广义调和级数及其推广应用研究

创建了广义调和级数和广义欧拉常数函数的概念,得到了广义调和级数和广义欧拉常数一系列十分重要的性质,得到了完全和公式,近似夹公式,讨论了广义欧拉常数与黎曼和的关系.利用复数的知识,找到了一种研究级数∑n=1∞b0+b1n+…+brnr/a0+a1n+…+aknk敛散性和求和的全新思路和方法—广义调和级数法.     

广义(m,n)超环

本文研究了广义(m,n)超环,n元正则关系以及n元强正则关系等的一些性质.利用广义(m,n)超环间的同态关系以及正则和强正则关系,得到了(m,n)子超环和(m,n)超理想的不变性,广义(m,n)超环的商结构,以及构成商超环和商环的充分必要条件,推广了文献[5]的一些结果.     

(N,1)-广义正定矩阵

本文对被屠伯埙称为亚正定的矩阵类进行了推广,即给出了(n,1)-广义正定矩阵的概念,进而得到了(n,1)-广义正定矩阵的一系列性质,最后将关于正定阵的Hadamard乘积的Schur定理及华罗庚定理推广到(n,1)-广义正定矩阵.     

关于近似广义极因子的误差界

设A是m×n阶复矩阵,分解式A=QH称为A的广义极分解,如果Q是m×n阶次酉短阵和H是n×n半正定的Hermite矩阵．本文给出了广义极分解的一些性质和推广了有关近似极因子的相关结论．     

方程φ_2(φ_3(n))=3~(ω(n))的可解性

φ_e(n)为广义Euler函数,探讨了含有e=2和e=3时的广义复合Euler函数的不定方程φ_2(φ_3(n))=3~(w(n))的可解性问题.基于广义Euler函数φ_e(n)的性质,借助初等方法给出方程φ_2(φ_3(n))=3~(w(n))的全部20组解.     

点到直线距离公式的n维推广

结合向量内积的性质和矩阵广义逆的知识,利用优化方法得到了n维空间中点到广义平面的距离公式.     

关于广义函数族的表示

<正> [1]中研究了(R)中的广义函数族的局部性质及其解析表示,要对(R~n)(n≥2)中的广义函数研究类似的问题,会遇到一些本质性的困难.当n≥2时,即使单个广义函数的解析表示也是一个没有完全解决的问题,过去只有这个问题在某些特殊情形下的解答(见[2],[3]). 本文采用与[1]中不同的方法来研究(R~n)中广义函数族的局部性质及其解析表示.第一节中,我们证明了广义函数族的局部结构定理,当n=1时,所得结果改进了     

广义Hamming重量的广义Griesmer界进一步分析

通过对q元线性码广义Hamming重量dr(·)的分析,应用支撑重量ωs(C)的性质,再次分析了q元[n,k]线性码广义Griesmer界n≥dr+sum from i=1 to k-r[(q-1)dr/qi(qr-1)].     

直线上的重心坐标公式应用几例

众所周知,用数学知识研究和计算力学中质点组的重心是十分方便的。反过来,力学中质点组的重心在数学中的应用,却常被人们所忽视。本文讨论了最简单的重心公式——直线上的重心坐标公式在中学数学中的有趣应用,给出四个典型例题。一、直线上的重心坐标公式为在数轴ax上坐标x_i的点A_i处放置质量m_i(i=1,2,…,n),设质点组(A_1,m_i),(A_2,m_2),…,(A_n,m_n)的重心坐标为C,C的坐标为x,则很明显地,这个结论是坐标平面上重心坐标公式的特殊情况,证明从略。     

R~n上的一类广义Poisson括号

利用任意一个n阶反对称矩阵,我们在欧氏空间P~n上定义一个广义Poisson括号,并讨论了相关的性质     

凸多边形上的Bezier曲面片

本文建立了正多边形上的重心坐标，并构造了多边形上的Ｂｅｚｉｅｒ曲面片、     

一般化凸空间上的一个连续选择定理(英文)

本文研究一般化凸空间上的连续选择定理.利用在D■X的条件下,一般化凸空间(X,D;Γ)上Γ-凸子集的概念,得到了两类一般化凸空间之间,以及φ映射和Γ-凸映射之间的关系,并且得到了一个连续选择定理.本文推广了一般化凸空间上凸子集的概念.     

Some Hermite–Hadamard type inequalities for functions of generalized convex derivative

We obtain some new inequalities of Hermite–Hadamard type. We consider functions that have convex or generalized convex derivative. Additional inequalities are proven for functions whose second derivative in absolute values are convex. Applications of the main results are presented.

     

Product numerical range in a space with tensor product structure

We study operators acting on a tensor product Hilbert space and investigate their product numerical range, product numerical radius and separable numerical range. Concrete bounds for the product numerical range for Hermitian operators are derived. Product numerical range of a non-Hermitian operator forms a subset of the standard numerical range containing the barycenter of the spectrum. While the latter set is convex, the product range needs not to be convex nor simply connected. The product numerical range of a tensor product is equal to the Minkowski product of numerical ranges of individual factors.     

Facet defining inequalities among graph invariants: The system GraPHedron

We present a new computer system, called GraPHedron, which uses a polyhedral approach to help the user to discover optimal conjectures in graph theory. We define what should be optimal conjectures and propose a formal framework allowing to identify them. Here, graphs with n nodes are viewed as points in the Euclidian space, whose coordinates are the values of a set of graph invariants. To the convex hull of these points corresponds a finite set of linear inequalities. These inequalities computed for a few values of n can be possibly generalized automatically or interactively. They serve as conjectures which can be considered as optimal by geometrical arguments.We describe how the system works, and all optimal relations between the diameter and the number of edges of connected graphs are given, as an illustration. Other applications and results are mentioned, and the forms of the conjectures that can be currently obtained with GraPHedron are characterized.     

The inverse Fourier transform of the filtering function and its integral powers

The convex coordinates of a point, P,in a closed triangular region are defined as weights which must be placed at the vertices of the region to shift the balance point of the weight distribution to P. The vertices of the triangle are interpreted as energy states of a physical system, and the convex coordinates become probabilities. The Maxwell‐Boltzmann distribution function is derived with the aid of convex coordinates.     

广义凸模糊映射

给出广义凸模糊映射、广义弱凸模糊映射等概念和若干特例。其次,构造集合Axf,y、Af,证明当f为下半连续广义弱凸模糊映射时Afx,y为闭弱凸集,进而得到广义凸模糊映射的充分条件。最后,给出广义凸模糊映射的性质,并指出半严格广义凸模糊映射成为严格广义凸模糊映射的条件。     

Noncompact simplexes in Banach spaces with the Radon-Nikodým property

It is well known that a compact convex subset C of a locally convex topological vector space is a simplex if and only if each point x of C admits a unique probability measure on the extreme points of C with barycenter x. An exact analog of this result is proved for a closed and bounded separable convex subset of a Banach space with the Radon-Nikodým Property, and a weaker analog is proved in the nonseparable case.     

On Some Generalized Polyhedral Convex Constructions

Generalized polyhedral convex sets, generalized polyhedral convex functions on locally convex Hausdorff topological vector spaces, and the related constructions such as sum of sets, sum of functions, directional derivative, infimal convolution, normal cone, conjugate function, subdifferential are studied thoroughly in this paper. Among other things, we show how a generalized polyhedral convex set can be characterized through the finiteness of the number of its faces. In addition, it is proved that the infimal convolution of a generalized polyhedral convex function and a polyhedral convex function is a polyhedral convex function. The obtained results can be applied to scalar optimization problems described by generalized polyhedral convex sets and generalized polyhedral convex functions.