Unavoidable Sets of Partial Words

The notion of an unavoidable set of words appears frequently in the fields of mathematics and theoretical computer science, in particular with its connection to the study of combinatorics on words. The theory of unavoidable sets has seen extensive study over the past twenty years. In this paper we extend the definition of unavoidable sets of words to unavoidable sets of partial words. Partial words, or finite sequences that may contain a number of ?do not know? symbols or ?holes,? appear naturally in several areas of current interest such as molecular biology, data communication, and DNA computing. We demonstrate the utility of the notion of unavoidability of sets of partial words by making use of it to identify several new classes of unavoidable sets of full words. Along the way we begin work on classifying the unavoidable sets of partial words of small cardinality. We pose a conjecture, and show that affirmative proof of this conjecture gives a sufficient condition for classifying all the unavoidable sets of partial words of size two. We give a result which makes the conjecture easy to verify for a significant number of cases. We characterize many forms of unavoidable sets of partial words of size three over a binary alphabet, and completely characterize such sets over a ternary alphabet. Finally, we extend our results to unavoidable sets of partial words of size k over a k-letter alphabet

On morphisms preserving palindromic richness

It is known that each word of length $n$ contains at most $n+1$ distinct palindromes. A finite rich word is a word with maximal number of palindromic factors. The definition of palindromic richness can be naturally extended to infinite words. Sturmian words and Rote complementary symmetric sequences form two classes of binary rich words, while episturmian words and words coding symmetric $d$-interval exchange transformations give us other examples on larger alphabets. In this paper we look for morphisms of the free monoid, which allow to construct new rich words from already known rich words. We focus on morphisms in Class $P_{ret}$. This class contains morphisms injective on the alphabet and satisfying a particular palindromicity property: for every morphism $\varphi$ in the class there exists a palindrome $w$ such that $\varphi(a)w$ is a first complete return word to $w$ for each letter $a$. We characterize $P_{ret}$ morphisms which preserve richness over a binary alphabet. We also study marked $P_{ret}$ morphisms acting on alphabets with more letters. In particular we show that every Arnoux-Rauzy morphism is conjugated to a morphism in Class $P_{ret}$ and that it preserves richness

An algorithm to test if a given circular HDOL-language avoids a pattern

To prove that a pattern p is avoidable on a given alphabet, one has to construct an infinite language L that avoids p. Usually, L is a DOLlanguage (obtained by iterating a morphism h) or a HDOL-language (obtained by coding a DOL-language with another morphism g). Our purpose is to find an algorithm to test, given a HDOL-system G, whether the language L(G) generated by this system avoids p. We first define the notions of circular morphism, circular DOL-system and circular HDOL-system, and we show how to compute the inverse image of a pattern by a circular morphism. Then we prove that by computing successive inverse images of p, we can decide whether the language L(G) avoids p for any fixed pattern p (which may even contain constants), provided that the HDOL-system G is circular and expansive. 1 Introduction The theory of avoidable patterns, introduced by Zimin [13] and Bean, Ehrenfeucht and McNulty [2], generalizes problems studied by Axel Thue [12] and many others, such as the ex..

Application of computational limit analysis to soil-structure interaction in masonry arch bridges.

For the assessment of Masonry Arch Bridges (MAB), many structural and material models have been applied, ranging from sophisticated non-linear finite element analysis models to much simpler rigid-block limit analysis models. i.e. elastic and plastic methods respectively. The application of elastic analysis to MAB suffers many drawbacks since it requires full mechanical characterization of ancient masonry structures. The mechanical characterization of ancient masonry is difficult since these structures have typically undergone a century or more of environmental deterioration and in many cases have been already subjected to extensive modification. Also, sophisticated material models generally require specialized parameters that are hard to assess, particularly if non-destructive tests are used. In these cases practicing engineers typically favour simpler material models, involving fewer parameters. Thus non-linear finite element methods or other sophisticated models may not be a good choice for the assessment of MAB, while simplified approaches for example based on limit analysis principles are likely to be more appropriate. In this research. a holistic computational limit analysis procedure is presented which involves modelling both soil and masonry components explicitly. Masonry bridge parts are discretized using rigid blocks whilst the soil fill is discretized using deformable triangular elements and modelled a.'i a Mohr-Coulomb material with a tension cut-off. Lower and upper bound estimates of the collapse load are obtained. Results are compared with those from recently performed bridge tests carried out in collaboration with the University of Salford. A key project finding is that the use of peak soil strength parameters in limit analysis models is inappropriate when the soil is modelled explicitly. However, use of mobilized strengths appears to be a promising way forward, yielding much closer correlation with experimental data