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统计代写|时间序列分析代写Time-Series Analysis代考|DATA DESCRIPTION

The data set used in this paper has been taken from covid-tracker source (COVID19 India Org Data Operations Group, 2020). The data set is tallied with the official government data source and is highly reliable. However, the escalation and fall in cases depends upon many factors. For example, person-to-person spreads were subsequently increased due to recent freedom of movement in metropolitan cities. The line plots in Figure 8.4 are based on the daily figures of confirmed and recovered cases in the entire Indian subcontinent, starting from the first reported case on February 3, 2020 to the tally as on October 20, 2020. For the training and evaluation of the forecasting models, the data has been split into training (from March 14, 2020 to October 20, 2020) and testing (from October 21, 2020 to November 4, 2020) sets. From Figure 8.4, it is evident that the virus spread exponentially in the months leading up to September 2020, upon which it hit a certain peak and a steep downfall is visible. On November 4,2020 , the recorded cases were $8,363,329$, recovered cases were $7,710,463$ and deaths were 123,765 . However, the number of cases is expected to increase in the coming days due to the complete relaxation of lockdown norms and extensive inter-state movement due to the upcoming festival.

统计代写|时间序列分析代写Time-Series Analysis代考|SeVERITY ANALYSIS

Figure 8.6 gives the severity analysis of the confirmed cases in all states and union territories of India. The geographical map is color-coded to show the comparative distribution of cases from the most severely affected to the least affected state. The five worst-affected states-Maharashtra with $19.6 \%$, Karnataka with $9.69 \%$, Kerala with 5.99\%, Andhra Pradesh with $9.60 \%$ and Tamil Nadu with $8.55 \%$ of total Indian cases-are on the highly affected zone of the spectrum of the heat map. These five states are together responsible for a majority of total cases reported in the entire nation. In contrast, states like Himachal Pradesh and the northeastern states of Sikkim, Meghalaya, Tripura, etc., have a negligible share of COVID-19 cases and are among the least-affected regions.

For the experiments, we used the Jupyter notebook environment with Python version 3.7. The computational specifications include an Intel Core i5 eighth generation processor with 8 GB RAM and a 64-bit Windows 10 operating system. Since none of the experiments involved graphical processing units (GPUs) for training deep learning models, the details for the same have been omitted. The data processing and model design is done with the help of open-source libraries like Numpy (Oliphant, 2006), Pandas (McKinney, 2010), Scikit-Learn (Pedregosa et al., 2011) and PyTorch (Paszke et al., 2017). All the deep learning models (RNN, LSTM, GRU) are manually designed using functions and classes of the PyTorch framework, and the machine learning models (SVR, PR, VAR) are implemented using the Scikit-Learn and StatsModel libraries.

Except for $\mathrm{PR}$, all the other forecasting models are trained on a univariate data set with the end goal of predicting daily confirmed cases for the next 15 days. All predictive models are trained and evaluated on the data of the ten states individually and their predictions, are compared based on the following performance metrics:
$$M A E=\frac{1}{n} * \sum_{k=1}^n\left|Y_k-\hat{Y}_k\right|$$

$$\begin{gathered} \text { RMSLE }=\sqrt{\frac{1}{2} \sum_{k=1}^n\left(\log Y_k-\log \hat{Y}k\right)^2} \ \text { MAPE }=\frac{100}{n} * \sum{k=1}^n\left|\frac{Y_k-\hat{Y}_k}{Y_k}\right| \ E V=1-\frac{\operatorname{Var}(Y-\hat{Y})}{\operatorname{Var}(Y)} \end{gathered}$$
where $Y$ is the actual value, $\hat{Y}$ the predicted value and $n$ denotes the total number of instances.

