Generalized trapezoidal words

The factor complexity function $C_w(n)$ of a finite or infinite word $w$ counts the number of distinct factors of $w$ of length $n$ for each $n \ge 0$. A finite word $w$ of length $|w|$ is said to be trapezoidal if the graph of its factor complexity $C_w(n)$ as a function of $n$ (for $0 \leq n \leq |w|$) is that of a regular trapezoid (or possibly an isosceles triangle); that is, $C_w(n)$ increases by 1 with each $n$ on some interval of length $r$, then $C_w(n)$ is constant on some interval of length $s$, and finally $C_w(n)$ decreases by 1 with each $n$ on an interval of the same length $r$. Necessarily $C_w(1)=2$ (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 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 $\mathbb{R}^2$ 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" (also known as "full words") - a wider family of finite and infinite words characterized by containing the maximal number of palindromes - studied in depth by the first author and others in 2009. In this paper, we introduce a natural generalization of trapezoidal words over an arbitrary finite alphabet $\mathcal{A}$, called generalized trapezoidal words (or GT-words for short). In particular, we study combinatorial and structural properties of this new class of words, and we show that, unlike the binary case, not all GT-words are rich in palindromes when $|\mathcal{A}| \geq 3$, but we can describe all those that are rich.Comment: Major revisio

Extremal properties of (epi)Sturmian sequences and distribution modulo 1

Starting from a study of Y. Bugeaud and A. Dubickas (2005) on a question in distribution of real numbers modulo 1 via combinatorics on words, we survey some combinatorial properties of (epi)Sturmian sequences and distribution modulo 1 in connection to their work. In particular we focus on extremal properties of (epi)Sturmian sequences, some of which have been rediscovered several times

Characterizations of finite and infinite episturmian words via lexicographic orderings

In this paper, we characterize by lexicographic order all finite Sturmian and episturmian words, i.e., all (finite) factors of such infinite words. Consequently, we obtain a characterization of infinite episturmian words in a "wide sense" (episturmian and episkew infinite words). That is, we characterize the set of all infinite words whose factors are (finite) episturmian. Similarly, we characterize by lexicographic order all balanced infinite words over a 2-letter alphabet; in other words, all Sturmian and skew infinite words, the factors of which are (finite) Sturmian.Comment: 18 pages; to appear in the European Journal of Combinatoric

A characterization of fine words over a finite alphabet

To any infinite word w over a finite alphabet A we can associate two infinite words min(w) and max(w) such that any prefix of min(w) (resp. max(w)) is the lexicographically smallest (resp. greatest) amongst the factors of w of the same length. We say that an infinite word w over A is "fine" if there exists an infinite word u such that, for any lexicographic order, min(w) = au where a = min(A). In this paper, we characterize fine words; specifically, we prove that an infinite word w is fine if and only if w is either a "strict episturmian word" or a strict "skew episturmian word''. This characterization generalizes a recent result of G. Pirillo, who proved that a fine word over a 2-letter alphabet is either an (aperiodic) Sturmian word, or an ultimately periodic (but not periodic) infinite word, all of whose factors are (finite) Sturmian.Comment: 16 pages; presented at the conference on "Combinatorics, Automata and Number Theory", Liege, Belgium, May 8-19, 2006 (to appear in a special issue of Theoretical Computer Science

American fundamentalism and nuclear deterrence a critique

This paper is an attempt to deal with two trends which have recently received a great deal of attention. One trend is an increasing questioning of the role which nuclear deterrence plays in securing a better world for our own and future generations. The other trend is the rise of the New Christian Right and its mixing of religion and American patriotism. These two trends have come face to face in the New Christian Right’s emphasis on restoring American superiority in its relative position to the Soviet Union—a country which they believe is the source of a monolithic Communist threat. The efforts of the New Christian Right to deal with nuclear deterrence are applauded but not accepted as representing the final word on how the gospel message is to be applied to such issues. The analysis of the New Christian Right’s position focuses on several areas. These include the following: anthropology, history, faith and reason, and biblical interpretation. It is concluded that their attempt to absolutize American values and America’s need for nuclear superiority does not square with the prophetic voice of the gospel, which stands in judgment on all human points of view, systems and structures. Thus the Christian approach to nuclear deterrence involves not an absolute faith commitment to the defense of the American point of view, but an emphasis on dialogue with those communities which have endorsed values which are sometimes in opposition to these. Such a dialogue can be pursued in economic terms, striving to discover the strength and weaknesses of both Capitalism and Communism. This attempt affirms that truth is not limited to any one particular community and that, through dialogue, values which point towards global peace and justice can be affirmed

On the critical exponent of generalized Thue-Morse words

For certain generalized Thue-Morse words t, we compute the "critical exponent", i.e., the supremum of the set of rational numbers that are exponents of powers in t, and determine exactly the occurrences of powers realizing it.Comment: 13 pages; to appear in Discrete Mathematics and Theoretical Computer Science (accepted October 15, 2007