第二章研究有界正则函数导数的估计问题。
In second chapter, we discuss the problems of estimating in the derivation of bounded functions.
证明了一条幂指函数的求导法则，并总结了幂指函数导数计算的常用方法。
The paper gives a derivation rule of the power exponential function and analyzes some common calculation methods.
当网络连接权值矩阵的最小特征值大于激活函数导数的倒数时，网络并行收敛。
When the minimal eigenvalue of connection weights matrix is greater than the reciprocal of derivation of its neuron activation function, the network will be convergent in parallel update mode.
本文在灰函数和灰极限的基础上，给出了灰函数的导数(或灰导数)的定义，并研究了简单灰函数导数。
Based on grey function and grey limit, this paper gives definition to derivate of grey function (or to grey derivate) and studies the derivate of simple grey function.
关于这个方法很直观的一点想法是，如果导数非常大，函数也就变化的非常快，因此我们想一小步一小步的来。
One way to think about this intuitively if the derivative is very large the function is changing quickly, and therefore we want to take small steps.
不是每个函数都有导数。
我们说一个函数是可微的，如果这些偏导数存在。
And, we say that a function is differentiable if these things exist.
因此，一个多变量的函数没有通常的导数。
So, a function of several variables doesn't have the usual derivative.
有些不是规则的函数，却有导数。
You have functions that are not regular enough to actually have a derivative.
未知量是一个函数，这个方程，将把函数的偏导数联系起来。
So the unknown is a function, and the equation will relate the partial derivatives of that function to each other.
如果我们想更加精确一点，我们知道当t变化的时候，乘以导数就出现了，well, t, times, the, derivative, comes, in，这是函数变化量的线性近似。
If we want to be a bit more precise, we know that when we change by t, t that's for linear approximation to how the function changes.
微积分基本定理，不是曲线积分的，告诉我们，如果对函数的导数积分，就会得回原函数。
So, the fundamental theorem of calculus, not for line integrals, tells you if you integrate a derivative, then you get back the function.
好，如果导数很小的话，函数就基本没什么变化，可能我们就想把步子迈大一点儿了，但是别为这个担心？
All right. If the derivative is small, it's not changing, maybe want to take a larger step, but let's not worry about that all right?
但是有一件事情，我必须指出来，在现实中，很多函数的偏导数之间，存在着很好的关系。
But one thing at a time. I wanted to point out to you that very often functions that you see in real life satisfy many nice relations between the partial derivatives.
对光学系统输出刀口像的一阶导数做傅里叶变换，可以得到该系统的光学传递函数（OTF）。
The OTF (Optical Transfer Function) of an optical system can be evaluated by the Fourier Transform of the first derivative of the optical image of an knife-edge (object).
在微积分中，在函数的极小值点上导数等于零或者不存在导数。
In calculus, the derivative equals zero or does not exist at a function's minimum point.
提出了一种新的NCP函数，并将它应用于自由导数方法，达到了提高算法速度的目的。
The paper gives out a NCP-function; in the paper it is applied to derivative-free method, we can increase the speed of arithmetic.
这对于需要目标函数的导数信息的传统优化方法是很困难的。
This presents difficulties for most conventional optimization methods, which need derivative information of the objective function.
科学计算及其应用常常需要多变量函数的有关偏导数问题的计算，通常使用的计算方法是符号微分或差分近似。
Evaluation relevant to the partial derivatives of the multivariable functions is often done in scientific computation, usually by means of the symbolic differentiation or the divided difference.
自动微分技术能以较低的成本精确计算中大规模问题函数的导数，在科学计算、工程计算及其应用领域中有着广泛的应用。
Automatic differentiation, by which the derivatives of the function can be evaluated both exactly and economically, is applied to the field of scientific and engineering computation extensively.
通过分析函数的一阶导数与二阶导数图像的特征，提出了一种含有权值的图像边缘锐化的新算法。
By analyzing the characteristics of the function's once and quadratic differential coefficient's image, a new algorithm of images' edge sharpening which contains weights.
在结点互异或结点重合时，将函数差商与其导数之间的关系式推广为关于两个函数的情形。
Thie relationship between divided difference and derivative of a function can be generalized for two functions which the knots are distinct or the knots are coincident.
本文推导了网络函数及其导数的求值公式，应用这些公式可使网络分析及优化算法更有效。
Evaluating formulae of network functions and their derivatives are introduced in this paper. The algorithm of network analysis and optimization will be more efficient by using these formulae.
给出了在无穷远处具有有穷极限的函数的有关导数的几个定理及其严格证明。
The paper gives several theorems for derivate of the function with finite limit at the infinite point and its strict proof.
而广义移动最小二乘近似要求近似函数及其导数在所有节点处的误差的平方和最小。
However, the generalized moving least squares approximation makes require least squares approximation with regard to functional and its derivative value on all nodes.
根据贝塞尔级数理论，建立H变换，导出函数及其导数的贝塞尔级数的完整表达式。
In this paper, H-transformation is formulated according to the theory of Bessel series. The complete series expressing a function and its derivative are derived.
用奇点分布法计算浮体在波浪上的运动需要多次计算自由水面格林函数及其导数。
The analysis of interaction between waves and floating bodies by singularity distribution method involves a great number of estimations for the Green Function with surface and its derivatives.
用奇点分布法计算浮体在波浪上的运动需要多次计算自由水面格林函数及其导数。
The analysis of interaction between waves and floating bodies by singularity distribution method involves a great number of estimations for the Green Function with surface and its derivatives.
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