### 数学代写|拓扑学代写Topology代考|MTH 3002

statistics-lab™ 为您的留学生涯保驾护航 在代写拓扑学Topology方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写拓扑学Topology代写方面经验极为丰富，各种代写拓扑学Topology相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|拓扑学代写Topology代考|Two-Dimensional Manifolds

According to the classification Theorem [9], a 2-manifold without boundary is compact if every open cover of it has a finite subcover. The compact 2-manifolds can be exhausted by the two infinite families $\left{S^{2}\right.$ (sphere), $\mathbb{T}^{2}$ (torus), $\mathbb{T}^{2} # \mathbb{T}^{2}$ (double torus), $. .}$ and $\left{\mathbb{P}^{2}\right.$ (projective plane), $\mathbb{P}^{2} # \mathbb{P}^{2}$ (Klein bottle),… $}$, where # denotes the connected sum of two manifolds. A 2 -manifold with boundary can be derived by removing an open disk from a 2 -manifold without boundary. All 2 -manifolds without boundary can be derived by gluing the basic 2 -manifolds with boundaries. Disk, cylinder, and Möbius strip are typical 2 -manifolds with boundaries. Structural surfaces can be described as the orientable 2-manifolds, on which the normal vector can be defined globally. Sphere is a typical orientable 2 -manifold. Structural inter-faces can include both the orientable and non-orientable 2 -manifolds. Non-orientable 2 -manifolds are the 2 -manifolds on which the normal vector can be defined locally instead of globally. Möbius strip is a typical non-orientable 2 -manifold.

Without loosing the generality, topology optimization in this chapter is implemented on both the orientable and non-orientable 2 -manifolds, which are homeomorphic to sphere, torus, Möbius strip, and their connected sum or glued counterparts (Fig.5.1)

## 数学代写|拓扑学代写Topology代考|PDEs for Physical Fields

To implement topology optimization on a 2 -manifold, a design variable, which is a relaxed binary distribution, is defined on this 2 -manifold to represent the geometrical configuration of a surface structure. This design variable is bounded in the typical interval $[0,1]$ with 0 and 1 representing two different material phases, respectively. An optimization problem can be formulated by minimizing or maximizing a design

objective used to evaluate the desired performance of the surface structure with the pattern implicitly expressed on the 2 -manifold. The physical field used to evaluate the performance of the surface structure can be described by a PDE. This optimization problem is a PDE constrained optimization problem. It is nonlinear and challenging to be solved directly. The iterative solution procedure is thus widely utilized.

To ensure the monolithic convergence of the iterative procedure, regularization based on a surface-PDE filter and threshold projection can be imposed on the design variable. The projected design variable is referred to as the material density. The iterative procedure including the surface-PDE filter and the threshold projection can control the minimum length scale of the surface structure and remove the gray regions from the derived pattern of the surface structure.

For optical waves, the physical field is defined in a three-dimensional volume domain and the design variable is defined on the 2-manifold corresponding to an interior interface of the volume domain, the second-order PDE used to describe the physical field can be expressed in a typical abstract form as
$$\left{\begin{array}{l} \nabla \times[p(\nabla \times \mathbf{u}, \mathbf{u})]=c_{s}, \text { in } \Omega \ \left\lfloor p\left(\nabla_{s} \times \mathbf{u}, \mathbf{u}\right) \rrbracket \times \mathbf{n}=\alpha(\bar{\gamma})\left(\mathbf{u}-\mathbf{u}{d}\right), \text { on } \Sigma{S} \hookrightarrow \Omega\right. \ \mathbf{u}=\mathbf{u}{d}, \text { at } \mathscr{P} \subset \Sigma{S} \ \mathbf{u}=\mathbf{u}{0}, \text { on } \partial \Omega \end{array}\right.$$ where I. ] denotes the local jump of a variable across $\Sigma{S}$ embedded in $\Omega ; \hookrightarrow$ is the embedding operator; $u_{d}$ is the known value of the physical variable on the surface structure defined on $\Sigma_{S} ; \Omega$ is the open and bounded volume domain with the boundary of Lipschitz type; $\alpha(\bar{\gamma}$ ) is the material interpolation used to implement the penalization between the interfacial and Dirichlet boundary types. Based on this material interpolation, the mixed boundary condition is formulated at $\Sigma_{S}$ as that in Eq. 5.1. When the material density $\bar{\gamma}$ takes on the value of $0, \alpha$ should be valued to be large enough to ensure the dominance of the Dirichlet term $\left(\mathbf{u}-\mathbf{u}{d}\right)$. When $\alpha$ is valued as 0 with $\bar{\gamma}$ taking on the value of 1 , the mixed boundary condition degenerates into the interfacial type. Sequentially, the pattern of the surface structure with the physical variable known as $u{d}$ can be determined.

