内积

Firstly,a involution on Universal Clifford algebra 〓 is introduced.then we equip a inner product and the corresponding two equivalent norms onUniversal Clifford algebra 〓,and then we prove that Universal Cliffordalgebra 〓 is a Banach algebra with one of the above norms,these resultsare very important for further study in this field.

第一，本文创造性地准确地给出了在少Clifford代数〓上的一个对合运算表示，由此，我们给出了在泛Clifford代数〓上的一个内积，然后给出了相应的在泛Clifford代数〓上的两个等价的范数，并证明了若在〓上赋予其中的一个范数，则〓是一个Banach代数，这为在泛Clifford分析中进一步开展工作奠定了初步的基础。

The content of this course is divided into four chapter, the normed linear space and bounded linear operator on the normed space; the character of finite dimensional linear space; the basic theorems on Banach space.

本课程主要分为四章，赋范线性空间与内积空间；赋范线性空间上的有界线性算子，有限维赋范线性空间的特征。Banach空间中的基本定理：泛函存在定理，一致有原理，开映象，闭图象、逆算子定理。

In this paper,quadratic form theory,the European space inner product nature and Jensen inequality are three ways to prove cauchy inequality,and bthe relation between Cauchy inequality and higher mathematics was illustrated briefly.

本文运用二次型理论、欧式空间中内积性质和詹森不等式三种方法证明柯西不等式，并简要说明柯西不等式与高等数学之间的联系。

The thesis consists of the followingmain results:the Cauchy's integral formula on certain distinguished boundaryfor LR regular functions with values in a universal Clifford algebra 〓,theCauchy's integral theorem,the mean value theorem,the maximum modulus the-orem.the Taylor's expansion,the Laurent's expansion and the residue theoremetc..

第一章叙述了泛Clifford代数基本理论，其中，我们首先准确而又富有创造性地给出了在泛Clifford代数〓上的一个对合运算表示，由此，我们给出了在泛Clifford代数〓上的一个内积，然后，我们借此给出了在泛Clifford代数〓上的相应的两个等价的范数，并证明了若在〓上赋予其中的一个范数，则〓是一个Banach代数。

In the system, any complex number can be uniquely expressed by a single digit string, and the subtraction involved in the original complex multiplication is eliminated essentially.

二、在矩阵-矢量运算算法方面，提出了以负二进制编码为代表的全并行内积算法，一步得到混合进制表示的乘积矢量结果。

The gradient of f; its dot product with is the directional derivative of f in the direction of w .

f的梯度；它与的内积为f在 w 方向上的方向导数。

The gradient of f; its dot product with uw is the directional derivative of f in the direction of w

f f 的梯度；它与 uw 的内积为 f 在 w 方向上的方向导数

The divergence of the vector field w; it is the dot product of the vector operator with the vector w , or .

向量场w的散度；为向量算符与向量 w 的内积，或。

For example,a generalization of linear algebra is developed in which "convex bifunctions" are the analogues of linear transformations,and "inner products" of convex sets and functions are defined in terms of the extremal values in Fenchel's Duality Theorem.

例如，一类线性代数的推广形式在凸双重函数中发展起来，而凸双重函数是线性变换的类似情况，凸集和凸函数的内积是根据 Fenchel 对偶理论的极值定义的。

It employs the complex-valued step function to build neural model, then constructs network structure and decides the number of neurons; it adopts complex-valued Hebbian rules and inner product way to calculate the net weight matrix; then it analyzes the convergence of the network by taking advantage of the definition of energy function. In the image pre-processing stage, 2-dimensional Fourier Transform and Euler's formula are used, respectively, to transfer a gray-scaled image into phase information needed by the complex-valued neural network when stores the traffic sign. In the image post-processing stage, phase inverse transformation is used to change the phase image to gray-scaled image.

采用复数阶跃函数构建了神经元模型，建立了复数Hopfield神经网络结构，并且确定了网络中神经元的个数；借助复数Hebbian学习规则和复数内积法给出了网络权值的确定方法；借用复数网络能量函数的定义，说明了复数识别网络的收敛性；在前期的图像数据处理部分，分别使用离散二维傅里叶变换和欧拉公式，给出了将灰度图像数据转化为复数网络所需要的相位信息的方法，实现了网络对路牌的记忆存储；在后期的图像还原部分利用相位逆变换的方法，实现了相位图到灰度图转化。

The split between the two groups can hardly be papered over.

这两个团体间的分歧难以掩饰。

This approach not only encourages a greater number of responses, but minimizes the likelihood of stale groupthink.

这种做法不仅鼓励了更多的反应，而且减少跟风的可能性。