Abstract:
We give a differential-geometric construction of Calabi-Yau fourfolds by the `doubling' method, which was introduced in \cite{DY14} to construct Calabi-Yau threefolds. We also give examples of Calabi-Yau fourfolds from toric Fano fourfolds. Ingredients in our construction are \emph{admissible pairs}, which were first dealt with by Kovalev in \cite{K03}. Here in this paper an admissible pair $(\overline{X},D)$ consists of a compact K\"{a}hler manifold $\overline{X}$ and a smooth anticanonical divisor $D$ on $\overline{X}$. If two admissible pairs $(\overline{X}_1,D_1)$ and $(\overline{X}_2,D_2)$ with $\dim_{\mathbb{C}}\overline{X}_i=4$ satisfy the \emph{gluing condition}, we can glue $\overline{X}_1\setminus D_1$ and $\overline{X}_2\setminus D_2$ together to obtain a compact Riemannian $8$-manifold $(M,g)$ whose holonomy group $\mathrm{Hol}(g)$ is contained in $\mathrm{Spin}(7)$. Furthermore, if the $\widehat{A}$-genus of $M$ equals $2$, then $M$ is a Calabi-Yau fourfold, i.e., a compact Ricci-flat K\"{a}hler fourfold with holonomy $\mathrm{SU}(4)$. In particular, if $(\overline{X}_1,D_1)$ and $(\overline{X}_2,D_2)$ are identical to an admissible pair $(\overline{X},D)$, then the gluing condition holds automatically, so that we obtain a compact Riemannian $8$-manifold $M$ with holonomy contained in $\mathrm{Spin}(7)$. Moreover, we show that if the admissible pair is obtained from \emph{any} of the toric Fano fourfolds, then the resulting manifold $M$ is a Calabi-Yau fourfold by computing $\widehat{A}(M)=2$.

Abstract:
We give a differential-geometric construction of compact manifolds with holonomy $\mathrm{Spin}(7)$ which is based on Joyce's second construction of compact $\mathrm{Spin}(7)$-manifolds in \cite{Joyce00} and Kovalev's gluing construction of $G_2$-manifolds in \cite{Kovalev03}. We also give some examples of compact $\mathrm{Spin}(7)$-manifolds, at least one of which is \emph{new}. Ingredients in our construction are \emph{orbifold admissible pairs with} a compatible antiholomorphic involution. Here in this paper we need orbifold admissible pairs $(\overline{X}, D)$ consisting of a four-dimensional compact K\"{a}hler orbifold $\overline{X}$ with isolated singular points modelled on $\mathbb{C}^4/\mathbb{Z}_4$, and a smooth anticanonical divisor $D$ on $\overline{X}$. Also, we need a compatible antiholomorphic involution $\sigma$ on $\overline{X}$ which fixes the singular points in $\overline{X}$ and acts freely on the anticanoncial divisor $D$. If two orbifold admissible pairs $(\overline{X}_1, D_1)$, $(\overline{X}_2, D_2)$ with $\dim_{\mathbb{C}} \overline{X}_i = 4$ and compatible antiholomorphic involutions $\sigma_i$ on $\overline{X}_i$ satisfy the \emph{gluing condition}, we can glue $(\overline{X}_1 \setminus D_1)/\braket{\sigma_1}$ and $(\overline{X}_2 \setminus D_2)/\braket{\sigma_2}$ together to obtain a compact Riemannian $8$-manifold $(M, g)$ whose holonomy group $\mathrm{Hol}(g)$ is contained in $\mathrm{Spin}(7)$. Furthermore, if the $\widehat{A}$-genus of $M$ equals $1$, then $M$ is a $\mathrm{Spin}(7)$-manifold, i.e., a compact Riemannian manifold with holonomy $\mathrm{Spin}(7)$. We shall investigate our gluing construction using $(\overline{X}_i,D_i)$ with $i=1,2$ when $D_1=D_2=D$ and $D$ is a complete intersection in a weighted projective space, as well as when $(\overline{X}_1,D_1)=(\overline{X}_2,D_2)$ and $\sigma_1=\sigma_2$ (the \emph{doubling} case).

Abstract:
We give a differential-geometric construction and examples of Calabi-Yau threefolds, at least one of which is {\it{new}}. Ingredients in our construction are {\it admissible pairs}, which were dealt with by Kovalev in \cite{K03} and further studied by Kovalev and Lee in \cite{KL11}. An admissible pair $(\overline{X},D)$ consists of a three-dimensional compact K\"{a}hler manifold $\overline{X}$ and a smooth anticanonical $K3$ divisor $D$ on $\overline{X}$. If two admissible pairs $(\overline{X}_1,D_1)$ and $(\overline{X}_2,D_2)$ satisfy the {\it gluing condition}, we can glue $\overline{X}_1\setminus D_1$ and $\overline{X}_2\setminus D_2$ together to obtain a Calabi-Yau threefold $M$. In particular, if $(\overline{X}_1,D_1)$ and $(\overline{X}_2,D_2)$ are identical to an admissible pair $(\overline{X},D)$, then the gluing condition holds automatically, so that we can {\it always} construct a Calabi-Yau threefold from a {\it single} admissible pair $(\overline{X},D)$ by {\it doubling} it. Furthermore, we can compute all Betti and Hodge numbers of the resulting Calabi-Yau threefolds in the doubling construction.

