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https://mathoverflow.net/questions/442150 | 3 | Does there exist a modal formula $φ$ with the following properties?
1. for every finite Kripke frame $F$ there is some ground substitution $\sigma$ such that for every point $w \in F$ we have $F, w \Vdash \sigma(φ)$,
2. there exists an infinite Kripke frame $F$ such that for every ground substitution $\sigma$ there i... | https://mathoverflow.net/users/484622 | Existence of certain formulas in modal logic K | $\def\eq{\leftrightarrow}\def\sset{\subseteq}$An example of such a formula is
$$\phi(p,q)=(\Box p\to p)\land(p\to(q\eq\neg\Box q)).$$
**Lemma.** The formula $\phi$ satisfies condition 1.
**Proof:**
Let $(F,R)$ be a finite frame. For all $i\in\omega$, let $W\_i=\{w\in F:w\vDash\Box^i\bot\}$, where $\Box^i=\underbrac... | 4 | https://mathoverflow.net/users/12705 | 442159 | 178,425 |
https://mathoverflow.net/questions/442160 | 5 | Let $f: [0, 1] \to \mathbb R$ be a function of bounded variation. We say that $g$ is a $C^0$ reparametrization if $g = f \circ s$ for $s$ a continuous increasing bijection from a finite interval $I$ to $[0, 1]$.
**Question:** Does a continuous function of bounded variation always admit a $C^0$ reparametrization that ... | https://mathoverflow.net/users/173490 | Can a continuous bounded variation function be $C^0$-reparametrized to be continuously differentiable? | Non-constant $C^1$ functions have intervals of monotonicity: if $f'(x\_0)\neq 0$ then $f$ is monotone on some interval $(x\_0-\epsilon,x\_0+\epsilon)$; this follows from the inverse function theorem. Evidently monotone continuous reparameterizations preserve this property. But a function of
bounded variation does not h... | 12 | https://mathoverflow.net/users/25510 | 442164 | 178,426 |
https://mathoverflow.net/questions/442155 | 9 | Let us assume that there is a non-trivial elementary embedding $j \colon L\_\gamma \to L\_\gamma$ and $\gamma \geq \omega\_1^V$. Can we conclude that $0^{\#}$ exists?
In general, it is known that if there is an elementary embedding $j \colon L\_\alpha \to L\_\beta$ such that $\mathrm{crit}\, j = \delta$, $(\delta^{+}... | https://mathoverflow.net/users/41953 | $0^{\#}$ and self embeddings of $L_\gamma$ | No, such embeddings can consistently exist in $L$. In the following, $\omega\_1$ and $\omega\_2$ will denote these cardinals as computed in $V$.
Let us assume that $0^\sharp$ exists in $V$. There is an increasing sequence $\langle \xi\_n\mid n<\omega\rangle\in V$ of order indiscernibles over the structure $(L\_{\omeg... | 10 | https://mathoverflow.net/users/125703 | 442166 | 178,428 |
https://mathoverflow.net/questions/442189 | 0 | For countable models, elementary equivalence is not equivalent to isomorphism.
For example, let $\frak{A}= \omega+\Bbb{Z}\*\omega$ and $\frak{B}= \omega+\Bbb{Z}\*\omega^\*$ ($ω^∗$ is the reverse of $ω$). Then $\frak{A}\equiv\frak{B}$ but $\frak{A}\not\cong\frak{B}$ for $\frak{A}$ and $\frak{B}$ are not atomic. (For two... | https://mathoverflow.net/users/nan | Need proof on a model being elementarily equivalent but non-isomorphic | To show that the models are not isomorphic assume an isomorphism $f:\frak{A}\to\frak{B}$, take $(a\_n)\_{n\in\omega}$ be an increasing sequence such $a\_i,a\_j$ are from a different copy of $\Bbb Z$ for each $i\ne j$.
Now look at $f(a\_i),f(a\_j)$. If they are from the same copy of $\Bbb Z$ then there exists some nat... | 1 | https://mathoverflow.net/users/113405 | 442190 | 178,433 |
https://mathoverflow.net/questions/442181 | 3 | Consider a set $N$ with elements $n\_1, n\_2, \dots, n\_k$ which are distinct integers. Introduce the notation $N\_{i=1,2,\dots,s}$ for the $s$ blocks of a set partition of $N$. Consider a supplementary integer $n\_{k+1}$.
We have the identity
$(\sum\_{i=1}^{k+1} n\_i -k)^{k-1}= \sum\_{\text{set partitions } N\_i \... | https://mathoverflow.net/users/70000 | Proof of a combinatorial identity for a sum over partitions of sets giving rise to a symmetric polynomial? | The notation and framing of the problem are not optimal, because the $n\_i$'s being integers is a bit of a distraction.
Let $[k]:=\{1,\ldots,k\}$, and let ${\rm Part}\_k$ be the set of set partitions of $[k]$. We also use $|A|$ to denote the cardinality of a set $A$.
Let $x\_1,\ldots,x\_k$ and $y$ be formal variables... | 5 | https://mathoverflow.net/users/7410 | 442192 | 178,434 |
https://mathoverflow.net/questions/442163 | 20 | One motivation for studying infinity categories is that the derived category does not satisfy Zariski descent, although the infinity categorical version does.
I would like to see an example of Zariski descent failing for the (one categorical) derived category of coherent sheaves on a scheme.
That is I would like an... | https://mathoverflow.net/users/153310 | The derived category does not satisfy descent - example | Consider the canonical functor $H$ from the homotopy category of homotopy coherent descent data in the ∞-category of coherent sheaves
to the category of descent data in the derived category of coherent sheaves.
Nonuniqueness of gluing amounts to $H$ not being a full functor
and is observed for covers of cardinality 3... | 11 | https://mathoverflow.net/users/402 | 442194 | 178,435 |
https://mathoverflow.net/questions/442183 | 9 | This is a reference question:
Let $\psi(x)$ be the psi-Chebyshev function. Is there any unconditional result in the literature that proves that there exists $0<a<2$ such that
$$
\int\_2^x (\psi(y)-y)^2 \mathrm dy =O(x^{a})
?$$
| https://mathoverflow.net/users/9232 | $\psi(x)-x$ on average | Impossible as this would imply that $\frac{\zeta'(s)}{\zeta(s)}+\frac1{s-1}$ is analytic on $\Re(s)\ge 1/2$.
Your bound
$$\int\_1^x |\psi(y)-y|^2 dy = O(x^a)$$ for some $a < 2$ implies (with Cauchy-Schwarz inequality) that $$\int\_x^{2x} (\psi(y)-y)dy = O(x^{a/2+1/2}) \implies\int\_1^2 (\psi(xt)-xt)dt = O(x^{a/2-1/2}... | 8 | https://mathoverflow.net/users/84768 | 442209 | 178,440 |
https://mathoverflow.net/questions/442198 | 3 | In my setting, $\mu$ and $\nu$ are probability measures on $\mathbb{R}^{2}$ with compact support. For any function $f\in{C^{2}\_{b}(\mathbb{R}^{2})}$, I have the estimate:
$$
|\mathbb{E}\_{\mu}(f)-\mathbb{E}\_{\nu}(f)|\leq{a\|Df\|\_{L^{\infty}}+b\|D^{2}f\|\_{L^{\infty}}}
$$
for some coefficients $a,b>0$. Does the estim... | https://mathoverflow.net/users/80052 | Getting Wasserstein closeness from a derivative estimate | Assuming the support of both measures is fixed, it is enough to bound $W\_1(\mu,\nu)$. That is, we want to bound $\vert \int f d\mu -d\nu \vert$ for any $1$-Lipschitz function $f$.
Take a triangulation of the plane with equilateral triangles of size $\sqrt b$ and let $f\_1$ be the linear interpolation of $f$. We have... | 2 | https://mathoverflow.net/users/112954 | 442210 | 178,441 |
https://mathoverflow.net/questions/442203 | 11 | The category of condensed sets is the colimit of the toposes of $\kappa$-condensed sets over all cardinals $\kappa$, or equivalently the category of "small sheaves" on the large site of all compact Hausdorff spaces. As such, it is not itself a topos, although it is an infinitary pretopos. I have encountered claims that... | https://mathoverflow.net/users/49 | Are condensed sets (locally) cartesian closed? | Condensed sets are indeed locally cartesian closed. On the other hand, for no cardinal $\kappa$ (no matter how inaccessible) the functor from $\kappa$-condensed sets to condensed sets preserves all internal Hom's. Indeed, consider the internal Hom from a discrete set $I$ towards the discrete set $\{0,1\}$: This is evid... | 13 | https://mathoverflow.net/users/6074 | 442211 | 178,442 |
https://mathoverflow.net/questions/442212 | 9 | Let $\mathcal{C}$ denote a category with a zero object. Let $\mathrm{N}(X)$ denote the norm of an object $X$ of a category $\mathcal{C}$, defined as $\mathrm{N}(X) = \,|\mathrm{Hom}(X,X)\,|$. From this terminology, we say a non-zero object $X$ is *finite* if $\mathrm{N}(X)$ is finite. Let $\mathrm{P}(\mathcal{C})$ deno... | https://mathoverflow.net/users/70508 | Properties of categorical zeta function | Consider the category $\def\Set{\mathbf{Set}}\Set\_\*$ of pointed sets.
Kurokawa defines the simple objects to be those nonzero $X$ for which the only morphisms $X\to Y$ are monomorphisms or zero, so the only simple object of $\Set\_\*$ is the two-element set $P = \{\*, 0\}$, whose norm is $N(P) = 2$.
Therefore,
$$
\... | 11 | https://mathoverflow.net/users/160838 | 442220 | 178,446 |
https://mathoverflow.net/questions/442230 | 1 | Let $W$ be a one-dimensional Brownian motion. Consider the stochastic differential equation (SDE)
$$dX\_t = C(t)(1-X\_t)dW\_t,\quad \forall t\ge 0,$$
where $C$ is a continuous and bounded function. Under which condition (on $C$),
1. the above SDE has an explicit solution?
2. any solution $X$ satisfies $\lim\_{t\t... | https://mathoverflow.net/users/493556 | On a martingale defined via some SDE | If $X\_0$ and $C$ are deterministic (or independent of $W$), then
$$X\_t = 1-(1-X\_0)\exp\left(-\int\_0^tC(s)\mathrm dW\_s-\frac12\int\_0^tC(s)^2\mathrm ds\right).$$
This will go to 1 if the argument of the exponential goes to $-\infty$. I claim that this is equivalent to the condition $I:=\int\_0^\infty C(s)^2\mathrm ... | 2 | https://mathoverflow.net/users/129074 | 442231 | 178,448 |
https://mathoverflow.net/questions/442188 | 5 | Let $\mathscr{P}$ be a polyhedral decomposition of a real vector space $V$. By that I mean that $\mathscr{P}$ is a finite set of polyhedra in $V$ satisfying the following three properties:
1. The union of the polyhedra in $\mathscr{P}$ is equal to $V$.
2. If $P\in \mathscr{P}$ and $Q$ is a face of $P$, then $Q\in\mat... | https://mathoverflow.net/users/10273 | The bounded complex of a polyhedral decomposition | Below I construct a deformation retract of $V$ onto the bounded sub-complex of $\mathscr P$, proving that the latter is contractible.
For a polyhedral complex $\mathscr P$ resp. a polyhedron $P\in\mathscr P$ I write $\mathscr P^b\subseteq\mathscr P$ resp. $P^b\subseteq P$ for its bounded sub-complex.
Also, by *a retr... | 1 | https://mathoverflow.net/users/108884 | 442236 | 178,450 |
https://mathoverflow.net/questions/442202 | 2 | Let $(P,\leq)$ be a finite poset that contains a (global) minimal element $0$ and a (global) maximal element $1$. We say that a subset $U \subset P$ is *upward closed* if $x \in U$ and $y \geq x$ forces $y \in U$. Given an arbitrary subset $A \subset P$, let $A^+$ be the smallest upward closed subset of $P$ containing ... | https://mathoverflow.net/users/18263 | Posets with cardinality bounds on upward-closed subsets | We have $A\subset A^+\cap D$ and $(A^+\cap D)^+=A^+$, so we can assume $A=A^+\cap D$, let $V=A^+$ the problem turn into $\frac{|V|-|V\cap U|}{|P|-|U|}=\frac{|V\cap(P-U)|}{|P|-|U|}\leq\frac{|V\cap U|}{|U|}\Leftrightarrow\frac{|V|}{|P|}\leq\frac{|V\cap U|}{|U|}\Leftrightarrow|V||U|\leq|P||V\cap U|$
for all upward close... | 4 | https://mathoverflow.net/users/432274 | 442239 | 178,451 |
https://mathoverflow.net/questions/442171 | 3 | Let $G$ be a $p$-group contained in $S\_{p^m}$ for some $m\in \mathbb{N}$, this is, the symmetric group of degree $p^m$. Assume that the lower central series, LCS for short, of $G$ is well understood.
