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https://mathoverflow.net/questions/311796 | 4 | It is well known that if $\Omega\subset\mathbb{R}^{N}$ open, $F\subset\Omega$ closed, such that $\mathcal{H}^{N−1}(F)=0$,where $\mathcal{H}^{N−1}$ denotes (N-1) dimensional Hausdorff measure, then $W^{1,p}(\Omega)=W^{1,p}(\Omega\backslash\Sigma)$ for $p\in (1,+\infty)$.
Question: Assume $F$ and $\Omega$ satisfy the c... | https://mathoverflow.net/users/123056 | Removable set for Sobolev space | This is not true and a counterexample is based on the example in the comment of Math777.
Note that $W^{1,p}\_0(\Omega\setminus F)\subset W^{1,p}(\mathbb{R}^n)$, because $W^{1,p}\_0$ is the closure of $C\_0^\infty(\Omega\setminus F)\subset C\_0^\infty(\mathbb{R}^n)$.
If $p>n$, then Sobolev functions $W^{1,p}\_0(\Omega... | 3 | https://mathoverflow.net/users/121665 | 311798 | 135,647 |
https://mathoverflow.net/questions/311802 | 0 | I have the following "argument" about Fourier series, which I know is wrong because it yields a ridiculous conclusion. However, I don't know where the mistake is, and need to know which step is the problem.
We consider $f \in L^2[0,2 \pi]$ and $R\_f (\theta) := \int\_0^{2 \pi} f(u) f(u +\theta) \, d \theta $. Here, w... | https://mathoverflow.net/users/125275 | A question about the convolution theorem | No contradiction; an arbitrary square integrable sequence **is** $\ell^4$. The sequence is decaying to zero, so its fourth powers decay *faster*. Indeed, we have $\ell^p \subset \ell^q$ for any $q > p$.
| 3 | https://mathoverflow.net/users/4832 | 311805 | 135,651 |
https://mathoverflow.net/questions/311774 | 1 | Let $M$ be a finitely generated module over a Noetherian ring $R$ such that $M$ is isomorphic with its double dual $M^{\*\*}=Hom(Hom(M,R),R)$.
Then is the natural map $j:M \to M^{\*\*}$ defined as $j(m)(f)=f(m),\forall m\in M, \forall f\in M^\*$, an isomorphism ?
| https://mathoverflow.net/users/127118 | Noetherian module, over Noetherian ring, which is isomorphic to its double dual | I asked this question (or at least a similar one) here:
[Characterisation of reflexive modules](https://mathoverflow.net/questions/288878/characterisation-of-reflexive-modules)
By an answer of Jeremy Rickard it is even true for noetherian rings that are not necessarily commutative.
It seems to be an open question for f... | 1 | https://mathoverflow.net/users/61949 | 311806 | 135,652 |
https://mathoverflow.net/questions/311720 | 3 | Let $K\subset R^d$ be a compact set. It is well known that its Fourier dimension is defined by
$$\dim\_F K=\sup\{s\ge 0: \exists \mu \in M\_1(K) s.t. \hat{\mu}(x)=O(|x|^{-s/2})\}(|x|\to\infty),$$
where $M\_1(K)$ denotes the set of probability measures with support in $K.$
My question is: Does the condition $\dim\_FK=... | https://mathoverflow.net/users/129565 | Does zero Fourier dimension imply there is no Rajchman measure? | No, this does not follow. Since the Hausdorff dimension dominates the Fourier dimension, it suffices to establish the existence of a compact set $K$ of Hausdorff dimension zero that supports a Rajchman measure.
The last theorem of section 3 (attributed to Ivashev-Musatov 1962) of [Lyons's survey](https://pdfs.semanti... | 4 | https://mathoverflow.net/users/48839 | 311815 | 135,653 |
https://mathoverflow.net/questions/311825 | 3 | Let $Z\_1,\dots,Z\_n$ be dependent Gaussian random variables. Is it true that $X=\max\{Z\_1,\dots,Z\_n\}$ has a log-concave distribution function? This is true for the independent case, but is it true in general?
| https://mathoverflow.net/users/24494 | Log concavity of the maximum of dependent Gaussians | $\newcommand{\R}{\mathbb{R}}
\renewcommand{\P}{\operatorname{\mathsf P}}
\newcommand{\ii}[1]{\operatorname{\mathsf I}\{#1\}}$
This is false in general. E.g., let $Z\_1=U$ and $Z\_2=|V|\,\text{sign}\,U$, where $U,V$ are iid standard normal random variables. Let $F$ be the cdf of $\max(Z\_1,Z\_2)$ and $L:=\ln F$.
Let... | 7 | https://mathoverflow.net/users/36721 | 311827 | 135,655 |
https://mathoverflow.net/questions/311824 | 8 | As the title suggests, I'm trying to find motivation on the definition of the Thom spectrum of $-\xi$, or more generally on the definition of the Thom spectrum of a virtual bundle.
In [this paper](https://archive.org/stream/arxiv-math0312523/math0312523#page/n5) by S. Bauer (middle of page 7) he defines the Thom spec... | https://mathoverflow.net/users/48216 | Why the Thom spectrum of $-\xi$ (or more generally of a virtual bundle) is defined as it is? | Just a quick answer to explain the original reason behind the definition and why our modern understanding of Thom spectra vindicates it.
Let $X$ be a space and $\xi$ a virtual vector bundle over $X$. The definition you give is the only possible definition of $X^\xi$ such that
* $X^\xi$ coincide with the classical n... | 13 | https://mathoverflow.net/users/43054 | 311834 | 135,659 |
https://mathoverflow.net/questions/311808 | 1 | Let $n\ge 1$ be an integer, $\mathcal P\_n$ be the vector space of all polynomial functions over $[a,b]$, of degree at most $n$.
My question is : Is it true that
$$\inf\_{x\_0,x\_1,...,x\_n\in[a,b], x\_0<x\_1<...<x\_n} \sup\_{x\in [a,b]} \prod\_{i=0}^n |(x-x\_i)|=\inf\_{P\in \mathcal P\_n } \sup\_{x\in [a,b]} |x^{... | https://mathoverflow.net/users/127118 | A min-max approximation | When $[a,b]=[-1,1]$, the $\inf$ on the right-hand side is attained (uniquely) by the monic Chebyshev polynomial $T\_{n+1}$. It is well known that its roots belong to $[-1,1]$ and are simple.
For a general interval $[a,b]$, using the linear map
$$f(x)=\left(\frac{2}{b-a}\right)x-\left(\frac{b+a}{b-a}\right)$$
that s... | 1 | https://mathoverflow.net/users/89429 | 311854 | 135,665 |
https://mathoverflow.net/questions/311448 | 2 | Suppose $\kappa$ is a large cardinal (strong cardinal seems to be enough). Is there a forcing notion $\mathbb{R}$ with the following properties:
$(1)$ Forcing with $\mathbb{R}$ adds a club $C$ into $\kappa$ of $V$-regular cardinals.
$(2)$ Forcing with $\mathbb{R}$ preserves cardinals and the regularity of $\kappa.$... | https://mathoverflow.net/users/11115 | A variant of Radin forcing | A mild variant of the standard Radin forcing has property (4).
Let $\mathbb{R}$ be the forcing notion with conditions $p = (p\_0, \dots, p\_n)$ such that $p\_i = (u\_i, A\_i)$, $u\_i$ is a measure sequence and if $\mathrm{len}( u\_i) > 0$ then $A\_i$ is large relative to all measures in $u\_i$. We require also that ... | 3 | https://mathoverflow.net/users/41953 | 311859 | 135,666 |
https://mathoverflow.net/questions/311807 | 4 | Let $A \colon V \to W$ be a surjective linear map
(defined over $\mathbb{Z}$),
inducing a projection
$\alpha \colon \mathbb{P}(V) \to \mathbb{P}(W)$.
Let $X \subseteq \mathbb{P}(V)$ and $Y \subseteq \mathbb{P}(W)$
be some absolutely irreducible projective varieties
(defined over $\mathbb{Z}$) that we know well.
Su... | https://mathoverflow.net/users/39055 | Submersion implies many rational points in image? | This is an attempt at an answer to what I think the question is (please tell me if anything isn't clear).
>
> Theorem
>
>
> Let $f:X\to Y $ be a dominant morphism of finite type schemes over $\mathbb{Z}$ with $X\_{\mathbb{Q}}, Y\_{\mathbb{Q}}$ geometrically integral of dimensions $n,m$, respectively. Then there e... | 5 | https://mathoverflow.net/users/5101 | 311864 | 135,667 |
https://mathoverflow.net/questions/311672 | 0 | I have a dynamical system in the form of an integro-differential equation which I want to analyze in terms of stability. To demonstrate my problem consider the following integro-differential equation:
$$\dot{x}(t)=ax(t)+b\int\_0^tx(\tau)\text{d}\tau,$$
where $\dot{x}(t)$ denotes the time derivative of $x(t)$. If we... | https://mathoverflow.net/users/129529 | Does differentiating an integro-differential equation results in equivalent stability of the solution? | What you are missing is that your system secretly has two degrees of freedom. Your integro-differential equation is equivalent to the first order linear differential system
$$ \begin{cases} \dot{p}(t) = bx(t) ,& p(0) = 0 \\
\dot{x}(t) = a x(t) + p (t),& x(0) = ? \end{cases} $$
The initial condition that $p(0)$ com... | 2 | https://mathoverflow.net/users/3948 | 311883 | 135,675 |
https://mathoverflow.net/questions/311860 | 2 | I am trying to understand direct limit in category of $C^\*$ algebras.
>
> Is it well known that direct limit behaves well with double dual and quotient of $C^\*$ algebras?
>
>
>
Any references or ideas?
P.S: This question was first asked on mathatack but there I did not get any answer!
| https://mathoverflow.net/users/129638 | Behaviour of Direct limit with quotient and double dual | It definitely does not work for double duals. Indeed, if you view compact operators as an inductive limit of matrix algebras, then the sequence of double duals is exactly the same, but the double dual of compacts is $B(H)$.
| 5 | https://mathoverflow.net/users/24953 | 311888 | 135,677 |
https://mathoverflow.net/questions/311885 | 1 | Let $k\leq n$ be positive integers with $n\geq 2$, and let $[n]=\{1,\ldots,n\}$. Let $V\_n=\{0,1\}^{[n]}$ be the set of all functions $f:[n]\to\{0,1\}$, and let
$$E\_{k,n} =\big\{\{f,g\}: f,g\in V\_n\text{ and } |\{m\in [n]:f(m)\neq g(m)\}|=k\big\}.$$
We call $G\_{k,n} = (V\_n,E\_{k,n})$ the $(k,n)$-*binary graph*. It... | https://mathoverflow.net/users/8628 | $(k,n)$-binary graphs | No, the coprimality of $k$ and $n$ does not imply that $G\_{k,n}$ is connected. For a counterexample we consider the case $(k,n)=(2,3)$.
