## 数学代写|表示论代写Representation theory代考|The Reflection a Source

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• 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|表示论代写Representation theory代考|The Reflection a Source

We assume that $j$ is a source of the quiver $Q^{\prime}$. For every representation $\mathcal{N}$ of $Q^{\prime}$ we will construct from $\mathcal{N}$ a representation of $\sigma_j Q^{\prime}$, denoted by $\Sigma_j^{-}(\mathcal{N})$. The idea is to keep the vector space $N(r)$ as it is, for any vertex $r \neq j$, and also to keep the linear map $N(\gamma)$ as it is, for any arrow $\gamma$ which does not start at $j$. We want to find a vector space $N^{-}(j)$, and for each arrow $\beta_i: j \rightarrow i$, we want to define a linear map $N^{-}\left(\bar{\beta}_i\right)$ from $N(i)$ to $N^{-}(j)$, to be constructed using only data from $\mathcal{N}$. We first fix some notation, and then we study small examples.

Definition 11.14. Let $j$ be a source in the quiver $Q^{\prime}$. We label the distinct arrows starting at $j$ by $\beta_1, \beta_2, \ldots, \beta_t$, say $\beta_i: j \rightarrow i$. Then we write $\bar{\beta}_i: i \rightarrow j$ for the arrows of $\sigma_j Q^{\prime}$ obtained by reversing the $\beta_i$.

(1) Let $t=1$, and take the quivers $Q^{\prime}$ and $\sigma_j Q^{\prime}$ as follows:
$$1 \stackrel{\beta_1}{\longleftarrow} j \text { and } 1 \stackrel{\bar{\beta}_1}{\longrightarrow} j$$
We start with a representation $\mathcal{N}$ of $Q^{\prime}$,
$$N(1) \stackrel{N\left(\beta_1\right)}{\longleftarrow} N(j)$$
and we want to define a representation of $\sigma_j Q^{\prime}$, that is,
$$N(1) \stackrel{N^{-}\left(\bar{\beta}_1\right)}{\longrightarrow} N^{-}(j)$$
and this should only use information from $\mathcal{N}$. There is not much choice, we take $N^{-}(j):=N(1) / \operatorname{im}\left(N\left(\beta_1\right)\right)$, which is a quotient space of $N(1)$, and we take $N^{-}\left(\bar{\beta}_1\right)$ to be the canonical surjection. This defines the representation $\Sigma_j^{-}(\mathcal{N})$ of $\sigma_j Q^{\prime}$.

## 数学代写|表示论代写Representation theory代考|Quivers of Infinite Representation Type

We will now prove that if the underlying graph of $Q$ is not a union of Dynkin diagrams then $Q$ has infinite representation type. This is one direction of Gabriel’s theorem. As we have seen in Lemma 9.27, it is enough to consider connected quivers, and we should deal with smallest connected quivers whose underlying graph is not a Dynkin diagram (see Lemma 9.26).

Proposition 11.27. Assume $Q$ is a connected quiver with no oriented cycles. If the underlying graph of $Q$ is not a Dynkin diagram, then $Q$ has infinite representation type.
The proof of Proposition 11.27 will take the entire section.
By Lemma 10.1 we know that a connected quiver $Q$ whose underlying graph is not a Dynkin diagram must have a subquiver $Q^{\prime}$ whose underlying graph is a Euclidean diagram. By Lemma 9.26, it suffices to show that the subquiver $Q^{\prime}$ has infinite representation type. We will do this case-by-case going through the Euclidean diagrams listed in Fig. 10.2.

We start with Euclidean diagrams of type $\widetilde{A}_n$, which has almost been done already.

Proposition 11.28. Assume $Q^{\prime}$ is a quiver without oriented cycles whose underlying graph is a Euclidean diagram of type $\widetilde{A}_n$. Then $Q^{\prime}$ is of infinite representation type.

Proof. Let $n=1$, then $Q^{\prime}$ is the Kronecker quiver, and we have seen in Example 9.30 that it has infinite representation type. Now assume $n>1$. We will stretch the Kronecker quiver repeatedly as described in Definition 9.20; and Exercise 9.4 shows that $Q^{\prime}$ can be obtained from the Kronecker quiver by finitely many stretches. Now Lemma 9.31 implies that $Q^{\prime}$ has infinite representation type.
We will now deal with quivers whose underlying graphs are other Euclidean diagrams as listed in Fig. 10.2. We observe that each of them is a tree. Therefore, by Corollary 11.26, in each case we only need to show that it has infinite representation type just for one orientation, which we can choose as we like.

We will use a more general tool. This is inspired by the indecomposable representation of the quiver with underlying graph a Dynkin diagram $D_4$ where the space at the branch vertex is 2-dimensional, which we have seen a few times. In Lemma 9.5 we have proved that for this representation, any endomorphism is a scalar multiple of the identity. The following shows that this actually can be used to produce many representations of a new quiver obtained by just adding one vertex and one arrow.

# 表示论代考

## 数学代写|表示论代写Representation theory代考|The Reflection a Source

(1) 让 $t=1$ ，并取箭袋 $Q^{\prime}$ 和 $\sigma_j Q^{\prime}$ 如下:
$$1 \stackrel{\beta_1}{\longleftarrow} j \text { and } 1 \stackrel{\bar{\beta}_1}{\longrightarrow} j$$

$$N(1) \stackrel{N\left(\beta_1\right)}{\longleftarrow} N(j)$$

$$N(1) \stackrel{N^{-}\left(\bar{\beta}_1\right)}{\longrightarrow} N^{-}(j)$$

## 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|表示论代写Representation theory代考|The Coxeter Transformation

statistics-lab™ 为您的留学生涯保驾护航 在代写表示论Representation theory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。我们的专家在代写表示论Representation theory代写方面经验极为丰富，各种代写表示论Representation theory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|表示论代写Representation theory代考|The Coxeter Transformation

In this section we will introduce a particular map, the Coxeter transformation, associated to a Dynkin diagram with standard labelling as in Example 10.4. This map will later be used to show that for Dynkin diagrams positive roots parametrize indecomposable representations.

Let $\Gamma$ be one of the Dynkin diagrams, with standard labelling. We have seen in Lemma 10.9 that each reflection $s_j$, where $j$ is a vertex of $\Gamma$, preserves the set $\Delta_{\Gamma}$ of roots. Then the set $\Delta_{\Gamma}$ of roots is also preserved by arbitrary products of reflections, that is, by any element in the group $W$, the subgroup of the automorphism group $\operatorname{Aut}\left(\mathbb{Z}^n\right)$ generated by the reflections $s_j$. The Coxeter transformation is an element of $W$ and it has special properties.

Definition 10.14. Assume $\Gamma$ is a Dynkin diagram with standard labelling as in Example 10.4. Let $s_j: \mathbb{Z}^n \rightarrow \mathbb{Z}^n, s_j(x)=x-\left(x, \varepsilon_j\right){\Gamma} \varepsilon_j$ be the reflections as in Definition 10.2. The Coxeter transformation $C{\Gamma}$ is the map
$$C_{\Gamma}=s_n \circ s_{n-1} \circ \ldots \circ s_2 \circ s_1: \mathbb{Z}^n \rightarrow \mathbb{Z}^n .$$
The Coxeter matrix is the matrix of $C_{\Gamma}$ with respect to the standard basis of $\mathbb{R}^n$.
Example 10.15 (Coxeter Transformation in Dynkin Type A). Let $\Gamma$ be the Dynkin diagram of type $A_n$ with standard labelling. We describe the Coxeter transformation and its action on the roots of $q_{\Gamma}$. To check some of the details, see Exercise 10.8 below. Let $s_j$ be the reflection, as defined in Definition 10.2. Explicitly, we have for $x=\left(x_1, x_2, \ldots, x_n\right) \in \mathbb{R}^n$ that
$$s_j(x)= \begin{cases}\left(-x_1+x_2, x_2, \ldots, x_n\right) & j=1 \ \left(x_1, \ldots, x_{j-1},-x_j+x_{j-1}+x_{j+1}, x_{j+1}, \ldots, x_n\right) & 2 \leq j \leq n-1 \ \left(x_1, \ldots, x_{n-1}, x_{n-1}-x_n\right) & j=n .\end{cases}$$

## 数学代写|表示论代写Representation theory代考|Reflecting Quivers and Representations

Gabriel’s theorem states implicitly that the representation type of a quiver depends only on the underlying graph but not on the orientation of the arrows. To prove this, we will use ‘reflection maps’, which relate representations of two quivers with the same underlying graph but where some arrows have different orientation. This construction will show that any two quivers with the same underlying graph $\Gamma$ have the same representation type, if $\Gamma$ is an arbitrary finite tree.
Throughout this chapter let $K$ be an arbitrary field.
Definition 11.2. Let $Q$ be a quiver. A vertex $j$ of $Q$ is called a sink if no arrows in $Q$ start at $j$. A vertex $k$ of $Q$ is a source if no arrows in $Q$ end at $k$.

