一阶导数

In this paper,we give the error expressions of piecewise linear Lagrange polynomial interpolation and piecewise cubic polynomial interpolation using the Taylor expansions.

中文摘要：本文利用Taylor展开得到三角形上线性Lagrange插值和三次Lagrange插值的导数余项公式，对这些余项公式进行分析，给出了两类能以四阶精度逼近被插函数在对称点的导数值的格式，一种是在均匀剖分时其分片线性插值的相邻单元的导数值的后处理格式，一种是在六片强正规剖分时的三次插值在对称点上的导数值的后处理格式，使得在已知原函数在各节点的值后，通过一个简单的线性计算就可得到原函数在对称点的导数的一个超逼近值，将以往提出的平均导数的二阶精度提高到四阶。

With the help of the notation of the skew-symmetric tensor, the expression for the 1st order absolute and relative derivatives of a tensor was presented, and the expression for the 2nd order absolute and relative derivatives of a tensor was offered further.

由此，并借助于反对称张量，给出了张量的一阶绝对导数和相对导数之间的关系式，并又进一步给出了张量的二阶绝对导数和相对导数之间的关系。

Still another form for the second derivative of f with respect to x.

f对x的二阶导数的另一种形式。

In the first section,the existence of solution for a class of non-linear systems with three -point boundary problem is obtained by applying the differential inequality theory .

在第一部分中，我们应用微分不等式理论证明了一类非线性系统三点边值问题解的存在性；在第二部分中，运用微分不等式理论研究了一类带有转向点的拟线性奇摄动边值问题的非单调内层解；在第三部分中，利用对角化方法研究了一类向量二点或者三点边值问题解的存在性，并获得解及它的一、二阶导数的渐近估计。

With the guide of non-linear program theory, by using interpolative steps and inexact line search, an improved conjugate gradient method was found, by which the training rate of BP networks increases by tens or hundreds of times. Moreover, the improved method is effective to solve non-linear equations for which Newton's method does not converge owing to the problems of the quadratic derivative and inverse matrix.

通过大量的数值模拟试验发现，在非线性规划理论的指导下采用间插步骤和不精确的一维搜索技术改进的共轭梯度法，是基于梯度和共轭方向的连续搜索算法中最有效的算法，这种算法使BP网络的训练速度提高几十到几百倍，使BP网络的实际应用效果大为改善；而且这种算法对于用牛顿法由于求二阶导数和求逆矩阵等问题难于收敛的非线性方程组的求解也是很有效的。

According to the objective of second-order derivative related to curvatures, it is proved that there is no difference whether we insert a knot or not. And it presents a characteristic of cubic Hermite interpolation curves.

如果以与曲率有关的二阶导数为目标，证明插入节点与不插入节点的情形是一样的，体现三次Hermite插值曲线的一种特性。

Then the spatial localization of the main well-known visual neuron's receptive field models such as DOG function, LOG function, higher order derivatives of Gaussian functions and 1—D, 2—D Gabot function models is quantitatively analysed.

如对通常被作为建立模型的主要根据的&最优的空间定域性&判据的物理意义和适用性也没有得到深入的论证，而且已有的主要几类感受野模型，如DOG函数、LOG函数、高斯函数的高阶导数和一维、二维Gabor函数等模型在空间分布形态上难以准确反映系统中神经元感受野的形态和信息处理机制等等。

According to the objective of third-order derivative related to torsions, the formula to calculate the cubic optimized Hermite interpolation curves is given. And then we propose a new method to construct curves.

如果以与挠率有关的三阶导数为目标，给出优化三次Hermite曲线的计算公式，从而提出一种新的曲线构造方法。

Starting with the zenith, a series of star fields at different altitude are observed by a telescope with larger field of view, the derivatives of various orders are calculated at different zenith distances, and finally the observational atmospheric refraction can be obtained by numerical integration.

利用一台较大视场的望远镜从天顶开始，在不同高度上对星空作一系列观测，计算不同天顶距处大气折射函数的各阶导数，最后经数值积分可给出大气折射实测值。

In the improved algorithm, the computation of three-order time derivative is replaced by convolution approximate calculation, which simplifies the computation process and reduces computation amount, and is high effective and robust.

改进算法用褶积近似运算代替时间3阶导数的求解，因此简化了计算过程，减少了计算量，是一种高效稳健的算法。

The split between the two groups can hardly be papered over.

这两个团体间的分歧难以掩饰。

This approach not only encourages a greater number of responses, but minimizes the likelihood of stale groupthink.

这种做法不仅鼓励了更多的反应，而且减少跟风的可能性。