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

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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。