函数代数

Applying multivariate spline theory which was established by Wang RenHong in ( [1] , [2] ), the Nother theorem of algebraic curve [71] is generalized to the piecewise algebraic curve.

1：利用王仁宏在文献（[1]，[2]）中关于多元样条函数的基本理论，本文将代数曲线的Nother定理（[71]）推广到分片代数曲线上，给出了剖分为〓，和星形剖分上的C〓分片Nother型定理。

Based on the partition of equivalent classes,the resolving of a linear system of equations and the calculation of the dual basis of the standard basis,three methodologies are presented.

以此解释了Rijndael算法S盒代数表达式复杂度低的本质原因，给出其分量函数间等价关系的一种直接证明方法。1 引言插入攻击[1]对代数次数和复杂程度低的分组密码尤其凑效。

By studying measure theory, the students are asked to understand σ-algebras, measure, measurable functions, integral, the measure theory of product spaces, Fubini theorem, and the representation theory of Boolean algebras.

通过本课程学习，使学生掌握σ-代数、测度、可测函数、积分，初步掌握乘积测度空间的构造及Fubini定理，以及布尔代数的表示理论。

We use algebra to describe the semantics. Algebra provides a common base for function, class and algebra specification.

它以代数作为数学基础，这使得函数、类和代数规范等具有统一的合成基础。

But , when we changeour trainofthought , we can find that sometimes using mathematical Analysis is a better solution to the linear algebraic problems.In thispaper, use the propertyofcontinuous function andimproper integralof infinite interval and to solve some matrix and quadratic form problems.Algebraic ideas andmethodstosolvemathematical analysisproblems are indicated by several specific examples. Close contactof differentmathematical analysis fields is shown.

该文探讨运用数学分析的有关知识（如：函数连续性、无穷区间的广义积分）和方法解决高等代数中的某些复杂、繁琐的问题；通过具体的实例，阐述高等代数的思想和方法在解决数学分析问题中的作用，揭示不同的数学领域之间的密切联系，例如：二次型与正交变换是代数学的基本内容，其用途十分广泛，而重积分的计算往往存在技术性的困难，若利用"二次型"与"正交变换"的有关理论去解决某些重积分的计算问题是颇有功效的。

The multivariate polynomial interpolation problem is a classical mathematical problem that is widely used in many fields, such as multivariate function list, surface design and finite element method. In recent years, multivariate polynomial interpolation has been focused by many people, of which the geometric topological struction of sets of interpolation nodes is also much concerned by us.

多元多项式插值问题是一个十分具有研究意义和实际应用价值的数学问题，它广泛应用于多元函数列表，以及曲面的外形设计和有限元法等诸多领域，近年来多元多项式插值越来越受到人们的广泛关注，其中有关插值结点组的几何拓扑结构问题也是人们十分关注的内容。1998年，梁学章和吕春梅在文献[2]中借助代数几何中的有关理论，进一步讨论了沿无重复分量平面代数曲线上的Lagrange插值问题，并应用Cayley ？

In numerical mathematics, it is interesting and of importance to study the theory of algebraic functions and its applications.

古典代数函数论的研究对象，是以代数关系式 f=0 (1)联系着的两个变量的有理函数φ。

In the POPF model, wind farm is modeled by the probabilistic wind farm model considering the reactive power-slip characteristic, and the inequality constraints include not only the unit output constraints, the ratio constraints, the voltage constraints and the line current constraints but also the reactive compensation capacity constraints in wind farm and the system climbing capacity constraints per minute. By introducing the NCP function, the KKT conditions of POPF system are transformed equivalently. Based on the transformed nonsmooth nonlinear algebraic equations, the FOSMM is used to determine the POPF model expressed by the numerical characteristic of variables. The model includes nonsmooth functions, so it can be solved by a semismooth Newton-type method based on the subdifferential.

概率最优潮流模型中，风电场采用考虑无功功率—滑加热器差特性的风电概率模型，不等式约束中除了机组出力约束、有载调压变压器变比约束、电压约束和支路电流约束，也考虑了风电场无功补偿容量约束、系统的分钟级爬坡能力约束；使用非线性互补函数将概率最优潮流的KKT条件转化为一组包含有不光滑函数的非线性代数方程组，然后基于一次二阶矩法确定了以待求量的数字特征表示的POPF模型，由于该模型包含不光滑函数，因此采用基于次微分的半光滑牛顿型方加热器法求解。

So, in this paper, a circular cavity witha large radius is used to replace the straight boundary of thehalf space, then the half space problem can be changed tothe scattering problem of two circular cavities to the steadyincident P-wave . Having the aid of the mature cylinderfunction theory, the general solutions of the wave functionscan be given, and an infinite linear algebraic equations ofthe unknown coefficients in the wave functions can be gottenwith the boundary conditions and the Fourier complex seriesexpansion technology, the infinite linear algebraic equationscan be approximately solved by the finite trunction withsatisfying some definite precision, at the basis of thissolution of the equation, the variations and the lay-outs ofthe DSCF at the circular cavity boundary vs. the differentincident angles, the different embedded depths of thecircular cavity as well as the different dimensionless wavenumber of the incident P-wave.

为此，本文采用一个半径很大的圆孔来代替半空间的直边界，将该半空间问题转化为一无限大空间中两个圆孔对稳态P波的共同散射问题，借助于成熟的柱函数理论，通过写出问题波函数的一般形式解，利用问题的边界条件，并采用复数傅立叶级数展开技术将其化为一个仅包含问题波函数中未知系数的一无穷线性代数方程组，在满足一定计算精度的前提下，通过有限项截断进行近似求解，进而讨论了圆孔边界处的动应力集中系数随不同入射角、不同的圆孔掩埋深度、入射波的不同无量纲波数以及介质的泊松比变化和分布情况。

In this paper, first we define a kind of transformation on the sequences of Boolean functions, and then analyze the changes of nonlinearity, algebraic degree and balancedness of Boolean functions under the transformation.

本文首先定义了对布尔函数的序列的一种变换，并分析了进行此种变换后布尔函数的非线性度、代数次数与平衡性的变化情况，最终在此基础上提出一种新的递归构造方法，可以构造出非线性度很高的n元m阶n-m-1次弹性函数。

I didn't watch TV last night, because it .

昨晚我没有看电视，因为电视机坏了。

Since this year, in a lot of villages of Beijing, TV of elevator liquid crystal was removed.

今年以来，在北京的很多小区里，电梯液晶电视被撤了下来。