数学代写|代数数论代写Algebraic number theory代考|What Is Number Theory

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  • 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|代数数论代写Algebraic number theory代考|What Is Number Theory

数学代写|代数数论代写Algebraic number theory代考|What Is Number Theory

Number Theory is the study of numbers, in particular the whole numbers $1,2,3, \ldots$, also called the natural numbers. The set of natural numbers is denoted by $\mathbb{N}$. Leaving aside the unit 1 , these numbers fall into two categories: The indivisible numbers $2,3,5,7, \ldots$ are the primes, and the rest $4,6,8,9,10, \ldots$ composed of primes, are the composite numbers. The following basic facts, with proofs, about these numbers were already known to Euclid around 300 B.C.
Theorem 1.1. There are infinitely many primes.
Theorem 1.2 (Fundamental Theorem of Arithmetic). Every natural number $n>1$ is a unique product
n=p_{1}^{e_{1}} \ldots p_{r}^{c_{r}} \quad(r \geq 1)
of powers of distinct primes $p_{1}, \ldots, p_{r}$, taken in some order.
By looking at the list of primes, one can ask several naive but still unanswered questions. For example, is there an endless supply of twin primes? We call a pair of primes $q, p$ twin primes if $p=q+2$. [This is the closest two odd primes can be to each other.] A glance at the list
3,5 ; 5,7 ; 11,13 ; 17,19 ; 29,31 ; \ldots
suggests that there are infinitely many pairs of twin primes, but no one has ever been able to prove this so far. Another big problem in number theory is the unproven conjecture of Goldbach, which asserts that every even number larger than 2 is a sum of two primes.

Many questions in number theory arise naturally in the study of geometry. The most fundamental fact in Euclidean geometry is the theorem of Pythagoras, which may be called the fundamental theorem of geometry. Actually, it was known to the Egyptians and Babylonians about two thousand years earlier, but they had no rigorous proof of it like Euclid did.

数学代写|代数数论代写Algebraic number theory代考|Methods of Proving Theorems in Number Theory

The method that has been used since antiquity is the unique factorization. Let us recall Euclid’s proof of Theorem 1.1.

It follows from the unique factorization (1.1) that any $n>1$ is either a prime or has a prime factor. To prove Theorem $1.1$ by contradiction, suppose there are only finitely many primes, say $p_{1}, \ldots, p_{r}$. Now consider the number $n=p_{1} \ldots p_{\mathrm{r}}+1$. It is not a prime because it is larger than every prime $p_{j}$. So, it has a prime factor, say $p_{1}$. Therefore $n=p_{1} a$ for an integer $a$. This implies that $1=p\left(a-p_{2} \ldots p_{r}\right)$. This is a contradiction because 1 has no prime factor.

Another example of such a proof is the proof below by Euler (1770) of the following claim of Fermat (1657): 27 is the only cube that exceeds a square by 2. In modern terminology, $(3, \pm 5)$ are the only points with integer coordinates on the elliptic curve
y^{2}=x^{3}-2 .
Proof. In the ring $\mathbb{Z}[\sqrt{-2}]={a+b \sqrt{-2} \mid a, b \in \mathbb{Z}}$, which is a UFD (see Exercise 8, Chapter 2), we use the factorization
x^{3}=y^{2}+2=(y+\sqrt{-2})(y-\sqrt{-2}) .
In general, in a UFD, if $\alpha, \beta$ have no common factor other than units, and $\alpha \beta=\gamma^{m}$ for an integer $m>0$, then $\alpha=\alpha_{1}^{m}$ and $\beta=\beta_{1}^{m}$ for some $\alpha_{1}, \beta_{1}$ in it. Therefore
y+\sqrt{-2}=(a+b \sqrt{-2})^{3} \text { for } a, b \in \mathbb{Z} .
By expanding $(a+b \sqrt{-2})^{3}$ and comparing the real/imaginary parts, we get
1=b\left(3 a^{2}-2 b^{2}\right), y=a^{3}-6 a b^{2} .
But the first equation in (1.7) can hold only if $b=1$ and $a=\pm 1$. This implies $y=\pm 5$.

