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https://mathoverflow.net/questions/284351 | 11 | In <https://www.math.dartmouth.edu/~carlp/ordertalkunder.pdf>
Carl Pomerance writes: "... Over two centuries ago, Gauss asked if this deal with the decimal for $1/p$ occurred for infinitely many primes $p$. I.e., do we have $l\_{10}(p) = p − 1$ for infinitely many primes $p$?"
Meanwhile in
<http://guests.mpim-bonn.mp... | https://mathoverflow.net/users/3621 | Gauss and primitive roots | (Not really an answer, but too long for a comment.)
I found many references writing that Gauss conjectured that there exst infinitely many primes, with $10$ a primitive root, some referring to Disquitiones, others without any reference.
Gauss studied the period of $1/p$ from his early days. A link to a manuscript (i... | 5 | https://mathoverflow.net/users/36707 | 284635 | 125,801 |
https://mathoverflow.net/questions/284633 | 3 | Let $f,g \in \mathbb{Z}[x],\deg(f),\deg(g)>1$ and $f$ is monic.
Assume $f$ and $g$ are coprime.
For integer $a$ is it possible $g(a) \mid f(a)$ many times?
Is it possible unbounded number of times for fixed degree?
| https://mathoverflow.net/users/12481 | $g(a)$ divides monic $f(a)$ many times, is this possible? | If the polynomials $f,g$ are coprime, then there are integer polynomials $f\_1,g\_1$ such that $d=ff\_1+gg\_1$ is a nonzero constant (it's true if we allow $f\_1,g\_1$ to have rational coefficients, then we just clear denominators). If $g(a)\mid f(a)$, then $g(a)\mid d$, from where it's clear that for fixed pair $f,g$ ... | 12 | https://mathoverflow.net/users/30186 | 284637 | 125,802 |
https://mathoverflow.net/questions/284592 | 17 | Today the word ***form*** can refer to (at least) three different kinds of mathematical object:
1. A homogeneous polynomial. This was apparently started by Gauss ([1801](https://archive.org/stream/werkecarlf01gausrich#page/n129)), renaming what others had called ***formulas***a. (See e.g. Bachmann [1922](https://www.... | https://mathoverflow.net/users/19276 | Tracing the word “form” | The evolution of the concept of a *form* from arithmetic to algebra is discussed on pp. 20, 21, 27 of F. Brechenmacher ([arXiv:0712.2566](https://arxiv.org/abs/0712.2566); revised version published in [2016](http://www.ams.org/mathscinet-getitem?mr=3617469)):
>
> Whereas such terms as “forms” and “transformations” ... | 6 | https://mathoverflow.net/users/11260 | 284640 | 125,803 |
https://mathoverflow.net/questions/283956 | 8 | In connection with [this question](https://mathoverflow.net/questions/283588/how-composite-can-2n-1-be-infinitely-often) and [its follow-up](https://mathoverflow.net/questions/283756/the-limit-superior-of-varphi2n-1-2n-1).
Suppose that $a\ge 2$ and $b\ne 0$ are integers, and $f$ is a monotonically increasing functio... | https://mathoverflow.net/users/9924 | How composite $a^n+b$ is? | This is an answer to the part of the question where the OP is asking if the image of the function
$$R\_{2,\varphi}: \mathbf N^+ \to \mathbf R: n \mapsto \frac{\varphi(2^n - 1)}{2^n-1}$$ is dense in the interval $[0,1]$. The short answer is yes, some more details follow.
---
This week, I met Carlo Sanna in Turin ... | 5 | https://mathoverflow.net/users/16537 | 284645 | 125,805 |
https://mathoverflow.net/questions/284612 | 3 | In [On the number of inscribed squares of a simple closed curve in the plane](https://arxiv.org/abs/0810.4806) it is shown that
>
> **Theorem:** For every positive integer $n$ there is a simple closed curve in the
> plane (which can be taken infinitely differentiable and convex) which has exactly $n$ inscribed sq... | https://mathoverflow.net/users/90655 | Can one find a Jordan curve which has exactly one inscribed rectangle? | As Wojowu writes in the comments, Vaughan's argument (which is a paragraph long and can be read [here on page 71](http://topo.math.auburn.edu/tp/reprints/v06/tp06107.pdf)) shows that there ought to be infinitely many rectangles inscribed in any Jordan curve. (The rectangles come from the double points of a real project... | 4 | https://mathoverflow.net/users/116537 | 284650 | 125,807 |
https://mathoverflow.net/questions/284625 | 8 | In topology when studying a space with non-trivial fundamental group it becomes important to consider homology and cohomology with coefficients in representations of the fundamental group, i.e. local systems. The general Serre spectral sequence for a fibration with non-simply connected base provides an example. Another... | https://mathoverflow.net/users/50948 | What are the uses of coefficient systems for arithmetic cohomology theories? | Well when you say:
>
> The general Serre spectral sequence for a fibration with non-simply connected base provides an example.
>
>
>
you've already pretty much got it - except in algebraic geometry we usually use the Leray spectral sequence instead of the Serre spectral sequence.
A lot of the foundational ... | 6 | https://mathoverflow.net/users/18060 | 284656 | 125,808 |
https://mathoverflow.net/questions/284654 | 1 | I once did some work on using orthogonal function expansions for fitting 3D distribution functions. To ensure completeness over $L^2$ (which was considered sufficient even though technically a distribution function is an element of $L^1$), we used eigenfunctions of a self-adjoint operator. Initially we used the bound s... | https://mathoverflow.net/users/103864 | Completeness of the solutions to the Schrödinger Hydrogen Atom | The attractive Coulomb potential has a bound spectrum for $E<0$ and a continuous spectrum for $E>0$, and if you consider the full spectrum you obtain a complete set of eigenfunctions, as proven for example in [Direct demonstration of the completeness of the eigenstates of the Schrödinger equation with local and non-loc... | 3 | https://mathoverflow.net/users/11260 | 284657 | 125,809 |
https://mathoverflow.net/questions/284662 | 8 | Does the following mathematical gadget have a standard name? Let $R$ be an $\mathbb{N}$-graded ring together with an $S\_n$ action on each $R\_n$ which are compatible in the following sense. Let $i:S\_a \times S\_b \rightarrow S\_{a+b}$ be the standard inclusion. If $x$ is in grade $a$ with $\sigma \in S\_a$ and $y$ is... | https://mathoverflow.net/users/22 | Graded rings with compatible S_n actions | Steven Sam and Andrew Snowden and their other collaborators call these `twisted commutative algebras' and have been having fun writing papers about properties of generators and similar.
But predating this, any topologist of a certain sort would call this a monoid in the category of symmetric abelian groups. If you d... | 6 | https://mathoverflow.net/users/102519 | 284669 | 125,813 |
https://mathoverflow.net/questions/284642 | 3 |
>
> Let $\mathrm{M}$ be a finitely generated submonoid of $\mathbb{Z}^{\oplus d}$ for some $d$, let $A := k[\mathrm{M}]$ be the associated monoid algebra over a field $k$, let $\mathfrak{m} \subset A$ be a maximal ideal corresponding to a $k$-point of $A$. Is the local ring $A\_{\mathfrak{m}}$ geometrically unibranch... | https://mathoverflow.net/users/15505 | Are local rings of monoid algebras geometrically unibranch? | **Edit.** As Friedrich Knop points out, these schemes are typically not even $S2$. I added an example at the end. (I believe it is the example that Friedrich Knop was suggesting.)
**Original answer.** This is not true. Let $d$ equal $2.$ Let $n\geq 2$ be an integer that is prime to the characteristic of $k.$ Inside $... | 3 | https://mathoverflow.net/users/13265 | 284671 | 125,814 |
https://mathoverflow.net/questions/284629 | 2 | Let $f$ be a transcendental entire function, we know that
$\log M(r, f)$, with $M(r,f)=\max\_{|z|=r}|f(z)|$, is a convex function with respect to $\log r$ and
$\lim\limits\_{r\rightarrow\infty}\frac{\log M(r,f)}{\log r}=\infty$.
My question is: for any function $\rho : [1, +\infty)\rightarrow [0, \infty)$
such that... | https://mathoverflow.net/users/11966 | A problem on the maximal modulus | It is easy to show that $M(r)$ is piece-wise analytic, so it cannot be an arbitrary function whose $\log$ is convex with respect to $\log$. On the other
hand, arbitrary such function can be approximated by $M(r)$ of an entire function,
in the sense that $\log M(r)\sim\log\rho(r)$:
MR0176075
Clunie, J.
On integral fun... | 3 | https://mathoverflow.net/users/25510 | 284673 | 125,815 |
https://mathoverflow.net/questions/284647 | 13 | **Context/background:**
I'm approaching this topic from the perspective of anyonic systems.
In the study of anyons, one works with fusion categories. Of course, for physicality, we demand that
i) The category is *unitary* (all $F$ and $R$ symbols are unitary) i.e. a *UTC*.
ii) The category is *modular* ($S$ mat... | https://mathoverflow.net/users/116558 | Classification of unitary modular tensor categories (UMTCs) | 1. Recently, Mignard and Schaunberg found a counter example to the conjecture (<https://arxiv.org/abs/1708.02796>). Counting $(S,T)$ pairs is still just a shorthand way of counting monoidal equivalence classes, which is not exactly the same as counting gauge classes (it's close though). Taking a note along that line th... | 13 | https://mathoverflow.net/users/25642 | 284674 | 125,816 |
https://mathoverflow.net/questions/284685 | 5 | I was wondering if there is an explicit estimate on the probability that the lowest eigenvalue of a $n \times n$ GOE matrix is larger than some number $x \in \mathbb{R}$. I am aware of the fact that there is in principle an explicit formula for that, but if $n$ becomes large, this event is really difficult to compute. ... | https://mathoverflow.net/users/114633 | Estimate on lowest eigenvalue in GOE | See [Extreme Value Statistics of Eigenvalues of Gaussian Random Matrices](https://arxiv.org/abs/0801.1730) (2008), in particular the large-$n$ result:
$$\text{Prob}(E\_{\rm smallest}\geq x)\rightarrow\exp\left[-n^2\Phi\left(\frac{x+\sqrt{2n}}{\sqrt{n}}\right)\right],\;\;-\sqrt{2n}<x<0,$$
$$\Phi(z)=S(-\sqrt{2})-S(-\sq... | 2 | https://mathoverflow.net/users/11260 | 284686 | 125,819 |
https://mathoverflow.net/questions/284397 | 2 | Given a possibly singular, connected, symplectic algebraic variety with a torus action, every fiber of the moment map admits a torus action. Is each fiber of this moment map connected? Any examples or counter-examples? Thanks!
| https://mathoverflow.net/users/7780 | connectedness of fibers of torus-equivariant moment maps | In the category of symplectic algebraic varieties moment maps have in general disconnected fibers. Easy example go as follows: Let $T={\bf G}\_m$ act on the affine plane ${\bf A}^2$ by $t\cdot(x,y)=(tx,t^{-1}y)$. Then symplectic form $\omega\_0=dx\wedge dy$ is $T$-invariant. The corresponding moment map is $m\_0(x,y)=x... | 2 | https://mathoverflow.net/users/89948 | 284692 | 125,823 |
https://mathoverflow.net/questions/284606 | 9 | I apologize if my question is trivial. I am a group theorist with a minor knowledge of topology. Suppose $(X, T\_X)$ and $(Y, T\_Y)$ are two topological spaces and there is an inclusion-reversing (inclusion preserving) bijection $\varphi:T\_X\to T\_Y$. What is the name of this kind of equivalence? What can be said abou... | https://mathoverflow.net/users/44949 | What is the name of this kind of equivalence between two topological spaces? | As far as I can see there are two possibilities where we can say more:
* The bijection is order preserving. As Qiaochu Yuan said in a comment, this is the same as asking that their associated [locales](https://ncatlab.org/nlab/show/locale) are isomorphic. Thanks to the [equivalence](https://en.wikipedia.org/wiki/Ston... | 11 | https://mathoverflow.net/users/43054 | 284702 | 125,827 |
https://mathoverflow.net/questions/284682 | 16 | What is the origin/motivation for the adjective "free" in the term "free object"?
