## 数学代写|偏微分方程代写partial difference equations代考|MATH4310

statistics-lab™ 为您的留学生涯保驾护航 在代写偏微分方程partial difference equations方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写偏微分方程partial difference equations代写方面经验极为丰富，各种代写偏微分方程partial difference equations相关的作业也就用不着说。

• 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|偏微分方程代写partial difference equations代考|Sobolev Spaces

Possibly the most important scales of distribution spaces consist of the Sobolev spaces. In this text we will solely make use of the Sobolev spaces based on $L^2$, which we shall denote by $H^s\left(\mathbb{R}^n\right)$ with $s \in \mathbb{R}: H^s\left(\mathbb{R}^n\right)$ is the linear space of tempered distributions $u$ whose Fourier transform $\widehat{u}$ is a square-integrable function in $\mathbb{R}^n$ with respect to the density $\left(1+|\xi|^2\right)^s \mathrm{~d} \xi$. The Hermitian product
$$(u, v)s=(2 \pi)^{-n} \int{\mathbb{R}^n} \widehat{u}(\xi) \overline{\widehat{v}(\xi)}\left(1+|\xi|^2\right)^s \mathrm{~d} \xi$$ defines a Hilbert space structure on $H^s\left(\mathbb{R}^n\right)$; we use the notation $|u|_s=\sqrt{(u, u)s}$. We have $H^0\left(\mathbb{R}^n\right)=L^2\left(\mathbb{R}^n\right)$; if $s^{\prime}{s^{\prime}} \leq|u|_{s^s}$. All the Hilbert spaces $H^s\left(\mathbb{R}^n\right)$ are isomorphic: it is immediate to see that the operators
$$\left(1-\Delta_x\right)^{t / 2} \varphi(x)=(2 \pi)^{-n} \int_{\mathbb{R}^n} \mathrm{e}^{-i x \cdot \xi}\left(1+|\xi|^2\right)^{t / 2} \widehat{\varphi}(\xi) \mathrm{d} \xi, t \in \mathbb{R},$$
form a group of (continuous linear) automorphisms of $\mathcal{S}\left(\mathbb{R}^n\right) ;(2.2 .2)$ extends as an isometry of $H^s\left(\mathbb{R}^n\right)$ onto $H^{s-t}\left(\mathbb{R}^n\right)$, whatever the real numbers $s, t$.

We mention a useful inequality, valid for all $s, t \in \mathbb{R}$ such that $a=s-t>0$, all $\varepsilon>0$ and $u \in H^s\left(\mathbb{R}^n\right)$
$$|u|_t^2 \leq \varepsilon|u|_s^2+\frac{1}{4 \varepsilon}|u|_{t-a}^2,$$
a direct consequence of the inequality $A^t \leq \varepsilon A^s+\frac{1}{4 \varepsilon} A^{t-a}, A=1+|\xi|^2$.

## 数学代写|偏微分方程代写partial difference equations代考|Distribution Kernels

We must now introduce distributions $F(x, y)$ on products $\Omega_1 \times \Omega_2$ with $\Omega_1 \subset$ $\mathbb{R}^{n_1}, \Omega_2 \subset \mathbb{R}^{n_2}$ open sets. Distributions belonging to $\mathcal{D}^{\prime}\left(\Omega_1 \times \Omega_2\right)$ are often referred to as kernels or distribution kernels. We can regard the product of two test-functions $\varphi \in C_{\mathrm{c}}^{\infty}\left(\Omega_1\right)$ and $\psi \in C_{\mathrm{c}}^{\infty}\left(\Omega_2\right)$ as an element of $C_{\mathrm{c}}^{\infty}\left(\Omega_1 \times \Omega_2\right)$, denoted by $\varphi \otimes \psi$, and evaluate $F \in \mathcal{D}^{\prime}\left(\Omega_1 \times \Omega_2\right)$ on it. Fixing $\psi$ defines a distribution in $\Omega_1$ :
$$C_{\mathrm{c}}^{\infty}\left(\Omega_1\right) \ni \varphi \mapsto\langle F, \varphi \otimes \psi\rangle \in \mathbb{C} .$$
To emphasize this partial action it is convenient to adopt the “Volterra notation”: to write $\int F(x, y) \psi(y)$ d $y$ rather than $\langle F(x, y), \psi(y)\rangle$. (Keep in mind, however, that $\int$ does not stand for a true integral!) In passing we point out that the Fubini formula is always true in distribution theory: $$\int\left(\int F(x, y) \psi(y) \mathrm{d} y\right) \varphi(x) \mathrm{d} x=\int\left(\int F(x, y) \varphi(x) \mathrm{d} x\right) \psi(y) \mathrm{d} y .$$
The map
$$C_{\mathrm{c}}^{\infty}\left(\Omega_2\right) \ni \psi \mapsto \mathfrak{I}F \psi(x)=\int F(x, y) \psi(y) \mathrm{d} y \in \mathcal{D}^{\prime}\left(\Omega_1\right)$$ is linear and continuous. The Schwartz Kernel Theorem states that, actually, every continuous linear map $C{\mathrm{c}}^{\infty}\left(\Omega_2\right) \longrightarrow \mathcal{D}^{\prime}\left(\Omega_1\right)$ is of the kind (2.3.1), and that the correspondence between continuous linear maps and distribution kernels is one-toone. This is a very special property of $\mathcal{D}^{\prime}$, obviously false for any infinite-dimensional Banach space (but true for $\mathcal{E}^{\prime}, C^{\infty}, C_{\mathrm{c}}^{\infty}$, if properly reformulated).

The composition $A_{1,2} \circ A_{2,3}$ of two linear operators $A_{1,2}: C_{\mathrm{c}}^{\infty}\left(\Omega_2\right) \longrightarrow \mathcal{D}^{\prime}\left(\Omega_1\right)$, $A_{2,3}: C_{\mathrm{c}}^{\infty}\left(\Omega_3\right) \longrightarrow \mathcal{D}^{\prime}\left(\Omega_2\right)$, puts requirements of regularity and support on the factors. For instance, we might require that $A_{2,3}$ maps $C_{\mathrm{c}}^{\infty}\left(\Omega_3\right)$ into $C_{\mathrm{c}}^{\infty}\left(\Omega_2\right)$, or else that $A_{1,2}$ extend as a continuous linear operator $\mathcal{D}^{\prime}\left(\Omega_2\right) \longrightarrow \mathcal{D}^{\prime}\left(\Omega_1\right)$, which is equivalent to requiring that the transpose $A_{1,2}^{\top}$ maps $C_{\mathrm{c}}^{\infty}\left(\Omega_1\right)$ into $C_{\mathrm{c}}^{\infty}\left(\Omega_2\right)$. These concerns are addressed in Definitions $2.3 .1$ and $2.3 .6$ below.

