Posted on Categories:Complex analys, 复分析, 数学代写

# 数学代写|复分析代写Complex analysis代考|Weak lineal convexity

avatest™

avatest复分析Complex analysis代写，免费提交作业要求， 满意后付款，成绩80\%以下全额退款，安全省心无顾虑。专业硕 博写手团队，所有订单可靠准时，保证 100% 原创。avatest™， 最高质量的复分析Complex analysis作业代写，服务覆盖北美、欧洲、澳洲等 国家。 在代写价格方面，考虑到同学们的经济条件，在保障代写质量的前提下，我们为客户提供最合理的价格。 由于统计Statistics作业种类很多，同时其中的大部分作业在字数上都没有具体要求，因此复分析Complex analysis作业代写的价格不固定。通常在经济学专家查看完作业要求之后会给出报价。作业难度和截止日期对价格也有很大的影响。

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

## 数学代写|复分析代写Complex analysis代考|Weak lineal convexity

Definition 9.5.3 An open subset $\Omega$ of $\mathbf{C}^n$ is said to be weakly lineally convex if there passes, through every point on the boundary of $\Omega$, a complex affine hyperplane which does not cut $\Omega$.
It is clear that every lineally convex open set is weakly lineally convex. The converse does not hold. This is not difficult to see if we allow sets that are not connected:

Example 9.5.4 Given a number $c$ with $0<c<1$, define an open set $\Omega_c$ in $\mathbf{C}^2$ as the union of the set
$$\left{z=\left(x_1+\mathrm{i} y_1, x_2+\mathrm{i} y_2\right) \in \mathbf{C}^2 ; c<\left|x_1\right|<1,\left|y_1\right|<1,\left|x_2\right|<c,\left|y_2\right|<c\right}$$
with the two sets obtained by permuting $x_1, x_2$ and $y_2$. Thus $\Omega_c$ consists of six boxes. It is easy to see that it is weakly lineally convex, but there are many points in its complement such that every complex line passing through that point hits $\Omega_c$.

Any complex line intersects the real hyperplane defined by $y_1=0$ in the empty set or in a real line or in a real two-dimensional plane, and the three-dimensional set $\left{z ; y_1=0\right} \cap \Omega_c$ is easy to visualize.
It is less easy to construct a connected set with these properties, but this has been done by Yužakov \& Krivokolesko (1971b:325, Example 2). See also an example due to Hörmander in the book by Andersson, Passare \& Sigurdsson (2004:20-21, Example 2.1.7).

However, the boundary of the constructed set is not of class $C^1$, and this is essential. Indeed, Yužakov \& Krivokolesko (1971b:323, Theorem 1) proved that a connected bounded open set with “smooth” boundary is locally weakly lineally convex in the sense of Definition 9.5.8 below if and only if it is lineally convex. It is then even C-convex (1971b:324, Assertion). See also Corollary 4.6.9 in (Hörmander 1994), which states that a connected bounded open set with boundary of class $C^1$ is locally weakly lineally convex if and only if it is $\mathbf{C}$-convex (and every $\mathbf{C}$ convex open set is lineally convex).

## 数学代写|复分析代写Complex analysis代考|Local weak lineal convexity

Definition 9.5.7 We shall say that an open set $\Omega \subset \mathbf{C}^n$ is locally weakly lineally convex if for every point $p$ there exists a neighborhood $V$ of $p$ such that $\Omega \cap V$ is weakly lineally convex.
Obviously, a weakly lineally convex open set has this property, but the converse does not hold, which is obvious for sets that are not connected: Take the union of two open balls whose closures are disjoint. Also for connected sets the converse does not hold as we showed in Example 9.4.8. In that example it is essential that the boundary is not smooth.

Zelinskij (1993:118, Example 13.1) constructs an open set which is locally weakly lineally convex but not weakly lineally convex. The set is not equal to the interior of its closure.

Definition 9.5.8 Let us say that an open set $\Omega$ is locally weakly lineally convex in the sense of Yužakov and Krivokolesko (1971b:323) if for every boundary point $p$ there exists a complex hyperplane $Y$ passing through $p$ and a neighborhood $V$ of $p$ such that $Y$ does not meet $V \cap \Omega$.

As we shall see, this property is strictly weaker than the local weak lineal convexity defined above in Definition 9.5.7.

Hörmander (1994:Proposition 4.6.4) and Andersson, Passare \& Sigurdsson (2004: Proposition 2.5.8) use this property only for open sets with boundary of class $C^1$. Then the hyperplane $Y$ is unique.

For all open sets, local weak lineal convexity obviously implies local weak lineal convexity in the sense of Yužakov and Krivokolesko. In the other direction, Hörmander’s Proposition 4.6.4 shows that for bounded open sets with boundary of class $C^1$, local weak lineal convexity in the sense of Yužakov and Krivokolesko implies local weak lineal convexity even weak lineal convexity if the set is connected.

## 数学代写|复分析代写Complex analysis代考|Weak lineal convexity

$\backslash$ left $\left{z=\backslash\right.$ left $\left(x_{-} 1+\backslash\right.$ mathrm ${i} y_{-} 1, x_{-} 2+\backslash$ mathrm ${i} y_{-} 2 \backslash$ right $) \backslash$ in $\backslash$ mathbf ${c} \wedge 2 ; c<\backslash$ left $\mid x_{-} 1 \backslash$ right $\mid<1, \backslash$ left $\mid y_{-}$

$\backslash$ 左 $\left{z ; y_{-} 1=0 \backslash\right.$ right $}$ lcap \Omega_c 很容易形象化。

Theorem 1) 证明了具有“平滑”边界的连通有界开集在下面定义 9.5 .8 的意义上是局部弱线性凸的，当且仅当它 是线性凸的。它甚至是 C 凸的 (1971b:324，Assertion)。另请参见 (Hörmander 1994) 中的推论 4.6.9，其 中指出具有类边界的连通有界开集 $C^1$ 是局部弱线性凸的当且仅当它是 $\mathbf{C}$-凸面 (并且每个 $\mathbf{口}$ 开集是线性凸 集) 。

## 数学代写|复分析代写Complex analysis代考|Local weak lineal convexity

Zelinskij (1993:118, Example 13.1) 构造了一个局部弱线性凸集但不是弱线性凸集的开集。该集合不 等于其闭包的内部。

Hörmander (1994:Proposition 4.6.4) 和Andersson, Passare $\mid \&$ Sigurdsson (2004:Proposition 2.5.8) 仅将此属性用于具有类边界的开集 $C^1$. 然后是超平面 $Y$ 是独特的。

## 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。