## 数学代写|数理逻辑代写Mathematical logic代考|MHF5306

statistics-lab™ 为您的留学生涯保驾护航 在代写数理逻辑Mathematical logic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。我们的专家在代写数理逻辑Mathematical logic代写方面经验极为丰富，各种代写数理逻辑Mathematical logic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|数理逻辑代写Mathematical logic代考|Terms and Formulas in First-Order Languages

Given a symbol set $S$, we call certain strings over $\mathbb{A}S$ formulas of the first-order language determined by $S$. For example, if $S=S{G r}$, we want the strings
$$e \equiv e, \quad e \circ v_1 \equiv v_2, \quad \exists v_1\left(e \equiv e \wedge v_1 \equiv v_2\right)$$
to be formulas, but not
$$\equiv \wedge e, \quad e \vee e$$

The formulas $e \equiv e$ and $e \circ v_1 \equiv v_2$ have the form of equations. Mathematicians call the strings to the left and to the right of the equality symbol terms. Terms are “meaningful” combinations of function symbols, variables, and constants (together with commas and parentheses). Clearly, to give a precise definition of formulas and thus, in particular, of equations, we must first specify more exactly what we mean by terms.

In mathematics, terms are written in different notation, such as $f(x), f x, x+e$, $g(x, e), g x e$. We choose a parenthesis-free notation, as with $f x$ and $g x e$.

To define the notion of term we give instructions (or rules) which tell us how to generate the terms. (Such a system of rules is often called a calculus.)
3.1 Definition. S-terms are precisely those strings in $\mathbb{A}_S^*$ which can be obtained by finitely many applications of the following rules:
(T1) Every variable is an $S$-term.
(T2) Every constant in $S$ is an $S$-term.
(T3) If the strings $t_1, \ldots, t_n$ are $S$-terms and $f$ is an $n$-ary function symbol in $S$, then $f t_1 \ldots t_n$ is also an $S$-term.
We denote the set of $S$-terms by $T^S$.

## 数学代写|数理逻辑代写Mathematical logic代考|Induction in the Calculi of Terms and of Formulas

Let $S$ be a set of symbols and let $Z \subseteq \mathbb{A}_S^*$ be a set of strings over $\mathbb{A}_S$. In the case where $Z=T^S$ or $Z=L^S$ we described the elements of $Z$ by means of a calculus. Each rule of such a calculus either says that certain strings belong to $Z$ (e.g., the rules (T1), (T2), (F1), and (F2)), or else permits the passage from certain strings $\zeta_1, \ldots, \zeta_n$ to a new string $\zeta$ in the sense that, if $\zeta_1, \ldots, \zeta_n$ all belong to $Z$, then $\zeta$ also belongs to $Z$. The way such rules work is made clear when we write them schematically, as follows:

By allowing $n=0$, the first sort of rules mentioned above (“premise-free” rules) is included in this scheme. Now we can write the rules for the calculus of terms as follows:
(T1) $\frac{}{x}$;
(T2) $\frac{}{c}$ if $c \in S$
(T3) $\frac{t_1, \ldots, t_n}{f t_1 \ldots t_n}$ if $f \in S$ and $f$ is $n$-ary.
When we define a set $Z$ of strings by means of a calculus $\mathcal{E}$ we can then prove assertions about elements of $Z$ by means of induction over $\mathfrak{C}$. This principle of proof corresponds to induction over the natural numbers. If one wants to show that all elements of $Z$ have a certain property $P$, then it is sufficient to show that

Hence in the case $n=0$ we must show that $\zeta$ has the property $P$.
This principle of proof is evident: In order to show that all strings derivable in $\mathfrak{C}$ have the property $P$, we show that everything derivable by means of a “premisefree” rule (i.e., $n=0$ in (I)) has the property $P$, and that $P$ is preserved under the application of the remaining rules. This method can also be justified using the principle of complete induction for natural numbers. For this purpose, one defines, in an obvious way, the length of a derivation in $\mathfrak{C}$ (cf. the examples of derivations in Section 3), and then argues as follows: If the condition (I) is satisfied for $P$, one shows by induction on $m$ that every string which has a derivation of length $m$ has the property $P$. Since every element of $Z$ has a derivation of some finite length, $P$ must hold for all elements of $Z$.

