In this note preconditioners for the Conjugate Gradient method are studied to solve the Newton system with a symmetric positive definite Jacobian. In particular, we define a sequence of preconditioners built by means of BFGS rank-two updates. Reasonable conditions are derived which guarantee that the preconditioned matrices are not far from the identity in a matrix norm. Some notes on the imple...

We prove that if X and S are smooth varieties and f : X → S is an elliptic fibration with singular fibers curves of types IN with N ≥ 1, II, III and IV, then the relative Jacobian f̂ : MX/S → S of f , defined as the relative moduli space of semistable pure dimension one sheaves of rank 1 and degree 0 on the fibers of f , is an elliptic fibration such that all its fibers are irreducible. This ext...

Necessary conditions in the form of maximum principles are derived for optimal control problems with mixed control and state constraints. Traditionally, necessary condtions for problems with mixed constraints have been proved under hypothesis which include the requirement that the Jacobian of the mixed constraint functional, with respect to the control variable, have full rank. We show that it ...

There has been important progress in constructing units and Sunits associated to curves of genus 2 or 3. These approaches are based mainly on the consideration of properties of Jacobian varieties associated to hyperelliptic curves of genus 2 or 3. In this paper, we construct a unit group of the ray class field k6 of Q(exp(2πi/5)) modulo 6 with full rank by special values of Siegel modular funct...

In this paper it is shown that, given n ∈ Z ≥3 and a, b, c ∈ Q × , there exists a polynomial d(t) ∈ Q[t] such that the curves over Q(t) given by y 2 = x n + ad(t) resp. y 2 = x n + bd(t) and y 2 = x n + cd(t) all have a Jacobian with positive Mordell-Weil rank over Q(t). Extensions of this result to sets of four curves are discussed, as well as the problem of demanding in addition that d(t) is ...

This report describes a new family of Newton-Krylov methods for solving nonlinear systems of equations arising from the solution of Richards’ equation and in fully implicit formulations in air-water systems. The basic approach is to perform secant (Broyden) updates restricted to the Krylov subspace generated by the GMRES iterative solver. This approach is introduced as Krylov-secant methods. On...

We extend the Bayesian Information Criterion (BIC), an asymptotic approximation for the marginal likelihood, to Bayesian networks with hidden variables. This approximation can be used to select models given large samples of data. The standard BIC as well as our extension punishes the complexity of a model according to the dimension of its parameters. We argue that the dimension of a Bayesian ne...

Let G be a simple Lie group of real rank one, with Iwasawa decomposition KAN̄ and Bruhat big cell NMAN̄ . Then the space G/MAN̄ may be (almost) identified with N and with K/M , and these identifications induce the (generalised) Cayley transform C : N → K/M . We show that C is a conformal map of Carnot–Caratheodory manifolds, and that composition with the Cayley transform, combined with multiplicat...

We analyze the convergence of the sequential quadratic programming (SQP) method for nonlinear programming for the case in which the Jacobian of the active constraints is rank deecient at the solution and/or strict complementarity does not hold for some or any feasible Lagrange multipliers. We use a nondiieren-tiable exact penalty function, and we prove that the sequence generated by the SQP is ...