## matlab代写|time series analysisEMET3007/8012 Assignment 2

statistics-lab™ 为您的留学生涯保驾护航 在代写时间序列分析time series analysis方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写时间序列分析time series analysis代写方面经验极为丰富，各种代写时间序列分析time series analysis相关的作业也就用不着说。

• 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

## Instructions:

This assignment is worth either 20% or 25% of the final grade, and is worth a total of 75 points. All working must be shown for all questions. For questions which ask you to write a program, you must provide the code you used. If you have found code and then modified it, then the original source must be cited. The assignment is due by 5pm Friday 1st of October (Friday of Week 8), using Turnitin on Wattle. Late submissions will only be accepted with prior written approval. Good luck.

[10 marks] In this exercise we will consider four different specifications for forecasting monthly Australian total employed persons. The dataset (available on Wattle) AUSEmp 1oy 2022. csv contains three columns; the first column contains the date; the second contains the sales figures for that month (FRED data series LFEMTTTTAUM647N), and the third contains Australian GDP for that month.1] The data runs from January 1995 to January $2022 .$

Let $M_{i t}$ be a dummy variable that denotes the month of the year. Let $D_{i t}$ be a dummy variable which denotes the quarter of the year. The four specifications we consider are
\begin{aligned} &S_1: y_t=a_0+a_1 t+\alpha_4 D_{4 t}+\epsilon_t \ &S_2: y_t=a_1 t+\sum_{i=1}^4 \alpha_i D_{i t}+\epsilon_t \ &S_3: y_t=a_0+a_1 t+\beta_{12} M_{12, t}+\epsilon_t \ &S_4: y_t=a_1 t+\sum_{i=1}^{12} \beta_i M_{i t}+\epsilon_t \end{aligned}
where $\mathbb{E} \epsilon_t=0$ for all $t$.

a) For each specification, describe this specification in words.
b) For each specification, estimate the values of the parameters, and compute the MSE, $\mathrm{AIC}$, and BIC. If you make any changes to the csv file, please describe the changes you make. As always, you must include your code.
c) For each specification, compute the MSFE for the 1-step and 5-step ahead forecasts, with the out-of-sample forecasting exercise beginning at $T_0=50$.
d) For each specification, plot the out-of-sample forecasts and comment on the results.

[10 marks] Now add to Question 1 the additional assumption that $\epsilon_t \sim \mathcal{N}\left(0, \sigma^2\right)$. One estimator ${ }^2$ for $\sigma^2$ is
$$\hat{\sigma}^2=\frac{1}{T-k} \sum_{t=1}^T\left(y_t-\hat{y}_t\right)^2$$
where $\hat{y}_t$ is the estimated value of $y_t$ in the model and $k$ is the number of regressors in the specification.
a) For each specification $\left(S_1, \ldots, S_4\right)$, compute $\hat{\sigma}^2$.
b) For each specification, make a $95 \%$ probability forecast for the sales in June $2021 .$
c) For each specification, compute the probability that the total employed persons in June 2022 will be greater than $13.5$ million. According to the FRED series LFEMTTTTAUM647N, what was the actual employment level for that month.
d) Do you think the assumption that $\epsilon_t$ is iid is a reasonable assumption for this data series.

[10 marks] Here we investigate whether adding GDP $\mathrm{Gs}^3$ as a predictor can improve our forecasts. Consider the following modified specifications:
\begin{aligned} &S_1^{\prime}: y_t=a_0+a_1 t+\alpha_4 D_{4 t}+\gamma x_{t-h}+\epsilon_t \ &S_2^{\prime}: y_t=a_1 t+\sum_{i=1}^4 \alpha_i D_{i t}+\gamma x_{t-h}+\epsilon_t \ &S_3^{\prime}: y_t=a_0+a_1 t+\beta_{12} M_{12, t}+\gamma x_{t-h}+\epsilon_t \ &S_4^{\prime}: y_t=a_1 t+\sum_{i=1}^{12} \beta_i M_{i t}+\gamma x_{t-h}+\epsilon_t \end{aligned}
where $\mathbb{E} \epsilon_t=0$ for all $t$, and $x_{t-h}$ is GDP at time $t-h$. For each specification, compute the MSFE for the 1-step ahead, and the 5-step ahead forecasts, with the out-of-sample forecasting exercise beginning at $T_0=50$. For each specification, plot the out-of-sample forecasts and comment on the results.

[15 marks] Here we investigate whether Holt-Winters smoothing can improve our forecasts. Use a Holt-Winters smoothing method with seasonality, to produce 1-step ahead and 5-step ahead forecasts and compute the MSFE for these forecasts. You should use smoothing parameters $\alpha=\beta=\gamma=0.3$ and start the out-of-sample forecasting exercise at $T_0=50$. Plot these out-of-sample forecasts and comment on the results.
Additionally, estimate the values for $\alpha, \beta$, and $\gamma$ which minimise the MSFE. Find the MSFE for these parameter vales and compare it to the baseline $\alpha=\beta=\gamma=0.3$.

[5 marks] Questions 1, 3 and 4 each provided alternative models for forecasting Australian Total Employment. Compare the efficacy of these forecasts. Your comparison should include discussions of MSFE, but must also make qualitative observations (typically based on your graphs).

[10 marks] Develop another model, either based on material from class or otherwise, to forecast Australian Total Employment. Your new model should perform better (have a lower MSFE or MAFE) than all models from Questions 1,3, and 4. As part of your response to this question you must provide:
a) a brief written explanation of what your model is doing,
b) a brief statement on why you think your new model will perform better,
c) any relevant equations or mathematics/statistics to describe the model,
d) the code to run the model, and
e) the MSFE and/or MAFE error found by your model, and a brief discussion of how this compares to previous cases.

[15 marks] Consider the ARX(1) model
$$y_t=\mu+a t+\rho y_{t-1}+\epsilon_t$$
where the errors follow an $\mathrm{AR}(2)$ process
$$\epsilon_t=\phi_1 \epsilon_{t-1}+\phi_2 \epsilon_{t-2}+u_t, \quad \mathbf{u} \sim \mathcal{N}\left(0, \sigma^2 I\right)$$
for $t=1, \ldots, T$ and $e_{-1}=e_0=0$. Suppose $\phi_1, \phi_2$ are known. Find (analytically) the maximum likelihood estimators for $\mu, a, \rho$, and $\sigma^2$.

Hint: First write $y$ and $\epsilon$ in vector/matrix form. You may wish to use different looking forms for each. Find the distribution of $\epsilon$ and $y$. Then apply some appropriate calculus. You may want to let $H=I-\phi_1 L-\phi_2 L^2$, where $I$ is the $T \times T$ identity matrix, and $L$ is the lag matrix.

## EMET3007/8012代写

matlab代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

## 数学代写|matlab代写|ENES206

MATLAB是一个编程和数值计算平台，被数百万工程师和科学家用来分析数据、开发算法和创建模型。

statistics-lab™ 为您的留学生涯保驾护航 在代写matlab方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写matlab代写方面经验极为丰富，各种代写matlab相关的作业也就用不着说。

• 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|matlab代写|TAYLOR AND LAURENT EXPANSIONS AND SINGULARITIES

In the previous section we showed what a crucial role singularities play in complex integration. Before we can find the most general way of computing a closed complex integral, our understanding of singularities must deepen. For this, we employ power series.

One reason why power series are so important is their ability to provide locally a general representation of a function even when its arguments are complex. For example, when we were introduced to trigonometric functions in high school, it was in the context of a right triangle and a real angle. However, when the argument becomes complex, this geometrical description disappears and power series provide a formalism for defining the trigonometric functions, regardless of the nature of the argument.

