Discrete Hardy-type Inequalities

This paper studies the Hardy-type inequalities on the discrete intervals. The first result is the variational formulas of the optimal constants. Using these formulas, one may obtain an approximating procedure and the known basic estimates of the optimal constants. The second result, which is the main innovation of this paper, is about the factor of basic upper estimates. An improved factor is presented, which is smaller than the known one and is best possible. Some comparison results are included for comparing the optimal constants on different intervals.Comment: 27 pages, no figur

Analytic Solutions of Von Karman Plate under Arbitrary Uniform Pressure --- Equations in Differential Form

The large deflection of a circular thin plate under uniform external pressure is a classic problem in solid mechanics, dated back to Von K{\'a}rm{\'a}n \cite{Karman}. {This problem is reconsidered in this paper using an analytic approximation method, namely the homotopy analysis method (HAM).} Convergent series solutions are obtained for four types of boundary conditions with rather high nonlinearity, even in the case of $w(0)/h>20$, where $w(0)/h$ denotes the ratio of central deflection to plate thickness. Especially, we prove that the previous perturbation methods for an arbitrary perturbation quantity (including the Vincent's [2] and Chien's [3] methods) and the modified iteration method [4] are only the special cases of the HAM. However, the HAM works well even when the perturbation methods become invalid. All of these demonstrate the validity and potential of the HAM for the Von K{\'a}rm{\'a}n's plate equations, and show the superiority of the HAM over perturbation methods for highly nonlinear problemsComment: 33 pages, 4 figure

CPCP Invariance Study of J/ψ→ΛΛˉJ/\psi\to\Lambda\bar\Lambda and Λ\Lambda Nonpleptonic Decays in Helicity Frame

We present the joint helicity amplitudes for $J/\psi \to \Lambda \bar{\Lambda}$, $\Lambda(\bar\Lambda)$ decays to different final states in the helicity frame. Two observables to search for $CP$ violation in $J/\psi\to\Lambda\bar\Lambda$ can be expressed with the information of helicity angles of baryon and antibaryon. Four decay parameters of $\Lambda$ and $\bar\Lambda$, namely, $\alpha_-,\alpha_+,\alpha_0$ and $\bar\alpha_0$, can be obtained with the joint helicity amplitude equations by the likelihood fit method. With the data sample of $10^{10}$ $J/\psi$ decays accumulated by BESIII, the precision of the measurements is estimated to be about $10^{-3}$

Analytic Solutions of Von Karman Plate under Arbitrary Uniform Pressure (II): Equations in Integral Form

In this paper, the homotopy analysis method (HAM) is successfully applied to solve the Von Karman's plate equations in the integral form for a circular plate with the clamped boundary under an arbitrary uniform external pressure. Two HAM-based approaches are proposed. One is for a given external load Q, the other for a given central deflection. Both of them are valid for an arbitrary uniform external pressure by means of choosing a proper value of the so-called convergence-control parameters c_1 and c_2 in the frame of the HAM. Besides, it is found that iteration can greatly accelerate the convergence of solution series. In addition, we prove that the interpolation iterative method is a special case of the HAM-based 1st-order iteration approach for a given external load Q when c_1=-theta and c_2=-1, where theta denotes the interpolation parameter of the interpolation iterative method. Therefore, like Zheng and Zhou}, one can similarly prove that the HAM-based approaches are valid for an arbitrary uniform external pressure, at least in some special cases such as c_1=-theta and c_2=-1. Furthermore, it is found that the HAM-based iteration approaches converge much faster than the interpolation iterative method. All of these illustrate the validity and potential of the HAM for the famous Von Karman's plate equations, and show the superiority of the HAM over perturbation methods.Comment: 26 pages, 8 figure

On the Limiting Stokes' Wave of Extreme Height in Arbitrary Water Depth

As mentioned by Schwartz (1974) and Cokelet (1977), it was failed to gain convergent results of limiting Stokes' waves in extremely shallow water by means of perturbation methods even with the aid of extrapolation techniques such as Pad\'{e} approximant. Especially, it is extremely difficult for traditional analytic/numerical approaches to present the wave profile of limiting waves with a sharp crest of $120^\circ$ included angle first mentioned by Stokes in 1880s. Thus, traditionally, different wave models are used for waves in different water depths. In this paper, by means of the homotopy analysis method (HAM), an analytic approximation method for highly nonlinear equations, we successfully gain convergent results (and especially the wave profiles) of the limiting Stokes' waves with this kind of sharp crest in arbitrary water depth, even including solitary waves of extreme form in extremely shallow water, without using any extrapolation techniques. Therefore, in the frame of the HAM, the Stokes' wave can be used as a unified theory for all kinds of waves, including periodic waves in deep and intermediate depth, cnoidal waves in shallow water and solitary waves in extremely shallow water.Comment: 26 pages, 9 figures, accepted by J. Fluid Mech. in Feb 201