For example, consider the quiver $1 \longrightarrow 2 \longleftarrow 3 \longleftarrow 4$. Then vertices 1 and 4 are sources, vertex 2 is a sink and vertex 3 is neither a sink nor a source.

Exercise 11.1. Let $Q$ be a quiver without oriented cycles. Show that $Q$ contains a sink and a source.

Definition 11.3. Let $Q$ be a quiver and let $j$ be a vertex in $Q$ which is a sink or a source. We define a new quiver $\sigma_j Q$, this is the quiver obtained from $Q$ by reversing all arrows adjacent to $j$, and keeping everything else unchanged. We call $\sigma_j Q$ the reflection of $Q$ at the vertex $j$. Note that if a vertex $j$ is a sink of $Q$ then $j$ is a source of $\sigma_j Q$, and if $j$ is a source of $Q$ then it is a sink of $\sigma_j Q$. We also have that $\sigma_j \sigma_j Q=Q$

Example 11.4. Consider all quivers whose underlying graph is the Dynkin diagram of type $A_4$. Up to labelling of the vertices, there are four possible quivers,
\begin{aligned} & Q_1: 1 \longleftarrow 2 \longleftarrow 3 \longleftarrow 4 \ & Q_2: 1 \longleftrightarrow 2 \longleftarrow 3 \longleftarrow 4 \ & Q_3: 1 \longleftarrow 2 \longleftrightarrow 3 \longleftarrow 4 \ & Q_4: 1 \longleftarrow 2 \longleftarrow 3 \longleftrightarrow 4 \end{aligned}
Then $\sigma_1 Q_1=Q_2$ and $\sigma_2 \sigma_1 Q_1=Q_3$; and moreover $\sigma_3 \sigma_2 \sigma_1 Q_1=Q_4$. Hence each $Q_i$ can be obtained from $Q_1$ by applying reflections.

# 表示论代考

## 数学代写|表示论代写Representation theory代考|The Coxeter Transformation

$$C_{\Gamma}=s_n \circ s_{n-1} \circ \ldots \circ s_2 \circ s_1: \mathbb{Z}^n \rightarrow \mathbb{Z}^n \text {. }$$
Coxeter 矩阵是 $C_{\Gamma}$ 关于标准基础 $\mathbb{R}^n$.

$$s_j(x)=\left{\left(-x_1+x_2, x_2, \ldots, x_n\right) \quad j=1\left(x_1, \ldots, x_{j-1},-x_j+x_{j-1}+x_{j+1}, x_{j+1}, \ldots, x_n\right)\right.$$

## 数学代写|表示论代写Representation theory代考|Reflecting Quivers and Representations

$$Q_1: 1 \longleftarrow 2 \longleftarrow 3 \longleftarrow 4 \quad Q_2: 1 \longleftrightarrow 2 \longleftarrow 3 \longleftarrow 4 Q_3: 1 \longleftarrow 2 \longleftrightarrow 3 \longleftarrow 4 \quad Q_4$$

## 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。

## MATH4080 Representation theory课程简介

This course is in the algebra course sequence, introducing representations of finite groups and modules over rings. One can think of the topics informally as “linear algebra over a group” and “linear algebra over a ring”. Both concepts are widely used in pure and applied mathematics.

The course will be administered using blackboard. Go there for more information.

## PREREQUISITES

This course is in the algebra course sequence, introducing representations of finite groups and modules over rings. One can think of the topics informally as “linear algebra over a group” and “linear algebra over a ring”. Both concepts are widely used in pure and applied mathematics.

The course will be administered using blackboard. Go there for more information.

## MATH4080 Representation theory HELP（EXAM HELP， ONLINE TUTOR）

If $A$ is a subalgebra of the algebra of $n \times n$-matrices $M_n(K)$, or a subalgebra of the algebra $\operatorname{End}_K(V)$ of $K$-linear maps on a vector space $V$ (see Example 1.3), then $A$ has a natural module, which we will now describe.

Let $A$ be a subalgebra of $M_n(K)$, and let $V=K^n$, the space of column vectors, that is, of $n \times 1$-matrices. By properties of matrix multiplication, multiplying an $n \times n$-matrix by an $n \times 1$-matrix gives an $n \times 1$-matrix, and this satisfies axioms (i) to (iv). Hence $V$ is an $A$-module, the natural $A$-module. Here $A$ could be all of $M_n(K)$, or the algebra of upper triangular $n \times n$-matrices, or any other subalgebra of $M_n(K)$.

Let $V$ be a vector space over the field $K$. Assume that $A$ is a subalgebra of the algebra $\operatorname{End}_K(V)$ of all $K$-linear maps on $V$ (see Example 1.3). Then $V$ becomes an $A$-module, where the action of $A$ is just applying the linear maps to the vectors, that is, we set
$$A \times V \rightarrow V,(\varphi, v) \mapsto \varphi \cdot v:=\varphi(v)$$
To check the axioms, let $\varphi, \psi \in A$ and $v, w \in V$, then we have
$$(\varphi+\psi) \cdot v=(\varphi+\psi)(v)=\varphi(v)+\psi(v)=\varphi \cdot v+\psi \cdot v$$
by the definition of the sum of two maps, and similarly
$$\varphi \cdot(v+w)=\varphi(v+w)=\varphi(v)+\varphi(w)=\varphi \cdot v+\varphi \cdot w$$
since $\varphi$ is $K$-linear. Moreover,
$$\varphi \cdot(\psi \cdot v)=\varphi(\psi(v))=(\varphi \psi) \cdot v$$
since the multiplication in $\operatorname{End}_K(V)$ is given by composition of maps, and clearly we have $1_A \cdot v=\operatorname{id}_V(v)=v$

Let $B$ be an algebra and $A$ a subalgebra of $B$. Then every $B$-module $M$ can be viewed as an $A$-module with respect to the given action. The axioms are then satisfied since they even hold for elements in the larger algebra $B$. We have already used this, when describing the natural module for subalgebras of $M_n(K)$, or of $\operatorname{End}_K(V)$.

## Textbooks

• An Introduction to Stochastic Modeling, Fourth Edition by Pinsky and Karlin (freely
available through the university library here)
• Essentials of Stochastic Processes, Third Edition by Durrett (freely available through
the university library here)
To reiterate, the textbooks are freely available through the university library. Note that
you must be connected to the university Wi-Fi or VPN to access the ebooks from the library
links. Furthermore, the library links take some time to populate, so do not be alarmed if
the webpage looks bare for a few seconds.

