w\_i$.
It can be shown that $(|w\_i| - |w\_{i+... | https://mathoverflow.net/users/505577 | Orthonormal matrices with columns that switch signs | This is not true.
For example, consider $H\_{12}$: the $12 \times 12$ [Hadamard matrix](https://en.wikipedia.org/wiki/Hadamard_matrix). Well, $H\_{12}/\sqrt{12}$ is an orthonormal matrix with the "abs property", but the matrices $U$ and $V$ cannot possibly exist since $\sqrt{6}U$ and $\sqrt{6}V$ would be $6 \times 6$... | 1 | https://mathoverflow.net/users/11236 | 447531 | 180,221 |
https://mathoverflow.net/questions/447527 | 7 | Let $C, D$ two Grothendieck sites. Since the corresponding toposes $E, F$ are locally presentable categories, then (by Gabriel-Ulmer duality) they correspond to limit theories $X, Y$ (that is, small categories having all $\lambda$-small limits; here $\lambda$ is chosen larger of two). Now we can consider $X$-objects in... | https://mathoverflow.net/users/148161 | Tensor product of sites | The category $H$ can be described as the category of $E$-valued sheaves on $D$, or $F$-valued sheaves on $C$.
You get a site by taking the category $C \times D$ and taking the topology generated by the $(c\_i,d) \to (c,d)$ for $c\_i \to c$ a covering family in $C$, and the $(c,d\_i) \to (c,d)$ for $d\_i \to d$ a cove... | 10 | https://mathoverflow.net/users/22131 | 447532 | 180,222 |
https://mathoverflow.net/questions/447519 | 3 | I have been looking at some papers on the "restricted Burnside problem". On page 4 of Vaughan-Lee and Zelmanov's survey, "[Bounds in the restricted Burnside problem](https://doi.org/10.1017/S144678870000121X)", I think they implicitly are invoking the claim that:
>
> If $G$ is a Lie algebra over $\Bbb{F}\_p$ with n... | https://mathoverflow.net/users/130484 | Bounding size of group by number of generators, order of elements, and nilpotency class (Restricted Burnside's) | Converting my [comment](https://mathoverflow.net/questions/447519/bounding-size-of-group-by-number-of-generators-order-of-elements-and-nilpotenc#comment1156329_447519) into an answer:
Let $G = G\_1 \ge G\_2 \ge \cdots$ be the lower central series. Then $G\_k/G\_{k+1}$ is spanned by the left-normed commutators $[x\_1,... | 3 | https://mathoverflow.net/users/20598 | 447534 | 180,223 |
https://mathoverflow.net/questions/447515 | 6 | **Question:** Does $\overline{\mathbb{CP}^2 \setminus B^4}$ (that is the closure of complex projective plane minus a 4-ball) embed (smoothly/topologically/piecewise linearly) in $\mathbb R^5$?
**Background:** The question I am really interested in is whether the connected sum $\#\_k(\mathbb{CP}^2 \# -\mathbb{CP}^2)$ ... | https://mathoverflow.net/users/15650 | Does $\overline{\mathbb{C}P^2 \setminus B^4}$ embed (smoothly/topologically) in $\mathbb R^5$? | $\def\RR{\mathbb{R}}\def\CC{\mathbb{C}}\def\PP{\mathbb{P}}$I think the answer is no. Topology isn't my strength, so check this argument.
Let $\infty$ be the point $[0:0:1]$ in $\CC \PP^2$ and put $U = \CC\PP^2 \setminus \{ \infty \}$. Note that $\{ [z\_1:z\_2:0] \} \cong \CC\PP^1 \cong S^2$, I'll always identify $S^2... | 6 | https://mathoverflow.net/users/297 | 447536 | 180,224 |
https://mathoverflow.net/questions/447524 | 1 | Let $(X,\sigma,\mu)$ be a measure space, $\mu(X)=1$. End let $T:X\to X$ be a measurable map such that
$$D\in\sigma\Longrightarrow \mu(T^{-1}D)=\mu(D).$$ Let $f:X\to\mathbb{R}\_+$ be an integrable function. Construct a measurable function as follows
$$\psi\_n:=\Big(\limsup\_{k\to\infty}\frac{1}{k}\sum\_{i=0}^{k-1}fT^i\B... | https://mathoverflow.net/users/486323 | near ergodic theory question | I believe you are referring to the function $\lambda$ defined in the first paragraph of page $250$ which carries an extra $-1/n$ (this is irrelevant for the invariance). If I am not missing anything, you have the following.
Define $g\_{\ell}\overset{\Delta}= \limsup\_{k\rightarrow \infty} \frac{1}{k} \sum\_{i=0}^{k-1... | 2 | https://mathoverflow.net/users/138242 | 447538 | 180,226 |
https://mathoverflow.net/questions/447548 | 1 | Let $V$ be a finite-dimensional vector space and
$$
U \subseteq W \subseteq V \otimes V
$$
be a proper inclusion of vector subspaces. Then take the tensor algebra
$$
T(V) = \bigoplus\_{i=1}^{\infty} V^{\otimes i}
$$
and the quotients
$$
A\_W = T(V)/\langle W \rangle, ~~~~~~~~~~~ A\_U = T(V)/\langle U \rangle
$$
by the ... | https://mathoverflow.net/users/126606 | A question about surjective maps between quadratic algebras | There are plentiful examples of $m$-generated quadratic algebras $A$ with $A\_3 = 0$, and dimension of relation subspace $R$ being as low as $m^2/2$. One of those algebras is $$\Bbb k\langle x\_1, y\_1, \dots, x\_n, y\_n \rangle / (x\_i y\_j, x\_i x\_j - y\_i y\_j).$$ Moreover, if $\operatorname{dim} R \geq 3m^2/4$, th... | 5 | https://mathoverflow.net/users/81055 | 447553 | 180,231 |
https://mathoverflow.net/questions/447545 | 14 | $\DeclareMathOperator\GL{GL}$Let $G$ be a subgroup of $\GL\_n(\mathbb{C})$ such that for every $g \in G$ there exists $c \in \GL\_n(\mathbb{C})$ for which $cgc^{-1}$ is unitary (or, which is the same, $g$ is diagonalizable with unimodular eigenvalues). Does there exist $c \in \GL\_n(\mathbb{C})$ such that $cGc^{-1}$ co... | https://mathoverflow.net/users/505475 | Group of matrices in which every matrix is similar to unitary | An example in $GL(3, {\mathbb C})$ was first given by Bass (answering a question by Kaplansky) in Example 1.10 of
*Bass, Hyman*, [**Groups of integral representation type**](https://doi.org/10.2140/pjm.1980.86.15), Pac. J. Math. 86, 15-51 (1980). [ZBL0444.20006](https://zbmath.org/?q=an:0444.20006).
In section 1 of... | 15 | https://mathoverflow.net/users/39654 | 447554 | 180,232 |
https://mathoverflow.net/questions/447563 | 2 | Let $M, N$ be smooth manifolds with $M$ orientable and compact. Let $\sigma$ be some volume form on $M$ and consider the set $\mathcal{M}$ of smooth maps from $M$ to $N$ in a fixed homotopy class. Now consider the group $G$ of volume-preserving diffeomorphisms on $M$. This group naturally acts on the whole space of smo... | https://mathoverflow.net/users/480923 | Does composition on the right by a volume-preserving diffeomorphism preserve homotopy class? | Let $X$ be a smooth, compact, orientable manifold and let $\omega$ be a choice of volume form. On $X\times X$, we have the natural volume form $\sigma = \pi\_1^\*\omega \wedge \pi\_2^\*\omega$ where $\pi\_i : X\times X \to X$ is projection onto the $i^{\text{th}}$ factor. The diffeomorphism $s : X\times X \to X\times X... | 6 | https://mathoverflow.net/users/21564 | 447564 | 180,235 |
https://mathoverflow.net/questions/447550 | 2 | Let $X$ and $Y$ be smooth projective complex varieties. Suppose we have a Fourier-Mukai equivalence
$$
\Phi\_\mathcal P :Perf X \to Perf Y
$$
with kernel $\mathcal P$. Moreover, suppose $\mathcal P$ locally corresponds to a graph of a birational map $\phi:X \supset U \cong V \subset Y$. Now, if there is a (necessarily ... | https://mathoverflow.net/users/177839 | A Fourier-Mukai kernel locally given by a graph of a birational map and compatibility with extension | This is not true, you can find a well-known counter example in
[Namikawa, Yoshinori. Mukai flops and derived categories. J. Reine Angew. Math. 560 (2003), 65–76].
| 2 | https://mathoverflow.net/users/4428 | 447565 | 180,236 |
https://mathoverflow.net/questions/447558 | 4 | I am a student learning Iwasawa theory. I am so sorry if this post is too trivial for this site. I posted it on math.stackexchange yesterday but obtained no responce.
