The basic methods of constructing the sets of mutually unbiased bases in the Hilbert space of an arbitrary finite dimension are reviewed and an emerging link between them is outlined. It is shown that these methods employ a wide range of important mathematical concepts like, e.g., Fourier transforms, Galois fields and rings, finite, and related projective geometries, and entanglement, to mention a few. Some applications of the theory to quantum information tasks are also mentioned.

An aperiodic (low frequency) spectrum may originate from the error term in the mean value of an arithmetical function such as Möbius function or Mangoldt function, which are coding sequences for prime numbers. In the discrete Fourier transform the analyzing wave is periodic and not well suited to represent the low frequency regime. In place we introduce a new signal processing tool based on the

No abstract

On explicite l'approche de Cohn (1955) de la théorie de Markoff. On montre en particulier comment l'arbre complet des solutions de l'équation diophantienne associée apparasît comme quotient du groupeGL (2,ℤ)des matrices2×2à coefficients entiers et de déterminant±1par un sous-groupe diédralD6à12éléments. Différents développements intermédiaires sont faits autour du groupeAut (F 2)des automorphismes du groupe libre engendré par deux élémentsF 2.We detail the approach followed by Cohn for the Markoff theory. We show particularly how appears the whole tree of solutions for the associated Diophantine equation as a quotient of the groupGL (2,ℤ)of matrices2×2with integer coefficients and determinant±1by its dihedral subgroupD6with12elements. Some developments are made with the groupAut (F 2)of automorphisms of the free groupF 2generated by two elements.

No abstract