Infinitely many solutions to linearly coupled Schr\"{o}dinger equations with non-symmetric potential

We study a linearly coupled Schr\"{o}dinger system in $\R^N(N\leq3).$ Assume that the potentials in the system are continuous functions satisfying suitable decay assumptions, but without any symmetry properties and the parameters in the system satisfy some suitable restrictions. Using the Liapunov-Schmidt reduction methods two times and combing localized energy method, we prove that the problem has infinitely many positive synchronized solutions, which extends the result Theorem 1.2 about nonlinearly coupled Schr\"{o}dinger equations in \cite{aw} to our linearly coupled problem.Comment: 27 pages. arXiv admin note: text overlap with arXiv:1210.8209 by other author

Infinitely many solutions for p-Laplacian equation involving double critical terms and boundary geometry

Let $1<p<N$, $p^{*}=Np/(N-p)$, $0<s<p$, $p^{*}(s)=(N-s)p/(N-p)$, and \Om\in C^{1} be a bounded domain in $\R^{N}$ with 0\in\bar{\Om}. In this paper, we study the following problem \begin{cases} -\Delta_{p}u=\mu|u|^{p^{*}-2}u+\frac{|u|^{p^{*}(s)-2}u}{|x|^{s}}+a(x)|u|^{p-2}u, & \text{in }\Om,\\ u=0, & \text{on }\pa\Om, \end{cases} where $\mu\ge0$ is a constant, \De_{p} is the $p$-Laplacian operator and a\in C^{1}(\bar{\Om}). By an approximation argument, we prove that if $N>p^{2}+p,a(0)>0$ and $\Omega$ satisfies some geometry conditions if $0\in\partial\Omega$, say, all the principle curvatures of $\partial\Omega$ at $0$ are negative, then the above problem has infinitely many solutions.Comment: 28pages,no figur

Infinitely many sign-changing solutions for an elliptic problem with double critical Hardy-Sobolev-Maz'ya terms

In this paper, we investigate the following elliptic problem involving double critical Hardy-Sobolev-Maz'ya terms: $$\left\{\begin{array}{ll} -\Delta u = \mu\frac{|u|^{2^*(t)-2}u}{|y|^t} + \frac{|u|^{2^*(s)-2}u}{|y|^s} + a(x) u, & {\rm in}\ \Omega,\\ \quad u = 0, \,\, &{\rm on}\ \partial \Omega, \end{array} \right.$$ where $\mu\geq0$, $a(x)>0$, $2^*(t)=\frac{2(N-t)}{N-2}$, $2^*(s) = \frac{2(N-s)}{N-2}$, $0\leq t<s<2$, $x = (y,z)\in \mathbb{R}^k\times \mathbb{R}^{N-k}$, $2\leq k<N$, $(0,z^*) \in \bar{\Omega}$ and $\Omega$ is an bounded domain in $\mathbb{R}^N$. Applying an abstract theorem in \cite{sz}, we prove that if $N>6+t$ when $\mu>0,$ and $N>6+s$ when $\mu=0,$ and $\Omega$ satisfies some geometric conditions, then the above problem has infinitely many sign-changing solutions. The main tool is to estimate Morse indices of these nodal solution.Comment: 11page

On modules for double affine Lie algebras

Imaginary Verma modules, parabolic imaginary Verma modules, and Verma modules at level zero for double affine Lie algebras are constructed using three different triangular decompositions. Their relations are investigated, and several results are generalized from the affine Lie algebras. In particular, imaginary highest weight modules, integrable modules, and irreducibility criterion are also studied.Comment: 15 pages, 15 ref

Infinitely many solutions for a nonlinear Schr\"{o}dinger equation with non-symmetric electromagnetic fields

In this paper, we study the nonlinear Schr\"{o}dinger equation with non-symmetric electromagnetic fields $$\Big(\frac{\nabla}{i}-A_{\epsilon} x)\Big)^2 u+V_{\epsilon}(x)u=f(u),\ u\in H^1 (\mathbb{R}^N,\mathbb{C}),$$ where $A_{\epsilon}(x)=(A_{\epsilon,1}(x),A_{\epsilon,2}(x),\cdots,A_{\epsilon,N}(x))$ is a magnetic field satisfying that $A_{\epsilon,j}(x)(j=1,\ldots,N)$ is a real $C^{1}$ bounded function on $\mathbb{R}^{N}$ and $V_{\epsilon}(x)$ is an electric potential. Both of them satisfy some decay conditions and $f(u)$ is a nonlinearity satisfying some nondegeneracy condition. Applying localized energy method, we prove that there exists some $\epsilon_{0 }> 0$ such that for $0 < \epsilon < \epsilon_{0 }$, the above problem has infinitely many complex-valued solutions.Comment: 39fages, 0 figures. arXiv admin note: text overlap with arXiv:1210.8209, arXiv:1209.2824 by other author

