We show that under the assumption of Artin’s Primitive Root Conjecture, for all primes p there exist ordinary elliptic curves over Fp(x) with arbitrary high rank and constant j-invariant. For odd primes p, this result follows from a theorem which states that whenever p is a generator of (Z/lZ)∗/〈−1〉 (l an odd prime) there exists a hyperelliptic curve over Fp whose Jacobian is isogenous to a pow...

We describe a slight modi cation of the well-known sequential quadratic programming method for nonlinear programming that attains superlinear convergence to a primal-dual solution even when the Jacobian of the active constraints is rank de cient at the solution. We show that rapid convergence occurs even in the presence of the roundo errors that are introduced when the algorithm is implemented ...

Suppose f = p/q is a quotient of two polynomials and that p has degree rp and q has degree rq . Assume that f(A) and f(A+ uv ) are defined where A ∈ R, u ∈ R, and v ∈ R are given and set r = max{rp, rq}. We show how to compute f(A+uv ) in O(rn2) flops assuming that f(A) is available together with an appropriate factorization of the “denominator matrix” q(A). The central result can be interprete...

This paper extends a recently proposed singularity analysis method to lower-mobility parallel manipulators having an articulated nacelle. Using screw theory, a twist graph is introduced in order to simplify the constraint analysis of such manipulators. Then, a wrench graph is obtained in order to represent some points at infinity on the Plücker lines of the Jacobian matrix. Using GrassmannCayle...

In this paper we deal with optimality conditions that can be verified by a nonlinear optimization algorithm, where only a single Lagrange multiplier is avaliable. In particular, we deal with a conjecture formulated in [R. Andreani, J.M. Mart́ınez, M.L. Schuverdt, “On second-order optimality conditions for nonlinear programming”, Optimization, 56:529–542, 2007], which states that whenever a local...

Many problems give rise to polynomial systems. These systems often have several parameters and we are interested to study how the solutions vary when we change the values for the parameters. Using predictor-corrector methods we track the solution paths. A point along a solution path is critical when the Jacobian matrix is rank deficient. The simplest case of quadratic turning points is well und...

A kinematic singularity occurs when the Jacobian matrix associated with the forward kinematic map loses rank. At a singular connguration, the end-eeector velocities are restricted to a subspace of the task space velocities. Also, rate control schemes that require the inversion of the Jacobian matrix fail at such singulari-ties. However, even at a singular connguration, if the kinematic map is g...

Many problems give rise to polynomial systems. These systems often have several parameters and we are interested to study how the solutions vary when we change the values for the parameters. Using predictor-corrector methods we track the solution paths. A point along a solution path is critical when the Jacobian matrix is rank deficient. The simplest case of quadratic turning points is well und...

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...

We prove an extension of Yuan’s Lemma to more than two matrices, as long as the set of matrices has rank at most 2. This is used to generalize the main result of [A. Baccari and A. Trad. On the classical necessary second-order optimality conditions in the presence of equality and inequality constraints. SIAM J. Opt., 15(2):394–408, 2005], where the classical necessary second-order optimality co...