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https://mathoverflow.net/questions/82536 | 6 | Recall:
-------
Given a category $A$, and two classes of morphisms $S,S'$, we say that $S$ is *right-cancellative with respect to $S'$* if for any pair of maps $f\in S, g\in S'$ such that $gf$ is defined, we have the implication $gf\in S \Rightarrow g\in S$.
Recall that the class of inner-anodyne morphisms in the ... | https://mathoverflow.net/users/1353 | Is the class of inner-anodyne morphisms right-cancellative with respect to the of the class of monomorphisms? | The fact that in $\operatorname{Set}\_{\Delta}$ the class of inner anodyne maps has the right cancellation property (within the class of monomorphisms) was proven by Danny Stevenson in his 2016 paper [*Stability for inner fibrations revisited*](https://arxiv.org/abs/1608.07699), which was also published in [*TAC*](http... | 3 | https://mathoverflow.net/users/25477 | 306376 | 133,560 |
https://mathoverflow.net/questions/304515 | 2 | I have something elementary to ask. Let $E\rightarrow X$ be a holomorphic line bundle over a Riemann surface. Then in general a section of $E$ is a meromorphic function on $X$, since $O\_{div(s)}\cong E$. But we also know that the first Dolbeault cohomology group be the group of holomorphic sections on $X$. The space o... | https://mathoverflow.net/users/124606 | holomorphic sections of line bundles on Riemann surfaces | I think you make some confusion, and also that this question is perhaps more appropriate for MSE.
Anyway, if you are given a holomorphic line bundle $\pi\colon E\to X$, a holomorphic section is a holomorphic map $s\colon X\to E$ such that $\pi\circ s=\operatorname{Id}\_X$.
More concretely, if your line bundle is d... | 4 | https://mathoverflow.net/users/9871 | 306377 | 133,561 |
https://mathoverflow.net/questions/306366 | 10 | Let A is a $n\times n$ matrix given by \begin{align\*} a\_{ij} = [\Gamma(\lambda\_{i}+\mu\_{j})] \end{align\*} where $0 < \lambda\_{1} < \ldots < \lambda\_{n}$ and $0 < \mu\_{1} < \ldots < \mu\_{n}$ are real positive numbers and $\Gamma$ denotes the Gamma function given by $\Gamma(z) = \int\_0^\infty t^{z-1}e^{-t}dt$ \... | https://mathoverflow.net/users/126770 | Prove that the matrix $[\Gamma(\lambda_{i}+\mu_{j})]$ is nonsingular | By Andreieff's identity:
$$
{\rm det}(A)=\frac{1}{n!}\int\_{(0,\infty)^{n}}
{\rm det}[e^{-\frac{t\_k}{2}}t\_k^{\lambda\_i-\frac{1}{2}}]\_{1\le i,k\le n}\ \times\
{\rm det}[e^{-\frac{t\_k}{2}}t\_k^{\mu\_j-\frac{1}{2}}]\_{1\le k,j\le n}\ \ dt\_1\cdots dt\_n
$$
$$
=\frac{1}{n!}\int\_{(0,\infty)^{n}}
\left(\prod\_{l=1}^{n... | 18 | https://mathoverflow.net/users/7410 | 306385 | 133,564 |
https://mathoverflow.net/questions/306368 | 3 | To understand a crucial example in representation theory, I need the explicit spectral decomposition of the differential operator
$$
Df(x)=(1+x^2)f''(x)+2xf'(x)
$$ on $L^2({\mathbb R})$. I'm not an expert, but at first glance, theory tells me the existence of a spectral measure, but not what it looks like. Is the spec... | https://mathoverflow.net/users/nan | Spectral decomposition of a specific operator | This is a Sturm-Liouville operator $(Df)(x)=(pf')'$, with $p=1+x^2$. These can be rewritten as Schrodinger equations, by using what I would call a Kummer-Liouville transformation. By some conspiracy, all reference to these on the internet seems to have disappeared, but see perhaps my answer [here.](https://mathoverflow... | 4 | https://mathoverflow.net/users/48839 | 306393 | 133,567 |
https://mathoverflow.net/questions/306310 | 1 | $\phi(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}$ is the pdf of a standard normal distribution.
$\Phi(x) = \int\_{- \infty}^x \phi(t) dt$ is the cdf of a standard normal distribution.
How does one calculate the following:
$$
T = \int\limits\_0^{\infty} \Phi^2(bx) \phi(x) dx
$$
In fact, I can find the result ... | https://mathoverflow.net/users/83321 | Integral of product of Gaussian pdf and cdf | Maybe this proof is too large, but you can deduce the result you want, sorry for that.
First of all:
* $\phi(x)= \frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}}$
* $\Phi(x)=\int\_{-\infty}^{x}\phi(t)dt=\frac{1}{2}(1+erf(\frac{x}{\sqrt{2}}))$
To prepare the proof, start with this facts:
1. $\int\_{0}^{\infty}xe^{-x^2a}d... | 3 | https://mathoverflow.net/users/126749 | 306405 | 133,571 |
https://mathoverflow.net/questions/262952 | 4 | This is a very basic question about the definition of a reproducing kernel Hilbert space (RKHS).
It seems the standard definition of a RKHS is as a Hilbert space $H$ of functions on some set $X$ where the evaluation functionals are continuous. It is then noted that the Riesz representation theorem yields a map $k$ fr... | https://mathoverflow.net/users/38085 | Abstract Definition of a Reproducing Kernel Hilbert Space | As an interested observer of the "reproducing kernel Hilbert space" world, it seems to me that, for purposes of "outsiders", the typical substance of a discussion about such spaces is an assertion that some naturally occurring space of functions (on a physical space with various further attributes) *is* a RKHS... *beca... | 2 | https://mathoverflow.net/users/15629 | 306414 | 133,573 |
https://mathoverflow.net/questions/306409 | 9 | Please help me to find a proper reference to the following infinite version of the Sunflower Lemma.
>
> **Lemma.** Let $n\in\mathbb N$. Every infinite family of $n$-element sets contains an infinite subfamily $\mathcal F$ such that $A\cap B=\bigcap\mathcal F$ for any distinct sets $A,B\in\mathcal F$.
>
>
>
Bro... | https://mathoverflow.net/users/61536 | A reference to infinite version of the Sunflower Lemma | I found it in the book Komjáth, Péter; Totik, Vilmos, [**Problems and theorems in classical set theory**](http://dx.doi.org/10.1007/0-387-36219-3), Problem Books in Mathematics. New York, NY: Springer (ISBN 0-387-30293-X/hbk). xii, 514 p. (2006). [ZBL1103.03041](https://zbmath.org/?q=an:1103.03041). It's stated on p. 1... | 13 | https://mathoverflow.net/users/43266 | 306416 | 133,575 |
https://mathoverflow.net/questions/306394 | 17 | I'm looking for a geometric or combinatorial depiction of the algebraic identity
$$
xyz = \frac{1}{24} \Big\{(x+y+z)^3 - (x-y+z)^3 - (x+y-z)^3 + (x-y-z)^3 \Big\}.
\label{\*}\tag{$\*$}
$$
Here is the kind of thing I'd like. For the simpler identity $xy = \frac{1}{4} \big\{(x+y)^2 - (x-y)^2 \big\}$ we can rearrange to ... | https://mathoverflow.net/users/88133 | Geometric/combinatorial depiction of algebraic identity? | The identity can be rewritten as
>
> $(a+b+c)^3=a^3+b^3+c^3+3(a+b)(a+c)(b+c)$
>
>
>
by means of a linear change of variables $a:=(−x+y+z)/2$, etc.
Let $T$ be a circle of length $a+b+c$, and let's chop it into three intervals $A$, $B$, $C$ of respective lengths $a$, $b$, $c$. Consider also the intervals $A':=... | 9 | https://mathoverflow.net/users/1516 | 306417 | 133,576 |
https://mathoverflow.net/questions/306420 | 3 | I am more familiar with Cartan geometry, and in this setting we have a notion of development of curves. As described in Cap & Slovak 1.5.17, on a Cartan geometry $(\mathcal{P} \to M, \omega)$ modelled on $G/P$, we have a "Cartan space" which is an associated bundle $S :=\mathcal{P} \times^P G/P \to M$ that comes equipp... | https://mathoverflow.net/users/56938 | Is the development map in Hyperbolic geometry related to development in Cartan geometry? | Of course it is the same. Any submanifold on which the curvature of a Cartan geometry vanishes has a developing map from its universal covering space. I don't know any great reference, but I have used this in many of my papers, for example: <https://arxiv.org/abs/1005.1472>
| 3 | https://mathoverflow.net/users/13268 | 306432 | 133,583 |
https://mathoverflow.net/questions/306438 | 9 | I'm attempting to understand the Bombieri-Lang Conjecture:
>
> If $X$ is a smooth projective variety of general type defined over a number field, then the set of rational points of $X$ is not dense.
>
>
>
I don't understand what it means for a variety to be ''of general type''. I know it's when the variety's K... | https://mathoverflow.net/users/126815 | Understanding what it means to be ''of general type'' | To start understanding this, it's probably best to start with some examples.
First, the conjecture says that if a curve has Zariski dense rational points, then it is genus zero or one. This is known (Faltings).
Second, the conjecture, plus the Enriques-Kodaira classification, says that if a surface has Zariski dens... | 10 | https://mathoverflow.net/users/18060 | 306441 | 133,584 |
https://mathoverflow.net/questions/306443 | 9 | In his seminal 1937 paper, Jones [1] proved the following result about [Moore spaces](https://en.wikipedia.org/wiki/Moore_space_(topology)):
>
> **Theorem. (Jones)** If $2^{\aleph\_0}<2^{\aleph\_1}$ then all *separable* normal Moore spaces are metrizable.
>
>
>
Then he came up with the idea that maybe the *sep... | https://mathoverflow.net/users/82843 | On the Large Cardinal Strength of Normal Moore Space Conjecture | If $\text{NMSC}$ is consistent, then so is $\text{NMSC}+\text{"there are no strongly inaccessible cardinals"}$.
This is because if $V \models \text{NMSC}$, then $V\_\kappa \models \text{NMSC}$ for any inaccessible cardinal $\kappa$. (A proof sketch is given below.) In particular, if $\kappa$ is the first strongly in... | 14 | https://mathoverflow.net/users/70618 | 306453 | 133,590 |
https://mathoverflow.net/questions/306389 | 6 | Let $M$ be a countable transitive model of (enough of) ZFC. I'm looking for notions of forcing $\mathbb{P}$ such that if $G$ is $M$-generic for $\mathbb{P}$, then $c$ is a Cohen real over $M$ if and only if $c$ is Cohen real over $M[G]$.
Ideally, $\mathbb{P}$ should be a "typical" poset for adding reals in an obvious... | https://mathoverflow.net/users/16107 | Cohen generics over the ground model still Cohen over other generic extensions? | Another example of a forcing like this is the forcing to add an infinitely equal real. I'll sketch the proof, although it is quite similar to the one for Sacks forcing.
Recall that conditions in the poset $\mathbb{P}$ are partial functions $p:\omega\to\omega$ with coinfinite domains and satisfying $p(n)\leq 2^n$, ord... | 7 | https://mathoverflow.net/users/1058 | 306458 | 133,592 |
https://mathoverflow.net/questions/306455 | 1 | Consider an orthonormal basis $(\varphi\_n)\_{n \in \mathbb N}$ of $L^2(\mathbb R).$
We consider the functionals $\Phi\_n$ given by $$ C^b(\mathbb R) \ni f \mapsto \left\langle \varphi\_n, f \varphi\_{n+1} \right\rangle$$
for any $n \in \mathbb N$ where $C^b(\mathbb R)$ are the continuous and bounded functions on $... | https://mathoverflow.net/users/126819 | Do functions exist and are they dense? Or does it depend on the basis? | The complement of $X$ may be empty.
