In this paper, decoherence in a system consisting of two Bose-Einstein condensates is investigated analytically. It is indicated that decoherence can be controlled through manipulating the interaction between the system and environment. The influence of the decoherence on quantum coherent atomic tunneling (AT) between two condensates with arbitrary initial states is studied in detail. Analytic expressions of the population difference (PD) and the AT current between two condensates are found. It is shown that the decoherence leads to the decay of the PD and the suppression of the AT current.
A graph is called $1$-planar if it admits a drawing in the plane such that each edge is crossed at most once. Let $G$ be a bipartite $1$-planar graph with $n$ ($n\ge 4$) vertices and $m$ edges. Karpov showed that $m\le 3n-8$ holds for even $n\ge 8$ and $m\le 3n-9$ holds for odd $n\ge 7$. Czap, Przybyło and Škrabul'áková proved that if the partite sets of $G$ are of sizes $x$ and $y$, then $m\le 2n+6x-12$ holds for $2\leq x\leq y$, and conjectured that $m\le 2n+4x-12$ holds for $x\ge 3$ and $y\ge 6x-12$. In this paper, we settle their conjecture and our result is even under a weaker condition $2\le x\le y$.
In [C. Thomassen, Tilings of the torus and the Klein bottle and vertextransitive graphs on a fixed surface, Trans. Amer. Math. Soc. 323 (1991) 605-635], Thomassen described completely all (except finitely many) regular tilings of the torus S 1 and the Klein bottle N 2 into (3,6)-tilings, (4,4)-tilings and (6,3)-tilings. Many authors made great efforts to investigate the crossing number (in the plane) of the Cartesian product of an m-cycle and an n-cycle, which is a special (4,4)-tiling. For other tilings, there are quite rare results concerning on their crossing numbers. This motivates us in the paper to determine the crossing number of a hexagonal graph H 3,n , which is a special kind of (3,6)-tilings.
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