乘积公式

The formulas calculating higher derivatives of parametric functions and composite functions are given by the chain rule and the product formula for derivatives.

用函数的链式法则和乘积公式给出了参数函数和复合函数的高阶导数的计算公式。

By the procedure, we first use the cosine values of the angle between view vector and normal vector on each vertex as the texture coordinates, through calculation of the dot product of the two vectors. The derivative of radial curvature on the model surface is used as the second pass texture mapping coordinates, by a calculation using Euler formula.

该方法首先通过模型顶点的视线向量和法线向量的点乘积计算其夹角的余弦值，作为第一步的纹理坐标进行映射；再通过欧拉公式计算径向曲率及其方向导数值，作为第二步映射的纹理坐标。

For example, basic tools from calculus such as Fermat theorem, Rolle theorem and the intermediate value theorem may not necessarily hold and fundamental concepts such as chain and product rules and certain smoothness properties all need to be modified.

例如，微积分中的基本工具诸如Fermat定理、Rolle定理以及介值定理不再成立，一些基本的概念诸如链式法则、乘积公式以及某些光滑性都需要做适当的修正。

By means of infinite product of evenly distributed probability spaces,the concept of truth degrees of propositions in 3-valued logic systems W_3,G_3,Π_3 and S_3 are introduced,and some properties of distribution of propositional truth degree and certain inference rules are obtained.

利用势为3的均匀概率空间的无穷乘积在W3、G3、Π3及S3系统中引入了公式的真度概念，得到了命题真度分布的一些性质，同时给出了三值真度推理规则。

Using the infinite product of unevenly distributed probability space with potential of 3, the concept of formula truth degree was introduced into the Gdel 3-valued propositional logic system. Moreover, it was proved that in the 3-value (1/4, 1/2, 1/4), measure the set of truth of degree of propositions was dense in [0, 1], and a general expression of truth degree was obtained, providing a basis for further developing approximate reasoning of 3-valued propositional logic system.

利用势为3的非均匀概率空间的无穷乘积，在Gdel三值命题逻辑系统中引入公式的真度概念，在三值逻辑(四分之一，二分之一，四分之一)测度下证明G3中全体公式的真度值之集在[0， 1]上是稠密的，并给出公式真度的表达通式，为进一步在三值命题逻辑系统中展开近似推理奠定基础。

Optimizing and Realization of the Finite Field Inversion Algorithm Based on FPGA;2. The stability and operation of the inversion formula for Toeplitz matrices are also considered.

利用两个线性方程组是否有解给出了Toeplitz矩阵可逆的条件，表明Toeplitz矩阵之逆阵可以表示为φ-循环矩阵与上三角Toeplitz矩阵的乘积之和，给出了其逆矩阵列的递推公式，得到了求Toeplitz矩阵之逆矩阵的快速算法。

Matrix element of coordinate operator ~l of harmonic oscillator is discussed by using coherent state and normal product.

利用相干态和正规乘积对谐振子任意次幂坐标算符^Xl矩阵元进行了讨论，导出了计算^Xl矩阵元的一般公式，为处理谐振子的微扰问题提供了一种新的方法。

There are not primitive roots,for mod n,by computing the order of "product number", for "product module",this artical has made a number(call generalized primitive root ),parametrically,which order is also the common multiple of the all orders.

对于没有原根的模n，通过乘积数对乘积模的次数的计算，该文构造性地得到一种整数，其次数也是所有次数的公倍数，并得到这个公倍数次数的一种计算公式。

By means of two different methods, the author derives a general equation for expanding the product of two Jacobis triple products, which can be considered as the common generalization of the quintuple, sextuple and septuple product identities.

利用两种不同的方法，作者建立一个关于两个Jacobi三重积之乘积展开的广义公式，它可以看作是五重积，六重积和七重积恒等式的统一形式。

In Chapter 2, we recapitulate some fundamental definitions and propositions of topological degrees and some fundamental results about the computation of topological degrees;in Chapter 3, applying relative definitions and propositions of topological degrees, we establish product formulae of topological degrees in product space, in particular, the product formula of fixed point index on product cone, moreover, combining with the fundamental propositions and the computation of fixed point index, we obtain some fixed point theorems on product cone.

为此，在第二章中，我们给出了拓扑度的基本概念和性质以及拓扑度计算方面的基本结果；在第三章中，运用拓扑度的有关知识，我们建立了乘积空间中各类映射的拓扑度的乘积公式，特别是乘积锥上的紧连续映射的不动点指数的乘积公式，进一步，结合不动点指数的基本性质和不动点指数计算方面的基本结果，得到了乘积锥上的若干不动点定理。

Neither the killing of Mr Zarqawi nor any breakthrough on the political front will stop the insurgency and the fratricidal murders in their tracks.

在对危险的南部地区访问时，他斥责什叶派民兵领导人对中央集权的挑衅行为。

In fact,I've got him on the satellite mobile right now.

实际上 我们已接通卫星可视电话了