Privileged Words and Sturmian Words

This dissertation has two almost unrelated themes: privileged words and Sturmian words. Privileged words are a new class of words introduced recently. A word is privileged if it is a complete first return to a shorter privileged word, the shortest privileged words being letters and the empty word. Here we give and prove almost all results on privileged words known to date. On the other hand, the study of Sturmian words is a well-established topic in combinatorics on words. In this dissertation, we focus on questions concerning repetitions in Sturmian words, reproving old results and giving new ones, and on establishing completely new research directions. The study of privileged words presented in this dissertation aims to derive their basic properties and to answer basic questions regarding them. We explore a connection between privileged words and palindromes and seek out answers to questions on context-freeness, computability, and enumeration. It turns out that the language of privileged words is not context-free, but privileged words are recognizable by a linear-time algorithm. A lower bound on the number of binary privileged words of given length is proven. The main interest, however, lies in the privileged complexity functions of the Thue-Morse word and Sturmian words. We derive recurrences for computing the privileged complexity function of the Thue-Morse word, and we prove that Sturmian words are characterized by their privileged complexity function. As a slightly separate topic, we give an overview of a certain method of automated theorem-proving and show how it can be applied to study privileged factors of automatic words. The second part of this dissertation is devoted to Sturmian words. We extensively exploit the interpretation of Sturmian words as irrational rotation words. The essential tools are continued fractions and elementary, but powerful, results of Diophantine approximation theory. With these tools at our disposal, we reprove old results on powers occurring in Sturmian words with emphasis on the fractional index of a Sturmian word. Further, we consider abelian powers and abelian repetitions and characterize the maximum exponents of abelian powers with given period occurring in a Sturmian word in terms of the continued fraction expansion of its slope. We define the notion of abelian critical exponent for Sturmian words and explore its connection to the Lagrange spectrum of irrational numbers. The results obtained are often specialized for the Fibonacci word; for instance, we show that the minimum abelian period of a factor of the Fibonacci word is a Fibonacci number. In addition, we propose a completely new research topic: the square root map. We prove that the square root map preserves the language of any Sturmian word. Moreover, we construct a family of non-Sturmian optimal squareful words whose language the square root map also preserves.This construction yields examples of aperiodic infinite words whose square roots are periodic.

Quest/ion of Identities in African American Feminist Postmodern Drama: A Study of Selected Plays by Suzan-Lori Parks

Innovative and unconventional, Pulitzer Prize-winning playwright Suzan-Lori Parks belongs to the continuum of African American playwrights who have contributed to the quest/ion – the quest for and question – of identities for African Americans. Her plays are sites in which the quest/ion of identities for African Americans is pursued, raised and enacted. She makes use of both page and stage to emphasize the exigency of reshaping African Americans’ identities through questioning the dominant ideologies and metanarratives, delegitimizing some of the prevailing stereotypes imposed on them, drawing out the complicity of the media in perpetuating racism, evoking slavery, lynching and their aftereffects, rehistoricizing African American history, catalyzing reflections on the various intersections of sex, race, class and gender orientations, and proffering alternative perspectives to help readers think more critically about issues facing African Americans. In my dissertation, I approach three plays by Parks – The Death of the Last Black Man in the Whole Entire World (1990), Venus (1996) and Fucking A (2000) – from the standpoints of postmodern drama and African American feminism with a focus on the terrains that reflect the quest/ion of identities for African Americans, especially African American women. I argue that postmodern drama and African American feminism provide the ground for Parks to promote the development of a political agenda in order to call into question a number of dominant ideologies and metanarratives with regard to African Americans and draw upon the roles of those metanarratives as a powerful apparatus of racial and sexual oppressions. I also explore how Parks engages with postmodern drama and African American feminism to incorporate her own mininarratives in the dominant discourses. I argue that Parks in these plays uses postmodern drama and African American feminism to encourage reflections on intersectionality in order to reveal the concerns of African Americans, particularly African American women. Her plays challenge the dominant order of hierarchy and patriarchy, while in some cases urging unity and solidarity between African American men and women by showing how unity and solidarity can help them confront race, class and gender oppressions. Furthermore, I discuss how the utilization of postmodern techniques and devices helps Parks to transform the conventional features of playwriting, to create incredulity toward the dominant systems of oppression and to incorporate her mininarratives within the context of dominant discourses.