数学代写|抽象代数作业代写abstract algebra代考|Sources of group theory

如果你也在 怎样代写抽象代数abstract algebra这个学科遇到相关的难题,请随时右上角联系我们的24/7代写客服。

抽象代数是代数的一组高级课题,涉及抽象代数结构而不是通常的数系。这些结构中最重要的是群、环和场。

statistics-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。

我们提供的抽象代数abstract algebra及其相关学科的代写,服务范围广, 其中包括但不限于:

  • 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|抽象代数作业代写abstract algebra代考|Sources of group theory

数学代写|抽象代数作业代写abstract algebra代考|Classical Algebra

The major problems in algebra at the time $(1770)$ that Lagrange wrote his fundamental memoir “Reflections on the solution of algebraic equations” concerned polynomial equations. There were “theoretical” questions dealing with the existence and nature of the roots-for example, does every equation have a root? how many roots are there? are they real, complex, positive, negative?-and “practical” questions dealing with methods for finding the roots. In the latter instance there were exact methods and approximate methods. In what follows we mention exact methods.

The Babylonians knew how to solve quadratic equations, essentially by the method of completing the square, around $1600 \mathrm{BC}$ (see Chapter 1). Algebraic methods for solving the cubic and the quartic were given around 1540 (Chapter 1). One of the major problems for the next two centuries was the algebraic solution of the quintic. This is the task Lagrange set for himself in his paper of 1770 .In this paper Lagrange first analyzed the various known methods, devised by Viète, Descartes, Euler, and Bezout, for solving cubic and quartic equations. He showed that the common feature of these methods is the reduction of such equations to auxiliary equations-the so-called resolvent equations. The latter are one degree lower than the original equations.

Lagrange next attempted a similar analysis of polynomial equations of arbitrary degree $n$. With each such equation he associated a resolvent equation, as follows: let $f(x)$ be the original equation, with roots $x_{1}, x_{2}, x_{3}, \ldots, x_{n}$. Pick a rational function $\mathbf{R}\left(x_{1}, x_{2}, x_{3}, \ldots, x_{n}\right)$ of the roots and coefficients of $f(x)$. (Lagrange described methods for doing this.) Consider the different values which $\mathbf{R}\left(x_{1}, x_{2}, x_{3}, \ldots, x_{n}\right)$ assumes under all the $n$ ! permutations of the roots $x_{1}, x_{2}, x_{3}, \ldots, x_{n}$ of $f(x)$. If these are denoted by $y_{1}, y_{2}, y_{3}, \ldots, y_{k}$, then the resolvent equation is given by $g(x)=\left(x-y_{1}\right)\left(x-y_{2}\right) \cdots\left(x-y_{k}\right) .$

It is imporotant to note that the coefficients of $g(x)$ aree symmetric functions in $x_{1}, x_{2}, x_{3}, \ldots, x_{n}$, hence they are polynomials in the elementary symmetric functions of $x_{1}, x_{2}, x_{3}, \ldots, x_{n}$; that is, they are polynomials in the coefficients of the original equation $f(x)$. Lagrange showed that $k$ divides $n$ !-the source of what we call Lagrange’s theorem in group theory.

For example, if $f(x)$ is a quartic with roots $x_{1}, x_{2}, x_{3}, x_{4}$, then $\mathbf{R}\left(x_{1}, x_{2}, x_{3}, x_{4}\right)$ may be taken to be $x_{1} x_{2}+x_{3} x_{4}$, and this function assumes three distinct values under the twenty-four permutations of $x_{1}, x_{2}, x_{3}, x_{4}$. Thus the resolvent equation of a quartic is a cubic. However, in carrying over this analysis to the quintic Lagrange found that the resolvent equation is of degree six.

Although Lagrange did not succeed in resolving the problem of the algebraic solvability of the quintic, his work was a milestone. It was the first time that an association was made between the solutions of a polynomial equation and the permutations of its roots. In fact, the study of the permutations of the roots of an equation was a cornerstone of Lagrange’s general theory of algebraic equations. This, he speculated, formed “the true principles of the solution of equations.” He was, of course, vindicated in this by Galois. Although Lagrange spoke of permutations without considering a “calculus” of permutations (e.g., there is no consideration of their composition or closure), it can be said that the germ of the group concept-as a group of permutations-is present in his work. For details see [12], [16], [19], [25], [33].

