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## avatest™帮您通过考试

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## 数学代写|几何组合代写Geometric Combinatorics代考|Sphere Theorems

As mentioned in our discussion at the end of Section 5, one can sometimes use discrete Morse theory to make statements about more than just the homotopy type of the simplicial complex. One can sometimes classify the complex up to homeomorphism or combinatorial equivalence. In this section we give some examples of such arguments. An interesting application of these ideas is presented in the next section. So far, we have not placed any restrictions on the simplicial complexes under consideration. The main idea of this section is that if our simplicial complex has some additional structure, then one may be able to strengthen the conclusion. This idea rests on some very deep work of J. H. C. Whitehead [95].

A simplicial complex $K$ is a combinatorial $d$-ball if $K$ and the standard $d$ simplex $\sigma_d$ have isomorphic subdivisions. A simplicial complex $K$ is a combinatorial $(d-1)$-sphere if $K$ and $\dot{\sigma}_d$ have isomorphic subdivisions (where $\dot{\sigma}_d$ denotes the boundary of $\sigma_d$ with its induced simplicial structure). A simplicial complex $K$ is a combinatorial d-manifold with boundary if the link of every vertex is either a combinatorial $(d-1)$-sphere or a combinatorial $(d-1)$-ball. The following is a special case of the powerful main theorem of $[\mathbf{9 5}]$.

Theorem 14. Let $K$ be a combinatorial d-manifold with boundary which simplicially collapses to a vertex. (That is, $K$ can be a reduced to a vertex by a sequence of elementary simplicial collapses.) Then $K$ is a combinatorial d-ball.

With this theorem, and its generalizations, one can sometimes strengthen the conclusion of Theorem 11 beyond homotopy equivalence. We present just one example.
Theorem 15. Let $X$ be a combinatorial d-manifold with a discrete gradient vector field with exactly two critical simplices. Then $X$ is a combinatorial d-sphere.

## 数学代写|几何组合代写Geometric Combinatorics代考|Our Second Example

In this section we demonstrate some of the ideas of the previous sections with a simple example from algebra. Fix a positive integer $n$, and consider the following $(n-2)$-dimensional simplicial complex, which we denote $M_n$. Starting with the following expression
$$\left(x_0 x_1 x_2 \ldots x_n\right)$$
consider all ways of adding legal pairs of parentheses. An expression resulting from adding $p+1$ pairs of parentheses will be a $p$-simplex in our complex. The faces of this $p$-simplex are all expressions that result from removing corresponding pairs of parentheses.

For example, consider the case $n=3$. The vertices of $M_3$ are the expressions
$$\begin{gathered} v_1=\left(\left(x_0 x_1\right) x_2 x_3\right), \quad v_2=\left(\left(x_0 x_1 x_2\right) x_3\right), \quad v_3=\left(x_0\left(x_1 x_2\right) x_3\right), \ v_4=\left(x_0\left(x_1 x_2 x_3\right)\right), \quad v_5=\left(x_0 x_1\left(x_2 x_3\right)\right) \end{gathered}$$
and the edges are the expressions
$$\begin{gathered} e_1=\left(\left(\left(x_0 x_1\right) x_2\right) x_3\right), \quad e_2=\left(\left(x_0\left(x_1 x_2\right)\right) x_3\right), \quad e_3=\left(x_0\left(\left(x_1 x_2\right) x_3\right)\right), \ e_4=\left(x_0\left(x_1\left(x_2 x_3\right)\right)\right), \quad e_5=\left(\left(x_0 x_1\right)\left(x_2 x_3\right)\right) . \end{gathered}$$
One can easily check the relations
$$\begin{gathered} e_1=\left{v_1, v_2\right}, \quad e_2=\left{v_2, v_3\right}, \quad e_3=\left{v_3, v_4\right} \ e_4=\left{v_4, v_5\right}, \quad e_5=\left{v_5, v_1\right} \end{gathered}$$
so that $M_3$ is a circle triangulated with 5 edges and 5 vertices.

