We study square-tiled tori, that is, tori obtained from a finite collection of unit squares by parallel side identifications. Square-tiled can be parametrized in natural way allows to count the number tiled given square tiles. There is $\mathrm{SL}(2,\mathbf{Z})$-action on and we classify $\mathrm{SL}(2,\mathbf{Z})$-orbits using two numerical invariants easily computed. deduce exact size every ...

A square trisection is a problem of assembling three identical squares from a larger square, using a minimal number of pieces. This paper presents an historical overview of the square trisection problem starting with its origins in the third century. We detail the reasoning behind some of the main known solutions. Finally, we give a new solution and three ruler-and-compass constructions. We con...

In this paper, we study the mean square convergence of the kernel least mean square (KLMS). The fundamental energy conservation relation has been established in feature space. Starting from the energy conservation relation, we carry out the mean square convergence analysis and obtain several important theoretical results, including an upper bound on step size that guarantees the mean square con...

We answer a question of Harju: For every n > 3 there is a square-free ternary word of length n with a square-free self-shuffle.