X-ray cavities in the hot corona of the lenticular galaxy NGC~4477

NGC 4477 is a low-mass lenticular galaxy in the Virgo Cluster, residing at 100\,kpc to the north of M87. Using a total of 116\,ks {\sl Chandra} observations, we study the interplay between its hot ($\sim$0.3\,keV) gas halo and the central supermassive black hole. A possible cool core is indicated by the short cooling time of the gas at the galaxy centre. We identify a pair of symmetric cavities lying 1.1\,kpc southeast and 0.9\,kpc northwest of the galaxy centre with diameters of 1.3\,kpc and 0.9\,kpc, respectively. We estimate that these cavities are newly formed with an age of $\sim$4\,Myr. No radio emission is detected at the positions of the cavities with the existing VLA data. The total energy required to produce the two cavities is $\sim$$10^{54}$\,erg, at least two orders of magnitude smaller than that of typical X-ray cavities. NGC 4477 is arguably far the smallest system and the only lenticular galaxy in which AGN X-ray cavities have been found. It falls on the scaling relation between the cavity power and the AGN radio luminosity, calibrated for groups and clusters. Our findings suggest that AGN feedback is universal among all cool core systems. Finally, we note the presence of molecular gas in NGC~4477 in the shape of a regular disk with ordered rotation, which may not be related to the feedback loop.Comment: 9 pages, 9 figures, accepted for publication in MNRA

Joint Channel Assignment and Opportunistic Routing for Maximizing Throughput in Cognitive Radio Networks

In this paper, we consider the joint opportunistic routing and channel assignment problem in multi-channel multi-radio (MCMR) cognitive radio networks (CRNs) for improving aggregate throughput of the secondary users. We first present the nonlinear programming optimization model for this joint problem, taking into account the feature of CRNs-channel uncertainty. Then considering the queue state of a node, we propose a new scheme to select proper forwarding candidates for opportunistic routing. Furthermore, a new algorithm for calculating the forwarding probability of any packet at a node is proposed, which is used to calculate how many packets a forwarder should send, so that the duplicate transmission can be reduced compared with MAC-independent opportunistic routing & encoding (MORE) [11]. Our numerical results show that the proposed scheme performs significantly better that traditional routing and opportunistic routing in which channel assignment strategy is employed.Comment: 5 pages, 4 figures, to appear in Proc. of IEEE GlobeCom 201

The AαA_\alpha-spectral radius of graphs with given degree sequence

Let $G$ be a graph with adjacency matrix $A(G)$, and let $D(G)$ be the diagonal matrix of the degrees of $G$. For any real $\alpha\in[0,1]$, write $A_\alpha(G)$ for the matrix $$A_\alpha(G)=\alpha D(G)+(1-\alpha)A(G).$$ This paper presents some extremal results about the spectral radius $\rho(A_\alpha(G))$ of $A_\alpha(G)$ that generalize previous results about $\rho(A_0(G))$ and $\rho(A_{\frac{1}{2}}(G))$. In this paper, we give some results on graph perturbation for $A_\alpha$-matrix with $\alpha\in [0,1)$. As applications, we characterize all extremal trees with the maximum $A_\alpha$-spectral radius in the set of all trees with prescribed degree sequence firstly. Furthermore, we characterize the unicyclic graphs that have the largest $A_\alpha$-spectral radius for a given unicycilc degree sequence

Boundary H\"{o}lder Regularity for Elliptic Equations

This paper investigates the relation between the boundary geometric properties and the boundary regularity of the solutions of elliptic equations. We prove by a new unified method the pointwise boundary H\"{o}lder regularity under proper geometric conditions. "Unified" means that our method is applicable for the Laplace equation, linear elliptic equations in divergence and non-divergence form, fully nonlinear elliptic equations, the $p-$Laplace equations and the fractional Laplace equations etc. In addition, these geometric conditions are quite general. In particular, for local equations, the measure of the complement of the domain near the boundary point concerned could be zero. The key observation in the method is that the strong maximum principle implies a decay for the solution, then a scaling argument leads to the H\"{o}lder regularity. Moreover, we also give a geometric condition, which guarantees the solvability of the Dirichlet problem for the Laplace equation. The geometric meaning of this condition is more apparent than that of the Wiener criterion.Comment: to appear in Journal de Math\'ematiques Pures et Appliqu\'ee

Improvement of the matching of the exact solution and variational approaches in an interacting two-fermion system

A more reasonable trial ground state wave function is constructed for the relative motion of an interacting two-fermion system in a 1D harmonic potential. At the boundaries both the wave function and its first derivative are continuous and the quasi-momentum is determined by a more practical constraint condition which associates two variational parameters. The upper bound of the ground state energy is obtained by applying the variational principle to the expectation value of the Hamiltonian of relative motion on the trial wave function. The resulted energy and wave function show better agreement with the analytical solution than the original proposal.Comment: 5 pages, 3 figures, submitted to PR

