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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, $R_{u v}=u\left(R_v\right)$.

## 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。