Exponential convergence of 1-graph of the solution semigroup of contact Hamilton-Jacobi equations

Under certain assumptions, we show that for the solution semigroup of evolutionary contact Hamilton-Jacobi equations, its 1-graph, as a pseudo Legendrian graph, converges exponentially to the 1-graph of the viscosity solution of stationary equations in the sense of certain Hausdorff metrics. This result reveals an essential difference between certain dissipative systems and conservative systems from weak KAM aspects

Snapshot light-field laryngoscope

The convergence of recent advances in optical fabrication and digital processing yields a new generation of imaging technology: light-field cameras, which bridge the realms of applied mathematics, optics, and high-performance computing. Herein for the first time, we introduce the paradigm of light-field imaging into laryngoscopy. The resultant probe can image the three-dimensional (3D) shape of vocal folds within a single camera exposure. Furthermore, to improve the spatial resolution, we developed an image fusion algorithm, providing a simple solution to a long-standing problem in light-field imaging.Comment: 15 pages, 6 figures, 1 tabl

X-ray Radiation Mechanisms and Beaming Effect of Hot Spots and Knots in Active Galactic Nuclear Jets

The observed radio-optical-X-ray spectral energy distributions (SEDs) of 22 hot spots and 45 knots in the jets of 35 active galactic nuclei are complied from literature and modeled with single-zone lepton models. It is found that the observed luminosities at 5 GHz (L_5GHz) and at 1 keV (L_1keV) are tightly correlated, and the two kinds of sources can be roughly separated with a division of L_1keV=L_5GHz. Our SED fits show that the mechanisms of the X-rays are diverse. Considering the sources at rest, the synchrotron-self-Compton (SSC) scattering would dominate the IC process. This model can interpret the X-rays of some hot spots with a magnetic field strength (B_ssc^delta=1) being consistent with the equipartition magnetic field (B_eq^delta=1) in one order of magnitude, but an unreasonably low B_ssc^delta=1 is required to model the X-rays for all knots. The ratio R_B=B_eq^delta=1/B_ssc^delta=1 is greater than 1 and it is tightly anti-correlated with ratio R_L= L_1keV/L_5GHz for both the knots and the hot spots. We propose that the deviation may be due to the neglect of the relativistic bulk motion for these sources. Considering this effect, we show that the IC/CMB model well explains the X-ray emission for most sources. Both B_eq' and delta are tentatively correlated with R_L. Corrected by the beaming effect, the L'_5GHz-L'_1keV relations for the two kinds of sources are even tighter than the observed ones. These facts suggest that, under the equipartition condition, the observational differences of the X-rays from the knots and hot spots may be mainly due the differences on the Doppler boosting effect and the co-moving magnetic field of the two kinds of sources. Our IC scattering models predict a prominent GeV-TeV component in the SEDs for some sources, which are detectable with H.E.S.S. and Fermi/LAT.Comment: 38 pages, including 2 tables and 9 figures. Accepted for publications in Ap

An Ensemble EM Algorithm for Bayesian Variable Selection

We study the Bayesian approach to variable selection in the context of linear regression. Motivated by a recent work by Rockova and George (2014), we propose an EM algorithm that returns the MAP estimate of the set of relevant variables. Due to its particular updating scheme, our algorithm can be implemented efficiently without inverting a large matrix in each iteration and therefore can scale up with big data. We also show that the MAP estimate returned by our EM algorithm achieves variable selection consistency even when $p$ diverges with $n$. In practice, our algorithm could get stuck with local modes, a common problem with EM algorithms. To address this issue, we propose an ensemble EM algorithm, in which we repeatedly apply the EM algorithm on a subset of the samples with a subset of the covariates, and then aggregate the variable selection results across those bootstrap replicates. Empirical studies have demonstrated the superior performance of the ensemble EM algorithm

