For x real, let {x} be the fractional part of x (i.e. {x} = x − bxc). In this paper we prove the k = 5 case of the following conjecture (the lonely runner conjecture): for any k positive reals v1, . . . , vk there exists a real number t such that 1/(k + 1) ≤ {vit} ≤ k/(k + 1) for i = 1, . . . , k.

This Science Fiction Prototype explores both the possibilities of Brain-Computer Interfaces with Artificial Intelligence and writing autobiographical Science Fiction. The paper proceeds in three parts. The context for the story is introduced in the first section, outlining the particularly personal circumstances from which the story has been written. In the second section the story is presented...

We improve upper bounds on the chromatic number proven independently in [1] and [5]. Our main lemma gives a sufficient condition for two paths in graph to be completely joined. Using this, we prove that if a graph has an optimal coloring with more than ω2 singleton color classes, then it satisfies χ ≤ ω+∆+1 2 . It follows that a graph satisfying n − ∆ < α + ω−1 2 must also satisfy χ ≤ ω+∆+1 2 ,...