乘法

By using the principle of least absolute deviation method , we will give a new priority method—least logarithm absolute deviation method.

利用最小一乘法原理，在层次分析中提出了一种新的排序方法——对数最小一乘法，并将其转化成线性规划问题求解，证明了对数最小一乘法的一些性质。

We then can verify the associative law and the commutative law for multiplication, and the uniqueness of the result of addition indicates the uniqueness and the cancellation law for multiplication.

我们容易验证乘法满足结合律和交换律，并且由加法结果的唯一性得出乘法结果的唯一性和乘法消去律。

So 2/4=1/2. Here we have used the associative law or the cancellation law for multiplication, which are satisfied for natural numbers, now we assume they are satisfied with fraction numbers.

2的数，所以2/4=1/2。这里我们用到了乘法结合律也可以用消去律，本来这些运算律是对正整数乘法适用的，但对于有分数参加的乘法我们也规定适用。

Derive recursive least squares method, recursive general least squares method, recursive augmented matrix method, recursive multiple-stage least squares method and recursive maximum likelihood method based on recursive Householder transformation by applying the three real-time solutions of overdetermined linear equation to difference model.

将线性矛盾方程组的三种实时解法应用于差分模型，导出了基于Househol-der变换的递推最小二乘法、递推广义最小二乘法、递推增广矩阵法、递推多级最小二乘法和递推极大似然法。

The details as follow: 1.Based on rules about form error in GB, mathematical models of form error are established including straightness error on the base of minimizing zone method、least-squares method and two dots linked method ; flatness error based on minimizing zone method、lease-squares method、diagonal line and three dots method; circularity and cylindricity error based on minimizing zone method、lease-squares method、minimum circumscribed circle and maximum inscribed circle.

具体内容如下：根据国标中形状误差的相关规定建立了各种要素评定方法的数学模型包括：直线度的最小区域法、最小二乘法、两端点连线法；平面度的最小二乘法、最小区域法、对角线法、三远点法；圆度和圆柱度的最小区域法、最小二乘法、最小外接圆法、最大内切圆法。

Firstly, some basic concepts about ECC are introduced;then the scalar multiplication under affine coordinate is discussed, we make a research and analysis of the side channel attack of scalar multiplication used in portable device, give out a new scalar multiplication algorithm which has the advantage of efficiency over other protected algorithm; and an improvement of the standard scalar multiplication algorithm proposed by IEEE1363 is made with the efficiency increasing by about 10%; at last, we deal with the using of complex multiplication in scalar multiplication algorithm, we generalize the methods and deduce a complete computing procedure, with proposing a new method which used several different fast endomorphism ,we give out an approach to boost the scalar multiplication with fast endomorphism.

本文首先介绍了椭圆曲线密码的有关基本概念；其次介绍了椭圆曲线上点的标量乘法在仿射坐标下的计算，对一般便携设备上的椭圆曲线点的标量乘法的边信道攻击做了研究与分析，给出了一种在效率上优于其它可抵抗边信道攻击的标量乘法的新算法，并对IEEE P1363 标准给出的标量乘法做了改进，使得标量乘法的运算效率提高了近10%；最后介绍了利用复乘计算标量乘法的方法，对已有的方法进行总结归纳，得出完整的计算过程，并针对两类超奇异椭圆曲线给出了一种利用多个可快速计算的复乘的标量乘法，得出一类普遍的结果，并给出了进一步用复乘加快标量乘法的思路。

Furthermore, it is proved that convex combination of judgment matrix addition and convex combination of Hadamard multiplication in group AHP can remove the influence of subjective factors, and keep and improve the consistency of judgment matrix.

文中定义了具有实际意义的矩阵"加法"、"乘法"、"指数"运算，指出了这些运算的性质；给出了判断矩阵"加法"凸组合及Hadamard"乘法"凸组合的概念，并证明了在群体AHP中判断矩阵的"加法"凸组合及Hadamard"乘法"凸组合不仅能消除主观因素的影响，而且还能保持和改善判断矩阵的一致性。

"A set of elements having two operations, designated addition and multiplication , satisfying the conditions that multiplication is distributive over addition and that both operations are associative and commutative for all elements of the set, with the exception of the additive identity element, which is not commutative in respect to multiplication ."

"一组元素，进行加法和乘法两种运算，满足条件：乘法运算需分配于加法之中，这组元素要将这两种运算结合且换进行，附加的就乘法而言不可交换的恒等元素除外。"

Mathematics A set of elements having two operations, designated addition and multiplication, satisfying the conditions that multiplication is distributive over addition and that both operations are associative and commutative for all elements of the set, with the exception of the additive identity element, which is not commutative in respect to multiplication.

域，体：进行加法和乘法两种运算的一组数，满足如下条件，即乘法分配律同样适用于加法分配律，各数在做加法和乘法两种运算中可相互结合并交换，但在乘法中不可交换的加性恒等幺元除外

Wehave obtained two formulas to express thire relation in different view;at the same time,wegive an explicit description of multiplicative unitary operators corresponding to compactquantum group 〓 and 〓 on concrete Hilbert spaces.

我们首先在第四章讨论了紧量子群对应的乘法酉算子和紧量子群与离散群的交叉积对应乘法酉算子的关系，我们得到了在不同观点下的它们之间的关系式；同时给出了紧量子群〓对应的乘法酉算子和〓对应的乘法酉算子在具体的Hilbert空间上它们之间的表达式。

The concept of equivalent rotationally rigidity is offered and the formula of rotationally rigidity is obtained.

主要做了如下几个方面的工作：对伸臂位于顶部的单层框架—筒体模型进行分析，提出了等效转动约束的概念和转动约束刚度的表达式。

Male cats normally do not need aftercare with the exception of the night after the anesthetic.

男猫通常不需要善后除了晚上的麻醉。