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• (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|数论作业代写number theory代考|Elliptic Curves

An elliptic curve is a set of all solution $x, y \in \mathbb{R}^{2}$ of the equation
$$y^{2}+a_{1} x y+a_{3} y=x^{3}+a_{2} x^{2}+a_{4} x+a_{6},$$
where the coefficients $a_{i}$ are rational numbers (sometimes $x, y$ are taken to be in $\mathbb{Z}$ or $\mathbb{Q}$ or $\mathbb{C}$ ). It is obvious that no elliptic curve can be an ellipse. Their name is only related to the use of these curves for calculation of the length of elliptic arcs, e.g. in the determination of trajectories of planets [215, p. 186]. Equations of type (2.16) with integer coefficients were already solved by the Greek mathematician Diophantus (and also Niels Henrik Abel) because of their interesting properties. In the proof of Fermat’s Last Theorem, the following special elliptic curve
$$y^{2}=x\left(x-a^{p}\right)\left(x+b^{p}\right)$$
was employed, where $p \geq 5$ is a prime.
By means of the linear substitution $y \mapsto y-a_{1} x / 2-a_{3} / 2$, the Eq. (2.16) can be simplified as follows
$$y^{2}=x^{3}+b_{2} x^{2}+2 b_{4} x+b_{6},$$
where $b_{i}$ are suitable rational numbers. By another linear substitution $x \mapsto x-b_{2} / 3$ representing only a shift, we can reduce (2.17) to
$$y^{2}=x^{3}+A x+B .$$
The associated discriminant
$$\Delta=-\left(\frac{A}{3}\right)^{3}-\left(\frac{B}{2}\right)^{2}=-\frac{4 A^{3}+27 B^{2}}{108}$$
has a very nice geometric interpretation. If $\Delta=A=0$, then the curve $y^{2}=x^{3}$ has the so-called cusp singularity and the corresponding graph is connected (see the left graph of Fig. 2.2). If $\Delta=0$ and $A \neq 0$, then the elliptic curve (2.18) crosses itself (seee the middle graph of Fig. 2.2). If $\Delta \neq 0$, then the polynomial $p(x)-x^{3}+A x+B$ has three different roots; otherwise it has a multiple root. If $\Delta<0$, then there are one real and two complex conjugate roots and the corresponding elliptic curve is connected. If $\Delta>0$, then there are only real roots and the elliptic curve has two components (see the right graph of Fig. 2.2).

## 数学代写|数论作业代写number theory代考|Fermat’s Last Theorem

In 1670 Samuel de Fermat (1630-1690), the son of Pierre de Fermat, published Diophantos’s Arithmetic extended by notes that his father wrote in the margins of his copy from 1621. In one of these notes, P. Fermat states without any proof the sentence that is today generally called Fermat’s Last Theorem: Cubem autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere …
Using modern terminology, the paragraph above can be rewritten in English as follows (see Fig. 2.4): It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second, into two like powers …

Theorem 2.11 (Fermat’s Last Theorem) There are no integers $x, y, z \in \mathbb{N}$ and $n \geq 3$ such that
$$x^{n}+y^{n}=z^{n} .$$
First let us introduce a simple and amusing application of this theorem. We shall prove that the number $\sqrt[n]{2}$ is irrational for $n \geq 3$. Assume to the contrary that
$$\sqrt[n]{2}=\frac{z}{y}$$
for some positive integers $y$ and $z$. Then $2=z^{n} / y^{n}$. From this
$$2 y^{n}=y^{n}+y^{n}=z^{n}$$
which is a special case of (2.25) that has no solution. This is a contradiction.
Fermat’s Last Theorem is one of the most famous mathematical problems of all times. The exponent $n \geq 3$ is either divisible by an odd prime, or $n$ is a power of 2 . Now we show that it is enough to prove the above theorem only for an arbitrary odd prime exponent or for $n=4$. Assume for a moment that (2.25) has a solution for some $n=p q$, where $p$ is an odd prime and $q>1$.

# 数论作业代写

## 数学代写|数论作业代写number theory代考|Elliptic Curves

$$y^{2}+a_{1} x y+a_{3} y=x^{3}+a_{2} x^{2}+a_{4} x+a_{6},$$

$$y^{2}=x\left(x-a^{p}\right)\left(x+b^{p}\right)$$

$$y^{2}=x^{3}+b_{2} x^{2}+2 b_{4} x+b_{6},$$

$$y^{2}=x^{3}+A x+B .$$

$$\Delta=-\left(\frac{A}{3}\right)^{3}-\left(\frac{B}{2}\right)^{2}=-\frac{4 A^{3}+27 B^{2}}{108}$$

## 数学代写|数论作业代写number theory代考|Fermat’s Last Theorem

1670 年，皮埃尔德·费马 (Pierre de Fermat) 的儿子塞缪尔德.费马 (Samuel de Fermat) (1630-1690) 发表了丢番图的《算术》，并增加了他父亲 1621 年在其副本 页边空白处写下的注释。在其中一篇注释中，P. Fermat 没有任何证据地陈述今天通常被称为费马大定理的句子: Cubem autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere … 使用现代术语，上面的段落可以改写为英文如下（见图2.4）：不可能将一个立方体分成两个立方体，或者将一个四次方分成两个四次方，或者一般来说，任何高 于二次方的幂都不能分成两个类似的幂……

$$x^{n}+y^{n}=z^{n} .$$

$$\sqrt[n]{2}=\frac{z}{y}$$

$$2 y^{n}=y^{n}+y^{n}=z^{n}$$

## 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。

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