## 数学代写|应用数学代写applied mathematics代考|MTH103

statistics-lab™ 为您的留学生涯保驾护航 在代写应用数学applied mathematics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。我们的专家在代写应用数学applied mathematics代写方面经验极为丰富，各种代写应用数学applied mathematics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|应用数学代写applied mathematics代考|The Lie bracket

In general, a Lie algebra $\mathfrak{g}$ is a vector space with a bilinear, skew-symmetric bracket operation
$$[\cdot, \cdot]: \mathfrak{g} \times \mathfrak{g} \rightarrow \mathfrak{g}$$
that satisfies the Jacobi identity
$$[u,[v, w]]+[v,[w, u]]+[w,[u, v]]=0$$
The Lie bracket of vector fields $\vec{v}, \vec{w}$ is defined by their commutator
$$[\vec{v}, \vec{w}]=\vec{v} \vec{w}-\vec{w} \vec{v}$$
where the vector fields are understood as differential operators. Explicitly, if
$$\vec{v}=\xi^i \partial_{x^i}, \quad \vec{w}=\eta^i \partial_{x^i}$$
then
$$[\vec{v}, \vec{w}]=\left(\xi^j \frac{\partial \eta^i}{\partial x^j}-\eta^j \frac{\partial \xi^i}{\partial x^j}\right) \partial_{x^i}$$
The Lie bracket of vector fields measures the non-commutativity of the corresponding flows:
$$\vec{v}, \vec{w}=\left.\frac{1}{2} \frac{d^2}{d \varepsilon^2}\left(e^{\varepsilon \vec{v}} e^{\varepsilon \vec{w}} e^{-\varepsilon \vec{v}} e^{-\varepsilon \vec{w}}\right) x\right|_{\varepsilon=0} .$$
One can show that the Lie bracket of any two vector field that generate elements of a Lie group of transformations also generates an element of the Lie group. Thus, the infinitesimal generators of the Lie group form a Lie algebra.

## 数学代写|应用数学代写applied mathematics代考|Transformations of the plane

As simple, but useful, examples of Lie transformation groups and their associated Lie algebras, let us consider some transformations of the plane.
The rotations of the plane $g(\varepsilon): \mathbb{R}^2 \rightarrow \mathbb{R}^2$ are given by
$$g(\varepsilon):\left(\begin{array}{l} x \ y \end{array}\right) \mapsto\left(\begin{array}{rr} \cos \varepsilon & -\sin \varepsilon \ \sin \varepsilon & \cos \varepsilon \end{array}\right)\left(\begin{array}{l} x \ y \end{array}\right) \quad \text { where } \varepsilon \in \mathbb{T} .$$
These transformations form a representation of the one-dimensional Lie group $S O(2)$ on $\mathbb{R}^2$. They are the flow of the ODE
$$\frac{d}{d \varepsilon}\left(\begin{array}{l} x \ y \end{array}\right)=\left(\begin{array}{c} -y \ x \end{array}\right)$$
The vector field on the right hand side of this equation may be written as
$$\vec{v}(x, y)=-y \partial_x+x \partial_y$$
and thus
$$g(\varepsilon)=e^{\varepsilon \vec{v}}$$
The Lie algebra $s \mathfrak{o}(2)$ of $S O(2)$ consists of the vector fields
$$\left{-\varepsilon y \partial_x+\varepsilon x \partial_y: \varepsilon \in \mathbb{R}\right}$$
The translations of the plane in the direction $(a, b)$
$$(x, y) \mapsto(x-\varepsilon a, y-\varepsilon b)$$ are generated by the constant vector field
$$a \partial_x+b \partial_y$$
The rotations and translations together form the orientation-preserving Euclidean group of the plane, denoted by $E^{+}(2)$. The full Euclidean group $E(2)$ is generated by rotations, translations, and reflections.

# 应用数学代考

## 数学代写|应用数学代写applied mathematics代考|The Lie bracket

$$[\cdot, \cdot]: \mathfrak{g} \times \mathfrak{g} \rightarrow \mathfrak{g}$$

$$[u,[v, w]]+[v,[w, u]]+[w,[u, v]]=0$$

$$[\vec{v}, \vec{w}]=\vec{v} \vec{w}-\vec{w} \vec{v}$$

$$\vec{v}=\xi^i \partial_{x^i}, \quad \vec{w}=\eta^i \partial_{x^i}$$

$$[\vec{v}, \vec{w}]=\left(\xi^j \frac{\partial \eta^i}{\partial x^j}-\eta^j \frac{\partial \xi^i}{\partial x^j}\right) \partial_{x^i}$$

$\$ \$$Ivec {v}, Ivec {w}=\backslash left. Ifrac {1}{2} \backslash \operatorname{|rac}\left{d^{\wedge} 2\right}{d Ivarepsilon^2 } \backslash \operatorname{left}\left(e^{\wedge}{\operatorname{lvarepsilon} \mid v e c{v}}\right. \mathrm{e}^{\wedge}{ Ivarepsilon Ivec {\mathrm{W}}} \mathrm{e}^{\wedge}{-Ivarepsilon Ivec {\mathrm{v}}} \mathrm{e}^{\wedge}{- Ivarepsilon Ivec {\mathrm{W}}} \backslash right) x right \left.\right|_{-}{lvarepsilon =0} 。 \ \$$

