Small eigenvalues of closed Riemann surfaces for large genus

In this article we study the asymptotic behavior of small eigenvalues of Riemann surfaces for large genus. We show that for any positive integer $k$, as the genus $g$ goes to infinity, the smallest $k$-th eigenvalue of Riemann surfaces in any thick part of moduli space of Riemann surfaces of genus $g$ is uniformly comparable to $\frac{1}{g^2}$ in $g$. In the proof of the upper bound, for any constant $\epsilon>0$, we will construct a closed Riemann surface of genus $g$ in any $\epsilon$-thick part of moduli space such that it admits a pants decomposition whose boundary curves all have length equal to $\epsilon$, and the number of separating systole curves in this surface is uniformly comparable to $g$.Comment: 22 pages, 8 figures, comments are welcom

Generalized thermal instability criterion of black hole accretion disks

The conventional thermal instability criterion can not be applied to the advection-dominated accretion disks around black holes where the radiative cooling is insufficient to balance the viscous heating. The surface density change associated with the temperature perturbations, which was usually neglected in deriving the conventional criterion, was recently shown to be much significant in the advection-dominated disks. Considering both advection and surface density change, I suggested a generalized thermal instability criterion. By applying it to the optically thin and optically thick advection-dominated disks, I found that the former one is thermally stable and the latter one is thermally unstable against short wavelength perturbations, which agrees well to those found recently by both analytic and quantitative stability analyses.Comment: 5 pages, LaTex file, accepted for publication in Chinese Physics Letter

Magnetic Energy Injection in GRB 080913

GRB 080913, with a spectroscopically determined redshift of z=6.7, was the record holder of the remotest stellar object before the discovery of the recent gamma-ray burst GRB 090423, whose redshift is about 8.2. The gradually accumulated high redshift GRB sample has shed light on the origin and physics of GRBs during the cosmic re-ionization epoch. We here present a detailed numerical fit to the multi-wavelength data of the optical afterglow of GRB 080913 and then constrain its circum-burst environment and the other model parameters. We conclude that the late optical/X-ray plateau at about one day since the burst is due to the Poynting-flux dominated injection from the central engine which is very likely a massive spinning black hole with super strong magnetic fields.Comment: 3 pages, pdf only. accepted for publication in Science in China Series

Generalized susceptibilities along the phase boundary of the three-dimensional, three-state Potts model

Through the Monte Carlo simulation of the three-dimensional, three-state Potts model, which is a paradigm of finite-temperature pure gauge QCD, we study the fluctuations of generalized susceptibilities near the temperatures of external fields of first-, second-order phase transitions and crossover. Similar peak-like fluctuation appears in the second order susceptibility at three given external fields. Oscillation-like fluctuation appears in the third and fourth order susceptibilities. We find that these non-monotonic fluctuations are not only associated with the second-order phase transition, but also the first-order one and crossover in a system of finite-size. We further present the finite-size scaling analysis of the second and fourth order susceptibilities, respectively. The exponent of the scaling characterizes the order of the transitions, or the crossover.Comment: 9 pages, 5 figure

TRPL+K: Thick-Restart Preconditioned Lanczos+K Method for Large Symmetric Eigenvalue Problems

The Lanczos method is one of the standard approaches for computing a few eigenpairs of a large, sparse, symmetric matrix. It is typically used with restarting to avoid unbounded growth of memory and computational requirements. Thick-restart Lanczos is a popular restarted variant because of its simplicity and numerically robustness. However, convergence can be slow for highly clustered eigenvalues so more effective restarting techniques and the use of preconditioning is needed. In this paper, we present a thick-restart preconditioned Lanczos method, TRPL+K, that combines the power of locally optimal restarting (+K) and preconditioning techniques with the efficiency of the thick-restart Lanczos method. TRPL+K employs an inner-outer scheme where the inner loop applies Lanczos on a preconditioned operator while the outer loop augments the resulting Lanczos subspace with certain vectors from the previous restart cycle to obtain eigenvector approximations with which it thick restarts the outer subspace. We first identify the differences from various relevant methods in the literature. Then, based on an optimization perspective, we show an asymptotic global quasi-optimality of a simplified TRPL+K method compared to an unrestarted global optimal method. Finally, we present extensive experiments showing that TRPL+K either outperforms or matches other state-of-the-art eigenmethods in both matrix-vector multiplications and computational time.Comment: 27 pages, 6 figures, 7 tables. Submitted to SIAM Journal on Scientific Computing, Minor Revisio

Precision Test of the Weak Equivalence Principle from Gamma-Ray Burst Polarization

If the weak equivalence principle (WEP) is broken, the measured values of the parametrized post-Newtonian parameter $\gamma$ from photons with left- and right-handed circular polarizations should differ slightly, leading to the arrival-time difference of these two circular components. Thus, the polarization vector of a linearly polarized light may rotate during the propagation. The rotation angle of the polarization vector depends on both the photon energy and the distance of the source. It is believed that if the rotation angle differs by more than $\pi/2$ over an energy range, then the net polarization of the signal would be significantly suppressed and could not be as high as the observed level. Thus, the detection of highly polarized photons implies that the relative rotation angle ($\Delta\Theta$) should not be too large. In this paper, we give a detailed calculation on the evolution of gamma-ray burst (GRB) polarization arising from a possible violation of the WEP, and we find that more than $60\%$ of the initial polarization degree can be conserved even if $\Delta\Theta$ is larger than $\pi/2$. In addition, to tightly constrain the WEP violation, GRBs with harder spectra and polarization observations in a wider energy range seem to be favored. Applying our formulas to the measurements of linear polarization from GRB 110721A and GRB 061122, we obtain strict limits on the differences of the $\gamma$ values as low as $\Delta\gamma<1.3\times10^{-33}$ and $\Delta\gamma<0.8\times10^{-33}$. These provide the most stringent limits to date on a deviation from the WEP, improving at least 6 orders of magnitude over previous bounds.Comment: 6 pages, 3 figures. Published in PR

