Let G be a directed graph on n vertices. A k-path in G is a path p = v1 → . . . → vk in G. Given an integer k ≤ n, the SIMPLE k-PATH problem asks whether there exists a simple k-path in G. In case G is weighted, the MIN-WT SIMPLE k-PATH problem asks for a simple k-path in G of minimal weight. The fastest currently known deterministic algorithm for MIN-WT SIMPLE k-PATH by Fomin, Lokshtanov and S...

A trapped-ion-based quantum computer that fits in two boxes, each the size of a studio apartment’s shower, can create fully entangled 24-particle state.

A prime gap is the difference between two successive numbers. Two smallest possible primes. twin a that has of two. The conjecture states there are infinitely many This been one great open problems in number theory for years. In May 2013, popular Yitang Zhang's paper was accepted by journal Annals Mathematics where it announced some integer N less than 70 million, pairs primes differ N. few mon...

Let $G=(V_G, E_G)$ be a simple and connected graph. A set $M\subseteq E_G$ is called matching of $G$ if no two edges $M$ are adjacent. The number in its size. maximal it cannot extended to larger $G$. smallest size the saturation In this paper we concerned with numbers lexicographic product graphs. We also address solve an open problem about maximum matchings graphs given degree $\Delta$.

Understanding the mechanical properties of nanoscale systems requires a range of measurement techniques and theoretical approaches to gather the relevant physical and chemical information. The arrangements of atoms in nanostructures and macroscopic matter can be different, principally due to the role of surface energy, but the interplay between atomic and electronic structure in association wit...