A singularity handling method is proposed in this paper. It is done by introducing virtual redundant joints into the Jacobian matrix to maintain the rank of the Jacobian matrix when singularity occurs. These additional joints do not exist physically. Therefore, although mathematically stable, the manipulator still can not perform tasks in the degenerate direction(s). This method is comparativel...

For any projective curve X let M d (X) be the Simpson moduli space of pure dimension one rank 1 degree d sheaves that are semistable with respect to a fixed polarization H on X . When X is a reduced curve the connected component of M d (X) that contains semistable line bundles can be considered as the compactified Jacobian of X . In this paper we give explicitly the structure of this compactifi...

Assuming finiteness of the Tate–Shafarevich group, we prove that Birch–Swinnerton–Dyer conjecture correctly predicts parity rank semistable principally polarised abelian surfaces. If surface in question is Jacobian a curve, require curve has good ordinary reduction at 2-adic places.

We show that the Weierstrass points of the generic curve of genus g over an algebraically closed field of characteristic 0 generate a group of maximal rank in the Jacobian. The Weierstrass points are a set of distinguished points on curves, which are geometrically intrinsic. In particular, the group these points generate in the Jacobian is a geometric invariant of the curve. A natural question ...

Recent developments in formulations for generating swept volumes have made a significant impact on the efficiency of employing such algorithms and on the extent to which formulations can be used in representing complex shapes. In this paper, we outline a method for employing the representation of implicit surfaces using the Jacobian rank deficiency condition presented earlier for the sweep of p...

This paper uses majorant techniques to study the convergence of Broyden's single-rank update method for nonlinear systems of equations. It also contains a very elementary proof of the local convergence of the method. The heart of the method is a procedure for generating an approximation to the Jacobian of the system using only information on hand and not requiring partial derivatives.

The main result of this note is that there are at most seven rational points (including the one at infinity) on the curve CA with the affine equation y = x+A (where A is a tenth power free integer) when the Mordell-Weil rank of the Jacobian of CA is one. This bound is attained for A = 18 .

In this paper, we give an exact expression for Clarke generalized Jacobian of the projection onto symmetric cones, which is linked to rank-1 matrices. As an application, we employ the projection operator to design a semismooth Newton algorithm for solving nonlinear symmetric cone programs. The algorithm is proved to be locally quadratically convergent without assuming strict complementarity of ...

We examine the local convergence of the Levenberg–Marquardt method for the solution of nonlinear least squares problems that are rank-deficient and have nonzero residual. We show that replacing the Jacobian by a truncated singular value decomposition can be numerically unstable. We recommend instead the use of subset selection. We corroborate our recommendations by perturbation analyses and num...