数学代写|抽象代数作业代写abstract algebra代考|Number Theory

In the Disquisitiones Arithmeticae (Arithmetical Investigations) of 1801 Gauss summarized and unified much of the number theory that preceded him. The work also suggested new directions which kept mathematicians occupied for the entire century. As for its impact on group theory, the Disquisitiones may be said to have initiated the theory of finite abelian groups. In fact, Gauss established many of the significant properties of these groups without using any of the terminology of group theory.
The groups appeared in four different guises: the additive group of integers modulo $m$, the multiplicative group of integers relatively prime to $m$, modulo $m$, the group of equivalence classes of binary quadratic forms, and the group of $n$-th roots of unity. And although these examples turned up in number-theoretic contexts, it is as abelian groups that Gauss treated them, using what are clear prototypes of modern algebraic proofs.

For example, considering the nonzero integers modulo $p$ ( $p$ a prime), he showed that they are all powers of a single element; that is, that the group $Z_{p}^{*}$ of such integers

is cyclic. Moreover, he determined the number of generators of this group, showing that it is equal to $\varphi(p-1)$, where $\varphi$ is Euler’s $\varphi$-function.

Given any element of $Z_{p}^{}$, he defined the order of the element (without using the terminology) and showed that the order of an element is a divisor of $p-1$. He then used this result to prove Fermat’s “little theorem,” namely that $a^{p-1} \equiv 1(\bmod p)$ if $p$ does not divide $a$, thus employing group-theoretic ideas to prove number-theoretic results. Next he showed that if $t$ is a positive integer which divides $p-1$, then there exists an element in $Z_{p}^{}$ whose order is $t$-essentially the converse of Lagrange’s theorem for cyclic groups.

Concerning the $n$-th roots of 1 , which he considered in connection with the cyclotomic equation, he showed that they too form a cyclic group. In relation to this group he raised and answered many of the same questions he raised and answered in the case of $Z_{p}^{*}$.

The problem of representing integers by binary quadratic forms goes back to Fermat in the early seventeenth century. (Recall his theorem that every prime of the form $4 n+1$ can be represented as a sum of two squares $x^{2}+y^{2}$.) Gauss devoted a large part of the Disquisitiones to an exhaustive study of binary quadratic forms and the representation of integers by such forms.

A binary quadratic form is an expression of the form $a x^{2}+b x y+c y^{2}$, with $a, b, c$ integers. Gauss defined a composition on such forms, and remarked that if $K_{1}$ and $K_{2}$ are two such forms, one may denote their composition by $K_{1}+K_{2}$. He then showed that this composition is associative and commutative, that there exists an identity, and that each form has an inverse, thus verifying all the properties of an abelian group.

Despite these remarkable insights, one should not infer that Gauss had the concept of an abstract group, or even of a finite abelian group. Although the arguments in the Disquisitiones are quite general, each of the various types of “groups” he considered was dealt with separately-there was no unifying group-theoretic method which he applied to all cases.
For further details see [5], [9], [25], [30], [33].

数学代写|抽象代数作业代写abstract algebra代考|Geometry

We are referring here to Klein’s famous and influential (but see [18]) lecture entitled “A Comparative Review of Recent Researches in Geometry,” which he delivered in 1872 on the occasion of his admission to the faculty of the University of Erlangen. The aim of this so-called Erlangen Program was the classification of geometry as the study of invariants under various groups of transformations. Here there appear groups such as the projective group, the group of rigid motions, the group of similarities, the hyperbolic group, the elliptic groups, as well as the geometries associated with them. (The affine group was not mentioned by Klein.) Now for some background leading to Klein’s Erlangen Program.

The nineteenth century witnessed an explosive growth in geometry, both in scope and in depth. New geometries emerged: projective geometry, noneuclidean geometries, differential geometry, algebraic geometry, $n$-dimensional geometry, and

Grassmann’s geometry of extension. Various geometric methods competed for supremacy: the synthetic versus the analytic, the metric versus the projective.