Represent a function $f\in V\_3$ by the sequence $(f(1),f(2),f(3))$. Then $V\_{2,3}$ is the disjoint union of two complete graphs with $4$ vertices each. The first one has vertices ... | 1 | https://mathoverflow.net/users/296 | 311891 | 135,678 |
https://mathoverflow.net/questions/310495 | 2 | **Background/Notation:**
Given the Iwahori-Hecke algebra $H\_{n}$ (over some ring commutative ring $R$ with identity) with generators $\{T\_{1},\ldots T\_{n-1}\}$, we know it has basis
$\{T\_{w}\}\_{w\in S\_{n}}$, where $S\_{n}$ is the permutation group and for some reduced expression $w=s\_{i\_{1}}\cdots s\_{i\_{r}... | https://mathoverflow.net/users/116558 | Basis for Annular Skein Algebra | The outlined proof almost works: only need to consider the weaker statement "*$\sigma\_{w}$ is conjugate to $\sigma\_{w'}$ in $B\_{n} \implies w$ is conjugate to $w'$ in $S\_{n}$*". (A useful discussion with Wade Bloomquist sent me in the right direction regarding conjugacy). One can easily show this:
Use the homomor... | 1 | https://mathoverflow.net/users/116558 | 311893 | 135,679 |
https://mathoverflow.net/questions/311897 | 2 | Suppose I have a symmetric tridiagonal (Jacobi) matrix in the following form:
$ \begin{pmatrix}
1 & a\_{1} & 0 & ... & 0 \\\
a\_{1} & 1 & a\_{2} & & ... \\\
0 & a\_{2} & 1 & ... & 0 \\\
... & & ... & & a\_{n-1} \\\
0 & ... & 0 & a\_{n-1} & 1
\end{pmatrix}, $
where all $0< a\_i < 1$ for $i = 1\ldots n-1$ but the ... | https://mathoverflow.net/users/34445 | Upper Bounds on the Largest Eigenvalue of Jacobi Matrices | The entries are nonnegative, so the dominant eigenvector has
all entries positive, and its eigenvalue is an increasing function of the $a\_i$.
If each $a\_i = 1$ then that eigenvalue is $1 + 2 \cos\frac\pi{n+1}$
if I did this right; since you allow only $a\_i < 1$, this bound
$1 + 2 \cos\frac\pi{n+1}$ is not attained, ... | 7 | https://mathoverflow.net/users/14830 | 311899 | 135,682 |
https://mathoverflow.net/questions/311862 | 19 | Let $G=(V,E)$ be a simple graph with $n$ vertices. The isoperimetric constant of $G$ is defined as
$$
i(G) := \min\_{A \subset V,|A| \leq \frac n2} \frac{|\partial A|}{|A|}
$$
where $\partial A$ is the set edges $uv \in E$, with $v \in A$ and $u \in A^c$.
>
> Is there a graph $G$ such that $$i(G) + i(G^c) < \... | https://mathoverflow.net/users/53059 | Is it possible that both a graph and its complement have small connectivity? | It is *almost* obvious. Let $i(A)=\frac{|\partial\_G A|}{|A|}$ and $j(B)$ be the same ratio for a vertex set $B$ and the complement of $G$. Let $x=|A\cap B|, y=|A\cap B^c|, z=|A^c\cap B|, t=|A^c\cap B^c|$. The edges going between $A\cap B$ and $A^c\cap B^c$ are boundary edges for both $A$ and $B$ and so are the edges b... | 15 | https://mathoverflow.net/users/1131 | 311902 | 135,683 |
https://mathoverflow.net/questions/311907 | 7 |
>
> Does there exist a local domain of infinite dimension in which every chain of prime ideals is finite?
>
>
>
Of course, such a ring must be neither noetherian nor catenary.
(This question arose while trying to compare different definitions of catenarity, and more precisely while trying to understand what ma... | https://mathoverflow.net/users/11025 | An infinite dimensional local domain whose chains of primes are finite | Choose a field $k$ and a Noetherian $k$-algebra $R$ of infinite dimension. (I know you know such a thing exists.) Let $R' \subset R[x]$ be the set of polynomials $f = \sum a\_i x^i$ whose constant term is constant, i.e., $a\_0 \in k \subset R$. Then we have
$$
\text{Spec}(R') = \text{Spec}(R[x]) \amalg\_{\text{Spec}(R)... | 11 | https://mathoverflow.net/users/129659 | 311918 | 135,689 |
https://mathoverflow.net/questions/311894 | 8 | Furstenberg–Sárközy's theorem states that if a set of positive integers has positive upper density, then there exists (infinitely many) pair of elements of the set, whose difference is a perfect square. Since prime numbers have zero density, this theorem does not apply to them. And I was wondering if there is a result ... | https://mathoverflow.net/users/112027 | Prime plus square equals prime | Tao and Ziegler extended the Green-Tao theorem to the polynomial setting. As a very special case we get that any subset of the primes with positive relative density contains a difference which is a square.
<https://arxiv.org/abs/math/0610050>
| 19 | https://mathoverflow.net/users/18698 | 311919 | 135,690 |
https://mathoverflow.net/questions/311921 | 22 | Hoping that my question is appropriate for MO, I would like to ask the following question: I have sent one of the editors of a very good math journal a paper of mine which contains a main result, call it theorem A.
The editor wrote me that he sent my paper to referee and will contact me when he will get the referee's r... | https://mathoverflow.net/users/72288 | Should I inform the editor about a generalized result of a result in a paper under review? | I myself would do (2). It adds information, enhances interest and acceptability, and is basic openness. The goal here is to enlighten and advance our understanding of the subject matter. Good luck! You should probably also prepare a careful revision for the referee's consideration.
| 24 | https://mathoverflow.net/users/9449 | 311924 | 135,693 |
https://mathoverflow.net/questions/311929 | 2 | For a finitely generated extension of fields $K/k$, let us define "$S\_{K/k}$" to be the minimum of the degrees $[K:\ell]$ where $\ell/k$ ranges over the purely transcendental subextensions of $K$ with $\mathrm{tr.deg}(K/k) = \mathrm{tr.deg}(\ell/k)$.
>
> Does there exist a tower $K\_{3}/K\_{2}/K\_{1}$ of finitely ... | https://mathoverflow.net/users/15505 | Transitivity of an invariant of finitely generated field extensions | Yes, I think so. Let $P\in \mathbb{C}[t]$ be a polynomial of degree $\geq 3$, with no multiple root and $P(0)\neq 0$.
Take $K\_1=\mathbb{C}$, $K\_2=\mathbb{C}(t)[x]/(x^2-P(t))$, $K\_3=K\_2[u]/(u^2-t)$. Then $S\_{K\_2/K\_1}=2$, $S\_{K\_3/K\_2}=[K\_3:K\_{2}]=2$, but $S\_{K\_3/K\_1}=2$ because $K\_3=\mathbb{C}(u)[x]/(x^2-... | 5 | https://mathoverflow.net/users/40297 | 311931 | 135,696 |
https://mathoverflow.net/questions/215821 | 2 | Define ${\cal L}$ as in [this question](https://mathoverflow.net/questions/215582/how-big-is-the-lattice-of-all-functions): the set of functions $f:\mathbb{N}\rightarrow\mathbb{N}$ such that $f(n)\leq f(n+1)\leq f(n)+1$, where two functions are considered equal if they differ at at most finitely many points, and $f\pre... | https://mathoverflow.net/users/8628 | Relationship of ${\cal P}(\omega)/fin$ and ${\cal L}$ | Look at your and my second answer to [the original question](https://mathoverflow.net/questions/215582/how-big-is-the-lattice-of-all-functions).
Take your map from $\mathcal{P}(\omega)$ into $\mathcal{L}$ (or rather the set of functions before identifying almost equal elements).
That is for $A\subseteq\omega$ define $f... | 3 | https://mathoverflow.net/users/5903 | 311938 | 135,697 |
https://mathoverflow.net/questions/311767 | 10 | Assume that $k$ is an infinite field. Let $G$ be a finite (constant) group, let $V$ be a faithful $G$-representation over $k$, $U$ a non-empty open subset of $V$ where the action is free. The map $\pi:U\to U/G$ is a $G$-torsor, and moreover, every $G$-torsor over $k$ arises as a fiber of $\pi$, and the set of $p\in (U/... | https://mathoverflow.net/users/129591 | Connecting torsors by a rational curve | Welcome new contributor. That is true, even for $G$ a geometrically reductive group scheme, not merely for finite groups. The proof is elementary, but it is a bit long. It is related to the problems of **weak approximation** and **strong approximation**, just for affine space in your original question. Strong approxima... | 6 | https://mathoverflow.net/users/13265 | 311944 | 135,700 |
https://mathoverflow.net/questions/311945 | 4 | If $R$ is a domain, and $M$ a (left) $R$-module, what are the different notions of dimension of $M$ and their respective assets, what do they measure?
I found out that if $\dim\_RM$ is the cardinal of any maximal independent subset of $M$, then it does not depend of maximal independent subset chosen if $R$ is a left-... | https://mathoverflow.net/users/18583 | Dimension of a module over a left-Ore domain | I do not know of an explicit reference, but the facts in your question follow from the exactness of Ore localization and the corresponding linear algebra facts. A possible reference for the exactness of Ore localization is Exercise 18 at the end of $\S$10 in Lam's "Lectures on Modules and Rings".
In more detail, writ... | 3 | https://mathoverflow.net/users/86006 | 311966 | 135,704 |
https://mathoverflow.net/questions/311765 | 21 | Experiments support the below identity.
>
> **Question.** Is this true? Combinatorial proof preferred if possible.
> $$\sum\_{m=0}^n\binom{n-\frac13}m\binom{n+\frac13}{n-m}(1+6m-3n)^{2n+1}
> =\left(\frac43\right)^n\frac{(3n+1)!}{n!}.$$
>
>
>
In View of MTyson's suggestion (see below), a generalized question c... | https://mathoverflow.net/users/66131 | A proof required for this identity | **"show that the a priori degree $\le n$ polynomial $\sum\_{i+j=n}{n−x\choose i}{n+x\choose j}(i−j)^k$ is only degree $k$".**
That is fairly obvious when $k<2n+1$ because it suffices to check it for integer $x\in\{-n,\dots,n\}$. We will just show that for every $a,b\ge 0$ with $a+b\le k$, the sum $\sum\_{i+j=n}{n−x\c... | 5 | https://mathoverflow.net/users/1131 | 311970 | 135,705 |
https://mathoverflow.net/questions/311887 | 12 | In their paper, Erdos and Renyi consider a random graph with a fixed number of edges, as opposed to the more modern approach of adding each edge independently with probability $p$. From what I understand (both intuitively and from searching around) their results on connectivity still hold if one use the more standard $... | https://mathoverflow.net/users/106377 | A Modern Proof of Erdos and Renyi's 1959 Random Graph Paper? | The most satisfying solution I've seen so far are [these](https://www.math.wisc.edu/~roch/teaching_files/833.f14/mdp-chap2-web.pdf) notes by Sabastian Roch (see section 2.2.3, and in particular claim 2.25). He first argues that the only components besides the giant one are isolated vertices, and he then shows that ther... | 5 | https://mathoverflow.net/users/106377 | 311989 | 135,713 |
https://mathoverflow.net/questions/311980 | 4 | Given a probability measure $\mu$ on $\mathbb{R}^n$, its Poincare constant is the least number $C$ such that:
$$
\int f^2 d\mu \leq C\int \|\nabla f\|^2 d\mu
$$
for all zero mean function $f$.
Is there a version of Cheeger's inequality that works in this setting? I would assume so but was only able to find references... | https://mathoverflow.net/users/112954 | Cheeger inequality for measures | Yes, the Cheeger's inequality is even known in the general framework of a Dirichlet space $(X,d,\mathcal{E},\mu)$, where $\mu$ is a probability measure. Indeed, assume that $\mathcal{E}$ is strictly local with a carre du champ $\Gamma$, that Lipschitz functions are in the domain of $\mathcal{E}$ and that $\sqrt{\Gamma(... | 5 | https://mathoverflow.net/users/48356 | 311993 | 135,716 |
https://mathoverflow.net/questions/312011 | 0 | Let $R$ be a Noetherian local ring with maximal ideal $I$.