Let us begin our analysis by considering the complex function $f(z)$, which is analytic everywhere on the boundary, and the interior of a circle whose center is at $z=z_{0}$. Then, if $z$ denotes any point within the circle, we have from Cauchy’s integral formula that
$$f(z)=\frac{1}{2 \pi i} \oint_{C} \frac{f(\zeta)}{\zeta-z} d \zeta=\frac{1}{2 \pi i} \oint_{C} \frac{f(\zeta)}{\zeta-z_{0}}\left[\frac{1}{1-\left(z-z_{0}\right) /\left(\zeta-z_{0}\right)}\right] d \zeta,$$
where $C$ denotes the closed contour. Expanding the bracketed term as a geometric series, we find that
$$f(z)=\frac{1}{2 \pi i}\left[\oint_{C} \frac{f(\zeta)}{\zeta-z_{0}} d \zeta+\left(z-z_{0}\right) \oint_{C} \frac{f(\zeta)}{\left(\zeta-z_{0}\right)^{2}} d \zeta+\cdots+\left(z-z_{0}\right)^{n} \oint_{C} \frac{f(\zeta)}{\left(\zeta-z_{0}\right)^{n+1}} d \zeta+\cdots\right]$$
Applying Cauchy’s integral formula to each integral in Equation 1.7.2, we finally obtain
$$f(z)=f\left(z_{0}\right)+\frac{\left(z-z_{0}\right)}{1 !} f^{\prime}\left(z_{0}\right)+\cdots+\frac{\left(z-z_{0}\right)^{n}}{n !} f^{(n)}\left(z_{0}\right)+\cdots$$
or the familiar formula for a Taylor expansion. Consequently, we can expand any analytic function into a Taylor series. Interestingly, the radius of convergence ${ }^{6}$ of this series may be shown to be the distance between $z_{0}$ and the nearest nonanalytic point of $f(z)$.

## 数学代写|matlab代写|EVALUATION OF REAL DEFINITE INTEGRALS

One of the important applications of the theory of residues consists of the evaluation of certain types of real definite integrals. Similar techniques apply when the integrand contains a sine or cosine.

• Example 1.9.1
Let us evaluate the integral
$$\int_{0}^{\infty} \frac{d x}{x^{2}+1}=\frac{1}{2} \int_{-\infty}^{\infty} \frac{d x}{x^{2}+1} .$$
This integration occurs along the real axis. In terms of complex variables, we can rewrite Equation $1.9 .1$ as
$$\int_{0}^{\infty} \frac{d x}{x^{2}+1}=\frac{1}{2} \int_{C_{1}} \frac{d z}{z^{2}+1},$$
where the contour $C_{1}$ is the line $\Im(z)=0$. However, the use of the residue theorem requires an integration along a closed contour. Let us choose the one pictured in Figure 1.9.1. Then
$$\oint_{C} \frac{d z}{z^{2}+1}=\int_{C_{1}} \frac{d z}{z^{2}+1}+\int_{C_{2}} \frac{d z}{z^{2}+1},$$
where $C$ denotes the complete closed contour and $C_{2}$ denotes the integration path along a semicircle at infinity. Clearly we want the second integral on the right side of Equation $1.9 .3$ to vanish; otherwise, our choice of the contour $C_{2}$ is poor. Because $z=R c^{\theta i}$ and $d z=i R e^{\theta i} d \theta$
$$\left|\int_{C_{2}} \frac{d z}{z^{2}+1}\right|=\left|\int_{0}^{\pi} \frac{i R \exp (\theta i)}{1+R^{2} \exp (2 \theta i)} d \theta\right| \leq \int_{0}^{\pi} \frac{R}{R^{2}-1} d \theta$$
which tends to zero as $R \rightarrow \infty$. On the other hand, the residue theorem gives
$$\oint_{C} \frac{d z}{z^{2}+1}=2 \pi i \operatorname{Res}\left(\frac{1}{z^{2}+1} ; i\right)=2 \pi i \lim _{z \rightarrow i} \frac{z-i}{z^{2}+1}=2 \pi i \times \frac{1}{2 i}=\pi .$$

## 数学代写|matlab代写|TAYLOR AND LAURENT EXPANSIONS AND SINGULARITIES

$$f(z)=\frac{1}{2 \pi i} \oint_{C} \frac{f(\zeta)}{\zeta-z} d \zeta=\frac{1}{2 \pi i} \oint_{C} \frac{f(\zeta)}{\zeta-z_{0}}\left[\frac{1}{1-\left(z-z_{0}\right) /\left(\zeta-z_{0}\right)}\right] d \zeta$$

$$f(z)=\frac{1}{2 \pi i}\left[\oint_{C} \frac{f(\zeta)}{\zeta-z_{0}} d \zeta+\left(z-z_{0}\right) \oint_{C} \frac{f(\zeta)}{\left(\zeta-z_{0}\right)^{2}} d \zeta+\cdots+\left(z-z_{0}\right)^{n} \oint_{C} \frac{f(\zeta)}{\left(\zeta-z_{0}\right)^{n+1}} d \zeta+\cdots\right]$$

$$f(z)=f\left(z_{0}\right)+\frac{\left(z-z_{0}\right)}{1 !} f^{\prime}\left(z_{0}\right)+\cdots+\frac{\left(z-z_{0}\right)^{n}}{n !} f^{(n)}\left(z_{0}\right)+\cdots$$

## 数学代写|matlab代写|EVALUATION OF REAL DEFINITE INTEGRALS

• 示例 $1.9 .1$
让我们评估积分
$$\int_{0}^{\infty} \frac{d x}{x^{2}+1}=\frac{1}{2} \int_{-\infty}^{\infty} \frac{d x}{x^{2}+1}$$
这种整合沿实轴发生。就复变量而言，我们可以重写方程1.9.1作为
$$\int_{0}^{\infty} \frac{d x}{x^{2}+1}=\frac{1}{2} \int_{C_{1}} \frac{d z}{z^{2}+1}$$
轮廓在哪里 $C_{1}$ 是线 $\Im(z)=0$. 然而，使用余数定理需要沿闭合轮廓进行积分。让我们选择图 $1.9 .1$ 中所示的那 个。然后
$$\oint_{C} \frac{d z}{z^{2}+1}=\int_{C_{1}} \frac{d z}{z^{2}+1}+\int_{C_{2}} \frac{d z}{z^{2}+1},$$
在哪里 $C$ 表示完整的闭合轮廓，并且 $C_{2}$ 表示沿无限远半圆的积分路径。显然我们想要方程右边的第二个积分 1.9.3消失; 否则，我们选择的轮廓 $C_{2}$ 很穷。因为 $z=R c^{\theta i}$ 和 $d z=i R e^{\theta i} d \theta$
$$\left|\int_{C_{2}} \frac{d z}{z^{2}+1}\right|=\left|\int_{0}^{\pi} \frac{i R \exp (\theta i)}{1+R^{2} \exp (2 \theta i)} d \theta\right| \leq \int_{0}^{\pi} \frac{R}{R^{2}-1} d \theta$$
趋向于零 $R \rightarrow \infty$. 另一方面，剩余定理给出
$$\oint_{C} \frac{d z}{z^{2}+1}=2 \pi i \operatorname{Res}\left(\frac{1}{z^{2}+1} ; i\right)=2 \pi i \lim _{z \rightarrow i} \frac{z-i}{z^{2}+1}=2 \pi i \times \frac{1}{2 i}=\pi .$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|matlab代写|CS1132

MATLAB是一个编程和数值计算平台，被数百万工程师和科学家用来分析数据、开发算法和创建模型。

statistics-lab™ 为您的留学生涯保驾护航 在代写matlab方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写matlab代写方面经验极为丰富，各种代写matlab相关的作业也就用不着说。

• 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|matlab代写|THE CAUCHY-RIEMANN EQUATIONS

In the previous two sections, we introduced complex arithmetic. We are now ready for the concept of function as it applies to complex variables.

We already defined the complex variable $z=x+i y$, where $x$ and $y$ are variable. We now introduce another complex variable $w-u+i v$ so that for each value of $z$ there corresponds a value of $w=f(z)$. From all of the possible complex functions that we might invent, we focus on those functions where for each $z$ there is one, and only one, value of $w$. These functions are single-valued. They differ from functions such as the square root, logarithm, and inverse sine and cosine, where there are multiple answers for each $z$. These multivalued functions do arise in various problems. However, they are beyond the scope of this book and we shall always assume that we are dealing with single-valued functions.