Does it refer to them coming "for free" (as being constructed from a set in a straight-forward manner) or does it refer some type of freedom they enjoy?
| https://mathoverflow.net/users/2082 | Why are free objects "free"? | Free objects were first defined\* by MacLane in [Duality for Groups.](https://projecteuclid.org/download/pdf_1/euclid.bams/1183515045) That paper gives "free" a curious political context, I quote from page 486:
>
> Call the dual (in this sense) of a free (nonabelian) group a fascist
> group. R . Baer has shown me ... | 20 | https://mathoverflow.net/users/11260 | 284706 | 125,829 |
https://mathoverflow.net/questions/283525 | 3 | Let $F \colon \mathbb{R}^2 \to \mathbb{R}^n$ be a $C^{1}$-function 1-periodic in each variable, so it can be considered as a function on the flat torus $\mathbb{T}^2 = \mathbb{R}^2 / \mathbb{Z}^2$. We say that a point $\theta \in \mathbb{T}^2$ is a critical point for $F$ if rank of the differential of $F$ at the point ... | https://mathoverflow.net/users/85336 | Smoothing a periodic function of two variables | Let's suppose that $n \geq 2$ and take a $C^{1}$-function $H \colon \mathbb{R}^2 \to \mathbb{R}^n$ such that the set of its critical points has measure zero. Consider the family of functions $\Phi\_{\alpha} = F + \alpha H$ for $\alpha \in \mathbb{R}$. It turns out that there exists a sequence $\alpha\_{n}$ tending to z... | 0 | https://mathoverflow.net/users/85336 | 284719 | 125,832 |
https://mathoverflow.net/questions/284717 | 2 | Let $k$ be a field and $I$ be an infinite set such that $|k| > |I|$ . Let $R := k [X\_i : i \in I ] $ and $m$ be a maximal ideal of $R$ ; then is it true that $m \cap k[X\_i] \ne 0 , \forall i \in I$ ?
I can show that $R/m$ is an algebraic extension of $k$ ; I don't know whether it is helpful here or not . Perhaps we... | https://mathoverflow.net/users/nan | On maximal ideals of $k [X_i : i \in I ] $ where $k$ is a field , $I$ is an infinite set with $|k| > |I|$ | You are right that the algebraicity of $R/\mathfrak m$ is important. Indeed, suppose $\mathfrak m \cap k[X\_i] = 0$. That means that the map $k[X\_i] \to R/\mathfrak m$ is injective. But then (the image of) $X\_i$ is a transcendental element, which is impossible. (More generally, this shows that the intersection with a... | 4 | https://mathoverflow.net/users/82179 | 284720 | 125,833 |
https://mathoverflow.net/questions/284715 | 3 | This question is probably very elementary but I don't know how to tackle the conversely part of the following result. Let $M(x,y)$ and $N(x,y)$ be two differentiable and **homogeneous functions of the same degree $d$ and such that $M(x,y)dx+N(x,y)dy$ is not exact** that is:
$$ \frac{\partial M}{\partial y}\neq\frac{\pa... | https://mathoverflow.net/users/72331 | General formula for integrating factor of an homogeneous differential 1 form | The last display asks whether (1) is the *unique* solution of (2). It isn’t: try $M=N=x$, $\mu=1/x$.
In fact an integrating factor is never unique: see e.g. Serret ([1886, thm 681](https://archive.org/stream/coursdecalculdif02serruoft#page/449)).
Now if you are asking for *heuristics*, then e.g. (ibid., §685) “deri... | 4 | https://mathoverflow.net/users/19276 | 284723 | 125,834 |
https://mathoverflow.net/questions/284714 | 6 | Let $\mathcal{K}$ be a category and $\mathcal{K}\_{\text{fin}}$ its free completion with finite limits.
>
> * Does the embedding $\mathcal{K} \hookrightarrow \mathcal{K}\_{\text{fin}}$ preserve some colimits?
>
>
>
I am especially interested in directed colimits.
| https://mathoverflow.net/users/104432 | Freely adding finite limits preserves some colimits? | Yes, all the colimits that exist in ${\cal K}$. Indeed, ${\cal K}\_{{\rm fin}}$ can be identified with the smallest **full** subcategory of ${\rm Fun}({\cal K},{\rm Set})^{{\rm op}}$ which contains the representable functors ${\rm Hom}(x,-)$ and is closed under finite limits, and the composition ${\cal K} \to {\cal K}\... | 8 | https://mathoverflow.net/users/51164 | 284729 | 125,837 |
https://mathoverflow.net/questions/284544 | 9 | Let $X$ be a compact metric space. Say that two compact subsets $E,F\subset X$ are *parallel* if
$$ dist(x,F) = dist(y,E)$$
for all $x\in E$ and $y\in F$. Here $ dist(y,E) = \inf\{d(y,z):z\in E\}.$
The overall question I would like to understand is the following:
>
> Let $X$ be a compact (Hausdorff, second counta... | https://mathoverflow.net/users/116515 | Making compact subsets "parallel" | The answer to this problem is affirmative (at least for covers).
**Definition.** A family $\mathcal C$ of subsets of a topological space $X$ is called
$\bullet$ *lower semicontinuous* if for any open set $U\subset X$ its $\mathcal C$-star $St(U;\mathcal C):=\bigcup\{C\in\mathcal C:C\cap U\ne\emptyset\}$ is open in ... | 5 | https://mathoverflow.net/users/61536 | 284730 | 125,838 |
https://mathoverflow.net/questions/284732 | 5 | There are many results about isometric embeddings of Riemannian manifolds but I haven't been able to find one that quite answers this question (which I believe must have some kind answer in the literature).
>
> **Question:** For $n \in \mathbb{N}$ what is the minimal integer $r$ (depending on $n$) such that every ... | https://mathoverflow.net/users/22810 | What are the minimal local models for Riemannian manifolds? A local question about isometric embeddings | The [Nash-Kuiper theorem](https://en.wikipedia.org/wiki/Nash_embedding_theorem) says that for $C^1$ isometric embeddings, $r$ just needs to be $n+1$, so in what follows smooth means $C^2$ or better.
These results are all in chapter 1 of Han and Hong's book "Isometric Embedding of Riemannian Manifolds in Euclidean Spa... | 6 | https://mathoverflow.net/users/353 | 284735 | 125,839 |
https://mathoverflow.net/questions/282139 | 4 | Assume that $\gamma$ is an analytic simple closed curve in $\mathbb{R}^2$ which surrounds origin.
>
> Is there a polynomial vector field on the plane which is tangent to $\gamma$? In the other word, can an arbitrary analytic simple closed curve be realized as a closed orbit or a limit cycle of a polynomial vector f... | https://mathoverflow.net/users/36688 | Polynomial vector field tangent to a given analytic simple closed curve | I see this as very unlikely. A polynomial vector field would have a slope function that is a rational function of two variables with a finite number of coefficients.
As a consequence, if you take the field generated over $\mathbb Q$ by the coordinates $(x\_k, y\_k), k=1,\ldots, n$ of the points on the curve, as well... | 3 | https://mathoverflow.net/users/38468 | 284738 | 125,841 |
https://mathoverflow.net/questions/284684 | 18 | for a ring $R$ with unity , let $U(R)$ denote the group of units of $R$ . Now there are lots of finite commutative rings, of arbitrarily high order, with exactly one unit ; indeed $U(R)=1$ for a finite commutative ring $R$ iff $a^2=a , \forall a \in R$ . Incidentally , I couldn't find any finite non-commutative ring wi... | https://mathoverflow.net/users/nan | Finite non-commutative ring with few invertible (unit) elements | This answer presents **an alternate proof** of *users*' negative answer by **proving directly** that a finite ring whose only unit is its identity must be a [Boolean ring](https://en.wikipedia.org/wiki/Boolean_ring), hence commutative. The proof given below is based on a result by Melvin Henriksen. It doesn't rely on t... | 15 | https://mathoverflow.net/users/84349 | 284739 | 125,842 |
https://mathoverflow.net/questions/284725 | 12 | A bit of plotting suggests that $\zeta^{(k)}(s) < 0$ for all $s\in (0,1)$ and all integers $k\geq 0$. (Or, what is the same: $\zeta^{(k)}(s)$ has no zeroes on $(0,1)$.) Is there a brief, clean proof of this apparent fact (and/or a reference for it)?
| https://mathoverflow.net/users/398 | $\zeta^{(k)}(s) < 0$ for $s\in (0,1)$ | The coefficients computed in the comments appear to imply that the Taylor expansion at $s=0$ of $\zeta(s)+\frac1{1-s}-\frac12$ has very small coefficients, which would imply the result.
Following section 2.1 in Titchmarsh Theory of the Riemann zeta function,
by integration/summation by parts (or one step of Euler-Ma... | 14 | https://mathoverflow.net/users/9849 | 284741 | 125,843 |
https://mathoverflow.net/questions/284740 | 15 | Let $f : S^2 \to \mathbb{R}$ be a continuous map such that $f(-x) = -f(x)$. Consider the set $Z = f^{-1}(0)$. Must $Z$ contain some path from some point to its antipode? Indeed, must $Z$ contain a continuous loop intersecting each "meridian", passing through antipodal points on antipodal meridians?
[I see this often ... | https://mathoverflow.net/users/3902 | Must any continuous odd map from $\mathbb{S}^2$ to $\mathbb{R}$ have a path of zeros between antipodal points? | Let $K\subset \mathbb S^2$ be the compact set obtained by modifying the equator of $\mathbb S^2$ around two antipodal points so that it locally looks like the adherence of the graph of $t\neq0\mapsto \sin \frac{1}{t}$. You can make these modifications so that $K$ is symmetric for the antipodal involution. It is compact... | 15 | https://mathoverflow.net/users/24309 | 284743 | 125,845 |
https://mathoverflow.net/questions/284733 | 3 | In the paper *Introduction to Extensive and Distributive Categories* by Carboni, Lack, and Walters, the authors write at the end of the introduction:
>
> The Burnside rig of a distributive category is well known [6]. It has as elements isomorphism classes of objects of the category, and its addition and multiplicat... | https://mathoverflow.net/users/69037 | References on extensivity as an essentially additive notion | This is not a complete answer, and in particular I don't know anything about a tensor product of extensive categories. But my understanding of the phrase in question is that it means that extensivity is a property of a category with coproducts only. One doesn't need to assume the category has *any* limits in order to s... | 2 | https://mathoverflow.net/users/49 | 284757 | 125,851 |
https://mathoverflow.net/questions/284756 | 5 | If $A$ is an abelian locally compact group, the Plancherel measure on $\hat A$ is a Haar measure, so, up to scaling it is the unique invariant Radon measure.