## 数学代写|几何组合代写Geometric Combinatorics代考|Our Second Example

$$\left(x_0 x_1 x_2 \ldots x_n\right)$$

$$v_1=\left(\left(x_0 x_1\right) x_2 x_3\right), \quad v_2=\left(\left(x_0 x_1 x_2\right) x_3\right), \quad v_3=\left(x_0\left(x_1 x_2\right) x_3\right), v_4=\left(x_0\left(x_1 x_2 x_3\right)\right), \quad v_5=\left(x_0 x_1\left(x_2 x_3\right)\right)$$

$$e_1=\left(\left(\left(x_0 x_1\right) x_2\right) x_3\right), \quad e_2=\left(\left(x_0\left(x_1 x_2\right)\right) x_3\right), \quad e_3=\left(x_0\left(\left(x_1 x_2\right) x_3\right)\right), e_4=\left(x_0\left(x_1\left(x_2 x_3\right)\right)\right), \quad e_5=\left(\left(x_0 x_1\right)\left(x_2 x_3\right)\right) .$$

$$\backslash \text { begin }{\text { gathered }} e_{-} 1=\backslash \text { left }\left{v_{-} l, v_{-} 2 \backslash \text { right }\right}, \backslash q u a d e_{-} 2=\backslash \text { left }\left{v_{-} 2, v_{-} 3 \backslash \text { right }\right}, \backslash q u a d e_{-} 3=\backslash \text { left }\left{v_{-} 3, v_{-} 4 \backslash \text { right }\right} \backslash e_{-} 4=\backslash \text { left }\left{v_{-} 4, v_{-} 5 \backslash \text { right }\right}, \backslash q$$

## 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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## 数学代写|几何组合代写Geometric Combinatorics代考|Cell Complexes and CW Complexes

The main theorems of discrete (and smooth) Morse theory are best stated in the language of CW complexes, so we begin with an overview of the basics of such complexes. J. H. C. Whitehead introduced CW complexes in his foundational work on homotopy theory, and all of the results in this section are due to him. The reader should consult $[\mathbf{6 8}]$ for a very complete introduction to this topic. In these notes we will consider only finite $\mathrm{CW}$ complexes, so many of the subtleties of the subject will not appear.

The building blocks of cell complexes are cells. Let $B^d$ denote the closed unit ball in $d$-dimensional Euclidean space. That is, $B^d=\left{x \in \mathbb{E}^d\right.$ s.t. $\left.|x| \leq 1\right}$. The boundary of $B^d$ is the unit $(d-1)$-sphere $S^{(d-1)}=\left{x \in \mathbb{E}^d\right.$ s.t. $\left.|x|=1\right}$. A d-cell is a topological space which is homeomorphic to $B^d$. If $\sigma$ is $d$-cell, then we denote by $\dot{\sigma}$ the subset of $\sigma$ corresponding to $S^{(d-1)} \subset B^d$ under any homeomorphism between $B^d$ and $\sigma$. A cell is a topological space which is a $d$-cell for some $d$.

The basic operation of cell complexes is the notion of attaching a cell. Let $X$ be a topological space, $\sigma$ a $d$-cell and $f: \dot{\sigma} \rightarrow X$ a continuous map. We let $X \cup_f \sigma$ denote the disjoint union of $X$ and $\sigma$ quotiented out by the equivalence relation that each point $s \in \dot{\sigma}$ is identified with $f(s) \in X$. We refer to this operation by saying that $X \cup_f \sigma$ is the result of attaching the cell $\sigma$ to $X$. The map $f$ is called the attaching map.

We emphasize that the attaching map must be defined on all of $\dot{\sigma}$. That is, the entire boundary of $\sigma$ must be “glued” to $X$. For example, if $X$ is a circle, then Figure 1(i) shows one possible result of attaching a 1-cell to $X$. Attaching a 1-cell to $X$ cannot lead to the space illustrated in Figure 1 (ii) since the entire boundary of the 1-cell has not been “glued” to $X$.

We are now ready for our main definition. A finite cell complex is any topological space $X$ such that there exists a finite nested sequence
(1)
$$\emptyset \subset X_0 \subset X_1 \subset \cdots \subset X_n=X$$
such that for each $i=0,1,2, \ldots, n, X_i$ is the result of attaching a cell to $X_{(i-1)}$.