# 数理逻辑代写

## 数学代写|数理逻辑代写Mathematical logic代考|Terms and Formulas in First-Order Languages

$$e \equiv e, \quad e \circ v_1 \equiv v_2, \quad \exists v_1\left(e \equiv e \wedge v_1 \equiv v_2\right)$$

$$\equiv \wedge e, \quad e \vee e$$

$3.1$ 定义。S-terms 正是那些字符串 $\mathbb{A}_S^*$ 可以通过以下规则的有限多次应用获得：
(T1) 每个变量都是一个 $S$-学期。
(T2) 中的每个常量 $S$ 是一个 $S$-学期。
(T3) 如果字符串 $t_1, \ldots, t_n$ 是 $S$-条款和 $f$ 是一个 $n$ – 中的二进制函数符号 $S$ ，然后 $f t_1 \ldots t_n$ 也是一个 $S$-学 期。

## 数学代写|数理逻辑代写Mathematical logic代考|Induction in the Calculi of Terms and of Formulas

(T1) $\bar{x}$;
$(\mathrm{T} 2)-\frac{x}{c}$ 如果 $c \in S$
(T3) $\frac{t_1, \ldots, t_n}{f t_1 \ldots t_n}$ 如果 $f \in S$ 和 $f$ 是 $n$ – 阿里。

## 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|数理逻辑代写Mathematical logic代考|MATH318

statistics-lab™ 为您的留学生涯保驾护航 在代写数理逻辑Mathematical logic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。我们的专家在代写数理逻辑Mathematical logic代写方面经验极为丰富，各种代写数理逻辑Mathematical logic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|数理逻辑代写Mathematical logic代考|A Preliminary Analysis

We now sketch some aspects which the two examples just given have in common.
In each case one starts from a system $\Phi$ of propositions which is taken to he a system of axioms for the theory in question (group theory, theory of equivalence relations). The mathematician is interested in finding the propositions which follow from $\Phi$, where the proposition $\psi$ is said to follow from $\Phi$ if $\psi$ holds in every structure which satisfies all propositions in $\Phi$. A proof of $\psi$ from a system $\Phi$ of axioms shows that $\psi$ follows from $\Phi$.

When we think about the scope of methods of mathematical proof, we are led to ask about the converse:
(*) Is every proposition $\psi$ which follows from $\Phi$ also provable from $\Phi$ ?
For example, is every proposition which holds in all groups also provable from the group axioms (G1), (G2), and (G3)?

The material developed in Chapters II through V and in Chapter VII yields an essentially positive answer to (). Clearly it is necessary to make the concepts “proposition”, “follows from”, and “provable”, which occur in (), more precise. We sketch briefly how we shall do this.
(1) The Concept “Proposition.” Usually mathematicians use their everyday language (e.g., English or German) to formulate their propositions. But since sentences in everyday language are not, in general, completely unambiguous in their meaning and structure, one cannot specify them by precise definitions. For this reason we shall introduce a formal language $L$ which reflects features of mathematical statements. Like programming languages used today, $L$ will be formed according to fixed rules: Starting with a set of symbols (an “alphabet”), we obtain so-called formulas as finite symbol strings built up in a standard way. These formulas correspond to propositions expressed in everyday language. For example, the symbols of $L$ will include $\forall$ (to be read “for all”), $\wedge$ (“and”), $\rightarrow$ (“if … then”), $\equiv($ “equal”) and variables like $x, y$ and $z$. Formulas of $L$ will be expressions like
$$\forall x x \equiv x, \quad x \equiv y, \quad x \equiv z, \quad \forall x \forall y \forall z((x \equiv y \wedge y \equiv z) \rightarrow x \equiv z) .$$

## 数学代写|数理逻辑代写Mathematical logic代考|The Alphabet of a First-Order Language

We wish to construct formal languages in which we can formulate, for example, the axioms, theorems, and proofs about groups and equivalence relations which we considered in Chapter I. In that context the connectives, the quantifiers, and the equality relation played an important role. Therefore, we shall include the following symbols in the first-order languages: $\neg$ (for “not”), $\wedge$ (for “and”), $\vee$ (for “or”), $\rightarrow$ (for “ifthen”), $\leftrightarrow$ (for “if and only if”), $\forall$ (for “for all”), $\exists$ (for “there exists”), 三 (as symbol for equality). To these we shall add variables (for elements of groups, elements of equivalence structures, etc.) and, finally, parentheses as auxiliary symbols.

To formulate the axioms for groups we also need certain symbols specific to group theory, e.g., a binary function symbol, say $\circ$, to denote the group multiplication, and a symbol, say $e$, to denote the identity element. We call $e$ a constant symbol, or simply a constant. For the axioms of the theory of equivalence relations we need a binary relation symbol, say $R$.