Now for a nonabelian locally compact group $G$. Can the Plancherel measure on the unitary dual $\hat G$ be characterized by any (invariance) property?
In a way... | https://mathoverflow.net/users/nan | Characterizing the Plancherel measure | If $G$ is a unimodular second countable Type I group, then the Plancherel measure is the unique measure $\mu$ such that
$$\|f\|\_2^2 = \int\_{\widehat{G}} \|\pi(f)\|\_{\mathrm{HS}}^2 \mathrm{d}\mu(\pi).$$
for every $f \in \mathrm{L}^1(G) \cap \mathrm{L}^2(G)$. This appears as Theorem 18.8.2 in Dixmier's book on $C^... | 7 | https://mathoverflow.net/users/99234 | 284762 | 125,854 |
https://mathoverflow.net/questions/284767 | 8 | How to write the covariant power set functor (restricted to finite sets for simplicity)
$$P : \mathsf{FinSet} \to \mathsf{Set}$$
concisely as a colimt of representable functors? There is an epimorphism $$\coprod\_{n \geq 0} \hom(\{1,\dotsc,n\},-) \to P,$$
mapping $f \in \hom(\{1,\dotsc,n\},X)$ to $\mathrm{im}(f) \in P(... | https://mathoverflow.net/users/98306 | Presentation of the covariant power set functor | It helps a lot that in this case $\hom(\{1,...,n\},-)\cong\hom(\{1\},-)^n$. Denote $\hom(\{1\},-)$ by $X$ and take more generally in any category with finite products and countable colimits distributing over each other the free monoid on $X$, that is, $1\sqcup X\sqcup X^2\sqcup X^3\sqcup\cdots$. Now in addition factor ... | 9 | https://mathoverflow.net/users/41291 | 284772 | 125,857 |
https://mathoverflow.net/questions/284771 | 2 | Let $p$ be a prime number, $n$ be a positive integer, and let ${\mathbb Z}\_p^{n\times n}$ denote the set of $n\times n$-matrices over ${\mathbb Z}/p{\mathbb Z}$.
Suppose we are given an integer $m>0$ and matrices ${\bf A}\_1,\ldots, {\bf A}\_m\in {\mathbb Z}\_p^{n\times n}.$ I am looking at the following problem: if... | https://mathoverflow.net/users/8628 | Decidability of matrix problem in ${\mathbb Z}/p{\mathbb Z}$ | Yes. If we conider the sequence $A\_i^n$, because it is a sequence inside a finite set, it must eventually repeat. After it repeats, the sequence won't take any new values, so we can assume $n\_i$ is less than this first repetition. This means that there are only finitely many possibilities, which of course reduces the... | 5 | https://mathoverflow.net/users/18060 | 284773 | 125,858 |
https://mathoverflow.net/questions/284703 | 2 | What is an example of a connected symplectic manifold $(M,\omega)$, with a Hamiltonian action of $G = U(1) =S^{1}$ with infinitely many stabiliser types?
Infinitely many stabiliser types means that infinitely many sub-groups of $G$ appear as stabilisers as points in $M$.
I am aware that $M$ is necessarily non-compa... | https://mathoverflow.net/users/99732 | Hamiltonian Group action with infinitely many stabiliser types | As an simple example with infinitely many different stabilizer subgroups you may take countably many disks. Since each disk may be rotated independently with a different speed, we obtain an action of $ U(1) $ such that all the subgroups $ \mathbb Z / n \mathbb Z$ with $ n \in \mathbb N $ occur as stabilizer.
Of cours... | 0 | https://mathoverflow.net/users/17047 | 284775 | 125,860 |
https://mathoverflow.net/questions/284783 | 5 | This question concerns quantum mechanics experiment. But I believe it belongs here, on MathOverflow.
So, we have two players. They play a simple game and either both win or both loose, so they cooperate.
A fair coin is thrown in front of each player. (Each player sees only "his" coin). Having seen the result of the... | https://mathoverflow.net/users/116642 | Is there an information exchange in this game? (Bell's inequality) | The protocol you describe satisfies 2 and 3 but not 1, so when that protocol is adopted, 1 is wrong.
The correct form of statement 1 is that 75% is a maximum win rate if each player must choose a strategy that is contingent on the realization of some classical random variable (so that in particular there exists a joi... | 2 | https://mathoverflow.net/users/10503 | 284785 | 125,864 |
https://mathoverflow.net/questions/284262 | 2 | Let $(T\_1,..., T\_n)\in \mathcal{L}(E)^n$ be a tuple of commuting normal operators (i.e. each $T\_k$ is normal and $T\_iT\_j=T\_jT\_i$ for all $i,j$), where $E$ is a complex Hilbert space.
I want to show that
$$\displaystyle\sup\_{\|x\|=1}\bigg(\displaystyle\sum\_{i=1}^n|\langle T\_ix,x\rangle|^2\bigg)\geq\displayst... | https://mathoverflow.net/users/113054 | Inequality for normal operators | Consider the case of finite dimensional Hilber space. Then may suppose that operators $T\_1,\dots,T\_d$ are diagonal in the same orthonormal basis $(e\_1,e\_2,\dots)$, denote the diagonal of $T\_i$ by $(p\_{i1},p\_{i2},\dots)$. The square of RHS is nothing but $\sup\_j |p\_{1j}|^2+|p\_{2j}|^2+\dots+|p\_{dj}|^2$. For fi... | 4 | https://mathoverflow.net/users/4312 | 284793 | 125,867 |
https://mathoverflow.net/questions/284802 | 8 | Let $X$ be a connected space. According to Getzler [BV-algebras and two-dimensional topologcial field theories](https://projecteuclid.org/download/pdf_1/euclid.cmp/1104254599) , page 271, we have and isomorphism
$
H\_\*(\Omega^2\Sigma^2X) \cong {\cal G}( \widetilde{H}\_\* X )
$
where ${\cal G}( V)$ means the free G... | https://mathoverflow.net/users/1246 | Is the homology of $\Omega^2\Sigma^2X$ free as a Gerstenhaber algebra? | Over $\mathbb{Z}\_p$ it is not true that $H\_\*(\Omega^2\Sigma^2X)$ is the free Gerstenhaber algebra. Instead, Cohen proves that $H\_\*(\Omega^n\Sigma^nX)$ is a free object in a more elaborate category involving some Dyer-Lashof operations. In the case $n=2$ there is only one Dyer-Lashof operation but it still creates ... | 11 | https://mathoverflow.net/users/10366 | 284803 | 125,870 |
https://mathoverflow.net/questions/284787 | 6 | I am familiar with the theory of modular forms and weight k Eisenstein series, and I am wondering if such a theory exists when the base field is not $\mathbb{Z}$.
Is there a theory of modular forms over $SL\_2(\mathcal{O\_k})$ where $k$ is a real or imaginary quadratic number field? Moreover, is there a nice Fourier... | https://mathoverflow.net/users/116646 | Eisenstein series for quadratic number fields | When $k$ is a real quadratic field (or more generally a totally real number field) the short answer is Hilbert modular forms. The corresponding Eisenstein series are called Hecke-Eisenstein series and are quite easily defined,
and yes they have a very nice Fourier expansion. For instance, this is what allowed Siegel to... | 9 | https://mathoverflow.net/users/81776 | 284813 | 125,872 |
https://mathoverflow.net/questions/284805 | 12 | One definition of (symmetric) [star-autonomous category](https://ncatlab.org/nlab/show/star-autonomous+category) is as a closed symmetric monoidal category $(C,\otimes,I,\multimap)$ equipped with an object $\bot$ such that all double-dualization maps $A \to ((A\multimap\bot)\multimap \bot)$ are isomorphisms. It follows... | https://mathoverflow.net/users/49 | Uniqueness of dualizing objects | If a dualizing object exists, there is a bijection between isomorphism classes of dualizing objects and isomorphism classes of $\otimes$-invertible objects (i.e. the Picard group), given by tensoring your favorite dualizing object by a $\otimes$-invertible object. So the groupoid of $\ast$-autonomous structures, if non... | 17 | https://mathoverflow.net/users/2362 | 284816 | 125,874 |
https://mathoverflow.net/questions/284807 | 11 | The divisibility relation "$a$ divides $b$", or concisely, $a \vert b$ defined over a commutative integral domain $R$ with identity induces a partial order on the multiplicative semigroup $R/R^{\times}$ where $R^{\times}$ is the unit group of $R$. The inclusion relation $I \subset J$ is a partial order on the set of pr... | https://mathoverflow.net/users/82839 | What is known about ideal and divisibility lattices of GCD domains and their generalizations? | Given a ring $R$, let us denote by $L(R)$ the [lattice](https://en.wikipedia.org/wiki/Lattice_(order)) of two-sided ideals of $R$ for which the infimum and supremum are given by $\inf(I, J) = I \cap J$ and $\sup(I, J) = I + J$.
Such lattices are complete and modular. If $R$ is a principal ideal domain (see [original ... | 12 | https://mathoverflow.net/users/84349 | 284819 | 125,875 |
https://mathoverflow.net/questions/284794 | 4 | Let $G$ be a locally compact, second countable group. We equip the unitary dual $\widehat{G}$ with the Fell topology. I am looking for conditions which guarantee that the topological space $\widehat{G}$ is uniformizable.
Here, a topological space $X$ is called uniformizable if there exists a uniform space whose under... | https://mathoverflow.net/users/116434 | When is the unitary dual of a lscs group uniformizable? | A topological space $X$ is uniformizable iff it is completely regular. Glimm's Theorem states that a second countable group is Type I iff $\widehat{G}$ is $T\_0$. Since a $T\_0$ uniformizable space is Hausdorff (which can be seen easily from the pseudometric definition of uniformity), all Type I examples will be Hausdo... | 1 | https://mathoverflow.net/users/99234 | 284821 | 125,876 |
https://mathoverflow.net/questions/284809 | 15 | I know that smooth Fano varieties over $\mathbb{C}$ may be classified into a finite number of families in each dimension (1 in dimension 1, 10 in dimension 2, 105 in dimension 3 ...).
I am interested in cases where the Family is non-trivial (i.e. variety is not rigid).
Suppose that we have such a family, I assume ... | https://mathoverflow.net/users/99732 | symplectic form on an algebraic family | Let $(X\_\alpha, \mathcal{L}\_\alpha)\_{\alpha\in A}=(M, L, J\_\alpha, \overline{\partial}\_\alpha)\_{\alpha\in A}$ be a family of polarized varieties, with $A$ the complex manifold parametrizing the family (assumed to be connected). Here is $M$ the common underlying smooth manifold of the varieties $X\_\alpha$, $J\_\a... | 1 | https://mathoverflow.net/users/2819 | 284829 | 125,878 |
https://mathoverflow.net/questions/284812 | 8 | Give an algebra $A$, a bialgebra $B$, and an action, that is, a bilinear map $\triangleright: B \times A \to A$ such that
$$
(b\_1b\_2) \triangleright a = b\_1\triangleright(b\_2 \triangleright a).
$$
When it also holds that
$$
b \triangleright(ac) = (b\_{(1)} \triangleright a)(b\_{(2)}\triangleright c)
$$
what do w... | https://mathoverflow.net/users/42100 | Name for the action of a bialgebra on an algebra | According to nLab, such an action is called a [Hopf action](https://ncatlab.org/nlab/show/Hopf+action) and your data specify a [left $B$-module algebra](https://ncatlab.org/nlab/show/module+algebra). Such a structure is also referred to in the literature as an [algebra in the category](https://mathoverflow.net/q/246881... | 10 | https://mathoverflow.net/users/85967 | 284842 | 125,882 |
https://mathoverflow.net/questions/284823 | 5 | Can we classify finite 2-generated groups $G$ satisfying the following property:
For any pair $x,y$ which generate $G$, the pair $x,yxy^{-1}$ also generates $G$.
