广义函数论

引言函数是数学分析里的基本概念.按照古典的定义,所谓函数就是对空间内或其中某点集内每一个点赋予一个数值的对应关系.几个世纪以来它是数学分析的主题,到了十九世纪末叶以后,由于实际的需要及数学科学自身的发展,有必要把函数的概念加以推广.人们开始提出和研究集合的函数以及函数的函数.函数的变元不仅是点而且可以是集合,甚至于可以也是函数.于是在这样的概念的基础上产生和发展了近代的实函数论、积分论以及泛函分析理论.     

趣说函数

函数是一种特殊的映射,当A,B是非空的数的集合时,映射f:A→B就叫做从A到B的函数,记作y=f(x),其中x∈A,y∈B.解析式y=f(x)表示,对于集合A中的任意一个x,在对应法则f的作用下,即可得到y,因此,f是使“对应”得以实现的方式和途径,是联系x与y的纽带,从而是函数的核心.f可用一个或多个解析式来表示,也可以用数表或图象等其他方式表示.原象集合A叫函数f(x)的定义域,象集合C叫函数f(x)的值域,很明显C B.“函数”概念是初中和高中阶段的重点和难点,有不少的同学直到高三也不能深刻理解这一概念.原因在于这一概念的抽象性,如果把“函数”与我们…     

BOREL方向上的充满圆

本文讨论了在 Banel 方向上存在充满圆序列的条件.把 Rauch 定理推广到包含无限级和部分零级的半纯函数.构造了具有 Banel 方向而无充满圆序列的半纯函数.本文用⊿(θ)表示一条从原点引出的半直线:angz=θ.用 n(E,α)表示函数 f(z)在集合 E 内的 α 值点个数,α可为有限,也可以为无穷,且都计算重数.我们用 C.K 表示正常数,且前后出现的可以表示不同的常数.     

平面向量及其运算

1　重、难点分析本单元学习的重点是 :1)向量的概念 ;2 )向量的运算及其性质 ;3)向量及其运算的坐标表示 .我们知道 ,在平面上取定一点O后 ,平面上的任意点P就与向量OP成一一对应 ,这样关于点的几何问题就与向量联系起来 ,由于向量可以进行运算 ,因此通过向量也就把代数运算引入到几何中 .所以 ,用代数的方法 (向量运算的方法 )处理几何问题是本单元内容中渗透的重要数学思想方法 .具体地 ,由向量的线性运算 (向量的加法、实数与向量的积 )可以得到两向量平行的充要条件及定比分点公式 ;由向量的数量积运算可以得到两向量垂直的充要条件及…     

多元指数磨光算子的构造和相关偏微分方程基本解与磨光核的升维解法

本文目的在于回答:δ分布的多元指数磨光函数,即磨光核函数的解析表示问题.从我们给出的多元指数磨光算子的定义出发,将磨光核函数的表示,归结为先求相应偏微分方程的基本解,再对它的广义差分.然后用我们提出的"升维方法",彻底解决了基本解的解析表达问题.从而也就回答了磨光核函数的解析表示.磨光核函数的支集既可以是高维立方体,也可以是高维单纯形.因此,多元指数箱(E-Box)和单纯形(E-Simplex)样条的表示,皆能用我们的统一方法解决.     

韦恩图及其应用

用平面上一条封闭的曲线所围成的图形来表示一个集合 ,这个图形就叫做韦恩图 (Ｖｅｎｎ) (也叫文恩图 ,简称文氏图 ) .因为用图形表示集合是欧拉所创 ,韦恩图是在欧拉图基础上改进而来 ,故又叫欧拉图 .用文氏图表示集合具有形象、直观的特点 .应该注意的是 ,文氏图的形状与集合的性质没有任何质的联系 .它不是几何学的图形 ,而仅仅是把集合中的元素都包围在圈内 (非该集合的元素不包括在内 )的直观表示 .因此 ,文氏图与封闭曲线的形状无关 ,如画成矩形、正方形、圆、椭圆乃至信手勾出任意的非规则封闭曲线 ,但要考虑数学美 ,所以通常都画成…     

