Numerical Complete Solution for Random Genetic Drift by Energetic Variational Approach

In this paper, we focus on numerical solutions for random genetic drift problem, which is governed by a degenerated convection-dominated parabolic equation. Due to the fixation phenomenon of genes, Dirac delta singularities will develop at boundary points as time evolves. Based on an energetic variational approach (EnVarA), a balance between the maximal dissipation principle (MDP) and least action principle (LAP), we obtain the trajectory equation. In turn, a numerical scheme is proposed using a convex splitting technique, with the unique solvability (on a convex set) and the energy decay property (in time) justified at a theoretical level. Numerical examples are presented for cases of pure drift and drift with semi-selection. The remarkable advantage of this method is its ability to catch the Dirac delta singularity close to machine precision over any equidistant grid.Comment: 22 pages, 11 figures, 2 table

The Tensor Rank of the Tripartite State ∣W⟩⊗n\ket{W}^{\otimes n}}

Tensor rank refers to the number of product states needed to express a given multipartite quantum state. Its non-additivity as an entanglement measure has recently been observed. In this note, we estimate the tensor rank of multiple copies of the tripartite state $\ket{W}=\tfrac{1}{\sqrt{3}}(\ket{100}+\ket{010}+\ket{001})$. Both an upper bound and a lower bound of this rank are derived. In particular, it is proven that the tensor rank of $\ket{W}^{\otimes 2}$ is seven, thus resolving a previously open problem. Some implications of this result are discussed in terms of transformation rates between $\ket{W}^{\otimes n}$ and multiple copies of the state $\ket{GHZ}=\tfrac{1}{\sqrt{2}}(\ket{000}+\ket{111})$.Comment: Comments: 3 pages (Revtex 4). Minor corrections to Theorem 1. Presentation refined. Main results unchanged. Comments are welcom

Path diversity improves the identification of influential spreaders

Identifying influential spreaders in complex networks is a crucial problem which relates to wide applications. Many methods based on the global information such as $k$-shell and PageRank have been applied to rank spreaders. However, most of related previous works overwhelmingly focus on the number of paths for propagation, while whether the paths are diverse enough is usually overlooked. Generally, the spreading ability of a node might not be strong if its propagation depends on one or two paths while the other paths are dead ends. In this Letter, we introduced the concept of path diversity and find that it can largely improve the ranking accuracy. We further propose a local method combining the information of path number and path diversity to identify influential nodes in complex networks. This method is shown to outperform many well-known methods in both undirected and directed networks. Moreover, the efficiency of our method makes it possible to be applied to very large systems.Comment: 6 pages, 6 figure

Transverse force on a moving vortex with the acoustic geometry

We consider the transverse force on a moving vortex with the acoustic metric using the $\phi$-mapping topological current theory. In the frame of effective spacetime geometry the vortex appear naturally by virtue of the vortex tensor in the Lorentz spacetime and we show that it is just the vortex derived with the order parameter in the condensed matter. With the usual Lagrangian we obtain the equation of motion for the vortex. At last, we show that the transverse force on the moving vortex in our equation is just the usual Magnus force in a simple model.Comment: 11 pages, no figur