By the comments, no nontrivial abelian group can satisfy this property, so I suppose the first question is: Do such groups $G$ exist?
| https://mathoverflow.net/users/15242 | Can we classify all finite 2-generated groups $G$ such that if $x,y$ generate $G$, then so does $x,yxy^{-1}$? | **Theorem** The only finite group satisfying the condition is the trivial group.
*Proof.* Let $G$ be a nontrivial finite group and $S$ be a simple quotient of $G$, which satisfies the condition by Gaschutz's lemma. Then $S$ is non-abelian as mentioned above. If $x\in S$ is any involution, then by well-known result of... | 10 | https://mathoverflow.net/users/40723 | 284853 | 125,887 |
https://mathoverflow.net/questions/284859 | 18 | This is a very naive question.
Let $X$ be a complex manifold. Let $\mathcal{O}\_X$ be the structure sheaf of $X$, a sheaf of rings whose sections over opens $U\subset X$ are just the holomorphic functions $U\rightarrow\mathbb{C}$.
A sheaf of $\mathcal{O}\_X$-modules $F$ is coherent if:
1. It is locally finitely g... | https://mathoverflow.net/users/15242 | Why is Oka's coherence theorem a deep result? | In scheme theory applied to complex geometry one usually does not encounter coherent rings which are not noetherian as well.
However if $X$ is (for example) a Stein manifold then the ring $R = \mathcal{O}(X)$ of global holomorphic functions is typically non-noetherian, which makes remarkable the fact that $R$ is noneth... | 23 | https://mathoverflow.net/users/21724 | 284860 | 125,888 |
https://mathoverflow.net/questions/284856 | 0 | Suppose $f\_n$ is a sequence of holomorphic functions taking $\mathbb{D} \to \mathbb{C}$ where $\mathbb{D}$ is the unit disk. Further, $f\_n$ has a continuous extension to $\overline{\mathbb{D}}$. We can assume $\sum\_{n=0}^\infty f\_n$ converges normally on compact subsets of $\mathbb{D}$ to a holomorphic function $f$... | https://mathoverflow.net/users/nan | Interchanging sums and integrals in a specific instance | Take $f\_n(z) = 3z^{3n} - z^n$ and $C = \{ e^{i\theta} \ | \ \theta \in (0,\pi) \}$. It satisfies all of your hypotheses with
$$
f(z) = \frac{2+z}{1+z+z^2}
$$
and $\int\_C f\_n =0$ for $n \geq 1$. But $\int\_C f$ is not a convergent integral.
Of course the stronger condition $\sum\_n \int\_C |f\_n| < \infty$ is suff... | 1 | https://mathoverflow.net/users/21724 | 284863 | 125,889 |
https://mathoverflow.net/questions/284824 | 10 | Let $R(X)$ be the region defined by
$$\displaystyle R(X) = \{(a,b,c) \in \mathbb{R}^3 \colon |b| \leq a \leq c, \, a,c \geq 1, \, a(4ac-b^2) \leq X\}.$$
I want to know how to estimate the sum
$$\displaystyle \sum\_{(a,b,c) \in R(X) \cap \mathbb{Z}^3} 2^{\omega(4ac-b^2)},$$
where $\omega(n)$ denotes the number o... | https://mathoverflow.net/users/10898 | Averaging $2^{\omega(n)}$ over a region | Your guess is correct. Since $2^{\omega(n)} = \sum\_{d | n} \mu^2(d)$ one has
$$
S = \sum\_{(a,b,c) \in R(X)} 2^{\omega(4ac-b^2)} = \sum\_{d \leq X} \mu^2(d) |R\_d(X)|
$$
where $R\_d(X)$ is the set of integer triples $(a,b,c) \in R(X)$ such that $4ac \equiv b^2 \pmod d$.
Let $d$ be a squarefree integer. Let $a,b$ be ... | 8 | https://mathoverflow.net/users/21724 | 284869 | 125,891 |
https://mathoverflow.net/questions/284855 | 20 | 4-dimensional Smale conjecture claims the following:
The inclusion $SO(5)$ → $SDiff(S^4)$ is a homotopy equivalence.
or Does $Diff(S^4)$ have the homotopy-type of $O(5)$ ?.
The inclusion $SO(n + 1$) → $SDiff(S^n)$ is a homotopy equivalence for n = 1 (trivial proof), n = 2 [1004,Smale,1959,Proc. Amer. Math. Soc.],... | https://mathoverflow.net/users/99280 | What is the status of the 4-dimensional Smale Conjecture? | This problem is completely open.
| 22 | https://mathoverflow.net/users/3460 | 284870 | 125,892 |
https://mathoverflow.net/questions/284868 | 13 | Let $M$ be a compact orientable manifold which is homeomorphic to its connected sum with itself $M\# M$. Must $M$ be homeomorphic to a sphere?
I will explain why I am interested (at the risk of being outright foolish or overambitious). The 'equation' $M = M\#M$ is the simplest conceivable equation in the category of ... | https://mathoverflow.net/users/37477 | A question on connected sum of compact manifolds | Let us suppose that $dim(M)\geq 3$ then we have that:
* $\pi\_1(M \# M)\cong \pi\_1(M)\*\pi\_1(M)$,
* $H\_\*(M;\mathbb{Z})\cong H\_\*(M;\mathbb{Z})\oplus H\_\*(M;\mathbb{Z})$ when $\*< dim(M)$.
As $M$ is compact this implies that $\pi\_1(M)$ is finitely presented thus that $M$ is simply connected (\*). Together wit... | 18 | https://mathoverflow.net/users/27816 | 284872 | 125,893 |
https://mathoverflow.net/questions/284839 | 10 | I consider the irrational rotation
$T\_\alpha(x) = x + \alpha \text{ mod } 1$ for given irrational $\alpha \in [0,1]$. For a given open interval $A \subset [0,1]$ with length $|A|>0$, I consider the recurrence times $I = \{n\in \mathbb{N}: T^n(0) \in A \}$. I want to show that
$\sum\_{i \in I} p\cdot(1-p)^i \to |A|$ as... | https://mathoverflow.net/users/116667 | Irrational rotation - recurrence times | You can recover this result in two steps:
* a variation on Birkhoff's ergodic theorem yields the Cesàro convergence of the sums;
* Cesàro convergence implies Abel convergence.
First step: $T\_\alpha$ preserves the Lebesgue measure and is uniquely ergodic. Hence, for all $f \in \mathcal{C} (\mathbb{S}\_1, \mathbb{R}... | 8 | https://mathoverflow.net/users/75670 | 284875 | 125,894 |
https://mathoverflow.net/questions/284867 | 1 | Is it true that every bounded holomorphic functions on a smooth analytic hypersurface $X$ of $\Bbb C^n$ is constant?
* Remark that if $X$ is algebraic, the answer is yes.
Otherwise can you provide counterexamples?
| https://mathoverflow.net/users/116680 | Bounded holomorphic functions on hypersurfaces of $\Bbb C^n$ | This is not true: there are smooth hypersurfaces (even curves in $C^2$)
which are holomorphically equivalent to the unit disk in the plane.
MR1359950
Globevnik, Josip, Stensønes, Berit,
Holomorphic embeddings of planar domains into $C^2$.
Math. Ann. 303 (1995), no. 4, 579–597.
| 3 | https://mathoverflow.net/users/25510 | 284881 | 125,896 |
https://mathoverflow.net/questions/284882 | 5 | Let $X$ be a simplicial abelian group. Let $NX$ be its normalised chain complex denoted
...$\rightarrow NX\_{K}$ $\rightarrow$ $NX\_{k-1}$ $\rightarrow$...
Now define a new chain complex $Y$ by shifting the normalised chain complex; i.e. $Y\_{k} = NX\_{k+1}$. Now applying the functor from Dold Kan correspondence o... | https://mathoverflow.net/users/110510 | Loop space of a Simplicial Abelian group | You can find such a map, but it goes the other way: $\Gamma(Y)\to \Omega X$. It is always wise to keep one's right adjoints on the right hand side.t
But first, let me note that every simplicial abelian group $Z$ is equivalent to $\prod\_{k\ge0} K(\pi\_kZ,k)$ (this is just a restatement of the classical fact that ever... | 8 | https://mathoverflow.net/users/43054 | 284886 | 125,897 |
https://mathoverflow.net/questions/284818 | 29 | Let $M(n,k)$ be the matroid on the ground set $\{\pm 1,\ldots,\pm n\}$ for which a set is independent if and only if it contains at most $k$ pairs $\pm i$. Note that the signed permutation group (the Coxeter group of type $B\_n$) acts on this matroid. Questions:
1. Does this matroid have a name?
2. Has it been studi... | https://mathoverflow.net/users/10273 | Have you seen my matroid? | Let $U$ be the uniform matroid of rank $k$ on $n$. Since $U$ is orientable one can consider the Lawrence oriented matroid $\Lambda(U)$ associated with any orientation of $U$ (the Lawrence construction doesn't care about which orientation you take). Then $M(n,k)$ is precisely the underlying unoriented matroid $\underlin... | 14 | https://mathoverflow.net/users/94968 | 284888 | 125,898 |
https://mathoverflow.net/questions/284891 | 3 | So, my question specifically pertains to $T^\*SO(3)$ but I guess adjusted it could be asked about Lie groups in general. The canonical symplectic form on the cotangent bundle is invariant under the cotangent lifted action of $SO(3)$ on $T^\*SO(3)$. Symplectic reduction by this mapping then sends this form to the usual ... | https://mathoverflow.net/users/86065 | Symplectic submanifolds of cotangent bundles of Lie groups | The preimage of a coadjoint orbit under a moment map is, under a mild transversality assumption, a *coisotropic* submanifold; so its null foliation $\smash{\ker(\omega\_{|\Phi^{-1}(\mathcal O)})}$ is not by symplectic but by *isotropic* leaves (symplectic orthogonals to coisotropic): see Guillemin & Sternberg ([1984](h... | 3 | https://mathoverflow.net/users/19276 | 284902 | 125,903 |
https://mathoverflow.net/questions/284876 | 9 | Let $K$ be a subset of the positive integers $\mathbb{N}$. For each $n\in \mathbb{N}$, $K\_{n}$ denotes the set $\{k\in K: k\leq n\}$ and $|K\_{n}|$ denotes the number of the elements in $K\_{n}$. The natural density of $K$ is defined by $$\delta(K)=\lim\_{n\rightarrow \infty}\frac{|K\_{n}|}{n}.$$
A sequence $(x\_{k})\... | https://mathoverflow.net/users/41619 | On statistical bases in Banach spaces | **Edit 26.04.2022**. The problem is now solved; please see [my other answer](https://mathoverflow.net/a/359434/15129).