A popular method for representing a complex function involves drawing some closed domain in the $z$-plane and then showing the corresponding domain in the $w$-plane. This procedure is called mapping and the $z$-plane illustrates the domain of the function while the $w$-plane illustrates its image or rangc. Figure 1.3.1 shows the $z$-plane and $w$-plane for $w=z^{2}$; a pie-shaped wedge in the $z$-plane maps into a semicircle on the $w$-plane.

## 数学代写|matlab代写|LINE INTEGRALS

So far, we discussed complex numbers, complex functions, and complex differentiation. We are now ready for integration.

Just as we have integrals involving real variables, we can define an integral that involves complex variables. Because the $z$-plane is two-dimensional, there is clearly greater freedom in what we mean by a complex integral. For example, we might ask whether the integral of some function between points $A$ and $B$ depends upon the curve along which we integrate. (In general it does.) Consequently, an important ingredient in any complex integration is the contour that we follow during the integration.

The result of a line integral is a complex number or expression. Unlike its counterpart in real variables, there is no physical interpretation for this quantity, such as area under a curve. Generally, integration in the complex plane is an intermediate process with a physically realizable quantity occurring only after we take its real or imaginary part. For example, in potential fluid flow, the lift and drag are found by taking the real and imaginary parts of a complex integral, respectively.

How do we compute $\int_{C} f(z) d z$ ? Let us deal with the definition; we illustrate the actual method by examples.

A popular method for evaluating complex line integrals consists of breaking everything up into real and imaginary parts. This reduces the integral to line integrals of real-valued functions, which we know how to handle. Thus, we write $f(z)=u(x, y)+i v(x, y)$ as usual, and because $z=x+i y$, formally $d z=d x+i d y$. Therefore,
\begin{aligned} \int_{C} f(z) d z &=\int_{C}[u(x, y)+i v(x, y)][d x+i d y] \ &=\int_{C} u(x, y) d x-v(x, y) d y+i \int_{C} v(x, y) d x+u(x, y) d y . \end{aligned}

## 数学代写|matlab代写|LINE INTEGRALS

$$\int_{C} f(z) d z=\int_{C}[u(x, y)+i v(x, y)][d x+i d y]=\int_{C} u(x, y) d x-v(x, y) d y+i \int_{C} v(x, y) d x+u(x$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|matlab代写|CSC113

MATLAB是一个编程和数值计算平台，被数百万工程师和科学家用来分析数据、开发算法和创建模型。

statistics-lab™ 为您的留学生涯保驾护航 在代写matlab方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写matlab代写方面经验极为丰富，各种代写matlab相关的作业也就用不着说。

• 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|matlab代写|COMPLEX NUMBERS

A complex number is any number of the form $a+b i$, where $a$ and $b$ are real and $i=\sqrt{-1}$. We denote any member of a set of complex numbers by the complex variable $z=x+i y$. The real part of $z$, usually denoted by $\Re(z)$, is $x$ while the imaginary part of $z, \Im(z)$, is $y$. The complex conjugate, $\bar{z}$ or $z^{*}$, of the complex number $a+b i$ is $a-b i$.

Complex numbers obey the fundamental rules of algebra. Thus, two complex numbers $a+b i$ and $c+d i$ are equal if and only if $a=c$ and $b=d$. Just as real numbers have the fundamental operations of addition, subtraction, multiplication, and division, so too do complex numbers. These operations are defined:
$$(a+b i)+(c+d i)=(a+c)+(b+d) i$$
Subtraction
$$(a+b i)-(c+d i)=(a-c)+(b-d) i$$

Multiplication
$$(a+b i)(c+d i)=a c+b c i+a d i+i^{2} b d=(a c-b d)+(a d+b c) i$$
Division
$$\frac{a+b i}{c+d i}=\frac{a+b i}{c+d i} \frac{c-d i}{c-d i}=\frac{a c-a d i+b c i-b d i^{2}}{c^{2}+d^{2}}=\frac{a c+b d+(b c-a d) i}{c^{2}+d^{2}} .$$
The absolute value or modulus of a complex number $a+b i$, written $|a+b i|$, equals $\sqrt{a^{2}+b^{2}}$. Additional properties include:
$$\begin{gathered} \left|z_{1} z_{2} z_{3} \cdots z_{n}\right|=\left|z_{1}\right|\left|z_{2}\right|\left|z_{3}\right| \cdots\left|z_{n}\right| \ \left|z_{1} / z_{2}\right|=\left|z_{1}\right| /\left|z_{2}\right| \quad \text { if } \quad z_{2} \neq 0 \ \left|z_{1}+z_{2}+z_{3}+\cdots+z_{n}\right| \leq\left|z_{1}\right|+\left|z_{2}\right|+\left|z_{3}\right|+\cdots+\left|z_{n}\right| \end{gathered}$$
and
$$\left|z_{1}+z_{2}\right| \geq\left|z_{1}\right|-\left|z_{2}\right| .$$

## 数学代写|matlab代写|FINDING ROOTS

The concept of finding roots of a number, which is rather straightforward in the case of real numbers, becomes more difficult in the case of complex numbers. By finding the roots of a complex number, we wish to find all of the solutions $w$ of the equation $w^{n}=z$, where $n$ is a positive integer for a given $z$.
We begin by writing $z$ in the polar form:
$$z=r e^{i \varphi},$$
while we write
$$w=R e^{i \Phi}$$
for the unknown. Consequently,
$$w^{n}=R^{n} e^{i n \Phi}=r e^{i \varphi}=z .$$
We satisfy Equation $1.2 .3$ if
$$R^{n}=r \quad \text { and } \quad n \Phi=\varphi+2 k \pi, \quad k=0, \pm 1, \pm 2, \ldots,$$
because the addition of any multiple of $2 \pi$ to the argument is also a solution. Thus, $R=r^{1 / n}$, where $R$ is the uniquely determined real positive root, and
$$\Phi_{k}=\frac{\varphi}{n}+\frac{2 \pi k}{n}, \quad k=0, \pm 1, \pm 2, \ldots$$

## 数学代写|matlab代写|COMPLEX NUMBERS

$$(a+b i)+(c+d i)=(a+c)+(b+d) i$$

$$(a+b i)-(c+d i)=(a-c)+(b-d) i$$

$$(a+b i)(c+d i)=a c+b c i+a d i+i^{2} b d=(a c-b d)+(a d+b c) i$$

$$\frac{a+b i}{c+d i}=\frac{a+b i}{c+d i} \frac{c-d i}{c-d i}=\frac{a c-a d i+b c i-b d i^{2}}{c^{2}+d^{2}}=\frac{a c+b d+(b c-a d) i}{c^{2}+d^{2}} .$$

$$\left|z_{1} z_{2} z_{3} \cdots z_{n}\right|=\left|z_{1}\right|\left|z_{2}\right|\left|z_{3}\right| \cdots\left|z_{n}\right| \quad\left|z_{1} / z_{2}\right|=\left|z_{1}\right| /\left|z_{2}\right| \quad \text { if } \quad z_{2} \neq 0\left|z_{1}+z_{2}+z_{3}+\cdots+z_{n}\right| \leq \mid z_{1}$$

$$\left|z_{1}+z_{2}\right| \geq\left|z_{1}\right|-\left|z_{2}\right|$$

## 数学代写|matlab代写|FINDING ROOTS

$$z=r e^{i \varphi},$$

$$w=R e^{i \Phi}$$

$$w^{n}=R^{n} e^{i n \Phi}=r e^{i \varphi}=z .$$

$$R^{n}=r \quad \text { and } \quad n \Phi=\varphi+2 k \pi, \quad k=0, \pm 1, \pm 2, \ldots,$$

$$\Phi_{k}=\frac{\varphi}{n}+\frac{2 \pi k}{n}, \quad k=0, \pm 1, \pm 2, \ldots$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|matlab代写|CS1132

MATLAB是一个编程和数值计算平台，被数百万工程师和科学家用来分析数据、开发算法和创建模型。

statistics-lab™ 为您的留学生涯保驾护航 在代写matlab方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写matlab代写方面经验极为丰富，各种代写matlab相关的作业也就用不着说。

