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# 数学代写|复分析代写Complex analysis代考|Lineal convexity viewed from mathematical morphology

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## 数学代写|复分析代写Complex analysis代考|Lineal convexity viewed from mathematical morphology

Lineal concavity is an example of $\mathscr{S}$-concavity, taking $\mathscr{S}$ equal to the family $\mathscr{Z}$ of all complex hyperplanes in $\mathbf{C}^n$ containing the origin. Weak lineal convexity means that $\kappa_{\mathscr{Z}}(\Omega)$ does not meet the boundary of $\Omega$.

There are also local variants of these definitions: we take $\mathscr{S}=\mathscr{Z}r$ as the family of all intersections $Z \cap B \leqslant(0, r)$, where $Z$ is a complex hyperplane passing through the origin. The corresponding $\mathbf{Z}_r$-convexity, for some positive $r$, can be called uniform local lineal convexity. Let us take again the family $\mathscr{S}$ of structuring elements in Definition 9.7 .10 as the set $\mathscr{Z} \subset \mathscr{P}\left(\mathscr{P}\left(\mathbf{C}^n\right)\right)$ of all complex affine hyperplanes in $\mathbf{C}^n$. We define a dilation $\psi: \mathscr{P}(\mathscr{Z}) \rightarrow \mathscr{P}\left(\mathbf{C}^n\right)$ by $$\psi(\mathscr{B})=\bigcup{Z \in \mathscr{B}} Z, \quad \mathscr{B} \in \mathscr{P}(\mathscr{Z})$$
Its lower inverse $\psi_{[-1]}: \mathscr{P}\left(\mathbf{C}^n\right) \rightarrow \mathscr{P}(\mathscr{Z})$ is defined by

$$\psi_{[-1]}(A)=\cup_{\mathscr{B} \in \mathscr{Z}}^{\cup}(\mathscr{B} ; \psi(\mathscr{B}) \subset A)={Z \in \mathscr{Z} ; Z \subset A}, \quad A \in \mathbf{P}\left(\mathbf{C}^n\right)$$
We note that $\varepsilon=\psi_{[-1]}$ is an erosion-as the lower inverse of a dilation, but also easily seen directly. There is a relation between $\Gamma_A$ and $\varepsilon$ :
$$\Gamma_A(b)={Z \in \varepsilon(\complement A) ; b \in Z}$$
The upper inverse $\varepsilon^{[-1]}: \mathscr{P}(\mathscr{Z}) \rightarrow \mathscr{P}\left(\mathbf{C}^n\right)$ of $\varepsilon$ is a dilation defined by
$$\varepsilon^{[-1]}(\mathscr{B})=\underset{A \in \mathscr{P}\left(\mathbf{C}^n\right)}{\cap}(A ; \varepsilon(A) \supset \mathscr{B})=\underset{Z \in \mathscr{B}}{\cup} Z=\psi(\mathscr{B}), \quad \mathscr{B} \in \mathscr{P}(\mathscr{Z}) .$$

## 数学代写|复分析代写Complex analysis代考|Exterior accessibility of Hartogs domains

We shall now study Hartogs domains in $\mathbf{C}^n \times \mathbf{C}$, where we write coordinates as $(z, t) \in \mathbf{C}^n \times \mathbf{C}$.

To define complete Hartogs sets, we may use either the function $R$, the function $h=R^2$, or the function $f=-\log R$. An open complete Hartogs set is then defined equivalently by $|t|<R(z) ;|t|^2<h(z) ;|t|<\mathrm{e}^{-f}$, and we are free to choose whichever is convenient for a specific calculation. We note that if $f$ is plurisubharmonic, then $\Omega$, defined by $\log |t|+f(z)<0$, is pseudoconvex.
Complex hyperplanes in $\mathbf{C}^n \times \mathbf{C}$ are of three kinds:

