19th Ave New York, NY 95822, USA

# 数学代写|凸分析代写Convex Analysis代考|MA542 Chapters 1–4 in the Special Case of Subspaces

1. 如果$0 \leq r \leq 1$是实数，并且$x, y\in C$，那么$r x+(1-r) y \in C$。[1]
2. 如果$0<r<1$是实数，并且$x, y\in C$有$x\neq y$，那么$r x+(1-r) y\in C$。

my-assignmentexpert™提供最专业的一站式服务：Essay代写，Dissertation代写，Assignment代写，Paper代写，Proposal代写，Proposal代写，Literature Review代写，Online Course，Exam代考等等。my-assignmentexpert™专注为留学生提供Essay代写服务，拥有各个专业的博硕教师团队帮您代写，免费修改及辅导，保证成果完成的效率和质量。同时有多家检测平台帐号，包括Turnitin高级账户，检测论文不会留痕，写好后检测修改，放心可靠，经得起任何考验！

## 数学代写|凸分析代写Convex Analysis代考|Chapters 1–4 in the Special Case of Subspaces

The aim of this section is to give some initial insight into the contents of Chaps. $1-4$ of this book. The aim of these chapters is to present the properties of convex sets and now we display the specializations of some of these properties from convex sets to special convex sets, subspaces. A subspace of $\mathbb{R}^n$ is a subset $L$ of $\mathbb{R}^n$ that is closed under sums and scalar multiples, $x, y \in L, \alpha, \beta \in \mathbb{R} \Rightarrow \alpha x+\beta y \in L$. A subspace can be viewed as the central concept from linear algebra: it is the solution set of a finite system of homogeneous linear equations in $n$ variables. A subspace results from Chaps. $1-4$ from convex sets to subspaces are standard results from linear algebra.

Chapter 1 A subspace of $X=\mathbb{R}^n$ is defined to be a nonempty subset $L \subseteq X$ that is closed under linear combinations of two elements:
$$a, b \in L, \rho, \sigma \in \mathbb{R} \Rightarrow \rho a+\sigma b \in L$$
Example 1.3.1 (False Equations and Homogenization) Now we illustrate homogenization of convex sets by means of a simple high school example. Before the use of the language of sets in high school, the following three cases were distinguished in some school books, for a system of two linear equations in two variables:
\begin{aligned} &a_{11} x_1+a_{12} x_2=b_1, \ &a_{21} x_1+a_{22} x_2=b_2 . \end{aligned}

## 数学代写|凸分析代写Convex Analysis代考|Convex Cones: Visualization by Models

The aim of this section is to present three models for convex cones $C \subseteq X=\mathbb{R}^n$ that lie in the upper halfspace $x_n \geq 0$ : the ray model, the hemisphere model and the top-view model.

Example 1.4.1 (Ray, Hemisphere and Top-View Model for a Convex Cone) Figure $1.5$ illustrates the three models in one picture for a convex cone $C$ in the plane $\mathbb{R}^2$ that lies above or on the horizontal axis.

The ray model visualizes this convex cone $C \subseteq \mathbb{R}^2$ by viewing $C \backslash\left{0_2\right}$ as a collection of open rays, and then considering the set whose elements are these rays; the hemisphere model visualizes $C$ by means of an arc: the intersection of $C$ with the upper half-circle $x_1^2+x_2^2=1, x_2 \geq 0$; the top-view model visualizes $C$ by looking from high above at the arc, or, to be precise, by taking the interval in $[-1,+1]$ that is the orthogonal projection of the arc onto the horizontal axis.

The three models are all useful for visualization in low dimensions. Fortunately, most, maybe all, properties of convex cones in $\mathbb{R}^n$ for general $n$ can be understood by looking at such low dimensional examples. Therefore, making pictures in your head or with pencil and paper by means of one of these models is of great help to understand properties of convex cones-and so to understand the entire convex analysis, as we will see.

## 数学代写|凸分析代写CONVEX ANALYSIS代考|CHAPTERS 1-4 IN THE SPECIAL CASE OF SUBSPACES

$$a, b \in L, \rho, \sigma \in \mathbb{R} \Rightarrow \rho a+\sigma b \in L$$

## 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。