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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
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|图论作业代写Graph Theory代考|Distance, Diameter, and Radius

Dijkstra’s Algorithm provides the method for determining the shortest path between two points on a graph, which we define as the distance between those

vertices. There are many theoretical implications for this distance. We will investigate a few of these below; further discussion will occur throughout this book, most notably in Chapter 3 when discussing trees and in Chapter 4 when discussing connectivity. In particular, we will begin with defining the diameter and radius of a graph and the eccentricity of a vertex.

Definition 2.25 Given two vertices $x, y$ in a graph $G$, define the distance $d(x, y)$ as the length of the shortest path from $x$ to $y$. The eccentricity of a vertex $x$ is the maximum distance from $x$ to any other vertex in $G$; that is $\epsilon(x)=\max _{y \in V(G)} d(x, y)$.

The diameter of $G$ is the maximum eccentricity among all vertices, and so measures the maximum distance hetween any two vertices; that. is $\operatorname{diam}(G)=\max {x, y \in V(G)} d(x, y)$. The radius of a graph is the minimum eccentricity among all vertices; that is $\operatorname{rad}(G)=\min {x \in V(G)} \epsilon(x)$.

If a graph is connected, all of these parameters will be nonnegative integers. What happens if the graph is disconnected? If $x$ and $y$ are in separate components of $G$ then there is no shortest path between them and $d(x, y)=\infty$. This would then make $\operatorname{diam}(G)=\operatorname{rad}(G)=\infty$ since $\epsilon(v)=\infty$ for all vertices in $G$. Conceptually, you can think of the diameter as the longest path you can travel between any two points on a graph and the radius as the shortest distance among all pairs of vertices.

## 数学代写|图论作业代写Graph Theory代考|Minimum Spanning Trees

In Chapter 2 we not only asked if a graph had a hamiltonian cycle, but also how to find the optimal one within a weighted graph. Just as knowing if a spanning tree exists within a graph was an easy question to answer, finding an optimal one is also straightforward.

Definition 3.3 Given a weighted graph $G=(V, E, w), T$ is a minimum spanning tree, or MST, of $G$ if it is a spanning tree with the least total weight.

There are many algorithms that can find a MST, but we will focus on only two. Additional algorithms will appear in the Exercises.

Similar to Dijkstra’s Algorithm studied in Section 2.3, Kruskal’s Algorithm is fairly modern, first published in 1956 [60]. Joseph Kruskal was an American mathematician best known for his work in statistics and computer science. This algorithm is unique in that it is both efficient and optimal while still easily implemented and understandable for a non-scientist. In fact, it is the preferred method for finding a minimum spanning tree when the edges can be easily sorted[40].
Kruskal’s Algorithm
Input: Weighted connected graph $G=(V, E)$.
Steps:

1. Choose the edge of least weight. Highlight it and add it to $T=\left(V, E^{\prime}\right)$.
2. Repeat Step (1) so long as no circuit is created. That is, keep picking the edges of least weight but skip over any that would create a cycle in $T$.
Output: Minimum spanning tree $T$ of $G$.
Kruskal’s Algorithm does not distinguish between two edges of the same weight, in part because it does not influence the outcome. If at any point there are two edges to choose from of the same weight, you can pick either one. In addition, at each step of the algorithm we are building a forest subgraph that will eventually result in a spanning tree.

# 图论代考

## 数学代写|图论作业代写图论代考|距离、直径和半径

Dijkstra算法提供了确定图上两点之间最短路径的方法，我们将其定义为之间的距离

## 数学代写|图论作业代写图论代考|最小生成树

. 在第二章中，我们不仅讨论了一个图是否具有哈密顿循环，而且还讨论了如何在一个加权图中找到最优的哈密顿循环。正如知道生成树是否存在于图中是一个容易回答的问题一样，寻找最优生成树也很简单 给定一个加权图$G=(V, E, w), T$是$G$的最小生成树，如果它是一棵总权值最小的生成树 有许多算法可以找到MST，但我们只关注两种算法。其他算法将在练习中出现。

Kruskal’s Algorithm

1. 选择权重最小的边。突出显示并将其添加到 $T=\left(V, E^{\prime}\right)$.
2. 直到没有电路产生，重复步骤(1)。也就是说，继续选择权重最小的边，但跳过任何会产生循环的边 $T$.
输出:最小生成树 $T$ 的 $G$.
Kruskal算法不区分权值相同的两条边，部分原因是它不影响结果。如果在任意一点上有两条权值相同的边可供选择，你可以任选一条。此外，在算法的每一步，我们都在构建一个森林子图，最终生成树。

## 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。

assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师