统计代写|时间序列分析代写Time-Series Analysis代考|SeVERITY ANALYSIS

$$\begin{gathered} M A E=\frac{1}{n} * \sum_{k=1}^n\left|Y_k-\hat{Y}k\right| \ \operatorname{RMSLE}=\sqrt{\frac{1}{2} \sum{k=1}^n\left(\log Y_k-\log \hat{Y} k\right)^2} \mathrm{MAPE}=\frac{100}{n} * \sum k=1^n\left|\frac{Y_k-\hat{Y}_k}{Y_k}\right| E V=1-\frac{\operatorname{Var}(Y-\hat{Y})}{\operatorname{Var}(Y)} \end{gathered}$$

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。

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统计代写|统计推断代考Statistical Inference代写|Multivariate case

The generalisation to the $n$-dimensional multivariate case does not introduce any new ideas. Suppose that $X_1, \ldots, X_n$ are jointly continuous random variables with cumulative distribution function $F_{X_1, \ldots, X_n}$. The joint density is the function $f_{X_1, \ldots, X_n}$ satisfying
$$F_{X_1, \ldots, X_n}\left(x_1, \ldots, x_n\right)=\int_{-\infty}^{x_n} \ldots \int_{-\infty}^{x_1} f_{X_1, \ldots, X_n}\left(u_1, \ldots, u_n\right) d u_1 \ldots d u_n$$
In order to generate the marginal density of $X_j$, we integrate the joint density with respect to all the other variables,
$$f_{X_j}\left(x_j\right)=\int_{-\infty}^{\infty} \ldots \int_{-\infty}^{\infty} f_{X_1, \ldots, X_n}\left(x_1, \ldots, x_n\right) d x_1 \ldots d x_{j-1} d x_{j+1} \ldots d x_n$$

统计代写|统计推断代考Statistical Inference代写|Expectation and joint moments

We often encounter situations where we are interested in a function of several random variables. Consider the following illustration: let $X_1, \ldots, X_5$ represent our models for the total rainfall in December at five locations around the UK. Functions that may be of interest include:

• the mean across locations, $\frac{1}{5} \sum_{i=1}^5 X_i$.
• the maximum across locations, $\max _i\left(X_i\right)$.
• the mean of the four rainiest locations, $\frac{1}{4}\left[\sum_{i=1}^5 X_i-\min _i\left(X_i\right)\right]$.
As a function of random variables, each of these is itself a random variable. In the situations that we will consider, if $X_1, \ldots, X_n$ are random variables and $g: \mathbb{R}^n \rightarrow \mathbb{R}$ is a function of $n$ variables, then $g\left(X_1, \ldots, X_n\right)$ is also a random variable. In many instances, the distribution of $g\left(X_1, \ldots, X_n\right)$ is of interest; this topic is tackled in section 4.6. We start with something more straightforward: calculation of the mean.

统计推断代写

统计代写|统计推断代考Statistical Inference代写|Multivariate case

$$F_{X_1, \ldots, X_n}\left(x_1, \ldots, x_n\right)=\int_{-\infty}^{x_n} \ldots \int_{-\infty}^{x_1} f_{X_{1, \ldots, X_n}}\left(u_1, \ldots, u_n\right) d u_1 \ldots d u_n$$

$$f_{X_j}\left(x_j\right)=\int_{-\infty}^{\infty} \ldots \int_{-\infty}^{\infty} f_{X_1, \ldots, X_n}\left(x_1, \ldots, x_n\right) d x_1 \ldots d x_{j-1} d x_{j+1} \ldots d x_n$$

统计代写|统计推断代考Statistical Inference代写|Expectation and joint moments

• 跨位畐的平圽值, $\frac{1}{5} \sum_{i=1}^5 X_i$.
• 跨位置的最大值， $\max _i\left(X_i\right)$.
• 四个最多雨地朢的平均值, $\frac{1}{4}\left[\sum_{i=1}^5 X_i-\min _i\left(X_i\right)\right]$. 更直接䪨事情开始: 计算均值。

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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。

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数学代写数值分析代写Numerical analysis代考|Pivoting