Consider a basis such that $\mathrm{supp}\,\varphi\_k \cap \mathrm{supp}\,\varphi\_{k+1}=\emptyset$ for some $k$ (for example, consisting of Walsh functions on each interval $[n, n+1]$). Then obviously $\mathrm{Ker}\,\Phi\_k = C^b(\mathbb{R})$.
| 3 | https://mathoverflow.net/users/88291 | 306459 | 133,593 |
https://mathoverflow.net/questions/306464 | 11 | If $G$ is a paracompact topological group, then is $G \times G$ paracompact?
This question is raised by [Gepner and Henriques](https://arxiv.org/abs/math/0701916) (first paragraph of 2.2). Of course, this is not true for arbitrary paracompact spaces, as shown by the [Sorgenfrey plane](https://en.wikipedia.org/wiki/Pa... | https://mathoverflow.net/users/2362 | If $G$ is a paracompact topological group, then is $G \times G$ paracompact? | I am at a [topology conference](https://sites.google.com/site/summertopology2018/) today, and among the many good topologists here is Jan van Mill, a leading expert on topological groups. I ran your question by him, thinking he might know the answer off the top of his head. He did -- the answer is that if $G$ is a para... | 12 | https://mathoverflow.net/users/70618 | 306479 | 133,600 |
https://mathoverflow.net/questions/306469 | 2 | $\newcommand{\al}{\alpha}
\newcommand{\be}{\beta}
\newcommand{\de}{\delta}
\newcommand{\De}{\Delta}
\newcommand{\ep}{\varepsilon}
\newcommand{\ga}{\gamma}
\newcommand{\Ga}{\Gamma}
\newcommand{\la}{\lambda}
\newcommand{\si}{\sigma}
\newcommand{\Si}{\Sigma}
\newcommand{\thh}{\theta}
\newcommand{\om}{\omega}
\newcommand{\... | https://mathoverflow.net/users/36721 | Symmetric orthogonal matrices with constant diagonal entries | No. A symmetric $M$ will satisfy $M^2=1$ if and only if the spectrum is contained in $\pm 1$, which is equivalent to $M=P-(1-P)=2P-1$ for some orthogonal projection $P$. Now you're asking if the extra condition that the diagonal is constant will give $P$ rank $1$ or $n-1$.
It's clear that this won't follow because we... | 6 | https://mathoverflow.net/users/48839 | 306481 | 133,601 |
https://mathoverflow.net/questions/306446 | 1 | Consider an orthonormal basis $(\varphi\_k)$ of $L^2(\mathbb R)$ with Lebesgue measure.
I came along a nice number theoretic question in analysis:
Write $$f\_k(x):=\int\_{\left\lvert y \right\rvert \ge x } \left\lvert \varphi\_k(z) \right\rvert^2 \ dz.$$
Clearly, $f\_k$ are continuous monotonically decreasing fu... | https://mathoverflow.net/users/126819 | Number theory on Banach space $L^2(\mathbb R)$ meets linear independence? | You can have an orthonormal basis of $L^2(\mathbb R)$ with all $f\_n$ equal.
Start with the Walsh functions, which are an orthonormal basis of $L^2([0,1])$ with absolute value $1$ everywhere. Then use an isometry of $L^2([0,1])$ to $L^2(\mathbb R)$ given by $Tf(t) = a(t) f(b(t))$ for suitable functions $a(t)$ and $b(t)... | 5 | https://mathoverflow.net/users/13650 | 306486 | 133,604 |
https://mathoverflow.net/questions/306251 | 16 |
>
> If $\mathcal{C}$ is a symmetric monoidal $(\infty,1)$-category with duals, then there should be a functor
> $$
> d: \mathcal{C} \longrightarrow \mathcal{C}^{op}
> $$
> such that $d(x)$ is dual to $x$ for all objects $x \in \mathcal{C}$.
> How can one construct such a functor?
>
>
>
Of course, the above i... | https://mathoverflow.net/users/91925 | How can I functorially dualise in a symmetric monoidal $(\infty,1)$-category with duals? | One way to construct the duality functor ${\cal C} \to {\cal C^{\rm op}}$ is through the notion of a *pairing* of $\infty$-categories (see HA, Definition 5.2.1.5). In particular, in this case we're talking about a self-pairing on ${\cal C}$, which by definition is a right fibration $\mu:{\cal M} \to {\cal C} \times {\c... | 11 | https://mathoverflow.net/users/51164 | 306491 | 133,605 |
https://mathoverflow.net/questions/306492 | 3 | I know that the squared distance function from a point $p$ on a Riemann manifold $M$ is smooth in a n-hood of $p$. Therefore for a smooth curve $c:\mathbb{R}\to M$ the concatenation $d(p,\cdot)^2\circ c$ is smooth near a point $t\_0$ for $p\in U$, where $U$ is a n-hood of $c(t\_0)$.
What I am trying to prove (and whe... | https://mathoverflow.net/users/126834 | Smoothness of a curve vs. smoothness of the squared distance from the curve to points on Riemann manifolds | Take an orthonormal basis in the tangent space of $M$ at some point $p\_0$ on your curve $c$. For each vector $u\_i$ in that basis, let $p\_i=\exp\_{p\_0}(-(\rho/2)e\_i)$, where $\rho$ is the injectivity radius at $p\_0$. Then the distance function $f\_i(p)=d(p\_i,p)$ from $p\_i$ has gradient $e\_i$ at $p\_0$. So $f\_1... | 6 | https://mathoverflow.net/users/13268 | 306494 | 133,607 |
https://mathoverflow.net/questions/306036 | 5 | I have come across an n-category cafe post where someone describes a monad that generates symmetric monoidal categories. Can someone give details, like what is the base category, what exactly is the endofunctor and natural isomorphisms for “free symmetric monoidal category” 2-monad? Could this generate the category of ... | https://mathoverflow.net/users/10007 | What is the “free symmetric monoidal category” 2-monad? | There is a 2-monad $P$ on $\mathrm{Cat}$ whose strict algebras are symmetric strict monoidal categories, and whose pseudo-algebras are "unbiased" symmetric monoidal categories. On objects, $PA$ is the category whose objects are finite lists of objects of $A$, and in which a morphism $(a\_1,\dots,a\_n)\to (b\_1,\dots,b\... | 7 | https://mathoverflow.net/users/49 | 306497 | 133,609 |
https://mathoverflow.net/questions/306472 | 18 | **Let $f:\mathbb{R}\to \mathbb{R}$ be continuous at $x$ for every $x\in I$ where $I\subset \mathbb R$ could be arbitrary. Does there always exist a function $F:\mathbb{R}\to \mathbb{R}$ differentiable on $I$ and $F'(x) = f(x)$ for every $x \in I$?**
The definition of a primitive is naturally defined on an interval. w... | https://mathoverflow.net/users/126827 | Existence of an antiderivative function on an arbitrary subset of $\mathbb{R}$ | Let's start with the case of a locally bounded function $f:J\to\mathbb{R}$ (say defined on some nonempty open interval $ J\subset\mathbb{R}$, with a fixed $x\_0\in J$). We may consider for any $x\in J$ the [upper Darboux integral](https://en.wikipedia.org/wiki/Darboux_integral#Darboux_integrals) of $f$ from $x\_0\in J$... | 17 | https://mathoverflow.net/users/6101 | 306502 | 133,612 |
https://mathoverflow.net/questions/306516 | 8 | Does the 6-element group $S\_3$ have a finite (balanced) **semigroup** presentation of the form $$\langle a\_1,...,a\_n\mid a\_1=u\_1, a\_2=u\_2,...,a\_n=u\_n\rangle$$ where $u\_1,u\_2,...,u\_n$ are semigroup words? Let us call such a semigroup presentation {\it tree-like}.
**Edit:** By a semigroup word, I mean a wor... | https://mathoverflow.net/users/nan | A balanced tree-like presentation of $S_3$ | I see that Jeremy has beaten me to it - but here is a solution found by computer by choosing random entries from the group multiplication table.
This one also generates $S\_3$ as a semigroup. The Magma command $\mathtt{RWSMonoid}$ applies the Knuth-Bendix algorithm to the presentation, and regards it as a monoid pres... | 6 | https://mathoverflow.net/users/35840 | 306520 | 133,617 |
https://mathoverflow.net/questions/306511 | 5 | What is known about the statistical independence of the eigenvectors of a real symmetric matrix with independent Gaussian entries with zero mean, and finite variance? The matrix elements are not assumed to have same variance.
I see some results for Wigner matrices in literature, where the entries are i.i.d. standard ... | https://mathoverflow.net/users/125930 | Statistical independence of eigenvectors of real symmetric Gaussian random matrices | A precise answer exists for the Gaussian Orthogonal Ensemble (all variances the same): then the eigenvectors are the columns of an orthogonal matrix which is uniformly distributed with the Haar measure; they are therefore not independent --- they cannot be because they must be orthogonal to one another.
In the limit ... | 2 | https://mathoverflow.net/users/11260 | 306527 | 133,621 |
https://mathoverflow.net/questions/306530 | 2 | Suppose that $1/2+it$ is not a zero of the Riemann zeta function $\zeta$, where $t \in \mathbb{R}$. Can $1/\zeta(1/2+it)$ be expressed as a Dirichlet series ?
| https://mathoverflow.net/users/480516 | On the Dirichlet series for $1/\zeta(s)$ at $\Re(s)=1/2$ | As Jarek Kuben remarked, the question is almost a duplicate of [this question](https://mathoverflow.net/questions/164874/is-it-possible-to-show-that-sum-n-1-infty-frac-mun-sqrtn-diverg). Lucia's response there can be adapted here. Briefly, for a fixed $t\in\mathbb{R}$, let us write $M\_0(x):=\sum\_{n\leq x}\mu(n)/n^{1/... | 10 | https://mathoverflow.net/users/11919 | 306534 | 133,625 |
https://mathoverflow.net/questions/306483 | 15 | I posted this question on MSE a few days ago, but got no response (despite a bounty). I hope it will get more answers here, but I'm afraid it might not be appropriate as I'm not sure it's actually research-level. Please do tell me if it's not appropriate and if possible tell me how to modify the question so that it may... | https://mathoverflow.net/users/102343 | Locales as spaces of ideal/imaginary points | I can only answer some of your questions.
Yes, the Zariski locale is extensively studied. It's one of the ways of setting up scheme theory in a constructive context: Don't define schemes as locally ringed spaces, but as locally ringed locales. The locally ringed locale $\mathrm{Spec}(A)$ always enjoys the universal p... | 16 | https://mathoverflow.net/users/31233 | 306536 | 133,627 |
https://mathoverflow.net/questions/306505 | 2 | Fix $n$, and consider the characteristic polynomials for all $C=2^{\frac{n(n-1)}{2}}$ adjacency matrices representing undirected, unweighted graphs on $n$ vertices.