Is there anything known about the LCS of $W(G):=G\wr C\_p$, where $C\_p$ is a cyclic group of order $p$, and $\wr$ is... | https://mathoverflow.net/users/500544 | Is there anything known about the lower central series of a group $G\wr C_p$? | Here is a solution in the special case in which each lower central factor $\gamma\_i(G) / \gamma\_{i+1}(G)$ is elementary abelian.
First consider the case in which $G$ is elementary abelian, written additively, so $G \cong \mathbf F\_p^d$ for some $d$. Let $x$ be a generator of $C\_p$. Then for $g \in G^p$ we have $[... | 2 | https://mathoverflow.net/users/20598 | 442240 | 178,452 |
https://mathoverflow.net/questions/441993 | 9 |
>
> Question 1:Is there a reference that lists all possible finite subgroups and their orders of $\mathrm{SO}(n)$ and $\mathrm{O}(n)$ for $n=4$ or even higher $n$ over the real numbers?
>
>
>
I can only find incomplete lists on wikipedia, mathoverflow or other sources in the internet.
>
> Question 2: Is ther... | https://mathoverflow.net/users/61949 | Finite subgroups of $\mathrm{SO}(n)$ and $\mathrm{O}(n)$ | Finite subgroups of $\operatorname{SO}(4)$ and $\operatorname{O}(4)$ are enumerated in Chapter 4 of the book:
*Conway, John H.; Smith, Derek A.*, **On quaternions and octonions: their geometry, arithmetic, and symmetry**, Natick, MA: A K Peters (ISBN 1-56881-134-9/hbk). xii, 159 p. (2003). [ZBL1098.17001](https://zbm... | 10 | https://mathoverflow.net/users/10266 | 442248 | 178,455 |
https://mathoverflow.net/questions/442243 | 0 | Currently, I'm reading the paper "Towards the fixed point property for superreflexive spaces" by Andrzej Wiśnicki. In this article, given $X\_1,\dots,X\_n$ Banach spaces, he defines $(X\_1\oplus\dots\oplus X\_n)\_{\infty}$ as the product $X\_1\times\dots\times X\_n$ with the norm
\begin{equation\*}
\lVert (x\_1,\dots... | https://mathoverflow.net/users/490148 | The ultrapower of the direct sum is the direct sum of ultrapowers | I'll attempt to record a careful proof here for your first question. My apologies for any errors or inefficiencies, as I am a bit rusty. I will write $\|\,\|\_j$ for the norm on $X\_j$ and $\|\,\|$ for the max-norm on $(X\_1\oplus\cdots\oplus X\_n)\_\infty$.
Writing $\Pi\_0$ for the space of bounded sequences in $(X\... | 3 | https://mathoverflow.net/users/349327 | 442258 | 178,456 |
https://mathoverflow.net/questions/442250 | 4 | Suppose that $f \colon [0, 1] \to \mathbb{R}$ is a $1$-Lipschitz function.
Define the uniform norm $\|f\|\_\infty = \sup\_{x} |f(x)|$.
Given $\{U\_j\}\_{j=1}^\infty$ independent and identically distributed uniform random variables on the unit interval $[0, 1]$, I am curious to know about the behavior of the random va... | https://mathoverflow.net/users/121486 | Rate of convergence of sample maximum, $\Big|\max_{j \leq n} |f(U_j)| - \|f\|_\infty\Big|$ | $\newcommand\De\Delta\newcommand{\Ga}{\Gamma}\newcommand\ga{\gamma}$Let $X\_j:=g(U\_j)$, where $g:=|f|$. Then the $X\_i$'s are i.i.d. random variables and $g$ is a $1$-Lipschitz function. We have
$$M:=\max\_{[0,1]}g=g(u)$$
for some $u\in[0,1]$.
For any real $h>0$,
\begin{align\*}
P(\De\_n(f)>h)&=P(M-\max\_1^n X\_j>h)... | 4 | https://mathoverflow.net/users/36721 | 442259 | 178,457 |
https://mathoverflow.net/questions/442262 | -1 | Is it possible to build a 1-priodic smooth function from a rapidly decreasing sequence such that the sequence be the Fourier coefficients of the function?
More precisely:
Let $\lbrace c\_k\rbrace \_{k \in \mathbb{Z}}$ be rapidly decreasing sequence. Is it possible to show that there is a 1-priodic smooth function $F:... | https://mathoverflow.net/users/137242 | Building a smooth function from a rapidly decreasing sequence | **Yes**. (Assuming that that **rapidly decreasing** means that $\sup\_{k\in \mathbb Z}\vert k^m c\_k \vert <\infty$ for all $m=0,1,2,\dots$.)
The function must be $f(t)=\sum\_{k\in \mathbb Z} c\_k \exp(2\pi i k t)$.
Just write $f\_N(t)=\sum\_{\vert k \vert \le N} c\_k \exp(2\pi i k t)$. As $\vert c\_k\vert = O(k^{-... | 1 | https://mathoverflow.net/users/126651 | 442264 | 178,459 |
https://mathoverflow.net/questions/441742 | 1 | $\newcommand{\EE}{\mathbb{E}}$
Let $A\in\mathbb{R}^{m\times n}$ and $X$ be an isotropic random vector in $\mathbb{R}^n$, i.e. it holds that $\EE(XX^T) = I\_n$.
How to calculate $$M = \EE\left(\tfrac{XX^T}{\|AX\|^2}\right)$$
and/or $$AMA^T = \EE\left(\tfrac{AXX^TA^T}{\|AX\|^2}\right) = \EE\left(\tfrac{AXX^TA^T}{X^TA^T... | https://mathoverflow.net/users/9652 | Calculating $\mathbb{E}\left(\tfrac{XX^T}{\|AX\|^2}\right)$ for isotropic random vectors $X$ | Assuming $X\sim N(0,I)$, writing $A^TA=\sum\_i L\_i u\_i u\_i^T$, since $sign(u\_i^TX)$ and $sign(u\_k^TX)$ are independent and mean-zero, $u\_i^TMu\_k=0$ if $i\le k$ because the denominator
$\|AX\|^2=\sum\_i L\_i Z\_i^2$ is independent of the signs, where $Z\_i=u\_i^TX$ are iid $N(0,1)$.
This shows that $M$ is diagona... | 1 | https://mathoverflow.net/users/141760 | 442268 | 178,460 |
https://mathoverflow.net/questions/442271 | 8 | I think this is a silly question, but I'm quite confused. In Lurie's "Rotation Invariance In Algebraic K-Theory" Notation 3.2.4. he defines a filtered spectrum $\mathbb{A}$ given by
$$\mathbb{A}\_n = \begin{cases}
\mathbb{S} &\text{ if } n =0 \\
0 &\text{ otherwise}\end{cases} $$
He then notes (example 3.2.9) that $$... | https://mathoverflow.net/users/131360 | Why isn't the anchor map in Lurie's "Rotation Invariance in Algebraic K-Theory" zero? | It's zero as a map of filtered spectra, but it is considered as a map of $\mathbb{A}$-bimodules.
| 12 | https://mathoverflow.net/users/39747 | 442273 | 178,461 |
https://mathoverflow.net/questions/441651 | 3 | Let $\Omega$ be an open subset of $\mathbb{C}^n$ and let $f$ be analytic in $\Omega$. Assume $P\in\mathbb{C}[z\_1,\ldots,z\_n]$ is a polynomial whose irreducible factors are all of multiplicity one.
If $f$ vanishes on the zero set $Z(P)$ of $P$, how to justify that $f/P$ is analytic in $\Omega$ ?
(asked on [MSE](ht... | https://mathoverflow.net/users/89429 | Factorization of an analytic function in $\mathbb{C}^n$ | *This is a copy of [my answer](https://math.stackexchange.com/a/4654372/312406) from math.SE*
I think you could argue stalk wise. Take a point $x \in \Omega$. By general theory, the ring $\mathcal O\_{\mathbb C^n, x}$ of convergent power series at $x$ is a UFD. Since $P$ is square-free, and $f$ vanishes at $\{P =0\}$... | 1 | https://mathoverflow.net/users/111897 | 442278 | 178,463 |
https://mathoverflow.net/questions/442253 | 4 | Let $\mathcal{A}$ be an abelian category, it is known that any complex $A^{\bullet}$ admits a distinguished triangle
$$B^{\bullet}\rightarrow A^{\bullet}\rightarrow C^{\bullet}\rightarrow B^{\bullet}[1]$$
in the unbounded derived category $D(\mathcal{A})$ for some $B^{\bullet}\in D^-(\mathcal{A})$ and $C^{\bullet}\in D... | https://mathoverflow.net/users/nan | Decompose an unbounded (cochain) complex in the homotopy category | Yes. Let
$$\tau^{\leq0}A^\bullet:= \cdots\to A^{-2}\to A^{-1}\to\ker(d^0)\to0\to\cdots$$
be the usual truncation of $A^\bullet$. Then the mapping cone of the inclusion map $\tau^{\leq0}A^\bullet\to A^\bullet$ is homotopy equivalent to
$$\rho^{>0}A^\bullet:=\cdots\to0\to\ker(d^0)\to A^0\to A^1\to\cdots,$$
so there is a ... | 6 | https://mathoverflow.net/users/22989 | 442279 | 178,464 |
https://mathoverflow.net/questions/442277 | 3 | The following theorem is well-known in the ordinary analysis textbook:
**Theorem**: Assume the function $f:U\to\Bbb R^n$ is Lipschitz continuous on an open set $U\subset\Bbb R^m$, then $f$ is almost everywhere differentiable on $U$.
My question:
**Question**: Assume the function $f:U\to\Bbb{R}^n$ is Lipschitz con... | https://mathoverflow.net/users/113353 | Is there any strengthened version of Rademacher's Theorem or any counterexample? | The best thing you can do is the following: for every $\epsilon > 0$ there exists a $C^1$ function $g: U \to \mathbf{R}^n$ so that
\begin{equation}
\mathcal{H}^m(\{ f \neq g \} \cup \{ Df \neq Dg \} ) < \epsilon.
\end{equation}
This can be deduced from the Whitney extension theorem; you can find a proof in Leon Simon's... | 7 | https://mathoverflow.net/users/103792 | 442281 | 178,465 |
https://mathoverflow.net/questions/442270 | 9 | nlab presents a [proof](https://ncatlab.org/nlab/show/locally+compact+topological+space#categorytheoretic_properties) that the category of locally compact Hausdorff spaces does not admit infinite products in general. In particular it shows that there is no infinite product of $\mathbb{R}$, since such a product would be... | https://mathoverflow.net/users/163483 | Does the category of locally compact Hausdorff spaces with proper maps have products? | Suppose $X,Y$ are locally compact Hausdorff spaces admitting a product $X\otimes Y$ in the category $\mathcal{H}$ of all such spaces. If one of $X,Y$ is empty, then $X\otimes Y$ exists, so I'll assume that neither is.
Then the points of $X\otimes Y$ are exactly the maps $\ast\rightarrow X\otimes Y$. Since $X\otimes Y... | 11 | https://mathoverflow.net/users/54788 | 442282 | 178,466 |
https://mathoverflow.net/questions/442111 | 11 | Consider the algebra $B=P(\omega)/\_{\mathrm{fin}}$ (the quotient of the power set of natural numbers modulo the ideal of finite sets). Is there an infinite strictly descending chain $\{A\_i\mid i\in I\}$ of subalgebras of $B$, such that there is an embedding of $A\_{i+1}$ into $A\_i$, but there is no embedding of $A\_... | https://mathoverflow.net/users/22019 | A strictly descending chain of subalgebras of $P(\omega)/_{\mathrm{fin}}$ | There is a family $\{K\_X:X\subseteq\mathfrak{c}\}$ of separable compact
zero-dimensional spaces such that there is a continuous surjection of $K\_X$
onto $K\_Y$ if and only if $X\subseteq Y$.
