Band rules for the frequency spectra of three kinds of aperiodic photonic crystals with negative refractive index materials

By means of the theory of electromagnetic wave propagation and transfer matrix method, this paper investigates the band rules for the frequency spectra of three kinds of one-dimensional （1D） aperiodic photonic crystals （PCs）, generalized Fibonacci GF（p, 1）, GF（1,2）, and Thue Morse （TM） PCs, with negative refractive index （NRI） materials. It is found that all of these PCs can open a broad zero-n gap, TM PC possesses the largest zero-n gap, and with the increase of p, the width of the zero-n gap for GF（p, 1） PC becomes smaller. This characteristic is caused by the symmetry of the system and the open position of the zero-n gap. It is found that for GF（p, 1） PCs, the possible limit zero-n gaps open at lower frequencies with the increase of p, but for GF（1,2） and TM PCs, their limit zero-n gaps open at the same frequency. Additionally, for the tbree bottom-bands, we find the interesting perfect self-similarities of the evolution structures with the increase of generation, and obtain the corresponding subband-number formulae. Based on 11 types of evolving manners Qi （i = 1, 2,... , 11） one can plot out the detailed evolution structures of the three kinds of aperiodic PCs for any generation.

Convolution of Absolute Value Sum of Coefficients on Chebyshev Polynomials of the Second Kind

The main purpose of this paper is in using the generating function of generalized Fibonacci polynomials and its partial derivative to study the convolution evaluation of the second-kind Chebyshev polynomials, and give an interesting formula.

McKay Matrices for Pointed Rank One Hopf Algebras of Nilpotent Type

Let H be a finite-dimensional pointed rank one Hopf algebra of nilpotent type over a finite group G.In this paper,we investigate the McKay matrix WV of H for tensoring with the 2-dimensional indecomposable H-module V:=M(2,0).It turns out that the characteristic polynomial,eigenvalues and eigenvectors of WV are related to the character table of the finite group G and a kind of generalized Fibonacci polynomial.Moreover,we construct some eigenvectors of each eigenvalue for WV by using the factorization of the generalized Fibonacci polynomial.As an example,we explicitly compute the characteristic polynomial and eigenvalues of WV and give all eigenvectors of each eigenvalue for WV when G is a dihedral group of order 4N+2.

All Good (Bad) Words Consisting of 5 Blocks

Generalized Fibonacci cube Qd（f）, introduced by Ilid, Klavzar and Rho, is the graph obtained from the hypercube Qd by removing all vertices that contain f as factor. A word f is good if Qd（f） is an isometric subgraph of Q4 for all d ≥ 1, and bad otherwise. A non-extendable sequence of contiguous equal digits in a word μ is called a block of μ.Ilic, Klavzar and Rho shown that all the words consisting of one block are good, and all the words consisting of three blocks are bad. So a natural problem is to study the words consisting of other odd number of blocks. In the present paper, a necessary condition for a word consisting of odd number of blocks being good is given, and all the good （bad） words consisting of 5 blocks is determined.