>
> Are the characteristic polynomials distributed somewhat uniformly among adjacency matrices? Or are some characteristic polynomials "much more popul... | https://mathoverflow.net/users/8927 | How are characterstic polynomials (resp. Alexander polynomials) distributed amongst adjacency matrices (resp. grid diagrams)? | The answer depends quite nontrivially on which model of random knots you consider. For a general description, see Even-Zohar's [very nice survey.](https://arxiv.org/abs/1711.10470) In many models a random knot is a very small knot (unknot, trefoil, figure eight dominating), with the resulting singularity of the distrib... | 1 | https://mathoverflow.net/users/11142 | 306549 | 133,634 |
https://mathoverflow.net/questions/306551 | 1 | Suppose $G=(V,E)$ is a simple, undirected graph with $|V|,|E|$ infinite. Is there $B\subseteq E$ with $|B| = |E|$ such that $(V,B)$ is bipartite?
| https://mathoverflow.net/users/8628 | Bipartite subgraphs with lots of edges | Yes. In each component Take a vertex $v$ and for every vertex $u$ let $d(v,u)$ be the shortest distance in $G$ between $v$ and $u$, where distance is defined as the fewest number of edges. Then every edge $\{u,u'\}$ in $G$ satisfies $|d(v,u)-d(v,u')| \le 1$, and the graph formed from $G$ by removing all edges between v... | 5 | https://mathoverflow.net/users/122188 | 306555 | 133,637 |
https://mathoverflow.net/questions/306557 | 4 | A "$n$-order matrix" $T\in M\_n(\mathbb F\_2)$ is a matrix such that there exists a partial ordered relation $\leq\_T\subset [1,n]^2$ such that :
$T\_{ij}=1\Leftrightarrow i\leq\_T j$
(where $T\_{ij}$ is the $i,j$ coefficient of $T$)
A "$n$-graph matrix" $S\in M\_n(\mathbb F\_2)$ is a symmetric matrix such that $... | https://mathoverflow.net/users/112382 | Is a simple graph the "sum" of a partial order and its dual? | Even with $0$'s on the diagonal it's still false. Consider the matrix $S = \left[\begin{matrix}0&1&0&0&1\cr 1&0&1&0&0\cr 0&1&0&1&0\cr 0&0&1&0&1\cr 1&0&0&1&0\end{matrix}\right]$. Suppose $S = T + T^t$ for some order matrix $T$. Then this order must have either $1 < 2$ or $2 < 1$ since $s\_{12} = 1$. Wlog say $1 < 2$. Si... | 8 | https://mathoverflow.net/users/23141 | 306558 | 133,638 |
https://mathoverflow.net/questions/306487 | 1 | Suppose $f: X \rightarrow Y$ is a finite, flat (hence locally free) morphism of curves (i.e. schemes of dimension 1, not smooth or even reduced). Suppose $L$ is a reflexive sheaf on $X$, locally free of rank 1 at each generic point of $X$.
Is the direct image $f\_\* L$ still reflexive on $Y$? (Better its top exterior... | https://mathoverflow.net/users/91935 | Direct image of reflexive sheaf via finite, flat map | Here is a (probably non-optimal) statement that may apply in your situation. In your situation with curves, the hypothesis says that you need $X$ and $Y$ to be Gorenstein.
**Claim.** *Let $X$ and $Y$ be noetherian schemes satisfying $G\_1$ and $S\_2$. If $f\colon X \to Y$ is a finite surjective morphism and $\mathscr... | 3 | https://mathoverflow.net/users/33088 | 306559 | 133,639 |
https://mathoverflow.net/questions/306541 | 6 | Suppose $M$ is an abelian von Neumann algebra, carrying a (point-ultraweakly) continuous action $G\curvearrowright M$ of a locally compact, second-countable group.
Let $y: G\to\cal U(M)$ be an ultraweakly continuous unitary cocycle, i.e., $y\_{gh}=y\_g\cdot (g.y\_h)$ for all $g,h\in G$. Suppose that $y$ is uniformly ... | https://mathoverflow.net/users/29404 | Trivializing unitary cocycles in abelian von Neumann algebras that are uniformly close to the trivial one | With your uniformly closeness assumption, $b(g) := \sqrt{-1}\log y\_g$ is a usual additive cocycle (i.e., $b(gh) = b(g) + g\cdot b(h)$) which is moreover real and bounded. I think any bounded cocycle into $L^\infty(X)$ is a coboundary. By exponentiating it, one gets the unitary element $v$. Here's a standard proof for ... | 7 | https://mathoverflow.net/users/7591 | 306563 | 133,641 |
https://mathoverflow.net/questions/306560 | 2 | This is the generalization of a question [Is a simple graph the "sum" of a partial order and its dual?](https://mathoverflow.net/questions/306557/is-a-simple-graph-the-sum-of-a-partial-order-and-its-dual)
Nik Weaver found a counterexample in a very nice, complete (and instantaneous!) answer, so I modify the question ... | https://mathoverflow.net/users/112382 | Is a simple graph matrix the sum of a "shiftordered" matrix and its transposed matrix | Expanded comment:
It seems that essentially the same counting argument applies as the one for the previous question (mathoverflow.net/q/306562):
We can make first $m$ rows and last $m$ columns of an $n\times n$ matrix zero in $n+1$ different ways, this increases the total number of matrices $S$ at most $(n+1)$ times.... | 3 | https://mathoverflow.net/users/24076 | 306565 | 133,642 |
https://mathoverflow.net/questions/304448 | 3 | Let $J$ be an interval of integers viewed as a linearly ordered set, and let $I \subseteq \mathbf{N}(J)$ be the subsimplicial set given by the union of the elementary edges $(x, x+1)$.
The inclusion $I \to \mathbf{N}(J)$ is a categorical equivalence, and so for a quasi-category $\mathcal{C}$, functors $\mathbf{N}(J) ... | https://mathoverflow.net/users/nan | Specifying complexes in quasicategories via squares | The specific example of presenting $J^{\Delta[1]}$ can be resolved by the trick of observing it is a retract of $J \times J$.
Let $K$ be the subsimplicial set of $I \times I$ consisting of:
* The vertices $(x,y)$ with $x \leq y$
* For each $x \leq y < \max(I)$, the elementary square with top-left vertex $(x,y)$
* F... | 1 | https://mathoverflow.net/users/nan | 306574 | 133,643 |
https://mathoverflow.net/questions/306573 | 0 | For a univariate distribution or a univariate random variable, we call it continuous/absolutely continuous if its cumulative distribution function (CDF) is continuous/absolutely continuous. Now I am trying to extend this concept to the multivariate case, and want the following holds: The continuity of a random vector d... | https://mathoverflow.net/users/126001 | Questions on a new definition of continuous multivariate distribution | 1. I believe the term *continuous distribution* (or a *continuous measure*) is often used to refer to a distribution without atoms.
2. Homeomorphisms of $\mathbb{R}^2$ can be *very singular*. For example, there is a homeomorphism $\Phi$ that maps the vertical interval $\{(0, y) : y \in [0, 1]\}$ into the [Osgood curve]... | 2 | https://mathoverflow.net/users/108637 | 306583 | 133,645 |
https://mathoverflow.net/questions/306539 | 4 | Let $f : X \to \operatorname{Spec}(R)$ be a flat, projective morphism with reduced fibers, where $R =\mathbb{C}[[t]]$. One may assume that the dualizing sheaf of the special fiber is trivial. Let $\mathcal{F}$ be a coherent sheaf on $X$, not necessarily flat over $R$. Let $\mathcal{G}$ be the sheaf of $t$-torsion secti... | https://mathoverflow.net/users/58651 | Cohomology of $t$-torsion subsheaf | $\def\C{\mathbb{C}}\def\OO{\mathcal{O}}\def\cL{\mathcal{E}}\def\cF{\mathcal{F}}$Let $X\_0$ be a variety over $\C$ and $X=X\_0\times\_{\C}R$. Denote by $i:X\_0\to X$ the obvious closed immersion.
Let $\cL$ be a coherent sheaf on $X\_0$. We will construct an extension $0\to i\_\*\cL\to \cF\xrightarrow{p} i\_\*\OO\to 0$... | 3 | https://mathoverflow.net/users/39304 | 306584 | 133,646 |
https://mathoverflow.net/questions/306566 | 10 | We know that there are non-triangulable 4-manifolds, such as the E$\_8$ manifold.
* Can E$\_8$ manifold be a boundary of some 5-manifold $M\_5$? Can such a $M\_5$ be triangulable or non-triangulable? What are the possible $M\_5$ (s)?
* I wonder whether the non-triangulable 4-manifold can always be a boundary of some ... | https://mathoverflow.net/users/27004 | Non-triangulable 4-manifold as a boundary of some 5 manifold | Your questions are answered by Hsu in his paper [4-Dimensional Topological Bordism](https://core.ac.uk/download/pdf/82491711.pdf).
In particular, associated to any closed oriented topological 4-manifold $X$ is a signature $\sigma(X) \in \Bbb Z$ and the Kirby-Siebenman class $\text{ks}(X) \in H^4(X;\Bbb Z/2) = \Bbb Z/... | 15 | https://mathoverflow.net/users/40804 | 306585 | 133,647 |
https://mathoverflow.net/questions/306578 | 11 | Is every complete Boolean algebra isomorphic to a quotient, as a Boolean algebra, of some powerset algebra $\wp(X)$?
It is not true for arbitrary Boolean algebras, see the comments, or see [my MathSE question](https://math.stackexchange.com/questions/2859214/is-every-boolean-algebra-isomorphic-to-the-quotient-of-a-po... | https://mathoverflow.net/users/126347 | Is every complete Boolean algebra isomorphic to the quotient of a powerset algebra? | This variation of the question comes from the comments on the original question.
**The question is whether all (complete) BAs are isomorphic in the category BA to a quotient of a powerset algebra.**
---
The Sikorski extension theorem guarantees that every complete BA is a quotient of a power set algebra.
**Th... | 20 | https://mathoverflow.net/users/75735 | 306586 | 133,648 |
https://mathoverflow.net/questions/306567 | 0 | Fix a natural number $n\geq 1.$
Let $\mu$ be a norm on $\mathbb{R}^n$ satisfying
$$\mu(0,...,0,\stackrel{i}{1},0,...,0) = 1 \quad\text{for all }1\leq i\leq n.$$
Let
$$B\_{\mu} = \{(a\_1,...,a\_n)\in \mathbb{R}^n : \mu(a\_1,...,a\_n)\leq 1\},$$
that is, ball of $\rho$-norm centered at origin.
Let
$$B\_{\ell^1} = \{(... | https://mathoverflow.net/users/42411 | Does there exists an extreme point $(a_1^*,...,a_n^*)$ of $B_{\mu^*}$ such that $a_i^*\neq 0$ for all $1\leq i\leq n$ and $\sum_{I=1}^n a_i^*a_i=1?$ | I think that you assumed that $a\_i\ne 0$ for all $i$, otherwise an easy "no" answer is given by the vector $(1,0,\dots,0)$ in the Euclidean space.
Assuming that this correction was made, consider the following space: $\ell\_\infty^2\oplus\_\infty\ell\_2^{n-2}$. This is the space of sequences of length $n$ such that ... | 1 | https://mathoverflow.net/users/37822 | 306589 | 133,649 |
https://mathoverflow.net/questions/306575 | 5 | Let $k$ be an algebraically closed field of char $p>0$ and $X$ be a proper smooth variety over $k$ that is simply connected. Then we know $Pic(X)$ does not contain any $\ell$-torsion for $\ell \not= p$, so Picard variety $Pic^0$ is trivial and $Pic(X)$ is a finitely generated abelian group.
Do we have $Pic(X)[p]=0$ a... | https://mathoverflow.net/users/102104 | p-torsion in Picard group of simply-connected variety of char p | What follows is more relevant to TKe's question in the comments than it is to the original question. Regarding the original question, I believe that Enriques surfaces in characteristic 2 provide the simplest and best-studied example of this phenomenon.
