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Closure of the product of subfunctors

Background:Let $X: textbf{CRing} to textbf{Set}$ be a presheaf on the category of affine schemes and $Z subseteq X$ a subfunctor. One defines $Z$ to be closed if...

Asked on 12/08/2021 by Nate Gallup

1 answer

Small-$r$ asymptotics of an integral of $1/log ^alpha r$

Let $f(r)=frac{1}{(-log r)^{alpha}}$ and let $xi_r$ be the unique value in $(0,r)$ such that$f(xi_r)=frac{1}{r}int_{0}^rf(t)dt$, where $alphain (0,1)$ and $r>0$. My question is about the...

Asked on 12/08/2021 by Yongpan Huang

1 answer

Does $mathcal{A}otimesmathbb{C}(t)congmathcal{D}otimesmathbb{C}(t)$ imply an isomorphism of Lie algebras?

Assume that $mathcal{A}$ is a Lie $mathbb{C}$-algebra. Also denote the Lie $mathbb{C}$-algebra $mathcal{D}=mathbb{C}[x_1,ldots, x_n]partial_{x_1}oplusldotsoplusmathbb{C}[x_1,ldots, x_n]partial_{x_n}$. My questions are as follows Let $mathcal{A}otimes_{mathbb{C}}mathbb{C}(t)$, $mathcal{D}otimes_{mathbb{C}}mathbb{C}(t)$ are...

Asked on 12/08/2021 by solver6

1 answer

When does $det begin{pmatrix} A & X \ X^T & A end{pmatrix} = (det A)^2 + (det X)^2$?

Let $A$ be an $n times n$ real symmetric matrix.Let$$M = begin{pmatrix} A & X \ X^T & A end{pmatrix}$$...

Asked on 12/08/2021

1 answer

When does cohomology of a pro-algebraic group commute with filtered colimits of coefficients?

Let $G$ be a pro-algebraic group, that is, a projective limit of algebraic groups $G_i$. Let $V_i$ be an inductive system of finite dimensional rational $G_i$-representations,...

Asked on 12/08/2021 by Patrick Elliott

1 answer

Bounded weak and weak-$star$ topologies and metrics

Let $X$ be a reflexive and separable Banach space. Let $(h_n)$ be a sequence dense in $overline{B^*_1}$ (the closed ball in $X^*$ of radius $1$)....

Asked on 12/08/2021 by Jorge E. Cardona

0 answer

Is semi-simplicity of Galois representations local?

Let $rho:G_{mathbb{Q}}rightarrow text{Gl}(V)$ be a finite dimensional $ell$-adic Galois representation. Then for each prime, by pre-composing $rho$ with the natural inclusion $G_{mathbb{Q}_p}rightarrow G_{mathbb{Q}}$, we get a...

Asked on 12/08/2021

0 answer

Maximum eigenvalue of a covariance matrix of Brownian motion

$$ A := begin{pmatrix}1 & frac{1}{2} & frac{1}{3} & cdots & frac{1}{n}\frac{1}{2} & frac{1}{2} & frac{1}{3} & cdots & frac{1}{n}\frac{1}{3} & frac{1}{3} & frac{1}{3} & cdots...

Asked on 12/06/2021 by Weiqiang Yang

2 answer

Identity theorem in $p$-adic geometry/analysis

If one wants to do $p$-adic analysis and geometry, it is often bad so adapt "naively" complex analytic ideas, basically because $mathbb{Q}_p$ is disconnected. The modern approach to...

Asked on 12/06/2021

1 answer

Updates to Stanley's 1999 survey of positivity problems in algebraic combinatorics?

[I am a co-moderator of the recently started Open Problems in Algebraic Combinatorics blog and as a result starting doing some searching for existing surveys of open...

Asked on 12/05/2021 by Sam Hopkins

1 answer

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