具有特殊坐标曲线网的B\''ezier曲面

在计算机辅助几何设计中, B\''ezier曲面是一类重要的参数曲面.在微分几何中,坐标曲线网也是重要的研究内容.本文中,我们对具有特殊坐标曲线网(如正交曲线网、曲率曲线网、共轭曲线网等)的B\''ezier曲面进行研究.此外,我们还构造了满足能量约束的特殊B\''ezier曲面,给出了基于控制结构的充分条件并给出具体实例.     

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.     

扰动Fej\'{e}r点上复Hermite插值的联合逼近

$D$是复平面中由闭Jordan曲线$\Ga$围成的单连区域. 考虑在$\Ga$上扰动Fej\''er点的 Hermite插值一致逼近、平均逼近和联合逼近于函数$f\in A^{(q)}(\o D)$. 该文中的逼近阶一般说来是不可再改进的, 区域的边界限制条件到目前为止是最少的. 以往的全部同类结果都包括在该文中作为特殊情形, 由于该文方法上的改进, 简化和省去了以往 某些证明过程.     

一类广义Li\'enard方程周期正解的存在性

本文证明一类广义Li\''enard方程周期正解的存在性及渐近稳定性. 我们讨论的非自治函数可满足超线性条件,克服了连续定理讨论超线性条件的困难. 文章的最后,我们给出两个例子和数值解以及对应的相图和时间序列图来验证我们的结论.     

R\''enyi公式的推广

在本文,作为著名的R\''enyi公式(其刻画了标号连通单圈图的计数显式)的自然推广,我们研究了标号匀称$(k+1)$秩$(p,~q)$超单圈的计数问题,给出了如下的计数显式:$$U_{p,~q}^{(k+1)}=\begin{cases} \frac{p!}{2[(k-1)!]^q}\cdot\sum_{t=2}^q \frac{q^{q-t-1}\cdot sgn(tk-2)}{(q-t)!}, & p=qk, \\ 0,& p\neq qk, \end{cases}$$其中$k,~p,~q$均为正整数.     

受L\''evy噪声扰动的Logistic方程的全局吸引性

本文考虑了受L\''evy噪声扰动的Logistic方程. 在合适的条件下, 我们得到了解的全局存在性与唯一性; 我们证明了当初始值小于环境的容纳量时, 唯一的正的平衡态具有全局吸引性.     

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     

线性超图的边染色

超图$H$的一个$k$-边染色是用$k$种颜色的边染色, 使得相交的边染不同的颜色. Erd\H{o}s-Faber-Lov\''asz猜想认为任一$n$个顶点的无环线性超图都有一个$n$-边染色. 2021年, Kang, Kelly, K\"uhn, Methuku和Osthus对充分大的$n$确认了该猜想成立. 在本文中, 我们证明该猜想对弱冲突的超图是成立的. 这严格拓展了Bretto, Faisant 和Hennecart在2020年的两个相关结果.     

带两个形状参数的四次Bézier曲线的扩展

给出了一组含有两个形状参数α,β的四次多项式基函数,是四次Bernstein基函数的扩展,分析了这组基的性质;基于这组基定义了带两个形状参数的多项式曲线,所定义的曲线不仅保留了四次Bézier曲线一些实用的几何特征,而且具有形状的可调性,在控制多边形不变的情况下,改变参数α,β的取值,可以生成不同的逼近控制多边形的曲线;通过分析该曲线与四次Bézier曲线之间的关系,给出了α和β的几何意义,并利用Bézier曲线递归分割算法给出了这种曲线的几何作图法,同时还讨论了曲线间的拼接问题.     

非齐度量测度空间上Morrey-Herz空间上的Calder\'{o}n-Zygmund算子及其交换子[英文]

设$(\mathcal{X},d,\mu)$ 是一个同时满足上双倍条件 和几何双倍条件的非齐度量测度空间, 本文中, 引进一类非齐度量测度空间上的Morrey-Herz 空间, 利用非齐度量测度空间的特征, 特别是$\eta$-弱逆倍条件, 证明 Calder\''{o}n-Zygmund算子及其交换子在Morrey-Herz空间上的有界性.     

三次Pythagorean Hodograph 曲线在给定总弧长下的$G^2$拼接

Pythagorean-hodograph (PH)曲线因其在弧长和等距线计算方面的优势而被广泛应用于曲线建模中.本文讨论了在总弧长约束下的三次PH曲线$G^2$连续拼接问题.具体地说,给定两个端点和一个拼接点,构造两条三次PH曲线,使其在指定总弧长下插值两个端点,并且在连接点处是$G^2$连续的.这也可以看作是一个曲线延拓问题.根据三次PH曲线的弧长公式和$G^2$连续条件,最终将问题转化为了一个带有约束的极小值问题,同时我们给出了几个具体例子来说明该方法.     

三角B\''{e}zier曲面和四边B\''{e}zier曲面之间的相互转化

B\'{e}zier曲面有两种不同的形式:三角B\'{e}zier曲面和四边B\'{e}zier曲面,它们有着不同的基底和不同的几何拓扑结构, 但是它们也有很多共同的性质,因此三角B\'{e}zier曲面和四边B\'{e}zier曲面之间的相互转化就成为CAGD 里一个重要研究课题.在本文中, 我们用函数复合的方法实现两者之间的相互转化.被复合的两个函数, 一个用Polar形式表示,另一个用常见的Bernstein基形式表示.     

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.     

Poincar{\''e} Bifurcation from an Elliptic Hamiltonian of Degree Four with Two-saddle Cycle

In this paper, we consider Poincar{\''e} bifurcation from an elliptic Hamiltonian of degree four with two-saddle cycle. Based on the Chebyshev criterion, not only one case in the Li{\''e}nard equations of type $(3, 2)$ is discussed again in a different way from the previous ones, but also its two extended cases are investigated, where the perturbations are given respectively by adding $\varepsilon y(d_0 + d_2 v^{2n})\frac{\partial }{{\partial y}}$ with $ n\in \mathbb{N^+}$ and $\varepsilon y(d_0 + d_4 {v^4}+ d_2 v^{2n+4})\frac{\partial }{{\partial y}}$ with $n=-1$ or $ n\in \mathbb{N^+}$, for small $\varepsilon > 0$. For the above cases, we obtain all the sharp upper bound of the number of zeros for Abelian integrals, from which the existence of limit cycles at most via the first-order Melnikov functions is determined. Finally, one example of double limit cycles for the latter case is given.     

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.     

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.     

具有BMO系数的拟线性椭圆方程解的BMO估计

We establish a global limiting case of nonlinear Calderon-Zygmund theory to quasilinear elliptic equations div A(x,Du) = div(|F|~(p-2)F) under the BMO smallness of the nonlinearity,that is |F|~(p-2)F∈BMO implies that Du ∈BMO.     

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.