We give a combinatorial characterization of nodal curves admitting a natural (i.e. compatible with and independent of specialization) d-th Abel map for any d ≥ 1. Let X be a smooth projective curve and d a positive integer; the classical d-th Abel map of X , αX : X d −→ PicX , associates to (p1, . . . , pd) ∈ X the class of the line bundle OX(p1 + . . .+ pd) in Pic d X . Such a morphism has goo...

New integrability properties of a family sequences ordinary differential equations, which contains the Riccati and Abel chains as most simple sequences, are studied. The determination n generalized symmetries nth-order equation in each chain provides, without any kind integration, n-1 functionally independent first integrals equation. A remaining integral arises by quadrature using Jacobi last ...

We perform the analytic study of buoyancy-drag equation with a time-dependent acceleration $\gamma(t)$ by two methods. first determine its equivalence class under point transformations Roger Liouville, and thus for some values define Hamiltonian from which can be derived. then Lie symmetries equation, only exist including previous ones, plus additional classes accelerations is reducible to an A...

The basic linear, Volterra integral equation of the first kind with a weakly singular kernel is solved via a Galerkin approximation. It is shown that the approximate solution is a sum with the first term being the solution of Abel's equation and the remaining terms computable as components of the solution of an initialvalue problem. The method represents a significant decrease in the normal num...

Tetration F as the analytic solution of equations F (z − 1) = ln(F (z)), F (0) = 1 is considered. The representation is suggested through the integral equation for values of F at the imaginary axis. Numerical analysis of this equation is described. The straightforward iteration converges within tens of cycles; with double precision arithmetics, the residual of order of 1.e14 is achieved. The nu...