### 数学代写|数值方法作业代写numerical methods代考| FUNCTIONS AND IMPLICIT FORMS

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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|数值方法作业代写numerical methods代考|FUNCTIONS AND IMPLICIT FORMS

Some problems use functions of two variables that are written in the implicit form:
$$f(x, y)=0 .$$
In this case we have an implicit relationship between the variables $x$ and $y$. We assume that $y$ is a function of $x$. The basic result for the differentiation of this implicit function is:
$$d f \equiv \frac{\partial f}{\partial x} d x+\frac{\partial f}{\partial y} d y=0$$
or:
$$\frac{d y}{d x}=-\frac{\partial f / \partial x}{\partial f / \partial y}$$
We now use this result by posing the following problem. Consider the transformation:
$$\left.\begin{array}{l} u=u(x, y) \ v=v(x, y) \end{array}\right} \text { original equations }$$
and suppose we wish to transform back:
$$\left.\begin{array}{l} x=x(u, v) \ y=y(u, v) \end{array}\right} \text { find } x, y \text { (inverse functions). }$$
To this end, we examine the following differentials:
\begin{aligned} &d u=\frac{\partial u}{\partial x} d x+\frac{\partial u}{\partial y} d y \ &d v=\frac{\partial v}{\partial x} d x+\frac{\partial v}{\partial y} d y \end{aligned}

Let us assume that we wish to find $d x$ and $d y$, given that all other quantities are known. Some arithmetic applied to Equation (1.13) (two equations in two unknowns!) results in:
\begin{aligned} &d x=\left(\frac{\partial v}{\partial y} d u-\frac{\partial u}{\partial y} d v\right) / J \ &d y=\left(-\frac{\partial v}{\partial x} d u+\frac{\partial u}{\partial x} d v\right) / J \end{aligned}
where $J$ is the Jacobian determinant defined by:
$$J=\left|\begin{array}{ll} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} \end{array}\right|=\frac{\partial(u, v)}{\partial(x, y)}$$
We can thus conclude the following result.
Theorem $1.1$ The functions $x=F(u, v)$ and $y=G(u, v)$ exist if:
$$\frac{\partial u}{\partial x}, \frac{\partial u}{\partial y}, \frac{\partial v}{\partial x}, \frac{\partial v}{\partial y}$$
are continuous at $(a, b)$ and if the Jacobian determinant is non-zero at $(a, b)$.
Let us take the example:
$$u=x^{2} / y, v=y^{2} / x$$
You can check that the Jacobian is given by:
$$\frac{\partial(u, v)}{\partial(x, y)}=\left|\begin{array}{cc} 2 x / y & -x^{2} / y^{2} \ -y^{2} / x^{2} & 2 y / x \end{array}\right|=3 \neq 0$$
Solving for $x$ and $y$ gives:
$$x=u^{2 / 3} v^{1 / 3}, y=u^{1 / 3} v^{2 / 3}$$
You need to be comfortable with partial derivatives. A good reference is Widder (1989).

## 数学代写|数值方法作业代写numerical methods代考|Metric Spaces

We work with sets and other mathematical structures in which it is possible to assign a so-called distance function or metric between any two of their elements. Let us suppose that $X$ is a set, and let $x, y$ and $z$ be elements of $X$. Then a metric $d$ on $X$ is a non-negative real-valued function of two variables having the following properties:
\begin{aligned} &D 1: d(x, y) \geq 0 ; \quad d(x, y)=0 \text { if and only if } x=y \ &D 2: d(x, y)=d(y, x) \ &D 3: d(x, y) \leq d(x, z)+d(z, y) \text { where } x, y, z \in X \end{aligned}
The concept of distance is a generalisation of the difference between two real numbers or the distance between two points in $n$-dimensional Euclidean space, for example.
Having defined a metric $d$ on a set $X$, we then say that the pair $(X, d)$ is a metric space. We give some examples of metrics and metric spaces:

