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Xia Huang

Publications and source records attributed to Xia Huang.

28 records · Page 2Linked to original sources
arXiv2020

Existence and nonexistence results for a weighted elliptic equation in exterior domains

We consider positive solution to the weighted elliptic problem \begin{equation*} \left \{ \begin{array}{ll} -{\rm div} (|x|^θ\nabla u)=|x|^\ell u^p \;\;\; \mbox{in $\mathbb{R}^N \backslash {\overline B}$},\\ u=0 \;\;\; \mbox{on $\partial B$}, \end{array} \right. \end{equation*} where $B$ is the standard unit ball of $\mathbb{R}^N$. We give a complete answer for the existence question when $N':=N+θ>2$. In particular, for $N' > 2$ and $τ:=\ell-θ>-2$, it is shown that the problem admits a unique positive radial solution for $p>p_s:=\frac{N'+2+2τ}{N'-2}$, while for any $ 0<p \leq p_s$, the only nonnegative solution is $u \equiv 0$.

math.AP↗
arXiv2019

Predictive Coding Networks Meet Action Recognition

Action recognition is a key problem in computer vision that labels videos with a set of predefined actions. Capturing both, semantic content and motion, along the video frames is key to achieve high accuracy performance on this task. Most of the state-of-the-art methods rely on RGB frames for extracting the semantics and pre-computed optical flow fields as a motion cue. Then, both are combined using deep neural networks. Yet, it has been argued that such models are not able to leverage the motion information extracted from the optical flow, but instead the optical flow allows for better recognition of people and objects in the video. This urges the need to explore different cues or models that can extract motion in a more informative fashion. To tackle this issue, we propose to explore the predictive coding network, so called PredNet, a recurrent neural network that propagates predictive coding errors across layers and time steps. We analyze whether PredNet can better capture motions in videos by estimating over time the representations extracted from pre-trained networks for action recognition. In this way, the model only relies on the video frames, and does not need pre-processed optical flows as input. We report the effectiveness of our proposed model on UCF101 and HMDB51 datasets.

cs.CV↗
arXiv2017

On Delaunay solutions of a biharmonic elliptic equation with critical exponent

We are interested in the qualitative properties of positive entire solutions $u \in C^4 (\mathbb{R}^n \backslash \{0\})$ of the equation \begin{equation} \label{0.0} Δ^2 u=u^{\frac{n+4}{n-4}} \;\;\mbox{in $\mathbb{R}^n \backslash \{0\}$ and 0 is a non-removable singularity of $u(x)$}. \end{equation} It is known from [Theorem 4.2] that any positive entire solution $u$ of \eqref{0.0} is radially symmetric with respect to $x=0$, i.e. $u(x)=u(|x|)$, and equation \eqref{0.0} also admits a special positive entire solution $u_s (x)=\Big(\frac{n^2 (n-4)^2}{16} \Big)^{\frac{n-4}{8}} |x|^{-\frac{n-4}{2}}$. We first show that $u-u_s$ changes signs infinitely many times in $(0, \infty)$ for any positive singular entire solution $u \not \equiv u_s$ in $\mathbb{R}^N \backslash \{0\}$ of \eqref{0.0}. Moreover, equation \eqref{0.0} admits a positive entire singular solution $u(x) \; (=u(|x|)$ such that the scalar curvature of the conformal metric with conformal factor $u^{\frac{4}{n-4}}$ is positive and $v(t):=e^{\frac{n-4}{2} t} u(e^t)$ is $2T$-periodic with suitably large $T$. It is still open that $v(t):=e^{\frac{n-4}{2} t} u(e^t)$ is periodic for any positive entire solution $u(x)$ of \eqref{0.0}.

math.AP↗
arXiv2015

Ensemble order parameter equations in star network

The OA ansatz has attracted much attention recently, infinite-dimensional Kuramoto model could collapses to a two-dimensional system of order differential equations with it. In this paper, we propose the ensemble order parameter (EOP) equations to describe the dynamics for networks with a finite size. To verify the effectiveness of this method, we apply it into the star network and star-connected network. In the star network, numerous phase transitions among different synchronous states are observed, three processes of synchronization, one process of de-synchronization and a group of hybrid phase transitions, the processes of those transitions are revealed by the EOP dynamics and other nolinear tools such as time reversibility analysis and linear stability analysis. Also in the star-connected network, the two-step synchronization transition is observed. The process of it is still be revealed by the similar methods in the single star network.

