19th Ave New York, NY 95822, USA

# 数学代写|组合学代写Combinatorics代考|MA1510 AW-relations

my-assignmentexpert™提供最专业的一站式服务：Essay代写，Dissertation代写，Assignment代写，Paper代写，Proposal代写，Proposal代写，Literature Review代写，Online Course，Exam代考等等。my-assignmentexpert™专注为留学生提供Essay代写服务，拥有各个专业的博硕教师团队帮您代写，免费修改及辅导，保证成果完成的效率和质量。同时有多家检测平台帐号，包括Turnitin高级账户，检测论文不会留痕，写好后检测修改，放心可靠，经得起任何考验！

## 数学代写|组合学代写Combinatorics代考|AW-relations

The relations of $A, A^$ in $\operatorname{End}(V)$ $$(\mathrm{AW})\left{\begin{array}{c} A^2 A^-\beta A A^* A+A^* A^2 \ =\gamma\left(A A^+A^ A\right)+\delta A^+\gamma^ A^2+\omega A+\eta^* \ \left(A^\right)^2 A-\beta A^ A A^+A\left(A^\right)^2 \ =\gamma^\left(A^ A+A A^\right)+\delta^ A+\gamma\left(A^\right)^2+\omega A^+\eta \end{array}\right.$$
are called Askey-Wilson relations (AW-relations). Here, $\beta, \gamma, \gamma^, \delta, \delta^, \omega, \eta, \eta^*$ are complex constants determined for each AW-relation. In this section, we first show that L-pairs satisfy AW-relations. Then, we consider the condition for pre-TD-pairs and preL-pairs to satisfy AW-relations. As an application, we show that if a TD-pair satisfies AW-relations, it becomes an L-pair.

Before the discussion of the main theory, we look at another expression of AWrelations (AW). In the first equation of $(\mathrm{AW})$, replace $\omega$ by $\omega^$ and denote it as $(\mathrm{AW})_1$, and denote the second equation of $(\mathrm{AW})$ as $(\mathrm{AW})_2$. Namely, if we set \begin{aligned} & C=A^2 A^-\beta A A^* A+A^* A^2-\gamma\left(A A^+A^ A\right)-\delta A^, \ & C^=\left(A^\right)^2 A-\beta A^ A A^+A\left(A^\right)^2-\gamma^\left(A^ A+A A^\right)-\delta^ A, \end{aligned}
then we have
$$\begin{array}{ll} (\mathrm{AW})_1: & C=\gamma^* A^2+\omega^* A+\eta^* \ (\mathrm{AW})_2: & C^=\gamma\left(A^\right)^2+\omega A^*+\eta \end{array}$$

## 数学代写|组合学代写Combinatorics代考|Classification

In this section, we prove the following theorem.
Theorem 6.69. A pre-L-system $\left(A, A^* ;\left{V_i\right}_{i=0}^d,\left{V_i^\right}_{i=0}^d\right)$ becomes an L-system if and only if the data $\left{\theta_i\right}_{i=0}^d,\left{\theta_i^\right}_{i=0}^d,\left{\lambda_i\right}_{i=0}^{d-1}$ are expressed by $A W$-parameters as in the table of Theorem $6.61$.

By the above theorem, isomorphism classes of L-systems are in one-to-one correspondence with the data in the table of Theorem 6.61, and in this sense, the classification of L-systems is completed. Note that if $\lambda_i \neq 0(0 \leq i \leq d-1)$, a pre-L-pair with given data $\left{\theta_i\right}_{i=0}^d,\left{\theta_i^\right}_{i=0}^d,\left{\lambda_i\right}_{i=0}^{d-1}$ uniquely exists up to isomorphism. In order to prove Theorem $6.69$, it suffices to show the following proposition. Proposition 6.70. Suppose the data $\left{\theta_i\right}_{i=0}^d,\left{\theta_i^\right}_{i=0}^d,\left{\lambda_i\right}_{i=0}^{d-1}$ of a pre-L-system $\left(A, A^\right.$; $\left{V_i\right}_{i=0}^d,\left{V_i^\right}_{i=0}^d$ ) are expressed by AW-parameters as in the table of Theorem 6.61. Then the following (1), (2) hold:

(1) $A, A^$ satisfy $A W$-relations;
(2) $V=\bigoplus_{i=0}^d V_i=\bigoplus_{i=0}^d V_i^$ is irreducible as an $\left\langle A, A^*\right\rangle$-module.

## 数学代与写组合学代写COMBINATORICS代考|AWRELATIONS

$\$ \

$\backslash$ begin ${$ array $}{|}}(\backslash m a t h r m{A W}) _1: \& C=\backslash g a m m a^{\wedge *} A^{\wedge} 2+\backslash$ omega ${ }^{\wedge *} A+\backslash$ eta ${ }^{\wedge \star} \backslash(\backslash m a t h r m{A W}) _2: \& C^{\wedge}=\backslash$ gamma $\backslash$ left $\left(\mathrm{A}^{\wedge} \backslash \text { right }\right)^{\wedge} 2+\backslash$ omega $A^{\wedge *}+\backslash$ eta $\backslash$ end ${$ array} $}$

## 数学代写|组合学代写COMBINATORICS代考|CLASSIFICATION

1 一个，一个 满足 $A W$-关系；
$2 \quad \mathrm{~V}=\langle\text { bigoplus_{i }=0}^{\wedge} \mathrm{d} V_{-} \mathrm{i}=\langle\text { bigoplus_{i }=0}^{\wedge} \mathrm{d} V_{-} \mathrm{i} \wedge$ 是不可约的 $\left\langle A, A^*\right\rangle$-模块。

## 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。