Scared 2 be lonely beat（当真爱降临）

In this note we develop a linear programming framework to produce upper and lower bounds for the lonely runner problem.

膝关节骨性关节炎临床发生率较高，患者通常具有膝骨关节疼痛、肿胀、下肢运动功能下降等症状。论文对目前膝关节骨性关节炎临床治疗相关形式进行了综合分析，通过学术资料、文献书籍总结膝关节骨性关节炎临床治疗措施，希望为膝关节骨性关节炎临床治疗提供参考。　

The Lonely Runner Conjecture originated in Diophantine approximation is turning 60. Even if the conjecture is still widely open, the flow of partial results, innovative tools and connections to different problems and applications has been steady on its long life. This survey attempts to give a panoramic view of the status of the problem, trying to highlight the contributions of the many papers that it has originated.

The Lonely Runner Conjecture of Wills and Cusick states that if $k+1$ runners start running at distinct constant speeds around a unit-length circular track, then for each runner there is a time when he/she is at least $1/(k+1)$ away from all other runners. Rosenfeld recently obtained a computer-assisted proof of the conjecture for $8$ runners. By refining his approach with a sieve, we obtain proofs (also computer-assisted) for $9$ and $10$ runners.