三调和三角B\'ezier曲面设计

偏微分方程曲面设计,是由给定边界条件出发构造满足偏微分方程的曲面.本文基于三调和方程,提出三类边界条件,分别通过求解线性方程组,给出三调和三角形B\''ezier曲面的设计方法.证明了在这些边界条件下,生成曲面的唯一性,并分别给出具体曲面设计算法.通过实例验证了本文结论的有效性,并对三种边界条件进行对比分析.     

Painlev\'{e} Analysis and Auto-B\"{a}cklund Transformation for a General Variable Coefficient Burgers Equation with Linear Damping Term

This paper investigates a general variable coefficient (gVC) Burgers equation with linear damping term. We derive the Painlev\''{e} property of the equation under certain constraint condition of the coefficients. Then we obtain an auto-B\"{a}cklund transformation of this equation in terms of the Painlev\''{e} property. Finally, we find a large number of new explicit exact solutions of the equation. Especially, infinite explicit exact singular wave solutions are obtained for the first time. It is worth noting that these singular wave solutions will blow up on some lines or curves in the $(x,t)$ plane. These facts reflect the complexity of the structure of the solution of the gVC Burgers equation with linear damping term. It also reflects the complexity of nonlinear wave propagation in fluid from one aspect.     

二部图形式的Erd\H{O}s-S\'{o}s猜想

二部图形式的Erd\H{O}s-S\''{o}s猜想     

Hopf Cyclicity of a Class of Li\'{e}nard-Type Systems

The Hopf cyclicities of some smooth polynomial, rational polynomial and piecewise smooth Li\''{e}nard systems are studied. For two Li\''{e}nard systems with the same damping term and different restoring (or potential) terms, we provide sufficient conditions that the two systems have the same Hopf cyclicity. Then, some examples are given to illustrate the efficiency and applicability of our results.     

Critical point approaches to Gradient-Type systems on the Sierpi\'{n}ski Gasket

We investigate the existence of multiple solutions for parametric quasi-linear systems of the gradient-type on the Sierpi\''{n}ski gasket. We give some new criteria to guarantee that the systems have at least three weak solutions by using a variational method and some critical points theorems due to Ricceri. We extend and improve some recent results. Finally, we give two examples to illustrate the main results.     

Exact travelling wave solutions for nonlinear Schr\"{o}dinger equation with variable coefficients

In this paper, two nonlinear Schr\"{o}dinger equations with variable coefficients in nonlinear optics are investigated. Based on travelling wave transformation and the extended $(\frac{G''}{G})$-expansion method, exact travelling wave solutions to nonlinear Schr\"{o}dinger equation with time-dependent coefficients are derived successfully, which include bright and dark soliton solutions, triangular function periodic solutions, hyperbolic function solutions and rational function solutions.     

Fej\'{e}r type inequalities for Harmonically-convex functions with applications

In this paper, a new weighted identity involving harmonically symmetric functions and differentiable functions is established. By using the notion of harmonic symmetricity, harmonic convexity and some auxiliary results, some new Fej\''{e}r type integral inequalities are presented. Applications to special means of positive real numbers are given as well.     

平面三次Pythagorean-Hodograph Hyperbolic曲线

本文基于Pythagorean-hodograph (PH)曲线和代数双曲线的良好几何特性,构造了Pythagorean-Hodograph Hyperbolic (PH-H)曲线,并给出了PH-H曲线的定义以及相应性质.同时,分别利用Hyperbolic基函数和Algebraic Hyperbolic (AH) B\''ezier基函数,得到了平面三次AH B\''ezier曲线为PH曲线的两个不同的充要条件.此外,三次PH-H曲线也被用于求解具有确定解的$G^1$ Hermite插值问题.文中给出了具体实例来说明我们的方法.     

Asymptotic Behavior of a Stochastic Predator-prey Model with Beddington-DeAngelis Functional Response and L\'{e}vy Jumps

A stochastic two-prey-one-predator model with Beddington-DeAngelis functional response and L\''{e}vy jumps is proposed and investigated in this paper. First of all, we prove the existence and uniqueness of the global positive solution, and stochastic ultimate boundedness of the solution. Next, under a simple assumption, by using It\^{o} formula and other important inequalities, some sufficient conditions are established to ensure the extinction and persistence in the mean of the system. The results show that neither strong white noise nor L\''{e}vy noise is conducive to the persistence of the population. Finally, the theoretical results are verified by numerical simulations.     

On the Main Aspects of the Inverse Conductivity Problem

We consider a nonlinear inverse problem for an elliptic partial differential equation known as the Calder{\''o}n problem or the inverse conductivity problem. Based on several results, we briefly summarize them to motivate this research field. We give a general view of the problem by reviewing the available results for $C^2$ conductivities. After reducing the original problem to the inverse problem for a Schr\"odinger equation, we apply complex geometrical optics solutions to show its uniqueness. After extending the ideas of the uniqueness proof result, we establish a stable dependence between the conductivity and the boundary measurements. By using the Carleman estimate, we discuss the partial data problem, which deals with measurements that are taken only in a part of the boundary.