Our aim is to stress the importance of Jacobian matrix conditioning for model validation. We also comment on Monari and Dreyfus (2002), where, following Rivals and Personnaz (2000), it is proposed to discard neural candidates that are likely to overfit and/or for which quantities of interest such as confidence intervals cannot be computed accurately. In Rivals and Personnaz (2000), we argued th...

We give a lower bound on the rank of the Cartier operator of Jacobian varieties of hyperelliptic and superelliptic curves in terms of their genus.

We analyse the elementwise convergence in the Jacobian of the classical Newton method as the solution of a non-linear system f(x) = 0 is approached. It is shown that each Jacobian element approaches a constant value at a rate determined by the nonlinearity of the relevant function with respect to the appropriate scalar variable. For many practical problems, this means that in the last steps of ...

Newton-Krylov methods are standard tools for solving nonlinear problems. A common approach is to “lag” the Jacobian when assembly or preconditioner setup is computationally expensive, in exchange for some degradation in the convergence rate and robustness. We show that this degradation may be partially mitigated by using the lagged Jacobian as an initial operator in a quasi-Newton method, which...

We consider the moduli space SUsC(2,OC) of rank 2 stable vector bundles with trivial determinant on a smooth projective curve C of genus g. We show that the Abel-Jacobi map on the rational Chow group CH1(SU s C(2,OC))hom ⊗ Q of one cycles which are homologous to zero, is an isomorphism onto the bottom weight intermediate Jacobian, which is identified with the Jacobian Jac(C)⊗ Q.

In this paper, some definitions and traditional formulas for calculating the mobility of mechanisms are represented, e.g. Grubler formula, Somov - Malyshev formula, and Buchsbaum - Freudenstei. It is discussed that there are certain cases in which they are too ambiguous and incorrect to use. However, a matrix method is suggested based on the rank of the Jacobian of the mechanism and its applica...

The variation of the rank of elliptic curves over Q in families of quadratic twists has been extensively studied by Gouvêa, Mazur, Stewart, Top, Rubin and Silverberg. It is known, for example, that any elliptic curve over Q admits infinitely many quadratic twists of rank ≥ 1. Most elliptic curves have even infinitely many twists of rank ≥ 2 and examples of elliptic curves with infinitely many t...

In this paper, we consider the moduli space SUC(r,OC) of rank r semistable vector bundles with trivial determinant on a smooth projective curve C of genus g. When the rank r = 2, F. Kirwan constructed a smooth log resolution X → SUC(2,OC). Based on earlier work of M. Kerr and J. Lewis, Lewis explains in the Appendix the notion of a relative Chow group (w.r.to the normal crossing divisor), and a...

When a manipulator suuers a joint failure, its performance can be signiicantly affected. If the failed joint is locked, the resulting manipulator Jacobian is given by the original Jacobian, except that the column associated with the failed joint is removed. The rank of the resulting Jacobian then determines if the manipulator still has the ability to perform arbitrary end-eeector motions. Unfor...

We summarise recent advances in techniques for solving Diophantine problems on hyperelliptic curves; in particular, those for finding the rank of the Jacobian, and the set of rational points on the curve.