Tootfinder

Sometimes you look at a graph and think that a particular value seems a bit odd. Then you look at the other numbers and they seem off, too. At that point, you realise that the graph is not intended to clarify, but to obscure. On German #EV charging prices, Quartets and diesel.

Sometimes you look at a graph and think that a particular value seems a… | Jaap Burger
Sometimes you look at a graph and think that a particular value seems a bit odd. Then you look at the other numbers and they seem off, too. At that point, you realise that the graph is not intended to clarify, but to obscure. Thank you, Felix Hamer, for drawing attention to this recent slide from BDEW Bundesverband der Energie- und Wasserwirtschaft e.V. The title suggests that EV charging would only cost the same as filling up a diesel or petrol car for the same distance in an 'Extremszenario…

Browns DC Jim Schwartz expects career year from Myles Garrett following record-setting contract https://www.nfl.com/news/browns-dc-jim-schwartz-expects-career-year-from-myles-garrett-following-record-setting-contra…

The number of primes not in a numerical semigroup
Yong-Gao Chen, Hui Zhu
https://arxiv.org/abs/2506.03625 https://arxiv.org/pdf/2506.…

The number of primes not in a numerical semigroup
For two coprime positive integers $a,b$, let $π^* (a, b)$ be the number of primes that cannot be represented as $au+bv$, where $u,v$ are nonnegative integers. In this paper, we prove that $π^* (a, b)\ge 0.014 π(ab-a-b)$ and pose following conjecture: $π^* (a, b)\ge \frac 12 π(ab-a-b)$, where $π(x)$ denotes the number of primes not exceeding $x$. This conjecture is confirmed for $1\le a\le 8$.

The number of Americans filing for jobless benefits last week rises to highest level in eight months (Matt Ott/Associated Press)
https://apnews.com/article/unemployment-benefits-jobless-claims-layoffs-labor-795ec07804ed3f35eb702af27c4983d0
http://www.memeorandum.com/250605/p53#a250605p53

Edinburgh's low emission zone seems to be working.
https://road.cc/content/news/ediburghs-lez-results-more-cycling-and-less-car-use-314269

Edinburgh sees “statistically significant” shift towards cycling and walking plus reduced car use following introduction of low emission zone, finds study
Within a year, the LEZ has contributed in improving air quality, with the number of non-compliant vehicles entering the zone also decreasing significantly

Star decompositions via orientations
Viktor Harangi
https://arxiv.org/abs/2506.05194 https://arxiv.org/pdf/2506.05194

Star decompositions via orientations
A $k$-star decomposition of a graph is a partition of its edges into $k$-stars (i.e., $k$ edges with a common vertex). The paper studies the following problem: given $k \leq d/2$, does the random $d$-regular graph have a $k$-star decomposition (asymptotically almost surely, provided that the number of edges is divisible by $k$)? Delcourt, Greenhill, Isaev, Lidický, and Postle proved the a.a.s. existence for every odd $k$ using earlier results regarding orientations satisfying certain degree co…

This https://arxiv.org/abs/2504.21059 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

A remark on the number of automorphisms of some algebraic structures
In these notes we look at the following question. Given a category $\mathcal C$ of algebraic structure (e.g. the category of groups, monoids, partial groups, ...) and a rational $r\in \mathbb Q$, does there exists an element $x\in \mathcal C$ such that the size of its automorphism group $\text{Aut}_{\mathcal C} (x)$ divided by the size of $x$ (whatever that would means) is equal to $r$\,? To our knowledge, this question was introduced by Tărnăuceanu in the category of groups. Here, we answer …

Optimized and regularly repeated lattice-based Latin hypercube designs for large-scale computer experiments
Xu He, Junpeng Gong, Zhaohui Li
https://arxiv.org/abs/2506.04582

Optimized and regularly repeated lattice-based Latin hypercube designs for large-scale computer experiments
Computer simulations serve as powerful tools for scientists and engineers to gain insights into complex systems. Less costly than physical experiments, computer experiments sometimes involve large number of trials. Conventional design optimization and model fitting methods for computer experiments are inefficient for large-scale problems. In this paper, we propose new methods to optimize good lattice point sets, using less computation to construct designs with enhanced space-filling properties …

Dynamic Space Filling
P. L. Krapivsky
https://arxiv.org/abs/2506.01128 https://arxiv.org/pdf/2506.01128

Dynamic Space Filling
Dynamic space filling (DSF) is a stochastic process defined on any connected graph. Each vertex can host an arbitrary number of particles forming a pile, with every arriving particle landing on the top of the pile. Particles in a pile, except for the particle at the bottom, can hop to neighboring vertices. Eligible particles hop independently and stochastically, with the overall hopping rate set to unity for every eligible particle at every vertex. When the number of vertices in a graph is equa…

Cardinalities of the total number of independent sets
Benedek Kov\'acs, Zolt\'an L\'or\'ant Nagy
https://arxiv.org/abs/2505.24794 https://

Cardinalities of the total number of independent sets
We study the set of numbers the total number of independent sets can admit in $n$-vertex graphs. In this paper, we prove that the cardinality of this set $\mathcal{N}i(n)$ is very close to $2^n$ in the following sense: $\mathcal{N}i(n)/2^n = O(n^{-1/5})$ while for infinitely many $n$, we have $\log_2(\mathcal{N}i(n)/2^n)\ge -2^{(1+o(1)\sqrt{\log_2 n}}$. This set is also precisely the set of possible values of the independence polynomial $I_G(x)$ at $x=1$ for $n$-vertex graphs $G$. As an applica…