James D. Currie https://hdl.handle.net/10680/1339 2023-03-28T08:36:31Z Characterization 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. 2020-05-19T00:00:00Z On 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). 2018-02-13T00:00:00Z Extremal 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. 2014-01-22T00:00:00Z Unary 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