A WENO Algorithm of the Temperature and Ionization Profiles around a Point Source

We develop a numerical solver for radiative transfer problems based on the weighted essentially nonoscillatory (WENO) scheme modified with anti-diffusive flux corrections, in order to solve the temperature and ionization profiles around a point source of photons in the reionization epoch. Algorithms for such simulation must be able to handle the following two features: 1. the sharp profiles of ionization and temperature at the ionizing front (I-front) and the heating front (T-front), and 2. the fraction of neutral hydrogen within the ionized sphere is extremely small due to the stiffness of the rate equations of atom processes. The WENO scheme can properly handle these two features, as it has been shown to have high order of accuracy and good convergence in capturing discontinuities and complicated structures in fluid as well as to be significantly superior over piecewise smooth solutions containing discontinuities. With this algorithm, we show the time-dependence of the preheated shell around a UV photon source. In the first stage the I-front and T-front are coincident, and propagate with almost the speed of light. In later stage, when the frequency spectrum of UV photons is hardened, the speeds of propagation of the ionizing and heating fronts are both significantly less than the speed of light, and the heating front is always beyond the ionizing front. In the spherical shell between the I- and T-fronts, the IGM is heated, while atoms keep almost neutral. The time scale of the preheated shell evolution is dependent on the intensity of the photon source. We also find that the details of the pre-heated shell and the distribution of neutral hydrogen remained in the ionized sphere are actually sensitive to the parameters used. The WENO algorithm can provide stable and robust solutions to study these details.Comment: 24 pages, 7 figures, accepted in New Astronom

Payment Barriers in China\u27s B2C Business

With the entry into the WTO, China has made a lot of progress in its e-business. However, there are still some barriers that limit the development of e-business in China. One of the most difficult hurdles might be the fragile online payment system. According to The first DHL Global E-commerce Report, in mature Internet countries, online payment is no longer a problem. For instance, six out of 10 companies in the US (60%), Australia (61%) and Finland (58%) say payment is not a barrier to ecommerce. But less mature markets including China consider payment a barrier. This paper mainly discusses the payment barrier to B2C Business in China, and then forwards some suggestions on how to remove those barriers on the basis of analyzing the case of BOLChina—A successful B2C model from Germany. Also, some conclusions are given to make it more clear that China should break the ice of payment barriers on the way to the bright future of its e-business

Approximation error of the Lagrange reconstructing polynomial

The reconstruction approach [Shu C.W.: {\em SIAM Rev.} {\bf 51} (2009) 82--126] for the numerical approximation of $f'(x)$ is based on the construction of a dual function $h(x)$ whose sliding averages over the interval $[x-\tfrac{1}{2}\Delta x,x+\tfrac{1}{2}\Delta x]$ are equal to $f(x)$ (assuming an homogeneous grid of cell-size $\Delta x$). We study the deconvolution problem [Harten A., Engquist B., Osher S., Chakravarthy S.R.: {\em J. Comp. Phys.} {\bf 71} (1987) 231--303] which relates the Taylor polynomials of $h(x)$ and $f(x)$, and obtain its explicit solution, by introducing rational numbers $\tau_n$ defined by a recurrence relation, or determined by their generating function, $g_\tau(x)$, related with the reconstruction pair of ${\rm e}^x$. We then apply these results to the specific case of Lagrange-interpolation-based polynomial reconstruction, and determine explicitly the approximation error of the Lagrange reconstructing polynomial (whose sliding averages are equal to the Lagrange interpolating polynomial) on an arbitrary stencil defined on a homogeneous grid.Comment: 31 pages, 1 table; revised version to appear in J. Approx. Theor

Forming an O Star via Disk Accretion?

We present a study of outflow, infall, and rotation in a ~10^5 Lsun (solar luminosity) star-forming region, IRAS 18360-0537, with Submillimeter Array (SMA) and IRAM 30m observations. The 1.3 mm continuum map shows a 0.5 pc dust ridge, of which the central compact part has a mass of ~80 Msun (solar mass) and harbors two condensations, MM1 and MM2. The CO (2--1) and SiO (5--4) maps reveal a biconical outflow centered at MM1, which is a hot molecular core (HMC) with a gas temperature of 320+/-50 K and a mass of ~13 Msun. The outflow has a gas mass of 54 Msun and a dynamical timescale of 8,000 yr. The kinematics of the HMC is probed by high-excitation CH3OH and CH3CN lines, which are detected at sub-arcsecond resolution and unveil a velocity gradient perpendicular to the outflow axis, suggesting a disk-like rotation of the HMC. An infalling envelope around the HMC is evidenced by CN lines exhibiting a profound inverse P-Cygni profile, and the estimated mass infall rate, 1.5x10^{-3} Msun/yr, is well comparable to that inferred from the mass outflow rate. A more detailed investigation of the kinematics of the dense gas around the HMC is obtained from the 13CO and C18O (2--1) lines; the position-velocity diagrams of the two lines are consistent with the model of a free-falling and Keplerian-like rotating envelope. The observations suggest that the protostar of a current mass ~10 Msun embedded within MM1 will develop into an O star via disk accretion and envelope infall.Comment: Accepted for publication in the Ap