However, this phenomenon does happen in every characteristic. For... | 7 | https://mathoverflow.net/users/13265 | 306591 | 133,650 |
https://mathoverflow.net/questions/306383 | 2 | Suppose $X$ and $Y$ are compact metric spaces. Let $\varphi\colon C(X)\to M\_{n}(C(Y))$ be any $\*$-homomorphism. If $\pi$ is an irreducible representation of $M\_{n}(C(Y))$, then $\pi$ is unitarily equivalent to a point evaluation $\textrm{ev}\_{y}$. The $\*$-homomorphism $\textrm{ev}\_{y}\circ\varphi\colon C(X)\to M\... | https://mathoverflow.net/users/126776 | Closeness of points in the irreducible decomposition of a C$^{*}$-algebra representation | Yes, it varies continuously. To see this, suppose it did not, i.e., ${\rm max}\_i {\rm min}\_j d(x^y\_i, x^{y\_m}\_j) \not\to 0$. Passing to a subsequence, we can assume that ${\rm min}\_j d(x^y\_i, x^{y\_m}\_j)$ is bounded away from $0$ for some fixed value of $i$. Now find $f \in C(X)$ such that $f(x\_i) = 1$ and $f(... | 2 | https://mathoverflow.net/users/23141 | 306594 | 133,651 |
https://mathoverflow.net/questions/306470 | 14 | I was giving a talk in a seminar, and I mistakenly said that the coskeleton tower of a quasi-category was its Postnikov tower. Someone corrected me, but a discussion then ensued about what, precisely, this tower is. It appears to be homotopy-invariant, and each $k$-coskeleton looks like it is somehow related to somethi... | https://mathoverflow.net/users/1353 | What is the coskeleton tower of a quasi-category? | It turns out the answer is yes: $k$-coskeletalization of a quasicategory models truncation of an $(\infty,1)$-category to a $(k-1,1)$-category.
Let's collect some easy observations.
1. We have an adjunction $sk\_k \dashv cosk\_k : sSet \to sSet$.
2. $cosk\_k$ preserves the property of being a quasicategory, i.e. de... | 14 | https://mathoverflow.net/users/2362 | 306596 | 133,653 |
https://mathoverflow.net/questions/306603 | 14 | Bockstein homomorphim and obstruction of spin-c structure: Let $w\_2$ be the Stiefel Whintney class of manifold $M$. Let the Bockstein homomorphim $\beta$ be the
$$
H^2(\mathbb{Z}\_2,M) \to H^3(\mathbb{Z},M),
$$
such that $\beta(w\_2)$ is the integral cohomology class.
>
> (1) Is this true that for certain dimensi... | https://mathoverflow.net/users/106497 | Obstruction of spin-c structure and the generalized Wu manifods | Define the Wu manifold $W(n) = SU(n)/SO(n)$, the inclusion $SO \to SU$ given by thinking of $\Bbb C^n = \Bbb R^n \otimes \Bbb C$ (that is, including real matrices into complex matrices). Note that $W(1) = \*$, $W(2) = S^2$, and $W(3)$ is what is usually called the Wu manifold.
There is a natural map $W(n) \to W(n+1)$... | 16 | https://mathoverflow.net/users/40804 | 306605 | 133,655 |
https://mathoverflow.net/questions/306568 | 2 | I am a physicist and I am aware that this forum is for professional mathematical questions, but please be not too hard on my notation.
I encountered the following integral equation for functions $f:[0,\infty) \rightarrow [0,\infty)$ that is
$$f(x) = \frac{C}{(1+x)} \int\_{0}^{\infty} \frac{f(y)}{1+x+y} \ dy$$
an... | https://mathoverflow.net/users/126870 | Uniqueness of solution depending on constant? | The answer to the question is as follows:
**Theorem 1.** For every $C > 0$ there is, up to scalar multiples, only one function $0 \le f \in L^1 := L^1((0,\infty))$ such that $Tf = f$ (where $T$ is the $C$-dependent operator from the question).
For the proof we need a few preparations.
First we quote the following... | 3 | https://mathoverflow.net/users/102946 | 306607 | 133,656 |
https://mathoverflow.net/questions/306598 | 6 | Let $G$ be a finite group and let $D(g)$ be a projective representation of $G$ i.e.
\begin{equation}
D(g) D(h) = e^{i \omega(g,h)} D(gh)
\end{equation}
These can be classified by the equivalence relation $\omega(g,h) \sim \omega(g,h)+\theta(g)+ \theta(h) - \theta(gh)$ subject to the condition $\omega(g,h)+\omega(gh,l)-... | https://mathoverflow.net/users/57270 | Relationship between irreducible representations of the Schur covering group and elements of $H^2(G,U(1))$ | The answer to your question is **Yes.** Consider your covering group $C$ as a central extension:
$$1 \to N \to C \to G \to 1$$
and suppose it is given by a 2-cocycle $\alpha \in H^2(G, N)$. Then for any homomorphism $f: N \to \mathbb{C}^\times$, we get a 2-cocycle $f \circ \alpha \in H^2(G,\mathbb{C}^\times)$ by compos... | 4 | https://mathoverflow.net/users/121 | 306609 | 133,657 |
https://mathoverflow.net/questions/306234 | 8 | I am looking for an example of a smooth Fano $3$-fold $X$ over $\mathbb{C}$, with a non-trival $\mathbb{C}^{\*}$-action, which satisfies the following properties:
1. There is a $\mathbb{C}^{\*}$-action such that the fixed point set is finite.
2. $Aut(X)$ contains no copy of $(\mathbb{C^{\*}})^{2}$.
3. The rank of the... | https://mathoverflow.net/users/99732 | $\mathbb{C}^{*}$-actions on Fano $3$-folds | Choose $X$ to be the blow-up of a smooth quadric $Q\subset \mathbb{P}^4$ at a curve $\Gamma\subset Q$ being a smooth normal rational quartic curve.
In coordinates, you can for instance choose $\Gamma$ to be the image of
$$\mathbb{P}^1\to \mathbb{P}^4, [u:v]\mapsto [u^4:u^3v:u^2v^2:uv^3:v^3]$$
and $Q$ to be given by... | 3 | https://mathoverflow.net/users/23758 | 306610 | 133,658 |
https://mathoverflow.net/questions/306181 | 3 | A type universe is a type of small types that is closed under the basic type formation operations (dependent product, sum, coproduct etc.), that is to say for example that from
* $A \colon U\_i$ and
* $x \colon A \vdash B[x] \colon U\_i$
derive $\prod(x\colon A).B[x] \colon U\_i$.
I haven't yet found anything in ... | https://mathoverflow.net/users/28145 | Can a type in a lower universe be formed from types in higher universes? | I expect you're right that if the only primitive rules for universes are "closure" ones such as
$$\frac{\vdash A:U\_i \qquad x:A \vdash B[x] : U\_i}{\vdash \prod(x:A). B[x] : U\_i}$$
then there should be a metatheorem that whenever $\prod(x:A). B[x] : U\_i$ it must have been derived by this rule so that we have $A:... | 4 | https://mathoverflow.net/users/49 | 306627 | 133,665 |
https://mathoverflow.net/questions/306633 | 13 | I want to learn about homotopy theory on number fields, and I heard that the theory of motives made it possible, so I want to know what is a good textbook for motive theory.
To be honest, I don’t know algebraic geometry (I only read Hartshorne).
So please tell me what is needed to get some knowledge about Algebraic G... | https://mathoverflow.net/users/118682 | What is the best reference for motives? | I would add this as a comment, but I do not have enough reputation to do so.
While there are certainly more contemporary references, Voevodsky's "Triangulated category of motives over a field" is a place where you can read about motives (<https://www.math.ias.edu/vladimir/sites/math.ias.edu.vladimir/files/s5.pdf>). ... | 8 | https://mathoverflow.net/users/113828 | 306634 | 133,667 |
https://mathoverflow.net/questions/306246 | 6 | $\require{AMScd}$Let $\cal K$ be a 2-category, and $j : A\to B$ one of its 1-cells. Assume that the induced map
$$
j^\* : {\cal K}(B,B)\to {\cal K}(A,B)
$$ precomposing with $j$ has a left adjoint $j\_!$, the *left extension along $j$*.
Given 1-cells $x,y\in {\cal K}(A,B)$, it is possible to obtain canonical maps
*... | https://mathoverflow.net/users/7952 | The skew monoidal structure induced by a functor | There is a fast and down to earth proof. We use the letters $R$ and $L$ to indicate the right and left adjoints of morphisms.
For the first one, just call $a=j\_!x\circ y$ and $b=j\_!x\circ j\_!y$, we have one morphism $\phi:a\to j^\*b$ with left adjoint $L\phi:j\_!a\to b$. Then your diagram becomes
$$\begin{CD}
j^\*... | 2 | https://mathoverflow.net/users/45660 | 306645 | 133,670 |
https://mathoverflow.net/questions/306637 | 22 |
>
> **Conjecture:** Let $f:\mathbb{R}→\mathbb{R}$ be an **everywhere** differentiable function and assume that $f(x)+f′(x)∈ \{-1,1\}$ **almost everywhere** and $f'(0)=0$. Then is $f$ necessarily a constant function?
>
>
>
Can you give me a counter-example? I have already asked the question [here on MathSE](https... | https://mathoverflow.net/users/126827 | Differential equation changing sign almost everywhere | Your conjecture is **true** and there is no counterexample.
---
Suppose, contrary to the above claim, that your conjecture is false. Define $$g(x) = f(x) + \int\_0^x f(y) dy,$$ so that $g'(x) = f'(x) + f(x)$. Thus, $g$ is everywhere differentiable, $g'(x) \in \{-1,1\}$ almost everywhere, and $g'$ is not a constan... | 38 | https://mathoverflow.net/users/108637 | 306647 | 133,671 |
https://mathoverflow.net/questions/306646 | 7 | Is there a non integrable $2$ dimensional distribution $D$ of a $3$ dimensional Riemannian manifold such that the distribution is totally geodesic in the following sense:
Every geodesic whose tangent vector of its intitial point is tangent to the distribution then the tangent vector at all its points is tangent to $D... | https://mathoverflow.net/users/36688 | A non integrable distribution which is totally geodesic | Yes, the standard contact structure on the unit three-sphere in $\mathbb{R}^4 = \mathbb{C}^2$, for instance. The Legendrian great circles are the intersections of the sphere with the Lagrangian two-planes.
| 9 | https://mathoverflow.net/users/21123 | 306655 | 133,675 |
https://mathoverflow.net/questions/306462 | -1 | Given a function of real numbers f(x), I can create approximations to arbitrary precision using Taylor polynomials.