  1. Analytic Methods
    Euler initiated what we call the analytic number theory. The study of infinite series (analysis) can lead to interesting results in number theory. Let us recall Euler’s proof of the infinitude of primes. Leaving aside the issue of convergence, by multiplying the infinite series formally, one sees that
    \sum_{n=1}^{\infty} \frac{1}{n}=& \sum_{n=1} \frac{1}{p_{1}^{e_{1}} \ldots p_{r}^{e_{r}}}=\prod_{p}\left(1+\frac{1}{p}+\frac{1}{p^{2}}+\cdots\right), \text { i.e. } \
    & \frac{1}{n}=\prod_{p}^{\infty}\left(1-\frac{1}{p}\right)^{-1}
    the product (called the Euler product) taken over all primes $p$. Note that the first equality is a consequence of the unique factorization (1.1).

The partial sums $\sum_{n=1}^{N-1} \frac{1}{n}$ of the series $\sum_{n=1}^{\infty} \frac{1}{n}$ are bounded from below by the area (cf. Figure 1.1) $\int_{1}^{N} \frac{d x}{x}=\ln N$, which goes to infinity as $N$ goes to infinity.

数学代写|代数数论代写Algebraic number theory代考|Techniques from Algebraic Geometry

Algebraic geometry is the study of the solutions of polynomial equations in a number of variables $x_{1}, \ldots, x_{n}$ with values of $x_{j}$ in a field $K$. Unless we assume $K$ to be algebraically closed, such as the field $\mathbb{C}$ of complex numbers, the subject is not satisfactory. For example, $x^{2}+y^{2}+1=0$ has no solution with $x, y$ even in such a big field as $\mathbb{R}$, the field of real numbers. Moreover, a line (equation of degree 1) is supposed to meet a circle (equation of degree 2) in two points. This rarely happens, but happens every time (in the projective plane $\mathbb{P}^{2}(\mathbb{C})$ ), thanks to Bezout’s Theorem: Two curves of degree $d_{1}, d_{2}$ with no component in common intersect in $d_{1} d_{2}$ points in the projective plane $\mathbb{P}^{2}(\mathbb{C})$, counted properly.

The arithmetic algebraic geometry is the subject in which algebraic geometric methods are used to answer questions in number theory. We illustrate it by finding the primitive Pythagorean triples, which is the same as finding the rational points (points with rational coordinates) on the unit circle
with the rational numbers $X, Y$ in the lowest form. A primitive Pythagorean triple $(x, y, z)$ gives such a rational point with $X=\frac{x}{z}, Y=\frac{y}{z}$, and vice versa.

To obtain an algorithm to find all the primitive Pythagorean triples $(x, y, z)$, we parameterize the unit circle (1.9) by the slope $t$ of the line through the fixed point $(-1,0)$ and a variable point $(X, Y)$ on this circle (cf. Figure 1.2).
Substituting for $X$ from the equation $X=t Y-1$ of this line in equation (1.9) of the unit circle, an easy calculation shows that
Y=\frac{2 t}{1+t^{2}} \text { and } X=t Y-1=\frac{1-t^{2}}{1+t^{2}} .
If we run $t$ through all rational numbers in the lowest form $t=\frac{a}{b}$, we get the following result:
FIGURE 1.2: Rational points on the unit circle.
Theorem 1.6. Every primitive Pythagorean triplet $(x, y, z)$ is of the form
x=a^{2}-b^{2}, y=2 a b, z=a^{2}+b^{2},
where $a, b(a>b)$ are positive integers of opposite parity (one odd, the other even) with no common factor.

Note that the condition of opposite parity is necessary because otherwise $x$, $y, z$ are all even, so $(x, y, z)$ is not primitive. We also remark that switching $x$ and $y$ does not produce a different Pythagorean triplet.

