Beltrami vector fields with an icosahedral symmetry

A vector field is called a Beltrami vector field, if $B\times(\nabla\times B)=0$. In this paper we construct two unique Beltrami vector fields $\mathfrak{I}$ and $\mathfrak{Y}$, such that $\nabla\times\mathfrak{I}=\mathfrak{I}$, $\nabla\times\mathfrak{Y}=\mathfrak{Y}$, and such that both have an orientation-preserving icosahedral symmetry. Both of them have an additional symmetry with respect to a non-trivial automorphism of the number field $\mathbb{Q}(\,\sqrt{5}\,)$.Comment: 22 pages, 9 figure

Planar 2-homogeneous commutative rational vector fields

In this paper we prove the following result: if two 2-dimensional 2-homogeneous rational vector fields commute, then either both vector fields can be explicitly integrated to produce rational flows with orbits being lines through the origin, or both flows can be explicitly integrated in terms of algebraic functions. In the latter case, orbits of each flow are given in terms of $1$-homogeneous rational functions $W$ as curves $W(x,y)=\textrm{const}$. An exhaustive method to construct such commuting algebraic flows is presented. The degree of the so-obtained algebraic functions in two variables can be arbitrarily high.Comment: 23 page

The Minkowski ?(x) function and Salem's problem

R. Salem (Trans. Amer. Math. Soc. 53 (3) (1943) 427-439) asked whether the Fourier-Stieltjes transform of the Minkowski question mark function ?(x) vanishes at infinity. In this note we present several possible approaches towards the solution. For example, we show that this transform satisfies integral and discrete functional equations. Thus, we expect the affirmative answer to Salem's problem. In the end of this note we show that recent attempt to settle this question (S. Yakubovich, C. R. Acad. Sci. Paris, Ser. I 349 (11-12) (2011) 633-636) is fallacious.Comment: 4 pages. C. R. Acad. Sci. Paris, Ser. I. (2012