On character sums over flat numbers

Let $q\geqslant2$ be an integer, $\chi$ be any non-principal character mod $q$, and $H=H(q)\leqslant q.$ In this paper the authors prove some estimates for character sums of the form $$\mathcal{W}(\chi,H;q)=\sum_{n\in\mathscr{F}(H)}\chi(n),$$ where \mathscr{F}(H)=\left\{n\in\mathbb{Z}|(n,q)=1,1\leqslant n,\bar{n}\leqslant q, |n-\bar{n}|\leqslant H\}, $\bar{n}$ is defined by $n\bar{n}\equiv1\pmod q.$Comment: 9 pages, with a complete proof of Theorem 3, Section 5. Accepted by J. Number Theor

Modelling the number counts of early-type galaxies by pure luminosity evolution

In this paper, we explore the plausible luminosity evolution of early-type galaxies in different cosmological models by constructing a set of pure luminosity evolution (PLE) models via the choices of the star formation rate (SFR) parameters and formation redshift $z_f$ of galaxies, with the observational constraints derived from the Hubble Space Telescope (HST) morphological number counts for elliptical and S0 galaxies of the Medium Deep Survey (MDS) and the Hubble Deep Field (HDF). We find that the number counts of early-type galaxies can be explained by the pure luminosity evolution models, without invoking exotic scenarios such as merging or introducing an additional population. But the evolution should be nearly passive, with a high $z_f$ assumed. The conclusion is valid in all of the three cosmological models we adopted in this paper. We also present the redshift distributions for three bins of observed magnitudes in F814w pass-band, to show at which redshift are the objects that dominate the counts at a given magnitude. The predictions of the redshift distribution of $22.5<b_j<24.0$ are also presented for comparison with future data.Comment: Plain tex, 15pages, 9 eps figures, plus an extra figure fig2c.eps, with the tex-macro mn.tex. MNRAS, accepte

The tensor renormalization group study of the general spin-S Blume-Capel model

We focus on the special situation of $D=2J$ of the general spin-S Blume-Capel model on the square lattice. Under the infinitesimal external magnetic field, the phase transition behaviors due to the thermal fluctuations are discussed by the newly developed tensor renormalization group method. For the case of the integer spin-S, the system will undergo $S$ first-order phase transitions with the successive symmetry breaking with the magnetization $M=S,S-1,...0$. For the half-integer spin-S, there are similar $S-1/2$ first order phase transition with $M=S,S-1,...1/2$ stepwise structure, in addition, there is a continuous phase transition due to the spin-flip $Z_2$ symmetry breaking. In the low temperature regions, all first-order phase transitions are accompanied by the successive disappearance of the optional spin-component pairs($s,-s$), furthermore, the critical temperature for the nth first-order phase transition is the same, independent of the value of the spin-S. In the absence of the magnetic field, the visualization parameter characterizing the intrinsic degeneracy of the different phases clearly demonstrates the phase transition process.Comment: 6 pages, 7 figure