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   Bobkov, S. G.; Ledoux, M. (January 2020, Journal of fourier analysis applications)

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   Scoccola, Luis; Perea, Jose A. (January 2023, Forum of Mathematics, Sigma)

   Abstract We introduce $$\varepsilon $$ -approximate versions of the notion of a Euclidean vector bundle for $$\varepsilon \geq 0$$ , which recover the classical notion of a Euclidean vector bundle when $$\varepsilon = 0$$ . In particular, we study Čech cochains with coefficients in the orthogonal group that satisfy an approximate cocycle condition. We show that $$\varepsilon $$ -approximate vector bundles can be used to represent classical vector bundles when $$\varepsilon> 0$$ is sufficiently small. We also introduce distances between approximate vector bundles and use them to prove that sufficiently similar approximate vector bundles represent the same classical vector bundle. This gives a way of specifying vector bundles over finite simplicial complexes using a finite amount of data and also allows for some tolerance to noise when working with vector bundles in an applied setting. As an example, we prove a reconstruction theorem for vector bundles from finite samples. We give algorithms for the effective computation of low-dimensional characteristic classes of vector bundles directly from discrete and approximate representations and illustrate the usage of these algorithms with computational examples. 

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