Note that this definition requires that X0 be a 0-cell. If X is a cell complex, we refer to any sequence of spaces as in (1) as a cell decomposition of X. Suppose that in the cell decomposition (1), of the $n+1$ cells that are attached, exactly $c_d$ are $d$-cells. Then we say that the cell complex $X$ has a cell decomposition consisting of $c_d d$-cells for every $d$.

## 数学代写|几何组合代写Geometric Combinatorics代考|The Morse Theory

In this section we introduce the main topic of the first three lectures, namely discrete Morse theory. Morse theory, in the standard setting of smooth manifolds, is usually described in the language of smooth functions on smooth manifolds (e.g. [71]). In practice, though, it is often useful to work with gradient vector fields rather than functions (e.g. $[\mathbf{7 2}],[\mathbf{8 2}]$ ). In the discrete setting, too, one can follow either path. In these notes, we will focus on the notion of a (discrete) gradient vector field. To see how discrete Morse theory can be presented from the function point of view, see $[\mathbf{3 1}]$ or $[\mathbf{3 2}]$

Let $K$ be a CW complex. (Most of our examples will be simplicial complexes, but in a few places, even when our object of study is a simplicial complex, it will be convenient to allow more general cell complexes.)

Definition 8. Let $\beta$ be a $(p+1)$-cell of $K$, with attaching map $h: S^p \rightarrow K_p$, where $K_p$ denotes the union of the cells of dimension $\leq p$.
(i) A cell $\alpha$ is a face of $\beta$, denoted by $\alpha<\beta$ (or $\beta>\alpha$ ) if $\beta \neq \alpha \subset \beta$ (where here we are identifying a cell with its image in $K$ ).
(ii) A face $\alpha$ of $\beta$ is said to be regular if
(a) $h^{-1}(\alpha)$ is homeomorphic to a ball, and
(b) $h$ restricted to $h^{-1}(\alpha)$ is a homeomorphism onto $\alpha$.
(iii) A regular $C W$ complex is a CW complex in which every face is regular. We note that every simplicial complex or polyhedron is a regular $\mathrm{CW}$ complex.

## 数学代写|几何组合代写Geometric Combinatorics代考|Cell Complexes and CW Complexes

$(1)$
$$\emptyset \subset X_0 \subset X_1 \subset \cdots \subset X_n=X$$

## 数学代写|几何组合代写Geometric Combinatorics代考|The Morse Theory

(i) 细胞 $\alpha$ 是一张脸 $\beta$ ，表示为 $\alpha<\beta$ (或者 $\beta>\alpha$ ) 如果 $\beta \neq \alpha \subset \beta$ (在这里我们用它的图像来识别一个单 元格 $K)$.
(二) 做 $\alpha$ 的 $\beta$ 如果
(a) $h^{-1}(\alpha)$ 与球同胚，并且
(b) $h$ 受限于 $h^{-1}(\alpha)$ 是同胚到 $\alpha$.
(iii) 常规 $C W$ complex 是一个 $\mathrm{CW}$ 复形，其中每个面都是规则的。我们注意到每个单纯复形或多面体都是规 则的CW复杂的。

## 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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## 数学代写|几何组合代写Geometric Combinatorics代考|Cluster Complexes and Generalized Associahedra

This section is based on $[\mathbf{1 9}, \mathbf{2 1}]$, except for the last statement in Theorem 4.11, which was proved in $[\mathbf{1 3}]$.

It can be shown that in a given cluster algebra of finite type, each seed is uniquely determined by its cluster. Consequently, the combinatorics of exchanges is encoded by the cluster complex, a simplicial complex (indeed, a pseudomanifold) on the set of all cluster variables whose maximal simplices (facets) are the clusters. See Figure 4.3. By Theorem 4.10, the cluster variables-hence the vertices of the cluster complex-can be naturally labeled by the set $\Phi_{\geq-1}$ of “almost positive roots” in the associated root system $\Phi$.

This dual graph of the cluster complex is precisely the exchange graph of the cluster algebra.

Theorem 4.11 below shows that the cluster complex is always spherical, and moreover polytopal.

Recall that $Q_{\mathbb{R}}$ denotes the $\mathbb{R}$-span of $\Phi$. The $\mathbb{Z}$-span of $\Phi$ is the root lattice, denoted by $Q$.