These spaces are continuous images of $\omega^\*$, the Stone space of
$\mathcal{P}(\omega)/\mathit{fin}$.
So Stone duality th... | 9 | https://mathoverflow.net/users/5903 | 442284 | 178,467 |
https://mathoverflow.net/questions/442272 | 3 | The motivation for this question comes from the study of lifts of an orbifold chart, which is simplified as the following:
Suppose that $ U $ is an open connected subset of $ \mathbb{R}^n $ and $ G $ is a finite group of diffeomorphisms of $ U $, which naturally acts on $ U $. Let $ \pi:U\rightarrow U/G $ denote the ... | https://mathoverflow.net/users/131015 | The difference between two lifts along an orbifold chart | Let $\mathbb Z/2$ act on $U=\mathbb R$ by reflection through the origin $x=0$. Let $V=\mathbb R$ as well.
Consider $f,g:\mathbb R\rightarrow \mathbb R$ given by
$$
f(x)=\begin{cases} e^{-1/x^2} \qquad \text{if }\qquad & x>0 \\
0 & x=0 \\
-e^{-1/x^2} & x<0
\end{cases}
$$
and $g(x)=e^{-1/x^2}$ for $x\not=0$ and $g(... | 3 | https://mathoverflow.net/users/12156 | 442286 | 178,468 |
https://mathoverflow.net/questions/442296 | 6 | If G is an almost simple group, then Aut(G) is complete?
Apologies - I meant to post this on Stack Exchange
Just wondering if anyone has a reference to the above - it's quoted on Wikipedia (so supposing that means it's true...) and seems a plausible result.
I've tried generalising the proof of the version when G ... | https://mathoverflow.net/users/499057 | If G is an almost simple group, then Aut(G) is complete? | This isn't true. Let $S$ be a simple group such as ${\rm PSL}(4,5)$ with outer automorphism group isomorphic to $D\_8$ (dihedral of order $8$), and let $G = S.2$ be an extension of $S$ by a non-central involution in $D\_8$.
Then ${\rm Aut}(G) = S.2^2$ and ${\rm Aut}({\rm Aut}(G)) = S.D\_8 = {\rm Aut}(S)$.
| 14 | https://mathoverflow.net/users/35840 | 442302 | 178,471 |
https://mathoverflow.net/questions/441891 | 4 | This question was first asked [here](https://math.stackexchange.com/questions/4648121/a-set-with-a-topology-and-a-partial-ordering-question), on math stack exchange, but wasn't able to attract any attention. Now that I am thinking more, it feels like the most suitable place for this question is here.
Suppose I have a... | https://mathoverflow.net/users/54507 | When is this topology compatible with the partial ordering? | One of the standard topologies to consider would be the lower-cone topology, whose basic open sets are the lower cones $i{\downarrow}=\{j\mid j\leq i\}$. In this topology, the open sets are exactly the downsets. If $i\in A\_i$ is open, then indeed $i{\downarrow}\subseteq A\_i$, and furthermore $(i{\downarrow})\cap(j{\d... | 3 | https://mathoverflow.net/users/1946 | 442305 | 178,472 |
https://mathoverflow.net/questions/442303 | 0 | Let $f(x)$ be a real-valued function defined in $(0, \infty)$. I am curious what kind of $f(x)$ has the following representations:
$$
f(x) = \sum\_{j=0}^\infty a\_j e^{-jx}, \quad \forall x \in (0, \infty).
$$
My initial thought was that consider the transformation $s = e^{-x}, s \in (0, 1)$, then we want to find $\{... | https://mathoverflow.net/users/165697 | What kind of functions can be represented as infinite linear combinations of exponential functions? | $\newcommand\R{\mathbb R}\newcommand\C{\mathbb C}$You were almost there.
Indeed, suppose that
$$f(x)=\sum\_{j=0}^\infty a\_j e^{-jx}\in\R \quad \forall x\in(0,\infty). \tag{1}\label{1}$$
Then
$$g\_f(s):=f(-\ln s)=\sum\_{j=0}^\infty a\_j s^j\in\R \quad \forall s\in(0,1).$$
So, the radius of convergence of the power se... | 1 | https://mathoverflow.net/users/36721 | 442307 | 178,473 |
https://mathoverflow.net/questions/442310 | 3 | [This has been edited in response to comments from Fedor Petrov]
Suppose that $n=dm$ with $d,m>1$. Consider an $n\times n$ matrix $M$ such that
1. All diagonal entries are equal to one
2. Each row has $d$ ones, and all other entries are zero.
This can be seen as the adjacency matrix of a certain kind of directed ... | https://mathoverflow.net/users/10366 | Lower bound on the rank of a graph | $m$ is a tight bound.
If the matrix be composed of $m$ blocks $d\times d$ with all 1's, its rank equals $m$.
For any $M$ satisfying your conditions, you may color $\{1, 2,\ldots,n\}$ in $d$ colors so that $M\_{i, j}=0$ for all $i<j$ of the same color (if $1, 2,\ldots,k-1$ are already colored, you may choose an appr... | 2 | https://mathoverflow.net/users/4312 | 442313 | 178,475 |
https://mathoverflow.net/questions/442309 | 2 | I have the following $(n+1)\times (n+1)$ matrix
$$P = \begin{bmatrix}
f(0) & g(0) & 0 & 0 & 0 & \dots & 0\\
f(1) & 0 & g(1) & 0 & 0 & \dots & 0\\
f(2) & 0 & 0 & g(2) & 0 & \dots & 0\\
f(3) & 0 & 0 & 0 & g(3) & \dots & 0\\
\vdots & \vdots & & & & \ddots & 0\\
f(n-1) & 0 & 0 & 0 & 0 &\dots & g(n-1)\\
f(n) & 0 & 0 & 0 &... | https://mathoverflow.net/users/477568 | Summation of rows of a matrix P^k is decreasing with the power k | A counterexample is given by
$$P=\left(
\begin{array}{ccc}
0 & 1 & 0 \\
0 & 0 & \frac{1}{2} \\
\frac{1}{2} & 0 & 0 \\
\end{array}
\right),$$
so that then
$$P^2=\left(
\begin{array}{ccc}
0 & 0 & \frac{1}{2} \\
\frac{1}{4} & 0 & 0 \\
0 & \frac{1}{2} & 0 \\
\end{array}
\right).$$
---
Here the conditions on $P$... | 3 | https://mathoverflow.net/users/36721 | 442321 | 178,477 |
https://mathoverflow.net/questions/442324 | 11 | Gödel's incompleteness theorem shows that there are undecidable statements, i.e., formal logical claims which neither have proofs nor disproofs.
In doing so, Gödel famously enumerated all well-formed formulas with his so-called "Gödel numbering". Thus, the following question is well posed:
>
> What proportion of ... | https://mathoverflow.net/users/159298 | Are 100% of statements undecidable, in Gödel's numbering? | This is going to depend sensitively on your exact choice of Gödel number, and the limit will often not be defined. Pick your favorite very short undecidable statement S. Then for any statement A, you have the nearly the same length statements like "A or S", "A and S", "A and True", "A or False". However, one can constr... | 16 | https://mathoverflow.net/users/127690 | 442325 | 178,479 |
https://mathoverflow.net/questions/442291 | 5 | Let $I=(0,1)$ and let $C>1$ be a constant. Let $L^2(I)$ and $H^1(I)$ be the standard Sobolev spaces on $I$. Suppose that $U$ is a **subspace** of $H^1(I)$ with the additional property that:
$$ \| u\|\_{H^1(I)} \leq C \|u\|\_{L^2(I)}\quad \forall\, u\in U.$$
By compact embedding of $H^1(I)\subset L^2(I)$, it is trivial ... | https://mathoverflow.net/users/50438 | Bounds on dimension of a subspace | Just use the Wirtinger's inequality that says that if $u=0$ at the center of an interval $I\subset \mathbb R$, then $\int\_I|u|^2\le (|I|/\pi)^2\int\_I|u'|^2$. Thus, if the dimension is $>n$, we can create a function that vanishes at the centers of $n$ subintervals of the interval $(0,1)$ of length $1/n$ and have $\|u\... | 6 | https://mathoverflow.net/users/1131 | 442326 | 178,480 |
https://mathoverflow.net/questions/442280 | 0 | I am dealing with a problem of the form ($a<b$)
$$
\displaystyle \max\_{v \in C^1([a, b])} \int\_a^b v(x)~\mathrm{d}x, \quad \mathrm{s.t.} \int^b\_a \big(-o'(x)v(x)-v'(x)o(x)\big)f(x)~\mathrm{d}x \leq 0,~ \frac{o(b)}{o(a)} \leq v \leq 1
$$
where $o \in C^\infty(\mathbb{R})$ with $o>0$ and $o'< 0$ on $(a, b)$ and $f \in... | https://mathoverflow.net/users/500621 | Constrained linear optimization problem on $C^1$ | As you suggest, let me consider the case $f \equiv 1$. Without loss of generality, assume also that $a = 0$ and $b = 1$. Let $\sigma := o(1)/o(0) \in (0,1)$. The problem becomes
$$
\sup\_{v \in C^1([0,1])} \int\_0^1 v(x) \mathrm{d}x
\quad \text{s.t.} \quad
v(0)\leq \sigma v(1) \quad \text{and} \quad \sigma \leq v(x) \l... | 1 | https://mathoverflow.net/users/50777 | 442328 | 178,481 |
https://mathoverflow.net/questions/442327 | 1 | Let $\{W(t),\,t \in [0,1]\}$ be a standard Brownian motion in $\mathbb{R}^d$, starting from $0$. Let $U$ be a non-empty open set such that $0 \in \partial U$.
Which conditions on $U$ are necessary and sufficient for
$
\mathbb{P}\{\exists t\in[0,1]:W(t) \in U\}=1
$?
Most likely, there are no simple conditions on $U$... | https://mathoverflow.net/users/499882 | Brownian motion hitting open set starting from its boundary |
>
> is it true that $\mathbb{P}\{\exists t\in[0,1]:W(t) \in U\}=1$ if and only if $0$ is a regular point for $U$?
>
>
>
Yes, some books even take the definition of regular points to be that eg. see online notes ["Classical potential theory and Brownian motion"](https://digitalassets.lib.berkeley.edu/math/ucb/tex... | 1 | https://mathoverflow.net/users/99863 | 442329 | 178,482 |
https://mathoverflow.net/questions/442315 | 1 | $\newcommand\SS{P}\newcommand\TT{Q}$I will call a Gaussian probability measure $\SS$ on $\mathbb{R}^d$ *isotropic* if its covariance matrix is diagonal with non-vanishing determinant; i.e. $\Sigma\_{i,i}>0$ for $i=1,\dots,d$ and $\Sigma\_{i,j}=0$ whenever $i\neq j$ for each $i,j=1,\dots,d$.
*Note: My definition of "... | https://mathoverflow.net/users/491352 | References: error and stability estimates for information projection | Assuming you want to minimize the Kullback–Leibler divergence
$$D(P\parallel Q)=\int dP\,\ln\frac{dP}{dQ}$$
over all isotropic Gaussian $P$, "the" minimizer is in general not unique and, accordingly, not Lipschitz even on the set of measures $Q$ where it is unique.
The idea of a counterexample is quite simple: Suppos... | 3 | https://mathoverflow.net/users/36721 | 442343 | 178,485 |
https://mathoverflow.net/questions/442317 | 3 | Let $A$ be a a two-sided noetherian semiperfect ring and assume that the injective dimension of the left and right regular modules are equal to $n \geq 1$.