Suppose we have a morphism of smooth $R$-algebras $f : A\to B$ such that its reduction modulo $I^n$
$$f\_n : A/I^n \to B/I^n$$
is an isomorphism for all $n\ge 1$. What can we say about $f$? Is $f$ smooth?
| https://mathoverflow.net/users/nan | Smooth loci and formal neighborhoods | No. Take for $R$ a discrete valuation ring with fraction field $K$, and for $f$ any non-smooth morphism of smooth $K$-algebras (viewed as a morphism of $R$-algebras).
| 3 | https://mathoverflow.net/users/7666 | 312014 | 135,721 |
https://mathoverflow.net/questions/312006 | 4 | More specifically, if $j:V\prec M$ has critical point $κ$ and for any $X\in M$ with $|X|=μ$, $|X|^M=μ$, does $κ$ necessarily have some form of $μ$-compactness? Is it related to strong compactness in any way?
The hypothesis is a weakening of supercompactness; it asserts that if there is *any* bijection from $X$ to $μ$... | https://mathoverflow.net/users/115951 | If $j:V\prec M$ has critical point $κ$ and for any $X\in M$ with $|X|=μ$, $|X|^M=μ$, what properties does $κ$ have? | Let me start by observing that this property is equivalent to a more standard property:
**Claim:** Let $M\subseteq V$ be a transitive model of $\mathrm{ZFC}$ and let $\mu \in M$ be a cardinal in $V$. The following are equivalent:
* For every $X\in M$, $|X|^V = \mu \iff |X|^M = \mu$.
* $(\mu^+)^V = (\mu^+)^M$.
**P... | 4 | https://mathoverflow.net/users/41953 | 312018 | 135,723 |
https://mathoverflow.net/questions/312022 | 4 | My question is what is the relation (if any) between the following two notions.
1. Starting from an augmented algebra $A$ over a field $k$, one way to compute the homology of $A$ is to find a projective resolution of $k$ as an $A$-module.
2. On the other hand, one could also see $A$ as a (let's say positively graded)... | https://mathoverflow.net/users/68468 | Algebras: Homology vs. Resolution as a dg-algebra | From your first description, I guess that what you call "the homology of $A$" is in fact the Hochschild homology of $A$ with constant coefficients, i.e. $HH\_\*(A;k)$. Please tell me if I got this wrong.
Then yes, it's possible to compute it from a projective resolution of $A$. Suppose that $R \xrightarrow{\sim} A$ i... | 8 | https://mathoverflow.net/users/36146 | 312025 | 135,724 |
https://mathoverflow.net/questions/311959 | 15 | Given a finite Borel measure $\mu$ on $\mathbb{S}^1 = \mathbb{R}/\mathbb{Z}$, define its Fourier coefficients by
$$ \hat\mu(n) = \int e^{2i\pi nx} d\mu(x) \qquad\forall n\in \mathbb{Z}.$$
Clearly, $(\hat\mu(n))\_n$ is bounded.
1. What sufficient conditions on a bounded sequence $(a\_n)\_n$ are known that ensure t... | https://mathoverflow.net/users/4961 | Which bounded sequence can be realized as the Fourier Series of a probability measure on the circle? | The most elegant solution exists for problem 2: the necessary and sufficient condition for $a\_n=\hat{\mu}(n)$ for a positive measure is that he sequence $(a\_n)$ is non-negative semi-definite, which means
that all Toplitz forms
$$\sum\_{i,j=0}^na\_{i-j}z\_i\overline{z}\_j\geq 0$$
for all integers $n$ and complex $z\... | 10 | https://mathoverflow.net/users/25510 | 312030 | 135,726 |
https://mathoverflow.net/questions/311915 | 9 | Consider a smooth closed curve $u\_0$ in a compact Riemannian manifold $(M,g)$. Let $u\_0$ evolve by harmonic map heat flow, $\partial\_tu=\nabla\_{\partial\_su}\partial\_su$, and call the result $u(t)$.
Since the circle is 1-dimensional, a miracle happens and we get a gradient estimate for free, so the flow exists ... | https://mathoverflow.net/users/90154 | Does harmonic map heat flow of a curve always fully converge to a geodesic? | The situation is actually quite complicated, it seems.
In the case where the target manifold is real analytic, Leon Simon's results in [*Asymptotics for a Class of Non-Linear Evolution Equations, with Applications to Geometric Problems*](https://www.jstor.org/stable/2006981?seq=1#metadata_info_tab_contents) implies ... | 10 | https://mathoverflow.net/users/3948 | 312037 | 135,730 |
https://mathoverflow.net/questions/311928 | 3 | What are the advantages of a hyperbolic program over a semi definite program? SDPs can be used to represent a wide variety of algebraic constraints. Are there constraints that can be represented in a hyperbolic program but not a semi definite program?
Is there a reference that describes an application or application... | https://mathoverflow.net/users/129665 | Advantages of hyperbolic programming over semidefinite programming? | *Disclaimer: I'm not an expert in the area, just a fellow curious.*
Update (2023): In the pre-print "[Sums of Squares Representations on Singular Loci](https://arxiv.org/abs/2303.05081)" by Ngoc Hoang Anh Mai and Victor Magron is claimed that every hyperbolic program is equivalent to a semidefinite program.
---
... | 4 | https://mathoverflow.net/users/22389 | 312039 | 135,731 |
https://mathoverflow.net/questions/312034 | 2 | So we have a 3x3 matrix and two players, a player that only puts in ones and a player that only puts in zeros. A coin flip is used to decide which player goes first. The first move is always to fill the upper left entry with the player's number, whoever won the coin flip. Then the players take turns filling in ones and... | https://mathoverflow.net/users/129718 | Matrix tic tac toe | 0 always wins (regardless of 1's strategy). This is because any configuration with (a) 0s all in one row or (b) 0s all in one column or (c) 0s all in a 2x2 minor will win.
For the case 1 goes first, let 0 play in the center square. After which up to symmetry there are only four cases that need to be considered after... | 6 | https://mathoverflow.net/users/3948 | 312041 | 135,733 |
https://mathoverflow.net/questions/312028 | 4 | Let $S$ denote the set of natural numbers $m$ with the property that for all prime powers $p^k || m$ we have $k \equiv 1 \pmod{2}$.
What is the asymptotic density of $S$?
Note that $S$ contains all prime numbers and more generally, all square-free numbers, so that $\liminf\_{X \rightarrow \infty} \frac{\# (S \cap ... | https://mathoverflow.net/users/10898 | Numbers whose prime factors all have odd exponent | The Dirichlet series $L (s)=\sum\_{n \in S} n^{-s} $ has the Euler product expression
$$L (s) = \prod\_p \frac{1+p^{-s}-p^{-2s}}{1-p^{-2s}}.$$
So $S $ has analytic density
$$\mu (S) = \operatorname{Res}\_{s=1} L(s) = \lim\_{s \to 1} \frac{L(s)}{\zeta(s)} = \prod\_p \frac{1+1/p-1/p^2}{1+1/p}$$
as can also be seen by an ... | 8 | https://mathoverflow.net/users/6506 | 312045 | 135,735 |
https://mathoverflow.net/questions/311978 | 6 | I'm having trouble understanding why the "essential image" is defined the way it is.
[The nlab article](https://ncatlab.org/nlab/show/essential+image) gives the following definition:
>
> (A concrete realization of) the essential image of a functor $F: A\to B$ between categories or $n$-categories is the smallest r... | https://mathoverflow.net/users/95265 | Why must the essential image break the principle of equivalence? | There are really two different issues here. In this answer I'm going to deal with the one which I think is mainly a distraction, namely the difference between giving a concrete construction and characterizing something only up to isomorphism.
As the nLab page says, one thing that violates the principle of equivalence... | 11 | https://mathoverflow.net/users/49 | 312053 | 135,737 |
https://mathoverflow.net/questions/312047 | 5 | This is probably a simple-minded question, but I haven't been able to prove it or find a counterexample. This [old question](https://mathoverflow.net/questions/26723/does-smooth-target-space-and-smooth-fibers-imply-smooth-total-space?rq=1) seems to dance around my question, but I don't think any of the answers address ... | https://mathoverflow.net/users/62154 | smooth equidimensional fibers over a smooth base | The flatness statement you would like follows from Theorem 3.3.27 of Schoutens's [book](https://www.springer.com/us/book/9783642133671) on ultraproducts.
| 4 | https://mathoverflow.net/users/10076 | 312054 | 135,738 |
https://mathoverflow.net/questions/312023 | 3 | At the right column of the page 654 of the paper,
[R. Palais, The Visualization of Mathematics: Towards a mathematical Exploratorium, Notice AMS](http://www.ams.org/notices/199906/fea-palais.pdf) it is written "There can be no eversion of a circle (or any odd dimensional spheres)".
In that paper the eversion is defi... | https://mathoverflow.net/users/36688 | A question on eversion of (odd) spheres | Let me summarize the comments. An "eversion" of $S^n$ is a regular homotopy between, on the one part the "identity" embedding $S^n \to \mathbb{R}^{n+1}$, and on the other part an embedding $S^n \to \mathbb{R}^{n+1}$ that turns the sphere inside out.
For an even sphere, the antipodal map turns the sphere inside out (A... | 7 | https://mathoverflow.net/users/36146 | 312056 | 135,739 |
https://mathoverflow.net/questions/301119 | 12 | I am studying Peter Scholze's paper [$p$-adic Hodge theory for rigid-analytic varieties](http://www.math.uni-bonn.de/people/scholze/pAdicHodgeTheory.pdf) and I am confused by the following.
Let $X$ be a finite type scheme over $\mathbb{C}\_p$ (proper and smooth if you want) and let $X\_{\mathrm{ét}}$ be the usual éta... | https://mathoverflow.net/users/124880 | (pro)Étale cohomology of adic spaces and inverse limit | This problem came up in a reading group yesterday, and the consensus was that the problem is exactly the one that user45878 pointed out in the comments. Thanks to Bogdan Zavyalov for explaining this issue to me and providing the example below. Let's see why these are different with an example:
Consider the scheme $X ... | 4 | https://mathoverflow.net/users/56878 | 312058 | 135,740 |
https://mathoverflow.net/questions/233754 | 7 | I am looking for any reference dealing explicitly with Eisenstein series on Siegel space (the simplest case of $\rm{SP}\_4$ is fine). Anything would be welcome, but in particular I'm interested in the Fourier-expansions and constant terms. Another preference is that reference be written in the language of the reals ins... | https://mathoverflow.net/users/4181 | Eisenstein Series on Siegel Space | For some reason I did just now remember that an appendix in R. Langlands' SLN 544 does show a classical, non-adelic method to treat (non-cuspidal data) Eisenstein series for $GL\_n$ and $Sp\_n$. The actual immediate point seemed/seems to be to show how to do this over number fields, paying attention to class numbers gr... | 1 | https://mathoverflow.net/users/15629 | 312062 | 135,742 |
https://mathoverflow.net/questions/312063 | 3 | I understand that there are sets of 7 points on a circle that can be fully
shattered using triangles.But, it is not clear to me why it cannot shatter 8 points.