A quite basic object is the Iwasawa algebra. A basic version is as follows:
Let $L/\mathbb{Q}\_p$ be a finite extension (i.e. a local number field) w... | https://mathoverflow.net/users/161208 | Geometric interpretation of Iwasawa algebras: $\mathbb{Z}_p[[T]]$ as a disk? | The correct viewpoint is not "$\Lambda$ is like a disc", but "$\Lambda$ is like the *functions on* a disc".
To see this, ask yourself: given an element $f \in \mathbb{Z}\_p[[T]]$, what values can we plug in for $T$ such that the series will converge? It's pretty easy to see that if we put $T = x$ for some $x \in \mat... | 7 | https://mathoverflow.net/users/2481 | 447570 | 180,239 |
https://mathoverflow.net/questions/447303 | 3 | I want to assign a finite number, $n(G)$, to a finite group $G$ such that if $H$ is a proper retract of $G$, then $n(H)\lneq n(G)$. By a retract of $G$, I mean a subgroup $H$ of $G$ for which there is an epimorphism $r:G\to H$ such that $r(x)=x$ for all $x\in H$. By a proper retract, I mean a retract $H$ such that $H\n... | https://mathoverflow.net/users/505169 | Assigning a finite number, $n(G)$, to a finite group $G$ with this property | As you said yourself, you could define $n(G) = |G|$, but you are not happy with that. I am guessing that you would like $n(G)$ to be as small as possible. The length of a chief series of $G$ would also work, and would be an improvement on $|G|$, but we can do better than that.
I propose to define $n(G)$ to be the max... | 3 | https://mathoverflow.net/users/35840 | 447589 | 180,242 |
https://mathoverflow.net/questions/447579 | 0 | For any set $S\subseteq \mathbb{Z}\times\mathbb{Z}= \mathbb{Z}^2$ and $a\in \mathbb{Z}^2$, we set $a+S = \{a+s: s\in S\}$, where $+$ is the componentwise addition in $\mathbb{Z}^2$. Moreover, for any collection of subsets ${\frak S}\subseteq {\cal P}(\mathbb{Z}^2)$ we let $a + {\frak S} = \{a+S: S\in{\frak S}\}$.
We ... | https://mathoverflow.net/users/8628 | Does $\mathbb{Z}\times\mathbb{Z}$ have an aperiodic monotile? | Yes. A $2$-by-$2$ square $\{0,1\}^2$ can tile $\mathbb{Z}^2$ with just one period. So $\{0,2\}^2$ can tile $2\mathbb{Z}^2 \leq \mathbb{Z}^2$ with just one period. Break other periods in the other cosets.
| 5 | https://mathoverflow.net/users/123634 | 447596 | 180,243 |
https://mathoverflow.net/questions/447593 | 7 | *Crossposted from [Math.SE 4698387](https://math.stackexchange.com/questions/4698387/).*
---
In the [rational sequence topology](https://topology.pi-base.org/spaces/S000057/properties), rationals are discrete and irrationals have a local base defined by choosing a Euclidean-converging sequence of rationals and de... | https://mathoverflow.net/users/73785 | Is every rational sequence topology homeomorphic? | There are lots of classes of RSTs (rational sequence topologies) in $\mathbb{R}$ up to homeomorphism. Note that, as mentioned in the answer by Will Brian, any homeomorphism $(\mathbb{R},T\_1)\to(\mathbb{R},T\_2)$ between RST spaces is induced by some bijection $\mathbb{Q}\to\mathbb{Q}$. So any homeomorphism class of RS... | 9 | https://mathoverflow.net/users/172802 | 447601 | 180,245 |
https://mathoverflow.net/questions/447587 | 3 | Let $F$ be a continuous convex function on $\mathbb{R}^n$.
If the subdifferential $\partial F(x)$ of $F(x)$ admits a continuous selection, for every $x \in \mathbb{R}^n$, does it mean that $F$ is differentiable on $ \mathbb{R}^n$ ?
I was trying to use theorem 25.5 and 25.6 (Rockafellar: Convex Analysis) but the nor... | https://mathoverflow.net/users/505680 | Subdifferential of a convex function admits a continuous selection | $\newcommand\p\partial\newcommand\R{\mathbb R}\newcommand\cl{\operatorname{cl}}\newcommand\conv{\operatorname{conv}}$The answer is yes, and you were almost there.
Indeed, suppose the contrary: that we we have a continuous function $f\colon\R^n\to\R^n$ such that $f(z)\in\p F(z)$ for all $z\in\R^n$ and yet
$F$ is not d... | 3 | https://mathoverflow.net/users/36721 | 447607 | 180,246 |
https://mathoverflow.net/questions/447582 | 0 | Let $G$ be a graph of $n$ arcs and let $x\in \mathbb{R}^n$. I want to compute the orthogonal projection of $x$ onto the set of radial graphs with $k$ roots contained in $G$ (or a forest with $k$ root) i.e. $$\Pi \left( x\right) =\underset{y\in S}{\arg \min }\left\Vert x-y\right\Vert
^{2},$$
where $S$ is the underlying ... | https://mathoverflow.net/users/334710 | Minimum spanning tree and projection | I think you mean that $y$ is the characteristic vector. That is, $y\_{ij}=1$ if edge $(i,j)$ is in the forest and $0$ otherwise. Let $E$ be the edge set of $G$. Given $x$, you want to find $y$ to minimize
$$
\sum\_{(i,j)\in E} (x\_{ij}-y\_{ij})^2
= \sum\_{(i,j)\in E} (x\_{ij}^2-2x\_{ij}y\_{ij}+y\_{ij}^2)
= \sum\_{(i,j)... | 2 | https://mathoverflow.net/users/141766 | 447608 | 180,247 |
https://mathoverflow.net/questions/447603 | 0 | Let $f: \mathbb{R} \to \mathbb{R}$ be a smooth function such that $f(x)$ is positive in a small punctured neighborhood of $x=0$ but $f(0)=0$.
Now, define a collection of centered Gaussian measures on $\mathbb{R}$ as
\begin{equation}
d\mu\_a(x):=\frac{1}{\sqrt{2\pi}f(a)} e^{-\frac{1}{2}\frac{x^2}{[f(a)]^2}}
\end{equat... | https://mathoverflow.net/users/56524 | Convergence of Gaussian measures $\{ d\mu_a \}$ whose variances depend smoothly on the index $a$ | $\newcommand\R{\mathbb R}$You need some restriction on the rate of growth of the smooth function $F$. Otherwise, if e.g. $F(x)=e^{|x|^p}$ for $p>2$, then $\int\_{\R}F(x)\mu\_a(dx)=\infty\not\to F(0)$ (as $a \to 0$) and similarly $\int\_\R(F(x)-F(0))^2\mu\_a(dx)=\infty$.
