### 数学代写|数论作业代写number theory代考|MAST90136

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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|数论作业代写number theory代考|Divisibility and Congruence

In one of the oldest Chinese books I-Ching (Book of Changes), which dates approximately from the 8 th century $\mathrm{BC}$, there is a picture (so-called hexagram) containing $8 \times 8$ boxes. Each box contains 6 broken or full horizontal lines (see Fig. 1.1). The broken line indicates the old Chinese principle yin and the full principle of yang, which are in opposition. Yin is associated with the Moon, humidity, darkness, Earth, woman, and passivity, yang on the other hand with the Sun, drought, light, heaven, man, and activity.

The prominent German mathematician Gottfried Wilhelm Leibniz (1646-1716) associated this hexagram with the discovery of a binary system. Considering zero instead of the broken line and one instead of the full line, the symbols in particular boxes from left to right (starting from the top line) can be interpreted as the numbers $0,1,2,3, \ldots$ written in the binary system. The first number in the upper left corner is therefore zero, even though this notation was not used for operations with numbers in the 8th century BC. The last number in the lower right corner corresponds to 63 , which is written as 111111 in the binary system. The use of zero nowadays seems completely natural, but its discovery and in particular, its symbolic representation signified great progress in mathematics over the entire world (cf. Fig. 1.2).

Although the ancient Chinese did not perform with the symbols yin-yang any arithmetic operations, we cannot deny they were the first to represent numbers by the binary system. The discovery of the binary system found practical application only in today’s computer age, i.e. almost three thousand years later. Computers display and process all information (including numbers) in the binary system. This is the easiest way in electronic circuits of a computer to process data. Thus, the functioning of e-mail, scanners, copiers, digital cameras, compact disks $\mathrm{CD}$ and DVD, cell phones, and the worldwide network of Internet is actually based on the ancient Chinese principles of yin $(=0)$ and yang $(=1)$.

## 数学代写|数论作业代写number theory代考|Natural Numbers

From ancient times people used the numbers $1,2,3, \ldots$ to express the number of some objects. The oldest use of zero was recorded in India. For a long time, zero was not even considered to be a number. Moreover, at present historians still do not have a year zero (but it is used by astronomers).

Sometimes we encounter the question of whether zero is or is not a natural number. Unfortunately, it is not possible to give a clear answer to this question of the YES/NO type, since whether or not we consider zero to be a natural number is a matter of definition. It is advisable to include zero in the set of natural numbers, for example, when determining the number of elements of finite sets, because the number of elements of the empty set is zero.

On the other hand, there are good reasons why it is sometimes advantageous not to include zero in the set of natural numbers. This is, for example, to avoid division by zero or when raising natural numbers to a natural power. In particular, the symbol $0^{0}$ cannot be unambiguously assigned to one value that would naturally correspond to standard arithmetical operations with real numbers. For example, for $n=1,2, \ldots$ we have $0^{n}=0$, while $n^{0}=1$. Archimedes’ axiom presented below could not be applied if 0 would be a natural number. It is also not possible to define reasonably the least common multiple of, for example, the numbers 0 and 3 , as we shall see in Sect. 1.4. Therefore, more often zero is not considered to be a natural number.
The set of natural numbers (positive integers) will be denoted by
$$\mathbb{N}={1,2,3, \ldots}$$
It took a long time for mathematicians to figure out how in fact, natural numbers should be introduced. Among several options, the following four axioms formulated around 1891 hy the Italian mathematician Giuseppe Peano (1858-1939) were defined. ‘They use a special function “successor”, truthfully characterize the set of natural numbers and are called Peano’s axioms after him:
(A1) There exists a unique natural number that is not a successor of any natural numbers. We will denote this number by the symbol 1 .
(A2) Each natural number has exactly one successor.

## 数学代写|数论作业代写number theory代考|Natural Numbers

ñ=1,2,3,…

(A1) 存在一个唯一的自然数，它不是任何自然数的后继。我们将用符号 1 来表示这个数字。
(A2) 每个自然数只有一个后继。

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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。