# 偏微分方程代写

## 数学代写|偏微分方程代写partial difference equations代考|Sobolev Spaces

$$(u, v) s=(2 \pi)^{-n} \int \mathbb{R}^n \widehat{u}(\xi) \overline{\hat{v}(\xi)}\left(1+|\xi|^2\right)^s \mathrm{~d} \xi$$

$$\left(1-\Delta_x\right)^{t / 2} \varphi(x)=(2 \pi)^{-n} \int_{\mathbb{R}^n} \mathrm{e}^{-i x \cdot \xi}\left(1+|\xi|^2\right)^{t / 2} \widehat{\varphi}(\xi) \mathrm{d} \xi, t \in \mathbb{R}$$

$$|u|t^2 \leq \varepsilon|u|_s^2+\frac{1}{4 \varepsilon}|u|{t-a}^2,$$

## 数学代写|偏微分方程代写partial difference equations代考|Distribution Kernels

$$C_{\mathrm{c}}^{\infty}\left(\Omega_1\right) \ni \varphi \mapsto\langle F, \varphi \otimes \psi\rangle \in \mathbb{C} .$$

$$\int\left(\int F(x, y) \psi(y) \mathrm{d} y\right) \varphi(x) \mathrm{d} x=\int\left(\int F(x, y) \varphi(x) \mathrm{d} x\right) \psi(y) \mathrm{d} y .$$

$$C_{\mathrm{c}}^{\infty}\left(\Omega_2\right) \ni \psi \mapsto \Im F \psi(x)=\int F(x, y) \psi(y) \mathrm{d} y \in \mathcal{D}^{\prime}\left(\Omega_1\right)$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|偏微分方程代写partial difference equations代考|Math462

statistics-lab™ 为您的留学生涯保驾护航 在代写偏微分方程partial difference equations方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写偏微分方程partial difference equations代写方面经验极为丰富，各种代写偏微分方程partial difference equations相关的作业也就用不着说。

• 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|偏微分方程代写partial difference equations代考|The wave-front set of a distribution

Let $\Omega \subset \mathbb{R}^n$ be an open set and let $x^{\circ} \in \Omega, \xi^{\circ} \in \mathbb{R}^n \backslash{0}$ be arbitrary. By a cone in $\mathbb{R}^n \backslash{0}$ we shall always mean a set invariant under all dilations $\xi \mapsto \lambda \xi, \lambda>0$ (i.e., a cone with vertex at the origin).
Lemma 2.1.4 Let $u \in \mathcal{D}^{\prime}(\Omega)$ have the following property:
(NWF) There exist an open set $U \subset \subset \Omega$ containing $x^{\circ}$ and $\varphi \in C_c^{\infty}(\Omega), \varphi(x)=1$ for every $x \in U$, and an open cone $\Gamma \subset \mathbb{R}^n \backslash{0}$ containing $\xi^{\circ}$ such that
$$\forall m \in \mathbb{Z}{+}, \sup {\xi \in \Gamma}\left((1+|\xi|)^m|\overline{(\varphi u)}(\xi)|\right)<+\infty .$$
Then, if $\Gamma^{\prime} \subset \mathbb{R}^n \backslash{0}$ is an open cone such that $\Gamma^{\prime} \cap \mathbb{S}^{n-1} \subset \subset \Gamma$, we have
$$\forall m \in \mathbb{Z}{+}, \sup {\xi \in \Gamma^{\infty}}\left((1+|\xi|)^m|\widehat{(\psi u)}(\xi)|\right)<+\infty$$
for every $\psi \in C_c^{\infty}(U)$
Proof Let $\varphi$ and $\psi$ be as in the statement; we have $\psi u=\psi \varphi u$ and therefore
$$\widehat{(\psi u)}(\xi)=(2 \pi)^{-n} \int \widehat{\psi}(\xi-\eta) \widehat{(\varphi u)}(\eta) \mathrm{d} \eta .$$
Here we shall use the notation, for $k \in \mathbb{Z}{+}$, $$|\psi|_k=\sup {\xi \in \mathbb{R}^n}\left((1+|\xi|)^k|\widehat{\psi}(\xi)|\right)$$
as well as
$$|\varphi u|{k, \Gamma}=\sup {\xi \in \Gamma}\left((1+|\xi|)^k|\overline{(\varphi u)}(\xi)|\right) .$$
Using the self-evident inequality $(1+|\xi|)^m \leq(1+|\eta|)^m(1+|\xi-\eta|)^m$ we get, for $\xi \in \Gamma^{\prime}$

## 数学代写|偏微分方程代写partial difference equations代考|Action of diferential operators on distributions

The action of a linear PDO on a distribution $u$ in $\Omega$ is defined by transposition:
$$\langle P(x, \mathrm{D}) u, \varphi\rangle=\left\langle u, P(x, \mathrm{D})^{\top} \varphi\right\rangle, \varphi \in \mathcal{C}{\mathrm{c}}^{\infty}(\Omega) .$$ When $u \in C^{\infty}(\Omega)$, (2.1.6) simply reflects integration by parts. Likewise, $$\langle P(x, \mathrm{D}) u, \bar{\varphi}\rangle=\left\langle u, \overline{P(x, \mathrm{D})^* \varphi}\right\rangle, \varphi \in C{\mathrm{c}}^{\infty}(\Omega) .$$
It follows directly from (2.1.6) that the inclusion (1.3.2), $\operatorname{supp} P(x, \mathrm{D}) f \subset$ supp $f$, remains valid when $f \in \mathcal{D}^{\prime}(\Omega)$. It is also obvious that
$$\text { singsupp } P(x, \text { D) } f \subset \operatorname{singsupp} f \text {, }$$
and if the coefficients of $P(x, \mathrm{D})$ are real-analytic, that
$$\text { singsupp }{\mathrm{a}} P(x, \mathrm{D}) f \subset \text { singsupp }{\mathrm{a}} f \text {. }^2$$
In other words, differential operators “decrease” the singular supports, just like they decrease the supports.

Every linear PDO maps $\mathcal{D}^{\prime}(\Omega)$ linearly and continuously into itself, and $\mathcal{E}^{\prime}(\Omega)$ into itself. In particular, $P(x, \mathrm{D}$ ) acts in the distribution sense (often called “the weak sense”) on a function $f \in L_{\text {loc }}^1(\Omega)$ :
$$\langle P(x, \mathrm{D}) f, \varphi\rangle=\int f P(x, \mathrm{D})^{\top} \varphi \mathrm{d} x, \varphi \in C_{\mathrm{c}}^{\infty}(\Omega) .$$
Actually [cf. (2.1.5)], every distribution $u \in \mathcal{D}^{\prime}(\Omega)$ can be represented locally as a finite sum of derivatives of continuous functions.

# 偏微分方程代写

## 数学代写|偏微分方程代写partial difference equations代考|The wave-front set of a distribution

(NWF) 存在一个开集 $U \subset \subset \Omega$ 含有 $x^0$ 和 $\varphi \in C_c^{\infty}(\Omega), \varphi(x)=1$ 每一个 $x \in U$ ，和一个开雉 $\Gamma \subset \mathbb{R}^n \backslash 0$ 含有 $\xi^{\circ}$ 这样
$$\forall m \in \mathbb{Z}+, \sup \xi \in \Gamma\left((1+|\xi|)^m|\overline{(\varphi u)}(\xi)|\right)<+\infty$$

$$\forall m \in \mathbb{Z}+, \sup \xi \in \Gamma^{\infty}\left((1+|\xi|)^m|\widehat{(\psi u)}(\xi)|\right)<+\infty$$

$$\widehat{(\psi u)}(\xi)=(2 \pi)^{-n} \int \widehat{\psi}(\xi-\eta) \widehat{(\varphi u)}(\eta) \mathrm{d} \eta .$$

$$|\psi|_k=\sup \xi \in \mathbb{R}^n\left((1+|\xi|)^k|\widehat{\psi}(\xi)|\right)$$

$$|\varphi u| k, \Gamma=\sup \xi \in \Gamma\left((1+|\xi|)^k|\overline{(\varphi u)}(\xi)|\right)$$

## 数学代写|偏微分方程代写partial difference equations代考|Action of diferential operators on distributions

$$\langle P(x, \mathrm{D}) u, \varphi\rangle=\left\langle u, P(x, \mathrm{D})^{\top} \varphi\right\rangle, \varphi \in \mathcal{C c}^{\infty}(\Omega) .$$

$$\langle P(x, \mathrm{D}) u, \bar{\varphi}\rangle=\left\langle u, \overline{P(x, \mathrm{D})^* \varphi}\right\rangle, \varphi \in C \mathrm{c}^{\infty}(\Omega) .$$

$$\text { singsupp a } P(x, \mathrm{D}) f \subset \text { singsupp a } f .{ }^2$$

$$\langle P(x, \mathrm{D}) f, \varphi\rangle=\int f P(x, \mathrm{D})^{\top} \varphi \mathrm{d} x, \varphi \in C_{\mathrm{c}}^{\infty}(\Omega) .$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|偏微分方程代写partial difference equations代考|MATH1470

statistics-lab™ 为您的留学生涯保驾护航 在代写偏微分方程partial difference equations方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写偏微分方程partial difference equations代写方面经验极为丰富，各种代写偏微分方程partial difference equations相关的作业也就用不着说。

