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Given two quasi-definite moment functionals, the corresponding orthogonal polynomial systems satisfy an algebraic differential relation(called an extended coherent pair). We study generalizing extended coherent pairs that unify extended coherent pairs and extended symmetric coherent pairs and find the related coefficients. When one of the moment functionals is (strongly) classical, we find another orthogonal polynomial system to find three-term recurrence coefficients. Moreover, we determine the companion moment functional as a rational modification of the classical one.

We present spectroscopy of 28 SNR candidates as well as one H II region in M81, and two SNR candidates in M82. Twenty six out of the M81 candidates turn out to be genuine SNRs, and two in M82 may be shocked condensations in the galactic outflow or SNRs. The distribution of [N II]/H{\alpha} ratios of M81 SNRs is bimodal. M81 SNRs are divided into two groups in the spectral line ratio diagrams: an [O III]-strong group and an [O III]-weak group. The latter have larger sizes, and may have faster shock velocity. [N II]/H{\alpha} ratios of the SNRs show a strong correlation with [S II]/H{\alpha} ratios. They show a clear radial gradient in [N II]/H{\alpha} and [S II]/H{\alpha} ratios: dLog ([N II]/H{\alpha})/dLog R = -0.018 {\pm} 0.008 dex/kpc and dLog ([S II]/H{\alpha})/dLog R = -0.016 {\pm} 0.008 dex/kpc where R is a deprojected galactocentric distance. We estimate the nitrogen and oxygen abundance of the SNRs from the comparison with shock-ionization models. We obtain a value for the nitrogen radial gradient, dLog(N/H)/dLogR = -0.023 {\pm} 0.009 dex/kpc, and little evidence for the gradient in oxygen. This nitrogen abundance shows a few times flatter gradient than those of the planetary nebulae and H II regions. We find that five SNRs are matched with X-ray sources. Their X-ray hardness colors are consistent with thermal SNRs., Comment: 19 pages, 24 figures, 5 tables, ApJ accepted

Narrow band images covering strong emission lines are efficient to survey supernova remnants (SNRs) in nearby galaxies. Using the narrow band images provided by the Local Group Galaxy Survey we searched for SNRs in M33. Culling the objects with enhanced [SII]/Ha and round morphology in the continuum-subtracted Ha and [SII] images, we produce a list of 199 sources. Among them 79 are previously unknown. Their progenitor and morphology types were classified. A majority of the sample (170 objects) are likely remnants of core-collapse SNe, and 29 are remnants of Type Ia SNe. The cumulative size distribution of these objects is found to be similar to that of the M31 remnants derived in the similar way. We obtain a power law slope, a = 2.38 + 0:05. Thus a majority of the sources are considered to be in the Sedov-Taylor phase, consistent with previous findings. The histogram of the emission line ratio ([SII]/Ha) of the remnants has two concentrations at [SII]/Ha ~0.55 and ~0.8, as in M31. Interestingly Lx (and L20cm) of the compact center-bright objects are correlated with their optical luminosity. The remnants with X-ray emission have brighter optical surface brightness and smaller diameters than those without X-ray emission., Comment: 38 pages, 19 figures, 4 tables, Accepted for publication in ApJ

We present a survey of optically emitting supernova remnants (SNRs) in M31 based on H$\alpha$ and [SII] images in the Local Group Survey. Using these images, we select objects that have [SII]:H$\alpha$ $>$ 0.4 and circular shapes. We find 76 new SNR candidates. We also inspect 234 SNR candidates presented in previous studies, finding that only 80 of them are SNR candidates according to our criteria. Combining them with the new candidates, we produce a master catalog of 156 SNR candidates in M31. We classify these SNR candidates according to two criteria: the SNR progenitor type [Type Ia and core-collapse (CC) SNRs] and the morphological type. Type Ia and CC SNR candidates make up 23% and 77%, respectively, of the total sample. Most of the CC SNR candidates are concentrated in the spiral arms, while the Type Ia SNR candidates are rather distributed over the entire galaxy, including the inner region. The CC SNR candidates are brighter in H$\alpha$ and [SII] than the Type Ia SNR candidates. We derive a cumulative size distribution of the SNR candidates, finding that the distribution of the candidates with 17 $< D <$ 50 pc is fitted well by a power law with the power law index $\alpha = 2.53\pm0.04$. This indicates that most of the SNR candidates identified in this study appear to be in the Sedov-Taylor phase. The [SII]:H$\alpha$ distribution of the SNR candidates is bimodal, with peaks at [SII]:H$\alpha$ $\sim$ 0.4 and $\sim$ 0.9. The properties of these SNR candidates vary little with the galactocentric distance. The H$\alpha$ and [SII] surface brightnesses show a good correlation with the X-ray luminosity of the SNR candidates that are center-bright. The SNR candidates with X-ray counterparts have higher surface brightnesses in H$\alpha$ and [SII] and smaller sizes than those without such counterparts., Comment: 55 pages, 23 figures, 6 tables, Accepted for publication in ApJ

We investigate the connection between the presence of bars and AGN activity, using a volume-limited sample of $\sim$9,000 late-type galaxies with axis ratio $b/a>0.6$ and $M_{r} < -19.5+5{\rm log}h$ at low redshift ($0.02\le z\lesssim 0.055$), selected from Sloan Digital Sky Survey Data Release 7. We find that the bar fraction in AGN-host galaxies (42.6%) is $\sim$2.5 times higher than in non-AGN galaxies (15.6%), and that the AGN fraction is a factor of two higher in strong-barred galaxies (34.5%) than in non-barred galaxies (15.0%). However, these trends are simply caused by the fact that AGN-host galaxies are on average more massive and redder than non-AGN galaxies because the fraction of strong-barred galaxies ($\bfrsbo$) increases with $u-r$ color and stellar velocity dispersion. When $u-r$ color and velocity dispersion (or stellar mass) are fixed, both the excess of $\bfrsbo$ in AGN-host galaxies and the enhanced AGN fraction in strong-barred galaxies disappears. Among AGN-host galaxies we find no strong difference of the Eddington ratio distributions between barred and non-barred systems. These results indicate that AGN activity is not dominated by the presence of bars, and that AGN power is not enhanced by bars. In conclusion we do not find a clear evidence that bars trigger AGN activity., Comment: 13 pages, 11 figures, accepted for publication in ApJ

We present a study of HII regions in M51 using the Hubble Space Telescope ACS images taken as part of the Hubble Heritage Program. We have catalogued about 19,600 HII regions in M51 with Ha luminosity in the range of L = 10^{35.5} erg/s to 10^{39.0} erg/s. The Ha luminosity function of HII regions (HII LF) in M51 is well represented by a double power law with its index alpha=-2.25\pm0.02 for the bright part and alpha=-1.42\pm0.01 for the faint part, separated at a break point L= 10^{37.1} erg/s. This break was not found in previous studies of M51 HII regions. Comparison with simulated HII LFs suggests that this break is caused by the transition of HII region ionizing sources, from low mass clusters (with ~ 10^3 M_sun, including several OB stars) to more massive clusters (including several tens of OB stars). The HII LFs with L < 10^{37.1} erg/s are found to have different slopes for different parts in M51: the HII LF for the interarm region is steeper than those for the arm and the nuclear regions. This observed difference in HII LFs can be explained by evolutionary effects that HII regions in the interarm region are relatively older than those in the other parts of M51., Comment: 36 pages, 12 figures, 2 tables, accepted to ApJ