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# 数学代写|微分拓扑作业代写differential topology代考|A Robot’s Arm

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## 数学代写|微分拓扑作业代写differential topology代考|Question

What would the space $S$ of positions look like if the telescope got stuck at $|y|=2$ ?
Partial answer to the question: since $y=(3,0)-x-z$ we could try to get an idea of what points of $T$ satisfy $|y|=2$ by means of inspection of the graph of $|y|$. Figure $1.5$ is an illustration showing $|y|$ as a function of $T$ given as a graph over $[0,2 \pi] \times[0,2 \pi]$, and also the plane $|y|=2$.

The desired set $S$ should then be the intersection shown in Figure 1.6. It looks a bit weird before we remember that the edges of $[0,2 \pi] \times[0,2 \pi]$ should be identified. On the torus it looks perfectly fine; and we can see this if we change our perspective a bit. In order to view $T$ we chose $[0,2 \pi] \times[0,2 \pi]$ with identifications along the boundary. We could just as well have chosen $[-\pi, \pi] \times[-\pi, \pi]$, and then the picture would have looked like Figure 1.7. It does not touch the boundary, so we do not need to worry about the identifications. As a matter of fact, $S$ is homeomorphic to the circle homeomorphic means that there is a bijection between $S$ and the circle, and both the function from the circle to S and its inverse are continuous. See Definition A.2.8.

## 数学代写|微分拓扑作业代写differential topology代考|Dependence on the Telescope’s Length

Even more is true: we notice that $S$ looks like a smooth and nice curve. This will not happen for all values of $|y|$. The exceptions are $|y|=1,|y|=3$ and $|y|=5$. The values 1 and 5 correspond to one-point solutions. When $|y|=3$ we get a picture like Figure $1.8$ (the solution really ought to touch the boundary).

We will learn to distinguish between such circumstances. They are qualitatively different in many aspects, one of which becomes apparent if we view the example shown in Figure $1.9$ with $|y|=3$ with one of the angles varying in $[0,2 \pi]$ while the other varies in $[-\pi, \pi]$. With this “cross” there is no way our solution space is homeomorphic to the circle. You can give an interpretation of the picture above: the straight line is the movement you get if you let $x=z$ (like two wheels of equal radius connected by a coupling rod $y$ on an old-fashioned train), whereas the curved line corresponds to $x$ and $z$ rotating in opposite directions (very unhealthy for wheels on a train).

Actually, this cross comes from a “saddle point” in the graph of $|y|$ as a function of $T$ : it is a “critical” value at which all sorts of bad things can happen.

## 数学代写|微分拓扑作业代写differential topology代考|Moral

The configuration space $T$ is smooth and nice, and we get different views on it by changing our “coordinates”. By considering a function on $T$ in our case the length of y and restricting to the subset of $T$ corresponding to a given value of our function, we get qualitatively different situations according to what values we are looking at.

## 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。