• 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|偏微分方程代写partial difference equations代考|Basics on Distributions in Euclidean Space

Let $\Omega$ be an open subset of $\mathbb{R}^n$, as before. If $u$ is a complex-valued linear functional on the vector space $C_{\mathrm{c}}^{\infty}(\Omega)$, i.e., if $u$ is a linear map $C_{\mathrm{c}}^{\infty}(\Omega) \longrightarrow \mathbb{C}$, we denote by $\langle u, \varphi\rangle$ its evaluation at the test-function $\varphi \in C_{\mathrm{c}}^{\infty}(\Omega)$. The linear functional $u$ is a distribution in $\Omega$ if $\left\langle u, \varphi_j\right\rangle \rightarrow 0$ whenever the sequence $\left{\varphi_j\right}_{j=0,1,2, \ldots} \subset C_{\mathrm{c}}^{\infty}(\Omega)$ converges to zero in the following sense:
(•) all derivatives $\partial^\alpha \varphi_j$ converge uniformly to zero and there is a compact set $K \subset \Omega$ such that $\operatorname{supp} \varphi_j \subset K$ whatever $j$.

The space of distributions in $\Omega$ is denoted by $\mathcal{D}^{\prime}(\Omega)$. The restriction of a distribution $u \in \mathcal{D}^{\prime}(\Omega)$ to an open subset $\Omega^{\prime}$ of $\Omega$ is simply the restriction of the linear functional $u$ to the linear subspace $C_{\mathrm{c}}^{\infty}\left(\Omega^{\prime}\right)$ of $C_{\mathrm{c}}^{\infty}(\Omega)$. By using partitions of unity in $C_{\mathrm{c}}^{\infty}(\Omega)$ it is readily proved that there is a smallest closed subset of $\Omega$, called the support of $u$ and denoted by supp $u$, such that $u$ vanishes (“identically”) in $\Omega \backslash F$. The subspace of distributions in $\Omega$ that have compact support (contained in $\Omega$ ) is denoted by $\mathcal{E}^{\prime}(\Omega)$; it can be identified with the dual of $C^{\infty}(\Omega)$.

The convergence of a sequence of distributions $u_j\left(j \in \mathbb{Z}{+}\right)$is to be understood in the “weak sense”: $u_j \rightarrow 0$ if $\left\langle u_j, \varphi\right\rangle \rightarrow 0$ for each $\varphi \in C{\mathrm{c}}^{\infty}(\Omega)$. For $u_j \in \mathcal{E}^{\prime}(\Omega)$ to converge to zero in $\mathcal{E}^{\prime}(\Omega)$ it is moreover required that there be a compact set $K \subset \Omega$ such that $\operatorname{supp} u_j \subset K$ for all $j$.

Every continuous linear map of $C_{\mathrm{c}}^{\infty}(\Omega)$ into itself defines, by transposition, a continuous linear map of $\mathcal{D}^{\prime}(\Omega)$ into itself. Most important among these are multiplication by smooth functions in $\Omega$ and partial derivatives. If $P\left(x, \mathrm{D}x\right)$ is a linear partial differential operator with smooth coefficients in $\Omega$ we define, for arbitrary $u \in \mathcal{D}^{\prime}(\Omega), \varphi \in C{\mathrm{c}}^{\infty}(\Omega)$,
$$\left\langle P\left(x, \mathrm{D}_x\right) u, \varphi\right\rangle=\left\langle u, P\left(x, \mathrm{D}_x\right)^{\top} \varphi\right\rangle,$$
where $P\left(x, \mathrm{D}_x\right)^{\top}$ is the transpose of $P\left(x, \mathrm{D}_x\right)$ [cf. (1.3.3)].

## 数学代写|偏微分方程代写partial difference equations代考|Tempered distributions and their Fourier transforms

As is customary, $\mathcal{S}\left(\mathbb{R}^n\right)$ stands for the (Schwartz) space of functions $\varphi \in C^{\infty}\left(\mathbb{R}^n\right)$ rapidly decaying at infinity: given arbitrary $\alpha \in \mathbb{Z}{+}^n$ and $m \in \mathbb{Z}{+}$,
$$\sup {x \in \mathbb{R}^n}\left(1+|x|^2\right)^{\frac{1}{2} m}\left|\partial_x^\alpha \varphi(x)\right|<+\infty .$$ A sequence of functions $\varphi \in \mathcal{S}\left(\mathbb{R}^n\right)$ converges to zero if the seminorms on the left in (2.1.1) converge to zero for all choices of $m$ and $\alpha ; \mathcal{S}\left(\mathbb{R}^n\right)$ is a Fréchet space and thus its topology can be defined by (equivalent) metrics that turn it into a complete metric space. The space $\mathcal{S}^{\prime}\left(\mathbb{R}^n\right)$ of tempered distributions in $\mathbb{R}^n$ is the subspace of $\mathcal{D}^{\prime}\left(\mathbb{R}^n\right)$ consisting of the distributions $u$ which can be written as finite sums of distribution derivatives $$u=\sum{|\alpha| \leq m} \mathrm{D}^\alpha\left(P_\alpha f_\alpha\right)$$
in which the $P_\alpha$ are polynomials and the $f_\alpha$ belong, say, to $L^1\left(\mathbb{R}^n\right)$. By transposing the dense injection $C_{\mathrm{c}}^{\infty}\left(\mathbb{R}^n\right) \hookrightarrow \mathcal{S}\left(\mathbb{R}^n\right)$ the dual of $\mathcal{S}\left(\mathbb{R}^n\right)$ is identified with $\mathcal{S}^{\prime}\left(\mathbb{R}^n\right)$. Below we often denote by $\int u(x) \varphi(x) \mathrm{d} x$ (rather than by $\langle u, \varphi\rangle$ ) the duality bracket between $u \in \mathcal{S}^{\prime}\left(\mathbb{R}^n\right)$ and $\varphi \in \mathcal{S}\left(\mathbb{R}^n\right)$.
The Fourier transform
$$\widehat{u}(\xi)=\int_{\mathbb{R}^n} \mathrm{e}^{-i x \cdot \xi} u(x) \mathrm{d} x$$
defines a Fréchet space isomorphism of $\mathcal{S}\left(\mathbb{R}x^n\right)$ onto $\mathcal{S}\left(\mathbb{R}{\xi}^n\right)$ whose inverse is given by
$$u(x)=(2 \pi)^{-n} \int_{\mathbb{R}^n} \mathrm{e}^{i x \cdot \xi} \widehat{u}(\xi) \mathrm{d} x .$$

# 偏微分方程代写

## 数学代写|偏微分方程代写partial difference equations代考|Basics on Distributions in Euclidean Space

(•) 所有导数 $\partial^\alpha \varphi_j$ 一致收敛于零且存在紧集 $K \subset \Omega$ 这样 $\operatorname{supp} \varphi_j \subset K$ 任何 $j$.

$$\left\langle P\left(x, \mathrm{D}_x\right) u, \varphi\right\rangle=\left\langle u, P\left(x, \mathrm{D}_x\right)^{\top} \varphi\right\rangle,$$