## 数学代写|几何组合代写Geometric Combinatorics代考|Polytopal Realizations of Generalized Associahedra

We now demonstrate how to explicitly describe each generalized associahedron by a set of linear inequalities.

Theorem 4.17. Suppose that a $\left(-w_{\circ}\right)$-invariant function $F:-\Pi \rightarrow \mathbb{R}$ satisfies the inequalities
$$\sum_{i \in I} a_{i j} F\left(-\alpha_i\right)>0 \quad \text { for all } j \in I .$$
Let us extend $F$ (uniquely) to a $\left\langle\tau_{-}, \tau_{+}\right\rangle$-invariant function on $\Phi_{\geq-1}$. The generalized associahedron is then given in the dual space $Q_{\mathbb{R}}^*$ by the linear inequalities
$$\langle\mathbf{z}, \alpha\rangle \leq F(\alpha), \text { for all } \alpha \in \Phi_{\geq-1}$$
An example of a function $F$ satisfying the conditions in Theorem 4.17 is obtained by setting $F\left(-\alpha_i\right)$ equal to the coefficient of the simple coroot $\alpha_i^{\vee}$ in the half-sum of all positive coroots. (Coroots are the roots of the “dual” root system; see $[\mathbf{9}, \mathbf{3 4}]$.

Example 4.18. In type $A_3$, Theorem 4.17 is illustrated in Figure 4.7, which shows a 3-dimensional associahedron given by the inequalities
\begin{aligned} \max \left(-z_1,-z_3, z_1, z_3, z_1+z_2, z_2+z_3\right) & \leq 3 / 2 \ \max \left(-z_2, z_2, z_1+z_2+z_3\right) & \leq 2 . \end{aligned}

## 数学代写|几何组合代写Geometric Combinatorics代考|Polytopal Realizations of Generalized Associahedra

$$\sum_{i \in I} a_{i j} F\left(-\alpha_i\right)>0 \quad \text { for all } j \in I .$$

$$\langle\mathbf{z}, \alpha\rangle \leq F(\alpha), \text { for all } \alpha \in \Phi_{\geq-1}$$

$$\max \left(-z_1,-z_3, z_1, z_3, z_1+z_2, z_2+z_3\right) \leq 3 / 2 \max \left(-z_2, z_2, z_1+z_2+z_3\right) \quad \leq 2$$

## 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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## avatest™帮您通过考试

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## 数学代写|几何组合代写Geometric Combinatorics代考|Other “Finite Type” Classifications

The classification of root systems is similar or identical to several other classifications of objects of “finite type,” briefly reviewed below.

Non-crystallographic root systems
Lifting the crystallographic restriction does not allow very many additional root systems. The only non-crystallographic irreducible finite root systems are those of types $H_3, H_4$ and $I_2(m)$ for $m=5$ or $m \geq 7$. See [34].
Coxeter groups and reflection groups
By Theorems $2.10$ and 2.11, the classification of finite Coxeter groups is parallel to the classification of reflection groups and is essentially the same as the classification of root systems. The difference is that the root systems $B_n$ and $C_n$ correspond to the same Coxeter group $B_n$. A Coxeter group is encoded by its Coxeter diagram, a graph whose vertex set is $S$, with an edge $s$ – $t$ whenever $m_{s t}>2$. If $m_{s t}>3$, the edge is labeled by $m_{s t}$. Figure $2.2$ shows the Coxeter diagrams of the finite irreducible Coxeter systems, including the non-crystallographic Coxeter groups $\mathrm{H}_3$, $H_4$ and $I_2(m)$. The group $G_2$ appears as $I_2(6)$. See $[34]$ for more details.

