Photometry of Variable Stars from THU-NAOC Transient Survey I: The First 2 Years

In this paper, we report the detections of stellar variabilities from the first 2-year observations of sky area of about 1300 square degrees from the Tsinghua University-NAOC Transient Survey (TNTS). A total of 1237 variable stars (including 299 new ones) were detected with brightness < 18.0 mag and magnitude variation >= 0.1 mag on a timescale from a few hours to few hundred days. Among such detections, we tentatively identified 661 RR Lyrae stars, 431 binaries, 72 Semiregular pulsators, 29 Mira stars, 11 slow irregular variables, 11 RS Canum Venaticorum stars, 7 Gamma Doradus stars, 5 long period variables, 3 W Virginis stars, 3 Delta Scuti stars, 2 Anomalous Cepheids, 1 Cepheid, and 1 nove-like star based on their time-series variability index Js and their phased diagrams. Moreover, we found that 14 RR Lyrae stars show the Blazhko effect and 67 contact eclipsing binaries exhibit the O'Connell effect. Since the period and amplitude of light variations of RR Lyrae variables depend on their chemical compositions, their photometric observations can be used to investigate distribution of metallicity along the direction perpendicular to the Galactic disk. We find that the metallicity of RR Lyrae stars shows large scatter at regions closer to the Galactic plane (e.g., -3.0 < [Fe/H] < 0) but tends to converge at [Fe/H]~ -1.7 at larger Galactic latitudes. This variation may be related to that the RRAB Lyrae stars in the Galactic halo come from globular clusters with different metallicity and vertical distances, i.e. OoI and OoII populations, favoring for the dual-halo model.Comment: 18 pages, 19 figures, published in AJ, 150, 10

An inventory of invasive alien species in China

Invasive alien species (IAS) are a major global challenge requiring urgent action, and the Strategic Plan for Biodiversity (2011–2020) of the Convention on Biological Diversity (CBD) includes a target on the issue. Meeting the target requires an understanding of invasion patterns. However, national or regional analyses of invasions are limited to developed countries. We identified 488 IAS in China’s terrestrial habitats, inland waters and marine ecosystems based on available literature and field work, including 171 animals, 265 plants, 26 fungi, 3 protists, 11 procaryots, and 12 viruses. Terrestrial plants account for 51.6% of the total number of IAS, and terrestrial invertebrates (104 species) for 21.3%. Of the total numbers, 67.9% of plant IAS and 34.8% of animal IAS were introduced intentionally. All other taxa were introduced unintentionally despite very few animal and plant species that invaded naturally. In terms of habitats, 64.3% of IAS occur on farmlands, 13.9% in forests, 8.4% in marine ecosystems, 7.3% in inland waters, and 6.1% in residential areas. Half of all IAS (51.1%) originate from North and South America, 18.3% from Europe, 17.3% from Asia not including China, 7.2% from Africa, 1.8% from Oceania, and the origin of the remaining 4.3% IAS is unknown. The distribution of IAS can be divided into three zones. Most IAS are distributed in coastal provinces and the Yunnan province; provinces in Middle China have fewer IAS, and most provinces in West China have the least number of IAS. Sites where IAS were first detected are mainly distributed in the coastal region, the Yunnan Province and the Xinjiang Uyghur Autonomous Region. The number of newly emerged IAS has been increasing since 1850. The cumulative number of firstly detected IAS grew exponentially

Stability of the Caffarelli-Kohn-Nirenberg inequality: the existence of minimizers

