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数学代写|几何组合代写Geometric Combinatorics代考|MA726 The Pentagon Recurrence

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数学代写|几何组合代写Geometric Combinatorics代考|The Pentagon Recurrence

Consider a sequence $f_1, f_2, f_3, \ldots$ defined recursively by $f_1=x, f_2=y$, and
(1)
$$f_{n+1}=\frac{f_n+1}{f_{n-1}} .$$
Thus, the first five entries are
$$x, y, \frac{y+1}{x}, \frac{x+y+1}{x y}, \frac{x+1}{y} .$$
Unexpectedly, the sixth and seventh entries are $x$ and $y$, respectively, so the sequence is periodic with period five! We will call (1) the pentagon recurrence. ${ }^1$
This sequence has another important property. A priori, we can only expect its terms to be rational functions of $x$ and $y$. In fact, each $f_i$ is a Laurent polynomial (actually, with nonnegative integer coefficients). This is an instance of what is called the Laurent phenomenon.

It will be helpful to represent this recurrence as the evolution of a “moving window” consisting of two consecutive terms $f_i$ and $f_{i+1}$ :
$$\left[\begin{array}{l} f_1 \ f_2 \end{array}\right] \stackrel{\tau_1}{\longrightarrow}\left[\begin{array}{l} f_3 \ f_2 \end{array}\right] \stackrel{\tau_2}{\longrightarrow}\left[\begin{array}{l} f_3 \ f_4 \end{array}\right] \stackrel{\tau_1}{\longrightarrow}\left[\begin{array}{l} f_5 \ f_4 \end{array}\right] \stackrel{\tau_2}{\longrightarrow}\left[\begin{array}{l} f_5 \ f_6 \end{array}\right] \longrightarrow \cdots,$$
where the maps $\tau_1$ and $\tau_2$ are defined by
$$\tau_1:\left[\begin{array}{l} f \ g \end{array}\right] \longmapsto\left[\begin{array}{c} \frac{g+1}{f} \ g \end{array}\right] \text { and } \tau_2:\left[\begin{array}{l} f \ g \end{array}\right] \longmapsto\left[\begin{array}{c} f \ \frac{f+1}{g} \end{array}\right] \text {. }$$
Both $\tau_1$ and $\tau_2$ are involutions: $\tau_1^2=\tau_2^2=1$, where 1 denotes the identity map. The 5-periodicity of the recurrence (1) translates into the identity $\left(\tau_2 \tau_1\right)^5=1$. That is, the group generated by $\tau_1$ and $\tau_2$ is a dihedral group with 10 elements.

Let us now consider a similar but simpler pair of maps. Throw away the $+1$ ‘s that occur in the definitions of $\tau_1$ and $\tau_2$, and take logarithms. We then obtain a pair of linear maps
$$s_1:\left[\begin{array}{l} x \ y \end{array}\right] \longmapsto\left[\begin{array}{c} y-x \ y \end{array}\right] \text { and } s_2:\left[\begin{array}{l} x \ y \end{array}\right] \longmapsto\left[\begin{array}{c} x \ x-y \end{array}\right] \text {. }$$

数学代写|几何组合代写Geometric Combinatorics代考|Reflection Groups

Our first goal will be to understand the finite groups generated by linear reflections in a vector space $V$. It turns out that for such a group, it is always possible to define a Euclidean structure on $V$ so that all of the reflections in the group are ordinary orthogonal reflections. The study of groups generated by orthogonal reflections is a classical subject, which goes back to the classification of Platonic solids by the ancient Greeks.

Let $V$ be a Euclidean space. In what follows, all reflecting hyperplanes pass through the origin, and all reflections are orthogonal. A finite reflection group is a finite group generated by some reflections in $V$. In other words, we choose a collection of hyperplanes such that the group of orthogonal transformations generated by the corresponding reflections is finite. Infinite reflection groups are also interesting, but in these lectures, “reflection group” will always mean a finite one.

The set of reflections in a reflection group $W$ is typically larger than a minimal set of reflections generating $W$. This is illustrated in Figure $1.1$, where $W$ is the group of symmetries of a regular pentagon. This 10-element group is generated by two reflections $s$ and $t$ whose reflecting lines make an angle of $\pi / 5$. It consists of 5 reflections, 4 rotations, and the identity element. In Figure 1.1, each of the 5 lines is labeled by the corresponding reflection.

数学代写|几何组合代写Geometric Combinatorics代考|The Pentagon Recurrence

(1)
$$f_{n+1}=\frac{f_n+1}{f_{n-1}} .$$

$$x, y, \frac{y+1}{x}, \frac{x+y+1}{x y}, \frac{x+1}{y} .$$

$$\left[f_1 f_2\right] \stackrel{{ }^{\top}}{\longrightarrow}\left[f_3 f_2\right] \stackrel{{ }^\tau}{\longrightarrow}\left[f_3 f_4\right] \stackrel{{ }^\tau}{\longrightarrow}\left[f_5 f_4\right] \stackrel{\tau^2}{\longrightarrow}\left[f_5 f_6\right] \longrightarrow \cdots,$$

$$\tau_1:[f g] \longmapsto\left[\frac{g+1}{f} g\right] \text { and } \tau_2:[f g] \longmapsto\left[f \frac{f+1}{g}\right] .$$

$$s_1:[x y] \longmapsto[y-x y] \text { and } s_2:[x y] \longmapsto[x x-y] .$$

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。