二元凸函数的判别条件

给出了二元凸函数的定义,导出了二元凸函数的判别条件,该判别条件由二元函数的二阶导数给出.用二元凸函数的判别条件和半正定的(半负定)矩阵的性质,得到了二元二次多项式凸性的简单判别形式.     

切触有理插值新方法

针对传统连分式插值,计算复杂度高,计算过程中分母为零的不可预知性及插值函数不满足某些给定条件,应用不方便等问题,利用已知节点、函数值、导数值,构造两个多项式,分别作为有理插值函数的分子和分母,得出各阶导数条件下切触有理插值的新公式,并给出特殊情形的表达式.若添加适当的参数,可任意降低插值函数次数.该方法计算简洁,应用方便,插值函数的分母在节点处不为零且满足全部插值条件.数值例子验证了新方法的可行性、有效性和实用性.     

有限离散函数的导数和性质

通过引入有限离散函数的导数概念,分别从几何直观和性质两个角度,比较了有限离散函数的导数概念和常规连续函数导数的相似性.结果表明,在局部情况下,有限离散函数导数近似等于连续情形下的导数.在运算性质上,有限离散函数导数的性质非常相似于连续情形时的导数性质.最后的例子给出了有限离散函数导数的一个应用.     

二元函数梯度存在的一个充要条件

本文首先拓展了二元函数方向导数的计算公式,并给出了该公式成立的充分必要条件,在此基础上也给出了二元函数梯度存在的一个充要条件.     

含导函数Stieltjes积分边界条件下二阶问题的正解

本文研究了一类含导函数Stieltjes积分边值条件下二阶边值问题的正解.由于边值条件中带有导数,导致讨论过程与已有文献不同,并且给出相应的格林函数.应用不动点指数理论证明非线性项f(x,y,z)关于x,y有超(次)线性增长情形下方程正解的存在性.通过两个具体例子进行说明理论结果的有效性,例子中边值条件包含积分型与多点型的形式.     

含有离散时滞及分布时滞分数阶神经网络的渐近稳定性分析

研究了含有离散时滞及分布时滞的分数阶神经网络在Caputo导数意义下的渐近稳定性问题.通过构造Lyapunov函数和利用分数阶Razumikhin定理给出了含有离散时滞和分布时滞的分数阶神经网络渐近稳定性的充分条件,并给出4个例子验证了定理条件的有效性.     

广义二阶导数和二阶混合偏导数之间的关系

给出并证明了函数在一点处广义二阶可导的一个充分条件,分析了二元函数在一点的广义二阶导数和二阶混合偏导数之间的关系.     

二元3带小波紧框架的构造

研究二元3带小波紧框架的结构.首先给出二元3带小波紧框架的充分条件.并给出这种小波紧框架的显式公式.若给定的尺度函数的符号函数是有理函数,则可以构造出符号函数为有理函数的小波紧框架.文中给出了数值例子,还给出了二元3带小波紧框架的分解和重构算法.     

依赖于一阶导数的(n-1,1)共轭m点边值问题正解的存在性

利用锥上的Krasnoselskii's不动点定理的推广定理,讨论了一类边值问题,给出了相应依赖于一阶导数的非线性n阶m点边值问题至少一个正解的存在性.对于n阶m点边值问题,给出了相应的Green 函数.证明了在非线性项厂满足一定的增长条件下至少存在1个正解.并且给出了一个例子来说明所获得的结果.     

谈谈多元函数的全微分与全增量的关系

<正> (一) 我们知道,若一元函数可导且导数不为零时,则其增量与微分是等价无穷小量,从而也就可以说微分是增量的线性主部。我们自然会想到,多元函数的全微分与全增量也有这样的关系吗?为说明这个问题(以二元函数为例),首先讨论两个例子。     

A spectral method for parabolic differential equations

We present a spectral method for parabolic partial differential equations with zero Dirichlet boundary conditions. The region Ω for the problem is assumed to be simply-connected and bounded, and its boundary is assumed to be a smooth surface. An error analysis is given, showing that spectral convergence is obtained for sufficiently smooth solution functions. Numerical examples are given in both ?² and ?³.     

