The auto-cross covariance matrix is defined as [mathbf{M}_n=frac{1} {2T}sum_{j=1}^Tbigl(mathbf{e}_jmathbf{e}_{j+tau}^*+mathbf{e}_{j+ tau}mathbf{e}_j^*bigr),] where $mathbf{e}_j$s are $n$-dimensional vectors of independent standard complex components
with a common mean 0, variance $sigma^2$, and uniformly bounded $2+eta$th moments and $tau$ is the lag. Jin et al. [Ann. Appl. Probab. 24 (2014) 1199-1225] has proved that the LSD of $mathbf{M}_n$ exists uniquely and nonrandomly, and independent of $tau$ for all $tauge 1$. And in addition they gave an analytic expression of the LSD. As a continuation of Jin et al. [Ann. Appl. Probab. 24 (2014) 1199-1225], this paper proved that under the condition of uniformly bounded fourth moments, in any closed interval outside the support of the LSD, with probability 1 there will be no eigenvalues of $mathbf{M}_n$ for all large $n$. As a consequence of the main theorem, the limits of the largest and smallest eigenvalue of $mathbf{M}_n$ are also obtained.
Two-dimensional electron gas in the integer quantum Hall regime is investigated numerically by studying the dynamics of an electron hopping on a square lattice subject to a perpendicular magnetic field and random on-site energy with white noise distr
ibution. Focusing on the lowest Landau band we establish an anti-levitation scenario of the extended states: As either the disorder strength $W$ increases or the magnetic field strength $B$ decreases, the energies of the extended states move below the Landau energies pertaining to a clean system. Moreover, for strong enough disorder, there is a disorder dependent critical magnetic field $B_c(W)$ below which there are no extended states at all. A general phase diagram in the $W-1/B$ plane is suggested with a line separating domains of localized and delocalized states.
Several experimental techniques have been introduced in recent years in attempts to measure spin transfer torque in magnetic tunnel junctions (MTJs). The dependence of spin torque on bias is important for understanding fundamental spin physics in mag
netic devices and for applications. However, previous techniques have provided only indirect measures of the torque and their results to date for the bias dependence are qualitatively and quantitatively inconsistent. Here we demonstrate that spin torque in MTJs can be measured directly by using time-domain techniques to detect resonant magnetic precession in response to an oscillating spin torque. The technique is accurate in the high-bias regime relevant for applications, and because it detects directly small-angle linear-response magnetic dynamics caused by spin torque it is relatively immune to artifacts affecting competing techniques. At high bias we find that the spin torque vector differs markedly from the simple lowest-order Taylor series approximations commonly assumed.
Coalescence overgrowth of pattern-grown GaN nanocolumns (NC) on c-plane sapphire substrate with metal organic chemical vapour deposition is demonstrated. The subsequent coalescence overgrowth opens a possibility for dislocation reduction due to the l
ateral strain relaxation in columnar geometry. We present further growth optimization and innovative characterization of MOCVD layers, overgrown on the columnar structure with varying diameter of colums. Nanoimprint lithography was applied to open circular holes of 250, 300, 450, 600 nm in diameter on the SiO2 layer, deposited on the GaN layer on c-plane sapphire template.
