### 数学代写|抽象代数作业代写abstract algebra代考|Math 417

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|抽象代数作业代写abstract algebra代考|First Principle of Mathematical Induction

So, to use induction to prove that a statement involving positive integers is true for every positive integer, we must first verify that the statement is true for the integer 1 . We then assume the statement is true for the integer $n$ and use this assumption to prove that the statement is true for the integer $n+1$.

Our next example uses some facts about plane geometry. Recall that given a straightedge and compass, we can construct a right angle.

IEXAMPLE 12 We use induction to prove that given a straightedge, a compass, and a unit length, we can construct a line segment of length $\sqrt{n}$ for every positive integer $n$. The case when $n=1$ is given. Now we assume that we can construct a line segment of length $\sqrt{n}$. Then use the straightedge and compass to construct a right triangle with height 1 and base $\sqrt{n}$. The hypotenuse of the triangle has length $\sqrt{n+1}$. So, by induction, we can construct a line segment of length $\sqrt{n}$ for every positive integer $n$.

## 数学代写|抽象代数作业代写abstract algebra代考|Second Principle of Mathematical Induction

To use this form of induction, we first show that the statement is true for the integer a. We then assume that the statement is true for all integers that are greater than or equal to $a$ and less than $n$, and use this assumption to prove that the statement is true for $n$.

EXAMPLE 14 We will use the Second Principle of Mathematical Induction with $a=2$ to prove the existence portion of the Fundamental Theorem of Arithmetic. Let $S$ be the set of integers greater than 1 that are primes or products of primes. Clearly, $2 \in S$. Now we assume that for some integer $n, S$ contains all integers $k$ with $2 \leq k<n$. We must show that $n \in S$. If $n$ is a prime, then $n \in S$ by definition. If $n$ is not a prime, then $n$ can be written in the form $a b$, where $1<a<n$ and $1<b<n$.

Since we are assuming that both $a$ and $b$ belong to $S$, we know that each of them is a prime or a product of primes. Thus, $n$ is also a product of primes. This completes the proof.

Notice that it is more natural to prove the Fundamental Theorem of Arithmetic with the Second Principle of Mathematical Induction than with the First Principle. Knowing that a particular integer factors as a product of primes does not tell you anything about factoring the next larger integer. (Does knowing that 5280 is a product of primes help you to factor 5281 as a product of primes?)

The following problem appeared in the “Brain Boggler” section of the January 1988 issue of the science magazine Discovery. ${ }^{2}$ Problems like this one are often called chicken McNugget problems, postage stamp problems, or Frobenius coin problems. Originally, McDonald’s sold its chicken nuggets in packs of 9 and 20 . The largest number of nuggets that could not have been bought with these packs is 151 .

## 数学代写|抽象代数作业代写abstract algebra代考| First Principle of Mathematical Induction

IEXAMPLE 12 我们使用归纳来证明，给定一个直尺、一个指南针和一个单位长度，我们可以构造一条长度的线 段。 $\sqrt{n}$ 对于每个正整数 $n$. 当 $n=1$ 给出。现在我们假设我们可以构造一条长度的线段 $\sqrt{n}$. 然后使用直尺和指南 针构建高度为 1 和底座的直角三角形 $\sqrt{n}$.三角形的斜边具有长度 $\sqrt{n+1}$. 因此，通过归纳，我们可以构造长度的 线段 $\sqrt{n}$ 对于每个正整数 $n$.

## 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。