At mid-century a major problem had arisen, namely the classification of the relations and inner connections among the different geometries and geometric methods. This gave rise to the study of “geometric relations,” focusing on the study of properties of figures invariant under transformations. Soon the focus shifted to a study of the transformations themselves. Thus the study of the geometric relations of figures became the study of the associated transformations.

Various types of transformations (e.g., collineations, circular transformations, inversive transformations, affinities) became the objects of specialized studies. Subsequently, the logical connections among transformations were investigated, and this led to the problem of classifying transformations, and eventually to Klein’s group-theoretic synthesis of geometry.

Klein’s use of groups in geometry was the final stage in bringing order to geometry. An intermediate stage was the founding of the first major theory of classification in geometry, beginning in the $1850 \mathrm{~s}$, the Cayley-Sylvester Invariant Theory. Here the objective was to study invariants of “forms” under transformations of their variables (see Chapter 8.1). This theory of classification, the precursor of Klein’s Erlangen Program, can be said to be implicitly group-theoretic. Klein’s use of groups in geometry was, of course, explicit. (For a thorough analysis of implicit group-theoretic thinking in geometry leading to Klein’s Erlangen Program see [33].)

In the next section we will note the significance of Klein’s Erlangen Program (and his other works) for the evolution of group theory. Since the Program originated a hundred years after Lagrange’s work and eighty years after Gauss’ work, its importance for group theory can best be appreciated after a discussion of the evolution of group theory beginning with the works of Lagrange and Gauss and ending with the period around 1870 .

数学代写|抽象代数作业代写abstract algebra代考|Sources of group theory

抽象代数代写

数学代写|抽象代数作业代写abstract algebra代考|Classical Algebra

当时代数的主要问题(1770)拉格朗日写了他的基础回忆录“对代数方程解的思考”,涉及多项式方程。有一些“理论”问题涉及根的存在和性质——例如,每个方程都有根吗?有多少根?它们是真实的、复杂的、积极的、消极的吗?以及处理寻找根源的方法的“实际”问题。在后一种情况下,有精确方法和近似方法。在下文中,我们将提到确切的方法。

巴比伦人知道如何解二次方程,主要是通过完成平方的方法,大约1600乙C(见第 1 章)。大约在 1540 年(第 1 章)给出了求解三次和四次的代数方法。接下来两个世纪的主要问题之一是五次元的代数解。这是拉格朗日在 1770 年的论文中为自己设定的任务。在这篇论文中,拉格朗日首先分析了由 Viète、笛卡尔、欧拉和贝祖设计的各种已知方法,用于求解三次方程和​​四次方程。他表明,这些方法的共同特点是将这些方程简化为辅助方程,即所谓的求解方程。后者比原始方程低一级。

拉格朗日接下来尝试对任意次数的多项式方程进行类似的分析n. 对于每一个这样的方程,他都关联了一个求解方程,如下所示:F(X)是原方程,有根X1,X2,X3,…,Xn. 选择一个有理函数R(X1,X2,X3,…,Xn)的根和系数F(X). (拉格朗日描述了这样做的方法。)考虑不同的值R(X1,X2,X3,…,Xn)假设在所有n!根的排列X1,X2,X3,…,Xn的F(X). 如果这些表示为是1,是2,是3,…,是ķ,则求解方程由下式给出G(X)=(X−是1)(X−是2)⋯(X−是ķ).

值得注意的是,系数G(X)是对称函数X1,X2,X3,…,Xn, 因此它们是基本对称函数中的多项式X1,X2,X3,…,Xn; 也就是说,它们是原方程系数中的多项式F(X). 拉格朗日表明ķ划分n!——群论中我们称之为拉格朗日定理的来源。

例如,如果F(X)是有根的四次方X1,X2,X3,X4, 然后R(X1,X2,X3,X4)可能被认为是X1X2+X3X4,并且该函数在 的 24 种排列下假定三个不同的值X1,X2,X3,X4. 因此,四次的求解方程是三次的。然而,在对五次拉格朗日进行这种分析时,发现求解方程是六次的。