## 数学代写|偏微分方程代写partial difference equations代考|Tempered distributions and their Fourier transforms

$$\sup x \in \mathbb{R}^n\left(1+|x|^2\right)^{\frac{1}{2} m}\left|\partial_x^\alpha \varphi(x)\right|<+\infty .$$

$$u=\sum|\alpha| \leq m \mathrm{D}^\alpha\left(P_\alpha f_\alpha\right)$$

$$\widehat{u}(\xi)=\int_{\mathbb{R}^n} \mathrm{e}^{-i x \cdot \xi} u(x) \mathrm{d} x$$

$$u(x)=(2 \pi)^{-n} \int_{\mathbb{R}^n} \mathrm{e}^{i x \cdot \xi} \widehat{u}(\xi) \mathrm{d} x$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|常微分方程代写ordinary differential equation代考|MATH2410

statistics-lab™ 为您的留学生涯保驾护航 在代写常微分方程ordinary differential equation方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写常微分方程ordinary differential equation代写方面经验极为丰富，各种代写常微分方程ordinary differential equation相关的作业也就用不着说。

• 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|常微分方程代写ordinary differential equation代考|Linear ODEs

Another important type of ODE which can be solved easily is the linear equation (both homogeneous and non-homogeneous). Let $J$ be a closed interval and $P: J \rightarrow \mathbb{R}$ be a continuous function. An equation of the form
$$y^{\prime}(x)+P(x) y(x)=0$$
is called a first order linear homogeneous ODE. If $Q$ is a nonzero continuous function on $J$, then
$$y^{\prime}(x)+P(x) y(x)=Q(x)$$
is called a first order linear non-homogeneous ODE. Any first order ODE that we consider in this chapter which is not in any of the forms (2.26) or (2.27) is called a nonlinear $O D E$.

There are many ways to solve (2.26). One of them is to apply the method of separation of variables. On comparing (2.26) with (2.1), we get
$$f(x)=-P(x), g(y)=\frac{1}{y} .$$
Therefore a solution to (2.26) is implicitly given by
$$\begin{gathered} \int^y \frac{d y}{y}=-\int^x P(x) d x+\tilde{c}, \tilde{c} \in \mathbb{R}, \ y=e^{\tilde{c}} e^{-\int^x P(x) d x} . \end{gathered}$$
From the previous relation, we directly obtain that
$$\phi(x)=c e^{-\int^x P(x) d x}, c \in \mathbb{R},$$
is a solution to (2.26). We now describe another way of obtaining the solution given in (2.28). Let $\phi$ be a solution to (2.26). On substituting $\phi$ in (2.26) and multiplying with $e^{\int^x P(x) d x}$ on both sides, we arrive at
or
$$\begin{gathered} e^{\int^x P(x) d x} \frac{d \phi(x)}{d x}+\frac{d}{d x}\left(e^{\int^x P(x) d x}\right) \phi(x)=0 \ \frac{d}{d x}\left(\phi(x) e^{\int^x P(x) d x}\right)=0 \end{gathered}$$

## 数学代写|常微分方程代写ordinary differential equation代考|Well-posedness

Throughout this chapter, we assume that every interval that we consider has a positive length ${ }^3$. We assume that $J$ and $\Omega$ are open intervals in $\mathbb{R}$. Let $\bar{J}$ and $\bar{\Omega}$ denote the smallest closed intervals containing $J$ and $\Omega$, respectively. Let $f: \bar{J} \times \bar{\Omega} \rightarrow \mathbb{R}$ be a function. Consider the problem
$$\left{\begin{array}{l} y^{\prime}(x)=f(x, y(x)), x \in J, \ y\left(x_0\right)=y_0 . \end{array}\right.$$
Definition 2.2.1. Let $J_1 \subseteq \bar{J}$ be an interval containing $x_0$. We say that a function $\phi: J_1 \rightarrow \mathbb{R}$ is said to be a solution to (2.34) if
(i) $\phi \in C\left(J_1\right) \cap C^1\left(J_1^o\right)$, where $J_1^o$ is the interval (inf $J_1, \sup J_1$ ),
(ii) $\phi(x) \in \Omega, x \in J_1$,
(iii) on substituting $y=\phi$ in (2.34) we get an identity in $J_1$.
Moreover, if $J_1 \backslash\left{x_0\right} \subset J \backslash\left{x_0\right}$, then we say that $\phi$ is a local solution. Otherwise it is called a global solution. If $J_1$ is of the form $\left[x_0, x_1\right]$ or $\left[x_0, x_1\right)$, then we say that $\phi$ is a right solution. If $J_1$ is of the form $\left[x_1, x_0\right]$ or $\left(x_1, x_0\right]$, then we say that $\phi$ is a left solution. If $x_0 \in J_1^o$ then we say that $\phi$ is a bilateral solution. If $J=\left(x_0, x_1\right)$ where $x_1 \in \mathbb{R} \cup{\infty}$, then (2.34) is said to be an initial value problem (IVP) and we deal with the right solutions in the study of IVPs. On the other hand, if $x_0 \in J$ then (2.34) is said to be a Cauchy problem. We usually seek bilateral solutions while studying Cauchy problems.
In fact, one of the main theorems of this chapter is to prove the existence of a bilateral (right) solutions to Cauchy problems (IVPs).

# 常微分方程代写

## 数学代写|常微分方程代写ordinary differential equation代考|Linear ODEs

$$y^{\prime}(x)+P(x) y(x)=0$$

$$y^{\prime}(x)+P(x) y(x)=Q(x)$$

(2.26)有多种求解方法。其中之一是应用变量分离法。将 (2.26) 与 (2.1) 进行比较，我们得到
$$f(x)=-P(x), g(y)=\frac{1}{y}$$

$$\int^y \frac{d y}{y}=-\int^x P(x) d x+\tilde{c}, \tilde{c} \in \mathbb{R}, y=e^{\bar{c}} e^{-\int^x P(x) d x}$$

$$\phi(x)=c e^{-\int^x P(x) d x}, c \in \mathbb{R}$$

$$e^{\int^x P(x) d x} \frac{d \phi(x)}{d x}+\frac{d}{d x}\left(e^{f^x P(x) d x}\right) \phi(x)=0 \frac{d}{d x}\left(\phi(x) e^{f^x P(x) d x}\right)=0$$

## 数学代写|常微分方程代写ordinary differential equation代考|Well-posedness

y^{\prime}(x)=f(x, y(x)), x \in J, y\left(x_0\right)=y_0
$$正确的。 \ \$$

(二) $\phi(x) \in \Omega, x \in J_1$,
(iii) 关于替代 $y=\phi$ 在 (2.34) 中我们得到一个恒等式 $J_1$. 决方案。如果 $J_1$ 是形式 $\left[x_0, x_1\right]$ 要么 $\left[x_0, x_1\right)$ ，那么我们说 $\phi$ 是一个正确的解决方案。如果 $J_1$ 是形式 $\left[x_1, x_0\right]$ 要么 $\left(x_1, x_0\right]$ ，那么我们说 $\phi$ 是左解。如果 $x_0 \in J_1^o$ 然后我们说 $\phi$ 是双边解决方案。如果
$J=\left(x_0, x_1\right)$ 在哪里 $x_1 \in \mathbb{R} \cup \infty$ ，那么 (2.34) 被称为初始值问题 (IVP) 并且我们在 IVP 的研究中处理正 确的解决方案。另一方面，如果 $x_0 \in J$ 则 (2.34) 被称为柯西问题。我们在研究柯西问题时通常寻求双边 解快方案。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|常微分方程代写ordinary differential equation代考|MATH3331

statistics-lab™ 为您的留学生涯保驾护航 在代写常微分方程ordinary differential equation方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写常微分方程ordinary differential equation代写方面经验极为丰富，各种代写常微分方程ordinary differential equation相关的作业也就用不着说。

• 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|常微分方程代写ordinary differential equation代考|Separation of variables

