相关论文: There are exponentially many ternary words that av…
In 1976, Dekking showed that there exists an infinite binary word that contains neither squares yy with y >= 4 nor cubes xxx. We show that `cube' can be replaced by any fractional power > 5/2. We also consider the analogous problem where…
A \emph{square} is a finite non-empty word consisting of two identical adjacent blocks. A word is \emph{square-free} if it does not contain a square as a factor. In any finite word one may delete the repeated block of a square, obtaining…
withdrawed due to a substantial error.
This paper has been withdrawn by the authors because M. Aschenbrenner pointed out that the proof of theorem 3.1 was incorrect.
The paper is withdrawn. The proof has an error and it requires a different approach.
A finite word $w$ is an abelian square if $w = xx^\prime$ with $x^\prime$ a permutation of $x$. In 1972, Entringer, Jackson, and Schatz proved that every binary word of length $k^2 + 6k$ contains an abelian square of length $\geq 2k$. We…
This paper has been withdrawn by the author, since the result was already known.
We revisit the question of classification of balanced circular words and focus on the case of a ternary alphabet. We propose a $3$-dimensional generalisation of the discrete approximation representation of Christoffel words. By considering…
This paper has been withdrawn by the author, due to an error in the proof of lemma 7.4. However, numerical evidence strongly suggest that this lemma is true.
This paper has been withdrawn by the author due to a crucial error in equation.
A word is square-free if it does not contain any square (a word of the form $XX$), and is extremal square-free if it cannot be extended to a new square-free word by inserting a single letter at any position. Grytczuk, Kordulewski, and…
In this note we present a characterisation of all unary and binary patterns that do not only contain variables, but also reversals of their instances. These types of variables were studied recently in either more general or particular…
This paper has been withdrawn by the author due to the presented idea is wrong.
An early result in the theory of Natural Dualities is that an algebra with a near unanimity (NU) term is dualizable. A converse to this is also true: if V(A) is congruence distributive and A is dualizable, then A has an NU term. An…
This paper has been withdrawn by the author due to an error in the proof of Proposition 4.8.
The set of all avoidable patterns in n or fewer letters can be avoided on an alphabet with 2(n+2) letters.
In this paper we study the enumeration and the construction, according to the number of ones, of particular binary words avoiding a fixed pattern. The growth of such words can be described by particular jumping and marked succession rules.…
In this short note we show, providing counterexamples, that the "two important theorems" in the recent paper [Y, Yuan, Global optimization solutions to a class of non-convex quadratic minimization problems with quadratic constraints, in…
This paper is wrong and is therefore withdrawn.
The paper has been withdrawn by the author due to a gap in Proof of Theorem 1.1.