Consider the Jacobian of a genus two curve defined over a finite field and with complex multiplication. In this paper we show that if the l-Sylow subgroup of the Jacobian is not cyclic, then the embedding degree of the Jacobian with respect to l is one.

Newton-Krylov methods, combination of Newton-like methods and Krylov subspace methods for solving the Newton equations, often need adequate preconditioning in order to be successful. Approximations of the Jacobian matrices are required to form preconditioners and this step is very often the dominant cost of Newton-Krylov methods. Therefore, working with preconditioners destroys in principle the...

A parametric, continuous-time Markov model for digraph panel data is considered. The parameter is estimated by the method of moments. A convenient method for estimating the variance–covariance matrix of the moment estimator relies on the delta method, requiring the Jacobian matrix—that is, the matrix of partial derivatives—of the estimating function. The Jacobian matrix was estimated hitherto b...

The objective of this paper is to present and make a comparative study of several inverse kinematics methods for serial manipulators, based on the Jacobian matrix. Besides the well-known Jacobian transpose and Jacobian pseudo-inverse methods, three others, borrowed from numerical analysis, are presented. Among them, two approximation methods avoid the explicit manipulability matrix inversion, w...

We propose a modification to Newton’s method for solving nonlinear equations, namely a Jacobian Computation-free Newton’s Method . Unlike the classical Newton’s method, the proposed modification neither requires to compute and store the Jacobian matrix, nor to solve a system of linear equations in each iteration. This is made possible by approximating the Jacobian inverse to a diagonal matrix w...

In this paper, we compare the effectiveness of three preconditioning strategies in simulations of variably saturated flow. Using Richards’ equation as our model, we solve the nonlinear system using a Newton-Krylov method. Since Krylov solvers can stagnate, resulting in slow convergence, we investigate different strategies of preconditioning the Jacobian system. Our work uses a multigrid method ...

The Jacobian of a graph, also known as the Picard Group, Sandpile Group, or Critical Group, is a discrete analogue of the Jacobian of an algebraic curve. It is known that the order of the Jacobian of a graph is equal to its number of spanning trees, but the exact structure is known for only a few classes of graphs. In this paper, we compute the Jacobian for graphs of the form Kn\E(H) where H is...

It is shown that the Jacobian Conjecture holds for all polynomial maps F : k → k of the form F = x + H , such that JH is nilpotent and symmetric, when n ≤ 4. If H is also homogeneous a similar result is proved for all n ≤ 5. Introduction Let F := (F1, . . . , Fn) : C → C be a polynomial map i.e. each Fi is a polynomial in n variables over C. Denote by JF := (i ∂xj )1≤i,j≤n, the Jacobian matrix ...

For redundant robot kinematics we construct an extended Jacobian inverse that approximates the Jacobian pseudo-inverse. Our construction relies on the approximation of codistributions associated with the Jacobian pseudo-inverse by an integrable codistribution, spanned by differentials of certain functions that serve as components of the augmenting kinematics map in the extended Jacobian inverse...

The integral curve of a system of ODEs with uncertain parameters will likewise be uncertain. Such a system can be reformulated into a system of Galerkin-projected ODEs that govern the coefficients of a (truncated) polynomial chaos (PC) representation of the (random) state variables. While this reformulated system is deterministic, it is of higher dimension. In this work we provide a framework f...