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# 数学代写|概率论代考Probability Theory代写|The sieve of Eratosthenes

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## 数学代写|概率论代考Probability Theory代写|The sieve of Eratosthenes

The classical applications of inclusion-exclusion were in the theory of numbers. Write $\operatorname{gcd}(a, b)$ for the greatest common divisor of two natural numbers $a$ and $b$, and for every real number $x$ let $\lfloor x\rfloor$ denote the greatest integer $\leq x$.

THEOREM 1 Let $\mathrm{N}$ be a natural number and $\mathrm{a}1, \ldots, \mathrm{a}{\mathrm{n}}$ natural numbers that are relatively prime, that is to say, $\operatorname{gcd}\left(a_i, a_j\right)=1$ if $i \neq j$. Let $R$ be a random number selected from ${1, \ldots, N}$. Then the probability that $R$ is divisible by none of the $a_i$ is given by
$$1-\sum_{1 \leq i \leq n} \frac{1}{N}\left\lfloor\frac{N}{a_i}\right\rfloor+\sum_{1 \leq i<j \leq n} \frac{1}{N}\left\lfloor\frac{N}{a_i a_j}\right\rfloor-\cdots+(-1)^n \frac{1}{N}\left\lfloor\frac{N}{a_1 a_2 \cdots a_n}\right\rfloor .$$
PrOof: Identify $A_i$ as the event that $a_i$ divides R. As the number of strictly positive integers $\leq N$ that are divisible by $a_i$ is $\left\lfloor N / a_i\right\rfloor$, the term $S_1$ in the inclusion-exclusion formula is given by $\sum_{1 \leq i \leq n} \frac{1}{N}\left\lfloor\frac{N}{a_i}\right\rfloor$. As $a_i$ and $a_j$ are relatively prime if $i \neq j$, the number of strictly positive integers $\leq N$ that are divisible both by $a_i$ and $a_j$ is $\left\lfloor N /\left(a_i a_j\right)\right\rfloor$ and the term $S_2$ in the inclusion-exclusion formula is given by $\sum_{1 \leq i<j \leq n} \frac{1}{N}\left\lfloor\frac{N}{a_i a_j}\right\rfloor$. Proceeding in this fashion, the kth term is given by $S_k=\sum_{1 \leq i_1<i_2<\cdots<i_k \leq n} \frac{1}{N}\left\lfloor\frac{N}{a_{i_1} a_{i_2} \cdots a_{i_k}}\right\rfloor$. The probability that none of the $a_i$ divide $R$ is then given by $1-S_1+S_2-\cdots+(-1)^n S_n$.

Multiplying the expression in (2.1) throughout by $\mathrm{N}$ yields the number of positive integers $\leq N$ that are relatively prime to each of $a_1, \ldots, a_n$ and the intrinsically combinatorial nature of the result is again manifest.

For each natural number $N$, Euler’s totient function $\varphi(N)$ is defined to be the number of positive integers $k$, no larger than $N$, which are relatively prime to $\mathrm{N}$. This function is of great importance in number theory.

## 数学代写|概率论代考Probability Theory代写|On trees and a formula of Cayley

A graph $\mathcal{G}=(\mathcal{V}, \mathcal{E})$ is a mathematical object consisting of an abstract set $\mathcal{V}$ of objects called vertices together with a collection $\varepsilon$ of unordered pairs of vertices called edges. We may view a graph pictorially by representing vertices by points on a sheet of paper and drawing lines between points to represent edges. In a vivid language we may then say that an edge ${i, j}$ is incident on the vertices $i$ and $j$ and say colloquially that $i$ and $j$ are neighbours or are adjacent. The degree of a vertex $i$, denoted $\operatorname{deg}(i)$, is the number of edges incident upon it, that is to say, the number of its neighbours.

A graph is connected if, starting from any vertex, we can move to any other vertex by traversing only along the edges of the graph. Formally, $\mathcal{G}$ is connected if, for every pair of distinct vertices $u$ and $v$, there exists $k \geq 0$ and a sequence of distinct vertices $u=i_0, i_1, \ldots, i_k, i_{k+1}=v$ so that $\left{i_j, i_{j+1}\right}$ is an edge of the graph for $0 \leq j \leq k$. Thus, we may traverse from $u$ to $v$ through the sequence of vertices $u=i_0 \rightarrow i_1 \rightarrow \cdots \rightarrow i_k \rightarrow i_{k+1}=v$ by progressing along the edges between succeeding vertices; we call such a progression a path. Thus, a graph is connected if, and only if, there is a path between any two vertices. An example of a (large) connected graph is the internet with users, computers, and routers representing vertices, and edges between entities that are directly linked.

Connected graphs can have fantastically complicated structures and a principle of parsimony may suggest that we begin with a consideration of the simplest kind of graph structure that is still connected. Suppose the graph contains a sequence of edges $\left{i_0, i_1\right},\left{i_1, i_2\right}, \ldots,\left{i_{k-1}, i_k\right},\left{i_k, i_0\right}$ where none of the vertices $i_1, \ldots, i_k$ repeat. Starting at vertex $i_0$ and traversing these edges of the graph in sequence returns us to $i_0$. Naturally enough, we will call such a sequence of edges a cycle. Now it is clear that if $\mathcal{G}$ is not already cycle-free, then we may remove any edge (but not the associated vertices) from any cycle of $\mathcal{G}$ without affecting connectivity (though the paths which originally involved the eliminated edge will now become longer). This pruning process has created a new connected graph with one fewer edge. Repeated iterations of the pruning procedure will terminate in a finite number of steps (as there are only a finite number of possible edges) in a cycle-free connected subgraph of $\mathcal{G}$ on the orig- inal set of vertices (that is to say, a connected graph on the $n$ vertices which contains no cycles and whose edges are all also edges of the parent graph $\mathcal{G}$ ).

# 概率论代写

## 数学代写|概率论代考Probability Theory代写|The sieve of Eratosthenes

$$1-\sum_{1 \leq i \leq n} \frac{1}{N}\left\lfloor\frac{N}{a_i}\right\rfloor+\sum_{1 \leq i<j \leq n} \frac{1}{N}\left\lfloor\frac{N}{a_i a_j}\right\rfloor-\cdots+(-1)^n \frac{1}{N}\left\lfloor\frac{N}{a_1 a_2 \cdots a_n}\right\rfloor .$$

$S_k=\sum_{1 \leq i_1<i_2<\cdots<i_k \leq n} \frac{1}{N}\left\lfloor\frac{N}{a_{i_1} a_{i 2} \cdots a_{i k}}\right\rfloor$. 没有一个的概率 $a_i$ 划分 $R$ 然后由 $1-S_1+S_2-\cdots+(-1)^n S_n$.

## 数学代写|概率论代考Probability Theory代写|On trees and a formula of Cayley

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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。