线性微分系统的广义反射函数

研究了线性微分系统的广义反射函数,作为应用,给出了线性微分系统具有三角型反射矩阵的充要条件.     

二次多项式微分系统的广义反射函数

利用广义反射函数理论,讨论多项式微分系统的广义反射函数的结构形式.并利用所得结论探讨二次多项式微分系统的周期解的几何性质.     

具有相同广义反射函数的微分系统的等价性

利用广义反射函数给出了微分系统等价的定义,研究了线性微分系统的相互等价性及其与非线性微分系统的等价性.     

广义函数的乘法

1950年,法国数学家L.Schwartz成功地引入了广义函数概念:具有紧支集的无穷次连续可微函数空间上的连续线性泛函。但如何定义任意两个广义函数的乘积问题一直未能得到圆满解决。从广义函数概念出发,易于定义无穷次可微函数与任意广义函数的乘积:     

广义差集与几乎完美序列

提出了广义差集的概念,并且给出了广义差集的一些初等性质.从应用的角度讲,广义差集就是使得其±1特征序列的自相关函数是(最多)三值的一种组合结构.因此,广义差集不仅仅是在概念(理论)上的推广,它还具有深层次的应用背景.事实上,给出了一些广义差集,它不是可分差集,也不是相对差集.同时也给出了一类广义差集存在的一些必要条件,使得这些广义差集对应的±1特征序列成为几乎完美序列.并举例说明本文中的方法是有效的.     

覆盖广义粗糙集的模糊性

在研究覆盖广义粗糙集的基础上,利用两个距离函数Hamming和Euclidean距离函数,结合模糊集的最近寻常集,引入了覆盖广义粗糙集模糊度的概念,给出了一种模糊度计算方法,并证明了该模糊度的一些重要性质。这些结果在覆盖广义粗糙集的理论研究和应用都发挥着一定作用。     

环Z2l上广义Bent函数的稳定性

文献「４」为研究密钥流序列的线性复杂度稳定性和使一些流密码能抗ＢＡＡ（最佳仿射逼近）攻击,提出Ｂｅｎｔ函数稳定性概念,文献「７」研究了素域Ｚｐ上广义Ｂｅｎｔ函数的稳定性及其构造,并指出当ｍ是合数时,ｍ值广义Ｂｅｎｔ函数并不都有稳定性,本文进一步在环Ｚ２＾ｌ（ｌ〉１）上提出了广义Ｂｅｎｔ函数稳定性的概念,综合应用谱、概率和代数数论的方法考察了稳定的概率意义,给出了稳定函数的概率判别条件,提供了构造稳     

白噪声分析中广义算子值函数的Bochner-Wick积分

白噪声广义算子在白噪声分析理论及其应用中起着十分重要的作用. 本文主要讨论了白噪声广义算子值函数的积分及相关问题. 主要工作有: 引入了广义算子值测度的概念, 分别讨论了这种测度在象征和算子p-范数意义下的变差及相互关系; 借助于广义算子的Wick积运算, 引入了广义算子值函数关于广义算子值测度的一种积分---Bochner-Wick积分, 讨论了这种积分的性质, 建立了相应的收敛定理并且展示了其在量子白噪声理论中的应用; 探讨了Bochner-Wick积分的Fubini定理及相关问题.     

凸域内定长线段运动测度公式的另一种表达式

本文研究了凸域内定长线段的运动测度.利用广义支持函数及径向函数的概念,获得了运动测度公式的另一种表达式,并改进了已有文献中的相关结果.     

