一类双重退化抛物方程组解的整体存在与爆破

研究一类具有非线性边界流的双重退化抛物方程组解的整体存在与爆破,通过构造自相似的上下解,得到了整体存在曲线.借助一些新的结果,获得了Fujita临界指数.其中一个有趣的现象是:整体存在曲线和Fujita临界曲线分别是由一个矩阵和线性方程组来决定.     

Algebraic criteria for global existence or blow-up for a boundary coupled system of nonlinear diffusion equations

This article deals with the critical curves for a degenerate parabolic system coupled via nonlinear boundary flux. By constructing the self-similar supersolution and subsolution, we obtain the critical global existence curve. The critical curve of Fujita type is conjectured with the aid of some new results.     

Critical curves for fast diffusive non-Newtonian equations coupled via nonlinear boundary flux

This paper deals with the critical curve for a nonlinear boundary value problem of a fast diffusive non-Newtonian system. We first obtain the critical global existence curve by constructing the self-similar supersolution and subsolution. And then the critical Fujita curve is conjectured with the aid of some new results.     

Global existence and blowing-up of solutions to a class of coupled reaction–convection–diffusion systems

This paper concerns the global existence and blowing-up of solutions to the homogeneous Neumann problem of a coupled reaction–convection–diffusion system. The critical Fujita curve is determined and blowing-up theorem of Fujita type is established. An interesting phenomenon is that the critical Fujita curve even could be the infinite due to the convection.     

Critical curves for fast diffusive polytropic filtration equations coupled through boundary

This article deals with the critical curve of a fast diffusive polytropic filtration system coupled at the boundary condition. By constructing the self-similar supersolution and subsolution, we obtain the critical global existence curve. The critical curve of Fujita type is conjectured with the aid of some new results.     

带有局部源的非局部扩散方程组的Fujita现象

本文研究了一类带有局部源的非局部扩散系统$u_t=J*u-u+a(x)v^{p}$, $v_t=J*v-v+a(x)u^q$的柯西问题,首先根据是否存在全局解建立了Fujita曲线$(pq)_c=1+\max \{p+1,q+1\}$,也即证明了:如果$1(pq)_c$时,则既存在全局解,也存在非全局解.然后我还根据初始值在无穷远处的衰减率建立了第二临界曲线.     

A note on a conjecture for the critical curve of a weakly coupled system of semilinear wave equations with scale-invariant lower order terms

In this note, two blow-up results are proved for a weakly coupled system of semilinear wave equations with distinct scale-invariant lower order terms both in the subcritical case and in the critical case when the damping and the mass terms make both equations in some sense “wave-like.” In the proof of the subcritical case, an iteration argument is used. This approach is based on a coupled system of nonlinear ordinary integral inequalities and lower bound estimates for the spatial integral of the nonlinearities. In the critical case, we employ a test function-type method that has been developed recently by Ikeda-Sobajima-Wakasa and relies strongly on a family of certain self-similar solutions of the adjoint linear equation. Therefore, as critical curve in the p−q plane of the exponents of the power nonlinearities for this weakly coupled system, we conjecture a shift of the critical curve for the corresponding weakly coupled system of semilinear wave equations.     

Monotonicity and phase diagram for multirange percolation on oriented trees

We consider Bernoulli bond percolation on oriented regular trees, where besides the usual short bonds, all bonds of a certain length are added. Independently, short bonds are open with probability p and long bonds are open with probability q. We study properties of the critical curve which delimits the set of pairs (p,q) for which there are almost surely no infinite paths. We also show that this curve decreases with respect to the length of the long bonds.     

Approximation of optimal stopping problems

We consider optimal stopping of independent sequences. Assuming that the corresponding imbedded planar point processes converge to a Poisson process we introduce some additional conditions which allow to approximate the optimal stopping problem of the discrete time sequence by the optimal stopping of the limiting Poisson process. The optimal stopping of the involved Poisson processes is reduced to a differential equation for the critical curve which can be solved in several examples. We apply this method to obtain approximations for the stopping of iid sequences in the domain of max-stable laws with observation costs and with discount factors.     

Critical Heegaard surfaces

In this paper we introduce critical surfaces, which are described via a 1-complex whose definition is reminiscent of the curve complex. Our main result is that if the minimal genus common stabilization of a pair of strongly irreducible Heegaard splittings of a 3-manifold is not critical, then the manifold contains an incompressible surface. Conversely, we also show that if a non-Haken 3-manifold admits at most one Heegaard splitting of each genus, then it does not contain a critical Heegaard surface. In the final section we discuss how this work leads to a natural metric on the space of strongly irreducible Heegaard splittings, as well as many new and interesting open questions.     

一类具有非线性边界条件的拟线性抛物方程解的全局存在性和爆破

研究了一类具有非线性边界条件的拟线性方程组解的整体存在性和爆破.通过构造不同类型的上、下解并利用M-矩阵的基本性质,给出了非负解整体存在性的充要条件.借助这些新结果,给出了Fuiita型临界曲线,把最近的结果推广到了更一般的方程.     

