A Characterization of Bispecial Sturmian Words

A finite Sturmian word w over the alphabet {a,b} is left special (resp. right special) if aw and bw (resp. wa and wb) are both Sturmian words. A bispecial Sturmian word is a Sturmian word that is both left and right special. We show as a main result that bispecial Sturmian words are exactly the maximal internal factors of Christoffel words, that are words coding the digital approximations of segments in the Euclidean plane. This result is an extension of the known relation between central words and primitive Christoffel words. Our characterization allows us to give an enumerative formula for bispecial Sturmian words. We also investigate the minimal forbidden words for the set of Sturmian words.Comment: Accepted to MFCS 201

Towards an S-matrix Description of Gravitational Collapse

Extending our previous results on trans-Planckian ($Gs \gg \hbar$) scattering of light particles in quantum string-gravity we present a calculation of the corresponding S-matrix from the region of large impact parameters ($b \gg G\sqrt{s}>\lambda_s$) down to the regime where classical gravitational collapse is expected to occur. By solving the semiclassical equations of a previously introduced effective-action approximation, we find that the perturbative expansion around the leading eikonal result diverges at a critical value $b = b_c = O(G\sqrt{s})$, signalling the onset of a new (black-hole related?) regime. We then discuss the main features of our explicitly unitary S-matrix -- and of the associated effective metric -- down to (and in the vicinity of) $b = b_c$, and present some ideas and results on its extension all the way to the $b \to 0$ region. We find that for $b<b_c$ the physical field solutions are complex-valued and the S-matrix shows additional absorption, related to a new production mechanism. The field solutions themselves are, surprisingly, everywhere regular, suggesting a quantum-tunneling -- rather than a singular-geometry -- situation.Comment: 39 pages, 5 figures; added discussion sect. 7, added references, acknowledgement