So, assume that
$$|F(x)|\le Ce^{Cx^2} \tag{1}\l... | 2 | https://mathoverflow.net/users/36721 | 447612 | 180,248 |
https://mathoverflow.net/questions/447541 | 3 | Given an integer partition $\lambda$, introduce the following quantities:
\begin{align\*}
c(\lambda)&=\sum\_{i\geq1}\left\lceil\frac{\lambda\_i}2\right\rceil, \qquad c\_o(\lambda)=\sum\_{i\geq1}\left\lceil\frac{\lambda\_{2i-1}}2\right\rceil, \qquad
c\_e(\lambda)=\sum\_{i\geq1}\left\lceil\frac{\lambda\_{2i}}2\right\rcei... | https://mathoverflow.net/users/66131 | Seeking for a combinatorial argument for partition identities | It's easy to check that all four identities are true when the sum is restricted to two terms indexed by a partition $\lambda$ and its conjugate $\lambda'$. (If $\lambda=\lambda'$ then the two terms are equal, so we only need one term.) From this (1)-(4) follow by grouping the terms in conjugate pairs (or just one term ... | 5 | https://mathoverflow.net/users/2807 | 447623 | 180,251 |
https://mathoverflow.net/questions/447615 | 1 | Finding the real positive solution of $x^n+x-c^2;\ 2\lt n\in\mathbb{N},\,c\in\mathbb{R},$ comes from a practical problem I encountered.
>
> **Question:**
>
>
> what can be said about exact solutions for specific values of $n$ and/or non-recursive functions $f(\xi,\eta)\mapsto x$ such that yields $\min\max\limits\... | https://mathoverflow.net/users/31310 | Real solutions of $x^n+x-c^2=0$ | Hardly anything interesting can be said here for small $n$. So, let $n$ be large. Let $x=x\_{c,n}$ be the unique nonnegative zero of $f(x):=x^n+x-c^2$. We have to distinguish the following three cases.
*Case 1: $c^2<1$.* Then $x<c^2$ and hence
$$c^2-c^{2n}<x\_{c,n}<c^2,$$
so that $x\_{c,n}=c^2+o(1)$ exponentially fas... | 6 | https://mathoverflow.net/users/36721 | 447624 | 180,252 |
https://mathoverflow.net/questions/428771 | 8 | Let $M$ be a complex manifold, and $Z \subset M$ a closed real analytic subvariety. Suppose that the set of smooth points in $Z$ is complex analytic in $M$. Will it follow that $Z$ is complex analytic? I can deduce this statement from Remmert-Stein theorem when $2\dim\_R S < \dim\_R Z$, where $S$ is the singular set of... | https://mathoverflow.net/users/3377 | Real analytic subvariety in complex manifold which is complex outside of its singular set | There is a paper by Hans-Jörg Reiffen called "Fastholomorphe Algebren" (1970) where he apparently proves something like this. [Here](https://link.springer.com/article/10.1007/BF01338660) is a link to the springer publication. I have not read the paper in detail so I can not really comment on the methods used there. Mor... | 4 | https://mathoverflow.net/users/109193 | 447630 | 180,254 |
https://mathoverflow.net/questions/447626 | 2 | If we have a real random variable $X$ such that $a\leq X\leq b$ almost surely, we can establish the following inequality:
\begin{align}
\mathbb{E}\left[\exp\Big(t(X-\mathbb{E}[X])\Big)\right]\leq\exp\left(\frac{1}{8}t^2(b-a)^2\right).
\end{align}
Is there a similar bound for a complex random variable $Y$ such that it s... | https://mathoverflow.net/users/68835 | Hoeffding's Lemma for bounded complex random variables? | Restricting $Y$ to an annulus doesn't seem useful as any bounds are likely to be satisfied also inside the annulus.
A bound with $Y$ restricted to a disk, or more generally to a region with bounded diameter, appears in Mikhail Isaev and Brendan McKay. "On a bound of Hoeffding in the complex case." Electron. Commun. P... | 3 | https://mathoverflow.net/users/9025 | 447634 | 180,257 |
https://mathoverflow.net/questions/447166 | 5 | In a previous [MO post](https://mathoverflow.net/a/447123/12905), H. Cohen suggested [Gorodetsky's 2021 paper](https://arxiv.org/abs/2102.11839) which discussed $6+6+3=15$ "*sporadic sequences*". The first 6 are Zagier's sporadic sequences, the second 6 are by Almkvist-Zudilin, while the last 3 is credited to Cooper.
... | https://mathoverflow.net/users/12905 | On the continued fractions using Cooper's sequences $s_7,\, s_{10},\, s_{18}$ and the Zudilin-Cohen sequence | *(This answers Question 2.)*
Thanks to [Cohen's 2022 paper](https://arxiv.org/abs/2212.01095), turns out *there* is a deg-$4$ and one can find polynomials $Q\_k(n)$ for general deg-$k$ such that,
$$(n+1)^k s\_{n+1} = Q\_k (n)\, s\_n - n^k s\_{n-1}\qquad\tag{eq.1}$$
is a 3-term recurrence relation. This can then b... | 0 | https://mathoverflow.net/users/12905 | 447642 | 180,259 |
https://mathoverflow.net/questions/368893 | 11 | I am trying to make my way into Homotopy Type Theory(HoTT) where a mathematician may view proofs as paths. Intuitively, this leads me to the idea of a metric on the space of mathematical propositions. Has this been developed?
Specifically, is there a way to analyse short proofs as geodesics within the space of
mathem... | https://mathoverflow.net/users/56328 | natural metrics for proof length | Inspired by the informal notion of *Cognitive distance*, in 2010 Charles Bennett, Peter Gács, Ming Li, Paul Vitanyí and Wojcech Zurech introduced the notion of *Information Distance* which was used in the seminal paper *Clustering by Compression*:
\begin{equation}
ID(x,y)=\min\{|p|:p(x)=y \land p(y)=x\} \tag{1}
\end{... | 2 | https://mathoverflow.net/users/56328 | 447652 | 180,263 |
https://mathoverflow.net/questions/447648 | 5 | Suppose $\mathcal{F} \subseteq \mathcal{P} (\omega) $ is an almost disjoint family and $\aleph\_0 < \vert \mathcal{F} \vert = \kappa < 2^{\aleph\_0} $. Is it consistent that for some such cardinal $\kappa$, we can uniformize every two-valued function on the family; that is, if $\mathcal{F} = \langle A\_i \mid i < \kapp... | https://mathoverflow.net/users/495743 | Uniformization of almost disjoint families | No, this is not consistent: there is (provably in ZFC) an almost disjoint family of size $\aleph\_1$ and a two-valued function on that family such that the function cannot be uniformized in the way you've described.
To get such a family and function, we'll use a [Hausdorff gap](https://en.wikipedia.org/wiki/Hausdorff... | 8 | https://mathoverflow.net/users/70618 | 447654 | 180,264 |
https://mathoverflow.net/questions/447641 | 1 | We know two basic random graph models:$G(n,p)$ and $G(n,m)$. $G(n,m)$ consists of all graphs with $n$ vertices and $m$ edges, in which the graphs have the same probability. We know that $G(n,p)$ and $G(n,m)$ are asymptotically equivalent when $C\_n^2 p=m$. So when we want to calculate the probability of some property i... | https://mathoverflow.net/users/178444 | Random graph uniformly sample from the special graphs | There is a notion of *contiguity* for models of graphs. Two sequences of probability spaces
$$\{(\Omega\_n, \mathcal{F}\_n, P\_n)\}\_{n\in\mathbb{N}} \text{ and } \{(\Omega\_n, \mathcal{F}\_n, \tilde{P}\_n)\}\_{n\in\mathbb{N}}$$
(with the same underlying measurable spaces $(\Omega\_n, \mathcal{F}\_n)$) are *contiguous*... | 1 | https://mathoverflow.net/users/75344 | 447656 | 180,265 |
https://mathoverflow.net/questions/447660 | 0 | We consider the heat kernel
$$
g :\mathbb R\_{>0} \times \mathbb R^d \to \mathbb R, (t, x) \mapsto \frac{1}{(4\pi t)^{d/2}} \exp \bigg ( - \frac{|x|^2}{4t} \bigg ).
$$
I have already proved that
$$
\begin{align}
g(t, x+y) &\le 2^{d/2} g(2t, x) e^{|y|^2/(4t)}, \\
\nabla g(t, x) &= \frac{-x}{2t} g(t, x).
