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• Foundations of Data Science 数据科学基础

## 数学代写|交换代数代写commutative algebra代考|About the Equality and the Membership Tests

We now introduce several constructive notions relating to the equality test and the membership test.

A set $E$ is well defined when we have indicated how to construct its elements and when we have constructed an equivalence relation which defines the equality of two elements in a set. We denote by $x=y$ the equality in $E$, or $x={ }_{E} y$ if necessary. The set $E$ is called discrete when the following axiom holds
$$\forall x, y \in E \quad x=y \text { or } \neg(x=y)$$
Classically, every set is discrete, as the “or” present in the definition is understood in an abstract manner. Constructively, this same “or”‘ is understood according to the usual language’s meaning: at least one of the two alternatives must occur. It is thus an “or” of an algorithmic nature. In short, a set is discrete if we have a test for the equality of two arbitrary elements of this set.

If we want to be more precise and explain in detail what comprises an equality test in the set $E$, we will say that it is a construction which, from two given elements of $E$, provides a “yes” or “no” answer to the posed question (are these elements equal?). However, we could not go into much further detail. In constructive mathematics the notions of integers and of construction are basic concepts. They can be explained and commented on, but not strictly speaking “defined.” The constructive meaning of the “or” and that of the “there exists” are as such directly dependent of the notion of construction, ${ }^{6}$ which we do not attempt to define.
A discrete field is simply a ring where the following axiom is satisfied:
$$\forall x \in \mathbf{A} \quad x=0 \text { or } x \in \mathbf{A}^{\times}$$
The trivial ring is a discrete field.

## 数学代写|交换代数代写commutative algebra代考|Strongly Discrete Coherent Rings and Modules

A ring (resp. a module) is said to be strongly discrete when the finitely generated ideals (resp. the finitely generated submodules) are detachable, i.e. if the quotients by the finitely generated ideals (resp. by the finitely generated submodules) are discrete.
This means that we have a test for deciding whether a linear equation $L X=c$ has a solution or not, and by computing one in the affirmative case.

A key result in constructive algebra and Computer Algebra states that $\mathbb{Z}\left[X_{1}, \ldots, X_{n}\right]$ is a strongly discrete coherent ring.

More generally, we have the following constructive version of the Hilbert theorem (see [MRR, Adams \& Loustaunau]).

If $\mathbf{A}$ is a strongly discrete Noetherian coherent ring, so is any finitely presented A-algebra.
The following proposition is proven similarly to Proposition 3.1.
3.7 Proposition Over a strongly discrete coherent module $M$, every system of linear equations $B X=C\left(B \in M^{k \times n}, C \in M^{k \times 1}, X \in \mathbf{A}^{n \times 1}\right)$ can be tested. In the affirmative case, a particular solution $X_{0}$ can be computed. Furthermore the solutions $X$ are all the elements of $X_{0}+N$ where $N$ is a finitely generated $\mathbf{A}$-submodule of $\mathbf{A}^{n \times 1}$.

# 交换代数代考

## 数学代写|交换代数代写commutative algebra代考|About the Equality and the Membership Tests

$$\forall x, y \in E \quad x=y \text { or } \neg(x=y)$$

$$\forall x \in \mathbf{A} \quad x=0 \text { or } x \in \mathbf{A}^{\times}$$

## 数学代写|交换代数代写commutative algebra代考|Strongly Discrete Coherent Rings and Modules

3.7 关于强离散相干模块的命题米, 每个线性方程组乙X=C(乙∈米ķ×n,C∈米ķ×1,X∈一个n×1)可以测试。在肯定的情况下，一个特定的解决方案X0可以计算。此外解决方案X是所有的元素X0+ñ在哪里ñ是一个有限生成的一个-子模块一个n×1.

## 有限元方法代写

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## MATLAB代写

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

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