| Summary: | 碩士 === 中原大學 === 應用數學研究所 === 94 === Let α=(√5-1)/2. Define G=…d-2d-1d0d1d2…, where dn=└ (n+1) α┘-└ nα┘; G'=…e-2e-1e0e1e2…, where en=┌(n+1) α┐-┌nα┐. G and G' are two-way infinite words, and are called two-way infinite Fibonacci binary words. For every nonnegative integer m,
let Gm=dm+1dm+2dm+3… and G'm=em+1em+2em+3…. Let u, v be nonempty alphabets,
h1=u, h2=v, hn=hn-1hn-2 (n≧3). The infinite word h1h2h3… is called a Fibonacci word pattern generated by u and v and is denoted by F(u,v), where the words u and v are called the seed words of F(u,v). In [5], W.F. Chuan, C.H. Chang, and Y.L. Chang proved that for each nonnegative integer m, Gm is a Fibonacci word pattern and
all pairs of seed words of Gm obtained are Fibonacci words. In this thesis, we prove that for each negative integer m, Gm and G'm are also Fibonacci word patterns and
all pairs of seed words of Gm and G'm obtained are Fibonacci words of consecutive order. Moreover, we use matrices to generate labels of seed words of Gm and G'm. Given any two binary strings r1r2…rn, s1s2…sn+1, we have a method to test whether F(w(r1r2…rn),w(s1s2…sn+1)) is a suffix of G or a suffix of G'. If it is a suffix of G (resp., G'), we determine an integer m such that F(w(r1r2…rn),w(s1s2…sn+1))= Gm (resp., G'm). On the other hand, let u and v be any two Fibonacci words of consecutive order, we also can test whether or not F(u,v) is a suffix of G or G', and identify such a suffix.
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