On partial sums of a gaussian sequence with stationary increments

We establish large increment properties for a Gaussian sequence with stationary increments under global conditions. Limit theorems for partial sums of the sequence with nonpositive correlation functions or the correlation functions on their speed of convergence to zero are proved via estimating a probability inequality on the supremum of Gaussian processes     

Global minimization of difference of quadratic and convex functions over box or binary constraints

In this paper, we present necessary as well as sufficient conditions for a given feasible point to be a global minimizer of the difference of quadratic and convex functions subject to bounds on the variables. We show that the necessary conditions become necessary and sufficient for global minimizers in the case of a weighted sum of squares minimization problems. We obtain sufficient conditions for global optimality by first constructing quadratic underestimators and then by characterizing global minimizers of the underestimators. We also derive global optimality conditions for the minimization of the difference of quadratic and convex functions over binary constraints. We discuss several numerical examples to illustrate the significance of the optimality conditions. The authors are grateful to the referees for their helpful comments and valuable suggestions which have contributed to the final preparation of the paper.     

Stability results for nonlinear functional differential equations using fixed point methods

We present new conditions for stability of the zero solution for three distinct classes of scalar nonlinear delay differential equations. Our approach is based on fixed point methods and has the advantage that our conditions neither require boundedness of delays nor fixed sign conditions on the coefficient functions. Our work extends and improves a number of recent stability results for nonlinear functional differential equations in a unified framework. A number of examples are given to illustrate our main results.     

On the distribution of zeros of functions from weighted Bergman spaces

We refine some well-known necessary characteristics of zero sets of functions from weighted Bergman spaces under sufficiently general conditions on weights.     

Partial stability analysis of some classes of nonlinear systems

A nonlinear differential equation system with nonlinearities of a sector type is studied. Using the Lyapunov direct method and the comparison method, conditions are derived under which the zero solution of the system is stable with respect to all variables and asymptotically stable with respect to a part of variables. Moreover, the impact of nonstationary perturbations with zero mean values on the stability of the zero solution is investigated. In addition, the corresponding time-delay system is considered for which delay-independent partial asymptotic stability conditions are found. Three examples are presented to demonstrate effectiveness of the obtained results.     

The L* Partial Order on the Set of Group Matrices

In this paper,we use the L(o)wner partial order and the star partial order to introduce a new partial order (denoted by "L*") on the set of group matrices,and get some characteristics and properties of the new partial order.In particular,we prove that the L* partial order is a special kind of the core par-tial order and it is equivalent to the star partial order under some conditions.We also illustrate its difference from other partial orders with examples and find out under what conditions it is equivalent to other partial orders.     

Functions of small growth with no unbounded Fatou components

We prove a form of the cos πρ theorem which gives strong estimates for the minimum modulus of a transcendental entire function of order zero. We also prove a generalisation of a result of Hinkkanen that gives a sufficient condition for a transcendental entire function to have no unbounded Fatou components. These two results enable us to show that there is a large class of entire functions of order zero which have no unbounded Fatou components. On the other hand, we give examples which show that there are in fact functions of order zero which not only fail to satisfy Hinkkanen’s condition but also fail to satisfy our more general condition. We also give a new regularity condition that is sufficient to ensure that a transcendental entire function of order less than 1/2 has no unbounded Fatou components. Finally, we observe that all the conditions given here which guarantee that a transcendental entire function has no unbounded Fatou components also guarantee that the escaping set is connected, thus answering a question of Eremenko for such functions.     

Distributional chaos in random dynamical systems

In this paper, we introduce the notion of distributional chaos and the measure of chaos for random dynamical systems generated by two interval maps. We give some sufficient conditions for a zero measure of chaos and examples of chaotic systems. We demonstrate that the chaoticity of the functions that generate a system does not, in general, affect the chaoticity of the system, i.e. a chaotic system can arise from two nonchaotic functions and vice versa. Finally, we show that distributional chaos for random dynamical system is, in some sense, unstable.