Extreme-Point Symmetric Mode Decomposition Method for Data Analysis

An extreme-point symmetric mode decomposition (ESMD) method is proposed to improve the Hilbert-Huang Transform (HHT) through the following prospects: (1) The sifting process is implemented by the aid of 1, 2, 3 or more inner interpolating curves, which classifies the methods into ESMD_I, ESMD_II, ESMD_III, and so on; (2) The last residual is defined as an optimal curve possessing a certain number of extreme points, instead of general trend with at most one extreme point, which allows the optimal sifting times and decompositions; (3) The extreme-point symmetry is applied instead of the envelop symmetry; (4) The data-based direct interpolating approach is developed to compute the instantaneous frequency and amplitude. One advantage of the ESMD method is to determine an optimal global mean curve in an adaptive way which is better than the common least-square method and running-mean approach; another one is to determine the instantaneous frequency and amplitude in a direct way which is better than the Hilbert-spectrum method. These will improve the adaptive analysis of the data from atmospheric and oceanic sciences, informatics, economics, ecology, medicine, seismology, and so on.Comment: 40 pages, 28 figure

A Variational Algorithm for Bayesian Variable Selection

There has been an intense development on the estimation of a sparse regression coefficient vector in statistics, machine learning and related fields. In this paper, we focus on the Bayesian approach to this problem, where sparsity is incorporated by the so-called spike-and-slab prior on the coefficients. Instead of replying on MCMC for posterior inference, we propose a fast and scalable algorithm based on variational approximation to the posterior distribution. The updating scheme employed by our algorithm is different from the one proposed by Carbonetto and Stephens (2012). Those changes seem crucial for us to show that our algorithm can achieve asymptotic consistency even when the feature dimension diverges exponentially fast with the sample size. Empirical results have demonstrated the effectiveness and efficiency of the proposed algorithm

A representation formula of viscosity solutions to weakly coupled systems of Hamilton-Jacobi equations with applications to regularizing effect

Based on a fixed point argument, we give a {\it dynamical representation} of the viscosity solution to Cauchy problem of certain weakly coupled systems of Hamilton-Jacobi equations with continuous initial datum. Using this formula, we obtain some regularity results related to the viscosity solution, including a partial extension of Lions' regularizing effect \cite{L} to the case of weakly coupled systems

Coalescing Majorana edge modes in non-Hermitian PT-symmetric Kitaev chain

A single unit cell contains all the information about the bulk system, including the topological feature. The topological invariant can be extracted from a finite system, which consists of several unit cells under certain environment, such as a non-Hermitian external field. We investigate a non- Hermitian finite-size Kitaev chain with PT-symmetric chemical potentials. Exact solution at the symmetric point shows that Majorana edge modes can emerge as the coalescing states at exceptional points and PT symmetry breaking states. The coalescing zero mode is the finite-size projection of the conventional degenerate zero modes in a Hermitian infinite system with the open boundary condition. It indicates a variant of the bulk-edge correspondence: The number of Majorana edge modes in a finite non-Hermitian system can be the topological invariant to identify the topological phase of the corresponding bulk Hermitian system.Comment: 7 pages, 2 figure

On class A Lorentzian 2-tori with poles I: Closed geodesics pass through poles

In this paper, by studying certain isometries on globally hyperbolic planes, we prove that if $p$ is a timelike pole on a class A Lorentzian 2-torus, then there exists a closed timelike geodesic passing through $p$ with any preassigned free homotopy class in the interior of the stable time cone. We also show a non-rigid result when timelike poles appear

Hybrid Interference Induced Flat Band Localization in Bipartite Optomechanical Lattices

The flat band localization, as an important phenomenon in solid state physics, is fundamentally interesting in the exploration of exotic ground property of many-body system. Here we demonstrate the appearance of a flat band in a general bipartite optomechanical lattice, which could have one or two dimensional framework. Physically, it is induced by the hybrid interference between the photon and phonon modes in optomechanical lattice, which is quite different from the destructive interference resulted from the special geometry structure in the normal lattice (e.g., Lieb lattice). Moreover, this novel flat band is controllable and features a special local density of states (LDOS) pattern, which makes it is detectable in experiments. This work offers an alternative approach to control the flat band localization with optomechanical interaction, which may substantially advance the fields of cavity optomechanics and solid state physics.Comment: 7 pages, 9 figures, Published in Scientifc Reports 7, 15188 (2017