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数学代写|超平面置换理论代写Hyperplane Arrangements代考|Math565 The number of regions

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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

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