We prove the existence of ratio asymptotic for a sequence of multiple orthogonal polynomials which share orthogonality relations with a collection of m finite Borel measures supported on a bounded interval of the real line and constitute a so called Nikishin system of measures. When m = 1 our result reduces to E. A. Rakhmanov's known Theorem on ratio asymptotic for orthogonal polynomials on a segment.

The corrosion of pure Ag, Cu, Ni, and Sn specimens exposed for 1 to 24 months in a simulated indoor environment, consisting of a rain sheltered atmospheric corrosion test chamber placed in an urban desert environment (Baja California) has been measured. The corrosion rates of the metals were determined by mass loss measurement and the environment was thus classified in the low to medium indoor corrosivity category (IC2-IC3) according to ISO. Silver and copper weight losses were found to be very similar, while the nickel and tin weight losses were several times lower. The silver surface was tarnished in a non-uniform manner, presenting Ag 2 S and AgCl corrosion products, while the copper specimens corrode uniformly, being covered with Cu 2 O corrosion product. Owing to the presence of chloride contamination, the nickel and tin oxide corrosion films show fracture and pitting corrosion, developed over the first few months of exposure.

Abstract. We investigate the asymptotic properties of orthogonal polynomials for a class of inner products including the discrete Sobolev inner products h, g = hg dµ + m j=1

For polynomials orthogonal with respect to a discrete Sobolev product, we prove that, for each n, Q n has at least n -m zeros on the convex hull of the support of the measure, where m denotes the number of terms in the discrete part. Interlacing properties of zeros are also described.

Suppose we have a Nikishin system of p measures with the kth generating measure of the Nikishin system supported on an interval ∆ k ⊂ R with ∆ k ∩ ∆ k+1 = ∅ for all k. It is well known that the corresponding staircase sequence of multiple orthogonal polynomials satisfies a (p + 2)-term recurrence relation whose recurrence coefficients, under appropriate assumptions on the generating measures, have periodic limits of period p. (The limit values depend only on the positions of the intervals ∆ k .) Taking these periodic limit values as the coefficients of a new (p + 2)-term recurrence relation, we construct a canonical sequence of monic polynomials {Pn} ∞ n=0 , the so-called Chebyshev-Nikishin polynomials. We show that the polynomials Pn themselves form a sequence of multiple orthogonal polynomials with respect to some Nikishin system of measures, with the kth generating measure being absolutely continuous on ∆ k . In this way we generalize a result of the third author and Rocha [22] for the case p = 2. The proof uses the connection with block Toeplitz matrices, and with a certain Riemann surface of genus zero. We also obtain strong asymptotics and an exact Widom-type formula for the second kind functions of the Nikishin system for {Pn} ∞ n=0 .

The objective of this study was to analyze the effect that the application of the personal and social responsibility model has on students’ perception of a teacher’s interpersonal style and on the perception of autonomy. A quasi-experimental design was used with a control group (n = 60) and an experimental group (n = 60) to which the intervention was applied. Participants were aged between 10 and 13 years. As the main results, the experimental group saw improvements in support for the autonomous interpersonal style, in the need for autonomy satisfaction and also in the perception of personal and social responsibility. Perception of the controlling style decreased. In conclusion, the use of this type of program in educational centers is recommended for its benefits with regard to students’ autonomy and personal and social responsibility.

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