1. A hyperplane can be given by an equation $\beta \cdot\left(z-z^0\right)=0$ for some $\beta \in \mathbf{C}^n \backslash{0}$ and some point $z^0 \in \mathbf{C}^n$ (we shall call it a vertical hyperplane ).
2. It can have the equation $t=c$ for some complex constant $c$ (we shall call it a horizontal hyperplane ).
3. Finally it can have the equation $t=\beta \cdot\left(z-z^0\right)$, where $\beta$ is nonzero. Such a hyperplane intersects the hyperplane $t=0$ in a hyperplane in $\mathbf{C}^n$ containing $z^0$.
4. 5. \begin{aligned} 6. \left|1+\beta \cdot\left(z-z^0\right)\right| & \leqslant 1+\operatorname{Re} \beta \cdot\left(z-z^0\right)+\gamma\left|\beta \cdot\left(z-z^0\right)\right|^2 \ 7. & \leqslant 1+\operatorname{Re} \beta \cdot\left(z-z^0\right)+\gamma|\beta|_2^2 \cdot\left|z-z^0\right|_2^2, 8. \end{aligned} 9.
10. with equality between the first and last expression only when $z=z^0$ or $\beta=0$. Therefore, if we choose $c>\frac{1}{2}|\beta|_2^2$
11. $$12. R(z) /\left|t^0\right| \leqslant 1+\operatorname{Re} \beta \cdot\left(z-z^0\right)+c\left|z-z^0\right|_2^2, \quad z \in \mathbf{C}^n 13.$$
14. with equality only when $z=z^0$.
15. So the set
16. $$17. U=\left{(z, t) ; \operatorname{Re}\left(t / t^0\right)>1+\operatorname{Re} \beta \cdot\left(z-z^0\right)+c\left|z-z^0\right|_2^2\right} 18.$$
19. taking $c>\frac{1}{2}|\beta|_2^2$, is a set with smooth boundary and the real hyperplane defined by $\operatorname{Re} t / t^0=1+\operatorname{Re} \beta \cdot\left(z-z^0\right)$ is an external tangent plane of class $C^2$ of $\Omega$ at $\left(z^0, t^0\right)$.

## 数学代写|复分析代写Complex analysis代考|Lineal convexity viewed from mathematical morphology

$$\psi_{[-1]}(A)=\cup_{\mathscr{B} \in \mathscr{Z}}^{\cup}(\mathscr{B} ; \psi(\mathscr{B}) \subset A)=Z \in \mathscr{Z} ; Z \subset A, \quad A \in \mathbf{P}\left(\mathbf{C}^n\right)$$

$$\Gamma_A(b)=Z \in \varepsilon(C A) ; b \in Z$$

$$\varepsilon^{[-1]}(\mathscr{B})=\prod_{A \in \mathscr{P}\left(\mathbf{C}^n\right)}(A ; \varepsilon(A) \supset \mathscr{B})=\bigcup_{Z \in \mathscr{B}}^{\cup} Z=\psi(\mathscr{B}), \quad \mathscr{B} \in \mathscr{P}(\mathscr{Z}) .$$

## 数学代写|复分析代写Complex analysis代考|Exterior accessibility of Hartogs domains

1. 超平面可以由方程给出 $\beta \cdot\left(z-z^0\right)=0$ 对于一些 $\beta \in \mathbf{C}^n \backslash 0$ 和一些点 $z^0 \in \mathbf{C}^n$ (我们称它为垂直超平面)。
2. 它可以有等式 $t=c$ 对于一些旿杂的常量 $c$ (我们称它为水平超平面)。
3. $\$ \$$4. \begin } { \text { 对齐 } } 5. \backslash left \mid 1+\backslash beta \backslash cdot \backslash left (z z \wedge 0 \backslash right ) \backslash right \mid \& \backslash leqsiant 1+ loperatorname { Re } \backslash beta lcodot \backslash left \mid z z \wedge 0 \backslash right \mid _2 \wedge 2, 6. \结束 { 对产 } 7. \ \$$
8. 仅当第一个和最后一个表达式相等时 $z=z^0$ 或者 $\beta=0$. 因此，如果我们选择 $c>\frac{1}{2}|\beta|_2^2$
9. \$\$
10. $\mathrm{R}(\mathrm{z}) / \backslash$ 左 $\mid \mathrm{t} \wedge 0 \backslash$ 右 $\mid \backslash$ leqslant $1+\backslash$ operatorname ${$ Re $} \backslash$ beta $\backslash \cot \backslash$ left $(z z \wedge 0 \backslash$ right $)+\mathrm{c} \backslash$ left $|z z \wedge 0 \backslash r i g h t| _2 \wedge 2$, $\backslash$ quad $z \backslash$ in $\backslash$ mathbf ${C} \wedge n$
11. $\$ \$$12. 只有当 z=z^0. 13. 所以奪合 14. \ \$$
15. $U=\backslash$ left ${(z, t)$; \operatorname ${\operatorname{Re}} \backslash$ left $(t / t \wedge 0 \backslash$ right $)>1+\backslash$ operatorname ${\mathrm{Re}} \backslash$ beta $\mid c$ dot $\backslash$ left $(z z \wedge 0 \backslash$ right $)+c \backslash$ left $\mid z z \wedge 0 \backslash$ right $\mid _2 \wedge 2 \backslash$ 右 $}$
16. $\$ \
17. 服用 $c>\frac{1}{2}|\beta|_2^2$ ，是一个边界平滑的集合，实超平面定义为 $\operatorname{Re} t / t^0=1+\operatorname{Re} \beta \cdot\left(z-z^0\right)$ 是类的外切平面 $C^2$ 的 $\Omega$ 在 $\left(z^0, t^0\right)$

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