Statistics-lab™可以为您提供cuhk.edu MATH4080 Representation theory表示论课程的代写代考辅导服务！ 请认准Statistics-lab™. Statistics-lab™为您的留学生涯保驾护航。

## 数学代写|表示论代写Representation theory代考|MTH4107

statistics-lab™ 为您的留学生涯保驾护航 在代写表示论Representation theory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。我们的专家在代写表示论Representation theory代写方面经验极为丰富，各种代写表示论Representation theory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|表示论代写Representation theory代考|Representation Type of Quivers

We translate modules over the path algebra to representations of quivers, and the Krull-Schmidt theorem translates as well. That is, every (finite-dimensional) representation of a quiver $Q$ is a direct sum of indecomposable representations, unique up to isomorphism and labelling. Therefore it makes sense to define the representation type of a quiver.

Recall that we have fixed a field $K$ and that we consider only finite-dimensional representations of quivers over $K$, see Definition 9.1. Moreover, we assume throughout that quivers have no oriented cycles; this allows us to apply the results of Sect. $9.3$.

Definition 9.25. A quiver $Q$ is said to be of finite representation type over $K$ if there are only finitely many indecomposable representations of $Q$, up to isomorphism. Otherwise, we say that the quiver has infinite representation type over $K$.

By our Definition 9.1, a representation of $Q$ always corresponds to a finitedimensional $K Q$-module. In addition, we assume $Q$ has no oriented cycles and hence $K Q$ is finite-dimensional. Therefore the representation type of $Q$ is the same as the representation type of the path algebra $K Q$, as in Definition $8.1$.

In most situations, our arguments will not refer to a particular field $K$, so we often just speak of the representation type of a quiver, without mentioning the underlying field $K$ explicitly.

For determining the representation type of quivers there are some reductions which follow from the work done in previous sections.

Given a quiver $Q$, since we have seen in Sect. $9.2$ that we can relate indecomposable representations of its subquivers to indecomposable representations of $Q$, we might expect that there should be a connection between the representation type of subquivers with that of $Q$.

Lemma 9.26. Assume $Q^{\prime}$ is a subquiver of a quiver $Q$. If $Q^{\prime}$ has infinite representation type then $Q$ also has infinite representation type.

## 数学代写|表示论代写Representation theory代考|Dynkin Diagrams and Euclidean Diagrams

Gabriel’s theorem (which will be proved in the next chapter) states that a connected quiver has finite representation type if and only if the underlying graph $\Gamma$ is one of the Dynkin diagrams of types $A_n$ for $n \geq 1, D_n$ for $n \geq 4, E_6, E_7, E_8$, which we define in Fig. 10.1.

We have seen some small special cases of Gabriel’s theorem earlier in the book. Namely, a quiver of type $A_1$ (that is, the one-vertex quiver) has only one indecomposable representation by Example 9.28; in particular, it is of finite representation type. Moreover, also in Example $9.28$ we have shown that the quiver $1 \longrightarrow 2$ has finite representation type; note that this quiver has as underlying graph a Dynkin diagram of type $A_2$.

To deal with the case when $\Gamma$ is not a Dynkin diagram, we will only need a small list of graphs. These are the Euclidean diagrams, sometimes also called extended Dynkin diagrams. They are shown in Fig. 10.2, and are denoted by $\widetilde{A}_n$ for $n \geq 1$, $\widetilde{D}_n$ for $n \geq 4$, and $\widetilde{E}_6, \widetilde{E}_7, \widetilde{E}_8$. For example, the Kronecker quiver is a quiver with underlying graph a Euclidean diagram of type $\widetilde{A}_1$; and we have seen already in Example $9.30$ that the Kronecker quiver has infinite representation type.

We refer to graphs in Fig. $10.1$ as graphs of type $A, D$, or $E$. We say that a quiver has Dynkin type if its underlying graph is one of the graphs in Fig. 10.1. Similarly, we say that a quiver has Euclidean type if its underlying graph belongs to the list in Fig. 10.2.

In analogy to the definition of a subquiver in Definition 9.13, a subgraph $\Gamma^{\prime}=\left(\Gamma_0^{\prime}, \Gamma_1^{\prime}\right)$ of a graph $\Gamma$ is a graph which consists of a subset $\Gamma_0^{\prime} \subseteq \Gamma_0$ of the vertices of $\Gamma$ and a subset $\Gamma_1^{\prime} \subseteq \Gamma_1$ of the edges of $\Gamma$.

The following result shows that we might not need any other graphs than Dynkin and Euclidean diagrams.

Lemma 10.1. Assume $\Gamma$ is a connected graph. If $\Gamma$ is not a Dynkin diagram then $\Gamma$ has a subgraph which is a Euclidean diagram.

Proof. Assume $\Gamma$ does not have a Euclidean diagram as a subgraph, we will show that then $\Gamma$ is a Dynkin diagram.

The Euclidean diagrams of type $\widetilde{A}_n$ are just the cycles; so $\Gamma$ does not contain a cycle; in particular, it does not have a multiple edge. Since $\Gamma$ is connected by assumption, it must then be a tree.

# 表示论代考

## 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|表示论代写Representation theory代考|MATH4314

statistics-lab™ 为您的留学生涯保驾护航 在代写表示论Representation theory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。我们的专家在代写表示论Representation theory代写方面经验极为丰富，各种代写表示论Representation theory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|表示论代写Representation theory代考|Representations of Subquivers

When studying representations of quivers it is often useful to relate representations of a quiver to representations of a ‘subquiver’, a notion we now define.
Definition 9.13. Assume $Q$ is a quiver with vertex set $Q_0$ and arrow set $Q_1$.
(a) A subquiver of $Q$ is a quiver $Q^{\prime}=\left(Q_0^{\prime}, Q_1^{\prime}\right)$ such that $Q_0^{\prime} \subseteq Q_0$ and $Q_1^{\prime} \subseteq Q_1$.
(b) A subquiver $Q^{\prime}$ of $Q$ as above is called a full subquiver if for any two vertices $i, j \in Q_0^{\prime}$ all arrows $i \stackrel{\alpha}{\longrightarrow} j$ of $Q$ are also arrows in $Q^{\prime}$.

Note that since $Q^{\prime}$ must be a quiver, it is part of the definition that in a subquiver the starting and end points of any arrow are also in the subquiver (see Definition 1.11). Thus one cannot choose arbitrary subsets $Q_1^{\prime} \subseteq Q_1$ in the above definition.
Example 9.14. Let $Q$ be the quiver
We determine the subquivers $Q^{\prime}$ of $Q$ with vertex set $Q_0^{\prime}={1,2}$. For the arrow set we have the following possibilities: $Q_1^{\prime}=\emptyset, Q_1^{\prime}={\alpha}, Q_1^{\prime}={\beta}$ and $Q_1^{\prime}={\alpha, \beta}$. Of these, only the last quiver is a full subquiver. However, by the preceding remark we cannot choose $Q_1^{\prime}={\alpha, \gamma}$ since the vertex 3 is not in $Q_0^{\prime}$.

Given a quiver $Q$ with a subquiver $Q^{\prime}$, we want to relate representations of $Q$ with representations of $Q^{\prime}$. For our purposes two constructions will be particularly useful. We first present the ‘restriction’ of a representation of $Q$ to a representation of $Q^{\prime}$. Starting with a representation of $Q^{\prime}$, we then introduce the ‘extension by zero’ which produces a representation of $Q$.

## 数学代写|表示论代写Representation theory代考|Stretching Quivers and Representations

There are further methods to relate representations of different quivers. We will now present a general construction which will be very useful later. This construction works for quivers without loops; for simplicity we consider from now on only quivers without oriented cycles. Recall that the corresponding path algebras are then finite-dimensional, see Exercise $1.2$.

Consider two quivers $Q$ and $\widetilde{Q}$ where $\widetilde{Q}$ is obtained from $Q$ by replacing one vertex $i$ of $Q$ by two vertices $i_1, i_2$ and one arrow, $i_1 \stackrel{\gamma}{\longrightarrow} i_2$, and by distributing the arrows adjacent to $i$ between $i_1$ and $i_2$. The following definition makes this construction precise.