数学代写|代数数论代写Algebraic number theory代考|What Is Number Theory


数学代写|代数数论代写Algebraic number theory代考|What Is Number Theory

数论是对数字的研究,尤其是整数1,2,3,…,也称为自然数。自然数集表示为ñ. 撇开单位 1 不谈,这些数字分为两类: 不可分割的数字2,3,5,7,…是素数,其余的4,6,8,9,10,…由质数组成,是合数。欧几里得在公元前 300 年左右就已经知道以下关于这些数字的基本事实和证明
。定理 1.1。有无穷多个素数。
定理 1.2(算术基本定理)。每个自然数n>1是独一无二的产品

通过查看素数列表,人们可以提出几个幼稚但仍然没有答案的问题。例如,是否有无穷无尽的孪生素数?我们称一对素数q,p孪生素数如果p=q+2. [这是最接近的两个奇数素数。] 一览表

表明存在无限多对孪生素数,但迄今为止没有人能够证明这一点。数论中的另一个大问题是未经证实的哥德巴赫猜想,它断言每个大于 2 的偶数都是两个素数的和。


数学代写|代数数论代写Algebraic number theory代考|Methods of Proving Theorems in Number Theory

自古以来一直使用的方法是独特的因式分解。让我们回顾一下欧几里得对定理 1.1 的证明。

从唯一的因式分解(1.1)可以得出,任何n>1要么是素数,要么有素因子。证明定理1.1通过矛盾,假设只有有限多个素数,比如说p1,…,pr. 现在考虑数字n=p1…pr+1. 它不是素数,因为它比所有素数都大pj. 所以,它有一个主要因素,比如说p1. 所以n=p1一个对于一个整数一个. 这意味着1=p(一个−p2…pr). 这是一个矛盾,因为 1 没有素因数。

这种证明的另一个例子是 Euler (1770) 对 Fermat (1657) 的以下主张的证明: 27 是唯一一个超过正方形 2 的立方体。在现代术语中,(3,±5)是椭圆曲线上唯一具有整数坐标的点

证明。在环中从[−2]=一个+b−2∣一个,b∈从,这是一个 UFD(参见练习 8,第 2 章),我们使用分解

通常,在 UFD 中,如果一个,b除了单位之外没有公因数,并且一个b=C米对于一个整数米>0, 然后一个=一个1米和b=b1米对于一些一个1,b1在里面。所以

是+−2=(一个+b−2)3 为了 一个,b∈从.

但是(1.7)中的第一个方程只有当b=1和一个=±1. 这意味着是=±5.

  1. 解析方法
    ∑n=1∞1n=∑n=11p1和1…pr和r=∏p(1+1p+1p2+⋯), IE  1n=∏p∞(1−1p)−1
    取所有素数的乘积(称为欧拉乘积)p. 请注意,第一个等式是唯一因式分解 (1.1) 的结果。

部分金额∑n=1ñ−11n该系列的∑n=1∞1n从下方以该区域为界(参见图 1.1)∫1ñdXX=ln⁡ñ, 无穷大为ñ走向无穷大。

数学代写|代数数论代写Algebraic number theory代考|Techniques from Algebraic Geometry

代数几何是研究多项式方程在多个变量中的解X1,…,Xn值为Xj在一个领域ķ. 除非我们假设ķ是代数封闭的,例如域C复数,题目不令人满意。例如,X2+是2+1=0没有解决方案X,是即使在这么大的领域R,实数域。此外,一条线(1 次方程)应该在两点与圆(2 次方程)相交。这很少发生,但每次都会发生(在投影平面磷2(C)),感谢 Bezout 定理:两条度数曲线d1,d2没有共同的组件相交d1d2投影平面上的点磷2(C),正确计算。



获得一种算法来找到所有原始毕达哥拉斯三元组(X,是,和),我们通过斜率参数化单位圆(1.9)吨通过固定点的线(−1,0)和一个可变点(X,是)在这个圆圈上(参见图 1.2)。

是=2吨1+吨2 和 X=吨是−1=1−吨21+吨2.
图 1.2:单位圆上的有理点。
定理 1.6。每个原始毕达哥拉斯三元组(X,是,和)是形式


请注意,相反奇偶性的条件是必要的,否则X, 是,和都是偶数,所以(X,是,和)不是原始的。我们还注意到,切换X和是不会产生不同的毕达哥拉斯三元组。

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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。



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