statistics-lab™ 为您的留学生涯保驾护航 在代写黎曼几何Riemannian geometry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。我们的专家在代写黎曼几何Riemannian geometry代写方面经验极为丰富，各种代写黎曼几何Riemannian geometry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
assignmentutor™您的专属作业导师

## 数学代写|黎曼几何代写Riemannian geometry代考|Extrinsic Criteria for Weighted Parabolicity

In this section we analyze the weighted parabolicity of a submanifold $P$ immersed in a weighted manifold $M$ as it was done, (in the Riemannian case), in Sections 14 , 18 and 19 . In this case we consider the restriction to $P$ of the distance function to the pole, (the extrinsic distance), and the family of capacitors in the submanifolds, constituted by the extrinsic balls.

Our first result is an extension to the weighted setting of previous theorems for Riemannian manifolds in [EP] by Esteve and the third author, and in [MaP4, MaP7]. We note that the particular situations of weighted $(w, f)$-model spaces were established in Theorems $3.2$ and $3.3$ of [HPR1].

In [HPR2] a parabolicity criterion is established assuming lower bounds on the $q$-weighted sectional curvatures of the ambient manifold and upper bounds on the derivatives of the weight and the radial weighted mean curvatures of the submanifold.

## 数学代写|黎曼几何代写Riemannian geometry代考|The Grigor’yan–Fernandez Criterion

Our aim in this subsection is to give a weighted version of the hyperbolicity criterion studied in Section 19 (see also [CHS] for a weighted version with a modified isoperimetric profile).

Let $M$ be a Riemannian manifold with a continuous density $f=e^{h}$. For any open set $\Omega \subseteq M$, the weighted isoperimetric profile of $\Omega$ is the function $I_{\Omega, h}$ : $\left[0, V_{h}(\Omega)\right] \rightarrow \mathbb{R}$ defined by $I_{\Omega, h}(0)=0$, and
$I_{\Omega, h}(v)=\inf \left{A_{h}(\Sigma) ; \Sigma \subseteq \bar{\Omega}\right.$ is a compact hypersurface enclosing weighted volume $v}$,
for any $\left.v \in\left(0, V_{h} \Omega\right)\right]$
Remark 27.1. Obviously we get the weighted isoperimetric inequality
$$A_{h}(\Sigma) \geq I_{\Omega, h}\left(V_{h}(E)\right) \geq I_{M, h}\left(V_{h}(E)\right)$$
for any open set $E \subseteq \Omega$ with smooth boundary $\Sigma$.
Remark 27.2. On the other hand, if we consider the function $h=0$, and the sets $\Omega_{0}={p}$ and $\Omega=M$ in Definition $19.1$, then, given $t \geq 0$, the weighted isoperimetric profile of $\Omega$ is the $\Omega_{0}={p}$-rooted isoperimetric profile of $M$, namely, $I_{\Omega, h}(t)=\phi_{M,{p}}(t) .$

We consider $\Omega$ a precompact domain in $M$ and $\psi: \Omega \rightarrow \mathbb{R}$ a smooth function such that $\psi(\Omega)=[a, b]$ with $a<b$. Following the notation of Section 6 , applying formula (6.2) with $u=g f=g e^{h}$, and assuming that the set of critical points of $\psi$, denoted as $\Omega_{0}$, has measure zero we deduce,
\begin{aligned} \int_{\Omega} g|\nabla \psi| d V_{h} &=\int_{a}^{b}\left(\int_{\Sigma_{t}} g d A_{h, t}\right) d t \ \frac{d}{d t} V_{h}(t) &=\int_{\Gamma(t)}|\psi(x)|^{-1} d A_{h, t} \end{aligned}

where $V_{h}(t)=\operatorname{Vol}{h}(\Omega(t))$. So we can state the following weighted version of the co-area formula: Theorem 27.3. i) For every integrable function $u$ on $\bar{\Omega}$ : $$\int{\Omega} u \cdot|\nabla \psi| d V_{h}=\int_{a}^{b}\left(\int_{\Gamma(t)} u d A_{h, t}\right) d t$$
where $d A_{h, t}$ is the weighted volume element of $\Gamma(t)$.
ii) The function $V_{h}(t):=\operatorname{Vol}{\mathrm{h}}(\Omega(t))$ is a smooth function on the regular values of $\psi$, where its derivative is given by (assuming that the set of critical points of $\psi$ has measure zero): $$\frac{d}{d t} V{h}(t)=\int_{\Gamma(t)}|\nabla \psi|^{-1} d A_{h, t}$$
Remark 27.4. If $g \geq 0$ on $\Omega$ and assuming that the set of critical points of $\psi, \Omega_{0}$, has null measure, we have, using the first equation in (27.2), that
\begin{aligned} \int_{\Omega} g d V_{h} &=\int_{\Omega_{0}} g d V_{h}+\int_{\Omega-\Omega_{0}}\left(g|\nabla \psi|^{-1}\right)|\nabla \psi| d V_{h} \ &=\int_{a}^{b}\left(\int_{\Sigma_{t}} g|\nabla \psi|^{-1} d A_{h, t}\right) d t \end{aligned}
We are now ready to prove a lower bound for the capacity via the isoperimetric profile function.