• 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

When the inner and outer walls of a body, for example the inner and outer walls of a house, are maintained at different constant temperatures, heat will flow from the warmer wall to the colder one. When each surface parallel to a wall has attained a constant temperature, the flow of heat has reached a steady state. In a steady-state flow of heat, each surface parallel to a wall, because its temperature is now constant, is referred to as an isothermal surface. Isothermal surfaces at different distances from an interior wall will have different temperatures. In many cases the temperature of an isothermal surface is only a function of its distance $x$ from the interior wall, and the rate of flow of heat $Q$ in a unit time across such a surface is proportional both to the area $A$ of the surface and to $d T / d x$, where $T$ is the temperature of the isothermal surface. Hence,
$$Q=-\kappa A \frac{d T}{d x},$$
where $\kappa$ is called the thermal conductivity of the material between the walls.
In place of a flat wall, let us consider a hollow cylinder whose inner and outer surfaces are located at $r=r_{1}$ and $r=r_{2}$, respectively. At steady state, Equation $1.2 .27$ becomes
$$Q_{r}=-\kappa A \frac{d T}{d r}=-\kappa(2 \pi r L) \frac{d T}{d r},$$
assuming no heat generation within the cylindrical wall.
We can find the temperature distribution inside the cylinder by solving Equation 1.2.28 along with the appropriate conditions on $T(r)$ at $r=r_{1}$ and $r=r_{2}$ (the boundary conditions). To illustrate the wide choice of possible boundary conditions, let us require that the inner surface is maintained at the temperature $T_{1}$. We assume that along the outer surface,heat is lost by convection to the environment, which has the temperature $T_{\infty}$. This heat loss is usually modeled by the equation
$$\left.\kappa \frac{d T}{d r}\right|{r=\mathrm{r}{2}}=-h\left(T-T_{\infty}\right),$$
where $h>0$ is the convective heat transfer coefficient. Upon integrating Equation $1.2 .28$,
$$T(r)=-\frac{Q_{r}}{2 \pi \kappa L} \ln (r)+C,$$
where $Q_{r}$ is also an unknown. Substituting Equation 1.2.30 into the boundary conditions, we obtain
$$T(r)=T_{1}+\frac{Q_{r}}{2 \pi \kappa L} \ln \left(r_{1} / r\right),$$
with
$$Q_{r}=\frac{2 \pi \kappa L\left(T_{1}-T_{\infty}\right)}{\kappa / r_{2}+h \ln \left(r_{2} / r_{1}\right)} .$$

## 数学代写|matlab代写|Logistic equation

The study of population dynamics yields an important class of first-order, nonlinear, ordinary differential equations: the logistic equation. This equation arose in Pierre François Verhulst’s (1804-1849) study of animal populations. ${ }^{3}$ If $x(t)$ denotes the number of species in the population and $k$ is the (constant) environment capacity (the number of species that can simultaneously live in the geographical region), then the logistic or Verhulst’s equation is
$$x^{\prime}=a x(k-x) / k,$$
where $a$ is the population growth rate for a small number of species.
To solve Equation 1.2.41, we rewrite it as
$$\frac{d x}{(1-x / k) x}=\frac{d x}{x}+\frac{x / k}{1-x / k} d x=r d t .$$
Integration yields
$$\ln |x|-\ln |1-x / k|=r t+\ln (C),$$
or
$$\frac{x}{1-x / k}=C e^{r t}$$
If $x(0)=x_{0}$,
$$x(t)=\frac{k x_{0}}{x_{0}+\left(k-x_{0}\right) e^{-r t}} .$$
As $t \rightarrow \infty, x(t) \rightarrow k$, the asymptotically stable solution.

## matlab代写

$$Q=-\kappa A \frac{d T}{d x},$$

$$Q_{r}=-\kappa A \frac{d T}{d r}=-\kappa(2 \pi r L) \frac{d T}{d r},$$

$$\kappa \frac{d T}{d r} \mid r=\mathrm{r} 2=-h\left(T-T_{\infty}\right),$$

$$T(r)=-\frac{Q_{r}}{2 \pi \kappa L} \ln (r)+C,$$

$$T(r)=T_{1}+\frac{Q_{r}}{2 \pi \kappa L} \ln \left(r_{1} / r\right),$$

$$Q_{r}=\frac{2 \pi \kappa L\left(T_{1}-T_{\infty}\right)}{\kappa / r_{2}+h \ln \left(r_{2} / r_{1}\right)} .$$

## 数学代写|matlab代写|Logistic equation

$$x^{\prime}=a x(k-x) / k,$$

$$\frac{d x}{(1-x / k) x}=\frac{d x}{x}+\frac{x / k}{1-x / k} d x=r d t$$

$$\ln |x|-\ln |1-x / k|=r t+\ln (C),$$

$$\frac{x}{1-x / k}=C e^{r t}$$

$$x(t)=\frac{k x_{0}}{x_{0}+\left(k-x_{0}\right) e^{-r t}} .$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|matlab代写|BMS13

MATLAB是一个编程和数值计算平台，被数百万工程师和科学家用来分析数据、开发算法和创建模型。

statistics-lab™ 为您的留学生涯保驾护航 在代写matlab方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写matlab代写方面经验极为丰富，各种代写matlab相关的作业也就用不着说。

• 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|matlab代写|Terminal velocity

As an object moves through a fluid, its viscosity resists the motion. Let us find the motion of a mass $m$ as it falls toward the earth under the force of gravity when the drag varies as the square of the velocity.
From Newton’s second law, the equation of motion is
$$m \frac{d v}{d t}=m g-C_{D} v^{2},$$
where $v$ denotes the velocity, $g$ is the gravitational acceleration, and $C_{D}$ is the drag coefficient. We choose the coordinate system so that a downward velocity is positive.

Equation 1.2.19 can be solved using the technique of separation of variables if we change from time $t$ as the independent variable to the distance traveled $x$ from the point of release. This modification yields the differential equation
$$m v \frac{d v}{d x}=m g-C_{D} v^{2},$$
since $v=d x / d t$. Separating the variables leads to
$$\frac{v d v}{1-k v^{2} / g}=g d x,$$
or
$$\ln \left(1-\frac{k v^{2}}{g}\right)=-2 k x,$$
where $k=C_{D} / m$ and $v=0$ for $x=0$. Taking the inverse of the natural logarithm, we finally obtain
$$v^{2}(x)=\frac{g}{k}\left(1-e^{-2 k x}\right) .$$
Thus, as the distance that the object falls increases, so does the velocity, and it eventually approaches a constant value $\sqrt{g / k}$, commonly known as the terminal velocity.

Because the drag coefficient $C_{D}$ varies with the superficial area of the object while the mass depends on the volume, $k$ increases as an object becomes smaller, resulting in a smaller terminal velocity. Consequently, although a human being of normal size will acquire a terminal velocity of approximately $120 \mathrm{mph}$, a mouse, on the other hand, can fall any distance without injury.

## 数学代写|matlab代写|Interest rate

Consider a bank account that has been set up to pay out a constant rate of $P$ dollars per year for the purchase of a car. This account has the special feature that it pays an annual interest rate of $r$ on the current balance. We would like to know the balance in the account at any time $t$.

Although financial transactions occur at regularly spaced intervals, an excellent approximation can be obtained by treating the amount in the account $x(t)$ as a continuous function of time governed by the equation
$$x(t+\Delta t) \approx x(t)+r x(t) \Delta t-P \Delta t,$$
where we have assumed that both the payment and interest are paid in time increments of $\Delta t$. As the time between payments tends to zero, we obtain the first-order ordinary differential equation
$$\frac{d x}{d t}=r x-P .$$
If we denote the initial deposit into this account by $x(0)$, then at any subsequent time
$$x(t)=x(0) e^{r t}-P\left(e^{r t}-1\right) / r .$$
Although we could compute $x(t)$ as a function of $P, r$, and $x(0)$, there are only three separate cases that merit our close attention. If $P / r>x(0)$, then the account will eventually equal zero at $r t=\ln {P /[P-r x(0)]}$. On the other hand, if $P / r<x(0)$, the amount of money in the account will grow without bound. Finally, the case $x(0)=P / r$ is the equilibrium case where the amount of money paid out balances the growth of money due to interest so that the account always has the balance of $P / r$.

