### 数学代写|数论作业代写number theory代考|MATH 1001

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|数论作业代写number theory代考|Coprime Numbers

Natural numbers $m$ and $n$ are called coprime, if $(m, n)=1$. An interesting real-life technical application of coprime numbers is depicted in Fig. 1.5. It shows two gear ratios. In the left part of the figure, the larger wheel has 20 teeth and the smaller one 10 teeth. If there is one tooth slightly damaged on the larger wheel (it is marked with a dot in Fig. 1.5), then it fits into exactly the same gap in the smaller wheel after each turn of the larger wheel. Just at this gap, the smaller wheel will be very quickly worn out. In the right part of Fig. $1.5$ we see two wheels with 25 and 12 teeth. Since $(25,12)=1$, there will be completely uniform wear. Let us still note that the ratio of the teeth is actually almost the same in both cases: 2 and $2.083 .$

Here is another practical use of coprime numbers. The fixed part of the caliper is equipped with a scale in which each centimeter is divided into $10 \mathrm{~mm}$. On the moving part, the so-called vernier is divided into 10 equally long pieces, which together have $9 \mathrm{~mm}$ (see Fig. 1.6). The fact that 9 and 10 are coprime allows us to determine the dimensions of small objects with precision to the nearest tenth of a millimeter. When measuring, we determine which line of the vernier merges with some line on the millimeter scale of the caliper. So many tenths of a millimeter is then added to the measured millimeters (i.e. to their largest integer value). If we were to choose instead of $9 \mathrm{~mm}$ another length that divides $10 \mathrm{~mm}$, then several lines could merge so we would not know which data applies.

## 数学代写|数论作业代写number theory代考|Euclidean Algorithm

To calculate the greatest common divisor $(m, n)$ of two large natural numbers $m \geq n$ the well-known Euclidean algorithm is often used. It can be briefly characterized as follows:

If $n$ divides $m$, then $(m, n)=n$, otherwise we have
$$(m, n)=(n, z),$$
where $z \geq 1$ is the remainder when dividing the number $m$ by the number $n$. Since $z<m$, larger problem is thus converted to a smaller one. The next steps of the algorithm then proceed similarly. The original problem is thus reduced to smaller and smaller parts until we get the remainder 0 .
For instance, if $m=54$ and $n=16$, then by the Euclidean algorithm we get
$$(54,16)=(16,6)=(6,4)=(4,2)=2 .$$
Now let us imagine that we have a squared paper with dimensions $54 \times 16$ (see Fig. 1.7). From this we will gradually cut off squares as large as possible and we will perform this as long as possible (see [188]), i.e., in the first step we cut off 3 squares $16 \times 16$, in the next step we cut 2 squares $6 \times 6$, etc. The length of the side of the square that we have left, is the result of the Euclidean algorithm, i.e. the largest common divisor of numbers 54 and 16 .

If the numbers $m$ and $n$ are coprime, the Euclidean algorithm ends with the least possible square $1 \times 1$. For example, two consecutive natural numbers are always coprime.

To calculate the least common multiple of $[m, n]$, it is also useful to apply the Euclidean algorithm first, because $(m, n) \leq[m, n]$, and then use the relation (1.2). We return to the Euclidean algorithm in Theorem 7.7.

## 数学代写|数论作业代写number theory代考|Euclidean Algorithm

$$(m, n)=(n, z)$$

$$(54,16)=(16,6)=(6,4)=(4,2)=2 .$$

## 有限元方法代写

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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。