Local symmetry dynamics in one-dimensional aperiodic lattices

A unifying description of lattice potentials generated by aperiodic one-dimensional sequences is proposed in terms of their local reflection or parity symmetry properties. We demonstrate that the ranges and axes of local reflection symmetry possess characteristic distributional and dynamical properties which can be determined for every aperiodic binary lattice. A striking aspect of such a property is given by the return maps of sequential spacings of local symmetry axes, which typically traverse few-point symmetry orbits. This local symmetry dynamics allows for a classification of inherently different aperiodic lattices according to fundamental symmetry principles. Illustrating the local symmetry distributional and dynamical properties for several representative binary lattices, we further show that the renormalized axis spacing sequences follow precisely the particular type of underlying aperiodic order. Our analysis thus reveals that the long-range order of aperiodic lattices is characterized in a compellingly simple way by its local symmetry dynamics.Comment: 15 pages, 12 figure

On the Combinatorics of Palindromes and Antipalindromes

We prove a number of results on the structure and enumeration of palindromes and antipalindromes. In particular, we study conjugates of palindromes, palindromic pairs, rich words, and the counterparts of these notions for antipalindromes.Comment: 13 pages/ submitted to DLT 201

On the origin of the mitochondrial genetic code: Towards a unified mathematical framework for the management of genetic information

The origin of the genetic code represents one of the most challenging problems in molecular evolution. The genetic code is an important universal feature of extant organisms and indicates a common ancestry of different forms of life on earth. Known variants of the genetic code can be mainly divided in mitochondrial and nuclear classes. Here we provide a new insight on the origin of the mitochondrial genetic code: we found that its degeneracy distribution can be explained by using a mathematical approach recently developed for the description of the Euplotes nuclear variant of the genetic code. The results point to a primeval mitochondrial genetic code composed of four base codons, which we call tesserae, that, among other features, exhibit outstanding error detection capabilities. The theoretical description suggests also a formulation of a plausible biological theory about the origin of protein coding. Such theory is based on the symmetry properties of hypothetical primeval chemical adaptors between nucleic acids and amino acids (ancient tRNA’s). Our paper provides a unified mathematical framework for different hypotheses on the origin of genetic coding. Also, it contributes to revisit our present view about the evolutionary steps that led to extant genetic codes by giving a new first-principles perspective on the difficult problem of the origin of the genetic code, and consequently, on the origin of life on earth

EERTREE: An Efficient Data Structure for Processing Palindromes in Strings

We propose a new linear-size data structure which provides a fast access to all palindromic substrings of a string or a set of strings. This structure inherits some ideas from the construction of both the suffix trie and suffix tree. Using this structure, we present simple and efficient solutions for a number of problems involving palindromes.Comment: 21 pages, 2 figures. Accepted to IWOCA 201