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# 数学代写|图论代考GRAPH THEORY代写|Excursion: Decision-Making

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## 数学代写|图论代写GRAPH THEORY代考|Excursion: Decision-Making

In the United States presidential election of 2000 , George W. Bush narrowly defeated Al Gore. Although Gore received a higher popular vote total than Bush, the winner of the election was Bush because his electoral vote total was higher than that of Gore.

During a typical year, there are numerous occasions when decisions are made by voting. Whether it’s electing a president, a prime minister, a senator, a governor, a mayor or a student representative on a committee, decisions must be made as to which individuals will hold these positions. Furthermore, a procedure must be in place to determine how this decision will be made. The answer may seem simple. The decision is made by voting. However, this is not as simple as it may first appear. If there are several candidates for a certain position, then there is a variety of ways of deciding the outcome of an election. It would seem that it is easy to decide the outcome of a twoperson election and in general this is true, with the aforementioned 2000 United States presidential election being a possible exception (even though there were more than two candidates for president). Making a decision among several choices is not restricted to governmental or college elections, however.

Example 7.12 Al, his wife Barbara and their three children Cassie, Donna and Edwin have discussed which new car they should purchase and have agreed that the choice should be made from a General Motors car $(G M)$, a Honda $(H)$, a Chrysler $(C), a$ Toyota $(T)$ and a Ford $(F)$. Al and Barbara also agreed that this should be a family decision and that each family member would have an equal voice in the decision.

Actually, Al’s preferences coincide exactly with the order of cars listed above. That is, $\mathrm{Al}$ prefers a General Motors car to a Honda, a Honda to a Chrysler and so on. Al’s preferences are given in the tournament shown in Figure 7.16. For example, the directed edge from $\mathrm{C}$ to $\mathrm{F}$ indicates that $\mathrm{Al}$ prefers a Chrysler to a Ford. The tournament in Figure 7.16 is called the tournament of paired comparisons for $\mathrm{Al}$ ‘s preferences as it indicates his preferred choices for each pair of cars.

## 数学代写|图论代写GRAPH THEORY代考|Exploration: Wine Bottle Problems

There are games and problems in which success is attained by proceeding through a sequence of steps. That is, in the process of playing the game or attempting to solve the problem, an individual may find himself or herself at one of a number of states and from that state it is possible to move to certain other states by a single (allowable) step. Such a situation can be modeled by a graph whose vertices are the states and where two states $\mathrm{A}$ and $\mathrm{B}$ are adjacent if it is possible to proceed from A to B by a single step. This is under the assumption that moving from A to B is reversible by a single reverse step. If, on the other hand, there are states $A$ and $B$ such that it is possible to proceed from $A$ to B by single step but not so from B to A, then this situation is more appropriately modeled by a digraph rather than a graph. We now look at a class of problems that can be modeled by digraphs.
Example 7.14 Three wine bottles $A, B$ and $C$ have capacities of 1,3 and 4 liters, respectively. These bottles are not graduated, however. That is, there are no markings on the bottles. So,
looking at a single bottle, it would be impossible to know exactly how much wine is in it, unless, of course, the bottle was full or empty. The largest bottle is filled with wine and the other two containers are empty. By a pouring, we mean that the contents of some bottle X containing wine is poured into a bottle $Y$ until either bottle $Y$ is filled or bottle $X$ is empty. We wish to divide the wine into two equal portions by pouring successively from one bottle to another. The problem then is: Can we obtain 2 liters of wine in the largest bottle and 2 liters in the medium-size bottle and if so, what is the fewest possible number of pourings needed to accomplish this?

At any particular time, suppose that bottle A contains $a$ liters of wine, B has $b$ liters of wine and C has $c$ liters of wine. Thus $a+b+c=4$ and initially $a=b=0$. Indeed, knowing only $a$ and $b$ tells us how much wine is in all three bottles. To help us answer this question, we construct a digraph $D$ such that
$$V(D)={(a, b): a \in{0,1}, b \in{0,1,2,3}},$$
where $\left(a_1, b_1\right)$ is adjacent to $\left(a_2, b_2\right)$ if we can proceed from $\left(a_1, b_1\right)$ to $\left(a_2, b_2\right)$ by a single pouring. The answer to the question is therefore the distance from the vertex $(0,0)$ to the vertex $(0,2)$ in $D$. The digraph $D$ is shown in Figure 7.20.

## 数学代写|图论代写GRAPH THEORY代考|Exploration: Wine Bottle Problems

$$V(D)={(a, b): a \in{0,1}, b \in{0,1,2,3}},$$

## 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。