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7% to 6.6%, and deteriorated the O/W emulsion stability and altered the silicate wettability toward less water wet state. Based on the experimental results, a possible mechanism of the interaction between surfactin and acetoin was proposed.We examine the closedness of sets of realized neural networks of a fixed architecture in Sobolev spaces. For an exactly m-times differentiable activation function ρ, we construct a sequence of neural networks [Formula see text] whose realizations converge in order-(m-1) Sobolev norm to a function that