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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

管理科学代写|决策论代写Management Science Models for Decision Making代考|Fourier Elimination Method for Linear Inequalities

By 1827, Fourier generalized the elimination method to solve a system of linear inequalities. The method now known as the Fourier or Fourier-Motzkin elimination method is one of the earliest methods proposed for solving systems of linear inequalities. It consists of successive elimination of variables from the system. We will illustrate one step in this method using an example in which we will eliminate the variable $x_1$ from the following system.
\begin{aligned} x_1-2 x_2+x_3 & \leq 6, \ 2 x_1+6 x_2-8 x_3 & \leq-6, \ -x_1-x_2-2 x_3 & \leq 2, \ -2 x_1-6 x_2+2 x_3 & \leq 2 \end{aligned}
$x_1$ appears with a positive coefficient in the first and second constraints and a negative coefficient in the third and fourth constraints. By making the coefficient of $x_1$ in each constraint into 1 , these constraints can be expressed as
$$\begin{gathered} x_1 \leq 6+2 x_2-x_3, \ x_1 \leq-3-3 x_2+4 x_3, \ -2-x_2-2 x_3 \leq x_1, \ -1-3 x_2+x_3 \leq x_1 . \end{gathered}$$

The remaining system after $x_1$ is eliminated and is therefore
\begin{aligned} &-2-x_2-2 x_3 \leq 6+2 x_2-x_3 \ &-2-x_2-2 x_3 \leq-3-3 x_2+4 x_3 \ &-1-3 x_2+x_3 \leq 6+2 x_2-x_3 \ &-1-3 x_2+x_3 \leq-3-3 x_2+4 x_3 \end{aligned}
and then $\max \left{-2-x_2-2 x_3,-1-3 x_2+x_3\right} \leq x_1 \leq \min \left{6+2 x_2-x_3,-3-\right.$ $\left.3 x_2+4 x_3\right}$ is used to get a value for $x_1$ in a feasible solution when values for other variables are obtained by applying the same steps on the remaining problem successively.

However, starting with a system of $m$ inequalities, the number of inequalities can jump to $O\left(\mathrm{~m}^2\right)$ after eliminating only one variable from the system, so this method is not practically viable except for very small problems.

管理科学代写|决策论代写Management Science Models for Decision Making代考|History of the Simplex Method for LP

In 1827, Fourier published a geometric version of the principle behind the simplex algorithm for a linear program (vertex-to-vertex descent along the edges to an optimum, a rudimentary version of the simplex method) in the context of a specific LP in three variables (an LP model for a Chebyshev approximation problem), but did not discuss how this descent can be accomplished computationally on systems stated algebraically. In 1910, De la Vallée Poussin designed a method for the Chebyshev approximation problem, which is an algebraic and computational analogue of this Fourier’s geometric version; this procedure is essentially the primal simplex method applied to that problem.

In a parallel effort, Gordan (1873), Farkas (1896), and Minkowski (1896) studied linear inequalities, and laid the foundations for the algebraic theory of polyhedra and derived necessary and sufficient conditions for a system of linear constraints, including linear inequalities to have a feasible solution.

Studying LP models for organizing and planning production (Kantorovich 1939) developed idcas of dual variables (resolving multipliers) and derived a dual=simplox type method for solving a general LP.

Full citations for references before 1939 mentioned so far can be seen from the list of references in Danizig (1963) or Schrijver (1986).

This work culminated in the mid-twentieth century with the development of the primal simplex method by Dantzig. This was the first complete, practically and computationally viable method for solving systems of linear inequalities. So, LP can be considered as the branch of mathematics, which is an extension of linear algebra to solve systems of linear inequalities. The development of LP is a landmark event in the history of mathematics, and its application brought our ability to solve general systems of linear constraints (including linear equations, inequalities) to a state of completion.

决策论代写

管理科学代写|决策论代写管理科学决策模型代考|线性不等式的傅里叶消除方法

.

\begin{aligned} x_1-2 x_2+x_3 & \leq 6, \ 2 x_1+6 x_2-8 x_3 & \leq-6, \ -x_1-x_2-2 x_3 & \leq 2, \ -2 x_1-6 x_2+2 x_3 & \leq 2 \end{aligned}
$x_1$在第一个和第二个约束条件中出现正系数，在第三和第四个约束条件中出现负系数。将每个约束条件$x_1$的系数设为1，则这些约束条件可表示为
$$\begin{gathered} x_1 \leq 6+2 x_2-x_3, \ x_1 \leq-3-3 x_2+4 x_3, \ -2-x_2-2 x_3 \leq x_1, \ -1-3 x_2+x_3 \leq x_1 . \end{gathered}$$

$x_1$后的剩余系统被消去，因此是
\begin{aligned} &-2-x_2-2 x_3 \leq 6+2 x_2-x_3 \ &-2-x_2-2 x_3 \leq-3-3 x_2+4 x_3 \ &-1-3 x_2+x_3 \leq 6+2 x_2-x_3 \ &-1-3 x_2+x_3 \leq-3-3 x_2+4 x_3 \end{aligned}
，然后是$\max \left{-2-x_2-2 x_3,-1-3 x_2+x_3\right} \leq x_1 \leq \min \left{6+2 x_2-x_3,-3-\right.$$\left.3 x_2+4 x_3\right}$，当其他变量的值通过对剩下的问题依次应用相同的步骤得到时，使用 得到$x_1$在可行解中的值

管理科学代写|决策论代写管理科学决策模型代考| LP的Simplex方法的历史

1827年，傅立叶发表了一个几何版本的线性程序的单纯形算法背后的原理(沿着边缘的顶点到顶点下降到最优，单纯形方法的一个基本版本)，在一个特定的三个变量的LP (Chebyshev近似问题的LP模型)的背景下，但没有讨论如何在代数表示的系统上完成这种下降计算。1910年，De la Vallée Poussin为切比雪夫近似问题设计了一种方法，这是傅里叶几何版本的代数和计算模拟;这个过程本质上是应用于该问题的原始单纯形方法

Gordan (1873)， Farkas(1896)和Minkowski(1896)在平行的努力中，研究了线性不等式，奠定了多面体代数理论的基础，并导出了线性约束系统(包括线性不等式)有可行解的充分必要条件

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。

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