Consider the ODE of the form
$$\frac{d}{d x} y(x)=\frac{f(x)}{g(y(x))} .$$
We assume that $f:\left(a_0, a_1\right) \rightarrow \mathbb{R}$ and $g:\left(b_0, b_1\right) \rightarrow(0, \infty)$ are continuous functions. Wè also assume that there exists $y_0$ in the interval $\left(b_0, b_1\right)$ such that
$$g\left(y_0\right) \neq 0 .$$
We define a function $F:\left(a_0, a_1\right) \times\left(b_0, b_1\right) \rightarrow \mathbb{R}$ by
$$F(x, y)=\int_{y_0}^y g(\xi) d \xi-\int_{x_0}^x f(s) d s, x \in\left(a_0, a_1\right), y \in\left(b_0, b_1\right) .$$
Since $f$ and $g$ are continuous, $F$ is a $C^1$-function. Moreover for every $x_0 \in$ $\left(a_0, a_1\right)$ we have
$$\frac{\partial F}{\partial y}\left(x_0, y_0\right)=g\left(y_0\right) \neq 0 \text {. }$$

Therefore by the implicit function theorem (see Appendix C) there exists $\delta>0$ and a $C^1$-function $\phi:\left(x_0-\delta, x_0+\delta\right) \rightarrow \mathbb{R}$ such that
$$F(x, \phi(x))=\int_{y_0}^{\phi(x)} g(\xi) d \xi-\int_{x_0}^x f(s) d s=F\left(x_0, y_0\right), x \in\left(x_0-\delta, x_0+\delta\right) .$$
One can easily prove that $\phi$ is a solution to (2.1). For, on differentiating (2.3) with respect to $x$ (using the Leibniz rule of differentiation ${ }^1$ ) we get
$$\phi^{\prime}(x) g(\phi(x))-f(x)=0, x \in\left(x_0-\delta, x_0+\delta\right) .$$
This proves that the function $\phi$ which is implicitly given by the relation $F(x, y)=F\left(x_0, y_0\right)$, is a solution to (2.1). In other words, the relation
$$\int^y g(y) d y=\int^x f(x) d x+c, c \in \mathbb{R},$$
where the above integrals are indefinite integrals, defines a solution to (2.1). We now present some examples where this technique is demonstrated.

## 数学代写|常微分方程代写ordinary differential equation代考|Exact cquations

In this subsection, we present another special form of differential equations called exact equations which can be solved easily. Let $M, N$ be continuous functions in a rectangle
$$R=\left{(x, y):\left|x-x_0\right| \leq a,\left|y-y_0\right| \leq b\right},$$
and $N$ does not vanish in $R$. An ODE of the form
$$N(x, y(x)) y^{\prime}(x)+M(x, y(x))=0,$$
is said to be exact if there exists a $C^1$-function $F: R \rightarrow \mathbb{R}$ such that
$$\frac{\partial F}{\partial x}(x, y)=M(x, y), \quad \frac{\partial F}{\partial y}(x, y)=N(x, y),(x, y) \in R .$$
Example 2.1.8. Show that $y(x) y^{\prime}(x)+x=0$ is an exact equation.
Solution. In order to prove this, we first compare the given equation with (2.18) to get $M(x, y)=x$ and $N(x, y)=y$. It is easy to verify that

$$F(x, y)=\frac{x^2+y^2}{2},$$
satisfies (2.19). Hence the given equation is exact.
We now establish the connection between $F$ and the solutions to (2.18). To this end, we suppose (2.18) is exact and $F$ is known to us. We observe that $\frac{\partial F}{\partial y}=N \neq 0$, in $R$. Let $(\tilde{x}, \tilde{y}) \in \mathbb{R}^2$ satisfy $\left|x_0-\tilde{x}\right|<a$ and $\left|y_0-\tilde{y}\right|<b$. Then by the implicit function theorem there exists an interval $(\tilde{x}-\delta, \tilde{x}+\delta)$, which is denoted by $J$, and a $C^1$-function $\phi: J \rightarrow \mathbb{R}$ such that
$$F(x, \phi(x))=F(\tilde{x}, \tilde{y}), x \in J .$$
Claim. The function $\phi$ is a solution to (2.18).
For, on differentiating (2.20) with respect to $x$ we get
$$\frac{\partial F}{\partial x}(x, \phi(x))+\frac{\partial F}{\partial y}(x, \phi(x)) \phi^{\prime}(x)=0, x \in J .$$
Thus we have
$$M(x, \phi(x))+N(x, \phi(x)) \phi^{\prime}(x)=0, x \in J,$$
which proves that $\phi$ is a solution to (2.18). Hence the claim is proved.
Now, we shall revisit Example 2.1.8 and solve the ODE therein.

# 常微分方程代写

## 数学代写|常微分方程代写ordinary differential equation代考|Separation of variables

$$\frac{d}{d x} y(x)=\frac{f(x)}{g(y(x))}$$

$$g\left(y_0\right) \neq 0 .$$

$$F(x, y)=\int_{y_0}^y g(\xi) d \xi-\int_{x_0}^x f(s) d s, x \in\left(a_0, a_1\right), y \in\left(b_0, b_1\right) .$$

$$\frac{\partial F}{\partial y}\left(x_0, y_0\right)=g\left(y_0\right) \neq 0 .$$

$$F(x, \phi(x))=\int_{y_0}^{\phi(x)} g(\xi) d \xi-\int_{x_0}^x f(s) d s=F\left(x_0, y_0\right), x \in\left(x_0-\delta, x_0+\delta\right)$$

$$\phi^{\prime}(x) g(\phi(x))-f(x)=0, x \in\left(x_0-\delta, x_0+\delta\right) .$$

$$\int^y g(y) d y=\int^x f(x) d x+c, c \in \mathbb{R}$$

## 数学代写|常微分方程代写ordinary differential equation代考|Exact cquations

$$N(x, y(x)) y^{\prime}(x)+M(x, y(x))=0,$$

$$\frac{\partial F}{\partial x}(x, y)=M(x, y), \quad \frac{\partial F}{\partial y}(x, y)=N(x, y),(x, y) \in R .$$

$$F(x, y)=\frac{x^2+y^2}{2}$$

$$F(x, \phi(x))=F(\tilde{x}, \tilde{y}), x \in J .$$

$$\frac{\partial F}{\partial x}(x, \phi(x))+\frac{\partial F}{\partial y}(x, \phi(x)) \phi^{\prime}(x)=0, x \in J .$$

$$M(x, \phi(x))+N(x, \phi(x)) \phi^{\prime}(x)=0, x \in J,$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|常微分方程代写ordinary differential equation代考|MATH53

statistics-lab™ 为您的留学生涯保驾护航 在代写常微分方程ordinary differential equation方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写常微分方程ordinary differential equation代写方面经验极为丰富，各种代写常微分方程ordinary differential equation相关的作业也就用不着说。

• 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|常微分方程代写ordinary differential equation代考|Ordinary differential equations

The term ‘equatio differentialis’ (differential equations) was first used by Leibniz in 1676 to denote a relationship between the differentials of two variables. Very soon, this restricted usage was abandoned. Roughly speaking, differential equations are the equations involving one or more dependent variables (unknowns) and their derivatives/partial derivatives. If the unknown in the differential equation is a function of only one variable, then such differential equation is called an ordinary differential equation (ODE).
Notation: Unless specified otherwise, the unknown in the differential equation is denoted by $y$. Let $\mathbb{R}$ denote the set of real numbers, and $J$ be an open interval in $\mathbb{R}$. Throughout the book we denote the derivative of the function $y: J \rightarrow \mathbb{R}$ with respect to $x$ by either
$$\frac{d}{d x} y(x) \text { or } \frac{d y}{d x}(x) \text { or } y^{\prime}(x) .$$
When there is no ambiguity regarding the argument in the function $y$, we denote the derivative simply with $\frac{d y}{d x}$ or $y^{\prime}$. Similarly, let $y^{\prime \prime}$ and $y^{\prime \prime \prime}$ denote the second and the third derivative of $y$, respectively. In general, for $k \in \mathbb{N}$, $y^{(k)}$ or $\frac{d^k y}{d x^k}$ denotes the $k$-th order derivative of $y$.
With this notation, examples of ODEs are
$$\begin{gathered} \frac{d}{d x} y(x)=\left(\frac{d^2}{d x^2} y(x)\right)^5+y^2(x), x \in(0,1), \ y^{\prime}=3 y^2+(\sin x) y+\log \left(\cos ^2 y\right), x \in \mathbb{R} . \end{gathered}$$
The order of an ODE is the largest number $k$ such that the $k$-th order derivative of the unknown is present in the ODE. For example, the order of (1.1) is two.
At the beginning, it may look like tools from the integral calculus are sufficient to study ODEs. But very soon one realizes that to develop methods to solve or analyze them, one needs notions from subjects like analysis, linear algebra, etc. In fact, the study of differential equations motivated crucial development of many areas of mathematics: the theory of Fourier series and more general orthogonal expansions, integral transformations, Hilbert spaces, and Lebesgue integration to name a few.

