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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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• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|泛函分析作业代写Functional Analysis代考|Integral Equations

As mentioned in the introduction, some of the first applications of operator theory were in the study of integral operators and the integral equations that they come from. We will here briefly discuss the so-called Fredholm and Volterra integral equations that occur frequently in problems related to mathematical physics. We will assume in the following that $K$ is a HilbertSchmidt operator on $L^{2}(I)$ where $I$ is a bounded interval. The kernel for $K$ will be denoted $k(x, t)$, and we have that $k \in L^{2}(I \times I)$.

DEFINITION 7.3 A Fredholm integral equation of the first kind is an integral equation of the form
$$K u=f, \quad f \in L^{2}(I)$$
A Fredholm integral equation of the second kind is an integral equation of the form
$$(K-\lambda I) u=f, \quad f \in L^{2}(I) \quad \lambda \neq 0 .$$
We have met the Fredholm integral equation of the second kind before as a special case of Theorem 6.10. Notice that putting $\lambda=0$ in an equation of the second kind gives an equation of the first kind, but it is convenient to distinguish between the two. The reason is that the solution operator $K^{-1}$ to the equation of the first kind (if it exists) must be unbounded since $K$ is compact. Therefore, it is not to be expected that the solution has a nice dependence on the data $f$. Recall that in applications a solution $u$ to $K u=f$ will often be found by an approximation procedure: we find an expression for $K^{-1}$ that works on a set of well-behaved functions $\left(f_{n}\right)$ chosen such that $f_{n} \rightarrow f$ in $L^{2}(I)$; the hope is then that the sequence $\left(K^{-1} f_{n}\right)$ converges to the solution $u$. This will always be true when $K^{-1}$ is bounded, but when $K^{-1}$ is unbounded we cannot conclude anything in general.

## 数学代写|泛函分析作业代写Functional Analysis代考|The Resolvent

Now we will explore the connection between the semigroup $S(t)$ and its infinitesimal generator $A$. More precisely, we will investigate the resolvent $R_{\lambda}(A)$ of $A$. Much of what we will do can be put into a more general framework, but we will limit our presentation to the case where $S(t)$ is a $C_{0}$-semigroup on a Hilbert space $H$ with infinitesimal generator $A$. We will denote the growth constant of $S(t)$ by $\alpha$ such that
$$|S(t)| \leq M e^{\alpha t}$$
for all $t \geq 0$. Then, for $\operatorname{Re}(\lambda)>\alpha$ we can define a bounded, linear operator $R(\lambda)$ on $H$ by
$$R(\lambda) x=\int_{0}^{\infty} e^{-\lambda s} S(s) x d s$$
and we will show that $R(\lambda)$ is exactly the resolvent $R_{\lambda}(A)$ of $A$. Notice for later use that it is obvious from the definition that
$$|R(\lambda)| \rightarrow 0$$
for $\operatorname{Re}(\lambda) \rightarrow \infty$.
First we will show that the range of $R(\lambda)$ is $D(A)$, for every $\lambda$ with $R e(\lambda)>\alpha$. The first step is to calculate, for $\epsilon>0$ :
\begin{aligned} \frac{1}{\epsilon}(S(\epsilon)-I) R(\lambda) x &=\frac{1}{\epsilon} \int_{0}^{\infty} e^{-\lambda s}(S(s+\epsilon) x-S(s) x) d s \ &=\frac{1}{\epsilon}\left(\int_{\epsilon}^{\infty} e^{-\lambda s} e^{\lambda \epsilon} S(s) x d s-\int_{0}^{\infty} e^{-\lambda s} S(s) x d s\right) \ &=-\frac{1}{\epsilon} \int_{0}^{\epsilon} e^{-\lambda s} S(s) x d s+\frac{1}{\epsilon}\left(e^{\lambda \epsilon}-1\right) \int_{\epsilon}^{\infty} e^{-\lambda s} S(s) x d s \ & \rightarrow-x+\lambda R(\lambda) x \end{aligned}

# 泛函分析代写

## 数学代写|泛函分析作业代写Functional Analysis代考|Integral Equations

$$K u=f, \quad f \in L^{2}(I)$$

$$(K-\lambda I) u=f, \quad f \in L^{2}(I) \quad \lambda \neq 0 .$$

## 数学代写|泛函分析作业代写Functional Analysis代考|The Resolvent

$$|S(t)| \leq M e^{\alpha t}$$

$$R(\lambda) x=\int_{0}^{\infty} e^{-\lambda s} S(s) x d s$$

$$|R(\lambda)| \rightarrow 0$$

$$\frac{1}{\epsilon}(S(\epsilon)-I) R(\lambda) x=\frac{1}{\epsilon} \int_{0}^{\infty} e^{-\lambda s}(S(s+\epsilon) x-S(s) x) d s \quad=\frac{1}{\epsilon}\left(\int_{\epsilon}^{\infty} e^{-\lambda s} e^{\lambda \epsilon} S(s) x d s-\int_{0}^{\infty} e^{-\lambda s} S(s) x d s\right)=-\frac{1}{\epsilon} \int_{0}^{\epsilon} e^{-\lambda s} S(s) x d s+\frac{1}{\epsilon}\left(e^{\lambda \epsilon}-\right.$$

## 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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