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https://mathoverflow.net/questions/378106 | 3 | For a reciprocal of a polynomial, $f = \frac{1}{p}$, we (presumably) may construct a sequence $(c\_n)\_{n=0}^\infty$ such that for all $N\ge 0$
$$f(k)k! = \sum\_{n=0}^{N-1} c\_n(k-n)! + O((k-N)!). $$
I ask about constructing such sequences in [Calculating "factorial sequence" of a rational function](https://mathoverflo... | https://mathoverflow.net/users/130484 | Why does this "factorial sequence" appear in the OEIS? | We have
$$\frac{1}{x}f(\frac1x) = \sum\_{n\geq 1} (F\_{2n-3}-1)x^n,$$
where $F\_{2n+1}$ are Fibonacci numbers.
Per [answers](https://mathoverflow.net/a/378165) to [your previous question](https://mathoverflow.net/questions/378103/calculating-factorial-sequence-of-a-rational-function), it follows that
$$c\_{n+1} = \sum\... | 3 | https://mathoverflow.net/users/7076 | 378215 | 157,591 |
https://mathoverflow.net/questions/378218 | 7 | The outer automorphism group of a topological group $G$ is constructed by the short exact sequence
$$
1\longrightarrow \operatorname{Inn}(G) \longrightarrow \operatorname{Aut}(G) \longrightarrow \operatorname{Out}(G) \longrightarrow 1.
$$
This sequence does not always split, see [Non-split Aut(G) $\to$ Out(G)?](https:/... | https://mathoverflow.net/users/169795 | Does Aut(G) → Out(G) always split for a compact, connected Lie group G? | Yes, $\operatorname{Aut}(G) \to \operatorname{Out}(G)$ always splits. The proof is just as in [my answer](https://mathoverflow.net/a/378141) to your question [Classification of (not necessarily connected) compact Lie groups](https://mathoverflow.net/questions/377981/classification-of-not-necessarily-connected-compact-l... | 5 | https://mathoverflow.net/users/2383 | 378220 | 157,594 |
https://mathoverflow.net/questions/378213 | 1 | Given a set $X$, we can promote it to a discrete category $\mathcal{C}\_X$ by considering a set containing identities on all the objects of $X$ and trivial composition/identity selecting functions.
>
> What is a natural way to view this process of adding identities?
>
>
>
Assuming we're working in $ZFC$ (or so... | https://mathoverflow.net/users/92164 | Categorical promotion | Let me preface this answer by noting that I don't know whether this is true all known models for $n$-categories; certainly it is true in the quasicategorical model for $(n,1)$-categories, and it seems to me like it should be true in all reasonable models and for $(n,n)$-categories as well.
The construction you descri... | 3 | https://mathoverflow.net/users/102343 | 378224 | 157,595 |
https://mathoverflow.net/questions/378221 | 6 | Let $A=\{a\_n|n\in\mathbb{N}\}$ be a sequence of positive integers with the following properties:
1. $1<a\_1<a\_2<\cdots<a\_n<\cdots$,
2. $\frac{1}{a\_1}+\frac{1}{a\_2}+\cdots+\frac{1}{a\_n}+\cdots=\infty$.
Does there always exist a subsequence $B\subseteq A$ with $B=\{b\_n|n\in\mathbb{N}\}$ such that
1. $\frac{1... | https://mathoverflow.net/users/27034 | Divergent sums of reciprocals with unique factorization property | No. Let $A$ be the sequence of semiprimes $pq$ with $p < q < p^2$. Since the sum of reciprocals of primes between $p$ and $p^2$ approaches $\ln \ln (p^2) - \ln \ln p = \ln 2$,
$$\sum \frac1A \sim \sum\_n \frac{1}{p\_n} \ln 2 = \infty.$$
But any $n$ elements of $B$ whose prime factors are all at most $p\_{n - 1}$ wo... | 10 | https://mathoverflow.net/users/68546 | 378249 | 157,604 |
https://mathoverflow.net/questions/378160 | 16 | This question is a follow-up to [Classification of (not necessarily connected) compact Lie groups](https://mathoverflow.net/questions/377981). In the answer to that question, @LSpice proved that any compact, *not necessarily connected* Lie group $G$ takes the form
$$
G = \frac{G\_0 \rtimes R}{P}
$$
where $G\_0$ is the ... | https://mathoverflow.net/users/169795 | Improved classification of compact Lie groups | @LSpice has already proven my revised conjecture in the updated answer to [Classification of (not necessarily connected) compact Lie groups](https://mathoverflow.net/questions/377981/classification-of-not-necessarily-connected-compact-lie-groups?noredirect=1#comment959636_377981), but let me give another, closely relat... | 2 | https://mathoverflow.net/users/169795 | 378257 | 157,607 |
https://mathoverflow.net/questions/310702 | 2 | Is the following conjecture correct?
>
> **Conjecture:**
>
>
> If $A,B,C,D$ are four points in general position in the euclidean plane, with
>
>
> $a:=\|C-B\|,\ \ b:=\|C-A\|,\ \ c:=\|B-A\|$
>
> $a':=\|D-A\|,\ b':=\|D-B\|,\ c':=\|D-C\|$ ,
>
>
> $\begin{align}
> a+b+c\ &>\ a+b'+c',\ a'+b+c',\ a'+b'+c\\
> ... | https://mathoverflow.net/users/31310 | Checking planar convexity of 4 points with Stewart's formula | The given expression is
$$\begin{multline\*}(a' + b')((a + c')(a - c') - a'b') + b'(b + a)(b − a) \\
= a'(a^2 - b'^2 - c'^2) + b'(b^2 - c'^2 - a'^2) = -2a'b'c'(\cos ∠BDC + \cos ∠CDA).\end{multline\*}$$
If the configuration is convex, the intersecting segments must be $AB, CD$ by the condition on $c + c'$, so $∠BDC ... | 1 | https://mathoverflow.net/users/68546 | 378260 | 157,608 |
https://mathoverflow.net/questions/378253 | 3 | If $A$ is an $n \times m$ matrix with full row rank (in general not square) and $B$ is invertible, then does there exist $C, D, E$ such that
$$AXA^T = B \iff CX^{-1}D = E$$
for any symmetric positive semidefinite invertible matrix $X$?
If not, is there another way to transform $AXA^T = B$ into a matrix equation i... | https://mathoverflow.net/users/170022 | Are there matrices $C,D,E$ such that $AXA^T = B \iff CX^{-1}D = E?$ | Converting previous comments into an answer, because I noticed they can fully answer the question.
First of all, one can change basis to assume $A = [I\,\, 0]$, and partition
$$
X = \begin{bmatrix}X\_{11} & X\_{12} \\ X\_{21} & X\_{22}\end{bmatrix}, \quad Y = X^{-1} = \begin{bmatrix}Y\_{11} & Y\_{12} \\ Y\_{21} & Y\_... | 3 | https://mathoverflow.net/users/1898 | 378264 | 157,609 |
https://mathoverflow.net/questions/378197 | 9 | I know all real forms of ${\rm SL}(2,{\Bbb C}$). They are ${\rm SL}(2,{\Bbb R})$ and ${\rm SU}(2)$.
Moreover, ${\rm SL}(2,{\Bbb R})$ is isomorphic to ${\rm SU}(1,1)$. Thus I can say that all real forms of ${\rm SL}(2,{\Bbb C})$
are of the form ${\rm SU}(2,F\_\lambda)$, where $F\_\lambda$ is the diagonal Hermitian form ... | https://mathoverflow.net/users/4149 | Forms of ${\rm SL}(2)$ | $\DeclareMathOperator\SL{SL}\DeclareMathOperator\PGL{PGL}\DeclareMathOperator\Br{Br}\DeclareMathOperator\U{U}\DeclareMathOperator\disc{disc}\DeclareMathOperator\Nm{Nm}\DeclareMathOperator\diag{diag}\DeclareMathOperator\SU{SU}\DeclareMathOperator\GL{GL}$The general theory tells us that forms of a reductive group $G$ are... | 15 | https://mathoverflow.net/users/2481 | 378267 | 157,610 |
https://mathoverflow.net/questions/378247 | 2 | Let $\Sigma$ be a compact surface of genus at least $1$ with one boundary component, equipped with a hyperbolic metric so that the boundary is geodesic. There is a version of the collar lemma that says there is a collar neighbourhood $C$ of the boundary such that no simple closed geodesic on $\Sigma$ enters $C$.
Give... | https://mathoverflow.net/users/158773 | Hyperbolic length of curve that does not enter a collar | There is no such constant $d$. The "reason" is that geodesic laminations exist, and are approximated by simple closed curves.
Thought process (which can be ignored):
>
> Suppose that $\tau$ is a train track in $\Sigma$. Let $R$ be a
> rectangle embedded in $\Sigma - \tau$ with one side running along
> $\partial \... | 4 | https://mathoverflow.net/users/1650 | 378268 | 157,611 |
https://mathoverflow.net/questions/378230 | 8 | I would like to know which $C^\*$-algebras are ideals in their second duals?
There is a [paper by S. Watanabe](https://projecteuclid.org/download/pdf_1/euclid.nihmj/1273778977) that claims in introduction that it is well known that a $C^\*$-algebra is an ideal in its second dual iff it is a dual $C^\*$-algebra. But I... | https://mathoverflow.net/users/19593 | When a $C^*$-algebra is an ideal in its second dual? | Warning: the following is just what I found from some work on MathSciNet, following Yemon's hint in the comments. It's not meant to be accurate historical notes.
A "dual" $C^\ast$-algebra is defined as follows. Let $A$ be an algebra and for a subset $M\subseteq A$ let $R(M) = \{ x\in A : Mx=\{0\}\}$; similarly define... | 9 | https://mathoverflow.net/users/406 | 378269 | 157,612 |
https://mathoverflow.net/questions/378185 | 8 | There are two corresponding posts [MSE](https://math.stackexchange.com/questions/3930167/simply-connected-non-compact-surface-with-boundary) and [MSE](https://math.stackexchange.com/questions/3931592/embedding-a-disc-in-a-simply-connected-surface) by me without any answers.
>
> **Problem:** Let $\Sigma$ be a non-co... | https://mathoverflow.net/users/129539 | All non-compact simply connected $2$-manifolds with boundary | Here is one proof, using the Uniformization Theorem. This proof will be easier in the setting of the "Primer" since the authors are considering universal covering spaces of complete hyperbolic surfaces with geodesic boundary.
Start with a simply-connected surface with boundary $S$ and let $DS$ denote the double of $S... | 10 | https://mathoverflow.net/users/39654 | 378283 | 157,615 |
https://mathoverflow.net/questions/4317 | 5 | I have a weird vision that comes from reading a [paper](http://www.informaworld.com/smpp/content~db=all~content=a780098125) by Raphael and Desrochers..
Let $R$ be commutative unitary semiprime ring such that for any integral and essential element $a$ of $R$, $R[a]$ is a projective $R$-module. I conjecture that for an... | https://mathoverflow.net/users/1245 | Integrally closed factor rings and projective modules | The answer is yes.
The key observation is that rational projective extensions are trivial.
Recall an extension of commutative rings $A \subseteq B$ is called *rational* when for each $b \in B$, the ideal $(A :\_A b)$ is dense(=has trivial annihilator) in $A$ and $(A :\_A b)b \not= 0$.
**Lemma** Let $A \subseteq B... | 1 | https://mathoverflow.net/users/97635 | 378288 | 157,616 |
https://mathoverflow.net/questions/378272 | 12 | Consider a set of strings $ {\mathcal S} \subset \{0, 1, 2\}^n $ satisfying the following two conditions: 1.) every string in $ {\mathcal S} $ has exactly $ k $ symbols from $ \{0, 1\} $ (i.e., $ \forall x=x\_1 \cdots x\_n \in {\mathcal S} \;\; |\{ i : x\_i = 2 \}| = n - k $), and 2.) for every two strings $ x, y \in {... | https://mathoverflow.net/users/83189 | Ternary sequences satisfying $ x_i + y_i = 1 $ for some $ i $ | I claim that $S\_{k,n}=2^k$ for all $n\geqslant k$. Moreover, $\sum 2^{-m\_i}\leqslant 1$ if the set of strings satisfies this condition, and $m\_i$ denotes the number of zeros/ones in $i$-th string.