Is there any intuitive way of arriving at the conclusion that it can't shatter 8 points? Is there a simple explanation without using much advanced geometry to... | https://mathoverflow.net/users/129729 | Why the VC dimension of triangles in 2D space is not greater than 7? | Let $x\_1, \dots, x\_8$ be $8$ points on a circle in this order. Then there is no triangle containing exactly $x\_2, x\_4, x\_6, x\_8$ in its interior and the other four points in its exterior. Intuitively, it would have to intersect the circle in at least $8$ points, but any triangle intersects a circle in at most $6$... | 4 | https://mathoverflow.net/users/24076 | 312066 | 135,743 |
https://mathoverflow.net/questions/312070 | 0 | It is quite general and elementary question.
Is it possible that some holomorphic functions $f\_1,\cdots,f\_m $ on a region $\Omega $ of $\mathbb C$ satisfies:
Whenever $(f\_1(z), \cdots, f\_m (z)) $ is a zero of some polynomial $p \in \mathbb Q [x\_1, \cdots, x\_m]$ for some $z \in \Omega $, then $p (f\_1,\cdots,f... | https://mathoverflow.net/users/123157 | Algebraic independence of certain values implies algebraic independence of functions? | EDITED:
No, it is not possible if $f\_1, \ldots, f\_m$ are not all constant.
Suppose wlog $f\_1$ is not constant. I'll ignore $f\_2, \ldots, f\_m$ and consider polynomials $p(f\_1(z))$. For convenience I'll omit the subscript and
call this $p(f(z))$. $f(\Omega)$ is an open subset of $\mathbb C$, so it contains
$\a... | 4 | https://mathoverflow.net/users/13650 | 312073 | 135,746 |
https://mathoverflow.net/questions/312078 | 5 | I'm studying this paper: <http://matwbn.icm.edu.pl/ksiazki/sm/sm73/sm7313.pdf>
At the top of page 36, it states the following Proposition:
>
> Let $S$ be a compact and $\mu$ a regular Borel measure on $S$ with total variation 1. If for every partition of $S$ into two Borel subsets the measure of the smaller one i... | https://mathoverflow.net/users/100740 | Regular Borel measures and the measure of a singleton | Let $\mu$ be a (positive, probability) measure satisfying the hypothesis. Note that there cannot be any Borel set $A$ with $1/3 \le \mu(A) \le 2/3$, since then in the partition $S = A \cup A^c$, the smaller set would have measure at least $1/3$. Thus it suffices to show there is an $x\_0$ with $\mu(\{x\_0\}) \ge 1/3$.
... | 8 | https://mathoverflow.net/users/4832 | 312081 | 135,749 |
https://mathoverflow.net/questions/65452 | 2 | Hi everyone,
I am looking for a way of simulating correlation matrices of fixed dimension in (at least) two ways.
First, I would like to determine the "uniform" distribution over the "correlation matrices space" and being able to sample from it (even defining what "uniform" means is not such an easy task !!!).
S... | https://mathoverflow.net/users/2642 | Uniform correlation matrix sampling and not so uniform laws | Even if this answer doesn't give a satisfying answer to the second point of my original question, [I found an article that answers exactly to the first point](https://arxiv.org/abs/1809.00351) untitled "A fast Metropolis-Hastings method for generating random correlation matrices" from Irene Córdoba, Gherardo Varando, C... | 0 | https://mathoverflow.net/users/2642 | 312085 | 135,750 |
https://mathoverflow.net/questions/312024 | 2 | Recently, I read a notes about Sakellaridis and Venkatesh conjecture. It mentions a technique called "unfolding" and gives an example:
>
> Let X=A\G, X'=N\G, where G=PGL(2), A={
> $\left[\begin{array}{cc}
> \* & 0 \\
> 0 & 1
> \end{array}\right]$ },
> N={
> $\left[\begin{array}{cc}
> 1 & \* \\
> 0 & 1
> \end{... | https://mathoverflow.net/users/121163 | $L^2(X) \cong L^2(X',\xi)$ | It is not difficult to see why the maps are inverse to one another, but I am not sure why it is called unfolding. Let $\phi\in L^2(X)$, i.e. $\phi$ is left $A$-invariant. We want to show that
$$\int\_{A}\int\_{N} \phi(nag)\overline{\xi(n)}dnd^\times a=\phi(g).$$
Writing $n=\begin{pmatrix}1&x\\&1\end{pmatrix}$ and $a=\b... | 1 | https://mathoverflow.net/users/36735 | 312091 | 135,753 |
https://mathoverflow.net/questions/312055 | 2 | Consider two morphisms $T\to Z$ and $Y\to Z$ of varieties over an algebraically closed field $k$ where $Z$ is an affine space. If $Y\to Z$ is flat, is it always true that the fiber product of $T\times\_Z Y$ is a complete intersection in $T\times\_k Y$?
The motivation comes from an argument of Knop in his paper [On th... | https://mathoverflow.net/users/62154 | Complete intersection argument | Let $Z=\mathbb{A}^n$. The map $T\times \_ZY\rightarrow T$ is the pull back over $T$ of $Y\rightarrow Z$, hence it is flat, of relative dimension $\dim(Y)-n$. Therefore $$\dim(T\times \_ZY)=\dim(T)+(\dim(Y)-n)=\dim(T\times Y)-n\, .$$ On the other hand,
$T\times \_ZY$ is defined in $T\times Y$ by the $n$ equations $f\_i... | 1 | https://mathoverflow.net/users/40297 | 312092 | 135,754 |
https://mathoverflow.net/questions/312106 | 0 | If a commutative ring with unity has finite Krull dimension, then it satisfies a.c.c. and d.c.c. on prime ideals. The converse is not true in general, as can be seen from here [An infinite dimensional local domain whose chains of primes are finite](https://mathoverflow.net/questions/311907/an-infinite-dimensional-local... | https://mathoverflow.net/users/127118 | Valuation ring satisfying either a.c.c. or d.c.c. on prime ideals | For any totally ordered set $W, R=k[x\_i/x\_j^n: i,j \in W, i \lt j, n \ge 0]$ is a valuation ring with value group $\oplus\_{i \in W} \mathbb{Z}$ and Spec($R$) = {initial segments of $W$}. Thus $W$=negative integers gives a counterexample to (1) and $W$=positive integers gives a counterexample to (2).;
| 2 | https://mathoverflow.net/users/59248 | 312109 | 135,758 |
https://mathoverflow.net/questions/312108 | 12 | Let $k$ be an algebraically closed field.
By a finitely generated subfield of $k$ I mean a subfield $k\_0\subset k$ that is finitely generated over the prime subfield of $k$ (that is, over $\mathbb Q$ or $\mathbb F\_p$).
I need a reference or a proof for the following (well-known? evident?) proposition:
>
> **Pro... | https://mathoverflow.net/users/4149 | Defining abstract varieties and their morphisms over a finitely generated subfield of the base field | This is treated (in much greater generality) in EGA IV$\_3$, **Théorème 8.8.2**. The existence of $X\_0$ and $Y\_0$ follows from part (ii), whereas the existence of $f\_0$ is part (i).
**References.**
[EGA IV$\_3$] A. Grothendieck, [*Éléments de géométrie algébrique. IV: Étude locale des schémas et des morphismes d... | 17 | https://mathoverflow.net/users/82179 | 312111 | 135,759 |
https://mathoverflow.net/questions/312118 | 17 | Disclaimer: I wasn't sure if this was an appropriate question for MathOverflow, and so I've also asked this on StackExchange.
There appears to be a discrepancy in the literature regarding the definition of a localisation of a category. Let $\mathcal{C}$ be a category and let $S$ be a class of morphisms. The classica... | https://mathoverflow.net/users/115048 | What is the correct definition of localisation of a category? | Actually, *both* of these definitions look weird to me.
I would say there are two natural ways to define the localization $C[S^{-1}]$ by a universal property, as follows. For any category $D$, let ${\rm Fun}\_S(C,D)$ denote the full subcategory of the functor category ${\rm Fun}(C,D)$ spanned by those functors that s... | 28 | https://mathoverflow.net/users/49 | 312123 | 135,760 |
https://mathoverflow.net/questions/312120 | 1 | Let $(\mu\_{n})$ sequence of probability measures of $\mathbb{R}^{d}$ converging to the prob measure $\mu$.
Then by definition we know that $\int f d\mu\_{n} \longrightarrow \int f d\mu $ for f continuous and bounded function.
I was wondering if I can write the formula above as $\int\int f(x-y) d\mu\_{n}(x)d\mu\_n(y)... | https://mathoverflow.net/users/128523 | A question on weak convergence of probability measures | $\newcommand{\eD}{\overset{\text{D}}\to}$
Let $X\_n$ and $X$ be any random vectors with distributions $\mu\_n$ and $\mu$, respectively. Let $Y\_n$ be an independent copy of $X\_n$, for each $n$, and let $Y$ be an independent copy of $X$. The questions can then be restated as follows:
>
> Q1: Does $X\_n\eD X$ impl... | 1 | https://mathoverflow.net/users/36721 | 312124 | 135,761 |
https://mathoverflow.net/questions/311850 | 3 | My question is as the title, i.e.,
Is there a metaLindelof nonLindelof space which has a dense hereditarily Lindelof subspace?
The question is related to the following result:
Every separable metaLindelof space is Lindelof.
Thank you!
| https://mathoverflow.net/users/129635 | Is there a metaLindelof nonLindelof space which has a dense hereditarily Lindelof subspace? | Suppose $X$ is metaLindelöf, and $A \subseteq X$ is hereditarily Lindelöf and dense.
Let $\mathcal{U}$ be an open cover of $X$ and let $\mathcal{V}$ be a point-countable refinement of $\mathcal{U}$. By a standard fact on covers, there is a discrete subset $D \subseteq A$ such that $\operatorname{st}(D,\mathcal{V}\_A)... | 3 | https://mathoverflow.net/users/2060 | 312127 | 135,762 |
https://mathoverflow.net/questions/287394 | 7 | Let $(V,0)\subset (\mathbb{C}^n,0)$, $n\geq 3$ a (germ of) hypersurface given by $V =\{f=0\}$, $f$ a germ of holomorphic function. A (germ of) holomorphic vector field $X$ on $(\mathbb{C}^n,0)$ is "tangent to $V$" if $X(f) = {\rm d}f(X)$ belongs to the ideal generated by $f$ in $\mathcal{O}\_{\mathbb{C}^n,0}$.
My qu... | https://mathoverflow.net/users/82672 | Holomorphic vector fields tangent to a hypersuface singularity | I think that the answer to this question is no for $\mathbb C^3$, for
$$f=zy(z-y)(z-xy).$$
I'll assume that $v$ is holomorphic and tangent to $f=0$ in a small neighbourhood of $(0,0,0)$.
*Proof.* Suppose by contradiction that $v$ is a holomorphic vector field tangent to $f=0$ near $(0,0,0)$ with an isolated zero at ... | 2 | https://mathoverflow.net/users/943 | 312137 | 135,767 |
https://mathoverflow.net/questions/311963 | 2 | I already created [this post on Math Stack Exchange](https://math.stackexchange.com/questions/2940155/local-factors-determine-weil-representations-proof-of-the-cyclic-case) but I was not so sure if this question fits better here. If it is not, I want to apologize in advance and feel free to delete my post.