## 数学代写|常微分方程代写ordinary differential equation代考|Applications of ODEs

Many laws in physics, chemistry, biology etc., can be easily expressed using differential equations. One of the reasons for this is the following. The quantity $y^{\prime}(x)$ can be interpreted as the rate of change of the quantity $y$ with respect to the quantity $x$. In many natural phenomena, there is a relationship between the unknowns (which are relatively difficult to measure), the rate of change of the unknowns with respect to a known quantity, and the other known quantities (which are easy to measure) that govern the process. When this relationship is expressed in mathematics, it turns out to be a (system of) differential equation(s). Therefore the study of ODEs is crucial in understanding physical sciences. In fact, much of the theory developed in ODEs owes to the questions/situations raised in the study of subjects like mechanics, astronomy, electronics etc.
Listing all the available ODE models in any branch of science is an impossible task. Therefore in this chapter, we present a few ODE models which arise from physics and biology which can be solved or analyzed using the material in the book. We begin with models from physics.

Example 1.2.1 (Radioactivity and half-life). Let $N(t)$ denote the number of radioactive active atoms in a substance of a fixed quantity at time $t$. Then a model for the decay of the number of radioactive atoms is
$$\begin{gathered} \frac{d}{d t} N(t)=-k N(t), t>0, \ N\left(t_0\right)=N_0, \end{gathered}$$
where $k>0$. Equation (1.3b) is known as the initial condition. This kind of models are studied in detail in Chapter 2, Subsection 2.1.3. One can easily verify that the solution to (1.3a) is
$$N(t)=N_0 e^{-k\left(t-t_0\right)}, t>t_0 .$$
The half-life of a specific radioactive isotope is defined as the time taken for half of its radioactive atoms to decay. In fact, the half-life is independent of the quantity of the radioactive material. We now calculate the half-life of an isotope using (1.3a) if $k$ is known explicitly. For, it is enough to find $T$ at which $N(T)=\frac{N_0}{2}$. From (1.4) we have
$$N(T)=N_0 e^{-k\left(T-t_0\right)}=\frac{N_0}{2}$$

# 常微分方程代写

## 数学代写|常微分方程代写ordinary differential equation代考|Ordinary differential equations

$$\frac{d}{d x} y(x) \text { or } \frac{d y}{d x}(x) \text { or } y^{\prime}(x) .$$

$$\frac{d}{d x} y(x)=\left(\frac{d^2}{d x^2} y(x)\right)^5+y^2(x), x \in(0,1), y^{\prime}=3 y^2+(\sin x) y+\log \left(\cos ^2 y\right), x \in \mathbb{R}$$
$\mathrm{ODE}$ 的阶数是最大数 $k$ 这样的 $k \mathrm{ODE}$ 中存在末知数的 -th 阶导数。例如，(1.1) 的阶数为二。

## 数学代写|常微分方程代写ordinary differential equation代考|Applications of ODEs

$$\frac{d}{d t} N(t)=-k N(t), t>0, N\left(t_0\right)=N_0,$$

$$N(t)=N_0 e^{-k\left(t-t_0\right)}, t>t_0 .$$

$$N(T)=N_0 e^{-k\left(T-t_0\right)}=\frac{N_0}{2}$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|matlab代写|BMS13

MATLAB是一个编程和数值计算平台，被数百万工程师和科学家用来分析数据、开发算法和创建模型。

statistics-lab™ 为您的留学生涯保驾护航 在代写matlab方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写matlab代写方面经验极为丰富，各种代写matlab相关的作业也就用不着说。

• 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|matlab代写|Applications of Deep Learning

Deep learning is used in many applications today. Here are a few:
Image recognition – This is arguably the best known and most controversial use of deep learning. A deep learning system is trained with pictures of people. Cameras are distributed everywhere, and images are captured. The system then identifies individual faces and matches them against its trained database. Even with variations in lighting, weather conditions, and clothing, the system can identify the people in the images.

Speech recognition – You hardly ever get a human being on the phone anymore. You are first presented with a robotic listener that can identify what you are saying, at least within the limited context of what it expects. When a human listens to another human, the listener is not just recording the speech, they are guessing what the person is going to say and filling in gaps of garbled words and confusing grammar. Robotic listeners have some of the same abilities. A robotic listener is an embodiment of the “Turing test.” Did you ever get one that you thought was a human being? Or for that matter, did you ever reach a human who you thought was a robot?

Handwriting analysis – A long time ago, you would get forms in which you had boxes in which to write numbers and letters. At first, they had to be block capitals! A robotic handwriting system could figure out the letters in those boxes reliably. Years later, though many years ago, the US Post Office introduced zip code reading systems. At first, you had to put the zip code on a specific part of the envelope. That system has evolved so that it can find zip codes anywhere. This made the zip $+4$ system valuable and a big productivity boost.

Machine translation – Google translate does a pretty good job considering it can translate almost any language in the world. It is an example of a system with online training. You see that when you type in a phrase and the translation has a checkmark next to it because a human being has indicated that it is correct. Figure $1.10$ gives an example. Google harnesses the services of free human translators to improve its product!

Targeting – By targeting, we mean figuring out what you want. This may be a movie, a clothing item, or a book. Deep learning systems collect information on what you like and decide what you would be most interested in buying. Figure $1.11$ gives an example. This is from a couple of years ago. Perhaps, ballet dancers like Star Wars!

Other applications include game playing, autonomous driving, medicine, and many others. Just about any human activity can be an application of deep learning.

## 数学代写|matlab代写|Organization of the Book

This book is organized around specific deep learning examples. You can jump into any chapter as they are pretty much independent. We’ve tried to present a wide range of topics, some of which, hopefully, align with your work or interests. The next chapter gives an overview of MATLAB products for deep learning. Besides the core MATLAB development environment, we only use three of their toolboxes in this book.
Each chapter except for this and the next is organized in the following order:

1. Modeling
2. Building the system
3. Training the system
4. Testing the system
Training and testing are often in the same script. Modeling varies with each chapter. For physical problems, we derive numerical models, usually sets of differential equations, and build simulations of the processes.

The chapters in this book present a range of relatively simple examples to help you learn more about deep learning and its applications. It will also help you learn the limitations of deep learning and areas for future research. All use the MATLAB Deep Learning Toolbox.

1. What Is Deep Learning? (this chapter).
2. MATLAB Machine Learning Toolboxes – This chapter gives you an introduction to MATLAB machine intelligence toolboxes. We’ll be using three of the toolboxes in this book.
3. Finding Circles with Deep Learning – This is an elementary example. The system will try to figure out if a figure is a circle. It will be presented with circles, ellipses, and other objects and trained to determine which are circles.
4. Classifying Movies – All movie databases try to guess what movies will be of most interest to their viewers to speed movie selection and reduce the number of disgruntled customers. This example creates a movie rating system and attempts to classify movies in the movie database as good or bad.
5. Algorithmic Deep Learning – This is an example of fault detection using a detection filter as an element of the deep learning system. It uses a custom deep learning algorithm, the only example that does not use the MATLAB Deep Learning Toolbox.
6. Tokamak Disruption Detection – Disruptions are a major problem with a nuclear fusion device known as a Tokamak. Researchers are using neural nets to detect disruptions before they happen so that they can be stopped. In this example, we use a simplified dynamical model to demonstrate deep learning.