与给定多边形相切的可调广义Ball闭曲线

讨论了与给定控制多边形相切的分段三次、五次和六次可调广义Ball曲线的构造方法,所构造的曲线分别是C1,C2和C3连续的,而且对切线多边形是保形的.曲线上的所有广义Ball曲线段的控制点由切线多边形的顶点直接计算产生.给出了在保持公共连接点处相应连续的情况下,内控制点的活动范围.曲线可以在一定范围内做局部修改.计算实例...     

Rational cubic/quartic Said-Ball conics

In CAGD, the Said-Ball representation for a polynomial curve has two advantages over the Bézier representation, since the degrees of Said-Ball basis are distributed in a step type. One advantage is that the recursive algorithm of Said-Ball curve for evaluating a polynomial curve runs twice as fast as the de Casteljau algorithm of Bézier curve. Another is that the operations of degree elevation and reduction for a polynomial curve in Said-Ball form are simpler and faster than in Bézier form. However, Said-Ball curve can not exactly represent conics which are usually used in aircraft and machine element design. To further extend the utilization of Said-Ball curve, this paper deduces the representation theory of rational cubic and quartic Said-Ball conics, according to the necessary and sufficient conditions for conic representation in rational low degree Bézier form and the transformation formula from Bernstein basis to Said-Ball basis. The results include the judging method for whether a rational quartic Said-Ball curve is a conic section and design method for presenting a given conic section in rational quartic Said-Ball form. Many experimental curves are given for confirming that our approaches are correct and effective.     

二次曲线与其任意两条切线所围面积为定常值的问题研究

本文得到了二次曲线的任意两条相交切线与曲线本身围成的面积如果为定常值，则切线交点的轨迹仍为同类型二次曲线．又若给定两条同类的二次曲线，由其中一条上的每一点向另一条引出两条切线，则这两条切线与另一条曲线围成的面积为定常值．     

Logistic模型的改进——双S形曲线模型的研究

在Logistic模型的基础上,对一类空间饱和容量会随着时间发生变化的问题进行了讨论,指出了Lo-gistic模型应用在这类问题上的局限性,建立了解决该类问题的双S形曲线模型.对一组实际数据进行了拟合,结果表明双S形曲线模型能够更好地刻画实际问题,相对准确地预测未来的发展趋势.     

带两个形状参数的五次Bézier曲线的扩展

给出了一组含有两个形状参数α,β的六次多项式基函数,是五次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,所定义的曲线具有五次Bézier曲线的性质,改变参数α,β的取值,曲线具有更灵活的形状可调性,而且能向上或从两侧逼近控制多边形.另外,经典的五次Bézier曲线和有关文献中带一个形状参数的曲线均是该文所定义曲线的特例.实例表明,定义的曲线为曲线/曲面的设计提供了一种有效的方法.     

与给定多边形相切的C2和C3 广义 Ball闭曲线

B样条曲线和Bezier曲线在CAD、CAGD及逼近中非常有用,例如,在服装CAD纸样的设计过程中,通常先根据尺寸要求确定控制点,再根据控制点用直线段绘制样片的大体轮廓,即绘制多边形,在多边形内用直线和曲线绘制封闭的图形,由于样片的形状不规则,即构成样片的曲线比较复杂,为了使曲线光滑、丰满、有弹性,经常遇到曲线与控制多边形相切     

Koch曲线的级数展开式

本文给出Koch曲线的级数展开式，利用这一展开式证明了Koch曲线满足指数为α=log3/log4的Hoelder条件和处处不可微。     

Discussion on relationship between minimal energy and curve shapes

Energy minimization has been widely used for constructing curve and surface in the fields such as computer-aided geometric design, computer graphics. However, our testing examples show that energy minimization does not optimize the shape of the curve sometimes. This paper studies the relationship between minimizing strain energy and curve shapes, the study is carried out by constructing a cubic Hermite curve with satisfactory shape. The cubic Hermite curve interpolates the positions and tangent vectors of two given endpoints. Computer simulation technique has become one of the methods of scientific discovery, the study process is carried out by numerical computation and computer simulation technique. Our result shows that： （1） cubic Hermite curves cannot be constructed by solely minimizing the strain energy; （2） by adoption of a local minimum value of the strain energy, the shapes of cubic Hermite curves could be determined for about 60 percent of all cases, some of which have unsatisfactory shapes, however. Based on strain energy model and analysis, a new model is presented for constructing cubic Hermite curves with satisfactory shapes, which is a modification of strain energy model. The new model uses an explicit formula to compute the magnitudes of the two tangent vectors, and has the properties： （1） it is easy to compute; （2） it makes the cubic Hermite curves have satisfactory shapes while holding the good property of minimizing strain energy for some cases in curve construction. The comparison of the new model with the minimum strain energy model is included.