数学代写|抽象代数作业代写abstract algebra代考|Development of “specialized” theories of groups

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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。

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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|Development of “specialized” theories of groups

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

As noted earlier, Lagrange’s work of 1770 initiated the study of permutations in connection with the study of the solution of equations. It was probably the first clear instance of implicit group-theoretic thinking in mathematics. It led directly to the $\mathrm{~ w o ̄ ̄ i k s ̄ ~ o ̂ f ~ K u ̈ f f i n ̃ i , ~ A ̊ b e ́ l , ~ a ̄ n đ ~ G a ̄ l o ̄ i s ~ đ u ̈}$ to the concept of a permutation group.

Ruffini and Abel proved the unsolvability of the quintic by building on the ideas of Lagrange concerning resolvents. Lagrange showed that a necessary condition for the solvability of the general polynomial equation of degree $n$ is the existence of a resolvent of degree less than $n$. Ruffini and Abel showed that such resolvents do not exist for $n>4$. In the process they developed elements of permutation theory. It was Galois, however, who made the fundamental conceptual advances, and who is considered by many as the founder of (permutation) group theory.

He was familiar with the works of Lagrange, Abel, and Gauss on the solution of polynomial equations. But his aim went well beyond finding a method for solvability of equations. He was concerned with gaining insight into general principles, dissatisfied as he was with the methods of his predecessors: “From the beginning of this century,” he wrote, “computational procedures have become so complicated that any progress by those means has become impossible.”

Galois recognized the separation between “Galois theory”-the correspondence between fields and groups-and its application to the solution of equations, for he wrote that he was presenting “the general principles and just one application” of the theory. “Many of the early commentators on Galois theory failed to recognize this distinction, and this led to an emphasis on applications at the expense of the theory” [19].

Galois was the first to use the term “group” in a technical sense-to him it signified a collection of permutations closed under multiplication: “If one has in the same group the substitutions $S$ and $T$, one is certain to have the substitution $S T$.” He recognized that the most important properties of an algebraic equation were reflected in certain properties of a group uniquely associated with the equation-“the group of the equation.” To describe these properties he invented the fundamental notion of normal subgroup and used it to great effect.

While the issue of resolvent equations preoccupied Lagrange, Ruffini, and Abel, Galois’ basic idea was to bypass them, for the construction of a resolvent required great skill and was not based on a clear methodology. Galois noted instead that the existence of a resolvent was equivalent to the existence of a normal subgroup of prime index in the group of the equation. This insight shifted consideration from the resolvent equation to the group of the equation and its subgroups.
Galois defined the group of an equation as follows:
Let an equation be given, whose $m$ roots are $a, b, c, \ldots .$ There will always be a group of permutations of the letters $a, b, c, \ldots$ which has the following property: (1) that every function of the roots, invariant under the substitutions of that group, is rationally known [i.e., is a rational function of the coefficients and any adjoined quantities]; (2) conversely, that every function of the roots, which can be expressed rationally, is invariant under these substitutions [19].

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

As noted earlier, the main source for abelian group theory was number theory, beginning with Gauss’ Disquisitiones Arithmeticae. (Note also implicit abelian group theory in Euler’s number-theoretic work [33].) In contrast to permutation theory, grouptheoretic modes of thought in number theory remained implicit until about the last third of the nineteenth century. Until that time no explicit use of the term “group” was made, and there was no link to the contemporary, flourishing theory of permutation groups. We now give a sample of some implicit group-theoretic work in number theory, especially in algebraic number theory.

