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# 数学代写|数学建模代写Mathematical Modeling代考|Qualitative Relations in Applied Mathematics

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## 数学代写|数学建模代写Mathematical Modeling代考|Qualitative Relations in Applied Mathematics

It has been stated that “Applied mathematics is nothing but the solution of differential equations.” This statement is wrong on many counts: (i) Applied mathematics also deals with solutions of difference, differential-difference, integral, integro-differential, functional, and algebraic equations; (ii) applied mathematics is equally concerned with inequations of all types; (iii) applied mathematics is also concerned with mathematical modeling, in fact mathematical modeling has to precede solution of equations; and $(i v)$ applied mathematics also deals with situations which cannot be modeled in terms of equations or inequations, one such set of situations is concerned with qualitative relations.

Mathematics deals with both quantitative and qualitative relationships. Typical qualitative relations are: $y$ likes $x, y$ hates $x, y$ is superior to $x, y$ is subordinate to $x, y$ belongs to same political party as $x$, set $y$ has a non-null intersection with set $x$; point $y$ is joined to point $x$ by a road, state $y$ can be transformed into state $x$, team $y$ has defeated team $x, y$ is father of $x$, course $y$ is a prerequisite for course $x$, operation $y$ has to be done before operation $x$, species $y$ eats species $x, y$ and $x$ are connected by an airline, $y$ has a healthy influence on $x$, any increase of $y$ leads to a decrease in $x, y$ belongs to same class as $x, y$ and $x$ have different nationalities, and so on.

Such relationships are very conveniently represented by graphs where a graph consists of a set of vertices and edges joining some or all pairs of these vertices. To illustrate the typical problem situations which can be modeled through graphs, we consider the first problem so historically modeled viz. the problem of the seven bridges of Konigsberg.

## 数学代写|数学建模代写Mathematical Modeling代考|The Seven Bridges Problem

There are four land masses $A, B, C, D$ which are connected by seven bridges numbered 1 to 7 across a river (Figure 7.1). The problem is to start from any point in one of the land masses, cover each of the seven bridges once and once only, and return to the starting point.

There are two ways of attacking this problem. One method is to try to solve the problem by walking over the bridges. Hundreds of people tried to do so in their evening walks and failed to find a path satisfying the conditions of the problem. A second method is to draw a scale map of the bridges on paper and try to find a path by using a pencil.

It is at this stage that concepts of mathematical modeling are useful. It is obvious that the sizes of the land masses are unimportant, and the lengths of the bridges or whether these are straight or curved are irrelevant. What is relevant information is that $A$ and $B$ are connected by two bridges 1 and 2, $B$ and $C$ are connected by two bridges 3 and $4, B$ and $D$ are connected by one bridge number $5, A$ and $D$ are connected by bridge number 6 , and $C$ and $D$ are connected by bridge number 7 . All these facts are represented by the graph with four vertices and seven edges in Figure 7.2. If we can trace this graph in such a way that we start with any vertex and return to the same vertex and trace every edge once and once only without lifting the pencil from the paper, the problem can be solved. Again the trial and error method cannot be satisfactorily used to show that no solution is possible.

The number of edges meeting at a vertex is called the degree of that vertex. We note that the degrees of $A, B, C, D$ are $3,5,3,3$ respectively and each of these is an odd number. If we have to start from a vertex and return to it, we need an even number of edges at that vertex. Thus it is easily seen that the Konigsberg bridges problem cannot be solved.

This example also illustrates the power of mathematical modeling. We have not only disposed of the seven bridges problem, but we have discovered a technique for solving many problems of the same type.

## 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。