Scared 2 be lonely beat（当真爱降临）

We prove that the lonely runner conjecture holds for eight runners. Our proof relies on a computer verification and on recent results that allow bounding the size of a minimal counterexample. We note that our approach also applies to the known cases with 4, 5, 6, and 7 runners. We expect that minor improvements to our approach could be enough to solve the cases of 9 or 10 runners.

We characterize semigroups in $\{0,1,2,\ldots\}$ of matricial dimension $2$ and produce a counterexample to the conjecture that a numerical semigroup whose small elements are lonely has matricial dimension at most $2$.

山东省的降水量处于我国降水充沛的南方和少雨干旱的北方的两者之间的地带,影响降水量的因素较多且复杂,这就造成山东省的年降水量和季节降水量变化较大,而且短期气候预测准确率较低。本文分析了山东省夏季(6-8月)的降水特征并对降水进行了预测分析,以期能够提升山东省短期气候预测能力的提升。

Motivated by colouring minimal Cayley graphs, in 1978, Babai conjectured that no-lonely-colour graphs have bounded chromatic number. We disprove this in a strong sense by constructing graphs of arbitrarily large girth and chromatic number that have a proper edge-colouring in which each cycle contains no colour exactly once.