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Using finite fields , the classical groups give rise to finite groups , intensively studied in relation to the finite simple groups ; and associated finite geometry , which has both combinatorial and algebro-geometric sides.

利用有限领域，古典的群体产生有限的群体，深入研究有关的有限简单群体；及相关有限几何，这两个组合和algebro几何双方。

The number of polynomial function over a finite commutative ring. This paper is on the basis of the research for the first question.With the structure of finite commutative ring,we know that a finite commutative ring can be expressed by direct sum of some finite commutative local rings.

本文主要基于对问题一的研究，并由有限交换环的构造可以知道，有限交换环可以由一些有限交换局部环的直和表示，于是就将问题简化到对有限交换局部环上多项式的判定。

The coordinate transformation and numerical integration was executed on the discretized tetrahedral elements based on which the 3-D MT vector-finite element method was implemented. A whole computation framework for 3-D vector-finite element method with unstructured mesh was given. Based on this some typical models were tested which has demonstrated that our algorithm could distinctively avoid problems caused by the fake solution and both the accuracy and efficiency were enhanced which made our algorithm has a bright future for further application.3. According to theory of Sobolev vector space and the discretization of Helmholtz space, the error estimate which was suitable for 3-D MT vector-finite element modeling was deduced by which the procedure of adaptive technique was guaranteed.4. Based on the fully unstructured tetrahedralization and optical strategy, the 3-D magnetotelluric h-adaptive vector-finite element method was presented through combining the error estimate. With this work, the accuracy and creditableness for 3-D MT complicatedly modeling was guaranteed.5. The 3-D magnetotelluric h-adaptive vector-finite element algorithm with unstructured mesh was implemented.

针对非结构化的四面体单元，采用坐标变换和数值积分方法，实现了MT三维矢量有限单元分析，建立起基于非结构化网格的三维MT矢量有限元计算流程，并对典型模型和国际标准电磁模型进行了数值模拟，结果对比和分析表明，基于非结构化网格的三维MT矢量有限元不仅消除了节点型有限元的伪解，而且具有很高的计算精度和速度，有广阔的应用前景。3、根据Sobolev函数的向量空间和Hemlholtz空间的分解，推导出基于残差的三维大地电磁矢量有限元后验误差估计公式，为三维大地电磁自适应矢量有限元数值模拟的实现奠定了基础。4、在完全非结构化四面体单元剖分及优化基础上，结合三维大地电磁矢量有限元后验误差估计公式，提出了基于非结构化网格的三维大地电磁h-型自适应矢量有限元计算策略，保证了对复杂大地电磁模型数值计算的精度和可靠性。5、实现了基于非结构化网格的三维大地电磁h-型自适应矢量有限元计算流程，对典型模型和国际标准电磁模型进行了数值模拟。

Prange first began to study cyclic codes over finite field in 1957 and Berkelamp first began to investigate negacyclic codes over finite field in 1968,but the codes over finite rings didnt received much attention until the 1990s due to the works of Nechaev in 1991 and Hammons in 1994,which showed that well-known nonlinear codes such as Kerdock and Preparata codes are constructed from linear codes over finite rings.

Prange(1957)和Berkelamp(1968)分别首先开始研究有限域上的循环码和负循环码，直到上世纪90年代，Nechaev(1991)和Hammons(1994)通过有限环上的线性码构造出著名的非线性码如Kerdock码和Preparata码，有限环上的码才受到广泛关注。

ECC system makes use of the finite group of elliptic curve on the finite field instead of finite cyclic group used in the discrete logarithm problem. The theory of ECC is being studied a lot and it is the focus of the cryptography and the industrial estate, especially the technique of the implementation of ECC.In this thesis, the meaning of the information technology is discussed firstly, then we analyze the classical cryptography algoritham. After that, the mathematics base of elliptic curve cryptosytems are introduced, especially the theory of the finite field. As for some algorithms, we also give out the program of reliazation.

随后，介绍了研究椭圆曲线密码系统所需要的数学理论基础，特别是有限域中的椭圆曲线理论，并对相关的方法给出相应的实例及其程序实现；论文第三章讨论给出了椭圆曲线密码体制的研究现状，综述了椭圆曲线与现有一些密码算法的结合，介绍了椭圆曲线的离散对数问题以及椭圆曲线面临的攻击，详细描述了椭圆曲线密码系统的实现细节及各个部分的实验数据。

150 Chapter 1 The History and Future of Computers 3.2 Boolean Algebra Table 3-2 Distributivity Idempotency Absorption laws 分配律同一律吸收律 a=ab+ac a+= a+a=a aa=a a+ab=a a=a'=a'b''=a'+b' DeMorgan's laws德摩根定理计算机专业英语 1-151 Chapter 1 The History and Future of Computers 3.2 Boolean Algebra Since a finite set of n elements has exactly 2n subsets, and it can be shown that the finite Boolean algebras are precisely the finite set algebras, each finite Boolean algebra consists of exactly 2n elements for some integer n.

