标量积

In this paper, we introduce the algorithm of Schoof-Elkies-Atkin to compute the order of elliptic curves over finite fields. We give out a fast algorithm to compute the division polynomial f〓 and a primitive point of order 2〓. This paper also gives an improved algorithm in computing elliptic curve scalar multiplication. Using the method of complex multiplication, we find good elliptic curves for use in cryptosystems, and implemented ElGamal public-key scheme based on elliptic curves. As a co-product, we also realized the algorithm to determine primes using Goldwasser-Kilian's theorem. Lastly, the elliptic curve method of integer factorization is discussed. By making some improvement and through properly selected parameters, we successfully factored an integer of 55 digits, which is the product of two 28-digit primes.

本文介绍了计算有限域上椭圆曲线群的阶的Schoof-Elkies-Atkin算法，在具体处理算法过程中，我们给出了计算除多项式f〓的快速算法和寻找2〓阶本原点的快速算法；标量乘法是有关椭圆曲线算法中的最基本运算，本文对[Koe96]中的椭圆曲线标量乘法作了改进，提高了其运算速度；椭圆曲线的参数的选择直接影向到椭圆曲线密码体的安全性，文中利用复乘方法构造了具有良好密码特性的椭圆曲线，并实现了椭圆曲线上ElGamal公钥体制；文中还给出了利用Goldwasser-Kilian定理和椭圆曲线的复乘方法进行素数的确定判别算法；最后讨论了利用椭圆曲线分解整数的方法并进行了某些改进，在PC机上分解了两个28位素数之积的55位整数。

If two vectors are associated with each other as a dot product, the result is a scalar.

如果两个向量以标量积结合，其结果为一标量。

Since scalars are independent of the coordinate system, the dot product of two vectors is called a scalar invariant

因为标量与坐标系无关，故两个矢量的点积称为标量不变量。

In case of high input dimension system model, taking norm of input vector as the input of wavelet network instead of using tensor product method to construct wavelet network ,which could solve the problems of high computation and curse of dimensionality.

在模型结构方面，对于多维输入输出系统建模，本文放弃选用传统的标量积确定小波基函数的方法，将输入矢量的范数作为小波网络基函数的输入变量，从而避免了高维输入所带来的可用小波基数量巨大和计算复杂的缺点。

This network's multidimensional wavelet basis function is built by substituting norm of the vector for that vector, not scalar product function of the vector, thus the cost of calculation of high dimensionalities is decreased nearly to the situation of one dimensionality.

此网直接用矢量的范数代替矢量来构造多维小波基函数，而不是采用通常的矢量的标量积函数的方法构造多维基函数，这样多维情况的运算量被降低到几乎和一维的情况基本相当的程度。

Vector algebra, Vector product, Scalar product, Gradient, Divergence, Rotation, and several theories

掌握矢量代数运算，矢量积，标量积，掌握标量场与矢量场的特性参数及相应的定理。

In this paper,it will be shown that by replacing the time integral of the Hamiltonian by another appropriate functional,and by substituting the scalar product by a semi-scalar product,an appropriate function space can be found in which classically non- conservative systems,i.e.,systems which are non-...

本文将指出用另一个适当的泛函代替啥密顿的时间积分以及用一个拟标量积代替标量积，可以建立一个适当的函数空间，在那个函数空间中，传统的非保守系统——即关于能量是非保守的系统具有保守系统的性质。

For the multisensor linear discrete time-varying stochastic control systems with the different local dynamic models, using the Kalman filtering method, under the optimal fusion criterion weighted by scalars, the unified and universal optimal information fusion white noise deconvolution estimators are presenced, and for the corresponding time-invariant systems, the steady-state optimal information fusion white noise deconvolution estimators are also presented.

对于带不同局部动态模型的多传感器线性离散时变随机控制系统，应用Kalman滤波方法，在按标量加权最优融合准则下，提出了统一和通用的最优信息融合白噪声反卷积估值器，并对定常系统提出了稳态最优信息融合白噪声反卷积估值器。

For the multisensor linear discrete time-varying stochastic control systems with the different local dynamic models, using the Kalman filtering method, under the optimal fusion criterion weighted by scalars, the unified and universal optimal information fusion white noise deconvolution estimators are presenced, and for the corresponding time-invariant systems, the steady-state optimal information fusion white noise deconvolution estimators are also presented.

黑龙江大学自动化系，哈尔滨150080中文摘要：对于带不同局部动态模型的多传感器线性离散时变随机控制系统，应用Kalman滤波方法，在按标量加权最优融合准则下，提出了统一和通用的最优信息融合白噪声反卷积估值器，并对定常系统提出了稳态最优信息融合白噪声反卷积估值器。

The dot product, being the product of three scalars, is a scalar.

点积是三个标量的乘积，所以是个标量。

But we don't care about Battlegrounds.

但我们并不在乎沙场中的显露。

Ah! don't mention it, the butcher's shop is a horror.

啊！不用提了。提到肉，真是糟透了。