Proof. Toss a coin for each location and consider the following events enumerated by your strings: if the string has 1... | 9 | https://mathoverflow.net/users/4312 | 378290 | 157,617 |
https://mathoverflow.net/questions/378289 | 0 | I have a finite measure space $(X, \mathcal{S}, \mu)$, and a transformation $f:X\rightarrow X$ that "preserves measure on average". That is, for $A \in \mathcal{S}$
$$
\lim\_{n\rightarrow \infty} \frac{1}{n}\sum\_{k=1}^n \mu(f^k(A)) = \mu(A)
$$
I would like to be able to apply some ergodic or recurrence theorem in this... | https://mathoverflow.net/users/129192 | Recurrence results for an "on average" measure preserving transformation | The phrase "a finite measure space $(X,\mathcal{S},\mu)$ where $\mu$ is the Lebesgue measure" can only reasonably mean that $\mu$ is a nonzero multiple of the counting measure, with $\mathcal{S}$ being the powerset of $X$. At least, we may assume that, if $A\subsetneq X$, then $\mu(A)<\mu(X)$. Therefore and because the... | 0 | https://mathoverflow.net/users/36721 | 378294 | 157,619 |
https://mathoverflow.net/questions/378291 | 0 | I asked this question of MSE, but to no avail; alas, here I am.
Let $k>0$, $C\geq 1$, $\alpha \in (0,1]$, and let $(x\_n)\_{n\geq 1}$, be a sequence of real numbers given by the recursion
$$
x\_{n+1} = k |C|^{\alpha} + |x\_{n}|^{\alpha} \qquad x\_0=0.
$$
Is there a simple "non-recursive" expression for $x\_{n}$, for... | https://mathoverflow.net/users/36886 | Closed-form for recursive "geometric-like" recursion | It seems extremely unlikely that a simple "non-recursive" expression for $x\_n$ is possible. However, let us obtain an exact upper bound on the $x\_n$'s.
Let $a:=\alpha\in(0,1]$ and $b:=k|C|^a\in(0,\infty)$. It is clear that $x\_n\ge0$ for all $n$. So,
$$x\_0=0,\quad x\_1=b,\tag{1}$$
and
$$x\_{n+1}=g(x\_n)\tag{2}$$
f... | 1 | https://mathoverflow.net/users/36721 | 378297 | 157,621 |
https://mathoverflow.net/questions/378316 | 12 | ### Question
Let $X$ be some reflexive Banach space. Suppose $x\_n$ is some sequence in $X$ that weak converges to some $y \neq 0$. Is it the case that
$$ \limsup \|x\_n - y\| < \limsup \|x\_n\| ?$$
### Positive with Hilbert spaces
In the Hilbert space case this is true, as
$$ \langle x\_n - y, x\_n -y \rangle = ... | https://mathoverflow.net/users/3948 | Subtracting the weak limit reduces the norm in the limit | The property you indicate is known as (strict) Opial’s Property (see <https://en.m.wikipedia.org/wiki/Opial_property>). It fails generally in reflexive spaces; in fact, it fails generally even for uniformly convex spaces where it is equivalent to Delta convergence (see <https://en.m.wikipedia.org/wiki/Delta-convergence... | 11 | https://mathoverflow.net/users/166628 | 378325 | 157,629 |
https://mathoverflow.net/questions/378332 | 0 | For each $m\in\mathbb{N}$ and fixed $a>0,\theta\in\mathbb{R}$, I want to factorizate the polynomial $p\_m(x) = x^{2m} - 2a^m\cos (m\theta)x^m + a^{2m}$ into $m$ polynomials of second order. Using basic algebra I get for $m=2$ the following decomposition
$$ p\_2 (x)= x^4 - 2a^2\cos(2\theta)x^2 + a^4 = (x^2+2a\cos(\the... | https://mathoverflow.net/users/137336 | Factorization of a polynomial involving cosine into $m$ second-order factors | At first, substitute $t=x^m$ in your equation to get
$t^2-2a^m\cos(m\theta)t+a^{2m}$
This is factorized as $(t-a^m e^{im\theta})(t-a^m e^{-im\theta})$. I did that using the quadratic formula, but there is probably a better way of seeing it.
But then, $x=t^{1/m}$, which is (accounting for all the roots of unity) $... | 2 | https://mathoverflow.net/users/114143 | 378335 | 157,630 |
https://mathoverflow.net/questions/378337 | 3 | For any groups $G,H$, we can define the category, in fact a groupoid,
$$\underline{\text{Hom}}(G,H)$$
whose objects are group morphisms $G\to H$ and morphisms $(f:G\to H)\to (g:G\to H)$ are elements of $H$ conjugating $f$ into $g$. Let $\{H\_i\}\_{i\in I}$ be an inverse system of groups. Then of course we have $\text{H... | https://mathoverflow.net/users/152554 | Limit of Hom-groupoid | Sure, this follows from the Yoneda lemma. Let $B : \text{Grp} \to \text{Grpd}$ be the functor sending any group $G$ to the groupoid with one element $\star$ such that $BG(\star, \star) = G$. For any groupoid $X$ we have
\begin{align}
\text{Grpd}(X, \underline{\text{Hom}}(G,\lim\_{i\in I} H\_i))
&\cong \text{Grpd}(X, \u... | 7 | https://mathoverflow.net/users/33143 | 378342 | 157,631 |
https://mathoverflow.net/questions/378121 | 6 | In Chapter 4 (page 23, subsection "Somos sequence update") of [his *Tracking the Automatic
Ant*](https://doi.org/10.1007/978-1-4612-2192-0_1), David Gale
discusses three families of recursively defined sequences of numbers, all
due to Dana Scott and inspired by the Somos sequences:
>
> **Sequence 1.** Fix a positiv... | https://mathoverflow.net/users/2530 | Is this Laurent phenomenon explained by invariance/periodicity? | Yes, we can. The argument for odd $k$ made in the Alman/Cuenca/Huang paper was a red herring. We can argue for arbitrary $k \geq 2$ as follows:
Let $n \geq k+2$. Then, the recursive definition of Sequence 3 yields
\begin{align\*}
a\_{n}=\dfrac{a\_{n-1}a\_{n-2}+a\_{n-2}a\_{n-3}+\cdots +a\_{n-k+2}a\_{n-k+1}}{a\_{n-k}}
... | 4 | https://mathoverflow.net/users/2530 | 378344 | 157,633 |
https://mathoverflow.net/questions/378345 | 1 | This is from the beginning of the first section of the paper
"Second Chern class and Riemann-Roch for vector bundles on resolutions of surfaces singularities" by J. Wahl.
$C$ is a smooth projective curve over $\mathbb{C}$. $E$ is a vector bundle of degree $e$ and rank $n$ (I believe $n$ should be at least $2$). If $E... | https://mathoverflow.net/users/46923 | Vanishing of a section of a vector bundle at more than e/n points | Here's a possible way to see this.
One chooses the smallest integer $d$ such that $e-nd<0$. Note that $nd \le e+n$.
Then if $D$ is effective of degree $d$, $h^0(E(-D))=0$ since $E(-D)$ is semistable with negative degree.
Since $h^0(E)\le h^0(E(-D))+nd$ (by using the exact sequence $0\to \mathcal O(-D)\to \mathcal... | 2 | https://mathoverflow.net/users/5659 | 378352 | 157,634 |
https://mathoverflow.net/questions/378351 | 10 | Context: an obvious necessary condition for a monoid to embed into a group (as submonoid) is to satisfy the left and right cancelation rules:
$$xy=xz \quad\Longrightarrow y=z;$$
$$yx=zx \quad\Longrightarrow y=z.$$
It's sufficient for commutative monoids, by an easy standard construction. However, in general it's known ... | https://mathoverflow.net/users/14094 | Generalized cancelation properties ensuring a monoid embeds into a group | The answer is no. What you call a generalized cancellation rule is called a quasi-identity in universal algebra. Malcev proved in 1939 that there is no finite basis of quasi-identities defining group embeddable monoids or equivalently defining the quasi-variety of monoids generated by groups.
You can find details in ... | 15 | https://mathoverflow.net/users/15934 | 378358 | 157,636 |
https://mathoverflow.net/questions/378328 | 1 | Fix a commutative ring $k;$ all dg-categories will be dg-categories over $k.$ Throughout the question, I will be following the notation and conventions of Toën's "[The homotopy theory of dg-categories and derived Morita theory](https://arxiv.org/abs/math/0408337)." For a dg-category $C,$ let $[C]$ be the category whose... | https://mathoverflow.net/users/29322 | Are dg-modules over a cofibrant dg-category cofibrant? |
>
> My question is: suppose that C is a cofibrant dg-category. Then are either of Ĉ or dgMod\_C^op cofibrant dg-categories?
>
>
>
A cofibrant object in a cofibrantly generated model category (such as dgCat)
is a retract of a transfinite composition of cobase changes of generating cofibrations.
Generating cofibra... | 1 | https://mathoverflow.net/users/402 | 378359 | 157,637 |
https://mathoverflow.net/questions/378356 | 1 | *The same question was posted on [StackExchange](https://math.stackexchange.com/q/3930845/631742).*
Informal problem description
============================
Assume that we have a stock whose price behaves exactly like a [Wiener process](https://en.wikipedia.org/wiki/Wiener_process). (There are multiple reasons why... | https://mathoverflow.net/users/129831 | Winning money from random walks? | The expectation of $S$ is indeed $0$. This follows by the optional stopping theorem; see e.g. [THM 29.11](https://www.math.wisc.edu/%7Eroch/grad-prob/gradprob-notes29.pdf) with $X=W$, $S=0$, and $T:=\inf\{t\ge0\colon W\_t\notin(-1,0.1)\}$.
Alternatively and more specifically, one can use Wald's lemma for the Brownian... | 6 | https://mathoverflow.net/users/36721 | 378361 | 157,638 |
https://mathoverflow.net/questions/378366 | 11 | In the same paper where [Milnor introduced](https://www.maths.ed.ac.uk/%7Ev1ranick/papers/micro001.pdf) the concept of microbundles, he gave the following definition. $M$ has a microbundle neighborhood in $N$ if there is a neighborhood $U$ of $M$ in $N$ and a retraction $r: U \to M$ so that $(M, U, i, r)$ is a microbun... | https://mathoverflow.net/users/99414 | Existence of normal microbundles | Not all locally flat submanifolds have a normal microbundle, but they do stably.
[Rourke-Sanderson](https://link.springer.com/article/10.1007/BF01402954) prove that there is a PL embedding $S^{19} \times I \to S^{29}$ with no topological normal microbundle.
Milnor, in Microbundles 1, showed that every submanifold $... | 12 | https://mathoverflow.net/users/40804 | 378368 | 157,641 |
https://mathoverflow.net/questions/378261 | 17 | A condensed set à la Clausen–Scholze is, as far as I understand it, a small sheaf on the large site of profinite spaces. In [Scholze's notes](http://www.math.uni-bonn.de/people/scholze/Condensed.pdf) they are described as being objects of a category that is a large sequential colimit of toposes, each of which is $\math... | https://mathoverflow.net/users/4177 | Why strong limit cardinals in the definition of condensed sets? | Let me write down an answer to mark the question as answered.
As David points out, the definition would not change if one used all cardinals $\kappa$ instead of the strong limit ones, as the latter are cofinal in the former.
