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数学代写|离散数学作业代写discrete mathematics代考|Graph Colouring and Four-Colour Problem

It is very common for maps to be coloured in such a way that neighbouring states or countries are coloured differently. This allows different states or countries to be easily distinguished as well as the borders between them. The question naturally arises as to how many colours are needed (or determining the least number of colours needed) to colour the entire map, as it might be expected that a large number of colours would be needed to colour a large complicated map.

However, it may come as a surprise that in fact very few colours are required to colour any map. A former student of the British logician, Augustus De Morgan, had noticed this in the mid-1800s, and he proposed the conjecture of the four-colour theorem. There were various attempts to prove that 4 colours were sufficient from the mid-1800s onwards, and it remaincd a famous unsolved problem in mathematics until the late twentieth century.

Kempe gave an erroneous proof of the four-colour problem in 1879, but his attempt led to the proof that five colours are sufficient (which was proved by Heawod in the late $1800 \mathrm{~s})$. Appel and Haken of the University of Illinois finally provided the proof that 4 colours are sufficient in the mid-1970s (using over $1000 \mathrm{~h}$ of computer time in their proof).

Each map in the plane can be represented by a graph, with each region of the graph represented by a vertex. Edges connect two vertices if the regions have a common border. The colouring of a graph is the assignment of a colour to each vertex of the graph so that no two adjacent vertices in this graph have the same colour.
Definition
Let $\mathrm{G}=(\mathrm{V}, \mathrm{E})$ be a graph and let $\mathrm{C}$ be a finite set called the colours. Then, a colouring of $\mathrm{G}$ is a mapping $\kappa: \mathrm{V} \rightarrow \mathrm{C}$ such that if $u v \in \mathrm{E}$ then $\kappa(u) \neq \kappa(v)$.
That is, the colouring of a simple graph is the assignment of a colour to each vertex of the graph such that if two vertices are adjacent then they are assigned a different colour. The chromatic number of a graph is the least number of colours needed for a colouring of the graph. It is denoted by $\chi(G)$.

数学代写|离散数学作业代写discrete mathematics代考|Cryptography

Cryptography was originally employed to protect communication of private information between individuals. Today, it consists of mathematical techniques that provide secrecy in the transmission of messages between computers, and its objective is to solve security problems such as privacy and authentication over a communications channel.

It involves enciphering and deciphering messages, and it employs theoretical results from number theory to convert the original message (or plaintext) into cipher text that is then transmitted over a secure channel to the intended recipient. The cipher text is meaningless to anyone other than the intended recipient, and the recipient uses a key to decrypt the received cipher text and to read the original message.

The origin of the word “cryptography” is from the Grcek ‘kryptos’ mcaning hidden, and ‘graphein’ meaning to write. The field of cryptography is concerned with techniques by which information may be concealed in cipher texts and made unintelligible to all but the intended recipient. This ensures the privacy of the information sent, as any information intercepted will be meaningless to anyone other than the recipient.

Julius Caesar developed one of the earliest ciphers on his military campaigns in Gaul. His objective was to communicate important messages safely to his generals. His solution is one of the simplest and widely known encryption techniques, and it involves the substitution of each letter in the plaintext (i.e., the original message) by a letter a fixed number of positions down in the alphabet. The Caesar cipher involves a shift of 3 positions and this leads to the letter $\mathrm{B}$ being replaced by E, the letter $\mathrm{C}$ by $\mathrm{F}$, and so on.

The Caesar cipher (Fig. 10.1) is easily broken, as the frequency distribution of letters may be employed to determine the mapping. However, the Gaulish tribes who were mainly illiterate, and it is likely that the cipher provided good security.

数学代写|离散数学作业代写discrete mathematics代考|MATH-UA120

离散数学代写

数学代写|离散数学作业代写离散数学代考|图着色和四色问题


地图上色的方式使邻国或国家的颜色不同是很常见的。这样就可以很容易地区分不同的州或国家,以及它们之间的边界。这个问题很自然地就产生了:给整个地图上色需要多少颜色(或确定所需颜色最少的数量),因为可以预期,给一张复杂的大地图上色需要大量的颜色


然而,令人惊讶的是,事实上任何地图都只需要很少的颜色。英国逻辑学家奥古斯都·德·摩根(Augustus De Morgan)以前的一个学生在19世纪中期就注意到了这一点,并提出了四色定理的猜想。从19世纪中期开始,人们进行了各种各样的尝试来证明4种颜色是充分的,直到20世纪后期,这仍然是数学中一个著名的未解决的问题


肯普在1879年对四色问题给出了一个错误的证明,但他的尝试证明了五种颜色是充分的(这是由hewood在$1800 \mathrm{~s})$晚期证明的。伊利诺斯大学的阿佩尔和哈肯终于在20世纪70年代中期证明了4种颜色是足够的(在他们的证明中使用了超过$1000 \mathrm{~h}$的计算机时间)


平面上的每个映射都可以用一个图表示,图的每个区域用一个顶点表示。如果区域有共同的边界,则边连接两个顶点。图的着色是给图的每个顶点指定一种颜色,这样图中的相邻顶点就没有相同的颜色。
定义
让 $\mathrm{G}=(\mathrm{V}, \mathrm{E})$ 是一个图,让 $\mathrm{C}$ 是一个叫做颜色的有限集合。然后,上色 $\mathrm{G}$ 是一个映射 $\kappa: \mathrm{V} \rightarrow \mathrm{C}$ 这样,如果 $u v \in \mathrm{E}$ 然后 $\kappa(u) \neq \kappa(v)$也就是说,简单图的着色就是给图的每个顶点分配一种颜色,这样,如果两个顶点相邻,则给它们分配一种不同的颜色。图形的色数是图形着色所需的最少颜色数。它被表示为 $\chi(G)$.

数学代写|离散数学作业代写discrete mathematics代考|Cryptography

. crypgraphy


密码学最初是用来保护个人之间私人信息的通信。今天,它由数学技术组成,在计算机之间的消息传输中提供保密性,其目标是解决诸如通信通道上的隐私和身份验证等安全问题


它涉及到对消息的加密和解密,它利用数论的理论结果将原始消息(或明文)转换为密文,然后通过安全通道传输给预定的接收者。除了指定的接收方,该密文对其他任何人都没有意义,接收方使用密钥解密所收到的密文并读取原始消息


“密码学”一词的起源来自希腊语“kryptos”,意为隐藏,“graphein”意为书写。密码学研究的是一种技术,通过这种技术,信息可以隐藏在密文中,除了预期的接受者之外,所有人都无法理解。这确保了所发送信息的隐私性,因为任何被截获的信息对于除收件人以外的任何人都是无意义的


尤里乌斯·凯撒在高卢的军事战役中发明了最早的密码之一。他的目标是安全地将重要信息传达给他的将军们。他的解决方案是最简单且广为人知的加密技术之一,它涉及到将明文(即原始信息)中的每个字母替换为字母表中下方固定位置的一个字母。凯撒密码包含3个位置的移动,这导致字母$\mathrm{B}$被E取代,字母$\mathrm{C}$被$\mathrm{F}$取代,以此类推


凯撒密码(图10.1)很容易被破解,因为可以利用字母的频率分布来确定映射。然而,高卢部落的人主要是文盲,密码很可能提供了良好的安全性

数学代写|离散数学作业代写discrete mathematics代考

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