A Note on Rate of Convergence in Probability to Semicircular Law

In the present paper, we prove that under the assumption of the finite sixth moment for elements of a Wigner matrix, the convergence rate of its empirical spectral distribution to the Wigner semicircular law in probability is $O(n^{-1/2})$ when the dimension $n$ tends to infinity.Comment: 13 page

Convergence of the empirical spectral distribution function of Beta matrices

Let $\mathbf{B}_n=\mathbf {S}_n(\mathbf {S}_n+\alpha_n\mathbf {T}_N)^{-1}$, where $\mathbf {S}_n$ and $\mathbf {T}_N$ are two independent sample covariance matrices with dimension $p$ and sample sizes $n$ and $N$, respectively. This is the so-called Beta matrix. In this paper, we focus on the limiting spectral distribution function and the central limit theorem of linear spectral statistics of $\mathbf {B}_n$. Especially, we do not require $\mathbf {S}_n$ or $\mathbf {T}_N$ to be invertible. Namely, we can deal with the case where $p>\max\{n,N\}$ and $p<n+N$. Therefore, our results cover many important applications which cannot be simply deduced from the corresponding results for multivariate $F$ matrices.Comment: Published at http://dx.doi.org/10.3150/14-BEJ613 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Quantum steering in a star network

In this work, we will consider the star network scenario where the central party is trusted while all the edge parties (with a number of $n$) are untrusted. Network steering is defined with an $n$ local hidden state model which can be viewed as a special kind of $n$ local hidden variable model. Three different types of sufficient criteria, nonlinear steering inequality, linear steering inequality, and Bell inequality, will be constructed to verify the quantum steering in a star network. Based on the linear steering inequality, it is found that the network steering can be demonstrated even though the trusted party performs a fixed measurement.Comment: 9 pages, 1 figure, comments welcom

Bacterial transport and deposition in an intact soil lysimeter and packed sand columns

Microbial transport in porous media is mainly facilitated by flowing water, whilst retention is due to adsorption to phase interfaces. Water flow in porous media depends greatly on water content and pore size distribution. Microbial adsorption to air-water interfaces is especially important in unsaturated porous media. Bacterial transport in unsaturated soils is much less well understood than in saturated conditions, especially for intact soils. The first experiment was designed to investigate the fate and transport of bacteria in intact soils with different water saturations, and particularly the effect of low suction (and hence removal of water flow in the largest macropores). An intact soil column (50 cm diameter x 70 cm depth) with a tension infiltrometer was used to investigate the transport and deposition of Bacillus subtilis endospores (i.e. dormant and persistent bacteria) during both saturated and unsaturated flows. Soil porosity and pore size distribution were measured. Porosity decreased with depth and macropores were concentrated in the topsoil. Three tensiometers and a temperature sensor were installed along the soil column to monitor matric suction and temperature. Breakthrough curves for bacteria and chemical tracer Br⁻ at 0 kPa and 0.5 kPa suction were obtained during the three-month leaching experiment. Bacterial breakthrough occurred earlier than the inert chemical tracer, which is consistent with effects of pore size exclusion. Also, saturated flow gave a significantly higher concentration and recovery ratio of leached bacteria, i.e. 51 % vs. 0.88%. Recovery of Br⁻ in leachate at both suctions reached above 85%. The column was destructively sampled for deposited endospores at the completion of leaching. Bacterial deposition was concentrated in the top ten centimeters, then decreased abruptly and was relatively constant with column depth, although showing some irregularity at the bottom of the column. To more thoroughly investigate the factors which influence bacterial transport in porous media, a sand column leaching system with well-controlled degree of water saturation and flow rate was built to investigate the effects of water content, particle size, and column length on bacterial transport. Adsorption of E. coli strain D to silica sands was measured in batch adsorption tests. The average percent of adsorption for coarse and fine sands was 45.9 ± 7.8 % and 96.9 ± 3.2%, respectively. The applicability of results from batch adsorption experiments to bacterial transport was limited because of the dynamic feature of bacteria in sand columns. The early breakthrough of E. coli relative to bromide was clear for all sand columns, namely c. 0.15~0.3 pore volume earlier. The column length had no significant effects on the E. coli peak concentration and total recovery in leachate, which supported the observation that bacteria were retained in top layer of sands. Tailing of breakthrough curves was more prominent for all fine sand columns than their coarse sand counterparts. Bacterial recovery in leachate from coarse and saturated sand columns was significantly higher than fine and unsaturated columns. Observed data were fitted by equilibrium, one-site, two-site and their AWI amended convection-dispersion kinetic models. Two-site + AWI model achieved constant high model efficiency for both coarse and fine sands, under either saturated or unsaturated flow conditions. However, ƒ in two-site model could not be physically measured and the fitted value might just be an artifact. Although the intrinsic flaw associated with two-site + AWI model, it is still a simple and effective modeling approach