Definition 9.20. Let $Q$ be a quiver without oriented cycles and $i$ a fixed vertex. Let $T$ be the set of all arrows adjacent to $i$, and suppose $T=T_1 \cup T_2$, a disjoint union. Define $\widetilde{Q}$ to be the quiver obtained from $Q$ as follows.
(i) Replace vertex $i$ by $i_1 \stackrel{\gamma}{\longrightarrow} i_2$ (where $i_1, i_2$ are different vertices);
(ii) Join the arrows in $T_1$ to $i_1$;
(iii) Join the arrows in $T_2$ to $i_2$.
In (ii) and (iii) we keep the original orientation of the arrows. We call the new quiver $\widetilde{Q}$ a stretch of $Q$.

By assumption, $Q$ does not have loops, so any arrow adjacent to $i$ either starts at $i$ or ends at $i$ but not both, and it belongs either to $T_1$ or to $T_2$. Note that if $T$ is large then there are many possible stretches of a quiver $Q$ at a given vertex $i$, coming from different choices of the sets $T_1$ and $T_2$.

We illustrate the general construction from Definition $9.20$ with several examples.

# 表示论代考

## 数学代写|表示论代写Representation theory代考|Representations of Subquivers

(a) 的子箭袋 $Q$ 是一个箭袋 $Q^{\prime}=\left(Q_0^{\prime}, Q_1^{\prime}\right)$ 这样 $Q_0^{\prime} \subseteq Q_0$ 和 $Q_1^{\prime} \subseteq Q_1$.
(b) 一个子箭袋 $Q^{\prime}$ 的 $Q$ 如果对于任何两个顶点，如上称为完全子箭袋 $i, j \in Q_0^{\prime}$ 所有箭头 $i \stackrel{\alpha}{\longrightarrow} j$ 的 $Q$ 也是 箭头 $Q^{\prime}$.

$Q_1^{\prime}=\emptyset, Q_1^{\prime}=\alpha, Q_1^{\prime}=\beta$ 和 $Q_1^{\prime}=\alpha, \beta$. 其中，只有最后一个箭袋是完整的子箭袋。然而，根据前面 的评论，我们不能选择 $Q_1^{\prime}=\alpha, \gamma$ 因为顶点 3 不在 $Q_0^{\prime}$.

## 数学代写|表示论代写Representation theory代考|Stretching Quivers and Representations

(i) 替换顶点 $i$ 经过 $i_1 \stackrel{\gamma}{\longrightarrow} i_2$ （在哪里 $i_1, i_2$ 是不同的顶点）；
(ii) 加入箭头 $T_1$ 到 $i_1$;
(iii) 加入箭头 $T_2$ 到 $i_2$.

## 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|表示论代写Representation theory代考|MAST90017

statistics-lab™ 为您的留学生涯保驾护航 在代写表示论Representation theory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。我们的专家在代写表示论Representation theory代写方面经验极为丰富，各种代写表示论Representation theory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|表示论代写Representation theory代考|Matrix Reduction Algorithm

There is a version of the algorithm PAIRPERSISTENCE that uses only matrix operations. First notice the following:

• The boundary operator $\partial_p: \mathbf{C}p \rightarrow \mathbf{C}{p-1}$ can be represented by a boundary matrix $D_p$ where the columns correspond to the $p$-simplices and rows correspond to $(p-1)$-simplices.
• It represents the transformation of a basis of $\mathrm{C}p$ given by the set of $p$-simplices to a basis of $C{p-1}$ given by the set of $(p-1)$-simplices:
$$D_p[i, j]= \begin{cases}1 & \text { if } \sigma_i \in \partial_p \sigma_j, \ 0 & \text { otherwise. }\end{cases}$$
• One can combine all boundary matrices into a single matrix $D$ that represents all linear maps $\bigoplus_p \partial_p-\bigoplus_p\left(\mathrm{C}p \rightarrow \mathrm{C}{p-1}\right)$, that is, transformation of a basis of all chain groups together to a basis of itself, but with a shift to a one lower dimension:
$$D[i, j]= \begin{cases}1 & \text { if } \sigma_i \in \partial_* \sigma_j, \ 0 & \text { otherwise. }\end{cases}$$
Definition 3.12. (Filtered boundary matrix) Let $\mathcal{F}: \varnothing=K_0 \hookrightarrow K_1 \hookrightarrow \cdots$ $\hookrightarrow K_n=K$ be a filtration induced by an ordering of simplices $\left(\sigma_1, \sigma_2, \ldots, \sigma_n\right)$ in $K$. Let $D$ denote the boundary matrix for simplices in $K$ that respects the ordering of the simplices in the filtration, that is, the simplex $\sigma_i$ in the filtration occupies column $i$ and row $i$ in $D$. We call $D$ the filtered boundary matrix for $\mathcal{F}$.

Given any matrix $A$, let row ${ }_A\lfloor i\rfloor$ and $\operatorname{col}_A\lfloor j\rfloor$ denote the $i$ th row and $j$ th column of $A$, respectively. We abuse notation slightly to let $\operatorname{col}_A[j]$ denote also the chain $\left{\sigma_i \mid A[i, j]=1\right}$, which is the collection of simplices corresponding to $1 \mathrm{~s}$ in the column $\operatorname{col}_A[j]$.

Definition 3.13. (Reduced matrix) Let $\operatorname{low}_A[j]$ denote the row index of the last 1 in the $j$ th column of $A$, which we call the low-row index of the column $j$. It is undefined for empty columns (marked with $-1$ in Algorithm 3). The matrix $A$ is reduced (or is in reduced form) if low $[j] \neq \operatorname{low}_A\left[j^{\prime}\right]$ for any $j \neq j^{\prime}$; that is, no two columns share the same low-row indices.

## 数学代写|表示论代写Representation theory代考|Efficient Implementation

The matrix reduction algorithm considers a column from left to right and reduces it by left-to-right additions. As we have observed, every addition to a column with index $j$ pushes $\operatorname{low}_D[j]$ upward. In the case that $\sigma_j$ is a positive simplex, the entire column is zeroed out. In general, positive simplices incur more cost than negative ones because $\operatorname{low}_D[\cdot]$ needs to be pushed all the way up for zeroing out the entire column. However, they do not participate in any future left-to-right column additions. Therefore, if it is known beforehand that the simplex $\sigma_j$ will be a positive simplex, then the costly step of zeroing out the column $j$ can be avoided.

Chen and Kerber [94] observed the following simple fact. If we process the input filtration backward in dimension, that is, process the boundary matrices $D_p, p=1, \ldots, d$, in decreasing order of dimensions, then a persistence pair $\left(\sigma^{p-1}, \sigma^p\right)$ is detected from $D_p$ before processing the column for $\sigma^{p-1}$ in $D_{p-1}$. Fortunately, we already know that $\sigma^{p-1}$ has to be a positive simplex because it cannot pair with a negative simplex $\sigma^p$ otherwise. So, we can simply ignore the column of $\sigma^{p-1}$ while processing $D_{p-1}$. We call it clearing out column $p-1$. In practice, this saves a considerable amount of computation in cases where a lot of positive simplices occur such as in Rips filtrations. Algorithm 4: ClearPersistence implements this idea.

We cannot take advantage of the clearing for the last dimension in the filtration. If $d$ is the highest dimension of the simplices in the input filtration, the matrix $D_d$ has to be processed for all columns because the pairings for the positive $d$-simplices are not available.

If the number of $d$-simplices is large compared to the number of simplices of lower dimensions, the incurred cost of processing their columns can still be high. For example, in a Rips filtration restricted to a certain dimension $d$, the number of $d$-simplices becomes usually much larger than the number of, say,

1-simplices. In those cases, the clearing can be more cost-effective if it can be applied forward.

