Jacobian matrices can be accumulated using either the forward or reverse mode of Automatic Di erentiation. Alternatively, derivative code can be generated to compute the Jacobian directly at the current argument. The minimisation of the corresponding number of arithmetic operations leads to a computationally hard combinatorial optimisation problem. A method for its approximate solution by dynam...

The supersingular K3 surface X in characteristic 2 with Artin invariant 1 admits several genus 1 fibrations (elliptic and quasi-elliptic). We use a bijection between fibrations and definite even lattices of rank 20 and discriminant 4 to classify the fibrations, and exhibit isomorphisms between the resulting models of X. We also study a configuration of (−2)-curves on X related to the incidence ...

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This paper is concerned with some Algebraic Geometry codes on Jacobians of genus 2 curves. We derive a lower bound for the minimum distance of these codes from an upper “Weil type” bound for the number of rational points on irreducible (possibly singular or non-absolutely irreducible) curves lying on an abelian surface over a finite field.

Let Pn := K[x1, . . . , xn] be a polynomial algebra over a field K of characteristic zero. The Jacobian algebra An is the subalgebra of EndK(Pn) generated by the Weyl algebra An := D(Pn) = K〈x1, . . . , xn, ∂1, . . . , ∂n〉 and the elements (∂1x1) −1, . . . , (∂nxn) −1 ∈ EndK(Pn). The algebra An appears naturally in study of the group of automorphisms of Pn. The algebra An is large since it cont...

The Jacobian Conjecture can be generalized and is established : Let S be a polynomial ring over a field of characteristic zero in finitely may variables. Let T be an unramified, finitely generated extension of S with T = k . Then T = S. Let k be an algebraically closed field, let k be an affine space of dimension n over k and let f : k −→ k be a morphism of algebraic varieties. Then f is given ...

Abstract: Derivatives are essential ingredients of a wide range of numerical algorithms. We focus on the accumulation of Jacobian matrices by Gaussian elimination on a sparse implementation of the extended Jacobian. A symbolic algorithm is proposed to determine the fill-in. Its runtime undercuts that of the original accumulation algorithm by a factor of ten. On the given computer architecture w...

We review various algorithms for finding apparent horizons in 3+1 numerical relativity. We then focus on one particular algorithm, in which we pose the apparent horizon equation H ≡ ∇in + Kijnn − K = 0 as a nonlinear elliptic (boundary-value) PDE on angular-coordinate space for the horizon shape function r = h(θ, φ), finite difference this PDE, and use Newton’s method or a variant to solve the ...

Genes are connected in regulatory networks, often modelled by ordinary differential equations. Changes in expression of a gene propagate to other genes along paths in the network. At a stable state, the system’s Jacobian matrix confers information about network connectivity. To disclose the functional properties of genes, knowledge of network connections is essential. We present a new method to...

In this paper, the coupling between Jacobian learning and task sequencing through the redundancy approach is studied. It is well known that visual servoing is robust to modeling errors in the jacobian matrices. This justifies why jacobian estimation does not usually degrade the system convergence. However, we show that this is not true anymore when the redundancy formalism is used. In this case...