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In this talk, I will discuss two related topic. 1. How to express the inverse mapping of the injective function f by using the function f only. 2. How to realize this inversion formula numerically. The part 1 will intend as the precise description of the problem and describing a satisfactory answer when n = 1, 2 and the part 2 is devoted to obtaining the smoothing formula by way of the motivation of the part 1.

Singular integral inequalities and natural regularizations