But strong limit cardinals are used to study the "individual layers" appearing in the defin... | 9 | https://mathoverflow.net/users/102343 | 378375 | 157,643 |
https://mathoverflow.net/questions/378080 | 8 | How much of the inner model project can be constructed without assuming the axiom of choice? I.e. which large cardinals provably have canonical inner models not assuming choice?
| https://mathoverflow.net/users/168572 | Inner model theory without choice | Addressing the first question: I should argue that Choice comes in almost at the beginning of the inner model project, if we regard proving Covering Lemmata as an integral part of that project: one defines a model under an anti-large cardinal assumption; proves that it is rigid; the Covering Lemma is then proven as hol... | 10 | https://mathoverflow.net/users/6942 | 378389 | 157,646 |
https://mathoverflow.net/questions/378381 | 7 | $\newcommand{\op}{\mathrm{op}}\newcommand{\Un}{\mathrm{Un}}$Lurie's relative nerve functor $\mathrm{N}^{F}\_{\bullet}$ is a simpler version of the unstraightening functor
$$\Un\_\phi:\mathrm{sPSh}(\mathcal{C}\_\bullet)\to\mathsf{sSets}\_{/S}$$
(or rather of its opposite; see HTT 3.2.5.1) which "unstraightens the homoto... | https://mathoverflow.net/users/130058 | Explicit description of the left adjoint $\mathfrak{F}$ to the relative nerve | There is indeed a simple, explicit description of the left adjoint to the relative nerve functor. This left adjoint sends a simplicial set over the category $C$, i.e. a morphism of simplicial sets $p \colon X \to C$, to the functor $C \to \mathbf{sSet}$ that sends an object $c \in C$ to the simplicial set $X/c$ defined... | 7 | https://mathoverflow.net/users/57405 | 378393 | 157,647 |
https://mathoverflow.net/questions/366738 | 4 | As a simple example, suppose we have a function $f: \mathbb{R}^3 \to \mathbb{R}$ defined on the set (and taking $+\infty$ everywhere else),
$$\{x \in \mathbb{R}^3| x\_1 \in [-1, 1], x\_2 \in [-1, 1], x\_3 = 0\}$$
The set has no interior but a relative interior given by $(-1,1) \times (-1,1) \times \{0\}$.
Similar... | https://mathoverflow.net/users/145572 | Can functions be differentiable on sets with empty interiors? | I think what you are looking for is the standard definition of once-differentiable manifold with boundary.
In order to define derivative, you need a normed vector space. You need a vector space because differentiation is ${\Bbb R}$-linear. You want to preserve the linearity because in a sense, differentiation is line... | 1 | https://mathoverflow.net/users/42454 | 378405 | 157,650 |
https://mathoverflow.net/questions/378008 | 2 |
My advisor has some vague ideas about the relation between discrete random walks and SDEs, and advise me to read a little bit about them.
To be more precise, ( if I understand correctly what my professor told me), for some classes of SDEs, we can approach the solution of those SDEs by scaled discrete random walk... | https://mathoverflow.net/users/168269 | Discrete random walk and SDEs | Bessel process is a bit different, because of the singularity in the equation.
For convergence of Bessel random walks you can have a look at Lamperti, J. (1962). A new class of probability limit theorems. J. Math. Mech. 11, 749–
In this paper Lamperti proves convergence to the Bessel process for a class of random walks... | 1 | https://mathoverflow.net/users/85303 | 378416 | 157,655 |
https://mathoverflow.net/questions/378341 | 6 | Consider $n\times k$ matrices with entries from $\{0,1,-1\}$ such that the sum in each row and each column is 0 and the non-zero numbers in each row/column alternate in sign (so, they alternate if we make a torus from an $n\times k$ table). Were such things studied? Enumerated?
| https://mathoverflow.net/users/4312 | Toroidal alternating sign matrices | Recall Kuperberg's proof of the ASM conjecture [[arXiv:math/9712207]](https://arxiv.org/abs/math/9712207) via a relation to a quantum-integrable two-dimensional lattice model from classical statistical mechanics, namely a special case of the six-vertex model with domain-wall boundaries. See also the excellent book by B... | 3 | https://mathoverflow.net/users/45956 | 378425 | 157,656 |
https://mathoverflow.net/questions/378382 | 1 | Let $X$ be a compact metric space and consider the sheaf cohomology group $H^1(X, \mathbb{T})$. From a class in $H^1(X, \mathbb{T})$, I can get a principal $\mathbb{T}$-bundle over $X$ and from this, an associated complex line bundle. What is the relationship between classes in $H^1(X, \mathbb{T})$ and complex line bun... | https://mathoverflow.net/users/153044 | Relationship between $H^1(X, \mathbb{T})$ and complex line bundles | At least if $X$ has the homotopy type of a CW complex, there is a natural isomorphism between $H^1(X; \mathbb T)$ and the group of isomorphism classes of line bundles on $X$ under tensor product.
The usual way this is phrased is that the first Chern class defines an isomorphism from the group of line bundles to $H^2(... | 2 | https://mathoverflow.net/users/97265 | 378433 | 157,658 |
https://mathoverflow.net/questions/378364 | 6 | I know from the the theory of Artin groups that, as the $K(\pi,1)$ conjecture is known for Braids group, that using Salvetti complexes we have a fairly explicit finite CW-complex presentation of the classyfing spaces of Braid groups, i.e. of the configuration space of $n$ unordered points in $\mathbb{R}^2$. (See for e.... | https://mathoverflow.net/users/22131 | CW-presentation of configurations of points in plane and space | The Fox-Neuwirth-Fuks stratification of ${\rm Conf}\_n ~ \mathbb R^2$ is constructed by considering the projection map $\mathbb R^2 \to \mathbb R^1$. The image of a configuration under this projection is a subset of the real line. Considering the number of preimages of each element of this subset, we obtain an ordered ... | 5 | https://mathoverflow.net/users/52918 | 378435 | 157,659 |
https://mathoverflow.net/questions/378418 | 5 | Problem
-------
Consider the locally convex spaces $C^\infty(\mathbb{R})$ and $C^\infty\_c(\mathbb{R})$, the former equipped with its standard Fréchet topology, the latter equipped with the inductive limit topology given by $C^\infty\_c(\mathbb{R}) = \varinjlim C^\infty\_c([-n,n])$ where each $C^\infty\_c([-n,n])$ is... | https://mathoverflow.net/users/126256 | A "proof" that all separately continuous maps on LF-spaces are continuous | Already the first sentence in the ''proof'' is doubtful: The characterization of continuity of maps $f:\lim\_\limits{n\to} E\_n \to F$, that all restrictions to $E\_n$ are continuous, holds for *linear maps* but there is no reason that this should be true (for colimits in the category of locally convex spaces) for *bil... | 6 | https://mathoverflow.net/users/21051 | 378447 | 157,664 |
https://mathoverflow.net/questions/378455 | 2 | A set $A\subseteq\omega$ is called a cohesive set if $C$ is finite for each recursively enumerable set $W\_e$, either $A\cap W\_e$ is finite or $A\cap(\omega\setminus W\_e)$ is finite. And a set $A\subseteq\omega$ is called simple if it is recursively enumerable and its complement has no infinite recursively enumerable... | https://mathoverflow.net/users/5017 | Is a cohesive set always an almost subset of a co-simple set? | No, not necessarily. Let $S\_n,n\geq 0$ be an enumeration of all simple sets. Then for every $k$ the intersection $S\_0\cap\dots\cap S\_k$ is infinite - indeed, if not, then taking the least such $k$ we would have that $S\_0\cap\dots\cap S\_{k-1}$ would be an infinite r.e. set which, up to finitely many elements, is co... | 3 | https://mathoverflow.net/users/30186 | 378457 | 157,667 |
https://mathoverflow.net/questions/376597 | 12 | Let $f: X \to Y$ be a separated morphism between $k$-varieties or more general schemes
of finite type. The most common way in standard literature on algebraic
geometry to define the sheaf of relative Kähler differentials $\Omega\_{X/Y}$
is to observe that the diagonal map $\Delta: X \to X \times\_Y X$ is a closed
embed... | https://mathoverflow.net/users/108274 | Sheaf of relative Kähler differentials intuitively | This question may have interest for many people. In general if $k \rightarrow A \rightarrow^f B$ is an arbitrary sequence of maps of commutative unital rings, there is a canonical right exact sequence of $B$-modules
S1. $B\otimes \Omega^1\_{A/k} \rightarrow \Omega^1\_{B/k} \rightarrow \Omega^1\_{B/A} \rightarrow 0$.
... | 3 | https://mathoverflow.net/users/nan | 378501 | 157,676 |
https://mathoverflow.net/questions/378507 | 10 | Define the recurrence
\begin{align\*}
n(2n+x-3)u\_n(x)
&=2(2n+x-2)(4n^2+4nx-8n-3x+3)u\_{n-1}(x) \\
&-4(n+x-2)(2n-3)(2n+2x-3)(2n+x-1)u\_{n-2}(x)
\end{align\*}
with initial conditions $u\_0(x)=0$ and $u\_1(x)=x+1$.
The subject of ["Laurent phenomenon"](https://arxiv.org/pdf/math/0104241.pdf) was motivated by [Somos seq... | https://mathoverflow.net/users/66131 | "Laurent phenomenon"? | In fact,
$$
u\_n(x) =
{2}^{n-1}\prod \_{k=0}^{n-1}(2x+2k+1)
-{2\,n-1\choose n-1}\prod \_{k=0}^{n-1}(x+k) ,
\tag1$$
which is a polynomial with integer coefficients.
P.S. the proof rests on a routine verification that (1) satisfies the given recurrence relation and initial conditions.
| 17 | https://mathoverflow.net/users/454 | 378513 | 157,682 |
https://mathoverflow.net/questions/378488 | 0 | Let $f:X\rightarrow [0,\infty)$ be (resp. weakly) lower semi-continuous on the reflexive Banach space $X$. Let $\ell^p(X)$ denote the space of $p$-summable sequences in $X$, i.e.: $\sum\_{n=1}^{\infty}\|x\_n\|\_x^p<\infty$; here $1\leq p<\infty$. Then, is the "induced" map:
$$
F:(x\_n)\mapsto \sum\_{n=1}^{\infty} f(x\_... | https://mathoverflow.net/users/36886 | Lower semi-continuity of induced function on sequences | Let $x = (x\_n) \in \ell^p(X)$ and $F\_N(x) := \sum\_{n=1}^N f(x\_n)$, $N \in \mathbb{N}$. First if each $F\_N$ is l.s.c. (weakly or not), then $F = \sup\_{N \in \mathbb{N}} F\_N$ is l.s.c., second if each $x \to f(x\_n)$ is l.s.c., then $F\_N$ is l.s.c. as a sum of finite many l.s.c. functions and finally $x \to x\_n$... | 1 | https://mathoverflow.net/users/100904 | 378519 | 157,684 |
https://mathoverflow.net/questions/378482 | 5 | I am trying to compute the Hochschild cohomology of a particular bound quiver path algebra. The quiver $Q$ consists of one vertex and four loops $x,y, h\_1,h\_2$, and the relations $I$ are generated by:
1. All paths of length greater than 3.
2. All paths of length 3, except $yh\_1x$ and $xh\_2y$, and $yh\_1x+xh\_2y$.... | https://mathoverflow.net/users/131868 | Checking if Hochschild cohomology $\mathit{HH}^2(A)=0$ | I believe that there is a 2-dimensional cocycle $g$ such that:
$$
g(h\_1 x \otimes h\_2) = g(h\_1 \otimes x h\_2) = y h\_1 x
$$
and $g(a \otimes b) = 0$ for all other paths $a$ and $b$.
