## 数学代写|组合数学代写Combinatorial mathematics代考|COMP418

2022年12月22日

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## 数学代写|组合数学代写Combinatorial mathematics代考|Solutions to Exercises

8.1. Let us denote by $x$ the distance between the given line $L$ and the constructed line $L_1$ that cuts $F$ into two parts (Figure 8.4). We denote by $S_1$ and $S_2$ the areas of these parts. Then $S_1-S_2$ is a continuous function of $x$ (we don’t give a proof of this visually clear assertion because we did not introduce the exact definition of continuity). But when $x$ is small, $S_1-S_2$ is equal to $S$ (Figure 8.4), and when $x$ is large $S_1-S_2=-S$. So, according to the Intermediate Value Theorem, there is a value $x=c$ for which $S_1-S_2=0$, that is, $S_1=S_2=\frac{1}{2} S$

8.2. Let $L$ be a directed line through $p$ that forms angle $\phi$ with a fixed initial ray $L_0$ (Figure $8.13$ ). We denote the areas of the parts into which $L$ divides the figure $F$ by $S_1$ and $S_2\left(S_1\right.$ is to the left of $\left.L\right)$. Then $S_1-S_2$ is a continuous function of $\phi$. But as $\phi$ runs through all the values from 0 to $\pi$, the areas $S_1$ and $S_2$ interchange their roles. So, if $S_1-S_2$ is negative for $\phi=0$, then it will be positive for $\phi=\pi$. Consequently, by the Intermediate Value Theorem, there exists a value $\phi=c$ for which $S_1-S_2=0$, that is $S_1=S_2=\frac{1}{2} S$.

8.3. We squeeze the figure $F$ between two parallel lines $L_1$ and $L_2$ that form angle $\phi$ with a fixed initial ray (Figure 8.14). Then we squeeze $F$ between two parallel lines $M_1$ and $M_2$ that form the angle $\phi+90^{\circ}$ with the initial ray (Figure $8.15$ ). The four lines define a rectangle circumscribed about $F$. Let $a$ be the length of the side parallel to $L_1$ and $b$ be the length of the side parallel to $M_1$. Then $a-b$ is a continuous function of $\phi$. But when $\phi$ runs through the values from 0 to $\pi$, the lengths $a$ and $b$ interchange their roles. This means that if $a-b>0$ for $\phi=0$, then $a-b<0$ for $\phi=\pi$. Consequently, there exists an angle $\phi=\mathrm{c}$ for which $a-b=0$, that is, the circumscribed rectangle turns into a square.

## 数学代写|组合数学代写Combinatorial mathematics代考|Combinatorics of Acquaintance, or an Introduction

In mathematics, it is sometimes possible to derive something out of seemingly nothing, as the following problem illustrates.

Example 9.1. A number of people (more than one) come to a party. Prove that at least two of them have an equal number of acquaintances at the party. (The notion of acquaintance is reflexive: if $\mathrm{A}$ is acquainted with $\mathrm{B}$, then $\mathrm{B}$ is acquainted with A.)

Solution. If $n$ people come to the party, then for each person the number of acquaintances is an integer ranging from 0 to $n-1$. In fact, 0 and $n-1$ may not both serve as numbers of acquaintances for people in our party because 0 implies the existence of a person not acquainted with anybody and $n-1$ implies the existence of a person acquainted with everyone.

Thus, we have $n-1$ possibilities for number of acquaintances and $n$ people in the party. The Pigeonhole Principle now proves the required result.

In the solution to Example $9.1$ we discovered that at most $n-1$ integers can appear as numbers of acquaintances: $0,1, \ldots, n-2$ or $1,2, \ldots, n-1$. Curiosity prompts the following question.
Exercise 9.1. Is there a party of $n$ such that
(a) every integer $0,1, \ldots, n-2$ appears as the number of acquaintances of a person at the party?
(b) every integer $1,2, \ldots, n-1$ appears as the number of acquaintances of a person at the party?

## 数学代写|组合数学代写Combinatorial mathematics代考|Solutions to Exercises

8.1. 让我们用X给定线之间的距离大号和构造线大号1那削减F分为两部分（图 8.4）。我们用小号1和小号2这些部分的区域。然后小号1−小号2是的连续函数X（我们没有给出这个视觉上清晰的断言的证明，因为我们没有引入连续性的确切定义）。但当X是小，小号1−小号2等于小号（图 8.4），以及何时X很大小号1−小号2=−小号. 所以，根据中值定理，有一个值X=C为了哪个小号1−小号2=0， 那是，小号1=小号2=12小号

8.2. 让大号是一条有向线p形成角度φ具有固定的初始光线大号0（数字8.13). 我们将零件的区域表示为大号分割图F经过小号1和小号2(小号1在的左边大号). 然后小号1−小号2是的连续函数φ. 但作为φ遍历从 0 到π, 地区小号1和小号2互换角色。因此，如果小号1−小号2对φ=0, 那么它将是积极的φ=π. 因此，根据中值定理，存在一个值φ=C为了哪个小号1−小号2=0， 那是小号1=小号2=12小号.

8.3. 我们挤压图F两条平行线之间大号1和大号2形成角度φ具有固定的初始光线（图 8.14）。然后我们挤F两条平行线之间米1和米2形成角度φ+90∘与初始光线（图8.15). 这四条线定义了一个外接的矩形F. 让一种是平行于的边的长度大号1和b是平行于的边的长度米1. 然后一种−b是的连续函数φ. 但当φ遍历从 0 到π, 长度一种和b互换角色。这意味着如果一种−b>0为了φ=0， 然后一种−b<0为了φ=π. 因此，存在一个角φ=C为了哪个一种−b=0，即外接矩形变成正方形。

## 数学代写|组合数学代写Combinatorial mathematics代考|Combinatorics of Acquaintance, or an Introduction

（a）每个整数0,1,…,n−2出现在聚会上的人有多少熟人？
(b) 每个整数1,2,…,n−1出现在聚会上的人有多少熟人？

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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。