Relative to normoxia-reared fish, hypoxia-reared zebrafish Danio rerio achieved 24Á9 and 21Á4% slower maximal swim velocities in both normoxic and hypoxic waters, respectively. Hypoxiareared fish also produced 26Á1 and 63Á9% less lactate during resting conditions across both normoxic and hypoxic waters, respectively. During exercise, this trend continued as hypoxiareared fish produced 68Á2 and 55Á1% less lactate across both normoxic and hypoxic waters, respectively. This reduction in performance, rather than representing a purely pathological (maladaptive) response to hypoxia, appears to represent a fundamental shift in the metabolic response to hypoxia.
Given a countable set X (usually taken to be N or Z), an infinite permutation π of X is a linear ordering ≺π of X, introduced in [5]. This paper investigates the combinatorial complexity of the infinite permutation on N associated with the well-known and well-studied Thue-Morse word. A formula for the complexity is established by studying patterns in subpermutations and the action of the Thue-Morse morphism on the subpermutations. ). symbols, suppose {0, 1, 2} with 0 < 1 < 2, then the permutation could be associated with any of 00, 01, 11, or 12.For binary words the subpermutations depend on the order on the symbols used to compose ω, but the permutation complexity does not depend on the order. For words over 3 or more symbols, not only do the subpermutations depend on the order on the alphabet but so does the permutation complexity. For example, consider the Fibonacci word t = 0100101001001010010100100101 . . ., defined by iterating the morphism 0 → 01, 1 → 0 on the letter 0, and suppose the 1s are replaced by alternating a's and b's to create the word:If the symbols int are ordered 0 < a < b there will be 5 distinct subpermutations of length 3, and if the symbols are ordered a < 0 < b there will be only 4 distinct subpermutations of length 3. The verification of this fact is left to the reader.In view of the notion of an infinite permutation associated to an aperiodic word, it is natural to compute the permutation complexity of well-known classes of words. In [9], Makarov computes the permutation complexity of Sturmian words. The goal of this paper is to determine the permutation complexity of the Thue-Morse word.The Thue-Morse word, T = T 0 T 1 T 2 · · · , is:which can be generated by the morphism: µ T : 0 → 01, 1 → 10, by iterating on the letter 0. Axel Thue introduced this word in his studies of repetitions in words, and proved that the word T is overlap-free ([12]). A word ω is said to be overlap-free if it does not contain a factor of the form vuvuv for words u and v, with v non-empty.The Thue-Morse word was again discovered independently by Marston Morse in 1921 [11] through his study of differential geometry, and used in the foundations of symbolic dynamics. For a more in depth look at further properties, independent discoveries, and applications of the Thue-Morse word see [2].The factor complexity of the Thue-Morse word was computed independently by two groups in 1989,Brlek [4] and de Luca and Varricchio [6]. Our proof of the permutation complexity of the Thue-Morse word does not use the factor complexity function.The permutation complexity of the Thue-Morse word can be found as follows. For any n ≥ 2, we can write n as n = 2 a +b, with 0 < b ≤ 2 a . Using this notation, it will shown that the formula for the permutation
In this paper we study the maximal pattern complexity of infinite words up to Abelian equivalence. We compute a lower bound for the Abelian maximal pattern complexity of infinite words which are both recurrent and aperiodic by projection. We show that in the case of binary words, the bound is actually achieved and gives a characterization of recurrent aperiodic words
Given a countable set X (usually taken to be the natural numbers or integers), an infinite permutation, π, of X is a linear ordering of X. This paper investigates the combinatorial complexity of infinite permutations on the natural numbers associated with the image of uniformly recurrent aperiodic binary words under the letter doubling map. An upper bound for the complexity is found for general words, and a formula for the complexity is established for the Sturmian words and the Thue-Morse word
Given a countable set X (usually taken to be N or Z), an infinite permutation π of X is a linear ordering ≺π of X, introduced in [6]. This paper investigates the combinatorial complexity of infinite permutations on N associated with the image of uniformly recurrent aperiodic binary words under the letter doubling map. An upper bound for the complexity is found for general words, and a formula for the complexity is established for the Sturmian words and the Thue-Morse word.
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