Factor properties and their structures are important themes in combinatorics on words.Let D be the infinite one-sided sequence over the alphabet{a,b}generated by the period-doubling substitutionσ(a)=ab andσ(b)=aa.In this paper,we determine the derived sequence D_(ω)(D)for any factorω■D,and study some factor spectra using the structures of derived sequences.We also prove the reflexivity property of derived sequences.
The localization characteristics of the in-plane elastic waves in locally resonant aperiodic phononic crystals are examined in this study. In particular, the phononic crystals generated according to the Thue-Morse, Rudin-Shapiro and Period-Doubling sequences are theoretically investigated by using the extended transfer matrix method. For comparison, the binary and ternary locally resonant systems are considered, and their band structures are characterized by using the localization factors. Moreover, the influences of structural arrangement, material combination, incidence angle, number of components, length ratio, and random disorder on the band structures are also discussed. Some novel and interesting phenomena are observed and discussed.
