We present some recent progresses on Heun functions, gathering results from classical analysis up to elliptic functions. We describe Picard’s generalization of Floquet’s theory for differential equations with doubly periodic coefficients and give the detailed forms of the level one Heun functions in terms of Jacobi theta functions. The finite-gap solutions give an interesting alternative integr...

The motion of a charged particle in straight magnetic field B=B(y)z? with constant perpendicular gradient is solved exactly terms elliptic functions and integrals. can be decomposed periodic along the y-axis drift x-axis. described as trapped symmetric quartic potential y. canonical transformation from coordinates (y,Py) to action–angle (?,J) explicitly generating function S(?,J) that expressed...

We present an innovative technique to add elliptic curve points with the form P Q ± , and discuss its application to the generation of precomputed tables for the scalar multiplication. Our analysis shows that the proposed schemes offer, to the best of our knowledge, the lowest costs for precomputing points on both single and multiple scalar multiplication and for various elliptic curve forms, i...

We prove the uniform convergence of the multigrid V -cycle on graded meshes for corner-like singularities of elliptic equations on a bounded domain Ω ⊂ IR. In particular, using some weighted Sobolev space K a (Ω) and the method of subspace corrections with the elliptic projection decomposition estimate on K a (Ω), we show that the multigrid V -cycle converges uniformly for piecewise linear func...

Using the dynamics of self rational maps of elliptic K3 surfaces together with deformation theory, we prove that the union of rational curves is dense on a very general K3 surface and that the union of elliptic curves is dense in the 1st jet space of a very general K3 surface, both in the strong topology. The techniques developed here also lend themselves to applications to Abel-Jacobi images, ...

The efficient support of cryptographic protocols based on elliptic curves is crucial when embedded processors are adopted as the target hardware platforms. The implementation of Elliptic Curve Cryptography (ECC) offers a variety of design options, mostly covering the specific family of curves and the related coordinate system. At the same time, theory shows that a limited set of solutions can a...