Posted on Categories:数学代写, 超平面置换理论

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## 数学代写|超平面置换理论代写Hyperplane Arrangements代考| The number of regions

The next result is perhaps the first major theorem in the subject of hyperplane arrangements, due to Thomas Zaslavsky in 1975.

Theorem 2.5. Let $\mathcal{A}$ be an arrangement in an $n$-dimensional real vector space. Then
\begin{aligned} & r(\mathcal{A})=(-1)^n \chi_{\mathcal{A}}(-1) \ & b(\mathcal{A})=(-1)^{\operatorname{rank}(\mathcal{A})} \chi_{\mathcal{A}}(1) \end{aligned}
First proof. Equation (11) holds for $\mathcal{A}=\emptyset$, since $r(\emptyset)=1$ and $\chi_{\emptyset}(t)=t^n$. By Lemmas $2.1$ and $2.2$, both $r(\mathcal{A})$ and $(-1)^n \chi_{\mathcal{A}}(-1)$ satisfy the same recurrence, so the proof follows.

Now consider equation (12). Again it holds for $\mathcal{A}=\emptyset$ since $b(\emptyset)=1$. (Recall that $b(\mathcal{A})$ is the number of relatively bounded regions. When $\mathcal{A}=\emptyset$, the entire ambient space $\mathbb{R}^n$ is relatively bounded.) Now
$$\chi_{\mathcal{A}}(1)=\chi_{\mathcal{A}^{\prime}}(1)-\chi_{\mathcal{A}^{\prime \prime}}(1)$$
Let $d(\mathcal{A})=(-1)^{\operatorname{rank}(\mathcal{A})} \chi_{\mathcal{A}}(1)$. If $\operatorname{rank}(\mathcal{A})=\operatorname{rank}\left(\mathcal{A}^{\prime}\right)=\operatorname{rank}\left(\mathcal{A}^{\prime \prime}\right)+1$, then $d(\mathcal{A})=$ $d\left(\mathcal{A}^{\prime}\right)+d\left(\mathcal{A}^{\prime \prime}\right)$. If $\operatorname{rank}(\mathcal{A})=\operatorname{rank}\left(\mathcal{A}^{\prime}\right)+1$ then $b(\mathcal{A})=0$ [why?] and $L\left(\mathcal{A}^{\prime}\right) \cong L\left(\mathcal{A}^{\prime \prime}\right)$ [why?]. Thus from Lemma $2.2$ we have $d(\mathcal{A})=0$. Hence in all cases $b(\mathcal{A})$ and $d(\mathcal{A})$ satisfy the same recurrence, so $b(\mathcal{A})=d(\mathcal{A})$.

Second proof. Our second proof of Theorem $2.5$ is based on Möbius inversion and some instructive topological considerations. For this proof we assume basic knowledge of the Euler characteristic $\psi(\Delta)$ of a topological space $\Delta$. (Standard notation is $\chi(\Delta)$, but this would cause too much confusion with the characteristic polynomial.) In particular, if $\Delta$ is suitably decomposed into cells with $f_i$ $i$-dimensional cells, then
$$\psi(\Delta)=f_0-f_1+f_2-\cdots$$

## 数学代写|超平面置换理论代写Hyperplane Arrangements代考|Graphical arrangements

There are close connections between certain invariants of a graph $G$ and an associated arrangement $\mathcal{A}G$. Let $G$ be a simple graph on the vertex set $[n]$. Let $E(G)$ denote the set of edges of $G$, regarded as two-element subsets of $[n]$. Write $i j$ for the edge ${i, j}$. Definition 2.5. The graphical arrangement $\mathcal{A}_G$ in $K^n$ is the arrangement $$x_i-x_j=0, i j \in E(G) .$$ Thus a graphical arrangement is simply a subarrangement of the braid arrangement $\mathcal{B}_n$. If $G=K_n$, the complete graph on $[n]$ (with all possible edges $i j$ ), then $\mathcal{A}{K_n}=\mathcal{B}_n$

Definition 2.6. A coloring of a graph $G$ on $[n]$ is a map $\kappa:[n] \rightarrow \mathbb{P}$. The coloring $\kappa$ is proper if $\kappa(i) \neq \kappa(j)$ whenever $i j \in E(G)$. If $q \in \mathbb{P}$ then let $\chi_G(q)$ denote the number of proper colorings $\kappa:[n] \rightarrow[q]$ of $G$, i.e., the number of proper colorings of $G$ whose colors come from $1,2, \ldots, q$. The function $\chi_G$ is called the chromatic polynomial of $G$.

For instance, suppose that $G$ is the complete graph $K_n$. A proper coloring $\kappa:[n] \rightarrow[q]$ is obtained by choosing a vertex, say 1 , and coloring it in $q$ ways. Then choose another vertex, say 2 , and color it in $q-1$ ways, etc., obtaining
$$\chi_{K_n}(q)=q(q-1) \cdots(q-n+1) .$$
A similar argument applies to the graph $G$ of Figure 5. There are $q$ ways to color vertex 1 , then $q-1$ to color vertex 2 , then $q-1$ to color vertex 3 , etc., obtaining
\begin{aligned} \chi_G(q) & =q(q-1)(q-1)(q-2)(q-1)(q-1)(q-2)(q-2)(q-3) \ & =q(q-1)^4(q-2)^3(q-3) \end{aligned}

