Oleg N. German

Transference inequalities for multiplicative diophantine exponents

We prove inequalities for multiplicative analogues of Diophantine exponents; these inequalities are similar to the ones known in the classical case. Particularly, we show that a matrix is badly approximable if and only if its transpose is badly approximable and establish some inequalities connecting multiplicative exponents with ordinary ones.

‘Minesweeper’ and spectrum of discrete Laplacians

The article is devoted to a problem inspired by the ‘Minesweeper’ computer game. It is shown that certain configurations of open cells guarantee the existence and the uniqueness of solution. Mathematically the problem is reduced to some spectral properties of discrete differential operators. It is shown how the uniqueness can be used to create a new game which preserves the spirit of ‘Minesweeper’ but does not require a computer.