I want to ... | https://mathoverflow.net/users/115902 | Local factors determine Weil representations - proof of the cyclic case | **Why is the map ${\rm Gal}(FL/L)\to {\rm Gal}(F/K)$ injective?**
Elements of ${\rm Gal}(FL/L)$ are automorphisms of $FL$ that act trivially on $L$. To be in the kernel of the above map means to also act trivially on $F$. A field automorphism that acts trivially on $F$ and $L$ acts trivially on $FL$ (by definition of... | 3 | https://mathoverflow.net/users/35416 | 312139 | 135,769 |
https://mathoverflow.net/questions/311889 | 15 | Is there an example of a bounded operator $T\in\mathcal{B}(H)$, where $H$ is a separable complex Hilbert space, such that no restriction to an infinite dimensional closed subspace is multiple of identity plus compact?
Edit: What I mean by "restriction" is $T\_{|M}:M\to H$, where $M$ is an infinite dimensional subspa... | https://mathoverflow.net/users/129564 | Multiple of identity plus compact | OK, let me try too. It is going to be a somewhat long story. WLOG, $\|T\|\le 1$.
**Step 1:** It is enough to show that for every finite-dimensional subspace $E$ and every $\delta>0$, there exists a unit vector $v\in E^\perp$ and $z\in\mathbb C$ such that $\|Tv-zv\|\le\delta$.
**Proof:** Suppose that the claim holds... | 10 | https://mathoverflow.net/users/1131 | 312143 | 135,771 |
https://mathoverflow.net/questions/312105 | 3 | I have a question regarding Goormaghtigh conjecture on the Diophantine equation
$$\frac{x^m-1}{x-1}=\frac{y^n-1}{y-1}.$$
Suppose that a positive integer $N$ is given. How many integer solutions are there to the equation
$$\frac{x^m-1}{x-1}=N=\frac{y^n-1}{y-1},$$
with $x$ and $y$ prime powers?
Observe that I am not ... | https://mathoverflow.net/users/45242 | A question regarding Goormaghtigh conjecture | Some simple observations, which are independent of x or y being prime powers.
m must be less than $\log N$. If there is a solution, $N$ has to factor into (about) d(m) factors which are values of cyclotomic polynomials of indices $c$ dividing m, and thus the sizes of the factors are close to $x^{\phi(c)}$ in size, so... | 2 | https://mathoverflow.net/users/3402 | 312146 | 135,772 |
https://mathoverflow.net/questions/312131 | 5 | Let $K$ be a number field of degree $n$ over the rationals. Under what conditions does there exist a non-rational algebraic integer $\alpha $ in $K$ such that the discriminant of $\alpha $ divides the norm of $\alpha$?
This question was first asked on Math StackExchange, Question [2923849](https://math.stackexchange... | https://mathoverflow.net/users/17053 | Does there always exist a non-rational algebraic integer in a number field whose discriminant divides its norm? | Consider the cyclotomic field $K=\mathbb{Q}(\zeta\_5)$.
**Proposition.** There is no $\alpha \in \mathcal{O}\_K$ of degree 4 such that $D(\alpha)$ divides $N(\alpha)$.
**Proof.** Assume such an $\alpha$ exists. Since the discriminant of $K$ is $\Delta\_K=5^3$, we must have $5^3|D(\alpha)$ and thus $5^3|N(\alpha)$. ... | 6 | https://mathoverflow.net/users/6506 | 312154 | 135,775 |
https://mathoverflow.net/questions/312133 | 4 | I'm looking for the name and some references for the proof of the inequality below. I founded that is due to T. Carleman but no reference was given.
Let $f(z)$ be an analytic function on a subdomain $D$ of $\mathbb{C}$ containing a segment $I$ of the real axis.
We assume that $|f(z)|<M$ in $D$ and $|f(x)|<m$ in $I$.
... | https://mathoverflow.net/users/124904 | An inequality of T. Carleman | This is the **two-constants theorem** :
Let $D$ be a domain of $\mathbb{C}$ with (non-polar) boundary $\partial D$, and let $B$ a Borel subset of $\partial D$. If $u$ is subharmonic on $D$ and satsifies
$$
u(z)\leq M,~z\in D\quad\text{ and }\quad\limsup\_{z\to\zeta}u(z)\leq m,~\zeta\in B,
$$
then
$$
u(z)\leq m\omega\... | 3 | https://mathoverflow.net/users/89429 | 312155 | 135,776 |
https://mathoverflow.net/questions/312150 | 11 | Let $\Phi$ be a universal Turing machine and let $S$ be the set on which it halts. I’m curious about if its decidable to check if a number is close to $S$. There are two notions of distance that come to mind: the additive distance and the Hamming distance.
The additive distance, $d\_+(x,S)$ is the smallest number $n... | https://mathoverflow.net/users/45118 | Is being close to a Halting set computable? | Let me make James's answer a bit more explicit to show how the answer depends on the coding of Turing machines. I will work with the additive distance.
As James pointed out, we can make sure that the halting set is very dense, for instance by having every even number encode a fixed machine that halts. This way the qu... | 10 | https://mathoverflow.net/users/1176 | 312166 | 135,778 |
https://mathoverflow.net/questions/312077 | 1 | A prominent idea found e.g. in Koellner (<http://logic.harvard.edu/koellner/ORP_final.pdf>) is that reflection principles in set theory are motivated by the idea that proper classes are so large as to be "ineffable" or unable to be characterized by any internal structural property of the universe (of which they are pro... | https://mathoverflow.net/users/116705 | Definition of ineffability behind reflection principles in set theory | Referring to proper classes by means of constants clearly doesn’t use the “internal properties” of the universe of *sets*. You’re literally reaching outside the universe and adding an artificial means of referring to a class.
Regarding (1), I think the intuition is that classes are not objects of the model, but rathe... | 5 | https://mathoverflow.net/users/11145 | 312170 | 135,780 |
https://mathoverflow.net/questions/312096 | 5 | Let $X$ be a fixed curve (e.g. a Noetherian, projective scheme of dimension 1, of finite type over an algebraically closed field $k$) and let $S$ be an arbitrary parameter scheme over $k$. Let $D \subset X \times S$ be a flat family over $S$ of subschemes of $X$, of relative dimension 0 and degree $d$, with ideal sheaf... | https://mathoverflow.net/users/91935 | Multiple of a flat family of subschemes is flat | I am just posting my comment as an answer. No, that is not true. Let $X$ be $\text{Spec}\ k[t,u]/\langle tu\rangle $. Let $S$ be $\text{Spec}\ k[ϵ]/\langle ϵ^2 \rangle.$ Let $I$ be $\langle t,u−ϵ\rangle.$ Then $\mathcal{O}\_{X×S}/I^2$ is a direct sum of two copies of $\mathcal{O}\_S$ (generated by $1$ and $u$) plus one... | 5 | https://mathoverflow.net/users/13265 | 312172 | 135,781 |
https://mathoverflow.net/questions/312134 | 1 | I had trouble in finding a closed form solution for the following series, so now I am trying to find a good approximation for it. The $\sqrt{i^2 + j^2}$ in the exponent comes from distances on the euclidean grid from the origin.
$x = \sum\_{i,j} e^{-a\sqrt{i^2 + j^2}}$
where $i,j$ range from $0$ to infinity.
It ... | https://mathoverflow.net/users/86078 | Approximate the following series on the euclidean grid | You're surely right that there cannot be a "closed form" for such a series;
but it can still be approximated to any desired precision.
The defining sum
$$
x = x(a)
= \sum\_{i=0}^\infty \sum\_{j=0}^\infty \exp\Bigl(-a \sqrt{i^2+j^2}\Bigr)
$$
requires only $O(N^2/a)$ terms to approximate to within $\exp(-N)$,
so it conv... | 6 | https://mathoverflow.net/users/14830 | 312183 | 135,785 |
https://mathoverflow.net/questions/250734 | 7 | Let $\mathcal P(X)$ denote the space of all probability measure defined on a measurable space $X$. We canonically endow the former with its own measurability structure, generated by evaluation maps. Let $P \subseteq \mathcal P([0,1])$ be a measurable subset of probability measures, and let $\hat p\notin P$ be such that... | https://mathoverflow.net/users/11768 | Convex representation of a measure | The answer is no.
I understand the condition on $p\_f$ as belonging to the closure $\bar{P}$ for the weak-$\star$ topologie. And one can then ask whether $[\int p d\nu(p),\nu\in \mathcal{P}(\mathcal{P}([0,1]))]$ contain this closure. Consider the following counter example $$P=[\lambda \delta\_0+(1-\lambda)\delta\_1:0... | 3 | https://mathoverflow.net/users/99045 | 312184 | 135,786 |
https://mathoverflow.net/questions/312177 | 20 | A [hypergraph](https://en.wikipedia.org/wiki/Hypergraph) is a pair $H=(V,E)$ where $V\neq \emptyset$ is a set and $E\subseteq{\cal P}(V)$ is a collection of subsets of $V$. We say two hypergraphs $H\_i=(V\_i, E\_i)$ for $i=1,2$ are *isomorphic* if there is a bijection $f:V\_1\to V\_2$ such that $f(e\_1) \in E\_2$ for a... | https://mathoverflow.net/users/8628 | Does the hypergraph of subgroups determine a group? | In the comments to the question, I notice something which might be an error, or at least is an incomplete response. It is pointed out in the comments that there exist nonisomorphic groups with isomorphic subgroup lattices. While true, that fact doesn't answer this question, since it is possible to have isomorphic subgr... | 20 | https://mathoverflow.net/users/75735 | 312189 | 135,787 |
https://mathoverflow.net/questions/312069 | 21 | It is well known now that Yitang Zhang's work on Jacobian conjecture collapsed because his advisor's work earlier contains unjustified claims. I am wondering what specifically is unclear about [his paper](https://eudml.org/doc/152524). From fellow researchers I heard his paper is unreadable, and the last claim in his p... | https://mathoverflow.net/users/128551 | Does T-T Moh's paper really contain a gap? | Yes, there appears to be a follow-up work by Yansong Xu on the (99,66)-case: [Intersection Numbers and the Jacobian Conjecture](https://arxiv.org/abs/1604.07683v2), in turn followed by [The Jacobian Conjecture: Approximate roots and intersection numbers](https://arxiv.org/abs/1708.09367) by Guccione-Guccione-Horruitine... | 7 | https://mathoverflow.net/users/1849 | 312191 | 135,788 |
https://mathoverflow.net/questions/312210 | 0 | I know that there is a formula where even 0 is treated as significant, but is there a formula where 1 and 0 cannot be parts?
Edit:The formula that I was referencing was (Excuse the presentation) $\frac{(n-1)!}{(k-1)!(n-k)!}$
This one discards the 0, but what I was looking for one that would discard 1 as well.
Also, ... | https://mathoverflow.net/users/129795 | Number of compositions of n into k parts (where 0 and 1 are not allowed as parts) | Here how to obtain the generating function:
Give a compsotion of $n$ with $k$ summands the weight $x^n y^k$. The compositions of $n$ into $k$ parts with terms $\geq 2$ are then obtained as $Sequ(\mathbb{N}\_{\geq 2})$, see for example 2.2.1. of <https://arxiv.org/pdf/1409.2562.pdf>.