Regular polytopes
By Theorem 1.5, the symmetry group of a regular polytope is a reflection group. In fact, it is a Coxeter group whose Coxeter diagram is linear: the underlying graph is a path with no branching points. This narrows down the possibilities, leading to the conclusion that there are no other regular polytopes besides the ones described in Section 1.2. In particular, there are no “exceptional” regular polytopes beyond dimension 4: only simplices, cubes, and crosspolytopes.
Lie algebras
The original motivation for the Cartan-Killing classification of root systems came from Lie theory. Complex finite-dimensional simple Lie algebras correspond naturally, and one-to-one, to finite irreducible crystallographic root systems. There exist innumerable expositions of this classical subject; see, e.g.,

## 数学代写|几何组合代写Geometric Combinatorics代考|Reduced Words and Permutohedra

Each element $w \in W$ can be written as a product of elements of $S$ :
$$w=s_{i_1} \cdots s_{i_{\ell}} .$$
A shortest factorization of this form (or the corresponding sequence of subscripts $\left.\left(i_1, \ldots, i_{\ell}\right)\right)$ is called a reduced word for $w$; the number of factors $\ell$ is called the length of $w$.

Any finite Coxeter group has a unique element $w_{\circ}$ of maximal length. In the symmetric group $\mathcal{S}{n+1}=A_n$, this is the permutation $w{\circ}$ that reverses the order of the elements of the set ${1, \ldots, n+1}$.

Example 2.12. Let $W=\mathcal{S}_4$ be the Coxeter group of type $A_3$. The standard choice of simple reflections yields $S=\left{s_1, s_2, s_3\right}$, where $s_1, s_2$ and $s_3$ are the transpositions which interchange 1 with 2,2 with 3 , and 3 with 4 , respectively. (Cf. Example 1.7.)

The word $s_1 s_2 s_1 s_3 s_2 s_3$ is a non-reduced word for the permutation that interchanges 1 with 3 and 2 with 4 . This permutation has two reduced words $s_2 s_1 s_3 s_2$ and $s_2 s_3 s_1 s_2$.

An example of a reduced word for $w_{\circ}$ is $s_1 s_2 s_1 s_3 s_2 s_1$. There are 16 such reduced words altogether. (Cf. Example $2.14$ and Theorem 2.15.)

Recall from Section $1.2$ that we label the regions $R_w$ of the Coxeter arrangement by the elements of the reflection group $W$, so that $R_w$ is the image of $R_1$ under the action of $w$. More generally, $R_{u v}=u\left(R_v\right)$.

