### 数学代写|离散数学作业代写discrete mathematics代考|CS3653

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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|离散数学作业代写discrete mathematics代考|RECURSIVE DEFINITIONS

The definition of the factorial function can be written as
$$n !=n \times(n-1) ! .$$
Such a definition is called recursive because at the second step, it returns to the same definition, but with a smaller value of the parameter. Indeed, we compute the $n$-factorial through the $(n-1)$-factorial. Recursive definitions are often used in computer science and mathematics. As another example, let us consider a recursive definition of integer powers $\operatorname{pow}(a, n), a \neq 0$, that can be defined for any natural $n$ as pow $(a, 0)=1$ and $\operatorname{pow}(a, n+1)=a \times \operatorname{pow}(a, n)$.

The definitions of well-known arithmetic and geometric progressions (sequences), namely,
$$a_{n+1}=a_{n}+d,$$
where $a_{0}$ is the initial term and $d$, called the difference, are given numbers, and
$$a_{n+1}=a_{n} \times q ; a_{0} \text { and } q \text { are given, }$$
are also recursive definitions.
Problem 15 List the first five terms of the arithmetic progression with the first term $a_{0}=1$ and the difference $d=-2$. Prove that the terms of any arithmetic progression satisfy $2 a_{n+1}=a_{n}+a_{n+2}$
$$\begin{gathered} a_{n+k}=a_{n}+k \cdot d \ \sum_{k=0}^{r} a_{n+k}=(r+1) a_{n}+\frac{1}{2} r(r+1) d . \end{gathered}$$

## 数学代写|离散数学作业代写discrete mathematics代考|ELEMENTARY FUNCTIONS

The next few pages contain a very brief survey of the basic elementary functions ${ }^{5}$ – Power, Exponential, Logarithmic, and Trigonometric Functions. If the reader is familiar with that material, she can safely skip it and go to the next lecture. However, we know from the experience that many students, especially at the community colleges, know (if any) this stuff insufficiently, that is why it is included here.

Consider a quadratic equation $x^{2}=3$. It has two real roots, $\pm \sqrt{3}$. A similar equation $x^{2}=-3$ has no real solution, but if we consider it over the larger set of complex numbers, the equation has two roots. The reason for that is that the map $y=x^{2}$ for real $x$ is not a surjection, that is, given a $y$, we not always can return to $x$. This is a very common problem, and we address it now.

First, we consider bijective maps and let $f: X \rightarrow Y$ be bijective. This means that for every element $y \in Y$ there exists one and only one element $x=x_{y} \in X$ such that $y=f(x)$. Now we construct the map $g: Y \rightarrow X$ as follows. For every $y \in Y$ we set $g(y)=x_{y}$, where $x_{y}$ has been just defined. Since the element $y_{x}$ was defined uniquely, we uniquely defined the map $g: Y \rightarrow X$. By our construction, the map $g$ has the following properties.
The domain of $g$ is $Y$ and the co-domain is $X$. For each $x \in X$,$g \circ f(x)=g(f(x))=g(y)=x$ and $f \circ g(y)=f(g(y))=f\left(x_{y}\right)=y$,
therefore,
$$g \circ f=I_{X} \text { and } f \circ g=I_{\gamma} .$$

## 数学代写|离散数学作业代写discrete mathematics代考| RECURSIVE DEFINITIONS

$$n !=n \times(n-1) !$$

$$a_{n+1}=a_{n}+d$$

$$a_{n+1}=a_{n} \times q ; a_{0} \text { and } q \text { are given, }$$

$$a_{n+k}=a_{n}+k \cdot d \sum_{k=0}^{r} a_{n+k}=(r+1) a_{n}+\frac{1}{2} r(r+1) d$$

## 数学代写|离散数学作业代写discrete mathematics代考| ELEMENTARY FUNCTIONS

$$g \circ f=I_{X} \text { and } f \circ g=I_{\gamma} .$$

## 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。