The critical exponent of the Arshon words

Generalizing the results of Thue (for n = 2) and of Klepinin and Sukhanov (for n = 3), we prove that for all n greater than or equal to 2, the critical exponent of the Arshon word of order $n$ is given by (3n-2)/(2n-2), and this exponent is attained at position 1.Comment: 11 page

Critical Exponents and Stabilizers of Infinite Words

This thesis concerns infinite words over finite alphabets. It contributes to two topics in this area: critical exponents and stabilizers. Let w be a right-infinite word defined over a finite alphabet. The critical exponent of w is the supremum of the set of exponents r such that w contains an r-power as a subword. Most of the thesis (Chapters 3 through 7) is devoted to critical exponents. Chapter 3 is a survey of previous research on critical exponents and repetitions in morphic words. In Chapter 4 we prove that every real number greater than 1 is the critical exponent of some right-infinite word over some finite alphabet. Our proof is constructive. In Chapter 5 we characterize critical exponents of pure morphic words generated by uniform binary morphisms. We also give an explicit formula to compute these critical exponents, based on a well-defined prefix of the infinite word. In Chapter 6 we generalize our results to pure morphic words generated by non-erasing morphisms over any finite alphabet. We prove that critical exponents of such words are algebraic, of a degree bounded by the alphabet size. Under certain conditions, our proof implies an algorithm for computing the critical exponent. We demonstrate our method by computing the critical exponent of some families of infinite words. In particular, in Chapter 7 we compute the critical exponent of the Arshon word of order n for n ≥ 3. The stabilizer of an infinite word w defined over a finite alphabet Σ is the set of morphisms f: Σ*→Σ* that fix w. In Chapter 8 we study various problems related to stabilizers and their generators. We show that over a binary alphabet, there exist stabilizers with at least n generators for all n. Over a ternary alphabet, the monoid of morphisms generating a given infinite word by iteration can be infinitely generated, even when the word is generated by iterating an invertible primitive morphism. Stabilizers of strict epistandard words are cyclic when non-trivial, while stabilizers of ultimately strict epistandard words are always non-trivial. For this latter family of words, we give a characterization of stabilizer elements. We conclude with a list of open problems, including a new problem that has not been addressed yet: the D0L repetition threshold

Finding the growth rate of a regular language in polynomial time

We give an O(n^3+n^2 t) time algorithm to determine whether an NFA with n states and t transitions accepts a language of polynomial or exponential growth. We also show that given a DFA accepting a language of polynomial growth, we can determine the order of polynomial growth in quadratic time

Excess cholesterol induces mouse egg activation and may cause female infertility

The HDL receptor scavenger receptor, class B type I (SR-BI) controls the structure and fate of plasma HDL. Female SR-BI KO mice are infertile, apparently because of their abnormal cholesterol-enriched HDL particles. We examined the growth and meiotic progression of SR-BI KO oocytes and found that they underwent normal germinal vesicle breakdown; however, SR-BI KO eggs, which had accumulated excess cholesterol in vivo, spontaneously activated, and they escaped metaphase II (MII) arrest and progressed to pronuclear, MIII, and anaphase/telophase III stages. Eggs from fertile WT mice were activated when loaded in vitro with excess cholesterol by a cholesterol/methyl-β-cyclodextrin complex, phenocopying SR-BI KO oocytes. In vitro cholesterol loading of eggs induced reduction in maturation promoting factor and MAPK activities, elevation of intracellular calcium, extrusion of a second polar body, and progression to meiotic stages beyond MII. These results suggest that the infertility of SR-BI KO females is caused, at least in part, by excess cholesterol in eggs inducing premature activation and that cholesterol can activate WT mouse eggs to escape from MII arrest. Analysis of SR-BI KO female infertility raises the possibility that abnormalities in cholesterol metabolism might underlie some cases of human female infertility of unknown etiology.National Institutes of Health (U.S.)National Institutes of Health (U.S.) (Pre-doctoral Training Grant T32GM007287)Massachusetts Institute of Technology (International Science and Technology Initiatives Chile Cooperative Grant

The critical exponent of the Arshon words

Generalizing the results of Thue (for n = 2) [Norske Vid. Selsk. Skr. Mat. Nat. Kl. 1 (1912) 1–67] and of Klepinin and Sukhanov (for n = 3) [Discrete Appl. Math. 114 (2001) 155–169], we prove that for all n ≥ 2, the critical exponent of the Arshon word of order n is given by (3n–2)/(2n–2), and this exponent is attained at position 1

On Critical exponents in fixed points of

Let w be an infinite fixed point of a binary k-uniform morphism f, and let Ew be the critical exponent of w. We give necessary and sufficient conditions for Ew to be bounded, and an explicit formula to compute it when it is. In particular, we show that Ew is always rational. We also sketch an extension of our method to non-uniform morphisms over general alphabets

Morphic and Automatic Words: Maximal Blocks and Diophantine Approximation

Let w be a morphic word over a finite alphabet Σ, and let ∆ be a nonempty subset of Σ. We study the behavior of maximal blocks consisting only of letters from ∆ in w, and prove the following: let (ik, jk) denote the starting and ending positions, respectively, of the k’th maximal ∆-block in w. Then lim supk→∞(jk/ik) is algebraic if w is morphic, and rational if w is automatic. As a result, we show that the same conclusion holds if (ik, jk) are the starting and ending positions of the k’th maximal zero 1 block, and, more generally, of the k’th maximal x-block, where x is an arbitrary word. This enables us to draw conclusions about the irrationality exponent of automatic and morphic numbers. In particular, we show that the irrationality exponent of automatic (resp., morphic) numbers belonging to a certain class that we define is rational (resp., algebraic).