Vol 51, No 4 (2016)

The paper considers a class of g-natural metrics G on the tangent bundle TM of a Riemannian manifold (M, g). We prove that the flatness of g is necessary and sufficient for a metric G to be weakly symmetric (recurrent or pseudo-symmetric). Also, it is shown that the weak symmetry and recurrent or pseudo-symmetry properties of Sasakian lift metric, studied by Bejan and Crasmareanu, and Binh and Tamássy, respectively, are special cases of our result.

We prove a new asymptotic formula for the eigenvalues of Sturm-Liouville problem with summable potential. The obtained result extends and make more precise previously known formulas, and takes into account the smooth dependence of the spectral data on boundary conditions.

The paper is devoted to the questions of invariance of the index of a semielliptical operator on the scale of anisotropic spaces. A necessary and sufficient condition for the invariance of the index of a linear bounded operator on the embedded Hilbert spaces is obtained. This condition is applied to study the invariance of the index of a semielliptical operator. For a semielliptical operator a sufficient condition for the invariance of the index on the scale of anisotropic spaces is obtained.

The paper deals with the uniqueness problems when two meromorphic functions f and g share three distinct values CM and f satisfies the first, second or fourth Painlevé transcendents.