## 数学代写|matlab代写|Terminal velocity

$$m \frac{d v}{d t}=m g-C_{D} v^{2},$$

$$m v \frac{d v}{d x}=m g-C_{D} v^{2},$$

$$\frac{v d v}{1-k v^{2} / g}=g d x$$

$$\ln \left(1-\frac{k v^{2}}{g}\right)=-2 k x$$

$$v^{2}(x)=\frac{g}{k}\left(1-e^{-2 k x}\right) .$$

## 数学代写|matlab代写|Interest rate

$$x(t+\Delta t) \approx x(t)+r x(t) \Delta t-P \Delta t$$

$$\frac{d x}{d t}=r x-P .$$

$$x(t)=x(0) e^{r t}-P\left(e^{r t}-1\right) / r .$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|matlab代写|CSC113

MATLAB是一个编程和数值计算平台，被数百万工程师和科学家用来分析数据、开发算法和创建模型。

statistics-lab™ 为您的留学生涯保驾护航 在代写matlab方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写matlab代写方面经验极为丰富，各种代写matlab相关的作业也就用不着说。

• 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|matlab代写|CLASSIFICATION OF DIFFERENTIAL EQUATIONS

Differential equations are classified three ways: by type, order, and linearity. There are two types: ordinary and partial differential equations, which have already been defined. Examples of ordinary differential equations include
$$\begin{gathered} \frac{d y}{d x}-2 y=x \ (x-y) d x+4 y d y=0 \ \frac{d u}{d x}+\frac{d v}{d x}=1+5 x \end{gathered}$$ and
$$\frac{d^{2} y}{d x^{2}}+2 \frac{d y}{d x}+y=\sin (x) .$$
On the other hand, examples of partial differential equations include
$$\begin{gathered} \frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}=0, \ y \frac{\partial u}{\partial x}+x \frac{\partial u}{\partial y}=2 u, \end{gathered}$$
and
$$\frac{\partial^{2} u}{\partial t^{2}}+2 \frac{\partial u}{\partial t}=\frac{\partial^{2} u}{\partial x^{2}} .$$
In the examples that we have just given, we have explicitly written out the differentiation operation. However, from calculus we know that $d y / d x$ can also be written $y^{\prime}$. Similarly, the partial differentiation operator $\partial^{4} u / \partial x^{2} \partial y^{2}$ is sometimes written $u_{x x y y}$. We will also use this notation from time to time.

## 数学代写|matlab代写|SEPARATION OF VARIABLES

The simplest method of solving a first-order ordinary differential equation, if it works, is separation of variables. It has the advantage of handling both linear and nonlinear problems, especially autonomous equations. ${ }^{1}$ From integral calculus, we already met this technique when we solved the first-order differential equation
$$\frac{d y}{d x}=f(x) \text {. }$$

By multiplying both sides of Equation $1.2 .1$ by $d x$, we obtain
$$d y=f(x) d x .$$
At this point we note that the left side of Equation 1.2.2 contains only $y$ while the right side is purely a function of $x$. Hence, we can integrate directly and find that
$$y=\int f(x) d x+C .$$
For this technique to work, we must be able to rewrite the differential equation so that all of the $y$ dependence appears on one side of the equation while the $x$ dependence is on the other. Finally, we must be able to carry out the integration on both sides of the equation.
One of the interesting aspects of our analysis is the appearance of the arbitrary constant $C$ in Equation 1.2.3. To evaluate this constant, we need more information. The most common method is to require that the dependent variable give a particular value for a particular value of $x$. Because the independent variable $x$ often denotes time, this condition is usually called an initial condition, even in cases when the independent variable is not time.

## 数学代写|matlab代写|CLASSIFICATION OF DIFFERENTIAL EQUATIONS

$$\frac{d y}{d x}-2 y=x(x-y) d x+4 y d y=0 \frac{d u}{d x}+\frac{d v}{d x}=1+5 x$$

$$\frac{d^{2} y}{d x^{2}}+2 \frac{d y}{d x}+y=\sin (x)$$

$$\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}=0, y \frac{\partial u}{\partial x}+x \frac{\partial u}{\partial y}=2 u$$

$$\frac{\partial^{2} u}{\partial t^{2}}+2 \frac{\partial u}{\partial t}=\frac{\partial^{2} u}{\partial x^{2}}$$

## 数学代写|matlab代写|SEPARATION OF VARIABLES

$$\frac{d y}{d x}=f(x) .$$

$$d y=f(x) d x .$$

$$y=\int f(x) d x+C$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|matlab代写|Reed-Solomon Codes

MATLAB是一个编程和数值计算平台，被数百万工程师和科学家用来分析数据、开发算法和创建模型。

statistics-lab™ 为您的留学生涯保驾护航 在代写matlab方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写matlab代写方面经验极为丰富，各种代写matlab相关的作业也就用不着说。

• 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|matlab代写|Construction

In this chapter, we will present a type of code called a Reed-Solomon code. Reed-Solomon codes, like BCH codes, have polynomial codewords, are linear, and can be constructed to be multiple-error correcting. However, ReedSolomon codes are significantly better than BCH codes in many situations because they are ideal for correcting error bursts. When a binary codeword is transmitted, the received vector is said to contain an error burst if it contains several bit errors very close together. In data transmitted through space, error bursts are frequently caused by very brief periods of intense solar energy. It was for this reason that a Reed-Solomon code was used in the Voyager 2 satellite when it transmitted photographs of several of the planets in our solar system back to Earth. We will briefly discuss the use of a Reed-Solomon code in the Voyager 2 satellite in Section 5.6. In addition, there are a variety of other reasons why errors in binary codewords often occur naturally in bursts, such as power surges in cable and telephone wires, various types of interference, and scratches on compact discs. As a result, Reed-Solomon codes have a rich assortment of applications, and are claimed to be the most frequently used digital error-correcting codes in the world. They are used extensively in the encoding of music and video on CDs, DVDs, and Blu-ray discs, have played an integral role in the development of high-speed supercomputers, and will be an important tool in the future for dealing with complex communication and information transfer systems.

To construct a Reed-Solomon code, we begin by choosing a primitive polynomial $p(x)$ of degree $n$ in $\mathbb{Z}{2}[x]$, and forming the field $F=\mathbb{Z}{2}[x] /(p(x))$ of order $2^{n}$. As we did in Chapter 4 , throughout this chapter we will denote the element $x$ in our finite fields by $a$. Like BCH codewords, Reed-Solomon codewords are then polynomials of degree less than $2^{n}-1$. However, unlike $\mathrm{BCH}$ codewords, which are elements in $\mathbb{Z}_{2}[x]$, Reed-Solomon codewords are elements in $F[x]$. To construct a $t$-error correcting Reed-Solomon code $C$, we use the generator polynomial $g(x)=(x-a)\left(x-a^{2}\right) \cdots\left(x-a^{2 t}\right)$ in $F[x]$. The codewords in $C$ are then all multiples of $g(x)$ in $F(x)$ of degree less than $2^{n}-1$. Theorem $4.2$ can easily be modified to show that $C$ will be $t$-error correcting. The codewords in $C$ have length $2^{n}-1$ positions, and form a vector space of dimension $2^{n}-1-2 t$. We will describe a Reed-Solomon code using the notation and parameters $R S\left(2^{n}-1, t\right)$ if the codewords in the code have length $2^{n}-1$ positions and the code is $t$-error correcting.

## 数学代写|matlab代写|Error Correction

We should begin by noting that the error correction method for BCH codes that we presented in Chapter 4 yields the same information when it is applied to a received Reed-Solomon polynomial as when it is applied to a received $\mathrm{BCH}$ polynomial. However, the $\mathrm{BCH}$ error correction method cannot generally be used to correct errors in a received Reed-Solomon polynomial. Recall that the last step in the BCH error correction method involves finding the roots of an error locator polynomial, which reveals the error positions in a received polynomial. Because there are only two possible coefficients for each term in a BCH polynomial, knowledge of the error positions alone is sufficient to correct the polynomial. The BCH error correction method can also be used to find the error positions in a received Reed-Solomon polynomial. However, because there is more than one possible coefficient for each term in a Reed-Solomon polynomial, knowledge of the error positions alone is not generally sufficient to correct the polynomial. The specific error present within each error position would also have to be determined.