---
This is an open problem due to Vladimir Kadets. I would not expect an easy answer here. The problem with statistical convergence is that this filter is not countably generated (in which case ... | 8 | https://mathoverflow.net/users/15129 | 284912 | 125,909 |
https://mathoverflow.net/questions/284910 | 4 | In this post, Joel David Hamkins (Is there a 'largest' second-order categorical axiomatization of set theory, extended from ZFC2, URL (version: 2015-05-10): <https://mathoverflow.net/q/206178>) answers a questions about categorical theories extending second-order ZFC.
The way he obtains the categoricity of a theory ... | https://mathoverflow.net/users/116705 | Categorical set theories that are not extensions of second-order ZFC | Answering your more specific question: yes, it$^\*$ can be so characterized, in the following way:
* First, start with the axioms of second-order Zermelo set theory with choice (without Replacement). Note that $V\_\mu$ is a model of this if $\mu$ is the first limit of inaccessibles - indeed, as long as $\lambda$ is a... | 5 | https://mathoverflow.net/users/8133 | 284920 | 125,913 |
https://mathoverflow.net/questions/284947 | 5 | Let $f(x)=\sum\_{n\geq 1} c\_n\cdot x^n$ be a function given by a power series. Further there is some $\alpha >1$ such that for all $n$, $c\_n = \Theta(1/n^{\alpha})$. What can one say about the asymptotics (in terms of n) of the coefficients of the Taylor expansion (around $0$) of the inverse of f assuming it exists? ... | https://mathoverflow.net/users/116726 | Asymptotic growth of the of Taylor coefficients of the inverse of a function | You can use the Faa-di-Bruno formula, like it is done on page 7 of [here](http://www.mat.univie.ac.at/~michor/diff-zoo.pdf)
or the proof of claim (b) on page 21 of [here](http://www.mat.univie.ac.at/~michor/exotic-zoo.pdf).
| 4 | https://mathoverflow.net/users/26935 | 284948 | 125,921 |
https://mathoverflow.net/questions/284953 | 2 | I would like to find explicit definitions of pseudo, or even lax, algebras for a 2-monad, and their lax morphisms, *with all the coherence diagrams included*.
Alternatively, coherent lax algebras for an operad would also be welcome.
What would be a reference?
| https://mathoverflow.net/users/30366 | Pseudo or lax algebras for a 2-monad, reference request | The original reference is Marta C. Bunge, "Coherent extensions and relational algebras", Trans. Amer. Math. Soc. 197 (1974), pp. 355-390.
| 4 | https://mathoverflow.net/users/1310 | 284955 | 125,923 |
https://mathoverflow.net/questions/284951 | 3 | If $R$ is a PID, then we know that any projective module over $R$ is free. Is there any graded version of this? That is, let us call a graded commutative ring $R=\oplus \_{g \in G} R\_g$ a graded PID if every homogeneous ideal (i.e. graded ideal) of $R$ is generated by a homogeneous element (I don't know if this is a s... | https://mathoverflow.net/users/nan | On graded projective modules | This is true. More generally, we have the following:
>
> Let $\psi\colon G\rightarrow H$ be an epimorphism of abelian groups. We denote by $\bullet\_{[\psi]}$ the $\psi$-coarsening functor from the category of $G$-graded $R$-modules to the category of $H$-graded $R\_{[\psi]}$-modules. Let $R$ be a $G$-graded ring, ... | 2 | https://mathoverflow.net/users/11025 | 284958 | 125,926 |
https://mathoverflow.net/questions/284894 | 6 | Let $X$ be a possibly singular projective scheme which admits a torus $T$ action and has finitely many $T$ fixed points and one-dimensional $T-$orbits. There are many such schemes in the Grassmannian/flag variety for an algebraic group/Kac-Moody group. Then some theorems by GKM(Goresky-Kottwitz-MacPherson) allow us to ... | https://mathoverflow.net/users/7780 | Bialynicki-Birula Decomposition and moment polytopes/graphs | The moment graph comes with an additional structure: every edge is labelled by a character of $T$. More precisely, every edge $e$ corresponds to a 1-dimensional $T$-orbit closure $C\_e$. Its normalization is isomorphic to $\mathbf P^1$. An orientation of $e$ determines which fixed point in $C$ corresponds to $0$ or $\i... | 4 | https://mathoverflow.net/users/89948 | 284963 | 125,927 |
https://mathoverflow.net/questions/284960 | 2 | Let $E$ be a complex Hilbert space, with inner product $\langle\cdot\;, \;\cdot\rangle$ and the norm $\|\cdot\|$. Let $T\in \mathcal{L}(E)$
be bounded linear operators from $E$ to $E$ and $M\in \mathcal{L}(E)^+$ (i.e. $M^\*=M$ and $\langle Mx\;, \;x\rangle\geq 0$ for all $x\in E$.
Assume that there exists a sequence ... | https://mathoverflow.net/users/116483 | Problem of convergence of the following sequence | If $M$ is invertible then $(y\_n)\_n$ must be a bounded sequence since
$$1 = \langle My\_n, y\_n\rangle = \|M^{1/2}y\_n\|^2 \geq \frac{\|y\_n\|^2}{\|M^{-1/2}\|^2}.
$$
Thus, $(\langle MTy\_n, y\_n\rangle)\_n$ is a bounded sequence by the Cauchy-Schwarz inequality and so it has a convergent subsequence.
The Cauchy-Schw... | 6 | https://mathoverflow.net/users/76593 | 284970 | 125,929 |
https://mathoverflow.net/questions/284974 | 6 | It is obvious that if the Frattini quotient of a finite group $G$ is abelian, then $G$ is abelian by nilpotent and that finite nilpotent groups have abelian Frattini quotient.
I wonder if there is an example of a non-nilpotent finite group with abelian Frattini quotient ?
A better thing would be to have a nice ch... | https://mathoverflow.net/users/2042 | Are finite nilpotent groups the only finite groups with abelian Frattini quotient? | This is well known, but here is a quick proof anyway.
Suppose that $G/\Phi(G)$ is nilpotent, and let $P \in {\rm Syl}\_p(G)$. Then $P\Phi(G) \unlhd G$ so, by the Frattini Argument, $G = N\_G(P)P\Phi(G)= N\_G(P)\Phi(G)$. Hence $G = N\_G(P)$ by the non-generator property of $\Phi(G)$. So all Sylow subgroups are normal,... | 8 | https://mathoverflow.net/users/35840 | 284979 | 125,931 |
https://mathoverflow.net/questions/284623 | 6 | Let $X\_1,...,X\_n$ be i.i.d. random variable with a uniform distribution on [0,1]. Denote by $X\_{(1)}\leq X\_{(2)} \leq \ldots \leq X\_{(n)}$ their order statistics.
Given $k\geq 1$ and $u\in[0,k]$, I want a simple formula for
$$
p\_k(u):=\mathbb{P}[X\_{(1)}+\ldots + X\_{(k)}\leq u],
$$
or at least a simple lower ... | https://mathoverflow.net/users/67002 | Bound on probabilities of the sum of uniform order statistics | As pointed out by user63957, the conditional distribution of $T\_k:=X\_{(1)}+\ldots + X\_{(k)}$ given $X\_{(k+1)}=x$ is that of $xS\_k$, where $S\_k:=U\_1+\dots+U\_k$ and the $U\_i$'s are iid random variables (r.v.'s) uniformly distributed on $[0,1]$. So, we have the key relation
$$T\_k\overset D=X\_{(k+1)}S\_k,$$
ass... | 5 | https://mathoverflow.net/users/36721 | 284981 | 125,932 |
https://mathoverflow.net/questions/284964 | 4 | For the chromatic number $\chi(G)$ of a simple, undirected graph, there is a ["compactness" theorem by Erdos and De Bruijn](https://en.wikipedia.org/wiki/De_Bruijn%E2%80%93Erd%C5%91s_theorem_(graph_theory)) stating that if an infinite graph $G$ has finite chromatic number, then there is a finite subgraph $G\_0\subseteq... | https://mathoverflow.net/users/8628 | "Compactness" theorem for the coloring number | A line infinite in both directions has minimal degree $2$, but any its proper subgraph has minimal degree $1$ or $0$. Thus, $\mathrm{col}(G)=3$, but $\mathrm{col}(G\_0)\le2$ for any proper subgraph $G\_0$ of $G$.
More generally, if $G$ is the $d$-regular infinite tree, then $\mathrm{col}(G)=d+1$, but $\mathrm{col}(G\... | 6 | https://mathoverflow.net/users/12705 | 284982 | 125,933 |
https://mathoverflow.net/questions/283693 | 10 | I believe that every skeletal monoidal category is monoidally equivalent to a skeletal monoidal category with strict units. Does anybody know a reference for this fact in the literature?
| https://mathoverflow.net/users/799 | Skeletal monoidal categories with strict units | See Theorem 3.2 in [Turning Monoidal Categories into Strict Ones](http://nyjm.albany.edu/j/2001/7-16p.pdf).
Thus, any monoidal category $(\mathcal{C},\otimes, I,\alpha, \lambda,\rho)$ is monoidally equivalent to a monoidal category $(\mathcal{C},\otimes', I,\alpha')$ with strict unit (note that $\mathcal{C}$ is the sam... | 5 | https://mathoverflow.net/users/6517 | 284992 | 125,937 |
https://mathoverflow.net/questions/284991 | 1 | For Banach spaces $E$ and $F$ we denote the approximate operators by $\mathcal A(E,F)$ and projective tensor product by $\hat\otimes$.
Consider the natural map
$$\Delta: \mathcal A(\ell^q,\ell^p)\hat\otimes\mathcal A(\ell^p,\ell^q)\rightarrow \mathcal A(\ell^p), \quad S\otimes T\mapsto ST$$
Can we write elements... | https://mathoverflow.net/users/84700 | About representations of some elements in $\mathcal A(\ell^p)$ | So I think this works. Fix $1<r<\infty$ and let $r'$ be the conjugate index to $r$ so $1/r + 1/r' = 1$.
Let
$$ \tau = \sum\_n S\_n \otimes T\_n \in \mathcal{A}(\ell^q,\ell^p) \widehat\otimes \mathcal{A}(\ell^p, \ell^q), $$
where by definition, $\sum \|S\_n\| \|T\_n\| < \infty$. By rescaling the $S\_n$ and $T\_n$ we m... | 3 | https://mathoverflow.net/users/406 | 285001 | 125,939 |
https://mathoverflow.net/questions/284959 | 7 | My problem can be described easily:
I have a sequence $(X\_l)\_{l \in \mathbb{N}}$ of r.v. adapted to some filtration $(\mathcal{F}\_l)\_{l \in \mathbb{N}}$, such that
1. $\left|X\_{l+1}-X\_l\right|\le R$ a. s.
2. $\mathbb{E}[X\_{l+1}-X\_l| \mathcal{F}\_l]\le \delta$ a. s.
