广义算子半群与广义分布参数系统的适定性

首先,针对广义分布参数系统的求解问题,提出了由Hilbert空间中有界线性算子所引导的广义算子半群和广义积分半群;其次,讨论了广义预解算子的性质、广义算子半群与广义积分半群的性质;最后,研究了广义分布参数系统的适定性问题.     

关于广义函数赋值问题的完善与积分问题

本文采用非标准分析方法，首先对广义函数的赋值问题进行了完善；接着界定了广义函数的积分；最后讨论了广义函数的广义连续性．     

Hilbert 空间中的广义正交基

广义正交基是Hilbert 空间中正交基的一个自然推广. 本文首先给出一个广义正交基存在的较弱的充要条件; 然后研究广义正交基的性质, 特别地, 得到广义正交基版本的一些有关正交基的经典性质, 如广义正交基的Bessel 等式和不等式等. 作为广义正交基的一个应用, 本文给出广义Riesz 基的一些新刻画. 最后本文讨论广义框架的冗余问题.     

一阶广义分布参数系统的极点配置问题

本文讨论了一阶广义分布参数系统的极点配置问题,应用算子的广义逆给出了问题的解及解的构造性表达式.     

几种广义非线性发展方程的新解

本文研究了构造了广义Kd V方程和广义KP-Burgers方程等几种广义非线性发展方程的新解的问题.利用三种辅助方程及其新解,获得了广义Kd V方程和广义KP-Burgers方程等几种广义非线性发展方程的新解.这些解由双曲余割函数、双曲正切函数、双曲正割函数、双曲余切函数和余割函数组成.     

广义Riesz基的双正交特征刻画

本文利用广义双正交序列研究广义Riesz基的等价刻画,得到了算子序列是广义Riesz基当且仅当该算子列是广义完备的广义Bessel序列,且它存在广义双正交序列及这个双正交序列也是广义完备的广义Bessel序列.进一步证明了等价刻画中两个广义Bessel序列的广义完备性条件可以去掉一个(或者任一个),并举例说明了广义双正交,广义完备与广义Bessel条件之间的关系.     

Panel模型中的精确检验和置信区间

利用广义p-值和广义置信区间的概念,研究了Panel模型中未知参数的检验和置信区间问题.对于回归系数,分别考虑了单个情形和多个线性无关情形下的检验和置信区间问题,得到了精确检验和置信区间.对于方差分量,研究了其任意线性组合的检验和置信区间问题,建立了精确检验和置信区间.基于广义p-值和广义置信区间,获取精确检验和置信区间的方法具有计算方便、易应用于小样本问题的特点.最后,分别从理论和数值上研究了这些精确检验和置信区间的统计性质.     

广义系统正则状态观测器的存在性和设计

水文借助于广义函数与矩阵广义逆理论,在前人工作的基础上证明了广义系统的(包括干扰解耦的)正则状态观测器的设计问题,等价于一个低于rank E阶的正则系统的相应问题,从理论上解决了广义系统正则状态观测器的设计问题。     

广义酉矩阵与广义Hermite矩阵

给出了广义酉矩阵与广义(斜)Hermite矩阵的概念，研究了它们的性质及其与酉阵、共轭辛阵、Hermite阵、Hamilton及广义逆矩阵之间的联系；取得了许多新的结果；推广了酉矩阵、Hermite阵与斜Hermite阵间的相应结果，特别将正交阵的广义Cayley分解推广到了广义酉矩阵上；将各类酉矩阵、Hermite矩阵及广义逆矩阵统一了起来．     

W-空间上的广义向量拟平衡问题

引入了广义向量拟平衡问题和广义集值拟平衡问题,得到了W-空间上广义向量拟平衡问题的存在性定理,作为推论,得到了W-空间上四类拟平衡问题的存在性定理,这些结果推广和改进了文献[1-7]的相应结果。     

Methods of judging shape of solitary wave and solution formulae for some evolution equations with nonlinear terms of high order

In this paper, we present several methods of judging shape of the solitary wave and solution formulae for some nonlinear evolution equations by means of Lienard equations. Then, using the judgement methods and solution formulae, we obtain solutions of the solitary wave for some of important nonlinear evolution equations, which include generalized modified Boussinesq, generalized nonlinear wave, generalized Fisher, generalized Klein-Gordon and generalized Zakharov equations. Some new solitary-wave solutions are found for the equations.     