GE breaks down at Step $i$ if the ith diagonal entry of the current (modified) coefficient matrix, referred to as the pivot, is zero (or close to 0 ), since there is no way to eliminate a nonzero entry using a zero pivot. A zero pivot may arise at any step of GE, making the algorithm fail, even if $\mathbf{A}$ is nonsingular and a unique solution to $\mathbf{A x}=\mathbf{b}$ exists. Consider, for example, the linear equation
$$\left(\begin{array}{ll} 0 & 1 \ 1 & 0 \end{array}\right)\left(\begin{array}{l} x_1 \ x_2 \end{array}\right)=\left(\begin{array}{l} 0 \ 2 \end{array}\right) .$$
There is a unique solution $\left(x_1=2\right.$ and $\left.x_2=0\right)$, but GE fails at the first step.

数学代写|数值分析代写Numerical analysis代考|Work in LU/GE

How many floating point operations (flops) are needed to perform PLU? At Step $i$, we need to compare $n-i+1$ entries $a_{i i}, \ldots, a_{n i}$ and choose the largest one in modulus. After pivoting, we need $n-i$ divisions to get the multiple factors used for row operations. To eliminate each subdiagonal $(j, i)$ entry $(i<j \leq n)$ in the $i$ th column, it takes $n-i$ scalar multiplications and $n-i$ additions to subtract a multiple of the new $i$ th row from the $j$ th row. In total, there will be $\sum_{i=1}^{n-1}(n-i+1)=\frac{1}{2}(n+2)(n-1) \approx \frac{n^2}{2}$ scalar comparisons, $\sum_{i=1}^{n-1}[(n-i)+(n-i)(n-i)]=\frac{1}{3}\left(n^3-n\right) \approx \frac{n^3}{3}$ multiplication/divisions, and $\sum_{i=1}^{n-1}(n-i)(n-i)=\frac{1}{6} n(n-1)(2 n-1) \approx \frac{1}{3} n^3$ addition/subtractions. If multiplications and additions take about the same time (which is the case nowadays on many platforms), we can combine them and simply say that the arithmetic cost for LU factorization is $\frac{2}{3} n^3$ flops plus $\frac{1}{2} n^2$ comparisons.

The above estimates assume that every entry of $\mathbf{A}$ is nonzero. If $\mathbf{A}$ has certain special nonzero structures, the estimate may become smaller. For instance, if all entries of $\mathbf{A}$ below the $k$ th subdiagonal are zero, where $k$ is independent of $n$, then the total arithmetic cost is at most $\mathcal{O}\left(k n^2\right)$. This is because at each step of the factorization, at most $k-1$ entries need to be eliminated, and hence at most $k-1$ rows will be updated. In addition, for such a matrix $\mathbf{A}$, if all entries above the $k$ th superdiagonal are zero, then the total work is at most $\mathcal{O}\left(k^2 n\right)$.

Finally, assume that we have completed the LU factorization and have $\mathbf{P}, \mathbf{L}$, and $\mathbf{U}$ such that $\mathbf{P A}=\mathbf{L U}$. To solve the linear system $\mathbf{A x}=\mathbf{b}$, note that it is equivalent to $\mathbf{L U x}=\mathbf{P A x}=\mathbf{P b}$, which leads to $\mathbf{x}=\mathbf{U}^{-1} \mathbf{L}^{-1} \mathbf{P b}$. This can be evaluated by (a) solving the lower triangular system $\mathbf{L y}=\mathbf{P b}$ by forward substitution, then (b) solving the upper triangular system $\mathbf{U x}=\mathbf{y}$ by back substitution. Note that if one needs to solve many linear systems $\mathbf{A x}_k=\mathbf{b}_k(k=1,2, \ldots)$ with the same matrix and different right-hand sides $\mathbf{b}_1, \mathbf{b}_2, \ldots$, then at the first step one does $O\left(n^3\right)$ work to create the LU factorization, but then reuses $\mathbf{L}$ and $\mathbf{U}$ so that each additional solve needs only $O\left(n^2\right)$ work.