## 数学代写|几何组合代写Geometric Combinatorics代考|Reduced Words and Permutohedra

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 {. }
$$## 数学代写|几何组合代写Geometric Combinatorics代考|Reflection Groups Our first goal will be to understand the finite groups generated by linear reflections in a vector space V. It turns out that for such a group, it is always possible to define a Euclidean structure on V so that all of the reflections in the group are ordinary orthogonal reflections. The study of groups generated by orthogonal reflections is a classical subject, which goes back to the classification of Platonic solids by the ancient Greeks. Let V be a Euclidean space. In what follows, all reflecting hyperplanes pass through the origin, and all reflections are orthogonal. A finite reflection group is a finite group generated by some reflections in V. In other words, we choose a collection of hyperplanes such that the group of orthogonal transformations generated by the corresponding reflections is finite. Infinite reflection groups are also interesting, but in these lectures, “reflection group” will always mean a finite one. The set of reflections in a reflection group W is typically larger than a minimal set of reflections generating W. This is illustrated in Figure 1.1, where W is the group of symmetries of a regular pentagon. This 10-element group is generated by two reflections s and t whose reflecting lines make an angle of \pi / 5. It consists of 5 reflections, 4 rotations, and the identity element. In Figure 1.1, each of the 5 lines is labeled by the corresponding reflection. ## 几何组合代写 ## 数学代写|几何组合代写Geometric Combinatorics代考|The Pentagon Recurrence 考虑一个序列 f_1, f_2, f_3, \ldots. 递归地定义为 f_1=x, f_2=y ，和 (1)$$
f_{n+1}=\frac{f_n+1}{f_{n-1}} .
$$因此，前五个条目是$$
x, y, \frac{y+1}{x}, \frac{x+y+1}{x y}, \frac{x+1}{y} .
$$没想到第六和第七个条目是 x 和 y ，因此序列是周期性的，周期为 5 ！我们称 (1) 五边形弟倠。 1 这个序列还有另一个重要的性质。先验地，我们只能期望它的项是以下的有理函数 x 和 y. 事实上，每个 f_i 是洛朗多项式（实际上， 具有非负整数系数）。这是所调的洛朗现象的一个例子。 将这种重昆表示为由两个连续项组成的 “移动䆬口” 的演变会很有邦助 f_i 和 f_{i+1} :$$
\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,
$$地图在哪里 \tau_1 和 \tau_2 由定义$$
\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] .
$$两个都 \tau_1 和 \tau_2 是对合: \tau_1^2=\tau_2^2=1 ，其中 1 表示恒等映射。递归 (1) 的 5 -周期性转化为恒等式 \left(\tau_2 \tau_1\right)^5=1. 也就是说，由 \tau_1 和 \tau_2 是一个有 10 个元拜的二面角群。 现在让我们考虑一对相似但更简单的地图。扔掉十1的定义中出现的 \tau_1 和 \tau_2 ，并取对数。然后我们得到一对线性映射$$
s_1:[x y] \longmapsto[y-x y] \text { and } s_2:[x y] \longmapsto[x x-y] .
$$## 数学代写|几何组合代写Geometric Combinatorics代考|Reflection Groups 我们的第一个目标是理解向量空间中线性反射生成的有限群 V. 事实证明，对于这样的群，总是可以定义一个欧几里德结构 V 这样 组中的所有反射都是普通的正交反射。研究正交反射产生的群是一门经典学科，可以追溯到古希腊人对柏拉图立体的分类。 让 V 成为欧氏空间。在下文中，所有反射超平面都通过原点，并且所有反射都是正交的。有限反射群是由一些反射产生的有限群 V . 换句话说，我们选择一组超平面，使得相应反射生成的正交音换组是有限的。无限反射群也很有趣，但在这些讲座中，”反射 群” 总是指有限的。 反射组中的一组反射 W 通常大于生成的最小反射集 W. 如图所示 1.1 ， 在哪里 W 是正五边形的对称群。这个 10 元嗉群是由两次反 射产生的 s 和 t 其反射线的角度为 \pi / 5. 它由 5 次反射、 4 次旋转和标识元嗉组成。在图 1.1 中， 5 条线中的每条线都标有相应的反 射。 数学代写|几何组合代写Geometric Combinatorics代考 请认准UprivateTA™. UprivateTA™为您的留学生涯保驾护航。 ## 微观经济学代写 微观经济学是主流经济学的一个分支，研究个人和企业在做出有关稀缺资源分配的决策时的行为以及这些个人和企业之间的相互作用。my-assignmentexpert™ 为您的留学生涯保驾护航 在数学Mathematics作业代写方面已经树立了自己的口碑, 保证靠谱, 高质且原创的数学Mathematics代写服务。