We study the variation of relative cohomology for a pair consisting of a smooth projective hypersurface and an algebraic subvariety in it. We construct an inhomogeneous Picard-Fuchs equation by applying a Picard-Fuchs operator to the holomorphic top form on a Calabi-Yau hypersurface in toric variety, and deriving a general formula for the d-exact form on one side of the equation. We also derive...

In this article, we propose a new multigrid-based algorithm for solving integral equations of the reference interactions site model (RISM). We also investigate the relationship between the parameters of the algorithm and the numerical accuracy of the hydration free energy calculations by RISM. For this purpose, we analyzed the performance of the method for several numerical tests with polar and...

Fix a holomorphic line bundle ξ over a compact connected Riemann surface X of genus g, with g ≥ 2, and also fix an integer r such that degree(ξ) > r(2g−1). Let Mξ(r) denote the moduli space of stable vector bundles overX of rank r and determinant ξ. The Fourier–Mukai transform, with respect to a Poincaré line bundle on X × J(X), of any F ∈Mξ(r) is a stable vector bundle on J(X). This gives an e...

Why should a number theorist be interested in algebraic geometry? In this course we hope to demonstrate one very good reason, by showing essentially geometric reasons why certain Diophantine equations fail to have solutions. But we will begin by placing the study of Diophantine equations into the context of algebraic geometry, to see how techniques from many different realms of mathematics can ...

The antenna is an important element of radiocommunication, remote sensing and radiolocalisation systems. The measurement of the antenna radiation pattern characteristics allows one to verify the conformity of the antenna. The simplest measurement method consists in the direct far-field measurement. For large antennas, the necessary measurement distance raises a problem which was resolved by the...

Let P be the Prym variety associated with a covering π : Y → X between non-singular irreducible projective curves. If P̃ is a principally polarized Prym-Tyurin variety associated with P , we prove that the induced Abel-Prym morphism ρ̃ : Y → P̃ is birational onto its image for genus gX > 2 and deg π 6= 2. We use such result to prove that the Picard bundle over the Prym variety is simple and moreov...

Non-linear magnetic problems are commonly solved by iterative methods. This paper discusses the application of the Picard-method and the Newton-method in the context of solving time-harmonic problems. It is shown that the overall computation time can be decreased by initiating the non-linear solution process using Picard-iterations and by switching to Newton-iterations as soon as an estimator i...

Picard groups of tensor categories play an important role in rational conformal field theory. The Picard group of the representation category C of a rational vertex algebra can be used to construct examples of (symmetric special) Frobenius algebras in C. Such an algebra A encodes all data needed to ensure the existence of correlators of a local conformal field theory. The Picard group of the ca...

In dimension d, Q-factorial Gorenstein toric Fano varieties with Picard number ρX correspond to simplicial reflexive polytopes with ρX+d vertices. Casagrande showed that any d-dimensional simplicial reflexive polytope has at most 3d vertices, if d is even, respectively, 3d − 1, if d is odd. Moreover, for d even there is up to unimodular equivalence only one such polytope with 3d vertices, corre...

Recently Bai and Yang in [On HSS-based iteration methods for weakly nonlinear systems, Appl. Numer. Math. 59 (2009) 2923–2936.] proposed the Picard-HSS iteration method to solve the system of nonlinear equations Ax = φ(x), where A ∈ Cn×n is a nonHermitian positive definite matrix and φ : D ⊂ C → C is continuously differentiable function defined on the open complex domain D in the n-dimensional ...