It is shown that monic orthogonal polynomials on the unit circle are the characteristic polynomials of certain five-diagonal matrices depending on the Schur parameters. This result is achieved through the study of orthogonal Laurent polynomials on the unit circle. More precisely, it is a consequence of the five term recurrence relation obtained for these orthogonal Laurent polynomials, and the one to one correspondence established between them and the orthogonal polynomials on the unit circle. As an application, some results relating the behavior of the zeros of orthogonal polynomials and the location of Schur parameters are obtained.

We consider quantum random walks (QRW) on the integers, a subject that has been considered in the last few years in the framework of quantum computation.We show how the theory of CMV matrices gives a natural tool to study these processes and to give results that are analogous to those that Karlin and McGregor developed to study (classical) birth-and-death processes using orthogonal polynomials on the real line.In perfect analogy with the classical case, the study of QRWs on the set of nonnegative integers can be handled using scalar-valued (Laurent) polynomials and a scalar-valued measure on the circle. In the case of classical or quantum random walks on the integers, one needs to allow for matrix-valued versions of these notions.We show how our tools yield results in the well-known case of the Hadamard walk, but we go beyond this translation-invariant model to analyze examples that are hard to analyze using other methods. More precisely, we consider QRWs on the set of nonnegative integers. The analysis of these cases leads to phenomena that are absent in the case of QRWs on the integers even if one restricts oneself to a constant coin. This is illustrated here by studying recurrence properties of the walk, but the same method can be used for other purposes.The presentation here aims at being self-contained, but we refrain from trying to give an introduction to quantum random walks, a subject well surveyed in the literature we quote. For two excellent reviews, see [1,19]. See also the recent notes [20].

In this paper we prove that the simplest band representations of unitary operators on a Hilbert space are five-diagonal. Orthogonal polynomials on the unit circle play an essential role in the development of this result, and also provide a parametrization of such five-diagonal representations which shows specially simple and interesting decomposition and factorization properties. As an application we get the reduction of the spectral problem of any unitary Hessenberg matrix to the spectral problem of a five-diagonal one. Two applications of these results to the study of orthogonal polynomials on the unit circle are presented: the first one concerns Krein's Theorem; the second one deals with the movement of mass points of the orthogonality measure under monoparametric perturbations of the Schur parameters.

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The CGMV method allows for the general discussion of localization properties for the states of a one-dimensional quantum walk, both in the case of the integers and in the case of the non negative integers. Using this method we classify, according to such localization properties, all the quantum walks with one defect at the origin, providing explicit expressions for the asymptotic return probabilities to the origin.2000 Mathematics Subject Classification. 81P68, 47B36, 42C05.

The aim of the study was to examine the relation between demographic variables, parental characteristics, and cognitive, language and motor skills development in children with intellectual disabilities (ID). A sample of 89 children with ID, aged 20–47 months, completed the Bayley Scales of Infant Development to measure cognitive, motor, and linguistic development. Parents were administered questionnaires about demographic information and parental anxiety, depression, parental stress, conjugality and familial functioning. Parenting behaviors (affection, responsiveness, encouragement, and teaching) were observed using the Spanish version of PICCOLO (Parenting Interactions with Children: Checklist of Observations Linked to Outcomes). A bivariate analysis showed that cognitive development in infants was significantly related to the mother’s and father’s responsiveness, and to the father’s teaching scores. Infant language development was related to a variety of maternal factors (educational level, anxiety, depression, maternal responsiveness) and to the father’s teaching scores. None of the factors were statistically related to child motor development. A multivariate regression analysis indicated that children’s cognitive development can be predicted by a linear combination of maternal responsiveness and paternal teaching scores. Language development can be predicted by a linear combination of maternal anxiety and responsiveness, and paternal teaching scores. The present study provides evidence of the importance of paternal involvement for cognitive and language development in children with intellectual disabilities, and contributes to the increasing literature about fathering. Gaining knowledge about parental contributions to children’s development is relevant for improving positive parenting in early intervention programs.

Background/Objective: The aim of this study was to explore the psychometric properties of the Spanish version of the Parenting Interactions with Children: Checklist of Observations Linked to Outcomes (PICCOLO; Roggman et al., 2013a ). This observational measure is composed of 29 items that assess the quality of four domains of parenting interactions that promote child development: affection, responsiveness, encouragement, and teaching. Methods: The sample included 203 mother-child dyads who had been video-recorded playing together. Fifty-six percent of the children were male, and 44% were female, aged from 10 to 47 months. Video-recorded observations were rated using PICCOLO items. Results: Confirmatory factor analysis supported that the instrument has four first-order factors corresponding to the hypothesized domains of parenting behaviors, and a second-order factor corresponding to a general factor of positive parenting. Construct validation evidence was compiled by examining the relationship between PICCOLO scores and child age. As expected, teaching domain and total PICCOLO scores were positively correlated with child age. The Spanish PICCOLO also demonstrated good inter-rater reliability (ranging from 0.69 to 0.84) and internal consistency reliability (ranging from 0.59 to 0.88) for the four domain scores and the total parenting score. Concurrent criterion-related validity was examined via correlations between parenting scores and child cognitive, language and motor skills outcomes, measured using the Bayley Scales of Infant Development. Conclusion: The Spanish version of the PICCOLO meets the criteria for a reliable and valid observational measurement of parenting interactions with children. The psychometric properties of the instrument make it appropriate for general research purposes, but also for program evaluation of Early Intervention and other parenting-support interventions. This measure, focused on parent strengths, could be used to facilitate family-centered practices in early intervention and other programs that have parenting as an outcome.

In this paper we consider a general sequence of orthogonal Laurent polynomials on the unit circle and we first study the equivalences between recurrences for such families and Szegő's recursion and the structure of the matrix representation for the multiplication operator in Λ when a general sequence of orthogonal Laurent polynomials on the unit circle is considered. Secondly, we analyze the computation of the nodes of the Szegő quadrature formulas by using Hessenberg and five-diagonal matrices. Numerical examples concerning the family of Rogers-Szegő q-polynomials are also analyzed.