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Artificial neural networks are functions depending on a finite number of parameters typically encoded as weights and biases. The identification of the parameters of the network from finite samples of input-output pairs is often referred to…

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This paper has been withdrawn by the author due to the inaccurate result.

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This paper has been withdrawn by the author, because it is now part of an enlarged version entitled "Time functions as utilities" arXiv:0909.0890

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This paper has been withdrawn by the authors, due to a crucial error in beta functions.

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This paper has been withdrawn by the author due to a crucial problem associated with Figs. 2 and 3.