## 数学代写|应用数学代写applied mathematics代考|Transformations of the plane

$$g(\varepsilon):(x y) \mapsto\left(\begin{array}{ccc} \cos \varepsilon \quad-\sin \varepsilon \sin \varepsilon \quad \cos \varepsilon \end{array}\right)(x y) \quad \text { where } \varepsilon \in \mathbb{T} .$$

$$\frac{d}{d \varepsilon}(x y)=(-y x)$$

$$\vec{v}(x, y)=-y \partial_x+x \partial_y$$

$$g(\varepsilon)=e^{\varepsilon \vec{v}}$$

Weft{-lvarepsilon $y \backslash$ \partial_x+Ivarepsilon $x \backslash$ |partial_y: Ivarepsilon \in $\backslash m a t h b b{R} \backslash$ right $}$

$$(x, y) \mapsto(x-\varepsilon a, y-\varepsilon b)$$

$$a \partial_x+b \partial_y$$

## 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。

## 数学代写|应用数学代写applied mathematics代考|MATH101

statistics-lab™ 为您的留学生涯保驾护航 在代写应用数学applied mathematics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。我们的专家在代写应用数学applied mathematics代写方面经验极为丰富，各种代写应用数学applied mathematics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|应用数学代写applied mathematics代考|Translational invariance

A transformation of the space $E$ of dependent and independent variables into itself is called a point transformation. The group $G$ in (2.43) does not include all the point transformations that leave (2.36) invariant. In addition to the scaling transformations (2.44), the space-time translations
$(2.46) \quad(x, t, u) \mapsto(x-\delta, t, u), \quad(x, t, u) \mapsto(x, t-\varepsilon, u)$
where $-\infty<\delta, \varepsilon<\infty$, also leave (2.36) invariant, because the terms in the equation do not depend explicitly on $(x, t)$.

As we will show in Section 9.8, the transformations (2.44) and (2.46) generate the full group of point symmetries of (2.36). Thus, the porous medium equation does not have any point symmetries beyond the obvious scaling and translational invariances. This is not always the case, however. Many equations have point symmetries that would be difficult to find without using the theory of Lie algebras.
Remark 2.13. The one-dimensional subgroups of the two-dimensional group of space-time translations are given by
$$(x, t, u) \mapsto(x-c \varepsilon, t-\varepsilon, u)$$
where $c$ is a fixed constant (and also the space translations $(x, t, u) \mapsto(x-\varepsilon, t, u))$. The similarity solutions that are invariant under this subgroup are the traveling wave solutions
$$u(x, t)=f(x-c t)$$

Dimensional analysis leads to scaling invariances of a differential equation. As we have seen in the case of the porous medium equation, these invariances form a continuous group, or Lie group, of symmetries of the differential equation.

The theory of Lie groups and Lie algebras provides a systematic method to compute all continuous point symmetries of a given differential equation; in fact, this is why Lie first introduced the the theory of Lie groups and Lie algebras.

Lie groups and algebras arise in many other contexts. In particular, as a result of the advent of quantum mechanics in the early $20^{\text {th }}$-century, where symmetry considerations are crucial, Lie groups and Lie algebras have become a central part of mathematical physics.

We will begin by describing some basic ideas about Lie groups of transformations and their associated Lie algebras. Then we will describe their application to the computation of symmetry groups of differential equations. See Olver $[\mathbf{4 0 ,} \mathbf{4 1}]$, whose presentation we follow, for a full account.

## 数学代写|应用数学代写applied mathematics代考|Lie groups and Lie algebras

A manifold of dimension $d$ is a space that is locally diffeomorphic to $\mathbb{R}^d$, although its global topology may be different (think of a sphere, for example). This means that the elements of the manifold may, locally, be smoothly parametrized by $d$ coordinates, say $\left(\varepsilon^1, \varepsilon^2, \ldots, \varepsilon^d\right) \in \mathbb{R}^d$. A Lie group is a space that is both a manifold and a group, such that the group operations (composition and inversion) are smooth functions.

Lie groups almost always arise in applications as transformation groups acting on some space. Here, we are interested in Lie groups of symmetries of a differential equation that act as point transformations on the space whose coordinates are the independent and dependent variables of the differential equation.

The key idea we want to explain first is this: the Lie algebra of a Lie group of transformations is represented by the vector fields whose flows are the elements of the Lie Group. As a result, elements of the Lie algebra are often referred to as ‘infinitesimal generators’ of elements of the Lie group.