Let $\Omega^n(mod A)$ be the category of $n$-th syzygy modules consisting of modules that are projective or a direct summand of a module of the form $\Omega^n(M)$ f... | https://mathoverflow.net/users/61949 | $\Omega$ for noetherian semiperfect rings | This seems to be answered in Proposition 4.2 of Kameyama, Kimura and Nishida, "On stable equivalences of module subcategories over a semiperfect Noetherian ring".
| 2 | https://mathoverflow.net/users/460592 | 442350 | 178,490 |
https://mathoverflow.net/questions/442298 | 4 | It is well known that complex projective three space $\mathbb{C}\mathbf{P}^3$ is a complex manifold. However it also possess a non-integrable almost-complex structure (as discussed in this [article](https://www.sciencedirect.com/science/article/pii/S0393044020301728#b12) for instance). Can somebody give an explicit con... | https://mathoverflow.net/users/491434 | Non-integrable almost complex structure for complex projective $3$-space | There are lots of non-integrable almost-complex structures on $\mathbb{CP}^3$, but the one you are looking for is, I suspect, the following.
First I will explain the twistor fibration, which is a submersion $t \colon \mathbb{CP}^3 \to S^4$ with fibres $S^2$. To do this, fix an identification $\mathbb{C}^4 \cong \math... | 8 | https://mathoverflow.net/users/380 | 442352 | 178,491 |
https://mathoverflow.net/questions/442336 | 0 | Let $X,Y$ be real separable Hilbert space, and let $HS(X,Y)$ be the space of Hilbert-Schmidt operators from $X$ to $Y$, endowed with the [Hilbert-Schmidt norm](https://en.wikipedia.org/wiki/Hilbert%E2%80%93Schmidt_operator). Let $x\in X$ and $y\in Y$. I am interested in finding the derivative of the map
$$
HS(X,Y)\ni A... | https://mathoverflow.net/users/91196 | Derivative with respect to a Hilbert-Schmidt operator | Let me denote by $F : HS(X,Y) \to \mathbb{R}$ the map which you defined as $F(A) := \|y-Ax\|\_Y^2$. By standard arguments, you indeed have that $F$ is $C^1$ and, for any $A \in HS(X,Y)$,
$$
\mathcal{L}(HS(X,Y),\mathbb{R}) \ni DF\_{\rvert A} = \begin{cases}
HS(X,Y) \to \mathbb{R}, \\
B \mapsto 2 \langle Bx , Ax - y \ran... | 2 | https://mathoverflow.net/users/50777 | 442358 | 178,494 |
https://mathoverflow.net/questions/442018 | 2 | Let $M\_1$ and $M\_2$ be von Neumann algebras acting on Hilbert spaces $H\_1,H\_2$ and consider $M=M\_1\overline\otimes M\_2$ acting on $H\_1\otimes H\_2$. Let $K$ be an $M$-invariant subspace (so that $P\_K\in M\_1'\overline\otimes M\_2'$).
Assume that $PMP = B(K)$.
Consider the von Neumann algebras $R\_1 = P M\_1\o... | https://mathoverflow.net/users/485160 | Question on tensor product of von Neumann algebras and subfactors | I would say you get $R\_1 = R\_2'$, and also $R\_1 = B(K')$ for some $K'$.
1. The projection $P$ is minimal in $M\_1' \mathbin{\bar\otimes} M\_2'$:
In general, if $M \subset B(H)$ is a von Neumann algebra and $P \in M'$ with range $K$, you have $(M P)' = P M' P$ on $K$.
(You can find this in "Takesaki I", Proposition... | 2 | https://mathoverflow.net/users/9942 | 442360 | 178,496 |
https://mathoverflow.net/questions/442361 | 2 | Let $x\_1,\ldots,x\_n$ be $n$ complex numbers, and define $x\_I:=\sum\_{i\in I}x\_i$ for any set $I\subseteq[n]$. Finally, declare the family $(x\_1,\ldots,x\_n)$ to be "sumset-distinct" if the $2^n$ numbers $(x\_I)\_{I\subseteq [n]}$ are pairwise distinct. My questions are:
1. Has this notion been studied and if yes... | https://mathoverflow.net/users/477827 | Sumset-distinct numbers | The set $\{x\_i\}$ is called in the literature a sum-distinct set.
There is a big open problem in the area, due to Erdős and Moser: determining $f(n)=\min\{ \max(S): |S|=n, S\subseteq \mathbb{N}, S \text{ sum-distinct}\}$. The current lower bounds on $f$ are of the form $f(n)\ge C 2^n/\sqrt{n}$. A trivial upper bound... | 7 | https://mathoverflow.net/users/31469 | 442367 | 178,500 |
https://mathoverflow.net/questions/442304 | 4 | I obtained the very strange formula above and at begining I was just wanted know how to interpretate it. But now when I know what is this with help of @Carlo Beenakker, I am leaving it as a proof. BTW I found some [article](https://www.google.com/url?q=https://www.peertechzpublications.com/articles/AMP-5-141.pdf&sa=U&v... | https://mathoverflow.net/users/500629 | $\frac {f (0)}{2}+ \sum_{k=1}^{\infty}f (k)=\sum_{n=-\infty}^{\infty} \mathcal{L} \{ f \} (2 \pi i n)$ | *This is an answer to the question as it was originally formulated, it has now been heavily edited.*
---
I consider this formula in the OP,
$$\sum\_{k=1}^{\infty} f (k) = \int\_{0}^{\infty}f(t)dt+ \sum\_{n=1}^{\infty}\int\_{0}^{\infty}f(t)(e^{-2 \pi i nt}+e^{2 \pi i nt})dt.\qquad\qquad(\ast)
$$
This is *almost* t... | 3 | https://mathoverflow.net/users/11260 | 442373 | 178,502 |
https://mathoverflow.net/questions/442376 | 5 | Given an infinite family $\{\mathcal{F}\_{\lambda}$, $\lambda <\kappa\}$, $\kappa \geq \omega\_0$, of (ultra)filters of a set $X$, how it is defined the infinite tensor/Fubini product $$\bigotimes\_{\lambda <\kappa}\mathcal{F}\_\lambda$$ as a family of subsets of $X^{\kappa}$?
Recall that given two ultrafilers $\math... | https://mathoverflow.net/users/117312 | Infinite tensor/Fubini product of ultrafilters | The product of ultafilters $F\_\lambda$ for $\lambda<\kappa$ is defined on $\kappa\times X$, not $X^\kappa$, and it is defined relative to a fixed ultrafilter $\mu$ on the index set $\kappa$. Namely, for $Y\subseteq\kappa\times X$, we denote $Y\_\lambda=\{x\in X\mid (\lambda,x)\in Y\}$ for the various sections of $Y$ a... | 4 | https://mathoverflow.net/users/1946 | 442378 | 178,504 |
https://mathoverflow.net/questions/442384 | 1 | I have a sequence of $p$-dimensional infinitely divisible random vectors $S\_n'$, such that $S\_n' \Longrightarrow X$, as $n \to \infty$.
Suppose the following assumptions
1. The characteristic functions are given by:
$$\varphi\_{S\_n'}(u)=\exp\left\{ \int\_{\mathbb R^p} \left[e^{iu'x} - 1 - i u'x \right] \, d\nu\_... | https://mathoverflow.net/users/479236 | How to show that $\int x \,d\nu = 0$ using a pseudo-weak convergence of measures? | $\newcommand\de\delta$A counterexample is given by $p=1$, $\nu(dx):=|x|^{-5/2}\,1(0<|x|<1)\,dx$, and $\nu\_n(dx):=|x|^{-5/2}\,1(1/n<|x|<1)\,dx$.
---
The OP has added certain conditions. The only consequence of those additional conditions that matters in this context is that $\int d\nu\_n=n$ for all $n$ (so that $... | 2 | https://mathoverflow.net/users/36721 | 442388 | 178,507 |
https://mathoverflow.net/questions/442389 | 2 | I have a question about the p.d.f. of exponential family. Suppose $(X,\mathcal{F})$ is a measurable space and $\{F\_{\theta},\theta\in \Theta\}$ is a distribution family on $(X,\mathcal{F})$. When $\{F\_{\theta},\theta\in \Theta\}$ is dominated by a $\sigma$ finite measure $\mu$, $\{F\_{\theta},\theta\in \Theta\}$ is a... | https://mathoverflow.net/users/500703 | p.d.f. of exponential family | $\newcommand\la\lambda$Let $F:=F\_\theta$ and $R:=\exp\{\eta(\theta)^\top T-\xi(\theta)\}$, so that
$$\frac{dF}{d\mu}=Rh,$$
which means that
$$\int g\,dF=\int gRh\,d\mu \tag{1}\label{1}$$
for all measurable functions $g$ such that either one (or, equivalently, both) of the latter two integrals exist (say in $[-\infty,\... | 3 | https://mathoverflow.net/users/36721 | 442390 | 178,508 |
https://mathoverflow.net/questions/442383 | 10 | [In a recent course in Bonn](https://people.mpim-bonn.mpg.de/scholze/SixFunctors.pdf), P. Scholze explains a formalization of a six-functor formalism due to L. Mann. In this axiomatization, three of the functors $f\_!,f^\*,\otimes$ are "constructed" (in the form of a lax monoidal functor) and then the other three are d... | https://mathoverflow.net/users/131975 | Can we use Mann's six-functor formalism with D-modules? | The six functor formalism applies to $D$-modules, but you need to extend the theory to possibly non-smooth schemes. For this, we see that hypersheaves on the site of pairs $(X,Z)$, with $Z$ a closed subscheme of a smooth scheme $X$, with maps the obvious commutative squares, with coverings those maps $(X,Z)\to (X',Z')$... | 3 | https://mathoverflow.net/users/1017 | 442419 | 178,519 |
https://mathoverflow.net/questions/442397 | 2 | I am trying to lower bound the following function, $n \ge 3$ is a natural number:
$$l(n):=\frac{\log(n)}{\log(n)-\frac{1}{n}(\tau(n)\log(\tau(n))+(n-\tau(n))\log(n-\tau(n)))}$$
where $\tau(n)$ counts the number of divisors. Is this function exponential or polynomial in terms of $\log(n)$?
Thanks for your help.
| https://mathoverflow.net/users/165920 | Lower bound on a function of the number of divisors? | Tried to write a comment but got too long, not sure if correct or what you were looking for but hopefully is useful otherwise let me know and I will delete it.
What I did was to rewrite the denominator as follows
$\ln(n)-\frac{1}{n}(\tau(n)\ln(\tau(n))+(n-\tau(n))\ln(n-\tau(n)))\\
=\ln(2)+\ln(n/2)-((\tau(n)/n)\ln(\ta... | 3 | https://mathoverflow.net/users/142708 | 442425 | 178,521 |
https://mathoverflow.net/questions/435315 | 1 | Theorem 1 of [LS] [Liebeck and Seitz - On the subgroup structure of exceptional groups](https://doi.org/10.1090/S0002-9947-98-02121-7) says:
>
> Let $X = X(q)$ be a quasisimple group of Lie type in characteristic $p$, and suppose that $X < G$, where $G$ is a simple adjoint algebraic group of exceptional type, also ... | https://mathoverflow.net/users/2383 | Generic finite subgroups, associated to small finite fields, of reductive algebraic groups | The state of the art for subgroups $\operatorname{PSL}\_2(q)$, which constitute most of the cases where $q > 9$ in [LS], is now given by David Craven's Memoir [Maximal $\operatorname{PSL}\_2$ Subgroups of Exceptional Groups of Lie Type](https://doi.org/10.1090/memo/1355) ([arxiv version](https://arxiv.org/abs/1610.0746... | 3 | https://mathoverflow.net/users/3935 | 442434 | 178,525 |
https://mathoverflow.net/questions/442222 | 5 | Let $G=(V, E)$ be an acyclic digraph (DAG) with all in- and out-degrees at most $k$. Is it true that the edges of $G$ may be always colored properly in $2k$ colors?
In the discussion of [this](https://mathoverflow.net/a/442170/4312) question it is proved (for $k=2$, but the proofs work verbatim for other values of $k... | https://mathoverflow.net/users/4312 | Chromatic index of an acyclic digraph | Here is a solution (obtained while discussing the question with my colleague András Sebő).