\end{align}
$$... | https://mathoverflow.net/users/477203 | An estimate of the gradient of heat kernel | By your computation, it suffices to show that
$$\frac{|x|}{2\sqrt{t}}\leq 2^{d/2}\frac{g(2t,x)}{g(t,x)}=e^{\frac{|x|^2}{4t}}$$
This follows from the fact that $y\leq e^{y^2}$ for $y\geq 0$.
| 3 | https://mathoverflow.net/users/161306 | 447665 | 180,267 |
https://mathoverflow.net/questions/447664 | 22 | In a recent preprint [On the invariant subspace problem in Hilbert spaces](https://arxiv.org/abs/2305.15442) *Per H. Enflo* claims to have solved the [invariant subspace problem](https://en.wikipedia.org/wiki/Invariant_subspace_problem), showing that every bounded linear operator on a separable complex Hilbert space ha... | https://mathoverflow.net/users/59033 | Understanding a simplifying assumption in Invariant Subspace Problem proof | (If a moderator wants to remove this/make it a comment, please do. For various reasons I do not want to have an account on this site) )
I believe one simply needs to know that the spectrum of $T$ is nonempty, so up to replacing $T$ by some $T - \lambda I$ one can assume wlog that $0$ is in the spectrum of $T$ (of cou... | 23 | https://mathoverflow.net/users/nan | 447666 | 180,268 |
https://mathoverflow.net/questions/447684 | 1 | Does anyone have any references for iterated exponential sums? That is, sums like
$$\sum\_{1\leq n\leq X} e(f(n))\sum\_{1\leq m\leq n} e(f(m)),$$
where $e(x)=e^{2\pi i x}$? I am looking for references that give bounds, limit theorems, anything like that.
| https://mathoverflow.net/users/479223 | Iterated exponential sums | This sum is equal to
$$
\frac{\left( \sum\_{n \leq x} e(f(n)) \right)^2 - \sum\_{n \leq x} e(2 f(n))}{2}
$$
In most situations, we do not expect there to be more than square root cancellation, and thus one would expect that this sum is asymptotically equal to
$$
\frac{1}{2} \left( \sum\_{n \leq x} e(f(n)) \right)^2
$$
... | 2 | https://mathoverflow.net/users/88679 | 447702 | 180,272 |
https://mathoverflow.net/questions/439480 | 5 | Brownian bridges are interpreted as Brownian motions conditioned to start and end at given points. However, I have not seen a source that makes this precise, though this may be due to my own lack of exposure.
Let $W$ be a standard one dimensional Brownian motion on $[0, T]$, started at $0$. Denote by $\nu\_y$ the con... | https://mathoverflow.net/users/173490 | Brownian bridges as conditioning | If $X$ and $Y$ are independent random variables taking values in arbitrary spaces $E$ and $F$, if $Z = f(X,Y)$ for any measurable map $f : E \times F \to G$, then the family of distributions of the random variables $(F(X,y))\_{y \in F}$ provides a version of conditional law $\mathcal{L}(Z|Y)$. Indeed, an application of... | 1 | https://mathoverflow.net/users/169474 | 447704 | 180,273 |
https://mathoverflow.net/questions/447610 | 2 | I'm interested in solving a first-order linear PDE with 2 dependent variables in 3 dimensions by the method of characteristics. Something of this general form:
$$
A \frac{\partial u}{\partial x} + B \frac{\partial u}{\partial y} + C \frac{\partial u}{\partial z} + D \frac{\partial v}{\partial x} + E \frac{\partial v}... | https://mathoverflow.net/users/505696 | Method of characteristics with 2 dependent variables in 3 dimensions | The method of characteristics is a bit strange here because the equation is underdetermined, so one can't expect to be able to specify a solution by fixing initial data for $u$ and $v$ along a surface in $xyz$ space. It would be more efficient to put the equation in normal form and use the integration method appropriat... | 3 | https://mathoverflow.net/users/13972 | 447710 | 180,276 |
https://mathoverflow.net/questions/447709 | 10 | How can I solve this equation?
$$7^{x} +2=y^{2}$$
$x$ and $y$ must be natural numbers.
| https://mathoverflow.net/users/495657 | Integer solutions of an exponential equation | A general method, not necessarily best for this particular equation, is to split into three cases by writing $x=3u+v$ with $v\in\{0,1,2\}$. Then rewrite your equation as
$$ A(7^u)^3 + 2 = y^2\quad\text{with $A\in\{1,7,49\}$.} $$
So any solution to your equation gives an integer solution to one of the three equations
$$... | 18 | https://mathoverflow.net/users/11926 | 447712 | 180,277 |
https://mathoverflow.net/questions/447693 | 4 | Let $X$ be a locally finite type algebraic stack $X$ (but feel free to pretend it's a scheme) with a presentation as the filtered colimit of finite type open substacks $U\_i$. By descent, at the level of stable derived categories of $\ell$-adic sheaves we have $$D(X) = \varprojlim D(U\_i).$$ Suppose I have two objects ... | https://mathoverflow.net/users/333154 | Gluing isomorphism in derived categories along filtered colimit | The hope is reasonable if the open cover really is indexed by $\mathbb N$. Indeed, letting $Spine(\mathbb N)$ denote the simplicial set which can be depicted as $0\to 1\to 2\to ...$ (where the only nondegenerate simplices are the ones I've drawn, in particular there is no $1$-simplex $0\to 2$ !), the inclusion $Spine(\... | 4 | https://mathoverflow.net/users/102343 | 447715 | 180,279 |
https://mathoverflow.net/questions/447711 | 1 | Let $G=2^{2n}{:}Sp\_{2n}(2)$ be the split extension, where the symplectic group $Sp\_{2n}(2)$ acts naturally on the vector space $2^{2n}$. With the aid of GAP it turns out that the automorphism group $\textrm{Aut}(G)\cong G{:}2\cong 2^{2n+1}{:}Sp\_{2n}(2)=G\_1$ for $n=2, 3, 4$. Furthermore, the group $G\_1$ is not isom... | https://mathoverflow.net/users/148317 | The automorphism group of $2^{2n}{:}Sp_{2n}(2)$ | Since ${\rm Sp}(2n,2)$ has trivial outer automorphism group and its natural module $M$ is absolutely irreducible, this follows from the fact that $|H^1({\rm Sp}(2n,2),M)| =2$.
You can find that result, for example, in Table 4.5 of
Cohomology of finite groups of Lie type, I,
Edward Cline; Brian Parshall; Leonard Sco... | 6 | https://mathoverflow.net/users/35840 | 447719 | 180,280 |
https://mathoverflow.net/questions/447677 | 0 | Let $Y$ denote a Gaussian random variable characterized by a mean $\mu$ and a variance $\sigma^2$. Consider $N$ independent and identically distributed (i.i.d.) copies of $Y$, denoted as $Y\_1, Y\_2, \ldots, Y\_N$. Now, let's examine the number of $N$ for which the probability satisfies the inequality:
\begin{align}
... | https://mathoverflow.net/users/68835 | Is it reasonable to consider the subgaussian property of the logarithm of the Gaussian pdf? | $\newcommand{\de}{\delta}\newcommand{\ep}{\epsilon}\newcommand{\si}{\sigma}\newcommand{\R}{\mathbb R}$We have
\begin{equation\*}
L\_i:=-\ln f(Y\_i) = c+Z\_i^2/2,
\end{equation\*}
where $c:=\frac12\ln(2\pi\si^2)$ and $Z\_i:=\frac{Y\_i-\mu}\si$, so that the $Z\_i$'s are independent standard normal random variables (r.v.... | 1 | https://mathoverflow.net/users/36721 | 447720 | 180,281 |
https://mathoverflow.net/questions/447718 | 2 | Let $f\colon [0, \infty)\to\mathbb{R}$ be nonnegative, nondecreasing, and concave. Prove the following claim or give a counter example: There is a sequence of functions $f\_n\colon [0, \infty)\to\mathbb{R}$ with following properties:
1. $f\_n$ is nonnegative, nondecreasing, and concave for all $n\in\mathbb N$.