1. We define the set $X$ of all continuous real-valued functions of one variable on the interval $[a, b]$ (we denote this space by $C[a, b])$ ), and we define the metric:
$$d(f, g)=\max {|f(t)-g(t)| ; t \in[a, b]}$$
Then $(X, d)$ is a metric space.
2. $n$-dimensional Euclidean space, consisting of vectors of real or complex numbers of the form:
$$x=\left(x_{1}, \ldots, x_{n}\right), y=\left(y_{1}, \ldots, y_{n}\right)$$
with metric: $d(x, y)=\max \left{\left|x_{j}-y_{j}\right| ; j=1, \ldots, n\right}$ or using the notation for a norm $d(x, y)=|x-y|_{\infty}$.
3. Let $L^{2}[a, b]$ be the space of all square-integrable functions on the interval $[a, b]$ :
$$\int_{a}^{b}|f(x)|^{2} d x<\infty .$$
We can then define the distance between two functions $f$ and $g$ in this space by the metric:
$$d(f, g)=|f-g|_{2} \equiv\left{\int_{a}^{b}|f(x)-g(x)|^{2}\right}^{1 / 2}$$
This metric space is important in many branches of mathematics, including probability theory and stochastic calculus.

## 数学代写|数值方法作业代写numerical methods代考|Cauchy Sequences

We define the concept of convergence of a sequence of elements of a metric space $X$ to some element that may or may not be in $X$. We introduce some definitions that we state for the set of real numbers, but they are valid for any ordered field, which is basically a set of numbers for which every non-zero element has a multiplicative inverse and there is a certain ordering between the numbers in the field.

Definition 1.4 A sequence $\left(a_{n}\right)$ of elements on the real line $\mathbb{R}$ is said to be convergent if there exists an element $a \in \mathbb{R}$ such that for each positive element $\varepsilon$ in $\mathbb{R}$ there exists a positive integer $n_{0}$ such that:
$$\left|a_{n}-a\right|<\varepsilon \text { whenever } n \geq n_{0} .$$ A simple example is to show that the sequence $\left{\frac{1}{n}\right}, n \geq 1$ converges to 0 . To this end, let $\varepsilon$ be a positive real number. Then there exists a positive integer $n_{0}>1 / \varepsilon$ such that $\left|\frac{1}{n}-0\right|=\frac{1}{n}<\varepsilon$ whenever $n \geq n_{0}$.

Definition $1.5$ A sequence $\left(a_{n}\right)$ of elements of an ordered field $F$ is called a Cauchy sequence if for each $\varepsilon>0$ in $F$ there exists a positive integer $n_{0}$ such that:
$$\left|a_{n}-a_{m}\right|<\varepsilon \text { whenever } m, n \geq n_{0} .$$
In other words, the terms in a Cauchy sequence get close to each other while the terms of a convergent sequence get close to some fixed element. A convergent sequence is always a Cauchy sequence, but a Cauchy sequence whose elements belong to a field $F$ does not necessarily converge to an element in $F$. To give an example, let us suppose that $F$ is the set of rational numbers; consider the sequence of integers defined by the Fibonacci recurrence relation:
\begin{aligned} &F_{0}=0 \ &F_{1}=1 \ &F_{n}=F_{n-1}+F_{n-2}, \quad n \geq 2 . \end{aligned}

It can be shown that:
\begin{aligned} &F_{n}=\frac{1}{\sqrt{5}}\left[\alpha^{n}-\beta^{n}\right] \ &\text { where } \alpha=\frac{1+\sqrt{5}}{2} \beta=\frac{1-\sqrt{5}}{2} . \end{aligned}
Now define the sequence of rational numbers by:
$$x_{n}=F_{n} / F_{n-1}, \quad n \geq 1 .$$
We can show that:
$$\lim {n \rightarrow \infty} x{n}=\alpha=\frac{1+\sqrt{5}}{2} \text { (the Golden Ratio) }$$
and this limit is not a rational number. The Fibonacci numbers are useful in many kinds of applications, such as optimisation (finding the minimum or maximum of a function) and random number generation.

We define a complete metric space $X$ as one in which every Cauchy sequence converges to an element in $X$. Examples of complete metric spaces are:

• Euclidean space $\mathbb{R}^{n}$.
• The metric space $C[a, b]$ of continuous functions on the interval $[a, b]$.
• By definition, Banach spaces are complete normed linear spaces. A normed linear space has a norm based on a metric, as follows $d(x, y)=|x-y|$.
• $L^{p}(0,1)$ is the Banach space of functions $f:[0,1] \rightarrow \mathbb{R}$ defined by the norm
$$|f|_{p}=\left(\int_{0}^{1}|f(x)|^{p} d x\right)^{1 / p}<\infty \text { for } 1 \leq p<\infty .$$
Definition 1.6 An open cover of a set $E$ in a metric space $X$ is a collection $\left{G_{j}\right}$ of open subsets of $X$ such that $E \subset \cup_{j} G_{j}$.

