Degenerate complex Hessian equations on compact K\"ahler manifolds

Let $(X,\omega)$ be a compact K\"ahler manifold of dimension $n$ and fix $m\in \mathbb{N}$ such that $1\leq m \leq n$. We prove that any $(\omega,m)$-sh function can be approximated from above by smooth $(\omega,m)$-sh functions. A potential theory for the complex Hessian equation is also developed which generalizes the classical pluripotential theory on compact K\"ahler manifolds. We then use novel variational tools due to Berman, Boucksom, Guedj and Zeriahi to study degenerate complex Hessian equations

Polarization Elements-A Group Theoretical Study

The Classification of Polarization elements, the polarization affecting optical devices which have a Jones matrix representation, according to the types of eigenvectors they possess, is given a new visit through the Group-theoretical connection of polarization elements. The diattenuators and retarders are recognized as the elements corresponding to boosts and rotations respectively. The structure of homogeneous elements other than diattenuators and retarders are identified by giving the quaternion corresponding to these elements. The set of degenerate polarization elements is identified with the so called `null' elements of the Lorentz Group. Singular polarization elements are examined in their more illustrative Mueller matrix representation and finally the eigenstructure of a special class of singular Mueller matrices is studied.Comment: 7 pages, 2 tables, submitted to `Optics Communications

Constraining the Star Formation Histories in Dark Matter Halos: I. Central Galaxies

Using the self-consistent modeling of the conditional stellar mass functions across cosmic time by Yang et al. (2012), we make model predictions for the star formation histories (SFHs) of {\it central} galaxies in halos of different masses. The model requires the following two key ingredients: (i) mass assembly histories of central and satellite galaxies, and (ii) local observational constraints of the star formation rates of central galaxies as function of halo mass. We obtain a universal fitting formula that describes the (median) SFH of central galaxies as function of halo mass, galaxy stellar mass and redshift. We use this model to make predictions for various aspects of the star formation rates of central galaxies across cosmic time. Our main findings are the following. (1) The specific star formation rate (SSFR) at high $z$ increases rapidly with increasing redshift [$\propto (1+z)^{2.5}$] for halos of a given mass and only slowly with halo mass ($\propto M_h^{0.12}$) at a given $z$, in almost perfect agreement with the specific mass accretion rate of dark matter halos. (2) The ratio between the star formation rate (SFR) in the main-branch progenitor and the final stellar mass of a galaxy peaks roughly at a constant value, $\sim 10^{-9.3} h^2 {\rm yr}^{-1}$, independent of halo mass or the final stellar mass of the galaxy. However, the redshift at which the SFR peaks increases rapidly with halo mass. (3) More than half of the stars in the present-day Universe were formed in halos with 10^{11.1}\msunh < M_h < 10^{12.3}\msunh in the redshift range $0.4 < z < 1.9$. (4) ... [abridged]Comment: 15 figures, 22 pages, Accepted for publication in Ap

Weakly Supervised Domain-Specific Color Naming Based on Attention

The majority of existing color naming methods focuses on the eleven basic color terms of the English language. However, in many applications, different sets of color names are used for the accurate description of objects. Labeling data to learn these domain-specific color names is an expensive and laborious task. Therefore, in this article we aim to learn color names from weakly labeled data. For this purpose, we add an attention branch to the color naming network. The attention branch is used to modulate the pixel-wise color naming predictions of the network. In experiments, we illustrate that the attention branch correctly identifies the relevant regions. Furthermore, we show that our method obtains state-of-the-art results for pixel-wise and image-wise classification on the EBAY dataset and is able to learn color names for various domains.Comment: Accepted at ICPR201

Where Are the Binaries? Results of a Long-Term Search for Radial Velocity Binaries in Proto-Planetary Nebulae

We present the results of an expanded, long-term radial velocity search (25 yrs) for evidence of binarity in a sample of seven bright proto-planetary nebulae (PPNe). The goal is to investigate the widely-held view that the bipolar or point-symmetric shapes of planetary nebulae (PNe) and PPNe are due to binary interactions. Observations from three observatories were combined from 2007-2015 to search for variations on the order of a few years and then combined with earlier observations from 1991-1995 to search for variations on the order of decades. All seven show velocity variations due to periodic pulsation in the range of 35-135 days. However, in only one PPN, IRAS 22272+5435, did we find even marginal evidence found for multi-year variations that might be due to a binary companion. This object shows marginally-significant evidence of a two-year period of low semi-amplitude which could be due to a low-mass companion, and it also displays some evidence of a much longer period of >30 years. The absence of evidence in the other six objects for long-period radial velocity variations due to a binary companion sets significant constraints on the properties of any undetected binary companions: they must be of low mass, 30 years. Thus the present observations do not provide direct support for the binary hypothesis to explain the shapes of PNe and PPNe and severely constrains the properties of any such undetected companions.Comment: 28 pages, 5 figure

Star Formation and Stellar Mass Assembly in Dark Matter Halos: From Giants to Dwarfs

The empirical model of Lu et al. 2014 is updated with recent data and used to study galaxy star formation and assembly histories. At $z > 2$, the predicted galaxy stellar mass functions are steep, and a significant amount of star formation is hosted by low-mass haloes that may be missed in current observations. Most of the stars in cluster centrals formed earlier than $z\approx 2$ but have been assembled much later. Milky Way mass galaxies have had on-going star formation without significant mergers since $z\approx 2$, and are thus free of significant (classic) bulges produced by major mergers. In massive clusters, stars bound in galaxies and scattered in the halo form a homogeneous population that is old and with solar metallicity. In contrast, in Milky Way mass systems the two components form two distinct populations, with halo stars being older and poorer in metals by a factor of $\approx 3$. Dwarf galaxies in haloes with $M_{\rm h} < 10^{11}h^{-1}M_{\odot}$ have experienced a star formation burst accompanied by major mergers at $z > 2$, followed by a nearly constant star formation rate after $z = 1$. The early burst leaves a significant old stellar population that is distributed in spheroids.Comment: 17 pages, 17 figure