多项式

This paper begin with a review of curves and surfces modeling, then we discuss the parametric polynomial minimal surface. This is followed by a discussion of the representations of catenoids and helicoids. Next, we discuss the numeric approximations to the minimal surfaces with piecewise polygons and B-Spline surfaces.

全文安排如下首先，我们简要回顾曲面造型历史和发展现状在第二和第三章，讨论参数多项式极小曲面；在第四章讨论悬链面和螺旋面的表示；在第五章讨论极小曲面的分片线性逼近；在第六章讨论极小曲面的B-样条逼近；最后，我们对全文做简单的总结。

Minimal polynomial.

极小多项式。

The principal objects used by the NTRU PKCS are polynomial of degree N-1 having integer coefficient which are operated by addition,subtraction,multiplication and modular arithmetic .

NTRU公开密钥体制算法主要对象是对N-1次多项式进行加、减、乘、模运算，所以该加密解密的速度比RSA等著名算法快得多，和椭圆曲线密码体制一起，成为新一代最优秀的公钥密码体制。

Define Ck =∪rCk,r, where r passesthrough all monic irreducible polynomials of degree k over Zn.

定义Ck=∪rCk，r，这里r跑遍Zn中所有k次首一不可约多项式。

Depending on the coefficient of the polynomial, we can easily compute the number of modular addition of A mod p by the formulas for any given irreducible monic trinomial and monic pentanomial.

根据该表达式，对任意给定的不可约首一三项式和首一五项式，可以根据该多项式的系数分别得出以广义Mersenne数为模数的取模运算所需要的模加法的次数。

If ris also irreducible in Z_nand f~n~κ≡fmodn,r(xfor all f∈Z_nthen we call n a Carmichael number of orderκmodulo r.Denote the set of all such numbers by C_κ,r(x.Define C_κ=∪_r(xC_κ,r(x,where rpasses through all monic irreducible polynomials over Z of degreeκ.We call elements of the set C_κCarmichael numbers of orderκ.ZHU and SUN[Journal of Sichuan University,42;1(2005), 47-51]first gave a necessary condition for Carmichael numbers of order 3.Then they gave a sufficient condition for Carmichael numbers of order 3,but did not find any such numbers less than 10~8 satisfying the sufficient condition.At last,they asked three questions;1.is the necessary condition also sufficient?

f∈Z_n，就称n是模r的κ阶Carmichael数，用C_κ，r(x表示所有这种数的集合，定义C_κ=∪_r(xC_κ，r(x，这里r跑遍Z中所有κ次首一不可约多项式，C_κ里面的元素就称为κ阶Carmichael数，C_κ称为κ阶Carmichael数集，朱文余和孙琦[四川大学学报，42；1(2005)，47-51]首先给出了3阶Carmichael数的一个必要条件；然后又给出了这种数的一个充分条件，并发现10~8内没有满足此充分条件的这种数；最后他们提出三个问题；一、问必要条件是否也是充分的？

Each term in a polynomial is also called a monomial.

多项式是一种代数的表现方法，由项组成。

Hence a monomial is a polynomial of exactly one term.

只有一项的多项式亦被称为单项式。

Controller constructed by full monomial base will be in numerous terms for relatively high-order systems and obstruct its application in practice.

由多项式全基构造的控制器在相对高阶系统中具有很多项单项式，阻碍了其实际应用性。

A monomial: a polynomial containing only one term.

单项式：只包含一项的多项式。

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。