To check that it's a cocycle, we have to verify that for all paths $a$, $b$, $c$, we have
$$
a g(b \otimes c) - g(a b \otimes c) + g... | 6 | https://mathoverflow.net/users/360 | 378522 | 157,685 |
https://mathoverflow.net/questions/378515 | 4 | $\DeclareMathOperator\PSL{PSL}$Let $G$ be finite group and $N$ be the unique minimal normal subgroup of $G$. Assume that $N$ is abelian and $G/N \cong \PSL(2,2^f)$. Is there any upper bound for $\lvert N\rvert$?
| https://mathoverflow.net/users/84911 | Irreducible and faithful $\operatorname{PSL}_2(q)$-module | The answer is "no", since for every sufficiently large prime $p$ there are simple non-trivial $\mathbb{F}\_p[{\rm PSL}\_2(\mathbb{F}\_{2^f})]$-modules. You can take $N$ to be any such module and form the semi-direct product of $N$ and ${\rm PSL}\_2(\mathbb{F}\_{2^f})$. The following argument shows that actually all $N$... | 5 | https://mathoverflow.net/users/35416 | 378531 | 157,690 |
https://mathoverflow.net/questions/378538 | 5 | Working over $\mathsf{ZF}$ with an embedding $j:V\prec V$ with a critical point $\kappa$. Take $\lambda=\sup\_{n<\omega}j^n(\kappa)$. (You may assume $\mathsf{DC}\_\lambda$ if you need, but I am not sure it is consistent with a Reinhardt cardinal.)
My question is: is there a family $\mathcal{A}\subseteq V\_{\lambda+2... | https://mathoverflow.net/users/48041 | Finding many subsets of $V_{\lambda+2}$ stable under $j:V\prec V$ | There can be no such set $\mathcal A$. In fact, no set $\mathcal A$ with $j(\mathcal A) = \mathcal A$ and $j(x) = x$ for all $x\in \mathcal A$ can surject onto $\kappa$, and this does not require that the codomain of $j$ is $V$. The reason is that if $f :\mathcal A\to \kappa$ were a surjection, then $j(f) : \mathcal A\... | 9 | https://mathoverflow.net/users/102684 | 378540 | 157,692 |
https://mathoverflow.net/questions/375838 | 4 | $\DeclareMathOperator{\MCG}{\operatorname{MCG}}$Consider a bordered, punctured, orientable surface $S$. Associated to it there is its mapping class group $\MCG(S)$. One way to concretely think about it is in terms of generators and relations, which can be done in many ways. Typically one chooses a set of closed simple ... | https://mathoverflow.net/users/48526 | Representation of the mapping class group in terms of flips on triangulations | Yes there is an entirely constructive process to produce a sequence of flips (and a relabelling) from a labelled triangulation $\mathcal{T}$ to its image $D\_\gamma(\mathcal{T})$ under a Dehn twist about a curve $\gamma$. You will need to specify $\gamma$ combinatorially, for example, by specifying the number of time i... | 2 | https://mathoverflow.net/users/3121 | 378542 | 157,693 |
https://mathoverflow.net/questions/378485 | 6 | Recall that one way of drawing closed 2-manifolds is to take a disk $D^2$, take a cellular decomposition of $\partial D^2$, pair the vertices in this cellular decomposition so that the pairing preserves edges, and then take $D$ together with this quotient of the boundary.
We can do this in other dimensions as well, f... | https://mathoverflow.net/users/99414 | low dimensional manifolds by gluing the boundary of a ball | This closed orientable 3-manifolds obtained by gluing faces of the Platonic solids were classified by [Everitt](https://doi.org/10.1016/j.topol.2003.08.025).
That is for regular polyhedra with equal dihedral angles, and the gluing is done geometrically. However, it is also possible to do the gluing topologically, and... | 5 | https://mathoverflow.net/users/126206 | 378553 | 157,697 |
https://mathoverflow.net/questions/378551 | 4 | Consider the 2D Dirac operator
$$H = \begin{pmatrix} 0 & \partial\_{\bar z} \\ \partial\_z & 0 \end{pmatrix}$$
where $\partial\_z = \partial\_x - i \partial\_y$ and $\partial\_{\bar z} = \partial\_x + i \partial\_y.$
This implies by using the Bloch transform that there exist functions $\psi\_{\lambda}$ such that
... | https://mathoverflow.net/users/119875 | Periodic eigenfunctions for 2D Dirac operator | $$
\left( \begin{array}{c} 1 \\ \frac{k\_x - ik\_y }{\sqrt{k\_x^2 + k\_y^2 } } \end{array} \right) e^{i(k\_x x + k\_y y)} \ \ \ \mbox{with eigenvalue} \ \ \ i\sqrt{k\_x^2 + k\_y^2 }
$$
and
$$
\left( \begin{array}{c} 1 \\ -\frac{k\_x - ik\_y }{\sqrt{k\_x^2 + k\_y^2 } } \end{array} \right) e^{i(k\_x x + k\_y y)} \ \ \ \m... | 8 | https://mathoverflow.net/users/134299 | 378560 | 157,698 |
https://mathoverflow.net/questions/378537 | 1 | I assume that this sort of question has already been considered at great length. Nevertheless, I could not find an answer to this question in the related literature.
Given a constant $a\in (0,2]$, what are the known upper bounds of the maximum number of points we can place on the $d$-dimensional unit sphere such that... | https://mathoverflow.net/users/115803 | Upper bounds for high-dimensional spherical codes given the covering radius | If I interpreted your question correctly this is really about *spherical codes*. This is a difficult problem that is unsolved in general, there are some bounds, however.
I will use $n$ for the dimension, since $d$ is used for minimum distance of related binary codes.
A spherical code is a collection of points on th... | 4 | https://mathoverflow.net/users/17773 | 378562 | 157,699 |
https://mathoverflow.net/questions/378509 | 6 | Cosider the [K-Bessel function](https://en.wikipedia.org/wiki/Bessel_function#Modified_Bessel_functions:_I%CE%B1,_K%CE%B1) $$K\_\nu(x):= \frac\pi 2 \frac{I\_{-\nu}(x)-I\_\nu(x)}{\sin(\nu\pi)}.$$
See also *Watson, G. N.*, [**A treatise on the theory of Bessel functions.**](https://zbmath.org/?q=an:48.0412.02), Cambrid... | https://mathoverflow.net/users/170238 | Upper bounds for Bessel functions | $\newcommand{\om}{\omega}\newcommand{\al}{\alpha}\newcommand{\la}{\lambda}\newcommand{\Ga}{\Gamma}$Inequality (14) in the paper by Glushak and Pokruchin referred to in your comment is equivalent to
\begin{align\*}
&\Big\|\int\_0^\infty t^{\nu+n}K\_{\nu-n+1}(t\sqrt\mu)Y\_k(t)x\,dt\Big\| \\
&\le M\|x\|\int\_0^\inft... | 4 | https://mathoverflow.net/users/36721 | 378584 | 157,706 |
https://mathoverflow.net/questions/378575 | 1 | Let $(Y,Z)$ be a solution the the BSDE on a stochastic base $(\Omega,\mathcal{F},(\mathcal{F}\_t)\_t,\mathbb{P})$:
$$
Y\_t = \int\_t^T f(s,Y\_s,Z\_s)ds + Z\_t dW\_t \qquad Y\_T = \xi \in \mathcal{F}\_T^W;
$$
adapted to the filtration $(\mathcal{F}\_t^W)\_{t\in [0,T]}$. Is it possible to write down an SDE
$$
X\_t = \xi ... | https://mathoverflow.net/users/36886 | Time-Reversal of BSDE = SDE | Probably not. It seems to me that $Y\_t \in \mathcal{G}\_0$ for all t and therefore so is $X\_t$ and $\xi + \int\_0^t F(s,X\_s)ds$. But in this case $$X\_t - \xi + \int\_0^t F(s,X\_s)ds = \int\_0^t \Sigma(s,X\_s)dV\_s$$ is a $\mathcal{G}\_0$ adapted martingale and so constant. That leaves $X\_t$ and therefore $Y\_t$ be... | 1 | https://mathoverflow.net/users/143907 | 378587 | 157,707 |
https://mathoverflow.net/questions/378590 | 10 | Recall that two 4-manifolds $M$ and $N$ are *stably diffeomorphic* if there exist $m,n$ such that
$$M \#\_n (S^2 \times S^2) \cong N \#\_n (S^2 \times S^2).$$
That is, they become diffeomorphic after taking sufficiently many connected sums with $S^2 \times S^2$.
I am interested to find examples $M$ and $N$ which are ... | https://mathoverflow.net/users/184 | Homotopy equivalent smooth 4-manifolds which are not stably diffeomorphic? | $\newcommand{\Z}{\mathbb Z}\newcommand{\RP}{\mathbb{RP}}$ $\RP^4$ and Capell-Shaneson's fake $\RP^4$, which I'll
denote $Q$, are an example with fundamental group $\Z/2$. I don't know if this generalizes, but I like this example
for TFT reasons: [David Reutter proved that](https://arxiv.org/abs/2001.02288) semisimple 4... | 11 | https://mathoverflow.net/users/97265 | 378594 | 157,708 |
https://mathoverflow.net/questions/378596 | 7 | I proposed [this question at MO](https://mathoverflow.net/questions/378507/laurent-phenomenon) which was resolved neatly by Gerald Edgar in the form
$$
u\_n(x) =
{2}^{n-1}\prod \_{k=0}^{n-1}(2x+2k+1)
-{2\,n-1\choose n-1}\prod \_{k=0}^{n-1}(x+k).$$
Now that we confirmed that $u\_n(x)$ are all polynomials. I would like ... | https://mathoverflow.net/users/66131 | Real-rooted polynomials | Yes, because $u\_n(x)$ switches sign between each consecutive pair in
$x=0,-1,-2,-3,\ldots,1-n$, and $u\_n(-n) = 0$.
In general, if $P,Q$ are continuous functions each with $n$ simple roots
in an interval $I$, and those roots interlace, then the same argument gives
at least $n-1$ roots of $P-Q$ in $I$, and then an ex... | 18 | https://mathoverflow.net/users/14830 | 378597 | 157,709 |
https://mathoverflow.net/questions/378434 | 2 | If we numerically differentiate a given time series data consisting of *N* points by finite forward difference method, we will have N-1 points corresponding to first derivative. If it is a second derivative, we will have N-2 points and so on.
Let us say for the first derivative
$$
\approx\frac{f(x+\Delta x)-f(x)}{\... | https://mathoverflow.net/users/142414 | Numerically differentiated values and their corresponding x-coordinates | The OP asks for a "reputable source", I would think that Press and Teukolsky's [Numerical Recipes](https://aip.scitation.org/doi/pdf/10.1063/1.4822971) [section 5.7 in [The Book](https://books.google.nl/books?id=1aAOdzK3FegC)] qualifies as such. As they explain, if you approximate $f'(x)\approx h^{-1}[f(x+h)-f(x)]$ the... | 2 | https://mathoverflow.net/users/11260 | 378605 | 157,711 |
https://mathoverflow.net/questions/378593 | 4 | I'm studying the paper "On the verbal width of finitely generated pro-p groups" by Andrei Jaikin-Zapirain ([link at ProjectEuclid](https://projecteuclid.org/euclid.rmi/1218475356)) and I cannot see a claim made in a proof. I don't know if the my question is appropriate to this site, I apologize if not.
>
> **Theore... | https://mathoverflow.net/users/123172 | Why is every nilpotent-by-finite finitely generated pro-p-group always $p$-adic analytic | This is indeed true: every finitely generated nilpotent-by-finite (= virtually nilpotent) pro-$p$-group is $p$-adic analytic.