Rather than combining the BCH error correction method for identifying error positions in received polynomials with a separate method for actually correcting errors, we will present an entirely new method for both identifying and correcting errors in Reed-Solomon polynomials. Before stating this new Reed-Solomon error correction method, we first note the following analogue to Theorem 4.1.

Theorem 5.1 Suppose that $F$ is a field of order $2^{n}$, and let $C$ be an $R S\left(2^{n}-1, t\right)$ code in $F[x]$. Then $c(x) \in F[x]$ of degree less than $2^{\mathrm{n}}-1$ is in $C$ if and only if $c\left(a^{i}\right)=0$ for $i=1,2, \ldots, 2 t$.

## 数学代写|matlab代写|Error Correction Method Proof

In this section, we will verify the Reed-Solomon error correction method that we summarized and illustrated in Section 5.2. ${ }^{2}$

Suppose $F$ is a field of order $2^{n}$, and let $C$ be an $R S\left(2^{n}-1, t\right)$ code in $F[x]$. If $c(x) \in C$ is transmitted and we receive the polynomial $r(x) \in F[x]$ of degree less than $2^{n}-1$, then $r(x)=c(x)+e(x)$ for some error polynomial $e(x)$ in $F[x]$ of degree less than $2^{n}-1$. We will denote this error polynomial by $e(x)=\sum_{j=0}^{m-1} e_{j} x^{j}$, with $m=2^{n}-1$ and $e_{j} \in F$. To determine $e(x)$, we begin by computing the first $2 t$ syndromes of $r(x)$, which we will denote as follows for $i=1,2, \ldots, 2 t$.
$$s_{i}=r\left(a^{i}\right)=e\left(a^{i}\right)=\sum_{j=0}^{m-1} e_{j} a^{i j}$$
Next, we use the preceding syndromes to form the syndrome polynomial $S(z)=\sum_{i=0}^{2 t-1} s_{i+1} z^{i}$. Note then that $S(z)$ can be expressed as follows.
$$S(z)=\sum_{i=0}^{2 t-1} \sum_{j=0}^{m-1} e_{j} a^{(i+1) j} z^{i}=\sum_{j=0}^{m-1} e_{j} a^{j} \sum_{i=0}^{2 t-1} a^{i j} z^{i}$$
Let $M$ be the set of integers that correspond to the error positions in $r(x)$. That is, let $M=\left{0 \leq j \leq m-1 \mid e_{j} \neq 0\right}$. Note also the following.
\begin{aligned} S(z) &=\sum_{j \in M} e_{j} a^{j} \sum_{i=0}^{2 t-1} a^{i j} z^{i} \ &=\sum_{j \in M} e_{j} a^{j}\left(\frac{1-a^{j(2 t)} z^{2 t}}{1-a^{j} z}\right) \ &=\sum_{j \in M} \frac{e_{j} a^{j}}{1-a^{j} z}-\sum_{j \in M} \frac{e_{j} a^{j(2 t+1)} z^{2 t}}{1-a^{j} z} \end{aligned}

## 数学代写|matlab代写|Error Correction Method Proof

s一世=r(一个一世)=和(一个一世)=∑j=0米−1和j一个一世j

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|matlab代写|BCH Codes

MATLAB是一个编程和数值计算平台，被数百万工程师和科学家用来分析数据、开发算法和创建模型。

statistics-lab™ 为您的留学生涯保驾护航 在代写matlab方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写matlab代写方面经验极为丰富，各种代写matlab相关的作业也就用不着说。

• 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|matlab代写|Construction

The most useful codes we presented in Chapter 3 were Hamming codes because they are linear and perfect. However, Hamming codes are not ideal if the occurrence of more than one bit error in a single codeword is likely. Since Hamming codes are only one-error correcting, if more than one bit error occurs during transmission of a Hamming codeword, the received vector will not be correctable to the codeword that was sent. Moreover, since Hamming codes are perfect, if more than one bit error occurs, the received vector will be uniquely correctable, but to the wrong codeword. In this chapter, we will present a type of code called a $B C H$ code that is linear and can be constructed to be multiple-error correcting. BCH codes are named for their creators, Bose, Chaudhuri, and Hocquenghem.

One way BCH codes differ from the codes we presented in Chapter 3 is that BCH codewords are polynomials rather than vectors. To construct a $\mathrm{BCH}$ code, we begin with the polynomial $f(x)=x^{m}-1 \in \mathbb{Z}{2}[x]$ for some positive integer $m$. Then $R=\mathbb{Z}{2}[x] /(f(x))$ is a ring that can be represented by all polynomials in $\mathbb{Z}{2}[x]$ of degree less than $m$. Suppose $g(x) \in \mathbb{Z}{2}[x]$ divides $f(x)$. Then the set $C$ of all multiples of $g(x)$ in $\mathbb{Z}{2}[x]$ of degree less than $m$ is a vector space in $R$ with dimension $m-\operatorname{deg}(g(x))$. Thus, the polynomials in $C$ are the codewords in an $[m, m-\operatorname{deg}(g(x))]$ linear code in $R$ with $2^{m-\operatorname{deg}(g(x))}$ codewords. The polynomial $g(x)$ is called a generator polynomial for the code, and we consider the codewords in the code to have length $m$ positions because we view each term in a polynomial codeword as a codeword position. A codeword $c(x) \in \mathbb{Z}{2}[x]$ with $m$ terms can then naturally be expressed as a unique vector in $\mathbb{Z}_{2}^{m}$ by listing the coefficients of $c(x)$ in order (including coefficients of zero) for increasing powers of $x$. In this chapter, we will assume $\mathrm{BCH}$ codewords are transmitted in this form.

## 数学代写|matlab代写|Error Correction

As we mentioned in Section 4.1, the generator polynomial for a BCH code is chosen in a special way because of how it allows errors to be corrected in the resulting code. In this section, we will present the BCH code error correction method. Before doing so, we first note the following theorem.
Theorem 4.1 Suppose $p(x) \in \mathbb{Z}{2}[x]$ is a primitive polynomial of degree $n$, and let $C$ be the $B C H$ code that results from the first $s$ powers of $a=x$ in the finite field $\mathbb{Z}{2}[x] /(p(x))$. Then $c(x) \in \mathbb{Z}_{2}[x]$ of degree less than $2^{\mathrm{n}}-1$ is in $C$ if and only if $c\left(a^{i}\right)=0$ for $i=1,2, \ldots, s$.

Proof. Let $m_{i}(x)$ be the minimum polynomial of $a^{i}$ in $\mathbb{Z}{2}[x]$ for every $i=1,2, \ldots, s$, and let $g(x)$ be the least common multiple in $\mathbb{Z}{2}[x]$ of the $m_{i}(x)$ for $i=1,2, \ldots, s$. If $c(x) \in C$, then $c(x)=g(x) \cdot h(x)$ for some $h(x) \in \mathbb{Z}{2}[x]$. Thus, $c\left(a^{i}\right)=g\left(a^{i}\right) \cdot h\left(a^{i}\right)=0 \cdot h\left(a^{i}\right)=0$ for $i=1,2, \ldots, s$. Conversely, if $c\left(a^{i}\right)=0$ for $i=1,2, \ldots, s$, then $m{i}(x)$ divides $c(x)$ for $i=1,2, \ldots, s$. Thus, $g(x)$ divides $c(x)$, and $c(x) \in C$.

We will now outline the $\mathrm{BCH}$ error correction method. Let $p(x) \in \mathbb{Z}_{2}[x]$ be a primitive polynomial of degree $n$, and let $C$ be the $\mathrm{BCH}$ code that

results from the first $2 t$ powers of $a=x$ in the finite field $\mathbb{Z}{2}[x] /(p(x))$. We will show in Theorem $4.2$ that $C$ is then $t$-error correcting. Suppose $c(x) \in C$ is transmitted, and we receive the polynomial $r(x) \in \mathbb{Z}{2}[x]$ of degree less than $2^{n}-1$. Then $r(x)=c(x)+e(x)$ for some error polynomial $e(x)$ in $\mathbb{Z}_{2}[x]$ of degree less than $2^{\mathrm{n}}-1$ that contains exactly and only the terms in which $r(x)$ and $c(x)$ differ. To correct $r(x)$, we must only determine $e(x)$, for we could then compute $c(x)=r(x)+e(x)$. However, Theorem $4.1$ implies $r\left(a^{i}\right)=e\left(a^{i}\right)$ for $i=1,2, \ldots, 2 t$. Thus, by knowing $r(x)$, we also know some information about $e(x)$. We will call the values of $r\left(a^{i}\right)$ for $i=1,2, \ldots, 2 t$ the syndromes of $r(x)$.