Consider the undirected graph $H$ obtained from $G$ as follows: the partite sets of $H$ are two copies $V\_1$ and $V\_2$ of $V(G)$, and for each arc $(u,v)$ in $G$ we add an edge in $H$ between the copy of $u$ in $V\_1$ and the... | 3 | https://mathoverflow.net/users/45855 | 442440 | 178,527 |
https://mathoverflow.net/questions/442416 | 1 | For a long time, I had a false belief that the space/stack $\text{Coh}^{tf}\_{c\_1,c\_2}S$ of torsion-free sheaves $\mathcal{E}$ on a smooth algebraic surface $S$ was not connected, since if you take its map into its double dual
$$0\ \to \ \mathcal{E}\ \to\ \mathcal{E}^{\vee\vee}\ \to \ \mathcal{Q}\ \to\ 0$$
the length... | https://mathoverflow.net/users/119012 | Families of torsion-free sheaves whose length jumps | Consider the sheaf $F$ on $\mathbb{P}^2 \times \mathbb{A}^1$ defined from the exact sequence
$$
0 \to
F \longrightarrow
\mathcal{O}(-1)^{\oplus 3} \stackrel{(x,y,tz)}\longrightarrow
\mathcal{O} \longrightarrow
\mathcal{O}\_P \to
0,
$$
where $(x,y,z)$ are the homogeneous coordinates on $\mathbb{P}^2$, $t$ is the coor... | 1 | https://mathoverflow.net/users/4428 | 442442 | 178,528 |
https://mathoverflow.net/questions/442456 | 4 | Let $X$ be an irreducible smooth projective variety of pure dimension $d$ over the complex numbers and $Z\subset X\times X$ a codimension $d$ irreducible smooth closed subvariety.
>
> Is there a smooth hyperplane section $H$ on $X$ such that $H\times X$ contains a cycle in the rational equivalence class of $Z$?
>
... | https://mathoverflow.net/users/497064 | Cycles contained in ample enough hypersurfaces | Let $X:=\Sigma\_g$ be a smooth curve of genus $g\geq 2$ and $Z=\Delta$, where $\Delta \subset \Sigma\_g \times \Sigma\_g$ is the diagonal.
Then $\Delta$ is not rationally equivalent to a union of fibres of $p\_1 \colon \Sigma\_g \times \Sigma\_g \to \Sigma\_g$, for instance because $\Delta^2=2-2g <0$.
| 8 | https://mathoverflow.net/users/7460 | 442458 | 178,533 |
https://mathoverflow.net/questions/442412 | 1 | Let $L\_1$, $L\_2$ be two subspaces in $\mathbb{R}^n$ and $\dim(L\_1) = \dim(L\_2) = s<n$. Let $$0 \leq \theta\_1 \leq \dotsb\leq \theta\_{s} \leq \pi/2$$ be the principal angles between $L\_1, L\_2$. Let there be two unit vectors $u\_1\in L\_1$ and $u\_2 \in L\_2$ and the principal angle between $\operatorname{span}\{... | https://mathoverflow.net/users/151115 | Principal angles between subspaces and angle between unit vectors | $\newcommand\si\sigma\newcommand\al\alpha\newcommand\be\beta$Let $U:=L\_1$ and $V:=L\_2$. Without loss of generality, $s=\dim U=\dim V\ge1$. By the section [Angles between subspaces](https://en.wikipedia.org/wiki/Angles_between_flats#Angles_between_subspaces) of the Wikipedia article, there are orthonormal bases $(a\_1... | 2 | https://mathoverflow.net/users/36721 | 442460 | 178,534 |
https://mathoverflow.net/questions/442449 | 1 | Random variable $X\geq 0$ and its variance exists. How to prove
$$\mathbb{P}(X\geq(1-t)\mathbb{E}(X))\geq \frac{t^2\mathbb{E}(X)^2}{\mathbb{E}(X^2)}\enspace\text{for}\enspace t\in(0,1]$$
$$\mathbb{E}(\exp(-\lambda(X-\mathbb{E}(X))))\geq\exp(\lambda^2\mathbb{E}(X^2)/2)\enspace\text{for}\enspace \lambda\geq0$$
$$\m... | https://mathoverflow.net/users/500967 | Prove inequality on expectation | As Alapan Das noted, the first inequality for $t\in[0,1]$ is the [Paley–Zygmund inequality](https://en.m.wikipedia.org/wiki/Paley%E2%80%93Zygmund_inequality). For $t<0$ or $t>1$, the first inequality is false if e.g. $X=1$.
The second inequality is false for all real $\lambda\ne0$ if e.g. $X=1$.
The third inequalit... | 2 | https://mathoverflow.net/users/36721 | 442462 | 178,535 |
https://mathoverflow.net/questions/442451 | 2 | The following is a heuristic for the situation where a decision algorithm or a human, might solve a problem by reducing the entropy of the "search space" at every computation step $i$:
Let $X\_{i+1} \subset X\_i$ for $i = 1,\cdots,t-1$ be finite subsets contained all in a finite set $X\_1$.
We assume that there are... | https://mathoverflow.net/users/165920 | Entropy reduction? | Suppose $X\_1=\{1,\dots,m+n\}$, where $m,n\to\infty$ and $\frac mn\,\log\_2 n\to0$. Suppose that $f\_1(x)=x$ for $x\in\{1,\dots,m\}$ and $f\_1(x)=m+1$ for $x\in\{m+1,\dots,m+n\}$, so that $X\_2=\{1,\dots,m+1\}$. Suppose also that $f\_2(x)=x$ for $x\in\{1,\dots,m-1\}$ and $f\_2(x)=m$ for $x\in\{m,m+1\}$. Then
$$H\_1=\fr... | 3 | https://mathoverflow.net/users/36721 | 442463 | 178,536 |
https://mathoverflow.net/questions/425318 | 5 | It is classical that Euclidean normal currents are dense in the space of all currents.
This can be achieved through mollification.
What I want to know if this is still true for metric currents.
In particular I am interested if the space of Normal 1-currents is always dense in the space of all 1-currents for any separ... | https://mathoverflow.net/users/56713 | Are normal metric currents dense in the space of all metric currents? | The answer is no, and it is pretty trivial apparently.
Let us look at $[0,1]\subseteq\mathbb R$ and define on it the simplest current
$$
T(fd\pi)=\int\_0^1f(t)\pi'(t)dt.
$$
Now we enumerate the rational numbers on it $\{q\_n\}\_{n\in\mathbb N}\subseteq[0,1]$ and given any positive $1>\varepsilon>0$ and any positive e... | 0 | https://mathoverflow.net/users/56713 | 442470 | 178,538 |
https://mathoverflow.net/questions/442472 | 6 | Given a set $S$, a tight apartness relation on $S$ is a relation $\#$ which is tight, irreflexive, symmetric, and a comparison, or more specifically, a relation $\#$ such that
* for all elements $a \in S$ and $b \in S$, $a = b$ if and only if $\neg a \# b$
* for all elements $a \in S$, $\neg a \# a$
* for all element... | https://mathoverflow.net/users/483446 | Does a tight apartness relation on a subobject classifier imply the elementary topos is Boolean? | Yes. The tightness axiom, $(a=b)\iff \neg (a\#b)$, implies
$$\neg\neg(a=b) \iff \neg\neg\neg(a\#b) \iff \neg(a\#b) \iff (a=b).$$
Taking $b=\top$, we find that $\neg\neg a \iff a$ for all $a: \Omega$. This is the law of double negation, which is equivalent to excluded middle.
| 9 | https://mathoverflow.net/users/49 | 442476 | 178,539 |
https://mathoverflow.net/questions/442468 | 6 | An elliptic curve is (for the purpose of this question) a cubic algebraic curve defined by an equation (short Weierstrass equation) of the form
$$\displaystyle E\_{a,b} : y^2 = x^3 + ax + b, a, b \in \mathbb{Z}, 4a^3 + 27b^2 \ne 0.$$
The elliptic curves with $a = 0$ are called *Mordell curves*.
Do we have example... | https://mathoverflow.net/users/10898 | Mordell curves with large rank | As far as I know, this record is still held by a 3-isogenous pair of
Mordell curves of rank $\bf 17$ that I found and announced in February 2016.
(This superseded a rank-16 pair from earlier that month, and curves of ranks
13, 14, 15 from October 2009.)
The curves $y^2 = x^3 + b$ and $y^2 = x^3 - 27 b$
are always rel... | 13 | https://mathoverflow.net/users/14830 | 442487 | 178,544 |
https://mathoverflow.net/questions/375991 | 2 | In David Mumford's book Algebraic Geometry I, Complex Projective Varieties the
proof of (3.25) **Specialization principle** on page 53 contains an argument
I not understand.
General assumptions: all our varieties are over $\mathbb{C}$. The statement is:
>
> (3.25) **Specialization principle.** Let $Z \subset \mat... | https://mathoverflow.net/users/108274 | Comparison of classical and Zariski topologies with constructible sets | First, for lower semi-continuity, we should consider sublevel set, i.e.
$$\{x\in X:\ \#\phi^{-1}(x)\leq n\}$$
Now, since $\#\phi^{-1}(x)$ is a constructible function, the above subset, denoted as A, satisfies that
\begin{equation}
A=\bigcup\_{i=1}^N (Z\_{i,1}-Z\_{i,2})
\end{equation}
where $Z\_{i,j}$'s are all Zariski ... | 1 | https://mathoverflow.net/users/134247 | 442489 | 178,545 |
https://mathoverflow.net/questions/442249 | 3 | Let $A$ be any set and $S$ be a subset of $A^2$ such that sets $\{a\in A\mid (x,a)\in S\}\mathbin\triangle\{a\in A\mid (y,a)\in S\}$ and $\{a\in A\mid (a,x)\in S\}\mathbin\triangle\{a\in A\mid (a,y)\in S\}$ are finite for any $x,y\in A$. What can be the set $S$?
There are two simple examples of the set $S$, when sets... | https://mathoverflow.net/users/144883 | An almost uniform subset of the Cartesian square | At first, let us introduce some relevant definitions.
**Definition.** A $S\subseteq X\times Y$ is called
$\bullet$ *horizontally finite* in $X\times Y$ if for every $y\in Y$ the set $\{x\in X:(x,y)\in S\}$ is finite;
$\bullet$ *horizontally cofinite* in $X\times Y$ if for every $y\in Y$ the set $\{x\in X:(x,y)\no... | 4 | https://mathoverflow.net/users/61536 | 442502 | 178,548 |
https://mathoverflow.net/questions/442495 | 5 | Let $X$ be a topological space and $x \in X$ a point. Let $\Omega$ be the set of open sets (viꝫ. the topology) of $X$, and $\Omega\_x$ the set of *germs* around $x$ of open sets, that is, $\Omega\_x = \Omega/{\sim}$ where $U\sim V$ means there exists a neighborhood $W$ of $x$ such that $U\cap W = V\cap W$. (Note that $... | https://mathoverflow.net/users/17064 | Do germs of open sets around a point form a frame? | I just cooked up a counterexample in my head. Joseph Van Name beat me, but still here it is - to make things completely explicit. I hope I did not make any mistakes.
I will reason with closed sets and intersections; to get the answer with open sets and unions, just take the complement of everything.
So let $x = 0$,... | 5 | https://mathoverflow.net/users/39348 | 442507 | 178,550 |
https://mathoverflow.net/questions/436049 | 4 | Arinkin-Gaitsgory have defined the notion of singular support for any quasismooth $Y$
$$\text{SS}(\mathcal{F})\ \subseteq\ \text{Sing}(Y)$$
and $\mathcal{F}$ any ind-coherent sheaf, where $\text{Sing}(Y)$ is the category of singularities of $Y$ (the classical truncation of the minus one shifted cotangent complex). Thus... | https://mathoverflow.net/users/119012 | Two notions of singular support? | When $X$ is a smooth scheme, the derived loop space (i.e. odd tangent bundle) $\mathcal{L}X\simeq\mathbb{T}\_X[-1]$ has $\mathrm{Sing}(\mathbb{T}\_X[-1]) = T^\*X$. Furthermore, there is a Koszul duality:
$$\mathrm{Coh}(\mathbb{T}\_X[-1])^{B\mathbb{G}\_a \rtimes \mathbb{G}\_m} \simeq F\mathcal{D}(X)$$
where the right-ha... | 2 | https://mathoverflow.net/users/6059 | 442510 | 178,551 |
https://mathoverflow.net/questions/258159 | 26 | I'm trying to understand how the six functor philosophy applies to representation theory. Consider the category of classifying stacks $BG$ (assume $G$ discrete for simplicity). To every stack we can assign a triangulated category $D(BG)=D(BG,k):= D(Rep(G))$ the (perhaps bounded) derived category of the abelian category... | https://mathoverflow.net/users/22810 | Yoga of six functors for group representations? | The accepted answer here is on a rather negative note -- I don't think that's fair! In fact, I think the correct answer is that all of this works, except that it is $\pi\_\ast$ that gives group cohomology, and $\pi\_!$ that gives group homology.