3. $X\_0 = x\_0$ a.s. where $x\_0 \in \math... | https://mathoverflow.net/users/116734 | Prove an anti-concentration inequality for a martingale | Basically, the proof goes along the following lines:
(1) Take a small $\varepsilon>0$ and show that the expected exit time from the interval $[-\varepsilon\sqrt{vl},\varepsilon\sqrt{vl}]$ is less than $\varphi l$ (this is standard, using the fact that your martingale squared becomes a submartingale with uniformly pos... | 7 | https://mathoverflow.net/users/81488 | 285010 | 125,942 |
https://mathoverflow.net/questions/284883 | 4 | Suppose $x(t)$ is differentiable on $(0,T)$ and continuous on $[0,T]$. How to find the minimum and the minimal value of the integral $$\int\_0^T\|\dot x(t)+x(t)\|^2dt$$ such that $m\le x(t)\le M$ on $[0,T]$?
| https://mathoverflow.net/users/105002 | How to find the minimum of the integral? | Consider the quadratic functional $J\_T$ on the Hilbert space $H^1(0,T)$
$$J\_T(u):=\int\_0^T(\dot u+u)^2dt\ ,$$
and let $0<m< M$ be given. The complete picture for the minimization problem of $J\_T$ on $\{u\in H^1(0,T)\ :\ m\le u\le M\}$, as $T$ varies, is as follows:
>
> * For $0\le T \le T\_0:=\log(M/m)$ the mi... | 4 | https://mathoverflow.net/users/6101 | 285012 | 125,944 |
https://mathoverflow.net/questions/284924 | 4 | For a convex polytope $\mathcal K$ in $\Bbb R^n$ presented by $O(n^c)$ linear inequalities is it true that for $|\mathcal K\cap \Bbb Z^n|=0$ it is necessary that at least one axis of John's ellipsoid have length smaller than $1$?
| https://mathoverflow.net/users/10035 | On necessary condition for no integer points in polytope | The answer is $no$.
In the plane, the circle of diameter $d$ greater than $1$ but smaller than $\sqrt2$ and centered at $({1\over2},{1\over2})$ contains no point with integer coordinates, and the square $K\_2$ circumscribed about it with diagonals parallel to the coordinate axes contains no such point either. Let $K\... | 3 | https://mathoverflow.net/users/36904 | 285019 | 125,947 |
https://mathoverflow.net/questions/285009 | 1 | The two-parameter Wright function <http://dlmf.nist.gov/10.46> is defined as the infinite series
$$
\phi (\alpha, \beta \, | z)=\sum\limits\_{k=0}^\infty \frac{z^k}{\Gamma(k+1) \Gamma(\alpha k + \beta) }
$$
It arises naturally as a solution of certain classes of differential equations in mathematical physics.
I woul... | https://mathoverflow.net/users/48672 | representation of the Wright function | Representations of $\phi(\alpha,\beta|z)$ in terms of more elementary functions exist only in special cases, see [arXiv:1703.01912](https://arxiv.org/abs/1703.01912) (2017) and [An Extension of Wright Function and Its Properties](https://www.hindawi.com/journals/jmath/2015/950728/) (2015):
$$\phi(1,\nu+1|-z^2/4)=2z^{... | 0 | https://mathoverflow.net/users/11260 | 285035 | 125,950 |
https://mathoverflow.net/questions/285029 | 6 | Let $f:X\to S$ be a smooth proper morphism of schemes with geometrically connected fibres. Assume $S$ is a smooth irreducible variety over $\mathbb{C}$. Assume that there is a sequence of closed points $(s\_i)\_{i=1}^\infty$ in $S$ such that the fibres $X\_{s\_i}$ are pairwise non-isomorphic over $\mathbb{C}$.
Is the... | https://mathoverflow.net/users/116758 | On non-isotriviality of families | Let $k$ be an algebraically closed field that is uncountable. Let $S$ be finite type $k$-scheme. Let $f:X\to S$ be a proper, flat morphism of schemes.
**Proposition.** For every closed subset $T\subset S$, if there exists an infinite subset of $T(k)$ such that the corresponding fibers of $f$ are pairwise non-isomorph... | 5 | https://mathoverflow.net/users/13265 | 285047 | 125,953 |
https://mathoverflow.net/questions/285048 | 10 | Consider the three following large cardinal axioms:
1. there exists a nontrivial elementary embedding $j:V\to V$.
2. there exists a n.e.e. $j:V\to M$ such that $M^{j^\omega(crit(j))}\subseteq M$.
3. there exists a n.e.e. $j:V\_{\lambda+2}\to V\_{\lambda+2}$ for some $\lambda$.
Kunen's inconsistency theorem states t... | https://mathoverflow.net/users/78441 | Are these large cardinals properties equivalent? | No.
This is a very recent work in progress of myself with Juan Aguilera.
>
> **Definition.** We say that $\kappa$ is a *Kunen cardinal* if there is a nontrivial elementary embedding $j\colon V\_{\lambda+2}\to V\_{\lambda+2}$ with $\lambda=\sup j^n(\kappa)$.
>
>
>
(Sometimes it is easier to talk about the cri... | 11 | https://mathoverflow.net/users/7206 | 285056 | 125,955 |
https://mathoverflow.net/questions/285039 | 6 | Let $K$ be a field of characteristic $0$; let $\ell$ be any prime; and let $\mathrm{Mot}(K, \mathbb{Q}\_{\ell})$ be a *Tannakian* category of motives over $K$ with coefficients in $\mathbb{Q}\_{\ell}$. So, we may assume some conjectures which imply that our category of motives (defined with respect to any equivalence r... | https://mathoverflow.net/users/24757 | actions of the absolute Galois group and the motivic Galois group on étale cohomology | The absolute Galois group is a quotient of the motivic Galois group. The $\ell$-adic cohomology defines a section to the quotient map only on $\mathbb{Q}\_{\ell}$-points. This section gives the action of the Galois group on the $\ell$-adic cohomology (by definition). Such things are discussed in Saavedra Rivano's book ... | 4 | https://mathoverflow.net/users/116771 | 285059 | 125,957 |
https://mathoverflow.net/questions/285069 | 6 | Let $M\_g$ be the moduli space of Riemann surfaces, as described for example in the book of Harris and Morrison - *Moduli of curves*. As a topological space, or better as orbifold, it has smooth points and singular points. Let $S\subseteq M\_g$ be the subset of **smooth points**. In the article by Cornalba ("On the loc... | https://mathoverflow.net/users/81220 | Homotopy groups of smooth part of moduli space | I'm not sure why you're talking about codimension $1$ components of $S$ since the smooth locus $S$ is actually dense in $\mathcal{M}\_g$.
Anyway, the fundamental group of the locus $S$ of smooth points is the mapping class group $Mod\_g$ of the surface $\Sigma\_g$. To see this, recall that $\mathcal{M}\_g$ is the quo... | 9 | https://mathoverflow.net/users/317 | 285072 | 125,963 |
https://mathoverflow.net/questions/285031 | 6 |
>
> Are the pure genus zero mapping class groups residually torsion-free nilpotent?
>
>
>
They are
-a quotient of the pure braid groups (which are residually torsion-free nilpotent).
-torsion-free.
| https://mathoverflow.net/users/41970 | Are the pure genus zero mapping class groups residually torsion-free nilpotent? | The answer is yes. This is because these groups are fundamental groups of complements of fiber-type hyperplane arrangements; the fact that such groups are residually torsion free nilpotent goes back to Falk and Randell.
Indeed we are considering the fundamental group of the moduli space $M\_{0,n}$ of genus zero Riem... | 7 | https://mathoverflow.net/users/1310 | 285082 | 125,968 |
https://mathoverflow.net/questions/285054 | 17 | I kindly ask about some references concerning the representation theory of the Langlands dual of a compact Lie group, and how it relates to things related to the original compact Lie group.
My background: I know some basic facts about Lie groups/algebras, such as their root systems, Weyl groups etc. I am not familiar... | https://mathoverflow.net/users/81645 | References for Langlands classification | The first source in which I really discovered quite explicitly the archimedean local Langlands classification is in [this](https://www.math.stonybrook.edu/~aknapp/pdf-files/motives.pdf) beautiful article of Knapp, reviewing it in some pages. Moreover, it has the appeal to give a short historical motivation, to deal wit... | 13 | https://mathoverflow.net/users/43737 | 285083 | 125,969 |
https://mathoverflow.net/questions/285079 | 2 | The [OEIS entry for Pascal’s triangle](https://oeis.org/A007318) contains the following intriguing remark:
>
> $C(n,k)$ = the number of Dyck paths of semilength $n$, with $k$ "u"'s in odd numbered positions and $k$ returns to the x-axis.
>
>
> Example: {`U` = u in odd position and `_` = return to x axis}
>
>
> ... | https://mathoverflow.net/users/8217 | Counting particular Dyck paths | The answer is pretty simple. Consider any of $k$ parts between consecutive touches of $x$-axis. Note that we can assign a unique `U` character to each of these parts, namely, the first character, which meets our quota of `U` characters. The last character of each part obviously has to be `d`. Between the first and the ... | 4 | https://mathoverflow.net/users/106512 | 285086 | 125,970 |
https://mathoverflow.net/questions/284932 | 1 | Say we have a finite [Moufang Loop](https://en.wikipedia.org/wiki/Moufang_loop) $Q$, $|Q|<\infty$.
There is a [theorem proved](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.561.4372&rep=rep1&type=pdf) by Thompson that states:
Group $G$, $|G|<\infty$ is solvable $\iff$ $\forall a, b \in G \langle a, b\r... | https://mathoverflow.net/users/116711 | Portability of Thompson theorem about solvability to Moufang loops | Well, I've constructed a chain of proposition which conclude Thompson's theorem being translated to the case of finite Moufang loops in general case. Here we go:
"$\Rightarrow$". Consider $Q$ be a Moufang loop. The solvability property inherits to subloops (thanks we have [Lagrange property for Moufang loops](https:... | 2 | https://mathoverflow.net/users/116711 | 285094 | 125,973 |
https://mathoverflow.net/questions/285030 | 4 | If $(P,\leq)$ is a poset and $p\in P$, then we say that $p$ *is the lower part of a gap* there is $q \in P$, $q>p$ such that $[p,q] = \{p,q\}$. (This is equivalent to the statement that $(\uparrow p) \setminus \{p\}$ contains a minimal element.)
