离散数学(Discrete mathematics)是研究离散量的结构及其相互关系的数学学科,是现代数学的一个重要分支。

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越来越多来自不同领域的计算机科学家使用离散数学结构来解释概念和问题。在教学经验的基础上,作者提供了一个容易理解的文本,强调了离散数学的基础及其高级课题。这篇文章展示了如何用清晰的数学语言表达精确的思想。学生发现离散数学在描述计算机科学结构和解决问题方面的重要性。他们还学习如何掌握离散数学将帮助他们发展重要的推理技能,这些技能将在他们的职业生涯中继续发挥作用。

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We propose a new method to characterize firms' inventive activities via topological data analysis (TDA) that represents high-dimensional data in a shape graph. Applying this method to 333 major firms' patents in 1976-2005 reveals substantial heterogeneity: some firms remain undifferentiated; others develop unique portfolios. Firms with unique trajectories, which we define graph-theoretically as "flares" in the Mapper graph, perform better. This association is statistically and economically significant, and continues to hold after we control for portfolio size and firm survivorship. Comparison with existing techniques suggests the method's usefulness for data visualization and exploratory empirical research more generally.

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We propose a new method to characterize firms' inventive activities via topological data analysis (TDA) that represents high-dimensional data in a shape graph. Applying this method to 333 major firms' patents in 1976-2005 reveals substantial heterogeneity: some firms remain undifferentiated; others develop unique portfolios. Firms with unique trajectories, which we define graph-theoretically as "flares" in the Mapper graph, perform better. This association is statistically and economically significant, and continues to hold after we control for portfolio size and firm survivorship. Comparison with existing techniques suggests the method's usefulness for data visualization and exploratory empirical research more generally.

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