数学代写数值分析代写Numerical analysis代考|Pivoting

GE 在 Step 崩溃 $i$ 如果当前（修改后的）系数矩阵的第 $\mathrm{i}$ 个对角线条目（称为主元）为零（或接近 0 ），因为无法使用零主元消除 非零条目。零主元可能出现在 $G E$ 的任何一步，使算法失败，即使 $\mathbf{A}$ 是非奇异的并且是唯一的解决方室 $\mathbf{A} \mathbf{x}=\mathbf{b}$ 存在。例如，考 虞线性方程

数学代写数值分析代写Numerical analysis代考|Work in LU/GE

$\sum_{i=1}^{n-1}[(n-i)+(n-i)(n-i)]=\frac{1}{3}\left(n^3-n\right) \approx \frac{n^3}{3}$ 乘法/除法，和
$\sum_{i=1}^{n-1}(n-i)(n-i)=\frac{1}{6} n(n-1)(2 n-1) \approx \frac{1}{3} n^3$ 加法/减法。如果兆法和加法花费的时间大致相同（如今在许㶴平台上 都是这种情况），我们可以将它们结合起来，简单地说 LU 分解的算术成本是 $\frac{2}{3} n^3$ 人字拖加 $\frac{1}{2} n^2$ 比较。 角线为零，其中 $k$ 独立于 $n$ ，那么总的算术成本至多为 $\mathcal{O}\left(k n^2\right)$. 这是因为在分解的每一步，至多 $k-1$ 需要删除条目，因此最多 $k-1$ 行将被更新。另外，对于这样的矩阵 $\mathbf{A}$ ，如果上面的所有条目 $k$ 第 th 超对角线为零，则总功最 $\boldsymbol{\beta} \mathcal{O}\left(k^2 n\right)$.

$\mathbf{L U x}=\mathbf{P A x}=\mathbf{P b}$ ，这导致 $\mathbf{x}=\mathbf{U}^{-1} \mathbf{L}^{-1} \mathbf{P b}$. 这可以通过 ( $a$ ) 求解下三角系统来评估 $\mathbf{L} \mathbf{y}=\mathbf{P b}$ 通过正向代换，然后 (b) 求解上三角系统 $\mathbf{U x}=\mathbf{y}$ 通过反向拍换。请注意，如果需要求解多个线性系统 $\mathbf{A} \mathbf{x}_k=\mathbf{b}_k(k=1,2, \ldots)$ 具有相同的矩阵和不同 的右手边 $\mathbf{b}_1, \mathbf{b}_2, \ldots$,然后在第一步做 $O\left(n^3\right)$ 努力创建 LU 分解，然后重复使用 $\mathbf{L}$ 和U这样每个额外的解决方客只需要 $O\left(n^2\right) 工$ 作。

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。

Posted on Categories:LU分解代写, Numerical analysis, 多项式插值方法代写, 数值分析, 数值积分代写, 数学代写, 最小二乘法代写

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数学代写数值分析代写Numerical analysis代考|Error estimation for Gauss quadrature

The next theorem provides a bound on the error that has been committed by approximating the integral on the left-hand side of (10.6) by the quadrature rule on the right.

Theorem 10.1 Suppose that $w$ is a weight function, defined, integrable, continuous and positive on $(a, b)$, and that $f$ is defined and continuous on $[a, b]$; suppose further that $f$ has a continuous derivative of order $2 n+2$ on $[a, b], n \geq 0$. Then, there exists a number $\eta$ in $(a, b)$ such that
$$\int_a^b w(x) f(x) \mathrm{d} x-\sum_{k=0}^n W_k f\left(x_k\right)=K_n f^{(2 n+2)}(\eta),$$
and
$$K_n=\frac{1}{(2 n+2) !} \int_a^b w(x)\left[\pi_{n+1}(x)\right]^2 \mathrm{~d} x .$$

