Some new formulas related to the well-known symmetric Lucas polynomials are primary focus of this article. Different approaches used for establishing these formulas. A matrix approach is followed in order obtain some fundamental properties. Particularly, recurrence relations and determinant forms determined by suitable Hessenberg matrices. Conjugate generating functions derived examined. Severa...

Apparently, the energy levels merge and disappear in many PT symmetric models. This interpretation is incorrect: In square-well model we demonstrate how the doublets of states in question continue to exist at complex conjugate energies in the strongly non-Hermitian regime. PACS 03.65.Fd,03.65.Ge,03.65.Nk

In the present paper, we introduce two new subclasses P sc (λ, α) and Q sc (λ, α) of analytic functions with respect to 2k-symmetric conjugate points. Such results as integral representations, convolution conditions and coefficient inequalities for these classes are provided. 2000 Mathematics Subject Classification. Primary 30C45.

A method for computing a sparse incomplete factorization of the inverse of a symmetric positive definite matrix A is developed, and the resulting factorized sparse approximate inverse is used as an explicit preconditioner for conjugate gradient calculations. It is proved that in exact arithmetic the preconditioner is well defined if A is an H-matrix. The results of numerical experiments are pre...

For the numerical solution of the Galerkin equations associated with linear ill-posed problems that are symmetric and positive semideenite, the method of conjugate residuals is considered. An a posteriori stopping rule is introduced, and associated estimates for the approximations are provided which are order-optimal with respect to noise in the right-hand side and with respect to the discretiz...

It is shown how pseudo-Hermiticity, a necessary condition satisfied by operators of PT symmetric systems can be introduced in the three Gaussian classes of random matrix theory. The model describes transitions from real eigenvalues to a situation in which, apart from a residual number, the eigenvalues are complex conjugate.

A striking result of Bouc gives the decomposition of the representation of the symmetric group on the homology of the matching complex into irreducibles that are self-conjugate. We show how the combinatorial Laplacian can be used to give an elegant proof of this result. We also show that the spectrum of the Laplacian is integral.

Current implementations of OFDM/OQAM are restricted to band-limited symmetric filters. To circumvent this, non-symmetric conjugate root (CR) filters are proposed for OQAM modulation. The system is applied to Generalized Frequency Division Multiplexing (GFDM) and a method for achieving transmit diversity with OQAM modulation is presented. The proposal reduces implementation complexity compared t...

Quantitative bounds are presented for the superlinear convergence of the MINRES method of Paige and Saunders [SIAM J. Numer. Anal., 1975] for the solution of sparse linear systems Ax = b, with A symmetric and indefinite. It is shown that the superlinear convergence is observed as soon as the harmonic Ritz values approximate well the eigenvalues of A that are either closest to zero or farthest f...