Heidi Burgiel

How to lose at tetris

Tetris is a computer game which has obsessed many computer users and attracted much attention, despite the simplicity of its rules. This paper addresses the question: ‘can you “win” the game Tetris?’ Designed by Soviet mathematician Alexey Pazhitnov in the late eighties and imported to the United States by Spectrum Holobyte, Tetris won a record number of software awards in 1989. Versions of Tetris are sold for most personal computers. There are Tetris arcade games, Tetris Nintendo cartridges, and hand-held Tetris games; Tetris has been played on machines ranging from mainframes to calculators. The games success has prompted the invention of several similar games, including Hextris, Welltris , and Wordtris .

Gabor orthonormal bases generated by indicator functions of parallelepiped-shaped sets

Abstract The main objective of the present work is to provide a procedure to construct Gabor orthonormal bases generated by indicator functions of parallelepiped-shaped sets. Given two full-rank lattices of the same volume, we investigate conditions under which there exists a common fundamental domain which is the image of a unit cube under an invertible linear operator. More precisely, we provide a characterization of pairs of full-rank lattices in ℝ d {\mathbb{R}^{d}} admitting common connected fundamental domains of the type N ⁢ [ 0 , 1 ) d {N[0,1)^{d}} , where N is an invertible matrix. As a byproduct of our results, we are able to construct a large class of Gabor windows which are indicator functions of sets of the type N ⁢ [ 0 , 1 ) d {N[0,1)^{d}} . We also apply our results to construct multivariate Gabor frames generated by smooth windows of compact support. Finally, we prove in the two-dimensional case that there exists an uncountable family of pairs of lattices of the same volume which do not admit a common connected fundamental domain of the type N ⁢ [ 0 , 1 ) 2 {N[0,1)^{2}} , where N is an invertible matrix.