Let $\text{NPU}(\omega)$ be the set of non-principal ultrafilters on $\o... | https://mathoverflow.net/users/8628 | "Gaps" in the Rudin-Keisler ordering | Yes. Take U, V non-isomorphic Ramsey ultrafilters. Then there is no W that is RK-strictly in between $U$ and $U\cdot V$. The reason is the same as that given in [Infima in the Rudin-Keisler ordering](https://mathoverflow.net/questions/283045/infima-in-the-rudin-keisler-ordering?rq=1)
| 6 | https://mathoverflow.net/users/23835 | 285096 | 125,974 |
https://mathoverflow.net/questions/285092 | 5 | It is stated in several sources on numerical analysis that the general problem of polynomial root-finding is ill-conditioned, but that it is well-conditioned if the roots are near the unit circle. (e.g. see [page 133 of *Approximation Theory and Approximation Practice*](https://books.google.com/books/?hl=en&id=h80N5JHm... | https://mathoverflow.net/users/61573 | Stability of root-finding near the unit circle | The issue is explained nicely in [Six Myths of Polynomial Interpolation and Quadrature](https://people.maths.ox.ac.uk/trefethen/mythspaper.pdf). It is not a stability problem of polynomial root finding, but a problem of finding the proper representation of the polynomial. If the roots are on or near the unit circle, yo... | 8 | https://mathoverflow.net/users/11260 | 285097 | 125,975 |
https://mathoverflow.net/questions/285091 | 3 | I'm reading [this](http://www.cmat.edu.uy/~wschebor/publications/2009%20cohen-mw%20fract%20B-motion.pdf) paper and on page 8 the authors state without proof an asymptotic expansion of a multivariate Gaussian integral in terms of the covariance obtained by applying what they call the "Lebesgue theorem" (my guess is that... | https://mathoverflow.net/users/56931 | Asymptotic expansion of nonlinear Gaussian transformation in terms of covariance | Let $r:=r\_n:=A(n)$, so that $r\to0$. The right-hand side of the asymptotic relation in question is
\begin{equation\*}
\text{rhs}=\frac{r^2}2\,Eg(X)(X^2-1)\;Eg(Y)(Y^2-1)\Big[=\frac{r^2}2\,(Eg(X)X^2)^2\Big],
\end{equation\*}
since $X,Y$ are iid [and $Eg(X)=0$].
The left-hand side of that asymptotic relation is ... | 4 | https://mathoverflow.net/users/36721 | 285104 | 125,977 |
https://mathoverflow.net/questions/285113 | 2 | Let $T = {\mathbb R}/{\mathbb Z}$ be the $1$-torus. Let $a\_{ij}$ be integer numbers, $1 \leq i \leq m$, $1 \leq j \leq n$ and $A$ the $m \times n$ matrix whose $(i,j)$ entry is $a\_{ij}$. Consider the following system of $m$ linear equations:
$$\left\{\begin{array}{rl}
a\_{11}x\_1 + a\_{12}x\_2 + \cdots + a\_{1n}x\_... | https://mathoverflow.net/users/116786 | Elementary question about linear algebra on a circle | I believe the magic words are ["Smith Normal Form"](https://www.wikiwand.com/en/Smith_normal_form) (of the matrix $A.$)
| 2 | https://mathoverflow.net/users/11142 | 285114 | 125,979 |
https://mathoverflow.net/questions/285005 | 7 | Let $k$ be an algebraically closed field. Let $X$ be an integral $k$-scheme, separated and of finite type over $k$. Let $d := \dim X$, let $T := (\mathbb{G}\_{m,k})^{d}$ be the $d$-dimensional torus, and suppose we have an action of $T$ on $X$ over $k$ given by the action morphism $\sigma : T \times\_{k} X \to X$. Let ... | https://mathoverflow.net/users/15505 | When are these definitions of "toric variety" equivalent? | As noticed by Dave Anderson, (4) does not imply (2) because the generic stabilizer might be finite but non-trivial. But still, let $E$ be the stabilzer subgroup scheme of $x$ as in (5). Because $T$ is abelian, $E$ is normal in $T$. Hence $E$ acts trivially on the open set $Tx$. The fixed point scheme of $E$ being close... | 5 | https://mathoverflow.net/users/89948 | 285127 | 125,985 |
https://mathoverflow.net/questions/285130 | 6 | Let $k$ be an algebraically closed field, $X, Y$ integral $k$-schemes and $Y$ proper over $k$. Let $U$ be a non-empty open subset $U \subset X$
and $f:U \to Y$ a morphism of finite-type. Suppose that for any closed point $x \in X \backslash U$, any DVR $R$ with fraction field $K$, residue field $k$ and any morphism $\... | https://mathoverflow.net/users/58203 | Valuative criterion to extend morphism of schemes | No. Take $X$ to be a projective curve with one cusp at $x$, $U:=X\smallsetminus\{x\}$, $Y=$ the normalization of $X$, $f=$ the section of $Y\to X$ over $U$.
[Edit after Ron's comment] The answer is yes if $X$ is normal. Let $X'\subset X\times Y$ be the closure of the graph of $f$. Then $\pi:X'\to X$ induced by the fi... | 8 | https://mathoverflow.net/users/7666 | 285131 | 125,986 |
https://mathoverflow.net/questions/285133 | 0 | [Hedetniemi's conjecture](https://en.wikipedia.org/wiki/Hedetniemi%27s_conjecture) is about the chromatic number of the [categorical product](https://en.wikipedia.org/wiki/Tensor_product_of_graphs) of simple, finite, undirected graphs $G, H$ : it claims that $\chi(G\times H) = \min \{\chi(G), \chi(H)\}$.
For any simp... | https://mathoverflow.net/users/8628 | Hedetniemi's conjecture for the coloring number | No. If $H=G=K\_d$, $\text{col}(G)=d$, but $G\times G$ is a regular graph of degree $(d-1)^2$, thus $\text{col}(G\times G)\geqslant (d-1)^2+1>d$ for $d\geqslant 3$.
| 3 | https://mathoverflow.net/users/4312 | 285136 | 125,989 |
https://mathoverflow.net/questions/285138 | 4 | [This is a duplicate of [this question](https://math.stackexchange.com/questions/2501484/convolution-of-l-adic-sheaves-is-commutative) on Stackexchange]
I am trying to figure out how to prove a very basic statement about convolution of $\ell$-adic/perverse sheaves in Katz's "Rigid local systems" (section 2.5.3, (1) )... | https://mathoverflow.net/users/106906 | Convolution of $\ell$-adic sheaves is commutative if the group is commutative | The thing that makes everything easy here is that the horizontal maps in your diagram are isomorphisms. For instance, every commutative square where the horizontal maps are isomorphisms is Cartesian.
The fact that the horziontal maps are isomorphisms also means the base change maps are isomorphisms. You can think of... | 5 | https://mathoverflow.net/users/18060 | 285142 | 125,992 |
https://mathoverflow.net/questions/285020 | 4 | (Note that the logic systems described in this question *only refer* to logic systems restricted to the language $\in$)
**1. Can $\mathcal{L}\_{\kappa,\kappa}$ express $n$-th order finitary logic?** It is clear that it can express first-order logic ($\Pi\_{<\omega}^0$), but it seems unlikely that $\mathcal{L}\_{\kapp... | https://mathoverflow.net/users/115951 | How expressive can $\mathcal{L}_{\kappa,\kappa}$ be? | The answer is no.
Suppose that $\kappa$ is a Mahlo cardinal. This is $\Pi^1\_1$ expressible in $V\_\kappa$, since it amounts to the assertion that every closed unbounded subset of $\kappa$ contains a regular cardinal. But it cannot be expressible by an assertion of $\mathcal{L}\_{\kappa,\kappa}$, since there is a fo... | 11 | https://mathoverflow.net/users/1946 | 285148 | 125,994 |
https://mathoverflow.net/questions/285123 | 7 | Let $X:=\mathbb P(a\_0,a\_1, \ldots, a\_n)$ be a **well formed** weighted projective variety. Let $-K\_X$ be its anticanonical divisor, then how to express its volume ${\rm vol}(-K\_X)=(-K\_X)^n$ in terms of $a\_0,\ldots, a\_n$?
In principle, this could be computed by toric geometry, but the data seems too complicate... | https://mathoverflow.net/users/29730 | Volume of $-K_X$ for a weighted projective variety | I am just rewriting my comment above as an answer.
Let $S=k[x\_0,\dots,x\_n]$ be the $\mathbb{Z}\_{\geq 0}$-graded $k$-algebra with every $x\_i$ homogeneous of degree $a\_i.$ Denote by $X$ the associated projective $k$-scheme, $X =\text{Proj}\ S.$ Denote by $a$ the least common multiple of $(a\_0,\dots,a\_n).$ For ev... | 9 | https://mathoverflow.net/users/13265 | 285155 | 125,997 |
https://mathoverflow.net/questions/285128 | 3 | Consider $f\in L^2(I)$, where $I$ is the unit interval and $L^2$ is w.r.t. Lebesgue measure, and consider an approximation of $f$ denoted by $\tilde{f}\in L^2$.
The error in approximated the moments of $f$ by those of $\tilde{f}$ can be readily bounded by $\|f-\tilde{f}\|\_2$, e.g., denoting the respective expectanci... | https://mathoverflow.net/users/42864 | Tight L2 bound on moments approximation and reference | Let $X:=f$ and $Y:=\tilde f$. Let $\mu:=EX$ and $\nu:=EY$. Let $\|\cdot\|:=\|\cdot\|\_2$.
Let $D:=Y-X$. For any random variable $V$, let $\tilde V:=V-EV$.
Then the first displayed inequality can be rewritten as $ED\le\|D\|$. The upper bound here is attained if e.g. $D=1$. So, this bound is tight.
The second disp... | 5 | https://mathoverflow.net/users/36721 | 285165 | 126,000 |
https://mathoverflow.net/questions/285160 | 3 | Is there an order-preserving surjective map $f: {\cal P}(\omega)/(fin) \to [0,1]$? Or from ${\cal P}(\omega)/(fin)$ onto $[0,1]\cap \mathbb{Q}$?
| https://mathoverflow.net/users/8628 | Order-preserving surjective map $f: {\cal P}(\omega)/(fin) \to [0,1]$ | Replace $\omega$ with $X:=\mathbb Q\cap [0,1]$, and map each $A\subseteq X$ to $\limsup A\in [0,1]$. $\limsup$ is weakly increasing, and invariant under finite changes. This map is onto $[0,1]$.
If you want to map onto the dyadic rationals, partition $\omega$ into countably many infinite sets $\omega=A\_1\cup A\_2\cu... | 8 | https://mathoverflow.net/users/14915 | 285168 | 126,002 |
https://mathoverflow.net/questions/285164 | 11 | Let $\mathcal{S}:= \mathcal{S}(\mathbb{R}^n)$ be the Schwartz space of smooth functions with rapid decay. The question is pretty simply stated in the title. Pseudo-differential act continuously on the space $\mathcal{S}$, it is therefore natural to wonder whether they are the only ones:
>
> Is there a continuous op... | https://mathoverflow.net/users/22810 | Is every continuous endomorphism of the Schwartz space a pseudo-differential operator? | No, the most obvious example is the reflection operator: $Rf(x) = f(-x)$ this is not pseudolocal (in fact the $\xi$-compotent of the wavefront set gets a sign flip). Also the Fourier transform.
More generally, every compactly supported Fourier integral operator with non-trivial Lagrangian (not the co-normal to the di... | 15 | https://mathoverflow.net/users/20155 | 285170 | 126,003 |
https://mathoverflow.net/questions/285095 | 5 | For a scheme $X$, let $LE(X)$ denote the lisse-etale site on $X$. This is the full subcategory of $\textbf{Sch}/X$ consisting of smooth morphisms to $X$, equipped with the etale topology. Let $\mathcal{O}\_X$ denote the presheaf on $LE(X)$ sending
$$(U\rightarrow X)\mapsto \Gamma(U,\mathcal{O}\_U)$$
This is in fact rep... | https://mathoverflow.net/users/15242 | Pulling back the lisse-etale structure sheaf via the inclusion of a point | **EDIT**
I just realized that my previous version of the answer was totally misleading. Here is what really is problematic in your argument:
Just because the images of $t$ and $t^2$ are distinct in $\mathcal{O}\_X(U)$ it does not mean that they map to distinct elements in $\varinjlim \mathcal{O}\_X(U)$.
