Assumption 2 (Smoothness of the control problem) We assume that the function f : Rnx×nq 7→ Rx is sufficiently smooth. Note that Lipschitz continuity guarantees the unique existence of a solution x(·) of the differential equations for fixed q by virtue of the Picard-Lindelöf theorem. As we assume unique sensitivities in the following, also the partial derivative functions ∂f ∂x , ∂f ∂q , ∂f ∂x2 ...

We test the methods for computing the Picard group of a K3 surface in a situation of high rank. The examples chosen are resolutions of quartics in P having 14 singularities of type A1. Our computations show that the method of R. van Luijk works well when sufficiently large primes are used.

and Applied Analysis 3 in 1.3 plays the role of a ”perturbation term” and its choice is, of course, not unique. The solution of problem 1.3 is sought for in an analytic form by the method of successive approximations similar to the Picard iterations. According to the formulas

The Picard scheme of a smooth curve and a smooth complex variety is reduced. In this note we discuss which classes of surfaces in terms of the Enriques–Kodaira classification can have non-reduced Picard schemes and whether there are restrictions on the characteristic of the ground field. It turns out that non-reduced Picard schemes are uncommon in Kodaira dimension κ ≤ 0, that this phenomenon c...

We compute class groups of very general normal surfaces in P3 C containing an arbitrary base locus Z, thereby extending the classic Noether-Lefschetz theorem (when Z is empty). Our method is an adaptation of Griffiths and Harris’ degeneration proof, simplified by a cohomology and base change argument. We give some applications to computing Picard groups. Dedicated to Robin Hartshorne on his 70t...

The goal of this paper is to give an efficient computation of the 3-point Gromov-Witten invariants of Fano hypersurfaces, starting from the Picard-Fuchs equation. This simplifies and to some extent explains the original computations of Jinzenji. The method involves solving a gauge-theoretic differential equation, and our main result is that this equation has a unique solution.

In contrast with the classical gauge group cases, any method to prove exactly the scaling relation which relates moduli and prepotential is not known in the case of exceptional gauge groups. This paper provides a direct method to establish this relation by using Picard-Fuchs equations. In particular, it is shown that the scaling relation found by Ito in N = 2 supersymmetric G 2 Yang-Mills theor...

Let Γ̄ be the Picard modular group of an imaginary quadratic number field k and let D be the associated symmetric space. Let Γ ⊂ Γ̄ be a congruence subgroup. We describe a method to compute the integral cohomology of the locally symmetric space Γ\D. The method is implemented for the cases k = Q(i) and k = Q( √ −3), and the cohomology is computed for various Γ.

For a class of quasi-contractive operators defined on an arbitrary Banach space, it has been shown that the Picard iteration technique converges faster than the Mann iteration technique. In this paper we make a comparison of the Picard and Mann iterations with respect to their convergence rate for a more general class of operators called quasi-contractions in metrizable topological vector space...

Let X, d be a complete metric space, and let T be a self-map of X. If T has a unique fixed point, which can be obtained as the limit of the sequence {pn}, where pn Tp0, p0 any point of X, then T is called a Picard operator see, e.g., 1 , and the iteration defined by {pn} is called Picard iteration. One of the most general contractive conditions for which a map T is a Picard operator is that of ...