Since every finitely generated nilpotent group has all its subgroup finitely generated, a finitely generated nilpotent profinite group $G$ has a composition series by closed subgroups in which... | 5 | https://mathoverflow.net/users/14094 | 378607 | 157,712 |
https://mathoverflow.net/questions/378631 | 4 | How can one demonstrate there is no sequence $X\_i$ of sets such that $X\_{i+1}' = X\_i$ (this is really equality as sets though Turing equivalence would be interesting too).
I know it fails if I relax equality to simply $X\_{i+1}' <\_T X\_i$ as in [this question](https://mathoverflow.net/questions/173685/infinite-de... | https://mathoverflow.net/users/23648 | Infinite descending chain of Turing jumps with equality | Let $\Theta(X, n)$ be a uniform process such that $\Theta(Z^{(n)}, n) = Z$, i.e. $\Theta$ undoes jumps. Using the recursion theorem, define a functional $\Phi\_e$ such that $\Phi\_e^X(e)\downarrow$ iff $\exists n\, \Theta(X, n)(e) = 0$.
Now suppose such a sequence $(X\_i)\_{i \in \omega}$ existed. Consider the sequen... | 7 | https://mathoverflow.net/users/32178 | 378641 | 157,720 |
https://mathoverflow.net/questions/377462 | 2 | So a common method used to construct non-zero $\omega$-REA *arithmetic* degrees with various properties is to build an $\omega$-REA operator $J$ satisfying the constraints that (for all $X$)
$$\tag{1} J(X') \equiv\_T J(X) \oplus X'$$
$$\tag{2} J(X) >\_T X$$
Inductively, 1 implies that $J(X^n) \equiv\_T J(\emptyset)... | https://mathoverflow.net/users/23648 | Generality of construction for $\omega$-REA arithmetic degrees | So after some careful thought I'm pretty sure it is fully general. Let $A$ be a non-arithmetic $\omega$-REA set and $K(X)$ some $\omega$-REA operator satisfying 1 and 2 (see Odifreddi volume 2 XIII.3.1 for an existence proof). We now define $J(X)$ so that $J(X) \equiv\_T X \oplus A$ if $X = \emptyset^n$ for some $n$ an... | 0 | https://mathoverflow.net/users/23648 | 378645 | 157,723 |
https://mathoverflow.net/questions/378516 | 0 | Given two real and irreducible matrices $A$ and $B$ of size $n \times n$. A matrix $A$ is irreducible if there is no permutation matrix $Q$ so that
$$
Q^{-1} A Q = \begin{bmatrix} E & G \\ 0 & F \end{bmatrix}
$$
where $E$ and $F$ are square.
Also, $A$ and $B$ have equal principal minors, i.e,
$$
det A(\alpha) = det B... | https://mathoverflow.net/users/51478 | Principal minors and similarity | $$A=\begin{bmatrix} 1& 0& 1\\1&1&1\\1&1&1
\end{bmatrix} \quad
B=\begin{bmatrix} 1& 1& 1\\0&1&1\\1&1&1
\end{bmatrix} \quad
P=\begin{bmatrix} 0& 1& 0\\1&0&0\\0&0&1
\end{bmatrix}$$
$P^{-1}=P$ and $B=PAP^{-1}$. $A$ is irreducible since conjugation by a
permutation matrix simply moves around the single zero in $A$ (all... | 2 | https://mathoverflow.net/users/170118 | 378651 | 157,725 |
https://mathoverflow.net/questions/378646 | 1 | Let $k$ be a field, $C$ a $k$-coalgebra, and $M$ a left $C$-comodule. Then, for a short exact sequence
$$
0 \rightarrow N \rightarrow M \rightarrow L \rightarrow 0
$$
of vector spaces, we have that $N$ and $L$ are also $C$-comodules. Does the converse hold?
I've tried using the correspondence of left $C$-comodule... | https://mathoverflow.net/users/170346 | $M$ comodule if and only if $N$ and $L$ comodules | For any (coassociative, counital) coalgebra $C$ over a field $k$, there is a fully faithful exact functor from the category of $C$-comodules to the category of $C^\*$-modules. The essential image of this functor is the full subcategory of rational $C^\*$-modules (hence one can identify $C$-comodules with rational $C^\*... | 3 | https://mathoverflow.net/users/2106 | 378669 | 157,729 |
https://mathoverflow.net/questions/378666 | 0 | I am cross-posting the question below, which I asked in Mathematics StackExchange a week ago and did not receive answers there. Thank you for your help!
It is well known that, if $x\mapsto f(x)$ is a slowly-varying function, then
$$
\lim\_{x\to\infty}\sup\_{\lambda\in K}\left|\frac{f(\lambda x)}{f(x)}-1\right| = 0,
$... | https://mathoverflow.net/users/56384 | Slowly-varying functions near zero | Not sure if this answers the question.
* For a given $f$ slowly varying at $\infty$, one can find $\lambda$ convergent to $0$ at $\infty$ such that $f(\lambda(x)) / f(x)$ converges to $1$ as $x \to \infty$. Indeed, for every $n = 1, 2, \ldots$ find $x\_n$ such that $$\biggl|\frac{f(x/n)}{f(x)} - 1\biggr| < \frac{1}{n... | 1 | https://mathoverflow.net/users/108637 | 378672 | 157,730 |
https://mathoverflow.net/questions/378673 | 5 | If $X,Y$ are smooth projective schemes, then if we have a surjection $f:X\to Y$, we have an injective map on étale cohomology, or more generally on any Weil cohomology (see <https://mathoverflow.net/q/172527>). The proof of this statement uses Poincare duality on the target $Y$.
I would like to understand the cohomol... | https://mathoverflow.net/users/152554 | Cohomology of resolution of singularity | In general, the answer is no. It already fails for a nodal curve. In fancier terms, you can understand the obstruction as follows: if $X$ has a resolution $\tilde X$, and the cohomology of $X$ injects the cohomology of $\tilde X$, then $H^i(X)$ would be pure of weight $i$ as a Galois module/mixed Hodge structure (when ... | 13 | https://mathoverflow.net/users/4144 | 378676 | 157,731 |
https://mathoverflow.net/questions/378660 | 1 | There seem to be two formulations of the Bott–Samelson resolution flowing around. For concreteness, let $ G = \mathrm{GL}\_{n} ( \mathbb{C} ) $ with the Borel subgroup $ B \subset G $ of upper triangular matrices, and denote by $ P\_i \subset G $ the $i$th parabolic. Fix a reduced expression $ w = s\_{i\_1} \cdots s\_{... | https://mathoverflow.net/users/80467 | Two different formulations of the Bott–Samelson resolution | I was being stupid. Of course, as @LSpice also [remarks](https://mathoverflow.net/questions/378660/two-different-formulations-of-the-bott-samelson-resolution#comment960977_378660), the real map is
\begin{align\*}
B\backslash P\_{i\_1} \times^B \dotsb \times^B P\_{i\_k}/B
& \longrightarrow
\bigl( G/B \times\_{G/P\_{i\_1... | 0 | https://mathoverflow.net/users/80467 | 378724 | 157,754 |
https://mathoverflow.net/questions/378725 | 16 | The [Ackermann function](https://en.wikipedia.org/wiki/Ackermann_function) $A(m,n)$ is a binary function on the natural numbers defined by a certain double recursion, famous for exhibiting extremely fast-growing behavior.
One finds various slightly different formulations of the Ackermann function, with slightly diffe... | https://mathoverflow.net/users/1946 | Are the vertical sections of the Ackermann function primitive recursive? | No, already $A(n,3)$ is not primitive recursive. Let me use the essentially equivalent [up-arrow notation](https://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation): $A(n,m)=2\uparrow^{n-1}m$, and argue why $f(n)=2\uparrow^n 3$ is not PR. I claim $f(2n-2)\geq 2\uparrow^{n+1}n=A(n,n)$. $n\mapsto A(n,n)$ outgrows all $n... | 22 | https://mathoverflow.net/users/30186 | 378727 | 157,756 |
https://mathoverflow.net/questions/378734 | 1 | In their book Analytic Number Theory, Iwaniec and Kowalski, on page 95, define the analytic conductor by the following formula:
$\displaystyle{{\frak{q}}\_{\infty}(s)=\prod\_{j=1}^{d}\left(\vert s+\kappa\_{j}\vert+3\right)}$
Where does this $+3$ come from? Is it related to the abscissa of convergence? If yes, how?
... | https://mathoverflow.net/users/13625 | What's the motivation for the $3$ appearing in Iwaniec and Kowalski's definition of the analytic conductor? | There is nothing special about the value "3", it is there to ensure that $\log{\frak{q}}\_{\infty}(s)>0$ when $s\rightarrow\infty$.
| 3 | https://mathoverflow.net/users/11260 | 378736 | 157,758 |
https://mathoverflow.net/questions/376259 | 7 | Let $G$ be a connected reductive group over $\mathbb{Z}$. Let $c\_{G(\mathbb{F}\_q)}$ be the number of conjugacy classes of $G(\mathbb{F}\_q)$.
Question: Is it true that $c\_{G(\mathbb{F}\_q)}$ is a [quasi-polynomial](https://en.wikipedia.org/wiki/Quasi-polynomial) in $q$? I.e. is it true that there exists an integer... | https://mathoverflow.net/users/41301 | Number of conjugacy classes of finite reductive groups | Assume that $G$ is adjoint split over $\mathbb F\_q$. Let $G^\*$ be the (Langlands) dual group; it is also split over $\mathbb F\_q$. For each semisimple $s \in G^\*( \mathbb F\_q)$ let $N\_s$ be the number of unipotent representations
of the centralizer $Z\_{G^\*}(s)(\mathbb F\_q)$. The number of conjugacy classes in ... | 15 | https://mathoverflow.net/users/170426 | 378738 | 157,760 |
https://mathoverflow.net/questions/378735 | 0 | We define a affine(concave), upper semi continuous function and bounded function $f:X \to \mathbb{R}$, where $X \subset \mathbb{R}^{k}$ is compact and convex set. Assume that $T$ is an affine and upper semi continuous function on $X$. Let $S$ be a concave function on a compact and convex set $A$ defined by $S(t)=\sup\{... | https://mathoverflow.net/users/127839 | A concave function as supremum of upper semi continuous is upper semi continuous | Yes. Your assumptions imply $f$ is continuous and $X$ is compact by assumption. It should not be too hard to prove $S$ is usc using the $\varepsilon$-$\delta$ definition of usc.
Here is a more geometric argument:
One generally answers these questions by going through epigraphs. Function is lsc ($-f$ here) if the ep... | 0 | https://mathoverflow.net/users/170430 | 378745 | 157,764 |
https://mathoverflow.net/questions/378231 | 0 | In this paper: Matching for teams by G. Carlier and I. Ekeland (2010), <https://www.jstor.org/stable/25619994?seq=1#metadata_info_tab_contents>: they claim that follows a standard argument in Ekeland and Teman (1999), we rewrite
$$\inf\_{\nu\in\mathcal{M}(Z)} \sum\_{j=0}^N F\_j^\*(\nu)=-(\sum\_{j=0}^N F\_j^\*)^\*(0) ... | https://mathoverflow.net/users/168083 | One question about how to get $\inf_{\nu\in\mathcal{M}(Z)} \sum_{j=0}^N F_j^*(\nu)=-(\sum_{j=0}^N F_j^*)^*(0)$? | Just use the definition of the Fenchel transform:
Define $G(\nu) = \sum\_{j=0}^N F^\ast\_j(\nu)$ and calculate $G^\ast(0)$.
| 0 | https://mathoverflow.net/users/170430 | 378750 | 157,767 |
https://mathoverflow.net/questions/378714 | 3 | Consider a polygon $\Omega \subseteq \mathbb{R}^2$, and let us consider the usual Laplacian operator $\Delta = \partial\_x^2 + \partial\_y^2$, with Dirichlet boundary conditions. My question comes from trying to understand certain things in Chavel's book "Eigenvalues in Riemannian geometry". Suppose $\phi\_2$ is a Diri... | https://mathoverflow.net/users/170399 | Laplace eigenfunction on a polygonal domain symmetric about an axis | There is no uniform bound, independent of the domain. Consider a rectangle with side lengths $\pi, \pi/K$, with $K\gg 1$. The eigenvalues are $m^2+n^2K^2$, $m,n\ge 1$. In particular, $\lambda\_2$ is obtained for $m=2$, $n=1$.
