Given a finite, flat morphism between embeddable noetherian schemes of pure dimension 1, we define the notion direct and inverse image for generalized divisors line bundles. In case when deal with (possibly reducible, non-reduced) projective curves over field codomain curve is smooth, introduce compactified Jacobians parametrizing torsion-free rank-1 sheaves study Norm maps Jacobians. Finally, ...

We present a decomposition theorem of the generic polar curves generalized curve foliation with only one separatrix and Hamiltonian foliations defined by approximate roots generatrix. This is generalization to Jacobians given García Barroso Gwoździewicz for plane branches.

The compactified Jacobian of any projective curve X is defined as the Simpson moduli space of torsion free rank one degree d sheaves that are semistable with respect to a fixed polarization H on X. In this paper we give explicitly the structure of this compactified Simpson Jacobian in the case where X is a generalized tree-like curve, i.e., a projective, reduced and connected curve such that th...

The problem of algebraic dependence solutions to (non-linear) first order autonomous equations over an algebraically closed field characteristic zero is given a ‘complete’ answer, obtained independently model theoretic results on differentially fields. Instead, the geometry curves and generalized Jacobians provides key ingredient. Classification formal are treated. applied answer question $...

The titles published in this series are listed at the end of this volume. 1 1 2 4 4 4 5 5 6 6 7 9 Multifunctions and Derivatives Particular Locally Lipschitz Functions and Related Definitions Generalized Jacobians of Locally Lipschitz Functions Pseudo-Smoothness and D°f Piecewise Functions NCP Functions 1.4 Definitions of Regularity Definitions of Lipschitz Properties Regularity Definitions Fun...

It is shown how functions that are defined by evaluation programs involving the absolute value function abs() (besides smooth elementals), can be approximated locally by piecewise-linear models in the style of algorithmic, or automatic, differentiation (AD). The model can be generated by a minor modification of standard AD tools [GJU96b] and it is Lipschitz continuous with respect to the base p...

Changes of variables giving the dual model are constructed explicitly for σ-models without isotropy. In particular, the jacobian is calculated to give the known results. The global aspects of the abelian case as well as some of those of the cases where the isometry group is simply connected are considered. Considering the anomalous case, we infer by a consistency argument that the ‘multiplicati...

This paper presents a parameterized Newton method using generalized Jacobians and a Broyden-like method for solving nonsmooth equations. The former ensures that the method is well-deened even when the generalized Jacobian is singular. The latter is constructed by using an approximation function which can be formed for nonsmooth equations arising from partial diierential equations and nonlinear ...