A Coloring Problem for Sturmian and Episturmian Words

We consider the following open question in the spirit of Ramsey theory: Given an aperiodic infinite word $w$, does there exist a finite coloring of its factors such that no factorization of $w$ is monochromatic? We show that such a coloring always exists whenever $w$ is a Sturmian word or a standard episturmian word

Palindromic Decompositions with Gaps and Errors

Identifying palindromes in sequences has been an interesting line of research in combinatorics on words and also in computational biology, after the discovery of the relation of palindromes in the DNA sequence with the HIV virus. Efficient algorithms for the factorization of sequences into palindromes and maximal palindromes have been devised in recent years. We extend these studies by allowing gaps in decompositions and errors in palindromes, and also imposing a lower bound to the length of acceptable palindromes. We first present an algorithm for obtaining a palindromic decomposition of a string of length n with the minimal total gap length in time O(n log n * g) and space O(n g), where g is the number of allowed gaps in the decomposition. We then consider a decomposition of the string in maximal \delta-palindromes (i.e. palindromes with \delta errors under the edit or Hamming distance) and g allowed gaps. We present an algorithm to obtain such a decomposition with the minimal total gap length in time O(n (g + \delta)) and space O(n g).Comment: accepted to CSR 201

On substitutions closed under derivation: examples

We study infinite words fixed by a morphism and their derived words. A derived word is a coding of return words to a factor. We exhibit two examples of sets of morphisms which are closed under derivation --- any derived word with respect to any factor of the fixed point is again fixed by a morphism from this set. The first example involves standard episturmian morphisms, and the second concerns the period doubling morphism.Comment: 10 pages, 1 figures, submitted to Words 201

Palindromic complexity of trees

We consider finite trees with edges labeled by letters on a finite alphabet $\varSigma$. Each pair of nodes defines a unique labeled path whose trace is a word of the free monoid $\varSigma^*$. The set of all such words defines the language of the tree. In this paper, we investigate the palindromic complexity of trees and provide hints for an upper bound on the number of distinct palindromes in the language of a tree.Comment: Submitted to the conference DLT201

On Words with the Zero Palindromic Defect

We study the set of finite words with zero palindromic defect, i.e., words rich in palindromes. This set is factorial, but not recurrent. We focus on description of pairs of rich words which cannot occur simultaneously as factors of a longer rich word

Specular sets

We introduce the notion of specular sets which are subsets of groups called here specular and which form a natural generalization of free groups. These sets are an abstract generalization of the natural codings of linear involutions. We prove several results concerning the subgroups generated by return words and by maximal bifix codes in these sets.Comment: arXiv admin note: substantial text overlap with arXiv:1405.352

On the Number of Closed Factors in a Word

A closed word (a.k.a. periodic-like word or complete first return) is a word whose longest border does not have internal occurrences, or, equivalently, whose longest repeated prefix is not right special. We investigate the structure of closed factors of words. We show that a word of length $n$ contains at least $n+1$ distinct closed factors, and characterize those words having exactly $n+1$ closed factors. Furthermore, we show that a word of length $n$ can contain $\Theta(n^{2})$ many distinct closed factors.Comment: Accepted to LATA 201

Sturmian morphisms, the braid group B_4, Christoffel words and bases of F_2

We give a presentation by generators and relations of a certain monoid generating a subgroup of index two in the group Aut(F_2) of automorphisms of the rank two free group F_2 and show that it can be realized as a monoid in the group B_4 of braids on four strings. In the second part we use Christoffel words to construct an explicit basis of F_2 lifting any given basis of the free abelian group Z^2. We further give an algorithm allowing to decide whether two elements of F_2 form a basis or not. We also show that, under suitable conditions, a basis has a unique conjugate consisting of two palindromes.Comment: 25 pages, 4 figure

Strain-Driven Mn-Reorganization in Overlithiated LixMn2O4 Epitaxial Thin-Film Electrodes

Lithium manganate LixMn2O4 (LMO) is a lithium ion cathode that suffers from the widely observed but poorly understood phenomenon of capacity loss due to Mn dissolution during electrochemical cycling. Here, operando X-ray reflectivity (low- and high-angle) is used to study the structure and morphology of epitaxial LMO (111) thin film cathodes undergoing lithium insertion and extraction to understand the inter-relationships between biaxial strain and Mn-dissolution. The initially strain-relieved LiMn2O4 films generate in-plane tensile and compressive strains for delithiated (x 1) charge states, respectively. The results reveal reversible Li insertion into LMO with no measurable Mn-loss for 0 1) reveals Mn loss from LMO along with dramatic changes in the intensity of the (111) Bragg peak that cannot be explained by Li stoichiometry. These results reveal a partially reversible site reorganization of Mn ions within the LMO film that is not seen in bulk reactions and indicates a transition in Mn-layer stoichiometry from 3:1 to 2:2 in alternating cation planes. Density functional theory calculations confirm that compressive strains (at x = 2) stabilize LMO structures with 2:2 Mn site distributions, therefore providing new insights into the role of lattice strain in the stability of LMO