Convergence of the largest eigenvalue of normalized sample covariance matrices when p and n both tend to infinity with their ratio converging to zero

Let $\mathbf{X}_p=(\mathbf{s}_1,...,\mathbf{s}_n)=(X_{ij})_{p \times n}$ where $X_{ij}$'s are independent and identically distributed (i.i.d.) random variables with $EX_{11}=0,EX_{11}^2=1$ and $EX_{11}^4<\infty$. It is showed that the largest eigenvalue of the random matrix $\mathbf{A}_p=\frac{1}{2\sqrt{np}}(\mathbf{X}_p\mathbf{X}_p^{\prime}-n\mathbf{I}_p)$ tends to 1 almost surely as $p\rightarrow\infty,n\rightarrow\infty$ with $p/n\rightarrow0$.Comment: Published in at http://dx.doi.org/10.3150/11-BEJ381 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Quasi-local energy and the choice of reference

A quasi-local energy for Einstein's general relativity is defined by the value of the preferred boundary term in the covariant Hamiltonian formalism. The boundary term depends upon a choice of reference and a time-like displacement vector field (which can be associated with an observer) on the boundary of the region. Here we analyze the spherical symmetric cases. For the obvious analytic choice of reference based on the metric components, we find that this technique gives the same quasi-local energy values using several standard coordinate systems and yet can give different values in some other coordinate systems. For the homogeneous-isotropic cosmologies, the energy can be non-positive, and one case which is actually flat space has a negative energy. As an alternative, we introduce a way to determine the choice of both the reference and displacement by extremizing the energy. This procedure gives the same value for the energy in different coordinate systems for the Schwarzschild space, and a non-negative value for the cosmological models, with zero energy for the dynamic cosmology which is actually Minkowski space. The timelike displacement vector comes out to be the dual mean curvature vector of the two-boundary.Comment: 21 pages; revised version to appear in CQ

Excess electron screening of remote donors and mobility in modern GaAs/AlGaAs herostructures

In modern GaAs/Al$_x$Ga$_{1-x}$As heterostructures with record high mobilities, a two-dimensional electron gas (2DEG) in a quantum well is provided by two remote donor $\delta$-layers placed on both sides of the well. Each $\delta$-layer is located within a narrow GaAs layer, flanked by narrow AlAs layers which capture excess electrons from donors but leave each of them localized in a compact dipole atom with a donor. Still excess electrons can hop between host donors to minimize their Coulomb energy. As a result they screen the random potential of donors dramatically. We numerically model the pseudoground state of excess electrons at a fraction $f$ of filled donors and find both the mobility and the quantum mobility limited by scattering on remote donors as universal functions of $f$. We repeat our simulations for devices with additional disorder such as interface roughness of the doping layers, and find the quantum mobility is consistent with measured values. Thus, in order to increase the quantum mobility this additional disorder should be minimized.Comment: arXiv admin note: text overlap with arXiv:1804.0693

Atomic electron correlation in nuclear electron capture

The effect of electron-electron Coulomb correlation on orbital electron capture by the nucleus was treated by the multiconfigurational Hartree-Fock approach. The theoretical Be-7 L/K capture ratio was found to be 0.086, and the Ar-37 M/L ratio, 0.102. Both ratios were smaller than the independent particle predictions. Measurements exist for the Ar M/L ratio, and agreement between theory and experiment was excellent

Nanocrystalline iron at high pressure

X-ray diffraction measurements were performed on nanocrystalline iron up to 46 GPa. For nanocrystalline epsilon-Fe, analysis of lattice parameter data provides a bulk modulus, K, of 179±8 GPa and a pressure derivative of the bulk modulus, K[prime], of 3.6±0.7, similar to the large-grained control sample. The extrapolated zero-pressure unit cell volume of nanocrystalline epsilon-Fe is 22.9±0.2 Å^3, compared to 22.3±0.2 Å^3 for large-grained epsilon-Fe. No significant grain growth was observed to occur under pressure

Multiplet effects on the L(2,3) fluorescence yield of multiply ionized Ar

The 2p fluorescence yield of argon in the presence of 0 to 6 3p holes was calculated by statistically averaging the fluorescence yields of initial state that consist of individual multiplet configurations. These configurations were formed by coupling the 2p vacancy to the partially filled 3p shell. Results agree reasonably well with experimental fluorescence yields deduced from ion-atom collision measurements

Continuous Dynamical Decoupling with Bounded Controls

We develop a theory of continuous decoupling with bounded controls from a geometric perspective. Continuous decoupling with bounded controls can accomplish the same decoupling effect as the bang-bang control while using realistic control resources and it is robust against systematic implementation errors. We show that the decoupling condition within this framework is equivalent to average out error vectors whose trajectories are determined by the control Hamiltonian. The decoupling pulses can be intuitively designed once the structure function of the corresponding SU(n) is known and is represented from the geometric perspective. Several examples are given to illustrate the basic idea. From the physical implementation point of view we argue that the efficiency of the decoupling is determined not by the order of the decoupling group but by the minimal time required to finish a decoupling cycle

Two-flux Colliding Plane Waves in String Theory

We construct the two-flux colliding plane wave solutions in higher dimensional gravity theory with dilaton, and two complementary fluxes. Two kinds of solutions has been obtained: Bell-Szekeres(BS) type and homogeneous type. After imposing the junction condition, we find that only Bell-Szekeres type solution is physically well-defined. Furthermore, we show that the future curvature singularity is always developed for our solutions.Comment: 16 pages, Latex; typoes corrected; references added, minor modification

Marginally Trapped Surfaces in the Nonsymmetric Gravitational Theory

We consider a simple, physical approach to the problem of marginally trapped surfaces in the Nonsymmetric Gravitational Theory (NGT). We apply this approach to a particular spherically symmetric, Wyman sector gravitational field, consisting of a pulse in the antisymmetric field variable. We demonstrate that marginally trapped surfaces do exist for this choice of initial data.Comment: REVTeX 3.0 with epsf macros and AMS symbols, 3 pages, 1 figur