More precisely, consider the category $C$ of locally compact Hausdorff s... | 12 | https://mathoverflow.net/users/6074 | 442524 | 178,556 |
https://mathoverflow.net/questions/442500 | 3 | The similar and more general question is asked [here](https://mathoverflow.net/questions/334335/lipschitz-function-of-independent-subgaussian-random-variables?noredirect=1&lq=1), whose setting is random vectors.
Let $X$ be $\sigma$-sub-Gaussian and $f$ is a Lipschitz function w.r.t. constant $L$. How to prove can the... | https://mathoverflow.net/users/500967 | Lipschitz function of subgaussian random variable | $\newcommand{\si}{\sigma}\newcommand\R{\mathbb R}$One of mutually equivalent definitions of a ($\si$)-sub-Gaussian random variable (r.v.) $X$ is as follows:
\begin{equation\*}
Ee^{X^2/\si^2}\le2. \tag{1}\label{1}
\end{equation\*}
It is now clear that the answer to the question is no. Indeed, suppose that $X\sim N(0,... | 1 | https://mathoverflow.net/users/36721 | 442530 | 178,558 |
https://mathoverflow.net/questions/442533 | 0 | One of my research problem can be reduced to a question of the following form
>
> Given a set family $\mathcal{F}$ of $[n]$ , such that every element of $[n]$ lies in exactly $K$ sets in $\mathcal{F}$, can w partition $\mathcal{F}$ into $K$ subfamilies $\mathcal{F}\_i$ such that each subfamily is a partition of $[n... | https://mathoverflow.net/users/475875 | "JigSaw Puzzle" on Set Family | No. E.g., let $n=3$ and $\mathcal F=\{\{1,2\},\{1,3\},\{2,3\}\}$, so that $K=2$. However, no subset of $\mathcal F$ is a partition of $[n]$.
| 3 | https://mathoverflow.net/users/36721 | 442534 | 178,559 |
https://mathoverflow.net/questions/442539 | 4 | Suppose we have an elementary embedding $j: M\to N$, a forcing notion $\mathbb{P}\in M$, and a strong master condition $q\in j(\mathbb{P})$. A strong master condition for $j$ and $\mathbb{P}$ is a condition $q\in j(\mathbb{P})$ such that for every dense set $D\subseteq \mathbb{P}$ with $D\in M$, there is a condition $p... | https://mathoverflow.net/users/498641 | Given an elementary embedding $j: M\to N$ and a strong master condition $q\in N$, how are we able to construct a generic over $M$ from $M$? | You’re right; this kind of strong master condition cannot exist in the usual context where $j : M \to N$ is a class of $M$. The relevance is when we’re in an intermediate stage of lifting an embedding. Suppose $j : V \to N$ is definable from parameters in $V$ and $\mathbb P$ is $\mathrm{crit}(j)=\kappa$-c.c. and has th... | 3 | https://mathoverflow.net/users/11145 | 442541 | 178,561 |
https://mathoverflow.net/questions/442551 | 0 | Are there infinitely many ones in the simple continued fraction for pi? I know that there’s a probability distribution given through Gauss-Kuzmin, but is there a proof that there’s infinitely many ones? How about proofs that there are infinitely many of any positive integer in pi’s simple continued fraction?
| https://mathoverflow.net/users/500847 | Infinitely many ones in continued fraction of pi? | Nothing like that is known. There is likewise no proof that the decimal expansion of $\pi$ has infinitely many 1's (or any other specific digit).
The continued fraction expansions of algebraic numbers of degree greater than $2$ are equally opaque: none are known to have any specific positive integer appearing in them... | 6 | https://mathoverflow.net/users/3272 | 442554 | 178,563 |
https://mathoverflow.net/questions/442092 | 3 | Let $(X, d)$ be a compact metric space.
* We say that $\{x\_1, \cdots, x\_n\} \subseteq X$ is an $\varepsilon$-**covering** of $X$ if for any $x \in X$, there exists $i \in \{1, \ldots, n\}$ such that $d(x, x\_i) \leq \varepsilon$. Let
$$
\operatorname{Cov} (X, \varepsilon) := \min \{n: \exists \varepsilon \text {-co... | https://mathoverflow.net/users/99469 | If $X,X'$ have the same $\varepsilon$-packing numbers and $f:X \to X'$ surjective $1$-Lipschitz, then $f$ is an isometry | Consider two metrics on $\{x,x',y\}$ defined by
$$|x-y|\_1=|x'-y|\_1=|x-y|\_2=3, \quad |x-x'|\_1=|x-x'|\_2=1, \quad |x'-y|\_2=2.$$
Denote by $X\_1$ and $X\_2$ the corresponding metric spaces.
Note that $\mathrm{pack}\_\varepsilon X\_1\equiv \mathrm{pack}\_\varepsilon X\_2$.
Indeed, for both spaces we have
* $\mathr... | 2 | https://mathoverflow.net/users/1441 | 442567 | 178,568 |
https://mathoverflow.net/questions/442571 | 2 | Consider the series
$$f(z)=\sum\_{n\ge1}\dfrac{z^n}{n^2\binom{2n}{n}}$$
which converges for $|z|\le4$.
One has $f(0)=0$, $f(1)=\pi^2/18$, $f(2)=\pi^2/8$, $f(3)=2\pi^2/9$, and $f(4)=\pi^2/2$.
Naive question: are there any other reasonably "explicit" evaluations (for instance for
$z=-1$, $-2$, $1/2$...) ? Note that the... | https://mathoverflow.net/users/81776 | Special values of $\sum_{n\ge1}z^n/(n^2\binom{2n}{n})$ | I interpret the request of the OP for an "explicit" evaluation of the series as a request for a "closed-form" expression, which exists (it seems to go back to Euler, [here](https://math.stackexchange.com/a/383181/87355) are several proofs):
$$f(z)=\sum\_{n\ge1}\dfrac{z^n}{n^2\binom{2n}{n}}=2 \arcsin^2\,(\tfrac{1}{2}\sq... | 8 | https://mathoverflow.net/users/11260 | 442573 | 178,570 |
https://mathoverflow.net/questions/442578 | 0 | Let $u(x) \in H^1\_0$ is a complex function in sobolev space extension by zero.
How to find $J'(u)$ for
$$
J(u)= \int\limits\_0^l |u(x)|^2\operatorname{d\!}x\;??
$$
In $L\_2$ it's easy:
$$
J'(u) = \left(\int\limits\_0^l|u(x)|^2 \operatorname{d\!}x\right)'=\big(\|u(x)\|^2\big)'= 2 u(x),
$$
but it does not work with $... | https://mathoverflow.net/users/492767 | Derivative in Sobolev space extended by zero | We shall assume that $l\in(0,\infty)$, so that for any $h\in H\_0^1$ we have
$$\int\_0^l|h|^2=\int\_0^l dx\,\Big|\int\_0^x h'\Big|^2
\le\int\_0^l dx\,\Big(\int\_0^l|h'|\Big)^2 \\
\le\int\_0^l dx\,l\,\int\_0^l|h'|^2=l^2\|h\|^2\_{H\_0^1}. \tag{1}\label{1}$$
Next, for any $u$ and $h$ in $H\_0^1$, in view of \eqref{1},... | 3 | https://mathoverflow.net/users/36721 | 442580 | 178,571 |
https://mathoverflow.net/questions/442399 | 5 | The completeness game $G\_{\gamma}(P)$ for a partial order $P$ has players COM and INC play alternatingly and descendingly elements of $P$ with player INC playing first and player COM playing at limit steps. INC wins the game if at some point $\alpha<\gamma$ (necessarily a limit ordinal) there are no legal moves left. ... | https://mathoverflow.net/users/138274 | Are the completeness Games $G_{\lambda+1}(P)$ and $G_{\lambda^+}(P)$ equivalent for INC? | The answer to the second question is no as well.
Suppose $\dot\tau$ is a $Q$-name for a strategy for the player INC in the game $G\_\kappa(P)$. Let us pretend that COM opens the game instead of INC and note that this is unproblematic. We will define by induction a descending sequence $\langle p\_\alpha\mid\alpha<\kap... | 4 | https://mathoverflow.net/users/125703 | 442582 | 178,572 |
https://mathoverflow.net/questions/442557 | 6 | Let $\mathsf{C}$ be a *possibly large* category with a Grothendieck topology satisfying the [*Weakly Initial Set of Covers*](https://ncatlab.org/nlab/show/WISC) condition: there is for each $X$ a *set* (not a proper class) of covering families of $X$ such that if $\{U\_i \to X\}$ is an arbitrary covering family, it is ... | https://mathoverflow.net/users/135175 | Subobject classifier for sheaves on large sites with WISC | To answer your question directly, WISC does not imply the existence of subobject classifiers.
Notice that when there are only trivial covers, WISC is trivially satisfied, so it suffices to find a category with a proper class of sieves.
Consider the ordered class of ordinals with a terminal object adjoined.
Then each or... | 6 | https://mathoverflow.net/users/11640 | 442587 | 178,574 |
https://mathoverflow.net/questions/442550 | 3 | Currently I am working on studying stochastic integrals of the form: $$Z\_\infty = \int\_0^\infty e^{-f(t)}\mathop{d}S\_t$$
where $S\_t$ is a Compound-Poisson process with Exponentially-distributed increments. We know that the integral converges so long as $f$ is continuous on $[0,\infty)$ and $\lim\_{t\to\infty}f(t)... | https://mathoverflow.net/users/500844 | Are there any known results on the probability distributions of perpetuities with power law discount rates? | This is more of an extended comment trying to study this integral.
In the similar spirit as [here](https://math.stackexchange.com/questions/2586336/stochastic-integral-of-poisson-process/2587999#2587999) we study the Laplace transform. As explained [here](https://personal.ntu.edu.sg/nprivault/MA5182/stochastic-calcul... | 1 | https://mathoverflow.net/users/99863 | 442588 | 178,575 |
https://mathoverflow.net/questions/442480 | 4 | Suppose $M = K\backslash G/\Gamma$ is a quotient of a symmetric space by a lattice. I don't know all of the proper adjectives to apply here (e.g. what should be said about $G$ and so on), but I wouldn't be upset if we just took $G=\operatorname{O}(1,n)$ and $K = \operatorname{O}(n)$ so that $M$ is a finite volume hyper... | https://mathoverflow.net/users/151664 | Inheritance of arithmeticity properties in orbifold strata | Here is what I think is the correct setup:
Let $X$ be a symmetric space of noncompact type, $\Gamma$ is a lattice in the isometry group of $X$. Then $\Gamma$ has finitely many $\Gamma$-conjugacy classes of finite subgroups. Let $\Phi$ be one of these finite subgroups (I will pick one from each conjugacy class). Then ... | 2 | https://mathoverflow.net/users/39654 | 442589 | 178,576 |
https://mathoverflow.net/questions/437141 | 4 | Let $G=GL(n, F)$, $B$ be a Borel subgroup and let $B=AN$ be the Langlands decomposition. Let $\nu \in \mathfrak{a}^\*\_{\mathbb{C}}$ be in the positive Weyl chamber. Consider the normalized induced representation $I = I^G\_B(\nu)$. Let $F=\mathbb{R}$ and $J$ be the unique generic irreducible subrepresentation of $I$ wh... | https://mathoverflow.net/users/58056 | Intertwining operators and induced representation | I will answer question 1 for when $\nu=\rho$ and $F$ is $p$-adic (since I am not too familiar with $F=\mathbb R$; I assume a similar argument works for $F=\mathbb R$ as well). It suffices to check that $I\_B^G(\rho)$ is not isomorphic to $I\_B^G(w\rho)$ for $w\ne1\in W$. But this is clear, since
$$\begin{align\*}
\hom\... | 1 | https://mathoverflow.net/users/123673 | 442596 | 178,578 |
https://mathoverflow.net/questions/442555 | 1 | $\newcommand{\fg}{\mathfrak g}\newcommand{\ee}{\varepsilon}$Let $\fg$ be the complex simple Lie algebra of type G$\_2$.
