数学代写|抽象代数作业代写abstract algebra代考|Cubic and quartic equations

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  • 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
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  • 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|Cubic and quartic equations

数学代写|抽象代数作业代写abstract algebra代考|Cubic and quartic equations

The Babylonians were solving quadratic equations by about $1600 \mathrm{BC}$, using essentially the equivalent of the quadratic formula. A natural question is therefore whether cubic equations could be solved using similar formulas (see below). Another three thousand years would pass before the answer would be known. It was a great event

in algebra when mathematicians of the sixteenth century succeeded in solving “by radicals” not only cubic but also quartic equations.

A solution by radicals of a polynomial equation is a formula giving the roots of the equation in terms of its coefficients. The only permissible operations to be applied to the coefficients are the four algebraic operations (addition, subtraction, multiplication, and division) and the extraction of roots (square roots, cube roots, and so on, that is, “radicals”). For example, the quadratic formula $x=\left(-b \pm \sqrt{b^{2}-4 a c}\right) / 2 a$ is a solution by radicals of the equation $a x^{2}+b x+c=0$.

A solution by radicals of the cubic was first published by Cardano in The Great Art (referring to algebra) of 1545 , but it was discovered earlier by del Ferro and by Tartaglia. The latter had passed on his method to Cardano, who had promised that he would not publish it, which he promptly did. What came to be known as Cardano’s formula for the solution of the cubic $x^{3}=a x+b$ was given by
x=\sqrt[3]{b / 2+\sqrt{(b / 2)^{2}-(a / 3)^{3}}}+\sqrt[3]{b / 2-\sqrt{(b / 2)^{2}-(a / 3)^{3}}}

数学代写|抽象代数作业代写abstract algebra代考|The cubic and complex numbers

Mathematicians adhered for centuries to the following view with respect to the square roots of negative numbers: since the squares of positive as well as of negative numbers are positive, square roots of negative numbers do not-in fact, cannot-exist. All this changed following the solution by radicals of the cubic in the sixteenth century.
Square roots of negative numbers arise “naturally” when Cardano’s formula (see p. 6) is used to solve cubic equations. For example, application of his formula to the equation $x^{3}=9 x+2$ gives $x=\sqrt[3]{2 / 2+\sqrt{(2 / 2)^{2}-(9 / 3)^{3}}}+$ $\sqrt[3]{2 / 2-\sqrt{(2 / 2)^{2}-(9 / 3)^{3}}}=\sqrt[3]{1+\sqrt{-26}}+\sqrt[3]{1-\sqrt{-26}}$. What is one to make of this solution? Since Cardano was suspicious of negative numbers, he certainly had no taste for their square roots, so he regarded his formula as inapplicable to equations such as $x^{3}=9 x+2$. Judged by past experience, this was not an unreasonable attitude. For example, to the Pythagoreans, the side of a square of area 2 was nonexistent (in today’s language, we would say that the equation $x^{2}=2$ is unsolvable).

All this was changed by Bombelli. In his important book Algebra (1572) he applied Cardano’s formula to the equation $x^{3}=15 x+4$ and obtained $x=\sqrt[3]{2+\sqrt{-121}}+$ $\sqrt[3]{2-\sqrt{-121}}$. But he could not dismiss the solution, for he noted by inspection that $x=4$ is a root of this equation. Moreover, its other two roots, $-2 \pm \sqrt{3}$, are also real numbers. Here was a paradox: while all three roots of the cubic $x^{3}=$ $15 x+4$ are real, the formula used to obtain them involved square roots of negative numbers-meaningless at the time. How was one to resolve the paradox?

Bombelli adopted the rules for real quantities to manipulate “meaningless” expressions of the form $a+\sqrt{-b}(b>0)$ and thus managed to show that $\sqrt[3]{2+\sqrt{-121}}=2+\sqrt{-1}$ and $\sqrt[3]{2-\sqrt{-121}}=2-\sqrt{-1}$, and hence that $x=\sqrt[3]{2+\sqrt{-121}}+\sqrt[3]{2-\sqrt{-121}}=(2+\sqrt{-1})+(2-\sqrt{-1})=4$. Bombëlli had given meaning to the “meaningless” by thinking the “unthinkable,” namely that square roots of negative numbers could be manipulated in a meaningful way to yield

significant results. This was a very bold move on his part. As he put it:
It was a wild thought in the judgment of many; and I too was for a long time of the same opinion. The whole matter seemed to rest on sophistry rather than on truth. Yet I sought so long until I actually proved this to be the case [11].
Bombelli developed a “calculus” for complex numbers, stating such rules as $(+\sqrt{-1})(+\sqrt{-1})=-1$ and $(+\sqrt{-1})(-\sqrt{-1})=1$, and defined addition and multiplication of specific complex numbers. This was the birth of complex numbers. But birth did not entail legitimacy. For the next two centuries complex numbers were shrouded in mystery, little understood, and often ignored. Only following their geometric representation in 1831 by Gauss as points in the plane were they accepted as bona fide elements of the number system. (The earlier works of Argand and Wessel on this topic were not well known among mathematicians.) See [1], [7], [13].

Note that the equation $x^{3}=15 x+4$ is an example of an “irreducible cubic,” namely one with rational coefficients, irreducible over the rationals, all of whose roots are real. It was shown in the nineteenth century that any solution by radicals of such a cubic (not just Cardano’s) must involve complex numbers. Thus complex numbers are unavoidable when it comes to finding solutions by radicals of the irreducible cubic. It is for this reason that they arose in connection with the solution of cubic rather than quadratic equations, as is often wrongly assumed. (The nonexistence of a solution of the quadratic $x^{2}+1=0$ was readily accepted for centuries.)

