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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|Definition and Examples of Congruences

Let $n \geq 2$ be a fixed integer. We define two integers $a$ and $b$ to be congruent modulo $n$ if $n$ divides the difference $a-b$. We will denote this by writing $a \equiv b(\bmod n)$. We call the integer $n$ the modulus of the congruence. We note that by the definition of “divides,” $a \equiv b(\bmod n)$ means that $a-b=n k$ for some integer $k$. Note that we require $n$ to be greater than 1 since if $n=1$ then every integer is equivalent to every other integer.

Probably without realizing it, you have already encountered congruences in everyday life. For example, the U.S. clock system works modulo 12 whereas the military clock systems work modulo 24. Days of the week are determined modulo 7 because if a given day is Monday, then seven days later we have another Monday. Similarly, except for leap years, our yearly calendars work modulo 365. Let’s look at some examples.

Example 3.1. (a) We know $27 \equiv 5(\bmod 11)$ since $27-5=22=$ $11(2)$. Note that 27 is also congruent to 5 modulo 2 since, again, $27-5=22=2(11)$
(b) One has to be a bit more careful with negative numbers, but the idea is the same. For example, $4 \equiv-21(\bmod 5)$ since $4-(-21)=$ $4+21=25=5(5)$

## 数学代写|数论作业代写number theory代考|The Finite Sets Zn

The idea of finding the two remainders upon division by the modulus $n$ leads us to an important point which follows directly from the Division Algorithm (Theorem 1.2): Every integer is congruent modulo $n$ to exactly one of $n$ ‘s possible remainders. For each $n>1$, this finite set of remainders, i.e., the set ${0,1, \ldots, n-1}$, turns out to be very important because we can do arithmetic inside this set provided that we do the arithmetic modulo $n$. This set is called the integers $\bmod n$ and is denoted $\mathbb{Z}{n}$. To emphasize, we repeat: $\mathbb{Z}{n}={0,1, \ldots, n-1}$ with arithmetic done modulo $n$.
If $a$ is any integer, we shall use the notation $a(\bmod n)$ to denote the unique remainder of $a$ divided by $n$, which is of course an element of $\mathbb{Z}_{n}$. This remainder is also referred to as the least non-negative residue of $a$ modulo $n$. We shall refer to this operation as reduction $\bmod n$. Note that ” $a(\bmod n) “$ is an object; ” $a \equiv b$ $(\bmod n) “$ is a statement.

Example 3.3. The least non-negative residue of 27 modulo 5 (which we are denoting as $27(\bmod 5))$ is 2 because, by the Division Algorithm, $27=(5)(5)+2$. Similarly, the least non-negative residue of $-27$ modulo 5 (i.e., $-27(\bmod 5)$ ) is 3 since $-27=$ $(-6)(5)+3$

# 数论作业代写

## 数学代写|数论作业代写number theory代考|Definition and Examples of Congruences

(b) 必须对负数更加小心，但想法是一样的。例如， $4 \equiv-21(\bmod 5)$ 自从 $4-(-21)=4+21=25=5(5)$

## 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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