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B) The eigenvalues of Fn are ˙1 and ˙i with multiplicities depending on n. c) The characteristic polynomials of Fn are ( ( 2 ( 1)2 ( i) ( C 1) (n 1)/4 4 1 1) (n 2)/4 4 1 1 4 1 i) 2 1 4 1 n 4 1 (n 3)/4 n D 0 mod(4) n D 1 mod(4) n D 2 mod(4) n D 3 mod(4) (13) Every n n circulant matrix A is diagonalized by Fn . Further, if PA is the polynomial deﬁned by Eq. ); : : : ; PA (! n 1 ) (14) hence the rth eigenvalue of A is PA (! r ). Deﬁne the n n matrix r D diag(0; : : : ; 0; 1; 0; : : : ; 0) with the 1 in the rth position and set ˘r D Fn r Fn .
Institute for Theoretical Physics NSC KIPT Kharkov Ukraine CHEN, CHUN-HAO National Cheng–Kung University Tainan Taiwan CHEN, GUANRONG City University of Hong Kong Hong Kong China CHEN, Z HENGXIN University of Nebraska at Omaha Omaha USA CHOPARD, BASTIEN University of Geneva Geneva Switzerland CHUNG, FAN University of California San Diego USA CONTE, ROSARIA CNR Rome Italy CPAŁKA , KRZYSZTOF Cz˛estochowa University of Technology Cz˛estochowa Poland Academy of Humanities and Economics Lodz Poland CREUTZ, MICHAEL Brookhaven National Laboratory Upton USA DARDZINSKA , AGNIESZKA Białystok Technical University Białystok Poland DAVIDSSON, PAUL Blekinge Institute of Technology Ronneby Sweden DEMURJIAN, STEVEN A.
If L is isomorphic to Zn , the integers mod(n), the boundary conditions are periodic and the lattice is circular (it is a p-adic necklace). This is called a cylindrical cellular automata  because evolution of the rule can be represented as taking place on a cylinder. If the lattice is isomorphic to f0; : : : ; n 1g, null, or Dirchlet boundary conditions are set [78,79,80]. That is, the symbol assigned to all sites in L outside of this set is the null symbol. When the lattice is isomorphic to the non-negative integers ZC , null boundary conditions are set at the left boundary.