Thus, in addition to the “logical” symbols such as ” $\neg$ ” and ” $\wedge$ “, we need a set $S$ of relation symbols, function symbols, and constants which varies from theory to theory. Each such set $S$ of symbols determines a first-order language. We summarize:

By $\mathbb{A}$ we denote the set of symbols listed in (a) through (e). Let $S$ be the (possibly empty) set of symbols from (f). The symbols listed under (f) must, of course, be distinct from each other and from the symbols in $\mathbb{A}$.

The set $S$ determines a first-order language (cf. Section 3). We call $\mathbb{A}_S:=\mathbb{A} \cup S$ the alphabet of this language and $S$ its symbol set.

We have already become acquainted with some symbol sets: $S_{\mathrm{gr}}:={0, e}$ for group theory and $S_{\mathrm{eq}}:={R}$ for the theory of equivalence relations. For the theory of ordered groups we could use ${0, e, R}$, where the binary relation symbol $R$ is now taken to represent the ordering relation. In certain theoretical investigations we shall use the symbol set $S_{\infty}$, which contains the constants $c_0, c_1, c_2, \ldots$, and for every $n \geq 1$ countably many $n$-ary relation symbols $R_0^n, R_1^n, R_2^n, \ldots$ and $n$-ary function symbols $f_0^n, f_1^n, f_2^n, \ldots$

Henceforth we shall use the letters $P, Q, R, \ldots$ for relation symbols, $f, g, h, \ldots$ for function symbols, $c, c_0, c_1, \ldots$ for constants, and $x, y, z, \ldots$ for variables.

# 数理逻辑代写

## 数学代写|数理逻辑代写Mathematical logic代考|A Preliminary Analysis

$\left(^*\right)$ 是否每个合题 $\psi$ 从 $\Phi$ 也可以证明 $\Phi$ ?

(1) 概念”命题”。通常数学家使用他们的日常语言（例如英语或德语) 来表达他们的命题。但是，由于日 常语言中的句子通常在含义和结构上并非完全没有歧义，因此无法通过精确的定义来指定它们。为此，我 们将引入一种形式语言 $L$ 反映了数学陈述的特点。就像今天使用的编程语言一样， $L$ 将根据固定规则形 成: 从一组符号 (“字母表”) 开始，我们获得所谓的公式，作为以标准方式构建的有限符号串。这些公式 对应于用日常语言表达的命题。例如，符号 $L$ 会包括 $\forall$ (读作”为所有人”)， $\wedge($ “和”)， $\rightarrow($ “如果……那 么”)，三(“等于”) 和变量，如 $x, y$ 和 $z$. 的公式 $L$ 会像这样的表达
$$\forall x x \equiv x, \quad x \equiv y, \quad x \equiv z, \quad \forall x \forall y \forall z((x \equiv y \wedge y \equiv z) \rightarrow x \equiv z)$$

## 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|数理逻辑代写Mathematical logic代考|MATH4810

statistics-lab™ 为您的留学生涯保驾护航 在代写数理逻辑Mathematical logic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。我们的专家在代写数理逻辑Mathematical logic代写方面经验极为丰富，各种代写数理逻辑Mathematical logic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|数理逻辑代写Mathematical logic代考|An Example from Group Theory

In this and the next section we present two simple mathematical proofs. They illustrate some of the methods of proof used by mathematicians. Guided by these examples, we raise some questions which lead us to the main topics of the book.
We begin with the proof of a theorem from group theory. We therefore require the axioms of group theory, which we now state. We use o to denote the group multiplication and $e$ to denote the identity element. The axioms may then be formulated as follows:
(G1) For all $x, y, z: \quad(x \circ y) \circ z=x \circ(y \circ z)$.
(G2) For all $x: \quad x \circ e=x$.
(G3) For every $x$ there is a $y$ such that $x \circ y=e$.
A group is a triple $\left(G, \circ^G, e^G\right)$ which satisfies (G1)-(G3). Here $G$ is a set, $e^G$ is an element of $G$, and $\circ^G$ is a binary function on $G$, i.e., a function defined on all ordered pairs of elements from $G$, the values of which are also elements of $G$. The variables $x, y, z$ range over elements of $G, \circ$ refers to $\circ^G$, and $e$ refers to $e^G$.

