As a consequence the level of exposition varies considerably. In particular, some proofs (marked by ¶) and the Appendix require more mathematical background than the main body of the text. The reader is advised to study the paper without paying too much attention to those technical details. This gives sufficient information to complete the exercises. We hope that this introduction might motivat...

Let C be an algebraically closed field with trivial derivation and let F denote the differential rational field C〈Y1, . . . , Y5〉, with Y1, . . . , Y5 differentially independent over C. We show that there is a Picard-Vessiot extension E ⊃ F for a matrix equation X ′ = XA(Yi), with differential Galois group SO3, with the property that if F is any differential field with field of constants C then...

We construct generic Picard-Vessiot extensions for linear algebraic groups which are isomorphic to the semidirect product of a connected group G by an arbitrary finite group H, where the adjoint H-action on the Lie algebra of G is faithful.

This paper studies stable sheaves on abelian surfaces of Picard number one. Our main tools are semi-homogeneous sheaves and Fourier-Mukai transforms. We introduce the notion of semi-homogeneous presentation and investigate the behavior of stable sheaves under Fourier-Mukai transforms. As a consequence, an affirmative proof is given to the conjecture proposed by Mukai in the 1980s. This paper al...

Article history: Received 27 August 2014 Received in revised form 9 June 2015 Accepted 5 July 2015 Available online xxxx Communicated by Dominique Picard MSC: 60G05 60G17 60G20 60G52 60H40 42C40 46E35

For a Poisson manifold M we develop systematic methods to compute its Picard group Pic(M), i.e., its group of self Morita equivalences. We establish a precise relationship between Pic(M) and the group of gauge transformations up to Poisson diffeomorphisms showing, in particular, that their connected components of the identity coincide; this allows us to introduce the Picard Lie algebra of M and...

We develop a general framework for the study of strong Morita equivalence in which C∗algebras and hermitian star products on Poisson manifolds are treated in equal footing. We compare strong and ring-theoretic Morita equivalences in terms of their Picard groupoids for a certain class of unital ∗-algebras encompassing both examples. Within this class, we show that both notions of Morita equivale...

In this paper, we first present the complete list of the singularity types of the Picard number one Gorenstein log del Pezzo surface and the number of the isomorphism classes with the given singularity type. Then we give out a method to find out all singularity types of Gorenstein log del Pezzo surface. As an application, we present the complete list of the Dynkin type of the Picard number two ...

In this paper, we continue the study of contractive conditions for mappings in complete partial metric spaces. Concretely, we present fixed point results for weakly contractive and weakly Kannan mappings in such a way that the classical metric counterpart results are retrieved as a particular case. Special attention to the cyclical case is paid. Moreover, the well-posedness of the fixed point p...

The Picard curves are genus three curves with a non trivial automorphism, which have been intensively studied due their connection with interesting number theoretic problems. In 1989, R. Pellikaan obtained an algorithm decoding geometric codes up to b(d ? 1)=2c-errors, where d is the designed distance of the code. His algorithm is not completely eeective, but recently some authors have given an...