Finally, we say that a subset $K$ of a metric space $X$ is compact if every open cover of $K$ contains a finite subcover, that is $K \subset \cup_{j=1}^{N} G_{j}$ for some finite $N$.

## 数学代写|数值方法作业代写numerical methods代考|FUNCTIONS AND IMPLICIT FORMS

F(X,是)=0.

dF≡∂F∂XdX+∂F∂是d是=0

d是dX=−∂F/∂X∂F/∂是

\left.\begin{array}{l} u=u(x, y) \ v=v(x, y) \end{array}\right} \text { 原始方程 }\left.\begin{array}{l} u=u(x, y) \ v=v(x, y) \end{array}\right} \text { 原始方程 }

\left.\begin{array}{l} x=x(u, v) \ y=y(u, v) \end{array}\right} \text { find } x, y \text { （反函数）。}\left.\begin{array}{l} x=x(u, v) \ y=y(u, v) \end{array}\right} \text { find } x, y \text { （反函数）。}

d在=∂在∂XdX+∂在∂是d是 d在=∂在∂XdX+∂在∂是d是

dX=(∂在∂是d在−∂在∂是d在)/Ĵ d是=(−∂在∂Xd在+∂在∂Xd在)/Ĵ

Ĵ=|∂在∂X∂在∂是 ∂在∂X∂在∂是|=∂(在,在)∂(X,是)

∂在∂X,∂在∂是,∂在∂X,∂在∂是

∂(在,在)∂(X,是)=|2X/是−X2/是2 −是2/X22是/X|=3≠0

X=在2/3在1/3,是=在1/3在2/3

## 数学代写|数值方法作业代写numerical methods代考|Metric Spaces

D1:d(X,是)≥0;d(X,是)=0 当且仅当 X=是 D2:d(X,是)=d(是,X) D3:d(X,是)≤d(X,和)+d(和,是) 在哪里 X,是,和∈X

1. 我们定义集合X区间上一个变量的所有连续实值函数[一种,b]（我们用这个空间来表示C[一种,b]))，我们定义度量：
d(F,G)=最大限度|F(吨)−G(吨)|;吨∈[一种,b]
然后(X,d)是度量空间。
2. n维欧几里得空间，由以下形式的实数或复数向量组成：
X=(X1,…,Xn),是=(是1,…,是n)
有公制：d(x, y)=\max \left{\left|x_{j}-y_{j}\right| ; j=1, \ldots, n\right}d(x, y)=\max \left{\left|x_{j}-y_{j}\right| ; j=1, \ldots, n\right}或使用规范的符号d(X,是)=|X−是|∞.
3. 让大号2[一种,b]是区间上所有平方可积函数的空间[一种,b] :
∫一种b|F(X)|2dX<∞.
然后我们可以定义两个函数之间的距离F和G在这个空间中的度量：
d(f, g)=|fg|_{2} \equiv\left{\int_{a}^{b}|f(x)-g(x)|^{2}\right}^{1 / 2}d(f, g)=|fg|_{2} \equiv\left{\int_{a}^{b}|f(x)-g(x)|^{2}\right}^{1 / 2}
这个度量空间在许多数学分支中都很重要，包括概率论和随机微积分。

## 数学代写|数值方法作业代写numerical methods代考|Cauchy Sequences

|一种n−一种|<e 每当 n≥n0.一个简单的例子是证明序列\left{\frac{1}{n}\right}, n \geq 1\left{\frac{1}{n}\right}, n \geq 1收敛到 0 。为此，让e为正实数。那么存在一个正整数n0>1/e这样|1n−0|=1n<e每当n≥n0.

|一种n−一种米|<e 每当 米,n≥n0.

F0=0 F1=1 Fn=Fn−1+Fn−2,n≥2.

Fn=15[一种n−bn]  在哪里 一种=1+52b=1−52.

Xn=Fn/Fn−1,n≥1.

• 欧几里得空间Rn.
• 度量空间C[一种,b]区间上的连续函数[一种,b].
• 根据定义，Banach 空间是完全范数线性空间。一个带范数的线性空间有一个基于度量的范数，如下d(X,是)=|X−是|.
• 大号p(0,1)是函数的巴拿赫空间F:[0,1]→R由规范定义
|F|p=(∫01|F(X)|pdX)1/p<∞ 为了 1≤p<∞.
定义 1.6 集合的开盖和在度量空间X是一个集合\left{G_{j}\right}\left{G_{j}\right}的开放子集X这样和⊂∪jGj.

## 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。