一种基于训练数据的迭代改进核函数

为提高支持向量机性能,提出一种支持向量机核函数的迭代改进新算法.利用与数据有关的保角映射,使核函数包含了全部学习样本的信息,即核函数具有数据依赖性.基本核函数的参数可取随机初值,通过对核函数进行多次迭代改进,直至得到满意的学习效果.与传统方法相比,新算法不需要筛选核函数的参数.对一元连续函数和强地震事件的仿真计算结果表明,改进SVR(support vector regression)的学习效果优于传统方法,并且随着迭代次数的增加,学习风险下降收敛,收敛速度依赖于传统方法的基本参数和改进方法的参数.     

角平分线的一种向量形式及其应用

引入向量这一工具后,我们可以用它解决许多平面几何里的一些问题．本文借助向量表示角平分线,以提示向量的工具性作用．命题设OC同∠AOB的角平分线,则(?)=λ((?))(λ≥0),把我们该形式称为∠AOB角平分线OC的向量形式．     

关于向量的复习

有一种说法,高考数学试题往往产生于知识网络的交汇处.什么知识才算交汇处呢?我们不必追究它的定义,但我们可以肯定,向量属于这样的知识:向量是沟通代数、几何与三角函数的一种工具,有着极其丰富的实际背景.用向量证明几何中有关平行、共线和垂直的命题,用向量计算角度和距离,用向量表示点的轨迹,以及用向量处理三角恒等变换和代数恒等变形等,较之传统方法更为简捷.关于向量的复习,我们可以把它归结为两点:一是把握程序化的思路,二是注重灵活性的方法.程序化的思路,是由向量本身的特点决定的.比如向量具有“形”的特征,我们进行向量运算时,…     

结构可靠性分析的支持向量机方法

针对结构可靠性分析中功能函数不能显式表达的问题,将支持向量机方法引入到结构可靠性分析中．支持向量机是一种实现了结构风险最小化原则的分类技术,它具有出色的小样本学习性能和良好的泛化性能,因此提出了两种基于支持向量机的结构可靠性分析方法．与传统的响应面法和神经网络法相比,支持向量机可靠性分析方法的显著特点是在小样本下高精度地逼近函数,并且可以避免维数灾难．算例结果也充分表明支持向量机方法可以在抽样范围内很好地逼近真实的功能函数,减少隐式功能函数分析（通常是有限元分析）的次数,具有一定的工程实用价值．     

Restricted linear constrained minimization of quadratic functionals

In this work, a linearly constrained minimization of a positive semidefinite quadratic functional is examined. We propose two different approaches to this problem. Our results are concerning infinite dimensional real Hilbert spaces, with a singular positive semidefinite operator related to the functional, and considering as constraint a singular operator. The difference between the proposed approaches for the minimization and previous work on this problem is that it is considered for all vectors belonging to the least squares solutions set, or to the vectors perpendicular to the kernel of the related operator or matrix.     

Regularized Kernel-Based Reconstruction in Generalized Besov Spaces

We present a theoretical framework for reproducing kernel-based reconstruction methods in certain generalized Besov spaces based on positive, essentially self-adjoint operators. An explicit representation of the reproducing kernel is given in terms of an infinite series. We provide stability estimates for the kernel, including inverse Bernstein-type estimates for kernel-based trial spaces, and we give condition estimates for the interpolation matrix. Then, a deterministic error analysis for regularized reconstruction schemes is presented by means of sampling inequalities. In particular, we provide error bounds for a regularized reconstruction scheme based on a numerically feasible approximation of the kernel. This allows us to derive explicit coupling relations between the series truncation, the regularization parameters and the data set.     

Welch bounds for cross correlation of subspaces and generalizations

Lower bounds on the maximal cross correlation between vectors in a set were first given by Welch and then studied by several others. In this work, this is extended to obtaining lower bounds on the maximal cross correlation between subspaces of a given Hilbert space. Two different notions of cross correlation among spaces have been considered. The study of such bounds is done in terms of fusion frames, including generalized fusion frames. In addition, results on the expectation of the cross correlation among random vectors have been obtained.     