If we now reflect about the long side, then the eigenvalues of the doubled rectangle are $m^... | 3 | https://mathoverflow.net/users/48839 | 378757 | 157,768 |
https://mathoverflow.net/questions/378770 | 9 | Let $k$ be a field of positive characteristic $p$. Is there necessarily a discrete valuation ring of characteristic $0$ with maximal ideal $(p)$ and residue field isomorphic to $k$?
| https://mathoverflow.net/users/83073 | Is every field the residue field of a discretely valued field of characteristic 0? | Yes, by Hasse-Schmidt ("Die Struktur diskret bewerteter Koerper", Crelle's Journal, 1934) for any field $k$ of characteristic $p$ there exists a strict Cohen ring $A$, which is a Noetherian, complete local, discrete valuation ring with maximal ideal $pA$ and residue field $A/pA = k$. See also Mac Lane (Theorem 2 in "Su... | 11 | https://mathoverflow.net/users/9684 | 378774 | 157,780 |
https://mathoverflow.net/questions/378775 | 10 | In ZF with the axiom of infinity removed, is the axiom scheme of collection provable?
Note that Collection does follow from the axiom of Transitive Containment, which states that everything belongs to a transitive set. Mancini gave a model of ZF minus Infinity where Transitive Containment fails: see the end of Sectio... | https://mathoverflow.net/users/170446 | Does ZF minus infinity imply collection? | ZF - Inf does imply Collection. Fix a set $X$ and a property $P$ (which can be formalized in terms of a formula and a parameter). Since we have separation, we may assume for all $x \in X,$ there is $y$ such that $P(x,y).$ Suppose $X$ is finite, with cardinality $n.$ A standard inductive argument shows there is a set $Y... | 11 | https://mathoverflow.net/users/109573 | 378784 | 157,783 |
https://mathoverflow.net/questions/378777 | 16 | Edward Nelson advocated weak versions of arithmetic (called predicative arithmetic) that couldn't prove the totality of exponentiation. Since his theory extends Robinson arithmetic, the incompleteness theorems apply to it. But if the incompleteness theorems are proven in theories stronger than those he accepts, he coul... | https://mathoverflow.net/users/163672 | Did Edward Nelson accept the incompleteness theorems? | Gödel’s second incompleteness theorem requires neither exponentiation nor “impredicative concepts”. The systems Nelson works in are fragments of arithmetic interpretable on definable cuts in $Q$; one such fragment is the bounded arithmetic $I\Delta\_0+\Omega\_1$ (this appears to be what Nelson calls $Q\_4$ in the *Pred... | 22 | https://mathoverflow.net/users/12705 | 378796 | 157,790 |
https://mathoverflow.net/questions/378781 | 3 | Robin's inequality
$$\sigma\_1(n)<e^\gamma n\log\log n$$
at integers $n>5040$ provides necessary and sufficient condition for Riemann Hypothesis where $\sigma\_1(n)=\sum\_{d|n}d$ is sum of divisors function applied to integer $n$.
Is there a necessary and sufficient condition involving sum of squares of divisors func... | https://mathoverflow.net/users/10035 | Is there a Riemann Hypothesis criterion utilizing sum of squares of divisors? | I would expect no: the big difference between $\sigma\_1$ and $\sigma\_k$ for
$k\ge2$ is that in the latter case $\sigma\_k(n)/n^k$ is trivially bounded by
$\zeta(k)$, which doesn't leave much room for a Robin type criterion.
| 4 | https://mathoverflow.net/users/81776 | 378799 | 157,791 |
https://mathoverflow.net/questions/375191 | 0 | Let $X$ be a smooth variety, and let $\operatorname{Vec} (X)$ denote the $\mathcal{O}\_X$-module of vector fields on $X$. It is stated in several books on D-modules, for example [here](http://people.math.harvard.edu/%7Egaitsgde/grad_2009/Dmod_brav.pdf) in Corollary 6.6, that there should me an evident morphism
$$ \oper... | https://mathoverflow.net/users/156537 | Natural map from vector fields to cotangent variety | Let $X:=Spec(A)$ be an affine scheme of finite type over a field $k$ of characteristic zero, and let $D(A):=Diff\_k(A)$ be the ring of $k$-linear polynomial differential operators on $A$. It has a filtration $D^l(A):=Diff^l\_k(A) \subseteq Diff\_k(A)$ respecting the multiplication, and $k$ is in the center of $D(A)$. T... | 1 | https://mathoverflow.net/users/nan | 378827 | 157,799 |
https://mathoverflow.net/questions/378813 | 5 | A module $M$ over a ring $R$ is called *semisimple* if it admits a direct sum decomposition into simple modules. If $M$ admits a *finite* decomposition $M=\bigoplus\_{i=1}^n S\_i$ into simple $R$-modules $S\_i$, then this decomposition is unique up to isomorphism and permutation of the factors because from such a decom... | https://mathoverflow.net/users/nan | Uniqueness of infinite direct sum decomposition | Yes. A semisimple module $M$ is *canonically* isomorphic to
$$M \cong \bigoplus\_i \text{Hom}\_R(S\_i, M) \otimes\_{\text{End}(S\_i)} S\_i$$
where $\text{Hom}\_R(S\_i, M)$ is what you might call the multiplicity space of the simple module $S\_i$. It is naturally a module over the division ring $\text{End}(S\_i)$, a... | 8 | https://mathoverflow.net/users/290 | 378829 | 157,801 |
https://mathoverflow.net/questions/378787 | 3 | A Banach space $X$ is called a Grothendieck space if $\text{weak}^{\*}$-null sequences in $X^{\*}$ are weakly null. Some of the classical Grothendieck spaces are the $C(\Omega)$ spaces if $\Omega$ is extremally disconnected. However, $C[0,1]$ is not a Grothendieck space. This fact can easily be deduced from some well-k... | https://mathoverflow.net/users/41619 | $C[0,1]$ is not a Grothendieck space | Consider $\delta\_{1/n}-\delta\_0$; this defines a weak$^\*$ null sequences which is not weakly null; e.g., $\langle \delta\_{1/n}-\delta\_0, \chi\_{\{0\}} \rangle \not\to 0$. So if $\Omega$ contains a nontrivial convergent sequence, $C(\Omega)$ cannot be a Grothendieck space.
| 12 | https://mathoverflow.net/users/127871 | 378835 | 157,804 |
https://mathoverflow.net/questions/378779 | 8 | Consider the theory ZFfin := ZF − axiom of infinity + “all ordinals are finite” (i.e., every ordinal is zero or successor).
Of course, if we add the axiom of choice, this is not very interesting: it's basically first-order arithmetic in disguise. Basically what I'd like to know is whether we can make something combin... | https://mathoverflow.net/users/17064 | Can we make ZF − infinity + “all ordinals are finite” as strong as ZFC? | ZFfin implies every set is finite, and in particular choice holds. Suppose there is an infinite set $X.$ By replacing every element of $\mathcal{P}\_{\text{fin}}(X)$ with its cardinality, we see $\omega$ exists, contradiction. So this theory is roughly arithmetic.
| 9 | https://mathoverflow.net/users/109573 | 378838 | 157,805 |
https://mathoverflow.net/questions/378851 | 5 | Given a 2-component link in $S^3$ whose components are trivial knots, is it always possible to find a homeomorphism of $S^3$ that exchanges the components?
I guess the answer is "no" (but I could not find a counterexample), so here is a second question:
Which link-invariant could prevent the existence of such an ex... | https://mathoverflow.net/users/9248 | Exchanging the components of a two-component link | Indeed, as you suspect, the answer is no. For instance, take the link $L$ obtained from the Hopf link by doing a $(2,1)$-cable of one component and a Whitehead double of the other.
A way of telling them apart is to look at the JSJ decomposition of the complement: each piece contains one component, and one component i... | 6 | https://mathoverflow.net/users/13119 | 378852 | 157,807 |
https://mathoverflow.net/questions/378849 | 0 | I have seen the prime number theorem and on the of versions I know is that $\sum\_{p\leq x} \log p=O(x)$ (I am counting over primes here and in the rest of the post). Are there any similar results for say, $\displaystyle \sum\_{\substack{p\in [x/2, \hspace{0.1em} x]\\p \text{ prime}}} \log p$ or or $\displaystyle \sum\... | https://mathoverflow.net/users/nan | Are there some results which count $\sum_{p\in [x/2,x]} \log p$ or $\sum_{p\in [x,y]} \log p$ for for $x$ and $y$ positive and real? | Using the various forms of Mertens' theorems and the prime number theorem, you can easily generate bounds for your sums. That is, if you have an approximation for $\sum\_{n\leq y}a(n)$ and $\sum\_{n\leq x}a(n)$, then subtracting one from the other, you get an approximation for $\sum\_{x<n\leq y}a(n)$. However, better r... | 6 | https://mathoverflow.net/users/11919 | 378853 | 157,808 |
https://mathoverflow.net/questions/377863 | 18 | Short version of my question: I'm interested in the following fact.
>
> If $m,n$ are odd integers, then $m/n$ can be written as the ratio of two numbers of the form $\sum\_{j=0}^\ell \epsilon\_j 4^j$, where $\epsilon\_j \in \{-1,0,1\}$.
>
>
>
---
More background:
Let $A = \{0,1,4,5,16,\ldots\}$ be the se... | https://mathoverflow.net/users/133880 | Characterizing the elements of $(A-A)/(A-A)$, where $A$ is a Cantor-like subset of the integers | You can find an elementary proof in this paper with a very honest title:
>
> J.H. Loxton, A.J. van der Poorten, "An awful problem about integers in base four"
> Acta Arith., 49 (1987), pp. 193-203
>
>
>
In the paper the authors prove that any integer $m$ with an even number of $2$'s in its prime factorization ... | 13 | https://mathoverflow.net/users/2384 | 378862 | 157,813 |
https://mathoverflow.net/questions/378832 | 5 |
>
> [[ALEXEY ELAGIN AND VALERY A. LUNTS](https://arxiv.org/pdf/1901.09461.pdf), p.4.] Recall that triangulated category $\text{Perf}(A)$ is defined as the full thick triangulated subcategory of $D(A)$ generated by the dg $A$-module $A$.