## 数学代写|matlab代写|Organization of the Book

1. 造型
2. 构建系统
3. 训练系统
4. 测试系统
训练和测试通常在同一个脚本中。建模因每一章而异。对于物理问题，我们推导出数值模型，通常是微分方程组，并建立过程模拟。

1. 什么是深度学习？（本章）。
2. MATLAB 机器学习工具箱——本章介绍 MATLAB 机器智能工具箱。我们将使用本书中的三个工具箱。
3. Finding Circles with Deep Learning——这是一个基本的例子。系统将尝试判断图形是否为圆形。它将与圆圈、椭圆和其他对象一起呈现，并接受训练以确定哪些是圆圈。
4. 对电影进行分类——所有电影数据库都试图猜测观众最感兴趣的电影是什么，以加快电影选择速度并减少不满客户的数量。此示例创建一个电影评级系统，并尝试将电影数据库中的电影分类为好或坏。
5. 算法深度学习——这是一个使用检测过滤器作为深度学习系统元素的故障检测示例。它使用自定义深度学习算法，这是唯一不使用 MATLAB 深度学习工具箱的示例。
6. 托卡马克中断检测——中断是称为托卡马克的核聚变装置的主要问题。研究人员正在使用神经网络在中断发生之前检测它们，以便可以阻止它们。在此示例中，我们使用简化的动力学模型来演示深度学习。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|matlab代写|CSC113

MATLAB是一个编程和数值计算平台，被数百万工程师和科学家用来分析数据、开发算法和创建模型。

statistics-lab™ 为您的留学生涯保驾护航 在代写matlab方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写matlab代写方面经验极为丰富，各种代写matlab相关的作业也就用不着说。

• 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|matlab代写|Neural Nets

Neural networks, or neural nets, are a popular way of implementing machine “intelligence.” The idea is that they behave like the neurons in a brain. In this section, we will explore how neural nets work, starting with the most fundamental idea with a single neuron and working our way up to a multi-layer neural net. Our example for this will be a pendulum. We will show how a neural net can be used to solve the prediction problem. This is one of the two uses of a neural net, prediction and classification. We’ll start with a simple classification example.

Let’s first look at a single neuron with two inputs. This is shown in Figure 1.2. This neuron has inputs $x_1$ and $x_2$, a bias $b$, weights $w_1$ and $w_2$, and a single output $z$. The activation function $\sigma$ takes the weighted input and produces the output. In this diagram, we explicitly add icons for the multiplication and addition steps within the neuron, but in typical neural net diagrams such as Figure 1.1, they are omitted.
$$z=\sigma(y)=\sigma\left(w_1 x_1+w_2 x_2+b\right)$$
Let’s compare this with a real neuron as shown in Figure 1.3. A real neuron has multiple inputs via the dendrités. Some of thẻse branchẻs mean thăt multiplé inputś cản connect to the cell body through the same dendrite. The output is via the axon. Each neuron has one output. The axon connects to a dendrite through the synapse.
There are numerous commonly used activation functions. We show three:
\begin{aligned} \sigma(y) & =\tanh (y) \ \sigma(y) & =\frac{2}{1-e^{-y}}-1 \ \sigma(y) & =y \end{aligned}
The exponential one is normalized and offset from zero so it ranges from $-1$ to 1 . The last one, which simply passes through the value of $\mathrm{y}$, is called the linear activation function. The following code in the script OneNeuron . m computes and plots these three activation functions for an input q. Figure $1.4$ shows the three activation functions on one plot.

## 数学代写|matlab代写|Types of Deep Learning

There are many types of deep learning networks. New types are under development as you read this book. One deep learning researcher joked that you will have the name for an existing deep learning algorithm if you randomly put together four letters.
The following sections briefly describe some of the major types.

A CNN has convolutional layers. It convolves a feature with the input matrix so that the output emphasizes that feature. This effectively finds patterns. For example, you might convolve an $\mathrm{L}$ pattern with the incoming data to find corners. The human eye has edge detectors, making the human vision system a convolutional neural network of sorts.

Recurrent neural networks are a type of recursive neural network. Recurrent neural networks are often used for time-dependent problems. They combine the last time step’s data with the data from the hidden or intermediate layer, to represent the current time step. A recurrent neural net has a loop. An input vector at time $k$ is used to create an output which is then passed to the next element of the network. This is done recursively in that each stage is identical to external inputs and inputs from the previous stage. Recurrent neural nets are used in speech recognition, language translation, and many other applications. One can see how a recurrent network would be useful in translation. The meaning of the latter part of an English sentence can be dependent on the beginning. Now, this presents a problem. Suppose we are translating a paragraph. Is the output of the first stage necessarily relevant to the 100 th stage? In standard estimation, old data is forgotten using a forgetting factor. In neural networks, we can use Long Short-Term Memory (LSTM) networks that have this feature.

## 数学代写|matlab代写|Neural Nets

$$z=\sigma(y)=\sigma\left(w_1 x_1+w_2 x_2+b\right)$$

$$\sigma(y)=\tanh (y) \sigma(y) \quad=\frac{2}{1-e^{-y}}-1 \sigma(y)=y$$

## 数学代写|matlab代写|Types of Deep Learning

CNN 具有卷积层。它将特征与输入矩阵进行卷积，以便输出强调该特征。这有效地找到了模式。例如，您可能会卷积一个大号模式与传入的数据来寻找角落。人眼具有边缘检测器，使人类视觉系统成为一种卷积神经网络。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|matlab代写|STA518

MATLAB是一个编程和数值计算平台，被数百万工程师和科学家用来分析数据、开发算法和创建模型。

statistics-lab™ 为您的留学生涯保驾护航 在代写matlab方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写matlab代写方面经验极为丰富，各种代写matlab相关的作业也就用不着说。

• 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|matlab代写|Deep Learning

Deep learning is a subset of machine learning which is itself a subset of artificial intelligence and statistics. Artificial intelligence research began shortly after World War II [35]. Early work was based on the knowledge of the structure of the brain, propositional logic, and Turing’s theory of computation. Warren McCulloch and Walter Pitts created a mathematical formulation for neural networks based on threshold logic. This allowed neural network research to split into two approaches: one centered on biological processes in the brain and the other on the application of neural networks to artificial intelligence. It was demonstrated that any function could be implemented through a set of such neurons and that a neural net could learn to recognize patterns. In 1948, Norbert Wiener’s book Cybernetics was published which described concepts in control, communications, and statistical signal processing. The next major step in neural networks was Donald Hebb’s book in 1949, The Organization of Behavior, connecting connectivity with learning in the brain. His book became a source of learning and adaptive systems. Marvin Minsky and Dean Edmonds built the first neural computer at Harvard in 1950.

The first computer programs, and the vast majority now, have knowledge built into the code by the programmer. The programmer may make use of vast databases. For example, a model of an aircraft may use multidimensional tables of aerodynamic coefficients. The resulting software, therefore, knows a lot about aircraft, and running simulations of the models may present surprises to the programmer and the users since they may not fully understand the simulation, or may have entered erroneous inputs. Nonetheless, the programmatic relationships between data and algorithms are predetermined by the code.

In machine learning, the relationships between the data are formed by the learning system. Data is input along with the results related to the data. This is the system training. The machine learning system relates the data to the results and comes up with rules that become part of the system. When new data is introduced, it can come up with new results that were not part of the training set.

Deep learning refers to neural networks with more than one layer of neurons. The name “deep learning” implies something more profound, and in the popular literature, it is taken to imply that the learning system is a “deep thinker.” Figure $1.1$ shows a single-layer and multi-layer network. It turns out that multi-layer networks can learn things that single-layer networks cannot. The elements of a network are nodes, where weighted signals are combined and biases added. In a single layer, the inputs are multiplied by weights and then added together at the end, after passing through a threshold function. In a multi-layer or “deep learning” network, the inputs are combined in the second layer before being output. There are more weights and the added connections allow the network to learn and solve more complex problems.