Suppose $e(x)=x^{m_{1}}+x^{m_{2}}+\cdots+x^{m_{p}}$ for some integer error positions $m_{1}<m_{2}<\cdots<m_{p}$ with $p \leq t$ and $m_{p}<2^{n}-1$. To correct $r(x)$, we must only find the error positions $m_{1}, m_{2}, \ldots, m_{p}$. To do this, we begin by computing the syndromes of $r(x)$, which we will denote by $s_{1}=r(a)$, $s_{2}=r\left(a^{2}\right), \ldots, s_{2 t}=r\left(a^{2 t}\right)$. Next, we introduce the following error locator polynomial $E(z)$, called so because its roots (unknown at this point) reveal the error positions in $r(x)$.

## 数学代写|matlab代写|Construction

Because some of the functions that we will use are in the Maple LinearAlgebra package, we will begin by including this package. In addition, we will enter the following interface command to cause Maple to display all matrices of size $200 \times 200$ and smaller throughout the remainder of this Maple session.

with(LinearAlgebra):
interface $($ rtablesize $=200)$ :
We will now define the primitive polynomial $p(x)=x^{4}+x+1 \in \mathbb{Z}_{2}[x]$ used to construct the code.
$>\mathrm{p}:=\mathrm{x}->\mathrm{x}^{\sim} 4+\mathrm{x}+1:$ $>\operatorname{Primitive}(\mathrm{p}(\mathrm{x})) \bmod 2 ;$
Next, we will use the Maple degree function to assign the number of elements in the underlying finite field as the variable $f s$, and use the Maple Vector function to create a vector in which to store the field elements.
We can then use the following commands to generate and store the field elements in the vector field. Since for BCH codes we denote the field element $x$ by $a$, we use the parameters $a$ and $p(a)$ in the following Powmod command.
$>$ for i from 1 to fs-1 do
$>\quad f i e l d[i]:=\operatorname{Powmod}(a, i, p(a), a)$ mod 2 :
$>$ od:

$$\text { field[fs] :=0: }$$
We can view the entries in the vector field by entering the following command.

## 数学代写|matlab代写|Construction

BCH 码与我们在第 3 章中介绍的码的一个不同之处在于 BCH 码字是多项式而不是向量。构建一个乙CH代码，我们从多项式开始F(X)=X米−1∈从2[X]对于一些正整数米. 然后R=从2[X]/(F(X))是一个可以用所有多项式表示的环从2[X]学位小于米. 认为G(X)∈从2[X]划分F(X). 然后是集C的所有倍数G(X)在从2[X]学位小于米是向量空间R有尺寸米−你⁡(G(X)). 因此，多项式在C是[米,米−你⁡(G(X))]线性码R和2米−你⁡(G(X))码字。多项式G(X)被称为代码的生成多项式，我们认为代码中的代码字具有长度米位置，因为我们将多项式码字中的每个项视为码字位置。一个码字C(X)∈从2[X]和米然后可以自然地将术语表示为唯一的向量从2米通过列出的系数C(X)为了（包括零系数）增加幂X. 在本章中，我们将假设乙CH码字以这种形式传输。

## 数学代写|matlab代写|Construction

with(LinearAlgebra):

>p:=X−>X∼4+X+1: >原始⁡(p(X))反对2;

>对于 i 从 1 到 fs-1 做
>F一世和ld[一世]:=战俘⁡(一个,一世,p(一个),一个)模式 2：
>从：

$$\text { field[fs] :=0: }$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|matlab代写|Hamming Codes with Maple

MATLAB是一个编程和数值计算平台，被数百万工程师和科学家用来分析数据、开发算法和创建模型。

statistics-lab™ 为您的留学生涯保驾护航 在代写matlab方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写matlab代写方面经验极为丰富，各种代写matlab相关的作业也就用不着说。

• 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|matlab代写|Hamming Codes with Maple

In this section, we will show how Maple can be used to construct and correct errors in Hamming codes. We will consider the $[15,11]$ Hamming code.
Because some of the functions that we will use are in the Maple ListTools, LinearAlgebra, and Modular packages, we will begin by including these packages. In addition, we will enter the following interface command to cause Maple to display all matrices of size $50 \times 50$ and smaller throughout the remainder of this Maple session.

with(ListTools):
$>$ with(LinearAl gebra):
$>$ with (Modular):
$>$ interface $(r$ tablesize=50) :
Next we will construct the parity check matrix $H$ for the code. We first enter the length $m=4$ of the vectors that form the columns of $H$.
$>m:=4:$
Recall that the columns of $H$ are binary expressions of the integers $1,2, \ldots, 2^{m}-1$. We can obtain the binary expression of an integer in Maple by using the Maple convert function. For example, we can obtain the binary expression of the integer 4 by entering the following command.
$>c b:=$ convert $(4$, base 2$)$;
$c b:=[0,0,1]$
The entries in the preceding result for $c b$ are the coefficients in the expression $0 \cdot\left(2^{0}\right)+0 \cdot\left(2^{1}\right)+1 \cdot\left(2^{2}\right)$ of the integer 4 . Note that $c b$ contains only three positions, whereas for the columns of $H$ we want binary vectors of length $m=4$ positions. That is, to be placed as the fourth column of $H$, we would want the number 4 to be converted to binary vector $[0,0,1,0]$. Furthermore, the binary digits in $c b$ are the reverse of how they should be expressed in the fourth column of $H$. To be directly placed as the fourth column of $H$, the number 4 should be converted to the binary vector $[0,1,0,0]$. We can use the following command to take care of these two problems.

## 数学代写|matlab代写|Hamming Codes with MATLAB

In this section, we will show how MATLAB can be used to construct and correct errors in Hamming codes. We will consider the [15, 11] Hamming code.

We will begin by constructing the parity check matrix $H$ for the code. We first enter the length $m=4$ of the vectors that form the columns of $H$.
$$\mathrm{m}=4$$
Recall that the columns of $H$ are binary expressions of the integers $1,2, \ldots, 2^{m}-1$. We can obtain the binary expression of an integer in MATLAB by using the MATLAB dec2bin function. For example, we can obtain the binary expression of the integer 4 by entering the following command.
The first parameter in the preceding command is the normal base 10 expression of an integer, while the second parameter is the number of bits that are to be used in the binary expression of the integer. The output displayed for $c b$ is a string containing the coefficients in the expression $4=0 \cdot\left(2^{3}\right)+1 \cdot\left(2^{2}\right)+0 \cdot\left(2^{1}\right)+0 \cdot\left(2^{0}\right)$ as characters. However, rather than having these coefficients stored as characters in a string, we want these coefficients to be stored as positions in a vector. We can take care of this by entering the following for loop.
$>$ for $i=1:$ length $(\mathrm{cb})$
$b v(i)=\operatorname{str} 2 \operatorname{num}(c b(i))$;
end
The MATLAB length function is designed to count the number of characters in a string. Thus, the preceding for loop takes each of the binary digits in the string $c b$, converts them from characters into numbers using the MATLAB str2num function, and stores the resulting integers as positions in the vector $b v$. To see the contents of $b v$, we can enter the following command.
$\gg \mathrm{bv}$
$$b v=$$

