算术函数

In Chapter 2, one kind of rational cubic spline interpolation with linear denominator has been proposed, whose derivatives are replaced with accurate derivative values, difference quotient and arithmetic average difference quotientrespectively.

第二章给出了一类分母为线性的有理三次插值函数，当其导数值分别取为导数的精确值，一阶差商，以及算术均差商时，便得到三种插值格式，分别为给定导数值的有理三次插值函数，基于函数值的有理三次插值函数，以及基于算数均差商的有理三次插值函数。

The concept of the weighted logarithmic power mean is introduced; its relation with two-parameter mean is given; the inequality for weighted logarithmic power mean is derived; the magnitude relation among upper bounds of geometric mean and arithmetic mean of geometrically convex functions are made certain.

建立了几何凸函数的对称拟算术平均不等式，对文献[1]提出的不等式进行了推广统一；引进加权对数幂平均的概念，建立起其与双参数平均之间的关系，得到加权对数平均不等式，从而确定了几何凸函数的几何平均、算术平均的上界的大小关系；最后，提出了几何凸函数的对称拟算术平均不等式的推广问题。

In this dissertation, we study the mean value problem of some important arithmetical function.

本文研究了一些算术函数的均值估计问题，获得了一些均值定理。

The study on the properties of arithmetical sequences is the kernel of the research in number theory.

一书。在该书中，他提出了105个关于特殊序列、算术函数等未解决的数学问题及猜想，随着这些问题的提出，许多学者对此进行了深入的研究，并获得了不少具有重要理论价值的研究成果。

Software Description: XCCalc is a Floating-Point calculator program with the following features:- Can calculate with complex numbers like z = x + i*y - Extra precision to millions of decimal places - Uses Fast Hartley Transform to speed-up long multiplies - Separate input and output notation, base 2 to base 36 - Efficient Transcendental functions - Hyperbolic functions - Greatest Common Divisor function.

XCCalc XPCalc 是一款带有下列功能的浮点计算器软件：可以计算诸如 z = x + i*y 这样的复杂算术公式；带有数百万小数位的额外精度；使用 Fast Hartley Transform 以加速长乘法计算；基于2到36个单独的输入和输出表记法；高效的验算函数；双曲线函数；最大公约数函数。

These constructional methods can be used to treat many problems, the numerical results show that the given interval extension has smaller excessive width relative to the range of the functions over a domain. For univariate differentiable functions, the definition of interval slope is extended from the first order to high order according to their differentiability, and iterative formula of high-order interval slope computation is also developed.

对多元函数，在不要求函数的可微性及有关Lipschitz常数的条件下，给出了分量函数区间展开的斜率算术运算规则；利用函数局部凸或凹的运算特点，通过递归计算建立了多元函数区间扩张的构造方法。

We defined two arithmetical functions and got mean value properties of the two functions by using elementary and analytic methods; we also studied the mean value property of Smarandache ceil function S_kn!

定义了两个新的算术函数，利用初等和解析方法研究了这两个函数的均值性质；研究了Smarandache ceil函数S_kn！

This is a vast collection of computational algorithms ranging from elementary functions like sum, sine, cosine, and complex arithmetic, to more sophisticated functions like matrix inverse, matrix eigenvalues, Bessel functions, and fast Fourier transforms.

MATLAB 的工具箱函数库收集了大量的数学函数，从求和，正弦，余弦以及复杂的算术运算等基本函数，到矩阵求逆，矩阵求特征值，贝塞尔函数和快速傅里叶变换等复杂函数。

Then, using the interval arithmetic tool, construction methods of several quasi convex-concave extension are presented for general functions which are provided with arithmetic expression.

其次，对由算术表达式给定的一般函数，利用区间算术工具给出了几种拟凸-凹扩张的构造方法。

It is well known that the mean value problems of arithmetical functions play an important role in the study of analytic number theory, and they relate to many famous number theoretic problems.

众所周知，算术函数的均值估计问题在解析数论研究中占有十分重要的位置，许多著名的数论难题都与之密切相关。

We have no common name for a mime of Sophron or Xenarchus and a Socratic Conversation; and we should still be without one even if the imitation in the two instances were in trimeters or elegiacs or some other kind of verse--though it is the way with people to tack on 'poet' to the name of a metre, and talk of elegiac-poets and epic-poets, thinking that they call them poets not by reason of the imitative nature of their work, but indiscriminately by reason of the metre they write in.

索夫农 、森那库斯和苏格拉底式的对话采用的模仿没有一个公共的名称；三音步诗、挽歌体或其他类型的诗的模仿也没有——人们把&诗人&这一名词和格律名称结合到一起，称之为挽歌体诗人或者史诗诗人，他们被称为诗人，似乎只是因为遵守格律写作，而非他们作品的模仿本质。

The relationship between communicative competence and grammar teaching should be that of the ends and the means.

交际能力和语法的关系应该是目标与途径的关系。