Thus the generating function is $\... | 1 | https://mathoverflow.net/users/61949 | 312212 | 135,791 |
https://mathoverflow.net/questions/312112 | 3 | Let $\mathcal G$ be an étale groupoid with a locally compact, Hausdorff unit space $\mathcal G^{(0)}$. If $U⊆\mathcal G$ is an open subset, which is Hausdorff in the induced topology, and if $f$ is any member of $C\_c(U)$ (continuous, compactly supported, complex valued functions on $U$), one may view $f$ as a function... | https://mathoverflow.net/users/97532 | Continuity of functions on étale groupoids | Here is a positive answer to my own question based on Ben's comment above. For each $f$ in $C\_c(\mathcal G)$, and for
each $x$ in $\mathcal G^{(0)}$, define $$ \Phi(f)|\_x = \sum\_{\gamma\in s^{-1}(x)} f(\gamma), $$ where $s$ denotes the source map. I
next claim that $\Phi(f)$ is continuous on $\mathcal G^{(0)}$.
By... | 1 | https://mathoverflow.net/users/97532 | 312214 | 135,793 |
https://mathoverflow.net/questions/311892 | 12 | As the question title asks for, how do others "visualize" the finiteness of class number with algebro-geometric insight? I just think of it as a result in algebraic number theory and not one in algebraic geometry. Bonus points for pictures.
| https://mathoverflow.net/users/126532 | How to visualize finiteness of class number? | The first question to ask is how you visualize a number field, and the second is how you visualize a class group. There's more than one way to do each of these.
Schemes
-------
What I think your question intended to refer to is the viewpoint of schemes. Let $K$ be a degree $d$ number field with ring of integers $\m... | 9 | https://mathoverflow.net/users/129803 | 312217 | 135,794 |
https://mathoverflow.net/questions/312201 | 7 | Let $f\in \mathbb{R}[x\_1,x\_2,x\_3,x\_4]$ defined by
$$f\_a(x\_1,x\_2,x\_3,x\_4)=\prod\_{1\leqslant i<j\leqslant4}(x\_i-x\_j)^{2a\_{ij}}$$
where $a=(a\_{12},a\_{13},a\_{14},a\_{23},a\_{24},a\_{34})\in \mathbb{N}^6$.
Define a $4\times4$ matrix $A\_f$ as follow:
$$A\_{f\_a}=\begin{bmatrix}
L(1) & L(\frac{x\_2}{x\_1}) &... | https://mathoverflow.net/users/58096 | The determinant of a $4\times4$ matrix associated to some specific polynomial as follow | For a given monomial $Y=\frac{x\_{i\_1}\cdots x\_{i\_k}}{x\_{j\_1}\cdots x\_{j\_k}}$ the coefficient $L(Y)$ multiplied by the constant $(-1)^{\sum\_{i<j} a\_{ij}}$ equals $$[Y]\prod\_{i,j}(1-x\_i/x\_j)^{a\_{ij}}=\int Y^{-1}d\mu,$$
where $d\mu$ is the measure on the $4$-dimensional torus $\mathbb{T}^4=\{(x\_1,x\_2,x\_3,... | 10 | https://mathoverflow.net/users/4312 | 312218 | 135,795 |
https://mathoverflow.net/questions/309915 | 7 | By a theorem of Mattuck [*Abelian Varieties over $p$-Adic Ground Fields*, Annals of Mathematics, Second Series, Vol. 62, No. 1 (Jul., 1955), pp. 92-119], for an Abelian variety $A$ of dimension $g$ over a $p$-adic local field $K$ with ring of integers $O\_K$, there is an exact sequence $0 \to O\_K^g \to A(K) \to \{\mat... | https://mathoverflow.net/users/nan | analogue of Theorem of Mattuck for Abelian varieties over $\mathbf{F}_q(\!(t)\!)$ | By [Serre, *Lie Algebras and Lie Groups*], p. 116, Theorem and p. 118, Corollary 2, there is an open subgroup of $A(K)$ which is a pro-$p$-group. Therefore, it suffices to show that the torsion subgroup of $A(K)$ is finite, because then there is an exact sequence (algebraically and topologically) $$0 \to P \to A(K) \to... | 4 | https://mathoverflow.net/users/nan | 312221 | 135,797 |
https://mathoverflow.net/questions/312215 | 21 | Gross and Zagier prove the following fantastic result in their paper "Singular Moduli":
Let $R$ be a discrete valuation ring over $\mathbb Z\_p$ with uniformizer $\pi$ such that $k = R/\pi$ is algebraically closed and normalize the valuation so that $v(\pi) = 1$.
Now, let $E\_1,E\_2$ be two distinct (ie, non isomor... | https://mathoverflow.net/users/58001 | The valuation of j-functions vs number of isomorphisms for an elliptic curve | Yes, let's use the fact that the moduli stack of elliptic curves is etale-locally a scheme. We could also use the formal deformation space of the elliptic curve mod $\pi$ (of course we may assume $E\_1 \cong E\_2 \mod \pi$). We pick an etale-local model of the moduli stack of pairs of elliptic curves that includes $(E\... | 17 | https://mathoverflow.net/users/18060 | 312238 | 135,802 |
https://mathoverflow.net/questions/312236 | 5 | I am aware of the theorem that $p\_{n+1}- p\_n \leq n^{0.525}$ which is true for all sufficiently large numbers due to Baker, but if i want to make the implicit "for all sufficiently large numbers" explicit, is it known that $p\_{n+1}-p\_n \leq c n^{\alpha}$ for all $n \geq 1$ and for small $c$, lets say $c \leq 2$ and... | https://mathoverflow.net/users/95470 | consecutive prime gaps and explicit bound | The result you quote is due to Baker-Harman-Pintz (2000). I am not aware of any concrete effective version of this result, but if you increase the exponent $0.525$ to $2/3$, then such a variant is available by the [work of Dudek](https://arxiv.org/abs/1401.4233). See also my response to [this MO question](https://matho... | 8 | https://mathoverflow.net/users/11919 | 312239 | 135,803 |
https://mathoverflow.net/questions/309643 | 6 | The characteristic varieties $V\_d^i(X)$ of a (sufficiently nice) space $X$ are the cohomology jumping loci for 1-dimensional (complex) local systems on $X$. Assume that $H\_1(X;\mathbb{Z}) \cong \mathbb{Z}^n$ for some $n > 0$. Then
$$V^i\_d(X) = \{\ \rho\in \text{Hom}(\pi\_1(X), \mathbb{C}^\*)\ \ |\ \ \text{dim}\ H^... | https://mathoverflow.net/users/98320 | The characteristic varieties of the complement of the braid arrangement | Let $T$ be an irreducible component of $V^1\_d(X\_k)$ with $d\ge 2$. If $T$ contains $\mathbb{1}$, then $T=\{\mathbb{1}\}$. It is probably the case that all components of $V^1\_d(X\_k)$ pass through the identity, but I don't think that has been established, except for small values of $k$.
| 4 | https://mathoverflow.net/users/17846 | 312243 | 135,805 |
https://mathoverflow.net/questions/312248 | 4 | Denote an integer partition of $n$ by $\lambda=(\lambda\_1\geq\lambda\_2\geq\dots\geq\lambda\_k)$ where $\lambda\_k>0$. Also recall the $q$-analogues of integer $n$ given by $[n]\_q=\frac{1-q^n}{1-q}$. Further, let
$$[n]\_q!=[n]\_q[n-1]\_q\cdots[2]\_q[1]\_q \qquad \text{and} \qquad [0]\_q!=1.$$
If $\lambda=(\lambda\_1... | https://mathoverflow.net/users/66131 | Partitions and $q$-integers | As expected, it is not about $q$-analogs: the multisets $\cup\_{\lambda\vdash n} \{\lambda\_1,\lambda\_2,\dots\}$ and $\cup\_{\lambda\vdash n}\cup\_i \{1,2,\dots,\lambda\_i-\lambda\_{i+1}\}$ coincide. To see this compute the multiplicity of a given integer $m$ in them both. For the first, it equals $p(n-m)+p(n-2m)+p(n-... | 4 | https://mathoverflow.net/users/4312 | 312251 | 135,810 |
https://mathoverflow.net/questions/312259 | 13 | This is of course a very-well known problem, but still let me ask the questions my way. Let $L(s)$ be a "motivic" $L$-function, whatever that means: in particular, it has an Euler product (including at "bad" primes), and a (possibly conjectural) functional equation of standard type with an arithmetic conductor $N$ occu... | https://mathoverflow.net/users/81776 | Euler factors of L-function at bad primes | 1) True.
2) True.
3) True.
4) They are equal in the case of "tame ramification", e.g. if the degree of $K(\mu\_p)$ over $K$, $K$ the coefficient field of the motive, is greater than $d$.
One just has to recall that for $V$ the $\ell$-adic Galois representation associated to the motive, $\dim V$ is the degree o... | 16 | https://mathoverflow.net/users/18060 | 312260 | 135,813 |
https://mathoverflow.net/questions/312265 | 17 | The following is a conjecture due to Littlewood.
>
> For any set of distinct non-zero integers $n\_1,\ldots,n\_k$ the inequality
> $$\int\_0^{2\pi}|1+e^{in\_1x}+\cdots+e^{in\_kx}| \, dx\geq C\log k$$ holds.
>
>
>
Has this proven to be true or false?
**Update 1.** An extension to finite fields can be found [... | https://mathoverflow.net/users/48438 | A conjecture of Littlewood | This was proved by S. Konyagin [7] and independently by McGehee, Pigno, and Smith [13] in 1981. A short proof is available in [5].
[7] S.V. Konjagin, On a problem of Littlewood, Mathematics of the USSR, Izvestia, 18 (1981), 205–225. <http://mi.mathnet.ru/eng/izv1556>
[5] R.A. DeVore and G.G. Lorentz, Constructive ... | 25 | https://mathoverflow.net/users/129840 | 312267 | 135,815 |
https://mathoverflow.net/questions/312268 | 4 | Let $X$ be a connected separable metrizable topological space. Call it a cut-point space if $X\setminus \{x\}$ is disconnected for every $x\in X$. Then does $X$ embed into the plane?
My thoughts:
(0) It is not difficult to see that $X$ must have dimension $1$, and therefore embeds into $\mathbb R ^3$.
(1) Every s... | https://mathoverflow.net/users/95718 | Does every cut-point space embed into the plane? | Here's an example that contradicts (0).
Take $n\in\mathbb{N}$ and let $f:[0,1]\to[0,1]^n$ be a function with the following property: whenever $F\subseteq[0,1]^{n+1}$ is closed and $\pi\_1[F]$ is uncountable then the graph of $f$ intersects $F$ ($\pi\_1$ is the projection onto the first coordinate). Such an $f$ is con... | 3 | https://mathoverflow.net/users/5903 | 312282 | 135,818 |
https://mathoverflow.net/questions/311544 | 0 | Let $u\in C^\infty(\mathbb{R}^n\times\mathbb{R}^n)$ be symmetric and of strictly positive type on some hypersurface $S \subset \mathbb{R}^n$ diffeomorphic to $\{0\}\times\mathbb{R}^{n-1}$. This means there is some $c>0$, s.t.
$$\forall N>0, \enspace x\_1,...,x\_N \in S,\enspace v\in\mathbb{R}^N: \quad \sum\_{i=1}^N \su... | https://mathoverflow.net/users/128963 | Strict positive type function on hypersurface also of positive type in neighborhood? | ($\*$) yields positivity of the matrices $(u(x\_i,x\_j)-c)\_{i,j=1,...,N}$ for all $N$ and $x\_1,...x\_N\in S$, from where it is not hard to show the following Cauchy-Schwarz-like inequality
$$|u(x,y)-c| \leq \sqrt{u(x,x)-c}\cdot\sqrt{u(y,y)-c}, \qquad x,y\in S.$$
Hence the diagonal somehow controls the off-diagonal an... | 0 | https://mathoverflow.net/users/128963 | 312287 | 135,820 |
https://mathoverflow.net/questions/312286 | 11 | In Neukirch--Schmidt--Wingberg, "Cohomology of Number Fields", Second edition, page 624, Exercise 2, it is stated the following fact.
