The maximal length of the period of a periodic word defined by restrictions
We determine the maximal length of the period of a periodic word defined by $n$ restrictions. It happens to be the corresponding Fibonacci number.
math.CO↗
arXiv subjects
Grigory R. Chelnokov
Publications and source records attributed to Grigory R. Chelnokov.
Finding a subset of nonnegative vectors with a coordinatewise large sum
Given a rational $a=p/q$ and $N$ nonnegative $d$-dimensional real vectors $u_1$, ..., $u_N$, we show that it is always possible to choose $(d-1)+\lceil (pN-d+1)/q\rceil$ of them such that their sum is (componentwise) at least $(p/q)(u_1+...+u_N)$. For fixed $d$ and $a$, this bound is sharp if $N$ is large enough. The method of the proof uses Carathéodory's theorem from linear programming.
math.CO↗