数学代写|数值分析代写Numerical analysis代考|Composite Gauss formulae

It is often useful to define composite Gauss formulae, just as we did for the trapezium rule and Simpson’s rule in Section 7.5. Let us suppose, for the sake of simplicity, that $w(x) \equiv 1$. We divide the range $[a, b]$ into $m$ subintervals $\left[x_{j-1}, x_j\right], j=1,2, \ldots, m, m \geq 2$, each of width $h=(b-a) / m$, and write
$$\int_a^b f(x) \mathrm{d} x=\sum_{j=1}^m \int_{x_{j-1}}^{x_j} f(x) \mathrm{d} x$$
where
$$x_j=a+j h, \quad j=0,1, \ldots, m .$$
We then map each of the subintervals $\left[x_{j-1}, x_j\right], j=1,2, \ldots, m$, onto the reference interval $[-1,1]$ by the change of variable
$$x=\frac{1}{2}\left(x_{j-1}+x_j\right)+\frac{1}{2} h t, \quad t \in[-1,1],$$
giving
$$\int_a^b f(x) \mathrm{d} x=\frac{1}{2} h \sum_{j=1}^m \int_{-1}^1 g_j(t) \mathrm{d} t=\frac{1}{2} h \sum_{j=1}^m I_j$$
where
$$g_j(t)=f\left(\frac{1}{2}\left(x_{j-1}+x_j\right)+\frac{1}{2} h t\right) \quad \text { and } \quad I_j=\int_{-1}^1 g_j(t) \mathrm{d} t$$

数学代写数值分析代写Numerical analysis代考|Error estimation for Gauss quadrature

$$\int_a^b w(x) f(x) \mathrm{d} x-\sum_{k=0}^n W_k f\left(x_k\right)=K_n f^{(2 n+2)}(\eta),$$

$$K_n=\frac{1}{(2 n+2) !} \int_a^b w(x)\left[\pi_{n+1}(x)\right]^2 \mathrm{~d} x .$$

数学代写|数值分析代写Numerical analysis代考|Composite Gauss formulae

$$\int_a^b f(x) \mathrm{d} x=\sum_{j=1}^m \int_{x_{j-1}}^{x_j} f(x) \mathrm{d} x$$

$$x_j=a+j h, \quad j=0,1, \ldots, m .$$

$$x=\frac{1}{2}\left(x_{j-1}+x_j\right)+\frac{1}{2} h t, \quad t \in[-1,1],$$

$$\int_a^b f(x) \mathrm{d} x=\frac{1}{2} h \sum_{j=1}^m \int_{-1}^1 g_j(t) \mathrm{d} t=\frac{1}{2} h \sum_{j=1}^m I_j$$

$$g_j(t)=f\left(\frac{1}{2}\left(x_{j-1}+x_j\right)+\frac{1}{2} h t\right) \quad \text { and } \quad I_j=\int_{-1}^1 g_j(t) \mathrm{d} t$$

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。

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数学代写|运筹学代写Operations Research代考|Model Formulation

We consider a firm that receives recoverable product from the market. The firm can manufacture new products and recover the value of a used product or return through remanufacturing with dismantling for parts. The firm provides product at a constant demand rate of $d$ items per time unit. Product consists of two parts, denoted as part 1 and part 2. Each part is manufactured separately and placed in inventory (SS1-serviceable stock inventory for part 1, SS2 – serviceable stock inventory for part 2), then two parts are assembled with the cost $c_A$ and are sold in a market. Products are returned to the firm according the rate $\beta$, other products are immediately disposed of at the rate $\alpha=1-\beta$. The dismantling operation costs $c_D$. Returned product is dismantled for parts, any part is inspected whether it is usable or not, and then is placed in inventory (RS1-inventory for returned stock of part 1 , RS2-inventory for returned stock of part 2). Part 1 is not usable at the rate $q_1$ and should be remanufactured, the rest $\beta_1-q_1$ are as good as new and directly reused, part 2 isn’t usable at the rate $q_2$. Figure 1 represents the integrated closed-loop supply chain inventory system. The sequence of production activities is the following: in any time cycle $[0, T]$ demand for part 1 and part 2 is satisfied firstly through usable parts, then through remanufacturing of used parts and at last manufacturing of new parts. All activities in the model are supposed to be instantaneous and lot-for-lot. The production activities of each part are evaluated on separate production lines (Fig. 2).
Assumptions
This paper assumes:
(1) production and recovery are instantaneous,
(2) remanufactured items are as good as new,
(3) demand is known, constant and independent,
(5) the product consists of two parts
(6) no shortages are allowed,
(7) unlimited storage, and
(8) infinite planning horizon.

