Tootfinder

$q$-Congruences for Z.-W. Sun's generalized polynomials $w^{(\alpha)}_k(x)$
Lin-Yue Li, Rong-Hua Wang
https://arxiv.org/abs/2507.04653 https://

$q$-Congruences for Z.-W. Sun's generalized polynomials $w^{(α)}_k(x)$
In 2022, Z.-W. Sun defined \begin{equation*} w_k^{(α)}{(x)}=\sum_{j=1}^{k}w(k,j)^αx^{j-1}, \end{equation*} where $k,α$ are positive integers and $w(k,j)=\frac{1}{j}\binom{k-1}{j-1}\binom{k+j}{j-1}$. Let $(x)_{0}=1$ and $(x)_{n}=x(x+1)\cdots(x+n-1)$ for all $n\geq 1$. In this paper, it is proved by $q$-congruences that for any positive integers ${α,β, m,n,r}$, we have \begin{equation*} \frac{(2,n)}{n(n+1)(n+2)}\sum_{k=1}^{n}k^r(k+1)^r(2k+1)w_{k}^{(α)}(x)^{m}\in\mathbb{Z}[x], \end{equation*…

„I wasn’t allowed to speak or offer legal advice to M in the courtroom, so without looking down for too long, I wrote in my notebook:
Ils vont essayer de terminer ton case—tu vas dire NON. Tu as besoin de temps pour trouver un avocat.“
https://www.nplusonemag.com/online-only/on

Smooth manifolds in $G_{n,2}$ and $\mathbb{C} P^{N}$ defined by symplectic reductions of $T^n$-action
Victor M. Buchstaber, Svjetlana Terzi\'c
https://arxiv.org/abs/2507.04582

Smooth manifolds in $G_{n,2}$ and $\mathbb{C} P^{N}$ defined by symplectic reductions of $T^n$-action
Plücker coordinates define the $T^n$-equivariant embedding $p : G_{n,2}\to \mathbb{C} P^{N}$ of a complex Grassmann manifold $G_{n,2}$ into the complex projective space $\mathbb{C} P^{N}$, $N=\binom{n}{2}-1$ for the canonical $T^n$-action on $G_{n,2}$ and the $T^n$-action on $\mathbb{C} P^{N}$ given by the second symmetric power representation $T^n\to T^{N}$ and the standard $T^{N}$-action. Let $μ: G_{n,2}\to Δ_{n,2}\subset \mathbb{R} ^{n}$ and $\tildeμ: \mathbb{C} P^{N}\to Δ_{n,2}\subset …

#ndwgoecountdown Meteoriten – Boten aus dem All
Immer wieder bahnen sich Gesteinsbrocken aus dem Weltall ihren Weg auf die Erde. Meteoriten sind interessant für Forschende. Denn die Brocken sind Zeitzeugen der Entstehung des Sonnensystems. Ihre genaue Zusammensetzung kann helfen zu verstehen, wie das Sonnensystem zu dem Ort wurde, den wir heute kennen. Mit echten Meteoriten zu…

Trump mobilizes Marines and more National Guard troops for duty in LA (Tom Bowman/NPR)
https://www.npr.org/2025/06/09/g-s1-71640/trump-mobilizes-marines-for-duty-in-los-angeles
http://www.memeorandum.com/250609/p161#a250609p161

On the Jan. 6 attack, Republican leaders keep failing basic tests of self-awareness (Steve Benen/MSNBC)
https://www.msnbc.com/rachel-maddow-show/maddowblog/jan-6-attack-republican-leaders-keep-basic-tests-self-awareness-rcna217527
http://www.memeorandum.com/250708/p92#a250708p92

Independent Dan Osborn launches another Nebraska Senate run (Bridget Bowman/NBC News)
https://www.nbcnews.com/politics/2026-election/independent-dan-osborn-launches-another-nebraska-senate-run-rcna215998
http://www.memeorandum.com/250708/p3#a250708p3

Team Trump struggles to control the Epstein 'client list' fire it helped create (Steve Benen/MSNBC)
https://www.msnbc.com/rachel-maddow-show/maddowblog/team-trump-struggles-control-epstein-client-list-fire-helped-create-rcna217441
http://www.memeorandum.com/250708/p91#a250708p91

Pointing to protesters, but not ICE agents, Trump demands arrests of those in masks (Steve Benen/MSNBC)
https://www.msnbc.com/rachel-maddow-show/maddowblog/pointing-protesters-not-ice-agents-trump-demands-arrests-masks-rcna211787
http://www.memeorandum.com/250609/p60#a250609p60

Trump's IRS says churches can now endorse political candidates from the pulpit (Steve Benen/MSNBC)
https://www.msnbc.com/rachel-maddow-show/maddowblog/trumps-irs-says-churches-can-now-endorse-political-candidates-pulpit-rcna217495
http://www.memeorandum.com/250708/p87#a250708p87