We consider its root system as follows (though it is probably not necessary to state the question):
\begin{equation\*}
\Delta^+(\mathfrak g,\mathfrak h) = \left\{
\begin{array}{ll}
\ee\_2-\ee\_3,&\e... | https://mathoverflow.net/users/20052 | About certain elements in the zero weight space of an irreducible representation of the complex simple Lie algebra of type G$_2$ | The answer is no. Here is some SAGE code:
```
sage: G2, A1xA1 = [WeylCharacterRing(x,style="coroots") for x in ["G2","A1xA1"]]
sage: b = G2.maximal_subgroup("A1xA1")
sage: G2(2,2).branch(A1xA1,rule=b) ... | 3 | https://mathoverflow.net/users/425 | 442598 | 178,579 |
https://mathoverflow.net/questions/442056 | 2 | My question is on Brascamp-Lieb-inequality on the Euclidean sphere (which can be viewed as an analogue of Young's inequality on the sphere) obtained in [1].
(See also this question:
[Brascamp-Lieb inequalities on the sphere](https://mathoverflow.net/questions/325211/brascamp-lieb-inequalities-on-the-sphere) )
The i... | https://mathoverflow.net/users/116555 | (simplified further) Is Brascamp-Lieb inequality on the sphere applicable for these functions for some $1\leq p<2$ | Unfortunately, the answer to the question as stated is "No" even if $N=2$. Let $k=1,\alpha\_1=\alpha\_2=\frac 12$. Choose $x=(x\_1,x\_2)$ on the unit circle (say $(3/5,4/5)$). Then the integrals on the right are finite but the integral on the left diverges (you basically integrate inverse distance to $x$). Perhaps, you... | 1 | https://mathoverflow.net/users/1131 | 442601 | 178,581 |
https://mathoverflow.net/questions/442503 | 20 | The six functor formalism in a given cohomology theory consists of for each space a derived category of sheaves and six different ways to construct functors between those categories (four involving a morphism and two only a single space). It then consists of many *coherences* - these are isomorphisms between certain co... | https://mathoverflow.net/users/18060 | When (or why) is a six-functor formalism enough? | When defining a homotopy-coherent structure, you have to strike the correct balance between supplying enough data (so that all isomorphisms (between isomorphisms, ...) that you need later are actually defined), and not supplying "too much" data (because if you include two isomorphisms between $A$ and $B$, you might lat... | 19 | https://mathoverflow.net/users/6074 | 442604 | 178,582 |
https://mathoverflow.net/questions/442147 | 4 | *This is a follow-up to [a recent question of mine](https://mathoverflow.net/questions/442104/example-of-trickiness-of-finite-lattice-representation-problem):*
For $n\in\mathbb{N}$ let $C(n)$ be the smallest $k$ such that every bounded lattice with cardinality $\le n$ which is isomorphic to the congruence lattice of ... | https://mathoverflow.net/users/8133 | How large must algebras with a given congruence lattice be? | **What do we know about the growth rate of $C(n)$?**
We know the exact value of $C(n)$ if, in its definition,
we restrict to the class of distributive lattices.
Otherwise we only have partial results. Let me say a few words.
**Claim.**
If $L$ is a finite distributive that has a maximal chain
$\ell\_1<\ell\_2<\cdots ... | 3 | https://mathoverflow.net/users/75735 | 442610 | 178,584 |
https://mathoverflow.net/questions/442572 | 1 | Let $G=\operatorname{SO}^{+}\_{2n}(2)$. I did some Magma computation and found there were $3$ orbits on the natural $G$-set when $n=2,3,4$. The orbit sizes are $1$-$9$-$6$, $1$-$35$-$28$, $1$-$135$-$120$.
Is there a formula for the orbit sizes in general? Thank you.
| https://mathoverflow.net/users/488802 | Orbit sizes of $G=\operatorname{SO}^{+}_{2n}(2)$ | As Mikko Korhonen said in the comments, the (non-zero) singular and non-singular vectors form single orbits by Witt's lemma.
So it remains to count the singular vectors, which are (row) vectors ${\mathbf v}$ over ${\mathbb F}\_2$ such that ${\mathbf v}M {\mathbf v}^{\mathsf T} = 0$, where $M$ is the degree $2n$ matri... | 4 | https://mathoverflow.net/users/35840 | 442615 | 178,586 |
https://mathoverflow.net/questions/442297 | 5 | Reading [Chapter V](https://drive.google.com/file/d/1x9JpMslhI87jr6Utl4gX7Qf-etGbs6To/view?usp=sharing), pages (73-97) in *Proof Theory* (Springer, 1977), by Kurt Schütte, I have encountered a peculiar problem which puzzles me.
On page 96, a map $\rm{Nr}:\overline{\rm{OT}}\rightarrow \mathbb{N}$ is defined using the ... | https://mathoverflow.net/users/23204 | A possible flaw in Theorem 14.17 in Kurt Schütte's -Proof Theory- | Fortunately, there appears to be no flaw in Schütte's construction. Only in my understanding of it.
Ordinal terms are defined on page 86. The term $\alpha=(0,((0,0),0))$ does NOT represent $\varepsilon\_{0}$. After a more careful look at the definitions on page 86 and the definition of $\psi$ on page 84 I see now tha... | 4 | https://mathoverflow.net/users/23204 | 442631 | 178,591 |
https://mathoverflow.net/questions/442540 | 2 | If $A\subset\mathbb{R}^2$ is a Borel measurable set and $p\_\theta$ is projection onto the line spanned by $(\cos\theta,\sin\theta)$, then it is well known that for almost every $\theta\in[0,2\pi]$, $p\_\theta(A)$ has Hausdroff dimension the min of 1 and the Hausdorff dimension of $A$.
My question is: let's say $A$ h... | https://mathoverflow.net/users/147078 | Exceptional set for Marstrand's projection theorem | Theorem 1.1 of Orponen's paper "On the packing dimension and category of exceptional sets of orthogonal projections" contains a (relatively short) construction of a compact set $K\subseteq\mathbb{R}^2$ with $\infty>\mathcal{H}^1(K)>0$ and a dense $G\_\delta$ set $\Omega\subseteq[0,2\pi)$ such that $p\_\theta(K)$ has ze... | 1 | https://mathoverflow.net/users/349327 | 442635 | 178,592 |
https://mathoverflow.net/questions/442638 | 2 | $\DeclareMathOperator\Mod{Mod}$I would like to compute the mapping class group (homeomorphism preserving orientation modulo those isotopic to the identity) of the sphere $S^2$ minus $n$ points $p\_1,\dots, p\_n$: $\Mod(S^2\setminus \lbrace p\_1, \dots, p\_n\rbrace)$. I already know that the mapping class group of the d... | https://mathoverflow.net/users/158821 | How to get a presentation of the mapping class group of the $n$-punctured sphere | $\DeclareMathOperator\Diff{Diff}\DeclareMathOperator\Emb{Emb}\DeclareMathOperator\fix{fix}$There's a variety of ways to do this. If you take "mapping class group" to mean "isotopy classes of diffeomorphisms" then a fairly natural approach is to consider the braid group on $n$ strands to be $\pi\_0$ of diffeomorphisms o... | 5 | https://mathoverflow.net/users/1465 | 442640 | 178,594 |
https://mathoverflow.net/questions/442622 | 3 | Let $M$ be a connected and simply connected compact Riemannian manifold (without boundaries). Fix a point $p\in M$.
The diameter set $D\_p$ of $p$ is the set of points that maximize the distance from $p$, i.e., $D\_p=\lbrace q : d(p,q)=\max\_r d(p,r)\rbrace$.
The cut locus $C\_p$ is the set of points at which geodesics... | https://mathoverflow.net/users/485160 | Cut locus for simply connected manifolds | The equality $D\_p = C\_p$ holds for compact symmetric spaces (CROSSes) of rank $1$, but not in general for higher rank symmetric spaces.
A rank $1$ CROSS is isometric to a round sphere or projective space with Fubini-Study metric. For a sphere, we of course have $D\_p = C\_p = \{-p\}$. On $\mathbb{K}P^n$ with $\math... | 5 | https://mathoverflow.net/users/1708 | 442641 | 178,595 |
https://mathoverflow.net/questions/442642 | 3 | Given an abstract simplicial complex $K$ on a set of vertices $V$, we can form a semi-simplicial set by $F(K)$ sending $F(K)\_n$ to be the set of ordered $(n+1)$-tuples of vertices in $V$ forming an $n$-simplex in $K$; face maps are defined in the obvious way. One can add in degeneracies by hand to make this a simplici... | https://mathoverflow.net/users/120548 | Simplicial set from all orderings of simplicial complex | If I understand this correctly, this construction shows up in various places in the study of homological stability. The one main result I know connecting the topology of $F(K)$ and $K$ is as follows. Say that $K$ is **weakly Cohen-Macaulay** of dimension $n$ if $K$ is $(n-1)$-connected and for all simplices $\sigma$ of... | 4 | https://mathoverflow.net/users/317 | 442644 | 178,596 |
https://mathoverflow.net/questions/442600 | 12 |
>
> For each real $A>0$, let $x\_A$ denote the positive root $x$ of the polynomial $x^5-3x-A$. Is the function $(0,\infty)\ni A\mapsto x\_A$ elementary?
>
>
>
~~[I am using this [definition of elementary functions](https://en.wikipedia.org/wiki/Elementary_function#Basic_examples), except for the algebraic functi... | https://mathoverflow.net/users/36721 | Can the positive root of this polynomial be expressed elementarily? | I believe that the class of functions that Iosif Pinelis is interested in is what I would call *exponential-logarithmic functions* or *EL functions*; that is, they are the functions that can be expressed using some finite combination of constant functions, the identity function, $\exp$, $\log$, composition, and arithme... | 19 | https://mathoverflow.net/users/3106 | 442656 | 178,601 |
https://mathoverflow.net/questions/442436 | 2 | We are given a vector $n$-dimensional random vector $\mathbf{X}$ whose components are the Bernoulli random variables $X\_1, X\_2, \ldots X\_n$, such that the probability $\mathbb{P}(X\_1=X\_2=\ldots=X\_n=0)$ is null.
---
**Question:** For $n>1$, is there any joint distribution of $\mathbf{X}$’s components and a v... | https://mathoverflow.net/users/115803 | On the mean value taken by Bernoulli random variables with joint distribution constraints | $\newcommand\X{\mathbf X}\newcommand\v{\mathbf v}$
Indeed, $1\le\sum\_{i=1}^n X\_i\le n$ and hence
$$n\sum\_{i=1}^n X\_i\ge n=En\ge E\sum\_{i=1}^n X\_i,$$
so that
$$\frac1n\,E\frac{\langle\X,\v\rangle}{\sum\_{i=1}^n X\_i}
=E\frac{\langle\X,\v\rangle}{n\sum\_{i=1}^n X\_i}\le
E\frac{\langle\X,\v\rangle}{E\sum\_{i=1}^n... | 4 | https://mathoverflow.net/users/36721 | 442664 | 178,605 |
https://mathoverflow.net/questions/438809 | 2 | I recently asked in [this thread](https://mathoverflow.net/questions/434320/lower-bounds-for-pattern-complexity-of-aperiodic-subshifts) about lower bounds on the complexity in the case where we have an aperiodic subshift. If I denote $c\_n(\Omega)$ as the number of possible patterns on $Q\_n=\{0,...,n−1\}^d$ and had ho... | https://mathoverflow.net/users/143153 | Lower bounds for pattern complexity of linearly repetitive aperiodic subshifts | Just to give closure to this question, I think an earlier paper by Boris Solomyak, [Nonperiodicity implies unique composition for self-similar translationally finite Tilings](https://link.springer.com/article/10.1007/PL00009386), actually explicitly solves my question in the desired setting. In the paper by David Lenz,... | 0 | https://mathoverflow.net/users/143153 | 442685 | 178,613 |
https://mathoverflow.net/questions/442686 | 11 | I am a new learner of Iwasawa theory and currently reading the famous [paper](https://link.springer.com/article/10.1007/s00222-013-0448-1) by Skinner-Urban in 2014, and the following-up works by many other people. When reading these papers, I found that some people tend to work over the *general* symplectic group $\mat... | https://mathoverflow.net/users/161208 | Do people prefer working on $\mathrm{GSp}$ and $\mathrm{GU}$ rather than $\mathrm{Sp}$ and $\mathrm{U}$, and why? | *Symplectic case*: Here are two reasons (not necessarily the only ones) why $\operatorname{GSp}\_{2n}$ is more convenient to work with than $\operatorname{Sp}\_{2n}$.