$\textbf{Claim}$: If $N$ is a normal subgroup, minimal among normal subgroups, of a group $G$ not contained in the Frattini's subgroup of $G$, then $G$ is a semi-direct product over $... | https://mathoverflow.net/users/129859 | About normal minimal subgroups not in the Frattini | I think it is the claim from exercise 2 of Neukirch et al which is incorrect. I presume that all groups are meant to be finite in this question. The claim from exercise 2 is OK when $N$ is solvable (for then $N$ is an elementary Abelian $p$-group for some prime $p$ and there is a maximal subgroup $H$ of $G$ which does ... | 9 | https://mathoverflow.net/users/14450 | 312288 | 135,821 |
https://mathoverflow.net/questions/311871 | 1 | I asked this question some while ago on Stack Exchange but didn't get an answer ([link](https://math.stackexchange.com/questions/2812523/why-is-the-flat-cotorsion-pair-actually-a-cotorsion-pair)), so I am trying it here as well.
Fix a ringed space $(X,\mathcal{O})$ and denote by $\mathcal{F}$ the class of flat module... | https://mathoverflow.net/users/82627 | Why is the flat cotorsion pair actually a cotorsion pair? | The category of sheaves on a ringed space is a Grothendieck category. Let $Q$ denote an injective cogenerator. We write $hom$ for internal and $Hom$ for external hom.
Let $F^+:=hom(F,Q)$. Then for any sheaf $F$ the sheaf $F^+$ is pure injective. This follows just from the adjunction $\otimes \dashv \hom$ and the fact... | 1 | https://mathoverflow.net/users/82627 | 312289 | 135,822 |
https://mathoverflow.net/questions/312293 | 7 | Given a complex elliptic K3 surface $\pi\colon X\rightarrow \mathbb P^1$, its *discriminant locus* is the divisor $$D = \sum\_{i = 1}^s n\_i P\_i$$ on $\mathbb P^1$ such that $n\_i$ is equal to the Euler-Poincaré characteristic of the fiber $\pi^{-1}(P\_i)$, where the sum runs over the points $P\_i \in \mathbb P^1$ suc... | https://mathoverflow.net/users/43951 | Discriminant locus of elliptic K3 surfaces | The minimal $s$ is $3$.
It is attained by several elliptic K3's,
including $y^2 = x^3 + (t^2-t)^4$ which has IV\* fibers at
$t = 0, 1, \infty$ and no other singular fibers.
The comment by **Ariyan Javanpeykar** gives one argument that
$s$ can be no smaller. (*See postscript.*
This uses characteristic zero; in small... | 9 | https://mathoverflow.net/users/14830 | 312303 | 135,825 |
https://mathoverflow.net/questions/312302 | -1 | $\log^k(n) = \log(\log(\log(...\log(n))));$
Let $f(n)$ = smallest $k$, s.t. $\log^k(n) \leq 1$
Is there known name for function $f$ ? Or it's an instance of some known function ?
Basically I want to know if anyone else also treating $O(f(n))$ as good as $O(1)$.
| https://mathoverflow.net/users/125003 | Is there special name for this function ?, $f(n)$ = smallest $k$, s.t. $\log^k(n) \leq 1$ | This is the [iterated logarithm](https://en.wikipedia.org/wiki/Iterated_logarithm) function, commonly denoted $\log^\*(n)$.
It happens to come up from time to time in the analysis of various algorithms; while it grows extremely slowly, and is basically a constant for most practical purposes, it is still an unbounded ... | 5 | https://mathoverflow.net/users/12705 | 312308 | 135,827 |
https://mathoverflow.net/questions/312307 | 10 | It is easily verifiable that
$$\sum\_{k\geq0}\binom{2k}k\frac1{2^{3k}}=\sqrt{2}.$$
It is not that difficult to get
$$\sum\_{k\geq0}\binom{4k}{2k}\frac1{2^{5k}}=\frac{\sqrt{2-\sqrt2}+\sqrt{2+\sqrt2}}2.$$
>
> **Question.** Is there something similarly "nice" in computing
> $$\sum\_{k\geq0}\binom{8k}{4k}\frac1{2^{10k... | https://mathoverflow.net/users/66131 | Adventure with infinite series, a curiosity | It is moderately nice, I would say. We have $\sum \binom{2k}k x^k=(1-4x)^{-1/2}$ for $|x|<1/4$. If we need only terms with $k$ divisible by 4 and $x=2^{-5/2}$, $4x=2^{-1/2}$, we get $$\sum \binom{8k}{4k}2^{-10k}=\frac14\sum\_{w^4=1}(1-w/\sqrt{2})^{-1/2}$$
and so on.
| 20 | https://mathoverflow.net/users/4312 | 312309 | 135,828 |
https://mathoverflow.net/questions/312278 | 4 | Is there a Riemannian metric on $\mathbb{R}^3$ for which the corresponding curvature tensor $R$ satisfies $R(X,Y)Z=(X\wedge Y)\wedge Z$?
I have already discussed this question in the following post and I recived very interesting partial answer but I search for a complete answer containing a classification of all metr... | https://mathoverflow.net/users/36688 | Realizing the cross product of $\mathbb{R}^3$ as the curvature tensor of a Riemannian metric on $\mathbb{R}^3$ | There are two interpretations of your question.
### Metric cross product
Assuming that you are looking for a Riemannian metric $g$ such that (by the triple-product formula)
$$ R\_g(X,Y)Z = g(Z,X)Y - g(Z,Y)X $$
this implies immediately that your metric has constant sectional curvature.
### Euclidean cross produ... | 8 | https://mathoverflow.net/users/3948 | 312313 | 135,829 |
https://mathoverflow.net/questions/312314 | 8 | Let $\Gamma\subset SL(2,\mathbb{R})$ be a Fuchsian group of the first kind. Let $c\_1, c\_2$ be inequivalent cusps of $\Gamma.$
Consider $f\in M\_k(\Gamma)$ a weight $k$ holomorphic automorphic form, and suppose the Fourier expansion of $f$ at the cusp $c\_1$ is known.
Given the above expansion, is there an algorit... | https://mathoverflow.net/users/129868 | Fourier expansion at inequivalent cusps | This can be done numerically. The $n$-th Fourier coefficient around the cusp is given by an integral of the form $$\int\_0^{1} f|\_{\sigma\_2} (z) e(-nz) dx,$$ where $\sigma\_2$ is a scaling matrix for the cusp $c\_2$. See p.43 of Iwaniec's Topics in Classical Automorphic Forms for definitions. In the above, $z=x+iy$ a... | 5 | https://mathoverflow.net/users/2627 | 312321 | 135,832 |
https://mathoverflow.net/questions/312325 | 4 | I just finished grad school, earning a Phd in mathematics. Currently I do not know if I want to continue my academic career or not, but in the meantime I have written an article containing some of the results of my thesis and I would like to submit it to a journal for publication.
The journal I have chosen (published... | https://mathoverflow.net/users/60675 | Question on academic affiliation when submitting paper | This question is more suited for Academia Stackexchange than this forum. You should ask there for the best answers based on experience.
In general, you should acknowledge who supported you during your research. Even if you have significant contributions after you graduated, your university that granted you the degree... | 1 | https://mathoverflow.net/users/3402 | 312328 | 135,834 |
https://mathoverflow.net/questions/312324 | 4 | Let $A$ and $B$ be two finite-dimensional C\*-algebras.
Let $\gamma$ denote the projective Banach space tensor product norm on the algebraic tensor product $A\odot B$, so $\gamma(t)=\inf\{\sum\_{i}\|a\_i\|\|b\_i\|:t=\sum\_i a\_i\otimes b\_i\}$. Then $\gamma(ts)\leq\gamma(t)\gamma(s)$ and $\gamma(t^\*)=\gamma(t)$, but ... | https://mathoverflow.net/users/100607 | Is the norm of the Banach space projective tensor product of finite-dimensional C*-algebras a C*-norm? | I think there are various reasons from abstract tensor norm theory why this is impossible, but I'll try to give a concrete example. First, a $C^\*$-norm on an algebra is uniquely determined; therefore, if $A=\ell\_\infty(2)=\mathbf C^2$ with the max-norm, then the tensor norm on $A\otimes A$ is the injective tensor nor... | 4 | https://mathoverflow.net/users/127871 | 312332 | 135,838 |
https://mathoverflow.net/questions/312323 | 3 | This is related to my previous question, but it is probably less scary and an expert in using Mathematica could figure out an answer easily.
I would like to estimate the asymptotic behaviour of the solution of the following ODE
$$ w''(r)+\frac{1}{r}w'(r)-k^{2}w(r)=-g(r)$$
where $g$ is a smooth positive function satis... | https://mathoverflow.net/users/127739 | ODE with Bessel decay | Maple 2018 solves the ODE under consideration:
```
dsolve(((D@@2)(w))(r)+(D(w))(r)/r-k^2*w(r) = -g(r))
```
$$w \left( r \right) ={{\ I}\_{0}\left(kr\right)}{\it \\_C2}+{{ K}\_{0
}\left(kr\right)}{\it \\_C1}-\int \!{{ K}\_{0}\left(kr\right)}g
\left( r \right) r\,{\rm d}r{{ I}\_{0}\left(kr\right)}+$$ $$\int \!{
{ I... | 2 | https://mathoverflow.net/users/35959 | 312340 | 135,842 |
https://mathoverflow.net/questions/312316 | 12 | It is a famous consequence of Tsen's theorem that a smooth curve over an algebraically closed field has trivial Brauer group. But what about curves over non algebraically closed fields?
Let us fix a smooth, projective curve $X$ over some field $k$. If $X$ has a rational point $x\in X(k)$, then the natural map $\text{... | https://mathoverflow.net/users/45660 | Brauer group of a curve over non-algebraically closed field | I don't think your map is injective. Here is an attempt at a recipe for constructing a counterexample.
The ingredients are a $C\_1$-field $F$ of characteristic zero and a smooth projective curve $X\_0$ over $F$ having non-trivial Brauer group. For a concrete example take $F=\mathbf{C}(t)$ and $X\_0$ to be an elliptic... | 8 | https://mathoverflow.net/users/3753 | 312341 | 135,843 |
https://mathoverflow.net/questions/230276 | 18 | I am interested in the orders of random permutations. Since the law of the logarithm of the order of a permutation converges to a normal law (for instance Erdös-Turan Statistical group theory III), one expects that the probability for two permutations of $\frak S\_n$ to have the same order goes to 0 as n goes to infini... | https://mathoverflow.net/users/86270 | What is the probability that two random permutations have the same order? | Nice problem! I claim that $\limsup n^2 p(n) = \infty$.
