Harmonic and gold Sturmian words

AbstractIn the combinatorics of Sturmian words an essential role is played by the set PER of all finite words w on the alphabet A={a,b} having two periods p and q which are coprime and such that |w|=p+q−2. As is well known, the set St of all finite factors of all Sturmian words equals the set of factors of PER. Moreover, the elements of PER have many remarkable structural properties. In particular, the relation Stand=A∪PER{ab,ba} holds, where Stand is the set of all finite standard Sturmian words. In this paper we introduce two proper subclasses of PER that we denote by Harm and Gold. We call an element of Harm a harmonic word and an element of Gold a gold word. A harmonic word w beginning with the letter x is such that the ratio of two periods p/q, with p<q, is equal to its slope, i.e., (|w|y+1)/(|w|x+1), where {x,y}={a,b}. A gold word is an element of PER such that p and q are primes. Some characterizations of harmonic words are given and the number of harmonic words of each length is computed. Moreover, we prove that St is equal to the set of factors of Harm and to the set of factors of Gold. We introduce also the classes Harm and Gold of all infinite standard Sturmian words having infinitely many prefixes in Harm and Gold, respectively. We prove that Gold∩Harm contain continuously many elements. Finally, some conjectures are formulated

On an involution of Christoffel words and Sturmian morphisms

There is a natural involution on Christoffel words, originally studied by the second author in [A. de Luca, Combinatorics of standard Sturmian words, Lecture Notes in Computer Science 1261 (1997) 249–267]. We show that it has several equivalent definitions: one of them uses the slope of the word, and changes the numerator and the denominator respectively in their inverses modulo the length; another one uses the cyclic graph allowing the construction of the word, by interpreting it in two ways (one as a permutation and its ascents and descents, coded by the two letters of the word, the other in the setting of the Fine and Wilf periodicity theorem); a third one uses central words and generation through iterated palindromic closure, by reversing the directive word. We show further that this involution extends to Sturmian morphisms, in the sense that it preserves conjugacy classes of these morphisms, which are in bijection with Christoffel words. The involution on morphisms is the restriction of some conjugation of the automorphisms of the free group. Finally, we show that, through the geometrical interpretation of substitutions of Arnoux and Ito, our involution is the same thing as duality of endomorphisms (modulo some conjugation)

Well-formed scales, non-well-formed words and the Christoffel duality

Tesis inédita de la Universidad Complutense de Madrid, Facultad de Ciencias Matemáticas, leída el 02-02-2016La presente tesis analiza las escalas musicales generadas desde la perspectiva y las técnicas que ofrece la combinatoria algebraica de palabras. La noción de escala musical es una de las más primitivas: intuitivamente se puede reducir a un conjunto de notas ordenadas seg un la frecuencia de su fundamental (altura del sonido). Ya desde tiempos de la Escuela Pitagórica se vio que al pulsar una cuerda tensa, los sonidos que mejor suenan juntos, los más consonantes, están determinados por unas longitudes de cuerda cuyas proporciones son números fraccionarios sencillos. El más consonante de ellos, la octava, tiene una relación de longitudes 2:1. Este intervalo es tan consonante, que muchas veces los sonidos cuyas frecuencias están separadas en una octava suenan indistinguibles. Es por ello por lo que al estudiar las escalas se suelen identificar las notas cuya distancia es de una o varias octavas. Como resultado, suele entenderse por escala un conjunto de notas dentro de un rango de una octava, transportando dicha secuencia al resto de octavas en caso de necesidad. La definición formal de escala se llevar a a cabo en la sección 2.2, donde se mostrar a como cada octava puede representarse geométricamente mediante una circunferencia unitaria o, aritméticamente, como el conjunto cociente R=Z, es decir, como el intervalo (0,1]. De esta forma, una escala queda determinada por un conjunto de números ordenados entre el 0 y el 1 o bien, geométricamente, por un polígono inscrito en el círculo unidad...Fac. de Ciencias MatemáticasTRUEunpu

