The gap distribution of directions in some Schottky groups
Xin Zhang

We prove the existence and some properties of the limiting gap distribution for the directions of some Schottky group orbits in the Poincaré disk. A key feature is that the fundamental domains for these groups have infinite area.

keywords: Fuchsian groups of the second kind Patterson-Sullivan measure gap distribution
Subspace trust-region algorithm with conic model for unconstrained optimization
Xin Zhang Jie Wen Qin Ni
In this paper, a new subspace algorithm is proposed for unconstrained optimization. In this new algorithm, the subspace technique is used in the trust region subproblem with conic model, and the dogleg method is modified to solve this subproblem. The global convergence of this algorithm under some reasonable conditions is established. Numerical experiment shows that this algorithm may be superior to the corresponding algorithm without using subspace technique especially for large scale problems.
keywords: conic model trust region method Unconstrained optimization subspace method global convergence.
Bogdanov-Takens bifurcation of codimension 3 in a predator-prey model with constant-yield predator harvesting
Jicai Huang Sanhong Liu Shigui Ruan Xinan Zhang
Recently, we (J. Huang, Y. Gong and S. Ruan, Discrete Contin. Dynam. Syst. B 18 (2013), 2101-2121) showed that a Leslie-Gower type predator-prey model with constant-yield predator harvesting has a Bogdanov-Takens singularity (cusp) of codimension 3 for some parameter values. In this paper, we prove analytically that the model undergoes Bogdanov-Takens bifurcation (cusp case) of codimension 3. To confirm the theoretical analysis and results, we also perform numerical simulations for various bifurcation scenarios, including the existence of two limit cycles, the coexistence of a stable homoclinic loop and an unstable limit cycle, supercritical and subcritical Hopf bifurcations, and homoclinic bifurcation of codimension 1.
keywords: homoclinic bifurcation. constant-yield harvesting Bogdanov-Takens bifurcation of codimension 3 Hopf bifurcaton Predator-prey model
Some united existence results of periodic solutions for non-quadratic second order Hamiltonian systems
Xingyong Zhang Xianhua Tang
In this paper, some existence theorems are obtained for periodic solutions of second order Hamiltonian systems under non-quadratic conditions by using the minimax principle. Our results unite, extend and improve those relative works in some known literature.
keywords: critical point linking theorem. Second order Hamiltonian system nonconstant periodic solution nontrivial periodic solution
Remarks on the blow-up criterion for smooth solutions of the Boussinesq equations with zero diffusion
Yan Jia Xingwei Zhang Bo-Qing Dong
This article is concerned with the blow-up criterion for smooth solutions of three-dimensional Boussinesq equations with zero diffusion. It is shown that if the velocity field $u(x,t)$ satisfies \begin{eqnarray*} u\in L^p(0,T_1;B^r_{q,\infty}(R^3)),\quad \frac{2}{p}+\frac{3}{q}=1+r,\quad \frac{3}{1+r}< q \leq \infty, \quad -1 < r \leq 1, \end{eqnarray*} then the solution can be continually extended to the interval $(0,T)$ for some $T>T_1$.
keywords: Boussinesq equations blow-up criterion Besov spaces. zero diffusion
The impact of releasing sterile mosquitoes on malaria transmission
Hongyan Yin Cuihong Yang Xin'an Zhang Jia Li

The sterile mosquitoes technique in which sterile mosquitoes are released to reduce or eradicate the wild mosquito population has been used in preventing the malaria transmission. To study the impact of releasing sterile mosquitoes on the malaria transmission, we first formulate a simple SEIR (susceptible-exposed-infected-recovered) malaria transmission model as our baseline model, derive a formula for the reproductive number of infection, and determine the existence of endemic equilibria. We then include sterile mosquitoes in the baseline model and consider the case of constant releases of sterile mosquitoes. We examine how the releases affect the reproductive numbers and endemic equilibria for the model with interactive mosquitoes and investigate the impact of releasing sterile mosquitoes on the malaria transmission.

keywords: Malaria mathematical modeling sterile mosquitoes reproductive number endemic equilibrium
Test of copositive tensors
Li Li Xinzhen Zhang Zheng-Hai Huang Liqun Qi

In this paper, an SDP relaxation algorithm is proposed to test the copositivity of higher order tensors. By solving finitely many SDP relaxations, the proposed algorithm can determine the copositivity of higher order tensors. Furthermore, for any copositive but not strictly copositive tensor, the algorithm can also check it exactly. Some numerical results are reported to show the efficiency of the proposed algorithm.

keywords: Symmetric tensor polynomial optimization SDP relaxation

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