We investigate a method of construction of Calabi–Yau manifolds, that is, by smoothing normal crossing varieties. We develop some theories for calculating the Picard groups of the Calabi–Yau manifolds obtained in this method. As an application, we construct more than two hundred new families of Calabi–Yau 3-folds with Picard number one that have different Hodge numbers (h’s). We also exhibit a ...

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...

Picard-Vessiot rings are present in many settings like differential Galois theory, difference Galois theory and Galois theory of Artinian simple module algebras. In this article we set up an abstract framework in which we can prove theorems on existence and uniqueness of Picard-Vessiot rings, as well as on Galois groups corresponding to the Picard-Vessiot rings. As the present approach restrict...

F is a differential field of characteristic zero with algebraically closed field of constants C. A Picard–Vessiot antiderivative closure of F is a differential field extension E ⊃ F which is a union of Picard–Vessiot extensions of F , each obtained by iterated adjunction of antiderivatives, and such that every such Picard– Vessiot extension of F has an isomorphic copy in E. The group G of diffe...

We prove some finiteness theorems for the Picard functor of an algebraic stack, in the spirit of SGA6, exp. XII and XIII. In particular, we give a stacky version of Raynaud’s relative representability theorem, we give sufficient conditions for the existence of the torsion component of the Picard functor, and for the finite generation of the Néron-Severi groups or of the Picard group itself. We ...

We investigate an iterative method for the solution of time-periodic parabolic PDE constrained optimization problems. It is an inexact Sequential Quadratic Programming (iSQP) method based on the Newton-Picard approach. We present and analyze a linear quadratic model problem and prove optimal mesh-independent convergence rates. Additionally, we propose a two-grid variant of the Newton-Picard met...

Frequentely it happens that isogenous (in the sense of Mukai) K3 surfaces are partners of each other and sometimes they are even isomorphic. This is due, in some cases, to the (too high, e.g. bigger then or equal to 12)) rank of the Picard lattice as showed by Mukai in [13]. In other cases this is due to the structure of the Picard lattice and not only on its rank. This is the case, for example...

In this paper we study the automorphisms group of some K3 surfaces which are double covers of the projective plane ramified over a smooth sextic plane curve. More precisely, we study some particlar case of a K3 surface of Picard rank two. Introduction K3 surfaces which are double covers of the plane ramified over a plane sextic are classical objects. In this paper we determine the automorphisms...

In a previous paper, [S2], we described six families of K3-surfaces with Picardnumber 19, and we identified surfaces with Picard-number 20. In these notes we classify some of the surfaces by computing their transcendental lattices. Moreover we show that the surfaces with Picard-number 19 are birational to a Kummer surface which is the quotient of a non-product type abelian surface by an involut...