Department of Mathematics and Statistics
http://hdl.handle.net/10680/289
Fri, 30 Oct 2020 17:42:03 GMT2020-10-30T17:42:03ZOn avoidability of formulas with reversal
http://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).
Tue, 13 Feb 2018 00:00:00 GMThttp://hdl.handle.net/10680/18342018-02-13T00:00:00ZSquares and overlaps in the Thue-Morse sequence and some variants
http://hdl.handle.net/10680/1833
Squares and overlaps in the Thue-Morse sequence and some variants
Brown, Shandy; Rampersad, Narad; Shallit, Jeffrey; Vasiga, Troy
We consider the position and number of occurrences of squares in the Thue-Morse sequence, and show that the corresponding sequences are 2-regular. We also prove that changing any finite but nonzero number of bits in the Thue-Morse sequence creates an overlap, and any linear subsequence of the Thue-Morse sequence (except those corresponding to decimation by a power of 2) contains an overlap.
Sun, 01 Jan 2006 00:00:00 GMThttp://hdl.handle.net/10680/18332006-01-01T00:00:00ZAvelines's Hole: An Unexpected Twist in the Tale
http://hdl.handle.net/10680/1825
Avelines's Hole: An Unexpected Twist in the Tale
Schulting, Rick J.; Booth, Tom; Brace, Selina; Diekmann, Yoan; Thomas, Mark; Barnes, Ian; Meiklejohn, Chris; Babb, Jeff; Budd, Chelsea; Charlton, Sophy; Van Der Plicht, Hans; Mullan, Graham; Wilson, Linda
Aveline’s Hole is the largest known Early Mesolithic cemetery in Britain, previously thought to have no evidence for subsequent burial activity. Thus, it came as some surprise when the results of a recent ancient human DNA study found that, of four individuals from the site yielding genomic data, two showed high levels of ancestry from Early Neolithic Aegean farmers. Radiocarbon dating confirmed that these two individuals were indeed British Early Neolithic in date, while the other two had the expected ‘Western Hunter-Gatherer’ ancestry genomic signatures, with the two groups separated in time by nearly five millennia. Moreover, the two Neolithic samples were both crania, while the two Mesolithic samples were long bones. Given the absence of Neolithic dates in the previous sizeable dating programme combined with the difficult history of the collection, i.e., the WWII bombing of its Bristol repository, this raised the question of whether the crania might in fact be from another site. As we show in this paper, a very strong case can be made that the crania do in fact originate from Aveline’s Hole. Additional radiocarbon dating (14 in total, including the above mentioned four) suggests that about half the cranial elements from the site fall within the Early Neolithic, though there is still no evidence for the deposition of post-cranial remains at this time, nor is there any burial evidence in the long intervening period between the Early Mesolithic and the Early Neolithic. Intriguingly, craniometric analyses of legacy data including three crania lost in the bombing suggest that one, Aveline’s Hole ‘A’, may be Upper Palaeolithicin date. As part of this re-investigation of the human remains from the site, we present new stable carbon and nitrogen isotope analyses that differ significantly from those originally reported for the Early Mesolithic, with the new results more in keeping with other isotopic data for this period. We also present new stable carbon and nitrogen isotope results on human remains from the nearby Early Mesolithic sites of Badger Hole and Greylake, and report new Early Mesolithic radiocarbon dates and isotopic data from Cannington Park Quarry. Clear isotopic differences between the Early Mesolithic and the Neolithic remains can be seen, but these are argued to relate primarily to shifts in the underlying ecological baselines, rather than to differences in types of foods consumed (with the caveat that terrestrial wild and domesticated foods will be isotopically similar). The genetic data are summarised, giving evidence not only of the ancestry of Mesolithic and Neolithic individuals from Aveline’s Hole, but also suggesting something of their physical appearance. The degree of population replacement now indicated by ancient DNA suggests that there was a substantial migration of farmers into Britain at the start of the Neolithic. This new information demonstrates the archaeological importance of Aveline's Hole for both the Mesolithic and Neolithic periods.
Tue, 01 Jan 2019 00:00:00 GMThttp://hdl.handle.net/10680/18252019-01-01T00:00:00ZSupporting Teachers in Times of Change: The Job Demands- Resources Model and Teacher Burnout During the COVID-19 Pandemic
http://hdl.handle.net/10680/1823
Supporting Teachers in Times of Change: The Job Demands- Resources Model and Teacher Burnout During the COVID-19 Pandemic
Sokal, Laura J.; Trudel, Lesley G. Eblie; Babb, Jeff C.