Giant non-linear susceptibility of hydrogenic donors in silicon and germanium

Implicit summation is a technique for the conversion of sums over intermediate states in multiphoton absorption and the high-order susceptibility in hydrogen into simple integrals. Here, we derive the equivalent technique for hydrogenic impurities in multi-valley semiconductors. While the absorption has useful applications, it is primarily a loss process; conversely, the non-linear susceptibility is a crucial parameter for active photonic devices. For Si:P, we predict the hyperpolarizability ranges from $\chi^{(3)}/n_{\text{3D}}=2.9$ to $580 \times 10^{-38}$ $\text{m}^5/\text{V}^2$ depending on the frequency, even while avoiding resonance. Using samples of a reasonable density, $n_{\text{3D}}$, and thickness, $L$, to produce third-harmonic generation at 9 THz, a frequency that is difficult to produce with existing solid-state sources, we predict that $\chi^{(3)}$ should exceed that of bulk InSb and $\chi^{(3)}L$ should exceed that of graphene and resonantly enhanced quantum wells

Development of a modular quantum-chemistry framework for the investigation of novel basis functions

State-of-the-art methods for the calculation of electronic structures of molecules predominantly use Gaussian basis functions. The algorithms employed inside existing code packages are consequently often highly optimised keeping only their numerical requirements in mind. For the investigation of novel approaches, utilising other basis functions, this is an obstacle, since requirements might differ. In contrast, this thesis develops the highly flexible program package molsturm, which is designed in order to facilitate rapid design, implementation and assessment of methods employing different basis function types. A key component of molsturm is a Hartree-Fock (HF) self-consistent field (SCF) scheme, which is suitable to be combined with any basis function type. First the mathematical background of quantum mechanics as well as some numerical techniques are reviewed. Care is taken to emphasise the often overlooked subtleties when discretising an infinite-dimensional spectral problem in order to obtain a finite-dimensional eigenproblem. Common quantum-chemical methods such as full configuration interaction and HF are discussed providing insight into their mathematical properties. Different formulations of HF are contrasted and appropriate SCF solution schemes formulated. Next discretisation approaches based on four different types of basis functions are compared both with respect to the computational challenges as well as their ability to describe the physical features of the wave function. Besides (1) Slater-type orbitals and (2) Gaussian-type orbitals, the discussion considers (3) finite elements, which are piecewise polynomials on a grid, as well as (4) Coulomb-Sturmians, which are the analytical solutions to a Schrödinger-like equation. A novel algorithmic approach based on matrix-vector contraction expressions is developed, which is able to adapt to the numerical requirements of all basis functions considered. It is shown that this ansatz not only allows to formulate SCF algorithms in a basis-function independent way, but furthermore improves the theoretically achievable computational scaling for finite-element-based discretisations as well as performance improvements for Coulomb-Sturmian-based discretisations. The adequacy of standard SCF algorithms with respect to a contraction-based setting is investigated and for the example of the optimal damping algorithm an approximate modification to achieve such a setting is presented. With respect to recent trends in the development of modern computer hardware the potentials and drawbacks of contraction-based approaches are evaluated. One drawback, namely the typically more involved and harder-to-read code, is identified and a data structure named lazy matrix is introduced to overcome this. Lazy matrices are a generalisation of the usual matrix concept, suitable for encapsulating contraction expressions. Such objects still look like matrices from the user perspective, including the possibility to perform operations like matrix sums and products. As a result programming contraction-based algorithms becomes similarly convenient as working with normal matrices. An implementation of lazy matrices in the lazyten linear algebra library is developed in the course of the thesis, followed by an example demonstrating the applicability in the context of the HF problem. Building on top of the aforementioned concepts the design of molsturm is outlined. It is shown how a combination of lazy matrices and a contraction-based SCF scheme separates the code describing the SCF procedure from the code dealing with the basis function type. It is discussed how this allows to add a new basis function type to molsturm by only making code changes in a single integral interface library. On top of that, we demonstrate by the means of examples how the readily scriptable interface of molsturm can be employed to implement and assess novel quantum-chemical methods or to combine the features of molsturm with existing third-party packages. Finally, the thesis discusses an application of molsturm towards the investigation of the convergence properties of Coulomb-Sturmian-based quantum-chemical calculations. Results for the convergence of the ground-state energies at HF level are reported for atoms of the second and the third period of the periodic table. Particular emphasis is put on a discussion about the required maximal angular momentum quantum numbers in order to achieve convergence of the discretisation of the angular part of the wave function. Some modifications required for a treatment at correlated level are suggested, followed by a discussion of the effect of the Coulomb-Sturmian exponent. An algorithm for obtaining an optimal exponent is devised and some optimal exponents for the atoms of the second and the third period of the periodic table at HF level are given. Furthermore, the first results of a Coulomb-Sturmian-based excited states calculation based on the algebraic-diagrammatic construction scheme for the polarisation propagator are presented