On Hypersurface Quotient Singularity of Dimension 4

We consider geometrical problems on Gorenstein hypersurface orbifolds of dimension $n \geq 4$ through the theory of Hilbert scheme of group orbits. For a linear special group $G$ acting on \CZ^n, we study the $G$-Hilbert scheme, \hl^G(\CZ^n), and crepant resolutions of \CZ^n/G for $G$=the $A$-type abelian group $A_r(n)$. For $n=4$, we obtain the explicit structure of \hl^{A_r(4)}(\CZ^4). The crepant resolutions of \CZ^4/A_r(4) are constructed through their relation with \hl^{A_r(4)}(\CZ^4), and the connections between these crepant resolutions are found by the "flop" procedure of 4-folds. We also make some primitive discussion on \hl^G(\CZ^n) for the $G$= alternating group {\goth A}_{n+1} of degree $n+1$ with the standard representation on \CZ^n; the detailed structure of \hl^{{\goth A}_4}(\CZ^3) is explicitly constructed.Comment: 27 pages, Latex, 11 figures, Some reorganizations and improvement of presentations, Typos corrected, Arguments of Theorem 1 of section 3 in the earlier version are refined with clearer explanation for the justification of contradicting statement appeared in a published journal literature by some other autho

1-String CZ-Representation of Planar Graphs

In this paper, we prove that every planar 4-connected graph has a CZ-representation---a string representation using paths in a rectangular grid that contain at most one vertical segment. Furthermore, two paths representing vertices $u,v$ intersect precisely once whenever there is an edge between $u$ and $v$. The required size of the grid is $n \times 2n$

Mass function and dynamical study of the open clusters Berkeley 24 and Czernik 27

We present a $UBVI$ photometric study of the open clusters Berkeley 24 (Be 24) and Czernik 27 (Cz 27). The radii of the clusters are determined as 2\farcm7 and 2\farcm3 for Be 24 and Cz 27, respectively. We use the Gaia Data Release 2 (GDR2) catalogue to estimate the mean proper motions for the clusters. We found the mean proper motion of Be 24 as $0.35\pm0.06$ mas yr$^{-1}$ and $1.20\pm0.08$ mas yr$^{-1}$ in right ascension and declination for Be 24 and $-0.52\pm0.05$ mas yr$^{-1}$ and $-1.30\pm0.05$ mas yr$^{-1}$ for Cz 27. We used probable cluster members selected from proper motion data for the estimation of fundamental parameters. We infer reddenings $E(B-V)$ = $0.45\pm0.05$ mag and $0.15\pm0.05$ mag for the two clusters. Analysis of extinction curves towards the two clusters show that both have normal interstellar extinction laws in the optical as well as in the near-IR band. From the ultraviolet excess measurement, we derive metallicities of [Fe/H]= $-0.025\pm0.01$ dex and $-0.042\pm0.01$ dex for the clusters Be 24 and Cz 27, respectively. The distances, as determined from main sequence fitting, are $4.4\pm0.5$ kpc and $5.6\pm0.2$ kpc. The comparison of observed CMDs with $Z=0.01$ isochrones, leads to an age of $2.0\pm0.2$ Gyr and $0.6\pm0.1$ Gyr for Be 24 and Cz 27, respectively. In addition to this, we have also studied the mass function and dynamical state of these two clusters for the first time using probable cluster members. The mass function is derived after including the corrections for data incompleteness and field star contamination. Our analysis shows that both clusters are now dynamically relaxedComment: 16 pages including 8 tables. 22 figures. Accepted by MNRA

A 3-D wavelet analysis of substructure in the Coma cluster: statistics and morphology