由于n个元素的有限集有且只有个子集由于个元素的有限集有且只有2n个子集，而且很显然有限布个元素的有限集有且只有个子集，尔代数一定是有限集合代数，所以对某个整数n而言而言，尔代数一定是有限集合代数，所以对某个整数而言，每个有限布尔代数也有且只有2n个元素。例如，上文定义的集合T的限布尔代数也有且只有个元素。

The main contributions of the second part of this dissertation are focused on the cryptographic properties of logical functions over finite field, with the help of the properties of trace functions, and that of p-polynomials, as well as the permutation theory over finite field: The new definition of Chrestenson linear spectrum is given and the relation between the new Chrestenson linear spectrum and the Chrestenson cyclic spectrum is presented, followed by the inverse formula of logical function over finite field; The distribution for linear structures of the logical functions over finite field is discussed and the complete construction of logical functions taking on all vectors as linear structures is suggested, which leads to the conception of the extended affine functions over finite field, whose cryptographic properties is similar to that of the affine functions over field GF (2) and prime field F〓; The relationship between the degeneration of logical functions and the linear structures, the degeneration of logical functions and the support of Chrestenson spectrum, as well as the relation between the nonlinearity and the linear structures are discussed; Using the relation of the logical functions over finite field and the vector logical functions over its prime field, we reveal the relationship between the perfect nonlinear functions over finite field and the vector generalized Bent functions over its prime field; The existence or not of the perfect nonlinear functions with any variables over any finite fields is offered, and some methods are proposed to construct the perfect nonlinear functions by using the balanced p-polynomials over finite field.

重新定义了有限域上逻辑函数的Chrestenson线性谱，考察了新定义的Chrestenson线性谱和原来的Chrestenson循环谱的关系，并利用一组对偶基给出了有限域上逻辑函数的反演公式；给出了有限域上随机变量联合分布的分解式，并利用随机变量联合分布的分解式对有限域上逻辑函数的密码性质进行了研究；给出了有限域上逻辑函数与相应素域上向量逻辑函数的关系，探讨了它们之间密码性质的联系，如平衡性，相关免疫性，扩散性，线性结构以及非线性度等；讨论了有限域上逻辑函数各类线性结构之间的关系，并给出了任意点都是线性结构的逻辑函数的全部构造，由此引出了有限域上的"泛仿射函数"的概念；考察了有限域上逻辑函数的退化性与线性结构的关系、退化性与Chrestenson谱支集的关系；给出了有限域逻辑函数非线性度的定义，利用有限域上逻辑函数的非线性度与相应素域上向量逻辑函数非线性度的关系，考察了有限域上逻辑函数的非线性度与线性结构的关系；利用有限域上逻辑函数与相应素域上向量逻辑函数的关系，揭示了有限域上的广义Bent函数与相应素域上的广义Bent函数的关系，以及有限域上的完全非线性函数与相应素域上向量广义Bent函数之间的关系；给出了任意有限域上任意n元完全非线性函数存在性与否的完整证明，并利用有限域上平衡的p-多项式的性质给出了有限域上完全非线性函数的一些基本构造方法。

In this paper, we consider hyperbolic conservation laws, and the following work has been done to the finite volume method on unstructured meshes :In preface , we introduce the backupground of the oringe of the finite volume method, and point out the limitation of the finite difference methods and the finite element methods, and import the finite volume method.

本文主要考虑双曲型守恒律方程，对二维非结构三角形网格有限体积方法作了如下工作：介绍了有限体积方法的产生背景，指出了有限差分方法和有限元方法中的特点，对有限体积方法的产生和发展给出了比较详细的分析。

In this paper, the space-time finite element theory and its application technologies are studied and discussed. Furthermore the space-time finite element method is used to analyze the transient characteristics of electromagnetism equipments. The work is important to develop the numerical method for the transient electromagnetic field. The detailed work is as follow.In the basis theory of finite element method used in the transient engineering problem, the functional theory and the finite element theory are applied to studied the existence condition of a generalized variation principle, which is the basis of the space-time finite element method.

本文在完善时空有限元法的基础理论和应用技术并将该方法应用于解决电磁装置的瞬态过程分析等相关方面进行了深入的研究与探讨，该工作对于发展瞬变电磁场问题的数值计算方法具有重要的意义，为此进行了如下几方面的工作：在工程瞬变问题的有限元方法的基础理论方面，本文应用泛函理论和有限元理论，系统讨论了一般工程瞬变问题的变分定理的存在条件及对应的变分定理。

We all want to make the world a better place but,given finite resources,we should look for the most costeffective ways of doing so.

本句是由用逗号和&but&连接的两个分句组成的并列句。在第二个分句中，过去分词短语&given finite resources&用做状语。

It is also known as one of the most poisonous naturally occurring substances.

它也被称为一个最自然发生的有毒物质。

The greatest stress is found at the location on the cross section where V is the largest.

最大应力出现在横截面上V为最大的地方。