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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, noncommutative residue, defined independently by M. Wodzicki (1984) and Guillemin (1985), is a certain trace on the algebra of pseudodifferential operators on a compact differentiable manifold that is expressed via a local density. In the case of the circle, the noncommutative residue had been studied earlier by M. Adler (1979) and Y. Manin (1978) in the context of one-dimensional integrable systems.