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

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|数理逻辑代写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

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。