Burnout in teachers has been broadly investigated, but no studies have investigated burnout in teachers during a pandemic. The current study is based on a survey of 1278 Canadian teachers and examined whether the Job Demands-Resources model was a useful lens for examining teacher burnout in this unprecedented context. Results supported the model in general terms in that most demands were most strongly correlated with the initial exhaustion stage of burnout. However, not all resources were most strongly associated with the later stages of burnout, suggesting that the examination of specific resources in the context of a pandemic as opposed to examining resources together as a latent variable contributes to development of a more refined model. Suggestions for supporting teachers’ welfare are provided.
Thu, 01 Oct 2020 00:00:00 GMThttp://hdl.handle.net/10680/18232020-10-01T00:00:00ZExtremal words in morphic subshifts
http://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.
Wed, 22 Jan 2014 00:00:00 GMThttp://hdl.handle.net/10680/17632014-01-22T00:00:00ZUnary patterns under permutations
http://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.
Mon, 04 Jun 2018 00:00:00 GMThttp://hdl.handle.net/10680/17622018-06-04T00:00:00ZAvoiding three consecutive blocks of the same size and same sum
http://hdl.handle.net/10680/1761
Avoiding three consecutive blocks of the same size and same sum
Currie, James D.; Cassaigne, Julien; Shallit, Jeffrey O.; Schaeffer, Luke
We show that there exists an inﬁnite word over the alphabet {0,1,3,4} containing no three consecutive blocks of the same size and the same sum. This answers an open problem of Pirillo and Varricchio from1994.
Tue, 01 Apr 2014 00:00:00 GMThttp://hdl.handle.net/10680/17612014-04-01T00:00:00ZGrowth rate of binary words avoiding xxxR
http://hdl.handle.net/10680/1760
Growth rate of binary words avoiding xxxR
Currie, James D.; Rampersad, Narad
Abstract
Consider the set of those binary words with no non-empty factors of the form
xxx^R. Du, Mousavi, Schaeffer, and Shallit asked whether this set of words grows
polynomially or exponentially with length. In this paper, we demonstrate the
existence of upper and lower bounds of the form n^{lg n+o(lg n)} on the number of
such words of length n, where lg n denotes the base-2 logarithm of n.
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/10680/17602016-01-01T00:00:00ZAbelian complexity of fixed point of morphism 0 -> 012, 1 -> 02, 2 -> 1
http://hdl.handle.net/10680/1759
Abelian complexity of fixed point of morphism 0 -> 012, 1 -> 02, 2 -> 1
Currie, James D.; Blanchet-Sadri, Francine; Fox, Nathan; Rampersad, Narad
We study the combinatorics of vtm, a variant of the Thue-Morse word generated by the non-uniform morphism 0 -> 012,1 -> 02,2 -> 1 starting with 0. This inﬁnite ternary sequence appears a lot in the literature and ﬁnds applications in several ﬁelds such as combinatorics on words; for example, in pattern avoidance it is often used to construct inﬁnite words avoiding given patterns. It has been shown that the factor complexity of vtm, i.e., the number of factors of length n, is \Theta(n); in fact, it is bounded by 10/3 n for all n, and it reaches that bound precisely when n can be written as 3 times a power of 2. In this paper, we show that the abelian complexity of vtm, i.e., the number of Parikh vectors of length n, is O(logn) with constant approaching 3/4 (assuming base 2 logarithm), and it is \Omega(1) with constant 3 (and these are the best possible bounds). We also prove some results regarding factor indices in vtm.
Sun, 14 Feb 2016 00:00:00 GMThttp://hdl.handle.net/10680/17592016-02-14T00:00:00ZBinary Words Avoiding xxRx and Strongly Unimodal Sequences
http://hdl.handle.net/10680/1758
Binary Words Avoiding xxRx and Strongly Unimodal Sequences
Currie, James D.; Rampersad, Narad
In previous work, Currie and Rampersad showed that the growth of the number
of binary words avoiding the pattern xxxR was intermediate between polynomial and
exponential. We now show that the same result holds for the growth of the number
of binary words avoiding the pattern xxRx. Curiously, the analysis for xxRx is much
simpler than that for xxxR. We derive our results by giving a bijection between the
set of binary words avoiding xxRx and a class of sequences closely related to the class
of “strongly unimodal sequences”.
Mon, 14 Sep 2015 00:00:00 GMThttp://hdl.handle.net/10680/17582015-09-14T00:00:00Z