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...

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...

We consider diagonal cubic surfaces defined by an equation of the form ax + by + cz + dt = 0. Numerically, one can find all rational points of height 6 B for B in the range of up to 10 , thanks to a program due to D. J. Bernstein. On the other hand, there are precise conjectures concerning the constants in the asymptotics of rational points of bounded height due to Manin, Batyrev and the author...

When addressing the question of variable saturation and density groundwater flow in coastal zones, highly nonlinear system coupled water-salt equations may deserve more attention. The classical Picard scheme is associated with slow calculation speeds low precision, which hardly meet actual needs users. Here, we developed a new numerical solution for issues based on Newton compared advantages di...