Algebraic number theory arose in connection with Fermat’s Last Theorem, the insolvability in nonzero integers of $x^{n}+y^{n}=z^{n}$ for $n>2$, Gauss’ theory of binary quadratic forms, and higher reciprocity laws (see Chapter 3.2). Algebraic number fields and their arithmetical properties were the main objects of study. In 1846 Dirichlet studied the units in an algebraic number field and established that (in our terminology) the group of these units is a direct product of a finite cyclic group and a free abelian group of finite rank. At about the same time Kummer introduced his “ideal numbers,” defined an equivalence relation on them, and derived, for cyclotomic fields, certain special properties of the number of equivalence classes, the so-called class number of a cyclotomic field-in our terminology, the order of the ideal class group of the cyclotomic field. Dirichlet had earlier made similar studies of quadratic fields.
In 1869 Schering, a former student of Gauss, investigated the structure of Gauss’ (group of) equivalence classes of binary quadratic forms (see Chapter 3 ). He found certain fundamental classes from which all classes of forms could be obtained by composition. In group-theoretic terms, Schering found a basis for the abelian group of equivalence classes of binary quadratic forms.

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

As in number theory, so in geometry and analysis, group-theoretic ideas remained implicit until the last third of the nineteenth century. Moreover, Klein’s (and Lie’s) explicit use of groups in geometry influenced the evolution of group theory concep tually rather than technically. It signified a genuine shift in the development of group theory from a preoccupation with permutation groups to the study of groups of transformations. (That is not to suggest, of course, that permutation groups were no longer studied.) This transition was also notable in that it pointed to a turn from finite groups to infinite groups.

Klein noted the connection of his work with permutation groups but also realized the departure he was making. He stated that what Galois theory and his own program have in common is the investigation of “groups of changes,” but added that “to be sure,

the objects the changes apply to are different: there [Galois theory] one deals with a finite number of discrete elements, whereas here one deals with an infinite number of elements of a continuous manifold.”‘ To continue the analogy, Klein noted that just as there is a theory of permutation groups, “we insist on a theory of transformations, a study of groups generated by transformations of a given type.”

Klein shunned the abstract point of view in group theory, and even his technical definition of a (transformation) group is deficient:
Now let there be given a sequence of transformations $A, B, C, \ldots .$ If this sequence has the property that the composite of any two of its transformations yields a transformation that again belongs to the sequence, then the latter will be called a group of transformations [33].
Klein’s work, however, broadened considerably the conception of a group and its applicability in other fields of mathematics. He did much to promote the view that group-theoretic ideas are fundamental in mathematics:
The special subject of group theory extends through all of modern mathematics. As an ordering and classifying principle, it intervenes in the most varied domains.

数学代写|抽象代数作业代写abstract algebra代考|Development of “specialized” theories of groups

抽象代数代写

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

如前所述,拉格朗日 1770 年的工作启动了与方程解研究相关的排列研究。这可能是数学中隐含群论思维的第一个明显例子。直接导致了đđ 在这̄̄一世ķs̄ 这̂F ķ在̈FF一世ñ一世, 一种̊b和́l, 一个̄nD G一个̄l这̄一世s D在̈置换群的概念。

Ruffini 和 Abel 通过建立拉格朗日关于分解的思想证明了五次方程的不可解性。拉格朗日证明了一般多项式度方程的可解性的必要条件n是否存在度数小于的解析器n. Ruffini 和 Abel 表明此类解决方案不存在n>4. 在这个过程中,他们发展了置换理论的元素。然而,是伽罗瓦在概念上取得了根本性的进步,并且被许多人认为是(置换)群论的创始人。

他熟悉拉格朗日、阿贝尔和高斯在求解多项式方程方面的著作。但他的目标远远超出了寻找方程可解性的方法。他关心的是获得对一般原理的洞察力,尽管他对前人的方法不满意:“从本世纪初开始,”他写道,“计算程序变得如此复杂,以致不可能通过这些方法取得任何进展。 ”

伽罗瓦认识到“伽罗瓦理论”(场和群之间的对应)与其在方程解中的应用之间的分离,因为他写道,他正在展示该理论的“一般原理和一种应用”。“许多伽罗瓦理论的早期评论家未能认识到这种区别,这导致了以牺牲理论为代价来强调应用”[19]。

