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 ...

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 ...

In this work we consider the regularization of vectorial data such as color images. Based on the observation that edge alignment across image channels is a desirable prior for multichannel image restoration, we propose a novel scheme of minimizing the rank of the image Jacobian and extend this idea to second derivatives in the framework of total generalized variation. We compare the proposed co...

In this paper we present a formula for the number of isomorphism classes of p-rank zero étale double covers of genus 2 curves over an algebraically closed field of characteristic p > 2. The formula is a byproduct of our search for complete subvarieties of moduli spaces of curves. Many moduli spaces are not complete because the objects that they parametrize can degenerate. Examples are the modul...

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...

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...

This paper addresses the approximation problem of Jacobian inverse kinematics algorithms for redundant robotic manipulators. Specifically, we focus on the approximation of the Jacobian pseudo inverse by the extended Jacobian algorithm. The algorithms are defined as certain dynamic systems driven by the task space error, and identified with vector field distributions. The distribution correspond...

Newton-Raphson State Estimation method using bus admittance matrix remains as an efficient and most popular method to estimate the state variables. Elements of Jacobian matrix are computed from standard expressions which lack physical significance. In this paper, elements of the state estimation Jacobian matrix are obtained considering the power flow measurements in the network elements. These ...