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# 数学代写|凸分析代写Convex Analysis代考|MA542 Ray Model for a Convex Cone

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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$r x+(1-r) y\in C$。

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## 数学代写|凸分析代写Convex Analysis代考|Ray Model for a Convex Cone

The role of nonzero elements of a convex cone $C \subseteq X=\mathbb{R}^n$ is usually to describe one-sided directions. Then, two nonzero elements of $C$ that differ by a positive scalar multiple, $a$ and $\rho a$ with $a \in \mathbb{R}^n$ and $\rho>0$, are considered equivalent. The equivalence classes of $C \backslash\left{0_n\right}$ are open rays of $C$, sets of all positive multiples of a nonzero element. So, what often only matters about a convex cone $C$ is its set of open rays, as this describes a set of one-sided directions. The set of open rays of $C$ will be called the ray model for the convex cone $C$.

The ray model is the most simple description of the one-sided directions of $C$ from a mathematical point of view, but it might be seen as inconvenient that a onesided direction is modeled by an infinite set. Moreover, it requires some preparations such as the definition of distance between two open rays: this is needed in order to define convergence of a sequence of rays. Suppose we have two open rays, ${\rho v \mid \rho>$ $0}$ and ${\rho w \mid \rho>0}$, where $v, w \in X$ are unit vectors, that is, their lengths or Euclidean norms, $|v|=\left(v_1^2+\cdots+v_n^2\right)^{\frac{1}{2}}$ and $|w|=\left(w_1^2+\cdots+w_n^2\right)^{\frac{1}{2}}$, are both equal to 1 . Then the distance between these two rays can be taken to be the angle $\varphi \in[0, \pi]$ between the rays, which is defined by $\cos \varphi=v \cdot w=v_1 w_1+\cdots+v_n w_n$, the dot product of $v$ and $w$. To be precise about the concept of distance, the concept of metric space would be needed; however, we will not consider this concept.

Example 1.4.2 (Ray Model for a Convex Cone) Figure $1.5$ above illustrates the ray model for convex cones in dimension two. A number of rays of the convex cone $C$ are drawn.

## 数学代写|凸分析代写Convex Analysis代考|Sphere Model for a Convex Cone

One often chooses a representative for each open ray, in order to replace open rayswhich are infinite sets of points-by single points. A convenient way to do this is to normalize: that is, to choose the unit vector on each ray. Thus the one-sided directions in the space $X=\mathbb{R}^n$ are modeled as the points on the standard unit sphere $S_X=S_n={x \in X \mid|x|=1}$ in $X$. The set of unit vectors in a convex cone $C \subseteq X=\mathbb{R}^n$ will be called the sphere model for $C$. The subsets of the standard unit sphere $S_X$ that one gets in this way are precisely the geodesically convex subsets of $S_X$. A subset $T$ of $S_X$ is called geodesically convex if for each two different points $p, q$ of $T$ that are not antipodes $(p \neq-q)$, the shortest curve on $S_X$ that connects them is entirely contained in $T$. This curve is called the geodesic connecting these two points. Note that for two different points $p, q$ on $S_X$ that are not antipodes, there is a unique great circle on $S_X$ that contains them. A great circle on $S_X$ is a circle on $S_X$ with center the origin, that is, it is the intersection of the sphere $S_X$ with a two dimensional subspace of $X$. This great circle through $p$ and $q$ gives two curves on $S_X$ on this circle connecting the two points, a short one and a long one. The short one is the geodesic connecting these points.
Example 1.4.3 (Sphere Model for a Convex Cone)

1. Figure $1.5$ above illustrates the sphere model for convex cones in dimension two. The convex cone $C$ is modeled by an arc.
2. Figure $1.6$ illustrates the sphere model for convex cones in dimension three, and it illustrates the concepts great circle and geodesic on $S_X$ for $X=\mathbb{R}^3$.

Two great circles are drawn. These model two convex cones that are planes through the origin. For the two marked points on one of these circles, the segment on this circle on the front of the sphere connecting the two points is their geodesic. Moreover, you see that two planes in $\mathbb{R}^3$ through the origin (and remember that a plane is a convex cone) are modeled in the sphere model for convex cones by two large circles on the sphere.

1. Figure $1.7$ is another illustration of the sphere model for a convex cone $C$ in dimension two.

Here the convex cone $C$ is again modeled by an arc. The sphere model is convenient for visualizing the one-sided directions determined by a convex cone $C \subseteq X=\mathbb{R}^n$ for dimension $n$ up to three: for $n=3$, a one-sided direction in

## 数学代写|凸分析代写Convex Analysis代考|Sphere Model for a Convex Cone

1. 数字 $1.5$ 上面说明了二维凸推的球体模型。凸倠 $C$ 由圆弝建模。
画了两个大圆击。这些模抚两个凸雉，它们是通讨原点的平面。对于其中一个圆上的两个标记点，连接这两个点的球体前面的圆上 的线段是它们的测地线。此外，你着到两架飞机在 $\mathbb{R}^3$ 通过原点 (记住平面是口雉) 在凸雉的球体模型中由球体上的两个大圆表建 模。
2. 数字 $1.7$ 是凸锥的球体槻型的另一个例子 $C$ 在二次元中。
这里的凸锥 $C$ 再次由弝建模。球体模型便于可视化由凸锥确定的单边方向 $C \subseteq X=\mathbb{R}^n$ 对于维度 $n$ 最多三个: 为了 $n=3$, 单边方 向

## 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。