In this respect, the following observation becomes helpful. Let $D_p^$ denote the anti-transpose of the matrix $D_p$, defined by the transpose of $D_p$ with the columns and rows being ordered in reverse. This means that if $D_p$ has row and column indices $1, \ldots, m$ and $1, \ldots, n$, respectively, then $D_p^(i, j)=D_p(n+$ $1-j, m+1-i)$. We call it the twisted matrix of $D_p$. Figure $3.13$ shows the twisted matrix $D^$ of the matrix $D$ in Figure $3.12$ where the rows and columns are marked with the indices of the original matrix. The following proposition guaranteés thăt wé cañ computê thê persistencee pairs in $D_P$ from the matrix $D_p^$

# 表示论代考

## 数学代写|表示论代写Representation theory代考|Matrix Reduction Algorithm

• 边界运算符 $\partial_p: \mathbf{C} p \rightarrow \mathbf{C} p-1$ 可以用边界矩阵表示 $D_p$ 其中列对应于 $p$-单纯形和行对应 $(p-1)$ 简单的
• 它代表了基础的转变 $\mathrm{C} p$ 由一组给出 $p$-单纯形的基础 $C p-1$ 由一组给出 $(p-1)$-简单的:
$$D_p[i, j]=\left{1 \quad \text { if } \sigma_i \in \partial_p \sigma_j, 0 \quad\right. \text { otherwise. }$$
• 可以将所有边界矩阵组合成一个矩阵 $D$ 表示所有线性映射 $\bigoplus_p \partial_p-\bigoplus_p(\mathrm{C} p \rightarrow \mathrm{C} p-1)$ ，也就 是说，将所有链组的基础一起转换为自身的基础，但转移到一个较低的维度:
$$D[i, j]=\left{1 \quad \text { if } \sigma_i \in \partial_* \sigma_j, 0 \quad\right. \text { otherwise. }$$
定义 3.12。 (过滤后的边界矩阵) 让 $\mathcal{F}: \varnothing=K_0 \hookrightarrow K_1 \hookrightarrow \cdots \hookrightarrow K_n=K$ 是由单纯形的排 序引起的过滤 $\left(\sigma_1, \sigma_2, \ldots, \sigma_n\right)$ 在 $K$. 让 $D$ 表示单纯形的边界矩阵 $K$ 尊重过滤中单纯形的顺序，即 单纯形 $\sigma_i$ 在过滤占列 $i$ 和行 $i$ 在 $D$. 我们称之为 $D$ 过滤后的边界矩阵 $\mathcal{F}$.
给定任何矩阵 $A$ ， 让行 $A\lfloor i\rfloor$ 和 $\operatorname{col}A\lfloor j\rfloor$ 表示 $i$ 行和 $j$ 第 列 $A$ ，分别。我们稍微滥用符号让 $\operatorname{col}_A[j]$ 也表示链 \eft{\sigma_i \mid A[i, j]=1\right } } \text { , 这是对应于 } 1 \text { s在专栏中 } \operatorname { c o l } { A } [ j ] \text { . }
定义 3.13。(简化矩阵) 让 $\operatorname{low}_A[j]$ 表示最后一个 1 的行索引 $j$ 第列 $A$ ，我们称之为列的低行索引 $j$. 它对 于空列是末定义的 (标有 $-1$ 在算法 3) 中。矩阵 $A$ 如果低，则减少 (或减少形式) $[j] \neq \operatorname{low}_A\left[j^{\prime}\right]$ 对于 任何 $j \neq j^{\prime}$ ；也就是说，没有两列共享相同的低行索引。

## 数学代写|表示论代写Representation theory代考|Efficient Implementation

Chen 和 Kerber [94] 观察到以下简单事实。如果我们对输入过滤进行维度逆向处理，即对边界矩阵进行 处理 $D_p, p=1, \ldots, d$ ，按维度降序排列，然后是持久性对 $\left(\sigma^{p-1}, \sigma^p\right)$ 从检测到 $D_p$ 在处理列之前 $\sigma^{p-1}$ 在 $D_{p-1}$. 幸运的是，我们已经知道 $\sigma^{p-1}$ 必须是正单纯形，因为它不能与负单纯形配对 $\sigma^p$ 除此以外。所 以，我们可以简单地忽略列 $\sigma^{p-1}$ 加工时 $D_{p-1}$. 我们称之为清除列 $p-1$. 实际上，在出现大量正单纯形的 情况下（例如在 Rips 过滤中），这可以节省大量计算。算法 4：ClearPersistence 实现了这个想法。

1-单纯形。在这些情况下，如果可以向前应用清算，则清算可能更具成本效益。

## 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|表示论代写Representation theory代考|MATH4314

statistics-lab™ 为您的留学生涯保驾护航 在代写表示论Representation theory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。我们的专家在代写表示论Representation theory代写方面经验极为丰富，各种代写表示论Representation theory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|表示论代写Representation theory代考|Stability of Persistence Diagrams

A persistence diagram $\operatorname{Dgm}_p\left(\mathcal{F}_f\right)$, as a set of points in the extended plane $\overline{\mathbb{R}}^2$, summarizes certain topological information of a simplicial complex (space) in relation to the function $f$ that induces the filtration $\mathcal{F}_f$. However, this is not useful in practice unless we can be certain that a slight change in $f$ does not change this diagram dramatically. In practice $f$ is seldom measured accurately, and if its persistence diagram can be approximated from a slightly perturbed version, it becomes useful. Fortunately, persistence diagrams are stable. To formulate this stability, we need a notion of distance between persistence diagrams.

Let $\operatorname{Dgm}_p\left(\mathcal{F}_f\right)$ and $\operatorname{Dgm}_p\left(\mathcal{F}_g\right)$ be two persistence diagrams for two functions $f$ and $g$. We want to consider bijections between points from $\operatorname{Dgm}_p\left(\mathcal{F}_f\right)$ and $\operatorname{Dgm}_p\left(\mathcal{F}_g\right)$. However, they may have different cardinality for off-diagonal points. Recall that persistence diagrams include the points on the diagonal $\Delta$ each with infinite multiplicity. This addition allows us to borrow points from the diagonal when necessary to define the bijections. Note that we are considering only filtrations of finite complexes which also make each homology group finite.

Definition 3.9. (Bottleneck distance) Let $\Pi=\left{\pi: \operatorname{Dgm}p\left(\mathcal{F}_f\right) \rightarrow\right.$ $\left.\operatorname{Dgm}_p\left(\mathcal{F}_g\right)\right}$ denote the set of all bijections. Consider the distance between two points $x=\left(x_1, x_2\right)$ and $y=\left(y_1, y_2\right)$ in $L{\infty}$-norm $|x-y|_{\infty}=$ $\max \left{\left|x_1-x_2\right|,\left|y_1-y_2\right|\right}$ with the assumption that $\infty-\infty=0$. The bottleneck distance between the two diagrams (see Figure $3.10$ ) is
$$\mathrm{d}b\left(\operatorname{Dgm}_p\left(\mathcal{F}_f\right), \operatorname{Dgm}_p\left(\mathcal{F}_g\right)\right)=\inf {\pi \in \Pi} \sup {x \in \operatorname{Dgm}_p\left(\mathcal{F}_f\right)}|x-\pi(x)|{\infty} .$$

## 数学代写|表示论代写Representation theory代考|Computing Bottleneck Distances

Let $A$ and $B$ be the nondiagonal points in two persistence diagrams $\operatorname{Dgm}_p\left(\mathcal{F}_f\right)$ and $\operatorname{Dgm}_p\left(\mathcal{F}_g\right)$, respectively. For a point $a \in A$, let $\bar{a}$ denote the nearest point of $a$ on the diagonal. Define $\bar{b}$ for every point $b \in B$ similarly. Let $\bar{A}={\bar{a}}$ and $\bar{B}={\bar{b}}$. Let $\tilde{A}=A \cup \bar{B}$ and $\tilde{B}=B \cup \bar{A}$. We want to bijectively match points in $\tilde{A}$ and $\tilde{B}$. Let $\Pi={\pi}$ denote such a matching. It follows from the definition that
$$\mathrm{d}b\left(\operatorname{Dgm}_p\left(\mathcal{F}_f\right), \operatorname{Dgm}_p\left(\mathcal{F}_g\right)\right)=\min {\pi \in \Pi} \sup {a \in \tilde{A}, \pi(a) \in \tilde{B}}|a-\pi(a)|{\infty} .$$
Then, the bottleneck distance we want to compute must be $L_{\infty}$ distance $\max \left{\left|x_a-x_b\right|,\left|y_a-y_b\right|\right}$ for two points $a \in \tilde{A}$ and $b \in \tilde{B}$. We do a binary search on all such possible $O\left(n^2\right)$ distances where $|\tilde{A}|=|\tilde{B}|=n$. Let $\delta_0, \delta_1, \ldots, \delta_{n^{\prime}}$ be the sorted sequence of these distances in a nondecreasing order.