Expectations for SZ cluster counts: mass function versus X-ray luminosity function

We present a comparison of the SZ cluster counts predicted by the Press-Schechter (PS) mass function (MF) and the X-ray luminosity function (XLF) of clusters. The employment of the cluster XLF, together with the observationally determined X-ray luminosity(Lx)-temperature(T) relation, may allow us to estimate the SZ cluster counts in a more realistic manner, although such an empirical approach depends sensitively on our current knowledge of the dynamical properties of intracluster gas and its cosmic evolution. Using both the non-evolving and evolving XLFs of clusters suggested by X-ray observations, we calculate the expectations for SZ surveys of clusters with X-ray luminosity Lx>3X10^{44} erg/s and Lx>1X10^{43} erg/s in the 0.5 - 2.0 band, respectively. The non-evolving XLF results in a significant excess of SZ cluster counts at high redshifts as compared with the evolving XLF, while a slightly steeper Lx-T relation than the observed one is needed to reproduce the distributions of SZ clusters predicted by the standard PS formalism. It is pointed out that uncertainties in the cosmological application of future SZ cluster surveys via the standard PS formalism should be carefully studied.Comment: 6 pages, 2 figures, accepted for publication in Ap

Optimal small data Scattering for the generalized derivative nonlinear Schr\"odinger equations

In this work, we consider the following generalized derivative nonlinear Schr\"odinger equation \begin{align*} i\partial_t u+\partial_{xx} u +i |u|^{2\sigma}\partial_x u=0, \quad (t,x)\in \mathbb R\times \mathbb R. \end{align*} We prove that when $\sigma\ge 2$, the solution is global and scattering when the initial data is small in $H^s(\mathbb R)$, $\frac 12\leq s\leq1$. Moreover, we show that when $0<\sigma<2$, there exist a class of solitary wave solutions $\{\phi_c\}$ satisfying $$\|\phi_c\|_{H^1(\mathbb R)}\to 0,$$ when $c$ tends to some endpoint, which is against the small data scattering statement. Therefore, in this model, the exponent $\sigma\ge2$ is optimal for small data scattering. We remark that this exponent is larger than the short range exponent and the Strauss exponent.Comment: 26 page

Reconstruction of radial temperature profiles of galaxy clusters

In this Letter we present the radial temperature profiles of three X-ray clusters (A119, A2255 and A2256) reconstructed from a combination of the X-ray surface brightness measurements and the universal density profile as the underlying dark matter distribution. Our algorithm is based on the hydrostatic equilibrium for intracluster gas and the universality of the total baryon fraction within the virial radius. The scaled temperature profiles of these three clusters appear to be remarkably similar in shape, reflecting the underlying structural regularity, although they are inconsistent with either isothermality or a significant decline with increasing radius. Nevertheless, we find a good agreement between our derived temperature profiles and the recent analysis of 11 clusters observed with BeppoSAX (Irwin & Bregman 2000), which provides a useful clue to resolving the temperature profile discrepancy raised recently in literature. A comparison of our derived temperature profiles with future spatially-resolved spectral measurements may constitute a critical test for the standard model of structure formation and the conventional scenario for dynamical properties of clusters.Comment: 4 pages, 1 figure, A&A, 2000, 360, L43-L4

Robust Limits on Photon Mass from Statistical Samples of Extragalactic Radio Pulsars

The photon zero-mass hypothesis has been investigated for a long time using the frequency-dependent time delays of radio emissions from astrophysical sources. However, the search for a rest mass of the photon has been hindered by the similarity between the frequency-dependent dispersions due to the plasma and nonzero photon mass effects. Considering the contributions to the observed dispersion measure from both the plasma and nonzero photon mass effects, and assuming the dispersion induced by the plasma effect is an unknown constant, we obtain a robust limit on the photon mass by directly fitting a combination of the dispersion measures of radio sources. Using the observed dispersion measures from two statistical samples of extragalactic pulsars, here we show that at the 68\% confidence level, the constraints on the photon mass can be as low as $m_{\gamma}\leq1.51\times10^{-48}~\rm kg\simeq8.47\times10^{-13}~{\rm eV}/c^{2}$ for the sample of 22 radio pulsars in the Large Magellanic Cloud and $m_{\gamma}\leq1.58\times10^{-48}~\rm kg\simeq8.86\times10^{-13} {\rm eV}/c^{2}$ for the other sample of 5 radio pulsars in the Small Magellanic Cloud, which are comparable with that obtained by a single extragalactic pulsar. Furthermore, the statistical approach presented here can also be used when more fast radio bursts with known redshifts are detected in the future.Comment: 11 pages, 4 figures, 1 table. Accepted for publication in JCA