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Article Citation - WoS: 4Citation - Scopus: 3The Euler-Lagrange Theory for Schur's Algorithm: Wall Pairs(Academic Press inc Elsevier Science, 2006) Khrushchev, SThis paper develops a techniques of Wall pairs for the study of periodic exposed quadratic irrationalities in the unit ball of the Hardy algebra. (C) 2005 Elsevier Inc. All rights reserved.Article Citation - WoS: 36Citation - Scopus: 40On the Improvement of Analytic Properties Under the Limit Q-Bernstein Operator(Academic Press inc Elsevier Science, 2006) Ostrovska, SLet B-n(f, q; x), n = 1, 2,... be the q-Bernstein polynomials of a function f is an element of C[0, 1]. In the case 0 < q < 1, a sequence {B-n(f, q; x)} generates a positive linear operator B-infinity = B-infinity,B-q on C[0, 1], which is called the limit q-Bernstein operator In this paper, a connection between the smoothness of a function f and the analytic properties of its image under Boo is studied. (c) 2005 Elsevier Inc. All rights reserved.Article Citation - WoS: 5Citation - Scopus: 5The Euler-Lagrange Theory for Schur's Algorithm: Algebraic Exposed Points(Academic Press inc Elsevier Science, 2006) Khrushchev, SIn this paper the ideas of Algebraic Number Theory are applied to the Theory of Orthogonal polynomials for algebraic measures. The transferring tool are Wall continued fractions. It is shown that any set of closed arcs on the circle supports a quadratic measure and that any algebraic measure is either a Szego measure or a measure supported by a proper subset of the unit circle consisting of a finite number of closed arcs. Singular parts of algebraic measures are finite sums of point masses. (C) 2005 Elsevier Inc. All rights reserved.Article Citation - WoS: 26Citation - Scopus: 26Classification Theorems for General Orthogonal Polynomials on the Unit Circle(Academic Press inc Elsevier Science, 2002) Khrushchev, SVThe set P of all probability measures a on the unit circle T splits into three disjoint subsets depending on properties of the derived set of {\phi(n)\(2) dsigma}(ngreater than or equal to0), denoted by Lim(sigma). Here {phi(n)}(ngreater than or equal to0) are orthogonal polynomials in L-2(dsigma). The first subset is the set of Rakhmanov measures, i.e., of sigma is an element of P with {m} = Lim(sigma), m being the normalized (m(T) = 1) Lebesgue measure on T. The second subset Mar(T) consists of Markoff measures, i.e., of sigma is an element of P with m is not an element of Lim(sigma), and is in fact the subject of study for the present paper. A measure sigma, belongs to Mar(T) iff there are epsilon > 0 and l > 0 such that sup{\a(n+j)\: 0 less than or equal to j less than or equal to l) > epsilon, n = 0, 1, 2,..., {a(n)} is the Geronimus parameters (= reflection coefficients) of sigma. We use this equivalence to describe the asymptotic behavior of the zeros of the corresponding orthogonal polynomials (see Theorem G). The third subset consists of sigma is an element of P with {m} not subset of or equal toLim(sigma). We show that sigma is ratio asymptotic iff either sigma is a Rakhmanov measure or sigma satisfies the Lopez condition (which implies sigma is an element of Mar(T)). Measures sigma satisfying Lim(sigma) = {v} (i.e., weakly asymptotic measures) are also classified. Either v is the sum of equal point masses placed at the roots of z(n) = lambda, lambda is an element of T, n = 1, 2,..., or v is the equilibrium measure (with respect to the logarithmic kernel) for the inverse image under an m-preserving endomorphism z -->z(n), = 1, 2,..., of a closed arc J (including J = T) with removed open concentric are J(0) (including J(0) = empty set). Next, weakly asymptotic measures are completely described in terms of their Geronimus parameters. Finally, we obtain explicit formulae for the parameters of the equilibrium measures v and show that these measures satisfy {v} = Lim(v). (C) 2002 Elsevier Science (USA).Article Citation - WoS: 12Citation - Scopus: 13Classification of Some Quadrinomials Over Finite Fields of Odd Characteristic(Academic Press inc Elsevier Science, 2023) Ozbudak, Ferruh; Temur, Burcu GulmezIn this paper, we completely determine all necessary and sufficient conditions such that the polynomial f(x) = x3 + axq +2 + bx2q +1 + cx3q, where a, b, c is an element of Fq*, is a permutation quadrinomial of Fq2 over any finite field of odd characteristic. This quadrinomial has been studied first in [25] by Tu, Zeng and Helleseth, later in [24] Tu, Liu and Zeng revisited these quadrinomials and they proposed a more comprehensive characterization of the coefficients that results with new permutation quadrinomials, where char(Fq) = 2 and finally, in [16], Li, Qu, Li and Chen proved that the sufficient condition given in [24] is also necessary and thus completed the solution in even characteristic case. In [6] Gupta studied the permutation properties of the polynomial x3 + axq +2 + bx2q +1 + cx3q, where char(Fq) = 3, 5 and a, b, c is an element of Fq* and proposed some new classes of permutation quadrinomials of Fq2 . In particular, in this paper we classify all permutation polynomials of Fq2 of the form f(x) = x3 + axq +2 + bx2q +1 + cx3q, where a, b, c is an element of Fq*, over all finite fields of odd characteristic and obtain several new classes of such permutation quadrinomials. (c) 2022 Elsevier Inc. All rights reserved.
