arXiv subjects
Publications and source records attributed to Hanxiong Zhang.
We study certain kind of polynomials associated with Lissajous curves, called Chebyshev-Lissajous polynomials. We investigate their irreducibilities over the real numbers and complex numbers, thus comfirming two conjectures proposed by Merino.
Let $ω$ be a factor of Doubling sequence $D_\infty=x_1x_2\cdots$, then it occurs in the sequence infinitely many times. Let $ω_p$ be the $p$-th occurrence of $ω$ and $G_p(ω)$ be the gap between $ω_p$ and $ω_{p+1}$. In this paper, we discuss the structure of the gap sequence $\{G_p(ω)\}_{p\geq1}$. We prove that all factors can be divided into two types, one type has exactly two distinct gaps $G_1(ω)$ and $G_2(ω)$, the other type has exactly three distinct gaps $G_1(ω)$, $G_2(ω)$ and $G_4(ω)$. We determine the expressions of gaps completely. And also give the substitution of each gap sequence. The main tool in this paper is "envelope word", which is a new notion, denoted by $E_{m,i}$. As an application, we determine the positions of all $ω_p$, discuss some combinatorial properties of factors, and count the distinct squares beginning in $D_\infty[1,N]$ for $N\geq1$.
In this paper, we consider the factor properties and gap sequence of a special type of cutting sequence with slope $θ=[0;\dot{d}]$, denoted by $F_{d,\infty}$. Let $ω$ be a factor of $F_{d,\infty}$, then it occurs in the sequence infinitely many times. Let $ω_p$ be the $p$-th occurrence of $ω$ and $G_p(ω)$ be the gap between $ω_p$ and $ω_{p+1}$. We define the $d$ types of kernel words and envelope words, give two versions of "uniqueness of kernel decomposition property". Using them, we prove the gap sequence $\{G_p(ω)\}_{p\geq1}$ has exactly two distinct elements for each $ω$, and determine the expressions of gaps completely. Furthermore, we prove that the gap sequence is $σ_i(F_{d,\infty})$, where $σ_i$ is a substitution depending only on the type of $Ker(ω)$, i.e. the kernel word of $ω$. We also determine the position of $ω_p$ for all $(ω,p)$. As applications, we study some combinatorial properties, such as the power, overlap and separate property between $ω_p$ and $ω_{p+1}$ for all $(ω,p)$, and find all palindromes in $F_{d,\infty}$.