As an example of a group we mention the additive group of the reals $(\mathbb{R},+, 0)$, where $\mathbb{R}$ is the set of real numbers, $+$ is the usual addition, and 0 is the real number zero. On the other hand, $(\mathbb{R}, \cdot, 1)$ is not a group (where – is the usual multiplication). For example, the real number 0 violates axiom (G3): there is no real number $r$ such that $0 \cdot r=1$.

We call triples such as $(\mathbb{R},+, 0)$ or $(\mathbb{R}, \cdot, 1)$ structures. In Chapter III we shall give an exact definition of the notion of “structure.”
Now we prove the following simple theorem from group theory:
1.1 Theorem on the Existence of a Left Inverse. For every $x$ there is a $y$ such that $y \circ x=e$.

## 数学代写|数理逻辑代写Mathematical logic代考|An Example from the Theory of Equivalence Relations

The thenry of equivalence relations is hased on the following three axions ( $x k y$ is to be read as ” $x$ is equivalent to $y$ “);
(E1) For all $x: x R x$.
(E2) For all $x, y$ : If $x R y$, then $y R x$.
(E3) For all $x, y, z$ : If $x R y$ and $y R z$, then $x R z$.
Let $A$ be a nonempty set, and let $R^A$ be a binary relation on $A$, i.e., $R^A \subseteq A \times A$. For $(a, b) \in R^A$ we also write $a R^A b$. The pair $\left(A, R^A\right)$ is another example of a structure. We call $R^A$ an equivalence relation on $A$, and the structure $\left(A, R^A\right)$ an equivalence structure, if (E1), (E2), and (E3) are satisfied. For example, $\left(\mathbb{Z}, R_5\right)$ is an equivalence structure, where $\mathbb{Z}$ is the set of integers and
$$R_5={(a, b) \mid a, b \in \mathbb{Z} \text { and } b-a \text { is divisible by } 5} .$$
We now prove a simple theorem about equivalence relations.

2.1 Theorem. If $x$ and $y$ are both equivalent to a third element, they are equivalent to the same elements. More formally: For all $x$ and $y$, if there is a $u$ such that $x R u$ and $y R u$, then for all $z, x R z$ if and only if $y R z$.
Proof. Let $x$ and $y$ be given arbitrarily; suppose that for some $u$ $x R u$ and $y R u$.
From (E2) we then obtain $u R x$ and $u R y$.
From $x R u$ and $u R y$ we get, using (E3),
$$x R y,$$
and from $y R u$ and $u R x$ we likewise get (using (E3))
$$y R x .$$
Now let $z$ be chosen arbitrarily. If
$$x R z \text {, }$$
then, using (E3), we obtain from (4) and (5)
$$y R z .$$
On the other hand, if
$$y R z \text {, }$$
then, using (E3), we get from (3) and (6)
$$x R z \text {. }$$
Thus the claim is proved for all $z$.
As in the previous example, this proof shows that every structure (of the form $\left(A, R^A\right)$ ) which satisfies the axioms (E1), (E2), and (E3), also satisfies Theorem 2.1, i.e., that Theorem $2.1$ follows from (E1), (E2), and (E3).

# 数理逻辑代写

## 数学代写|数理逻辑代写Mathematical logic代考|An Example from Group Theory

(G1) 对于所有 $x, y, z:(x \circ y) \circ z=x \circ(y \circ z)$.
(G) 对所有人 $x: \quad x \circ e=x$.
(G3) 对于每个 $x$ 有一个 $y$ 这样 $x \circ y=e$.

$1.1$ 左逆的存在性定理。对于每一个 $x$ 有一个 $y$ 这样 $y \circ x=e$.

## 数学代写|数理逻辑代写Mathematical logic代考|An Example from the Theory of Equivalence Relations

(E1) 对于所有人 $x: x R x$.
(E2) 对于所有人 $x, y$ : 如果 $x R y$ ，然后 $y R x$.
(E3) 对于所有人 $x, y, z$ : 如果 $x R y$ 和 $y R z$ ， 然后 $x R z$.

. 这对 $\left(A, R^A\right)$ 是结构的另一个例子。我们称之为 $R^A$ 上的等价关系 $A$, 和结构 $\left(A, R^A\right)$ 如果满足 (E1)、
(E2) 和 (E3)，则为等价结构。例如， $\left(\mathbb{Z}, R_5\right)$ 是一个等价结构，其中 $\mathbb{Z}$ 是整数集，并且
$R_5=(a, b) \mid a, b \in \mathbb{Z}$ and $b-a$ is divisible by 5.