Is there something equivalent in the discrete case when I have a sequence of integers that I want to approximate to arbitrary precision.
| https://mathoverflow.net/users/126788 | Create approximations of finite integer sequence | One reason for approximating a real function by Taylor polynomials is to use properties of polynomials (and the real numbers) to estimate the function at an unknown value. In order for this to be useful, f has to behave consistently with the assumptions needed to use the approximations. In particular, f has to be a cer... | 1 | https://mathoverflow.net/users/3402 | 306660 | 133,678 |
https://mathoverflow.net/questions/306653 | 7 | Let $A$ be a real square matrix of size $n \times n$. Is there an upper bound on the minimum spectral norm under diagonal similarity, i.e.,
$$
s(A) = \min\_{D} \lVert D^{-1} A D\rVert\_2,
$$
where $D$ is a non-singular, diagonal real matrix. Also, is there are a relation between $s(A)$ and the spectral radius $\rho(... | https://mathoverflow.net/users/51478 | Minimize spectral norm under diagonal similarity | You cannot get an upper bound in general in terms of the spectral radius $\rho(A)$. Counterexample: if
$$
A = \begin{bmatrix} x & 1 \\ -x^2 & -x \end{bmatrix}
$$
then $\rho(A) = 0$ and $s(A) = 2|x|$. (This $A$ is essentially the most general $2\times 2$ matrix whose eigenvalues are both $0$.)
| 5 | https://mathoverflow.net/users/1044 | 306661 | 133,679 |
https://mathoverflow.net/questions/306663 | 14 | It is well known that as the negative discriminant $-D$ goes to infinity, the number of quadratic forms of discriminant $-D$ belonging to the principal genus also goes to infinity. Can we say anything about the asymptotic equidistribution of the corresponding CM points on the classical modular curve? I know that Duke '... | https://mathoverflow.net/users/422 | Equidistribution of CM points in the principal genus | This is known, and follows from Theorem 2 in Harcos and Michel's paper *The subconvexity problem for Rankin-Selberg $L$-functions and equidistribution of Heegner points. II* (Invent. math., vol. **163**, 2006, pp. 581--655). This result states, more generally, that there exists an absolute positive constant $\epsilon >... | 11 | https://mathoverflow.net/users/26522 | 306666 | 133,681 |
https://mathoverflow.net/questions/306359 | 2 | Let algebras be finite dimensional over a field $K$ and let $J$ denote the Jacobson radical (this is the intersection of all maximal right ideals) of an algebra. Being hereditary means that the algebra has global dimension at most 1.
Question:
>
> For the class of algebras $A$ with $Ext\_A^1(J,J) \neq 0$, can we ... | https://mathoverflow.net/users/61949 | $Ext_A^1(J,J)$ for the Jacobson radical $J$ of an algebra $A$ | Assume $J$ is not projective as a right $A$-module, or equivalently that $A$ is not hereditary. Then $\mbox{Ext}^1\_A(J,J) \neq 0$ and it is possible to write down a somewhat explicit non-split extension of direct summands of $J$ as follows.
Let $e\_1, \ldots, e\_n$ be a complete set of pairwise orthogonal primitive id... | 3 | https://mathoverflow.net/users/11791 | 306677 | 133,684 |
https://mathoverflow.net/questions/306111 | 2 | Let $B\_h (x)$ be the ball of radius $0<h \ll 1$ centered at $x\in \mathbb{R}^d$.
Let $I=[0,1]^{d-1}$ be the unit cube in $\mathbb{R}^{d-1}$, and let $f:I \to \mathbb{R}$ be a $C^2$ function. Then $$M:\,=\left\{(x,f(x)) ~ ~ | ~ ~ x\in I \right\} \, ,$$
is a $d-1$ dimensional manifold embedded in $\mathbb{R}^d$.
*... | https://mathoverflow.net/users/42864 | Projection of a ball in the ambient space to a manifold | If the *induced Riemannian metric* refers to the (Euclidean) length $d(x, y)$ of the shortest path contained in $M$ with given endpoints $x, y$, then this is a completely elementary question. Or, I have misunderstood the problem completely: in this case let me know and I will delete this answer.
---
Clearly, $d(x... | 1 | https://mathoverflow.net/users/108637 | 306688 | 133,688 |
https://mathoverflow.net/questions/306490 | 2 | Let $M$ be a simply connected closed Riemannian manifold. How does one find a necessary condition going both ways that may be imposed on $M$ (perhaps on the curvature of $M$ and on torsion) which guarantees that the rational cohomology ring $H^\*(M;\mathbb{Q})$ needs at least two generators? That is, how does one force... | https://mathoverflow.net/users/98896 | Show that the rational cohomology ring $H^*(M;\mathbb{Q})$ needs at least two generators | If $M$ is simply-connected and has reducible holonomy, then a [theorem of de Rham implies that $M$ is a product](https://en.wikipedia.org/wiki/Holonomy#Reducible_holonomy_and_the_de_Rham_decomposition), and hence does not have homology generated by one element.
| 11 | https://mathoverflow.net/users/1345 | 306690 | 133,690 |
https://mathoverflow.net/questions/306715 | 3 | Let $\kappa >\aleph\_0$ be a cardinal. Is there a connected space $(X,\tau)$ with $|X| = \kappa$ such that for every [dense set](https://en.wikipedia.org/wiki/Dense_set) $D\subseteq X$ we have $|D|=|X|$?
| https://mathoverflow.net/users/8628 | Connected spaces where every dense set is large | Yes. Take any countable connected Hausdorff space $C$, fix any point $c\in C$, and consider the quotient space $X=C\times \kappa/ \{c\}\times \kappa$. Here the cardinal $\kappa$ is endowed with the discrete topology.
It is easy to see that the space $X$ is connected and Hausdorff and each dense subset of $X$ has car... | 6 | https://mathoverflow.net/users/61536 | 306716 | 133,696 |
https://mathoverflow.net/questions/306724 | 4 | Let $X$ and $Y$ be two quandles and $f: X \rightarrow Y$ be a quandle homomorphism. Then we can define a map $\bar f: Inn(X) \rightarrow Inn(Y)$ as $\bar f(S\_a)=S\_{f(a)}$, where $a \in X$. Then $\bar f$ may not be a group homomorphism. But I am not able to construct such example.
I have posted this question on Mat... | https://mathoverflow.net/users/126897 | Quandle homomorphism does not always induces group homomorphism on inner automorphism groups of quandles | Let's make sure we agree on definitions.
A **quandle** is an algebraic structure $(A,\*)$ for which
each right multiplication map $S\_a(x)= x\*a$ by an element $a\in A$
is an automorphism fixing $a$. These right multiplications
are called **inner automorphisms** of the quandle, and **$\textit{Inn}(A)$**
is the group ... | 3 | https://mathoverflow.net/users/75735 | 306726 | 133,699 |
https://mathoverflow.net/questions/297870 | 13 | The setup is similar to [this question](https://mathoverflow.net/questions/296056/some-binomial-coefficient-determinants), but generalizes the size of the Hankel matrix. We'll define
$$d(n,k,r):=\det\left(\binom{2i+2j+k+r}{i+j}\right)\_{i,j=0}^{kn-1}.$$
*Edit: Thanks to Johann Cigler for checking, which made me dis... | https://mathoverflow.net/users/29783 | Some more binomial coefficient determinants | Johann Cigler and I have posted a proof of many of these observations on arXiv:
["An interesting class of Hankel determinants"](https://arxiv.org/abs/1807.08330), arXiv:1807.08330.
Let $d\_r(N)=\det\left({2i+2j+r\choose i+j}\right)\_{i,j=0}^{N-1}$. We show that for $k,n\ge 1$,
\begin{align}
&d\_{2k+1}((2k+1)n)=d\_{... | 5 | https://mathoverflow.net/users/112641 | 306728 | 133,700 |
https://mathoverflow.net/questions/306729 | 3 | I'm currently trying to figure out the following inequality. It looks like an inequality for the exponential sum, but I can't verify it or find a source explaining it any further. Most likely it has to do with the remainder I guess...
$$|E[\exp(itX\_{n,k})|F\_{n,k-1}]-1-\frac{1}{2}t^2E[X\_{n,k}^2|F\_{n,k-1}]|\\
\leq \f... | https://mathoverflow.net/users/126914 | Inequality for exponential sum in Dvoretzky 1972 | First here, there is a typo in the Dvoretzky paper: there must be $-1+\frac{1}{2}t^2E[X\_{n,k}^2|F\_{n,k-1}]$ instead of $-1-\frac{1}{2}t^2E[X\_{n,k}^2|F\_{n,k-1}]$ there. Otherwise, the inequality will not be true in general. Indeed, let, for brevity, $X:=X\_{n,k}$, $F:=F\_{n,k-1}$, $E\_F Z:=E(Z|F)$, and $c:=\epsilon$... | 4 | https://mathoverflow.net/users/36721 | 306730 | 133,701 |
https://mathoverflow.net/questions/304554 | 17 | Let $G$ be a finite group, and let $M$ be a group on which $G$ acts (via a homomorphism $G\to \operatorname{Aut}(M)$).
If $M$ is abelian, hence a $\mathbb{Z}G$-module, there is a primary decomposition
$$
H^k(G;M)=\bigoplus\_p H^k(G;M)\_{(p)}
$$
for each $k>0$, where $p$ ranges over the primes dividing $|G|$ and $H^k... | https://mathoverflow.net/users/8103 | primary decomposition for nonabelian cohomology of finite groups | It turns out the answer is no. Here I'll sketch a counter-example in which $H^1(G;M)$ is non-trivial (in fact infinite), while $H^1(H;M)$ is trivial for all proper subgroups $H<G$.
Let $G=A\_5$, the alternating group on $5$ letters. Then $G$ acts on the $2$-spine of the punctured Poincaré $3$-sphere without fixed poi... | 7 | https://mathoverflow.net/users/8103 | 306738 | 133,704 |
https://mathoverflow.net/questions/306686 | 13 | I have read in [these lecture notes](http://webpages.math.luc.edu/~ptingley/oldseminars/QuantumGroupsSpring2011/lecture1.pdf) that every deformation $U\_h(\mathfrak{g})$ of $U(\mathfrak{g})$ is trivial, i.e. isomorphic to $U(\mathfrak{g})[[h]]$ as associative $\mathbb{C}[[h]]$-algebras. Why is this true? The reason the... | https://mathoverflow.net/users/117053 | Why is every deformation of the universal enveloping algebra of a complex semisimple Lie algebra trivial? | The article [Deformation par quantification et rigidite des
algebres enveloppantes](https://arxiv.org/abs/math/0211416v1) by M. Bordemann, A. Makhlouf, T. Petit addresses these questions. They call Lie algebras $\mathfrak{g}$ with $HH^2(U(\mathfrak{g}),U(\mathfrak{g}))=0$ *strongly rigid*, and show that then
every form... | 10 | https://mathoverflow.net/users/32332 | 306740 | 133,705 |
https://mathoverflow.net/questions/306654 | 6 | In homotopy theory, the word "norm" is commonly used in two different ways (well, surely there are other ways, but these two have a particular familial resemblence).
1. Let $G$ be a finite group. A $G$-spectrum $E$ can be restricted to an $H$-spectrum for any subgroup $H \subseteq G$, and there is an "inclusion of fi... | https://mathoverflow.net/users/2362 | Is there a relationship between norms/transfers in equivariant homotopy theory and norms in the Tate construction / ambidexterity? | Here's a bit more of an organized answer:
First you have to decide how you'd like to model the notion of a genuine $G$-spectrum. There are at least ways that it's currently in vogue to do this, each of which is convenient for different purposes. (My groups are finite below, and I won't write down functors that aren't... | 4 | https://mathoverflow.net/users/6936 | 306750 | 133,706 |
https://mathoverflow.net/questions/301224 | 6 | I have been trying to learn about [congruence groups](https://en.wikipedia.org/wiki/Congruence_subgroup). Here is an example:
\begin{eqnarray\*} \Gamma\big(1+2i\big) &=& \text{SL}\_2\big(\mathbb{Z}[i]\big)(1+2i) \\ \\
&=& \left\{ \left( \begin{array}{cc} a & b \\ c & d \end{array} \right)
: ad-bc = 1 \text{ and }
\l... | https://mathoverflow.net/users/1358 | What are the $2 \times 2$ matrix generators of $\text{SL}_2\big(\mathbb{Z}[i]\big)(2+i)$? | $PSL\_2(\mathbb{O}\_d)$ acts on the upper half-space model of hyperbolic 3-space in a nice way, namely the quotient can be viewed as a finite volume 3-orbifold. Since all principal congruence subgroups are finite index in $PSL\_2(\mathbb{O}\_d)$, all principal congruence subgroups are finitely generated and correspond ... | 2 | https://mathoverflow.net/users/27453 | 306754 | 133,707 |
https://mathoverflow.net/questions/306723 | 2 | I want to compute $$\max\_{\frac{1}{5 \theta }\leq \alpha \leq \frac{1}{2}} \left(\frac{\alpha\log (\alpha)}{1-\alpha} + \log \left( 1 - \alpha\right) + \frac{1}{1-\alpha} \cdot \left( f\left(\frac{1-\alpha}{2\alpha}\right) + 1 + \frac{41}{15\theta} \right)\right)$$ for
\begin{align\*}
f(x) = \begin{cases}
- 1.3- \log... | https://mathoverflow.net/users/126942 | Computing minimum / maximum of strange two variable funcion | Let $a:=\alpha$ and $u:=\frac1{5\theta}$. The condition $\theta\ge1$ (now added in the question) means that $0<u\le1/5$, which will be assumed henceforth.
