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The minimum number of critical points of a small codimension smooth map between two manifolds is computed. Some partial results for the case of higher codimension when the manifolds are spheres are also given.

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While any nil-clean ring is clean, the last eight years, it was not known whether nil-clean elements in a ring are clean. We give an example of nil-clean element in the matrix ring ℳ2(Z) which is not clean.

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