2. $f\... | https://mathoverflow.net/users/108436 | Smooth approximation of nonnegative, nondecreasing, concave functions | Replace the convolution on $\mathbb{R}$ by a convolution on the group $\mathbb{R}\_+^\*$, endowed with the invariant measure $dx/x$, namely set
$$f\_n(x) := \int\_0^\infty \varphi\_n(y) f(x/y) \frac{dy}{y},$$
where $(\varphi\_n)\_{n \ge 1}$ is a sequence of non-negative $\mathcal{C}^\infty$ functions on $\mathbb{R}\_+^... | 2 | https://mathoverflow.net/users/169474 | 447722 | 180,282 |
https://mathoverflow.net/questions/447696 | 2 | On his book "Introduction to transcendental numbers", page 99-100, Lang wrote
"Finally, we note that Lindemann actually proves something slightly stronger than the algebraic independenceof $e^{\alpha\_1},\cdots,e^{\alpha\_n}$ if $\alpha\_1,\cdots,\alpha\_n$ are linearly independent over $\mathbb Q$. He proved that if $... | https://mathoverflow.net/users/33128 | Lang's remark on Lindemann-Weierstrass theorem | For brevity of notation, if $I = (i\_1,\ldots,i\_n) \in \mathbf N^n$, write $x^I = x\_1^{i\_1}\cdots x\_n^{i\_n}$. Write $\boldsymbol \alpha$ for $(\alpha\_1,\ldots,\alpha\_n)$, and set $I \cdot \boldsymbol \alpha = i\_1\alpha\_1 + \ldots + i\_n\alpha\_n$.
If $e^{\alpha\_1},\ldots,e^{\alpha\_n}$ are algebraically dep... | 3 | https://mathoverflow.net/users/82179 | 447727 | 180,284 |
https://mathoverflow.net/questions/447629 | 3 | Let $$z(t)=\begin{pmatrix}x(t) \\ y(t)\end{pmatrix}~,~ A=\begin{pmatrix}0 & -\beta \\ \alpha & -(\alpha + \beta)\end{pmatrix}$$ satisfies the following system of linear differential inequalities $$z'(t) \leq A\,z(t)$$ where $\alpha, \beta > 0$ are fixed constants, and $x(t), y(t) \in [0,\infty)$, I am wondering if ther... | https://mathoverflow.net/users/163454 | Gronwall lemma for a $2$-dimensional system of linear differential inequalities | **Counter-example.** Let $\left(z(t)\right)\_{t\geq 0}$ be the solution to
$$\dot{z}(t)=A z(t)-\varepsilon \mathbf{e}\_2,$$
with $\mathbf{e}\_2:=\left[0 \,\,\,\,1\right]^{\top}$ and initial condition $z(0)$. Then, $\left(z(t)\right)\_{t\geq 0}$ obeys the inequality $\dot{z}(t)\leq Az(t)$ for all $t$.
Let $\left(x... | 1 | https://mathoverflow.net/users/138242 | 447728 | 180,285 |
https://mathoverflow.net/questions/447731 | 8 | Let $C$ be a locally finitely presented category, $A, B$ are two finitely presented (synonym: compact) objects in it. Is it true that $A \times B$ is finitely representable?
At least I have looked at a number of examples of categories of algebras of (finitary) algebraic theories and this seems to be true (although, s... | https://mathoverflow.net/users/148161 | Is the Cartesian product of two finitely presented objects finitely presentable? | No. For counterexamples, see Theorems 3.8, 3.9, and 3.10 of
Finiteness properties of direct products
of algebraic structures
Peter Mayr, Nik Ruškuc
Journal of Algebra 494 (2018) 167-187.
These theorems show that products of finitely presented loops, idempotent magmas, or lattices may fail to be finitely presenta... | 16 | https://mathoverflow.net/users/75735 | 447732 | 180,287 |
https://mathoverflow.net/questions/447513 | 5 | Let $W$ be a standard Brownian motion, and let $X$ be the solution to the one dimensional SDE
$$dX\_t = \sigma(t, X\_t) \, dW\_t$$
with initial condition $X\_0 = x\_0$ a.s. for some $x\_0 \in \mathbb R$. We assume $\sigma$ is Lipschitz continuous and uniformly bounded away from $0$.
Suppose that $X\_t$ admits a d... | https://mathoverflow.net/users/173490 | Does the entropy of a SDE with nondegenerate noise always increase? | $\newcommand{\si}{\sigma}\newcommand{\R}{\mathbb R}\newcommand{\pa}{\partial}$The answer is no. The idea is to get a diffusion version of my [two-state Markov chain example](https://mathoverflow.net/questions/447513/does-the-entropy-of-a-sde-with-nondegenerate-noise-always-increase#comment1156881_447513).
Indeed, for... | 2 | https://mathoverflow.net/users/36721 | 447735 | 180,289 |
https://mathoverflow.net/questions/447452 | 5 | Let $\kappa$ be an infinite cardinal number and by $\mathcal{B}(\kappa)$ denote the class of all Banach spaces of density $\kappa$.
My question reads as follows:
*Does there exist $\kappa$ for which there is $X\in\mathcal{B}(\kappa)$ such that for every $Y\in\mathcal{B}(\kappa)$ the dual space $X^\ast$ contains a c... | https://mathoverflow.net/users/15860 | Complemented subspaces of a dual Banach space | The answer is positive for $\kappa = \omega$. The space $X$ is the $\ell\_1$ sum of a sequence $(E\_n)$ of finite dimensional spaces such that for every $\epsilon>0$ and every finite dimensional $E$, $E$ is $1+\epsilon$-isomorphic to one of the $E\_n$. Basically the same argument works for every $\kappa$--use the $\ell... | 5 | https://mathoverflow.net/users/2554 | 447736 | 180,290 |
https://mathoverflow.net/questions/447609 | 1 | Let $(E, |\cdot|)$ be an infinite-dimensional Banach space. Assume that
* $T:E\to E$ is a compact (bounded linear) operator, and
* $(\lambda\_n)$ is a sequence of distinct eigenvalues of $T$.
Let $E\_n$ be the corresponding eigenspace of $\lambda\_n$. Then $(E\_n)$ is a sequence of finite-dimensional subspaces of $... | https://mathoverflow.net/users/477203 | Is there a bounded sequence $(e_n)$ such that $e_n \in E_n$ and that $(e_n)$ does not have any convergent subsequence? | The answer is negative. Take a biorthogonal sequence $(x\_n,x\_n^\*)$ in $E$ such that $(x\_n)$ converges to some unit norm vector $x\_0$. Set
$T=\sum\_n 2^{-n} \|x\_n\|^{-1} x\_n^\* \otimes x\_n$.
Such a sequence exists in every infinite dimensional space. For example, in $\ell\_2$ let $x\_n = e\_0 + n^{-1} e\_n$,... | 6 | https://mathoverflow.net/users/2554 | 447738 | 180,291 |
https://mathoverflow.net/questions/447689 | 0 | Let $T:=[-1,1]^{n-1}\times (0,1]$. Let
$$f\_n(x\_1,\cdots,x\_n,w\_1,\cdots,w\_n):=g(x\_1,w\_1)+\cdots+g(x\_n,w\_n)=\sum\_{i=1}^ng(x\_i,w\_i),$$
where
(i) $w\_1,\cdots,w\_n$ are i.i.d. Gaussian random variables
(ii) $g(x\_i,w\_i)$ is a smooth function ($g\in C^\infty$)
(iii) $\mathbb{E}g(x\_i,w\_i)>0$
(iv) $\i... | https://mathoverflow.net/users/505491 | Difference between $P(f(x,w)>0)→1$ at any $x$ and $P(\inf(f(x,w))>0)\to1$ when dimension grows | A trivial counterexample:
$$g(x,w):=x.$$
Then convergence (1) holds but convergence (2) does not.