## 数学代写|matlab代写|Computer Exercises

1. Use a Hadamard matrix to construct the codewords in a $(31,16)$ code with 32 codewords. What is the maximum number of bit errors that are guaranteed to be uniquely correctable in this code? Correct the vector (0011110000101100001011000011110) to a codeword in this code.
2. As we mentioned in Section $3.3$, the $(32,16)$ Reed-Muller code was used in the Mariner 9 space probe when it transmitted photographs of Mars back to Earth.
(a) Construct the codewords in the $(32,16)$ Reed-Muller code. What is the maximum number of bit errors that are guaranteed to be uniquely correctable in this code?
(b) Correct the vector (11100101011010011110101101101001) to a codeword in the $(32,16)$ Reed-Muller code.
3. Find a parity check matrix for the code in Example 3.4.Find a parity check matrix for the code for which you constructed a generator matrix in Exercise 7 .
4. Let $C$ be the $[31,26]$ Hamming code.
5. (a) Construct the parity check matrix $H$ and a generator matrix $G$ for $C$.
6. (b) Find the number of codewords in $C$.
7. (c) Construct the codeword $w G$ in $C$ that results from the vector $\mathbf{w}=(10110101110110111110111000)$
8. (d) Correct the vector (1101011100110110110101011110111) to a codeword in $C$.
9. (e) Correct the vector (1101010110111010110011000011101) to a codeword in $C$.
10. Let $C$ be the $[63,57]$ Hamming code.
11. (a) Construct the parity check matrix and a generator matrix for $C$.
12. (b) Find the number of codewords in $C$.
13. (c) Construct two of the codewords in $C$.
14. Consider the Maple command on page 83 in which we used the convert function to convert the syndrome of $r$ from the binary expression of an integer into the normal base 10 expression of the integer, thereby revealing the position in $r$ that contained an error. Recall that we could have also identified the position in $r$ that contained an error by finding the number of the column in $H$ that matched the syndrome of $r$. Write a routine or sequence of commands in Maple to replace the convert command on page 83 in which you find the position in $r$ that contains an error by finding the number of the column in $H$ that matches the syndrome of $r$.
15. Consider the MATLAB command on page 88 in which we used the bin2dec function to convert the syndrome of $r$ from the binary expression of an integer into the normal base 10 expression of the integer, thereby revealing the position in $r$ that contained an error. Recall that we could have also identified the position in $r$ that contained an error by finding the number of the column in $H$ that matched the syndrome of $r$. Write a routine or sequence of commands in MATLAB to replace the bin2dec command on page 88 in which you find the position in $r$ that contains an error by finding the number of the column in $H$ that matches the syndrome of $r$.

## 数学代写|matlab代写|Hamming Codes with Maple

>与（线性代数）：
>与（模块化）：
>界面(rtablesize=50) ：

>米:=4:

>Cb:=兑换(4, 基数 2);
Cb:=[0,0,1]

## 数学代写|matlab代写|Hamming Codes with MATLAB

$$\数学{m} = 4 R和C一个ll吨H一个吨吨H和C这l在米ns这FH一个r和b一世n一个r是和Xpr和ss一世这ns这F吨H和一世n吨和G和rs1,2,…,2米−1.在和C一个n这b吨一个一世n吨H和b一世n一个r是和Xpr和ss一世这n这F一个n一世n吨和G和r一世n米一个吨大号一个乙b是在s一世nG吨H和米一个吨大号一个乙d和C2b一世nF在nC吨一世这n.F这r和X一个米pl和,在和C一个n这b吨一个一世n吨H和b一世n一个r是和Xpr和ss一世这n这F吨H和一世n吨和G和r4b是和n吨和r一世nG吨H和F这ll这在一世nGC这米米一个nd.吨H和F一世rs吨p一个r一个米和吨和r一世n吨H和pr和C和d一世nGC这米米一个nd一世s吨H和n这r米一个lb一个s和10和Xpr和ss一世这n这F一个n一世n吨和G和r,在H一世l和吨H和s和C这ndp一个r一个米和吨和r一世s吨H和n在米b和r这Fb一世吨s吨H一个吨一个r和吨这b和在s和d一世n吨H和b一世n一个r是和Xpr和ss一世这n这F吨H和一世n吨和G和r.吨H和这在吨p在吨d一世spl一个是和dF这rCb一世s一个s吨r一世nGC这n吨一个一世n一世nG吨H和C这和FF一世C一世和n吨s一世n吨H和和Xpr和ss一世这n4=0⋅(23)+1⋅(22)+0⋅(21)+0⋅(20)一个sCH一个r一个C吨和rs.H这在和在和r,r一个吨H和r吨H一个nH一个在一世nG吨H和s和C这和FF一世C一世和n吨ss吨这r和d一个sCH一个r一个C吨和rs一世n一个s吨r一世nG,在和在一个n吨吨H和s和C这和FF一世C一世和n吨s吨这b和s吨这r和d一个sp这s一世吨一世这ns一世n一个在和C吨这r.在和C一个n吨一个ķ和C一个r和这F吨H一世sb是和n吨和r一世nG吨H和F这ll这在一世nGF这rl这这p.>F这r一世=1:l和nG吨H(Cb)$$b在(一世)=字符串⁡2在一个⁡(Cb(一世))$;和nd吨H和米一个吨大号一个乙l和nG吨HF在nC吨一世这n一世sd和s一世Gn和d吨这C这在n吨吨H和n在米b和r这FCH一个r一个C吨和rs一世n一个s吨r一世nG.吨H在s,吨H和pr和C和d一世nGF这rl这这p吨一个ķ和s和一个CH这F吨H和b一世n一个r是d一世G一世吨s一世n吨H和s吨r一世nG$Cb$,C这n在和r吨s吨H和米Fr这米CH一个r一个C吨和rs一世n吨这n在米b和rs在s一世nG吨H和米一个吨大号一个乙s吨r2n在米F在nC吨一世这n,一个nds吨这r和s吨H和r和s在l吨一世nG一世n吨和G和rs一个sp这s一世吨一世这ns一世n吨H和在和C吨这r$b在$.吨这s和和吨H和C这n吨和n吨s这F$b在$,在和C一个n和n吨和r吨H和F这ll这在一世nGC这米米一个nd.$≫b在$bv=$\$

## 数学代写|matlab代写|Computer Exercises

1. 使用 Hadamard 矩阵构造码字(31,16)32 个码字的代码。在此代码中保证唯一可纠正的最大误码数是多少？将向量 (0011110000101100001011000011110) 更正为此代码中的一个码字。
2. 正如我们在章节中提到的3.3， 这(32,16)水手 9 号太空探测器将火星照片传回地球时使用了 Reed-Muller 代码。
(a) 在(32,16)里德-穆勒码。在此代码中保证唯一可纠正的最大误码数是多少？
(b) 将向量 (11100101011010011110101101101001) 修正为(32,16)里德-穆勒码。
3. 为示例 3.4 中的代码找到一个奇偶校验矩阵。为您在练习 7 中构造生成器矩阵的代码找到一个奇偶校验矩阵。
4. 让C成为[31,26]汉明码。
5. (a) 构造奇偶校验矩阵H和一个生成矩阵G为了C.
6. (b) 找出码字的个数C.
7. (c) 构造码字在G在C由向量产生的在=(10110101110110111110111000)
8. (d) 将向量 (1101011100110110110101011110111) 修正为C.
9. (e) 将向量 (1101010110111010110011000011101) 修正为C.
10. 让C成为[63,57]汉明码。
11. (a) 构造奇偶校验矩阵和生成矩阵C.
12. (b) 找出码字的个数C.
13. (c) 构造其中的两个码字C.
14. 考虑第 83 页的 Maple 命令，其中我们使用了 convert 函数来转换r从整数的二进制表达式转换为整数的正常基数 10 表达式，从而揭示在r包含一个错误。回想一下，我们也可以确定在r通过查找中的列号包含错误H符合的综合症r. 在 Maple 中编写一个例程或命令序列来替换第 83 页上的转换命令，您可以在其中找到位置r通过查找中的列号包含错误H匹配的综合症r.
15. 考虑第 88 页上的 MATLAB 命令，其中我们使用 bin2dec 函数来转换r从整数的二进制表达式转换为整数的正常基数 10 表达式，从而揭示在r包含一个错误。回想一下，我们也可以确定在r通过查找中的列号包含错误H符合的综合症r. 在 MATLAB 中编写一个例程或命令序列来替换第 88 页上的 bin2dec 命令，您可以在其中找到位置r通过查找中的列号包含错误H匹配的综合症r.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。