$2.1$ 定理。如果 $x$ 和 $y$ 都等价于第三个元素，它们等价于相同的元素。更正式地说：对于所有人 $x$ 和 $y$ ，如果 有 $u$ 这样 $x R u$ 和 $y R u$ ，那么对于所有 $z, x R z$ 当且仅当 $y R z$.

$x R y$,

$y R x$

$$x R z$$

$$y R z$$

$$y R z$$

$$x R z$$

## 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|傅里叶分析代写Fourier analysis代考|AMTH246

statistics-lab™ 为您的留学生涯保驾护航 在代写傅里叶分析Fourier analysis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。我们的专家在代写傅里叶分析Fourier analysis代写方面经验极为丰富，各种代写傅里叶分析Fourier analysis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|傅里叶分析代写Fourier analysis代考|Continuous, Discrete, and Digital Signals

This type of classification characterizes the type of sampling of the dependent and independent variables. Sampling the amplitude is called quantization. Table $1.1$ shows the signal classification based on sampling the amplitude and time. When both the variables of a signal can assume continuum of values, it is called a continuous signal, such as the ambient temperature. Most of the naturally occurring signals are of this type. The temperature measured by a digital thermometer is a quantized continuous signal. This type of signals occurs in the reconstruction of a continuous signal from its sampled version. Sampled continuous-valued signal is a discrete signal. This type of signals, shown in Fig. 1.4c, d, is used in the analysis of discrete signals and systems. A quantized discrete signal is called a digital signal, used in the digital systems.

The sinusoidal signals are defined by the values of the coordinates on a circle in Fig. 1.3. In each rotation of a point on the circle, the same set of values are produced indefinitely. This type of signals, such as the sine and cosine functions, is periodic signals. While only one period of a periodic signal contains new information, periodicity is required to represent signals such as power and communication signals. In communication engineering, the message signal is aperiodic and the carrier signal is periodic. Finite duration signals are represented, by the practically most often used version of the Fourier analysis, assuming periodic extension. The finite signal is considered as the values of one period and concatenation of it indefinitely on either side yields a periodic signal. A signal $x(t)$ is said to be periodic, if $x(t)=x(t+T)$, for all values of $t$ from $-\infty$ to $\infty$ and $T>0$ is a positive constant. The minimum value of $T$ that satisfies the constraint is the period. A periodic signal shifted by an integral number of its period remains unchanged. A signal that is not periodic is aperiodic, such as the impulse, step and ramp signals shown in Fig. 1.1 and the real exponential. The period is infinity, so that there is no indefinite repetition. The everlasting definition of a periodic signal is for mathematical convenience. In practice, physical devices are switched on at some time and the response reaches a steady state, after the transient response dies down.

## 数学代写|傅里叶分析代写Fourier analysis代考|Even- and Odd-Symmetric Signals

Any signal can be decomposed into its even and odd components. Knowing whether a signal is even or odd may reduce computational and storage requirements in its processing. If a signal $x(t)$ satisfies the condition
$$x(-t)=x(t) \text { for all } t$$ then it is said to be even. The plot of such a signal is symmetrical about the vertical axis at the origin. For example, the cosine waveforms, shown in Figs. 1.4a and 1.6b, are even. For the signal in Fig. 1.6b,
$$0.5 \cos \left(\frac{2 \pi}{32}(-n)\right)=0.5 \cos \left(\frac{2 \pi}{32} n\right)$$
If a signal $x(t)$ satisfies the condition
$$x(-t)=-x(t) \text { for all } t,$$
then it is said to be odd. The plot of such a signal is antisymmetrical about the vertical axis at the origin. For example, the sine waveforms, shown in Figs. 1.4b and 1.6c, are odd. For the signal in Fig. 1.6c,
$$\frac{\sqrt{3}}{2} \sin \left(\frac{2 \pi}{32}(-n)\right)=-\frac{\sqrt{3}}{2} \sin \left(\frac{2 \pi}{32} n\right)$$
Any function can be decomposed into its even and components. Let the even and odd components of $x(t)$ be $x_e(t)$ and $x_o(t)$, respectively. Then,
$$x(t)=x_e(t)+x_o(t) \text { and } x(-t)=x_e(t)-x_o(t)$$

