Minimizing Rational Functions by Exact Jacobian SDP Relaxation Applicable to Finite Singularities

This paper considers the optimization problem of minimizing a rational function. We reformulate this problem as polynomial optimization by the technique of homogenization. These two problems are shown to be equivalent under some generic conditions. The exact Jacobian SDP relaxation method proposed by Nie is used to solve the resulting polynomial optimization. We also prove that the assumption of nonsingularity in Nie's method can be weakened as the finiteness of singularities. Some numerical examples are given to illustrate the efficiency of our method.Comment: 23 page

Multiparty quantum secret splitting and quantum state sharing

A protocol for multiparty quantum secret splitting is proposed with an ordered $N$ EPR pairs and Bell state measurements. It is secure and has the high intrinsic efficiency and source capacity as almost all the instances are useful and each EPR pair carries two bits of message securely. Moreover, we modify it for multiparty quantum state sharing of an arbitrary $m$-particle entangled state based on quantum teleportation with only Bell state measurements and local unitary operations which make this protocol more convenient in a practical application than others.Comment: 7 pages, 1 figure. The revision of the manuscript appeared in PLA. Some procedures for detecting cheat have been added. Then the security loophole in the original manuscript has been eliminate

Fractal and multifractal properties of a family of fractal networks

In this work, we study the fractal and multifractal properties of a family of fractal networks introduced by Gallos {\it et al.} ({\it Proc. Natl. Acad. Sci. U.S.A.}, 2007, {\bf 104}: 7746). In this fractal network model, there is a parameter $e$ which is between $0$ and $1$, and allows for tuning the level of fractality in the network. Here we examine the multifractal behavior of these networks, dependence relationship of fractal dimension and the multifractal parameters on the parameter $e$. First, we find that the empirical fractal dimensions of these networks obtained by our program coincide with the theoretical formula given by Song {\it et al.} ( {\it Nat. Phys}, 2006, {\bf 2}: 275). Then from the shape of the $\tau(q)$ and $D(q)$ curves, we find the existence of multifractality in these networks. Last, we find that there exists a linear relationship between the average information dimension $$ and the parameter $e$.Comment: 12 pages, 7 figures, accepted by J. Stat. Mec