我们的专家在图论代写Graph Theory代写方面经验极为丰富，各种图论代写Graph Theory相关的作业也就用不着 说。 ## 线性代数代写 线性代数是数学的一个分支，涉及线性方程，如：线性图，如：以及它们在向量空间和通过矩阵的表示。线性代数是几乎所有数学领域的核心。 ## 博弈论代写 现代博弈论始于约翰-冯-诺伊曼（John von Neumann）提出的两人零和博弈中的混合策略均衡的观点及其证明。冯-诺依曼的原始证明使用了关于连续映射到紧凑凸集的布劳威尔定点定理，这成为博弈论和数学经济学的标准方法。在他的论文之后，1944年，他与奥斯卡-莫根斯特恩（Oskar Morgenstern）共同撰写了《游戏和经济行为理论》一书，该书考虑了几个参与者的合作游戏。这本书的第二版提供了预期效用的公理理论，使数理统计学家和经济学家能够处理不确定性下的决策。 ## 微积分代写 微积分，最初被称为无穷小微积分或 “无穷小的微积分”，是对连续变化的数学研究，就像几何学是对形状的研究，而代数是对算术运算的概括研究一样。 它有两个主要分支，微分和积分；微分涉及瞬时变化率和曲线的斜率，而积分涉及数量的累积，以及曲线下或曲线之间的面积。这两个分支通过微积分的基本定理相互联系，它们利用了无限序列和无限级数收敛到一个明确定义的极限的基本概念 。 ## 计量经济学代写 什么是计量经济学？ 计量经济学是统计学和数学模型的定量应用，使用数据来发展理论或测试经济学中的现有假设，并根据历史数据预测未来趋势。它对现实世界的数据进行统计试验，然后将结果与被测试的理论进行比较和对比。 根据你是对测试现有理论感兴趣，还是对利用现有数据在这些观察的基础上提出新的假设感兴趣，计量经济学可以细分为两大类：理论和应用。那些经常从事这种实践的人通常被称为计量经济学家。 ## 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。 Posted on Categories:几何组合, 数学代写 ## 数学代写|几何组合代写Geometric Combinatorics代考|MAT361 Identities in the Algebra of Polyhedra 如果你也在 怎样代写几何组合Geometric combinatorics MAT361学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。几何组合Geometric combinatorics 是数学的一个分支，尤其是组合学。它包括一些子领域，如多面体组合学（研究凸多面体的面），凸几何学（研究凸集，特别是其交叉点的组合学），以及离散几何学，这又在计算几何学方面有许多应用。 几何组合Geometric combinatorics其他重要领域包括多面体的度量几何，如关于凸多面体刚性的考奇定理。对规则多面体、阿基米德实体和接吻数的研究也是几何组合学的一部分。特殊的多面体也被考虑在内，如全等面体，协和面体和伯克霍夫多面体。 几何组合Geometric combinatorics 代写，免费提交作业要求， 满意后付款，成绩80\%以下全额退款，安全省心无顾虑。专业硕 博写手团队，所有订单可靠准时，保证 100% 原创。 最高质量的几何组合Geometric combinatorics 作业代写，服务覆盖北美、欧洲、澳洲等 国家。 在代写价格方面，考虑到同学们的经济条件，在保障代写质量的前提下，我们为客户提供最合理的价格。 由于作业种类很多，同时其中的大部分作业在字数上都没有具体要求，因此几何组合Geometric combinatorics 作业代写的价格不固定。通常在专家查看完作业要求之后会给出报价。作业难度和截止日期对价格也有很大的影响。 ## avatest™帮您通过考试 avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！ 在不断发展的过程中，avatest™如今已经成长为论文代写，留学生作业代写服务行业的翘楚和国际领先的教育集团。全体成员以诚信为圆心，以专业为半径，以贴心的服务时刻陪伴着您， 用专业的力量帮助国外学子取得学业上的成功。 •最快12小时交付 •200+ 英语母语导师 •70分以下全额退款 想知道您作业确定的价格吗? 免费下单以相关学科的专家能了解具体的要求之后在1-3个小时就提出价格。专家的 报价比上列的价格能便宜好几倍。 我们在数学Mathematics代写方面已经树立了自己的口碑, 保证靠谱, 高质且原创的数学Mathematics代写服务。我们的专家在金融数学Financial Mathematics代写方面经验极为丰富，各种金融数学Financial Mathematics相关的作业也就用不着 说。 ## 数学代写|几何组合代写Geometric Combinatorics代考|Identities in the Algebra of Polyhedra What can we do with polyhedra? One important observation is that the image of a polyhedron under a linear transformation is a polyhedron. Theorem 1. Let P \subset \mathbb{R}^d be a polyhedron and let T: \mathbb{R}^d \longrightarrow \mathbb{R}^k be a linear transformation. Then T(P) \subset \mathbb{R}^k is a polyhedron. Furthermore, if P is a rational polyhedron and T is a rational linear transformation (that is, the matrix of T is rational), then T(P) is a rational polyhedron. The crucial step in the proof. Let us consider the following particular case: k=d-1 and T is the projection onto the first (d-1) coordinates: \left(x_1, \ldots, x_d\right) \longmapsto \left(x_1, \ldots, x_{d-1}\right). Suppose that the polyhedron P is defined by a system of linear inequalities:$$
\sum_{j=1}^d a_{i j} x_j \leq b_i \quad \text { for } \quad i=1, \ldots, m
$$Let us look at the coefficients of x_d. Let I_{+}=\left{i: a_{i d}>0\right}, I_{-}=\left{i: a_{i d}<0\right}, and I_0=\left{i: a_{i d}=0\right}. Then a point y=\left(x_1, \ldots, x_{d-1}\right) belongs to T(P) if and only if$$
