Attractivity of Cycles in Hamming Spheres

• Main Authors: Yung-Yih Lur, 盧永毅
• Other Authors: Mau-hsiang Shih
• Format: Others
• Language: en_US
• Published: 1999
• Online Access: http://ndltd.ncl.edu.tw/handle/41249375899226830948

博士 === 中原大學 === 數學系 === 87 === This thesis consists of the following subjects ; (i) Develop a theory of products of Boolean matrices; (ii) Give a characterization of a Boolean contraction from the view point of products of Boolean matrices; (iii) Computation of the number of equivalence classes of hypercubic mappings; (iv) A study of the attractivity of a cycle; (v) Give a partial solution of the packing von Neumann neighbourhoods problem. This paper is a study of the motions of hypercube mappings $F:\{0,1 \}^n \to \{0,1 \}^n$. It is a simple fact that the iteration $x^{r+1}=F(x^r)$ is either convergent to a fixed point or convergent to a cycle with cycle length $\ge 2$. However, if we consider the attractivity in the Hamming spheres, the iteration behavior could be quite chaotic. Thus the consideration of the attractivity in the Hamming spheres becomes a significant point. The problems we shall study are the following : {\bf Problem 1. } Find necessary and sufficient conditions such that $C$ is attractive in $V_m(C)$. {\bf Problem 2. } Find sufficient conditions such that $C$ is strongly attractive in $V_m(C)$ or in the whole $n$-cube.\ There are two classes of mappings such that $F$ is simple. One is called a Boolean contraction which was introduced by Robert [14]. To study the Boolean contraction, it leads to a study of products of Boolean matrices which was motivated by Daubechies-Lagarias[3] and Shih [17] for a set of complex matrices. This is the main contexts of Chapters 2, 3 and 4. In the study the attractivity of a cycle, the following packing von Neumann neighbourhoods problems was raised by Robert [14], p.122. The packing problem is stated as follows : {\bf Packing von Neumann neighbourhoods Problem} : How many distinct points $a_i$ can be placed on the $n$-cube $\{0,1 \}^n$ such that $V_1(a_i)$ are mutually disjoint ? Equivalently, what is the maximum length of a cycle $C$ of $F:\{0,1 \}^n \to \{0,1 \}^n$ such that $V_1(a)\ (a \in C)$ are mutually disjoint ? We shall give a partial answer in Chapter 7. Thus we give an answer for $n=1,2,3,4,5,6$ or $n=2^r-1$ for $r \ge 1$. Since there are $(2^n)^{2^n}$ mappings of $\{0,1 \}^n$ into itself,the number $(2^n)^{2^n}$ grows astronomically as $n$ increases. Thus we shall see from {\it P\''lya theory of enumeration} that the number of mappings can be reduced, but still out of the question from the computer when $n$ is large. Let us say that two hypercube mappings $F$ and $G$ are {\it equivalent} if there is a permutation $\pi$ on $\{0,1 \}^n$ such that \[ F(x)=G(\pi(x)) \; \mbox{ for all } \; x \in \{0,1 \}^n. \hspace{5cm} \] It is shown that if $n=3$, there are 6435 numbers of equivalence classes; $n=4$, there are 300540195 numbers of equivalence classes. This thesis is organized as follows. In Chapters 2 and 3 we study the products of Boolean matrices. Chapter 4 is a characterization of Boolean contraction in terms of the Boolean derivatives. Chapter 5 is a study of the numbers of equivalence classes of hypercube mappings. Chapter 6, the main theme of this thesis, gives a study of the attractivity of a fixed point or a cycle of length $\ge 2$, mentioning in the previous problems. Finally, in Chapter 7, we give a partial solution of the packing von Neumann neighbourhoods problem.