OpenAI Gym1 is a toolkit for reinforcement learning research. It includes a growing collection of benchmark problems that expose a common interface, and a website where people can share their resultsâ€¦ (More)

A universal cycle is a compact listing of a class of combinatorial objects. In this paper, we prove the existence of universal cycles of classes of labeled graphs, including simple graphs, trees,â€¦ (More)

Small TiO2 crystals in the anatase phase are in high demand as photocatalysts. Stable TiO2 crystals in the anatase phase were obtained using a silica nanoparticle as a support. The focus of thisâ€¦ (More)

The English translation of Leo Tolstoyâ€™s novel War and Peace has the following notable property: it contains this paragraph as a subsequence. If one were to write the letters and spaces that appearâ€¦ (More)

Let D be a simple digraph without loops or digons (i.e. if (u, v) âˆˆ E(D), then (v, u) 6âˆˆ E(D)). For any v âˆˆ V (D) let N1(v) be the set of all vertices at out-distance 1 from v and let N2(v) be theâ€¦ (More)

Let D be a simple digraph without loops or digons. For any v âˆˆ V (D) let N1(v) be the set of all nodes at out-distance 1 from v and let N2(v) be the set of all nodes at out-distance 2. We provideâ€¦ (More)

The Ducci map applied to vectors in R is considered. It is shown that for any starting vector, the corresponding sequence of iterates either eventually becomes periodic or, if not eventuallyâ€¦ (More)

The Ducci map is considered applied to vectors in R. It is shown that for certain starting vectors of odd length, the corresponding sequence of iterates asymptotically approaches a periodic vector,â€¦ (More)