We need to compute
\begin{equation}
\inf\_{u\in(0,1/2)}\sup\_{a\in[u,1/2]}F(u,a),
\end{equation}
where
\begin{equation\*}
F(u,a):=\begin{cases}
F\_1(u,a)&\text... | 3 | https://mathoverflow.net/users/36721 | 306757 | 133,709 |
https://mathoverflow.net/questions/306739 | 3 | I have asked this in MSE [here](https://math.stackexchange.com/questions/2854782/random-complex-eigenvalues-and-averages-of-traces), but got no interesting answers.
Suppose I have a random matrix $M$ of dimension $N$ which is real, but not symmetric. Suppose I know that, for large $N$, the marginal distribution of it... | https://mathoverflow.net/users/83671 | Random complex eigenvalues and averages of traces | Summary: I don't think the average $\langle {\rm Tr}(M^n)\rangle \to 0$ for $N\rightarrow\infty$, at least for $n$ even I will argue this average is $\propto\sqrt N$ because of the contributions from eigenvalues on the real axis.
---
• Consider first the eigenvalues $\lambda\_p=r\_p e^{i\phi\_p}$ of $M$ off the r... | 5 | https://mathoverflow.net/users/11260 | 306761 | 133,711 |
https://mathoverflow.net/questions/306752 | 2 | Virtual large cardinals belong to a relatively new breed of strong axioms of infinity. They often appear as statements of the following form:
>
> **Definition.** Suppose $A$ is a large cardinal property characterized by the existence of suitable embeddings. A cardinal is *virtually* $A$ if the embeddings characteri... | https://mathoverflow.net/users/82843 | On the Actual Potential of Virtual Large Cardinals | In the context of virtual large cardinals, it doesn't matter whether you consider arbitrary extensions or just forcing extensions.
The basic situation is that the virtual large cardinal properties are generally witnessed by the existence of an elementary embedding $j:M\to N$ between two structures $M$ and $N$ of the... | 6 | https://mathoverflow.net/users/1946 | 306762 | 133,712 |
https://mathoverflow.net/questions/306765 | 7 | If $V$ is a real vector space, then the **complexification** of $V$ is formally defined as $V^{\mathbb{C}}=V\otimes\_{\mathbb{R}}\mathbb{C}$. Is there an analogous complexification operation for a real $n$-dimensional Riemannian manifold $(M,g)$?
*Idea:* The notion of complexification exists for Lie groups, so perha... | https://mathoverflow.net/users/98896 | How does one complexify a real $n$-dimensional Riemannian manifold $(M,g)$? | I believe the following is meant:
Every smooth (real) manifold $M$ has a (unique) real-analytic structure compatible with the smooth structure. So, cover $M$ with real-analytic charts, i.e. whose transition functions are real-analytic diffeomorphisms
$$
\phi\_{ij}:=\phi\_j^{-1}\circ\phi\_i: U\_{ij}:=\phi\_i^{-1}(\phi... | 10 | https://mathoverflow.net/users/1849 | 306767 | 133,715 |
https://mathoverflow.net/questions/306759 | 7 | I'm teaching a second course in advanced linear algebra, following the second half of Hoffman-Kunze. I have come across what I believe to be an error, but I want confirmation (or refutation) by experts. Here is the setup:
Let $V$ be an $n$-dimensional inner product space over either $\mathbb C$ or $\mathbb R$. [Here ... | https://mathoverflow.net/users/6871 | Error in Hoffman-Kunze (normal operators on finite-dimensional inner product space with a cyclic vector) | I haven't thought carefully about your argument, but I agree that the corollary must be false. Suppose that $A$ is a normal $2 \times 2$ real matrix. The corollary claims that $A$ is orthogonally similar to a matrix $B$ in rational canonical form. Then $B$ is also normal, and is either diagonal (hence $A$ is symmetric)... | 7 | https://mathoverflow.net/users/1044 | 306769 | 133,716 |
https://mathoverflow.net/questions/306768 | 5 | This is a follow up to [my previous question](https://mathoverflow.net/questions/306752/on-the-actual-potential-of-virtual-large-cardinals) concerning virtual large cardinals, that are generally weaker axioms of infinity obtained from ordinary large cardinals through the so-called *virtualization* process.
>
> If $... | https://mathoverflow.net/users/82843 | What are examples of non-equivalent virtualizations of a large cardinal? | An important feature which separates the notion of virtual large cardinals from the related notion of generic large cardinals is that we only consider embeddings on set-sized structures. Since most large cardinals are characterized by embeddings of the entire universe, there will always be some arbitrary choices made i... | 15 | https://mathoverflow.net/users/5984 | 306771 | 133,718 |
https://mathoverflow.net/questions/306775 | 2 | What is the relationship between the Betti numbers $b\_i(M;\mathbb{Q})=rkH\_i(M;\mathbb{Q})$ of a simply connected closed Riemannian manifold $M$ and the dimension of rational homotopy $\dim\_{\mathbb{Q}}\pi\_i(M)\otimes\mathbb{Q}$ (if there is any)?
[Cross positing on MSE](https://math.stackexchange.com/questions/28... | https://mathoverflow.net/users/98896 | Relationship between the Betti numbers $b_i(M;\mathbb{Q})$ and the dimension of rational homotopy $\dim_{\mathbb{Q}}\pi_i(M)\otimes\mathbb{Q}$ | If $M$ is a closed, simply connected smooth manifold, then the rank of the rational homotopy groups $\pi\_i(M)\otimes Q$ equals the number of degree $i$ generators introduced in the construction of the minimal model of $M$. For certain manifolds that are *formal* (e.g., if $M$ is Kahler), their cohomology ring is quasi... | 8 | https://mathoverflow.net/users/113061 | 306778 | 133,720 |
https://mathoverflow.net/questions/306600 | 3 | Let $f$ be an arbitrary function in $L^2(0,\infty)$ and consider the function
$$(g\_f)(y) = \frac{1}{y-x\_0} \int\_{0}^{\infty} f(x) \left(\frac{xy}{(x^2+y^2+1)}\right)^2 \ dx$$
where $x\_0$ is an arbitrary point in $(0,\infty).$
I ask: Is the function $g\_f$ ever a function in $L^2(0,\infty)?$
In other words, ... | https://mathoverflow.net/users/126870 | Function square-integrable | Yes. Indeed set
$$
G\_f(y) = (y-x\_0) g\_f(y)
= \int\_0^\infty f(x) \, \left(\frac{xy}{(x^2+y^2+1)}\right)^{\!2} \, dx.
$$
It is easy to find linearly independent $f\_1,f\_2 \in L^2(0,\infty)$
such that each $G\_{f\_i}(y)$ is also in $L^2(0,\infty)$ and is differentiable
on $(0,\infty)$; for instance, $f\_1,f\_2$ can ... | 1 | https://mathoverflow.net/users/14830 | 306779 | 133,721 |
https://mathoverflow.net/questions/282743 | 7 | I'm trying to figure out the time-complexity of the problem I describe below, which I call the semi-ordered Eulerian tour problem or the SOET problem. Either finding an efficient algorithm for this problem or proving that this is actually in NP-hard would settle this question for me. Below I describe the problem, give ... | https://mathoverflow.net/users/115354 | Complexity of finding semi-ordered Eulerian tours in a 4-regular graph | The SOET problem is NP-Complete as we show in the following paper:
Axel Dahlberg, Jonas Helsen, Stephanie Wehner, **How to transform graph states using single-qubit operations: computational complexity and algorithms**,
[CoRR abs/1805.05306 (2018)](https://arxiv.org/abs/1805.05306).
See Corollary 3.4.1
| 1 | https://mathoverflow.net/users/115354 | 306783 | 133,722 |
https://mathoverflow.net/questions/306659 | 6 | An algebraic curve defined by a polynomial of degree n can have at most (n−1)(n−2)/2 singularities. What is the maximal number of singularities of a curve in a k-dimensional space in terms of the degree of the k-1 polynomial equations defining the curve?
P.S.: The comment of Mohan posted below gives an upper bound on... | https://mathoverflow.net/users/4274 | Maximal number of singularities of an algebraic curve | The maximum number of singularities, properly counted, is equal to the arithmetic genus. For example, take the curve $x\_1^2=x\_2,\, x\_2^2=x\_3,\dots,x\_{k-1}^2=x\_k$. This curve has a rational parametrization. Thus, the geometric genus is equal to zero and the number of singularities (at infinity) is equal to the ari... | 0 | https://mathoverflow.net/users/4274 | 306806 | 133,728 |
https://mathoverflow.net/questions/306803 | 9 | Let $k$ be a field of characteristic $0$ and let $\mathfrak{g}$ be a finite dimensional Lie algebra over $k$. $\mathfrak{g}$ corresponds to a formal group scheme $\mathcal{G} = \text{Spf} (U(\mathfrak{g})^{\*})$. In fact, there is an equivalence of categories between finite-dimensional Lie algebras and infinitesimal fo... | https://mathoverflow.net/users/30211 | Exponential map of a Formal Group Scheme | $\newcommand{\g}{\mathfrak{g}}$
In a way this is tautological, in the sense that $\mathcal G$ can be seen as a formal exponentation of $\g$, although I don't think there is an actual exponential map from one to the other in general. Rather, there is a map (in fact an isomorphism of formal schemes) from the formal compl... | 9 | https://mathoverflow.net/users/13552 | 306807 | 133,729 |
https://mathoverflow.net/questions/306802 | 8 | This is a cross-post from a MSE [question](https://math.stackexchange.com/q/2842085/26141) which received no answers. Beware that the notation here is a little different.
Consider the following lifting problem(s):
$\require{AMScd}$
\begin{CD}
& & & & E\\
& & & @VV{p}V\\
Y @>{g}>> X @>{f}>> B
\end{CD}
Let's work ... | https://mathoverflow.net/users/21848 | Obstructions for the lifting problem after a pull-back | First, note that since you are assuming $F$ is $d-1$-connected, the primary obstruction lies in $H^{d+1}$, not $H^d$.
Now, consider the diagram $\require{AMScd}$
\begin{CD}
& & & & S^5\\
& & & @VV{p}V\\
S^4 @>{g}>> S^3 @>{id}>> S^3
\end{CD}
where $p$ represents the nontrivial element in $\pi\_5(S^3) \cong \mathbb{Z... | 11 | https://mathoverflow.net/users/104342 | 306808 | 133,730 |
https://mathoverflow.net/questions/306827 | 5 | Let $F$ be an algebraically closed field and $\mathbb{P}^1$ the projective line over $F$. Suppose $V\_1, V\_2$ are two 1-dimensional subvarieties of the 2-dimensional variety $\mathbb{P}^1\times\mathbb{P}^1$.