# 傅里叶分析代写

## 数学代写|傅里叶分析代写Fourier analysis代考|Even- and Odd-Symmetric Signals

$$x(-t)=x(t) \text { for all } t$$

$$0.5 \cos \left(\frac{2 \pi}{32}(-n)\right)=0.5 \cos \left(\frac{2 \pi}{32} n\right)$$

$$x(-t)=-x(t) \text { for all } t,$$

$$\frac{\sqrt{3}}{2} \sin \left(\frac{2 \pi}{32}(-n)\right)=-\frac{\sqrt{3}}{2} \sin \left(\frac{2 \pi}{32} n\right)$$

$$x(t)=x_e(t)+x_o(t) \text { and } x(-t)=x_e(t)-x_o(t)$$

## 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|傅里叶分析代写Fourier analysis代考|MAT3105

statistics-lab™ 为您的留学生涯保驾护航 在代写傅里叶分析Fourier analysis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。我们的专家在代写傅里叶分析Fourier analysis代写方面经验极为丰富，各种代写傅里叶分析Fourier analysis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|傅里叶分析代写Fourier analysis代考|Sinusoids and Complex Exponentials

The impulse and the sinusoid are the two most important signals in signal and system analysis. The impulse is the basis for convolution and the sinusoid is the basis for transfer function. The cosine and sine functions are two of the most important functions in trigonometry. As these functions are the basis functions in Fourier analysis, we have study them in detail.

The unit circle, defined by $x^2+y^2=1$ and shown in Fig. 1.3, is a circle with its center located at the origin and radius 1 . For each point on the circle defined by the coordinates $(x, y)$, starting at $(1,0)$ and moving in the counterclockwise direction, with $\theta \geq 0$ (the angle subtended by the $x$-axis and the line joining the point and the origin), the sine (sin) and cosine (cos) functions are defined in terms of its coordinates $(x, y)$ as
$$\cos (\theta)=x \quad \text { and } \quad \sin (\theta)=y$$
If the point lies on a circle of radius $r$, then
$$\cos (\theta)=x / r \text { and } \sin (\theta)=y / r, \quad r=\sqrt{x^2+y^2}$$
Clearly, the sinusoids are of periodic nature. Any function defined on a circle will be a periodic function of an angular variable $\theta$. Therefore, the trigonometric functions are also called circular functions. The argument $\theta$ is measured in radians or degrees. The radian is defined as the angle subtended between the $x$-axis and the line between the point and the origin on the unit circle. One radian is defined as the angle subtended by unit arc length. Since the circumference of the unit circle is $2 \pi$, one complete revolution is $2 \pi \mathrm{rad}$. In degree measure, $2 \pi=360^{\circ}$ and $\pi=180^{\circ}$. One radian is approximately $180 / \pi=57.3^{\circ}$.

A linear combination of sine and cosine functions is a sinusoid, in rectangular form, given by
$$a \cos (\theta)+b \sin (\theta)$$
where $a$ and $b$ are real numbers with $a \neq 0$ or $b \neq 0$. With $c=\sqrt{a^2+b^2}$, and $\cos (d)=a / c$ and $\sin (d)=b / c$,
$$a \cos (\theta)+b \sin (\theta)=c \cos (\theta-d)$$
is called the polar form of the sinusoid.

## 数学代写|傅里叶分析代写Fourier analysis代考|Exponential Signal

By using sine and cosine functions, signals can be represented. But it involves two basic functions and the two associated constants. It is found that an equivalent representation of signals is obtained using the complex exponential function, in which only one basic function and one associated constant is involved. The compact representation and the ease of manipulating the exponential functions make its use mandatory in the analysis of signals and systems. However, practical devices generate sine and cosine functions. Euler’s formula is the bridge between the theory and the practice. With $b$ any positive real number except 1 ,
$$x(t)=b^t$$
is called the exponential function with base $b$. Our primary interest, in this book, is the complex exponential function of the form
$$x(\theta)=A e^{j \theta}$$
The base is $e$, which is approximately $2.71828$. The exponent is a complex number with its real part zero (pure imaginary number). The coefficient of the exponential $A$ is a complex number.

The exponential $e^{j \theta}$, shown in Fig. 1.5, is a unit rotating vector, rotating in the counterclockwise direction. The exponential carries the same information about a sinusoid in an equivalent form, which is advantageous in the analysis of signals and systems. In combination with the exponential $e^{-j \theta}$, which rotates in the clockwise direction, a real sinusoidal waveform can be obtained. Since
$$e^{j \theta}=\cos (\theta)+j \sin (\theta) \text { and } e^{-j \theta}=\cos (\theta)-j \sin (\theta),$$
solving for $\cos (\theta)$ and $\sin (\theta)$ results in
$$\cos (\theta)=\frac{e^{j \theta}+e^{-j \theta}}{2} \text { and } \sin (\theta)=\frac{e^{j \theta}-e^{-j \theta}}{j 2}$$

