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

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 ${\cal A}$

Asymptotic behavior of the number of solutions for non-Archimedean Diophantine approximations with restricted denominators

AbstractWe consider metric results for the asymptotic behavior of the number of solutions of Diophantine approximation inequalities with restricted denominators for Laurent formal power series with coefficients in a finite field. We especially consider approximations by rational functions whose denominators are powers of irreducible polynomials, and study the strong law of large numbers for the number of solutions of the inequalities under consideration

Pairwise Well-Formed Modes and Transformations

One of the most significant attitudinal shifts in the history of music occurred in the Renaissance, when an emerging triadic consciousness moved musicians towards a new scalar formation that placed major thirds on a par with perfect fifths. In this paper we revisit the confrontation between the two idealized scalar and modal conceptions, that of the ancient and medieval world and that of the early modern world, associated especially with Zarlino. We do this at an abstract level, in the language of algebraic combinatorics on words. In scale theory the juxtaposition is between well-formed and pairwise well-formed scales and modes, expressed in terms of Christoffel words or standard words and their conjugates, and the special Sturmian morphisms that generate them. Pairwise well-formed scales are encoded by words over a three-letter alphabet, and in our generalization we introduce special positive automorphisms of $F3$, the free group over three letters.Comment: 12 pages, 3 figures, paper presented at the MCM2017 at UNAM in Mexico City on June 27, 2017, keywords: pairwise well-formed scales and modes, well-formed scales and modes, well-formed words, Christoffel words, standard words, central words, algebraic combinatorics on words, special Sturmian morphism

Enumerating Abelian Returns to Prefixes of Sturmian Words

We follow the works of Puzynina and Zamboni, and Rigo et al. on abelian returns in Sturmian words. We determine the cardinality of the set $\mathcal{APR}_u$ of abelian returns of all prefixes of a Sturmian word $u$ in terms of the coefficients of the continued fraction of the slope, dependingly on the intercept. We provide a simple algorithm for finding the set $\mathcal{APR}_u$ and we determine it for the characteristic Sturmian words.Comment: 19page

Critical connectedness of thin arithmetical discrete planes

An arithmetical discrete plane is said to have critical connecting thickness if its thickness is equal to the infimum of the set of values that preserve its $2$-connectedness. This infimum thickness can be computed thanks to the fully subtractive algorithm. This multidimensional continued fraction algorithm consists, in its linear form, in subtracting the smallest entry to the other ones. We provide a characterization of the discrete planes with critical thickness that have zero intercept and that are $2$-connected. Our tools rely on the notion of dual substitution which is a geometric version of the usual notion of substitution acting on words. We associate with the fully subtractive algorithm a set of substitutions whose incidence matrix is provided by the matrices of the algorithm, and prove that their geometric counterparts generate arithmetic discrete planes.Comment: 18 pages, v2 includes several corrections and is a long version of the DGCI extended abstrac

Alumni Presentation and Panel: Engaging the Past

The Alumni Panel featured three black, Dayton-area, UD alumni: LaShea Smith, B.A. International Studies, 1991; Veronica Morris, B.A. Communications Management, 1992; and J.W. Terry, B.S. Business Economics, 2010, Master’s of Public Administration, 2013. The alumni offered insightful perspectives on UD and race from their positions as graduates, as local business people, and, for one, as the mother of a UD student graduating in May 2016. The panelists were asked to prepare a short set of responses to two questions: 1) What were your most salient experience of race at UD? 2) Now, as a graduate of the university, what reflections about race on campus can you offer current students?https://ecommons.udayton.edu/afs_symp/1016/thumbnail.jp

The finite index basis property

We describe in this paper a connection between bifix codes, symbolic dynamical systems and free groups. This is in the spirit of the connection established previously for the symbolic systems corresponding to Sturmian words. We introduce a class of sets of factors of an infinite word with linear factor complexity containing Sturmian sets and regular interval exchange sets, namely the class of tree sets. We prove as a main result that for a uniformly recurrent tree set S, a finite bifix code X on the alphabet A is S-maximal of S-degree d if and only if it is the basis of a subgroup of index d of the free group on

The Entropy of Square-Free Words

Finite alphabets of at least three letters permit the construction of square-free words of infinite length. We show that the entropy density is strictly positive and derive reasonable lower and upper bounds. Finally, we present an approximate formula which is asymptotically exact with rapid convergence in the number of letters.Comment: 18 page