Admissible fundamental operators

Let $F$ and $G$ be two bounded operators on two Hilbert spaces. Let their numerical radii be no greater than one. This note investigate when there is a $Gamma$-contraction $(S,P)$ such that $F$ is the fundamental operator of $(S,P)$ and $G$ is the fundamental operator of $(S^*,P^*)$. Theorem 1 puts a necessary condition on $F$ and $G$ for them to be the fundamental operators of $(S,P)$ and $(S^*,P^*)$ respectively. Theorem 2 shows that this necessary condition is sufficient too provided we restrict our attention to a certain special case. The general case is investigated in Theorem 3. Some of the results obtained for $Gamma$-contractions are then applied to tetrablock contractions to figure out when two pairs $(F_1, F_2)$ and $(G_1, G_2)$ acting on two Hilbert spaces can be fundamental operators of a tetrablock contraction $(A, B, P)$ and its adjoint $(A^*, B^*, P^*)$ respectively. This is the content of Theorem 4.

Main-sequence variable stars in young open cluster NGC 1893

In this paper we present time series photometry of 104 variable stars in the cluster region NGC 1893. The association of the present variable candidates to the cluster NGC 1893 has been determined by using $(U-B)/(B-V)$ and $(J-H)/(H-K)$ two colour diagrams, and $V/(V-I)$ colour magnitude diagram. Forty five stars are found to be main-sequence variables and these could be B-type variable stars associated with the cluster. We classified these objects as $beta$ Cep, slowly pulsating B stars and new class variables as discussed by Mowlavi et al. (2013). These variable candidates show $sim$0.005 to $sim$0.02 mag brightness variations with periods of $<$ 1.0 d. Seventeen new class variables are located in the $H-R$ diagram between the slowly pulsating B stars and $delta$ Scuti variables. Pulsation could be one of the causes for periodic brightness variations in these stars. The X-ray emission of present main-sequence variables associated with the cluster lies in the saturated region of X-ray luminosity versus period diagram and follows the general trend by Pizzolato et al. (2003).

Variable Stars in the Globular Cluster NGC 4590 (M68

We present results of time series photometry to search for variable stars in the field of metal-poor globular cluster NGC 4590 (M68). Periods have been revised for 40 known variables and no significant changes were found. A considerable change in Blazhko effect for V25 has been detected. Among nine newly discovered variable candidates, 5 stars are of RRc Bailey type variables while 4 stars are unclassified. The variable stars V10, V21, V50 and V51 are found to be cluster members based on the radial velocity data taken from literature.

Photometric study of five open star clusters

$UBVRI$ photometry of the five open clusters Czernik 4, Berkeley 7, NGC 2236, NGC 7226 and King 12 has been carried out using ARIES 104 cm telescope, Nainital. Fundamental cluster parameters such as foreground reddening $E(B-V)$, distance, and age have been derived by means of the observed two colour and colour-magnitude diagrams, coupled to comparisons with theoretical models. $E(B-V)$ values range from 0.55 to 0.74 mag, while ages derived for these clusters range from $sim$10 to $sim$500 Myr. We have also studied the spatial structure, mass function and mass segregation effects. The present study shows that evaporation of low mass stars from the halo of the clusters increases as they evolve.

Pre-main sequence variable stars in young open cluster NGC 1893

We present results of multi-epoch (fourteen nights during 2007-2010) $V$-band photometry of the cluster NGC 1893 region to identify photometric variable stars in the cluster. The study identified a total of 53 stars showing photometric variability. The members associated with the region are identified on the basis of spectral energy distribution, $J-H/H-K$ two colour diagram and $V/V-I$ colour-magnitude diagram. The ages and masses of the majority of pre-main-sequence sources are found to be $lesssim$ 5 Myr and in the range 0.5 $lesssim$ $M/M_{odot}$ $lesssim$ 4, respectively. These pre-main-sequence sources hence could be T Tauri stars. We also determined the physical parameters like disk mass and accretion rate from the spectral energy distribution of these T Tauri stars. The periods of majority of the T Tauri stars range from 0.1 to 20 day. The brightness of Classical T Tauri stars is found to vary with larger amplitude in comparison to Weak line T Tauri stars. It is found that the amplitude decreases with increase in mass, which could be due to the dispersal of disks of massive stars.

Photometric search for variable stars in young open cluster Berkeley 59

We present time-series photometry of stars located in the extremely young open cluster Berkeley 59. Using the 1.04 m telescope at ARIES, Nainital, we have identified 42 variables in a field of 13x13 around the cluster. The probable members of the cluster are identified using (V, V-I) colour-magnitude diagram and (J-H, H-K) colour-colour diagram. Thirty one variables are found to be pre-main sequence stars associated with the cluster. The ages and masses of pre-main sequence stars are derived from colour-magnitude diagram by fitting theoretical models to the observed data points. The ages of the majority of the probable pre-main sequence variable candidates range from 1 to 5 Myrs. The masses of these pre-main sequence variable stars are found to be in the range of ~0.3 to ~3.5 Msun and these could be T Tauri stars. The present statistics reveal that about 90% T Tauri stars have periods < 15 days. The classical T Tauri stars are found to have larger amplitude in comparison to the weak line T Tauri stars. There is an indication that the amplitude decreases with increase of the mass, which could be due to the dispersal of disk of relatively massive stars.

The Feichtinger Conjecture and Reproducing Kernel Hilbert Spaces

We prove two new equivalences of the Feichtinger conjecture that involve reproducing kernel Hilbert spaces. We prove that if for every Hilbert space, contractively contained in the Hardy space, each Bessel sequence of normalized kernel functions can be partitioned into finitely many Riesz basic sequences, then a general bounded Bessel sequence in an arbitrary Hilbert space can be partitioned into finitely many Riesz basic sequences. In addition, we examine some of these spaces and prove that for these spaces bounded Bessel sequences of normalized kernel functions are finite unions of Riesz basic sequences.