Consider a Lie group $G$ acting on a vector space $E$. In other words, each $g \in G$ is a map $g: E \rightarrow E$. Often, one considers Lie groups of linear maps, which are a subgroup of the general linear group $G L(E)$, but we do not assume linearity here.
Suppose that $E=\mathbb{R}^n$, and write the coordinates of $x \in E$ as $\left(x^1, x^2, \ldots, x^n\right)$. We denote the unit vectors in the coordinate directions by
$$\partial_{x^1}, \partial_{x^2}, \ldots, \partial_{x^n}$$
That is, we identify vectors with their directional derivatives.
Consider a vector field
$$\vec{v}(x)=\xi^i(x) \partial_{x^i}$$
where we use the summation convention in which we sum over repeated upper and lower indices. The associated flow is a one-parameter group of transformations obtained by solving the system of ODEs
$$\frac{d x^i}{d \varepsilon}=\xi^i\left(x^1, x^2, \ldots, x^n\right) \quad \text { for } 1 \leq i \leq n$$
Explicitly, if $x(\varepsilon)$ is a solution of this ODE, then the flow $g(\varepsilon): x(0) \mapsto x(\varepsilon)$ maps the initial data at $\varepsilon=0$ to the solution at ‘time’ $\varepsilon$.
We denote the flow $g(\varepsilon)$ generated by the vector field $\vec{v}$ by
$$g(\varepsilon)=e^{\varepsilon \vec{v}}$$

# 应用数学代考

## 数学代写|应用数学代写applied mathematics代考|Translational invariance

$$(2.46) \quad(x, t, u) \mapsto(x-\delta, t, u), \quad(x, t, u) \mapsto(x, t-\varepsilon, u)$$

$$(x, t, u) \mapsto(x-c \varepsilon, t-\varepsilon, u)$$

$$u(x, t)=f(x-c t)$$

## 数学代写|应用数学代写applied mathematics代考|Lie groups and Lie algebras

$$\partial_{x^1}, \partial_{x^2}, \ldots, \partial_{x^n}$$

$$\vec{v}(x)=\xi^i(x) \partial_{x^i}$$

$$\frac{d x^i}{d \varepsilon}=\xi^i\left(x^1, x^2, \ldots, x^n\right) \quad \text { for } 1 \leq i \leq n$$

$$g(\varepsilon)=e^{\varepsilon \vec{v}}$$

## 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。

## 数学代写|应用数学代写applied mathematics代考|MTH103

statistics-lab™ 为您的留学生涯保驾护航 在代写应用数学applied mathematics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。我们的专家在代写应用数学applied mathematics代写方面经验极为丰富，各种代写应用数学applied mathematics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|应用数学代写applied mathematics代考|Traveling waves

One of the principal features of the KPP equation is the existence of traveling waves which describe the invasion of an unpopulated region (or a region whose population does not possess the favorable allele) from an adjacent populated region.
A traveling wave is a solution of the form
(1.9) $u(x, t)=f(x-c t)$
where $c$ is a constant wave speed. This solution consists of a fixed spatial profile that propagates with velocity $c$ without changing its shape.

For definiteness we assume that $c>0$. The case $c<0$ can be reduced to this one by a reflection $x \mapsto-x$, which transforms a right-moving wave into a left-moving wave.
Use of (1.9) in (1.8) implies that $f(x)$ satisfies the ODE
(1.10) $\quad f^{\prime \prime}+c f^{\prime}+f(1-f)=0$.
The equilibria of this $\mathrm{ODE}$ are $f-0, f-1$.
Note that (1.10) describes the spatial dynamics of traveling waves, whereas (1.6) describes the temporal dynamics of uniform solutions. Although these equations have the same equilibrium solutions, they are different ODEs (for example, one is second order, and the other first order) and the stability of their equilibrium solutions means different things.
The linearization of $(1.10)$ at $f=0$ is
$$f^{\prime \prime}+c f^{\prime}+f=0 .$$
The characteristic equation of this $\mathrm{ODE}$ is
$$\lambda^2+c \lambda+1=0$$
with roots
$$\lambda=\frac{1}{2}\left{-c \pm \sqrt{c^2-4}\right} .$$
Thus, the equilibrium $f=0$ is a stable spiral point if $0<c<2$, a degenerate stable node if $c=2$, and a stable node if $2<c<\infty$.
The linearization of $(1.10)$ at $f=1$ is
$$f^{\prime \prime}+c f^{\prime}-f=0$$

## 数学代写|应用数学代写applied mathematics代考|The existence of traveling waves

Let us discuss the existence of positive traveling waves in a little more detail. If $c=5 / \sqrt{6}$, there is a simple explicit solution for the traveling wave [1]:
$$F(x)=\frac{1}{\left(1+e^{x / \sqrt{6}}\right)^2} .$$
Although there is no similar explicit solution for general values of $c$, we can show the existence of traveling waves by a qualitative argument.

Writing (1.10) as a first order system of ODEs for $(f, g)$, where $g=f^{\prime}$, we get
\begin{aligned} &f^{\prime}=g, \ &g^{\prime}=-f(1-f)-c g . \end{aligned}

## 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。