尽管拉格朗日没有成功解决五次的代数可解性问题,但他的工作是一个里程碑。这是第一次在多项式方程的解与其根的排列之间建立关联。事实上,对方程根的排列的研究是拉格朗日代数方程的一般理论的基石。他推测,这形成了“方程解的真正原理”。当然,伽罗瓦在这方面证明了他是正确的。尽管拉格朗日在谈到排列时没有考虑排列的“演算”(例如,没有考虑它们的组成或闭合),但可以说,群概念的萌芽——作为一组排列——存在于他的作品中. 有关详细信息,请参阅 [12]、[16]、[19]、[25]、[33]。

数学代写|抽象代数作业代写abstract algebra代考|Number Theory

在 1801 年的 Disquisitiones Arithmeticae(算术研究)中,高斯总结并统一了他之前的大部分数论。这项工作还提出了新的方向,使数学家们占据了整个世纪。至于它对群论的影响,可以说《研究》开创了有限阿贝尔群的理论。事实上,高斯在没有使用任何群论术语的情况下建立了这些群的许多重要性质。
这些群以四种不同的形式出现:整数模的加法群米, 与 互质的整数的乘法群米, 模块米,二元二次形式的等价类群,以及n-th 统一的根源。尽管这些例子出现在数论环境中,但高斯将它们视为阿贝尔群,使用现代代数证明的清晰原型。

例如,考虑非零整数模p ( p素数),他证明了它们都是单一元素的幂;也就是说,该组从p∗这样的整数

是循环的。此外,他确定了这组发电机的数量,表明它等于披(p−1), 在哪里披是欧拉的披-功能。

给定任何元素从p,他定义了元素的顺序(不使用术语)并表明元素的顺序是p−1. 然后他用这个结果证明了费马的“小定理”,即一个p−1≡1(反对p)如果p不分一个,因此采用群论思想来证明数论结果。接下来他表明,如果吨是一个正整数,除以p−1,则存在一个元素从p谁的顺序是吨-本质上是循环群的拉格朗日定理的反面。

关于n1 的 -th 根,他将其与分圆方程结合起来考虑,他表明它们也形成了一个循环群。关于这个群体,他提出并回答了许多与他在案例中提出和回答的问题相同的问题。从p∗.

用二元二次形式表示整数的问题可以追溯到 17 世纪初的费马。(回想一下他的定理,形式的每个素数4n+1可以表示为两个平方的和X2+是2.) Gauss 将大部分论文用于详尽研究二元二次形式和用这种形式表示的整数。

二元二次形式是形式的表达式一个X2+bX是+C是2, 和一个,b,C整数。高斯在这种形式上定义了一个组合,并指出如果ķ1和ķ2是两种这样的形式,一种可以表示它们的组成ķ1+ķ2. 然后他证明了这个组合是结合和交换的,存在一个恒等式,并且每个形式都有一个逆,从而验证了阿贝尔群的所有性质。

尽管有这些非凡的见解,人们不应该推断高斯有抽象群的概念,甚至是有限阿贝尔群的概念。虽然论文中的论点很笼统,但他认为的各种“群”都是分开处理的——没有统一的群论方法,他将其应用于所有案例。
有关详细信息,请参阅 [5]、[9]、[25]、[30]、[33]。

数学代写|抽象代数作业代写abstract algebra代考|Geometry

我们在这里指的是克莱因著名且有影响力的(但请参阅 [18])讲座,题为“最近几何研究的比较回顾”,这是他在 1872 年进入埃尔兰根大学时发表的。这个所谓的埃尔兰根纲领的目的是将几何分类为研究各种变换组下的不变量。这里出现了诸如射影群、刚体运动群、相似群、双曲群、椭圆群等群,以及与它们相关的几何。(Klein 没有提到仿射群。)现在介绍导致 Klein 的 Erlangen 程序的一些背景。

十九世纪见证了几何学在范围和深度上的爆炸式增长。出现了新的几何:射影几何、非几里得几何、微分几何、代数几何、n维几何,和

格拉斯曼的外延几何。各种几何方法争夺霸权:综合与分析,度量与投影。

世纪中叶出现了一个重大问题,即不同几何和几何方法之间的关系和内在联系的分类。这引发了对“几何关系”的研究,重点研究图形在变换下的不变性。很快,焦点转移到了对转换本身的研究。因此,对图形几何关系的研究变成了对相关变换的研究。