# 傅里叶分析代写

## 数学代写|傅里叶分析代写Fourier analysis代考|Sinusoids and Complex Exponentials

$$\cos (\theta)=x \quad \text { and } \quad \sin (\theta)=y$$

$$\cos (\theta)=x / r \text { and } \sin (\theta)=y / r, \quad r=\sqrt{x^2+y^2}$$

$$a \cos (\theta)+b \sin (\theta)$$

$$a \cos (\theta)+b \sin (\theta)=c \cos (\theta-d)$$

## 数学代写|傅里叶分析代写Fourier analysis代考|Exponential Signal

$$x(t)=b^t$$

$$x(\theta)=A e^{j \theta}$$

$$e^{j \theta}=\cos (\theta)+j \sin (\theta) \text { and } e^{-j \theta}=\cos (\theta)-j \sin (\theta),$$

$$\cos (\theta)=\frac{e^{j \theta}+e^{-j \theta}}{2} \text { and } \sin (\theta)=\frac{e^{j \theta}-e^{-j \theta}}{j 2}$$

## 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|傅里叶分析代写Fourier analysis代考|MAST20026

statistics-lab™ 为您的留学生涯保驾护航 在代写傅里叶分析Fourier analysis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。我们的专家在代写傅里叶分析Fourier analysis代写方面经验极为丰富，各种代写傅里叶分析Fourier analysis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|傅里叶分析代写Fourier analysis代考|Unit-Impulse Signal

The unit-impulse and the sinusoidal signals are the most important signals in the study of signals and systems. The continuous unit-impulse $\delta(t)$ is a signal with a shape and amplitude such that its integral at the point $t=0$ is unity. It is defined, in terms of an integral, as
$$\int_{-\infty}^{\infty} x(t) \delta(t) d t=x(0)$$
It is assumed that $x(t)$ is continuous at $t=0$ so that the value $x(0)$ is distinct. The product of $x(t)$ and $\delta(t)$ is
$$x(t) \delta(t)=x(0) \delta(t)$$
since the impulse exists only at $t=0$. Therefore,
$$\int_{-\infty}^{\infty} x(t) \delta(t) d t=x(0) \int_{-\infty}^{\infty} \delta(t) d t=x(0)$$
The value of the function $x(t)$, at $t=0$, is sifted out or sampled by the defining operation. By using shifted impulses, any value of $x(t)$ can be sifted.

It is obvious that the integral of the unit-impulse is the unit-step. Therefore, the derivative of the unit-step signal is the unit-impulse signal. The value of the unit-step is zero for $t<0$ and 1 for $t>0$. Therefore, the unit area of the unit-impulse, as the derivative of the unit-step, must occur at $t=0$. The unit-impulse and the unitstep signals enable us to represent and analyze signals with discontinuities as we do with continuous signals. For example, these signals model the commonly occurring situations such as opening and closing of switches.

The continuous unit-impulse $\delta(t)$ is difficult to visualize and impossible to realize in practice. However, the approximation of it by some functions is effective in practice and can be used to visualize its effect on signals and its properties. While there are other functions that approach an impulse in the limit, the rectangular function is often used to approximate the impulse. The unit-impulse, for all practical purposes, is essentially a narrow rectangular pulse with unit area. Suppose we compress it by a factor of 2 , the area, called its strength, becomes $1 / 2=0.5$. The scaling property of the impulse is given as
$$\delta(a t)=\frac{1}{|a|} \delta(t), a \neq 0$$
With $a=-1, \delta(-t)=\delta(t)$ implying that the impulse is an even-symmetric signal. For example,
$$\delta(3 t-1)=\delta\left(3\left(t-\frac{1}{3}\right)\right)=\frac{1}{3} \delta\left(t-\frac{1}{3}\right)$$
The discrete unit-impulse signal, shown in Fig. 1.1a, is defined as
$$\delta(n)=\left{\begin{array}{l} 1 \text { for } n=0 \ 0 \text { for } n \neq 0 \end{array}\right.$$