Least-squares orthogonalization using semidefinite programming

We consider the problem of constructing an optimal set of orthogonal vectors from a given set of vectors in a real Hilbert space. The vectors are chosen to minimize the sum of the squared norms of the errors between the constructed vectors and the given vectors. We show that the design of the optimal vectors, referred to as the least-squares (LS) orthogonal vectors, can be formulated as a semidefinite programming (SDP) problem. Using the many well-known algorithms for solving SDPs, which are guaranteed to converge to the global optimum, the LS vectors can be computed very efficiently in polynomial time within any desired accuracy.By exploiting the connection between our problem and a quantum detection problem we derive a closed form analytical expression for the LS orthogonal vectors, for vector sets with a broad class of symmetry properties. Specifically, we consider geometrically uniform (GU) sets with a possibly non-abelian generating group, and compound GU sets which consist of subsets that are GU.     

Support vector machine classification with indefinite kernels

We propose a method for support vector machine classification using indefinite kernels. Instead of directly minimizing or stabilizing a nonconvex loss function, our algorithm simultaneously computes support vectors and a proxy kernel matrix used in forming the loss. This can be interpreted as a penalized kernel learning problem where indefinite kernel matrices are treated as noisy observations of a true Mercer kernel. Our formulation keeps the problem convex and relatively large problems can be solved efficiently using the projected gradient or analytic center cutting plane methods. We compare the performance of our technique with other methods on several standard data sets.     

Generalized Korn's inequality and conformal Killing vectors

Korn's inequality plays an important role in linear elasticity theory. This inequality bounds the norm of the derivatives of the displacement vector by the norm of the linearized strain tensor. The kernel of the linearized strain tensor are the infinitesimal rigid-body translations and rotations (Killing vectors). We generalize this inequality by replacing the linearized strain tensor by its trace free part. That is, we obtain a stronger inequality in which the kernel of the relevant operator are the conformal Killing vectors. The new inequality has applications in General Relativity.     

The nucleolus and kernel for simple games or special valid inequalities for 0–1 linear integer programs

An equivalence between simplen-person cooperative games and linear integer programs in 0–1 variables is presented and in particular the nucleolus and kernel are shown to be special valid inequalities of the corresponding 0–1 program. In the special case of weighted majority games, corresponding to knapsack inequalities, we show a further class of games for which the nucleolus is a representation of the game, and develop a single test to show when payoff vectors giving identical amounts or zero to each player are in the kernel. Finally we give an algorithm for computing the nucleolus which has been used successfully on weighted majority games with over twenty players.     

A numerical procedure for computing the Moore-Penrose inverse

An algorithm for computing the Moore-Penrose inverse of an arbitraryn×m real matrixA is presented which uses a Gram-Schmidt like procedure to form anA-orthogonal set of vectors which span the subspace perpendicular to the kernel ofA. This one procedure will work for any value ofn andm, and for any value of rank (A).     

基于Kernel Discriminant Canonical Correlation(KDCC)的多观测样本分类算法

针对多观测样本分类问题,提出一种基于Kernel Discriminant CanonicalCorrelation(KDCC)来实现多观测样本分类的模型.该算法首先把原空间样本非线性的投影到高维特征空间,通过KPCA得到核子空间,然后在高维特征空间定义一个使类内核子空间的相关性最大,同时使类间核子空间的相关性最小的KDCC矩阵,通过迭代法训练出最优的KDCC矩阵,把每个核子空间投影到KDCC矩阵上得到转换核子空间,采用典型相关性作为转换核子空间之间的相似性度量,并采用最近邻准则作为多观测样本的分类决策,从而实现多观测样本的分类.在三个数据库上进行了一系列实验,实验结果表明提出的方法对于多观测样本分类具有可行性和有效性.