广义部分Bent函数和广义Bent函数的关系

Bent函数是一类特殊的布尔函数,因其非线性性和稳定性在密码学和通信等领域有很重要的应用,但它们数量少,不平衡且无相关免疫性,为了弥补Bent函数的不足,Claud Carlet提出了部分Bent函数的概念,部分Bent函数是包含Bent函数的更大的函数类,后来,人们又将这两种函数概念先后都拓广到了环zm^n(m为正整数）上,分别被称为zm^n上的广义Bent函数和广义部分Bent函数,本文利用zp^n(p为素数）上广义部分Bent函数的Chrestenson循环谱特征讨论了zp^n上的广义部分Bent函数和广义Bent函数之间的关系,给出了这两种函数之间的函数关系式和谱值关系式。     

二次微分系统的反射函数及其周期解

本文给出了二镒多项式微分系统具有满足特定关系式的反射函数和存在周期解的充要条件，以及在此条件下反射函数的具体表达式及周期解的稳定性态。     

On the relationship between quadratic polynomial differential system and the Bernoulli equation

In this article, we have established a relationship between a quadratic polynomial differential system and a Bernoulli equation by using the method of reflective function. And applied the results to discuss the qualitative behavior of solutions of this quadratic polynomial differential system.     

Ulam-type stability for differential equations driven by measures

Based on a notion of Stieltjes derivative of a function with respect to another function, we provide Ulam–Hyers type stability results for nonlinear differential equations driven by measures on compact or on unbounded intervals, in the lack of Lipschitz continuity assumptions. In particular, one can deduce stability results for generalized differential equations, dynamic equations on time scales or impulsive differential equations (including the case of an infinite number of impulses that accumulate in the considered interval, thus allowing the study of Zeno hybrid systems).     

Sampling theorem and multi-scale spectrum based on non-linear Fourier atoms

This study concerns some new developments of unit analytic signals with non-linear phase. It includes ladder-shaped filter, generalized Sinc function based on non-linear Fourier atoms, generalized sampling theorem for non-bandlimited signals and the notion of multi-scale spectrum for discrete sequences. We first introduce the ladder-shaped filter and show that the impulse response of its corresponding linear time-shift invariant system is the generalized Sinc function as a product of periodic Poisson kernel and Sinc function. Secondly, we establish a Shannon-type sampling theorem based on generalized Sinc function for this type of non-bandlimited signal. We further prove that this type of signal may be holomorphically extended to strips in the complex plane containing the real axis. Finally, we introduce the notion of multi-scale spectrums for discrete sequences and develop the related fast algorithm.     

Differential calculus and integration of generalized functions over membranes

In this paper we continue the development of the differential calculus started in Aragona et al. (Monatsh. Math. 144:13–29, 2005). Guided by the so-called sharp topology and the interpretation of Colombeau generalized functions as point functions on generalized point sets, we introduce the notion of membranes and extend the definition of integrals, given in Aragona et al. (Monatsh. Math. 144:13–29, 2005), to integrals defined on membranes. We use this to prove a generalized version of the Cauchy formula and to obtain the Goursat Theorem for generalized holomorphic functions. A number of results from classical differential and integral calculus, like the inverse and implicit function theorems and Green’s theorem, are transferred to the generalized setting. Further, we indicate that solution formulas for transport and wave equations with generalized initial data can be obtained as well.     

微分系统的反射函数与周期解

本文主要介绍了微分系统反射函数与周期解理论的产生与发展动态．论述了一些基本问题，并提出了一些可供研究的新课题。     

A Discontinuous Colombeau Differential Calculus

Starting from the Colombeau Generalized Functions, the sharp topologies and the notion of generalized points, we introduce a new kind of differential calculus (for functions between totally disconnected spaces). We also define here the notions of holomorphic generalized functions (in this new framework) and generalized manifold. Finally we give an answer to a question raised in [6].     

A Discontinuous Colombeau Differential Calculus

Starting from the Colombeau Generalized Functions, the sharp topologies and the notion of generalized points, we introduce a new kind of differential calculus (for functions between totally disconnected spaces). We also define here the notions of holomorphic generalized functions (in this new framework) and generalized manifold. Finally we give an answer to a question raised in [6].Research partially supported by CNPq (Proc 300652/95-0).