# 数学代写|代数学代写Algebra代考|MATH412

#### Doug I. Jones

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
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## 数学代写|代数学代写Algebra代考|Volumes

We can also extend these ideas slightly to learn about the volume of parallelepipeds in $\mathbb{R}^3$, but we need to make use of both the cross product and the dot product in order to make it work:

Let $\mathbf{v}, \mathbf{w}, \mathbf{x} \in \mathbb{R}^3$ be vectors. Then the volume of the parallelepiped with sides $\mathbf{v}, \mathbf{w}$, and $\mathbf{x}$ is $|\mathbf{v} \cdot(\mathbf{w} \times \mathbf{x})|$.

Proof. We first expand the expression $|\mathbf{v} \cdot(\mathbf{w} \times \mathbf{x})|$ in terms of the lengths of $\mathbf{v}$ and $\mathbf{w} \times \mathbf{x}$. If $\theta$ is the angle between $\mathbf{v}$ and $\mathbf{w} \times \mathbf{x}$, then Definition $1.2 .3$ tells us that $|\mathbf{v} \cdot(\mathbf{w} \times \mathbf{x})|=|\mathbf{v}||\mathbf{w} \times \mathbf{x}||\cos (\theta)|$.

Our goal now becomes showing that this quantity equals the volume of the parallelepiped with sides $\mathbf{v}, \mathbf{w}$, and $\mathbf{x}$. This parallelepiped’s volume is equal to the area of its base times its height. However, its base (when oriented as in Figure 1.26) is a parallelogram with sides $\mathbf{w}$ and $\mathbf{x}$, and thus has area $|\mathbf{w} \times \mathbf{x}|$ according to Theorem 1.A.2.

It follows that all we have left to do is show that the height of this parallelepiped is $|\mathbf{v}||\cos (\theta)|$. To verify this claim, recall that $\mathbf{w} \times \mathbf{x}$ is perpendicular to each of $\mathbf{w}$ and $\mathbf{x}$, and is thus perpendicular to the base parallelogram. The height of the parallelepiped is then the amount that $\mathbf{v}$ points in the direction of $\mathbf{w} \times \mathbf{x}$, which we can see is indeed $|\mathbf{v}||\cos (\theta)|$ by drawing a right-angled triangle with hypotenuse $\mathbf{v}$, as in Figure $1.26$.

## 数学代写|代数学代写Algebra代考|Undirected Graphs

A graph is a finite set of vertices, together with a set of edges connecting those vertices. Vertices are typically drawn as dots (sometimes with labels like $A, B, C, \ldots$ ) and edges are typically drawn as (not necessarily straight) lines connecting those dots, as in Figure 1.27.

We think of the vertices as objects of some type, and edges as representing the existence of a relationship between those objects. For example, a graph might represent

• A collection of cities (vertices) and the roads that connect them (edges),
• People (vertices) and the friendships that they have with other people on a social networking website (edges), or
• Satellites (vertices) and communication links between them (edges).
We emphasize that a graph is determined only by which vertices and edges between vertices are present – the particular locations of the vertices and methods of drawing the edges are unimportant. For example, the two graphs displayed in Figure $1.28$ are in fact the exact same graph, despite looking quite different on the surface.

A problem that comes up fairly frequently is how to count the number of paths of a certain length between different vertices on a graph. For example, this could tell us how many ways there are to drive from Toronto to Montréal, passing through exactly three other cities on the way, or the number of friends of friends that we have on a social networking website. For small graphs, it is straightforward enough to count the paths of small lengths by hand.

Count the number of paths of the indicated type in the graph displayed in Figure 1.27(a):
a) Of length 2 from $A$ to $B$, and
b) of length 3 from $A$ to $D$.
Solutions:
a) We simply examine the graph and notice that there is only one such path: $A-D-B$.
b) Paths of length 3 are a bit more difficult to eyeball, but some examination reveals that there are 4 such paths:
$A-B-A-D, \quad A-D-A-D, \quad A-D-B-D, \quad$ and $A-D-C-D$.
As the size of the graph or the length of the paths increases, counting these paths by hand becomes increasingly impractical. Even for paths just of length 3 or 4 , it’s often difficult to be sure that we have found all paths of the indicated length.

# 代数学代写

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## 数学代写|代数学代写Algebra代考|无向图

• 城市(顶点)和连接它们的道路的集合(边)，
• 人(顶点)和他们在社交网站上与他人建立的友谊(边)，或
• 卫星(顶点)和它们之间的通信链接(边)。我们强调，一个图仅仅由顶点和顶点之间的边的存在来决定——顶点的特定位置和绘制边缘的方法并不重要。例如，图$1.28$中显示的两个图实际上是完全相同的图，尽管表面看起来非常不同一个经常出现的问题是如何计算图上不同顶点之间一定长度的路径的数量。例如，它可以告诉我们从多伦多开车到Montréal有多少条路，途中正好经过另外三个城市，或者我们在社交网站上有多少朋友的朋友。对于小图，手工计算小长度的路径是很简单的
计算图1.27(a)中显示的图中指定类型的路径数量:
a)长度为2的路径从$A$到$B$，
b)长度为3的路径从$A$到$D$。
解决方案:
a)我们简单地检查图并注意到只有一条这样的路径:$A-D-B$ .
b)长度为3的路径有点难以观察，但一些检查显示有4条这样的路径:
$A-B-A-D, \quad A-D-A-D, \quad A-D-B-D, \quad$和$A-D-C-D$ .
随着图的大小或路径的长度的增加，用手计算这些路径变得越来越不切实际。即使对于长度为3或4的路径，通常也很难确定是否找到了所有指定长度的路径。

## 有限元方法代写

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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。

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