cohomology

简明释义

英[/koʊˈhɑːmələdʒi/]美[/koʊˈhɑːmələdʒi/]

n. [数] 上同调

英英释义

单词用法

同义词

反义词

例句

1.Now, we obtain expectative results by apply cohomology to discussion of Analytical dynamics.

本文将上同调用于讨论动力学方程，得到了预期的结果。

2.The theory of homology and cohomology is very important in mathematics.

同调与上同调理论在数学中占有很重要的地位。

3.We give a sufficient condition for a finite dimensional symmetric self-dual Lie color algebras to be a double extension, thus we solve its classification in the sense of cohomology;

我们给出了有限维对称自对偶色李代数可以双扩张的充分条件，从而在上同调意义下解决了这类色李代数的分类问题；

4.We give a sufficient condition for a finite dimensional symmetric self-dual Lie color algebras to be a double extension, thus we solve its classification in the sense of cohomology;

我们给出了有限维对称自对偶色李代数可以双扩张的充分条件，从而在上同调意义下解决了这类色李代数的分类问题；

5.The language is purposefully kept on an elementary level, avoiding sheaf theory and cohomology theory.

该语言是有意不断初级程度，避免层论的上同调理论。

6.In this paper, we mainly discuss the procedure for computing the rational cohomology of quotients group actions in symplectic geometry.

本文主要讨论辛几何中群作用的商的有理上同调的计算方法。

7.And we also give some finiteness and vanishing results for generalized local cohomology modules under some conditions.

且在一定条件下得到了有关广义局部上同调模的有限生成性和零化性质的几个结果。

8.It discusses the relationship between left-supersymmetric structures on Lie superalgebra and its 1 th cohomology group.

讨论了李超代数上的左超对称结构与其上的1维上同调群的关系。

9.Researchers often use cohomology 上同调 theories to study complex manifolds.

研究人员经常使用 cohomology 上同调 理论来研究复杂流形。

10.One application of cohomology 上同调 is in the classification of fiber bundles.

一个 cohomology 上同调 的应用是在纤维束的分类中。

11.The cohomology 上同调 groups can reveal information about the holes in a given topological space.

该空间的 cohomology 上同调 群可以揭示关于给定拓扑空间中孔的信息。

12.In algebraic topology, the study of cohomology 上同调 provides important tools for understanding the properties of spaces.

在代数拓扑学中，cohomology 上同调 的研究提供了理解空间性质的重要工具。

13.The relationship between homology and cohomology 上同调 is a fundamental aspect of algebraic topology.

同调与 cohomology 上同调 之间的关系是代数拓扑学的一个基本方面。

作文

In the realm of mathematics, particularly in algebraic topology, the concept of cohomology plays a crucial role in understanding the properties of topological spaces. Cohomology is a way to associate a sequence of abelian groups or modules with a topological space, providing a powerful tool for analyzing its structure and features. The significance of cohomology lies in its ability to capture information about the shape and connectivity of spaces, which is essential for various applications in mathematics and physics.The roots of cohomology can be traced back to the work of mathematicians like Henri Poincaré and Émile Picard in the early 20th century. They developed the foundational ideas that would later evolve into the modern theory of cohomology. This theory allows mathematicians to study not just the points within a space, but also the relationships and higher-dimensional structures that exist between them.One of the key features of cohomology is its duality with homology. While homology provides information about the 'holes' in a space, cohomology focuses on the functions defined on these spaces. This duality is encapsulated in the famous Universal Coefficient Theorem, which links homology and cohomology groups in a profound way. Understanding this relationship deepens our insight into the topological characteristics of spaces, making cohomology an indispensable part of algebraic topology.Another important aspect of cohomology is its applicability to various branches of mathematics. For instance, in algebraic geometry, cohomology theories such as sheaf cohomology are used to study the properties of algebraic varieties. These tools allow mathematicians to derive significant results about the solutions to polynomial equations and their geometric interpretations.Moreover, cohomology has found applications in theoretical physics, particularly in string theory and gauge theory. In these contexts, cohomology helps physicists understand the underlying symmetries and conservation laws that govern physical systems. The interplay between cohomology and physics exemplifies the unity of mathematics and science, showcasing how abstract concepts can lead to tangible insights in the physical world.As one delves deeper into the study of cohomology, it becomes evident that it is not merely a theoretical construct. The computational techniques developed to calculate cohomology groups have practical implications in various fields, including data analysis and robotics. By employing these techniques, researchers can extract meaningful information from complex datasets, paving the way for advancements in technology and innovation.In conclusion, the concept of cohomology is a fundamental pillar of modern mathematics, bridging gaps between different areas of study and providing a framework for understanding the intricate nature of topological spaces. Its historical development, theoretical richness, and practical applications underscore the importance of cohomology in both mathematical research and real-world problem-solving. As we continue to explore the depths of this fascinating field, the potential for new discoveries remains vast, promising to unveil even more connections between mathematics and the world around us.

在数学的领域中，特别是在代数拓扑学中，概念共形性在理解拓扑空间的性质方面发挥着至关重要的作用。共形性是一种将一系列阿贝尔群或模与拓扑空间关联起来的方法，为分析其结构和特征提供了强有力的工具。共形性的重要性在于它能够捕捉有关空间形状和连通性的信息，这对于数学和物理中的各种应用至关重要。共形性的根源可以追溯到20世纪早期亨利·庞加莱和埃米尔·皮卡尔等数学家的工作。他们发展了后来演变为现代共形性理论的基础思想。这一理论使数学家能够研究不仅是空间内的点，还有它们之间存在的关系和更高维度的结构。共形性的一个关键特征是它与同调的对偶性。虽然同调提供了有关空间“孔”的信息，但共形性则侧重于定义在这些空间上的函数。这种对偶性在著名的普遍系数定理中得到了体现，该定理以深刻的方式将同调和共形性群联系在一起。理解这种关系加深了我们对空间拓扑特征的洞察，使得共形性成为代数拓扑学中不可或缺的一部分。共形性的另一个重要方面是它在各个数学分支中的适用性。例如，在代数几何中，像层共形性这样的共形性理论被用来研究代数簇的性质。这些工具使数学家能够推导出关于多项式方程及其几何解释的显著结果。此外，共形性还在理论物理学中找到了应用，特别是在弦理论和规范理论中。在这些背景下，共形性帮助物理学家理解支配物理系统的基本对称性和守恒定律。共形性与物理学之间的相互作用展示了数学与科学的统一，展示了抽象概念如何导致对物理世界的切实见解。随着人们深入研究共形性，显然这不仅仅是一个理论构造。为计算共形性群而开发的计算技术在各个领域具有实际意义，包括数据分析和机器人技术。通过运用这些技术，研究人员可以从复杂的数据集中提取有意义的信息，为技术和创新的进步铺平道路。总之，共形性的概念是现代数学的一个基本支柱，连接着不同研究领域之间的差距，并提供了理解拓扑空间复杂性质的框架。它的历史发展、理论丰富性和实际应用突显了共形性在数学研究和现实问题解决中的重要性。随着我们继续探索这一迷人领域的深度，新的发现潜力依然巨大，承诺揭示数学与我们周围世界之间更多的联系。