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Full-Text Articles in Physical Sciences and Mathematics
Foliations And Global Inversion, Eduardo C. Balreira
Foliations And Global Inversion, Eduardo C. Balreira
Eduardo Cabral Balreira
We consider topological conditions under which a locally invertible map admits a global inverse. Our main theorem states that a local diffeomorphism f : M → Rn is bijective if and only if Hn−1(M) = 0 and the pre-image of every affine hyperplane is non-empty and acyclic. The proof is based on some geometric constructions involving foliations and tools from intersection theory. This topological result generalizes in finite dimensions the classical analytic theorem of Hadamard-Plastock, including its recent improvement by Nollet-Xavier. The main theorem also relates to a conjecture of the aforementioned authors, involving the well known Jacobian Conjecture in …
A Generalization Of The Fujisawa–Kuh Global Inversion Theorem, Marius Radulescu, Sorin Radulescu, Eduardo C. Balreira
A Generalization Of The Fujisawa–Kuh Global Inversion Theorem, Marius Radulescu, Sorin Radulescu, Eduardo C. Balreira
Eduardo Cabral Balreira
We discuss the problem of global invertibility of nonlinear maps defined on the finitedimensional Euclidean space via differential tests. We provide a generalization of theFujisawa-Kuh global inversion theorem and introduce a generalized ratio conditionwhich detects when the pre-image of a certain class of linear manifolds is non-emptyand connected. In particular, we provide conditions that also detect global injectivity.