On the other hand, of course, convergence (2) always implies convergence (1).
| 1 | https://mathoverflow.net/users/36721 | 447741 | 180,292 |
https://mathoverflow.net/questions/447661 | 15 | I have a very indirect proof of the following property involving a parametrized integral. If $a,a\_1,\ldots,a\_n\in\mathbb R^n$ (here $n\ge2$), let me denote $V(a,a\_1,\ldots,a\_n)$ the volume of the simplex spanned by these $n+1$ points. Then
$$\sup\_{a\_1,\ldots,a\_n}\int\_{\mathbb R^n}\left[\frac{V(x,a\_1,\ldots,a\_... | https://mathoverflow.net/users/8799 | Integral inequality: an elementary proof? | By dyadic decomposition, it suffices to obtain bounds on the quantities
$$
\int\_{|x-a\_i| \sim R\_i \forall i} \left[ \frac{V(x,a\_1,\dots,a\_n)^2}{(R\_1 \cdots R\_n)^{n+1}}\right]^{\frac{1}{n-1}}\ dx$$
which, when summed over dyadic powers of two $R\_1,\dots,R\_n > 0$, becomes bounded by $O(1)$, uniformly over $a\_1,... | 18 | https://mathoverflow.net/users/766 | 447760 | 180,297 |
https://mathoverflow.net/questions/447692 | 10 | Let a *fuzzy prime* be an ideal of a commutative unital ring whose radical is prime (I'm not sure if this kind of ideal already has a name). Is the intersection of all fuzzy primes $\{0\}$?
Note this is the same as those ideals $I$ for which $ab\in I$ implies $a\in\sqrt{I}$ or $b\in\sqrt{I}$: a fuzzy prime will have ... | https://mathoverflow.net/users/134554 | What is the intersection of all ideals whose radicals are prime? | **Is the intersection of all fuzzy primes $\{0\}$?**
Not always. Let me describe a commutative ring where the intersection of the fuzzy primes is nonzero.
**Plan.** The idea will be to construct a commutative ring $R$ that has a nonzero element $c$ such that $c^2=0$ and $c$ is contained in every nonzero ideal of $R$. I... | 9 | https://mathoverflow.net/users/75735 | 447763 | 180,299 |
https://mathoverflow.net/questions/447775 | 1 | $\newcommand{\cM}{{\mathcal M}}$
For an integer $n>0$, let $\cM\_n$ denote the set of all matrices with three rows and $n$ columns such that every column is obtained by permitting the coordinates of one of the vectors
$$ \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix},\ %
\begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix},\ %
\begin{... | https://mathoverflow.net/users/9924 | The vertex-covering number of a particular hypergraph | I am afraid that $f(n)$ does not grow, even if you use only permutations of $(0,1,2)$.
I claim that the minimal size of $S$ is $3^{n-1}$. As an example, you may take all rows with first coordinate 0.
Assume that $|S|<3^{n-1}$. You may partition all rows onto $3^{n-1}$ triples of the form $(x,x+u,x+2u)$, where $u=(1... | 1 | https://mathoverflow.net/users/4312 | 447778 | 180,303 |
https://mathoverflow.net/questions/447776 | 1 | Let $\mathbb{G}$ be a compact (quantum group) with function algebra $(C(\mathbb{G}), \Delta)$ and Haar state $\varphi\_{\mathbb{G}}$. Consider the associated GNS-representation $\pi\_{\mathbb{G}}: C(\mathbb{G})\to B(L^2(\mathbb{G}))$ with GNS-vector $\xi\_\mathbb{G}$. Let $V \in M(B\_0(L^2(\mathbb{G}))\otimes C(\mathbb... | https://mathoverflow.net/users/216007 | Show that if $V\in M(B_0(H)\otimes A)$, then $V(B(H)\otimes 1)V^*\not\subseteq B(H)\otimes A$ where $A$ is a specific unital $C^*$-algebra | Take for $\mathbb{G}$ the compact group $S^1$ viewed as the unit circle in the plane. Denote by $(\lambda\_y)\_{y \in S^1}$ the regular representation on $L^2(S^1)$. Your question then becomes if the map $y \mapsto \lambda\_y T \lambda\_y^\*$ is continuous from $S^1$ to $B(L^2(S^1))$ equipped with the operator norm. Th... | 5 | https://mathoverflow.net/users/159170 | 447783 | 180,305 |
https://mathoverflow.net/questions/447788 | 4 | I'm reading Takayuki Tamura's article "On the recent results in the study of power semigroups", pp. 191-200 in Goberstein & Higgins' *Semigroups and Their Applications*, Kluwer, 1987 (the volume is the proceedings of the international conference "Algebraic Theory of Semigroups and Its Applications" held at the Californ... | https://mathoverflow.net/users/16537 | Three preprints and one manuscript of Tamura on power semigroups | The list of publications of Takayuki Tamura is [here](https://doi.org/10.1007/s00233-009-9156-y), compiled on the occasion of his 90th birthday, so we can safely assume it is complete. None of these preprints is listed as a publication.
| 6 | https://mathoverflow.net/users/11260 | 447791 | 180,306 |
https://mathoverflow.net/questions/446213 | 2 | For a closed Riemann surface $\Sigma$ of genus $g \geq 2$, the space of holomorphic quadratic differentials on $\Sigma$ can be identified with the cotangent space $T\_\Sigma^\* \mathcal{T}\_g$: in other words, "holomorphic quadratic differentials parametrize the infinitesimal deformations of complex structures on $\Sig... | https://mathoverflow.net/users/103792 | Teichmuller interpretation of unbounded holomorphic quadratic differentials | These can be seen as global or infinitesimal deformations of harmonic maps $\mathbb{C} \to \mathbb{H}$ with ideal polygonal image. In particular, Han-Tam-Treibergs-Wan [show](https://www.intlpress.com/site/pub/files/_fulltext/journals/cag/1995/0003/0001/CAG-1995-0003-0001-a003.pdf) that a harmonic injective immersion $... | 3 | https://mathoverflow.net/users/136267 | 447804 | 180,311 |
https://mathoverflow.net/questions/447733 | 2 | The problem I have is pretty simple, however I cannot find an answer.
I need an *efficient* algorithm to sample integral points within an m-dimensional ball of radius r around the origin (Euclidean norm) uniformly. I want to sample uniformly from the set
$$
S = \{x \in \mathbb{Z}^m | x \in \mathcal{B}^m(r)\}.
$$
The ob... | https://mathoverflow.net/users/505799 | Sample integral points in m-Ball | If $r$ is large compared to $\sqrt{m}$ you may proceed as follows.
You may sample a vector $(x\_1,\ldots,x\_m)$ of $m$ independent normal-distributed variables, and divide each coordinate $x\_i$ of that vector by the length $\sqrt{x\_1^2 + \ldots + x\_m^2}$ of the vector. This gives you a uniform-distributed vector o... | 2 | https://mathoverflow.net/users/105705 | 447807 | 180,312 |
https://mathoverflow.net/questions/285603 | 8 | I am trying to understand how to obtain the Picard group for general toric varieties. So far, I have been using information found in <https://arxiv.org/pdf/1003.5217.pdf> .
Here, a toric variety has homogeneous coordinates $H:=\{x\_i : i=1,\ldots, I\}$ equipped with a number $R$ of equivalence relations
$$ (x\_1,\dot... | https://mathoverflow.net/users/99595 | Picard group of toric varieties | Edit: I have elaborated on this approach to the Picard group in Section 2 of my [preprint](https://arxiv.org/abs/2308.08879).
