In this paper we introduce the persistent magnitude, a new numerical invariant of (sufficiently nice) graded persistence modules. It is a weighted and signed count of the bars of the persistence module, in which a bar of the form $[a,b)$ in degree $d$ is counted with weight $(e^{-a}-e^{-b})$ and sign $(-1)^d$. Persistent magnitude has good formal properties, such as additivity with respect to exact sequences and compatibility with tensor products, and has interpretations in terms of both the associated graded functor, and the Laplace transform. Our definition is inspired by Otter's notion of blurred magnitude homology: we show that the magnitude of a finite metric space is precisely the persistent magnitude of its blurred magnitude homology. Turning this result on its head, we obtain a strategy for turning existing persistent homology theories into new numerical invariants by applying the persistent magnitude. We explore this strategy in detail in the case of persistent homology of Morse functions, and in the case of Rips homology.

中文翻译：

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持续震级

在本文中，我们介绍了持久性幅度，这是（足够好）分级持久性模块的新数值不变量。它是持久性模块的柱形的加权和有符号计数，其中以 $d$ 为单位的 $[a,b)$ 形式的柱形以权重 $(e^{-a}-e^{ -b})$ 并签署 $(-1)^d$。持久幅度具有良好的形式属性，例如关于精确序列的可加性和与张量积的兼容性，并且在相关的分级函子和拉普拉斯变换方面都有解释。我们的定义受到 Otter 的模糊幅度同调概念的启发：我们证明有限度量空间的幅度恰好是其模糊幅度同调的持久幅度。扭转这个结果，我们获得了一种策略，通过应用持久幅度将现有的持久同源理论转化为新的数值不变量。我们在 Morse 函数的持久同源性和 Rips 同源性的情况下详细探讨了这种策略。