一变量多项式

An algebraic expression consisting of one or more summed terms, each term consisting of a constant multiplier and one or more variables raised to integral powers.

多项式包含一个或更多总项的代数表达法，每一项都含有一个常乘数和一个或更多整幂的变量。

Dirac equation can be separated into an angular equation and a radial equation with the help of separation variable.

通过分离变量得到Dirac方程相应的角向方程和径向方程，得出了用广义连带勒让德多项式表示的归一化角向波函数和用合流超几何函数表示的归一化径向波函数；获得了精确的束缚态能谱方程并对结果作适当讨论与结论。

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。

Polynomial smooth techniques are applied to SVM model and replace x+ by a very accurate smooth approximation that is Hermite Interpolation polynomial,thus the undifferential model is converted into a differential model.The deduction procedure of Hermite Interpolation polynomial smoothing x+ is extended.

三次Hermite插值多项式光滑的支持向量机模型采用的是一种多项式光滑技术，用三次Hermite插值多项式代替单变量函数x+，将原来不可微的模型变为可微的模型，并且给出了三次Hermite插值多项式光滑化单变量函数x+的推导过程。

This problem of planar parametric polynomial curve of degree 3 had been discussed perfectly many years ago, making use of invariant of affine transformation.

关于三次代数多项式参数曲线的这一问题早已解决，然而所用的几何不变量方法对于非代数曲线失效。

In the computational methods of TPBVP, in order to reduce some difficulties involved in solving a TPBVP via adjoint variables, we discuss a direct method in which state and control variables are indirectly parameterized, The method employs a recently developed direct optimization technique that uses a piecewise polynomial representation for the state and control variables, thus converting the optimal control problem into a nonlinear programming problem, which can be solved numerically.

在数值解法中，为了减少解决两点边值问题共轭变量带来的困难，主要讨论了将状态变量和控制变量进行参数化的一种直接方法，这种方法采用了近段发展起来的使用分段的多项式来代替状态和控制变量的直接优化方法，然后最优控制问题就转化成可以用数值方法解决的非线性规划问题，使得迭代初值更加容易选取。

Since the separable element algorithm of the rational univariate representation for solving zerodimensional systems presented by Rouillier makes the size of the coefficients in this representation too long, an improved algorithm for finding separating elements is presented in this paper, which is implemented by confirmingthe separating element of the coordinates step by step.

针对零维多项式系统中Rouillier计算可分元的算法使其相应的有理单变量表示中整系数过长的问题，提出一种改进的可分元选取算法，新算法通过逐步确定坐标可分元得以实现。

In spherical polar coordinates, DRSC potential have supersymmetry and shape invariance for θ and r coordinates. By using the method of supersymmetry and shape invariance, exact bound state solutions of Schrodinger equation with double ring-shaped Coulomb potential are presented. The normalized angular wave function expressed in terms of Jacobi polynomials and the normalized radial wave function expressed in terms of the Laguerre polynomials are presented. Energy spectrum equations are obtained.

本文研究了双环形Coulomb势Schrdinger方程的束缚态精确解，所采用的方法是首先对双环形Coulomb势的Schrdinger方程在球坐标系中进行分离变量，得到相应的角向方程和径向方程；证明双环形Coulomb势在角向和径向具有超对称性和形不变性；根据超对称性和形不变性的性质，获得了角动量量子化条件和束缚态的能谱方程，并将归一化角向波函数用Jacobi多项式表示，将归一化径向波函数用Laguerre多项式函数表示。

As one of the leading branches of the modern statistics, the riouparametric regression analysis is widely applied to explore the relationship between the the response variable y and the covariable A'.

对于非参数回归人们提出了许多估计方法，如核估计，局部多项式估计，光滑样条估计，级数估计(傅里叶级数估计，小波级数估计)等。这些方法本质上讲都是局部估计或局部光滑，当回归变量X为一维变量时，非参数回归函数用这些方法一般都能得到很好的估计。

As she looked at Warrington's manly face, and dark, melancholy eyes, she had settled in her mind that he must have been the victim of an unhappy attachment.

每逢看到沃林顿那刚毅的脸，那乌黑、忧郁的眼睛，她便会相信，他一定作过不幸的爱情的受害者。

Maybe they'll disappear into a pothole.

也许他们将在壶穴里消失