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# 数学代写|数学物理方法代写Mathematical Methods代考|PCS622 Orders of magnitude

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## 数学代写|数学物理方法代写Mathematical Methods代考|Orders of magnitude

Orders of magnitude. If as $x$ tends to a limit $\phi(x)$ tends to 0 or $\infty$, and $f(x) / \phi(x)$ is bounded, we say that $f(x)=O{\phi(x)}$, or that $f(x)$ is of the same order of magnitude as $\phi(x)$. If $f(x) / \phi(x) \rightarrow 0$ as $\phi(x) \rightarrow 0$ we write $f(x)=o{\phi(x)}$. If $f(x)$ is bounded we can write $f(x)=O(1)$. This notation must be distinguished from the common usage in physics, where we may say that the masses of Jupiter and Saturn are of the same order of magnitude, meaning roughly that they differ by not more than a factor 10 without there being any question of a limit. In the physical sense the quantities compared must have the same dimensions. This is not necessary in the mathematical sense. $x$ may, for instance, be a time-interval and $f(x)$ the distance travelled by a sound wave. Then $f(x)=O(x)$ because $f(x) / x$ is the velocity of sound and is supposed finite.
Note that
$$O\left(x^m\right) O\left(x^n\right)=O\left(x^{m+n}\right), \quad o\left(x^m\right) O\left(x^n\right)=o\left(x^{m+n}\right) .$$

## 数学代写|数学物理方法代写Mathematical Methods代考|Functions of bounded variation

Functions of bounded variation. If the function $f(x)$ is defined in the closed interval $(a, b)$, and there is a number $M$ such that
$$v=\left|f\left(x_1\right)-f\left(x_0\right)\right|+\left|f\left(x_2\right)-f\left(x_1\right)\right|+\ldots+\left|f\left(x_n\right)-f\left(x_{n-1}\right)\right| \leqslant M,$$
for every subdivision $a=x_0<x_1<x_2<\ldots<x_{n-1}<x_n=b, f(x)$ is said to be of bounded variation in the interval $(a, b)$; and the upper bound of the sums $v$ for all possible selections of the subdivisions is called the total variation of $f(x)$ in the interval.* The total variation is of interest since it is related to the condition for existence of a Stieltjes integral (1.102), and to the existence of the length of a curve, and it is useful in the theory of Fourier series and Fourier integrals.

We assume repeatedly that the sum and product of two continuous functions (and therefore of any finite number) are continuous, and that those of two functions of bounded variation are of bounded variation. The proofs are simple: for the last, notice that
\begin{aligned} f\left(x_{r+1}\right) g\left(x_{r+1}\right)-f\left(x_r\right) g\left(x_r\right)=\frac{1}{2}\left{f\left(x_r\right)+f\left(x_{r+1}\right)\right} & \left{g\left(x_{r+1}\right)-g\left(x_r\right)\right} \ & +\frac{1}{2}\left{g\left(x_r\right)+g\left(x_{r+1}\right)\right}\left{f\left(x_{r+1}\right)-f\left(x_r\right)\right} \end{aligned}
and it follows that if $M, N$ are the upper bounds of $|f(x)|,|g(x)|$, and $U, V$ the total variations of $f(x), g(x)$ in the interval, the total variation of $f(x) g(x)$ is not greater than $M V+N U$.

# 数学物理方法代写

## 数学代写|数学物理方法代写Mathematical Methods代考|Orders of magnitude

$\$ \$$O \backslash left (x \wedge m \backslash right ) o left (x \wedge n \backslash right )=0 \backslash left (x \wedge{m+n} \backslash right ), \quad o left (x \wedge m \backslash 右 ) O \backslash left (x \wedge n \backslash right )=o \backslash left (x \wedge{m+n} \backslash right ) 。 \ \$$

## 数学代写|数学物理方法代写Mathematical Methods代考|Functions of bounded variation

$\$ \$$对于每个细分 \ a=x _0<x _l<x _2<\backslash l d o t s<x _{n-1}<x _n=b ， f(x) \$$ 被称为区间 $\$(a, b) \$$中的有界变化; 并且所有可能的细分 选择的总和 \ v \$$ 的上限称为区间内 $\$ f(x) \$$的总变差。总变差很有趣，因为它与存在的条件有关一个 Stieltjes 积分 (1.102)，以 及曲线长度的存在性，它在傅里叶级数和傅里叶积分理论中很有用。 我们反㵊假设两个连续函数（因此任何有限数）的和与积是连㩺的，并且两个有界变差函数的和与积是有界变差。证明很简单：最 后，注意 \ \$$
lbegin ${$ aligned $}$
$f \backslash$ left $\left(x_{-}{r+1} \backslash\right.$ right $) g \backslash$ left $\left(x_{-}{r+1} \backslash\right.$ right $)-f \backslash$ left $\left(x_{-} r \backslash r i g h t\right) g \backslash$ left $\left(x_{-} r \backslash\right.$ right $)=\backslash$ frac ${1}$
${2} \backslash$ left $\left{f \backslash\right.$ left $\left(x_{-} r \backslash\right.$ right $)+f \backslash$ left $\left(x_{-}{r+1} \backslash r i g h t\right) \backslash$ right $} \& \backslash$ 左 $\left{g \backslash\right.$ left $\left(x_{-}{r+1} \backslash r i g h t\right)-g \backslash$ left $\left.\left(x_{-} r \backslash r i g h t\right) \backslash r i g h t\right} \backslash \&$
$+\backslash$ frac ${1}{2} \backslash$ left $\left{g \backslash\right.$ left $\left(x_{-} r \backslash\right.$ right $)+g \backslash$ left $\left(x_{-}{r+1} \backslash\right.$ right $) \backslash$ right $} \backslash$ left $\left{f \backslash l e f t\left(x_{-}{r+1} \backslash\right.\right.$ right $)-$
$f \backslash$ left $\left.\left(x _r \backslash r i g h t\right) \backslash r i g h t\right} \backslash$ lend ${$ aligned
}
$\$ \$$并且它菖砕如果 \ M, N \$$ 是 $\$|f(x)|,|g(x)| \$$的上限，并且 \ U, V \$$ 是 $\$ f(x)$的总变化，$g(x) \$区 i ⿵ 冂 人 ， \$ f(x) g(x) \$$的总变差不大 于 \ M V+N U \$$ 。

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。