Zeros of a family of hypergeometric para-orthogonal polynomials on the unit circle

Para-orthogonal polynomials derived from orthogonal polynomials on the unit circle are known to have all their zeros on the unit circle. In this note we study the zeros of a family of hypergeometric para-orthogonal polynomials. As tools to study these polynomials, we obtain new results which can be considered as extensions of certain classical results associated with three term recurrence relations and differential equations satisfied by orthogonal polynomials on the real line. One of these results which might be considered as an extension of the classical Sturm comparison theorem, enables us to obtain monotonicity with respect to the parameters for the zeros of these para-orthogonal polynomials. Finally, a monotonicity of the zeros of Meixner-Pollaczek polynomials is proved. © 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

Orthogonal polynomials on the unit circle and chain sequences

Szego{double acute} has shown that real orthogonal polynomials on the unit circle can be mapped to orthogonal polynomials on the interval [-1,1] by the transformation 2x=z+z-1. In the 80's and 90's Delsarte and Genin showed that real orthogonal polynomials on the unit circle can be mapped to symmetric orthogonal polynomials on the interval [-1,1] using the transformation 2x=z1/2+z-1/2. We extend the results of Delsarte and Genin to all orthogonal polynomials on the unit circle. The transformation maps to functions on [-1,1] that can be seen as extensions of symmetric orthogonal polynomials on [-1,1] satisfying a three-term recurrence formula with real coefficients {cn} and {dn}, where {dn} is also a positive chain sequence. Via the results established, we obtain a characterization for a point w(|w|=1) to be a pure point of the measure involved. We also give a characterization for orthogonal polynomials on the unit circle in terms of the two sequences {cn} and {dn}. © 2013 Elsevier Inc.

On computational aspects of discrete Sobolev inner products on the unit circle

In this paper, we show how to compute in O(n2) steps the Fourier coefficients associated with the Gelfand-Levitan approach for discrete Sobolev orthogonal polynomials on the unit circle when the support of the discrete component involving derivatives is located outside the closed unit disk. As a consequence, we deduce the outer relative asymptotics of these polynomials in terms of those associated with the original orthogonality measure. Moreover, we show how to recover the discrete part of our Sobolev inner product. © 2013 Elsevier Inc. All rights reserved.

A Favard type theorem for orthogonal polynomials on the unit circle from a three term recurrence formula

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Basic hypergeometric polynomials with zeros on the unit circle

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Sieved para-orthogonal polynomials on the unit circle

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COINCIDENCES OF FIBREWISE MAPS BETWEEN SPHERE BUNDLES OVER THE CIRCLE

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Signs, translation, and life in the Bakhtin circle and in Welby's significs

“Significs” is provisionally defined by Welby (1911: vii) as the study of the nature of significance in all its forms and relationships, of its workings in all spheres of human life and knowledge. Considering “significs” as a movement highlighting significance, Welby explores the action of signs in life; and more than the Saussurean sign composed of signifier and signified, the sign as understood by Welby refers to meaning as generated through signs in motion. This notion of “significs” empowers the study of signs when it considers the sign not in terms of the Saussurean structural representation of the union of the concept and acoustic image, but as (responsive and responsible) sign action in the world, in life. This also means to take into account the “extra-linguistic referent” (translinguistic and transdiscursive character of significs), history (space-time), subjectivity, the architecture of values connected to language, their communicative function. We believe that a dialogue can be established between Welby’s vision of significs and the notion of ideological sign proposed by Vološinov in Marxism and the Philosophy of Language, expanding the notions of “meaning” and “sense.”

Quantum Discord for d circle times 2 Systems

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Szegö polynomials: quadrature rules on the unit circle and on [-1, 1]

We consider some of the relations that exist between real Szegö polynomials and certain para-orthogonal polynomials defined on the unit circle, which are again related to certain orthogonal polynomials on [-1, 1] through the transformation x = (z1/2+z1/2)/2. Using these relations we study the interpolatory quadrature rule based on the zeros of polynomials which are linear combinations of the orthogonal polynomials on [-1, 1]. In the case of any symmetric quadrature rule on [-1, 1], its associated quadrature rule on the unit circle is also given.

New fermions and a vectorlike third generation in SU(3)(C)circle times SU(3)(L)circle times U(1)(N) models

We study two 3-3-1 models with (i) five (four) charge 2/3 (-1/3) quarks and (ii) four (five) charge 2/3 (-1/3) quarks and a vectorlike third generation. Possibilities beyond these models are also briefly considered.

Book Review of "Will the Circle Be Unbroken?": Aboriginal Communities, Restorative Justice, and the Challenges of Conflict and Change by Jane Dickson-Gilmore and Carol La Prairie

Over the past decade or two, restorative justice has become a popular approach for the criminal justice system to take in Canada, New Zealand, and Australia. In part, this is due in all three countries to an appalling disproportionality in the incarceration rates for racialized minorities. As the authors of "Will the Circle Be Unbroken?" point out, however, governments have been attracted to restorative justice for cost-cutting reasons as well. A burning question, therefore, is whether restorative justice works.

Rigidity for piecewise smooth homeomorphisms on the circle

We find conditions for two piecewise 'C POT.2+V' homeomorphisms f and g of the circle to be 'C POT.1' conjugate. Besides the restrictions on the combinatorics of the maps (we assume that the maps have bounded combinatorics), and necessary conditions on the one-side derivatives of points where f and g are not differentiable, we also assume zero mean-nonlinearity for f and g.

A forgotten little chapter on isoperimetric inequalities. On the fraction of a convex and closed plane area lying outside a circle with which it shares a diameter

[EN]Often some interesting or simply curious points are left out when developing a theory. It seems that one of them is the existence of an upper bound for the fraction of area of a convex and closed plane area lying outside a circle with which it shares a diameter, a problem stemming from the theory of isoperimetric inequalities. In this paper such a bound is constructed and shown to be attained for a particular area. It is also shown that convexity is a necessary condition in order to avoid the whole area lying outside the circle

Thaumatrope (Magic Circle)

The thaumatrope consists of a circle of cardstock, 2.5 inches in diameter with 2 strings attached, one each at opposite points of the diameter. There were 2 images painted on the cardstock, one on each side, with their positions inverted. The outline of the image was usually printed and the color hand-painted in (Barnes 7).