Pointwise densities of homogeneous Cantor measure and critical values

Let $N\ge 2$ and $\rho\in(0,1/N^2]$. The homogenous Cantor set $E$ is the self-similar set generated by the iterated function system $$\left\{f_i(x)=\rho x+\frac{i(1-\rho)}{N-1}: i=0,1,\ldots, N-1\right\}.$$ Let $s=\dim_H E$ be the Hausdorff dimension of $E$, and let $\mu=\mathcal H^s|_E$ be the $s$-dimensional Hausdorff measure restricted to $E$. In this paper we describe, for each $x\in E$, the pointwise lower $s$-density $\Theta_*^s(\mu,x)$ and upper $s$-density $\Theta^{*s}(\mu, x)$ of $\mu$ at $x$. This extends some early results of Feng et al. (2000). Furthermore, we determine two critical values $a_c$ and $b_c$ for the sets $$E_*(a)=\left\{x\in E: \Theta_*^s(\mu, x)\ge a\right\}\quad\textrm{and}\quad E^*(b)=\left\{x\in E: \Theta^{*s}(\mu, x)\le b\right\}$$ respectively, such that $\dim_H E_*(a)>0$ if and only if $a<a_c$, and that $\dim_H E^*(b)>0$ if and only if $b>b_c$. We emphasize that both values $a_c$ and $b_c$ are related to the Thue-Morse type sequences, and our strategy to find them relies on ideas from open dynamics and techniques from combinatorics on words.Comment: 30 pages, 1 figure and 1 tabl

Semi-groups of stochastic gradient descent and online principal component analysis: properties and diffusion approximations

We study the Markov semigroups for two important algorithms from machine learning: stochastic gradient descent (SGD) and online principal component analysis (PCA). We investigate the effects of small jumps on the properties of the semi-groups. Properties including regularity preserving, $L^{\infty}$ contraction are discussed. These semigroups are the dual of the semigroups for evolution of probability, while the latter are $L^{1}$ contracting and positivity preserving. Using these properties, we show that stochastic differential equations (SDEs) in $\mathbb{R}^d$ (on the sphere $\mathbb{S}^{d-1}$) can be used to approximate SGD (online PCA) weakly. These SDEs may be used to provide some insights of the behaviors of these algorithms

The convexity of inclusions and gradient's concentration for Lam\'e systems with partially infinite coefficients

It is interesting to study the stress concentration between two adjacent stiff inclusions in composite materials, which can be modeled by the Lam\'e system with partially infinite coefficients. To overcome the difficulty from the lack of maximum principle for elliptic systems, we use the energy method and an iteration technique to study the gradient estimates of the solution. We first find a novel phenomenon that the gradient will not blow up any more once these two adjacent inclusions fail to be locally relatively strictly convex, namely, the top and bottom boundaries of the narrow region are partially "flat". This is contrary to our expectation. In order to further explore the blow-up mechanism of the gradient, we next investigate two adjacent inclusions with relative convexity of order m and finally reveal an underlying relationship between the blow-up rate of the stress and the order of the relative convexity of the subdomains in all dimensions.Comment: 61 page

Self-Attention Recurrent Network for Saliency Detection

Feature maps in deep neural network generally contain different semantics. Existing methods often omit their characteristics that may lead to sub-optimal results. In this paper, we propose a novel end-to-end deep saliency network which could effectively utilize multi-scale feature maps according to their characteristics. Shallow layers often contain more local information, and deep layers have advantages in global semantics. Therefore, the network generates elaborate saliency maps by enhancing local and global information of feature maps in different layers. On one hand, local information of shallow layers is enhanced by a recurrent structure which shared convolution kernel at different time steps. On the other hand, global information of deep layers is utilized by a self-attention module, which generates different attention weights for salient objects and backgrounds thus achieve better performance. Experimental results on four widely used datasets demonstrate that our method has advantages in performance over existing algorithms

Center and isochronous center of a class of quasi-analytic switching systems

In this paper, we study the integrability and linearization of a class of quadratic quasi-analytic switching systems. We improve an existing method to compute the focus values and periodic constants of quasi-analytic switching systems. In particular, with our method, we demonstrate that the dynamical behaviors of quasi-analytic switching systems are more complex than that of continuous quasi-analytic systems, by showing the existence of six and seven limit cycles in the neighborhood of the origin and infinity, respectively, in a quadratic quasi-analytic switching system. Moreover, explicit conditions are obtained for classifying the centers and isochronous centers of the system.Comment: 24 pages, 3 figure