Evidence for clustering within the Coma cluster is found by means of a multiscale analysis of the combined angular-redshift distribution. We have compiled a catalogue of 798 galaxy redshifts from published surveys from the region of the Coma cluster. We examine the presence of substructure and of voids at different scales ranging from $\sim 1 to \sim 16 h^{-1}$ Mpc, using subsamples of the catalogue, ranging from $cz=3000$ km/s to $cz=28000$ km/s. Our substructure detection method is based on the wavelet transform and on the segmentation analysis. The wavelet transform allows us to find out structures at different scales and the segmentation method allows us a quantitative statistical and morphological analysis of the sample. From the whole catalogue we select a subset of 320 galaxies, with redshifts between cz=5858 km/s and cz=8168 km/s that we identify as belonging to the central region of Coma and on which we have performed a deeper analysis, on scales ranging from $180 h^{-1}$ kpc to $1.44 h^{-1}$ Mpc. Our results are expressed in terms of the number of structures or voids and their sphericity for different values of the threshold detection and at all the scales investigated. According to our analysis, there is strong evidence for multiple hierarchical substructure, on scales ranging from a few hundreds of kpc to about $4 h^{-1}$ Mpc. The morphology of these substructures is rather spherical. On the scale of $720 h^{-1}$ kpc we find two main subclusters which where also found before, but our wavelet analysis shows even more substructures, whose redshift position is approximatively marked by these bright galaxies: NGC 4934 & 4840, 4889, 4898 & 4864, 4874 & 4839, 4927, 4875.Comment: 24 pages, 6 figures. ApJ (Main Journal), accepted for publication. Added one section on statistical tests and slightly modified text and abstrac

Chinese space and aviation industries score major breakthroughs

An overview of the current status of China's aviation and aerospace industries is presented, as well as planned future development and areas of importance for China's future space programs. The development of China's CZ-1, CZ-2 and CZ-3 rocket program is discussed, as well as China's satellite launch capabilities. China's first geostationary communications satellite STW-1 is also mentioned, and further development of the second and third communications satellites to be launched in 1987 are shown. Other developments include a seventh low Earth orbiting photographic reconnaissance satellite, plans for an image transmitting remote sensing satellite to be launched in 1988 to 1990, and other satellite developments. The Chinese-designed Y-10 transport aircraft is discussed, as well as the TU-16 bomber aircraft and the co-production agreement with McDonnell Douglas for the MD-82 passenger aircraft

BVRI Light Curves for 22 Type Ia Supernovae

We present 1210 Johnson/Cousins B,V,R, and I photometric observations of 22 recent type Ia supernovae (SNe Ia): SN 1993ac, SN 1993ae, SN 1994M, SN 1994S, SN 1994T, SN 1994Q, SN 1994ae, SN 1995D, SN 1995E, SN 1995al, SN 1995ac, SN 1995ak, SN 1995bd, SN 1996C, SN 1996X, SN 1996Z, SN 1996ab, SN 1996ai, SN 1996bk, SN 1996bl, SN 1996bo, and SN 1996bv. Most of the photometry was obtained at the Fred Lawrence Whipple Observatory (FLWO) of the Harvard-Smithsonian Center for Astrophysics in a cooperative observing plan aimed at improving the data base for SN Ia. The redshifts of the sample range from $cz$=1200 to 37000 km s$^{-1}$ with a mean of $cz$=7000 km s$^{-1}$.Comment: Accepted to the Astronomical Journal, 41 pages, 8 figure

Basic vector valued Siegel modular forms of genus two

We consider the space of all vector-valued holomorphic modular forms f:H_n→Z of transformation type f(MZ)=vr(M)det(CZ+D)ϱ(CZ+D)f(Z). ϱ:GL(n,C)→GL(Z) is a rational representation on a finite dimensional complex vector space Z. These spaces can be collected in a graded A(Γ)-module We treat in this paper some special cases in genus 2. The first one is essentially due to Wieber. Thus we consider a variant of this case and a new example. In this final case the starting weight is 1/2, the starting multiplier system is the theta multiplier system v_ϑ and for ϱ we take the standard representation. In all these cases we will determine the structure of the spaces