Cultural Differences in E-Commerce: A Comparison between the U.S. and China

Every firm must identify and account cultural problems related to their online globalization. In order to accomplish this, they must identify the characteristics and trends of the Internet and ECommerce in their target market. Next, any firm must differentiates the lessons of e-commerce in the U.S. with the target market by analyzing differences in end user behavior. Finally, they must formulate a plan to discuss how to remove cultural barriers to enhance net growth. This paper specifically presents a study of the ECommerce Market of China. Naturally, the two primary factors that these firms must consider are securing a payment system and overcoming any language barriers. As stated above, this paper will present an overview of the Chinese market, differentiate the Chinese market with the United States, and supply solutions to the E-Commerce problems that are particular to China

Representation of the Lagrange reconstructing polynomial by combination of substencils

The Lagrange reconstructing polynomial [Shu C.W.: {\em SIAM Rev.} {\bf 51} (2009) 82--126] of a function $f(x)$ on a given set of equidistant (\Delta x=\const) points $\bigl\{x_i+\ell\Delta x;\;\ell\in\{-M_-,...,+M_+\}\bigr\}$ is defined [Gerolymos G.A.: {\em J. Approx. Theory} {\bf 163} (2011) 267--305] as the polynomial whose sliding (with $x$) averages on $[x-\tfrac{1}{2}\Delta x,x+\tfrac{1}{2}\Delta x]$ are equal to the Lagrange interpolating polynomial of $f(x)$ on the same stencil. We first study the fundamental functions of Lagrange reconstruction, show that these polynomials have only real and distinct roots, which are never located at the cell-interfaces (half-points) $x_i+n\tfrac{1}{2}\Delta x$ ($n\in\mathbb{Z}$), and obtain several identities. Using these identities, by analogy to the recursive Neville-Aitken-like algorithm applied to the Lagrange interpolating polynomial, we show that there exists a unique representation of the Lagrange reconstructing polynomial on $\{i-M_-,...,i+M_+\}$ as a combination of the Lagrange reconstructing polynomials on the $K_\mathrm{s}+1\leq M:=M_-+M_+>1$ substencils $\{i-M_-+k_\mathrm{s},...,i+M_+-K_\mathrm{s}+k_\mathrm{s}\}$ ($k_\mathrm{s}\in\{0,...,K_\mathrm{s}\}$), with weights $\sigma_{R_1,M_-,M_+,K_\mathrm{s},k_\mathrm{s}}(\xi)$ which are rational functions of $\xi$ ($x=x_i+\xi\Delta x$) [Liu Y.Y., Shu C.W., Zhang M.P.: {\em Acta Math. Appl. Sinica} {\bf 25} (2009) 503--538], and give an analytical recursive expression of the weight-functions. We then use the analytical expression of the weight-functions $\sigma_{R_1,M_-,M_+,K_\mathrm{s},k_\mathrm{s}}(\xi)$ to obtain a formal proof of convexity (positivity of the weight-functions) in the neighborhood of $\xi=\tfrac{1}{2}$, under the condition that all of the substencils contain either point $i$ or point $i+1$ (or both).Comment: final corrected version; in print J. Comp. Appl. Mat

A comparison of troubled-cell indicators for Runge-Kutta discontinuous Galerkin methods using weighted essentially nonoscillatory limiters

In [SIAM J. Sci. Comput., 26 (2005), pp. 907-929], we initiated the study of using WENO (weighted essentially nonoscillatory) methodology as limiters for the RKDG (Runge-Kutta discontinuous Galerkin) methods. The idea is to first identify "troubled cells," namely, those cells where limiting might be needed, then to abandon all moments in those cells except the cell averages and reconstruct those moments from the information of neighboring cells using a WENO methodology. This technique works quite well in our one- and two-dimensional test problems [SIAM J. Sci. Comput., 26 (2005), pp. 907-929] and in the follow-up work where more compact Hermite WENO methodology is used in the troubled cells. In these works we used the classical minmod-type TVB (total variation bounded) limiters to identify the troubled cells; that is, whenever the minmod limiter attempts to change the slope, the cell is declared to be a troubled cell. This troubled-cell indicator has a TVB parameter M to tune and may identify more cells than necessary as troubled cells when M is not chosen adequately, making the method costlier than necessary. In this paper we systematically investigate and compare a few different limiter strategies as troubled-cell indicators with an objective of obtaining the most efficient and reliable troubled-cell indicators to save computational cost