We describe the Dark Energy Survey (DES) Deep Fields, a set of images and associated multiwavelength catalogue (ugrizJHKs) built from Dark Energy Camera (DECam) and Visible and Infrared Survey Telescope for Astronomy (VISTA) data. The DES Deep Fields comprise 11 fields (10 DES supernova fields plus COSMOS), with a total area of ∼30 sq. deg. in ugriz bands and reaching a maximum i-band depth of 26.75 (AB, 10σ, 2 arcsec). We present a catalogue for the DES 3-yr cosmology analysis of those four fields with full 8-band coverage, totalling 5.88 sq. deg. after masking. Numbering 2.8 million objects (1.6 million post-masking), our catalogue is drawn from images coadded to consistent depths of r = 25.7, i = 25, and z = 24.3 mag. We use a new model-fitting code, built upon established methods, to deblend sources and ensure consistent colours across the u-band to Ks-band wavelength range. We further detail the tight control we maintain over the point-spread function modelling required for the model fitting, astrometry and consistency of photometry between the four fields. The catalogue allows us to perform a careful star–galaxy separation and produces excellent photometric redshift performance (NMAD = 0.023 at i < 23). The Deep-Fields catalogue will be made available as part of the cosmology data products release, following the completion of the DES 3-yr weak lensing and galaxy clustering cosmology work.

In statistical mechanics, for a system with fixed number of particles, e.g., a finite-size system, strictly speaking, the thermodynamic quantity needs to be calculated in the canonical ensemble. Nevertheless, the calculation of the canonical partition function is difficult. In this paper, based on the mathematical theory of the symmetric function, we suggest a method for the calculation of the canonical partition function of ideal quantum gases, including ideal Bose, Fermi, and Gentile gases. Moreover, we express the canonical partition functions of interacting classical and quantum gases given by the classical and quantum cluster expansion methods in terms of the Bell polynomial in mathematics. The virial coefficients of ideal Bose, Fermi, and Gentile gases is calculated from the exact canonical partition function. The virial coefficients of interacting classical and quantum gases is calculated from the canonical partition function by using the expansion of the Bell polynomial, rather than calculated from the grand canonical potential.

In order to obtain morphological information of unlabeled galaxies, we present an unsupervised machine-learning (UML) method for morphological classification of galaxies, which can be summarized as two aspects: (1) the methodology of convolutional autoencoder (CAE) is used to reduce the dimensions and extract features from the imaging data; (2) the bagging-based multiclustering model is proposed to obtain the classifications with high confidence at the cost of rejecting the disputed sources that are inconsistently voted. We apply this method on the sample of galaxies with H < 24.5 in CANDELS. Galaxies are clustered into 100 groups, each contains galaxies with analogous characteristics. To explore the robustness of the morphological classifications, we merge 100 groups into five categories by visual verification, including spheroid, early-type disk, late-type disk, irregular, and unclassifiable. After eliminating the unclassifiable category and the sources with inconsistent voting, the purity of the remaining four subclasses are significantly improved. Massive galaxies (M * > 1010 M ⊙) are selected to investigate the connection with other physical properties. The classification scheme separates galaxies well in the U − V and V − J color space and Gini–M 20 space. The gradual tendency of Sérsic indexes and effective radii is shown from the spheroid subclass to the irregular subclass. It suggests that the combination of CAE and multiclustering strategy is an effective method to cluster galaxies with similar features and can yield high-quality morphological classifications. Our study demonstrates the feasibility of UML in morphological analysis that would develop and serve the future observations made with China Space Station telescope.

A distance-deviation consistency and model-independent method to test the cosmic distance–duality relation (CDDR) is provided. This method is worthy of attention for two reasons. First, a distance-deviation consistency method is used to pair subsamples: instead of pairing subsamples with redshift deviations smaller than a value of, say, . The redshift deviation between subsamples decreases with the redshift to ensure the distance deviation stays the same. The method selects more subsamples at high redshift, up to z = 2.16, and provides 120 subsample pairs. Second, the model-independent method involves the latest data set of 1048 SNe Ia and 205 strong gravitational lensing systems (SGLS), which are used to obtain the luminosity distances D L and the ratio of angular diameter distance D A , respectively. With the model-independent method, parameters of the CDDR, the SNe Ia light curve, and the SGLS are fitted simultaneously. The result shows that and CDDR is validated at the 1σ confidence level in the form of .

A method for calculating the eigenvalue of a many-body system without solving the eigenfunction is suggested. In many cases, we only need the knowledge of eigenvalues rather than eigenfunctions, so we need a method solving only the eigenvalue, leaving alone the eigenfunction. In this paper, the method is established based on statistical mechanics. In statistical mechanics, calculating thermodynamic quantities needs only the knowledge of eigenvalues and then the information of eigenvalues is embodied in thermodynamic quantities. The method suggested in the present paper is indeed a method for extracting the eigenvalue from thermodynamic quantities. As applications, we calculate the eigenvalues for some many-body systems. Especially, the method is used to calculate the quantum exchange energies in quantum many-body systems. Using the method, we also calculate the influence of the topological effect on eigenvalues. Moreover, we improve the result of the relation between the counting function and the heat kernel in literature. * daiwusheng@tju.edu.cn.