Suppose $k < n/2$ is such that $n-k$ is divisible by $L\_k = \text{lcm}(1,2,\dots,k)$. Then if $\pi \in S\_n$ has a cycle of length $n-k$ (this happens with probability $1/(n-k)$) then $\text{ord}(\pi) = n-k$, so the probability gets a contribution of $1/(n-k)^2... | 6 | https://mathoverflow.net/users/20598 | 312352 | 135,846 |
https://mathoverflow.net/questions/312351 | 13 | If topology were invented for algebraic geometry or logic, in ignorance of Euclidean space, we might reasonably regard connected compact Hausdorff spaces as pathological, or even doubt their existence. After all, Hausdorffness is a "separation" condition whereas connectedndess is a "nonseparation" condition -- it would... | https://mathoverflow.net/users/2362 | Are there continua in $\infty$-topoi? | Every contractible finite CW complex $X$ satisfies these conditions. This follows from results in Section 7.3 of HTT and Appendix A of HA: we have $Shv(X) \otimes Shv(X)=Shv(X\times X)$ since $X$ is locally compact, proper morphisms between locally compact Hausdorff spaces give proper morphisms of $\infty$-topoi (so $S... | 12 | https://mathoverflow.net/users/20233 | 312354 | 135,847 |
https://mathoverflow.net/questions/312320 | 6 | This is a sort-of follow up to [this question](https://mathoverflow.net/questions/312047/smooth-equidimensional-fibers-over-a-smooth-base), which I asked before I became confused about if things were reduced.
More specifically, suppose $\varphi:X\to Y$ is a surjective morphism of finite presentation between algebrai... | https://mathoverflow.net/users/62154 | When are all the fibers of a morphism reduced? | Let me repeat the assumptions to make sure we agree. Say $f : X \to Y$ is a morphism of varieties over an algebraically closed field $k$ such that (a) $Y$ is affine and smooth of dimension $m$, (b) $(X\_y)\_{red}$ is smooth of fixed dimension $n$ for all $y \in Y(k)$, (c) the maximal open $V\_y \subset X\_y$ which is a... | 5 | https://mathoverflow.net/users/129891 | 312356 | 135,848 |
https://mathoverflow.net/questions/312343 | 3 | I would like to know whether the following kind of Noether--Lefschetz statement is true, and if so, to have a reference.
>
> Fix natural numbers $d\_1,\ldots,d\_n$ such that $\sum\_i d\_i \geq n+3$. Denote by $d$ the product $\prod\_i d\_i$. Let $C\_1, \ldots C\_k$ be smooth disjoint curves in $\mathbf P^{n+2}$, wi... | https://mathoverflow.net/users/121595 | Conditional Noether--Lefschetz theorems | I think this is false. Consider a very general quartic surface $S\subset\mathbb{P}^3$ containing a line $\ell$. Projecting from $\ell$ defines an elliptic fibration $S\rightarrow \mathbb{P}^1$. Choose some fibers $C\_1,\ldots ,C\_k$, with $k\geq 6$. For degree reasons $S$ is the only quartic surface containing all the ... | 5 | https://mathoverflow.net/users/40297 | 312361 | 135,850 |
https://mathoverflow.net/questions/306695 | 5 | The definition of a (pre)regular skeletal Reedy category in the sense of Cisinski generalizes the intuition behind the Eilenberg-Zilber factorization for simplicial sets, specifically the property that the image of any section of a representable is again representable, or equivalently that all nondegenerate sections of... | https://mathoverflow.net/users/1353 | Products of representables are regular on a regular skeletal Reedy category? | I asked Cisinski by e-mail if this was true, and he said that it isn't. A counterexample is the category of planar trees ($\Omega\_{\mathbf{pl}}$, not $\Omega$, which is not even normal) used in the theory of Dendroidal sets. He said that this property can be axiomatized by asking that the class of regular presheaves i... | 1 | https://mathoverflow.net/users/1353 | 312365 | 135,852 |
https://mathoverflow.net/questions/312366 | 4 | **Question.** Is it true that each uncountable group $G$ contains an uncountable subgroup $A$ and an infinite subgroup $B$ such that $A\cap B=\{1\}$? What will be the answer if we additionally require that $ab=ba$ for all $a\in A$ and $b\in B$?
| https://mathoverflow.net/users/61536 | Hereditarily indecomposable groups | This is not rue, by Shelah's construction of a Jonsson group
of cardinality $\aleph\_1$ (a group of size $\aleph\_1$ for which every proper subgroup is countable). See [On a problem of Kurosh, Jónsson groups, and applications](https://www.sciencedirect.com/science/article/pii/S0049237X08713466)
| 9 | https://mathoverflow.net/users/11115 | 312367 | 135,853 |
https://mathoverflow.net/questions/312363 | 10 | I know the Murnaghan–Nakayama rule, but I am wondering if there is any closed formulas for the character of the symmetric group. I know the following:
$$\chi\_{n}(\sigma) = 1$$
$$\chi\_{11...1}(\sigma) = sgn(\sigma)$$
$$\chi\_{n-1,1}(\sigma) = fix(\sigma)-1$$
$$\chi\_{21...1}(\sigma) = sgn(\sigma)(fix(\sigma) - 1)$$
... | https://mathoverflow.net/users/73667 | Closed formulas for the character of the symmetric group | For generalizing the formulas in your question, see <http://www.combinatorics.org/ojs/index.php/eljc/article/view/v16i2r19> and Examples 1.7.13 and 1.7.14 in Macdonald's *Symmetric Functions and Hall Polynomials*, 2nd ed.
For a different formula, see <https://www.researchgate.net/publication/227299451_Stanley%27s_Formu... | 7 | https://mathoverflow.net/users/2807 | 312368 | 135,854 |
https://mathoverflow.net/questions/312336 | 18 | A conjecture by Milnor state that if $G$ is a Lie group, then the map $B(G^{disc})\to BG$ sending the classifying space of $G$ endowed with the discrete topology to the classifying space of the topological group $G$ induces an isomorphism on homology with $\mod p$ coefficients.
In chromatic homotopy theory, there ar... | https://mathoverflow.net/users/115052 | Milnor Conjecture on Lie groups for Morava K-theory | Consider a map $f\colon X\to Y$ of spaces (such as $B(G^{\text{disc}})\to B(G)$). Say that $f$ is a $K(n)$-equivalence if $K(n)^\*(f)\colon K(n)^\*(Y)\to K(n)^\*(X)$ is an isomorphism. We will allow the case $n=\infty$ (corresponding to $K(\infty)^\*(X)=H^\*(X;\mathbb{F}\_p)$) but not the case $n=0$ (corresponding to $... | 7 | https://mathoverflow.net/users/10366 | 312370 | 135,855 |
https://mathoverflow.net/questions/312319 | 2 | Let $\alpha$ be an ordinal, and let $a\subseteq\alpha$ such that $\alpha$ is countable in $L[a]$. Moreover, let $\beta>\alpha$ be an ordinal such that, in $L[a]$, $\alpha$ and $\beta$ have the same cardinality and such that $\beta$ is "reasonably closed", say $L\_{\beta}[a]\models\text{ZF}^{-}$. Denote by $P\_{\alpha}$... | https://mathoverflow.net/users/49491 | Encoding sets in locally generic sets | No. Suppose $a$ is a random real over $L$. Then $a$ is random over any $L\_\alpha$ satisfying $ZF^-$ plus "$\mathbb R$ exists." If there were $x$ as in your question, then it would be $Add(\omega,\alpha)$-generic over $L\_\beta$. But by the well-known orthogonality of random and Cohen forcing, $L[x]$ thinks there are n... | 5 | https://mathoverflow.net/users/11145 | 312373 | 135,857 |
https://mathoverflow.net/questions/312349 | 7 | Let $K$ be a knot in the boundary of a compact smooth 4-manifold $X$, and suppose that $K$ is the the kernel of $\pi\_1(\partial X) \to \pi\_1(X)$. Then $K$ is the boundary of some immersed disk $D \to X$ that has some fine number of double points. I am interested in knowing how to compute minimum number of such double... | https://mathoverflow.net/users/99414 | Minimum number of double points over all immersed disks | For knots in $S^3=\partial B^4$ this is called the [four-ball crossing number or clasp number](http://www.indiana.edu/~knotinfo/descriptions/clasp_number.html).
For knots in $S^3$ you have the following inequalities relating $I$ to the $4$-ball genus $g\_4$ and the unknotting number $u$
$$g\_4\leq I \leq u.$$
In ... | 4 | https://mathoverflow.net/users/84120 | 312374 | 135,858 |
https://mathoverflow.net/questions/312389 | 12 | "floating point arithmetic" is a terminology that refer to the arithmetic perform over (finite) representation of real number. See the wikipedia [article](https://en.wikipedia.org/wiki/Floating-point_arithmetic) for more details.
In the [formal specification](https://en.wikipedia.org/wiki/IEEE_754) of floating poin... | https://mathoverflow.net/users/38822 | Terminology: algebraic structure for "floating point" arithmetic | The abstract picture here is that of adjoining an undefined or bottom element to a (partial) algebra. Doing this to a total algebra is boring, but it is a useful trick to turn partial algebras into total ones.
One just picks an element $\bot$ not in the carrier set, and extends the operations as follows: Any operatio... | 12 | https://mathoverflow.net/users/15002 | 312393 | 135,862 |
https://mathoverflow.net/questions/312362 | 6 | For a nonunital ring $R$ (or "rng") one has to be a little bit careful when considering the category of (left or right) $R$-module and, furthermore, the standard equivalent definitions of projective modules are in general not equivalent anymore (see e.g.<https://math.stackexchange.com/questions/120458/projective-module... | https://mathoverflow.net/users/15488 | Equivalence of idempotents and projective modules over nonunital rings | If $R$ is a ring (not necessarily unital) and $e\in R$ is an idempotent, then it is still the case that $Hom\_R(eR,M)\cong Me$ for any right $R$-module $M$ via $\phi\mapsto \phi(e)$. It immediately follows that, if $e,f\in R$ are two idempotents, then $eR\cong fR$ if and only if there exists $a\in eRf$ and $b\in fRe$ w... | 7 | https://mathoverflow.net/users/15934 | 312405 | 135,865 |
https://mathoverflow.net/questions/312400 | 6 | Let $BV(\mathbb R^n; \mathbb R^n)$ be the space of (vector-valued) [functions of bounded variation](https://www.encyclopediaofmath.org/index.php/Function_of_bounded_variation) and let $BD(\mathbb R^n;\mathbb R^n)$ the space of [functions with bounded deformation](https://en.wikipedia.org/wiki/Bounded_deformation). They... | https://mathoverflow.net/users/111164 | Bounded deformation vs bounded variation | Example 7.7 in
**L. Ambrosio, A. Coscia, Alessandra, G. Dal Maso**,
Fine properties of functions with bounded deformation.
*Arch. Rational Mech. Anal.* 139 (1997), no. 3, 201–238.
| 7 | https://mathoverflow.net/users/121665 | 312407 | 135,866 |
https://mathoverflow.net/questions/312402 | 5 | If $G$ is a finite group and its conjugacy classes are known, can the conjugacy classes of the wreath product $G \wr S\_n \cong G^n \rtimes S\_n$ be determined?
| https://mathoverflow.net/users/128120 | Determining the conjugacy classes of a wreath product $G \wr S_n$ | This is indeed handled in Section 4.2 of James and Kerber's book on Representations of the Symmetric group.
We also considered this problem in Section 2 our paper describing a computer algorithm for computing conjugacy class representatives in permutation groups.
J. Cannon and D. Holt, Computing conjugacy class rep... | 5 | https://mathoverflow.net/users/35840 | 312410 | 135,868 |
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