Amy Glen

Bio: Amy Glen is an academic researcher from Murdoch University. The author has contributed to research in topics: Sturmian word & Combinatorics on words. The author has an hindex of 14, co-authored 49 publications receiving 692 citations. Previous affiliations of Amy Glen include Université du Québec & University of Adelaide.

Palindromic richness

Abstract: In this paper, we study combinatorial and structural properties of a new class of finite and infinite words that are 'rich' in palindromes in the utmost sense. A characteristic property of the so-called rich words is that all complete returns to any palindromic factor are themselves palindromes. These words encompass the well-known episturmian words, originally introduced by the second author together with Droubay and Pirillo in 2001 [X. Droubay, J. Justin, G. Pirillo, Episturmian words and some constructions of de Luca and Rauzy, Theoret. Comput. Sci. 255 (2001) 539-553]. Other examples of rich words have appeared in many different contexts. Here we present the first unified approach to the study of this intriguing family of words. Amongst our main results, we give an explicit description of the periodic rich infinite words and show that the recurrent balanced rich infinite words coincide with the balanced episturmian words. We also consider two wider classes of infinite words, namely weakly rich words and almost rich words (both strictly contain all rich words, but neither one is contained in the other). In particular, we classify all recurrent balanced weakly rich words. As a consequence, we show that any such word on at least three letters is necessarily episturmian; hence weakly rich words obey Fraenkel's conjecture. Likewise, we prove that a certain class of almost rich words obeys Fraenkel's conjecture by showing that the recurrent balanced ones are episturmian or contain at least two distinct letters with the same frequency. Lastly, we study the action of morphisms on (almost) rich words with particular interest in morphisms that preserve (almost) richness. Such morphisms belong to the class of P-morphisms that was introduced by Hof, Knill, and Simon in 1995 [A. Hof, O. Knill, B. Simon, Singular continuous spectrum for palindromic Schrodinger operators, Comm. Math. Phys. 174 (1995) 149-159].

Episturmian words : a survey

Abstract: In this paper, we survey the rich theory of infinite episturmian words which generalize to any finite alphabet, in a rather resembling way, the well-known family of Sturmian words on two letters. After recalling definitions and basic properties, we consider episturmian morphisms that allow for a deeper study of these words. Some properties of factors are described, including factor complexity, palindromes, fractional powers, frequencies, and return words. We also consider lexicographical properties of episturmian words, as well as their connection to the balance property, and related notions such as finite episturmian words, Arnoux-Rauzy sequences, and "episkew words" that generalize the skew words of Morse and Hedlund.

A new characteristic property of rich words

Abstract: Originally introduced and studied by the third and fourth authors (A. Glen and L.Q. Zamboni) together with J. Justin and S. Widmer (2008), "rich words" constitute a new class of finite and infinite words characterized by containing the maximal number of distinct palindromes. Several characterizations of rich words have already been established. A particularly nice characteristic property is that all 'complete returns' to palindromes are palindromes. In this note, we prove that rich words are also characterized by the property that each factor is uniquely determined by its longest palindromic prefix and its longest palindromic suffix.

Proceedings of the National Academy of Sciences

Abstract: where A represents the magnetic vector potential, is an integral of the hydromagnetic equations. This -integral made it possible to formulate a variational principle for the force-free magnetic fields. The integral expresses the fact that motions cannot transform a given field in an entirely arbitrary different field, if the conductivity of the medium isconsidered infinite. In this paper we shall show that the full set of hydromagnetic equations admit five more integrals, besides the energy integral, if dissipative processes are absent. These integrals, as we shall presently verify, are I2 =fbHvdV, (2)

Automatic sequences. Theory, applications, generalizations.

Abstract: What do you do to start reading automatic sequences theory applications generalizations? Searching the book that you love to read first or find an interesting book that will make you want to read? Everybody has difference with their reason of reading a book. Actuary, reading habit must be from earlier. Many people may be love to read, but not a book. It's not fault. Someone will be bored to open the thick book with small words to read. In more, this is the real condition. So do happen probably with this automatic sequences theory applications generalizations.

Distribution Modulo One and Diophantine Approximation

Abstract: This book presents state-of-the-art research on the distribution modulo one of sequences of integral powers of real numbers and related topics. Most of the results have never before appeared in one book and many of them were proved only during the last decade. Topics covered include the distribution modulo one of the integral powers of 3/2 and the frequency of occurrence of each digit in the decimal expansion of the square root of two. The author takes a point of view from combinatorics on words and introduces a variety of techniques, including explicit constructions of normal numbers, Schmidt's games, Riesz product measures and transcendence results. With numerous exercises, the book is ideal for graduate courses on Diophantine approximation or as an introduction to distribution modulo one for non-experts. Specialists will appreciate the inclusion of over 50 open problems and the rich and comprehensive bibliography of over 700 references.

Palindromic richness

Abstract: In this paper, we study combinatorial and structural properties of a new class of finite and infinite words that are 'rich' in palindromes in the utmost sense. A characteristic property of the so-called rich words is that all complete returns to any palindromic factor are themselves palindromes. These words encompass the well-known episturmian words, originally introduced by the second author together with Droubay and Pirillo in 2001 [X. Droubay, J. Justin, G. Pirillo, Episturmian words and some constructions of de Luca and Rauzy, Theoret. Comput. Sci. 255 (2001) 539-553]. Other examples of rich words have appeared in many different contexts. Here we present the first unified approach to the study of this intriguing family of words. Amongst our main results, we give an explicit description of the periodic rich infinite words and show that the recurrent balanced rich infinite words coincide with the balanced episturmian words. We also consider two wider classes of infinite words, namely weakly rich words and almost rich words (both strictly contain all rich words, but neither one is contained in the other). In particular, we classify all recurrent balanced weakly rich words. As a consequence, we show that any such word on at least three letters is necessarily episturmian; hence weakly rich words obey Fraenkel's conjecture. Likewise, we prove that a certain class of almost rich words obeys Fraenkel's conjecture by showing that the recurrent balanced ones are episturmian or contain at least two distinct letters with the same frequency. Lastly, we study the action of morphisms on (almost) rich words with particular interest in morphisms that preserve (almost) richness. Such morphisms belong to the class of P-morphisms that was introduced by Hof, Knill, and Simon in 1995 [A. Hof, O. Knill, B. Simon, Singular continuous spectrum for palindromic Schrodinger operators, Comm. Math. Phys. 174 (1995) 149-159].