On a theorem concerning partially overlapping subpalindromes of a binary word

Younger children with a short memory span and older children with a long memory span recalled binary word sequences of different structure. It was found that the difficulty of recall was mainly determined by the run-structure but not by the cyclicity of the sequences. Incorrectly recalled sequences were systematically analysed in terms of two supposed coding mechanisms, i.e. “feature extraction” and “pattern imposition”. It could be demonstrated that certain features of the presented sequences were preserved while others usually were not. While the extracted feature served as a basis for reconstruction of the sequence in recall, the obtained sequences showed in addition certain characteristics which appeared to be unrelated to the input. Some differences in the efficiency of coding were found between 4- and 6-year-olds. However, it appeared that the coding strategies were similar at lower and higher levels of memory capacity and development.

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Algorithm for binary word recognition suited to ultrafast nonlinear optics

Electronics Letters ◽

10.1049/el:19930630 ◽

1993 ◽

Vol 29 (11) ◽

pp. 945-947 ◽

Cited By ~ 18

Author(s):

D. Cotter ◽

S.C. Cotter

Keyword(s):

Nonlinear Optics ◽

Word Recognition ◽

Ultrafast Nonlinear Optics ◽

Binary Word

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Extremal Infinite Overlap-Free Binary Words

The Electronic Journal of Combinatorics ◽

10.37236/1365 ◽

1998 ◽

Vol 5 (1) ◽

Cited By ~ 13

Author(s):

Jean-Paul Allouche ◽

James Currie ◽

Jeffrey Shallit

Keyword(s):

Fixed Point ◽

Binary Word ◽

Infinite Word

Let $\overline{\bf t}$ be the infinite fixed point, starting with $1$, of the morphism $\mu: 0 \rightarrow 01$, $1 \rightarrow 10$. An infinite word over $\lbrace 0, 1 \rbrace$ is said to be overlap-free if it contains no factor of the form $axaxa$, where $a \in \lbrace 0,1 \rbrace$ and $x \in \lbrace 0,1 \rbrace^*$. We prove that the lexicographically least infinite overlap-free binary word beginning with any specified prefix, if it exists, has a suffix which is a suffix of $\overline{\bf t}$. In particular, the lexicographically least infinite overlap-free binary word is $001001 \overline{\bf t}$.

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How Many Square Occurrences Must a Binary Sequence Contain?

The Electronic Journal of Combinatorics ◽

10.37236/1705 ◽

2003 ◽

Vol 10 (1) ◽

Cited By ~ 5

Author(s):

Gregory Kucherov ◽

Pascal Ochem ◽

Michaël Rao

Keyword(s):

Word Length ◽

Binary Sequence ◽

Binary Word ◽

Necessary And Sufficient

Every binary word with at least four letters contains a square. A. Fraenkel and J. Simpson showed that three distinct squares are necessary and sufficient to construct an infinite binary word. We study the following complementary question: how many square occurrences must a binary word contain? We show that this quantity is, in the limit, a constant fraction of the word length, and prove that this constant is $0.55080...$.

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Extremal Overlap-Free and Extremal $\beta$-Free Binary Words

The Electronic Journal of Combinatorics ◽

10.37236/9703 ◽

2020 ◽

Vol 27 (4) ◽

Author(s):

Lucas Mol ◽

Narad Rampersad ◽

Jeffrey Shallit

Keyword(s):

Real Number ◽

Single Letter ◽

Binary Word ◽

Free Word

An overlap-free (or $\beta$-free) word $w$ over a fixed alphabet $\Sigma$ is extremal if every word obtained from $w$ by inserting a single letter from $\Sigma$ at any position contains an overlap (or a factor of exponent at least $\beta$, respectively). We find all lengths which admit an extremal overlap-free binary word. For every "extended" real number $\beta$ such that $2^+\leqslant\beta\leqslant 8/3$, we show that there are arbitrarily long extremal $\beta$-free binary words.

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