数学代写|抽象代数作业代写abstract algebra代考|Algebraic notation: Viète and Descartes

Mathematical notation is now taken for granted. In fact, mathematics without a welldeveloped symbolic notation would be inconceivable. It should be noted, however, that the subject evolved for about three millennia with hardly any symbols. The introduction and perfection of symbolic notation in algebra occurred for the most part in the sixteenth and early seventeenth centuries, and is due mainly to Viète and Descartes.

The decisive step was taken by Viète in his Introduction to the Analytic Art (1591). He wanted to breathe new life into the method of analysis of the Greeks, a method of discovery used to solve problems, to be contrasted with their method of synthesis, used to prove theorems. The former method he identified with algebra. He saw it as “the science of correct discovery in mathematics,” and had the grand vision that it would “leave no problem unsolved.”

Viète’s basic idea was to introduce arbitrary parameters into an equation and to distinguish these from the equation’s variables. He used consonants $(B, C, D, \ldots)$ to denote parameters and vowels $(A, E, I, \ldots)$ to denote variables. Thus a quadratic equation was written as $B A^{2}+C A+D=0$ (although this was not exactly Viète’s notaation’, seé bẽlow). Tó us this āppeears to bé a simplé and natüral ideá, but it was a fundamental departure in algebra: for the first time in over three millennia one could speak of a general quadratic equation, that is, an equation with (arbitrary) literal coefficients rather than one with (specific) numerical coefficients.

数学代写|抽象代数作业代写abstract algebra代考|Cubic and quartic equations


数学代写|抽象代数作业代写abstract algebra代考|Cubic and quartic equations


在代数中,当 16 世纪的数学家成功地“通过根式”求解三次方程和​​四次方程时。


Cardano 于 1545 年在 The Great Art(指代数)中首次发表了由三次根式的解法,但较早时由 del Ferro 和 Tartaglia 发现。后者将他的方法传给了卡尔达诺,卡尔达诺承诺他不会发表,他很快就做到了。什么后来被称为卡尔达诺解立方的公式X3=一个X+b由


数学代写|抽象代数作业代写abstract algebra代考|The cubic and complex numbers

当卡尔达诺公式(参见第 6 页)用于求解三次方程时,负数的平方根“自然地”出现。例如,将他的公式应用于方程X3=9X+2给X=2/2+(2/2)2−(9/3)33+ 2/2−(2/2)2−(9/3)33=1+−263+1−−263. 这个解决方案有什么用?由于卡尔达诺对负数持怀疑态度,他当然不喜欢平方根,所以他认为他的公式不适用于诸如X3=9X+2. 以以往的经验来看,这并不是不合理的态度。例如,对于毕达哥拉斯学派来说,面积为 2 的正方形的边是不存在的(在今天的语言中,我们会说等式X2=2是无法解决的)。

这一切都被邦贝利改变了。在他的重要著作《代数》(1572)中,他将卡尔达诺公式应用于方程X3=15X+4并获得X=2+−1213+ 2−−1213. 但是他不能忽视这个解决方案,因为他通过检查注意到X=4是这个方程的根。此外,它的另外两个根,−2±3, 也是实数。这是一个悖论:而三次方的所有三个根X3= 15X+4是真实的,用于获得它们的公式涉及负数的平方根 – 当时没有意义。如何解决这个悖论?

Bombelli 采用实数规则来操纵形式的“无意义”表达式一个+−b(b>0)并因此设法证明2+−1213=2+−1和2−−1213=2−−1,因此X=2+−1213+2−−1213=(2+−1)+(2−−1)=4. Bombëlli 通过思考“不可想象”赋予“无意义”以意义,即负数的平方根可以以有意义的方式被操纵以产生

Bombelli 为复数开发了一种“微积分”,阐述了如下规则(+−1)(+−1)=−1和(+−1)(−−1)=1, 并定义了特定复数的加法和乘法。这就是复数的诞生。但出生并不意味着合法性。在接下来的两个世纪里,复数笼罩在神秘之中,鲜为人知,而且经常被忽视。只有在 1831 年高斯将它们的几何表示为平面中的点之后,它们才被接受为数系的真正元素。(Argand 和 Wessel 在这个主题上的早期著作在数学家中并不为人所知。)参见 [1]、[7]、[13]。

请注意,等式X3=15X+4是“不可约三次”的一个例子,即具有有理系数,在有理数上不可约,其所有根都是实数。19 世纪的情况表明,这种立方(不仅仅是卡尔达诺的)的任何根式解决方案都必须涉及复数。因此,在通过不可约三次的根式求解时,复数是不可避免的。正是由于这个原因,它们与三次方程的解而不是二次方程的解有关,正如经常被错误地假设的那样。(二次解的不存在X2+1=0几个世纪以来一直很容易被接受。)

数学代写|抽象代数作业代写abstract algebra代考|Algebraic notation: Viète and Descartes


Viète 在他的《分析艺术导论》(1591 年)中迈出了决定性的一步。他想为希腊人的分析方法注入新的活力,这是一种用来解决问题的发现方法,与他们用来证明定理的综合方法形成对比。他用代数确定了前一种方法。他将其视为“数学中正确发现的科学”,并拥有“没有任何问题未解决”的宏伟愿景。

Viète 的基本思想是将任意参数引入方程,并将这些参数与方程的变量区分开来。他用辅音(乙,C,D,…)表示参数和元音(一种,和,一世,…)来表示变量。因此,一个二次方程写为乙一种2+C一种+D=0(尽管这并不完全是 Viète 的符号,参见 bẽlow)。对我们来说,这似乎是一个简单和自然的想法,但它是代数的一个根本性转变:三千年以来,人们第一次可以谈论一般的二次方程,即具有(任意)字面系数的方程,而不是比具有(特定)数字系数的。

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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。





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。


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 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。