FermiFermi-LAT detection of a transient γ\gamma-ray source in the direction of a distant blazar B3 1428+422 at z=4.72z =4.72

We report the detection of a transient $\gamma$-ray source in the direction of B3 1428+422 ($z=4.72$) by analyzing the 110-month {\it Fermi}-LAT Pass 8 data. The new transient $\gamma-$ray source is far away from the Galactic plane and has a rather soft spectrum, in agreement with being a high redshift blazar. We suggest that the newly discovered transient is the $\gamma$-ray counterpart of B3 1428+422, which could be the {\it most distant} GeV source detected so far. The detection of a group of such distant $\gamma-$ray blazars will be helpful to reconstruct the evolution of the luminosity function and to study the extragalactic background light at such high redshifts.Comment: 9 pages, 4 figures, ApJL in pres

Two transient X-ray Quasi-Periodic Oscillations separated by an intermediate state in 1H 0707-495

In the narrow-line Seyfert 1 galaxy 1H 0707-495, recently a transient quasi-periodic oscillation (QPO) signal with a frequency of $\sim 2.6\times 10^{-4}$ Hz has been detected at a high statistical significance. Here, we reanalyze the same set of XMM-Newton data observed on 2008 February 4 with the Weighted-Wavelet Z-transform (WWZ) method. In addition to confirming the previous findings, we also find another QPO signal with a frequency of $\sim 1.2\times 10^{-4}$ Hz in a separated X-ray emission phase at the significance level of $\sim 3.7\sigma$. The signal is also found fitting an auto-regressive model though at a lower significance. The frequency ratio between these two signals is $\sim 2:1$. The analysis of other XMM-Newton measurements of 1H 0707-495 also reveals the presence of the $\sim 2.6\times 10^{-4}$ Hz ($\sim 1.2\times 10^{-4}$ Hz) QPO signal on 2007 May 14 (2010 September 17) at the significance level of $\sim 4.2\sigma$ ($\sim 3.5\sigma$). The QPO frequency found in this work follows the $f_{QPO}-M_{BH}$ relation reported in previous works spanning from stellar-mass to supermassive black holes. This is the first time to observe two separated transient X-ray QPO signals in active galactic nuclei (AGNs), which sheds new light on the physics of accreting supermassive black holes.Comment: 9 pages, 9 figures, 1 table; Accepted for publication in Ap

Global existence and blow-up of solutions to a class of non-Newton filtration equations with singular potential and logarithmic nonlinearity

In this paper, a class of non-Newton filtration equations with singular potential and logarithmic nonlinearity under initial-boundary condition is investigated. Based on potential well method and Hardy-Sobolev inequality, the global existence of solutions is derived when the initial energy $J(u_0)$ is subcritical($J(u_0)<d$), critical($J(u_0)=d$) with $d$ being the mountain-pass level. Finite time blow-up results are obtained as well when the initial energy $J(u_0)$ satisfies specific conditions. Moreover, the upper and lower bounds of the blow-up time are given.Comment: 18 page

Circinus Galaxy Revisted with 10 Years of Fermi-LAT Data

Circinus galaxy is a nearby composite starburst/AGN system. In this work we re-analyze the GeV emission from Circinus with 10 years of {\it Fermi}-LAT Pass 8 data. In the energy range of 1-500 GeV, the spectrum can be well fitted by a power-law model with a photon index of $\Gamma$ = $2.20\pm0.14$, and its photon flux is $(5.90\pm1.04) \times 10^{-10}$ photons cm$^{-2}$ s$^{-1}$. Our 0.1-500 GeV flux is several times lower than that reported in the previous literature, which is roughly in compliance with the empirical relation for star-forming and local group galaxies and might be reproduced by the interaction between cosmic rays and the interstellar medium. The ratio between the $\gamma$-ray luminosity and the total infrared luminosity is near the proton calorimetric limit, indicating that Circinus may be a proton calorimeter. However, marginal evidence for variability of the $\gamma$-ray emission is found in the timing analysis, which may indicate the activity of AGN jet. More {\it Fermi}-LAT data and future observation of CTA are required to fully reveal the origin of its $\gamma$-ray emission.Comment: 11 pages, 7 figures, 4 tables, accepted for publication by Ap

Estimate the exponential convergence rate of f-ergodicity via spectral gap

This paper studies the f-ergodicity and its exponential convergence rate for continuous-time Markov chain. Assume f is square integrable, for reversible Markov chain, it is proved that the exponential convergence of f-ergodicity holds if and only if the spectral gap of the generator is positive. Moreover, the convergence rate is equal to the spectral gap. For irreversible case, the positivity of spectral gap remains a sufficient condition of f-ergodicity. The effectiveness of these results are illustrated by some typical examples.Comment: 11 page