nlin.AO↗
arXiv2015

Jet/accretion and Unification for FSRQs/FRII and BL Lac objects/FRI radio galaxies

We compiled a sample of 24 flat-spectrum radio quasars (FSRQs), 21 BL Lacs, 13 FRI and 12 FRII radio galaxies to study the jet power and broad-line luminosity relation, and the jet power and black hole mass relation. Furthermore, we obtain the histograms of key parameters. Our main results are as the follows:(i) We found that the FSRQs are roughly separated from BL Lacs by the Ledlow-Owen FRI/FRII dividing line in the $\rm{\log{P_{jet}}-\log{M}}$ plane. This result supports the unification model that BL Lacs unify with FRI radio galaxies and FSRQs unify with FRII radio galaxies.(ii)We found that the coefficient of the best-fit linear regression equation of $\rm{\log{L_{BLR}}-\log{P_{jet}}}$ relation is very close to 1 for our sample. The correlation between broad line luminosity and jet power is significant which supports that jet power has a close link with accretion.

astro-ph.HE↗
arXiv2015

The jets-accretion relation, mass-luminosity relation in Fermi blazars

A sample of 111 Fermi blazars each with a well-established radio core luminosity, broad-line luminosity, bolometric luminosity and black hole mass has been compiled from the literatures.We present a significant correlation between radio core and broad-line emission luminosities that supports a close link between accretion processes and relativistic jets. Analysis reveals a relationship of $\rm{LogL_{BLR}\sim(0.81\pm0.06)LogL_{R}^{C}}$ which is consistant with theoretical predicted coefficient and supports that blazar jets are powered by energy extraction from a rapidly spinning Kerr black hole through the magnetic field provided by the accretion disk. Through studying the correlation between the intrinsic bolometric luminosity and the black hole mass, we find a relationship of $\rm{{Log}\frac{L_{in}}{L_{\odot}}=(0.95\pm0.26){Log}\frac{M}{M_{\odot}}+(3.53\pm2.24)}$ which supports mass-luminosity relation for Fermi blazars derived in this work is a powerlaw relation similar to that for main-sequence stars. Finally, evolutionary sequence of blazars is discussed.

astro-ph.HE↗
arXiv2015

Conformal metrics in ${\mathbb R}^{2m}$ with constant $Q$-curvature and arbitrary volume

We study the polyharmonic problem $Δ^m u = \pm e^u$ in ${\mathbb R}^{2m}$, with $m \geq 2$. In particular, we prove that {\sl for any} $V > 0$, there exist radial solutions of $Δ^m u = -e^u$ such that $$\int_{{\mathbb R}^{2m}} e^u dx = V.$$ It implies that for $m$ odd, given arbitrary volume $V > 0$, there exist conformal metrics $g$ on ${\mathbb R}^{2m}$ with positive constant $Q$-curvature and vol$(g) =V$. This answers some open questions in Martinazzi's work.

math.AP↗
arXiv2014

Explosive or Continuous: Incoherent state determines the route to synchronization

Collective behaviors of coupled oscillators have attracted much attention. In this Letter, we propose an ensemble order parameter(EOP) equation that enables us to grasp the essential low-dimensional dynamical mechanism of the explosive synchronization in heterogeneous networks. Dif- ferent solutions of the EOP equation build correspondences with diverse collective states, and different bifurcations reveal various transitions among these collective states. The structural relationship between the incoherent state and synchronous state leads to different routes of transitions to synchronization, either continuous or discontinuous. The explosive synchronization is determined by the bistable state where the measure of each state and the critical points are obtained analytically by using the EOP equation. Our method and results hold for heterogeneous networks with star graph motifs such as scale-free networks, and hence, provide an effective approach in understanding the routes to synchronization in more general complex networks.

nlin.AO↗