Abstract:
We report new improved photometric redshifts of 1048 galaxies in the Hubble Deep Field (HDF). A standard chi^2 minimizing method is applied to seven-color UBVIJHK photometry by Fernandez-Soto, Lanzetta, & Yahil (1999). We use 187 template SEDs representing a wide variety of morphology and age of observed galaxies based on a population synthesis model by Kodama & Arimoto (1997). We introduce two new recipes. First, the amount of the internal absorption is changed as a free parameter in the range of E(B-V)=0.0 to 0.5 with an interval of 0.1. Second, the absorption due to intergalactic HI clouds is also changed by a factor of 0.5, 1.0, and 1.5 around the opacity given by Madau (1995). The total number of template SEDs is thus 187x6x3=3,366, except for the redshift grid. The dispersion sigma_z of our photometric redshifts with respect to spectroscopic redshifts is sigma_z=0.08 and 0.24 for z<2 and z>2, respectively, which are smaller than the corresponding values (sigma_z=0.09 and 0.45) by Fernandez-Soto et al. Improvement is significant, especially in z>2. This is due to smaller systematic errors which are largely reduced mainly by including three opacities due to intergalactic HI clouds. We discuss redshift distribution N(z) and cosmic star formation rate based on our new photometric redshifts.

Abstract:
We describe an objective and automated method for detecting clusters of galaxies from optical imaging data. This method is a variant of the so-called `matched-filter' technique pioneered by Postman et al. (1996). With simultaneous use of positions and apparent magnitudes of galaxies, this method can, not only find cluster candidates, but also estimate their redshifts and richnesses as byproducts of detection. We examine errors in the estimation of cluster's position, redshift, and richness with a number of Monte Carlo simulations. No systematic discrepancies between the true and estimated values are seen for either redshift or richness. Spurious detection rate of the method is about less than 10% of those of conventional ones which use only surface density of galaxies. A cluster survey in the North Galactic Pole is executed to verify the performance characteristics of the method with real data. Two known real clusters are successfully detected. We expect these methods based on `matched-filter' technique to be essential tools for compiling large and homogeneous optically-selected cluster catalogs.

Abstract:
We search for stars with proper motions in a set of deep Subaru images, covering about 0.48 square degrees to a depth of $i' \simeq 26$, taken over a span of five and a half years. We follow the methods described in \citet{Richmond2009} to reduce and analyze this dataset. We present a sample of 69 stars with motions of high significance, and discuss briefly the populations from which they are likely drawn. Based on photometry and motions alone, we expect that 14 of the candidates may be white dwarfs. Our candidate with the largest proper motion is surprisingly faint and likely to prove interesting: its colors and motions suggest that it might be an M dwarf moving at over 500 km/sec or an L dwarf in the halo.

Abstract:
We study the photometric properties of stars in the data archive of the Sloan Digital Sky Survey (SDSS), the prime aim being to understand the photometric calibration over the entire data set. It is confirmed that the photometric calibration for point sources has been made overall tightly against the SDSS standard stars. We have also confirmed that photometric synthesis of the SDSS spectrophotometric data gives broad band fluxes that agree with broad band photometry with errors no more than 0.04 mag and little tilt along the wide range of colours, verifying that the response functions of the SDSS 2.5 m telescope system are well characterised. We locate stars in the SDSS photometric system, so that stars can roughly be classified into spectral classes from the colour information. We show how metallicity and surface gravity affect colours, and that stars contained in the SDSS general catalogue, plotted in colour space, show the distribution that matches well with what is anticipated from the variations of metallicity and surface gravity. The colour-colour plots are perfectly consistent among the three samples, stars in the SDSS general catalogue, SDSS standard stars and spectrophotometric stars of Gunn & Stryker, especially when some considerations are taken into account of the differences (primarily metallicity) of the samples. We show that the g-r - inverse temperature relation is tight and can be used as a good estimator of the effective temperature of stars over a fairly wide range of effective temperatures. We also confirm that the colours of G2V stars in the SDSS photometric system match well with the Sun.

Abstract:
We report $^{121,123}$Sb nuclear quadrupole resonance (NQR) measurements in the filled-skutterudite superconductor PrOs$_4$Sb$_{12}$ in the temperature range of 0.05-30 K. The electric field gradients (EFG), $V_{zz}$ and $V_{xx}-V_{yy}$, at the Sb site exhibit unusual temperature dependence below 30 K. To explain these features, we discuss the coupling between the Sb nuclear quadrupole moment and Pr $4f^2$-derived multipole moments. The observed $T$ dependence of EFG is well explained by the CEF quasi-quartet consisted of $\Gamma_1$ singlet and $\Gamma_4^{(2)}$ triplet states in the cubic point group $T_h$. These results, in turn, are indicative of the importance of the coupling between the $^{121,123}$Sb quadrupole moments and the hexadecapole moment caused by the quasi-quartet.

Abstract:
The earth has been getting smaller and narrower with the expansion of human activities. Now, it is an urgent task for social scientists to explore and study new socio-economic thoughts for this small and narrow earth. This article provides a normative theory and a descriptive theory for the present and future earth. The former is to provide, viewing the earth and human community as unity, evaluations of possible events and of designs of social institutions. The normative theory helps us think about where we should direct the earth. The latter discusses social sciences for practical management of the earth. Since, however, great diversity of cultures will remain, a unified management of the earth is practically impossible. We are required to rethink and develop new socio-economic thoughts in radical manners.