Given a $\delta=\delta_i \geq 0$ where $i$ is the median of the index in the hinary search interval $[\ell, u]$, we construct a bipartite graph $G=(\tilde{A} \cup \tilde{B}, E)$ where an edge $e=(a, b){{a \in \tilde{A}, b \in \tilde{B}}}$ is in $E$ if and only if either both $a \in \bar{A}$ and $b \in \bar{B}$ (weight $(e)=0$ ) or $|a-b|{\infty} \leq \delta$ (weight $(e)=|a-b|_{\infty}$ ). A complete matching in $G$ is a set of $n$ edges so that every vertex in $\tilde{A}$ and $\tilde{B}$ is incident to exactly one edge in the set. To determine if $G$ has a complete matching, one can use an $O\left(n^{2.5}\right)$ algorithm of Hopcroft and Karp [198] for complete matching in a bipartite graph. However, exploiting the geometric embedding of the points in the persistence diagrams, we can apply an $O\left(n^{1.5}\right)$ time algorithm of Efrat et al. [154] for the purpose. If such an algorithm affirms that a complete matching exists, we do the following: If $\ell=u$ we output $\delta$, otherwise we set $u=i$ and repeat. If no matching exists, we set $\ell=i$ and repeat. Observe that matching has to exist for some value of $\delta$, in particular for $\delta_{n^{\prime}}$ and thus the binary search always succeeds. Algorithm 1: BoTTLENECK lays out the pseudocode for this matching. The algorithm runs in $O\left(n^{1.5} \log n\right)$ time accounting for the $O(\log n)$ probes for binary search each applying an $O\left(n^{1.5}\right)$ time matching algorithm. However, to achieve this complexity, we have to avoid sorting $n^{\prime}=O\left(n^2\right)$ values taking $O\left(n^2 \log n\right)$ time. Again, using the geometric embedding of the points, one can perform the binary probes without incurring the cost for sorting. For details and an efficient implementation of this algorithm, seee [209].

## 数学代写|表示论代写Representation theory代考|Stability of Persistence Diagrams

$$\mathrm{d} b\left(\operatorname{Dgm}_p\left(\mathcal{F}_f\right), \operatorname{Dgm}_p\left(\mathcal{F}_g\right)\right)=\inf \pi \in \Pi \sup x \in \operatorname{Dgm}_p\left(\mathcal{F}_f\right)|x-\pi(x)| \infty$$

## 数学代写|表示论代写Representation theory代考|Computing Bottleneck Distances

$(e)=0$ ) 要么 $|a-b| \infty \leq \delta$ (重量 $(e)=|a-b|{\infty}$ ). 一个完整的匹配 $G$ 是一组 $n$ 边使得每个顶点在 $\tilde{A}$ 和 $\tilde{B}$ 恰好与集合中的一条边相关。确定是否 $G$ 有一个完整的匹配，一个可以使用 $O\left(n^{2.5}\right)$ Hopcroft 和 Karp [198] 的算法用于二分图中的完全匹配。然而，利用持久性图中点的几何嵌入，我们可以应用 $O\left(n^{1.5}\right)$ Efrat 等人的时间算法。[154] 的目的。如果这样的算法确认存在完全匹配，我们将执行以下操 作: 如果 $\ell=u$ 我们输出 $\delta$ ，否则我们设置 $u=i$ 并重复。如果不存在匹配项，我们设置 $\ell=i$ 并重复。观 察到对于某些值必须存在匹配 $\delta$ ，特别是对于 $\delta{n^{\prime}}$ 因此二分查找总是成功的。算法 1: BOTTLENECK 列出 了此匹配的伪代码。该算法运行于 $O\left(n^{1.5} \log n\right)$ 时间占 $O(\log n)$ 用于二进制搜索的探针每个应用一个 $O\left(n^{1.5}\right)$ 时间匹配算法。然而，为了实现这种复杂性，我们必须避免排序 $n^{\prime}=O\left(n^2\right)$ 取值
$O\left(n^2 \log n\right)$ 时间。同样，使用点的几何嵌入，可以在不产生排序成本的情况下执行二元探测。有关此 算法的详细信息和有效实现，请参阅 [209]。

## 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|表示论代写Representation theory代考|MTH4107

statistics-lab™ 为您的留学生涯保驾护航 在代写表示论Representation theory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。我们的专家在代写表示论Representation theory代写方面经验极为丰富，各种代写表示论Representation theory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|表示论代写Representation theory代考|Persistence

In both cases of space and simplicial filtration $\mathcal{F}$, we arrive at a homology module:
$$\mathrm{H}p \mathcal{F}: 0=\mathrm{H}_p\left(X_0\right) \rightarrow \mathrm{H}_p\left(X_1\right) \rightarrow \cdots \rightarrow \mathrm{H}_p\left(X_i\right) \rightarrow \stackrel{h_p^{h_j}}{ } \rightarrow \mathrm{H}_p\left(X_j\right) \rightarrow \cdots \rightarrow \mathrm{H}_p\left(X_n\right)=\mathrm{H}_p(X),$$ where $X_i=\mathbb{T}{a_i}$ if $\mathcal{F}$ is a space filtration of a topological space $X=\mathbb{T}$ or $X_i=K_i$ if $\mathcal{F}$ is a simplicial filtration of a simplicial complex $X=K$. Persistent homology groups for a homology module are algebraic structures capturing the survival of the homology classes through this sequence. In general, we will call homology modules persistence modules in Section $3.4$ recognizing that we can replace homology groups with vector spaces.

Definition 3.4. (Persistent Betti number) The $p$-th persistent homology groups are the images of the homomorphisms: $\mathrm{H}_p^{i, j}=\operatorname{im} h_p^{i, j}$, for $0 \leq i \leq$ $j \leq n$. The $p$-th persistent Betti numbers are the dimensions $\beta_p^{i, j}=\operatorname{dim} \mathrm{H}_p^{i, j}$ of the vector spaces $\mathrm{H}_p^{i, j}$.

The $p$-th persistent homology groups contain the important information of when a homology class is born or when it dies. The issue of birth and death of a class becomes more subtle because when a new class is born, many other classes that are the sum of this new class and any other existing class are also born. Similarly, when a class ceases to exist, many other classes may cease to exist along with it. Therefore, we need a mechanism to pair births and deaths canonically. Figure $3.7$ illustrates the birth and death of a class, though the pairing of birth and death events is more complicated as stated in Fact 3.3.
Observe that the nontrivial elements of $p$-th persistent homology groups $\mathrm{H}_p^{i, j}$ consist of classes that survive from $X_i$ to $X_j$, that is, the classes which do not get “quotiented out” by the boundaries in $X_j$. So, one can observe the following.