General solutions of three-dimensional problems for two-dimensional quasicrystals

Three-dimensional problems are systematically investigated for the coupled equations in two-dimensional hexagonal quasicrystals, and two new general solutions, which are called generalized Lekhnitskii–Hu–Nowacki (LHN) solutions and generalized Elliott–Lodge (E–L) solutions, are presented, respectively. By introducing two higher-order displacement functions, an operator analysis technique is applied in a novel way to obtain generalized LHN solutions. For further simplification, a decomposition and superposition procedure is taken to replace the higher-order displacement functions with five quasi-harmonic displacement functions, and then generalized E–L solutions are simplified in terms of these functions. In consideration of different cases of characteristic roots, generalized E–L solutions take different forms, but all are in simple forms that are conveniently applied. To illustrate the application of the general solutions obtained, the closed form solution is obtained for an infinite quasicrystal medium subjected to a point force at an arbitrary point.     

关于GMP和GMT问题的最优解

进一步研究模糊推理的非模糊形式，在几个重要的逻辑系统中形式地讨论GMP(广义取式)和GMT(广义拒取式)问题的最优解。结果表明，GMP和GMT问题的三I解和一种新的三I解都是某种意义下的最优解。还讨论所给算法的还原性问题。     

Extending the generalized tanh method to the generalized Hamiltonian equations: New soliton-like solutions

To certain nonlinear evolution equations, the tanh method has been generalized for constructing not only solitary-wave but also soliton-like solutions. In this paper, no loss of conciseness, we further extend the generalized tanh method with computerized symbolic computation to a pair of generalized Hamiltonian equations. A new family of soliton-like analytical solutions is obtained, of which the solitary waves and previously-claimed soliton-like solutions are shown to be the special cases.     

Numerical solution of fuzzy differential equations under generalized differentiability

In this paper, we interpret a fuzzy differential equation by using the strongly generalized differentiability concept. Utilizing the Generalized Characterization Theorem, we investigate the problem of finding a numerical approximation of solutions. Then we show that any suitable numerical method for ODEs can be applied to solve numerically fuzzy differential equations under generalized differentiability. The generalized Euler approximation method is implemented and its error analysis, which guarantees pointwise convergence, is given. The method’s applicability is illustrated by solving a linear first-order fuzzy differential equation.     

An existence result for generalized vector equilibrium problems without pseudomonotonicity

In this work, we introduce and study a class of generalized vector equilibrium problems for multifunctions which includes a number of generalized vector variational inequality problems and generalized vector variational-like inequality problems as special cases. By using the KKM–Fan theorem and Nadler’s result, we prove an existence theorem for solutions for this class of generalized vector equilibrium problems in Banach spaces. Applications to generalized vector variational-like inequalities are given.     

Generalized Higher-Order Optimality Conditions for Set-Valued Optimization under Henig Efficiency

In this paper, generalized mth-order contingent epiderivative and generalized mth-order epiderivative of set-valued maps are introduced, respectively. By virtue of the generalized mth-order epiderivatives, generalized necessary and sufficient optimality conditions are obtained for Henig efficient solutions to a set-valued optimization problem whose constraint set is determined by a fixed set. Generalized Kuhn–Tucker type necessary and sufficient optimality conditions are also obtained for Henig efficient solutions to a set-valued optimization problem whose constraint set is determined by a set-valued map.     

On generalized vector equilibrium problems

In this paper, we study the generalized vector equilibrium problems in real Hausdorff topological vector space settings. The concepts of weak solutions and strong solutions are introduced. Several new results of existence for the weak solutions and strong solutions of generalized vector equilibrium problems are derived. The new results extend and modify various existence theorems for similar problems.     

Connectedness of the Set of Efficient Solutions for Generalized Systems

We introduce the concept of positive proper efficient solutions to the generalized system in this paper. We show that, under some suitable conditions, the set of positive proper efficient solutions is dense in the set of efficient solutions to the generalized system. We discuss also the connectedness of the set of efficient solutions for the generalized system with monotone bifunctions in real locally convex Hausdorff topological vector spaces. This research was partially supported by the National Natural Science Foundation of China (10561007), the Natural Science Foundation of Jiangxi Province, China, and a grant from the National Science Council of ROC.     

The Riemann Problem with Delta Data for Zero-Pressure Gas Dynamics

In this paper, the Riemann problem with delta initial data for the one-dimensional system of conservation laws of mass, momentum and energy in zero-pressure gas dynamics is considered. Under the generalized Rankine-Hugoniot conditions and the entropy condition, we constructively obtained the global existence of generalized solutions which contains delta-shock. Moreover, the author obtains the stability of generalized solutions by making use of the perturbation of the initial data.