## 数学代写|傅里叶分析代写Fourier analysis代考|Unit-Step Signal

The discrete unit-step signal, shown in Fig. 1.1b, is defined as
$$u(n)=\left{\begin{array}{l} 1 \text { for } n \geq 0 \ 0 \text { for } n<0 \end{array}\right.$$ For positive values of its argument, the value of the unit-step signal is unity and it is zero otherwise. An arbitrary function can be expressed in terms of appropriately scaled and shifted unit-step or impulse signals. By this way, any signal can be specified, for easier mathematical analysis, by a single expression, valid for all $n$. For example, a pulse signal, shown in Fig. 1.2a, with its only nonzero values defined as $\{x(1)=1, x(2)=1, x(3)=1\}$ can be expressed as the sum of the two delayed unitstep signals shown in Fig. 1.2b, $x(n)=u(n-1)-u(n-4)$. The pulse can also be represented as a sum of delayed impulses. $$x(n)=u(n-1)-u(n-4)=\sum_{k=1}^3 \delta(n-k)=\delta(n-1)+\delta(n-2)+\delta(n-3)$$ The continuous unit-step signal is defined as $$u(t)= \begin{cases}1 & \text { for } t>0 \ 0 & \text { for } t<0 \\ \text { undefined for } t=0\end{cases}$$ The value $u(0)$ is undefined and can be assigned a suitable value from 0 to 1 to suit a specific problem. In Fourier analysis, $u(0)=0.5$. A common application of the unit-step signal is that multiplying a signal with it yields the causal form of the signal. For example, the continuous signal $\sin (t)$ is defined for $-\infty0$.

The discrete unit-ramp signal, shown in Fig. 1.1c, is also often used in the analysis of signals and systems. It is defined as
$$r(n)=\left{\begin{array}{l} n \text { for } n \geq 0 \ 0 \text { for } n<0 \end{array}\right.$$
It linearly increases for positive values of its argument and is zero otherwise.
The three signals, the unit-impulse, the unit-step, and the unit-ramp, are related by the operations of sum and difference. The unit-impulse signal $\delta(n)$ is equal to $u(n)-u(n-1)$, the first difference of the unit-step. The unit-step signal $u(n)$ is equal to $\sum_{k=0}^{\infty} \delta(n-k)$, the running sum of the unit-impulse. The shifted unit-step signal $u(n-1)$ is equal to $r(n)-r(n-1)$. The unit-ramp signal $r(n)$ is equal to
$$r(n)=n u(n)=\sum_{k=0}^{\infty} k \delta(n-k) .$$
Similar relations hold for continuous type of signals.

# 傅里叶分析代写

## 数学代写|傅里叶分析代写Fourier analysis代考|Unit-Impulse Signal

$$\int_{-\infty}^{\infty} x(t) \delta(t) d t=x(0)$$

$$x(t) \delta(t)=x(0) \delta(t)$$

$$\int_{-\infty}^{\infty} x(t) \delta(t) d t=x(0) \int_{-\infty}^{\infty} \delta(t) d t=x(0)$$

$$\delta(a t)=\frac{1}{|a|} \delta(t), a \neq 0$$

$$\delta(3 t-1)=\delta\left(3\left(t-\frac{1}{3}\right)\right)=\frac{1}{3} \delta\left(t-\frac{1}{3}\right)$$

$\$ \$$Idelta(n)=Veft { 1 for n=00 for n \neq 0 正确的。 ## 数学代写|傅里叶分析代写Fourier analysis代考|Unit-Step Signal 如图 1.1b 所示，离散单位阶跃信号定义为 \ \$$
$\mathrm{u}(\mathrm{n})=\backslash \mathrm{left}{$
1 for $n \geq 00$ for $n<0$ 、正确的。 Forpositivevaluesofitsargument, thevalueoftheunit – stepsignalisunityanditiszer $x(n)=u(n-1)-u(n-4)=\backslash$ sum_ ${k=1}^{\wedge} 3$ \delta(nk)=ldelta(n-1)+ldelta(n-2)+ \三角洲 $(n-3)$ Thecontinuousunit – stepsignalisdefinedas $u(t)=$ $$\left{\begin{array}{l} 1 \ \text { undefined for } t=0 \end{array} \text { for } t>00 \quad \text { for } t<0\right.$$
$\$ \$$价值 u(0) 是末定义的，可以分配一个从 0 到 1 的合适值以适应特定问题。在傅立叶分析中， u(0)=0.5. 单位阶跃信号的一个常见应用是将一个信号与其相乘产生信号的因果形式。例如，连续信 号 \sin (t) 被定义为 -\infty 0. 离散单位斜坡信号，如图 1.1c 所示，也经常用于信号和系统的分析。它被定义为 \ \$$
$$r(n)=\backslash l \text { eft }{$$
$n$ for $n \geq 00$ for $n<0$

## 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|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。