Now $V\_1$ and $V\_2$ do not necessarily intersect, as the simple example $V\_i=\{x\_i\}\times\mathbb{P}^1$ ... | https://mathoverflow.net/users/127000 | Intersections in $\mathbb{P}^1\times\mathbb{P}^1$ | The curve $V\_i$ is given by the vanishing of a polynomial $F\_i(x\_1,x\_2,y\_1,y\_2)$ that is homogeneous in $x\_1,x\_2$ of degree $d\_{i,1}$ and homogeneous in $y\_1,y\_2$ of degree $d\_{i,2}$. Then counting intersection points with multiplicities,
$$
V\_1 \cdot V\_2 = d\_{1,1}d\_{2,1} + d\_{1,2}d\_{2,2}.
$$
So $V\_1... | 11 | https://mathoverflow.net/users/11926 | 306830 | 133,735 |
https://mathoverflow.net/questions/306820 | 5 | Let $\Gamma$ be a countable discrete group and $\lambda$ the left regular representation of $\Gamma$ on $l^2(\Gamma)$. Let $\rho:\Gamma\rightarrow U(H)$ be a unitary representation of $\Gamma$ on some separable infinite-dimensional Hilbert space $H$, and consider the representation $\lambda\otimes\rho$ of $\Gamma$ on $... | https://mathoverflow.net/users/78729 | Fell's trick for Lie groups | Fell absorption works for all locally compact groups and all strongly continuous unitary representations; the intertwining map that you give in the discrete case can be generalized to
$$ W: L^2(G)\otimes\_2 H \to L^2(G)\otimes\_2 H $$
given by
$$ \langle W(\xi\otimes f), (\eta\otimes g) \rangle := \int\_G f(s)\ov... | 7 | https://mathoverflow.net/users/763 | 306837 | 133,737 |
https://mathoverflow.net/questions/306833 | 3 | Suppose that $G:M\leftrightarrow N: U$ is a Quillen equivalence between two model categories. Suppose that $a\in M$ is a fibrant object, is it true that there exists always a fibrant object $b\in N$ and weak equivalence $a\rightarrow U(b)$ in $M$ ?
We are not assuming cofibrancy property on $a$ and $b$.
| https://mathoverflow.net/users/127002 | Quillen equivalence, fibrant objects | Here is a counter-example to the dual assertion (so that you can get a counter-example to your original question by taking the opposite model categories). Consider the category ${\rm Set\_\Delta}$ of simplicial sets with the Kan-Quillen model structure. Since ${\rm Set}\_{\Delta}$ is right proper and the inclusion $i\_... | 8 | https://mathoverflow.net/users/51164 | 306838 | 133,738 |
https://mathoverflow.net/questions/306811 | 8 | On a 2n-dimensional phase-space with coordinates $x$ and $p$, the Moyal product can be written explicitly as
$$g(x,p) \star h(x,p) = g(x,p) e^{\frac{i}{2}\left( \overleftarrow{\partial\_x} \cdot \overrightarrow{\partial\_p} - \overrightarrow{\partial\_x} \cdot \overleftarrow{\partial\_p}\right)} h(x,p) \, .$$
Assum... | https://mathoverflow.net/users/37773 | Moyal $\star$-product inverse? | The inversion is conveniently described in terms of the Fourier transform
$$g(x,p)=\int dy\,e^{-iyp}G(x+y/2,x-y/2).$$ Then the composition $f(x,p)=g(x,p)\star h(x,p)$ is a matrix multiplication [1],
$$F(x,y)=\int dz\, G(x,z)H(z,y).\qquad(\ast)$$
So to find $h$ if $f$ and $g$ are given one would first calculate the Four... | 4 | https://mathoverflow.net/users/11260 | 306843 | 133,739 |
https://mathoverflow.net/questions/306127 | 6 | Let $X$ be a smooth variety over a field $k$, and let $Y$ be a smooth subvariety. In the literature, I've seen two versions of the **deformation to the normal cone**:
**Verdier's version:** $\tilde{X}\_Y^\mathrm{Ver} := \operatorname{Bl}\_{Y\times \{0\}}(X\times \Bbb A^1\_k) - \operatorname{Bl}\_Y(X)$
**Fulton's ve... | https://mathoverflow.net/users/36720 | Fulton's deformation to the normal cone vs Verdier's | *The following answer was emailed to me by Claude Sabbah. I have received his permission to post it here.*
Whenever you need to get some object on $Y$ from an object existing on the normal cone (in fact normal bundle in your case), you need to proceed by pushforward. As it is better to use a proper pushforward, the v... | 5 | https://mathoverflow.net/users/36720 | 306844 | 133,740 |
https://mathoverflow.net/questions/306857 | 14 | This is inspired by the Alexander Shen's post here: <https://www.facebook.com/groups/mathpuz/permalink/1058782384297603/> (the post is in Russian, but it is easy Russian, and google translate should work fine).
All proofs of irrationality of $\sqrt{n}$ where $n$ is not a perfect square that I know are using (explicit... | https://mathoverflow.net/users/nan | Can the Induction axiom in the Peano arithmetic be replaced by the irrationality of $\sqrt{2}$? | One can prove that $\sqrt{2}$ is irrational and indeed that any particular $\sqrt{n}$ is irrational (if $n$ is not a perfect squares) assuming only the principle of induction for $\Delta\_0$ formulas, which is much weaker than the full strength of PA. In this sense, the answer is negative.
The classical proof of the ... | 23 | https://mathoverflow.net/users/1946 | 306860 | 133,744 |
https://mathoverflow.net/questions/8534 | 15 | Let $M$ be Riemannian manifold and $\tilde M$ be its universal cover (with induced metric).
What is the upper bound for $k=\mathop{diam}\tilde M/\mathop{diam} M$ in terms of $m=|\pi\_1(M)|$ (or $\pi\_1(M)$)?
**Comments:**
* There is a similar answered question [here](https://mathoverflow.net/questions/7732/diameter... | https://mathoverflow.net/users/1441 | Diameter of universal cover | As both Anton and Greg pointed out, it is enough to look at the Cayley graph of a presented finite group $G$ with only quadratic and cubic relators and study how large $diam(G)$ can be in terms of $\vert G \vert$. Note that the property that all relators are quadratic or cubic implies that $G$ is the $1$-skeleton of a ... | 5 | https://mathoverflow.net/users/113851 | 306869 | 133,747 |
https://mathoverflow.net/questions/306801 | 6 | Let $p$ be a prime with $p\equiv 3 \mod 4$, for any $\mathcal{I} \subset \lbrace 0,...,p-1 \rbrace $ and any $\mathcal{J} \subset \lbrace 0,...,p-1 \rbrace $ with $\vert\mathcal{I}\vert \leq \sqrt{p} $ and $\vert\mathcal{J}\vert \leq \sqrt{p} $, I am looking for an upperbound on the following expression
\begin{align\*}... | https://mathoverflow.net/users/114045 | Upperbounding a sum of Legendre-Symbols | If $A$ and $B$ are two subsets of ${\Bbb Z}/p{\Bbb Z}$ with $|A|$ and $|B|$ being bigger than $p^{\alpha}$ (for some $\alpha>0$) then one expects that
$$
\Big| \sum\_{a\in A} \sum\_{b\in B} \chi(a+b) \Big| \le p^{-\delta} |A||B|,
$$
for some $\delta >0$ (depending only on $\alpha$). Here $\chi$ denotes a non-princi... | 10 | https://mathoverflow.net/users/38624 | 306870 | 133,748 |
https://mathoverflow.net/questions/306871 | 3 | Is there an example of a metric space $(X,d)$ whose corresponding [path metric](https://en.wikipedia.org/wiki/Intrinsic_metric), $d^\prime$ generates a strictly finer topology compared to the topology generated by $d$?
| https://mathoverflow.net/users/125490 | Path Metric Topology | Consider "the topologist's sine"
$$X = \left\{ \left(x, \sin \frac 1 x \right) \mid x>0 \right\} \cup \Big( \{0\} \times [-1,1] \Big) \subset \mathbb R^2$$
endowed with the distance induced by its natural embedding in $\mathbb R^2$. Clearly the sequence given by $s\_n = (\frac 1 {2n \pi}, 0)$ converges to $(0,0)$ i... | 4 | https://mathoverflow.net/users/54780 | 306877 | 133,750 |
https://mathoverflow.net/questions/306852 | 5 | 1) Is it possible to construct a $\mathbb{P}^1$-bundle $P\to B$ where $B$ is a proper variety and $P$ is not $\mathbb{P}(V)$ for a rank 2 vector bundle $V\to B$?
If we drop the properness assumption on $B$, I only know of a couple examples.
2) Does a Zariski locally trivial projective bundle come from the projectiv... | https://mathoverflow.net/users/37650 | $\mathbb{P}^1$-bundle over compact base | 1) There are many examples. As abx mentioned, they are called Severi-Brauer varieties, and they are related to 2-torsion classes in the Brauer group of $B$. If you want a geometric construction, to get a relatively simple example consider a general cubic 4-fold $X \subset \mathbb{P}^5$ containing a plane $\Pi \subset X... | 4 | https://mathoverflow.net/users/4428 | 306885 | 133,754 |
https://mathoverflow.net/questions/306883 | 10 | The modular equation $\Phi\_n(X,Y)$ is a polynomial in $\mathbf Z[X,Y]$ relating the modular invariant $j$ and the functions
$j\left(\frac{a\tau+b}{c\tau+d}\right)$, where $ad-bc=n$.
For example, we have identically
$$\Phi\_n(j(n\tau),j(\tau))=0.$$
It is often asserted that the coefficients of $\Phi\_n(X,Y)$ are... | https://mathoverflow.net/users/122104 | Why are the coefficients of the modular equation so large? | Paula Cohen, [On the coefficients of the transformation polynomials for the elliptic modular
function,](https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/article/on-the-coefficients-of-the-transformation-polynomials-for-the-elliptic-modular-function/6D1315068F3E83B0... | 15 | https://mathoverflow.net/users/11260 | 306897 | 133,756 |
https://mathoverflow.net/questions/306895 | 0 | Let $\{X\_i^n\}\_{i\in \mathbb{N}}$ and $\{Y\_i^n\}\_{i\in \mathbb{N}}$ be sequences of connected closed submanifolds of $\mathbb{S}^{n+2}$, with $n> 5$. Suppose that $\{X\_i^n\}\_{i\in \mathbb{N}}$ (resp. $\{Y\_i^n\}\_{i\in \mathbb{N}}$) has $X\_{\infty}\subset \mathbb{S}^{n+2}$ (resp. $Y\_{\infty}\subset \mathbb{S}^{... | https://mathoverflow.net/users/73454 | Hausdorff convergence of submanifolds in $\mathbb{S}^m$ | One cannot expect homology equivalence of limits. Here is a standard example for surfaces in $\mathbb R^3$ which can be easily adapted to your situation. Start from the unit sphere in $\mathbb R^3$ and attach a handle (still embedded in $\mathbb R^3$). Then look at the sequence with the handle getting smaller and disap... | 2 | https://mathoverflow.net/users/1573 | 306901 | 133,757 |
https://mathoverflow.net/questions/306909 | 5 | I am currently reading in [Boundary Conditions for Topological Quantum Field Theories, Anomalies and Projective Modular Functors](https://arxiv.org/abs/1409.5723), and have a (I guess) pretty basic question for my understanding of the (∞, n) category of Cobordisms...