## 数学代写|表示论代写Representation theory代考|Persistence Diagram

Fact $3.3$ provides a qualitative characterization of the pairing of births and deaths of classes. Now we give a quantitative characterization which helps to draw a visual representation of this pairing called a persistence diagram; see Figure 3.8(a). Consider the extended plane $\overline{\mathbb{R}}^2:=(\mathbb{R} \cup{\pm \infty})^2$ on which we represent the birth at $a_i$ paired with the death at $a_j$ as a point $\left(a_i, a_j\right)$. This pairing uses a persistence pairing function $\mu_p^{i, j}$ defined below. Strictly positive values of this function correspond to multiplicities of points in the persistence diagram (Definition 3.8). In what follows, to account for classes that never die, we extend the induced module in Eq. (3.3) on the right end by assuming that $\mathrm{H}p\left(X{n+1}\right)=0$.
Definition 3.6. For $0<i<j \leq n+1$, define
$$\mu_p^{i, j}=\left(\beta_p^{i, j-1}-\beta_p^{i, j}\right)-\left(\beta_p^{i-1, j-1}-\beta_p^{i-1, j}\right) .$$
The first difference on the right-hand side counts the number of independent classes that are born at or before $X_i$ and die entering $X_j$. The second difference counts the number of independent classes that are born at or before $X_{i-1}$ and die entering $X_j$. The difference between the two differences thus counts the number of independent classes that are born at $X_i$ and die entering $X_j$. When $j-n+1, \mu_p^{i, n+1}$ counts the number of independent classes that are born at $X_i$ and die entering $X_{n+1}$. They remain alive till the end in the original filtration without extension, or we say that they never die. To emphasize that classes which exist in $X_n$ actually never die, we equate $n+1$ with $\infty$ and take $a_{n+1}=$ $a_{\infty}=\infty$. Observe that, with this assumption, we have $\beta^{i, n+1}=\beta^{i, \infty}=0$ for every $i \leq n$.

## 数学代写|表示论代写Representation theory代考|Persistence

$$\mathrm{H} p \mathcal{F}: 0=\mathrm{H}_p\left(X_0\right) \rightarrow \mathrm{H}_p\left(X_1\right) \rightarrow \cdots \rightarrow \mathrm{H}_p\left(X_i\right) \rightarrow h_p^{h_j} \rightarrow \mathrm{H}_p\left(X_j\right) \rightarrow \cdots \rightarrow \mathrm{H}_p\left(X_n\right)=\mathrm{H}_p$$

## 数学代写|表示论代写Representation theory代考|Persistence Diagram

$$\mu_p^{i, j}=\left(\beta_p^{i, j-1}-\beta_p^{i, j}\right)-\left(\beta_p^{i-1, j-1}-\beta_p^{i-1, j}\right) .$$

## 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|表示论代写Representation theory代考|MAST90017

statistics-lab™ 为您的留学生涯保驾护航 在代写表示论Representation theory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。我们的专家在代写表示论Representation theory代写方面经验极为丰富，各种代写表示论Representation theory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|表示论代写Representation theory代考|Some Algebras of Small Dimensions

One might like to know how many $K$-algebras there are of a given dimension, up to isomorphism. In general there might be far too many different algebras, but for small dimensions one can hope to get a complete overview. We fix a field $K$, and we consider $K$-algebras of dimension at most 2. For these, there are some restrictions.
Proposition 1.28. Let $K$ be a field.
(a) Every 1 -dimensional $K$-algebra is isomorphic to $K$.
(b) Every 2-dimensional $K$-algebra is commutative.
Proof. (a) Let $A$ be a 1-dimensional $K$-algebra. Then $A$ must contain the scalar multiples of the identity element, giving a subalgebra $U:=\left{\lambda 1_{A} \mid \lambda \in K\right} \subseteq A$. Then $U=A$, since $A$ is 1-dimensional. Moreover, according to axiom (Alg) from Definition $1.1$ the product in $U$ is given by $\left(\lambda 1_{A}\right)\left(\mu 1_{A}\right)=(\lambda \mu) 1_{A}$ and hence the map $A \rightarrow K, \lambda 1_{A} \mapsto \lambda$, is an isomorphism of $K$-algebras.

(b) Let $A$ be a 2-dimensional $K$-algebra. We can choose a basis which contains the identity element of $A$, say $\left{1_{A}, b\right}$ (use from linear algebra that every linearly independent subset can be exlended to a basis). The basis elements clearly commute; but then also any linear combinations of basis elements commute, and therefore $A$ is commutative.

We consider now algebras of dimension 2 over the real numbers $\mathbb{R}$. The aim is to classify these, up to isomorphism. The method will be to find suitable bases, leading to ‘canonical’ representatives of the isomorphism classes. It will turn out that there are precisely three $\mathbb{R}$-algebras of dimension 2, see Proposition $1.29$ below.

So we take a 2-dimensional $\mathbb{R}$-algebra $A$, and we choose a basis of $A$ containing the identity. say $\left{1_{A}, b\right}$, as in the above proof of Proposition $1.28$. Then $b^{2}$ must be a linear combination of the basis elements, so there are scalars $\gamma, \delta \in \mathbb{R}$ such that $b^{2}=\gamma 1_{A}+\delta b$. We consider the polynomial $X^{2}-\delta X-\gamma \in \mathbb{R}[X]$ and we complete squares,
$$X^{2}-\delta X-\gamma=(X-\delta / 2)^{2}-\left(\gamma+(\delta / 2)^{2}\right)$$

## 数学代写|表示论代写Representation theory代考|Definition and Examples

A vector space over a field $K$ is an abelian group $V$ together with a scalar multiplication $K \times V \rightarrow V$, satisfying the usual axioms. If one replaces the field $K$ by a ring $R$, then one gets the notion of an $R$-module. Although we mainly deal with algebras over fields in this book, we slightly broaden the perspective in this chapter by defining modules over rings. We always assume that rings contain an identity element.

Definition 2.1. Let $R$ be a ring with identity element $1_{R}$. A left $R$-module (or just $R$-module ) is an abelian group $(M,+)$ together with a map
$$R \times M \rightarrow M, \quad(r, m) \mapsto r \cdot m$$
such that for all $r, s \in R$ and all $m, n \in M$ we have
(i) $(r+s) \cdot m=r \cdot m+s \cdot m$;
(ii) $r \cdot(m+n)=r \cdot m+r \cdot n$;
(iii) $r \cdot(s \cdot m)=(r s) \cdot m$;
(iv) $1_{R} \cdot m=m$.

Exercise 2.1. Let $R$ be a ring (with zero element $0_{R}$ and identity element $1_{R}$ ) and $M$ an $R$-module with zero element $0_{M}$. Show that the following holds for all $r \in R$ and $m \in M$ :
(i) $0_{R} \cdot m=0_{M}$
(ii) $r \cdot 0_{M}=0_{M}$;
(ii) $-(r \cdot m)=(-r) \cdot m=r \cdot(-m)$, in particular $-m=\left(-1_{R}\right) \cdot m$.
Remark 2.2. Completely analogous to Definition $2.1$ one can define right $R$-modules, using a map $M \times R \rightarrow M,(m, r) \mapsto m \cdot r$. When the ring $R$ is not commutative the behaviour of left modules and of right modules can be different; for an illustration see Exercise $2.22$. We will consider only left modules, since we are mostly interested in the case when the ring is a $K$-algebra, and scalars are usually written to the left.

Before dealing with elementary properties of modules we consider a few examples.
Example 2.3.
(1) When $R=K$ is a field, then $R$-modules are exactly the same as $K$-vector spaces. Thus, modules are a true generalization of the concept of a vector space.
(2) Let $R=\mathbb{Z}$, the ring of integers. Then every abelian group can be viewed as a $\mathbb{Z}$-module: If $n \geq 1$ then $n \cdot a$ is set to be the sum of $n$ copies of $a$, and $(-n) \cdot a:=-(n \cdot a)$, and $0_{\mathbb{Z}} \cdot a=0$. With this, conditions (i) to (iv) in Definition $2.1$ hold in any abelian group.
(3) Let $R$ be a ring (with 1 ). Then every left ideal $I$ of $R$ is an $R$-module, with $R$-action given by ring multiplication. First, as a left ideal, $(I,+)$ is an abelian group. The properties (i)-(iv) hold even for arbitrary elements in $R$.
(4) A very important special case of $(3)$ is that every ring $R$ is an $R$-module, with action given by ring multiplication.

