The Schrödinger equation for a quantum particle in a two-dimensional triangular billiard can be written as the Helmholtz equation with a Dirichlet boundary condition. We numerically explore the quantum entanglement of the eigenfunctions of the triangle billiard and its relation to the irrationality of the triangular geometry. We also study the entanglement dynamics of the coherent state with it...

The elliptic norm curve Qn in space S„_i admits a group G2ni of collineations, and in fact there is a single infinity of such curves which admit the same group. A particular Qn of the family is distinguished by a value of the parameter t , itself an elliptic modular function defined by the modular group congruent to identity (mod. n). In the group G2n2, there are certain involutory collineation...

Let w be a morphic word over a finite alphabet Σ, and let ∆ be a nonempty subset of Σ. We study the behavior of maximal blocks consisting only of letters from ∆ in w, and prove the following: let (ik, jk) denote the starting and ending positions, respectively, of the k’th maximal ∆-block in w. Then lim supk→∞(jk/ik) is algebraic if w is morphic, and rational if w is automatic. As a result, we s...