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Approximations and error bounds for computing the inverse mapping

Lucas Jódar, Enrique Ponsoda, G. Rodríguez Sánchez (1997)

Applications of Mathematics

In this paper we propose a procedure to construct approximations of the inverse of a class of 𝒞 m differentiable mappings. First of all we determine in terms of the data a neighbourhood where the inverse mapping is well defined. Then it is proved that the theoretical inverse can be expressed in terms of the solution of a differential equation depending on parameters. Finally, using one-step matrix methods we construct approximate inverse mappings of a prescribed accuracy.

Change of variables formula under minimal assumptions

Piotr Hajłasz (1993)

Colloquium Mathematicae

In the previous papers concerning the change of variables formula (in the form involving the Banach indicatrix) various assumptions were made about the corresponding transformation (see e.g. [BI], [GR], [F], [RR]). The full treatment of the case of continuous transformation is given in [RR]. In [BI] the transformation was assumed to be continuous, a.e. differentiable and with locally integrable Jacobian. In this paper we show that none of these assumptions is necessary (Theorem 2). We only need...

Differential conditions to verify the Jacobian Conjecture

Ludwik M. Drużkowski, Halszka K. Tutaj (1992)

Annales Polonici Mathematici

Let F be a polynomial mapping of ℝ², F(O) = 0. In 1987 Meisters and Olech proved that the solution y(·) = 0 of the autonomous system of differential equations ẏ = F(y) is globally asymptotically stable provided that the jacobian of F is everywhere positive and the trace of the matrix of the differential of F is everywhere negative. In particular, the mapping F is then injective. We give an n-dimensional generalization of this result.

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