We prove the equivalence of the Jacobian Conjecture (JC(n)) and the Conjecture on the cardinality of the set of fixed points of a polynomial nilpotent mapping (JN(n)) and prove a series of assertions confirming JN(n).

In this paper we consider the Jacobian conjecture for a map f of complex affine spaces of dimension n. It is well-known that if f is proper then the conjecture will hold. Using topological arguments, specifically Smith theory, we show that the conjecture holds if and only if f is proper onto its image.

It turns out that various algebraic computations can be reduced to the same type of computations: one has to study the series of integrals ∫ K f(k)g(k) dk, where f, g are complex valued K-finite functions on a compact Lie group K. So it is tempting to state a general conjecture about the behavior of such integrals, and to investigate the consequences of the conjecture. Main conjecture: Let K be...

We give a full proof of the two dimensional Jacobian conjecture.

We give a shorter proof to a recent result by Neuberger [11], in the real case. Our result is essentially an application of the global asymptotic stability Jacobian Conjecture. We also extend some of the results presented in [11].

The paper is devoted to the proof of equivalence of Jacobian and Dixmier conjectures. We show that 2n-dimensional Jacobian conjecture implies Dixmier conjecture for Wn. The proof uses “antiquantization”: positive characteristics and Poisson brackets on the center of Weyl algebra in characteristic p. 2000 Math. Subj. Class. 16S32, 16S80, 14R15.

We introduce ABC-schemes, a new class of linearly implicit one-step methods for numerical integration of stiff ordinary differential equation systems. Formulas of ABC-schemes invoke the Jacobian of differential system similary to the methods of Rosenbrock type, but unlike the latter they include also the square of the Jacobian matrix.

The Jacobian Conjecture would follow if it were known that real polynomial maps with a unipotent Jacobian matrix are injective. The conjecture that this is true even for C maps is explored here. Some results known in the polynomial case are extended to the C context, and some special cases are resolved.

In an effort to increase the versatility of finite element codes, we explore the possibility of automatically creating the Jacobian matrix necessary for the gradient-based solution of nonlinear systems of equations. Particularly, we aim to assess the feasibility of employing the automatic differentiation tool TAPENADE for this purpose on a large Fortran codebase that is the result of many years...

By exploiting a model of environmental distortion, model adaptation based on vector Taylor series (VTS) approaches have been shown to significantly improve the robustness of speech recognizers to environmental noise. However, the computational cost of VTS model adaptation (MVTS) methods hinders them from being more widely used. In this paper, we propose to reduce the computational cost of MVTS ...