Here is a... | 3 | https://mathoverflow.net/users/1084 | 285187 | 126,006 |
https://mathoverflow.net/questions/285185 | 4 | Let $E$ be a given elliptic curve over a number field $F$. For each prime $p$, one has the Galois representation $\mathrm{Gal}(\bar{F}/F)\to GL(E[p])$ where $\bar{F}$ is a fixed algebraic closure of $F$. Is $E[p]$ irreducible for infinitely many prime $p$? I suspect this might be well-known and elementary in the theory... | https://mathoverflow.net/users/44005 | Irreducibility of residual Galois representations attached to an elliptic curve | As Will Sawin points out, the following is mostly an answer to the question "Is it true that the set of primes such that $\rho\_p$ is reducible is finite?" which is related but strictly stronger than the question actually asked.
That question admits a negative answer, but a positive answer if $E$ has no complex multi... | 5 | https://mathoverflow.net/users/2284 | 285190 | 126,008 |
https://mathoverflow.net/questions/285189 | 7 | **Edit:** According to the interesting comments of Michael Albanese and Nick L we revise the question as follows:
By manifold compactification of a manifold $M$ we mean a compact manifold $\tilde{M}$ which contains $M$ as an open dense subset.
Assume that $M$ is an open connected manifold which admits a manifold co... | https://mathoverflow.net/users/36688 | Compactification of open manifolds in the form of a manifold( with zero Euler characteristic) | First note that odd dimensions the question of Euler characteristic $0$ is automatic, $M$ will embed in the orientable double cover of $\tilde{M}$, which will have $\chi = 0$ by Poincare Duality.
In even dimension = $2n$ (we assume $n > 1$), we recall the following fact. If $M\_{1},M\_{2}$ are compact connected manif... | 8 | https://mathoverflow.net/users/99732 | 285200 | 126,012 |
https://mathoverflow.net/questions/285175 | 5 | Suppose $\kappa$ is $\kappa^{+\omega}$-supercompact. If $U$ is a normal measure on $\mathcal P\_\kappa(\kappa^{+\omega})$, and $j : V \to M$ is the derived embedding, then it is easy to see that $j[\mathcal P\_\kappa(\kappa^{+\omega})] \in M$, and thus $M$ is closed under $\kappa^{+\omega+1}$-sequences by the cardinal ... | https://mathoverflow.net/users/11145 | supercompactness measure projections | I think about this a little more abstractly using what I call seed theory. See for example my article,
*Hamkins, Joel David*, [**Canonical seeds and Prikry trees**](http://dx.doi.org/10.2307/2275538), J. Symb. Log. 62, No.2, 373-396 (1997). [ZBL0890.03024](https://zbmath.org/?q=an:0890.03024), which provides an elem... | 4 | https://mathoverflow.net/users/1946 | 285208 | 126,014 |
https://mathoverflow.net/questions/285178 | 4 | Consider a non-singular, completely positive, unital map $\Psi: \mathbf M\_k(\mathbb C) \to \mathbf M\_h(\mathbb C)$. This map will have one or more retractions. Does $\Psi$ admit a retraction $\Phi: \mathbf M\_h(\mathbb C) \to \mathbf M\_k(\mathbb C)$, such that $\lVert \Phi \rVert = \lVert \Phi \rVert\_{\mathrm{cb}}$... | https://mathoverflow.net/users/3723 | Retractions for completely positive unital maps, with particularly nice norms | The answer is no again. When $h=k$, non-singular is the same as bijective so any $\Psi$ has only one map $\Phi$ such that $\Phi\circ\Psi = I\_{M\_k}$, namely the inverse. In the example below we find a bijective, unital, completely positive $\Phi$ such that its inverse $\Phi = \Psi^{-1}$ is unital, completely bounded b... | 4 | https://mathoverflow.net/users/76593 | 285209 | 126,015 |
https://mathoverflow.net/questions/281576 | 9 | A $\ast$-autonomous category is [usually defined](https://ncatlab.org/nlab/show/star-autonomous+category) as a *monoidal* category with some extra structure, including that it is closed monoidal (depending on the definition this may be assumed or proved). However, in many examples (such as sup-lattices) it seems that t... | https://mathoverflow.net/users/49 | star-autonomous closed categories | Barr's "Definition D" in [non-symmetric $\ast$-autonomous categories](http://www.math.mcgill.ca/barr/papers/asymm.pdf) is a closed category $\mathcal{A}$ together with an $\mathcal{A}$-enriched equivalence $(-)^\* : \mathcal{A}^{\mathrm{op}} \simeq \mathcal{A}$ and an $\mathcal{A}$-enriched natural isomorphism $[A,[B^\... | 2 | https://mathoverflow.net/users/49 | 285213 | 126,018 |
https://mathoverflow.net/questions/285197 | 8 | Let $0<\sigma\leq 1$. Let $T$ be large. How can we give good explicit $L^2$ bounds on the tails of $\zeta(\sigma+it)$? That is, we want to bound the quantity $$\int\_{\sigma-i\infty}^{\sigma-iT} + \int\_{\sigma+iT}^{\sigma+i\infty} \frac{|\zeta(s)|^2}{|s|^2} ds.$$
Morally, we should expect something about as good as $1... | https://mathoverflow.net/users/398 | $L_2$ bounds for tails of $\zeta(s)$ on a vertical line | Let $\sigma>0$ be fixed, and let $T\geq 2$ be a parameter.
By Theorem 4.11 in Titchmarsh: The theory of the Riemann zeta-function,
$$ \zeta(\sigma+it)=\sum\_{n\leq T}n^{-\sigma-it}+O(T^{-\sigma}),\qquad T<|t|\leq 2T.$$
It follows that
$$ \int\_{\sigma+iT}^{\sigma+2T}|\zeta(s)|^2\,ds\ll T^{1-2\sigma}+\int\_T^{2T}\left... | 9 | https://mathoverflow.net/users/11919 | 285215 | 126,019 |
https://mathoverflow.net/questions/285186 | 31 | I have been interested in fractional calculus for some time now, and I have seen "lots" of definitions of the $\frac {d^\alpha} {dx^\alpha}$ operator.
I started with the book *The Fractional Calculus* by Oldham and Spanier, and it comes as no surprise that I favor the Grünwald-Leitnikov derivative. It seems to me a g... | https://mathoverflow.net/users/114143 | Why are there so many fractional derivatives? | The reason is that the fractional derivative is not a local operator. The usual derivative is a local derivative in the sense that the value of the derivative at one point only depends on the value of the function in a neighborhood of that point. This is not the case for the fractional derivative and that cannot be due... | 44 | https://mathoverflow.net/users/6129 | 285224 | 126,021 |
https://mathoverflow.net/questions/285109 | 2 | The mean value theorem for vector-valued function in the real domain $f: \mathcal{R}^n \rightarrow \mathcal{R}^d$ can be expressed as
\begin{equation}
f(x)-f(y)=\int\_{0}^{1}\nabla f(x(\tau))d \tau \cdot (x -y),
\end{equation}
where $x(\tau) = y + \tau (x - y)$.
I wonder if there exists similar results for vector-va... | https://mathoverflow.net/users/90066 | Mean value theorem in terms of Wirtinger calculus? | The desired relation can be written in components as
$$f\_n(\mathbf{z}^{(1)},\mathbf{\bar z}^{(1)})-f\_n(\mathbf{z}^{(2)},\mathbf{\bar z}^{(2)})=\int\_0^1 d\tau\,\sum\_{m}\left(\frac{\partial f\_n}{\partial z\_m}(z\_m^{(1)}-z\_m^{(2)})+\frac{\partial f\_n}{\partial \bar{z}\_m}(\bar{z}\_m^{(1)}-\bar{z}\_m^{(2)})\right),... | 1 | https://mathoverflow.net/users/11260 | 285227 | 126,023 |
https://mathoverflow.net/questions/285232 | -1 | This problem actually arose when designing a computer network.
Let $G=(V,E)$ be a finite, simple, undirected graph such that every vertex has degree at least $2$. Given $n\in\mathbb{N}$, a map $c:E \to \{1,\ldots, n\}$ is said to be a *weak coloring* if for every $v\in V$ the edges adjacent to $v$ do not all have the... | https://mathoverflow.net/users/8628 | Different form of edge coloring | No. For a cycle $C\_n$ of odd length the condition is equivalent to bipartiteness of the line graph, which is itself $C\_n$ and is not bipartite, hence there is no weak coloring.
Odd cycles are the only counter-examples among connected graphs. To see that, suppose that there is a vertex $v$ of degree at least 3. Add ... | 5 | https://mathoverflow.net/users/106512 | 285233 | 126,026 |
https://mathoverflow.net/questions/285235 | 0 | This is a follow-up to [this question](https://mathoverflow.net/questions/285232/different-form-of-edge-coloring).
Let $G=(V,E)$ be a finite, simple, undirected graph such that every vertex has degree at least $2$. Given $n\in\mathbb{N}$, a map $c:E \to \{1,\ldots, n\}$ is said to be a *weak coloring* if for every $v... | https://mathoverflow.net/users/8628 | Upper bound for weak edge colorings? | According to the answers to the previous question, the only connected graphs which require more than two colors are odd cycles, which are trivially weak 3-colorable. Therefore, $n\_0 = 3$ always suffices.
| 1 | https://mathoverflow.net/users/106512 | 285236 | 126,028 |
https://mathoverflow.net/questions/285211 | 1 | Let $\rho\_1$, $\rho\_2$ be two measures(not necessarily nonnegative) on $(\Omega,\mathcal{F})$, where $\Omega$ is a set, and $\mathcal{F}$ is a $\sigma$-field in $\Omega$. Let $\mathcal{F}\_0$ be a field in $\Omega$ and assume that $\mathcal{F}$ is the $\sigma$-field generated by $\mathcal{F}\_0$. Suppose that $\rho\_... | https://mathoverflow.net/users/114158 | Extension preserves the relation between two measures | This is tailor-made for the [monotone class theorem](http://www.math.ubc.ca/~feldman/m420/monotone.pdf).
Suppose first that $\rho\_1, \rho\_2$ are finite signed measures. Then the collection $\mathcal{M} = \{A \in \mathcal{F} : \rho\_1(A) \le \rho\_2(A)\}$ is easily seen to be a monotone class (because countably add... | 1 | https://mathoverflow.net/users/4832 | 285242 | 126,030 |
https://mathoverflow.net/questions/284973 | 18 | Everybody knows how insightful are David Ben-Zvi talks (and comments/answers here on mathoverflow). I was trying to watch the [LMS 2007 Lecture Series on Geometric Langlands](http://www.ma.utexas.edu/users/benzvi/GRASP/Oxford.html) by David, supposedly made available via [GRASP Lectures](https://www.ma.utexas.edu/users... | https://mathoverflow.net/users/43101 | LMS Lectures on Geometric Langlands | The videos from the LMS lectures and all of the GRASP videos are now available again from the links you gave (for download, not streaming). Many apologies for their long hiatus offline and many thanks for your enthusiasm and persistence!! Please do email me for broken links etc., and I will update the site with some mo... | 17 | https://mathoverflow.net/users/582 | 285245 | 126,033 |
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