## 数学代写|超平面置换理论代写Hyperplane Arrangements代考| The number of regions

$$r(\mathcal{A})=(-1)^n \chi_{\mathcal{A}}(-1) \quad b(\mathcal{A})=(-1)^{\operatorname{rank}(\mathcal{A})} \chi_{\mathcal{A}}(1)$$

$$\chi_{\mathcal{A}}(1)=\chi_{\mathcal{A}^{\prime}}(1)-\chi_{\mathcal{A}^{\prime \prime}}(1)$$

$$\psi(\Delta)=f_0-f_1+f_2-\cdots$$

## 数学代写|超平面置换理论代写Hyperplane Arrangements代考|Graphical arrangements

$$\chi_G(q)=q(q-1)(q-1)(q-2)(q-1)(q-1)(q-2)(q-2)(q-3) \quad=q(q-1)^4(q-2)^3(q-3)$$

## 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:数学代写, 测度与积分

## avatest™帮您通过考试

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## 数学代写|测度与积分代写Measure And Integration代考|Polar Coordinates and Surface Measure

Polar Coordinates and Surface Measure. Let
$$S^{d-1}=\left{x \in \mathbb{R}^d:|x|^2:=\sum_{i=1}^d x_i^2=1\right}$$
be the unit sphere in $\mathbb{R}^d$. Let $\Phi: \mathbb{R}^d \backslash(0) \rightarrow(0, \infty) \times S^{d-1}$ and $\Phi^{-1}$ be the inverse map given by
$$\Phi(x):=\left(|x|, \frac{x}{|x|}\right) \text { and } \Phi^{-1}(r, \omega)=r \omega$$
respectively. Since $\Phi$ and $\Phi^{-1}$ are continuous, they are Borel measurable.
Consider the measure $\Phi_* m$ on $\mathcal{B}{(0, \infty)} \otimes \mathcal{B}{S^{d-1}}$ given by
$$\Phi_* m(A):=m\left(\Phi^{-1}(A)\right)$$
for all $A \in \mathcal{B}{(0, \infty)} \otimes \mathcal{B}{S^{d-1}}$. For $E \in \mathcal{B}{S^{d-1}}$ and $a>0$, let $$E_a:={r \omega: r \in(0, a] \text { and } \omega \in E}=\Phi^{-1}((0, a] \times E) \in \mathcal{B}{\mathbb{R}^d} \text {. }$$
Noting that $E_a=a E_1$, we have for $0<a<b, E \in \mathcal{B}_{S^{d-1}}, E$ and $A=(a, b] \times E$ that
\begin{aligned} \Phi^{-1}(A) & ={r \omega: r \in(a, b] \text { and } \omega \in E} \ & =b E_1 \backslash a E_1 . \end{aligned}

## 数学代写|测度与积分代写Measure And Integration代考|Hahn Decomposition Theorem

Definition 13.4. Let $\nu$ be a signed measure on $(X, \mathcal{M})$ and $E \in \mathcal{M}$, then
(1) $E$ is positive if for all $A \in \mathcal{M}$ such that $A \subset E, \nu(A) \geq 0$, i.e. $\left.\nu\right|{\mathcal{M}_E} \geq 0$. (2) $E$ is negative if for all $A \in \mathcal{M}$ such that $A \subset E, \nu(A) \leq 0$, i.e. $\left.\nu\right|{\mathcal{M}E} \leq 0$. (3) $E$ is null if for all $A \in \mathcal{M}$ such that $A \subset E$, i.e. $\left.\nu\right|{\mathcal{M}_E}=0$.
Here $\mathcal{M}_E \equiv{A \cap E: A \in \mathcal{M}}=$ trace of $M$ on $E$.
Lemma 13.5. Suppose that $\nu$ is a signed measure on $(X, \mathcal{M})$. Then
(1) Any subset of a positive set is positive.
(2) The countable union of positive (negative or null) sets is still positive (negative or null).
(3) Let us now further assume that $\nu(\mathcal{M}) \subset[-\infty, \infty)$ and $E \in \mathcal{M}$ is a set such that $\nu(E) \in(0, \infty)$. Then there exists a positive set $P \subseteq E$ such that $\nu(P) \geq \nu(E)$.

Proof. The first assertion is obvious. If $P_j \in \mathcal{M}$ are positive sets, let $P=$ $\bigcup_{n=1}^{\infty} P_n$. By replacing $P_n$ by the positive set $P_n \backslash\left(\bigcup_{j=1}^{n-1} P_j\right)$ we may assume that the $\left{P_n\right}_{n=1}^{\infty}$ are pairwise disjoint so that $P=\bigcup_{n=1}^{\infty} P_n$. Now if $E \subset P$ and $E \in \mathcal{M}$, $E=\coprod_{n=1}^{\infty}\left(E \cap P_n\right)$ so
$$\nu(E)=\sum_{n=1}^{\infty} \nu\left(E \cap P_n\right) \geq 0$$

## 数学代写测度与积分代写Measure And Integration代考|Polar Coordinates and Surface Measure

\left 缺分或无法识别的分隔符

$$\Phi(x):=\left(|x|, \frac{x}{|x|}\right) \text { and } \Phi^{-1}(r, \omega)=r \omega$$

$$\Phi_* m(A):=m\left(\Phi^{-1}(A)\right)$$

$$E_a:=r \omega: r \in(0, a] \text { and } \omega \in E=\Phi^{-1}((0, a] \times E) \in \mathcal{B}^d$$

## 数学代写|测度与积分代写Measure And Integration代考|Hahn Decomposition Theorem

left 缺少或无法识别的分隔符

$$\nu(E)=\sum_{n=1}^{\infty} \nu\left(E \cap P_n\right) \geq 0$$

## 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。