Quasiperiodic Sturmian words and morphisms

AbstractWe characterize all quasiperiodic Sturmian words: A Sturmian word is not quasiperiodic if and only if it is a Lyndon word. Moreover, we study links between Sturmian morphisms and quasiperiodicity

On Quasiperiodic Morphisms

Weakly and strongly quasiperiodic morphisms are tools introduced to study quasiperiodic words. Formally they map respectively at least one or any non-quasiperiodic word to a quasiperiodic word. Considering them both on finite and infinite words, we get four families of morphisms between which we study relations. We provide algorithms to decide whether a morphism is strongly quasiperiodic on finite words or on infinite words.Comment: 12 page

A Characterization of Infinite LSP Words

G. Fici proved that a finite word has a minimal suffix automaton if and only if all its left special factors occur as prefixes. He called LSP all finite and infinite words having this latter property. We characterize here infinite LSP words in terms of $S$-adicity. More precisely we provide a finite set of morphisms $S$ and an automaton ${\cal A}$ such that an infinite word is LSP if and only if it is $S$-adic and all its directive words are recognizable by ${\cal A}$

Unambiguous 1-Uniform Morphisms

A morphism h is unambiguous with respect to a word w if there is no other morphism g that maps w to the same image as h. In the present paper we study the question of whether, for any given word, there exists an unambiguous 1-uniform morphism, i.e., a morphism that maps every letter in the word to an image of length 1.Comment: In Proceedings WORDS 2011, arXiv:1108.341

Propofol requirement and EEG alpha band power during general anesthesia provide complementary views on preoperative cognitive decline

Background: Although cognitive decline (CD) is associated with increased post-operative morbidity and mortality, routinely screening patients remains difficult. The main objective of this prospective study is to use the EEG response to a Propofol-based general anesthesia (GA) to reveal CD. Methods: 42 patients with collected EEG and Propofol target concentration infusion (TCI) during GA had a preoperative cognitive assessment using MoCA. We evaluated the performance of three variables to detect CD (MoCA < 25 points): age, Propofol requirement to induce unconsciousness (TCI at SEF95: 8–13 Hz) and the frontal alpha band power (AP at SEF95: 8–13 Hz). Results: The 17 patients (40%) with CD were significantly older (p < 0.001), had lower TCI (p < 0.001), and AP (p < 0.001). We found using logistic models that TCI and AP were the best set of variables associated with CD (AUC: 0.89) and performed better than age (p < 0.05). Propofol TCI had a greater impact on CD probability compared to AP, although both were complementary in detecting CD. Conclusion: TCI and AP contribute additively to reveal patient with preoperative cognitive decline. Further research on post-operative cognitive trajectory are necessary to confirm the interest of intra operative variables in addition or as a substitute to cognitive evaluation

Morphic Primitivity and Alphabet Reductions

An alphabet reduction is a 1-uniform morphism that maps a word to an image that contains a smaller number of dfferent letters. In the present paper we investigate the effect of alphabet reductions on morphically primitive words, i. e., words that are not a fixed point of a nontrivial morphism. Our first main result answers a question on the existence of unambiguous alphabet reductions for such words, and our second main result establishes whether alphabet reductions can be given that preserve morphic primitivity. In addition to this, we study Billaud's Conjecture - which features a dfferent type of alphabet reduction, but is otherwise closely related to the main subject of our paper - and prove its correctness for a special case

Sublinear Algorithms for Approximating String Compressibility

We raise the question of approximating the compressibility of a string with respect to a fixed compression scheme, in sublinear time. We study this question in detail for two popular lossless compression schemes: run-length encoding (RLE) and a variant of Lempel-Ziv (LZ77), and present sublinear algorithms for approximating compressibility with respect to both schemes. We also give several lower bounds that show that our algorithms for both schemes cannot be improved significantly. Our investigation of LZ77 yields results whose interest goes beyond the initial questions we set out to study. In particular, we prove combinatorial structural lemmas that relate the compressibility of a string with respect to LZ77 to the number of distinct short substrings contained in it (its ℓth subword complexity , for small ℓ). In addition, we show that approximating the compressibility with respect to LZ77 is related to approximating the support size of a distribution.National Science Foundation (U.S.) (Award CCF-1065125)National Science Foundation (U.S.) (Award CCF-0728645)Marie Curie International Reintegration Grant PIRG03-GA-2008-231077Israel Science Foundation (Grant 1147/09)Israel Science Foundation (Grant 1675/09

Directive words of episturmian words

Episturmian morphisms constitute a powerful tool to study episturmian words. Indeed, any episturmian word can be infinitely decomposed over the set of pure episturmian morphisms. Thus, an episturmian word can be defined by one of its morphic decompositions or, equivalently, by a certain directive word. Here we characterize pairs of words directing a common episturmian word. As a consequence, we characterize episturmian words having a unique directive word

On a generalisation of trapezoidal words

The factor complexity function Cw(n) of a finite or infinite word w associates to each integer n ≥ 0 the number of distinct factors (i.e., blocks of consecutive letters) in w of length n. A finite word w of length │w│ is said to be trapezoidal if the graph of its factor complexity Cw(n) as a function of n (for 0 6 n 6 │w│) is that of a regular trapezoid (or possibly an isosceles triangle); that is, Cw(n) increases by 1 with each n on some interval of length r, then Cw(n) is constant on some interval of length s, and finally Cw(n) decreases by 1 with each n on an interval of the same length r. Necessarily Cw(1) = 2 (since there is one factor of length 0, namely the empty word), so any trapezoidal word is on a binary alphabet. Trapezoidal words were first introduced by A. de Luca (1999) when studying the behaviour of the factor complexity of finite Sturmian words, i.e., factors of infinite “cutting sequences", obtained by coding the sequence of cuts in an integer lattice over the positive quadrant of R2 made by a line of irrational slope. Every finite Sturmian word is trapezoidal, but not conversely. However, both families of words (trapezoidal and Sturmian) are special classes of so-called rich words - a new (wider) class of finite and infinite words characterised by containing the maximal number of palindromes { recently introduced by the speaker, together with J. Justin, S. Widmer, and L.Q. Zamboni (2009). In this talk, I will introduce a natural generalisation of trapezoidal words over an arbitrary finite alphabet A consisting of at least two distinct letters, called generalised trapezoidal words (or GT-words for short). In particular, I will discuss some combinatorial properties of this new class of words when │A│ ≥ 3 and I will show that, unlike in the binary case (│A│ = 2), not all GT-words are rich in palindromes, but we do have a neat characterisation of those that are. This work was inspired by a question of Ian Wanless at the 54th Annual Conference of the Australian Mathematical Society last year (2010)