A connection between orthogonal polynomials on the unit circle and matrix orthogonal polynomials on the real line

Szego's procedure to connect orthogonal polynomials on the unit circle and orthogonal polynomials on [-1,1] is generalized to nonsymmetric measures. It generates the so-called semi-orthogonal functions on the linear space of Laurent polynomials L, and leads to a new orthogonality structure in the module LxL. This structure can be interpreted in terms of a 2x2 matrix measure on [-1,1], and semi-orthogonal functions provide the corresponding sequence of orthogonal matrix polynomials. This gives a connection between orthogonal polynomials on the unit circle and certain classes of matrix orthogonal polynomials on [-1,1]. As an application, the strong asymptotics of these matrix orthogonal polynomials is derived, obtaining an explicit expression for the corresponding Szego's matrix function.Comment: 28 page

A Statistical Study of Photospheric Magnetic Field Changes During 75 Solar Flares

Abrupt and permanent changes of photospheric magnetic fields have been observed during solar flares. The changes seem to be linked to the reconfiguration of magnetic fields, but their origin is still unclear. We carried out a statistical analysis of permanent line-of-sight magnetic field ($B_{\rm LOS}$) changes during 18 X-, 37 M-, 19 C- and 1 B-class flares using data from Solar Dynamics Observatory/Helioseismic and Magnetic Imager. We investigated the properties of permanent changes, such as frequency, areas, and locations. We detected changes of $B_{\rm LOS}$ in 59/75 flares. We find that strong flares are more likely to show changes, with all flares $\ge$ M1.6 exhibiting them. For weaker flares, permanent changes are observed in 6/17 C-flares. 34.3\% of the permanent changes occurred in the penumbra and 18.9\% in the umbra. Parts of the penumbra appeared or disappeared in 23/75 flares. The area where permanent changes occur is larger for stronger flares. Strong flares also show a larger change of flux, but there is no dependence of the magnetic flux change on the heliocentric angle. The mean rate of change of flare-related magnetic field changes is 20.7 Mx cm$^{-2}$ min$^{-1}$. The number of permanent changes decays exponentially with distance from the polarity inversion line. The frequency of the strength of permanent changes decreases exponentially, and permanent changes up to 750 Mx cm$^{-2}$ were observed. We conclude that permanent magnetic field changes are a common phenomenon during flares, and future studies will clarify their relation to accelerated electrons, white light emission, and sunquakes to further investigate their origin.Comment: Piblished in Ap

Multipliers of Laplace Transform Type for Laguerre and Hermite Expansions

We present a new criterion for the weighted $L^p-L^q$ boundedness of multiplier operators for Laguerre and Hermite expansions that arise from a Laplace-Stieltjes transform. As a special case, we recover known results on weighted estimates for Laguerre and Hermite fractional integrals with a unified and simpler approach.Comment: 22 pages; new section added, corrected typos, new references adde

Quadratic Maps in Two Variables on Arbitrary Fields

Let $\mathbb{F}$ be a field of characteristic different from $2$ and $3$, and let $V$ be a vector space of dimension $2$ over $\mathbb{F}$. The generic classification of homogeneous quadratic maps $f\colon V\to V$ under the action of the linear group of $V$, is given and efficient computational criteria to recognize equivalence are provided.Comment: 12 pages, no figure

Interface growth in two dimensions: A Loewner-equation approach

The problem of Laplacian growth in two dimensions is considered within the Loewner-equation framework. Initially the problem of fingered growth recently discussed by Gubiec and Szymczak [T. Gubiec and P. Szymczak, Phys. Rev. E 77, 041602 (2008)] is revisited and a new exact solution for a three-finger configuration is reported. Then a general class of growth models for an interface growing in the upper-half plane is introduced and the corresponding Loewner equation for the problem is derived. Several examples are given including interfaces with one or more tips as well as multiple growing interfaces. A generalization of our interface growth model in terms of ``Loewner domains,'' where the growth rule is specified by a time evolving measure, is briefly discussed.Comment: To appear in Physical Review

Kinematic study of planetary nebulae in NGC 6822

By measuring precise radial velocities of planetary nebulae (which belong to the intermediate age population), H II regions, and A-type supergiant stars (which are members of the young population) in NGC 6822, we aim to determine if both types of population share the kinematics of the disk of H I found in this galaxy. Spectroscopic data for four planetary nebulae were obtained with the high spectral resolution spectrograph Magellan Inamori Kyocera Echelle (MIKE) on the Magellan telescope at Las Campanas Observatory. Data for other three PNe and one H II region were obtained from the SPM Catalog of Extragalactic Planetary Nebulae which employed the Manchester Echelle Spectrometer attached to the 2.1m telescope at the Observatorio Astron\'omico Nacional, M\'exico. In the wavelength calibrated spectra, the heliocentric radial velocities were measured with a precision better than 5-6 km s$^{-1}$. Data for three additional H II regions and a couple of A-type supergiant stars were collected from the literature. The heliocentric radial velocities of the different objects were compared to the velocities of the H i disk at the same position. From the analysis of radial velocities it is found that H II regions and A-type supergiants do share the kinematics of the H I disk at the same position, as expected for these young objects. On the contrary, planetary nebula velocities differ significantly from that of the H I at the same position. The kinematics of planetary nebulae is independent from the young population kinematics and it is closer to the behavior shown by carbon stars, which are intermediate-age members of the stellar spheroid existing in this galaxy. Our results are confirming that there are at least two very different kinematical systems in NGC 6822