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• 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

## 数学代写|抽象代数作业代写abstract algebra代考|The Dihedral Groups

The analysis carried out above for a square can similarly be done for an equilateral triangle or regular pentagon or, indeed, any regular $n$-gon $(n \geq 3)$. The corresponding group is denoted by $D_n$ and is called the dihedral group of order $2 n$.

The dihedral groups arise frequently in art and nature. Many of the decorative designs used on floor coverings, pottery, and buildings have one of the dihedral groups as a group of symmetry. Corporation logos are rich sources of dihedral symmetry. Chrysler’s logo has $D_5$ as a symmetry group, and that of Mercedes-Benz has $D_3$. The ubiquitous five-pointed star has symmetry group $D_5$.

The phylum Echinodermata contains many sea animals (such as starfish, sea cucumbers, feather stars, and sand dollars) that exhibit patterns with $D_5$ symmetry. Snowflakes have $D_6$ symmetry (see Exercise 19).

Chemists classify molecules according to their symmetry. Moreover, symmetry considerations are applied in orbital calculations, in determining energy levels of atoms and molecules, and in the study of molecular vibrations. The symmetry group of a pyramidal molecule such as ammonia $\left(\mathrm{NH}_3\right)$, depicted in Figure $1.2$, is $D_3$.

Mineralogists determine the internal structures of crystals (i.e., rigid bodies in which the particles are arranged in threedimensional discrete patterns – table salt and table sugar are two examples) by studying two-dimensional $x$-ray projections of the atomic makeup of the crystals. The symmetry present in the projections reveals the internal symmetry of the crystals themselves. Commonly occurring symmetry patterns are $D_4$ and $D_6$ (see Figure 1.3). Interestingly, it is mathematically impossible for a crystal to possess a $D_n$ repeating symmetry pattern with $n=5$ or $n>6$.

The dihedral group of order $2 n$ is often called the group of symmetries of a regular $n$-gon. A plane symmetry of a figure $F$ in a plane is a function from the plane to itself that carries $F$ onto $F$ and preserves distances; that is, for any points $p$ and $q$ in the plane, the distance from the image of $p$ to the image of $q$ is the same as the distance from $p$ to $q$. (The term symmetry is from the Greek word symmetros, meaning “of like measure.”)

## 数学代写|抽象代数作业代写abstract algebra代考|Definition and Examples of Groups

The term group was used by Galois around 1830 to describe sets of one-to-one functions on finite sets that could be grouped together to form a set closed under composition. As is the case with most fundamental concepts in mathematics, the modern definition of a group that follows is the result of a long evolutionary process. Although this definition was given by both Heinrich Weber and Walther von Dyck in 1882, it did not gain universal acceptance until the 20 th century.
Definition Binary Operation
Let $G$ be a set. A binary operation on $G$ is a function that assigns each ordered pair of elements of $G$ an element of $G$.
A binary operation on a set $G$, then, is simply a method (or formula) by which the members of an ordered pair from $G$ combine to yield a new member of $G$. This condition is called closure. The most familiar binary operations are ordinary addition, subtraction, and multiplication of integers. Division of integers is not a binary operation on the integers because an integer divided by an integer need not be an integer.

The binary operations addition modulo $n$ and multiplication modulo $n$ on the set ${0,1,2, \ldots, n-1}$, which we denote by $Z_n$, play an extremely important role in abstract algebra. In certain situations we will want to combine the elements of $Z_n$ by addition modulo $n$ only; in other situations we will want to use both addition modulo $n$ and multiplication modulo $n$ to combine the elements. It will be clear from the context whether we are using addition only or addition and multiplication. For example, when multiplying matrices with entries from $Z_n$, we will need both addition modulo $n$ and multiplication modulo $n$.

# 抽象代数代写

## 数学代写|抽象代数作业代写abstract algebra代考|Definition and Examples of Groups

1830 年左右，Galois 使用术语群来描述有限集上的一对一函数的集合，这些函数可以组合在一起形成一个在合成下闭合的集合。与数学中大多数基本概念的情况一样，现代对群的定义是长期进化过程的结果。尽管这个定义是由 Heinrich Weber 和 Walther von Dyck 在 1882 年给出的，但直到 20 世纪才得到普遍接受。

LetG成为一个集合。二元运算G是一个函数，它分配的每个有序元素对G的一个元素G.

## 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。

assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师