In this paper, we consider the following variational problem: \begin{eqnarray*} \inf_{u\in D^{1,2}_a(\bbr^N)\backslash\mathcal{Z}}\frac{\|u\|^2_{D^{1,2}_a(\bbr^N)}-C_{a,b,N}^{-1}\|u\|^2_{L^{p+1}(|x|^{-b(p+1)},\bbr^N)}}{dist_{D^{1,2}_{a}}^2(u, \mathcal{Z})}:=c_{BE}, \end{eqnarray*} where $N\geq2$, $b_{FS}(a)<b<a+1$ with $b_{FS}(a)$ being the Felli-Schneider curve, $a<\frac{N-2}{2}$, $p=\frac{N+2(1+a-b)}{N-2(1+a-b)}$, \mathcal{Z}= \{ c \tau^{\frac{N-2-2a}{2}}W(\tau x)\mid c\in\bbr\backslash\{0\}, \tau>0\} and up to dilations and scalar multiplications (also up to translations in the special case $a=b=0$), $W(x)$, which is positive and radially symmetric, is the unique extremal function of the following classical Caffarelli-Kohn-Nirenberg (CKN for short) inequality \begin{eqnarray*} \bigg(\int_{\bbr^N}|x|^{-b(p+1)}|u|^{p+1}dx\bigg)^{\frac{2}{p+1}}\leq C_{a,b,N}\int_{\bbr^N}|x|^{-2a}|\nabla u|^2dx \end{eqnarray*} with $C_{a,b,N}$ being the optimal constant. It is known in \cite{WW2022} that $c_{BE}>0$. In this paper, we prove that the above variational problem has a minimizer for $N\geq3$, which extends the result of Konig in \cite{K2023} for the Sobolev inequality to the CKN inequality. Moreover, we believe that our restriction on dimensions is optimal for the existence of minimizers of the above variational problem and believe that there is no minimizers of the above variational problem for $N=2$ with $c_{BE}=\frac{2(p-1)}{3p-1}$.Comment: This is the second version of the preprint arXiv:2205.08937v1 and any comments are welcom

Sharp stability of the logarithmic Sobolev inequality in the critical point setting

In this paper, we consider the Euclidean logarithmic Sobolev inequality \begin{eqnarray*} \int_{\mathbb{R}^d}|u|^2\log|u|dx\leq\frac{d}{4}\log\bigg(\frac{2}{\pi d e}\|\nabla u\|_{L^2(\mathbb{R}^d)}^2\bigg), \end{eqnarray*} where $u\in W^{1,2}(\mathbb{R}^d)$ with $d\geq2$ and $\|u\|_{L^2(\mathbb{R}^d)}=1$. It is well known that extremal functions of this inequality are precisely the Gaussians \begin{eqnarray*} \mathfrak{g}_{\sigma,z}(x)=(\pi\sigma)^{-\frac{d}{2}}\mathfrak{g}_{*}\bigg(\sqrt{\frac{\sigma}{2}}(x-z)\bigg)\quad\text{with}\quad \mathfrak{g}_{*}(x)=e^{-\frac{|x|^2}{2}}. \end{eqnarray*} We prove that if $u\geq0$ satisfying $(\nu-\frac12)c_0<\|u\|_{H^1(\mathbb{R}^d)}^2<(\nu+\frac12)c_0$ and $\|-\Delta u+u-2u\log |u|\|_{H^{-1}}\leq\delta$, where $c_0=\|\mathfrak{g}_{1,0}\|_{H^1(\mathbb{R}^d)}^2$, $\nu\in \mathbb{N}$ and $\delta>0$ sufficiently small, then \begin{eqnarray*} \text{dist}_{H^1}(u, \mathcal{M}^\nu)\lesssim\|-\Delta u+u-2u\log |u|\|_{H^{-1}} \end{eqnarray*} which is optimal in the sense that the order of the right hand side is sharp, where \begin{eqnarray*} \mathcal{M}^\nu=\{(\mathfrak{g}_{1,0}(\cdot-z_1), \mathfrak{g}_{1,0}(\cdot-z_2), \cdots, \mathfrak{g}_{1,0}(\cdot-z_\nu))\mid z_i\in\bbr^d\}. \end{eqnarray*} Our result provides an optimal stability of the Euclidean logarithmic Sobolev inequality in the critical point setting.Comment: 22 pages; contribution to Special Volume of AAG on Potentials & PDEs in memory of David R. Adam

Type IIn Supernova SN 2010jl: Optical Observations for Over 500 Days After Explosion