>
>
>
>
> [[Kontsevich](https://www.ihes.fr/%7E/maxim/TEXTS/Symplectic_AT2... | https://mathoverflow.net/users/118028 | Questions about $\text{Perf}(A)$ of dg algebra $A$ | As explained a little bit further in Elagin and Lunts' paper, the category $Perf(A)$ consists of the compact objects of $D(A)$, this is exactly what happens in the usual situation in algebraic geometry, for example for a nice space and certainly for a commutative ring, the perfect complexes as locally quasi-isomorphic ... | 6 | https://mathoverflow.net/users/44499 | 378865 | 157,814 |
https://mathoverflow.net/questions/378845 | 5 | $\newcommand\R{\mathbb R}\newcommand\Q{\mathcal Q}$For mutually orthogonal vectors unit vectors $a=[a\_1,\dots,a\_n]^T$ and $b=[b\_1,\dots,b\_n]^T$ in $\R^n=\R^{n\times1}$ (so that $n\ge2$) and for all $x=[x\_1,x\_2]^T\in\R^2$, let
$$\|x\|\_{a,b}:=\sum\_{i=1}^n|a\_i x\_1+b\_i x\_2|\quad\text{and}\quad \|x\|\_1:=|x\_1|+... | https://mathoverflow.net/users/36721 | On a certain norm of the identity operator on $\mathbb R^2$ | Simply observe that
$$\|x\|\_{a,b}=\|x\_1a+x\_2b\|\_1\,.$$
Thus, by orthogonality of $a,b$ and the easily-derived inequality $\|y\|\_2\le\|y\|\_1\le\sqrt{n}\|y\|\_2$ for any $y\in\mathbb{R}^n$, we have
\begin{eqnarray\*}
\|x\|\_{a,b} & = & \|x\_1a+x\_2b\|\_1 \\
& \ge & \|x\_1a+x\_2b\|\_2=\sqrt{x\_1^2+x\_2^2}=\|x\|\_2 ... | 7 | https://mathoverflow.net/users/166628 | 378867 | 157,816 |
https://mathoverflow.net/questions/378885 | 5 | Recall that a separable C\*-algebra $A$ is *quasi-diagonal* if there are completely positive and contractive maps $\varphi\_k \colon A \rightarrow M\_{n(k)}$ such that $||\varphi\_k(ab) - \varphi\_k(a)\varphi\_k(b)|| \rightarrow 0$ and $||\varphi\_k(a)|| \rightarrow ||a||$ for every $a, b \in A$, where $M\_{n(k)}$ deno... | https://mathoverflow.net/users/147609 | Faithful traces on quasi-diagonal C*-algebras | No, separable (unital) quasi-diagonal $C^\ast$-algebras do not necessarily admit a faithful tracial state. For instance, the $C^\ast$-algebra
\begin{equation}
A= \{ f\in C([0,1], \mathcal O\_2) : f(0) \in \mathbb C 1\_{\mathcal O\_2}\}
\end{equation}
(where $\mathcal O\_2$ is the Cuntz algebra with two canonical genera... | 12 | https://mathoverflow.net/users/126109 | 378887 | 157,820 |
https://mathoverflow.net/questions/378881 | 2 | What is the measure of the following set of infinite binary words?
>
> $S=\{w\in\{0,1\}^\omega\ \text{such that},\ \text{for every}\ N\in\mathbb{N},\, w\ \text{has a prefix of the form}\ pp\ \text{with}\ |p|\ge N \}$.
>
>
>
| https://mathoverflow.net/users/167834 | Binary words starting with arbitrarily long squares | For any given $p\in\{0,1\}^\*$ (finite word), the measure of the set of $w\in\{0,1\}^\omega$ starting with $pp$ is $2^{-2|p|}$. So, summing over all the $p$ with $|p|=L$ (there are $2^L$ of them), given $L\in\mathbb{N}$, the measure of the set $S\_L$ of $w\in\{0,1\}^\omega$ starting with $pp$ for some $p$ with $|p|=L$ ... | 4 | https://mathoverflow.net/users/17064 | 378888 | 157,821 |
https://mathoverflow.net/questions/378882 | 10 | I started reading the [Lectures on Condensed Mathematics](https://www.math.uni-bonn.de/people/scholze/Condensed.pdf). I am looking at the material at page 32-34. I have three fundamental computation questions:
---
1. At the last line of pg 32 - it seems to imply that for finite sets $S$, $\Bbb Z[S] \simeq \underl... | https://mathoverflow.net/users/97321 | Computations in condensed mathematics, page 32-34 | 1. Correct, as both sides are the $S$-indexed direct sum of copies of $\mathbb{Z}$. For the LHS this holds by the universal property of $\mathbb{Z}[S]$, and for the RHS note that $C(S,\mathbb{Z}) = \prod\_S \mathbb{Z} = \oplus\_S \mathbb{Z}$ which lets you calculate.
2. By definition, one could say. It's reasonable to ... | 13 | https://mathoverflow.net/users/3931 | 378889 | 157,822 |
https://mathoverflow.net/questions/378844 | 5 | I am curious about the topology of the space of simple closed curves in $S^2$.
The entire free loop space $LS^2$ admits an explicit description using the Morse theory of the energy functional for the round metric. The energy functional is Morse-Bott, with a sequence of critical manifolds corresponding to iterates of ... | https://mathoverflow.net/users/43158 | Space of simple closed curves in $S^2$ | Yes; I find this easier than the whole Morse theory package needed for $LS^2$.
*Thm*. The projection map $\text{Emb}(S^1, S^2) \to (TS^2 \setminus 0)$, given by $\gamma \mapsto \gamma'(0)$, is a weak homotopy equivalence.
A section sends a nonzero tangent vector $v$ above a point $p$ to the geodesic through $p$ wit... | 5 | https://mathoverflow.net/users/40804 | 378898 | 157,824 |
https://mathoverflow.net/questions/378895 | 1 | I know the following statement ture.
>
> Let $T \in B(L^1(\mathbb{R}^d), L^\infty(\mathbb{R}^d))$, where $B(X, Y)$ denotes all bounded linear operoters from $X$ to $Y$.
> Then, $T$ has the integral kernel $k(x,y) \in L^\infty(\mathbb{R}^d \times \mathbb{R}^d)$,
> that is, for all $f \in L^1(\mathbb{R}^d)$ and for a... | https://mathoverflow.net/users/170537 | Existence of integral kernel | If $f(x,y)=\sum\_i c\_i \chi\_{E\_i \times F\_i}(x,y)=\sum\_i c\_i \chi\_{E\_i}(x) \chi\_{F\_i}(y)$ with $(E\_i\times F\_i)\cap (E\_j \times F\_j) =\emptyset$ for $i\neq j$, define
$$\phi (f)=\sum\_i c\_i \int\_{R^d} (T\chi\_{E\_i})(y) \chi\_{F\_i}(y)\, dy.
$$
Then one checks that $\phi$ is well defined and continuous ... | 2 | https://mathoverflow.net/users/150653 | 378901 | 157,825 |
https://mathoverflow.net/questions/378902 | 10 | Consider the following cubic hypersurface in $\mathbb{P}^5$:
$$
X = \{z\_0z\_3z\_5-z\_1^2z\_5-z\_0z\_4^2+2z\_1z\_2z\_4-z\_2^2z\_3 = 0\}\subset\mathbb{P}^5
$$
The singular locus of $X$ is the Veronese surface $V\subset X$. I would like to ask if it is known what is the Picard group of $X\setminus V$?
Thank you ver... | https://mathoverflow.net/users/nan | Picard group of a cubic hypersurface | It is cyclic, generated by $\mathscr{O}(1)$. Indeed this is true for $X$ by the Lefschetz theorem (SGA2, Exp. XII, Cor. 3.7), and the restriction map $\operatorname{Pic}(X)\rightarrow \operatorname{Pic}(X\smallsetminus V) $ is an isomorphism, because the local rings of $X$ are parafactorial by SGA2, Exp. XI, Thm. 3.13)... | 7 | https://mathoverflow.net/users/40297 | 378903 | 157,826 |
https://mathoverflow.net/questions/378890 | 7 | Let $A = \bigoplus\_{k \in \mathbb{Z}} A\_k$ be a not necessarily commutative $\mathbb{Z}$-graded unital algebra over a field $\mathbb{K}$, and assume that it is **strongly graded**:
$$
A\_kA\_l = A\_{k+l}.
$$
In general can it happen that the multiplication does **not** give an isomorphism
$$
A\_k \otimes\_{A\_0} A\_l... | https://mathoverflow.net/users/170526 | $\mathbb{Z}$-graded algebras and tensor products | No it cannot happen.
And not only for strongly $\mathbb{Z}$-graded rings; this is always the case for any strongly $G$-graded ring, where $G$ is a group. $A\_k \otimes\_{A\_0} A\_l \simeq A\_{k+l}$ is an isomorphism of $A\_0$-bimodules.
(See: Corollary 3.1.2, p.82, from [Methods of Graded Rings](https://www.springe... | 8 | https://mathoverflow.net/users/85967 | 378907 | 157,827 |
https://mathoverflow.net/questions/378919 | -4 | This might seem a bit easy but I still like to ask it for pedagogical reasons.
>
> **QUESTION.** Is this inequality true for non-negative integers $n$?
> $$\frac{\pi}2\int\_0^1x^n\sin\left(\frac{\pi}2x\right)dx\geq\frac1{n+1}.$$
>
>
>
| https://mathoverflow.net/users/66131 | An elementary-looking integral inequality | For two increasing functions $f,g$ we have $\int\_0^1 fg\geqslant \int\_0^1 f\cdot \int\_0^1 g$ (Chebyshev's inequality). Apply this for $f(x)=x^n$ and $g(x)=\frac{\pi}2 \sin (\frac{\pi}2 x)$.
| 10 | https://mathoverflow.net/users/4312 | 378920 | 157,832 |
https://mathoverflow.net/questions/378913 | 4 | Let $A$, $B$, $C$ be closed irreducible subvarieties of $\mathbb{A}^n$. Let $V\_1$ be an irreducible component of $B\cap C$, and $V$ an irreducible component of $A\cap V\_1$. Must there
necessarily be an irreducible component $W\_1$ of $A\cap B$ such that $V$ equals some irreducible component of $W\_1\cap C$?
| https://mathoverflow.net/users/398 | Irreducible components: associativity for intersections? | No.
This is an example with irreducible (and nonsingular) $A,B,C$: Consider $\mathbb{A}^4$ with coordinates $(w,x, y, z)$. Let $B$ be the $(x,y)$-plane (i.e. the set $w = z = 0$), $C$ be the hypersurface $z = xy$, and $A$ be the $(y,z)$-plane. Then $B \cap C$ is the union of the "$x$-axis" and "$y$-axis". Let $V\_1$ ... | 6 | https://mathoverflow.net/users/1508 | 378925 | 157,836 |
https://mathoverflow.net/questions/378918 | 16 | *I forgot to mention originally: this was motivated by [this old MSE question](https://math.stackexchange.com/q/3947955/28111).*
It's not hard to show in $\mathsf{ZFC}$ that there are nontrivial elementary embeddings from $(Ord; \in)$ to itself - or rather, it's not hard to write down specific formulas which $\mathsf... | https://mathoverflow.net/users/8133 | Can $Ord$ have nontrivial second-order elementary self-embeddings? | Answering this question would either require refuting choiceless large cardinals or getting close to refuting Woodin's HOD Conjecture.