## 数学代写|黎曼几何代写Riemannian geometry代考|Graphs and Flows

In the two final sections of these notes we now briefly indicate how many of the concepts from the previous sections can be carried over almost verbatim to locally finite graphs and thus give a fruitful alternative understanding of what is going on. We first recall that the dimension of $P^{m}$ has so far been assumed to be $m>1$, partly because the intrinsic geometry of geodesic segments is completely trivial. This viewpoint is, of course, altered considerably if we allow geodesic segments to be joined in such a way as to form a metric geodesic graph in the ambient space. The analysis of restricted distance functions on minimal geodesic graphs in Riemannian manifolds has been studied in [Ma7] following essentially the same lines of reasoning – and the same set of goals – as in the first edition of the present notes.

In view of the new results for (sub-)manifolds that we have reported above, there are then several interesting and pertinent challenges and questions that are calling for similar results to hold true on (minimal) metric graphs in (weighted) manifolds with suitable bounds on their curvatures. Indeed, the notions of Dirichlet spectrum, mean exit time moment spectrum, the weighted capacities, the type problem, etc. are all well defined on such graph-structures. Therefore the quest for finding “good” relations between the Dirichlet spectrum and the moment spectrum in that setting is quite natural. This also holds for the quest of answering the Kac question for either one of the two spectra. These questions can thus be studied following the same lines – or other refined lines with graph theoretic and other discrete tools at our disposal – as in the previous smooth geometric settings. We refer to the papers [McT)Ma, McПMMh, CKT], which contain interesting results in this direction.

Here, however, we will only consider the purely combinatorial structure of graphs, which already by itself can be considered as carrier of such fundamental notions as potential functions, capacity, isoperimetric inequalities, etc., and for which we may also state the type problem, i.e., whether random walk on the graph is transient or recurrent. This latter question will be addressed and discussed in detail via a concrete example in the next section.

We let $G=(V, E)$ denote a finite graph with edge set $E$ and vertex set $V$. If we associate the resistance value of $1(\mathrm{Ohm})$ to every edge and consider the current from one vertex $a$ to another vertex $b$ in $G$, then we are studying potential theory on finite networks. We refer to [DoyS], [So], and [Wo]. Following [DoyS] we show explicitly the relation between the capacity energy and the harmonic potential in the case of a finite graph.

## 数学代写|黎曼几何代写Riemannian geometry代考|The Grigor’yanFernandez Criterion

\eft 的分隔符缺失或无法识别

$$A_{h}(\Sigma) \geq I_{\Omega, h}\left(V_{h}(E)\right) \geq I_{M, h}\left(V_{h}(E)\right)$$

$$\int_{\Omega} g|\nabla \psi| d V_{h}=\int_{a}^{b}\left(\int_{\Sigma_{t}} g d A_{h, t}\right) d t \frac{d}{d t} V_{h}(t) \quad=\int_{\Gamma(t)}|\psi(x)|^{-1} d A_{h, t}$$

$$\int \Omega u \cdot|\nabla \psi| d V_{h}=\int_{a}^{b}\left(\int_{\Gamma(t)} u d A_{h, t}\right) d t$$

ii) 功能 $V_{h}(t):=\operatorname{Vol} \mathrm{h}(\Omega(t))$ 是一个关于正则值的平滑函数 $\psi$ ，其中它的导数由下式给出（假设临界点的集合 测量为零) :

$$\frac{d}{d t} V h(t)=\int_{\Gamma(t)}|\nabla \psi|^{-1} d A_{h, t}$$

$$\int_{\Omega} g d V_{h}=\int_{\Omega_{0}} g d V_{h}+\int_{\Omega-\Omega_{0}}\left(g|\nabla \psi|^{-1}\right)|\nabla \psi| d V_{h} \quad=\int_{a}^{b}\left(\int_{\Sigma_{t}} g|\nabla \psi|^{-1} d A_{h, t}\right) d t$$

## 有限元方法代写

assignmentutor™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。