数学基础

A notable merit of parallel coordinates is its well-founded mathematical foundation.Its projective geometrical interpretation and duality properties facilitate the simultaneous projection information representation of multiple low dimension subspaces.

平行坐标的一个显著优点是其具有良好的数学基础，其射影几何解释和对偶特性使它可以很好地同时表达高维数据在多个低维子空间的投影信息。

In the future network and information age,it can be expected that Error-Correcting Code will gain its widerap placation,and the research and develop work of it will have high practicalv alue and profound meaning.

本文在介绍近世代数群、环、域等概念的数学基础上，学习并总结了纠错码译码算法的发展和现状。

Philosophy, indeed, can be viewed as a general Platonism and equally detrimental to our efforts to ground mathematics in social life.

实际上，哲学可以被看作一种普遍的或总体的柏拉图主义，其对于我们在社会生活中奠定数学基础的努力等于是有害的。

This paper first analyses and summarizes the ststus quo and evolution trend of encryption, some common used cryptograph are introduced, including the algorithms used in symmetric cryptosystem and asymmetirc cryptosystem. We describe the theory of each algorithms and compare the elliptic curve cryptosystem with the other two asymmetric cryptosystems to show the advantages of this algorithm. Second, the principle of ECC is discussed, including the math foundation of ECC, basic conception of elliptic curves, constructiong idea of ECC, operation on the elliptic curve and so on. Third, the current attacks of ECC were analyzed deeply, and an algorithm based on limited prime number field was constructed. We analyzed its realizability in theory, and implement it by using certain function of MIRACL software package. Latter half in this paper, the implementation model of a simple elliptic curve encryption system which based on GF has been introduced. The paper also put a deep analysis on the algorithm of point addition and point multiplication.

本文首先对密码技术的发展现状及其发展趋势进行了分析和综述，详细的介绍了私钥密码系统和公钥密码系统的发展，说明各种算法的原理和优缺点，并给出了一些典型的密码体制的简要分析，重点将椭圆曲线算法与其它几种公钥密码算法比较，说明椭圆曲线算法的优势；其次，探讨了椭圆曲线密码体制的原理，包括椭圆曲线密码的数学基础、基本概念、椭圆曲线密码体制的构造思想等问题；第三作者对椭圆曲线的攻击现状作了详细的分析，针对所使用的大素数域F_p，设计了素数域上安全椭圆曲线产生的算法，从理论上做了可实施性分析，从软件上做了具体实现；在本文的后半部分，提出了一个简单的基于有限素数域上的椭圆曲线加密方按算实现模型，并对SECES中设计的点加和点乘运算进行了深入分析。

The development of Montague's ideas can be divided into three stages: In the first stage (1955-1958), Montague was influenced by Frege, Russell, Carnap and Tarski, focusing on Mathematic Logic, Mathematical Foundation and Recursion Theory.

蒙太格思想的发展分为三个阶段：第一段（1955-1958）受弗雷格、罗素、卡尔纳普、塔斯基问题意识的支配，关注数理逻辑、数学基础、递归函数论等问题。

Fractal dimension structure of the Cosmos are explored, and the mathematical foundation, which include the expressions of fractal dimension differential and calculus, regular space integral solutions of fractal dimension differential equations, the fractal calculus definitions of fractal measure as well as the measure computational equation of self-similar fractal, of fractal dimension calculus and fractal measure are given.

摘 要：探讨了宇宙分维构造的形式，给出了分维微积分及分形测度的数学基础，包括分维导数及分维微积分的表述形式、分维微分方程的规整空间积分解、分形测度的分维微积分定义及自相似分形的测度计算方程。

Each kind of theory and technology about health safeguard can be integrated into a grading describable system. The 5 kinds basic space, which is derived sequentially from topological space, regular space, metric space, measured space and continuous space, are the mathematical base to structure the method of health safeguard.

从泛函分析的角度全面总结了各种健康保障理论与技术，将其统一到一个递级的描述系统中，由拓扑空间、规则空间、度量空间、测度空间和连续空间逐步继承而来的5类基本空间是构造健康保障方法的数学基础。

In Chapter two, it is given of the three-dimensional boundary element and the boundary element space, Sobolev space for the contact boundary element problem with friction; it is proved of existence and uniqueness of the solution by variation inequality for the contact Boundary Element Method matrix equation under two kinds of condition that the contact zone is fixed and the contact zone is increasing along with the load; it is given of the expressionfor the error of the exact solution and the boundary element solution.

第2章，给出三维边界单元及边界元空间、有摩擦接触边界元问题的Sobolev空间，利用变分不等式证明了不变接触区和随载荷可变接触区两种情况下的接触边界元法矩阵方程解的存在唯一性，给出了准确解与边界元解的误差表达式，还证明了三维弹塑性摩擦接触问题凝缩矩阵解的存在唯一性，为三维弹塑性摩擦接触边界元法奠定数学基础。

Then, we introduce the fundamental control theory in terms of mathematics, including the state space s...

其次介绍了控制理论的数学基础。

Application predicate logic on data sublanguage as its mathematical foundation would make the study of data sublanguage become the research of predicate logic and the optimization of data sublanguage become the matters of simplification of predicate logic.

本文旨在研究将谓词逻辑及公理化理论应用于关系数据库中表示数据子语言，应用谓词逻辑作为它的数学基础，使得对这些语言的研究成为对谓词逻辑的研究，优化数据子语言的表示成为对谓词逻辑的化简问题。

Singer Leona Lewis and former Led Zeppelin guitarist Jimmy Page emerged as the bus transformed into a grass-covered carnival float, and the pair combined for a rendition of "Whole Lotta Love".

歌手leona刘易斯和前率领的飞艇的吉他手吉米页出现巴士转化为基层所涵盖的嘉年华花车，和一双合并为一移交&整个lotta爱&。

This is Kate, and that's Erin.

这是凯特，那个是爱朗。