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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|离散数学作业代写discrete mathematics代考|MATRICES AND DETERMINANTS

We will also have several occasions to use matrices, even though they are usually studied more in Linear Algebra than in Discrete Mathematics. A $k \times l$ matrix is a rectangular table of $k$ rows by $l$ columns, whose elements are numbers or some other items, for example, functions; an $a_{x, y}$ stands for the item at the intersection of the $x$ th row and $y$ th column; the first element $x$ always stands for the rows, while the second for the columns; for example, the $2 \times 3$ matrix
$$A=\left(\begin{array}{ccc} 0 & 1 & -2 \ \pi & a+6 & 0 \end{array}\right)$$
Thus, each row and each column of a matrix is a vector, either a rowvector or a column-vector. We discuss here only some basic features of the matrices, needed in this course. The matrix $A$ in the example above is $a_{i, j}, i=1,2 ; j=1,2,3$, in particular, $a_{1,2}=1$ and $a_{2,1}=\pi$.
Problem 121. What are the other elements $a_{x, y}$ in the matrix $\Lambda$ above?
When we introduce any new entities in mathematics, we are, first of all, interested in what we can do with them; thus, first, we consider arithmetic operations with the matrices. Their addition/subtraction is done element-wise; therefore, all the familiar addition/subtraction properties of numbers, like the commutativity, associativity, existence of neutral elements are preserved. To multiply a matrix by a number (a scalar) $b$, we also do that component-wise, as
$$-3 A=\left(\begin{array}{ccc} 0 & -3 & 6 \ -3 \pi & -3 a-18 & 0 \end{array}\right) \text {. }$$

## 数学代写|离散数学作业代写discrete mathematics代考|FINITE GROUPS

Groups were introduced above. Finite groups often occur in cryptography, and we go into some details here. A group $G$ is finite if it consists of finitely many elements; their number is called the order $\rho=\operatorname{ord}(G)=|G|$ of $G$. For example, the two-element set $\mathbb{B}={0,1}$ is a group with respect to the binary addition. Its neutral element is 0 , and every element of $\mathbb{B}$ is inverse to itself. This two-element set has a rich algebraic structure, it is a ring and even a field. First of all, we discuss some examples.

Example 10. Consider any finite set $A=\left{a_{1}, a_{2}, \ldots, a_{d}\right}$ of cardinality $d$, and all the permutations of its elements; we know that there are $d !$ of these permutations. As a group operation *, we consider a superposition of two permutations, that is, $\rho_{2} * \rho_{1}=\rho_{2}\left(\rho_{1}\right)$, meaning that we first apply the permutation $\rho_{1}$ to A, and then $\rho_{2}$ to the image of the previous operation.

Problem 128. Prove that this is a non-commutative group, often denoted as $\mathbb{Z}{d}$. Find its neutral element and the inverse elements for each permutation. This finite group of permutations is called the symmetric group $S{d}$ of order ord $\left(S_{d}\right)=d !$ It is useful to consider some special cases. For instance, write explicitly all the permutations of three and four elements.

Let us consider a natural number $d$ and the set $\mathbb{Z}{d}={0,1, \ldots, d-1}$ of natural numbers modulo $d$, see Chapter 9 . This congruence relation is the equivalence relation on the symmetric group $\mathbb{Z}{d}$, and its $d$ equivalence classes are the arithmetic progressions of the integers having the same remainder after dividing by $d$. By $\mathbb{Z}{d}^{}$ we denote the set of natural numbers $k$, which are mutually prime with $d$, that is, $k \in{0,1,2, \ldots, d-1}$ and $\operatorname{gcd}(k, d)=1$. For example, $\mathbb{Z}{8}={0,1,2,3,4,5,6,7}$, and $\mathbb{Z}_{8}^{}={1,3,5,7}$. The next claim will be proven in Chapter 8 .

## 数学代写|离散数学作业代写discrete mathematics代考| MATRICES AND DETERMINANTS

$$A=\left(\begin{array}{lllll} 0 & 1 & -2 \pi & a+6 & 0 \end{array}\right)$$

$$-3 A=\left(\begin{array}{llllll} 0 & -3 & 6 & -3 \pi & -3 a-18 & 0 \end{array}\right) .$$

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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。

assignmentutor™您的专属作业导师
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