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A persistence module is a mathematical structure in persistent homology and topological data analysis that formally captures the persistence of topological features ...
A persistence module over a partially ordered set is a set of vector spaces indexed by , with a linear map : whenever , with equal to the identity and = for . Equivalently, we may consider it as a functor from P {\displaystyle P} considered as a category to the category of vector spaces (or R {\displaystyle R} -modules ).
In topological data analysis, a persistence barcode, sometimes shortened to barcode, is an algebraic invariant associated with a filtered chain complex or a persistence module that characterizes the stability of topological features throughout a growing family of spaces. [1]
The persistence module, installed as part of system BIOS/UEFI, detects when the Absolute Home & Office software has been removed. It ensures the software is automatically reinstalled even if the hard drive is replaced, or the firmware is flashed.
A persistence module indexed by is a vector space for each , and a linear map : whenever , such that = for all and = whenever . [13] An equivalent definition is a functor from considered as a partially ordered set to the category of vector spaces.
A 1-interleaving between two -indexed persistence modules M and N, represented as a diagram of vector spaces and linear maps between them.. In topological data analysis, the interleaving distance is a measure of similarity between persistence modules, a common object of study in topological data analysis and persistent homology.
Another round of fire-fueling Santa Ana winds is lashing Southern California after a brief weekend reprieve helped crews make more progress battling the deadly infernos in Los Angeles County.
Thus, the persistence of each class can be represented as an interval in the extended real line {} of either the form [,) or [′,). Since, in the case of an infinite field, the infinite number of classes always have the same persistence, the collection over all classes of such intervals does not give meaningful multiplicities for a multiset of ...