The question was answered in the comments above, but only for the case of torsion-free Picard group. However, for non-complete toric varieties, the Picard group may have torsion (see example b... | 4 | https://mathoverflow.net/users/505848 | 447811 | 180,316 |
https://mathoverflow.net/questions/447797 | 1 | I want to know if $\left\{\frac{(1-\cos \alpha x)} {x^2}\right\}\_{\alpha>0}$ is dense in $C\_0(\mathbb{R})=\{f\in C(\mathbb{R})\mid \lim\_{|x|\to\infty}f(x)=0\}$? That is, for any $f\in C\_0(\mathbb{R})$ and $\varepsilon >0,$ there is some $\alpha\_1>0,\alpha\_2>0,\cdots,\alpha\_n>0$ and $a\_1,a\_2,\cdots,a\_n\in \mat... | https://mathoverflow.net/users/484728 | Dense subset for $C_0(\mathbb{R})$ | The functions $f\_a$ considered are (up to a multiplicative constant) the Fourier transforms of even tent functions. The real linear combinations of even tent functions yield all even continuous and piecewise affine real functions with compact support, which are dense in the even integrable real functions for the $L^1$... | 2 | https://mathoverflow.net/users/169474 | 447813 | 180,317 |
https://mathoverflow.net/questions/447754 | 0 | $n,i\in\mathbb N$.
The summation in question is
$$\sum\_{k=1}^n\prod\_{l=1}^k\binom{2^n}{2^l}^i.$$
How fast does this grow? I am specifically looking at $i=1,2$.
| https://mathoverflow.net/users/10035 | How fast does this summation grow? | Let us obtain the asymptotic of $\log S\_n$, where $\log$ denotes the base-$2$ logarithm,
\begin{equation\*}
S\_n:=\sum\_{k=1}^n\prod\_{l=1}^k\binom{2^n}{2^l}^i
=\sum\_{k=1}^n a\_{n,k},
\end{equation\*}
and
\begin{equation\*}
a\_{n,k}:=\prod\_{l=1}^k\binom{2^n}{2^l}^i.
\end{equation\*}
For $k=1,\dots,n-1$,
\begi... | 3 | https://mathoverflow.net/users/36721 | 447814 | 180,318 |
https://mathoverflow.net/questions/447773 | 4 | I'm seeking help with a question regarding the space of bounded and uniformly continuous functions $C\_u(T,d)$, where $(T,d)$ is a metric space. In this context, $C\_u(T)$ is a closed subspace of $C\_b(T)$, therefore it is a Banach space as well.
In the second edition of Giné and Nickl's *Mathematical Foundations of ... | https://mathoverflow.net/users/505468 | Is $T$ totally bounded when $C_u(T)$ is separable? | What I realized from Mr. Ozawa's comment is as follows:
**Step1. There exists an $\epsilon>0$ and a sequence $\{x\_n\}\subset T$ such that the open balls $\{U\_\epsilon(x\_n)\}\_{n\in\mathbb{N}}$ are disjoint.**
This is because if there exists an $N\in\mathbb{N}$ such that $T\setminus\cup\_{n=0}^NU\_\epsilon(x\_n)$... | 3 | https://mathoverflow.net/users/505468 | 447817 | 180,319 |
https://mathoverflow.net/questions/447812 | 7 | I am reposting a [question that I had asked on stackexachage](https://math.stackexchange.com/questions/4694206/faithful-representations-of-integral-models) three weeks ago.
Let $G/\mathbb{Q}$ be a connected reductive group, and $\mathcal{G}/\mathbb{Z}$ be an integral model (i.e. flat affine group scheme of finite typ... | https://mathoverflow.net/users/157428 | Faithful representations of integral models | Yes, there exists a closed immersion $\mathcal{G}\to \mathrm{GL}\_n$ over
$\mathbb{Z}$. This is folklore. For a proof, see for example Proposition 3 of arXiv:2012.05708v3
| 7 | https://mathoverflow.net/users/503441 | 447828 | 180,322 |
https://mathoverflow.net/questions/447578 | 9 | Let $S \subset (0, \frac{1}{3}) \times [0, 1]$, be the set such that for each $0 < t < \frac{1}{3}$, $S \cap (\{ t \} \times [0, 1])$ is the standard [Smith-Volterra Cantor set](https://en.wikipedia.org/wiki/Smith%E2%80%93Volterra%E2%80%93Cantor_set) of parameter $t$.
**Question:** Do there exist measurable subsets $... | https://mathoverflow.net/users/173490 | Does the family of fat Cantor sets contain a measurable rectangle? | No. The intuition here is that sets that are missing a lot of "diagonal strips" cannot contain large product sets (even if the strips are very narrow).
For technical reasons it is convenient to work with the left half $S\_l = S \cap (0,\frac{1}{3}) \times [0,1/2]$ of $S$, so that all the missing intervals in the Smit... | 11 | https://mathoverflow.net/users/766 | 447831 | 180,323 |
https://mathoverflow.net/questions/447835 | 8 | Is there an example of a finitely generated subgroup $\Gamma \subset \mathrm{GL}\_n(\mathbb{C})$ such that the group of outer automorphisms $\mathrm{Out}(\Gamma)$ contains finite subgroups of unbounded orders?
| https://mathoverflow.net/users/45793 | Outer automorphisms of finitely generated linear groups | Yes. Furthermore, every recursively presented countable group embeds in such a group. Indeed, first Higman's embedding reduces to proving that every finitely presented group embeds into such a group (if you want all finite groups, it is enough to use a single known f.p. group with all finite groups in it, e.g., Thompso... | 11 | https://mathoverflow.net/users/14094 | 447836 | 180,325 |
https://mathoverflow.net/questions/446595 | 1 | $
\DeclareMathOperator\*{\argmax}{arg\,max}
\DeclareMathOperator\*{\argmin}{arg\,min}
\DeclareMathOperator\*{\cov}{cov}
\DeclareMathOperator\*{\supp}{supp}
\DeclareMathOperator\*{\dom}{dom}
\newcommand{\diff}{ \, \mathrm d}
\DeclareMathOperator\*{\EE}{\mathbb E}
\DeclareMathOperator\*{\PP}{\mathbb P}
\DeclareMathOperat... | https://mathoverflow.net/users/477203 | How to upper bound the difference between these two Gaussian-like densities? | $\newcommand{\si}{\sigma}\newcommand{\al}{\alpha}\newcommand\la\lambda$We have
\begin{equation\*}
|\al|=h^{-1-d/2}|g(1)-g(0)|\le h^{-1-d/2}\sup\_{s\in[0,1]}|g'(s)|,
\end{equation\*}
where
\begin{equation\*}
g(s):=\frac{G((\sqrt h\si(s))^{-1}y(s))}{\sqrt{\det a(s)}},
\end{equation\*}
$y(s):=u-hb(s)$, $u:=x'-x$,
$\si(... | 3 | https://mathoverflow.net/users/36721 | 447837 | 180,326 |
https://mathoverflow.net/questions/447847 | 4 | Let $$f(x)= \sum\_{n=1}^{\infty}a\_n x^n$$ where $a\_n\in \{0,1\}$, and the $f(x)$ has a natural boundary. By the way, $$f(x)= \sum\_{n=1}^{\infty}a\_n x^n$$ is rational function or transcendental one on $\mathbb{C}$。
Is $f(x)$ at $\frac{1}{10}$ transcendental? Or is the value of such a transcendental function of pow... | https://mathoverflow.net/users/14024 | Is the value of the power series at 0.1 transcendental? | This question is likely open.
We can tell whether $f(1/10)$ is rational or irrational (by asking whether $a\_n$ is eventually periodic); in this case, definitely irrational.
Can we have an **algebraic** irrational whose expansion base $10$ consists only of $0$s and $1$s? It is believed that is impossible, but has n... | 8 | https://mathoverflow.net/users/454 | 447854 | 180,333 |
https://mathoverflow.net/questions/447671 | 1 | I am reading the paper "On Convolutions" (1958) and have encountered the notion of "Admissible" and "Permitted" spaces.