Their (informal) definition of Bord(n) is
>
>... | https://mathoverflow.net/users/127040 | 2-morphisms for Bord(n) | The endomorphisms of the interval are, at this informal level of discussion, surfaces with $S^1$ boundary. More generally, if you want to talk about $\hom(M,N)$, where $M$ and $N$ are $k$-dimensional bordisms, then certainly $M$ and $N$ had better have the same domain and codomain $B = \partial M = \partial N$ (since t... | 3 | https://mathoverflow.net/users/78 | 306914 | 133,760 |
https://mathoverflow.net/questions/146632 | 3 | Does the set of bi-invariant Finsler metrics on $SU(N)$ exactly coincide the set of Finsler metrics with the one-parameter subgroups as their geodesics through the identity?
I know that being bi-invariant implies that the geodesics through the identity are exactly the one parameter subgroups. That is to say, every el... | https://mathoverflow.net/users/41654 | A property of bi-invarient Finsler metrics on SU(N) | I am re-editing this response because I got it wrong the first time around.
The following result, due to myself and José Barbosa Gomes, is (a small) part of the paper [Periodic solutions of Hilbert's fourth problem.](https://arxiv.org/abs/1809.02783)
**Theorem.** If a C2 reversible Finsler metric $F$ on a compact, ... | 5 | https://mathoverflow.net/users/21123 | 306915 | 133,761 |
https://mathoverflow.net/questions/306898 | 8 | In the textbook <https://www.springer.com/gp/book/9783034851688> (Klassische elementare Analysis, by M. Koecher) the following elegant recurrence relation is proved for $\zeta(2n)$ (on p. 157):
$$\left(n+\frac{1}{2}\right)\zeta(2n)=\sum\limits\_{m=1}^{n-1}\zeta(2m)\,\zeta(2n-2m). \tag{1}$$
In fact (1) is equivalent ... | https://mathoverflow.net/users/32389 | A recurrence relation for $\zeta(2n)$ - reference request | I have this recurrence in my collection of problems for my lessons in Analytic Number Theory. I have there the reference:
P. Ribenboim, Classical Theory of Algebraic Numbers, Springer, New York, 2001, p. 503.
I add a note that related formulas can be found in
S. Sekatskii, Novel integral representations of the R... | 8 | https://mathoverflow.net/users/7402 | 306918 | 133,762 |
https://mathoverflow.net/questions/297487 | 5 | Consider the measures on the circle, $M(\mathbb T)$, endowed with the convolution product which makes it a unital Banach algebra under the total variation norm. Denote by $\Delta$ the maximal ideal space of $M(\mathbb T)$. This space is quite large–recently it was shown that $\Delta$ [is non-separable](https://projecte... | https://mathoverflow.net/users/15129 | Are there any non-trivial convergent sequences in the maximal ideal space of the measure algebra? | Norbert is right (see the paper *[The Structure of Convolution Measure Algebras)](http://The%20Structure%20of%20Convolution%20Measure%20Algebras)*. Actually, I had known the result but it didn't occur to me that it would have solved the problem.
The maximal ideal space of $M(\mathbb T)$ contains an analytic disk, wh... | 1 | https://mathoverflow.net/users/15129 | 306924 | 133,765 |
https://mathoverflow.net/questions/306893 | 3 | I'm in the following situation. I have a self-injective finite-dimensional basic algebra $\Lambda$ (hence Frobenius) over a perfect field and two finite-dimensional **invertible** $\Lambda$-bimodules $M$ and $N$ which are isomorphic in the **stable category** of $\Lambda$-bimodules, i.e. there are finite-dimensional pr... | https://mathoverflow.net/users/12166 | Invertible bimodules which are isomorphic in the stable module category | Here is an easy counterexample: let $\Lambda =F\oplus F$ (sum of 2 copies of the base field). Then any $\Lambda-$bimodule is projective, so everything is isomorphic in the stable category. However non-isomorphic invertible bimodules do exist: take $M=\Lambda$ and $N=M$ but with the right action twisted by an automorphi... | 3 | https://mathoverflow.net/users/4158 | 306925 | 133,766 |
https://mathoverflow.net/questions/306932 | 5 | A subset $B$ of a group $G$ is called *balanced* if $gBg^{-1}=B$ for all $g\in G$.
An action of a group $G$ on a metric space $X$ is called *ballanced* if for each non-empty balanced subset $B\subset G$ and each $x\in X$ the set $Bx$ coincides with an open or closed ball centered at $x$.
This means that the topolo... | https://mathoverflow.net/users/61536 | Is the action of $SO(n)$ on the sphere $S^{n-1}$ ballanced? | If $n$ is even, then the action is not locally ballanced. You can choose $B$ to be the conjugacy class of an element of $SO(n)$, as close as you like to the identity, that does not have $1$ as an eigenvalue. Now whatever $x$ is, $x$ will not belong to $Bx$.
On the other hand, if $n$ is odd then every element of $SO(n... | 8 | https://mathoverflow.net/users/6666 | 306945 | 133,769 |
https://mathoverflow.net/questions/306859 | 3 | I'm reading Akhil Mathew's [blog post](https://amathew.wordpress.com/2012/06/23/formal-lie-theory-in-characteristic-zero/) on Formal Lie Theory in Characteristic Zero.
Let $H$ be cocommutative Hopf algebra over a field $k$. We can form $\mathfrak{g}$, the Lie algebra over $k$ consisting of the primitive elements of $... | https://mathoverflow.net/users/30211 | When is this map of Hopf algebras Surjective? | Some thoughts, regarding question (a):
In the case of a [pointed](http://library.msri.org/books/Book43/files/andrus.pdf), cocommutative hopf algebra $H$ over a field $k$ of characteristic $0$, by the Cartier-Konstant-Milnor-Moore theorem (see: [Classification of quasitriangular Hopf algebras](https://mathoverflow.ne... | 6 | https://mathoverflow.net/users/85967 | 306949 | 133,772 |
https://mathoverflow.net/questions/306936 | 6 | Recently I read some results about derived categories of coherent sheaves, and see one use Fourier-Mukai transforms to prove that the derived categories of coherent sheaves of a scheme, under some conditions, determine the scheme. More precisely, I have seen
1. (Bondal and Orlov) Let $X$ and $Y$ be smooth projective ... | https://mathoverflow.net/users/105980 | Information from the derived categories of coherent sheaves | If I understand the question correctly, it is about what information about a smooth projective variety you can extract from its derived category of coherent sheaves. Some result in this direction (besides those mentioned in the question statement)
* the dimension of a variety is uniquely determined by its derived cat... | 6 | https://mathoverflow.net/users/nan | 306951 | 133,773 |
https://mathoverflow.net/questions/306947 | 5 | I wonder if the following Kunneth formula for semidirect product is valid
$$ H^n(N\rtimes\_\phi G;\mathbb{Z}) = \sum\_{i+j=n} H^i(G; H^j(N;\mathbb{Z})),$$
where $H^\*$ is the group cohomology and $G$ has a proper action on $H^j(N;\mathbb{Z})$ as induced by $\phi$.
(For direct product, $G$ has no action on $H^j(N;\mathb... | https://mathoverflow.net/users/17787 | Kunneth formula for semidirect product | Just to collect the references I wrote in comments, and more. They each give (in)finite (non)abelian counterexamples, as well as general explanations for failure of collapse of the LHS spectral sequence with semi-direct products:
1. *Charlap, L. S.; Vasquez, A. T.*, [**The cohomology of group extensions**](http://dx... | 7 | https://mathoverflow.net/users/12310 | 306952 | 133,774 |
https://mathoverflow.net/questions/297483 | 7 | As is well known, quantized enveloping algebras $U\_q(\frak{g})$ admit far fewer sub-Hopf algebras than classical enveloping algebras $U(\frak{g})$. As one can check directly, for appropriate subsets of the (simple root) standard generators $E\_i,F\_i,K\_i$, a Hopf subalgebras are seen to be generated. However, I do no... | https://mathoverflow.net/users/121660 | Hopf Subalgebras of Quantized Algebras | Since we know from Etingof-Kazhdan that quantization is functorial we can safely say that the classification of sub-Lie bialgebra, which was obtained in the standard case, implies classification of sub-Hopf algebras.
The paper in which classification of Lie bialgebras is obtained is J. Stokman in "*the quantum orbit ... | 5 | https://mathoverflow.net/users/6032 | 306958 | 133,775 |
https://mathoverflow.net/questions/306934 | 3 | Consider a large $N\times N$ square lattice, where each cell has a probability $p$ of being "occupied" (let's call denote them as "black") and a probability $1-p$ of being empty (let's denote them as "white"). Cells in the [Moore neighbourhood](https://en.wikipedia.org/wiki/Moore_neighborhood) of any central cell and h... | https://mathoverflow.net/users/nan | Why is number of single cell clusters always greatest in a random matrix? | Here is a revised answer that might be clearer:
You define white clusters but I'll just look at black clusters since that is what your data does, and it implies the other interpretation (counts of monochrome clusters.)
Strictly speaking your claim is not totally accurate. For $p=1-\frac1{10^6}$ and a $1000 \times ... | 0 | https://mathoverflow.net/users/8008 | 306959 | 133,776 |
https://mathoverflow.net/questions/299616 | 6 | A *pencil* is a collection of some lines through a point, called the *center* of the pencil.
If the points of the plane are colored, then call a pencil *bichromatic* if there is a color that is present on all the lines of the pencil such that this color is different from the color of the center of the pencil.
Given a... | https://mathoverflow.net/users/955 | Bichromatic pencils | Observe that we are done if there is a monochromatic circle $C$, say blue: If a point inside the circle is not blue, then any pencil with that centre is bichromatic, so we may assume that the whole disk bounded by $C$ is blue. If a point outside is close enough and not blue, then we can find a bichromatic pencil with t... | 5 | https://mathoverflow.net/users/127060 | 306966 | 133,777 |
https://mathoverflow.net/questions/306962 | -1 | Assume that $f$ is smooth function defined in the unit disk $D: x^2+y^2\le 1$, and consider the integral $$I=\int\_D f dxdy=\int\_0^1r \int\_0^{2\pi} f(re^{it})dt.$$
Then it is clear that for $r\in[0,1]$ there is $t\_r\in [0,2\pi]$ so that $$I=2\pi \int\_0^1 r f(re^{it\_r})dr.$$ My question is, can we choose $t\_r$ t... | https://mathoverflow.net/users/124426 | A smooth curve and mean value theorem | You can even choose $t\_r$ to be independent of $r$. Indeed,
$$g(t):=\int\_0^1 rf(re^{it})\,dr,\qquad t\in[0,2\pi],$$
is a continuous function satisfying
$$I=\int\_0^{2\pi}g(t)\,dt.$$
Therefore, there exists $t\_0\in[0,2\pi]$ such that
$$I=2\pi\, g(t\_0)=2\pi \int\_0^1 r f(re^{it\_0})\,dr.$$
| 3 | https://mathoverflow.net/users/11919 | 306967 | 133,778 |
https://mathoverflow.net/questions/306957 | 4 | The equation $x\_1x\_2+y\_1y\_2=n$ is well-studied (Ingham, Heath-Brown, Deshouillers & Iwaniec, Ismoilov) because it arises in an *additive divisor problem.* The number of solutions in positive integers is $n(c\_2\log^2n+c\_1\log n+c\_0)+O(n^{1-\delta})$ where $c\_i$'s are some explicit arithmetic functions of $n$.
... | https://mathoverflow.net/users/5712 | The number of solutions of the equation $ax_1x_2+by_1y_2=n$ | For a smoothened version of your sum, an asymptotic formula can be found in Duke-Friedlander-Iwaniec: A quadratic divisor problem (Inventiones, 1994), see (1)-(5) there. It is straightforward to "unsmooth" this formula, much like it is done in (6)-(7) of the paper.
| 7 | https://mathoverflow.net/users/11919 | 306971 | 133,779 |
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