伽罗瓦是第一个在技术意义上使用“群”一词的人——对他来说,这意味着在乘法下封闭的排列集合:“如果一个人在同一个群中,则小号和吨, 一个肯定有替代品小号吨。” 他认识到代数方程最重要的性质反映在与方程唯一相关的群的某些性质上——“方程的群”。为了描述这些性质,他发明了正规子群的基本概念,并将其运用到了极好的效果。

虽然求解方程的问题一直困扰着拉格朗日、鲁菲尼和阿贝尔,但伽罗瓦的基本思想是绕过它们,因为构造求解方程需要高超的技巧,而且不是基于明确的方法论。相反,伽罗瓦指出,解算子的存在等价于方程组中素数索引的正规子群的存在。这种见解将考虑从求解方程转移到方程组及其子组。
伽罗瓦将方程组定义如下:
给定一个方程,其米根是一个,b,C,….总会有一组字母的排列一个,b,C,…它具有以下性质:(1)根的每个函数,在该组的替换下不变,是有理已知的[即,是系数和任何相邻量的有理函数];(2) 相反,可以合理表达的根的每个函数在这些替换下都是不变的[19]。

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

如前所述,阿贝尔群论的主要来源是数论,从高斯的算术研究开始。(还要注意欧拉数论著作中隐含的阿贝尔群论[33]。)与置换论相反,数论中的群论思想模式一直隐含到大约 19 世纪最后三分之一。在那之前,没有明确使用“群”这个词,也没有与当代蓬勃发展的置换群理论有任何联系。我们现在给出数论中一些隐含的群论工作的样本,特别是在代数数论中。

代数数论与费马大定理有关,即非零整数的不可解性Xn+是n=和n为了n>2,高斯的二元二次形式理论和更高的互易定律(见第 3.2 章)。代数数域及其算术性质是主要研究对象。1846 年,Dirichlet 研究了代数数域中的单位,并确定(用我们的术语)这些单位的群是有限循环群和有限秩的自由阿贝尔群的直接乘积。大约在同一时间,Kummer 介绍了他的“理想数”,在它们上定义了等价关系,并为分圆域导出了等价类数的某些特殊性质,即所谓的分圆域的类数——在我们的术语,分圆域的理想类群的顺序。狄利克雷早些时候对二次场进行了类似的研究。
1869 年,高斯的前学生先灵研究了二元二次形式的高斯(组)等价类的结构(见第 3 章)。他发现了某些基本类别,从这些基本类别中可以通过组合获得所有形式的类别。在群论方面,Schering 为二元二次形式的等价类的阿贝尔群找到了基础。

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

就像在数论中一样,在几何和分析中,群论的思想直到 19 世纪下半叶仍然是隐含的。此外,克莱因(和李)在几何学中明确使用群影响了群论在概念上而非技术上的演变。它标志着群论发展的真正转变,从专注于置换群转向研究变换群。(当然,这并不是说不再研究置换群。)这种转变也值得注意,因为它指出了从有限群到无限群的转变。

克莱因注意到他的工作与排列群的联系,但也意识到他正在做出的离开。他表示,伽罗瓦理论和他自己的计划的共同点是对“变化群”的调查,但补充说:“可以肯定的是,

变化适用的对象是不同的:[伽罗瓦理论]一个处理有限数量的离散元素,而这里处理一个连续流形的无限个元素。”’为了继续类比,克莱因指出由于有置换群理论,“我们坚持变换理论,研究由给定类型的变换产生的群。”

克莱因回避了群论中的抽象观点,甚至他对(变换)群的技术定义也是有缺陷的:
现在给定一个变换序列一种,乙,C,….如果这个序列具有这样的性质,即它的任意两个变换的复合产生一个再次属于该序列的变换,那么后者将被称为一组变换[33]。
然而,克莱因的工作大大拓宽了群的概念及其在其他数学领域的适用性。他做了很多工作来宣传群论思想是数学的基础:
群论这一特殊学科贯穿了所有现代数学。作为一种排序和分类原则,它介入了最多样化的领域。

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

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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。

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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

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