乘法的

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。

Furthermore, we define a convolution multiplication between characteristic functions of constructible subsets by using push-forward functor from the category of algebraic varieties over C to the category of spaces of constructible functions. We construct geometric model for "intrinsic symmetry" of the octahedral axiom in a triangulated category. Using it, we deduce the multiplication satisfies the Jacobi identity of Lie algebra and then realize infinite dimensional Lie algebras.

进一步，我们使用复代数簇范畴到可构函数空间范畴的pushforward函子，给出了可构集上特征函数的卷积乘法，并构造了三角范畴八面体公理的内蕴对称性的几何模型，最终证明了对于不可分解支撑有界可构集的特征函数，乘法满足李代数定义的Jacobi恒等式，从而给出了无限维李代数的实现。

Weak bialgebras and Hopf algebras are generalizations of ordinary bial-gebras and Hopf algebras in the following sense: the defining axioms are the same, but the multiplicativity of the counit and comultiplicativity of the unit, are replaced by weaker axioms.

中文摘要：弱双代数和弱Hopf代数是通常的双代数和Hopf代数的推广：定义的公理不变，但是余单位的乘法和单位的余乘法变得更弱一些了。本文主要将上环理论及其Galois理论运用于弱Hopf代数，得到弱Hopf代数的相应结果。

Based on a new modified Wallace multiplier structure, named bi-forked Wallace multiplier structure, this paper presents the design of a highly re-configurable Multiplier for VLIW multimedia-processor. We found a reusable design method to extend multiplier with full adder array. The Structure can enable 32/16/8 bit operands, and has been optimized for speed and area.

在VLIW多媒体芯片的设计过程中，针对传统乘法器与加法器的不足，提出了一种新的分叉华莱氏树结构的乘法器模型，采用可重用的模块化设计思想，通过重用一位全加器阵列对乘法器进行扩展，处理器可以在一个乘法器单元内部同时支持多个32/16/8位的乘法运算，同时使乘法单元的速度和面积均得以优化。

But because it don't consider geometric meaning that the over-determined equation set is represent for,the precision of point object measured is low.

针对空间点目标的三维重构，传统的方法是采用最小二乘法进行的，但是由于最小二乘法没有考虑所建立的超定方程组所代表的几何意义，所以计算出的点坐标精度不高。

Aiming at three-dimensional construction of point object,lease square-method is usually used in traditional method.But because it don't consider geometric meaning that the over-determined equation set is represent for,the precision of point object measured is low.

针对空间点目标的三维重构，传统的方法是采用最小二乘法进行的，但是由于最小二乘法没有考虑所建立的超定方程组所代表的几何意义，所以计算出的点坐标精度不高。

There are five important soft sensor models of moisture have been established in this paper: the moisture model of The Recursive Least Square based on Lethe gene, the moisture model of Principal Component Regression , the moisture model of Partial Least Square, the moisture model of BP Nerve Network, the moisture model based on PLS—BP. All of these soft sensor models of moisture are systematically analysed by using Compound Correlative Coefficient. The factors which influence control of moisture are discussed on detail in this paper.

建立了五个重要的水分软测量模型，即渐消记忆递推最小二乘法水分模型；主成分分析法水分模型；偏最小二乘法水分模型；BP 神经网络水分模型；偏最小二乘BP 网络水分模型；对以上水分软测量模型用统一的评价指标复相关系数对它们的优劣进行了详细的分析，选出最适合水分建模的建模方法，同时对影响水分的各个因素进行了深入的分析和探讨。

Linear algebra matrix multiplication is the most common operation, it is in scientific research and engineering development have a decisive role in raising the efficiency of its operations for many large academic research of the contents of papers here is to analyze the main two a matrix multiplication algorithm.

详细说明：矩阵乘法是线性代数运算中最常见的运算，它在科学研究以及工程开发中都具有举足轻重的作用，提高其运算的效率是很多大型的学术研究中的内容，这边论文是分析主要的两个矩阵乘法算法。

P×P and P were two major non-Bayesian algorithms. 6 Less than 25% of the participants used frequency. In those who got other results except P , much more people used probability. Experiment 2 had a randomized multigroup posttest design. There was only one factor named implied condition which had five levels: not imply, imply P, imply addition, imply division and imply all. The results showed that: 1 All the implied conditions significantly improved the participants' performance. When under the condition of implying division, the participants derived the best results. 2 31.9% of the participants got correct answer, 73% of who used frequency to rewrite the information. 3 78% of the participants could apply one or several of the four concepts: probability of "not the event", multiplication rule, addition rule and condition probability.

结果表明：1 被试的估计受问题内容的影响，权威型问题情境容易使被试高估，非权威型问题情境使被试的高估现象大大减少；数据结构和提问形式对被试估计不产生影响。2 有近5%的被试能正确估计P，其中 73%使用频数对信息进行再表征。3 贝叶斯推理中各分步骤的困难程度由低到高分别是：乘法、P、加法、除法。4 有50%的被试能运用对立事件概率、概率乘法、概率加法、条件概率这四个概念中的一种或者几种。5 使用最多的两种非贝叶斯算法是P×P和P。6 不到25%的被试使用频数；求得除P以外各类结果的被试中，使用概率的人数远多于使用频数的。

Taking operation of rational numbers as the carrier, this paper starts the research right from the study on multiplication of decimal, division of fraction and multiplication of rational numbers and studies the "understanding levels of operation of rational numbers and the teaching and learning strategies that matches them."

本论文以有理数运算作为载体，选取小数乘法、分数除法和有理数乘法作为研究的切入点，研究了&有理数运算的理解水平及其教与学的策略&，具体说来，研究了3个问题：(1)学生对有理数运算的理解达到怎样的水平；(2)学生达到某种理解水平的原因是什么；(3)基于学生的理解水平，有效的教与学的策略是什么。

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。