* Firstly: there is **no Shimura datum with underlying group $\operatorname{Sp}\_{2n}$**; you need the "extra room" provided by $\operatorname{GSp}\_{2... | 13 | https://mathoverflow.net/users/2481 | 442695 | 178,616 |
https://mathoverflow.net/questions/442674 | 0 | From the definition of sub-Gaussian distribution $X$ w.r.t. $\sigma$ i.e.
$$\mathbb{P}(|X-\mathbb{E}(X)|\geq t) \leq 2 \exp(-\frac{t^2}{2\sigma^2}).$$
It's natural that when $X \sim \mathcal{N}(\mu, \sigma^2)$,
$$\mathbb{P}(X-\mu\geq t) \leq \exp(-\frac{t^2}{2\sigma^2}).$$
But this bound is too loose since when $... | https://mathoverflow.net/users/500967 | Upper bound about Gaussian tail bound | Letting
$Z:=\frac{X-\mu}\sigma$ and $z:=\frac t\sigma$, rewrite the inequality in question as
$$d(z):=P(Z\ge z)-e^{-z^2/2}/2\le0 \tag{1}\label{1}$$
for real $z\ge0$.
Note that
$$d'(z)=(z-\sqrt{2/\pi})e^{-z^2/2}/2.$$
So, $d$ is decreasing on the interval $[0,\sqrt{2/\pi}]$ and increasing on the interval $[\sqrt{2/\pi}... | 1 | https://mathoverflow.net/users/36721 | 442699 | 178,619 |
https://mathoverflow.net/questions/442603 | 2 | Let $X$ be a smooth intersection of two quadrics in $\mathbb{P}^{2n+1}$.
Question: Does there exist a smooth projective variety $Y$ (different from $X$) of dimension $2n-1$ such that $Y$ is birational to $X$ and if $U$ is the maximal open subset of $Y$ which is isomophic to an open subset of $X$, then $Y \setminus U$... | https://mathoverflow.net/users/174161 | Examples of projective manifold birational to intersection of two quadrics | When n=1 they are elliptic curves, so the answer is no.
When n>1, then $Y$ and $X$ have dimension at least 3. By the Lefschetz hyperplane theorem for $X$, we have $\mathrm{Pic}(X) \cong \mathbb{Z}$. As the codimension of $Y\setminus U$ is 2 we have $\mathrm{Pic}(Y) \cong \mathrm{Pic}(U)$ and the latter is a quotient ... | 5 | https://mathoverflow.net/users/17630 | 442700 | 178,620 |
https://mathoverflow.net/questions/442676 | 5 | How many monotone mappings $[n] \times [n] \to [n]$ exist? Here:
* $[n]$ denotes $\{1, 2, \ldots, n\}$,
* Monotone means that if $x\_1 \le x\_2$ and $y\_1 \le y\_2$, then $f(x\_1, y\_1) \le f(x\_2, y\_2)$.
I'm interested in the answer up to $2^{\Theta(\cdot)}$-notation. To give an example, I would be absolutely hap... | https://mathoverflow.net/users/151271 | Number of monotone functions on a square grid | Thanks to [Alex Lazar](https://mathoverflow.net/users/128452/alex-lazar) for referring me to the right keywords.
This is called something like "the number of [plane partitions](https://en.wikipedia.org/wiki/Plane_partition) that fit inside the $n \times n \times n$ box", and [can be computed as](https://mathworld.wol... | 3 | https://mathoverflow.net/users/151271 | 442711 | 178,623 |
https://mathoverflow.net/questions/442385 | 1 | Let $q = p^s$ and $r = q^m$, where $p$ is a prime, $s$ and $m$ are positive integers. Let $N>1$ be an integer dividing $r - 1$, and put $n = (r - 1)/N$.
Let $\alpha$ be a primitive element of $\mathbb{F}\_r$, $\theta = \alpha^N$, and $Tr\_{r/q}$ be the trace function from $\mathbb{F}\_r$ to $\mathbb{F}\_q$. The set
... | https://mathoverflow.net/users/269936 | Weight of a codeword in a cyclic code as a function of the number of solutions of an equation involving the trace function | The codeword $c(\beta)$ in your question is defined on a set via the trace map that does not include the zero element. Thus its number of nonzero elements is simply the length *minus* the number of $k\in \{0,1,\ldots,n-1\}$ which give $Tr(\beta \theta^k)=0.$
From the equidistribution properties of the trace function ... | 1 | https://mathoverflow.net/users/17773 | 442715 | 178,624 |
https://mathoverflow.net/questions/442714 | 7 | Recently, I have been discussing **inverses** with a tenth grade class and **integrals** with an eleventh/twelfth grade class, and this has led me to the following wonder:
>
> **Wonder.** Is there a "reasonable" way to quantify which of finding an inverse and finding an antiderivative is more "difficult"? Meaning, ... | https://mathoverflow.net/users/22971 | Quantifying difficulty of integrals versus inverses | Finding integrals is easier, and finding inverses is more difficult, in many ways.
We can consider both integrability and invertibility in two ways:
* *Global integrability*: Is there a bound on $\int\_a^b f(x)\,dx$?
* *Elementary integrability*: Is there an elementary $F$ with $F'=f$?
* *Global invertibility*: Is ... | 8 | https://mathoverflow.net/users/nan | 442717 | 178,625 |
https://mathoverflow.net/questions/442709 | 2 | I would appreciate it if a reference could be given for the following claim.
Let $g$ be a Riemannian metric on $\mathbb R^n$, $n\geq 2$ that is equal to the Euclidean metric outside some compact set. Let us define for each $\alpha\in (0,1)$ the fractional Laplace--Beltrami operator on $\mathbb R^n$ via
$$ (-\Delta\_g... | https://mathoverflow.net/users/50438 | On the Fractional Laplace-Beltrami operator | This is false even for the euclidean fractional Laplacian. For a cheap proof, the Fourier transform of $(-\Delta)^{a/2}u$ is $|\xi|^a\widehat u(\xi)$ which is not smooth at 0. For a more solid proof, it is not difficult to prove that $(-\Delta)^{a/2} u$ for $u$ a compactly supported function decays like $\sim|x|^{-n-a}... | 2 | https://mathoverflow.net/users/7294 | 442721 | 178,626 |
https://mathoverflow.net/questions/442694 | 2 | My understanding is that convex hull of n points in 4D could have O(n²) edges in the worst case. Source: [https://sites.cs.ucsb.edu/~suri/cs235/ConvexHull.pdf](https://sites.cs.ucsb.edu/%7Esuri/cs235/ConvexHull.pdf)
This same source writes
>
> In 4D, there are n points in general
> position so that the edge joini... | https://mathoverflow.net/users/23064 | Example of worst case distributions for 4D convex hull | I'm making my comment an answer so this doesn't appear on unanswered lists.
The convex hull of points on the [moment curve](https://en.wikipedia.org/wiki/Moment_curve)
$$
(t, t^2, t^3, t^4)
$$
form what is known as a [cyclic polytope](https://en.wikipedia.org/wiki/Cyclic_polytope).
This is a "neighborly polytope" wit... | 3 | https://mathoverflow.net/users/6094 | 442728 | 178,629 |
https://mathoverflow.net/questions/442726 | 9 | Let $F$ by a finite field, and $R(x\_1,x\_2,\ldots,x\_n) := r\_1(x\_1,x\_2,\ldots,x\_n)/r\_2(x\_1,x\_2,\ldots,x\_n)$ a rational function in $n$ variables, frequently in analytic number theory or harmonic analysis one comes across the need to estimate the exponential sum$$ S:= \sum\_{x \in F^n} = e( R(x\_1,x\_2,\ldots,x... | https://mathoverflow.net/users/630 | Why are Deligne-type exponential sum estimates so hard to use? | There are a lot of subtle reasons such exponential sums can fail to exhibit square-root cancellation. First let me comment on two reasons suggested in your answer:
>
> (1) trying to have an explicit dependence of the implied constant on $r$
>
>
>
This should never be a problem. Katz proved a while ago simple e... | 15 | https://mathoverflow.net/users/18060 | 442729 | 178,630 |
https://mathoverflow.net/questions/442740 | -2 | Let $f$ be a polynomial with real coefficients in several indeterminates $x\_1, \dots, x\_n$. Suppose that
$$ f = g^2 $$
for some polynomial $g$.
Is it true that we can find polynomials $h\_1, \dots, h\_m$ which only involve monomials of even degree such that $f = {h\_1}^2 + \dots + {h\_m} ^2$?
| https://mathoverflow.net/users/136356 | can the square of a polynomial be written as a sum of squares of polynomials with only even degree terms? | There do not necessarily exist polynomials $h\_1,\ldots,h\_m$ which only involve monomials of even degree such that $f = h\_1^2 + \cdots + h\_m^2$, even in the case when $f$ is a univariate polynomial.
A polynomial $h$ that only involves monomials of even degree is an even polynomial: it satisfies $h(x) = h(-x)$ for ... | 2 | https://mathoverflow.net/users/8049 | 442741 | 178,634 |
https://mathoverflow.net/questions/442727 | 6 | **Question:** Let $k$ be an algebraically closed field, and $A,B$ abelian varieties over $k$. Suppose there exists an isogeny $A\to B$. Does this imply existence of a *separable* isogeny $A\to B$?
---
**Known Cases:** The answer is yes when $k$ has characteristic zero (because every isogeny is separable). More in... | https://mathoverflow.net/users/404359 | Is there a separable isogeny between any two isogenous abelian varieties? | The answer is no. I think Asvin's example in the comments is correct but there may be a few things to check. I will give a different example that is easier to check, using Moret-Bailly's famous example of a nonisotrivial supersingular abelian surface.
Let $E$ be a supersingular elliptic curve over $F$, the algebraic ... | 9 | https://mathoverflow.net/users/2290 | 442749 | 178,636 |
https://mathoverflow.net/questions/442747 | 4 | Summary
-------
The probability distribution (pdf) of a random walk in 1 dimension is represented by a Bessel function. On the other hand, the pdf of a Brownian motion in free space is represented by a Gaussian distribution. While it is possible to derive the diffusion equation from the master equation of a random wa... | https://mathoverflow.net/users/420641 | Derive the solution of the diffusion equation from the solution of a random walk | To carry out the limit, it helps to start from an integral representation of the Bessel function,
$$P\_n(T)=e^{-T}I\_n(T)=\frac{1}{2\pi}\int\_{-\pi}^\pi \exp [i k n+T \cos k-T]\,dk.$$
For $T\gg 1$ this may be approximated by expansion of the exponent to second order in $k$,
$$P\_n(T)\approx\frac{1}{2\pi}\int\_{-\infty}... | 6 | https://mathoverflow.net/users/11260 | 442750 | 178,637 |
https://mathoverflow.net/questions/442696 | 3 | My question concerns differences between the spectral *radius* $\rho$ and *norm* $\| \cdot \|$ of Markov operators in infinite-dimensional Banach spaces. This is far from my area of expertise, that is more "mixing for finite, typically reversible, Markov chains", where spectral properties are often simpler.
I have a ... | https://mathoverflow.net/users/59264 | Spectral Radius and Spectral Norm for Markov Operators | If I understood correctly, you are asking whether the spectral gap $\gamma=1-\rho$ of a non-reversible Markov chain $P$ provides any universal control on the Poincaré constant (which is the spectral gap of the additive reversibilization of $P$, or in your notation, $1-\|P\|$). The answer is no, even on finite state spa... | 2 | https://mathoverflow.net/users/477827 | 442753 | 178,638 |
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