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A word defined over an alphabet $𝒜$ is -balanced ( $\mathbb{N}$) if for all pairs of factors , of of the same length and for all letters $𝒜$, the difference between the number of letters in and is less or equal to . In this paper we consider a ternary alphabet $𝒜$ = {, , } and a class of substitutions ${\varphi }_p$ defined by ${\varphi }_p$() = , ${\varphi }_p$() = , ${\varphi }_p$() = where 1. We prove that the fixed point of ${\varphi }_p$, formally written as ${\varphi }_p^{\infty }$(), is 3-balanced and that...

In this paper we will deal with the balance properties of the infinite binary words associated to -integers when is a quadratic simple Pisot number. Those words are the fixed points of the morphisms of the type $\varphi \left(A\right)=A^{p}B$, $\varphi \left(B\right)=A^q$ for $p\in \mathbb{N}$, $q\in \mathbb{N}$, $p\ge q$, where $\beta =\frac{p+\sqrt{p^2+4q}}{2}$. We will prove that such word is -balanced with $t=1+\left[(p-1)/(p+1-q)\right]$. Finally, in the case that it is known [B. Adamczewski, (2002) 197–224] that the fixed point of the substitution $\varphi \left(A\right)=A^{p}B$, $\varphi \left(B\right)=A^q$ is not -balanced for any . We exhibit an infinite sequence of pairs of words...

We study some arithmetical and combinatorial properties of -integers for being the larger root of the equation . We determine with the accuracy of 1 the maximal number of -fractional positions, which may arise as a result of addition of two -integers. For the infinite word coding distances between the consecutive -integers, we determine precisely also the balance. The word is the only fixed point of the morphism → and → . In the case , the corresponding infinite word is sturmian, and, therefore,...

It is known that a real symmetric circulant matrix with diagonal entries $d\ge 0$, off-diagonal entries $\pm 1$ and orthogonal rows exists only of order $2d+2$ (and trivially of order $1$) [Turek and Goyeneche 2019]. In this paper we consider a complex Hermitian analogy of those matrices. That is, we study the existence and construction of Hermitian circulant matrices having orthogonal rows, diagonal entries $d\ge 0$ and any complex entries of absolute value $1$ off the diagonal. As a particular case, we consider matrices whose...