Communications in Mathematics |

In this note, we prove that there is no transcendental entire function f(z) ∈ ℚ[[z]] such that f(ℚ) ⊆ ℚ and den f(p/q) = F(q), for all sufficiently large q, where F(z) ∈ ℤ[z].

In a classic paper [14], W.G. Spohn established the to-date sharpest estimates from below for the simultaneous Diophantine approximation constants for three and more real numbers. As a by-result of his method which used Blichfeldt’s Theorem and the calculus of variations, he derived a bound for the critical determinant of the star body |x1|(|x1|3 + |x2|3 + |x3|3 ≤ 1. In this little note, after a brief exposition of the basics of the geometry of numbers and its significance for Diophantine approximation, this latter result is improved and extended to the star body |x1|(|x1|3 + |x2 2 + x3 2)3/2≤ 1.

Let Ω ⊂ ℝn be a bounded starshaped domain and consider the (p; q)-Laplacian problem -∆pu - ∆pu = λ(x)|u|p*-2u + μ|u|r-2u where μ is a positive parameter, 1 < q ≤ p < n, r ≥ p* and is the critical Sobolev exponent. In this short note we address the question of non-existence for non-trivial solutions to the (p; q)-Laplacian problem. In particular we show the non-existence of non-trivial solutions to the problem by using a method based on Pohozaev identity.

The main result establishes that a weak solution of degenerate semilinear elliptic equations can be approximated by a sequence of solutions for non-degenerate semilinear elliptic equations.

Let be the triangular algebra consisting of unital algebras A and B over a commutative ring R with identity 1 and M be a unital (A; B)-bimodule. An additive subgroup L of A is said to be a Lie ideal of A if [L;A] ⊆ L. A non-central square closed Lie ideal L of A is known as an admissible Lie ideal. The main result of the present paper states that under certain restrictions on A, every generalized Jordan triple higher derivation of L into A is a generalized higher derivation of L into A.

In this work, we study the existence and uniqueness of weak solu- tions of fourth-order degenerate parabolic equation with variable exponent using the di erence and variation methods.

We correct misprints in a formula in the last sentence at the end of page 129; the first paragraph of subsection 4:1; misprints at the end of page 132 and in Proposition 1 at page 133 of the paper ‘Variations by generalized symmetries of local Noether strong currents equivalent to global canonical Noether currents’, Communications in Mathematics 24 (2016), 125-135. DOI: 10.1515/cm-2016-0009