\sum_{j=1}^{d-1} a_{i j} x_j \leq b_j \quad \text { for } \quad i \in I_0
$$and there exists x_d such that (2)$$
\begin{aligned}
&x_d \leq \frac{b_i}{a_{i d}}-\sum_{j=1}^{d-1} \frac{a_{i j}}{a_{i d}} x_j \quad \text { for } \quad i \in I_{+} \
&x_d \geq \frac{b_i}{a_{i d}}-\sum_{j=1}^{d-1} \frac{a_{i j}}{a_{i d}} x_j \quad \text { for } \quad i \in I_{-}
\end{aligned}
$$## 数学代写|几何组合代写Geometric Combinatorics代考|A plausible argument A plausible argument. We don’t really prove this important theorem, although we come very close. We start by showing that the theorem is not obviously false. We notice that if P is non-empty and does not contain vertices then P contains a line and hence we can choose g=[P]. Suppose we have been sloppy and included in the sum not only all vertices v of P but also some non-vertices v \in P. No harm done: if v \in P is a non-vertex then \operatorname{co}(P, v) contains a line and so we just have to adjust g. This shows that the formula is robust enough. Suppose that the theorem holds for some polyhedron P \subset \mathbb{R}^d and let T: \mathbb{R}^d \longrightarrow \mathbb{R}^k be a sufficiently generic linear transformation. We claim that the theorem holds for the image T(P). Indeed, by Theorem 2 the transformation T gives rise to the transformation \mathcal{T} on the algebra of polyhedra. Let us apply \mathcal{T} to both sides of the identity. We have \mathcal{T}[P]=[T(P)] and \mathcal{T}[\operatorname{co}(P, v)]=[T(\operatorname{co}(P, v))]= [\operatorname{co}(T(P), T(v))], cf. Review Problem 10. We have to be somewhat careful with g : we know that g is a linear combination of indicators of polyhedra with lines. If we are unlucky, the kernel of T may “eat up” some of those lines and \mathcal{T}(g) will not lie in \mathcal{P}_0\left(\mathbb{R}^k\right). This is the reason why we chose T to be “generic”. Thus if we prove the theorem for some “model” polyhedra P, we can extend it (with some care) to polyhedra obtained from P by linear transformations. ## 几何组合代写 ## 数学代写|几何组合代写Geometric Combinatorics代考|Identities in the Algebra of Polyhedra 我们可以用多面体做什么? 一个重要的观察是，线性音换下的多面体图像是多面体。 定理 1.让 P \subset \mathbb{R}^d 是一个多面体，让 T: \mathbb{R}^d \longrightarrow \mathbb{R}^k 是一个线性音换。然后 T(P) \subset \mathbb{R}^k 是 个多面体。此外，如果 P 是 个有 理多面体并且 T 是有理线性音换（即矩阵 T 是有理数），那么 T(P) 是有理多面体。 证明的关键步骙。让我们考虑以下特殊情况: k=d-1 和 T 是投影到第一个 (d-1) 坐标: \left(x_1, \ldots, x_d\right) \longmapsto\left(x_1, \ldots, x_{d-1}\right). 假设客面体 P 由线性不等式秒统定义:$$
\sum_{j=1}^d a_{i j} x_j \leq b_i \quad \text { for } \quad i=1, \ldots, m
$$让我们看一下悉数 x_d. 让left 缺少或无法识别的分隔符，，和 left 缺少或无法识别的分隔符 . 然后一个点 y=\left(x_1, \ldots, x_{d-1}\right) 属于 T(P) 当且仅当$$
\sum_{j=1}^{d-1} a_{i j} x_j \leq b_j \quad \text { for } \quad i \in I_0
$$并且存在 x_d 这样 (2)$$
x_d \leq \frac{b_i}{a_{i d}}-\sum_{j=1}^{d-1} \frac{a_{i j}}{a_{i d}} x_j \quad \text { for } \quad i \in I_{+} \quad x_d \geq \frac{b_i}{a_{i d}}-\sum_{j=1}^{d-1} \frac{a_{i j}}{a_{i d}} x_j \quad \text { for } \quad i \in I_{-}


## 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。