James D. Currie
https://hdl.handle.net/10680/1339
2021-07-29T12:17:53ZCharacterization of the lengths of binary circular words containing no squares other than 00, 11, and 0101
https://hdl.handle.net/10680/1850
Characterization of the lengths of binary circular words containing no squares other than 00, 11, and 0101
Currie, James D.; Johnson, Jesse T.
We characterize exactly the lengths of binary circular words containing no squares other than 00, 11, and 0101.
2020-05-19T00:00:00ZOn avoidability of formulas with reversal
https://hdl.handle.net/10680/1834
On avoidability of formulas with reversal
Currie, James D.; Mol, Lucas; Rampersad, Narad
While a characterization of unavoidable formulas (without reversal) is well-known, little
is known about the avoidability of formulas with reversal in general. In this article, we characterize the unavoidable formulas with reversal that have at most two one-way variables (x is a one-way variable in formula with reversal φ if exactly one of x and x^R appears in φ).
2018-02-13T00:00:00ZExtremal words in morphic subshifts
https://hdl.handle.net/10680/1763
Extremal words in morphic subshifts
Zamboni, Luca Q.; Saari, Kalle; Rampersad, Narad; Currie, James D.
Given an infinite word x over an alphabet A, a letter b occurring in
x, and a total order \sigma on A, we call the smallest word with respect to \sigma
starting with b in the shift orbit closure of x an extremal word of x. In this
paper we consider the extremal words of morphic words. If x = g(f^\omega(a))
for some morphisms f and g, we give two simple conditions on f and
g that guarantees that all extremal words are morphic. This happens,
in particular, when x is a primitive morphic or a binary pure morphic
word. Our techniques provide characterizations of the extremal words of
the Period-doubling word and the Chacon word and give a new proof of
the form of the lexicographically least word in the shift orbit closure of
the Rudin-Shapiro word.
2014-01-22T00:00:00ZUnary patterns under permutations
https://hdl.handle.net/10680/1762
Unary patterns under permutations
Currie, James D.; Nowotka, Dirk; Manea, Florin; Reshadi, Kamellia
Thue characterized completely the avoidability of unary patterns. Adding function variables gives a general setting capturing avoidance of powers, avoidance of patterns with palindromes, avoidance of powers under coding, and other questions of recent interest. Unary patterns with permutations have been previously analysed only for lengths up to 3.
Consider a pattern $p=\pi_{i_1}(x)\ldots \pi_{i_r}(x)$, with $r\geq 4$, $x$ a word variable over an alphabet $\Sigma$ and $\pi_{i_j}$ function variables, to be replaced by morphic or antimorphic permutations of $\Sigma$. If $|\Sigma|\ge 3$, we show the existence of an infinite word avoiding all pattern instances having $|x|\geq 2$. If $|\Sigma|=3$ and all $\pi_{i_j}$ are powers of a single morphic or antimorphic $\pi$, the length restriction is removed. For the case when $\pi$ is morphic, the length dependency can be removed also for $|\Sigma|=4$, but not for $|\Sigma|=5$, as the pattern $x\pi^2(x)\pi^{56}(x)\pi^{33}(x)$ becomes unavoidable. Thus, in general, the restriction on $x$ cannot be removed, even for powers of morphic permutations. Moreover, we show that for every positive integer $n$ there exists $N$ and a pattern $\pi^{i_1}(x)\ldots \pi^{i_n}(x)$ which is unavoidable over all alphabets $\Sigma$ with at least $N$ letters and $\pi$ morphic or antimorphic permutation.
2018-06-04T00:00:00Z