generating set

简明释义

发电机组

英英释义

例句

1.Our facility requires a reliable generating set for emergency situations.

我们的设施需要一个可靠的发电机组以应对紧急情况。

2.The engineers designed a new generating set to provide backup power during outages.

工程师们设计了一套新的发电机组以在停电期间提供备用电力。

3.They installed a solar-powered generating set to reduce their carbon footprint.

他们安装了一个太阳能发电机组以减少碳足迹。

4.The generating set can produce enough electricity to power an entire neighborhood.

这个发电机组可以产生足够的电力来为整个社区供电。

5.The construction site was equipped with a portable generating set to power tools and equipment.

建筑工地配备了一个便携式发电机组来为工具和设备供电。

作文

In the field of mathematics and abstract algebra, the concept of a generating set is crucial for understanding the structure of various algebraic systems. A generating set can be defined as a subset of a given mathematical object such that every element of the object can be expressed as a combination of the elements in this subset. This idea is particularly significant in the study of groups, rings, and vector spaces. For instance, in group theory, a group can be generated by a specific set of elements, meaning that any element of the group can be written as a product of these generators and their inverses. To illustrate this concept further, consider the group of integers under addition, denoted as (Z, +). The integer 1 serves as a generating set for this group because any integer can be obtained by adding or subtracting 1 multiple times. Therefore, we can express any integer n as n = 1 + 1 + ... + 1 (n times) or n = -1 + -1 + ... + -1 (-n times). This simple example highlights how a single element can generate an entire group.In linear algebra, the notion of a generating set takes on a similar form. A vector space can have a generating set consisting of vectors that span the space. For example, in a two-dimensional space, any two non-collinear vectors can serve as a generating set. By taking linear combinations of these vectors, we can reach any point in that space. This property is vital in various applications, including computer graphics, where transformations are often represented using matrices derived from generating sets of vectors.The importance of generating sets extends beyond pure mathematics; it has practical implications in areas such as coding theory, cryptography, and network design. In coding theory, for instance, a generating set can determine the codes used for error detection and correction. The ability to generate different codewords from a generating set allows for efficient communication over unreliable channels.Moreover, in the context of graph theory, a generating set can refer to a set of vertices from which all other vertices can be reached through a series of edges. This concept is essential in network design, where ensuring connectivity among nodes is a primary concern. By identifying a suitable generating set, we can optimize network efficiency and reduce redundancy.In conclusion, the concept of a generating set is a fundamental aspect of various mathematical disciplines, providing a framework for constructing and analyzing complex structures. Whether in algebra, geometry, or applied fields, understanding how to identify and utilize generating sets can lead to deeper insights and more effective problem-solving strategies. As we continue to explore the vast landscape of mathematics, the role of generating sets will undoubtedly remain a cornerstone of our understanding, enabling us to connect disparate ideas and uncover new relationships within the mathematical universe.

在数学和抽象代数领域，生成集的概念对于理解各种代数系统的结构至关重要。生成集可以定义为给定数学对象的一个子集，使得该对象的每个元素都可以表示为该子集中元素的组合。这个想法在群、环和向量空间的研究中尤为重要。例如，在群论中，一个群可以由一组特定的元素生成，这意味着群中的任何元素都可以表示为这些生成元及其逆的乘积。为了进一步说明这个概念，考虑整数加法群，记作(Z, +)。整数1作为这个群的一个生成集，因为任何整数都可以通过多次加或减1来获得。因此，我们可以将任意整数n表示为n = 1 + 1 + ... + 1（n次）或n = -1 + -1 + ... + -1（-n次）。这个简单的例子突显了单个元素如何生成整个群。在线性代数中，生成集的概念具有类似的形式。一个向量空间可以有一个生成集，由能够覆盖该空间的向量组成。例如，在二维空间中，任何两个不共线的向量都可以作为生成集。通过对这些向量进行线性组合，我们可以到达该空间中的任何点。这个性质在各种应用中至关重要，包括计算机图形学，其中变换通常使用源自生成集的矩阵表示。生成集的重要性超越了纯数学，它在编码理论、密码学和网络设计等领域具有实际意义。在编码理论中，例如，生成集可以确定用于错误检测和纠正的代码。从生成集中生成不同的代码字的能力使得在不可靠通道上高效通信成为可能。此外，在图论的背景下，生成集可以指一个顶点集，从中可以通过一系列边到达所有其他顶点。这个概念在网络设计中至关重要，因为确保节点之间的连通性是主要关注点。通过识别合适的生成集，我们可以优化网络效率并减少冗余。总之，生成集的概念是各种数学学科的基本方面，为构建和分析复杂结构提供了框架。无论是在代数、几何还是应用领域，理解如何识别和利用生成集可以带来更深刻的见解和更有效的问题解决策略。随着我们继续探索数学的广阔领域，生成集的角色无疑将继续成为我们理解的基石，使我们能够连接不同的思想，并在数学宇宙中揭示新的关系。