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assignmentutor-lab™ 为您的留学生涯保驾护航 在代写抽象代数abstract algebra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。我们的专家在代写抽象代数abstract algebra代写方面经验极为丰富，各种代写抽象代数abstract algebra相关的作业也就用不着说。

• 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|Symmetries of a Square

Suppose we remove a square region from a plane, move it in some way, then put the square back into the space it originally occupied. Our goal in this chapter is to describe all possible ways in which this can be done. More specifically, we want to describe the possible relationships between the starting position of the square and its final position in terms of motions. However, we are interested in the net effect of a motion, rather than in the motion itself. Thus, for example, we consider a $90^{\circ}$ rotation and a $450^{\circ}$ rotation as equal, since they have the same net effect on every point. With this simplifying convention, it is an easy matter to achieve our goal.

To begin, we can think of the square region as being transparent (glass, say), with the corners marked on one side with the colors blue, white, pink, and green. This makes it easy to distinguish between motions that have different effects. With this marking scheme, we are now in a position to describe, in simple fashion, all possible ways in which a square object can be repositioned. See Figure 1.1. We now claim that any motion-no matter how complicated – is equivalent to one of these eight. To verify this claim, observe that the final position of the square is completely determined by the location and orientation (i.e., face up or face down) of any particular corner. But, clearly, there are only four locations and two orientations for a given corner, so there are exactly eight distinct final positions for the corner.

Let’s investigate some consequences of the fact that every motion is equal to one of the eight listed in Figure 1.1. Suppose a square is repositioned by a rotation of $90^{\circ}$ followed by a flip about the horizontal axis of symmetry.

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 $\mathrm{D}_3$.

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.”)

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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