Tootfinder

Splitting Families, Reaping Families, and Families of Permutations Associated with Asymptotic Density
David Valderrama
https://arxiv.org/abs/2506.21059 htt…

Splitting Families, Reaping Families, and Families of Permutations Associated with Asymptotic Density
We investigate several relations between cardinal characteristics of the continuum related with the asymptotic density of the natural numbers and some known cardinal invariants. Specifically, we study the cardinals of the form $\mathfrak{s}_X$, $\mathfrak{r}_X$ and $\mathfrak{dd}_{X,Y}$ introduced in arXiv:2304.09698 and arXiv:2410.21102, answering some questions raised in these papers. In particular, we prove that $\mathfrak{s}_0=$ cov$(\mathcal{M})$ and $\mathfrak{r}_0=$ non$(\mathcal{M})$. W…

The high precision time nuts, a.k.a. the “Time Lords” had a pretty good demonstration at #Hamvention. They built an LLM that had ingested 10 years of papers and mailing lists and could answer questions reliably

Citizen science illuminates the nature of city lights: #LightPollution

Delighted to see these two new papers come out in Nature (they've been on bioRxiv for a while).
How does Pavlov's dog learn that the bell predicts the food? One answer is that the bell appears ``close'' in time to the food and that enables learning. We're certain that dopamine has something to do with learning these kinds of associations. But the definition of ``close'' in time is actually really difficult to pin down. You can get associations over prett…

Extension of algebroids Part I: The Construction
Simon-Raphael Fischer
https://arxiv.org/abs/2506.01772 https://arxiv.org/pdf/2506.01…

Extension of algebroids Part I: The Construction
In this series of two papers we will generalise the concept of extending a Lie algebroid by a Lie algebra bundle, leading to a notion of extending a Lie algebroid by another Lie algebroid whose orbits lie in the orbits of the former algebroid. The resulting Lie algebroid's anchor will be the sum of the two initial anchors such that the constructions will be similar to matched pairs of Lie algebroids, but with the major difference that we will allow curvatures. In this part of this series we wil…