各种类型的变换(例如,准线、循环变换、逆变换、亲和)成为专门研究的对象。随后,变换之间的逻辑联系被研究,这导致了变换的分类问题,最终导致了克莱因对几何的群论综合。

克莱因在几何中使用群是给几何带来秩序的最后阶段。中间阶段是几何分类的第一个主要理论的建立,始于1850 s,Cayley-Sylvester 不变理论。这里的目标是研究“形式”在变量变换下的不变量(见第 8.1 章)。这种分类理论是克莱因埃尔兰根纲领的先驱,可以说是隐含的群论。当然,克莱因在几何中对群的使用是明确的。(有关导致 Klein 的 Erlangen 纲领的几何中隐含的群论思维的彻底分析,请参阅 [33]。)

在下一节中,我们将注意到克莱因的 Erlangen 纲领(以及他的其他著作)对群论演化的意义。由于该纲领起源于拉格朗日工作一百年和高斯工作八十年后,因此在讨论了从拉格朗日和高斯的工作开始到大约1870 年。

数学代写|抽象代数作业代写abstract algebra代考 请认准statistics-lab™

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

金融工程是使用数学技术来解决金融问题。金融工程使用计算机科学、统计学、经济学和应用数学领域的工具和知识来解决当前的金融问题,以及设计新的和创新的金融产品。

非参数统计代写

非参数统计指的是一种统计方法,其中不假设数据来自于由少数参数决定的规定模型;这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型(GLM)归属统计学领域,是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

术语 广义线性模型(GLM)通常是指给定连续和/或分类预测因素的连续响应变量的常规线性回归模型。它包括多元线性回归,以及方差分析和方差分析(仅含固定效应)。

有限元方法代写

有限元方法(FEM)是一种流行的方法,用于数值解决工程和数学建模中出现的微分方程。典型的问题领域包括结构分析、传热、流体流动、质量运输和电磁势等传统领域。

有限元是一种通用的数值方法,用于解决两个或三个空间变量的偏微分方程(即一些边界值问题)。为了解决一个问题,有限元将一个大系统细分为更小、更简单的部分,称为有限元。这是通过在空间维度上的特定空间离散化来实现的,它是通过构建对象的网格来实现的:用于求解的数值域,它有有限数量的点。边界值问题的有限元方法表述最终导致一个代数方程组。该方法在域上对未知函数进行逼近。[1] 然后将模拟这些有限元的简单方程组合成一个更大的方程系统,以模拟整个问题。然后,有限元通过变化微积分使相关的误差函数最小化来逼近一个解决方案。

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。

随机分析代写


随机微积分是数学的一个分支,对随机过程进行操作。它允许为随机过程的积分定义一个关于随机过程的一致的积分理论。这个领域是由日本数学家伊藤清在第二次世界大战期间创建并开始的。

时间序列分析代写

随机过程,是依赖于参数的一组随机变量的全体,参数通常是时间。 随机变量是随机现象的数量表现,其时间序列是一组按照时间发生先后顺序进行排列的数据点序列。通常一组时间序列的时间间隔为一恒定值(如1秒,5分钟,12小时,7天,1年),因此时间序列可以作为离散时间数据进行分析处理。研究时间序列数据的意义在于现实中,往往需要研究某个事物其随时间发展变化的规律。这就需要通过研究该事物过去发展的历史记录,以得到其自身发展的规律。

回归分析代写

多元回归分析渐进(Multiple Regression Analysis Asymptotics)属于计量经济学领域,主要是一种数学上的统计分析方法,可以分析复杂情况下各影响因素的数学关系,在自然科学、社会和经济学等多个领域内应用广泛。

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。

R语言代写问卷设计与分析代写
PYTHON代写回归分析与线性模型代写
MATLAB代写方差分析与试验设计代写
STATA代写机器学习/统计学习代写
SPSS代写计量经济学代写
EVIEWS代写时间序列分析代写
EXCEL代写深度学习代写
SQL代写各种数据建模与可视化代写

发表回复

您的电子邮箱地址不会被公开。