数学代写|运筹学代写Operations Research代考|Solution of the Model

Instead of solving the problem (9) the function $L(m, n)$ can be minimized subject to $m_j \geq 1, n_i \geq 1$, i.e., the following two-dimensional nonlinear integer optimization problem is relevant
$$\begin{gathered} \min {(m, n)} L(m, n)=\min {(m, n)}\left(P+\sum_{j=1}^l R_j m_j+\sum_{i=1}^k S_i n_i\right) \cdot\left(h_1+\sum_{j=1}^l \frac{h_2^j}{m_j}+\sum_{i=1}^k \frac{h_3^i}{n_i}\right), \ m=\left(m_1, m_2, \ldots, m_l\right), n=\left(n_1, n_2, \ldots, n_k\right) \ m_j, n_i \in{1,2, \ldots} \end{gathered}$$
For the solution of the problem (10), consider the following two-dimensional nonlinear integer optimization problem
\begin{aligned} & \min {\left(x_1, x_2, \ldots, x_n\right)} K\left(x_1, x_2, \ldots, x_n\right)=\min {\left(x_1, x_2, \ldots, x_n\right)}\left(b_0+\sum_{i=1}^i b_i x_i\right) \cdot\left(a_0+\sum_{i=1}^n \frac{a_i}{x_i}\right), \ & x_i \in{1,2, \ldots}, i=1,2, \ldots n . \end{aligned}
First, let us consider the following continuous auxiliary problem:
\begin{aligned} & \min {\left(x_1, x_2, \ldots, x_n\right)} K\left(x_1, x_2, \ldots, x_n\right)=\min {\left(x_1, x_2, \ldots, x_n\right)}\left(b_0+\sum_{i=1}^i b_i x_i\right) \cdot\left(a_0+\sum_{i=1}^n \frac{a_i}{x_i}\right) \ & x_i \geq 1, i=1,2, \ldots, n \end{aligned}

数学代可|运营管理学代写运营研究代考|模型的提出

(1) 生产和回收是瞬时的。
(2)再制造的物品和新的一样好。
(3) 需求是已知的、恒定的和独立的。
(4) 准备时间为零。
(5) 产品由两部分组成
(6)不允许出现短缺。
(7) 无限储存，以及
(8)无限的计划范围。

数学代写|运筹学代写运营研究代考|模型的解决方法

$$\L(m, n)=\min (m, n)\left(P+sum_{j=1}^l R_j m_j+\sum_{i=1}^k S_i n_i\right) \cdot\left(h_1+sum_{j=1}^l\frac{h_2^j}{m_j}+\sum_{i=1}^k \frac{h_3^i}{n_i}\right) 。m=left(m_1, m_2, \ldots, m_l\right), n=left(n_1, n_2, \ldots, n_k\right) m_j, n_i\in 1,2, .$$

$$K\left(x_1, x_2, \ldots, x_n\right)=\min \left(x_1, x_2, \ldots, x_n\right)=\min \left(x_1, x_2, \ldots, x_n\right)\left(b_0+sum_{i=1}^i b_i x_i\right) cdot\left(a_0+sum_{i=1}^n frac{a_i}{x_i}\right), \quad x_i \in 1,2, \ldots, i=1,2, \ldots n 。$$

$$K\left(x_1, x_2, \ldots, x_n\right)=\min \left(x_1, x_2, \ldots, x_n\right)=\min \left(x_1, x_2, \ldots, x_n\right)\left(b_0+sum_{i=1}^i b_i x_i\right) cdot\left(a_0+sum_{i=1}^n \frac{a_i}{x_i}\right) \quad x_i \geq 1, i=1, 2, \ldots, n$$

