Tootfinder

On the $4$-clique cover number of graphs
Yihan Chen, Jialin He, Tianying Xie
https://arxiv.org/abs/2506.10478 https://arxiv.org/pdf/2…

On the $4$-clique cover number of graphs
In 1966, Erdős, Goodman, and Pósa proved that $\lfloor n^2/4 \rfloor$ cliques are sufficient to cover all edges in any $n$-vertex graph, with tightness achieved by the balanced complete bipartite graph. This result was generalized by Dau, Milenkovic, and Puleo, who showed that at most $\lfloor \frac n 3 \rfloor \lfloor \frac {n+1} 3 \rfloor \lfloor \frac {n+2} 3 \rfloor$ cliques are needed to cover all triangles in any $n$-vertex graph $G$, and the bound is best possible as witnessed by the b…

Safety, Relative Tightness and the Probabilistic Frane Rule
Janez Ignacij Jereb, Alex Simpson
https://arxiv.org/abs/2506.01626 https://

Safety, Relative Tightness and the Probabilistic Frane Rule
Probabilistic separation logic offers an approach to reasoning about imperative probabilistic programs in which a separating conjunction is used as a mechanism for expressing independence properties. Crucial to the effectiveness of the formalism is the frame rule, which enables modular reasoning about independent probabilistic state. We explore a semantic formulation of probabilistic separation logic, in which the frame rule has the same simple formulation as in separation logic, without furthe…

Cowboys' Micah Parsons on latest contract talk, if he will attend training camp without new deal https://www.nytimes.com/athletic/6417336/2025/06/10/dallas-cowboys-micah-parsons-contract-minicamp/

Invariant measures for stochastic Burgers equation on unbounded domains
Zhenxin Liu, Zhiyuan Shi
https://arxiv.org/abs/2506.07119 https://

Invariant measures for stochastic Burgers equation on unbounded domains
In this paper, we investigate the stochastic damped Burgers equation with multiplicative noise defined on the entire real line. We demonstrate the existence and uniqueness of a mild solution to the stochastic damped Burgers equation and establish that the solution is uniformly bounded in time. Furthermore, by employing the uniform estimates on the tails of the solution, we obtain the tightness of a family of probability distributions of the solution. Subsequently, by applying the Krylov-Bogolio…

This https://arxiv.org/abs/2407.01101 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Packing Density of Sets With Only Two Nonmixed Gaps
For a finite set of integers such that the first few gaps between its consecutive elements equal $a$, while the remaining gaps equal $b$, we study dense packings of its translates on the line. We obtain an explicit lower bound on the corresponding optimal density, conjecture its tightness, and prove it in case one of the gap lengths, $a$ or $b$, appears only once. This is equivalent to a Motzkin problem on the independence ratio of certain integer distance graphs.

This https://arxiv.org/abs/2506.01626 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_csLO_…

Safety, Relative Tightness and the Probabilistic Frane Rule
Probabilistic separation logic offers an approach to reasoning about imperative probabilistic programs in which a separating conjunction is used as a mechanism for expressing independence properties. Crucial to the effectiveness of the formalism is the frame rule, which enables modular reasoning about independent probabilistic state. We explore a semantic formulation of probabilistic separation logic, in which the frame rule has the same simple formulation as in separation logic, without furthe…