On Symmetric (1, 1)-Coherent Pairs and Sobolev Orthogonal polynomials: an algorithm to compute Fourier coefficients

In the pioneering paper [13], the concept of Coherent Pair was introduced by Iserles et al. In particular, an algorithm to compute Fourier Coefficients in expansions of Sobolev orthogonal polynomials defined from coherent pairs of measures supported on an infinite subset of the real line is described. In this paper we extend such an algorithm in the framework of the so called Symmetric (1, 1)-Coherent Pairs presented in [8].

Sailing to Sicily

Chapter 2 addresses the partnership between Odysseus and the seer Theoklymenos in the Odyssey. I argue that the seer and the hero operate as a coherent pair within the framework of colonization and that Theoklymenos aids Odysseus-as-oikist in effecting the metaphorical refoundation of Ithaka. Theoklymenos and Odysseus reveal that the pairing of the seer and oikist is a possible and productive construct. Yet Homer’s presentation of Theoklymenos as a doublet of Odysseus also suggests the potential for competition between the seer and oikist in post-Homeric foundation tales.

Generalized coherent pairs on the unit circle and Sobolev orthogonal polynomials

A pair of regular Hermitian linear functionals (U, V) is said to be an (M,N)-coherent pair of order m on the unit circle if their corresponding sequences of monic orthogonal polynomials {?n(z)}n>0 and {?n(z)}n?0 satisfy ?Mi=0 ai,n?(m) n+m?i(z) = ?Nj=0 bj,n?n?j(z), n ? 0, where M,N,m ? 0, ai,n and bj,n, for 0 ? i ? M, 0 ? j ? N, n > 0, are complex numbers such that aM,n ? 0, n ? M, bN,n ? 0, n ? N, and ai,n = bi,n = 0, i > n. When m = 1, (U, V) is called a (M,N)-coherent pair on the unit circle. We focus our attention on the Sobolev inner product p(z), q(z)?= (U,p(z)q(1/z))+ ?(V, p(m)(z)q(m)(1/z)), ? > 0, m ? Z+, assuming that U and V is an (M,N)-coherent pair of order m on the unit circle. We generalize and extend several recent results of the framework of Sobolev orthogonal polynomials and their connections with coherent pairs. Besides, we analyze the cases (M,N) = (1, 1) and (M,N) = (1, 0) in detail. In particular, we illustrate the situation when U is the Lebesgue linear functional and V is the Bernstein-Szeg? linear functional. Finally, a matrix interpretation of (M,N)-coherence is given.

(1,1)-Coherent Pairs on the Unit Circle

A pair(𝒰,𝒱)of Hermitian regular linear functionals on the unit circle is said to be a(1,1)-coherent pair if their corresponding sequences of monic orthogonal polynomials{ϕn(x)}n≥0and{ψn(x)}n≥0satisfyϕn[1](z)+anϕn-1[1](z)=ψn(z)+bnψn-1(z),an≠0,n≥1, whereϕn[1](z)=ϕn+1′(z)/(n+1). In this contribution, we consider the cases when𝒰is the linear functional associated with the Lebesgue and Bernstein-Szegő measures, respectively, and we obtain a classification of the situations where𝒱is associated with either a positive nontrivial measure or its rational spectral transformation.

COHERENT PAIR PRODUCTION IN CRYSTALS IN PRESENCE OF ACOUSTIC WAVES

Coherent electron-positron pair production by high-energy photons is investigated in a periodically deformed single crystal with a complex base. The formula for the corresponding differential cross-section is derived for an arbitrary deformation field. The conditions are specified under which the influence of the deformation is considerable. The case is considered in detail when the photon enters into the crystal at small angles with respect to a crystallographic axis. The results of the numerical calculations are presented for SiO 2 single crystal in the case of the deformation field generated by the acoustic wave of the S type. It is shown that, in dependence of the parameters, the presence of deformation can either enhance or reduce the pair creation cross-section.