First, if choiceless cardinals are consistent, one cannot rule out the existence of a second-order elementary embedding. Corollary 4.8 in Usuba's ["A note on Lowenheim-Skolem cardinal... | 17 | https://mathoverflow.net/users/102684 | 378935 | 157,841 |
https://mathoverflow.net/questions/378899 | 6 | Suppose that $f: \mathbb{R} \to \mathbb{C}$ is a function which has support in $[-1,1]$. Let $g = g\_\sigma$ be a centered Gaussian with variance $\sigma^2$. Is there a way to reconstruct the convolution of $f$ with $g$ (on whole $\mathbb R$) if only sampled values of this convolution are given, i.e.
$$
\{ (f \*g)(a) :... | https://mathoverflow.net/users/170539 | Is there a way to reconstruct the convolution $(f * g)(x)$ of $f$ with a Gaussian $g$ from sampled values, $(f*g)(a), a \in A$? | $\newcommand\Z{\mathbb Z}\newcommand\R{\mathbb R}\newcommand\N{\mathbb N}\newcommand\C{\mathbb C}$Suppose indeed that $A=c\mathbb Z$ for some $c>0$. By rescaling, the problem reduces to the following: for given real $a$ and $b$ such that $a<b$, recover the (say continuous) function $f\colon[a,b]\to\C$ given the numbers... | 8 | https://mathoverflow.net/users/36721 | 378940 | 157,844 |
https://mathoverflow.net/questions/378941 | 1 | I've just started reading John Milne's book ''Etale Cohomology". Prop. 1.1 of Sec 1 in Ch1 reads as
follows: *If $X$ is a normal scheme and $X'\to X$ is its normalization in a certain finite separable extension of the field $R(X)$ of rational functions on $X$, then $X'\to X$ is a finite
morphism.*
I have an issue wit... | https://mathoverflow.net/users/170573 | Finiteness of the integral closure of an integral domain in its field of fractions | $B$ is a submodule of a finitely generated module over a Noetherian ring $A$. So $B$ is finitely generated over $A$.
| 2 | https://mathoverflow.net/users/170380 | 378942 | 157,845 |
https://mathoverflow.net/questions/377964 | 12 | (I originally asked this question [here](https://math.stackexchange.com/questions/3930126/convergence-of-the-double-series-sum-dn-mud-x-n), but the problem appears much more difficult than I think after a moment of thought, so I think it might be more suitable to post it here. Please tell me if this is not the right pl... | https://mathoverflow.net/users/121404 | Convergence of the series involving Mobius functions $\sum_{k,d} \mu(d) x_{kd}$ | This question was considered by [Wintner (1945)](https://archive.org/details/AnArithmeticalApproachToOrindaryFourierSeries/page/n1/mode/2up). On pages 16-18 of the linked document, he observes that $\sum\_n 2^{\omega(n)}|x\_n|<\infty$ implies the absolute convergence of the second display (here $\omega(n)=\sum\_{p\mid ... | 10 | https://mathoverflow.net/users/11919 | 378944 | 157,846 |
https://mathoverflow.net/questions/201114 | 9 | As it is well known, a balanced (and in particular ribbon) monoidal category is an algebra over the framed little 2-discs operad. The latter is homotopy equivalent to the operad of moduli space of genus 0 surfaces with boundaries, hence has a natural cyclic structure.
>
> What additional structure correspond to a c... | https://mathoverflow.net/users/13552 | Cyclic structure on a balanced (or ribbon) monoidal category | In <https://arxiv.org/abs/2010.10229> we characterize cyclic framed
little 2-disks algebras in any symmetric monoidal bicategory.
In the case that this symmetric monoidal bicategory is given by finite
categories, left exact functors and their transformations, it turns out
that they amount precisely to balanced Grothe... | 5 | https://mathoverflow.net/users/119240 | 378953 | 157,849 |
https://mathoverflow.net/questions/378956 | 1 | The Magnus expansion $\Omega(t, t\_0) = \sum^\infty\_k \Omega\_k(t,t\_0)$ is so that the solution
$$
Y(t) = e^{\Omega(t,t\_0)}\,Y\_0,
$$
solves an ODE
$$
Y'(t) = A(t)\,Y(t), \qquad Y(t\_0) = Y\_0.
$$
Consequently, we know that $d e^{\Omega(t,t\_0)}/dt = A(t)\, e^{\Omega(t,t\_0)}$.
I am asking if the following... | https://mathoverflow.net/users/170508 | Properties of Magnus expansion | To answer these questions it it helpful to write
$$e^{\Omega(t,t\_0)}={\cal T}\exp\left(\int\_{t\_0}^t A(t')dt'\right),\;\;t\geq t\_0,$$
in terms of the time ordering operator ${\cal T}$, such that a product of noncommuting operators $A(t\_1)A(t\_2)\ldots$ is ordered in the way that $t\_1\geq t\_2\geq\cdots$. By constr... | 3 | https://mathoverflow.net/users/11260 | 378958 | 157,851 |
https://mathoverflow.net/questions/378951 | 14 | In Coolidge's venerable *Treatise on algebraic plane curves*, Theorem 28 p. 392 states:
* Let $S=\{P\_1,\ldots ,P\_{10}\} \subset \mathbb{P}^2$ such that for any $i$, there is a sextic curve (**edit**: integral) singular along $S\smallsetminus P\_i$; then there is a sextic curve singular along $S$.
I am probably mi... | https://mathoverflow.net/users/40297 | Rational sextic plane curves | Let me try to reconstruct Coolidge's argument at pages 390-392.
**Note:** there is a misprint at line 4 of p. 392, where "with double points" should be "with triple points".
>
> **Proposition 1 [Coolodge, p. 390-391].** Assume that $P\_1, \ldots, P\_8$ are points on the plane, not three on a line and not six on a... | 10 | https://mathoverflow.net/users/7460 | 378969 | 157,856 |
https://mathoverflow.net/questions/378971 | 6 | **EDIT:** In general relativity given a manifold $M$ one can consider a functional on (pseudo-) Riemannian metrics $g$ $$\int\_M R\,\, dvol\_g,$$
where $R$ is the scalar curvature and $vol\_g$ is the (pseudo-) Riemannian measure. The extremal metrics for this functional (solutions of the Euler-Lagrange equation) satisf... | https://mathoverflow.net/users/16183 | In what sense exactly are the Einstein metrics distinguished? | If I understood your question correctly, the answer indeed is due to Lovelock. I think it's important to state all the hypotheses clearly, because they are not always reported accurately.
**Theorem.** (Lovelock, 1971) Given a metric $g\_{ab}$ and a covariantly constructed symmetric 2-tensor $T^{ab}(g,\partial g, \par... | 9 | https://mathoverflow.net/users/2622 | 378979 | 157,859 |
https://mathoverflow.net/questions/378976 | 6 | $\DeclareMathOperator\Var{Var}\DeclareMathOperator\CRings{CRings}\DeclareMathOperator\Grp{Grp}\DeclareMathOperator\Sets{Sets}$I'm not a logician/set theorist, and I have some questions on set theory and references that may seem "trivial" for experts. Still I ask the question – if you have references this would be inter... | https://mathoverflow.net/users/nan | Zermelo-Frankel set theory for algebraists | We say that a mathematical theory is [**categorical**](https://en.wikipedia.org/wiki/Categorical_theory) if it has exactly one model, up to isomorphism.
We *intend* some theories to be categorical, for instance the Peano axioms for natural numbers, Euclid's planar geometry, and set theory. Other theories are designed... | 29 | https://mathoverflow.net/users/1176 | 378991 | 157,865 |
https://mathoverflow.net/questions/378995 | 4 | For a vector space $V$ and a linear operator $R:V \otimes V \to V \otimes V$, we say that $R$ satisfies the **Yang--Baxter equation** if
$$(R\otimes id)(id\otimes R)(R\otimes id)
= (id\otimes R)(R\otimes id)(id\otimes R).$$
If instead $R$ satisfies
$$R\_{12}R\_{13}R\_{23} = R\_{23}R\_{13}R\_{12}$$
we say that $R$ sati... | https://mathoverflow.net/users/153228 | What is the difference between the Yang--Baxter equation and the quantum Yang--Baxter equation? | Your two equations are equivalent, and are both versions of the quantum YBE. (The [question](https://math.stackexchange.com/questions/29054/what-are-the-differences-between-classical-yang-baxter-equation-and-quantum-yang) from the comments does a good job of answering your classical versus quantum question.)
Write the ... | 9 | https://mathoverflow.net/users/113402 | 378998 | 157,867 |
https://mathoverflow.net/questions/378994 | 1 | Let $E$ be an elliptic curve defined over an algebraic number field $K$ and $X$ be a cyclic subgroup of $E(K)$ of order $p^n$, then we have the isogeny $E\rightarrow E/X$ and the dual isogeny $ E/X\rightarrow E$.
I want to ask why the kernel of the dual isogeny $ E/X\rightarrow E$ is also a cyclic group of order $p^n... | https://mathoverflow.net/users/nan | Kernel of dual isogeny of elliptic curve | Let $\phi$ be the map $$\phi:E \rightarrow E/X$$ then we have to use $\phi\circ \hat{\phi} = [p^{n}]$, and so there's an exact sequence given by $$0\rightarrow X \rightarrow E[p^{n}] \rightarrow \text{ker}\hat{\phi} \rightarrow 0 .$$ This gives an exact sequence $$0\rightarrow \mathbb{Z}\_{p^n} \rightarrow \mathbb{Z}\_... | 2 | https://mathoverflow.net/users/165827 | 379001 | 157,868 |
https://mathoverflow.net/questions/378999 | 10 | Suppose that $G$ is a group such that every subgroup $H \subseteq G$ (including $G$ itself) is either free or a non-trivial free product, i.e. $H = H\_1 \* H\_2$ with $H\_1, H\_2$ both non-trivial. Is there an example of such a $G$ which is not free?
If $G$ is finitely generated then Grushko's theorem implies $G$ mus... | https://mathoverflow.net/users/112378 | Is there a non-free group $G$ whose subgroups are all freely decomposable? | Yes, there's an example.
Kurosh proved that the group $G$ with presentation
$$\langle (a\_n)\_{n\ge 0},(b\_n)\_{n\ge 1}\mid a\_nb\_na\_n^{-1}b\_n^{-1}=a\_{n-1},\;\forall n\ge 1\rangle$$
has the following properties: $G$ is torsion-free, isomorphic to $G\ast\mathbf{Z}$, all freely indecomposable subgroups of $G$ are... | 20 | https://mathoverflow.net/users/14094 | 379002 | 157,869 |
https://mathoverflow.net/questions/378989 | 15 | In the paper "Self-duality in Four-dimensional Riemannian Geometry" (1978), Atiyah, Hitchin and Singer present a proof that the space of self-dual irreducible Yang-Mills connections is a Hausdorff manifold, and if it is not the empty set, then the dimension is given by
$$p\_1(\text{Ad}(P))-\frac{1}{2}\dim G(\chi(M)-\ta... | https://mathoverflow.net/users/168781 | Atiyah's proof of the moduli space of SD irreducible YM connections | Hopefully I remember this well. My adviser explained this computation to me I don't even want to think how many years ago.
The deformation complex of the SD equation is $\DeclareMathOperator{\Ad}{Ad}$
$$L=d\_A^-\oplus d\_A^\*:\Omega^1\big(\, \Ad(P)\,\big)\to\Omega^2\_-\big(\; \Ad(P)\;\big)\oplus \Omega^0\big(\;\Ad(... | 14 | https://mathoverflow.net/users/20302 | 379003 | 157,870 |
https://mathoverflow.net/questions/378909 | 3 | Let $K\_1 \subsetneq K\_2$ be two non-empty compact sets and let $D = (d\_n)\_{n \in \mathbb{N}}$ be a dense sequence on $K\_2\smallsetminus K\_1.$ Consider $f\_n : \mathbb{C}\smallsetminus K\_1 \rightarrow \mathbb{C}$ to be a sequence of analytic functions. Assume there exists an analytic function $f : \mathbb{C} \sma... | https://mathoverflow.net/users/145367 | Looking for a sequence of analytic functions with strange behaviour | Here is an attempt to construct an example.
I am going to let $K\_1$ and $K\_2$ be compact subsets in the sphere $\hat{\mathbb{C}}$, rather than the plane (of course, we can change coordinates to move infinity to a point outside of $K\_2$).
I will let $K\_1$ be the complement of the unit disc, and $K\_2$ the union ... | 2 | https://mathoverflow.net/users/3651 | 379004 | 157,871 |
https://mathoverflow.net/questions/373130 | 3 | The [half-transitive graphs](https://en.wikipedia.org/wiki/Half-transitive_graph) form a curious class of graphs with some kind of intermediate symmetry that is non-trivial to achieve. More precisely, a graph is *half-transitive* if its symmetry group is
* transitive on vertices,
* transitive on edges, but
* *not* tr... | https://mathoverflow.net/users/108884 | Spectral properties of half-transitive graphs | I was looking for some work on half-transitive graphs when I stumbled across this question, to which the answer is "No" (at least in the original form where you want to bound the multiplicity by 2).
Here is some SageMath code that will construct a 48-vertex graph, verify that it is vertex-, edge-, but not arc-transit... | 4 | https://mathoverflow.net/users/1492 | 379022 | 157,873 |
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