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...

Abstract Childless older people face many challenges resulting from their lack of children that may have an impact on subjective well-being, especially in a country where family ties are strong, and the main source support. In this paper, we examine coping strategies childless seniors Poland employ order to mitigate various risks related having no offspring. We use qualitative approach, drawing...

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 ,...

We feel lonely from time to and these feelings of loneliness sometimes have a serious effect on our mental health. However, in this pandemic feeling was long scenario. It has been seen that the long-term is associated with an increased risk certain health problems, including depression, anxiety stress. During Pandemic we different situation change impact everyone’s life. Due Covid-19 there lock...