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。

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数学代写数值分析代写Numerical analysis代考|Normed linear spaces

In order to be able to talk about ‘best approximation’ in a rigorous manner we need to recall from Chapter 2 the concept of norm; this will allow us to compare various approximations quantitatively and select the one which has the smallest approximation error. The definition given in Section $2.7$ applies to a linear space consisting of functions in the same way as to the finite-dimensional linear spaces considered in Chapter 2.
Definition 8.1 Suppose that $\mathcal{V}$ is a linear space over the field $\mathbb{R}$ of real numbers. A nonnegative function $|\cdot|$ defined on $\mathcal{V}$ whose value at $f \in \mathcal{V}$ is denoted by $|f|$ is called a norm on $\mathcal{V}$ if it satisfies the following axioms:
(1) $|f|=0$ if, and only if, $f=0$ in $\mathcal{V}$;
(2) $|\lambda f|=|\lambda||f|$ for all $\lambda \in \mathbb{R}$, and all $f$ in $\mathcal{V}$;
(3) $|f+g| \leq|f|+|g|$ for all $f$ and $g$ in $\mathcal{V}$ (the triangle inequality). A linear space $\mathcal{V}$, equipped with a norm, is called a normed linear space.

数学代写|数值分析代写Numerical analysis代考|Best approximation in the ∞-norm

According to the Weierstrass Approximation Theorem any function $f$ in $\mathrm{C}[a, b]$ can be approximated arbitrarily well from the set of all polynomials. Clearly, if instead of the set of all polynomials we restrict ourselves to the set of polynomials $\mathcal{P}n$ of degree $n$ or less, with $n$ fixed, then it is no longer true that, for any $f \in \mathrm{C}[a, b]$ and any $\varepsilon>0$, there exists $p_n \in \mathcal{P}_n$ such that $$\left|f-p_n\right|{\infty}<\varepsilon .$$
Consider, for example, the function $x \mapsto \sin x$ defined on the interval $[0, \pi]$ and fix $n=0$; then $|f-q|_{\infty} \geq 1 / 2$ for any $q \in \mathcal{P}0$, and therefore there is no $q$ in $\mathcal{P}_0$ such that $|f-q|{\infty}<1 / 2$. A similar situation will arise if $\mathcal{P}_0$ is replaced by $\mathcal{P}_n$, with the polynomial degree $n$ fixed. ${ }^1$

It is therefore relevant to enquire just how well a given function $f$ in $\mathrm{C}[a, b]$ may be approximated by polynomials of a fixed degree $n \geq 0$. This question leads us to the following approximation problem.
(A) Given that $f \in \mathrm{C}[a, b]$ and $n \geq 0$, fixed, find $p_n \in \mathcal{P}n$ such that $$\left|f-p_n\right|{\infty}=\inf {q \in \mathcal{P}_n}|f-q|{\infty} ;$$
such a polynomial $p_n$ is called a polynomial of best approximation of degree $n$ to the function $f$ in the $\infty$-norm.

数学代写数值分析代写 数学分析代写|规范线性空间

(3) 对于$mathcal{V}$中的所有$f$和$g$，$|f+g| \leq|f|+|g|$（三角不等式）。一个线性空间$mathcal{V}$，配备了一个规范，被称为规范化线性空间。

数学代写|数值分析代写|Best approximation in the $infty$-norm

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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。