## 数学代写|拓扑学代写Topology代考|Regularization

The material density is derived by sequentially implementing the surface-PDE filter and threshold projection operations on the design variable. Inspired by the PDE filter developed in [8], the surface-PDE filter for the design variable is implemented by solving the following surface-PDE defined on the 2-manifold:
$$\left{\begin{array}{l} \nabla_{s} \cdot\left(-r^{2} \nabla_{s} \bar{\gamma}\right)+\bar{\gamma}=\gamma, \text { on } \Sigma_{S} \ -r^{2} \nabla_{s} \bar{\gamma} \cdot \tau=0, \text { at } \partial \Sigma_{S} \end{array},\right.$$
where $r$ is the filter radius; $\gamma \in \mathscr{L}^{2}\left(\Sigma_{S}\right)$ is the design variable; $\bar{\gamma}$ is the filtered design variable; $\hat{\bar{\gamma}}$ is the test function of $\bar{\gamma}$. The variational formulation of the surface-PDE filter in Eq. $5.3$ is
Find $\bar{\gamma} \in \mathscr{H}(\Omega)$ such that
$$f(\bar{\gamma} ; \gamma):=\int_{\Sigma_{s}} r^{2} \nabla_{s} \bar{\gamma} \cdot \nabla_{s} \hat{\gamma}+\bar{\gamma} \hat{\gamma}-\gamma \hat{\gamma} \mathrm{d} s=0, \forall \hat{\gamma} \in \mathscr{H}\left(\Sigma_{S}\right) .$$
The filtered design variable is projected by using the threshold projection method [ 5 , 12], and the material density is derived as
$$\bar{\gamma}=\frac{\tanh (\beta \xi)+\tanh (\beta(\bar{\gamma}-\xi))}{\tanh (\beta \xi)+\tanh (\beta(1-\xi))}$$
where $\beta$ and $\xi$ are the projection parameters and their values are chosen based on numerical experiments. For more details on the choice of the projection parameters, one can refer to Ref. [5].

## 数学代写|拓扑学代写Topology代考|Nanostructures for Localized Surface Plasmonic

Ĵ=1F和0|和|4和04|X=X0=1F和0∫Ω|和|4和04d(距离⁡(X,X0))dΩ其中增强因子归一化为F和0; 和F和0是增强因子X0，对应于金属材料的纳米结构完全填充了设计域；X0是合理选择的增强位置Ω;d(∗)是狄拉克函数；距离(X,X0) 是点之间的欧几里得距离∀X∈Ω和指定的位置X0. 增强位置X0应该呈现在纳米结构的表面或耦合位置，因为在局部表面等离子体共振中，最大增强因子必须在金属表面或耦合位置。

## 数学代写|拓扑学代写Topology代考|Modeling

∇⋅[er−1∇(H和s+H和一世)]+ķ02μr(H和s+H和一世)=0, 在 Ω er−1∇H和s⋅n+jķ0er−1μrH和s=0, 上 Γ一世∪Γ○ H和s(X+一个)=H和s(X)和−jķ⋅一个,n(X+一个)⋅∇H和s(X+一个)=−和−jķ⋅一个n(X)⋅∇H和s(X)  为了 ∀X∈Γps,X+一个∈Γpd

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。