Interlacing of zeros of orthogonal polynomials under modification of the measure

We investigate the mutual location of the zeros of two families of orthogonal polynomials. One of the families is orthogonal with respect to the measure dμ (x), supported on the interval (a, b) and the other with respect to the measure |x -c|τ|x -d|γdμ (x), where c and d are outside (a, b) We prove that the zeros of these polynomials, if they are of equal or consecutive degrees, interlace when either 0 < τ, γ ≤ 1 or γ = 0 and 0 < τ ≤ 2. This result is inspired by an open question of Richard Askey and it generalizes recent results on some families of orthogonal polynomials. Moreover, we obtain further statements on interlacing of zeros of specific orthogonal polynomials, such as the Askey-Wilson ones. © 2013 Elsevier Inc.

Polinômios ortogonais e L-ortogonais associados a medidas relacionadas

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Polinômios de Szegö e análise de frequência

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Parâmetros genéticos para produção de leite e persistência de lactações múltiplas na raça Gir

Pós-graduação em Genética e Melhoramento Animal - FCAV

Efeito da seleção para crescimento em diferentes idades sobre o consumo de energia em animais da raça Nelore

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Application of continuous positive airway pressure in the delivery room: a multicenter randomized clinical trial

This study evaluated whether the use of continuous positive airway pressure (CPAP) in the delivery room alters the need for mechanical ventilation and surfactant during the first 5 days of life and modifies the incidence of respiratory morbidity and mortality during the hospital stay. The study was a multicenter randomized clinical trial conducted in five public university hospitals in Brazil, from June 2008 to December 2009. Participants were 197 infants with birth weight of 1000-1500 g and without major birth defects. They were treated according to the guidelines of the American Academy of Pediatrics (APP). Infants not intubated or extubated less than 15 min after birth were randomized for two treatments, routine or CPAP, and were followed until hospital discharge. The routine (n=99) and CPAP (n=98) infants studied presented no statistically significant differences regarding birth characteristics, complications during the prenatal period, the need for mechanical ventilation during the first 5 days of life (19.2 vs 23.4%, P=0.50), use of surfactant (18.2 vs 17.3% P=0.92), or respiratory morbidity and mortality until discharge. The CPAP group required a greater number of doses of surfactant (1.5 vs 1.0, P=0.02). When CPAP was applied to the routine group, it was installed within a median time of 30 min. We found that CPAP applied less than 15 min after birth was not able to reduce the need for ventilator support and was associated with a higher number of doses of surfactant when compared to CPAP applied as clinically indicated within a median time of 30 min.

The Positive Effects of Priming Exercise on Oxygen Uptake Kinetics and High-Intensity Exercise Performance Are Not Magnified by a Fast-Start Pacing Strategy in Trained Cyclists

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Polynomials to model the growth of young bulls in performance tests

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Development of diagnostic methods and study of the immunoreactivity of a mixture of recombinant core and E2 proteins fused to GST with control serum positive for hepatitis C

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Diastolic function and functional capacity after a single session of continuous positive airway pressure in patients with compensated heart failure

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Zeros of classical orthogonal polynomials of a discrete variable

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Basic hypergeometric polynomials with zeros on the unit circle

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Monotonicity, interlacing and electrostatic interpretation of zeros of exceptional Jacobi polynomials

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Stieltjes functions and discrete classical orthogonal polynomials

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Massless scattering at special kinematics as Jacobi polynomials

We study the scattering equations recently proposed by Cachazo, He and Yuan in the special kinematics where their solutions can be identified with the zeros of the Jacobi polynomials. This allows for a non-trivial two-parameter family of kinematics. We present explicit and compact formulas for the n-gluon and n-graviton partial scattering amplitudes for our special kinematics in terms of Jacobi polynomials. We also provide alternative expressions in terms of gamma functions. We give an interpretation of the common reduced determinant appearing in the amplitudes as the product of the squares of the eigenfrequencies of small oscillations of a system whose equilibrium is the solutions of the scattering equations.