Towards End-to-end Text Spotting with Convolutional Recurrent Neural Networks

In this work, we jointly address the problem of text detection and recognition in natural scene images based on convolutional recurrent neural networks. We propose a unified network that simultaneously localizes and recognizes text with a single forward pass, avoiding intermediate processes like image cropping and feature re-calculation, word separation, or character grouping. In contrast to existing approaches that consider text detection and recognition as two distinct tasks and tackle them one by one, the proposed framework settles these two tasks concurrently. The whole framework can be trained end-to-end, requiring only images, the ground-truth bounding boxes and text labels. Through end-to-end training, the learned features can be more informative, which improves the overall performance. The convolutional features are calculated only once and shared by both detection and recognition, which saves processing time. Our proposed method has achieved competitive performance on several benchmark datasets.Comment: 14 page

A New Dust Budget In The Large Magellanic Cloud

The origin of dust in a galaxy is poorly understood. Recently, the surveys of the Large Magellanic Cloud (LMC) provide astrophysical laboratories for the dust studies. By a method of population synthesis, we investigate the contributions of dust produced by asymptotic giant branch (AGB) stars, common envelope (CE) ejecta and type II supernovae (SNe II) to the total dust budget in the LMC. Based on our models, the dust production rates (DPRs) of AGB stars in the LMC are between about $2.5\times10^{-5}$ and $4.0\times10^{-6}M_\odot{\rm yr^{-1}}$. The uncertainty mainly results from different models for the dust yields of AGB stars. The DPRs of CE ejecta are about $6.3\times10^{-6}$(The initial binary fraction is 50\%). These results are within the large scatter of several observational estimates. AGB stars mainly produce carbon grains, which is consistent with the observations. Most of dust grains manufactured by CE ejecta are silicate and iron grains. The contributions of SNe II are very uncertain. Compared with SNe II without reverse shock, the DPRs of AGB stars and CE ejecta are negligible. However, if only 2 \% of dust grains produced by SNe II can survive after reverse shock, the contributions of SNe II are very small. The total dust masses produced by AGB stars in the LMC are between $2.8\times10^4$ and $3.2\times10^5M_\odot$, and those produced by CE ejecta are about $6.3\times10^4$. They are much lower than the values estimated by observations. Therefore, there should be other dust sources in the LMC.Comment: 13 pages, 11 figures, 1 table, Accepted for publication in MNRA

Uniqueness of positive solutions with Concentration for the Schr\"odinger-Newton problem

We are concerned with the following Schr\"odinger-Newton problem \begin{equation} -\varepsilon^2\Delta u+V(x)u=\frac{1}{8\pi \varepsilon^2} \big(\int_{\mathbb R^3}\frac{u^2(\xi)}{|x-\xi|}d\xi\big)u,~x\in \mathbb R^3. \end{equation} For $\varepsilon$ small enough, we show the uniqueness of positive solutions concentrating at the nondegenerate critical points of $V(x)$. The main tools are a local Pohozaev type of identity, blow-up analysis and the maximum principle. Our results also show that the asymptotic behavior of concentrated points to Schr\"odinger-Newton problem is quite different from those of Schr\"odinger equations

Donors of Persistent Neutron-star Low-mass X-ray Binaries

Properties of X-ray luminosities in low-mass X-ray binaries (LMXBs) mainly depend on donors. We have carried out a detailed study of donors in persistent neutron-star LMXBs (PLMXBs) by means of a population synthesis code. PLMXBs with different donors have different formation channels. Our numerical simulations show that more than 90% of PLMXBs have main sequence (MS) donors, and PLMXBs with red giant (RG) donors via stellar wind (Wind) are negligible. In our model, most of neutron stars (NSs) in PLMXBs with hydrogen-rich donors form via core-collapse supernovae, while more than 90% of NSs in PLMXBs with naked helium star (He) donors or white dwarf (WD) donors form via an evolution-induced collapse via helium star ($1.4 \leq M_{\rm He}/M_\odot \leq 2.5$) or an accretion-induced collapses for an accreting ONeMg WD.Comment: 9 pages, 4 figure

Population Synthesis of Millisecond X-ray Pulsars

As the evolutionary link between the radio millisecond pulsars (MSPs) and the low mass X-ray binaries or intermediate mass X-ray binaries, the millisecond X-ray pulsars (MSXPs) are important objects in testing theories of pulsar formation and evolution. In general, neutron stars in MSXPs can form via core collapse supernova (CC channel) of massive stars or accretion induced collapse (AIC channel) of an accreting ONeMg WD whose mass reaches the Chandrasekhar limit. Here, in addition to CC and AIC channels we also consider another channel, i.e., evolution induced collapse (EIC channel) of a helium star with mass between $1.4$ and $2.5 M_{\odot}$. Using a population synthesis code, we have studied MSXPs arising from three different evolutionary channels. We find that the Galactic birthrates of transient MSXPs and persistent MSXPs are about 0.7---$1.4\times 10^{-4}$ yr$^{-1}$. Our population synthesis calculations have shown that about 50\%---90\% of the MSXPs have undergone CC channel, about 10\%---40\% of them have undergone EIC channel, and the MSXPs via AIC channel are the least.Comment: 12 pages, 7 figures, 3 tables, Accepted for publication in MNRA