We present extensive optical observations of a Type IIn supernova (SN) 2010jl for the first 1.5 years after the discovery. The UBVRI light curves demonstrated an interesting two-stage evolution during the nebular phase, which almost flatten out after about 90 days from the optical maximum. SN 2010jl has one of the highest intrinsic H_alpha luminosity ever recorded for a SN IIn, especially at late phase, suggesting a strong interaction of SN ejecta with the dense circumstellar material (CSM) ejected by the progenitor. This is also indicated by the remarkably strong Balmer lines persisting in the optical spectra. One interesting spectral evolution about SN 2010jl is the appearance of asymmetry of the Balmer lines. These lines can be well decomposed into a narrow component and an intermediate-width component. The intermediate-width component showed a steady increase in both strength and blueshift with time until t ~ 400 days after maximum, but it became less blueshifted at t ~ 500 days when the line profile appeared relatively symmetric again. Owing to that a pure reddening effect will lead to a sudden decline of the light curves and a progressive blueshift of the spectral lines, we therefore propose that the asymmetric profiles of H lines seen in SN 2010jl is unlikely due to the extinction by newly formed dust inside the ejecta, contrary to the explanation by some early studies. Based on a simple CSM-interaction model, we speculate that the progenitor of SN 2010jl may suffer a gigantic mass loss (~ 30-50 M_sun) in a few decades before explosion. Considering a slow moving stellar wind (e.g., ~ 28 km/s) inferred for the preexisting, dense CSM shell and the extremely high mass-loss rate (1-2 M_sun per yr), we suggest that the progenitor of SN 2010jl might have experienced a red supergiant stage and explode finally as a post-red supergiant star with an initial mass above 30-40 M_sun.Comment: 34 pages, 9 figures, accepted for publication in A

F-stability, entropy and energy gap for supercritical Fujita equation

We study some problems on self similar solutions to the Fujita equation when $p>(n+2)/(n-2)$, especially, the characterization of constant solutions by the energy. Motivated by recent advances in mean curvature flows, we introduce the notion of $F-$functional, $F$-stability and entropy for solutions of supercritical Fujita equation. Using these tools, we prove that among bounded positive self similar solutions, the constant solution has the lowest entropy. Furthermore, there is also a gap between the entropy of constant and non-constant solutions. As an application of these results, we prove that if $p>(n+2)/(n-2)$, then the blow up set of type I blow up solutions is the union of a $(n-1)-$ rectifiable set and a set of Hausdorff dimension at most $n-3$.Comment: 59 pages; comments welcom

Infinitely many nonradial positive solutions for multi-species nonlinear Schr\"odinger systems in RN{\mathbb R}^N

In this paper, we consider the multi-species nonlinear Schr\"odinger systems in \bbr^N: \begin{equation*} \left\{\aligned&-\Delta u_j+V_j(x)u_j=\mu_ju_j^3+\sum_{i=1;i\not=j}^d\beta_{i,j} u_i^2u_j\quad\text{in }\bbr^N, &u_j(x)>0\quad\text{in } {\mathbb R}^N, &u_j(x)\to0\quad\text{as }|x|\to+\infty,\quad j=1,2,\cdots,d,\endaligned\right. \end{equation*} where $N=2,3$, $\mu_j>0$ are constants, $\beta_{i,j}=\beta_{j,i}\not=0$ are coupling parameters, $d\geq2$ and $V_j(x)$ are potentials. By Ljapunov-Schmidt reduction arguments, we construct infinitely many nonradial positive solutions of the above system under some mild assumptions on potentials $V_j(x)$ and coupling parameters $\{\beta_{i,j}\}$, {\it without any symmetric assumptions on the limit case of the above system}. Our result, giving a positive answer to the conjecture in Pistoia and Vaira \cite{PV22} and extending the results in \cite{PW13,PV22}, reveals {\it new phenomenon} in the case of $N=2$ and $d=2$ and is {\it almost optimal} for the coupling parameters $\{\beta_{i,j}\}$.Comment: 49 pages; any comment welcom

Constraining Type Ia supernova models: SN 2011fe as a test case

The nearby supernova SN 2011fe can be observed in unprecedented detail. Therefore, it is an important test case for Type Ia supernova (SN Ia) models, which may bring us closer to understanding the physical nature of these objects. Here, we explore how available and expected future observations of SN 2011fe can be used to constrain SN Ia explosion scenarios. We base our discussion on three-dimensional simulations of a delayed detonation in a Chandrasekhar-mass white dwarf and of a violent merger of two white dwarfs-realizations of explosion models appropriate for two of the most widely-discussed progenitor channels that may give rise to SNe Ia. Although both models have their shortcomings in reproducing details of the early and near-maximum spectra of SN 2011fe obtained by the Nearby Supernova Factory (SNfactory), the overall match with the observations is reasonable. The level of agreement is slightly better for the merger, in particular around maximum, but a clear preference for one model over the other is still not justified. Observations at late epochs, however, hold promise for discriminating the explosion scenarios in a straightforward way, as a nucleosynthesis effect leads to differences in the 55Co production. SN 2011fe is close enough to be followed sufficiently long to study this effect.Comment: Accepted for publication in The Astrophysical Journal Letter