In p.17-18 of the above paper, it says that an admissible space $E$ is a locally convex space of distributions with $\mathcal{D} \subset E \subset \mathcal{D}'$ with
1. the inclusions being contin... | https://mathoverflow.net/users/56524 | The notion of "Admissible" and "Permitted" in the context of convolution with distributions and hypocontinuity | The authors Yoshinaga and Ogata of *On concolutions* use Schwartz's notation for spaces of functions and distributions. The requirements for the sequence of smooth functions $\alpha\_k$ with values in $[0,1]$ are convergence to the constant function $1$ in the space $(\mathcal E)$ of all smooth functions, which means $... | 3 | https://mathoverflow.net/users/21051 | 447860 | 180,336 |
https://mathoverflow.net/questions/447829 | 2 | During my Master's thesis I encountered the theory of holonomy for the first time. Unluckily it was only tangentially related to the topic of my thesis, so I couldn't dive into it.
The book I was using is [Differential Geometry -
Cartan's Generalization of Klein's Erlangen Program](https://link.springer.com/book/978038... | https://mathoverflow.net/users/498097 | Learning roadmap for holonomy theory | Dominic Joyce has two relevant books: [Compact Manifolds with Special Holonomy](https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=&cad=rja&uact=8&ved=2ahUKEwio347d5Zz_AhUVOsAKHbxqCuEQFnoECA0QAQ&url=https%3A%2F%2Fwww.amazon.co.uk%2FCompact-Manifolds-Holonomy-Mathematical-Monographs%2Fdp%2F0198506015&usg=AOvV... | 5 | https://mathoverflow.net/users/13268 | 447862 | 180,338 |
https://mathoverflow.net/questions/447787 | 2 | By the circulant matrix $C$ in $M\_n(\mathbb{R})$, we mean that
$$C=[e\_n|e\_1|\cdots|e\_{n-1}]$$ where $e\_1,\cdots,e\_n$ are the standard basis vectors in $\mathbb{R}^n$. It is well-known that
$$C=\mathbf{F}\operatorname{Diag}(1,w,\cdots,w^{n-1})\mathbf{F}^{-1}$$
where $\mathbf{F}$ is just the discrete Fourier mat... | https://mathoverflow.net/users/84390 | Is the sum of the circulant matrix with a super upper triangular matrix diagonalizable? | Here's a counterexample: For $U$ take
$\left(\begin{smallmatrix}
0& 0 &0& 0& 1& 1& 1& 1\\
0& 0 &0 &1& 0& 1& 1& 1\\
0& 0 &0& 0& 1& 0& 0& 1\\
0& 0 &0& 0& 0& 0& 0& 1\\
0& 0 &0& 0& 0& 0& 1& 0\\
0& 0 &0& 0& 0& 0& 0& 0\\
0& 0 &0& 0& 0& 0& 0& 0\\
0& 0 &0& 0& 0& 0& 0& 0
\end{smallmatrix}\right)$.
Then the rational canonical ... | 2 | https://mathoverflow.net/users/460592 | 447871 | 180,340 |
https://mathoverflow.net/questions/447870 | 4 | Let $B$ a discrete valuation ring, say for simplicity with residue field of characteristic $0$, and $K$ its quotient field. Assume that I have an abelian variety $A$ over $K$ and let $A'$ be its Néron model over $B$. Let $n>1$ an integer. The $n$-torsion points $A[n]$ of $A$ are a finite étale scheme over $K$, i.e. of ... | https://mathoverflow.net/users/36563 | Néron model, torsion and ramification | If, for example, $A[n]$ is already contained in $A(K)$, then there will be no ramification, regardless of whether or not $A'$ is an abelian scheme (although that may well force the special fiber to be semi-stable, i.e., totally multiplicative). On the other hand, if you assume that there is no ramification for **all** ... | 7 | https://mathoverflow.net/users/11926 | 447873 | 180,342 |
https://mathoverflow.net/questions/447846 | 6 | Let $(M^n,g)$ be a Riemannian manifold with a fixed point $p$. Can we find a local coordinate system $(x\_1,x\_2,\cdots, x\_n)$ around $p$ satisfying the following two at the same time:
(1) $(x\_1,x\_2,\cdots, x\_n)$ are harmonic coordinates, i.e., $\Delta\_g x\_i=0$;
(2) $(x\_1,x\_2,\cdots, x\_n)$ are normal coord... | https://mathoverflow.net/users/16323 | Existence of normal and harmonic coordinates around a point | Yes. This follows from a standard fact about the Laplacian $\Delta\_g$, (because it is a second-order elliptic operator): The fact is this:
If $u$ is a smooth function on an open neighborhood $U$ of $p$ such that $\Delta\_gu$ vanishes to order $k$ at $p$, then there is a (possiblly smaller) $p$-neighborhood $V\subset... | 10 | https://mathoverflow.net/users/13972 | 447882 | 180,345 |
https://mathoverflow.net/questions/447878 | 6 | I am examining the roots of the equation in $x$, $\sum\_{q=0}^{2k-1} (-1)^{q} {2k+1 \choose q+1} x^{2k-q} m^{q}+r=0$ where $m$ and $r$ are positive integers.
I want to know whether the roots of this can always be fully expressed in radicals (which I suspect that it does), and if so, what can be said regarding the dis... | https://mathoverflow.net/users/265714 | Roots of this equation in x | Upon dividing the given polynomial by $m^{2k}$ and replacing $x$ with $mx+m$, it assumes the simpler form
\begin{equation}
s+(x+1)^{2k+1}-x^{2k+1}
\end{equation}
for some $s$.
Generically its Galois group is the wreath product $C\_2^k\rtimes S\_k$. This follow from Hilbert's irreducibility theorem together with the f... | 7 | https://mathoverflow.net/users/18739 | 447885 | 180,346 |
https://mathoverflow.net/questions/447868 | 1 | Suppose $X\_n\sim N(0,1) $ is iid, then it is easy to see that
$$\sum\_{n=1}^{\infty}\frac{X\_n}{n}\cos nx$$
converges a.s. for any $x$ since
$$\sum\_{n=1}^{\infty}var(\frac{X\_n}{n}\cos nx)<\infty$$
but how to show that the convergence is uniform? That is, for a mesurable set $A\subset \Omega, \mathbb{P}(A)=1$, the se... | https://mathoverflow.net/users/484728 | convergence for series of random variables | Let
\begin{equation\*}
F(x):=\sum\_{n=1}^{\infty}\frac{X\_n}n\,\cos nx. \tag{1}\label{1}
\end{equation\*}
For $j=0,1,\dots$, let
\begin{equation\*}
s\_j:=\sqrt{\sum\_{2^j\le n<2^{j+1}}E\Big(\frac{X\_n}n\Big)^2}=\sqrt{\sum\_{2^j\le n<2^{j+1}}\frac1{n^2}}.
\end{equation\*}
For any integer $N>0$ and
\begin{equatio... | 2 | https://mathoverflow.net/users/36721 | 447897 | 180,349 |
https://mathoverflow.net/questions/447899 | 1 | Put $P\_j=\frac{\partial}{\partial \xi\_j}$ et $Q\_j=2 i \xi\_j$ with$\xi=\left(\xi\_1, \ldots, \xi\_n\right)$ et $x=\left(x\_1, \ldots, x\_n\right)$. How to prove :
1. $\exp \left(\sum\_{j=1}^n x\_j P\_j\right)(f)(\xi)=f(\xi+x)$.
2. $\exp \left(\sum\_{j=1}^n x\_j Q\_j\right)(f)(\xi)=\exp (2 i(x, \xi)) f(\xi)$.
where... | https://mathoverflow.net/users/172078 | Differential operators in $\Bbb R^n$ | I want first to change your notations, sticking to the usual variables $x,\xi$ in the phase space. As a general statement about pseudo-differential operator with a symbol $a(x,\xi)$, I wish to write
$$
\bigl(\text{Op}(a) f\bigr)(x)=\int e^{2iπ x\cdot \xi} a(x,\xi) \hat f(\xi) \, d\xi.
$$
Let $y\in \mathbb R^n$ be given... | 4 | https://mathoverflow.net/users/21907 | 447902 | 180,351 |
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