## 数学代写|matlab代写|History of Deep Learning

Minsky wrote the book Perceptrons with Seymour Papert in 1969 , which was an early analysis of artificial neural networks. The book contributed to the movement toward symbolic processing in AI. The book noted that single-layer neurons could not implement some logical functions such as exclusive or (XOR) and implied that multi-layer networks would have the same issue. It was later found that three-layer networks could implement such functions. We give the XOR solution in this book.

Multi-layer neural networks were discovered in the 1960 s but not studied until the 1980 s. In the 1970 s, self-organizing maps using competitive learning were introduced [15]. A resurgence in neural networks happened in the 1980s. Knowledge-based, or “expert,” systems were also introduced in the 1980s. From Jackson [18]
An expert system is a computer program that represents and reasons with knowledge of some specialized subject to solve problems or give advice.
-Peter Jackson, Introduction to Expert Systems
Backpropagation for neural networks, a learning method using gradient descent, was reinvented in the 1980 s leading to renewed progress in this field. Studies began with both human neural networks (i.e., the human brain) and the creation of algorithms for effective computational neural networks. This eventually led to deep learning networks in machine learning applications.

Advances were made in the 1980 s as AI researchers began to apply rigorous mathematical and statistical analysis to develop algorithms. Hidden Markov Models were applied to speech. A Hidden Markov Model is a model with unobserved (i.e., hidden) states. Combined with massive databases, they have resulted in vastly more robust speech recognition. Machine translation has also improved. Data mining, the first form of machine learning as it is known today, was developed.

In the early 1990s, Vladimir Vapnik and coworkers invented a computationally powerful class of supervised learning networks known as support-vector machines (SVM). These networks could solve problems of pattern recognition, regression, and other machine learning problems.

## 数学代写|matlab代写|History of Deep Learning

Minsky 于 1969 年与 Seymour Papert 合着了《感知器》一书，这是对人工神经网络的早期分析。这本书推动了 AI 中符号处理的发展。该书指出，单层神经元无法实现一些逻辑功能，例如异或（XOR），并暗示多层网络也会有同样的问题。后来发现三层网络可以实现这样的功能。我们在本书中给出了 XOR 的解决方案。

-Peter Jackson，专家系统简介

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|偏微分方程代写partial difference equations代考|MATH4310

statistics-lab™ 为您的留学生涯保驾护航 在代写偏微分方程partial difference equations方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写偏微分方程partial difference equations代写方面经验极为丰富，各种代写偏微分方程partial difference equations相关的作业也就用不着说。

• 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|偏微分方程代写partial difference equations代考|Second Order Differential Equations

A second order differential equation in two variables $x$ and $y$ is given by
$$F\left(x, y ; u ; p, q ; u_{x x}, u_{x y}, u_{y y}\right)=0, \quad \text { for } z=u(x, y) \in C^2(\Omega),$$
where function $F$ is sufficiently smooth with respect to all involved variables, and $F_p^2+F_q^2 \not \equiv 0$ over $\Omega$. In particular, a second order quasilinear equation is given by
$$a u_{x x}+b u_{x y}+c u_{y y}+F_1\left(x, y ; u ; u_x, u_y\right)=0,$$
where the coefficients $a, b, c$ are functions of the independent variables $x, y$, and also of the dependent variable $z=u(x, y)$. As said earlier, (5.1.26) is a semilinear equation when functions $a, b, c$ depend on variables $x$ and $y$ only. Also, a general second order linear equation for a function $u \in C^2(\Omega)$ is given by
$$a u_{x x}+b u_{x y}+c u_{y y}+d u_x+e u_y+f u+g=0,$$
where the coefficients $a, \ldots, g$ are functions of the independent variables $x$ and $y$ only. As in the case of a first order differential equation in two variables, we say (5.1.27) is a homogeneous equation if the $g \equiv 0$. Otherwise, it is called a nonhomogeneous equation.

The next two examples illustrate that the second order differential equations of simpler linearity types arise naturally in mathematics, and also in practical situations. The main idea is to eliminate all parameters from the given functional relation. For convenience, we may write the second order partial derivatives of a $C^2$-function $u=u(x, y)$ as
$$r=u_{x x}=\frac{\partial^2 u}{\partial x^2}, \quad s=u_{x y}=\frac{\partial^2 u}{\partial x \partial y}, \quad t=u_{y y}=\frac{\partial^2 u}{\partial y^2} .$$

## 数学代写|偏微分方程代写partial difference equations代考|Classification and Canonical Forms

Let $\Omega \subseteq \mathbb{R}^2$ be an open set, and consider the general second order linear differential equation for a function $u \in C^2(\Omega)$ given by
$$a u_{x x}+2 b u_{x y}+c u_{y y}+F_1(x, y ; u ; p, q)=0, \quad \text { for } z=u(x, y),$$
where the coefficients $a, b, c \in C^2(\Omega)$ are such that the condition $a^2+b^2+c^2 \not \equiv 0$ holds over $\Omega$. In this section, our main concern is the principle part given by
$$a u_{x x}+2 b u_{x y}+c u_{y y},$$
because only it participates in the classification procedure described below. When the coefficients $a, b, c$ are constants, the geometry type of Eq. (5.2.1) remains uniform over a domain $\Omega$. However, in the general case, the equation may be of different types across various regions of $\Omega$. We will study Eq. (5.2.1) over a domain $\Omega_1 \subseteq \Omega$ such that the discriminant given by
$$D:=b^2-a c$$
has the same sign at each point of $\Omega_1$. We show that, for $\left(x_0, y_0\right) \in \Omega_1$, there exists a neighbourhood $U_0$ of the point $\left(x_0, y_0\right)$ and sufficiently smooth functions $\varphi, \phi$ such that the transformation $(x, y) \mapsto(\xi, \eta)$ given by
$$\xi=\varphi(x, y) \quad \text { and } \quad \eta=\phi(x, y),$$
changes Eq. (5.2.1) to a differential equation that has one of the three geometry types ${ }^3$ such as given below:

1. A hyperbolic type such as the wave equation (5.1.38).
2. A parabolic type such as the heat equation (5.1.40).
3. An elliptic type such as the Laplace equation (5.1.42).

# 偏微分方程代写

## 数学代写|偏微分方程代写partial difference equations代考|Second Order Differential Equations

$$F\left(x, y ; u ; p, q ; u_{x x}, u_{x y}, u_{y y}\right)=0, \quad \text { for } z=u(x, y) \in C^2(\Omega),$$

$$a u_{x x}+b u_{x y}+c u_{y y}+F_1\left(x, y ; u ; u_x, u_y\right)=0,$$

$$a u_{x x}+b u_{x y}+c u_{y y}+d u_x+e u_y+f u+g=0,$$

$$r=u_{x x}=\frac{\partial^2 u}{\partial x^2}, \quad s=u_{x y}=\frac{\partial^2 u}{\partial x \partial y}, \quad t=u_{y y}=\frac{\partial^2 u}{\partial y^2} .$$

## 数学代写|偏微分方程代写partial difference equations代考|Classification and Canonical Forms

$$a u_{x x}+2 b u_{x y}+c u_{y y}+F_1(x, y ; u ; p, q)=0, \quad \text { for } z=u(x, y),$$

$$a u_{x x}+2 b u_{x y}+c u_{y y},$$

$$D:=b^2-a c$$

$$\xi=\varphi(x, y) \quad \text { and } \quad \eta=\phi(x, y),$$

1. 双曲线类型如波动方程 (5.1.38)。
2. 抛物线类型，例如热方程 (5.1.40)。
3. 椭圆类型，例如拉普拉斯方程 (5.1.42)。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。