## 数学代写|表示论代写Representation theory代考|Some Algebras of Small Dimensions

(a) 每一维 $K$-代数同构于 $K$.
(b) 每个二维 $K$-代数是可交换的。

$\mathrm{~ U : = I l e f t { l l a m b d a ~ 1 _ { A } ~ \ m i d ~ \ l a m b d a ~ \ i n ~ K}$ 公理 (Alg) 1.1产品在 $U$ 是 (谁) 给的 $\left(\lambda 1_{A}\right)\left(\mu 1_{A}\right)=(\lambda \mu) 1_{A}$ 因此地图 $A \rightarrow K, \lambda 1_{A} \mapsto \lambda$ ，是一个同构 $K$-代 数。
(b) 让 $A$ 是二维的 $K$-代数。我们可以选择一个包含恒等元素的基 $A$ ，说】left{1_{A}, b\right } （使用线性代数，每个 线性独立的子集都可以扩展为一个基)。基本元素明显通勤；但随后基元素的任何线性组合也可以通勤，因此 $A$ 是 可交换的。

$$X^{2}-\delta X-\gamma=(X-\delta / 2)^{2}-\left(\gamma+(\delta / 2)^{2}\right)$$

## 数学代写|表示论代写Representation theory代考|Definition and Examples

$$R \times M \rightarrow M, \quad(r, m) \mapsto r \cdot m$$

(-) $(r+s) \cdot m=r \cdot m+s \cdot m$;
(二) $r \cdot(m+n)=r \cdot m+r \cdot n$;
$(\xi) r \cdot(s \cdot m)=(r s) \cdot m$
(四) $1_{R} \cdot m=m$.

(二) $r \cdot 0_{M}=0_{M}$;
(二) $-(r \cdot m)=(-r) \cdot m=r \cdot(-m)$ ，尤其是 $-m=\left(-1_{R}\right) \cdot m$.

(1) 当 $R=K$ 是一个场，那么 $R$-modules 与 $K$-向量空间。因此，模块是向量空间概念的真正概括。
(2) 让 $R=\mathbb{Z}$, 整数环。那么每个阿贝尔群都可以看成一个 $\mathbb{Z}$-模块: 如果 $n \geq 1$ 然后 $n \cdot a$ 被设置为总和 $n$ 的副本 $a$ ，和 $(-n) \cdot a:=-(n \cdot a)$ ，和 $0_{\mathbb{Z}} \cdot a=0$. 这样，定义中的条件 (i) 至 (iv) $2.1$ 在任何阿贝尔群中成立。
(3) 让 $R$ 是一个环（芇有 1) 。那么每一个左理想 $I$ 的 $R$ 是一个 $R$-模块，与 $R$-由环乘法给出的动作。首先，作为左派理 想， $(I,+)$ 是一个阿贝尔群。属性 (i)-(iv) 甚至适用于 $R$.
(4) 一个非常重要的特例 $(3)$ 是每一个环 $R$ 是一个 $R$-模块，通过环乘法给出动作。

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## 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|表示论代写Representation theory代考|MATH4314

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• (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|表示论代写Representation theory代考|Group Algebras

Let $G$ be a group and $K$ a field. We define a vector space over $K$ which has basis the set ${g \mid g \in G}$, and we call this vector space $K G$. This space becomes a $K$-algebra if one defines the product on the basis by taking the group multiplication, and extends it to linear combinations. We call this algebra $K G$ the group algebra.
Thus an arbitrary element of $K G$ is a finite linear combination of the form $\sum_{g \in G} \alpha_{g} g$ with $\alpha_{g} \in K$. We can write down a formula for the product of two elements, following the recipe in Remark 1.4. Let $\alpha=\sum_{g \in G} \alpha_{g} g$ and $\beta=\sum_{h \in G} \beta_{h} h$ be two elements in $K G$; then their product has the form
$$\alpha \beta=\sum_{x \in G}\left(\sum_{g h=x} \alpha_{g} \beta_{h}\right) x$$
Since the multiplication in the group is associative, it follows that the multiplication in $K G$ is associative. Furthermore, one checks that the multiplication in $K G$ is distributive. The identity element of the group algebra $K G$ is given by the identity element of $G$.

Note that the group algebra $K G$ is finite-dimensional if and only if the group $G$ is finite, in which case the dimension of $K G$ is equal to the order of the group $G$. The group algebra $K G$ is commutative if and only if the group $G$ is abelian.

Example 1.10. Let $G$ be the cyclic group of order 3 , generated by $y$, so that $G=\left{1_{G}, y, y^{2}\right}$ and $y^{3}=1_{G}$. Then we have
$$\left(a_{0} 1_{G}+a_{1} y+a_{2} y^{2}\right)\left(b_{0} 1_{G}+b_{1} y+b_{2} y^{2}\right)=c_{0} 1_{G}+c_{1} y+c_{2} y^{2},$$
with
$$c_{0}=a_{0} b_{0}+a_{1} b_{2}+a_{2} b_{1}, c_{1}=a_{0} b_{1}+a_{1} b_{0}+a_{2} b_{2}, c_{2}=a_{0} b_{2}+a_{1} b_{1}+a_{2} b_{0}$$

## 数学代写|表示论代写Representation theory代考|Path Algebras of Quivers

Path algebras of quivers are a class of algebras with an easy multiplication formula, and they are extremely useful for calculating examples. They also have connections to other parts of mathematics. The underlying basis of a path algebra is the set of paths in a finite directed graph. It is customary in representation theory to call such a graph a quiver. We assume throughout that a quiver has finitely many vertices and finitely many arrows.

Definition 1.11. A quiver $Q$ is a finite directed graph. We sometimes write $Q=\left(Q_{0}, Q_{1}\right)$, where $Q_{0}$ is the set of vertices and $Q_{1}$ is the set of arrows.

We assume that $Q_{0}$ and $Q_{1}$ are finite sets. For any arrow $\alpha \in Q_{1}$ we denote by $s(\alpha) \in Q_{0}$ its starting point and by $t(\alpha) \in Q_{0}$ its end point.

A non-trivial path in $Q$ is a sequence $p=\alpha_{r} \ldots \alpha_{2} \alpha_{1}$ of arrows $\alpha_{i} \in Q_{1}$ such that $t\left(\alpha_{i}\right)=s\left(\alpha_{i+1}\right)$ for all $i=1, \ldots, r-1$. Note that our convention is to read paths from right to left. The number $r$ of arrows is called the length of $p$, and we denote by $s(p)=s\left(\alpha_{1}\right)$ the starting point, and by $t(p)=t\left(\alpha_{r}\right)$ the end point of $p$.
For each vertex $i \in Q_{0}$ we also need to have a trivial path of length 0 , which we call $e_{i}$, and we set $s\left(e_{i}\right)=i=t\left(e_{i}\right)$.

We call a path $p$ in $Q$ an oriented cycle if $p$ has positive length and $s(p)=t(p)$.

## 数学代写|表示论代写Representation theory代考|Group Algebras

$$\alpha \beta=\sum_{x \in G}\left(\sum_{g h=x} \alpha_{g} \beta_{h}\right) x$$

$$\left(a_{0} 1_{G}+a_{1} y+a_{2} y^{2}\right)\left(b_{0} 1_{G}+b_{1} y+b_{2} y^{2}\right)=c_{0} 1_{G}+c_{1} y+c_{2} y^{2},$$

$$c_{0}=a_{0} b_{0}+a_{1} b_{2}+a_{2} b_{1}, c_{1}=a_{0} b_{1}+a_{1} b_{0}+a_{2} b_{2}, c_{2}=a_{0} b_{2}+a_{1} b_{1}+a_{2} b_{0}$$

## 有限元方法代写

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## 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。